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Spectroscopic notation provides a way to specify atomic ionization states , atomic orbitals , and molecular orbitals .
Spectroscopists customarily refer to the spectrum arising from a given ionization state of a given element by the element's symbol followed by a Roman numeral . The numeral I is used for spectral lines associated with the neutral element, II for those from the first ionization state, III for those from the second ionization state, and so on. [ 1 ] For example, "He I" denotes lines of neutral helium , and "C IV" denotes lines arising from the third ionization state, C 3+ , of carbon . This notation is used for example to retrieve data from the NIST Atomic Spectrum Database .
Before atomic orbitals were understood, spectroscopists discovered various distinctive series of spectral lines in atomic spectra, which they identified by letters. These letters were later associated with the azimuthal quantum number , ℓ . The letters, "s", "p", "d", and "f", for the first four values of ℓ were chosen to be the first letters of properties of the spectral series observed in alkali metals . Other letters for subsequent values of ℓ were assigned in alphabetical order, omitting the letter "j" [ 2 ] [ 3 ] [ 4 ] because some languages do not distinguish between the letters "i" and "j": [ 5 ] [ 6 ]
This notation is used to specify electron configurations and to create the term symbol for the electron states in a multi-electron atom. When writing a term symbol, the above scheme for a single electron's orbital quantum number is applied to the total orbital angular momentum associated to an electron state. [ 4 ]
The spectroscopic notation of molecules uses Greek letters to represent the modulus of the orbital angular momentum along the internuclear axis.
The quantum number that represents this angular momentum is Λ.
For Σ states, one denotes if there is a reflection in a plane containing the nuclei (symmetric), using the + above. The − is used to indicate that there is not.
For homonuclear diatomic molecules, the index g or u denotes the existence of a center of symmetry (or inversion center) and indicates the symmetry of the vibronic wave function with respect to the point-group inversion operation i . Vibronic states that are symmetric with respect to i are denoted g for gerade (German for "even"), and unsymmetric states are denoted u for ungerade (German for "odd").
For mesons whose constituents are a heavy quark and its own antiquark ( quarkonium ) the same notation applies as for atomic states. However, uppercase letters are used.
Furthermore, the first number is (as in nuclear physics) n = N + 1 {\displaystyle n=N+1} where N {\displaystyle N} is the number of nodes in the radial wave function, while in atomic physics n = N + ℓ + 1 {\displaystyle n=N+\ell +1} is used. Hence, a 1P state in quarkonium corresponds to a 2p state in an atom or positronium . | https://en.wikipedia.org/wiki/Spectroscopic_notation |
Spectroscopy is the field of study that measures and interprets electromagnetic spectra . [ 1 ] [ 2 ] In narrower contexts, spectroscopy is the precise study of color as generalized from visible light to all bands of the electromagnetic spectrum.
Spectroscopy, primarily in the electromagnetic spectrum, is a fundamental exploratory tool in the fields of astronomy , chemistry , materials science , and physics , allowing the composition, physical structure and electronic structure of matter to be investigated at the atomic, molecular and macro scale, and over astronomical distances .
Historically, spectroscopy originated as the study of the wavelength dependence of the absorption by gas phase matter of visible light dispersed by a prism . Current applications of spectroscopy include biomedical spectroscopy in the areas of tissue analysis and medical imaging . Matter waves and acoustic waves can also be considered forms of radiative energy, and recently gravitational waves have been associated with a spectral signature in the context of the Laser Interferometer Gravitational-Wave Observatory (LIGO). [ 3 ]
Spectroscopy is a branch of science concerned with the spectra of electromagnetic radiation as a function of its wavelength or frequency measured by spectrographic equipment, and other techniques, in order to obtain information concerning the structure and properties of matter. [ 4 ] Spectral measurement devices are referred to as spectrometers , spectrophotometers , spectrographs or spectral analyzers . Most spectroscopic analysis in the laboratory starts with a sample to be analyzed, then a light source is chosen from any desired range of the light spectrum, then the light goes through the sample to a dispersion array (diffraction grating instrument) and captured by a photodiode . For astronomical purposes, the telescope must be equipped with the light dispersion device. There are various versions of this basic setup that may be employed.
Spectroscopy began with Isaac Newton splitting light with a prism; a key moment in the development of modern optics . [ 5 ] Therefore, it was originally the study of visible light that we call color that later under the studies of James Clerk Maxwell came to include the entire electromagnetic spectrum . [ 6 ] Although color is involved in spectroscopy, it is not equated with the color of elements or objects that involve the absorption and reflection of certain electromagnetic waves to give objects a sense of color to our eyes. Rather spectroscopy involves the splitting of light by a prism, diffraction grating, or similar instrument, to give off a particular discrete line pattern called a "spectrum" unique to each different type of element. Most elements are first put into a gaseous phase to allow the spectra to be examined although today other methods can be used on different phases. Each element that is diffracted by a prism-like instrument displays either an absorption spectrum or an emission spectrum depending upon whether the element is being cooled or heated. [ 7 ]
Until recently all spectroscopy involved the study of line spectra and most spectroscopy still does. [ 8 ] Vibrational spectroscopy is the branch of spectroscopy that studies the spectra. [ 9 ] However, the latest developments in spectroscopy can sometimes dispense with the dispersion technique. In biochemical spectroscopy, information can be gathered about biological tissue by absorption and light scattering techniques. Light scattering spectroscopy is a type of reflectance spectroscopy that determines tissue structures by examining elastic scattering. [ 10 ] In such a case, it is the tissue that acts as a diffraction or dispersion mechanism.
Spectroscopic studies were central to the development of quantum mechanics , because the first useful atomic models described the spectra of hydrogen, which include the Bohr model , the Schrödinger equation , and Matrix mechanics , all of which can produce the spectral lines of hydrogen , therefore providing the basis for discrete quantum jumps to match the discrete hydrogen spectrum. Also, Max Planck 's explanation of blackbody radiation involved spectroscopy because he was comparing the wavelength of light using a photometer to the temperature of a Black Body . [ 11 ] Spectroscopy is used in physical and analytical chemistry because atoms and molecules have unique spectra. As a result, these spectra can be used to detect, identify and quantify information about the atoms and molecules. Spectroscopy is also used in astronomy and remote sensing on Earth. Most research telescopes have spectrographs. The measured spectra are used to determine the chemical composition and physical properties of astronomical objects (such as their temperature , density of elements in a star, velocity , black holes and more). [ 12 ] An important use for spectroscopy is in biochemistry. Molecular samples may be analyzed for species identification and energy content. [ 13 ]
The underlying premise of spectroscopy is that light is made of different wavelengths and that each wavelength corresponds to a different frequency. The importance of spectroscopy is centered around the fact that every element in the periodic table has a unique light spectrum described by the frequencies of light it emits or absorbs consistently appearing in the same part of the electromagnetic spectrum when that light is diffracted. This opened up an entire field of study with anything that contains atoms. Spectroscopy is the key to understanding the atomic properties of all matter. As such spectroscopy opened up many new sub-fields of science yet undiscovered. The idea that each atomic element has its unique spectral signature enabled spectroscopy to be used in a broad number of fields each with a specific goal achieved by different spectroscopic procedures. The National Institute of Standards and Technology maintains a public Atomic Spectra Database that is continually updated with precise measurements. [ 14 ]
The broadening of the field of spectroscopy is due to the fact that any part of the electromagnetic spectrum may be used to analyze a sample from the infrared to the ultraviolet telling scientists different properties about the very same sample. For instance in chemical analysis, the most common types of spectroscopy include atomic spectroscopy, infrared spectroscopy, ultraviolet and visible spectroscopy, Raman spectroscopy and nuclear magnetic resonance . [ 15 ] In nuclear magnetic resonance (NMR), the theory behind it is that frequency is analogous to resonance and its corresponding resonant frequency. Resonances by the frequency were first characterized in mechanical systems such as pendulums , which have a frequency of motion noted famously by Galileo . [ 16 ]
Spectroscopy is a sufficiently broad field that many sub-disciplines exist, each with numerous implementations of specific spectroscopic techniques. The various implementations and techniques can be classified in several ways.
The types of spectroscopy are distinguished by the type of radiative energy involved in the interaction. In many applications, the spectrum is determined by measuring changes in the intensity or frequency of this energy. The types of radiative energy studied include:
The types of spectroscopy also can be distinguished by the nature of the interaction between the energy and the material. These interactions include: [ 2 ]
Spectroscopic studies are designed so that the radiant energy interacts with specific types of matter.
Atomic spectroscopy was the first application of spectroscopy. Atomic absorption spectroscopy and atomic emission spectroscopy involve visible and ultraviolet light. These absorptions and emissions, often referred to as atomic spectral lines, are due to electronic transitions of outer shell electrons as they rise and fall from one electron orbit to another. Atoms also have distinct x-ray spectra that are attributable to the excitation of inner shell electrons to excited states.
Atoms of different elements have distinct spectra and therefore atomic spectroscopy allows for the identification and quantitation of a sample's elemental composition. After inventing the spectroscope, Robert Bunsen and Gustav Kirchhoff discovered new elements by observing their emission spectra. Atomic absorption lines are observed in the solar spectrum and referred to as Fraunhofer lines after their discoverer. A comprehensive explanation of the hydrogen spectrum was an early success of quantum mechanics and explained the Lamb shift observed in the hydrogen spectrum, which further led to the development of quantum electrodynamics .
Modern implementations of atomic spectroscopy for studying visible and ultraviolet transitions include flame emission spectroscopy , inductively coupled plasma atomic emission spectroscopy , glow discharge spectroscopy , microwave induced plasma spectroscopy, and spark or arc emission spectroscopy. Techniques for studying x-ray spectra include X-ray spectroscopy and X-ray fluorescence .
The combination of atoms into molecules leads to the creation of unique types of energetic states and therefore unique spectra of the transitions between these states. Molecular spectra can be obtained due to electron spin states ( electron paramagnetic resonance ), molecular rotations , molecular vibration , and electronic states. Rotations are collective motions of the atomic nuclei and typically lead to spectra in the microwave and millimetre-wave spectral regions. Rotational spectroscopy and microwave spectroscopy are synonymous. Vibrations are relative motions of the atomic nuclei and are studied by both infrared and Raman spectroscopy . Electronic excitations are studied using visible and ultraviolet spectroscopy as well as fluorescence spectroscopy . [ 2 ] [ 19 ] [ 20 ] [ 21 ] [ 22 ]
Studies in molecular spectroscopy led to the development of the first maser and contributed to the subsequent development of the laser .
The combination of atoms or molecules into crystals or other extended forms leads to the creation of additional energetic states. These states are numerous and therefore have a high density of states. This high density often makes the spectra weaker and less distinct, i.e., broader. For instance, blackbody radiation is due to the thermal motions of atoms and molecules within a material. Acoustic and mechanical responses are due to collective motions as well.
Pure crystals, though, can have distinct spectral transitions, and the crystal arrangement also has an effect on the observed molecular spectra. The regular lattice structure of crystals also scatters x-rays, electrons or neutrons allowing for crystallographic studies.
Nuclei also have distinct energy states that are widely separated and lead to gamma ray spectra. Distinct nuclear spin states can have their energy separated by a magnetic field, and this allows for nuclear magnetic resonance spectroscopy .
Other types of spectroscopy are distinguished by specific applications or implementations:
There are several applications of spectroscopy in the fields of medicine, physics, chemistry, and astronomy. Taking advantage of the properties of absorbance and with astronomy emission , spectroscopy can be used to identify certain states of nature. The uses of spectroscopy in so many different fields and for so many different applications has caused specialty scientific subfields. Such examples include:
The history of spectroscopy began with Isaac Newton 's optics experiments (1666–1672). According to Andrew Fraknoi and David Morrison , "In 1672, in the first paper that he submitted to the Royal Society , Isaac Newton described an experiment in which he permitted sunlight to pass through a small hole and then through a prism. Newton found that sunlight, which looks white to us, is actually made up of a mixture of all the colors of the rainbow." [ 38 ] Newton applied the word "spectrum" to describe the rainbow of colors that combine to form white light and that are revealed when the white light is passed through a prism.
Fraknoi and Morrison state that "In 1802, William Hyde Wollaston built an improved spectrometer that included a lens to focus the Sun's spectrum on a screen. Upon use, Wollaston realized that the colors were not spread uniformly, but instead had missing patches of colors, which appeared as dark bands in the spectrum." [ 38 ] During the early 1800s, Joseph von Fraunhofer made experimental advances with dispersive spectrometers that enabled spectroscopy to become a more precise and quantitative scientific technique. Since then, spectroscopy has played and continues to play a significant role in chemistry, physics, and astronomy. Per Fraknoi and Morrison, "Later, in 1815, German physicist Joseph Fraunhofer also examined the solar spectrum, and found about 600 such dark lines (missing colors), are now known as Fraunhofer lines, or Absorption lines." [ 38 ] [ better source needed ]
In quantum mechanical systems, the analogous resonance is a coupling of two quantum mechanical stationary states of one system, such as an atom , via an oscillatory source of energy such as a photon . The coupling of the two states is strongest when the energy of the source matches the energy difference between the two states. The energy E of a photon is related to its frequency ν by E = hν where h is the Planck constant , and so a spectrum of the system response vs. photon frequency will peak at the resonant frequency or energy. Particles such as electrons and neutrons have a comparable relationship, the de Broglie relations , between their kinetic energy and their wavelength and frequency and therefore can also excite resonant interactions.
Spectra of atoms and molecules often consist of a series of spectral lines, each one representing a resonance between two different quantum states. The explanation of these series, and the spectral patterns associated with them, were one of the experimental enigmas that drove the development and acceptance of quantum mechanics. The hydrogen spectral series in particular was first successfully explained by the Rutherford–Bohr quantum model of the hydrogen atom. In some cases spectral lines are well separated and distinguishable, but spectral lines can also overlap and appear to be a single transition if the density of energy states is high enough. Named series of lines include the principal , sharp , diffuse and fundamental series .
Spectroscopy has emerged as a growing practice within the maker movement , enabling hobbyists and educators to construct functional spectrometers using readily available materials. [ 39 ] Utilizing components like CD/DVD diffraction gratings, smartphones, and 3D-printed parts, these instruments offer a hands-on approach to understanding light and matter interactions. Smartphone applications [ 40 ] [ 41 ] along with open-source tools [ 42 ] facilitate integration, greatly simplify the capturing and analysis of spectral data. While limitations in resolution, calibration accuracy, and stray light management exist compared to professional equipment, DIY spectroscopy provides valuable educational experiences [ 43 ] and contributes to citizen science initiatives, fostering accessibility to spectroscopic techniques. | https://en.wikipedia.org/wiki/Spectroscopy |
A spectrum ( pl. : spectra or spectrums ) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum . [ 1 ] The word spectrum was first used scientifically in optics to describe the rainbow of colors in visible light after passing through a prism . In the optical spectrum, light wavelength is viewed as continuous, and spectral colors are seen to blend into one another smoothly when organized in order of their corresponding wavelengths. As scientific understanding of light advanced, the term came to apply to the entire electromagnetic spectrum , including radiation not visible to the human eye.
Spectrum has since been applied by analogy to topics outside optics. Thus, one might talk about the " spectrum of political opinion ", or the "spectrum of activity" of a drug, or the " autism spectrum ". In these uses, values within a spectrum may not be associated with precisely quantifiable numbers or definitions. Such uses imply a broad range of conditions or behaviors grouped together and studied under a single title for ease of discussion. Nonscientific uses of the term spectrum are sometimes misleading. For instance, a single left–right spectrum of political opinion does not capture the full range of people's political beliefs. Political scientists use a variety of biaxial and multiaxial systems to more accurately characterize political opinion.
In most modern usages of spectrum there is a unifying theme between the extremes at either end. This was not always true in older usage.
In Latin , spectrum means "image" or " apparition ", including the meaning " spectre ". Spectral evidence is testimony about what was done by spectres of persons not present physically, or hearsay evidence about what ghosts or apparitions of Satan said. It was used to convict a number of persons of witchcraft at Salem, Massachusetts in the late 17th century. The word "spectrum" [Spektrum] was strictly used to designate a ghostly optical afterimage by Goethe in his Theory of Colors and Schopenhauer in On Vision and Colors .
The prefix "spectro-" is used to form words relating to spectra. For example, a spectrometer is a device used to record spectra and spectroscopy is the use of a spectrometer for chemical analysis .
In the physical sciences , the term spectrum was introduced first into optics by Isaac Newton in the 17th century, referring to the range of colors observed when white light was dispersed through a prism . [ 2 ] [ 3 ] Soon the term referred to a plot of light intensity or power as a function of frequency or wavelength , also known as a spectral density plot .
Antibiotic spectrum of activity is a component of antibiotic classification . A broad-spectrum antibiotic is active against a wide range of bacteria, [ 4 ] whereas a narrow-spectrum antibiotic is effective against specific families of bacteria. [ 5 ] An example of a commonly used broad-spectrum antibiotic is ampicillin . [ 5 ] An example of a narrow spectrum antibiotic is Dicloxacillin , which acts on beta-lactamase -producing Gram-positive bacteria such as Staphylococcus aureus . [ 6 ]
In psychiatry, the spectrum approach uses the term spectrum to describe a range of linked conditions, sometimes also extending to include singular symptoms and traits . For example, the autism spectrum describes a range of conditions classified as neurodevelopmental disorders .
In mathematics , the spectrum of a matrix is the multiset of the eigenvalues of the matrix.
In functional analysis , the concept of the spectrum of a bounded operator is a generalization of the eigenvalue concept for matrices.
In algebraic topology , a spectrum is an object representing a generalized cohomology theory .
In social science , economic spectrum is used to indicate the range of social class along some indicator of wealth or income. In political science , the term political spectrum refers to a system of classifying political positions in one or more dimensions, for example in a range including right wing and left wing. | https://en.wikipedia.org/wiki/Spectrum |
In the physical sciences , the term spectrum was introduced first into optics by Isaac Newton in the 17th century, referring to the range of colors observed when white light was dispersed through a prism . [ 1 ] [ 2 ] Soon the term referred to a plot of light intensity or power as a function of frequency or wavelength , also known as a spectral density plot .
Later it expanded to apply to other waves , such as sound waves and sea waves that could also be measured as a function of frequency (e.g., noise spectrum , sea wave spectrum ). It has also been expanded to more abstract " signals ", whose power spectrum can be analyzed and processed . The term now applies to any signal that can be measured or decomposed along a continuous variable, such as energy in electron spectroscopy or mass-to-charge ratio in mass spectrometry . Spectrum is also used to refer to a graphical representation of the signal as a function of the dependent variable.
In Latin , spectrum means "image" or " apparition ", including the meaning " spectre ". Spectral evidence is testimony about what was done by spectres of persons not present physically, or hearsay evidence about what ghosts or apparitions of Satan said. It was used to convict a number of persons of witchcraft at Salem, Massachusetts in the late 17th century. The word "spectrum" [Spektrum] was strictly used to designate a ghostly optical afterimage by Goethe in his Theory of Colors and Schopenhauer in On Vision and Colors .
Electromagnetic spectrum refers to the full range of all frequencies of electromagnetic radiation [ 3 ] and also to the characteristic distribution of electromagnetic radiation emitted or absorbed by that particular object. Devices used to measure an electromagnetic spectrum are called spectrograph or spectrometer . The visible spectrum is the part of the electromagnetic spectrum that can be seen by the human eye . The wavelength of visible light ranges from 390 to 700 nm . [ 4 ] The absorption spectrum of a chemical element or chemical compound is the spectrum of frequencies or wavelengths of incident radiation that are absorbed by the compound due to electron transitions from a lower to a higher energy state. The emission spectrum refers to the spectrum of radiation emitted by the compound due to electron transitions from a higher to a lower energy state.
Light from many different sources contains various colors, each with its own brightness or intensity. A rainbow, or prism , sends these component colors in different directions, making them individually visible at different angles. A graph of the intensity plotted against the frequency (showing the brightness of each color) is the frequency spectrum of the light. When all the visible frequencies are present equally, the perceived color of the light is white, and the spectrum is a flat line. Therefore, flat-line spectra in general are often referred to as white , whether they represent light or another type of wave phenomenon (sound, for example, or vibration in a structure).
In radio and telecommunications, the frequency spectrum can be shared among many different broadcasters. The radio spectrum is the part of the electromagnetic spectrum corresponding to frequencies lower below 300 GHz, which corresponds to wavelengths longer than about 1 mm. The microwave spectrum corresponds to frequencies between 300 MHz (0.3 GHz ) and 300 GHz and wavelengths between one meter and one millimeter. [ 5 ] [ 6 ] Each broadcast radio and TV station transmits a wave on an assigned frequency range, called a channel . When many broadcasters are present, the radio spectrum consists of the sum of all the individual channels, each carrying separate information, spread across a wide frequency spectrum. Any particular radio receiver will detect a single function of amplitude (voltage) vs. time. The radio then uses a tuned circuit or tuner to select a single channel or frequency band and demodulate or decode the information from that broadcaster. If we made a graph of the strength of each channel vs. the frequency of the tuner, it would be the frequency spectrum of the antenna signal.
In astronomical spectroscopy , the strength, shape, and position of absorption and emission lines, as well as the overall spectral energy distribution of the continuum, reveal many properties of astronomical objects. Stellar classification is the categorisation of stars based on their characteristic electromagnetic spectra. The spectral flux density is used to represent the spectrum of a light-source, such as a star.
In radiometry and colorimetry (or color science more generally), the spectral power distribution (SPD) of a light source is a measure of the power contributed by each frequency or color in a light source. The light spectrum is usually measured at points (often 31) along the visible spectrum , in wavelength space instead of frequency space, which makes it not strictly a spectral density. Some spectrophotometers can measure increments as fine as one to two nanometers and even higher resolution devices with resolutions less than 0.5 nm have been reported. [ 7 ] the values are used to calculate other specifications and then plotted to show the spectral attributes of the source. This can be helpful in analyzing the color characteristics of a particular source.
A plot of ion abundance as a function of mass-to-charge ratio is called a mass spectrum. It can be produced by a mass spectrometer instrument. [ 8 ] The mass spectrum can be used to determine the quantity and mass of atoms and molecules. Tandem mass spectrometry is used to determine molecular structure.
In physics, the energy spectrum of a particle is the number of particles or intensity of a particle beam as a function of particle energy. Examples of techniques that produce an energy spectrum are alpha-particle spectroscopy , electron energy loss spectroscopy , and mass-analyzed ion-kinetic-energy spectrometry .
Oscillatory displacements , including vibrations , can also be characterized spectrally.
In acoustics , a spectrogram is a visual representation of the frequency spectrum of sound as a function of time or another variable.
A source of sound can have many different frequencies mixed. A musical tone 's timbre is characterized by its harmonic spectrum . Sound in our environment that we refer to as noise includes many different frequencies. When a sound signal contains a mixture of all audible frequencies, distributed equally over the audio spectrum, it is called white noise . [ 12 ]
The spectrum analyzer is an instrument which can be used to convert the sound wave of the musical note into a visual display of the constituent frequencies. This visual display is referred to as an acoustic spectrogram . Software based audio spectrum analyzers are available at low cost, providing easy access not only to industry professionals, but also to academics, students and the hobbyist . The acoustic spectrogram generated by the spectrum analyzer provides an acoustic signature of the musical note. In addition to revealing the fundamental frequency and its overtones, the spectrogram is also useful for analysis of the temporal attack , decay , sustain , and release of the musical note.
In the physical sciences , the spectrum of a physical quantity (such as energy ) may be called continuous if it is non-zero over the whole spectrum domain (such as frequency or wavelength ) or discrete if it attains non-zero values only in a discrete set over the independent variable , with band gaps between pairs of spectral bands or spectral lines . [ 13 ]
The classical example of a continuous spectrum, from which the name is derived, is the part of the spectrum of the light emitted by excited atoms of hydrogen that is due to free electrons becoming bound to a hydrogen ion and emitting photons, which are smoothly spread over a wide range of wavelengths, in contrast to the discrete lines due to electrons falling from some bound quantum state to a state of lower energy. As in that classical example, the term is most often used when the range of values of a physical quantity may have both a continuous and a discrete part, whether at the same time or in different situations. In quantum systems , continuous spectra (as in bremsstrahlung and thermal radiation ) are usually associated with free particles, such as atoms in a gas, electrons in an electron beam , or conduction band electrons in a metal . In particular, the position and momentum of a free particle has a continuous spectrum, but when the particle is confined to a limited space its spectrum becomes discrete.
Often a continuous spectrum may be just a convenient model for a discrete spectrum whose values are too close to be distinguished, as in the phonons in a crystal .
The continuous and discrete spectra of physical systems can be modeled in functional analysis as different parts in the decomposition of the spectrum of a linear operator acting on a function space , such as the Hamiltonian operator.
The classical example of a discrete spectrum (for which the term was first used) is the characteristic set of discrete spectral lines seen in the emission spectrum and absorption spectrum of isolated atoms of a chemical element , which only absorb and emit light at particular wavelengths . The technique of spectroscopy is based on this phenomenon.
Discrete spectra are seen in many other phenomena, such as vibrating strings , microwaves in a metal cavity , sound waves in a pulsating star , and resonances in high-energy particle physics . The general phenomenon of discrete spectra in physical systems can be mathematically modeled with tools of functional analysis , specifically by the decomposition of the spectrum of a linear operator acting on a functional space .
In classical mechanics , discrete spectra are often associated to waves and oscillations in a bounded object or domain. Mathematically they can be identified with the eigenvalues of differential operators that describe the evolution of some continuous variable (such as strain or pressure ) as a function of time and/or space.
Discrete spectra are also produced by some non-linear oscillators where the relevant quantity has a non- sinusoidal waveform . Notable examples are the sound produced by the vocal cords of mammals. [ 14 ] [ 15 ] : p.684 and the stridulation organs of crickets , [ 16 ] whose spectrum shows a series of strong lines at frequencies that are integer multiples ( harmonics ) of the oscillation frequency .
A related phenomenon is the appearance of strong harmonics when a sinusoidal signal (which has the ultimate "discrete spectrum", consisting of a single spectral line) is modified by a non-linear filter ; for example, when a pure tone is played through an overloaded amplifier , [ 17 ] or when an intense monochromatic laser beam goes through a non-linear medium . [ 18 ] In the latter case, if two arbitrary sinusoidal signals with frequencies f and g are processed together, the output signal will generally have spectral lines at frequencies | mf + ng |, where m and n are any integers.
In quantum mechanics , the discrete spectrum of an observable refers to the pure point spectrum of eigenvalues of the operator used to model that observable. [ 19 ] [ 20 ]
Discrete spectra are usually associated with systems that are bound in some sense (mathematically, confined to a compact space ). [ citation needed ] The position and momentum operators have continuous spectra in an infinite domain, but a discrete (quantized) spectrum in a compact domain and the same properties of spectra hold for angular momentum , Hamiltonians and other operators of quantum systems.
The quantum harmonic oscillator and the hydrogen atom are examples of physical systems in which the Hamiltonian has a discrete spectrum. In the case of the hydrogen atom the spectrum has both a continuous and a discrete part, the continuous part representing the ionization . | https://en.wikipedia.org/wiki/Spectrum_(physical_sciences) |
Spectrum Pharmaceuticals, Inc. is an American biopharmaceutical company located in Boston, MA . It develops and markets drugs for treatments in hematology and oncology .
After a deal valued at $248 million was announced in April 2023, [ 1 ] [ 2 ] Assertio Holdings announced that it had completed its acquisition of the Massachusetts -based company on July 31. [ 3 ]
In January 2019, Spectrum Pharmaceuticals sold its entire portfolio of hematology and oncology related products to Acrotech Biopharma USA, Inc., a New Jersey-based subsidiary of India's Aurobindo Pharma Ltd. The seven drugs had a combined sales of $76.4 million in the first three quarters of 2018.
The products sold were Fusilev (levoleucovorin), Folotyn (pralatexate injection), Zevalin (ibritumomab tiuxetan), Marqibo (vincristine sulfate LIPOSOME injection), Beleodaq (belinostat) for injection, Evomela (melphalan) for injection, and Khapzory (levoleucovorin).
The divestiture of seven marketed products infused non-dilutive capital back into the organization and enabled Spectrum to further advance its two cornerstone, value driving assets.
As of 2023, Spectrum has one FDA approved drug (ROLVEDON™ (eflapegrastim) is also formerly known as Rolontis) and one drug in advanced development (Poziotinib).
On September 9, 2022, FDA approved ROLVEDON™ (formerly known as Rolontis)(eflapegrastim-xnst) injection indicated to decrease the incidence of infection, as manifested by febrile neutropenia, in adult patients with non-myeloid malignancies receiving myelosuppressive anti-cancer drugs associated with clinically significant incidence of febrile neutropenia.
Development timeline for Rolvedon:
Sep 9, 2022 Spectrum announced FDA Approval of Rolvedon (eflapegrastim-xnst) Injection to Decrease the Incidence of Chemotherapy-Induced Neutropenia
April 11, 2022 FDA announced acceptance of Spectrum's re-submission of its Rolontis Biologics License Application (BLA)to FDA.
Aug 6, 2021
Spectrum received a Complete Response Letter (CRL) from FDA for Rolontis (eflapegrastim).
FDA cited to manufacturing deficiencies that required the manufacturing facilities to be re-inspected by FDA.
Oct. 24, 2019
Spectrum re-submitted an updated version of its Biologics License Application (BLA)for ROLONTIS (eflapegrastim) to FDA based on 643 early-stage breast cancer patients. This BLA version included additional information in the Chemistry, Manufacturing and Controls (CMC) section.
Mar 18, 2019
After receiving FDA's request for additional (CMC) chemistry-manufacturing-control related information for Rolontis (eflapegrastim), Spectrum voluntarily withdrew its 2018 Biologics License Application (BLA) from FDA as more than 60 days time would have beenneeded to provide the additional CMC-related information FDA required before March 29, 2019, the end date of FDA's initial 60-day review period.
Dec 27, 2018
Spectrum announced its submission of the Biologics License Application to the FDA for Rolontis (eflapegrastim) as a Treatment for Chemotherapy-Induced Neutropenia based on 643 early-stage breast cancer patients.
Poziotinib is a tyrosine kinase inhibitor in development for use in patients with previously treated locally advanced or metastatic non-small cell lung cancer (NSCLC) with HER2 exon 20 insertion mutations.
Nov 28, 2022
Spectrum announced the lay off 75% of R&D staff after FDA New Drug Application (NDA) rejection. Spectrum also announced plans to discontinue development of poziotinib, and to explore “strategic alternatives” for the Poziotinib program.
Nov 25, 2022
Spectrum received a Complete Response Letter from FDA for Poziotinib.
Sep 22, 2022
Spectrum provided an update on Poziotinib following the FDA Oncologic Drugs Advisory Committee (ODAC) Meeting regarding Spectrum's New Drug Application (NDA).
The ODAC voted 9–4 against a Poziotinib approval as the advisers judged the drug's risks outweighed its benefits.
Feb 11, 2022
Spectrum announced FDA acceptance of the New Drug Application (NDA) submitted for Poziotinib
Dec 6, 2021
Spectrum submitted the New Drug Application (NDA) for Poziotinib.
Mar 11, 2021
FDA Grants Fast Track Designation (FTD) to Spectrum for Poziotinib.
Dec 19, 2018
Spectrum announced that based on a subset of data from MD Anderson's ongoing Phase 2 study, FDA did not grant Breakthrough Therapy Designation (BTD) to Poziotinib for the treatment of patients with metastatic non-small cell lung cancer (NSCLC) whose tumors have EGFR exon 20 mutations.
Spectrum's BTD application included data from 30 patients from MD Anderson's Phase 2 study who had failed platinum-based chemotherapy.
Spectrum reported that the MDAnderson data demonstrated a confirmed objective response rate of 40% and median duration of response of 6.6 months.
Spectrum stated the safety profile in this subset was consistent with historical data published on poziotinib and other tyrosine kinase inhibitors. The historical objective response rates for mutation specific NSCLC patients range between 0% and 8% with tyrosine kinase inhibitors and for non-mutation specific NSCLC patients range between 0.8% and 22.9% with other treatments.
2018 Spectrum submits Poziotinib application for Breakthrough Therapy Designation (BTD) to FDA based on data reported on patients treated by John Heymach, MD and medical staff at the MD Anderson Cancer Center.
Sep 24, 2018
Spectrum Announced the release of updated Poziotinib data From MD Anderson's Phase 2 Study in Non-Small Cell Lung Cancer Patients.
May 3, 2018
Patent Licensing Agreement announced by and between The University of Texas MD Anderson Cancer Center (MDACC) and Spectrum Pharmaceuticals to cover discoveries by John Heymach, M.D., Ph.D., professor and chair of Thoracic/Head and Neck Medical Oncology, and Heymach's research staff at MDACC in the future.
The filed patents, if granted, are expected to extend the intellectual property protection to 2037.
Oct 27, 2017
Spectrum highlights Poziotinib Data in Non-Small-Cell Lung Cancer (NSCLC) from research results reported by Dr. John Heymach and The University of Texas MD Anderson Cancer Center.
At the 18th IASLC World Conference on Lung Cancer in Japan, Spectrum presented data reported by Dr. Heymach and the MD Anderson Cancer Center research team illustrating that Poziotinib demonstrated evidence of significant antitumor activity in NSCLC patients with EGFR exon 20 insertion mutations, with interim data showing an Objective Response Rate of 73%.
Evidence of central nervous system (CNS) activity in a patient with CNS metastasis and another with leptomeningeal disease (LMD).
March 17, 2017
First Poziotinib clinical trial patient enrolled by Dr. John Heymach of MD Anderson Cancer Center into the Investigator Study Protocol 2016–0783.
Feb. 23, 2017
Dr. John Heymach of M.D. Anderson Cancer Center initiated the Investigator IND as the Study Sponsor to of the clinical research, Protocol 2016–0783, “A Phase II Study of Poziotinib in EGFR or HER2 Mutant Advanced Solid Tumors” in collaboration with Spectrum and National Cancer Institute (NCI). | https://en.wikipedia.org/wiki/Spectrum_Pharmaceuticals |
A spectrum analyzer measures the magnitude of an input signal versus frequency within the full frequency range of the instrument. The primary use is to measure the power of the spectrum of known and unknown signals. The input signal that most common spectrum analyzers measure is electrical; however, spectral compositions of other signals, such as acoustic pressure waves and optical light waves, can be considered through the use of an appropriate transducer . Spectrum analyzers for other types of signals also exist, such as optical spectrum analyzers which use direct optical techniques such as a monochromator to make measurements.
By analyzing the spectra of electrical signals, dominant frequency, power , distortion , harmonics , bandwidth , and other spectral components of a signal can be observed that are not easily detectable in time domain waveforms . These parameters are useful in the characterization of electronic devices, such as wireless transmitters.
The display of a spectrum analyzer has frequency displayed on the horizontal axis and the amplitude on the vertical axis. To the casual observer, a spectrum analyzer looks like an oscilloscope , which plots amplitude on the vertical axis but time on the horizontal axis. In fact, some lab instruments can function either as an oscilloscope or a spectrum analyzer.
The first spectrum analyzers, in the 1960s, were swept-tuned instruments. [ 1 ]
Following the discovery of the fast Fourier transform (FFT) in 1965, the first FFT-based analyzers were introduced in 1967. [ 2 ]
Today, there are three basic types of analyzer: the swept-tuned spectrum analyzer, the vector signal analyzer, and the real-time spectrum analyzer. [ 1 ]
Spectrum analyzer types are distinguished by the methods used to obtain the spectrum of a signal. There are swept-tuned and fast Fourier transform (FFT) based spectrum analyzers:
Spectrum analyzers tend to fall into four form factors: benchtop, portable, handheld and networked.
This form factor is useful for applications where the spectrum analyzer can be plugged into AC power, which generally means in a lab environment or production/manufacturing area. Bench top spectrum analyzers have historically offered better performance and specifications than the portable or handheld form factor. Bench top spectrum analyzers normally have multiple fans (with associated vents) to dissipate heat produced by the processor . Due to their architecture, bench top spectrum analyzers typically weigh more than 30 pounds (14 kg). Some bench top spectrum analyzers offer optional battery packs , allowing them to be used away from AC power . This type of analyzer is often referred to as a "portable" spectrum analyzer.
This form factor is useful for any applications where the spectrum analyzer needs to be taken outside to make measurements or simply carried while in use. Attributes that contribute to a useful portable spectrum analyzer include:
This form factor is useful for any application where the spectrum analyzer needs to be very light and small. Handheld analyzers usually offer a limited capability relative to larger systems. Attributes that contribute to a useful handheld spectrum analyzer include:
This form factor does not include a display and these devices are designed to enable a new class of geographically-distributed spectrum monitoring and analysis applications. The key attribute is the ability to connect the analyzer to a network and monitor such devices across a network. While many spectrum analyzers have an Ethernet port for control, they typically lack efficient data transfer mechanisms and are too bulky or expensive to be deployed in such a distributed manner. Key applications for such devices include RF intrusion detection systems for secure facilities where wireless signaling is prohibited. As well cellular operators are using such analyzers to remotely monitor interference in licensed spectral bands. The distributed nature of such devices enable geo-location of transmitters, spectrum monitoring for dynamic spectrum access and many other such applications.
Key attributes of such devices include:
As discussed above in types , a swept-tuned spectrum analyzer down-converts a portion of the input signal spectrum to the center frequency of a band-pass filter by sweeping the voltage-controlled oscillator through a range of frequencies, enabling the consideration of the full frequency range of the instrument.
The bandwidth of the band-pass filter dictates the resolution bandwidth, which is related to the minimum bandwidth detectable by the instrument. As demonstrated by the animation to the right, the smaller the bandwidth, the more spectral resolution. However, there is a trade-off between how quickly the display can update the full frequency span under consideration and the frequency resolution, which is relevant for distinguishing frequency components that are close together. For a swept-tuned architecture, this relation for sweep time is useful:
Where ST is sweep time in seconds, k is proportionality constant, Span is the frequency range under consideration in hertz, and RBW is the resolution bandwidth in Hertz. [ 3 ] Sweeping too fast, however, causes a drop in displayed amplitude and a shift in the displayed frequency. [ 4 ]
Also, the animation contains both up- and down-converted spectra, which is due to a frequency mixer producing both sum and difference frequencies. The local oscillator feedthrough is due to the imperfect isolation from the IF signal path in the mixer .
For very weak signals, a pre-amplifier is used, although harmonic and intermodulation distortion may lead to the creation of new frequency components that were not present in the original signal.
With an FFT based spectrum analyzer, the frequency resolution is Δ ν = 1 / T {\displaystyle \Delta \nu =1/T} , the inverse of the time T over which the waveform is measured and Fourier transformed.
With Fourier transform analysis in a digital spectrum analyzer, it is necessary to sample the input signal with a sampling frequency ν s {\displaystyle \nu _{s}} that is at least twice the bandwidth of the signal, due to the Nyquist limit . [ 5 ] A Fourier transform will then produce a spectrum containing all frequencies from zero to ν s / 2 {\displaystyle \nu _{s}/2} . This can place considerable demands on the required analog-to-digital converter and processing power for the Fourier transform, making FFT based spectrum analyzers limited in frequency range.
Since FFT based analyzers are only capable of considering narrow bands, one technique is to combine swept and FFT analysis for consideration of wide and narrow spans. This technique allows for faster sweep time.
This method is made possible by first down converting the signal, then digitizing the intermediate frequency and using superheterodyne or FFT techniques to acquire the spectrum.
One benefit of digitizing the intermediate frequency is the ability to use digital filters , which have a range of advantages over analog filters such as near perfect shape factors and improved filter settling time. Also, for consideration of narrow spans, the FFT can be used to increase sweep time without distorting the displayed spectrum.
A realtime spectrum analyser does not have any blind time—up to some maximum span, often called the "realtime bandwidth". The analyser is able to sample the incoming RF spectrum in the time domain and convert the information to the frequency domain using the FFT process. FFT's are processed in parallel, gapless and overlapped so there are no gaps in the calculated RF spectrum and no information is missed.
In a sense, any spectrum analyzer that has vector signal analyzer capability is a realtime analyzer. It samples data fast enough to satisfy Nyquist Sampling theorem and stores the data in memory for later processing. This kind of analyser is only realtime for the amount of data / capture time it can store in memory and still produces gaps in the spectrum and results during processing time.
Minimizing distortion of information is important in all spectrum analyzers. The FFT process applies windowing techniques to improve the output spectrum due to producing less side lobes. The effect of windowing may also reduce the level of a signal where it is captured on the boundary between one FFT and the next. For this reason FFT's in a Realtime spectrum analyzer are overlapped. Overlapping rate is approximately 80%. An analyzer that utilises a 1024-point FFT process will re-use approximately 819 samples from the previous FFT process. [ 6 ]
This is related to the sampling rate of the analyser and the FFT rate. It is also important for the realtime spectrum analyzer to give good level accuracy.
Example: for an analyser with 40 MHz of realtime bandwidth (the maximum RF span that can be processed in realtime) approximately 50 Msample/second (complex) are needed. If the spectrum analyzer produces 250 000 FFT/s an FFT calculation is produced every 4 μs. For a 1024 point FFT a full spectrum is produced 1024 x (1/50 x 10 6 ), approximately every 20 μs. This also gives us our overlap rate of 80% (20 μs − 4 μs) / 20 μs = 80%.
Realtime spectrum analyzers are able to produce much more information for users to examine the frequency spectrum in more detail. A normal swept spectrum analyzer would produce max peak, min peak displays for example but a realtime spectrum analyzer is able to plot all calculated FFT's over a given period of time with the added colour-coding which represents how often a signal appears. For example, this image shows the difference between how a spectrum is displayed in a normal swept spectrum view and using a "Persistence" view on a realtime spectrum analyzer.
Realtime spectrum analyzers are able to see signals hidden behind other signals. This is possible because no information is missed and the display to the user is the output of FFT calculations. An example of this can be seen on the right.
In a typical spectrum analyzer there are options to set the start, stop, and center frequency. The frequency halfway between the stop and start frequencies on a spectrum analyzer display is known as the center frequency . This is the frequency that is in the middle of the display's frequency axis. Span specifies the range between the start and stop frequencies. These two parameters allow for adjustment of the display within the frequency range of the instrument to enhance visibility of the spectrum measured.
As discussed in the operation section, the resolution bandwidth filter or RBW filter is the bandpass filter in the IF path. It's the bandwidth of the RF chain before the detector (power measurement device). [ 7 ] It determines the RF noise floor and how close two signals can be and still be resolved by the analyzer into two separate peaks. [ 7 ] Adjusting the bandwidth of this filter allows for the discrimination of signals with closely spaced frequency components, while also changing the measured noise floor. Decreasing the bandwidth of an RBW filter decreases the measured noise floor and vice versa. This is due to higher RBW filters passing more frequency components through to the envelope detector than lower bandwidth RBW filters, therefore a higher RBW causes a higher measured noise floor.
The video bandwidth filter or VBW filter is the low-pass filter directly after the envelope detector . It's the bandwidth of the signal chain after the detector. Averaging or peak detection then refers to how the digital storage portion of the device records samples—it takes several samples per time step and stores only one sample, either the average of the samples or the highest one. [ 7 ] The video bandwidth determines the capability to discriminate between two different power levels. [ 7 ] This is because a narrower VBW will remove noise in the detector output. [ 7 ] This filter is used to "smooth" the display by removing noise from the envelope. Similar to the RBW, the VBW affects the sweep time of the display if the VBW is less than the RBW. If VBW is less than RBW, this relation for sweep time is useful:
Here t sweep is the sweep time, k is a dimensionless proportionality constant, f 2 − f 1 is the frequency range of the sweep, RBW is the resolution bandwidth, and VBW is the video bandwidth. [ 8 ]
With the advent of digitally based displays, some modern spectrum analyzers use analog-to-digital converters to sample spectrum amplitude after the VBW filter. Since displays have a discrete number of points, the frequency span measured is also digitised. Detectors are used in an attempt to adequately map the correct signal power to the appropriate frequency point on the display. There are in general three types of detectors: sample, peak, and average
The Displayed Average Noise Level (DANL) is just what it says it is—the average noise level displayed on the analyzer. This can either be with a specific resolution bandwidth (e.g. −120 dBm @1 kHz RBW), or normalized to 1 Hz (usually in dBm/Hz) e.g. −150 dBm(Hz).This is also called the sensitivity of the spectrum analyzer. If a signal level equal to the average noise level is fed there will be a 3 dB display. To increase the sensitivity of the spectrum analyzer a preamplifier with lower noise figure may be connected at the input of the spectrum analyzer. [ 9 ]
Spectrum analyzers are widely used to measure the frequency response , noise and distortion characteristics of all kinds of radio-frequency (RF) circuitry, by comparing the input and output spectra. For example, in RF mixers, spectrum analyzer is used to find the levels of third order inter-modulation products and conversion loss. In RF oscillators, spectrum analyzer is used to find the levels of different harmonics.
In telecommunications , spectrum analyzers are used to determine occupied bandwidth and track interference sources. For example, cell planners use this equipment to determine interference sources in the GSM frequency bands and UMTS frequency bands .
In EMC testing , a spectrum analyzer is used for basic precompliance testing; however, it can not be used for full testing and certification. Instead, an EMI receiver is used.
A spectrum analyzer is used to determine whether a wireless transmitter is working according to defined standards for purity of emissions. Output signals at frequencies other than the intended communications frequency appear as vertical lines (pips) on the display. A spectrum analyzer is also used to determine, by direct observation, the bandwidth of a digital or analog signal.
A spectrum analyzer interface is a device that connects to a wireless receiver or a personal computer to allow visual detection and analysis of electromagnetic signals over a defined band of frequencies. This is called panoramic reception and it is used to determine the frequencies of sources of interference to wireless networking equipment, such as Wi-Fi and wireless routers.
Spectrum analyzers can also be used to assess RF shielding. RF shielding is of particular importance for the siting of a magnetic resonance imaging machine since stray RF fields would result in artifacts in an MR image. [ 10 ]
Spectrum analysis can be used at audio frequencies to analyse the harmonics of an audio signal. A typical application is to measure the distortion of a nominally sinewave signal; a very-low-distortion sinewave is used as the input to equipment under test, and a spectrum analyser can examine the output, which will have added distortion products, and determine the percentage distortion at each harmonic of the fundamental. Such analysers were at one time described as "wave analysers". Analysis can be carried out by a general-purpose digital computer with a sound card selected for suitable performance [ 11 ] and appropriate software. Instead of using a low-distortion sinewave, the input can be subtracted from the output, attenuated and phase-corrected, to give only the added distortion and noise, which can be analysed. [ 12 ]
An alternative technique, total harmonic distortion measurement , cancels out the fundamental with a notch filter and measures the total remaining signal, which is total harmonic distortion plus noise; it does not give the harmonic-by-harmonic detail of an analyser.
Spectrum analyzers are also used by audio engineers to assess their work. In these applications, the spectrum analyzer will show volume levels of frequency bands across the typical range of human hearing , rather than displaying a wave. In live sound applications, engineers can use them to pinpoint feedback .
An optical spectrum analyzer uses reflective or refractive techniques to separate out the wavelengths of light. An electro-optical detector is used to measure the intensity of the light, which is then normally displayed on a screen in a similar manner to a radio- or audio-frequency spectrum analyzer.
The input to an optical spectrum analyzer may be simply via an aperture in the instrument's case, an optical fiber or an optical connector to which a fiber-optic cable can be attached.
Different techniques exist for separating out the wavelengths. One method is to use a monochromator , for example a Czerny–Turner design, with an optical detector placed at the output slit. As the grating in the monochromator moves, bands of different frequencies (colors) are 'seen' by the detector, and the resulting signal can then be plotted on a display. More precise measurements (down to MHz in the optical spectrum) can be made with a scanning Fabry–Pérot interferometer along with analog or digital control electronics, which sweep the resonant frequency of an optically resonant cavity using a voltage ramp to piezoelectric motor that varies the distance between two highly reflective mirrors. A sensitive photodiode embedded in the cavity provides an intensity signal, which is plotted against the ramp voltage to produce a visual representation of the optical power spectrum. [ 13 ]
The frequency response of optical spectrum analyzers tends to be relatively limited, e.g. 800–1600 nm (near-infrared), depending on the intended purpose, although (somewhat) wider-bandwidth general purpose instruments are available.
A vibration spectrum analyzer allows to analyze vibration amplitudes at various component frequencies, In this way, vibration occurring at specific frequencies can be identified and tracked. Since particular machinery problems generate vibration at specific frequencies, machinery faults can be detected or diagnosed. Vibration Spectrum Analyzers use the signal from different types of sensor, such as: accelerometers , velocity transducers and proximity sensors . The uses of a vibration spectrum analyzer in machine condition monitoring allows to detect and identify machine faults such as: rotor imbalance, shaft misalignment, mechanical looseness, bearing defects, among others. Vibration analysis can also be used in structures to identify structural resonances or to perform modal analysis. | https://en.wikipedia.org/wiki/Spectrum_analyzer |
A spectrum auction is a process whereby a government uses an auction system to sell the rights to transmit signals over specific bands of the electromagnetic spectrum and to assign scarce spectrum resources. Depending on the specific auction format used, a spectrum auction can last from a single day to several months from the opening bid to the final winning bid. With a well-designed auction, resources are allocated efficiently to the parties that value them the most, the government securing revenue in the process. [ 1 ] Spectrum auctions are a step toward market-based spectrum management and privatization of public airwaves, and are a way for governments to allocate scarce resources.
Alternatives to auctions include administrative licensing, [ clarification needed ] such as the comparative hearings conducted historically (sometimes referred to as "beauty contests" ), [ citation needed ] or lotteries.
In the past decade, telecommunications has turned into a highly competitive industry where companies are competing to buy valuable spectrum. This competition has been triggered by technological advancements, privatization, and liberalization . [ 2 ] Mobile communication in particular has made many transitions since 2000, mobile technology has moved from second generation (2G) to third generation (3G) to fourth generation (4G) and is now in transition to fifth generation (5G) technology.
With more providers in the mobile industry, the competition during spectrum auctions has increased due to more demand from consumers. When the United States made the transition in June 2009 from analog to digital broadcast television signals, [ 3 ] the valuable 700 MHz spectrum became available because it was no longer being used by analog TV signals.
In 2007, search giant Google announced that they would be entering the mobile business with their highly popular Android operating system and plans for a mobile broadband system. [ 4 ] Google said that they planned to bid for the "C" block of the spectrum auction which correspond to channels 54, 55, and 59 of the lower 700 MHz spectrum and channels 60, 61, 65, and 66 of the upper spectrum 700 MHz which are normally used to construct nationwide broadband services. Around the time of Google's announcement, AT&T and Verizon also announced plans to enter the spectrum auction in order to purchase "C" block spectrum. [ 4 ]
Advantages of auctions: [ 5 ]
As telecommunications is an area of federal jurisdiction, Innovation, Science and Economic Development Canada (ISED; formerly Industry Canada) holds nationwide spectrum auctions on behalf of the Canadian federal government. As per existing policy frameworks and statutes like the Competition Act , auctions are designed to encourage competition among telecom companies, and to not concentrate too much regional or economic power in the hands of single or a handful of firms. [ 8 ] [ 9 ] As such, auctions are generally set up to favour smaller telecom providers, such as setting aside certain wavelengths that the "Big Three" firms— Bell , Rogers and TELUS —are precluded from bidding on, or reserving certain valuable wavelengths (e.g. those that can more easily penetrate elevators or tunnels) for small firms. [ 10 ]
Canada held its first spectrum auction in 1999, for Broadband Wireless Access (BWA) spectrum in the 24 GHz and 38 GHz bands. [ 11 ]
In May 2008, ISED commenced an auction for 105 MHz of spectrum with 40 MHz reserved for new entrants. The auction concluded on July 23, 2008, after 331 rounds and raised $4.25 billion. [ 12 ]
In August 2011, Canada made the switch from analog to digital television, freeing up spectrum in the 700 MHz band for other uses. In February 2014, the country auctioned additional spectrum in the 700 MHz and 2500 MHz bands to the four major telecommunications players in the country and raised over $5.3 billion. [ 13 ] Christian Paradis , Minister of Industry at the time, was quoted as hoping that the auctioning of these two bands (sometimes referred to as "prime location") would help foster more competition in the telecom sector, particularly the wireless sector, where Canada is just beginning to feel the effects of competition from new wireless companies from the 2008 auction. [ 14 ]
In order to maximize compatibility and prevent cross-border interference, the FCC in the United States (America's broadcast regulator) and ISED agreed on August 14, 2015, to coordinate their frequencies in their border zone. [ 15 ] [ 16 ]
From 2000 to 2007–31 to 2000-08-18, the German government conducted an auction for 12 frequency blocks for the new UMTS mobile telephony standard. The total of the bids exceeded expectations by reaching the staggering amount of DEM 98.8 billions ( EUR 50.8 billions). (See de:Versteigerung der UMTS-Lizenzen in Deutschland )
In 2010, the highest bid in the German spectrum auction was 1,213 Million Euros for two blocks in the 800 MHz band [ 17 ]
India was among the early adopters of spectrum auctions beginning auctions in 1991. Despite the early start in auctions, services have been slow to roll out caused by unforeseen problems with the design and rules of the auction. [ 18 ] Potential service providers were required to seek foreign partners, as the Department of Telecom (DoT) felt that no Indian company alone had the financial means to enter the industry. Bidding for all licenses required a two-stage screening process.
Auctions did occur in 2015, where the government has obtained a revenue of ₹ 110,000 crore from spectrum allocation. [ 19 ]
On 2012-11-15 the Commission for Communications Regulation (ComReg) announced the results of its multi-band spectrum auction (Primarily for 4G (LTE)). [ 20 ] This auction awarded spectrum rights of use in the 800 MHz, 900 MHz and 1800 MHz bands in Ireland from 2013 to 2030. The winners of spectrum were 3 , Meteor , O 2 Ireland and Vodafone . All of the winning bidders in the auction have indicated that they intend to move rapidly to deploy advanced services. [ 21 ]
Licences were issued in respect of two time slices, the first ending contemporaneously with the expiry of the last existing licence in the 900 MHz band and close in time to the expiry of existing licences in the 1800 MHz band.
The auction consisted of: [ 22 ]
The existing licences in the 900 MHz and 1800 MHz bands were restricted to GSM use only. As the licences to be issued on foot of the auction are liberalised licences permitting use of the spectrum for UMTS , 4G and other technologies, existing licensees were permitted to bid to win their existing spectrum holdings on a liberalised basis and a rebate is payable in respect of the residual value of existing licences where this was done.
New Zealand's 1989 Radiocommunications Act of 1989 authorized Radio Spectrum Management (RSM) to create private property rights for spectrum and to use market-driven allocation mechanisms for the granting of these newly created licenses. Initially, spectrum licenses were sold using a tender system, but the first New Zealand spectrum auction was held in 1996, "making New Zealand the first country to sell rights to use spectrum in this way." [ 24 ] An internet-based computer system was developed for the second auction, held in 1998. [ 25 ]
During the year 2013 Telecommunications Regulatory Authority of the Slovak Republic concluded a CCA electronic auction for spectrum licences from the 800 MHz, 1800 MHz and 2600 MHz frequency bands. [ 26 ] These frequencies are reserved for operation of 4G networks (especially LTE technology).
The auction was accompanied by strict information embargo. Neither the public nor the auctioneers did not know who are the auctioneers nor how many auctioneers participates in the auction. During the primary clock rounds the auctioneers knew only a limited aggregate demand at the end of auction day.
Before the auction the Slovak Republic executed the process of releasing the 790–862 MHz frequency band (e.g. digital television transition ), defined as a digital dividend for broadband networks to provide electronic telecommunication services. The process of releasing formed a free range of 2 x 30 MHz, which after splitting into 6 blocks (each of size 2 x 5 MHz) was the subject of the auction. The maximum frequency spectrum that could be assigned to one company on the 800 MHz band was 2 x 10 MHz. Reserve price for each block was set at EUR 19 million.
Most frequencies from the 1800 MHz band had been already used to provide public electronic communication services in the Slovak republic. Before the auction three existing national mobile operators had leased 2 x 15.2 MHz each. Remaining fragments of frequencies with a total size of 2 x 20.4 MHz became the subject of the auction. These fragments ranged from 2 x 0.4 MHz to 2 x 10.6 MHz. The Authority created a total of 8 blocks in 7 categories with the largest blocks of 2 x 5 MHz. The reserve price ranged from EUR 200,000 to EUR 2,200,000 per block. The maximum frequency spectrum that could be assigned to one company on the 1800 MHz band was 2 x 20 MHz, thus existing mobile operators could gain only 2 x 4.8 MHz each.
According to ECC decision 2600 MHz band was split into two categories: FDD with 14 blocks of 2 x 5 MHz and TDD with 10 blocks of 1 x 5 MHz each. Reserve price was set at EUR 1.1 million per FDD block and EUR 400,000 per TDD block. In 2600 MHz frequency range no operator had leased the spectrum before the auction. The maximum frequency spectrum that could be assigned to one company on the 2600 MHz band was not limited.
Most of the frequencies were sold to the three existing national providers (Orange, Slovak Telekom, Telefónica Slovakia). [ 27 ] The auction brought also a new mobile operator, company called Swan. Total revenue of auction has been EUR 163.9 million that is 15% above the sum of reserve prices. All of the auctioned blocks were sold. The sold licences are valid till 2028.
Successful auctioneers undertook the obligation to enter into a contract with any parties interested in national roaming or wholesale offer.
On 2008-05-08 the Swedish Post and Telecom Authority (PTS) concluded an electronic 16-day simultaneous multiple-round ascending auction for nine 15-year 4G-licenses; for a total bandwidth of 190 MHz in the 2.6 GHz band. The total required minimum bids were SEK 50 million, but the total winning bids were SEK 2,099,450,000 (approx US$38.60 per inhabitant). [ 28 ]
In 2000, the Radiocommunications Agency of the UK government (now Ofcom ) raised £22.5 billion ( EUR 36.9 billion (2000)) from an auction of five licences for radio spectrum to support the 3G mobile telephony standard. [ 29 ] [ 30 ] The auction was conducted in a simultaneous ascending auction, similar to the US format with a slight deviation. In the UK's version of the simultaneous individual auction, each high bidder is only allowed to win one of the five auctions whereas in the US, many regions have multiple licences which multiple bidders can win. [ 31 ] After the auction in the UK, a severe recession in the telecom development industry was seen. [ citation needed ]
The 4G auction took place in 2013. [ 32 ] The results were:
MLL Telecom Ltd HKT (UK) Company Ltd was not a winning bidder.
On 13 April 2018, the UK telecoms regulator, Ofcom , announced the results of a spectrum auction of the 2.3 GHz band (for improved 4G capacity) and the 3.4 GHz band for future 5G mobile services. The results are:
The total of £1,369,879,000 will be paid to HM Treasury . [ 33 ]
In the United States , the Federal Communications Commission (FCC) conducts auctions of licenses for electromagnetic spectrum. The FCC has been conducting competitive auctions since 1994 rather than assigning spectra through comparative hearings under which the specific merits of each applicant is litigated, or through lotteries . [ 34 ] [ 35 ] Since July 1994, the FCC has conducted 87 spectrum auctions, which raised over $60 billion for the U.S. Treasury (not all of which has been collected). When initially planning and designing the spectrum auction, major telephone companies and the federal government relied on the input of various theorists including Paul Milgrom, Charles Plott, Barry Nalebuff, Preston McAfee, and John McMillan among others. [ 36 ] The auctions assigned thousands of licenses to hundreds of licensees. The auction approach is widely emulated throughout the world. To be considered a qualified [bidder] by the commission, companies or individuals have to submit an application and an upfront downpayment. FCC auctions are conducted electronically and are accessible over the Internet. [ 37 ] Bidders can follow the progress of an auction and view the results of each round. [ 34 ]
The FCC auctions have used a Simultaneous Multiple Round Auction (SMRA, also referred to as the Simultaneous Ascending Auction) in which groups of related licenses are auctioned simultaneously over many rounds of bidding. At the start of each round, bidders simultaneously make sealed bids for any licenses in which they are interested. When the bidding for the round has concluded, round results are posted, which include the identities of the new bids and bidders along with the standing high bid and the corresponding bidder. The initial standing high bid at the start of an auction is zero ($0) and the corresponding bidder is the auctioneer. As the auction progresses, the standing high bid changes to highest new bid and the corresponding bidder is the person who makes said bid. In addition to posting the round results, minimum bids for the next round are also posted. A minimum bid is computed from taking the standard high bid and adding a predetermined bid increment, such as 5% or 10%. [ 38 ] For an auction to come to a close there are several different options. McAfee, suggested that auctions should come to a close after a predetermined number of rounds, in which the license receives no new bids. [ 38 ] Wilson and Paul Milgrom of Stanford University proposed that all auctions should end simultaneously, when there is no new bid on a license. To date, the latter is used in the spectrum auctions. [ 38 ]
The US Congress set multiple goals for FCC when spectrum auction was first launched: "In designing auctions for spectrum licenses, the FCC is required by law to meet multiple goals and not focus simply on maximizing receipts. Those goals include ensuring efficient use of the spectrum, promoting economic opportunity and competition, avoiding excessive concentration of licenses, preventing the unjust enrichment of any party, and fostering the rapid deployment of new services, as well as recovering for the public a portion of the value of the spectrum." [ 39 ]
Despite the apparent success of spectrum auctions, important disadvantages limiting efficiency and revenues are demand reduction and collusive bidding.
The information and flexibility in the process of auction can be used to reduce auction prices by tacit collusion. When bidder competition is weak and one bidder holds an apparent advantage to win the auction for specific licenses, other bidders will often choose not to bid for higher prices, hence reducing the final revenue generated by the auction. [ citation needed ] In this case, the auction is best thought of as a negotiation among the bidders, who agree on who should win the auction for each discrete bit of spectrum. [ citation needed ] The foregoing notwithstanding, due to the complicated structure of spectrum auctions, it is not easy to identify collusive from non-collusive bidding, although simple analyses of bid behavior can provide an important basis for recommended changes to the rules and structure of the auctions. [ 40 ]
In certain cases, the FCC has imposed conditions on specific blocks of spectrum being auctioned. In the 2007 700 MHz auction, the FCC required the winning bidder of the C Block to comply with open platform conditions, "allow[ing] customers, device manufacturers, third-party application developers, and others to use or develop the devices and applications of their choice, subject to certain conditions." [ 41 ]
The FCC held the 700 MHz band spectrum auction on July 19, 2011. The auction, entitled "auction 92" featured 16 licenses which were left over from auction 73, as either unsold or not paid for. [ 42 ] Licenses available were from block A and B of the spectrum and included:
In mid 2015, the FCC began the 600 MHz incentive auction . This spectrum is "valuable" because of "high quality wireless airwaves which penetrate walls" and work better over long-distance than higher frequency spectra. The FCC said that a portion will be reserved for smaller carriers. [ 43 ] AT&T and Verizon control the majority of the high quality spectrum less than 1 Gigahertz, while Sprint and T-Mobile hold much less.
Broadcasters and mobile phone companies agreed on re-allocating 84 MHz of UHF TV broadcast spectrum, which is everything above UHF Channel 37. [ 44 ] Shown in the image to the right, the UHF band will be re-allocated to cell phones using 5-MHz-wide uplink and downlink blocks, with guard bands of varying width (3 MHz between UHF 37 and Uplink Block A and 11 MHz between Uplink Block G and Downlink Block A) to protect from adjacent-channel interference . Upon conclusion of the auction, television stations that choose to not accept a monetary offer to go off the air permanently (that is, to stay on the air) will have 39 months to migrate to a new, lower frequency (either on the UHF band, or the VHF-High or VHF-Low bands). If there are delays with preparing their broadcast towers for the new channel frequency, stations are eligible for a single 180-day (six-month) extension (similar to a Silent STA ), but they will have to cease broadcasts on their auction channel until they move. [ 45 ] [ 46 ]
In order to maximize compatibility and prevent cross-border interference, the FCC and Industry Canada (Canada's broadcast regulator) agreed on August 14, 2015, to coordinate their frequencies in their border zone. [ 15 ] [ 16 ]
On April 13, 2017, the FCC released the results of the 600 MHz incentive auction. T-Mobile US was the largest bidder and obtained nationwide low-band spectrum licenses for the first time in their history, with Dish Network , Comcast , and AT&T also gaining licenses in parts of the country. Verizon, who participated in the auction, ultimately purchased no licenses. [ 47 ] Lawrence Chu , an advisor to the FCC during the bidding process, considered the auction a success while admitting that "there will be some people disappointed on the broadcaster side." [ 48 ]
On November 14, 2018, the Federal Communications Commission (FCC) began auctioning spectrum for 5G services for the first time. Bidding opened for spectrum in the 28 GHz band, with a total of a little more than 3,000 country-based licenses up for grabs, and will be followed by an auction of spectrum in the 24 GHz band. [ 49 ] In total the FCC hopes to auction off around 6,000 licenses. [ 49 ] FCC Chairman Ajit Pai said the auction constituted "more spectrum than is currently used for terrestrial mobile broadband by all wireless service providers combined.” [ 50 ] The auction covered about a quarter of the U.S. airwaves. [ 51 ] Cox Communications was the only major U.S. cable operator to enter into the FCC's first auction of spectrum devoted to next-generation 5G services, with Comcast, Charter Communications and Altice USA not filing for the auctions. Along with Cox Communications, Dish Network, AT&T, Verizon, T-Mobile, Windstream and other telecom companies bid for licenses in the 24 GHz auction. Dish Network, AT&T, Verizon, T-Mobile, Windstream and Frontier Communications were among the companies bidding for 28 GHz licenses in the FCC's so-called “millimeter-wave” auction. [ 51 ]
Each spectrum auction has established rules and regulations, as outlined by the FCC. For example, the FCC published a Public Notice establishing procedures for their first 5G Spectrum Auctions in 2018. [ 52 ]
In a simultaneous ascending multiple-round (SMR) auction, all licenses are available for bidding at the same time throughout the entire auction, thus the term "simultaneous." Unlike most auctions in which bidding is continuous, SMR auctions have discrete, successive rounds, with the length of each round announced in advance by the commission.
Participants may bid on multiple licenses at once. After each round closes, round results are processed and made public. Only then do bidders learn about the bids placed by other bidders. This provides information about the value of the licenses to all bidders and increases the likelihood that the licenses will be assigned to the bidders who value them the most. The period between auction rounds also allows bidders to take stock of, and perhaps adjust, their bidding strategies.
In an SMR auction, there is no preset number of rounds. All licenses will continue to be for sale until there are none left or there are no buyers. In other words, bidding continues round after round, until a round occurs in which all bidder activity ceases. That round becomes the closing round of the auction. [ 53 ]
Information about anonymous bidding can be found under the section “Information Procedures During the Auction Process” within the FCC's Public Notice for any particular auction. For example, the established procedures for the FCC's 2018 5G Spectrum Auction 101 state that information to be made public after each round of bidding will include, for each license, the number of bidders that placed a bid on the license, the amount of every bid placed, whether a bid was withdrawn, the minimum acceptable bid amount for the next round, and whether the license has a provisionally winning bid. [ 54 ]
The amount of information available about the participants in the auctions is limited until the auction is officially over. This means that the amount of money a company has bid (or withdrawn) on a license will not be available to the public until the auction is closed. The names of the companies participating and what licenses they were going for when they filled out their applications is also protected. [ citation needed ] This rule also states that bidders are not allowed to cooperate with one another, or share bidding strategies, or have discussions on bids and what they would do with them in the market.
For each auction, the initial schedule for bidding rounds is typically released by the FCC in the Public Notice listing the qualified bidders before bidding in the auction starts. A Simultaneous Multiple-Round Auction offers every license for bid at the same time and consists of successive bidding rounds in which qualified bidders may place bids on individual licenses. Unless otherwise announced, bids will be accepted on all licenses in each round of the auction until bidding stops on every license. Each round of sequential bidding rounds is followed by the release of that round's results. Multiple bidding rounds may be conducted each day. [ 54 ]
Round results are released within approximately 15 minutes after each round closes. They are available for downloading, both to bidders and to the general public. Interested parties may perform detailed analysis by loading these data files into a spreadsheet program or the Auction Tracking Tool, which is provided by the FCC for most auctions. [ 37 ]
This rule was enacted to end the bidding at a reasonable time. When the FCC employs a simultaneous stopping rule approach to an auction, as it has in Auction 101 of its 2018 5G Spectrum Auctions, this means all licenses remain available for bidding until bidding stops on every license. Specifically, bidding will close on all licenses after the first round in which no bidder submits any new bids, applies a proactive waiver, or withdraws any provisionally winning bids. Bidding will remain open on all licenses until bidding stops on every license. [ 54 ] Also, in the first round, if participants don't make a bid, the auction will be closed. [ 37 ]
The FCC's Public Notice for any particular auction includes a section titled “Auction Delay, Suspension, or Cancellation” that outlines the cases in which an auction may be delayed, suspended, or cancelled. For example, the established procedures for the FCC's 2018 5G Spectrum Auction 101 state that at any time before or during the bidding process, the Bureau may delay, suspend, or cancel bidding in the auction in the event of a natural disaster, technical obstacle, network interruption, administrative or weather necessity, evidence of an auction security breach or unlawful bidding activity, or for any other reason that affects the fair and efficient conduct of competitive bidding. [ 54 ] The Bureau will notify participants of any such delay, suspension, or cancellation by public notice and/or through the FCC auction bidding system's announcement function. If the bidding is delayed or suspended, the Bureau may, in its sole discretion, elect to resume the auction starting from the beginning of the current round or from some previous round, or cancel the auction in its entirety. [ 54 ]
In order to prevent network congestion, FCC chairman Julius Genachowski sought companies who would voluntarily surrender their unused spectrum in exchange for a share of the money made from the spectrum auction. [ 55 ] With the growing demands for wireless services, the Obama Administration approved a plan, called the National Broadband Plan of making 500 MHz of spectrum, below 3 GHz, available over the next 10 years. The majority of the spectrum being examined by the FCC and NTIA are federally owned or federally shared bands. Regulators and carriers have also been considering blocks of the 300 MHz spectrum which is normally used for television broadcasters. [ 56 ] If a company agrees to volunteer their spectrum, the FCC will ask for 120 MHz of it. Also, the FCC has been thinking about spectrum sharing which would allow wireless ISPs to purchase DTV licenses
In January 2011, Clearwire agreed to sell off its unused spectrum in order to raise money for company spectrum and to seemingly allow other companies to pick up on some unused space. [ 57 ] | https://en.wikipedia.org/wiki/Spectrum_auction |
The Spectrum Commons theory states that the telecommunication radio spectrum should be directly managed by its users rather than regulated by governmental or private institutions. Spectrum management is the process of regulating the use of radio frequencies to promote efficient use and gain a net social benefit. [ 1 ] The theory of Spectrum Commons argues that there are new methods and strategies that will allow almost complete open access to this currently regulated commons with unlimited number of persons to share it without causing interference. This would eliminate the need for both a centralized, governmental management of the spectrum and the allocation of specific portions of the spectrum to private actors. [ 2 ]
The Spectrum Commons theory was developed to open up the spectrum to everyone. Users can share a spectrum as a commons without prior authorization from higher governance or regime. Proponents of spectrum commons theory believe government allocation of the spectrum is inefficient, and to be a true commons one must open up the spectrum to the users and minimize both government and private control. [ 3 ] The promise of the commons approach as one technologist, George Gilder once put it, "You can use the spectrum as much as you want as long as you don't collide with anyone else or pollute it with high-powered noise or other nuisances." [ 2 ]
The most basic characteristic of spectrum commons theory is the unlimited access to spectrum resources, but as most modern theorists point out, there is a need for some constraint of those resources. [ 4 ] A commons by definition is a resource that is owned or controlled jointly by a group of individuals. In order for a commons to be viable, someone must control the resource and set orderly sharing rules to govern its use. [ 2 ]
The radio spectrum is a shared resource that perhaps most strikingly affects the well being of society. Its use is governed by a set of rules and narrow restrictions, designed to limit interference, whose origins go back nearly a century. While in recent years some of those rules have been replaced by more flexible market like arrangements, the fundamental approach of this institution remains essentially unchanged. [ 5 ]
The early days of radio communication had no regulations, and everyone could use the spectrum without limitation. When a particular spectrum was filled up or overused, it created harmful interference. In order to manage the spectrum and prevent harmful interference, the NRA began to regulate the use of the spectrum. The period without regulation only lasted a few years, but this concept guided Spectrum Commons Theory. [ 4 ]
In the 1950s, economist Ronald Coase pointed out that the radio spectrum was no scarcer than wood or wheat, yet government did not routinely ration those items. Coase instead proposed the private ownership of, and a market in, spectrum, which would lead to a better allocation of the resource and avoid rent-seeking behavior by would be users of the spectrum. In the late 1990s, it seemed like the property rights view might carry the day as Congress finally allowed the FCC to auction licenses to use spectrum.
Radio spectrum is doled out to users by what the Federal Communications Commission calls a “command-and-control” process. The [FCC] first carves out a block of spectrum and decides to what use it will be put (e.g., television, mobile telephony). Then, the agency gives away, at no charge, the right to use the spectrum to applicants it deems appropriate. The FCC makes its choices based largely on a public record generated by a regulatory proceeding. The rationale for such a system has been that the radio spectrum is a scarce resource, that there are more people who would like to use it than there is space available, and thus that the government must apportion it lest there be chaos. [ 2 ]
Spectrum Commons Theory although conceptually tries to focus on functioning as a completely free and open environment, facts points to this idea as flawed. Complete open commons, is a regime under which anyone has access to an unowned resource without limitation; no one controls access to the resource under open access. As previously mentioned however, in order for a commons to be viable, someone must control the resource and set orderly sharing rules to govern its use. [ 2 ] While it is true that access to a commons can be open, this does not mean there is no central rule-setting authority. [ 3 ]
Complete open commons is not a feasible regime for spectrum because, as a scarce resource, it will be subject to tragedy. Even given new spectrum-sharing technologies, a controller is still needed because these technologies require standards setting and enforcement in order to function. [ 3 ]
Economists, who have long been skeptical about the ability of government agencies to allocate resources efficiently by “picking winners,” have preponderantly favored a market approach to the allocation of resources generally, and to the allocation of the spectrum in particular. As early as 1959, Ronald Coase wrote that spectrum was a fixed factor of production, like land or labor, and should be treated in the same way, with its use determined by the pricing system and awarded to the highest bidder. Coase concluded that government allocation of spectrum-use rights was not necessary to prevent interference and that, in fact, by preempting market allocation of spectrum, regulation was the source of extreme inefficiency.
Economists since Coase have favored a market-based approach if there is profit to be made from the charge of an entrance fee to such a park, then private enterprise and the profit motive can be relied upon to lead firms to carry out the necessary arrangements. And if entry into the commons is sufficiently beneficial to the entrants, there will indeed be profits to be made by giving them the opportunity to do so. [ 5 ]
Another way to expand on the Spectrum Commons Theory is looking at it as a supercommons. As Werbach points out, a supercommons can operate alongside the property and commons regimes, which are just different configurations of usage rights associated with spectrum. In other words, the commons would be the baseline, with property encompassed within it, rather than the reverse. Bandwidth would not need to be infinite to justify a fundamental reconceptualization of the spectrum debate. Even with real-world scarcity and transaction-cost constraints, a default rule allowing unfettered wireless communication would most effectively balance interests to maximize capacity. [ 6 ] The initial legal rule for this spectrum should be universal access. Anyone would be permitted to transmit anywhere, at any time, in any manner, so long as they did not impose an excessive burden on others. [ 6 ]
Which enables different communicate with one another and to choose nonconflicting frequencies or access points that will adjust their power levels to eliminate overlap. If this technology were able to reach a critical mass of adoption, even in localized areas, it could conceivably minimize those transaction costs necessary to adapt to neighboring uses of commons access spectrum. For neighboring buildings with scores of Wi-Fi transmitters , such technologies could prove very important, ensuring that different signals did not overlap and interfere with each other-thereby slowing data transmission and possibly triggering the destructive cycle of behavior noted above. Moreover, a logical extension of the swarm logic software is a function that could enable neighbors to identify those who deviated from accepted social norms in using commons access spectrum and, concomitantly, lower enforcement costs. Indeed, collective efforts-such as the Broadband Access Network Coordination ("BANC")-have already taken root to facilitate joint and controlled efforts to limit interference. [ 7 ] | https://en.wikipedia.org/wiki/Spectrum_commons_theory |
Spectrum continuation analysis (SCA) is a generalization of the concept of Fourier series to non-periodic functions of which only a fragment has been sampled in the time domain.
Recall that a Fourier series is only suitable to the analysis of periodic (or finite-domain) functions f ( x ) with period 2π. It can be expressed as an infinite series of sinusoids:
where F n {\displaystyle F_{n}} is the amplitude of the individual harmonics.
In SCA however , one decomposes the spectrum into optimized discrete frequencies. As a consequence, and as the period of the sampled function is supposed to be infinite or not yet known, each of the discrete periodic functions that compose the sampled function fragment can not be considered to be a multiple of the fundamental frequency:
As such, SCA does not necessarily deliver 2 π {\displaystyle 2\pi } periodic functions, as would have been the case in Fourier analysis.
For real-valued functions, the SCA series can be written as:
where A n and B n are the series amplitudes. The amplitudes can only be solved if the series of values ω n {\displaystyle \omega _{n}} is previously optimized for a desired objective function (usually least residuals ). C ( x ) {\displaystyle C(x)} is not necessarily the average value over the sampled interval: one might prefer to include predominant information on the behavior of the offset value in the time domain.
SCA deals with the prediction problem of continuing a frequency spectrum beyond a sampled (usually stochastic ) time series fragment. Unlike ordinary Fourier analysis that infinitely repeats an observed function period or time domain, SCA filters the exact composing frequencies out of the observed spectrum and let them continue (resp. precede) in the time domain.
In the scientific terminology, therefore preference is given to the term continuation rather than for instance extrapolation .
An algorithm is required to cope with several problems: detrending, decomposition, frequency resolution optimization, superposition, transformation and computational efficiency.
Since discrete Fourier transform is inherently related to Fourier analysis, this type of spectral analysis is by definition not suitable for spectrum decomposition in SCA. DFT (or FFT ) may provide however an initial approximation, which often speeds up the decomposition.
After decomposition of a discrete frequency, it should be filtered for optimal resolution (i.e. varying three parameters: frequency value, amplitude and phase).
Compared to DFT (or FFT ), which is characterized by perfect spectral resolution, but poor temporal information, SCA favours temporal information, but yields higher spectrum dispersion. This property shows where the analytic strength of SCA is located. For instance, discrete composing frequency resolution is by definition far better in SCA than in DFT. | https://en.wikipedia.org/wiki/Spectrum_continuation_analysis |
In model theory , a branch of mathematical logic , the spectrum of a theory is given by the number of isomorphism classes of models in various cardinalities . More precisely,
for any complete theory T in a language we write I ( T , κ ) for the number of models of T (up to isomorphism) of cardinality κ . The spectrum problem is to describe the possible behaviors of I ( T , κ ) as a function of κ . It has been almost completely solved for the case of a countable theory T .
In this section T is a countable complete theory and κ is a cardinal.
The Löwenheim–Skolem theorem shows that if I ( T , κ ) is nonzero for one infinite cardinal then it is nonzero for all of them.
Morley's categoricity theorem was the first main step in solving the spectrum problem: it states that if I ( T , κ ) is 1 for some uncountable κ then it is 1 for all uncountable κ .
Robert Vaught showed that I ( T ,ℵ 0 ) cannot be 2. It is easy to find examples where it is any given non-negative integer other than 2. Morley proved that if I ( T ,ℵ 0 ) is infinite then it must be ℵ 0 or ℵ 1 or 2 ℵ 0 . It is not known if it can be ℵ 1 if the continuum hypothesis is false: this is called the Vaught conjecture and is the main remaining open problem (in 2005) in the theory of the spectrum.
Morley's problem was a conjecture (now a theorem) first proposed by Michael D. Morley that I ( T , κ ) is nondecreasing in κ for uncountable κ . This was proved by Saharon Shelah . For this, he proved a very deep dichotomy theorem.
Saharon Shelah gave an almost complete solution to the spectrum problem. For a given complete theory T , either I ( T , κ ) = 2 κ for all uncountable cardinals κ , or I ( T , ℵ ξ ) < ℶ ω 1 ( | ξ | + ℵ 0 ) {\displaystyle \textstyle I(T,\aleph _{\xi })<\beth _{\omega _{1}}(|\xi |+\aleph _{0})} for all ordinals ξ (See Aleph number and Beth number for an explanation of the notation), which is usually much smaller than the bound in the first case. Roughly speaking this means that either there are the maximum possible number of models in all uncountable cardinalities, or there are only "few" models in all uncountable cardinalities. Shelah also gave a description of the possible spectra in the case when there are few models.
By extending Shelah's work, Bradd Hart, Ehud Hrushovski and Michael C. Laskowski gave the following complete solution to the spectrum problem for countable theories in uncountable cardinalities.
If T is a countable complete theory, then the number I( T , ℵ α ) of isomorphism classes of models is given for ordinals α>0 by the minimum of 2 ℵ α and one of the following maps:
Moreover, all possibilities above occur as the spectrum of some countable complete theory.
The number d in the list above is the depth of the theory.
If T is a theory we define a new theory 2 T to be the theory with an equivalence relation such that there are infinitely many equivalence classes each of which is a model of T . We also define theories ℶ n ( T ) {\displaystyle \beth _{n}(T)} by ℶ 0 ( T ) = T {\displaystyle \beth _{0}(T)=T} , ℶ n + 1 ( T ) = 2 ℶ n ( T ) {\displaystyle \beth _{n+1}(T)=2^{\beth _{n}(T)}} . Then I ( ℶ n ( T ) , λ ) = min ( ℶ n ( I ( T , λ ) ) , 2 λ ) {\displaystyle I(\beth _{n}(T),\lambda )=\min(\beth _{n}(I(T,\lambda )),2^{\lambda })} . This can be used to construct examples of theories with spectra in the list above for non-minimal values of d from examples for the minimal value of d . | https://en.wikipedia.org/wiki/Spectrum_of_a_theory |
Specula Melitensis Encyclica (“The Maltese Watchtower”) is a 1638 book by Fra Salvatore Imbroll, describing a machine invented by a Jesuit scholar Athanasius Kircher . It was printed in Naples by Secundino Roncagliolo [ 1 ] and dedicated to Giovanni Paolo Lascaris , Grand Master of the Knights of Malta . [ 2 ]
The book describes a machine that Kircher had devised while on a trip to Malta as the confessor of Friedrich of Hesse-Darmstadt . The machine was a combination mechanical calculator and a reference of contemporary scientific knowledge related primarily to astronomy, astrology, and medicine. The author of the document, Fra Salvatore Imbroll, seemed to have completed the project begun by Kircher. [ 3 ]
Kircher's work in 1638 predates that of Blaise Pascal , but comes after the mechanical calculators devised by John Napier and Wilhelm Schickard .
The full title reads
Specula Melitensis Encyclica , that is, a Syntagma of new Physico-Mathematical Instruments, in which anything pertaining to either Astronomy, or Physics, or disciplines related to them, by a new order and method, with utmost facility and brevity via wheels and dials artfully disposed, is seen orderly arranged. [ 2 ]
The Latin word specula means a watchtower. [ 4 ] The similarity of the words specula and speculum ("mirror") has led to the common mistranslation [ 5 ] [ 6 ] of the title as The Maltese Mirror or a variation of that. The intended meaning is explained in the first paragraph of the book:
This machine of ours is called Specula Melitensis both because of the similarity of its shape and form to that of a watchtower, and because, as from any watchtower, all in the circumference of the Horizon, being exposed far and wide, is revealed to an encounter, likewise as from this Specula, quicker than word, anything pertaining to Astronomy, Geography, Hydrography, Physics, and Medicine is revealed to the eye. [ 2 ]
The book opens with a dedication to Johannes (Giovanni) Paulus Lascaris , and a credit to the machine's creator, "the most erudite" Athanasius Kircher . It lauds the machine as the greatest of all the machines and instruments Kircher had invented. The introduction is dated January 6, 1638.
The work proper begins with a five-page Synopsis of That, Which in this Machine, or Syntagma, is Contained . The Synopsis gives a description of the machine's overall appearance and organization, its primary parts of interest being the Cube and the Pyramid.
The synopsis is followed by the Usage section, which makes up the bulk of the work. The section consists of 125 Propositions organized in six sections, one for each side of the Cube and one for the Pyramid. Each Proposition explains a specific use of the machine. [ 2 ]
The book does not include any illustrations or schematics. The machine is described as consisting of three principal parts: the Circle, the Cube, and the Pyramid.
The Circle forms the base, and features a representation of the immobile Horizon, and the information on the winds and the art of navigation.
The Cube is the middle part, with five of its sides (excluding the bottom) each dedicated to a specific Mathematical or "Physico-Logical" discipline as follows.
The first side, The Universal Chronoscope , contains eight wheels displaying Julian and Gregorian calendars.
The second side, The Cosmographical Mirror , includes a Horoscopium (not to be confused with a horoscope) which, given the current time in Malta, displays the time in any part of the world, and two planispheres showing the movement of the Primum Mobile and the eight spheres and the fixed stars, as well as the tides in all parts of the world.
The third side has in its middle a large wheel displaying the sunrise and sunset times, the sun's declination , star culminations , and other astronomical data by year, month, and day. The large wheel is surrounded by three additional wheels, and four more smaller wheels are arranged in the corners of that side of the Cube. The three wheels display Moon phases and astrological signs with their application to matters of agriculture, medicine and navigation, and physiognomical information. The four smaller wheels in the corners provide predictions based on planetary conjunctions .
The fourth side presents "the entire Medicine—Botanical, Chemical, Spagyrical , Hermetic , and Sympathetic ". It features four wheels. The first wheel contains information on the "simple and compound" medications of both mineral and animal origin. The second and the third wheels offer guidance in disease diagnostics. The fourth wheel, titled The Cabalistic Mirror , indicates which body part accommodates which medicine, and suggests the appropriate times for phlebotomy and the application of medicines.
The fifth side, which is the horizontal upper side of the Cube, shows the movement of the Sun, Moon, the planets, Caput and Cauda Draconis, and Zodiacal constellations, as well as their position at any given time.
The Pyramid consists of four parts corresponding to the four parts of the world, and describes the nature and languages of those parts. [ 2 ]
Specula Melitensis features prominently in Umberto Eco 's novel The Island of the Day Before . The description of the machine in the novel borrows liberally from Imbroll's book, however Kircher is not identified as the inventor. Instead, one of the characters, a Jesuit Father Caspar, finds a description of the machine in papers of his deceased brother, who in turn had learned about it from another brother who had traveled to Malta. The description is very brief and lacks any schematics or specifics, just like Imbroll's book. Father Caspar manages to construct a working machine from his understanding of the description. [ 6 ]
Importantly for the plot, in the novel the third side of the Cube contains a Horologium Catholicum (not found in the original), which shows the local time at any Jesuit mission in the world. The Horologium makes it possible to determine the longitude of the current location. [ 6 ] | https://en.wikipedia.org/wiki/Specula_Melitensis_Encyclica |
Speculative design is a design practice concerned with future design proposals of a critical nature. The term was popularised by Anthony Dunne and Fiona Raby as a subsidiary of critical design . The aim is not to present commercially-driven design proposals but to design proposals that identify and debate crucial issues that might happen in the future. [ 1 ] Speculative design is concerned with future consequences and implications of the relationship between science, technology, and humans. It problematizes this relation by proposing provocative future design scenarios where technology and design implications are accentuated. [ 2 ] These design proposals are meant to trigger debates about the future rather than marketing products. [ 3 ]
Dunne and Raby , the researchers who coined the term speculative design, describe it as:
“an activity where conjecture is as good as knowledge, where futuristic and alternative scenarios convey ideas, and where the goal is to emphasize implications of “mindless” decisions for mankind.”
Speculative design is used to challenge preconceptions, raise questions and to provoke debate. [ 4 ] It opens the door for designers to imagine possible futures.
James Auger claims speculative design "combines informed, hypothetical extrapolations of an emerging technology’s development with a deep consideration of the cultural landscape into which it might be deployed, to speculate on future products, systems and services”. [ 5 ]
Speculative designers develop alternative presents to ask why things are the way they are so that they can project the future. James Auger explains that these alternative presents can make radical interventions to the current practices and evolving technologies [ 5 ] by applying different ideologies and practices. [ 3 ]
Speculative design emphasizes the “philosophical inquiry into technological application”; it tends to take the discussion on technology beyond the experts to a broad population of the audience. [ 5 ] The resulting artifacts often appear subversive and irreverent in nature; they look different to the public, and this is the key behind triggering discussions and stimulating questions. [ 3 ]
Speculative design can be distinguished from design that operates within commercial borders where the aim of designing is profitability. Speculative design is an exploratory design genre and a Research through Design (RtD) approach. [ 6 ]
Anti-design and Italian radical design could be considered as ancestors of speculative design. [ 1 ] [ 2 ] However, the format of speculative design as we know it today is derived from the critical design practice. Both are connected and use similar approaches. Dunne and Raby described critical design as a practice that “uses speculative design proposals to challenge narrow assumptions, preconceptions, and givens about the role products play in everyday life”. [ citation needed ] Critical design is a form of design that uses design tools and process not to solve a problem but to rethink the borders and parameters of a problem from a critical point of view Dunne and Raby explained the term further in their book ‘Design Noir: The Secret Life of Electronic Objects,’ as “Instead of thinking about appearance, user-friendliness or corporate identity, industrial designers could develop design proposals that challenge conventional values”. [ citation needed ]
The relationship between speculative design and critical design can be seen from Matt Malpass identification of the current contemporary design practices into three classifications; the first is associative design, the second is speculative design, and the third is critical design. [ 2 ] Speculative design is a form of critical design that is concerned with future proposals. It examines future scenarios to ask the question of “what if?”.
Some attempts of the Italian radical design can be considered as speculative design. For instance, Ettore Sottsass worked on “The planet as a festival” in 1973. [ 7 ] Speculative design is inspired by the attitude and position of the Italian radical design, [ 1 ] yet does not necessarily imitate its format and motivations. [ 1 ]
Speculative design aims to defy capitalist-driven design directions and showcase their negative impacts on design practice. Dunne and Raby note that hyper-commercialization of design during the 1980s drove this practice. [ 1 ] [ 2 ] Designers struggled to find a social model to align with outside of the capitalist economy. However, after the financial crash of 2008 , the interest in finding other alternatives to the current design models was triggered. In this sense, the role of design is to be a catalyst in producing alternative visions rather than being the source of vision itself. [ 8 ] [ 9 ]
Speculative designers' motivation is to take a position or an attitude towards the current design practice and propose alternatives. [ 1 ] [ 10 ] Designers might have different points of view about how they would present a design idea or focal issue. Bruce and Stephanie Tharp identify the different positions designers could take towards their projects; these could be: declarative, suggestive, inquisitive, facilitative, and disruptive. [ 10 ]
Auger extends this discussion on explaining what speculative design should do by mentioning aspects for it:
Aiming at:
Speculative design relies on speculation and proposition; its value comes from speculating about future scenarios where design is used in a particular context to showcase a notion or an idea of debate. [ 2 ] The most significant aim of speculative design is to enact change rather than conforming to the status quo. According to Johannessen, Keitsch and Pettersen the change aspects can be segmented into three elements:
Speculative designers do not suggest what a preferable future is; they let society decide what is a preferable future for them, whereas affirmative design, government, and industries actually decide on their preferable future and create it. [ 1 ] It encourages the audience to suggest their preferable future that has no direct relevance with today’s perspective of how the future should be and this raises the awareness for society on how they could influence their choices for the future; [ 8 ] the logic of the ‘laws’ of future implies that if we strive for something, we can eventually turn it into reality, even if it seems incredible now. [ 13 ]
Speculative design triggers the debate about the actions we take today (in the present) that build future events. It encourages the users to be the change of today. It questions technology at early stages; it is concerned with the domestication of technology and upstream engagement. It poses societal and ethical implications to interrogate them. It questions the role of industrial and product design in delivering new science and technology. [ 2 ] Speculative design as a subsidiary of critical design is built on the fundamentals Frankfurt school of criticism. Therefore, critical thinking is an essential aspect of speculative design. [ 14 ] Critiquing norms, values and why we design is what motivates speculative designers.
Design is a future-oriented practice by nature. [ 15 ] [ 16 ] However, the issue lies in the fact that vast majority of designers tend to abide by technological advancements without interrogating them or questioning the implications of such technology. [ 8 ] An example of this is the wide adoption of social media and how this affected society (for example, the social dilemma ).
Designers, in this case, do not attempt to change the future, but rather they tend to adapt their design towards what they can see as a probable future. In this sense, they see it as something that they cannot change. In this context, speculative design aims to influence change by raising questions and provoking debates by implementing designed objects. Speculative design uses objects or prototypes that do imply implicit meanings about complex social and technological issues. [ citation needed ]
To highlight the differences between affirmative design and speculative design, Dunne and Raby introduced the A/B Manifesto to contrast their meanings and to highlight what does it mean to be critical or speculative in design. [ 1 ]
Speculative design can be seen as an attitude, stance, or position instead of a process or methodology.
Tactics, methods, and strategies for speculative design have wide variation. It depends on the designer’s intention and the careful management of the outcome of the design project. [ 14 ]
Speculative design needs a “perceptual bridge” between what the audience identifies as their reality and the fictional elements in the speculative concept. [ 5 ]
Tactics and strategies of speculative design:
and the outcome of speculative design can be a project in the form of:
Speculative design has many adjacent practices including critical design , discursive design, [ 10 ] and design fiction . They share similar motivations but different purposes or target areas.
The most significant criticism for critical and speculative design would be based on the understanding that design is not functional or useful, so it cannot be considered as design. The grounds for criticism are built on the basic understanding of design as a problem-solving activity. In contrast, speculative design is concerned with problem finding. It does not create functional objects at the end but rather problematizes an issue or social implication. [ 17 ] [ 2 ]
Other criticism would be directed towards speculative design as it does sometimes present dystopian futures that do resemble the lives of other parts of the world. It can sometimes be considered as a niche practice that is only presented in highly intellectual venues such as MOMA and V&A Museum , as pointed out by Prado & Oliveira in 2014. [ 18 ] [ 8 ]
Another criticism for speculative design is dissemination and reflection. The format and venues of presenting speculative design proposals do not imply a methodological approach for engaging with the audience and broader society. This is what Bruce and Stephanie Tharp call (a message in a bottle). [ 10 ] [ 19 ] | https://en.wikipedia.org/wiki/Speculative_design |
A speech droplet or speaking droplet is a particle of saliva involuntarily expelled from the mouth during speech, especially during vigorous articulation or the pronunciation of explosive consonants (such as /p/, /b/, /t/). These droplets, distinct from respiratory droplets , are produced by fluid dynamics in the oral cavity rather than by pulmonary exhalation. [ 1 ]
They form when tongue movements, lip bursts, or airflow turbulence disrupt the thin saliva film coating the mouth, ejecting droplets into the air. [ 2 ]
Speach droplets are typically: [ 3 ]
Modern studies (Anfinrud et al., 2020) have quantified their role in disease spread, particularly during the COVID-19 pandemic, where masks reduced their dispersion by 99 %. [ 4 ]
The study of speach droplets spans centuries, with early observations rooted in public hygiene, theater performance, and later, germ theory. | https://en.wikipedia.org/wiki/Speech_droplet |
Speed bumps (also called traffic thresholds , speed breakers or sleeping policemen ) are a class of traffic calming devices that use vertical deflection to slow motor-vehicle traffic in order to improve safety conditions. Variations include the speed hump , speed cushion , and speed table .
The use of vertical deflection devices is widespread around the world, and they are most commonly used to enforce a speed limit under 40 km/h (25 mph).
Although speed bumps are effective in keeping vehicle speeds down, their use is sometimes controversial—as they can increase traffic noise, may damage vehicles if traversed at too great a speed (despite that being the point), and slow emergency vehicles. Poorly-designed speed bumps that stand too tall or with too-sharp an angle can be disruptive for drivers, and may be difficult to navigate for vehicles with low ground clearance , even at very low speeds. Many sports cars have this problem with such speed bumps. Speed bumps can also pose serious hazards to motorcyclists and bicyclists if they are not clearly visible, though in some cases a small cut across the bump allows those vehicles to traverse without impediment.
Each of these devices can be made from a variety of materials, including asphalt , concrete , recycled plastic , metal , or vulcanized rubber . Several trade-offs must be made when selecting the material for a new speed cushion. Traditionally most vertical deflection devices have been constructed of asphalt or concrete. Due to the rigidity and durability of these materials, they have more permanence and are more effective at slowing traffic. However, they can be difficult to shape and form into consistent forms and precise dimensions.
Rubber products are pre-shaped to standard sizes to meet industry standards. Preformed rubber products are typically bolted down, making them easier to install or remove. Temporary bolt-down installations can be ideal for planners in testing the use and positioning of speed bumps before implementing them in a larger project. Bolt-down products can also be removed or relocated during winter snow periods—where speed bumps are easily concealed and may be damaged by snowplows.
On June 7, 1906, The New York Times reported on an early implementation of what might be considered speed bumps in Chatham, New Jersey , which planned to raise its crosswalks five inches (13 cm) above the road level: "This scheme of stopping automobile speeding has been discussed by different municipalities, but Chatham is the first place to put it in practice". [ 1 ] The average automobile's top speed at the time was around 50 km/h (30 mph), but braking was poor by modern standards. [ citation needed ]
Arthur Holly Compton was a physicist and winner of the Nobel Prize in physics in 1927 for his discoveries resulting in major changes in electromagnetic theory . He is commonly known for his work on the Compton Effect with X-rays . He also invented what he called "traffic control bumps", the basic design for the speed hump, in 1953. Compton began designs on the speed bump after noticing the speed at which motorists passed Brookings Hall at Washington University in St. Louis, Missouri , where he was chancellor . [ 2 ]
The British Transport and Road Research Laboratory published a comprehensive report in 1973 examining vehicle behavior for a large variety of different bump geometries. [ 3 ] At the time speed humps were not permitted on public roads but had been installed on private roads.
According to a publication by the Institute of Transportation Engineers , the first speed bump in Europe was built in 1970 in the city of Delft in the Netherlands . [ 4 ]
A speed bump is also known as a sleeping policeman in British English , Maltese English and Caribbean English , a judder bar in New Zealand English , and a lying-down policeman in Colombia , Dominican Republic , Hungary , Croatia , Serbia , Estonia , Lithuania , Slovenia , Bulgaria and Russia . A speed bump is a bump in a roadway with heights typically ranging between 8 and 10 centimetres (3 and 4 in). The traverse distance of a speed bump is typically less than or near to 0.3 m (1 ft); contrasting with the wider speed humps, which typically have a traverse distance of 3.0 to 4.3 m (10 to 14 ft). [ 5 ] [ 6 ]
Speed bumps are used in parking lots and on small-neighborhood roads where space and cost are limited. They are being replaced by Speed Humps (discussed in this Wikipedia section) in higher-traffic areas where speed bumps would be ineffective because bumps are mere blips to law-breaking speeders, while law-abiding drivers must slow to far below the speed limit to avoid large vehicle accelerations and displacements. These are the counter-productive results produced by unavoidable dynamic vehicle response. (See https://lindberglce.com/tech/Worst_Roads.PDF ).
Because of these counter-productive responses, speed bumps traversals of at least 2 m (about 5 ft.) with smooth approach and exit should be used wherever possible.
Speed bumps vary in length, but it is typical to leave space between the bump and either edge of an enclosed road (i.e. with curbs and gutters) to allow for drainage. Spaces on either side may also allow more expedient passage for emergency vehicles, though effectiveness will depend on the type of vehicle and specific road design.
Local authorities have cited disadvantages to speed bumps:
Other sources argue that speed bumps:
In 2003, the chairman of the London Ambulance Service, Sigurd Reinton claimed that delays caused by speed bumps were responsible for up to 500 avoidable deaths from cardiac arrest each year. He later denied the statement. [ 10 ]
In Sweden, an evaluation of spinal stress in bus drivers against ISO 2631-5 required on health grounds that: [ 11 ]
Speed bumps can also have adverse environmental impact. A study found that in one north London street with a speed limit of 20 miles per hour (32 km/h; 8.9 m/s) and fitted with road humps, a petrol driven car produced 64 per cent more nitrogen dioxide (NO 2 ) than in a similar 20 miles per hour (32 km/h; 8.9 m/s) street fitted with road cushions. It also produced 47 per cent more particulate matter (PM) and nearly 60 per cent more carbon monoxide (CO) emissions. [ 12 ] Another study estimated that, for a private automobile, the increase in fuel consumption due a pass over a speed bump is responsible for fuel waste of 10ml. [ 13 ] This multiplied with the number of vehicles going over a particular speed bump every day suggests significant annual fuel wastage for a single speed bump.
Dynamic speed bumps differ from conventional speed bumps in that they only activate if a vehicle is traveling above a certain speed. Vehicles traveling below this speed will not experience the discomfort of a conventional speed bump. Dynamic speed bumps may allow the passage of emergency vehicles at higher speeds.
The Actibump system, successfully used in Sweden, is based on powered equipment integrated into the road surface, which operates a platform that is lowered a few centimeters when a speeding vehicle approaches. Any vehicle approaching at or under the speed limit will pass on a level road. The system measures the speed of an oncoming vehicle by using radar. [ 14 ]
In another design, a rubber housing is fitted with a pressure relief valve that determines the speed of a vehicle. If the vehicle is traveling below the set speed, the valve opens allowing the bump to deflate as the vehicle drives over it, but it remains closed if the vehicle is traveling too fast. The valve can also be set to allow heavy vehicles, such as fire trucks , ambulances , and buses to cross at higher speeds. [ 15 ] [ 16 ]
A speed hump (also called a road hump , or undulation , [ 17 ] and speed ramp ) is a rounded traffic calming device used to reduce vehicle speed and thus sound volume on residential streets. Humps are placed across the road to slow traffic and are often installed in a series of several humps to prevent cars from speeding before and after the hump. Common speed hump shapes are parabolic , circular, and sinusoidal . [ 17 ] In Norway , speed humps are often placed at pedestrian crossings.
Generally, speed humps have a traverse distance of about 3.7 to 4.3 m (12 to 14 ft) and span the width of the road. The height of each hump ranges from 8 to 10 cm (3 to 4 in). [ 17 ] The traverse distance and height of each hump determines the speed at which traffic will travel over the devices. Shorter traverse lengths and greater heights slow cars most drastically. When placed in a series 110–170 m (350–550 ft) apart, humps will reduce 85th percentile speeds by 13–16 km/h (8–10 mph). [ 18 ]
Warning signs should be used to notify approaching motorists of upcoming humps. Humps generally have pavement markings to enhance visibility and a taper edge near the curb to allow a gap for drainage. [ 17 ]
Speed humps are used in locations where low speeds are desired and suitable for the surrounding traffic environment. [ 19 ] Speed humps are typically placed on residential roads and are not used on major roads, bus routes, or primary emergency response routes. Placement is generally mid-block between intersections.
Speed humps typically limit vehicle speeds to about 25–30 km/h (15–20 mph) at the hump and 40–50 km/h (25–30 mph) at the midpoint between humps, depending on spacing. Studies show an average 18% reduction in traffic volume and an average 13% reduction in collisions. [ 17 ]
While similar to speed bumps, humps are less aggressive than speed bumps at low speeds. Humps are often used on streets, while bumps are used more in parking lots. [ 20 ] While speed bumps generally slow cars to 10–15 km/h (5–10 mph), humps slow cars to 25–30 km/h (15–20 mph). The narrow traverse distance of speed bumps often allows vehicles to pass over them at high speed with only mild disturbance to the wheels and suspension, and hardly affecting the vehicle cab and its occupants. The relatively long slopes of speed humps are less disruptive at low–moderate speeds, but they create a greater, more sustained vertical deflection; at higher speeds, a more sustained deflection is less-absorbed by vehicle suspensions and has a greater effect on the vehicle as a whole. [ 21 ]
One problematic aspect of speed humps is their effect on emergency vehicles . Response time is slowed by 3–5 seconds per hump for fire trucks and fire engines and up to 10 seconds for ambulances with patients on board. [ 17 ] Speed humps are thus usually not placed on primary response routes. Speed cushions may be placed on these routes instead.
Occasionally, there is an increase in traffic noise from braking and acceleration of vehicles on streets with speed humps, particularly from buses and trucks. Other effects include increased fuel consumption and emissions [ citation needed ] as well as increased wear and tear on brakes, engine and suspension components.
Damage caused by snow plows during the winter months is an additional concern.
Heavy sedans , trucks , and SUVs are less affected by speed humps, and may not have to slow down as dramatically.
Thin cuts are sometimes placed in the middle of a hump in order to allow bicycle traffic to pass through. However, forcing cyclists to take a particular line on the road compromises their ability to position themselves safely according to the other traffic on the road at the time.
Speed cushions are a type of speed hump installation designed to alleviate the negative impacts that vertical deflections have on emergency vehicle response times. Speed cushions installations are typically made up of several small speed humps installed across the width of the road with spaces between them. They force normal cars to slow down as they ride with one or both wheels over the humps. Meanwhile, they allow fire engines (and other large vehicles) with wider axles to straddle the cushions without slowing down. [ 22 ] [ 23 ]
Wider, American-style ambulances might also be able to straddle speed cushions. However, in Europe and Australia, where vehicles like the Mercedes-Benz Sprinter are used most frequently as ambulances, there is no advantage. In these jurisdictions, narrower speed cushions are sometimes placed between lanes to allow ambulances to pass unobstructed while driving over the centre line during an emergency.
Speed cushions have several distinct advantages over similar traffic calming devices. Many municipalities are challenged by opposition to speed humps and speed tables since they slow down emergency vehicles and buses. Speed cushions address this problem by allowing larger vehicles to straddle the cushion without slowing down. This is also an advantage for buses , as lower floor vehicles can sometimes ground out on traditional humps. [ 24 ] Speed cushions are often less costly than speed humps or tables, and most cities report them to be just as effective. In some jurisdictions, narrower speed cushions are placed at more frequent intervals to allow ambulances to pass while driving over the centre line. Large trucks are also not slowed down.
Development of speed cushions has focused primarily on the European context. European vehicles typically have a narrower track width than American vehicles, meaning their left and right wheels are closer together. Emergency vehicles still feature a wide track width, and the difference between them makes speed cushions more applicable.
In North America, however, consumer vehicles have a track width of 1,300–1,500 millimetres (50–59 in). Many emergency vehicles are also equipped with dual tires on their rear axles. The additional tires limit track width to as narrow as 1,200 millimetres (48 in), meaning speed cushions may not be suitable for their intended use. [ citation needed ]
A speed table (also called a bus-friendly hump , flat top hump , or raised pedestrian crossing ) is designed as a long speed hump with a flat section in the middle. Speed tables are generally long enough for the entire wheelbase of a passenger car to rest on top. [ 25 ] The long, flat design allows cars to pass without slowing as significantly as with speed humps or cushions. [ 26 ] Because they slow cars less than similar devices, speed tables are often used on roads with typical residential speed limits.
Speed tables can also be signed as pedestrian crossings , namely zebra crossings . A raised zebra crossing is referred to as a wombat crossing in Australia. [ 27 ] Other road features may be included, such as junctions , or even mini-roundabouts . Speed tables are used with zebra crossings repeatedly in Leighton Buzzard .
Typical speeds resulting from 7-metre (22 ft) speed tables are 32–48 kilometres per hour (20–30 mph). One sample of 8 sites found a 45% decrease in accidents per year with the use of speed tables. [ 26 ] Wombat crossings may reduce casualties by 63%. [ 27 ]
Speed tables are effective in calming traffic on streets where the speed limit needs to be maintained rather than slowing cars more significantly. Traffic speed, volumes, and accidents have been shown to decrease with the use of tables. Although not as responsive to emergency vehicles as speed cushions, speed tables cause less of a delay than humps and are typically preferred by fire departments over speed humps. [ 25 ]
In the UK, vertical deflection in highways for the purpose of traffic calming typically takes one of the following forms:
The Department for Transport defines the regulations for the design and use of road humps. [ 28 ]
Speed bumps in some areas have been removed after protests by local residents. Such protests cite the lack of any consultation as one factor. [ 29 ] For example, complaints from Derby residents prompted the removal of 146 speed bumps from streets at a cost of £460,000. Similar incidents have been reported elsewhere in the UK. [ 30 ] UK news sources reported a cyclist being killed in a crash while attempting to avoid a speed bump. [ 31 ] | https://en.wikipedia.org/wiki/Speed_bump |
In telecommunication, speed of service is the time for a message to be received. For example:
This article related to telecommunications is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Speed_of_service |
The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elastic medium. More simply, the speed of sound is how fast vibrations travel. At 20 °C (68 °F), the speed of sound in air is about 343 m/s (1,125 ft/s ; 1,235 km/h ; 767 mph ; 667 kn ), or 1 km in 2.92 s or one mile in 4.69 s . It depends strongly on temperature as well as the medium through which a sound wave is propagating.
At 0 °C (32 °F), the speed of sound in dry air (sea level 14.7 psi) is about 331 m/s (1,086 ft/s; 1,192 km/h; 740 mph; 643 kn). [ 1 ]
The speed of sound in an ideal gas depends only on its temperature and composition. The speed has a weak dependence on frequency and pressure in dry air, deviating slightly from ideal behavior.
In colloquial speech, speed of sound refers to the speed of sound waves in air . However, the speed of sound varies from substance to substance: typically, sound travels most slowly in gases , faster in liquids , and fastest in solids .
For example, while sound travels at 343 m/s in air, it travels at 1481 m/s in water (almost 4.3 times as fast) and at 5120 m/s in iron (almost 15 times as fast). In an exceptionally stiff material such as diamond , sound travels at 12,000 m/s (39,370 ft/s), [ 2 ] – about 35 times its speed in air and about the fastest it can travel under normal conditions.
In theory, the speed of sound is actually the speed of vibrations. Sound waves in solids are composed of compression waves (just as in gases and liquids) and a different type of sound wave called a shear wave , which occurs only in solids. Shear waves in solids usually travel at different speeds than compression waves, as exhibited in seismology . The speed of compression waves in solids is determined by the medium's compressibility , shear modulus , and density. The speed of shear waves is determined only by the solid material's shear modulus and density.
In fluid dynamics , the speed of sound in a fluid medium (gas or liquid) is used as a relative measure for the speed of an object moving through the medium. The ratio of the speed of an object to the speed of sound (in the same medium) is called the object's Mach number . Objects moving at speeds greater than the speed of sound ( Mach 1 ) are said to be traveling at supersonic speeds .
In Earth's atmosphere, the speed of sound varies greatly from about 295 m/s (1,060 km/h; 660 mph) at high altitudes to about 355 m/s (1,280 km/h; 790 mph) at high temperatures.
Sir Isaac Newton 's 1687 Principia includes a computation of the speed of sound in air as 979 feet per second (298 m/s). This is too low by about 15%. [ 3 ] The discrepancy is due primarily to neglecting the (then unknown) effect of rapidly fluctuating temperature in a sound wave (in modern terms, sound wave compression and expansion of air is an adiabatic process , not an isothermal process ). This error was later rectified by Pierre-Simon Laplace . [ 4 ]
During the 17th century there were several attempts to measure the speed of sound accurately, including attempts by Marin Mersenne in 1630 (1,380 Parisian feet per second), Pierre Gassendi in 1635 (1,473 Parisian feet per second) and Robert Boyle (1,125 Parisian feet per second). [ 5 ] In 1709, the Reverend William Derham , Rector of Upminster, published a more accurate measure of the speed of sound, at 1,072 Parisian feet per second. [ 5 ] (The Parisian foot was 325 mm . This is longer than the standard "international foot" in common use today, which was officially defined in 1959 as 304.8 mm , making the speed of sound at 20 °C (68 °F) 1,055 Parisian feet per second).
Derham used a telescope from the tower of the church of St. Laurence, Upminster to observe the flash of a distant shotgun being fired, and then measured the time until he heard the gunshot with a half-second pendulum. Measurements were made of gunshots from a number of local landmarks, including North Ockendon church. The distance was known by triangulation , and thus the speed that the sound had travelled was calculated. [ 6 ]
In a gas or liquid, sound consists of compression waves. In solids, waves propagate as two different types. A longitudinal wave is associated with compression and decompression in the direction of travel, and is the same process in gases and liquids, with an analogous compression-type wave in solids. Only compression waves are supported in gases and liquids. An additional type of wave, the transverse wave , also called a shear wave , occurs only in solids because only solids support elastic deformations. It is due to elastic deformation of the medium perpendicular to the direction of wave travel; the direction of shear-deformation is called the " polarization " of this type of wave. In general, transverse waves occur as a pair of orthogonal polarizations.
These different waves (compression waves and the different polarizations of shear waves) may have different speeds at the same frequency. Therefore, they arrive at an observer at different times, an extreme example being an earthquake , where sharp compression waves arrive first and rocking transverse waves seconds later.
The speed of a compression wave in a fluid is determined by the medium's compressibility and density . In solids, the compression waves are analogous to those in fluids, depending on compressibility and density, but with the additional factor of shear modulus which affects compression waves due to off-axis elastic energies which are able to influence effective tension and relaxation in a compression. The speed of shear waves, which can occur only in solids, is determined simply by the solid material's shear modulus and density.
The speed of sound in mathematical notation is conventionally represented by c , from the Latin celeritas meaning "swiftness".
For fluids in general, the speed of sound c is given by the Newton–Laplace equation: c = K s ρ , {\displaystyle c={\sqrt {\frac {K_{s}}{\rho }}},} where
K s = ρ ( ∂ P ∂ ρ ) s {\displaystyle K_{s}=\rho \left({\frac {\partial P}{\partial \rho }}\right)_{s}} , where P {\displaystyle P} is the pressure and the derivative is taken isentropically, that is, at constant entropy s . This is because a sound wave travels so fast that its propagation can be approximated as an adiabatic process , meaning that there isn't enough time, during a pressure cycle of the sound, for significant heat conduction and radiation to occur.
Thus, the speed of sound increases with the stiffness (the resistance of an elastic body to deformation by an applied force) of the material and decreases with an increase in density. For ideal gases, the bulk modulus K is simply the gas pressure multiplied by the dimensionless adiabatic index , which is about 1.4 for air under normal conditions of pressure and temperature.
For general equations of state , if classical mechanics is used, the speed of sound c can be derived [ 7 ] as follows:
Consider the sound wave propagating at speed v {\displaystyle v} through a pipe aligned with the x {\displaystyle x} axis and with a cross-sectional area of A {\displaystyle A} . In time interval d t {\displaystyle dt} it moves length d x = v d t {\displaystyle dx=v\,dt} . In steady state , the mass flow rate m ˙ = ρ v A {\displaystyle {\dot {m}}=\rho vA} must be the same at the two ends of the tube, therefore the mass flux j = ρ v {\displaystyle j=\rho v} is constant and v d ρ = − ρ d v {\displaystyle v\,d\rho =-\rho \,dv} . Per Newton's second law , the pressure-gradient force provides the acceleration: d v d t = − 1 ρ d P d x → d P = ( − ρ d v ) d x d t = ( v d ρ ) v → v 2 ≡ c 2 = d P d ρ {\displaystyle {\begin{aligned}{\frac {dv}{dt}}&=-{\frac {1}{\rho }}{\frac {dP}{dx}}\\[1ex]\rightarrow dP&=(-\rho \,dv){\frac {dx}{dt}}=(v\,d\rho )v\\[1ex]\rightarrow v^{2}&\equiv c^{2}={\frac {dP}{d\rho }}\end{aligned}}}
And therefore:
c = ( ∂ P ∂ ρ ) s = K s ρ , {\displaystyle c={\sqrt {\left({\frac {\partial P}{\partial \rho }}\right)_{s}}}={\sqrt {\frac {K_{s}}{\rho }}},}
If relativistic effects are important, the speed of sound is calculated from the relativistic Euler equations .
In a non-dispersive medium , the speed of sound is independent of sound frequency , so the speeds of energy transport and sound propagation are the same for all frequencies. Air, a mixture of oxygen and nitrogen, constitutes a non-dispersive medium. However, air does contain a small amount of CO 2 which is a dispersive medium, and causes dispersion to air at ultrasonic frequencies (greater than 28 kHz ). [ 8 ]
In a dispersive medium , the speed of sound is a function of sound frequency, through the dispersion relation . Each frequency component propagates at its own speed, called the phase velocity , while the energy of the disturbance propagates at the group velocity . The same phenomenon occurs with light waves; see optical dispersion for a description.
The speed of sound is variable and depends on the properties of the substance through which the wave is travelling. In solids, the speed of transverse (or shear) waves depends on the shear deformation under shear stress (called the shear modulus ), and the density of the medium. Longitudinal (or compression) waves in solids depend on the same two factors with the addition of a dependence on compressibility .
In fluids, only the medium's compressibility and density are the important factors, since fluids do not transmit shear stresses. In heterogeneous fluids, such as a liquid filled with gas bubbles, the density of the liquid and the compressibility of the gas affect the speed of sound in an additive manner, as demonstrated in the hot chocolate effect .
In gases, adiabatic compressibility is directly related to pressure through the heat capacity ratio (adiabatic index), while pressure and density are inversely related to the temperature and molecular weight, thus making only the completely independent properties of temperature and molecular structure important (heat capacity ratio may be determined by temperature and molecular structure, but simple molecular weight is not sufficient to determine it).
Sound propagates faster in low molecular weight gases such as helium than it does in heavier gases such as xenon . For monatomic gases, the speed of sound is about 75% of the mean speed that the atoms move in that gas.
For a given ideal gas the molecular composition is fixed, and thus the speed of sound depends only on its temperature . At a constant temperature, the gas pressure has no effect on the speed of sound, since the density will increase, and since pressure and density (also proportional to pressure) have equal but opposite effects on the speed of sound, and the two contributions cancel out exactly. In a similar way, compression waves in solids depend both on compressibility and density—just as in liquids—but in gases the density contributes to the compressibility in such a way that some part of each attribute factors out, leaving only a dependence on temperature, molecular weight, and heat capacity ratio which can be independently derived from temperature and molecular composition (see derivations below). Thus, for a single given gas (assuming the molecular weight does not change) and over a small temperature range (for which the heat capacity is relatively constant), the speed of sound becomes dependent on only the temperature of the gas.
In non-ideal gas behavior regimen, for which the Van der Waals gas equation would be used, the proportionality is not exact, and there is a slight dependence of sound velocity on the gas pressure.
Humidity has a small but measurable effect on the speed of sound (causing it to increase by about 0.1%–0.6%), because oxygen and nitrogen molecules of the air are replaced by lighter molecules of water . This is a simple mixing effect.
In the Earth's atmosphere , the chief factor affecting the speed of sound is the temperature . For a given ideal gas with constant heat capacity and composition, the speed of sound is dependent solely upon temperature; see § Details below. In such an ideal case, the effects of decreased density and decreased pressure of altitude cancel each other out, save for the residual effect of temperature.
Since temperature (and thus the speed of sound) decreases with increasing altitude up to 11 km , sound is refracted upward, away from listeners on the ground, creating an acoustic shadow at some distance from the source. [ 9 ] The decrease of the speed of sound with height is referred to as a negative sound speed gradient .
However, there are variations in this trend above 11 km . In particular, in the stratosphere above about 20 km , the speed of sound increases with height, due to an increase in temperature from heating within the ozone layer . This produces a positive speed of sound gradient in this region. Still another region of positive gradient occurs at very high altitudes, in the thermosphere above 90 km .
For an ideal gas, K (the bulk modulus in equations above, equivalent to C , the coefficient of stiffness in solids) is given by K = γ ⋅ p . {\displaystyle K=\gamma \cdot p.} Thus, from the Newton–Laplace equation above, the speed of sound in an ideal gas is given by c = γ ⋅ p ρ , {\displaystyle c={\sqrt {\gamma \cdot {p \over \rho }}},} where
Using the ideal gas law to replace p with nRT / V , and replacing ρ with nM / V , the equation for an ideal gas becomes c i d e a l = γ ⋅ p ρ = γ ⋅ R ⋅ T M = γ ⋅ k ⋅ T m , {\displaystyle c_{\mathrm {ideal} }={\sqrt {\gamma \cdot {p \over \rho }}}={\sqrt {\gamma \cdot R\cdot T \over M}}={\sqrt {\gamma \cdot k\cdot T \over m}},} where
This equation applies only when the sound wave is a small perturbation on the ambient condition, and the certain other noted conditions are fulfilled, as noted below. Calculated values for c air have been found to vary slightly from experimentally determined values. [ 10 ]
Newton famously considered the speed of sound before most of the development of thermodynamics and so incorrectly used isothermal calculations instead of adiabatic . His result was missing the factor of γ but was otherwise correct.
Numerical substitution of the above values gives the ideal gas approximation of sound velocity for gases, which is accurate at relatively low gas pressures and densities (for air, this includes standard Earth sea-level conditions). Also, for diatomic gases the use of γ = 1.4000 requires that the gas exists in a temperature range high enough that rotational heat capacity is fully excited (i.e., molecular rotation is fully used as a heat energy "partition" or reservoir); but at the same time the temperature must be low enough that molecular vibrational modes contribute no heat capacity (i.e., insignificant heat goes into vibration, as all vibrational quantum modes above the minimum-energy-mode have energies that are too high to be populated by a significant number of molecules at this temperature). For air, these conditions are fulfilled at room temperature, and also temperatures considerably below room temperature (see tables below). See the section on gases in specific heat capacity for a more complete discussion of this phenomenon.
For air, we introduce the shorthand R ∗ = R / M a i r . {\displaystyle R_{*}=R/M_{\mathrm {air} }.}
In addition, we switch to the Celsius temperature θ = T − 273.15 K , which is useful to calculate air speed in the region near 0 °C ( 273 K ). Then, for dry air, c a i r = γ ⋅ R ∗ ⋅ T = γ ⋅ R ∗ ⋅ ( θ + 273.15 K ) , c a i r = γ ⋅ R ∗ ⋅ 273.15 K ⋅ 1 + θ 273.15 K . {\displaystyle {\begin{aligned}c_{\mathrm {air} }&={\sqrt {\gamma \cdot R_{*}\cdot T}}={\sqrt {\gamma \cdot R_{*}\cdot (\theta +273.15\,\mathrm {K} )}},\\c_{\mathrm {air} }&={\sqrt {\gamma \cdot R_{*}\cdot 273.15\,\mathrm {K} }}\cdot {\sqrt {1+{\frac {\theta }{273.15\,\mathrm {K} }}}}.\end{aligned}}}
Substituting numerical values R = 8.314 462 618 153 24 J / ( m o l ⋅ K ) {\displaystyle R=8.314\,462\,618\,153\,24~\mathrm {J/(mol{\cdot }K)} } M a i r = 0.028 964 5 k g / m o l {\displaystyle M_{\mathrm {air} }=0.028\,964\,5~\mathrm {kg/mol} } and using the ideal diatomic gas value of γ = 1.4000 , we have c a i r ≈ 331.3 m / s × 1 + θ 273.15 K . {\displaystyle c_{\mathrm {air} }\approx 331.3\,\mathrm {m/s} \times {\sqrt {1+{\frac {\theta }{273.15\,\mathrm {K} }}}}.}
Finally, Taylor expansion of the remaining square root in θ {\displaystyle \theta } yields c a i r ≈ 331.3 m / s × ( 1 + θ 2 × 273.15 K ) , ≈ 331.3 m / s + θ × 0.606 ( m / s ) / ∘ C . {\displaystyle {\begin{aligned}c_{\mathrm {air} }&\approx 331.3\,\mathrm {m/s} \times \left(1+{\frac {\theta }{2\times 273.15\,\mathrm {K} }}\right),\\&\approx 331.3\,\mathrm {m/s} +\theta \times 0.606\,\mathrm {(m/s)/^{\circ }C} .\end{aligned}}}
A graph comparing results of the two equations is to the right, using the slightly more accurate value of 331.5 m/s (1,088 ft/s) for the speed of sound at 0 °C . [ 11 ] : 120 -121
The speed of sound varies with temperature. Since temperature and sound velocity normally decrease with increasing altitude, sound is refracted upward, away from listeners on the ground, creating an acoustic shadow at some distance from the source. [ 9 ] Wind shear of 4 m/(s · km) can produce refraction equal to a typical temperature lapse rate of 7.5 °C/km . [ 12 ] Higher values of wind gradient will refract sound downward toward the surface in the downwind direction, [ 13 ] eliminating the acoustic shadow on the downwind side. This will increase the audibility of sounds downwind. This downwind refraction effect occurs because there is a wind gradient; the fact that sound is carried along by the wind is not important. [ 14 ]
For sound propagation, the exponential variation of wind speed with height can be defined as follows: [ 15 ] U ( h ) = U ( 0 ) h ζ , d U d H ( h ) = ζ U ( h ) h , {\displaystyle {\begin{aligned}U(h)&=U(0)h^{\zeta },\\{\frac {\mathrm {d} U}{\mathrm {d} H}}(h)&=\zeta {\frac {U(h)}{h}},\end{aligned}}} where
In the 1862 American Civil War Battle of Iuka , an acoustic shadow, believed to have been enhanced by a northeast wind, kept two divisions of Union soldiers out of the battle, [ 16 ] because they could not hear the sounds of battle only 10 km (six miles) downwind. [ 17 ]
In the standard atmosphere :
In fact, assuming an ideal gas , the speed of sound c depends on temperature and composition only, not on the pressure or density (since these change in lockstep for a given temperature and cancel out). Air is almost an ideal gas. The temperature of the air varies with altitude, giving the following variations in the speed of sound using the standard atmosphere— actual conditions may vary . [ citation needed ]
Given normal atmospheric conditions, the temperature, and thus speed of sound, varies with altitude:
The medium in which a sound wave is travelling does not always respond adiabatically, and as a result, the speed of sound can vary with frequency. [ 18 ]
The limitations of the concept of speed of sound due to extreme attenuation are also of concern. The attenuation which exists at sea level for high frequencies applies to successively lower frequencies as atmospheric pressure decreases, or as the mean free path increases. For this reason, the concept of speed of sound (except for frequencies approaching zero) progressively loses its range of applicability at high altitudes. [ 10 ] The standard equations for the speed of sound apply with reasonable accuracy only to situations in which the wavelength of the sound wave is considerably longer than the mean free path of molecules in a gas.
The molecular composition of the gas contributes both as the mass (M) of the molecules, and their heat capacities, and so both have an influence on speed of sound. In general, at the same molecular mass, monatomic gases have slightly higher speed of sound (over 9% higher) because they have a higher γ ( 5/3 = 1.66 ...) than diatomics do ( 7/5 = 1.4 ). Thus, at the same molecular mass, the speed of sound of a monatomic gas goes up by a factor of c g a s , m o n a t o m i c c g a s , d i a t o m i c = 5 / 3 7 / 5 = 25 21 = 1.091 … {\displaystyle {c_{\mathrm {gas,monatomic} } \over c_{\mathrm {gas,diatomic} }}={\sqrt {{5/3} \over {7/5}}}={\sqrt {25 \over 21}}=1.091\ldots }
This gives the 9% difference, and would be a typical ratio for speeds of sound at room temperature in helium vs. deuterium , each with a molecular weight of 4. Sound travels faster in helium than deuterium because adiabatic compression heats helium more since the helium molecules can store heat energy from compression only in translation, but not rotation. Thus helium molecules (monatomic molecules) travel faster in a sound wave and transmit sound faster. (Sound travels at about 70% of the mean molecular speed in gases; the figure is 75% in monatomic gases and 68% in diatomic gases).
In this example we have assumed that temperature is low enough that heat capacities are not influenced by molecular vibration (see heat capacity ). However, vibrational modes simply cause gammas which decrease toward 1, since vibration modes in a polyatomic gas give the gas additional ways to store heat which do not affect temperature, and thus do not affect molecular velocity and sound velocity. Thus, the effect of higher temperatures and vibrational heat capacity acts to increase the difference between the speed of sound in monatomic vs. polyatomic molecules, with the speed remaining greater in monatomics.
By far, the most important factor influencing the speed of sound in air is temperature. The speed is proportional to the square root of the absolute temperature, giving an increase of about 0.6 m/s per degree Celsius. For this reason, the pitch of a musical wind instrument increases as its temperature increases.
The speed of sound is raised by humidity. The difference between 0% and 100% humidity is about 1.5 m/s at standard pressure and temperature, but the size of the humidity effect increases dramatically with temperature.
The dependence on frequency and pressure are normally insignificant in practical applications. In dry air, the speed of sound increases by about 0.1 m/s as the frequency rises from 10 Hz to 100 Hz . For audible frequencies above 100 Hz it is relatively constant. Standard values of the speed of sound are quoted in the limit of low frequencies, where the wavelength is large compared to the mean free path. [ 19 ]
As shown above, the approximate value 1000/3 = 333.33... m/s is exact a little below 5 °C and is a good approximation for all "usual" outside temperatures (in temperate climates, at least), hence the usual rule of thumb to determine how far lightning has struck: count the seconds from the start of the lightning flash to the start of the corresponding roll of thunder and divide by 3: the result is the distance in kilometers to the nearest point of the lightning bolt. Or divide the number of seconds by 5 for an approximate distance in miles.
Mach number, a useful quantity in aerodynamics, is the ratio of air speed to the local speed of sound. At altitude, for reasons explained, Mach number is a function of temperature.
Aircraft flight instruments , however, operate using pressure differential to compute Mach number, not temperature. The assumption is that a particular pressure represents a particular altitude and, therefore, a standard temperature. Aircraft flight instruments need to operate this way because the stagnation pressure sensed by a Pitot tube is dependent on altitude as well as speed.
A range of different methods exist for the measurement of the speed of sound in air.
The earliest reasonably accurate estimate of the speed of sound in air was made by William Derham and acknowledged by Isaac Newton . Derham had a telescope at the top of the tower of the Church of St Laurence in Upminster , England. On a calm day, a synchronized pocket watch would be given to an assistant who would fire a shotgun at a pre-determined time from a conspicuous point some miles away, across the countryside. This could be confirmed by telescope. He then measured the interval between seeing gunsmoke and arrival of the sound using a half-second pendulum. The distance from where the gun was fired was found by triangulation, and simple division (distance/time) provided velocity. Lastly, by making many observations, using a range of different distances, the inaccuracy of the half-second pendulum could be averaged out, giving his final estimate of the speed of sound. Modern stopwatches enable this method to be used today over distances as short as 200–400 metres, and not needing something as loud as a shotgun.
The simplest concept is the measurement made using two microphones and a fast recording device such as a digital storage scope. This method uses the following idea.
If a sound source and two microphones are arranged in a straight line, with the sound source at one end, then the following can be measured:
Then v = x / t .
In these methods, the time measurement has been replaced by a measurement of the inverse of time ( frequency ).
Kundt's tube is an example of an experiment which can be used to measure the speed of sound in a small volume. It has the advantage of being able to measure the speed of sound in any gas. This method uses a powder to make the nodes and antinodes visible to the human eye. This is an example of a compact experimental setup.
A tuning fork can be held near the mouth of a long pipe which is dipping into a barrel of water . In this system it is the case that the pipe can be brought to resonance if the length of the air column in the pipe is equal to (1 + 2 n ) λ /4 where n is an integer. As the antinodal point for the pipe at the open end is slightly outside the mouth of the pipe it is best to find two or more points of resonance and then measure half a wavelength between these.
Here it is the case that v = fλ .
The effect of impurities can be significant when making high-precision measurements. Chemical desiccants can be used to dry the air, but will, in turn, contaminate the sample. The air can be dried cryogenically, but this has the effect of removing the carbon dioxide as well; therefore many high-precision measurements are performed with air free of carbon dioxide rather than with natural air. A 2002 review [ 20 ] found that a 1963 measurement by Smith and Harlow using a cylindrical resonator gave "the most probable value of the standard speed of sound to date." The experiment was done with air from which the carbon dioxide had been removed, but the result was then corrected for this effect so as to be applicable to real air. The experiments were done at 30 °C but corrected for temperature in order to report them at 0 °C . The result was 331.45 ± 0.01 m/s for dry air at STP, for frequencies from 93 Hz to 1,500 Hz .
In a solid, there is a non-zero stiffness both for volumetric deformations and shear deformations. Hence, it is possible to generate sound waves with different velocities dependent
on the deformation mode. Sound waves generating volumetric deformations (compression) and shear deformations (shearing) are called pressure waves (longitudinal waves) and shear waves (transverse waves), respectively. In earthquakes , the corresponding seismic waves are called P-waves (primary waves) and S-waves (secondary waves), respectively. The sound velocities of these two types of waves propagating in a homogeneous 3-dimensional solid are respectively given by [ 11 ] c s o l i d , p = K + 4 3 G ρ = E ( 1 − ν ) ρ ( 1 + ν ) ( 1 − 2 ν ) , {\displaystyle c_{\mathrm {solid,p} }={\sqrt {\frac {K+{\frac {4}{3}}G}{\rho }}}={\sqrt {\frac {E(1-\nu )}{\rho (1+\nu )(1-2\nu )}}},} c s o l i d , s = G ρ , {\displaystyle c_{\mathrm {solid,s} }={\sqrt {\frac {G}{\rho }}},} where
The last quantity is not an independent one, as E = 3K(1 − 2ν) . The speed of pressure waves depends both on the pressure and shear resistance properties of the material, while the speed of shear waves depends on the shear properties only.
Typically, pressure waves travel faster in materials than do shear waves, and in earthquakes this is the reason that the onset of an earthquake is often preceded by a quick upward-downward shock, before arrival of waves that produce a side-to-side motion. For example, for a typical steel alloy, K = 170 GPa , G = 80 GPa and p = 7700 kg/m 3 , yielding a compressional speed c solid,p of 6,000 m/s . [ 11 ] This is in reasonable agreement with c solid,p measured experimentally at 5,930 m/s for a (possibly different) type of steel. [ 21 ] The shear speed c solid,s is estimated at 3,200 m/s using the same numbers.
Speed of sound in semiconductor solids can be very sensitive to the amount of electronic dopant in them. [ 22 ]
The speed of sound for pressure waves in stiff materials such as metals is sometimes given for "long rods" of the material in question, in which the speed is easier to measure. In rods where their diameter is shorter than a wavelength, the speed of pure pressure waves may be simplified and is given by: [ 11 ] : 70 c s o l i d = E ρ , {\displaystyle c_{\mathrm {solid} }={\sqrt {\frac {E}{\rho }}},} where E is Young's modulus . This is similar to the expression for shear waves, save that Young's modulus replaces the shear modulus . This speed of sound for pressure waves in long rods will always be slightly less than the same speed in homogeneous 3-dimensional solids, and the ratio of the speeds in the two different types of objects depends on Poisson's ratio for the material.
In a fluid, the only non-zero stiffness is to volumetric deformation (a fluid does not sustain shear forces).
Hence the speed of sound in a fluid is given by c f l u i d = K ρ , {\displaystyle c_{\mathrm {fluid} }={\sqrt {\frac {K}{\rho }}},} where K is the bulk modulus of the fluid.
In fresh water, sound travels at about 1481 m/s at 20 °C (see the External Links section below for online calculators). [ 23 ] Applications of underwater sound can be found in sonar , acoustic communication and acoustical oceanography .
In salt water that is free of air bubbles or suspended sediment, sound travels at about 1500 m/s ( 1 500 .235 m/s at 1000 kilopascals , 10 °C and 3% salinity by one method). [ 24 ] The speed of sound in seawater depends on pressure (hence depth), temperature (a change of 1 °C ~ 4 m/s ), and salinity (a change of 1 ‰ ~ 1 m/s ), and empirical equations have been derived to accurately calculate the speed of sound from these variables. [ 25 ] [ 26 ] Other factors affecting the speed of sound are minor. Since in most ocean regions temperature decreases with depth, the profile of the speed of sound with depth decreases to a minimum at a depth of several hundred metres. Below the minimum, sound speed increases again, as the effect of increasing pressure overcomes the effect of decreasing temperature (right). [ 27 ] For more information see Dushaw et al. [ 28 ]
An empirical equation for the speed of sound in sea water is provided by Mackenzie: [ 29 ] c ( T , S , z ) = a 1 + a 2 T + a 3 T 2 + a 4 T 3 + a 5 ( S − 35 ) + a 6 z + a 7 z 2 + a 8 T ( S − 35 ) + a 9 T z 3 , {\displaystyle c(T,S,z)=a_{1}+a_{2}T+a_{3}T^{2}+a_{4}T^{3}+a_{5}(S-35)+a_{6}z+a_{7}z^{2}+a_{8}T(S-35)+a_{9}Tz^{3},} where
The constants a 1 , a 2 , ..., a 9 are a 1 = 1 , 448.96 , a 2 = 4.591 , a 3 = − 5.304 × 10 − 2 , a 4 = 2.374 × 10 − 4 , a 5 = 1.340 , a 6 = 1.630 × 10 − 2 , a 7 = 1.675 × 10 − 7 , a 8 = − 1.025 × 10 − 2 , a 9 = − 7.139 × 10 − 13 , {\displaystyle {\begin{aligned}a_{1}&=1,448.96,&a_{2}&=4.591,&a_{3}&=-5.304\times 10^{-2},\\a_{4}&=2.374\times 10^{-4},&a_{5}&=1.340,&a_{6}&=1.630\times 10^{-2},\\a_{7}&=1.675\times 10^{-7},&a_{8}&=-1.025\times 10^{-2},&a_{9}&=-7.139\times 10^{-13},\end{aligned}}} with check value 1 550 .744 m/s for T = 25 °C , S = 35 parts per thousand , z = 1,000 m . This equation has a standard error of 0.070 m/s for salinity between 25 and 40 ppt . See [1] for an online calculator.
(The Sound Speed vs. Depth graph does not correlate directly to the MacKenzie formula.
This is due to the fact that the temperature and salinity varies at different depths.
When T and S are held constant, the formula itself is always increasing with depth.)
Other equations for the speed of sound in sea water are accurate over a wide range of conditions, but are far more complicated, e.g., that by V. A. Del Grosso [ 30 ] and the Chen-Millero-Li Equation. [ 28 ] [ 31 ]
The speed of sound in a plasma for the common case that the electrons are hotter than the ions (but not too much hotter) is given by the formula (see here ) c s = ( γ Z k T e m i ) 1 / 2 = ( γ Z T e μ ) 1 / 2 × 90.85 m / s , {\displaystyle c_{s}=\left({\frac {\gamma ZkT_{\mathrm {e} }}{m_{\mathrm {i} }}}\right)^{1/2}=\left({\frac {\gamma ZT_{e}}{\mu }}\right)^{1/2}\times 90.85~\mathrm {m/s} ,} where
In contrast to a gas, the pressure and the density are provided by separate species: the pressure by the electrons and the density by the ions. The two are coupled through a fluctuating electric field.
The speed of sound on Mars varies as a function of frequency. Higher frequencies travel faster than lower frequencies. Higher frequency sound from lasers travels at 250 m/s (820 ft/s), while low frequency sound travels at 240 m/s (790 ft/s). [ 32 ]
When sound spreads out evenly in all directions in three dimensions, the intensity drops in proportion to the inverse square of the distance. However, in the ocean, there is a layer called the 'deep sound channel' or SOFAR channel which can confine sound waves at a particular depth.
In the SOFAR channel, the speed of sound is lower than that in the layers above and below. Just as light waves will refract towards a region of higher refractive index , sound waves will refract towards a region where their speed is reduced. The result is that sound gets confined in the layer, much the way light can be confined to a sheet of glass or optical fiber . Thus, the sound is confined in essentially two dimensions. In two dimensions the intensity drops in proportion to only the inverse of the distance. This allows waves to travel much further before being undetectably faint.
A similar effect occurs in the atmosphere. Project Mogul successfully used this effect to detect a nuclear explosion at a considerable distance. | https://en.wikipedia.org/wiki/Speed_of_sound |
Speedify is a software-based channel bonding service developed by Connectify Inc. Launched in 2014, Speedify enhances internet speed, reliability, and security by combining multiple internet connections at the same time into a single, faster, and more stable connection. It is available on various platforms, including Windows , macOS , Linux , iOS , Android and OpenWrt .
Speedify's core technology is channel bonding, which allows users to utilize multiple internet connections simultaneously. This technique splits internet traffic at the packet level across available connections—such as Wi-Fi, cellular, Ethernet, and satellite—to increase bandwidth and provide redundancy. Unlike traditional load balancing , which distributes traffic based on sessions, channel bonding operates at a more granular level ( network packet level), enhancing performance and reliability. [ 1 ]
Speedify's implementation of channel bonding is software -based, enabling it to run on standard consumer devices without specialized hardware. The application interacts with Speedify's cloud servers to manage and distribute data packets across the combined connections. [ 2 ]
The Speedify protocol is a proprietary, software-based solution operating at the packet level. It distributes data across various interfaces such as Wi-Fi , Ethernet , and 4G / 5G cellular networks , increasing bandwidth and providing redundancy . This approach ensures that if one connection experiences issues like packet loss or latency spikes, the others can compensate.
Unlike traditional VPN protocols like WireGuard , IPSec and OpenVPN , Speedify dynamically selects the optimal transport protocol— TCP , UDP , or HTTPS —based on current network conditions. [ 3 ] This flexibility allows it to adapt to various scenarios, such as using multiple parallel TCP connections to maximize throughput on high-speed networks.
Comparatively, Multipath TCP (MPTCP) is an extension of the standard TCP protocol that enables the use of multiple paths for a single connection. While MPTCP can distribute traffic across different interfaces, it typically utilizes only one TCP socket per connection, [ 4 ] limiting its ability to fully exploit multiple networks simultaneously. Additionally, MPTCP's performance can degrade in the presence of high latency or packet loss, as it is sensitive to such network variances.
Speedify includes a feature called Pair & Share that allows users to share internet connections between nearby devices. [ 5 ] The functionality is designed to enhance internet connectivity by enabling devices to bond not only their own internet interfaces—such as Wi-Fi, 4G / 5G cellular, or Ethernet—but also those of paired devices. [ 6 ]
The feature works by allowing two or more Speedify-enabled devices in proximity to connect directly via a secure local link. Once paired, each device contributes its internet connections to the shared pool. This is useful in environments with limited bandwidth, such as during live streams , remote work sessions, or in disaster recovery scenarios where robust connectivity is critical. [ 7 ]
Speedify’s proprietary network bonding protocol is used to encrypt and manage the shared traffic, ensuring that all data remains secure during transit. [ 8 ]
Independent reviews speak of Speedify in terms of getting stable and reliable internet in the context of live streaming, holding a steady latency and minimizing packet loss for Starlink connections. Miri Tech’s Powered by Speedify network bonding router [ 9 ] was also used in a live stream with Kai Cenat and MrBeast , to provide uninterruptible internet access to an off-grid location. [ 10 ]
Speedify offers individual plans that allow users to connect up to five devices simultaneously. [ 11 ]
Family plans extend Speedify's benefits to multiple users within a household, allowing family members to each have a Speedify account and use it on their devices. [ 12 ]
Speedify's enterprise solutions cater to organizations, offering tools and services to manage and optimize internet connectivity. [ 13 ]
Speedify Teams provides businesses with centralized account management through a dedicated dashboard, allowing administrators to manage billing, add or remove users, and monitor usage statistics. Additional features include:
The Speedify Software Development Kit (SDK) allows developers to integrate Speedify's channel bonding and VPN functionalities into their applications. [ 16 ] It offers flexible deployment options, which allows the use of either the Speedify Cloud, dedicated managed servers, or self-hosted servers for network bonding.
Powered by Speedify [ 17 ] refers to the integration of Speedify's technology into third-party hardware and software solutions. An example is the Miri X510 Bonding Router, [ 18 ] which incorporates Speedify's channel bonding technology.
Speedify's enterprise solutions have been adopted by various organizations:
For the Speedify SDK:
For Powered by Speedify: | https://en.wikipedia.org/wiki/Speedify |
SpeedOf.Me is an internet speed test service which uses browser capabilities such as HTML5 and JavaScript to test the internet speed of the user. SpeedOf.Me utilizes multiple servers around the world, with the server used being chosen automatically based on location. [ 1 ] [ 2 ] It is financed through its paid API [ 3 ] as well as an advertising.
This website-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Speedof.me |
A speedometer or speed meter is a gauge that measures and displays the instantaneous speed of a vehicle. Now universally fitted to motor vehicles , they started to be available as options in the early 20th century, and as standard equipment from about 1910 onwards. [ 1 ] Other vehicles may use devices analogous to the speedometer with different means of sensing speed, eg. boats use a pit log , while aircraft use an airspeed indicator .
Charles Babbage is credited with creating an early type of a speedometer, which was usually fitted to locomotives . [ 2 ]
The electric speedometer was invented by the Croat Josip Belušić [ 3 ] in 1888 and was originally called a velocimeter.
The speedometer was originally patented by Josip Belušić (Giuseppe Bellussich) in 1888. He presented his invention at the 1889 Exposition Universelle in Paris . His invention had a pointer and a magnet, using electricity to work. [ 4 ] [ 5 ] [ 6 ] German inventor Otto Schultze patented his version (which, like Belušić's, ran on eddy currents) on 7 October 1902. [ 7 ]
Many speedometers use a rotating flexible cable driven by gearing linked to the vehicle's transmission . The early Volkswagen Beetle and many motorcycles, however, use a cable driven from a front wheel.
Some early mechanical speedometers operated on the governor principle where a rotating weight acting against a spring moved further out as the speed increased, similar to the governor used on steam engines. This movement was transferred to the pointer to indicate speed.
This was followed by the Chronometric speedometer where the distance traveled was measured over a precise interval of time (Some Smiths speedometers used 3/4 of a second) measured by an escapement. This was transferred to the speedometer pointer. The chronometric speedometer is tolerant of vibration and was used in motorcycles up to the 1970s.
When the vehicle is in motion, a speedometer gear assembly turns a speedometer cable, which then turns the speedometer mechanism itself. A small permanent magnet affixed to the speedometer cable interacts with a small aluminium cup (called a speedcup ) attached to the shaft of the pointer on the analogue speedometer instrument. As the magnet rotates near the cup, the changing magnetic field produces eddy current in the cup, which itself produces another magnetic field. The effect is that the magnet exerts a torque on the cup, "dragging" it, and thus the speedometer pointer, in the direction of its rotation with no mechanical connection between them. [ 1 ]
The pointer shaft is held toward zero by a fine torsion spring . The torque on the cup increases with the speed of rotation of the magnet. Thus an increase in the speed of the car will twist the cup and speedometer pointer against the spring. The cup and pointer will turn until the torque of the eddy currents on the cup are balanced by the opposing torque of the spring, and then stop. Given the torque on the cup is proportional to the car's speed, and the spring's deflection is proportional to the torque, the angle of the pointer is also proportional to the speed, so that equally spaced markers on the dial can be used for gaps in speed. At a given speed, the pointer will remain motionless and point to the appropriate number on the speedometer's dial.
The return spring is calibrated such that a given revolution speed of the cable corresponds to a specific speed indication on the speedometer. This calibration must take into account several factors, including ratios of the tail shaft gears that drive the flexible cable, the final drive ratio in the differential , and the diameter of the driven tires .
One of the key disadvantages of the eddy current speedometer is that it cannot show the vehicle speed when running in reverse gear since the cup would turn in the opposite direction – in this scenario, the needle would be driven against its mechanical stop pin on the zero position.
Many modern speedometers are electronic . In designs derived from earlier eddy-current models, a rotation sensor mounted in the transmission delivers a series of electronic pulses whose frequency corresponds to the (average) rotational speed of the driveshaft , and therefore the vehicle's speed, assuming the wheels have full traction. The sensor is typically a set of one or more magnets mounted on the output shaft or (in transaxles) differential crown wheel, or a toothed metal disk positioned between a magnet and a magnetic field sensor . As the part in question turns, the magnets or teeth pass beneath the sensor, each time producing a pulse in the sensor as they affect the strength of the magnetic field it is measuring. [ 1 ] Alternatively, particularly in vehicles with multiplex wiring, some manufacturers use the pulses coming from the ABS wheel sensors which communicate to the instrument panel via the CAN Bus . Most modern electronic speedometers have the additional ability over the eddy current type to show the vehicle's speed when moving in reverse gear.
A computer converts the pulses to a speed and displays this speed on an electronically controlled, analogue-style needle or a digital display . Pulse information is also used for a variety of other purposes by the ECU or full-vehicle control system, e.g. triggering ABS or traction control, calculating average trip speed, or increment the odometer in place of it being turned directly by the speedometer cable.
Another early form of electronic speedometer relies upon the interaction between a precision watch mechanism and a mechanical pulsator driven by the car's wheel or transmission. The watch mechanism endeavours to push the speedometer pointer toward zero, while the vehicle-driven pulsator tries to push it toward infinity. The position of the speedometer pointer reflects the relative magnitudes of the outputs of the two mechanisms.
Virtual speedometers typically approximate speed based on distance traveled over time with the help of a satellite radio navigation system, such as GPS . Virtual speedometers tend to be less accurate than their analog counterparts and are affected by environmental factors such as weather conditions, terrain, and obstructions in the way of the signal.
Typical bicycle speedometers measure the time between each wheel revolution and give a readout on a small, handlebar-mounted digital display. The sensor is mounted on the bike at a fixed location, pulsing when the spoke-mounted magnet passes by. In this way, it is analogous to an electronic car speedometer using pulses from an ABS sensor, but with a much cruder time/distance resolution – typically one pulse/display update per revolution, or as seldom as once every 2–3 seconds at low speed with a 26-inch (660 mm) wheel. However, this is rarely a critical problem, and the system provides frequent updates at higher road speeds where the information is of more importance. The low pulse frequency also has little impact on measurement accuracy, as these digital devices can be programmed by wheel size, or additionally by wheel or tire circumference to make distance measurements more accurate and precise than a typical motor vehicle gauge. However, these devices carry some minor disadvantages in requiring power from batteries that must be replaced every so often in the receiver (and sensor, for wireless models), and, in wired models, the signal is carried by a thin cable that is much less robust than that used for brakes, gears, or cabled speedometers.
Other, usually older bicycle speedometers are cable driven from one or other wheel, as in the motorcycle speedometers described above. These do not require battery power, but can be relatively bulky and heavy, and may be less accurate. The turning force at the wheel may be provided either from a gearing system at the hub (making use of the presence of e.g. a hub brake, cylinder gear, or dynamo) as per a typical motorcycle, or with a friction wheel device that pushes against the outer edge of the rim (same position as rim brakes, but on the opposite edge of the fork) or the sidewall of the tire itself. The former type is quite reliable and low maintenance but needs a gauge and hub gearing properly matched to the rim and tire size, whereas the latter requires little or no calibration for a moderately accurate readout (with standard tires, the "distance" covered in each wheel rotation by a friction wheel set against the rim should scale fairly linearly with wheel size, almost as if it were rolling along the ground itself) but are unsuitable for off-road use, and must be kept properly tensioned and clean of road dirt to avoid slipping or jamming.
Most speedometers have tolerances of some ±10%, mainly due to variations in tire diameter. [ citation needed ] Sources of error due to tire diameter variations are wear, temperature, pressure, vehicle load, and nominal tire size. Vehicle manufacturers usually calibrate speedometers to read high by an amount equal to the average error, to ensure that their speedometers never indicate a lower speed than the actual speed of the vehicle, to ensure they are not liable for drivers violating speed limits. [ citation needed ]
Excessive speedometer errors after manufacture can come from several causes, but most commonly is due to nonstandard tire diameter, in which case the error is: Percentage error = 100 × ( 1 − new diameter standard diameter ) {\displaystyle {\mbox{Percentage error}}=100\times \left(1-{\frac {\mbox{new diameter}}{\mbox{standard diameter}}}\right)}
Nearly all tires now have their size is shown as "T/A_W" on the side of the tire (See: Tire code ), and the tires.
Diameter in millimetres = 2 × T × A / 100 + W × 25.4 {\displaystyle {\mbox{Diameter in millimetres}}=2\times T\times A/100+W\times 25.4} Diameter in inches = T × A / 1270 + W {\displaystyle {\mbox{Diameter in inches}}=T\times A/1270+W}
For example, a standard tire is "185/70R14" with diameter = 2*185*(70/100)+(14*25.4) = 614.6 mm (185x70/1270 + 14 = 24.20 in). Another is "195/50R15" with 2*195*(50/100)+(15*25.4) = 576.0 mm (195x50/1270 + 15 = 22.68 in). Replacing the first tire (and wheels) with the second (on 15" = 381 mm wheels), a speedometer reads 100 * ((614.6/576) - 1) = 100 * (24.20/22.68 - 1) = 6.7% higher than the actual speed. At an actual speed of 100 km/h (60 mph), the speedometer will indicate 100 x 1.067 = 106.7 km/h (60 * 1.067 = 64.02 mph), approximately.
In the case of wear, a new "185/70R14" tire of 620 mm (24.4 inch) diameter will have ≈8 mm tread depth, at legal limit this reduces to 1.6 mm, the difference being 12.8 mm in diameter or 0.5 inches which is 2% in 620 mm (24.4 inches).
In many countries the legislated error in speedometer readings is ultimately governed by the United Nations Economic Commission for Europe (UNECE) Regulation 39, [ 8 ] which covers those aspects of vehicle type approval that relate to speedometers. The main purpose of the UNECE regulations is to facilitate trade in motor vehicles by agreeing on uniform type approval standards rather than requiring a vehicle model to undergo different approval processes in each country where it is sold.
European Union member states must also grant type approval to vehicles meeting similar EU standards. The ones covering speedometers [ 9 ] [ 10 ] [ 11 ] are similar to the UNECE regulation in that they specify that:
The standards specify both the limits on accuracy and many of the details of how it should be measured during the approvals process. For example, the test measurements should be made (for most vehicles) at 40, 80 and 120 km/h (25, 50 and 75 mph), and at a particular ambient temperature and road surface. There are slight differences between the different standards, for example in the minimum accuracy of the equipment measuring the true speed of the vehicle.
The UNECE regulation relaxes the requirements for vehicles mass-produced following type approval. At Conformity of Production Audits the upper limit on indicated speed is increased to 110 percent plus 6 km/h (3.7 mph) for cars, buses, trucks, and similar vehicles, and 110 percent plus 8 km/h (5.0 mph) for two- or three-wheeled vehicles that have a maximum speed above 50 km/h (31 mph) (or a cylinder capacity, if powered by a heat engine , of more than 50 cm 3 (3.1 cu in)). European Union Directive 2000/7/EC, which relates to two- and three-wheeled vehicles, provides similar slightly relaxed limits in production.
There were no Australian Design Rules in place for speedometers in Australia before July 1988. They had to be introduced when speed cameras were first used. This means there are no legally accurate speedometers for these older vehicles. All vehicles manufactured on or after 1 July 2007, and all models of vehicle introduced on or after 1 July 2006, must conform to UNECE Regulation 39. [ 12 ]
The speedometers in vehicles manufactured before these dates but after 1 July 1995 (or 1 January 1995 for forward control passenger vehicles and off-road passenger vehicles) must conform to the previous Australian design rule. This specifies that they need only display the speed to an accuracy of ±10% at speeds above 40 km/h, and there is no specified accuracy at all for speeds below 40 km/h.
All vehicles manufactured in Australia or imported for supply to the Australian market must comply with the Australian Design Rules. [ 13 ] The state and territory governments may set policies for the tolerance of speed over the posted speed limits that may be lower than the 10% in the earlier versions of the Australian Design Rules permitted, such as in Victoria. [ 14 ] This has caused some controversy since it would be possible for a driver to be unaware that they are speeding should their vehicle be fitted with an under-reading speedometer. [ 15 ]
The amended Road Vehicles (Construction and Use) Regulations 1986 permits the use of speedometers that meet either the requirements of EC Council Directive 75/443 (as amended by Directive 97/39) or UNECE Regulation 39. [ 16 ]
The Motor Vehicles (Approval) Regulations 2001 [ 17 ] permits single vehicles to be approved. As with the UNECE regulation and the EC Directives, the speedometer must never show an indicated speed less than the actual speed. However, it differs slightly from them in specifying that for all actual speeds between 25 mph and 70 mph (or the vehicles' maximum speed if it is lower than this), the indicated speed must not exceed 110% of the actual speed, plus 6.25 mph.
For example, if the vehicle is actually traveling at 50 mph, the speedometer must not show more than 61.25 mph or less than 50 mph.
Federal standards in the United States allow a maximum 5 mph error at a speed of 50 mph on speedometer readings for commercial vehicles. [ 18 ] Aftermarket modifications, such as different tire and wheel sizes or different differential gearing, can cause speedometer inaccuracy.
Starting with U.S. automobiles manufactured on or after 1 September 1979, the NHTSA required speedometers to have a special emphasis on 55 mph (90 km/h) and display no more than a maximum speed of 85 mph (136 km/h). On 25 March 1982, the NHTSA revoked the rule because no "significant safety benefits" could come from maintaining the standard. [ 19 ]
GPS devices can measure speeds in two ways:
As mentioned in the satnav article, GPS data has been used to overturn a speeding ticket; the GPS logs showed the defendant traveling below the speed limit when they were ticketed. That the data came from a GPS device was likely less important than the fact that it was logged; logs from the vehicle's speedometer could likely have been used instead, had they existed. | https://en.wikipedia.org/wiki/Speedometer |
The speed of sound in any chemical element in the fluid phase has one temperature-dependent value. In the solid phase , different types of sound wave may be propagated, each with its own speed: among these types of wave are longitudinal (as in fluids), transversal , and (along a surface or plate) extensional . [ 1 ]
As quoted at http://www.webelements.com/ from this source:
As quoted from various sources in an online version of:
As quoted from this source in an online version of: David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition . CRC Press. Boca Raton, Florida, 2003; Section 6, Fluid Properties; Thermal Properties of Mercury
Dwight E. Gray (ed), American Institute of Physics Handbook . McGraw-Hill. Boca Raton, Florida, New York, 1957. | https://en.wikipedia.org/wiki/Speeds_of_sound_of_the_elements |
Speedtest.net , also known as Speedtest by Ookla , is a web service that provides free analysis of Internet access performance metrics, such as connection data rate and latency . It is the flagship product of Ookla, a web testing and network diagnostics company founded in 2006, and based in Seattle, Washington , United States . [ 5 ] [ 6 ]
The service measures the data throughput ( speed ) and latency ( connection delay ) of an Internet connection against one of over 16,000 geographically dispersed servers (as of December 2023). [ 7 ] Each test measures the data rate for the download direction, i.e. from the server to the user computer, and the upload data rate, i.e. from the user's computer to the server. The tests are performed within the user's web browser or within mobile apps . As of 17 February 2024 [update] , over 52.3 billion Internet speed tests have been completed. [ 8 ]
Tests were previously performed over HTTP . To improve accuracy, Speedtest.net now performs tests via a custom protocol over TCP sockets.
The site also offers detailed statistics based on test results. This data has been used by numerous publications in the analysis of Internet access data rates around the world. [ 9 ] [ 10 ] [ 11 ]
The owner and operator of Speedtest.net, Ookla, was established in 2006 by partners Mike Apgar and Doug Suttles. Suttles suggested the name Ookla because he already owned the Ookla.com domain name in honor of his pet cat, who was in turn named for a character on the TV series Thundarr the Barbarian . [ 5 ] The domain speedtest.net has been used to host a speed test since 2000, and was acquired by Ookla in 2006. [ 12 ]
As of 2011, Ookla claimed 80% market share and was one of the top 1000 most popular websites. At the time, Ookla derived its revenue primarily from fees paid by companies to license custom speed test and proprietary testing software. Clients reportedly included media companies like CNN and Disney , and telecommunications providers like AT&T , Verizon , and CenturyLink . [ 5 ]
Ookla was acquired by Ziff Davis in 2014. [ 13 ]
Speedtest.net started as a Flash -based broadband speed test service. After Adobe deprecated Flash, and announced its End-Of-Life (EOL) , Ookla ported the speed test from Flash to HTML5 . The new HTML5 based speed test went out of beta on January 9, 2018. [ 17 ] [ 18 ] | https://en.wikipedia.org/wiki/Speedtest.net |
2.1.5Isomeric SMILES
Spegatrine is an α 1 - and α 2 -adrenergic receptor antagonist isolated from Rauvolfia verticillata . Its dimer dispegatrine has greater antagonist affinity for α-adrenergic receptors . [ 1 ]
This organic chemistry article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Spegatrine |
Speisses are alloys of heavy metals like iron, cobalt, nickel and copper [ 1 ] with arsenic, antimony and, occasionally, tin. [ 2 ] The latter elements lower the melting point to around 1000 °C. [ 2 ] Speisses commonly occur in lead smelting operations [ 3 ] and copper smelting operations. [ 2 ]
Speisses are only partially miscible with mattes , and if there is enough arsenic or antimony in the copper feed to a matte smelting furnace, a separate speiss melt can form. [ 2 ] Speisses show high affinities for platinum group metals and gold . [ 2 ] The mass concentration of platinum group metals in the speiss phase is about 1000 times that of the concentration in the matte phase, while the ratio for gold is about 100 times. [ 2 ]
Speisses are also immiscible in liquid lead and flow out of lead blast furnaces as a separate phase. [ 2 ]
This industry -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Speiss |
Spektr-M [ 2 ] (Russian: Спектр-M) is a proposed Russian scientific satellite with a 10 m (33 ft) sub-millimeter to far infrared space telescope. It is designed to be a successor to the Herschel Space Observatory , covering similar wave bands, and to look into chemical evolution in the universe, black hole horizon radiation, and dark energy investigation. [ 3 ] Spacecraft design documentation and prototyping is currently underway and expected to continue until 2023. Due to budget cuts in 2019, launch is not expected until 2030. [ 4 ] [ 1 ]
The purpose of this mission is to study the universe in millimeter to far infra-red wavelengths. The Herschel mission did a similar job with a smaller dish of 3.5 m (11 ft), and this is a follow-up mission. The instruments are to be cooled with liquid helium to 4.5K for part of the mission, but sun shields will allow it to continue in a degraded mode once the coolant evaporates.
It will be placed in a halo orbit around the Sun–Earth L 2 Lagrangian point. [ 5 ]
This spacecraft or satellite related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Spektr-M |
Spektr-R [ 6 ] (part of RadioAstron program) ( Russian : Спектр-Р) was a Russian scientific satellite with a 10 m (33 ft) radio telescope on board. It was launched on 18 July 2011 [ 7 ] on a Zenit-3F launcher from Baikonur Cosmodrome , and was designed to perform research on the structure and dynamics of radio sources within and beyond the Milky Way . Together with some of the largest ground-based radio telescopes, the Spektr-R formed interferometric baselines extending up to 350,000 km (220,000 mi).
On 11 January 2019, the spacecraft stopped responding to ground control, but its science payload was described as "operational". The mission never recovered from the January 2019 incident, and the mission was declared finished (and spacecraft operations ended) on 30 May 2019.
The Spektr-R project was funded by the Astro Space Center of Russia, and was launched into Earth orbit on 18 July 2011, [ 3 ] with a perigee of 10,000 km (6,200 mi) and an apogee of 390,000 km (240,000 mi), about 700 times the orbital height of the Hubble Space Telescope at its highest point and 20 times at its lowest. [ 8 ] [ 9 ] In comparison, the average distance from Earth to the Moon is 384,400 km (238,900 mi). [ 10 ] As of 2018, the satellite has a much more stable orbit with a perigee of 57,000 km (35,000 mi) and an apogee of 320,000 km (200,000 mi), with its orbit no longer intersecting the Moon's orbit and being stable for possibly hundreds or even thousands of years.
The main scientific goal of the mission was the study of astronomical objects with an angular resolution up to a few millionths of an arcsecond . This was accomplished by using the satellite in conjunction with ground-based observatories and interferometry techniques. [ 3 ] Another purpose of the project was to develop an understanding of fundamental issues of astrophysics and cosmology . This included star formations , the structure of galaxies , interstellar space , black holes and dark matter .
Spektr-R was one of the instruments in the RadioAstron program , an international network of observatories led by the Astro Space Center of the Lebedev Physical Institute . [ 8 ]
The telescope was intended for radio-astrophysical observations of extragalactic objects with ultra-high resolution, as well as researching of characteristics of near-Earth and interplanetary plasma. The very high angular resolving power was achieved in conjunction with a ground-based system of radio-telescopes and interferometrical methods , operating at wavelengths of 1.35–6.0, 18.0 and 92.0 cm. [ 11 ] Once in space, the flower-like main dish was to open its 27 'petals' within 30 minutes. [ citation needed ]
There was a science payload of opportunity on board, PLASMA-F, which consists of four instruments to observe solar wind and the outer magnetosphere. These instruments are the energetic particle spectrometer MEP-2, the magnetometer MMFF, the solar wind monitor BMSW, and the data collection and processing unit SSNI-2. [ 12 ]
At launch the mass of the spacecraft was 3,660 kg (8,070 lb). It was launched from the Baikonur Cosmodrome on 18 July 2011 at 02:31 UTC by a Zenit-3F launch vehicle, which is composed of a Zenit-2M with a Fregat -SB upper stage. [ 3 ] [ 4 ]
On 11 January 2019, the spacecraft stopped responding to ground control. It was unknown whether the issue could be fixed, or whether the spacecraft's mission would be ended. [ 13 ] With Spektr-R's status unknown and the problems hitting the Mikhailo Lomonosov satellite, the Russian space program had no operational space observatories as of 12 January 2019. This changed with the launch of the Spektr-RG satellite in July 2019.
The mission was declared as finished on 30 May 2019. [ 14 ]
The external tank of the Fregat upper stage that delivered the Spektr-R observatory into orbit exploded on May 8, 2020, generating at least 65 trackable debris in orbit around Earth. [ 15 ]
At the beginning of the 1980s, one of the USSR 's leading developers of scientific space probes had completed a preliminary design of revolutionary, new-generation spacecraft, 1F and 2F. The main purpose of Spektr was to develop a common platform that could be used for future deep-space missions.
NPO Lavochkin hoped to use the designs of the 1F as the standard design for space telescopes . In 1982, NPO Lavochkin had completed technical blueprints for RadioAstron, a space-based radio telescope . The expectation was that the 1F and 2F spacecraft would follow the expectations of the RadioAstron mission (also known as Astron-2).
Early on, many criticized the 1F platform for its questionable astrophysics missions, even when compared to the older 4V spacecraft bus . Although the attitude control system of the 1F seemed to have little issues navigating planetary probes, its accuracy was much below the standard requirements for a high-precision telescope . To add to 1F's technical issues, the spacecraft seemed to lack electrically driven fly-wheels, which critics believed would have increased its stabilization in space. The spacecraft also failed to have a moveable solar panel system, which could track the position of the Sun without requiring the entire satellite to reposition, eventually disrupting the observations process.
It was one of three competing Spectrum missions, the others being Spektr X-Gamma
and Spektr-UV [ 16 ]
On 1 August 1983, VPK, the Soviet Military Industrial Commission commissioned an official decision (number 274) titled, "On works for creation of automated interplanetary vehicles for the exploration of planets of the Solar System , the Moon and cosmic space ". This document outlined a new impetus for the development of satellites . The new technical proposals submitted in mid-1984 included a gamma-ray telescope designated to register radio waves in the millimetre range. Both of these satellites incorporated rotating solar panels , a highly sensitive star-tracking operating system and fly wheels.
By the end of the 1980s, NPO Lavochkin Designer General, Vyacheslav Kovtunenko ( ru ) , proposed to design all future astrophysics satellites on the current Oko-1 spacecraft model, designed originally to track incoming ballistic missiles. According to this plan, Oko-1 (a missile-watching infrared telescope ) would eventually be replaced with scientific instruments where the satellite would be pointed towards space rather than Earth .
Using a technique called very-long-baseline interferometry , it was anticipated that ground telescopes in Australia, Chile , China, India, Japan, Korea , Mexico , Russia, South Africa , Ukraine and the United States would jointly make observations with the RadioAstron spacecraft.
The RadioAstron satellite's main 10-metre radio telescope would communicate in four different bands of radio waves with the international ground telescopes. It can also locate sources from two frequencies simultaneously. [ 17 ] The Spektr-R was also planned to include a secondary BMSV within the Plazma-F experiment, the goal of which was to measure the directions and intensity of solar wind. In May 2011, the news agency RIA Novosti reported that the BMSV instrument would indeed be on board. It was also reported that the BMSV would carry a micrometeoroid counter made in Germany.
The RadioAstron was expected to extend into a highly elliptical orbit in the Fregat state of the Zenit rocket's launch. Spektr-R's closest point ( perigee ) would be 500 kilometres (310 mi) above the Earth 's surface, with its apogee 340,000 kilometres (210,000 mi) away. The operational orbit would last at least nine years, with the RadioAstron never being in the Earth's shadow for more than two hours.
With its apogee as far as the orbit of the Moon , Spektr-R could be considered a deep-space mission. In fact, the gravitational pull of the Moon was expected to fluctuate the satellite's orbit in three-year cycles, with its apogee travelling between 265,000 and 360,000 kilometres (220,000 mi) from Earth and its perigee between 400 and 65,000 kilometres (250 and 40,390 mi). Each orbit would take RadioAstron around eight to nine days. This drift would vastly augment the telescope's range of vision. It was estimated that the satellite would have upwards of 80% of its potential targets within view at any one point in its orbit. The first 45 days of Spektr-R's orbit were scheduled to consist of engineering commissioning, that is, the launch of the main antenna , various systems checks and communications tests.
Spektr-R's tracking was to be handled by the RT-22 radio telescope in Pushchino , Russia. Flight control would be operated by ground stations in Medvezhi Ozera [ ru ] near Moscow and Ussuriysk in Russia's Far East. Other Spektr-R joint observations would be handled by ground telescopes in Arecibo, Badary, Effelsberg, Green Bank, Medicina, Noto, Svetloe, Zelenchukskaya and Westerbork.
The Spektr-R project was led by the Russian Academy of Sciences 's Astro Space Center of the Lebedev Physics Institute . The radio receivers on Spektr-R were to be built in India and Australia. In earlier plans, two additional receivers were to be provided by firms under contract with the European VLBI Consortium, the EVN . These additional payloads were eventually cancelled, with the project citing old age. Similar Russian materials replaced the Indian and Australian instruments. | https://en.wikipedia.org/wiki/Spektr-R |
Spektr-RG ( Russian : Спектр-РГ, Spectrum + Röntgen + Gamma ; also called Spectrum-X-Gamma , SRG , SXG ) is a Russian–German high-energy astrophysics space observatory which was launched on 13 July 2019. [ 4 ] It follows on from the Spektr-R satellite telescope launched in 2011. [ 5 ]
The original idea for this X-ray observatory satellite orbiting above Earth's atmosphere, which filters X-rays, was first proposed in the 1980s by Rashid Sunyaev of the Space Research Institute of the USSR Academy of Sciences . Twenty institutions from twelve countries came together to design a large observatory with five telescopes. However, after the collapse of the Soviet Union , the mission was abandoned due to cost-cutting from the Russian space program Roscosmos . The project was resurrected in 2003 with a scaled-down design. [ 6 ]
The primary instrument of the mission is eROSITA , built by the Max Planck Institute for Extraterrestrial Physics (MPE) in Germany. It is designed to conduct a seven-year X-ray survey, [ 7 ] the first in the medium X-ray band less than 10 keV energies, and the first to map an estimated 100,000 galaxy clusters. [ 8 ] This survey may detect new clusters of galaxies and active galactic nuclei . The second instrument, ART-XC , is a Russian high-energy X-ray telescope capable of detecting supermassive black holes . [ 8 ]
The Spektr-RG mission concept was published in 2005. [ 9 ] Construction was finished in 2016, and by mid-2018 it was under integration and testing. It was scheduled to be launched in June 2019 but was delayed to 12 July, before the flight was postponed at the last moment. It launched the next day, 13 July 2019, from Baikonur Site 81/24 . [ 1 ] The observatory was integrated into a Navigator satellite bus , [ 10 ] produced by NPO Lavochkin . [ 11 ]
The spacecraft entered an orbit around the Sun, circling the Sun-Earth L 2 Lagrangian point in a halo orbit , about 1.5 million kilometres away from Earth. Cruise to that location took three months, during which the two telescopes were checked out and calibrated. The next four years were planned to be spent performing eight all-sky surveys. As a goal, the three years after that are planned for observations of selected galaxy clusters and AGNs
( Active Galactic Nuclei ). [ 12 ]
On Monday 21 October 2019, Spektr-RG completed a 100-day cruise to L2-point. On 17 October 2019, the main eROSITA instrument achieved first light . [ 13 ] The first light image of ART-XC was taken on July 30, 2019. [ 14 ]
The operations of eROSITA were suspended on 26 February 2022 after the Russian invasion into Ukraine upon request from Germany. At the time, eROSITA had completed four of its planned eight full-sky surveys. [ 15 ]
In March 2022, Russia said they turned off one of the two telescopes aboard Spektr-RG (presumably eROSITA) upon request from Germany. [ 16 ] In June, the head of Roscosmos threatened to unilaterally seize control of the German telescope, citing German officials' "pro-fascist views". [ 17 ]
In 2023 it was published that Spektr-RG found 17 new AGNs. [ 18 ] In 2025 it was published that Spektr-RG found additional 11 new AGNs. [ 19 ] | https://en.wikipedia.org/wiki/Spektr-RG |
Spektr-UV , also known as World Space Observatory-Ultraviolet ( WSO-UV ), is a proposed ultraviolet space telescope intended for work in the 115 nm to 315 nm wavelength range. [ 3 ] [ 4 ] It is an international project led by Russia (Roscosmos), with participation from Spain and Japan. The launch had initially been planned for 2007, but has since been continually delayed; [ 5 ] as of December 2023 [update] , the launch is expected to take place no earlier than 2030 [ 1 ] atop an Angara A5M rocket from Vostochny Cosmodrome . [ 2 ]
The main instrument of the observatory is a 1.7-metre Ritchey–Chrétien telescope . The telescope will be equipped with the following instruments:
The WUVS spectrographs assembly consists of four channels:
The FCU has two channels, each fed by an independent pick off mirror:
[ 6 ] [ 7 ]
In October 2012, tests of antennas for the space telescope were completed. [ 8 ]
In July 2019, INASAN selected the first seven experiments to be performed by the observatory. [ 9 ]
Spektr-UV is an international project led by Russia ( Roscosmos ).
At present the international cooperation includes three basic participants: Russia (will provide the telescope, spacecraft, launch facilities , ground segment ); Spain (FCU detectors, ground segment); Japan (UVSPEX). | https://en.wikipedia.org/wiki/Spektr-UV |
Disc and disk are both variants of the English word for objects of a generally thin and cylindrical geometry. The differences in spelling correspond both with regional differences and with different senses of the word. For example, in the case of flat, rotational data storage media the convention is that the spelling disk is used for magnetic storage (e.g., hard disks ) while disc is used for optical storage (e.g., compact discs , better known as CDs). When there is no clear convention, the spelling disk is more popular in American English , while the spelling disc is more popular in British English .
The earlier word is disk , which came into the English language in the middle of the 17th century. In the 19th century, disk became the conventional spelling for audio recordings made on a flat plate, such as the gramophone record . Early BBC technicians differentiated between disks (in-house transcription records) and discs (the colloquial term for commercial gramophone records, or what the BBC dubbed CGRs). [ 1 ]
By the 20th century, the "k" spelling was more popular in the United States, while the "c" variant was preferred in the UK. [ 2 ] In the 1950s, when the American company IBM pioneered the first hard disk drive storage devices, it used the "k" spelling. Consequently, in computer terminology today it is common for the "k" word to refer mainly to magnetic storage devices [ 3 ] (particularly in British English, where the term disk is sometimes regarded as a contraction of diskette , a much later word and actually a diminutive of disk ). Kodak 's 1982 disc film used the -c variant. The RISC OS developed by the British Acorn Computers in 1987, and was subsequently forked when Acorn stopped working on it, uses the spelling 'disc' for magnetic media.
Some latter-day competitors to IBM prefer the c -spelling. In 1979, the Dutch company Philips , along with Sony , developed and trademarked the compact disc using the "c" spelling. The "c" spelling is now used consistently for optical media such as the compact disc and similar technologies. [ 4 ]
The words disc and disk can appear frequently in medical journals and textbooks, especially those in ophthalmology and orthopedics , and thus style guides often foster consistency by giving rules for which contexts take which spelling. AMA style for this topic is used by many publications. AMA says, "For ophthalmologic terms, use disc (e.g., optic disc ); for other anatomical terms, use disk (e.g., lumbar disk ). In discussions related to computers, use disk (e.g., floppy disk, disk drive, diskette ) (exceptions: compact disc, videodisc )." [ 5 ]
Disc sports , or disc games, are a category of activities which involve throwing and/or catching a flying disc . Participants of disc sports consistently use the "c" spelling when describing the sports equipment used in these activities, which includes team sports such as ultimate or individual sports such as disc golf . This is a parallel to the spelling of "discus," the flat and round weight thrown in the track and field sport discus throw . | https://en.wikipedia.org/wiki/Spelling_of_disc |
Spendor is a British loudspeaker manufacturing company founded in 1969 by audio engineer Spencer Hughes (1924–1983) and his wife Dorothy. [ 1 ] [ 2 ] It is located in East Sussex . [ 1 ] [ 3 ] [ 4 ] The name was derived from the first names of both. [ 4 ]
Spencer Hughes worked in an investigation team of the BBC research department in the 1960s. Though journeying into Television, this was a time period when the BBC's licence budget meant the main transmission output was still radio, the imperative behind that of a BBC licensed loudspeaker was that (within the physical confines of a small bookshelf speaker) the principle objective of the loudspeaker was to reproduce an output audio signal with an acoustic fidelity to an original radio presenter's voice; principally within the entire spoken vocal range. To this end the resultant, and historically important BBC LS35A loudspeaker hit the retail market, the optional license meant any manufacture could procure a license to produce the LS35A design only if they were able to build a speaker which matched the same high fidelity standard the BBC had worked to achieve. The goal of the BBC R&D team had been reached with the original LS35A standard was now available to consumers who had the means to buy an amplification and loudspeaker system guaranteed to bring in to their homes the exact same sonic signature BBC sound engineers heard while recording and during replay. An alternative, the unwieldily, but sonically superior Quad Acoustics Electrostatic Loudspeaker ("ELS") were simply too large and pricey for most audiophiles.
One resulting offshoot of the research was a membrane made from a polystyrene ("Bextrene") for mid-range speakers or woofers . [ 5 ]
In the first days Dorothy assisted with coil winding expertise, later she took over the general management. [ 6 ]
The first product was the BC1, which Spencer designed while still working for the BBC. [ 6 ] Several other designs followed, the BC2, BC3, SA1, SP1 and other. Spendor also made the BBC LS3/5a under licence from the BBC. [ 7 ]
The BC1 is bigger than a LS3/5a and also uses a Bextrene-membrane [ 1 ] [ 2 ] and was used in many radio stations, too. As a consequence many UK speaker designs are influenced by this improvement of sound quality through reduced colouration and greater consistency [ 5 ] as well as the stereo imaging. [ 8 ] The BC1 was built with smaller modifications until 1994. [ 9 ]
The present Head of Engineering, Terry Miles, started as Spencer Hughes’ assistant in 1975. [ 2 ]
Derek Hughes, son of Spencer and his wife Dorothy, worked at Spendor (in his letter from 1980 Spencer mentioned him as assist with research and development and general running of the factory). [ 6 ] After the untimely death of Spencer in 1983 he worked with his mother in the capacity of Technical Director, producing the original versions of what is now the Classic Series, most notably the SP1/2, SP2 and the S100 and he did amongst others the redesign of the 3/5 1998 [ 11 ] and is still working at loudspeakers as freelance consultant designer. [ 12 ] [ 13 ]
Since the year 2000 the company is owned by Philip Swift a speaker designer and co-founder of Audiolab , who personally knew Spencer Hughes. [ 3 ] then sold to Ajay Shirke. Spendor develops and manufactures all components in UK. [ 2 ] [ 4 ] [ 14 ]
Round about these days Spendor enlarged their product range by floorstanding loudspeakers, first with models of a S line, [ 15 ] currently be found as loudspeakers of A line and higher end D line. [ 2 ] | https://en.wikipedia.org/wiki/Spendor |
Spent caustic is a waste industrial caustic solution that has become exhausted and is no longer useful (or spent). Spent caustics are made of sodium hydroxide or potassium hydroxide , water , and contaminants. The contaminants have consumed the majority of the sodium (or potassium) hydroxide and thus the caustic liquor is spent, for example, in one common application H 2 S ( gas ) is scrubbed by the NaOH ( aqueous ) to form NaHS ( aq ) and H 2 O ( l ), thus consuming the caustic.
Spent caustics are malodorous wastewaters that are difficult to treat in conventional wastewater processes. Typically the material is disposed of by high dilution with biotreatment , deep well injection, incineration , wet air oxidation , Humid Peroxide Oxidation or other speciality processes. Most ethylene spent caustics are disposed of through wet air oxidation. | https://en.wikipedia.org/wiki/Spent_caustic |
Spent shale or spent oil shale (also known as retorted shale ) is a solid residue from the shale oil extraction process of producing synthetic shale oil from oil shale . It consists of inorganic compounds ( minerals ) and remaining organic matter known as char —a carbonaceous residue formed from kerogen . Depending on the extraction process and the amount of remaining organic matter, spent shale may be classified as oil shale coke, semi-coke or coke-ash residue, known also as oil shale ash. [ 1 ] [ 2 ] According to the European Union waste list all these types of spent shale are classified as hazardous waste . [ 2 ]
Oil shale coke is created by chamber ovens which were used for oil shale gas production. [ 1 ] Vertical retorts create mainly semi-coke. Most solid heat carrier processes create coke-ash residue as the semi-coke created during the process is combusted for the process's energy needs. [ 3 ]
Spent shale can be used as ingredients in cement or brick manufacture. [ 1 ] [ 4 ] [ 5 ] In Jordan , usage of spent shale for the production of sodium carbonate , ammonium sulfate , and potassium sulfate has been studied. [ 6 ] | https://en.wikipedia.org/wiki/Spent_shale |
Sperm-mediated gene transfer (SMGT) is a transgenic technique that transfers genes based on the ability of sperm cells to spontaneously bind to and internalize exogenous DNA and transport it into an oocyte during fertilization to produce genetically modified animals . 1 Exogenous DNA refers to DNA that originates outside of the organism. Transgenic animals have been obtained using SMGT, but the efficiency of this technique is low. Low efficiency is mainly due to low uptake of exogenous DNA by the spermatozoa, reducing the chances of fertilizing the oocytes with transfected spermatozoa. 2 In order to successfully produce transgenic animals by SMGT, the spermatozoa must attach the exogenous DNA into the head and these transfected spermatozoa must maintain their functionality to fertilize the oocyte. 2 Genetically modified animals produced by SMGT are useful for research in biomedical, agricultural, and veterinary fields of study. SMGT could also be useful in generating animals as models for human diseases or lead to future discoveries relating to human gene therapy.
The method for SMGT uses the sperm cell, a natural vector of genetic material, to transport exogenous DNA. The exogenous DNA molecules bind to the cell membrane of the head of the sperm cell. This binding and internalization of the DNA is not a random event. The exogenous DNA interacts with the DNA-binding proteins (DBPs) that are present on the surface of the sperm cell. 3 Spermatozoa are naturally protected against the intrusion of exogenous DNA molecules by an inhibitory factor present in mammals’ seminal fluid. This factor blocks the binding of sperm cells and exogenous DNA because in the presence of the inhibitory factor, DBPs lose their ability to bind to exogenous DNA. In the absence of this inhibitory factor, DBPs on sperm cells are able to interact with DNA and can then translocate the DNA into the cell. Therefore, the seminal fluid must be removed from the sperm samples by extensive washing immediately after ejaculation . 3 After the DNA is internalized, the exogenous DNA must be integrated into the genome. There are various mechanisms suggested for DNA integration, including integrating DNA at oocyte activation, at nucleus decondensation, or at the formation of the pronuclei, but all of these suggested mechanisms imply that the integration of DNA happens after the penetration of the sperm cell into the oocyte. 3
Sperm-mediated gene transfer is considered controversial because despite the successes, it has not yet become established as a reliable form of genetic manipulation. Skepticism arises based on the assumption that evolutionary chaos could arise if sperm cells could act as vectors for exogenous DNA. 4 Reasonable assumption tells us that because reproductive tracts contain free DNA molecules, sperm cells should be highly resistant to the risk of picking up exogenous DNA molecules. SMGT has been demonstrated experimentally and followed the assumption that nature has barriers against SMGT. These barriers are not always absolute and could explain the inconsistent experimental outcomes of SMGT. 4 If there are natural barriers against SMGT, then the successes may only represent unusual cases in which the barriers failed. Two barriers have been identified; the inhibitory factor in seminal fluid that prevents binding to foreign DNA molecules and a sperm endogenous nuclease activity that is triggered upon interaction of sperm cells with foreign DNA molecules. 4 These protections give reason to believe that unintentional interactions between sperm and exogenous genetic sequences is kept to a minimal. These barriers allow for protection against the threat that every fertilization event could become a potentially mutagenic one. 4
Transgenic animals have been produced successfully using gene transfer techniques such as sperm-mediated gene transfer. Though this production has been successful, the efficiency of the process is low. Low efficiency of SMGT in the production of transgenic animals is mainly due to poor uptake of the exogenous DNA by the sperm cells, thus reducing the number of fertilized oocytes with transfected spermatozoa. 5 From 1989 to 2004, there were over 30 claims for the production of viable transgenic animals using SMGT, but only about 25 percent of these demonstrated a transmission of the transgenes beyond the F 0 generation. 4 This transmission is required in order to claim usable animal transgenesis. According to previous studies, numerous animal species, including mammals, birds, insects, and fish, have been found susceptible to SMGT techniques, thus indicating that SMGT has broad applicability across a wide variety of Metazoan species. 4 Currently, despite the low frequency of transmission of transgenes, the frequency of phenotype modifications and overall animal transgenesis has been as high as 80 percent in some experiments. 4
The potential use of sperm-mediated gene transfer for embryo somatic gene therapy is a possibility for future research. Embryo somatic gene therapy would be advantageous because there seems to be an inverse correlation between the age of the patient and the effectiveness of gene therapy. Therefore, the possibility of gene therapy treatment before irreversible damage occurs would be ideal. 4 A majority of the experiments that report successful SMGT provide evidence of post-fertilization transfer and maintenance of transgenes. 6 SMGT has potential advantages of being a simple and cost-effective method of gene therapy, especially in contrast with pronuclear microinjection, another transgenic technique. Nevertheless, despite some successes and its potential utility, SMGT is not yet established as a reliable form of genetic modification. 6
1. Lavitrano M, Giovannoni R, Cerrito MG. 2013. Methods for sperm-mediated gene transfer. Methods Molecular Biology. 927:519-529.
2. García-Vázquez FA, Ruiz S, Grullón LA, Ondiz AD, Gutiérrez-Adán A, Gadea J. 2011. Factors affecting porcine sperm mediated gene transfer. Research in Veterinary Science. 91(3):446-53.
3. Lavitrano M, Busnelli M, Cerrito MG, Giovannoni R, Manzini S, Vargiolu A. 2006. Sperm-mediated gene transfer. Reproduction, Fertility and Development. 18:19-23.
4. Smith K, Spadafora C. 2005. Sperm-mediated gene transfer: applications and implications. BioEssays. 27(5):551-562.
5. Collares T, Campos VF, de Leon PM, Moura, Cavalcanti PV, Amaral, MG, et al. 2011. Transgene transmission in chickens by sperm-mediated gene transfer after seminal plasma removal and exogenous DNA treated with dimethylsulfoxide or N,N-dimethylacetamide. Journal of Biosciences. 36(4):613-620.
6. Smith K. 2004. Gene therapy: the potential applicability of gene transfer technology to the human germline. International Journal of Medical Sciences. 1(2):76-91. | https://en.wikipedia.org/wiki/Sperm-mediated_gene_transfer |
Sperm Chromatin Structure Assay (SCSA) is a diagnostic approach that detects sperm abnormality with a large extent of DNA fragmentation . [ 1 ] First described by Evenson in 1980, the assay is a flow cytometric test that detects the vulnerability of sperm DNA to acid-induced denaturation DNA in situ . [ 2 ] SCSA measures sperm DNA fragmentation attributed to intrinsic and extrinsic factors and reports the degree of fragmentation in terms of DNA Fragmentation Index (DFI). The use of SCSA expands from evaluation of male infertility and subfertility , toxicology studies and evaluation of quality of laboratory semen samples . Notably, SCSA outcompetes other convention sperm DNA fragmentation (sDF) assays such as TUNEL and COMET in terms of efficiency, objectivity, and repeatability.
Before the development of SCSA, diagnosis or prognosis of male infertility / subfertility was principally referenced the World Health Organisation (WHO) manual-based semen parameters, [ 3 ] including semen concentration , motility , and morphology . Yet, several reports of pregnancy failure had the parameters within normal range, suggesting that none of these measurements has drawn a reliable conclusion to reflect chance of fertility of a couple. [ 4 ] Furthermore, such parameters are often associated with high labour intensity and lack of statistical power .
In the late 1970s, Donald P. Evenson at Memorial Sloan Kettering Cancer Centre in the United States received an NIH Research Project Grant (RO1) for mammalian sperm chromatin structure study. [ 5 ] [ 6 ] Various techniques have since been adopted to gain access to sperm DNA integrity. In particular, transmission electron microscopy reflected a significant amount of sperm chromatin heterogeneity. [ 4 ] [ 7 ]
The heterogeneity was then confirmed through flow cytometry by contrasting AO staining results between human and mouse sperm nuclei . Homogeneous results were observed in the mouse sample while heterogeneous fluorescence intensity varied among the human sample. A hypothesis was proposed “single-stranded/double-stranded DNA breaks-induced sperm DNA fragmentation is correlated to male infertility .” [ 4 ] [ 5 ] In 1980, Evenson et al. published papers that synthesise this knowledge into clinical tests and found SCSA.
Initially, utilization of thermal energy in buffer (100 °C, 5 min) was proposed and used for denaturation of DNA at sites DNA damage. [ 8 ] However, the heated sperm protocol was time-consuming and induced random loss of sperm sample. Therefore, acid-induced denaturation has replaced heat-induced denaturation due to greater convenience of low pH technique and similarity in results. [ 5 ]
SCSA is a widespread diagnostic tool in detection of sperm samples with a high degree of DNA fragmentation and absence of histone -to- protamine proteins exchange in sperm nuclei . [ 9 ] SCSA defines sperm abnormality as an increased vulnerability of sperm DNA to in-situ heat/acid-induced denaturation . [ 4 ] Theoretically, a completely mature and healthy sperm nuclei , which is rich in disulfide bond (S-S) , shall have its DNA preserved in double-stranded form. [ 5 ] A low pH treatment opens up defective sperm DNA at the sites of damage. Through acridine orange (AO) staining , AO molecules are intercalated into double-stranded DNA in intact sperms while aggregation of AO molecules occurs at single-stranded DNA in defective sperms. [ 4 ] [ 5 ] Undergoing flow cytometry (blue light), green (native DNA) and red (damaged DNA) fluorescence will be emitted from intact and defective sperms respectively. [ 2 ] [ 4 ] [ 10 ] Signals will be analysed with software programming in examination of both sperm DNA fragmentation (sDF) and atypical chromatin structure.
The integrity of sperm DNA is in close correlation with the transfer of paternal DNA into the oocyte during fertilisation . The etiology of sperm DNA damage can be subdivided into intrinsic and extrinsic factors. The former is attributed to a series of pathophysiological phenomena during spermatogenesis ; the latter is caused by postnatal exposure to endogenous sources of DNA breaks.
Currently, only the SCSA protocol developed by Evenson et al. has received trademark protection in achievement of clinical relevance between different laboratories. [ 4 ] The individual steps of SCSA are as follows:
SCSA consists of a fixed flow cytometry protocol and a specific computing program, SCSAsoft ®. Measurements include DNA fragmentation index (DFI) and High DNA Stainable (HDS) fraction, which represent the percentage of sperm with DNA breaks / protamine defects and immature spermatozoa without full protamination respectively. [ 10 ]
Also known as Cells Outside the Main Peak of αt (COMPαt), DFI can be further sub-classified into mean DFI (X DFI) and standard deviation DFI (SD DFI). [ 5 ] The index has been determined as the most sensitive criteria for fertility assessment in reflection of sperm DNA integrity. Normal DFI implies no measurable value; moderate DFI sample infers normal sperm morphology ; and high DFI fractions exhibited elongated nuclei and signs of apoptosis . In general, the greater the DFI, the higher the chance of infertility or subfecundity.
Within DFI of 0-20%, the occurrence of spontaneous pregnancy remains consistent; [ 2 ] when DFI exceeds 20%, the rate of natural fertility gradually declines; [ 2 ] when DFI exceeds 30%, the odds ratio for natural or Intrauterine insemination (IUI) fertility is greatly reduced by 8-10 folds, suggesting a close-to-zero chance of pregnancy . [ 2 ]
The HDS sperm population has a remarkably high degree of DNA staining by AO molecules due to the presence of unprocessed P2 protamines . [ 9 ] [ 25 ] Determination of HDS value reflects structural chromatin abnormalities. A high HDS value is indicative of immature sperm morphology and hence pregnancy failure. [ 25 ] [ 26 ]
Since the SCSA can be performed to assess the sperm abnormality, it is a valid instrument to determine male infertility or subfertility .
Although the causes and events that actuate sperm DNA damage and fragmentation are not yet fathomed, Sperm DNA fragmentation has been shown to be closely correlated with fertility and subfertility in not only humans, but also bulls, boars, and stallions. [ 5 ] [ 27 ] [ 28 ] [ 29 ] Such finding asserts the DFI determined by SCSA to be a strong independent predictor of in vivo pregnancy and a clinically useful technique. [ 13 ] [ 23 ] [ 30 ] [ 31 ] [ 8 ]
Currently, 25% DFI is the established clinical threshold in classifying males into statistical probability of: 1) increased time for natural pregnancy, 2) lower chance of Intrauterine insemination (IUI) success, 3) more miscarriage , or 4) infertility . High HDS values are in positive correlation to pregnancy failures.
In such cases, other assisted reproductive technologies (ART) may be performed, including intracytoplasmic sperm injection (ICSI) (for sperm sample with DFI>25%) or testicular sperm extraction (TESE) (for sperm sample with DFI>50%). [ 9 ]
Sperm DNA damage can be attributed to exposure chemotherapy , radiotherapy , or other environmental toxicants . SCSA is highly dose-responsive to sperm DNA fragmentation induced by chemical toxicants. [ 13 ] Therefore, SDαt is the most important variable for toxicology studies .
SCSA is also performed to assess the quality of laboratory sperm samples that have been stored for at least 24 hours. Semen samples that have been stored at appropriate conditions will have essentially no change, while greater change in DNA quality indicates an improper handling. [ 32 ]
SCSA has numerous advantages when compared to other sperm DNA fragmentation (sDF) assays [ TUNEL assay , COMET assay , and Sperm Chromatin Dispersion (SCD)], which include:
Despite the objective data and advantages offered, the efficacy of SCSA in fertility assessment remains doubted clinically. Suggested limitations include: | https://en.wikipedia.org/wiki/Sperm_Chromatin_Structure_Assay |
A sperm bank , semen bank, or cryobank is a facility or enterprise which purchases, stores and sells human semen . The semen is produced and sold by men who are known as sperm donors . [ 1 ] The sperm is purchased by or for other persons for the purpose of achieving a pregnancy or pregnancies other than by a sexual partner. Sperm sold by a sperm donor is known as donor sperm .
A sperm bank may be a separate entity supplying donor sperm to individuals or to fertility centers or clinics, or it may be a facility which is run by a clinic or other medical establishment mainly or exclusively for their patients or customers.
A pregnancy may be achieved using donor sperm for insemination with similar outcomes to sexual intercourse . [ 2 ] By using sperm from a donor rather than from the sperm recipient's partner, the process is a form of third party reproduction . In the 21st century artificial insemination with donor sperm from a sperm bank is most commonly used for individuals with no male partner, i.e. single women and coupled lesbians. [ 3 ]
A sperm donor must generally meet specific requirements regarding age and screening for medical history . [ 4 ] In the United States, sperm banks are regulated as Human Cell and Tissue or Cell and Tissue Bank Product (HCT/Ps) [ 5 ] establishments by the Food and Drug Administration . [ 6 ] Many states also have regulations in addition to those imposed by the FDA. [ 7 ] In the European Union a sperm bank must have a license according to the EU Tissue Directive . In the United Kingdom, sperm banks are regulated by the Human Fertilisation and Embryology Authority .
The first sperm banks began as early as 1964 in Iowa, USA and Tokyo, Japan and were established for a medical therapeutic approach to support individuals who were infertile. As a result, over 1 million babies were born within 40 years. [ 4 ]
Sperm banks provide the opportunity for individuals to have a child who otherwise would not be able to conceive naturally. This includes, but is not limited to, single women, same-sexed couples, and couples where one partner is infertile. [ 3 ]
Where a sperm bank provides fertility services directly to a recipient woman, it may employ different methods of fertilization using donor sperm in order to optimize the chances of a pregnancy. Sperm banks do not provide a cure for infertility in individuals who produce non-viable sperm. Nevertheless, the increasing range of services available through sperm banks enables people to have choices over challenges with reproduction.
Individuals may choose an anonymous donor who will not be a part of family life, or they may choose known donors who may be contacted later in life by the donor children. People may choose to use a surrogate to bear their children, using eggs provided by the person and sperm from a donor. Sperm banks often provide services which enable an individual to have subsequent pregnancies by the same donor, but equally, people may choose to have children by a number of different donors. Sperm banks sometimes enable an individual to choose the sex of their child, enabling even greater control over the way families are planned. Sperm banks increasingly adopt a less formal approach to the provision of their services thereby enabling people to take a relaxed approach to their own individual requirements.
Men who donate semen through a sperm bank provide an opportunity for others who cannot have children on their own. Sperm donors may or may not have legal obligations or responsibilities to the child conceived through this route. Whether a donor is anonymous or not, this factor is important in allowing sperm banks to recruit sperm donors and to use their sperm to produce whatever number of pregnancies from each donor as are permitted where they operate, or alternatively, whatever number they decide.
In many parts of the world sperm banks are not allowed to be established or to operate. Where sperm banks are allowed to operate they are often controlled by local legislation which is primarily intended to protect the unborn child, but which may also provide a compromise between the conflicting views which surround their operation. A particular example of this is the control which is often placed on the number of children which a single donor may father and which may be designed to protect against consanguinity . However, such legislation usually cannot prevent a sperm bank from supplying donor sperm outside the jurisdiction in which it operates, and neither can it prevent sperm donors from donating elsewhere during their lives. [ 8 ] There is an acute shortage of sperm donors in many parts of the world and there is obvious pressure from many quarters for donor sperm from those willing and able to provide it to be made available as safely and as freely as possible. [ 9 ]
The finding of a potential sperm donor and motivating them to donate sperm is typically called recruitment. A sperm bank can recruit donors by advertising, often in colleges, in local newspapers, and also on the internet. [ 10 ]
A donor must be a fit and healthy male , normally between 18 and 45 years of age, and willing to undergo frequent and rigorous testing. The donor must also be willing to donate their sperm so that it can be used to impregnate people who are unrelated to and unknown by them. Some sperm banks require two screenings and a laboratory screening before a donor is eligible. [ 11 ] The donor must agree to relinquish all legal rights to all children which result from their donations. The donor must produce their sperm at the sperm bank thus enabling the identity of the donor, once proven, always to be ascertained, and also enabling fresh samples of sperm to be produced for immediate processing. Some sperm banks have been accused of heightism due to minimum height requirements. [ 12 ]
A sperm bank will aim to provide donor sperm which is safe by checking and screening donors and of their semen. A sperm donor must generally meet specific requirements regarding age and medical history. Requirements for sperm donors are strictly enforced, as in a study of 24,040 potential sperm donors, only 5620, or 23.38% were eligible to donate their sperm. [ 13 ]
Sperm banks typically screen potential donors for a range of diseases and disorders, including genetic diseases , chromosomal abnormalities and sexually transmitted infections that may be transmitted through sperm. The screening procedure generally also includes a quarantine period, in which the samples are frozen and stored for at least six months after which the donor will be re-tested for the STIs. This is to ensure no new infections have been acquired or have developed during the period of donation. Providing the result is negative, the sperm samples can be released from quarantine and used in treatments. Common reasons for sperm rejection include suboptimal semen quality and STDs. [ 13 ] Chromosomal abnormalities are also a cause for semen rejection, but are less common. [ 13 ] Children conceived through sperm donation have a birth defect rate of almost a fifth compared with the general population. [ 14 ]
A sperm bank takes a number of steps to ensure the health and quality of the sperm which it supplies and it will inform customers of the checks which it undertakes, providing relevant information about individual donors. A sperm bank will usually guarantee the quality and number of motile sperm available in a sample after thawing. They will try to select men as donors who are particularly fertile and whose sperm will survive the freezing and thawing process. Samples are often sold as containing a particular number of motile sperm per milliliter, and different types of samples may be sold by a sperm bank for differing types of use, e.g. ICI or IUI .
The sperm will be checked to ensure its fecundity and also to ensure that motile sperm will survive the freezing process. If a man is accepted onto the sperm bank's program as a sperm donor, his sperm will be constantly monitored, the donor will be regularly checked for infectious diseases, and samples of his blood will be taken at regular intervals. A sperm bank may provide a donor with dietary supplements containing herbal or mineral substances such as maca , zinc , vitamin E and arginine which are designed to improve the quality and quantity of the donor's semen, [ 15 ] as well as reducing the refractory time [ 16 ] (i.e. the time between viable ejaculations). All sperm is frozen in straws or vials and stored for as long as the sperm donor may and can maintain it.
Donors are subject to tests for infectious diseases such as human immunoviruses HIV (HIV-1 and HIV-2), human T-cell lymphotropic viruses (HTLV-1 and HTLV-2), syphilis , chlamydia , gonorrhea , hepatitis B virus , hepatitis C virus , cytomegalovirus (CMV), Trypanosoma cruzi and malaria as well as hereditary diseases such as cystic fibrosis , sickle cell anemia , familial Mediterranean fever , Gaucher's disease , thalassaemia , Tay–Sachs disease , Canavan's disease , familial dysautonomia , congenital adrenal hyperplasia , carnitine transporter deficiency . Some sperm banks may also use karyotyping to ensure donors are 46XY.
A sperm donor may also be required to produce their medical records and those of their family, often for several generations. A sperm sample is usually tested micro-biologically at the sperm bank before it is prepared for freezing and subsequent use. A sperm donor's blood group may also be registered to ensure compatibility with the recipient.
Some sperm banks may disallow sexually active gay men from donating sperm due to the population's increased risk of HIV and hepatitis B . Modern sperm banks have also been known to screen out potential donors based on genetic conditions and family medical history. [ 17 ]
The majority of sperm donors who donate their sperm through a sperm bank receive some kind of payment, although this is rarely a significant amount. A review including 29 studies from nine countries came to the result that the amount of money actual donors received for their donation varied from $10 to €70 per donation or sample. [ 18 ] The payments vary from the situation in the United Kingdom where donors are only entitled to their expenses in connection with the donation, to the situation with some US sperm banks where a donor receives a set fee for each donation plus an additional amount for each vial stored. At one prominent California sperm bank for example, TSBC , donors receive roughly $50 for each donation (ejaculation) which has acceptable motility/survival rates both at donation and at a test-thaw a couple of days later. Because of the requirement for the two-day celibacy period before donation, and geographical factors which usually require the donor to travel, it is not a viable way to earn a significant income—and is far less lucrative than selling human eggs. Some private donors may seek remuneration although others donate for altruistic reasons. According to the EU Tissue Directive donors in EU may only receive compensation, which is strictly limited to making good the expenses and inconveniences related to the donation.
A sperm donor will usually be required to enter into a contract with a sperm bank to supply their semen, typically for a period of six to twenty-four months depending on the number of pregnancies which the sperm bank intends to produce from the donor. If a sperm bank has access to world markets e.g. by direct sales, or sales to clinics outside their own jurisdiction, a man may donate for a longer period than two years, as the risk of consanguinity is reduced (although local laws vary widely). Some sperm banks with access to world markets impose their own rules on the number of pregnancies which can be achieved in a given regional area or a state or country, and these sperm banks may permit donors to donate for four or five years, or even longer.
The contract may also specify the place and hours for donation, a requirement to notify the sperm bank in the case of acquiring a sexual infection, and the requirement not to have intercourse or to masturbate for a period of usually 2–3 days before making a donation. [ 19 ]
The contract may also describe the types of treatment for which the donated sperm may be used, such as artificial insemination and IVF, and whether the donor's sperm may be used in surrogacy arrangements. It may also stipulate whether the sperm may be used for research or training purposes. In certain cases, a sperm donor may specify the maximum number of offspring or families which may be produced from the donor's sperm. 'Family' may be defined as a couple who may each bear children from the same donor. The contract may also require consent if the donor's samples are to be exported. In the United Kingdom, for example, the maximum number of families for which a donor is permitted to bear children is ten, but a sperm bank or fertility center in the UK may export sperm to other fertility centers so that this may be used to produce more pregnancies abroad. Where this happens, consent must be provided by the donor. Faced with a growing demand for donor sperm, sperm banks may try to maximize the use of a donor whilst still reducing the risk of consanguinity. In legislations with a national register of sperm donors or a national regulatory body, a sperm donor may be required to fill in a separate form of consent which will be registered with the regulatory authority. In the United Kingdom this body is the HFEA.
A sperm donor generally produces and collects sperm at a sperm bank or clinic by masturbation in a private room or cabin, known as a 'men's production room' (UK), 'donor cabin' (DK) or a masturbatorium (US). Many of these facilities contain pornography such as videos/DVD, magazines, and/or photographs which may assist the donor in becoming aroused in order to facilitate production of the ejaculate, also known as the "semen sample" but the increasing usage of porn in the U.S. has dulled many men to its effects. [ 20 ] Often, using any type of personal lubricant, saliva, oil or anything else to lubricate and stimulate the genitals is prohibited as it can contaminate the semen sample and have negative impacts on the quality and health of sperm. [ 21 ] In some circumstances, it may also be possible for semen from donors to be collected during sexual intercourse with the use of a collection condom which results in higher sperm counts. [ 22 ]
After collection, sperm must be processed for storage. According to the Sperm Bank of California, sperm banks and clinics can use the 'unwashed' or 'wash' method to process sperm samples. The 'wash' method includes removing unwanted particles and adding buffer solutions to preserve viable sperm. However, this approach can contribute to further stress on the sperm cells and decrease the survival of sperm after freezing. The 'unwashed' approach allows for more flexibility to freeze the semen sample and increases the number of sperm survival. [ 23 ] One sample can produce 1–20 vials or straws, depending on the quantity of the ejaculate and whether the sample is 'washed' or 'unwashed'. 'Unwashed' samples are used for intracervical insemination (ICI) treatments, and 'washed' samples are used in intrauterine insemination (IUI) and for in-vitro fertilization (IVF) or assisted reproduction technologies (ART) procedures.
A cryoprotectant semen extender is conducted if the semen sample is placed in the freezer for storage. Semen extenders play a key factor in protecting sperm sample from 'freeze and osmotic shock, oxidative stress, and cell injury' due to the formation of ice crystal during frozen storage. The collection of semen is preserved by stabilizing the properties of the sperm cells such as the membrane, motility, and 'DNA integrity' in order to create a sustainable viable environment. [ 24 ] There are two common forms of medium for sperm cyropreservation, one containing egg yolk from hens and glycerol, and the other containing just glycerol. [ 25 ] One study compared media supplemented with egg yolk and media supplemented with soy lecithin, finding that there was no significance between sperm motility, morphology, chromatin decondensation, or binding between the two, indicating that soy lecithin may be a viable alternative to egg yolk. [ 25 ]
After the sample has been processed for cryoprotection, the sperm is stored in small vials or straws holding between 0.4 and 1.0 ml of sperm and then cryogenically preserved in liquid nitrogen tanks. Two approaches for sperm cryoperservation include conventional freezing and vitrification. The conventional technique consists of a slow freezing process that is most commonly used for assisted reproduction technologies (ART). Whereas the vitrification method is a faster approach for sperm cryopreservation in converting liquid to solid state. The disadvantage of this latter process is increase in contamination from the liquid nitrogen and smaller sperm sample size to improve the speed for 'high cooling rate'. [ 26 ]
It has been proposed that there should be an upper limit on how long frozen sperm can be stored; however, a baby has been conceived in the United Kingdom using sperm frozen for 21 years [ 27 ] and andrology experts believe sperm can be frozen indefinitely. [ 28 ] The UK government places an upper limit for storage of 55 years. [ 29 ]
Following the necessary quarantine period, which is usually six months, a sample will be thawed. To thaw a sperm sample, the vial or straw is left at room temperature for approximately 30 minutes, and then brought to body temperature by holding it in the hands of the person performing the insemination. [ 30 ] Once a sperm sample is thawed, it cannot be frozen again, and should be used to artificially inseminate a recipient or used for another assisted reproduction technologies (ART) treatment immediately. [ 30 ]
Freeze-drying is another promising alternative for storing semen for its accessibility with regular refrigerator. This method has been successfully replicated in animal species. However, DNA can be damaged in this process, therefore further research is warranted to determine factors that can effect the efficacy of this method. [ 31 ]
One study conducted by investigators at the University of North Carolina Chapel Hill looked into donated sperm utilization within the United States from 1995 to 2017. Cross-sectional studies recorded that an estimated 170,701 individuals during 1995 used donated sperm, while the 2011 to 2013 cohort had a decreased amount of donated sperm use of 37,385. Most recently, in the 2015 to 2017 cohort, 440,986 individuals were reported to use donated sperm. [ 32 ] When looking at 200,197 individuals across 2011–2017, 76% had a 4-year college degree or further while 24% had high school or 2-year college degree. In terms of household percent of poverty, 71% of the sperm bank users were at or above 400% of the household poverty level while only 11% were between 200 and 399% of the household poverty levels. Although the household income levels were not explicit, there seems to be an obvious trend that higher education level attainment (such as finishing college or higher) and being at much higher income level above the household poverty levels were the common tendencies in the sperm bank users. [ 32 ]
Based on the statistics presented in earlier discussions, there is controversy with regard to a perceived lack of diversity within the donor sperm pool of many sperm banks. This includes, but is not limited to, height requirements implemented by some sperm banks. [ 13 ] As a result, it is alleged that potential sperm recipients often encounter very limited sperm donor pool options. Lack of diversity results in very limited choices especially among ethnic minorities within the United States. [ 33 ] Whenever an individual chooses to specify their preferred donor background, the number of available options (sperm donors that meet the particular individual's criteria) can dwindle down to the low single digits. [ 34 ] Scott Brown from California Cryobank admitted: "We don't get as many minority applicants as we [would] like." Even after numerous attempts to reach out to numerous ethnic communities, the response can be nearly nonexistent. [ 34 ]
At the California Cryoback, Brown mentions that one out of 100 would be able to become final sperm donor while Ottey from the Fairfax Cryobank mentions one out of 200 would be able to become ultimate sperm donors. [ 35 ] [ 34 ] In addition, locations of the California Cryobank are in Los Angeles, Los Altos, California; mid-Manhattan, and Cambridge Massachusetts. These locations are known to have a population with higher socioeconomic latitude and being more likely to afford the services. Moreover, one of the requirements includes the potential sperm donor to be able to live nearby the sperm bank in order to provide samples once to twice a month for at least a term of six months. [ 34 ] This could create potential barriers for populations who are at socioeconomic disadvantage and do not have their own forms of transportation; often having to rely on multiple forms of public transportation to reach certain places. This factor could cause a significant decrease in the sperm donor pool and less diverse availability for sperm recipients.
Some controversy stems from the fact that donors father children for others, in the majority of cases, for single people or same-sex couples, but usually take no part in the upbringing of such children. The issue of sperm banks providing fertility services to single women and coupled lesbians so that they can have their own biological children by a donor is itself often controversial in some jurisdictions, but in many countries where sperm banks operate, this group form the main body of recipients. Donors usually do not have a say in who may be a recipient of their sperm. [ 36 ]
Another controversy centers around the use of sperm posthumously, or after the death of the sperm donor, as pioneered by California Cryobank . [ 37 ] Within the United States, there were differences when it came to a child conceived after the father's death and the eligibility for survivor's benefits. Under California law, there was one court case (Vernoff vs. Astrue) in which the mother's child (conceived after the father's death) was not eligible for the survivor's benefits. [ 38 ] However, Arizona courts had a different approach when it came to children who were born after father's death that the children are eligible for the survivors benefits. There were numerous other stories of similar situations across different states in the United States and even the United Kingdom. Canada, France, Germany, and Sweden do not permit the retrieval use of sperm posthumously. [ 37 ]
Subject to any regulations restricting who can obtain donor sperm, donor sperm is available to all people who, for whatever reason, wish to have a child. These regulations vary significantly across jurisdictions, and some countries do not have any regulations. When an individual finds that they are barred from receiving donor sperm within their jurisdiction, they may travel to another jurisdiction to obtain sperm. Regulations change from time to time. In most jurisdictions, donor sperm is available to an individual if their partner is infertile or where they have a genetic disorder. However, the categories of individuals who may obtain donor sperm is expanding, with its availability to single persons and to same-sex couples becoming more common, and some sperm banks supply fertility centers which specialize in the treatment of such people.
Frozen vials of donor sperm may be shipped by the sperm bank to a recipient's home for self-insemination, or they may be shipped to a fertility clinic or physician for use in fertility treatments. The sperm bank will rely on the recipient woman or medical practitioner to report the outcome of any use of the sperm to the sperm bank. This enables a sperm bank to adhere to any national limits of pregnancy numbers. The sperm bank may also impose its own worldwide limit on numbers.
Sperm is introduced into the recipient by means of artificial insemination or by IVF . The most common technique is conventional artificial insemination which consists of a catheter to put the sperm into the vagina where it is deposited at the entrance to the cervix. In biological terms, this is much the same process as when semen is ejaculated from the penis during sexual intercourse. Owing to its simplicity, this method of insemination is commonly used for home and self inseminations principally by single women and lesbians. Other types of uses include intrauterine insemination ( IUI ) and deep intrauterine artificial insemination where 'washed' sperm must be used. These methods of insemination are most commonly used in fertility centers and clinics mainly because they produce better pregnancy rates than ICI insemination especially where the woman has no underlying fertility issues. [ 39 ]
Men may also store their own sperm at a sperm bank for future use particularly where they anticipate traveling to a war zone or having to undergo chemotherapy which might damage the testes.
Sperm from a sperm donor may also be used in surrogacy arrangements and for creating embryos for embryo donation . Donor sperm may be supplied by the sperm bank directly to the recipient to enable a woman to perform her own artificial insemination which can be carried out using a needleless syringe or a cervical cap conception device . The cervical cap conception device allows the donor semen to be held in place close to the cervix for between six and eight hours to allow fertilization to take place. Alternatively, donor sperm can be supplied by a sperm bank through a registered medical practitioner who will perform an appropriate method of insemination or IVF treatment using the donor sperm in order for the woman to become pregnant.
In the United States, sperm banks maintain lists or catalogs of donors which provide basic information about the donor such as racial origin, skin color, height, weight, color of eyes, and blood group. [ 17 ] Some of these catalogs are available for browsing on the Internet, while others are made available to patients only when they apply to a sperm bank for treatment. Some sperm banks make additional information about each donor available for an additional fee, and others make additional basic information known to children produced from donors when those children reach the age of 18. Some clinics offer "exclusive donors" whose sperm is used to produce pregnancies for only one recipient woman. How accurate this is, or can be, is not known, and neither is it known whether the information produced by sperm banks, or by the donors themselves, is true. Many sperm banks will, however, carry out whatever checks they can to verify the information they request, such as checking the identity of the donor and contacting his own doctor to verify medical details.
In the United Kingdom, most donors are anonymous at the point of donation and recipients can see only non-identifying information about their donor (height, weight, ethnicity etc.). Donors need to provide identifying information to the clinic and clinics will usually ask the donor's doctor to confirm any medical details they have been given. Donors are asked to provide a pen portrait of themselves which is held by the HFEA and can be obtained by the adult conceived from the donation at the age of 18, along with identifying information such as the donor's name and last known address. Known donation is permitted and it is not uncommon for family or friends to donate to a recipient couple.
Qualities that potential recipients typically prefer in donors include the donors being tall, college educated, and with a consistently high sperm count. [ 40 ] A review came to the result that 68% of donors had given information to the clinical staff regarding physical characteristics and education but only 16% had provided additional information such as hereditary aptitudes and temperament or character. [ 18 ]
Sperm banks make information available about the sperm donors whose donations they hold to enable customers to select the donor whose sperm they wish to use. This information is often available by way of an online catalog. Subscription fees to be able to view the sperm donor through California Cryobank, for example, start at $145. [ 41 ] This cost could potentially be a barrier for many on limited income and may not have discretionary income to spend on sperm donor services.
A sperm bank will also usually have facilities to help customers to make their choice and they will be able to advise on the suitability of donors for individual donors and their partners.
Where the recipient has a partner, they may prefer to use sperm from a donor whose physical features are similar to those of their partner if they have one. In some cases, the choice of a donor with the correct blood group will be paramount, with particular considerations for the protection of recipients with negative blood groups. If a surrogate is to be used, such as where the customer is not intending to carry the child, considerations about their blood group etc. will also need to be taken into account. Similar considerations will apply where both partners in a lesbian couple intend to have a child using the same donor.
Information made available by a sperm bank will usually include the race, height, weight, blood group, health and eye color of the donor. Sometimes information about the donor's age, family history and educational achievements will also be given. Some sperm banks make a 'personal profile' of a donor available and occasionally more information may be purchased about a donor, either in the form of a DVD or in written form. Catalogs usually state whether samples supplied by a particular donor have already given rise to pregnancies, but this is not necessarily a guide to the fecundity of the sperm since a donor may not have been in the program long enough for any pregnancies to have been recorded. The donor's educational qualification is also taken into account when choosing a donor. [ 21 ]
If an individual intends to have more than one child, they may wish to have the additional child or children by the same donor. Sperm banks will usually advise whether sufficient stocks of sperm are available from a particular donor for subsequent pregnancies, and they normally have facilities available so that the woman may purchase and store additional vials from that donor on payment of an appropriate fee. These will be stored until required for subsequent pregnancies or they may be on-sold if they become surplus to the woman's requirements.
The catalogue will also state whether samples of sperm are available for ICI, IUI, or IVF use.
Some sperm banks enable recipients to choose the sex of their child, through methods of sperm sorting . Although the methods used do not guarantee 100% success, the chances of being able to select the gender of a child are held to be considerably increased. [ 42 ]
One of the processes used is the 'swim up' method, whereby a sperm extender is added to the donor's freshly ejaculated sperm and the test-tube is left to settle. After about half-an-hour, the lighter sperm, containing the male chromosome pair (XY), will have swum to the top, leaving the heavier sperm, containing the female chromosome pair (XX), at the bottom, thus allowing selection and storage according to sex.
The alternative process is the Percoll Method which is similar to the 'swim up' method but involves additionally the centrifuging of the sperm in a similar way to the washing of samples produced for IUI inseminations, or for IVF purposes.
There is a market for vials of processed sperm and for various reasons a sperm bank may sell-on stocks of vials which it holds known as 'onselling'. The costs of screening of donors and storage of frozen donor sperm vials are not insignificant and in practice most sperm banks will try to dispose of all samples from an individual donor. The onselling of sperm therefore enables a sperm bank to maximize the sale and disposal of sperm samples which it has processed. The reasons for onselling may also be where part of, or even the main business of, a particular sperm bank is to process and store sperm rather than to use it in fertility treatments, or where a sperm bank is able to collect and store more sperm than it can use within nationally set limits. In the latter case a sperm bank may onsell sperm from a particular donor for use in another jurisdiction after the number of pregnancies achieved from that donor has reached its national maximum.
Sperm banks may supply other sperm banks or a fertility clinic with donor sperm to be used for achieving pregnancies.
Sperm banks may also supply sperm for research or educational purposes.
In the United States, sperm banks are regulated as Human Cell and Tissue or Cell and Tissue Bank Product (HCT/Ps) establishments by the Food and Drug Administration (FDA) with new guidelines in effect May 25, 2005. [ 43 ] Many states also have regulations in addition to those imposed by the FDA, including New York and California.
In the European Union a sperm bank must have a license according to the EU Tissue Directive which came into effect on April 7, 2006. In the United Kingdom, sperm banks are regulated by the Human Fertilisation and Embryology Authority .
In countries where sperm banks are allowed to operate, the sperm donor will not usually become the legal father of the children produced from the sperm he donates, but he will be the 'biological father' of such children. In cases of surrogacy involving embryo donation , a form of ' gestational surrogacy ', the 'commissioning mother' or the 'commissioning parents' will not be biologically related to the child and may need to go through an adoption procedure.
As with other forms of third party reproduction , the use of donor sperm from a sperm bank gives rise to a number of moral, legal, and ethical issues, including, but not limited to the right of the sperm donor remaining anonymous, and the child's right to know their familial background. [ 44 ]
Furthermore, as local regulations reduce the size of the donor pool and, in some cases, exclude entire classes of potential buyers such as single women and lesbian couples, restricting donations to only heterosexual couples who are married. Some customers choose to buy abroad or on the internet, having the samples delivered at home. [ 45 ]
There have been reports of incidents of abuse regarding forced insemination with sperm samples bought online. [ 46 ]
Further abuse of sperm banks comes from the fertility clinic staff themselves. There have been a number of reports of staff at sperm banks and fertility clinics providing their own sperm in place of donor sperm. There have also been cases in which men have claimed their sperm sample was used by a clinic to inseminate a woman without his consent. [ 47 ] [ 48 ] This has led to cases of malpractice, and in some states, lobbying to create fertility fraud laws. [ 49 ] These incidents have also led to outcry by people who had been conceived by such incidents, raising concerns of consanguinity, as well as the simple right to know who their siblings and biologic parents are. [ 49 ] | https://en.wikipedia.org/wiki/Sperm_bank |
Sperm chemotaxis is a form of sperm guidance , in which sperm cells ( spermatozoa ) follow a concentration gradient of a chemoattractant secreted from the oocyte and thereby reach the oocyte.
Since the discovery of sperm attraction to the female gametes in ferns over a century ago, [ 1 ] sperm guidance in the form of sperm chemotaxis has been established in a large variety of species [ 2 ] Although sperm chemotaxis is prevalent throughout the Metazoa kingdom, from marine species with external fertilization such as sea urchins and corals , to humans, [ 2 ] [ 3 ] [ 4 ] much of the current information on sperm chemotaxis is derived from studies of marine invertebrates, primarily sea urchin and starfish . [ 5 ] As a matter of fact, until not too long ago, the dogma was that, in mammals, guidance of spermatozoa to the oocyte was unnecessary. This was due to the common belief that, following ejaculation into the female genital tract, large numbers of spermatozoa 'race' towards the oocyte and compete to fertilize it.
Research during the 1980s [ 6 ] caused this belief to be taken apart when it became clear that only few of the ejaculated spermatozoa — in humans, only ~1 of every million spermatozoa — succeed in entering the oviducts ( fallopian tubes ) [ 4 ] [ 7 ] and when more recent studies showed that mammalian spermatozoa do respond chemotactically. [ 8 ]
In sperm chemotaxis, the oocyte secretes a chemoattractant , which, as it diffuses away, forms a concentration gradient : a high concentration close to the egg, and a gradually lower concentration as the distance from the oocyte increases. Spermatozoa can sense this chemoattractant and orient their swimming direction up the concentration gradient towards the oocyte. Sperm chemotaxis was demonstrated in a large number of non-mammalian species, from marine invertebrates [ 2 ] [ 3 ] to frogs. [ 9 ]
The sperm chemoattractants in non-mammalian species vary to a large extent. Some examples are shown in Table 1. So far, most sperm chemoattractants that have been identified in non-mammalian species are peptides or low-molecular-weight proteins (1–20 kDa ), which are heat stable and sensitive to proteases . [ 2 ] [ 3 ] Exceptions to this rule are the sperm chemoattractants of corals, ascidians , plants such as ferns, and algae (Table 1).
The variety of chemoattractants raises the question of species specificity with respect to the chemoattractant identity. There is no single rule for chemoattractant-related specificity. Thus, in some groups of marine invertebrates (e.g., hydromedusae and certain ophiuroids ), the specificity is very high; in others (e.g., starfish), the specificity is at the family level and, within the family, there is no specificity. [ 2 ] [ 3 ] [ 22 ] In mollusks , there appears to be no specificity at all. Likewise, in plants, a unique simple compound [e.g., fucoserratene — a linear, unsaturated alkene (1,3-trans 5-cis-octatriene)] might be a chemoattractant for various species. [ 10 ]
Here, too, there is no single rule. In some species (for example, in hydroids like Campanularia or tunicate like Ciona ), the swimming direction of the spermatozoa changes abruptly towards the chemoattractant source. In others (for example, in sea urchin, hydromedusa, fern, or fish such as Japanese bitterlings), the approach to the chemoattractant source is indirect and the movement is by repetitive loops of small radii. In some species (for example, herring or the ascidian Ciona) activation of motility precedes chemotaxis. [ 2 ] [ 3 ] [ 23 ] [ 24 ] In chemotaxis, cells may either sense a temporal gradient of the chemoattractant, comparing the occupancy of its receptors at different time points (as do bacteria [ 25 ] ), or they may detect a spatial gradient, comparing the occupancy of receptors at different locations along the cell (as do leukocytes [ 26 ] ). In the best-studied species, sea urchin, the spermatozoa sense a temporal gradient and respond to it with a transient increase in flagellar asymmetry. The outcome is a turn in the swimming path, followed by a period of straight swimming, [ 27 ] leading to the observed epicycloid-like movements directed towards the chemoattractant source. [ 28 ] The particular mechanism by which sea urchin sperm cells sense the temporal gradient has been recently identified as a natural implementation of the well-known adaptive controller known as extremum seeking. [ 29 ]
The molecular mechanism of sperm chemotaxis is still not fully known. The current knowledge is mainly based on studies in the sea urchin Arbacia punctulata , where binding of the chemoattractant resact (Table 1) to its receptor, a guanylyl cyclase , activates cGMP synthesis (Figure 1). The resulting rise of cGMP possibly activates K + -selective ion channels . The consequential hyperpolarization activates hyperpolarization-activated and cyclic nucleotide-gated (HCN) channels. The depolarizing inward current through HCN channels possibly activates voltage-activated Ca 2+ channels, resulting in elevation of intracellular Ca 2+ . This rise leads to flagellar asymmetry and, consequently, a turn of the sperm cell. [ 23 ]
A model of the signal-transduction pathway during sperm chemotaxis of the sea urchin Arbacia punctulata . Binding of a chemoattractant (ligand) to the receptor — a membrane-bound guanylyl cyclase (GC) — activates the synthesis of cGMP from GTP. Cyclic GMP possibly opens cyclic nucleotide-gated (CNG) K + -selective channels, thereby causing hyperpolarization of the membrane. The cGMP signal is terminated by the hydrolysis of cGMP through phosphodiesterase (PDE) activity and inactivation of GC. On hyperpolarization, hyperpolarization-activated and cyclic nucleotide-gated (HCN) channels allow the influx of Na + that leads to depolarization and thereby causes a rapid Ca 2+ entry through voltage-activated Ca 2+ channels (Ca v ), Ca 2+ ions interact by unknown mechanisms with the axoneme of the flagellum and cause an increase of the asymmetry of flagellar beat and eventually a turn or bend in the swimming trajectory. Ca 2+ is removed from the flagellum by a Na + /Ca 2+ exchange mechanism. (Taken from ref. [ 23 ] )
Following the findings that human spermatozoa accumulate in follicular fluid [ 30 ] [ 31 ] and that there is a remarkable correlation between this in vitro accumulation and oocyte fertilization, [ 30 ] chemotaxis was substantiated as the cause of this accumulation. [ 8 ] Sperm chemotaxis was later also demonstrated in mice [ 32 ] and rabbits. [ 33 ] In addition, sperm accumulation in follicular fluid (but without substantiating that it truly reflects chemotaxis) was demonstrated in horses [ 34 ] and pigs. [ 35 ] A key feature of sperm chemotaxis in humans is that this process is restricted to capacitated cells [ 36 ] [ 37 ] — the only cells that possess the ability to penetrate the oocyte and fertilize it. [ 38 ] This raised the possibility that, in mammals, chemotaxis is not solely a guidance mechanism but it is also a mechanism of sperm selection. [ 36 ] [ 37 ] Importantly, the fraction of capacitated (and, hence, chemotactically responsive) spermatozoa is low (~10% in humans), the life span of the capacitated/chemotactic state is short (1–4 hours in humans), a spermatozoon can be at this state only once in its lifetime, and sperm individuals become capacitated/chemotactic at different time points, resulting in continuous replacement of capacitated/chemotactic cells within the sperm population, i.e., prolonged availability of capacitated cells. [ 36 ] [ 39 ] These sperm features raised the possibility that prolonging the time period, during which capacitated spermatozoa can be found in the female genital tract, is a mechanism, evolved in humans, to compensate for the lack of coordination between insemination and ovulation. [ 7 ] [ 36 ] [ 37 ] [ 40 ]
In humans, there are at least two different origins of sperm chemoattractants. One is the cumulus cells that surround the oocyte, and the other is the mature oocyte itself. [ 41 ] The chemoattractant secreted from the cumulus cells is the steroid progesterone , shown to be effective at the picomolar range. [ 42 ] [ 43 ] [ 44 ] The chemoattractant secreted from the oocyte is even more potent. [ 41 ] It is a hydrophobic non-peptide molecule which, when secreted from the oocyte, is in complex with a carrier protein [ 45 ] Additional compounds have been shown to act as chemoattractants for mammalian spermatozoa. They include the chemokine CCL20 , [ 46 ] atrial natriuretic peptide (ANP), [ 47 ] specific odorants , [ 48 ] natriuretic peptide type C (NPPC), [ 49 ] and allurin, [ 50 ] to mention a few. It is reasonable to assume that not all of them are physiologically relevant.
Species specificity was not detected in experiments that compared the chemotactic responsiveness of human and rabbit spermatozoa to follicular fluids or egg-conditioned media obtained from human, bovine, and rabbit. [ 51 ] The subsequent findings that cumulus cells of both human and rabbit (and, probably, of other mammals as well) secrete the chemoattractant progesterone [ 42 ] [ 43 ] [ 44 ] is sufficient to account for the lack of specificity in the chemotactic response of mammalian spermatozoa.
Mammalian spermatozoa, like sea-urchin spermatozoa, appear to sense the chemoattractant gradient temporally (comparing receptor occupancy over time) rather than spatially (comparing receptor occupancy over space). This is because the establishment of a temporal gradient in the absence of spatial gradient, achieved by mixing human spermatozoa with a chemoattractant [ 52 ] or by photorelease of a chemoattractant from its caged compound, [ 53 ] results in delayed transient changes in swimming behavior that involve increased frequency of turns and hyperactivation events. On the basis of these observations and the finding that the level of hyperactivation events is reduced when chemotactically responsive spermatozoa swim in a spatial chemoattractant gradient [ 53 ] it was proposed that turns and hyperactivation events are suppressed when capacitated spermatozoa swim up a chemoattractant gradient, and vice versa when they swim down a gradient. [ 52 ] [ 53 ] In other words, human spermatozoa approach chemoattractants by modulating the frequency of turns and hyperactivation events, similarly to Escherichia coli bacteria. [ 25 ]
As in non-mammalian species, the end signal in chemotaxis for changing the direction of swimming is Ca 2+ . [ 54 ] The discovery of progesterone as a chemoattractant [ 42 ] [ 43 ] [ 44 ] led to the identification of its receptor on the sperm surface – CatSper , a Ca 2+ channel present exclusively in the tail of mammalian spermatozoa. [ 55 ] [ 56 ] (Note, though, that progesterone only stimulates human CatSper but not mouse CatSper. [ 56 ] Consistently, sperm chemotaxis to progesterone was not found in mice. [ 57 ] ) However, the molecular steps subsequent to CatSper activation by progesterone are obscure, though the involvement of trans-membrane adenylyl cyclase , cAMP and protein kinase A as well as soluble guanylyl cyclase , cGMP , inositol trisphosphate receptor and store-operated Ca 2+ channel was proposed. [ 58 ]
Chemotaxis is a short-range guidance mechanism. As such, it can guide spermatozoa for short distances only, estimated at the order of millimeters. [ 59 ] It is, therefore, believed that, in mammals, sperm chemotaxis occurs in the oviduct, close to the oocyte. First spermatozoa may be chemotactically guided to the oocyte-cumulus complex by the gradient of progesterone, secreted from the cumulus cells. [ 42 ] [ 43 ] [ 44 ] In addition, progesterone may inwardly guide spermatozoa, already present within the periphery of the cumulus oophorus. [ 42 ] Spermatozoa that are already deep within the cumulus oophorus may sense the more potent chemoattractant that is secreted from the oocyte [ 41 ] [ 45 ] and chemotactically guide themselves to the oocyte according to the gradient of this chemoattractant. It should be borne in mind, however, that this scenario may be an oversimplification. In view of the increasing number of different chemoattractants that are being discovered, the physiology of chemotaxis in vivo might be much more complex. | https://en.wikipedia.org/wiki/Sperm_chemotaxis |
Sperm competition is the competitive process between spermatozoa of two or more different males to fertilize the same egg [ 1 ] during sexual reproduction . Competition can occur when females have multiple potential mating partners. Greater choice and variety of mates increases a female's chance to produce more viable offspring. [ 2 ] However, multiple mates for a female means each individual male has decreased chances of producing offspring.
Sperm competition is an evolutionary pressure on males, and has led to the development of adaptations to increase male's chance of reproductive success . [ 3 ] Sperm competition results in a sexual conflict between males and females. [ 2 ] Males have evolved several defensive tactics including: mate-guarding, mating plugs , and releasing toxic seminal substances to reduce female re-mating tendencies to cope with sperm competition. [ 4 ] Offensive tactics of sperm competition involve direct interference by one male on the reproductive success of another male, for instance by mate guarding or by physically removing another male's sperm prior to mating with a female. [ 5 ] [ 6 ] For an example, see Gryllus bimaculatus .
Sperm competition is often compared to having tickets in a raffle ; a male has a better chance of winning (i.e. fathering offspring) the more tickets he has (i.e. the more sperm he inseminates a female with). However, sperm are not free to produce, [ 7 ] and as such males are predicted to produce sperm of a size and number that will maximize their success in sperm competition. By making many spermatozoa, males can buy more "raffle tickets", and it is thought that selection for numerous sperm has contributed to the evolution of anisogamy with very small sperm (because of the energy trade-off between sperm size and number). [ 8 ] Alternatively, a male may evolve faster sperm to enable his sperm to reach and fertilize the female's ovum first. Dozens of adaptations have been documented in males that help them succeed in sperm competition.
Mate-guarding is a defensive behavioral trait that occurs in response to sperm competition; males try to prevent other males from approaching the female (and/or vice versa) thus preventing their mate from engaging in further copulations . [ 2 ] Precopulatory and postcopulatory mate-guarding occurs in insects, lizards, birds and primates. Mate-guarding also exists in the fish species Neolamprologus pulcher , as some males try to "sneak" matings with females in the territory of other males. In these instances, the males guard their female by keeping her in close enough proximity so that if an opponent male shows up in his territory he will be able to fight off the rival male which will prevent the female from engaging in extra-pair copulation with the rival male. [ 9 ]
Organisms with polygynous mating systems are controlled by one dominant male. In this type of mating system, the male is able to mate with more than one female in a community. [ 10 ] The dominant males will reign over the community until another suitor steps up and overthrows him. [ 10 ] The current dominant male will defend his title as the dominant male and he will also be defending the females he mates with and the offspring he sires. The elephant seal falls into this category since he can participate in bloody violent matches in order to protect his community and defend his title as the alpha male. [ 11 ] If the alpha male is somehow overthrown by the newcomer, his children will most likely be killed and the new alpha male will start over with the females in the group so that his lineage can be passed on. [ 12 ]
Strategic mate-guarding occurs when the male only guards the female during her fertile periods. This strategy can be more effective because it may allow the male to engage in both extra-pair paternity and within-pair paternity. [ 13 ] This is also because it is energetically efficient for the male to guard his mate at this time. There is a lot of energy that is expended when a male is guarding his mate. For instance, in polygynous mate-guarding systems, the energetic costs of males is defending their title as alpha male of their community. [ 11 ] Fighting is very costly in regards to the amount of energy used to guard their mate. These bouts can happen more than once which takes a toll on the physical well-being of the male. Another cost of mate-guarding in this type of mating system is the potential increase of the spread of disease. [ 14 ] If one male has an STD, he can pass that on to the females that he's copulating with, potentially resulting in a depletion of the harem. This would be an energetic cost towards both sexes for the reason that instead of using the energy for reproduction, they are redirecting it towards ridding themselves of this illness. Some females also benefit from polygyny because extra pair copulations in females increase the genetic diversity with the community of that species. [ 15 ] This occurs because the male is not able to watch over all of the females and some will become promiscuous.
Eventually, the male will not have proper nutrition, which makes the male unable to produce sperm. [ 16 ] For instance, male amphipods will deplete their reserves of glycogen and triglycerides only to have it replenished after the male is done guarding that mate. [ 17 ] Also, if the amount of energy intake does not equal the energy expended, then this could be potentially fatal to the male. Males may even have to travel long distances during the breeding season in order to find a female, which significantly drains their energy supply. Studies were conducted to compare the cost of foraging of fish that migrate and animals that are residential. The studies concluded that fish that were residential had fuller stomachs containing higher quality of prey compared to their migrant counterparts. [ 18 ] With all of these energy costs that go along with guarding a mate, timing is crucial so that the male can use the minimal amount of energy. This is why it is more efficient for males to choose a mate during their fertile periods. [ 13 ] Also, males will be more likely to guard their mate when there is a high density of males in the proximity. [ 2 ] Sometimes, organisms put in all this time and planning into courting a mate in order to copulate and she may not even be interested. There is a risk of cuckoldry of some sort, since a rival male can successfully court the female that the male originally courting her could not do. [ 19 ]
However, there are benefits that are associated with mate-guarding. In a mate-guarding system, both parties, male and female, are able to directly and indirectly benefit from this. For instance, females can indirectly benefit from being protected by a mate. [ 20 ] The females can appreciate a decrease in predation and harassment from other males while being able to observe her male counterpart. [ 20 ] This will allow her to recognize particular traits that she finds ideal so that she'll be able to find another male that emulates those qualities. In polygynous relationships, the dominant male of the community benefits because he has the best fertilization success. [ 12 ] Communities can include 30 up to 100 females and, compared to the other males, will greatly increase his chances of mating success. [ 11 ]
Males who have successfully courted a potential mate will attempt to keep them out of sight of other males before copulation. One way organisms accomplish this is to move the female to a new location. Certain butterflies, after enticing the female, will pick her up and fly her away from the vicinity of potential males. [ 21 ] In other insects, the males will release a pheromone in order to make their mate unattractive to other males, or to mask her scent completely. [ 21 ] The male of certain cricket species will court a female loudly, until she accepts his gesture, when he suddenly becomes silent. [ 22 ] Some insects, prior to mating, will assume tandem positions to their mate or position themselves in a way to prevent other males from attempting to mate with that female. [ 21 ] The male checkerspot butterfly has developed a clever method in order to attract and guard a mate. He will situate himself near an area that possesses valuable resources that the female needs. He will then drive away any males that come near and this will greatly increase his chances of copulation with any female that comes to that area. [ 23 ]
In post-copulatory mate-guarding males are trying to prevent other males from mating with the female that they have mated with already. For example, male millipedes in Costa Rica will ride on the back of their mate letting the other males know that she's taken. [ 24 ] Japanese beetles will assume a tandem position to the female after copulation. [ 25 ] This can last up to several hours allowing him to ward off any rival males giving his sperm a high chance to fertilize that female's egg. These, and other, types of methods have the male playing defense by protecting his mate. Elephant seals are known to engage in bloody battles in order to retain their title as dominant male so that they are able to mate with all the females in their community. [ 11 ]
Copulatory plugs are frequently observed in insects, reptiles, some mammals, and spiders. [ 2 ] Copulatory plugs are inserted immediately after a male copulates with a female, which reduce the possibility of fertilization by subsequent copulations from another male, by physically blocking the transfer of sperm. [ 2 ] Bumblebee mating plugs, in addition to providing a physical barrier to further copulations, contain linoleic acid , which reduces re-mating tendencies of females. [ 26 ] A species of Sonoran desert Drosophila , Drosophila mettleri , uses copulatory plugs to enable males to control the sperm reserve space females have available. This behavior ensures males with higher mating success at the expense of female control of sperm (sperm selection).
Similarly, Drosophila melanogaster males release toxic seminal fluids, known as ACPs (accessory gland proteins), from their accessory glands to impede the female from participating in future copulations. [ 27 ] These substances act as an anti-aphrodisiac causing a dejection of subsequent copulations, and also stimulate ovulation and oogenesis. [ 5 ] Seminal proteins can have a strong influence on reproduction, sufficient to manipulate female behavior and physiology. [ 28 ]
Another strategy, known as sperm partitioning, occurs when males conserve their limited supply of sperm by reducing the quantity of sperm ejected. [ 2 ] In Drosophila , ejaculation amount during sequential copulations is reduced; this results in half filled female sperm reserves following a single copulatory event, but allows the male to mate with a larger number of females without exhausting his supply of sperm. [ 2 ] To facilitate sperm partitioning, some males have developed complex ways to store and deliver their sperm. [ 29 ] In the blue headed wrasse, Thalassoma bifasciatum , the sperm duct is sectioned into several small chambers that are surrounded by a muscle that allows the male to regulate how much sperm is released in one copulatory event. [ 30 ]
A strategy common among insects is for males to participate in prolonged copulations. By engaging in prolonged copulations, a male has an increased opportunity to place more sperm within the female's reproductive tract and prevent the female from copulating with other males. [ 31 ]
It has been found that some male mollies ( Poecilia ) have developed deceptive social cues to combat sperm competition. Focal males will direct sexual attention toward typically non-preferred females when an audience of other males is present. This encourages the males that are watching to attempt to mate with the non-preferred female. This is done in an attempt to decrease mating attempts with the female that the focal male prefers, hence decreasing sperm competition. [ 32 ]
Offensive adaptation behavior differs from defensive behavior because it involves an attempt to ruin the chances of another male's opportunity in succeeding in copulation by engaging in an act that tries to terminate the fertilization success of the previous male. [ 5 ] This offensive behavior is facilitated by the presence of certain traits, which are called armaments. [ 33 ] An example of an armament are antlers . Further, the presence of an offensive trait sometimes serves as a status signal . The mere display of an armament can suffice to drive away the competition without engaging in a fight, hence saving energy. [ 33 ] A male on the offensive side of mate-guarding may terminate the guarding male's chances at a successful insemination by brawling with the guarding male to gain access to the female. [ 2 ] In Drosophila , males release seminal fluids that contain additional toxins like pheromones and modified enzymes that are secreted by their accessory glands intended to destroy the sperm that have already made their way into the female's reproductive tract from a recent copulation. [ 5 ] However, this proved to be wrong because Drosophila melanogaster seminal fluid can actually protect the sperm of other males. [ 34 ] Based on the " last male precedence " idea, some males can remove sperm from previous males by ejaculating new sperm into the female; hindering successful insemination opportunities of the previous male. [ 35 ] An example of this behavior is seen in the beetle species Carabus insulicola and Onymacris unguicularis . In the former, the second male to mate with a female is able to use his hook-like genitalia to dislodge the spermatophore deposited by the previous male to increase his chances of fertilizing the most eggs. [ 36 ] In the latter species, the second male's spermatophore works to essentially push out the last male's spermatophore out of the female's body, although it has been shown that not all previous sperm are completely removed. [ 37 ]
The "good sperm hypothesis" is very common in polyandrous mating systems. [ 38 ] The "good sperm hypothesis" suggests that a male's genetic makeup will determine the level of his competitiveness in sperm competition. [ 38 ] When a male has "good sperm" he is able to father more viable offspring than males that do not have the "good sperm" genes. [ 38 ] Females may select males that have these superior "good sperm" genes because it means that their offspring will be more viable and will inherit the "good sperm" genes which will increase their fitness levels when their sperm competes. [ 39 ]
Studies show that there is more to determining the competitiveness of the sperm in sperm competition in addition to a male's genetic makeup. A male's dietary intake will also affect sperm competition. An adequate diet consisting of increased amounts of diet and sometimes more specific ratio in certain species will optimize sperm number and fertility. Amounts of protein and carbohydrate intake were tested for its effects on sperm production and quality in adult fruit flies (Diptera: Tephritidae). Studies showed these flies need to constantly ingest carbohydrates and water to survive, but protein is also required to attain sexual maturity. [ 40 ] In addition, The Mediterranean fruit fly, male diet has been shown to affect male mating success, copula duration, sperm transfer, and male participation in leks. [ 41 ] These all require a good diet with nutrients for proper gamete production as well as energy for activities, which includes participation in leks.
In addition, protein and carbohydrate amounts were shown to have an effect on sperm production and fertility in the speckled cockroach. Holidic diets were used which allowed for specific protein and carbohydrate measurements to be taken, giving it credibility. A direct correlation was seen in sperm number and overall of food intake. More specifically, optimal sperm production was measured at a 1:2 protein to carbohydrate ratio. Sperm fertility was best at a similar protein to carbohydrate ratio of 1:2. This close alignment largely factors in determining male fertility in Nauphoeta cinerea . [ 42 ] Surprisingly, sperm viability was not affected by any change in diet or diet ratios. It is hypothesized that sperm viability is more affected by the genetic makeup, like in the "good sperm hypothesis". These ratios and results are not consistent with many other species and even conflict with some. It seems there cannot be any conclusions on what type of diet is needed to positively influence sperm competition but rather understand that different diets do play a role in determining sperm competition in mate choice .
One evolutionary response to sperm competition is the variety in penis morphology of many species. [ 43 ] [ 44 ] For example, the shape of the human penis may have been selectively shaped by sperm competition. [ 45 ] The human penis may have been selected to displace seminal fluids implanted in the female reproductive tract by a rival male. [ 45 ] Specifically, the shape of the coronal ridge may promote displacement of seminal fluid from a previous mating [ 46 ] via thrusting action during sexual intercourse . [ 45 ] A 2003 study by Gordon G. Gallup and colleagues concluded that one evolutionary purpose of the thrusting motion characteristic of intense intercourse is for the penis to “upsuck” another man's semen before depositing its own. [ 47 ]
Evolution to increase ejaculate volume in the presence of sperm competition has a consequence on testis size. Large testes can produce more sperm required for larger ejaculates, and can be found across the animal kingdom when sperm competition occurs. [ 48 ] Males with larger testes have been documented to achieve higher reproductive success rates than males with smaller testes in male yellow pine chipmunks . [ 48 ] In cichlid fish, it has been found that increased sperm competition can lead to evolved larger sperm numbers, sperm cell sizes, and sperm swimming speeds. [ 49 ]
In some insects and spiders, for instance Nephila fenestrate , the male copulatory organ breaks off or tears off at the end of copulation and remains within the female to serve as a copulatory plug. [ 50 ] This broken genitalia is believed to be an evolutionary response to sperm competition. [ 50 ] This damage to the male genitalia means that these males can only mate once. [ 51 ]
Female factors can influence the result of sperm competition through a process known as "sperm choice". [ 52 ] Proteins present in the female reproductive tract or on the surface of the ovum may influence which sperm succeeds in fertilizing the egg. [ 52 ] During sperm choice, females are able to discriminate and differentially use the sperm from different males. One instance where this is known to occur is inbreeding; females will preferentially use the sperm from a more distantly related male than a close relative. [ 52 ]
Inbreeding ordinarily has negative fitness consequences ( inbreeding depression ), and as a result species have evolved mechanisms to avoid inbreeding. Inbreeding depression is considered to be due largely to the expression of homozygous deleterious recessive mutations. [ 53 ] Outcrossing between unrelated individuals ordinarily leads to the masking of deleterious recessive mutations in progeny. [ 54 ]
Numerous inbreeding avoidance mechanisms operating prior to mating have been described. However, inbreeding avoidance mechanisms that operate subsequent to copulation are less well known. In guppies , a post-copulatory mechanism of inbreeding avoidance occurs based on competition between sperm of rival males for achieving fertilization. [ 55 ] In competitions between sperm from an unrelated male and from a full sibling male, a significant bias in paternity towards the unrelated male was observed. [ 55 ]
In vitro fertilization experiments in the mouse, provided evidence of sperm selection at the gametic level. [ 56 ] When sperm of sibling and non-sibling males were mixed, a fertilization bias towards the sperm of the non-sibling males was observed. The results were interpreted as egg-driven sperm selection against related sperm.
Female fruit flies ( Drosophila melanogaster ) were mated with males of four different degrees of genetic relatedness in competition experiments. [ 57 ] Sperm competitive ability was negatively correlated with relatedness.
Female crickets ( Teleogryllus oceanicus ) appear to use post-copulatory mechanisms to avoid producing inbred offspring. When mated to both a sibling and an unrelated male, females bias paternity towards the unrelated male. [ 58 ]
It has been found that because of female choice (see sexual selection ), morphology of sperm in many species occurs in many variations to accommodate or combat (see sexual conflict ) the morphology and physiology of the female reproductive tract . [ 59 ] [ 60 ] [ 61 ] However, it is difficult to understand the interplay between female and male reproductive shape and structure that occurs within the female reproductive tract after mating that allows for the competition of sperm. Polyandrous females mate with many male partners. [ 62 ] Females of many species of arthropod , mollusk and other phyla have a specialized sperm-storage organ called the spermatheca in which the sperm of different males sometimes compete for increased reproductive success. [ 60 ] Species of crickets, specifically Gryllus bimaculatus , are known to exhibit polyandrous sexual selection. Males will invest more in ejaculation when competitors are in the immediate environment of the female.
Evidence exists that illustrates the ability of genetically similar spermatozoa to cooperate so as to ensure the survival of their counterparts thereby ensuring the implementation of their genotypes towards fertilization. Cooperation confers a competitive advantage by several means, some of these include incapacitation of other competing sperm and aggregation of genetically similar spermatozoa into structures that promote effective navigation of the female reproductive tract and hence improve fertilization ability. Such characteristics lead to morphological adaptations that suit the purposes of cooperative methods during competition. For example, spermatozoa possessed by the wood mouse ( Apodemus sylvaticus ) possess an apical hook which is used to attach to other spermatozoa to form mobile trains that enhance motility through the female reproductive tract. [ 63 ] Spermatozoa that fail to incorporate themselves into mobile trains are less likely to engage in fertilization. Other evidence suggests no link between sperm competition and sperm hook morphology. [ 64 ]
Selection to produce more sperm can also select for the evolution of larger testes . Relationships across species between the frequency of multiple mating by females and male testis size are well documented across many groups of animals. For example, among primates, female gorillas are relatively monogamous, so gorillas have smaller testes than humans , which in turn have smaller testes than the highly promiscuous bonobos . [ 65 ] Male chimpanzees that live in a structured multi-male, multi-female community, have large testicles to produce more sperm, therefore giving them better odds to fertilize the female. Whereas a community of gorillas consists of one alpha male and two or three females; when the female gorillas are ready to mate, normally only the alpha male is their partner.
Regarding sexual dimorphism among primates, humans fall into an intermediate group with moderate sex differences in body size but relatively large testes. This is a typical pattern of primates where several males and females live together in a group and the male faces an intermediate number of challenges from other males compared to exclusive polygyny and monogamy but frequent sperm competition. [ 66 ]
Other means of sperm competition could include improving the sperm itself or its packaging materials ( spermatophore ). [ 67 ]
The male black-winged damselfly provides a striking example of an adaptation to sperm competition. Female black-winged damselflies are known to mate with several males over the span of only a few hours and therefore possess a receptacle known as a spermatheca which stores the sperm. During the process of mating the male damselfly will pump his abdomen up and down using his specially adapted penis which acts as a scrub brush to remove the sperm of another male. This method proves quite successful and the male damselfly has been known to remove 90-100 percent of the competing sperm. [ 68 ]
A similar strategy has been observed in the dunnock , a small bird. Before mating with the polyandrous female, the male dunnock pecks at the female's cloaca in order to peck out the sperm of the previous male suitor. [ 69 ]
In the fly Dryomyza anilis , females mate with multiple males. It benefits the male to attempt to be the last one to mate with a given female. [ 70 ] This is because there seems to be a cumulative percentage increase in fertilization for the final male, such that the eggs laid in the last oviposition bout are the most successful.
A notion emerged in 1996 that in some species, including humans, a significant fraction of sperm cannot fertilize the egg; rather these sperm were theorized to stop the sperm from other males from reaching the egg, e.g. by killing them with enzymes or by blocking their access. This type of sperm specialization became known popularly as "kamikaze sperm" or "killer sperm", but most follow-up studies to this popularized notion have failed to confirm the initial papers on the matter. [ 71 ] [ 6 ] While there is also currently little evidence of killer sperm in any non-human animals [ 72 ] certain snails have an infertile sperm morph ("parasperm") that contains lysozymes , leading to speculation that they might be able to degrade a rival's sperm. [ 73 ]
In the parasitoid wasp Nasonia vitripennis , mated females can choose whether or not to lay a fertilized egg (which develops into a daughter) or an unfertilized egg (which develops into a son), therefore females suffer a cost from mating, as repeated matings constrain their ability to allocate sex in their offspring. The behaviour of these kamikaze-sperm is referred to in academic literature as "sperm-blocking", using basketball as a metaphor. [ 74 ]
Sperm competition has led to other adaptations such as larger ejaculates , prolonged copulation , deposition of a copulatory plug to prevent the female re-mating, or the application of pheromones that reduce the female's attractiveness.
The adaptation of sperm traits, such as length, viability and velocity might be constrained by the influence of cytoplasmic DNA (e.g. mitochondrial DNA ); [ 75 ] mitochondrial DNA is inherited from the mother only and it is thought that this could represent a constraint in the evolution of sperm. | https://en.wikipedia.org/wiki/Sperm_competition |
Sperm guidance is the process by which sperm cells ( spermatozoa ) are directed to the oocyte (egg) for the aim of fertilization . In the case of marine invertebrates the guidance is done by chemotaxis . In the case of mammals , it appears to be done by chemotaxis , thermotaxis and rheotaxis .
Since the discovery of sperm attraction to the female gametes in ferns over a century ago, [ 1 ] sperm guidance in the form of sperm chemotaxis has been established in a large variety of species [ 2 ] Although sperm chemotaxis is prevalent throughout the Metazoa kingdom, from marine species with external fertilization such as sea urchins and corals , to humans, [ 2 ] [ 3 ] [ 4 ] much of the current information on sperm chemotaxis is derived from studies of marine invertebrates, primarily sea urchin and starfish . [ 5 ] As a matter of fact, until not too long ago, the dogma was that, in mammals, guidance of spermatozoa to the oocyte was unnecessary. This was due to the common belief that, following ejaculation into the female genital tract, large numbers of spermatozoa 'race' towards the oocyte and compete to fertilize it. This belief was taken apart when it became clear that only few of the ejaculated spermatozoa — in humans, only ~1 of every million spermatozoa — succeed in entering the oviducts ( fallopian tubes ) [ 4 ] [ 6 ] and when more recent studies showed that mammalian spermatozoa employ at least three different mechanisms, each of which can potentially serve as a guidance mechanism: [ 7 ] chemotaxis, [ 8 ] thermotaxis [ 9 ] and rheotaxis. [ 10 ]
Sperm guidance in non-mammalian species is performed by chemotaxis. The oocyte secretes a chemoattractant , which, as it diffuses away, forms a concentration gradient : a high concentration close to the egg, and a gradually lower concentration as the distance from the oocyte increases. Spermatozoa can sense this chemoattractant and orient their swimming direction up the concentration gradient towards the oocyte. Sperm chemotaxis was demonstrated in a large number of non-mammalian species, from marine invertebrates [ 2 ] [ 3 ] to frogs. [ 11 ]
The sperm chemoattractants in non-mammalian species vary to a large extent. Some examples are shown in Table 1. So far, most sperm chemoattractants that have been identified in non-mammalian species are peptides or low-molecular-weight proteins (1–20 kDa ), which are heat stable and sensitive to proteases . [ 2 ] [ 3 ] Exceptions to this rule are the sperm chemoattractants of corals, ascidians , plants such as ferns, and algae (Table 1).
The variety of chemoattractants raises the question of species specificity with respect to the chemoattractant identity. There is no single rule for chemoattractant-related specificity. Thus, in some groups of marine invertebrates (e.g., hydromedusae and certain ophiuroids ), the specificity is very high; in others (e.g., starfish), the specificity is at the family level and, within the family, there is no specificity. [ 2 ] [ 3 ] [ 24 ] In mollusks , there appears to be no specificity at all. Likewise, in plants, a unique simple compound [e.g., fucoserratene — a linear, unsaturated alkene (1,3-trans 5-cis-octatriene)] might be a chemoattractant for various species. [ 12 ]
Here, too, there is no single rule. In some species (for example, in hydroids like Campanularia or tunicate like Ciona ), the swimming direction of the spermatozoa changes abruptly towards the chemoattractant source. In others (for example, in sea urchin, hydromedusa, fern, or fish such as Japanese bitterlings), the approach to the chemoattractant source is indirect and the movement is by repetitive loops of small radii. In some species (for example, herring or the ascidian Ciona) activation of motility precedes chemotaxis. [ 2 ] [ 3 ] [ 25 ] [ 26 ] In chemotaxis, cells may either sense a temporal gradient of the chemoattractant, comparing the occupancy of its receptors at different time points (as do bacteria [ 27 ] ), or they may detect a spatial gradient, comparing the occupancy of receptors at different locations along the cell (as do leukocytes [ 28 ] ). In the best-studied species, sea urchin, the spermatozoa sense a temporal gradient and respond to it with a transient increase in flagellar asymmetry. The outcome is a turn in the swimming path, followed by a period of straight swimming, [ 29 ] leading to the observed epicycloid-like movements directed towards the chemoattractant source. [ 30 ]
The molecular mechanism of sperm chemotaxis is still not fully known. The current knowledge is mainly based on studies in the sea urchin Arbacia punctulata , where binding of the chemoattractant resact (Table 1) to its receptor, a guanylyl cyclase , activates cGMP synthesis (Figure 1). The resulting rise of cGMP possibly activates K + -selective ion channels . The consequential hyperpolarization activates hyperpolarization-activated and cyclic nucleotide-gated (HCN) channels. The depolarizing inward current through HCN channels possibly activates voltage-activated Ca 2+ channels, resulting in elevation of intracellular Ca 2+ . This rise leads to flagellar asymmetry and, consequently, a turn of the sperm cell. [ 25 ]
Figure 1. A model of the signal-transduction pathway during sperm chemotaxis of the sea urchin Arbacia punctulata . Binding of a chemoattractant (ligand) to the receptor — a membrane-bound guanylyl cyclase (GC) — activates the synthesis of cGMP from GTP. Cyclic GMP possibly opens cyclic nucleotide-gated (CNG) K + -selective channels, thereby causing hyperpolarization of the membrane. The cGMP signal is terminated by the hydrolysis of cGMP through phosphodiesterase (PDE) activity and inactivation of GC. On hyperpolarization, hyperpolarization-activated and cyclic nucleotide-gated (HCN) channels allow the influx of Na + that leads to depolarization and thereby causes a rapid Ca 2+ entry through voltage-activated Ca 2+ channels (Ca v ), Ca 2+ ions interact by unknown mechanisms with the axoneme of the flagellum and cause an increase of the asymmetry of flagellar beat and eventually a turn or bend in the swimming trajectory. Ca 2+ is removed from the flagellum by a Na + /Ca 2+ exchange mechanism. [Taken from ref. [ 25 ] ]
Three different guidance mechanisms have been proposed to occur in the mammalian oviduct: thermotaxis, [ 9 ] rheotaxis, [ 10 ] and chemotaxis. [ 8 ] [ 31 ] [ 32 ] Indeed, due to obvious restrictions, all these mechanisms were demonstrated in vitro only. However, the discoveries of proper stimuli in the female – an ovulation-dependent temperature gradient in the oviduct, [ 33 ] [ 34 ] [ 35 ] post-coitus oviductal fluid flow in female mice, [ 10 ] and sperm chemoattractants secreted from the oocyte and its surrounding cumulus cells, [ 36 ] respectively – strongly suggest the mutual occurrence of these mechanisms in vivo .
Following the findings that human spermatozoa accumulate in follicular fluid [ 37 ] [ 38 ] and that there is a remarkable correlation between this in vitro accumulation and oocyte fertilization, [ 37 ] chemotaxis was substantiated as the cause of this accumulation. [ 8 ] Sperm chemotaxis was later also demonstrated in mice [ 31 ] and rabbits. [ 32 ] In addition, sperm accumulation in follicular fluid (but without substantiating that it truly reflects chemotaxis) was demonstrated in horses [ 39 ] and pigs. [ 40 ] A key feature of sperm chemotaxis in humans is that this process is restricted to capacitated cells [ 41 ] [ 42 ] — the only cells that possess the ability to penetrate the oocyte and fertilize it. [ 43 ] This raised the possibility that, in mammals, chemotaxis is not solely a guidance mechanism but it is also a mechanism of sperm selection. [ 41 ] [ 42 ] Importantly, the fraction of capacitated (and, hence, chemotactically responsive) spermatozoa is low (~10% in humans), the life span of the capacitated/chemotactic state is short (1–4 hours in humans), a spermatozoon can be at this state only once in its lifetime, and sperm individuals become capacitated/chemotactic at different time points, resulting in continuous replacement of capacitated/chemotactic cells within the sperm population, i.e., prolonged availability of capacitated cells. [ 41 ] [ 44 ] These sperm features raised the possibility that prolonging the time period, during which capacitated spermatozoa can be found in the female genital tract, is a mechanism, evolved in humans, to compensate for the lack of coordination between insemination and ovulation. [ 6 ] [ 41 ] [ 42 ] [ 45 ]
Chemotaxis is a short-range guidance mechanism. As such, it can guide spermatozoa for short distances only, estimated at the order of millimeters. [ 7 ]
In humans, there are at least two different origins of sperm chemoattractants. One is the cumulus cells that surround the oocyte, and the other is the mature oocyte itself. [ 36 ] The chemoattractant secreted from the cumulus cells is the steroid progesterone , shown to be effective at the picomolar range. [ 46 ] [ 47 ] [ 48 ] The chemoattractant secreted from the oocyte is even more potent. [ 36 ] It is a hydrophobic non-peptide molecule which, when secreted from the oocyte, is in complex with a carrier protein [ 49 ] Additional compounds have been shown to act as chemoattractants for mammalian spermatozoa. They include the chemokine CCL20 , [ 50 ] atrial natriuretic peptide (ANP), [ 51 ] specific odorants , [ 52 ] natriuretic peptide type C (NPPC), [ 53 ] and allurin, [ 54 ] to mention a few. It is reasonable to assume that not all of them are physiologically relevant.
Species specificity was not detected in experiments that compared the chemotactic responsiveness of human and rabbit spermatozoa to follicular fluids or egg-conditioned media obtained from human, bovine, and rabbit. [ 55 ] The subsequent findings that cumulus cells of both human and rabbit (and, probably, of other mammals as well) secrete the chemoattractant progesterone [ 46 ] [ 47 ] [ 48 ] is sufficient to account for the lack of specificity in the chemotactic response of mammalian spermatozoa.
Mammalian spermatozoa, like sea-urchin spermatozoa, appear to sense the chemoattractant gradient temporally (comparing receptor occupancy over time) rather than spatially (comparing receptor occupancy over space). This is because the establishment of a temporal gradient in the absence of spatial gradient, achieved by mixing human spermatozoa with a chemoattractant [ 56 ] or by photorelease of a chemoattractant from its caged compound, [ 57 ] results in delayed transient changes in swimming behavior that involve increased frequency of turns and hyperactivation events. On the basis of these observations and the finding that the level of hyperactivation events is reduced when chemotactically responsive spermatozoa swim in a spatial chemoattractant gradient [ 57 ] it was proposed that turns and hyperactivation events are suppressed when capacitated spermatozoa swim up a chemoattractant gradient, and vice versa when they swim down a gradient. [ 56 ] [ 57 ] In other words, human spermatozoa approach chemoattractants by modulating the frequency of turns and hyperactivation events, similarly to Escherichia coli bacteria. [ 27 ]
As in non-mammalian species, the end signal in chemotaxis for changing the direction of swimming is Ca 2+ . [ 58 ] The discovery of progesterone as a chemoattractant [ 46 ] [ 47 ] [ 48 ] led to the identification of its receptor on the sperm surface – CatSper , a Ca 2+ channel present exclusively in the tail of mammalian spermatozoa. [ 59 ] [ 60 ] (Note, though, that progesterone only stimulates human CatSper but not mouse CatSper. [ 60 ] Consistently, sperm chemotaxis to progesterone was not found in mice. [ 61 ] ) However, the molecular steps subsequent to CatSper activation by progesterone are obscure, though the involvement of trans-membrane adenylyl cyclase , cAMP and protein kinase A as well as soluble guanylyl cyclase , cGMP , inositol trisphosphate receptor and store-operated Ca 2+ channel was proposed. [ 62 ]
The realization that sperm chemotaxis can guide spermatozoa for short distances only, [ 7 ] triggered a search for potential long-range guidance mechanisms. The findings that, at least in rabbits [ 33 ] and pigs, [ 34 ] a temperature difference exists within the oviduct, and that this temperature difference is established at ovulation in rabbits due to a temperature drop in the oviduct near the junction with the uterus , creating a temperature gradient between the sperm storage site and the fertilization site in the oviduct, [ 35 ] led to a study of whether mammalian spermatozoa can respond to a temperature gradient by thermotaxis.
Mammalian sperm thermotaxis was, hitherto, demonstrated in three species: humans, rabbits, and mice. [ 9 ] [ 63 ] This was done by two methods. One involved a Zigmond chamber , modified to make the temperature in each well separately controllable and measurable. A linear temperature gradient was established between the wells and the swimming of spermatozoa in this gradient was analyzed. A small fraction of the spermatozoa (at the order of ~10%), shown to be the capacitated cells, biased their swimming direction according to the gradient, moving towards the warmer temperature. [ 9 ] The other method involved two [ 64 ] [ 65 ] - or three [ 63 ] -compartment separation tube placed within a thermoseparation device that maintains a linear temperature gradient. Sperm accumulation at the warmer end of the separation tube was much higher than the accumulation at the same temperature but in the absence of a temperature gradient. [ 63 ] This gradient-dependent sperm accumulation was observed over a wide temperature range (29-41 °C). [ 65 ] Since temperature affects almost every process, much attention has been devoted to the question of whether the measurements, mentioned just above, truly demonstrate thermotaxis or whether they reflect another temperature-dependent process. The most pronounced effect of temperature in liquid is convection , which raised the concern that the apparent thermotactic response could have been a reflection of a passive drift in the liquid current or a rheotactic response [ 10 ] to the current (rather than to the temperature gradient per se). Another concern was that the temperature could have changed the local pH of the buffer solution in which the spermatozoa are suspended. This could generate a pH gradient along the temperature gradient, and the spermatozoa might have responded to the formed pH gradient by chemotaxis. However, careful experimental examinations of all these possibilities with proper controls demonstrated that the measured responses to temperature are true thermotactic responses and that they are not a reflection of any other temperature-sensitive process, including rheotaxis and chemotaxis. [ 7 ] [ 65 ]
The behavioral mechanism of sperm thermotaxis has been so far only investigated in human spermatozoa. [ 66 ] Like the behavioral mechanisms of bacterial chemotaxis [ 27 ] and human sperm chemotaxis , [ 57 ] the behavioral mechanism of human sperm thermotaxis appears to be stochastic rather than deterministic. Capacitated human spermatozoa swim in rather straight lines interrupted by turns and brief episodes of hyperactivation. Each such episode results in swimming in a new direction. When the spermatozoa sense a decrease in temperature, the frequency of turns and hyperactivation events increases due to increased flagellar-wave amplitude that results in enhanced side-to-side head displacement. With time, this response undergoes partial adaptation. The opposite happens in response to an increase in temperature. This suggests that when capacitated spermatozoa swim up a temperature gradient, turns are repressed and the spermatozoa continue swimming in the gradient direction. When they happen to swim down the gradient, they turn again and again until their swimming direction is again up the gradient. [ 66 ]
The response of spermatozoa to temporal temperature changes even when the temperature is kept constant spatially [ 66 ] suggests that, as in the case of human sperm chemotaxis, [ 56 ] [ 57 ] sperm thermotaxis involves temporal gradient sensing. In other words, spermatozoa apparently compare the temperature (or a temperature-dependent function) between consecutive time points. This, however, does not exclude the occurrence of spatial temperature sensing in addition to temporal sensing.
Human spermatozoa can respond thermotactically within a wide temperature range (at least 29–41 °C). [ 65 ] Within this range they preferentially accumulate in warmer temperatures rather than at a single specific, preferred temperature. Amazingly, they can sense and thermotactically respond to temperature gradients as low as <0.014 °C/mm. [ 65 ] This means that when human spermatozoa swim a distance that equals their body length (~46 μm) they respond to a temperature difference of <0.0006 °C!
The molecular mechanism underlying thermotaxis, in general, and thermosensing with such extreme sensitivity, in particular, is obscure. It is known that, unlike other recognized thermosensors in mammals, the thermosensors for sperm thermotaxis do not seem to be temperature-sensitive ion channels . They are rather opsins , [ 63 ] known to be G-protein-coupled receptors that act as photosensors in vision . The opsins are present in spermatozoa at specific sites, which depend on the species and the opsin type. [ 63 ] They are involved in sperm thermotaxis via at least two signaling pathways: a phospholipase C signaling pathway and a cyclic-nucleotide pathway. The former was shown by pharmacological means in human spermatozoa to involve the enzyme phospholipase C, an inositol trisphosphate receptor located on internal calcium stores, the calcium channel TRPC3 , and intracellular calcium. [ 64 ] [ 63 ] The cyclic-nucleotide pathway was, hitherto, shown to involve phosphodiesterase . [ 63 ] Blocking both pathways fully inhibits sperm thermotaxis. [ 63 ]
When human and mouse spermatozoa are exposed to a fluid flow, roughly one half of them (i.e., both capacitated and noncapacitated spermatozoa) reorient and swim against the current. [ 10 ] The flow, which is prolactin -triggered oviductal fluid secretion, is generated in female mice within 4 h of sexual stimulation and coitus. Thus, rheotaxis orients spermatozoa towards the fertilization site. It was proposed that capacitated spermatozoa might detach from the oviductal surface faster than non-capacitated spermatozoa, enabling them to swim into the main current. [ 10 ] To understand the mechanism of sperm turning in rheotaxis, quantitative analysis of human sperm flagellar behavior during rheotaxis turning was carried out. The results revealed, both at the single cell and population levels, that there is no significant difference in flagellar beating between rheotaxis turning spermatozoa and free-swimming spermatozoa. [ 67 ] This finding taken together with the constant internal Ca 2+ signal, measured during rheotaxis turning, demonstrated that, in contrast to the active process of chemotaxis and thermotaxis, human sperm rheotaxis is a passive process and no flow sensing is involved. [ 67 ]
Like in any other highly essential system in biology, mammalian sperm guidance is expected to involve redundancy. Indeed, at least three guidance mechanisms are likely to act in the female genital tract, two active mechanisms — chemotaxis and thermotaxis, and a passive mechanism — rheotaxis. When one of these mechanisms is not functional for any reason, guidance is not expected to be lost and the cells should still be able to navigate to the oocyte. This resembles guidance of migrating birds, where the birds' navigation is unaffected when one of the guidance mechanisms is not functional. [ 68 ] It has been suggested that capacitated spermatozoa, released from the sperm storage site at the isthmus , [ 69 ] may be first actively guided by thermotaxis from the cooler sperm storage site towards the warmer fertilization site [ 9 ] (Figure 2). Two passive processes, rheotaxis [ 10 ] and contractions of the oviduct [ 70 ] may assist the spermatozoa to reach there. At this location, the spermatozoa may be chemotactically guided to the oocyte-cumulus complex by the gradient of progesterone, secreted from the cumulus cells. [ 46 ] [ 47 ] [ 48 ] In addition, progesterone may inwardly guide spermatozoa, already present within the periphery of the cumulus oophorus. [ 46 ] Spermatozoa that are already deep within the cumulus oophorus may sense the more potent chemoattractant that is secreted from the oocyte [ 36 ] [ 49 ] and chemotactically guide themselves to the oocyte according to the gradient of this chemoattractant. It should be borne in mind, however, that this is only a model.
Figure 2. A simplified scheme describing the suggested sequence of active sperm guidance mechanisms in mammals. In addition, two passive processes, sperm rheotaxis and contractions of the oviduct, may assist sperm movement towards the fertilization site.
A number of observations point to the possibility that chemotaxis and thermotaxis also occur at lower parts of the female genital tract. For example, small, gradual estrus cycle-correlated temperature increase was measured in cows from the vagina towards the uterine horns, [ 71 ] and a gradient of natriuretic peptide precursor A, shown to be a chemoattractant for mouse spermatozoa, was found, in decreasing concentration order, in the ampulla, isthmus, and uterotubal junction. [ 72 ] The physiological functions, if any, of these chemical and temperature gradients are yet to be resolved.
Sperm guidance by either chemotaxis or thermotaxis can potentially be used to obtain sperm populations that are enriched with capacitated spermatozoa for in vitro fertilization procedures. Indeed, sperm populations selected by thermotaxis were recently shown to have much higher DNA integrity and lower chromatin compaction than unselected spermatozoa and, in mice, to give rise to more and better embryos through intracytoplasmic sperm injection (ICSI), doubling the number of successful pregnancies. [ 73 ] Chemotaxis and thermotaxis can also be exploited possibly as a diagnostic tool to assess sperm quality. In addition, these processes can potentially be used, in the long run, as a means of contraception by interfering with the normal process of fertilization. [ 6 ] [ 74 ] | https://en.wikipedia.org/wiki/Sperm_guidance |
Sperm sorting is a means of choosing what type of sperm cell is to fertilize an egg cell using several conventional techniques of centrifugation or swim-up . Newly applied methods such as flow cytometry expand the possibilities of sperm sorting and new techniques of sperm sorting are being developed.
It can be used to sort out sperm that are most healthy, as well as for determination of more specific traits, such as sex selection in which spermatozoa are separated into X- (female) and Y- (male) chromosome bearing populations based on their difference in DNA content. The resultant 'sex-sorted' spermatozoa are then able to be used in conjunction with other assisted reproductive technologies such as artificial insemination or in-vitro fertilization (IVF) to produce offspring of the desired sex - in farming animals but also in human medical practice.
Several methods have been used to sort sperm before the advent of flow cytometry. Density gradient centrifugation (in a continuous or discontinuous gradient) can concentrate semen samples with low concentration of sperm, using the density of sperm as a measure of their quality. [ 1 ] [ 2 ] Similarly, so-called swim-up techniques apply a centrifugation step and then sperm is allowed to swim up into a medium, thus enriching a subpopulation of motile sperm. However, use of sperm centrifugation is detrimental to the sperm viability and elicits production of reactive oxygen species. [ 1 ] Conventional techniques are routinely used in assisted reproductive technology. [ 3 ]
Flow cytometry is another method used to sort sperm and adaptations of this technique opens new opportunities in sperm sorting. However, because flow cytometry-based sperm sorting often uses fluorescent dyes that often stain DNA, the safety of this technique in human reproductive medicine is a matter of scientific discussion. [ 4 ] [ 5 ]
However, flow cytometry is the only currently used technique able to determine the sex of future progeny by measuring DNA content of individual sperm cells. It evaluates if they contain the larger X chromosome (giving rise to a female offspring) or smaller Y chromosome (leading to male progeny). It then allows separation of X and Y sperm. [ 6 ] The so-called Beltsfield Sperm Sexing Technology was developed by USDA in conjunction with Lawrence Livermore National Laboratories, relying on the DNA difference between the X- and Y- chromosomes. [ 7 ] Prior to flow cytometric sorting, semen is labeled with a fluorescent dye called Hoechst 33342 which binds to the DNA of each spermatozoon. As the X chromosome is larger (i.e. has more DNA) than the Y chromosome, the "female" (X-chromosome bearing) spermatozoa will absorb a greater amount of dye than its male (Y-chromosome bearing) counterpart. As a consequence, when exposed to UV light during flow cytometry, X spermatozoa fluoresce brighter than Y- spermatozoa. As the spermatozoa pass through the flow cytometer in single file, each spermatozoon is encased by a single droplet of fluid and assigned an electric charge corresponding to its chromosome status (e.g. X-positive charge, Y-negative charge). The stream of X- and Y- droplets is then separated by means of electrostatic deflection and collected into separate collection tubes for subsequent processing. [ 8 ]
Another cytometric technique used in sperm sorting is magnetic-activated cell sorting (MACS) which is routinely applied in assisted reproduction hospitals to sort out sperm with fragmented DNA. This is achieved using antibodies to surface markers of programmed cell death ( apoptosis ) such as annexin V , coupled with magnetic beads. Following the binding of these antibodies, spermatozoa which undergo apoptosis are sorted by applying magnetic field to the sperm suspension. [ 9 ] MACS obviates the need for fluorescent DNA binding molecules.
DNA damage in sperm cells may be detected by using Raman spectroscopy . [ 10 ] It is not specific enough to detect individual traits, however. [ 10 ] The sperm cells having least DNA damage may subsequently be injected into the egg cell by intracytoplasmic sperm injection (ICSI). [ 10 ] Many other methods for sperm sorting have been proposed or are currently tested. [ 1 ] [ 3 ]
To select spermatozoa with low DNA damage index the population of sperm could be enriched with spermatozoa with non-fragmented DNA, with techniques like electrophoresis , [ 11 ] Z method [ 12 ] and MACS (Magnetic Activating Cell Sorting), which in combination with density gradient centrifugation in single sperm preparation protocols results in spermatozoa with superior quality. [ 13 ]
Hyaluronic acid (HA) binding sites on the sperm plasma membrane are an indicator of sperm maturity (Huszar et al., 2003, Yudin et al.,1999). There are two methods based on this fact: physiological intracytoplasmic sperm injection (PICSI), and a sperm slow procedure; both methods require sperm preparation via sperm washing or centrifugation.
Sperm undergoes a process of natural selection when millions of sperm enter the vagina but only few reach the egg cell and then only one is usually allowed to fertilize it. The sperm is selected not only by its highest motility but also by other factors such as DNA integrity , production of reactive oxygen species and viability . This selection is largely circumvented in case of in-vitro fertilization which leads to higher incidence of birth defects associated with assisted reproductive techniques . Egg cells are often fertilized by sperm which would have low chance of fertilizing it in natural conditions. [ 1 ] Sperm sorting could thus be used to decrease risks associated with assisted reproduction. Additionally, there is ongoing debate about using sperm sorting for choosing the child's sex.
Conventional methods of sperm sorting have been widely used to assess quality of sperm before subsequent artificial insemination or in-vitro fertilization . It has been verified that sperm sorted using these techniques is of superior quality than unsorted. [ 14 ] [ 15 ] However, important characteristics of sperm such as DNA integrity remain untested by these conventional methods. New flow-cytometry based techniques such as YO-PRO staining can discriminate apoptotic and dead spermatozoa from the viable ones. [ 2 ] For example, annexin V staining followed by MACS can significantly improve pregnancy rates in couples with previous assisted reproduction failure. [ 9 ]
Sperm sorting by flow cytometry is an established technique in veterinary practice, and in the dairy industry most female cows are artificially inseminated with sorted semen to increase the number of female calves (using sperm sorting is less common in other species of farm animals, however artificial insemination is common). [ 16 ] Artificial insemination of farm animals with sorted sperm is recognized by the Food and Agriculture Organization (FAO) as a promising way of increasing efficiency of agriculture needed to produce enough food for the growing human population. Utilizing artificial insemination with sorted sperm is seen as a way to create an optimal ratio of male and female calves to increase dairy milk production . [ 16 ]
Choosing the sex of children might help prevent sex-associated heritable diseases such as Duchene muscular dystrophy or haemophilia in families with a history of these diseases. On the other hand, sperm sorting in humans raises the ethical concerns implicit to the idea of sex selection . If applied large-scale, it has a potential to elicit a sex-ratio imbalance . It could also have implications on gender equality if parents consistently choose to have a boy as their first-born (first-borns were shown to be more likely to succeed in life). [ 17 ]
There is no country in the world which explicitly permits sex selection for non-medical purposes. There were 31 countries in 2009 which allowed sex selection in case of sex-linked disease risk or other medical purpose. [ 18 ] In the US, for humans, the application of sperm sorting in sex selection is tightly regulated by the FDA . After the establishment of the MicroSort technique, it was offered to parents as a part of a clinical trial . The procedure was made available to a limited number of participants each month, in addition to fulfilling certain criteria, such as having a disease with sex linkage or having at least one child (for family balancing). [ 19 ] There are currently MicroSort laboratories and collaborating physicians in several countries (some for general purposes, some only offering service in case of genetic disease risks associated with one sex). [ 20 ]
While highly accurate, sperm sorting by flow cytometry will not produce two completely separate populations. That is to say, there will always be some "male" sperm among the "female" sperm and vice versa. The exact percentage purity of each population is dependent on the species being sorted and the 'gates' which the operator places around the total population visible to the machine. In general, the larger the DNA difference between the X and Y chromosome of a species, the easier it is to produce a highly pure population. In sheep and cattle, purities for each sex will usually remain above 90% depending on 'gating', while for humans these may be reduced to 90% for "female" spermatozoa and 70% for "male" spermatozoa. [ 19 ] | https://en.wikipedia.org/wiki/Sperm_sorting |
Sperm thermotaxis is a form of sperm guidance, in which sperm cells (spermatozoa) actively change their swimming direction according to a temperature gradient, swimming up the gradient. Thus far this process has been discovered in mammals only.
The discovery of mammalian sperm chemotaxis and the realization that it can guide spermatozoa for short distances only, estimated at the order of millimeters, [ 1 ] triggered a search for potential long-range guidance mechanisms. The findings that, at least in rabbits [ 2 ] and pigs, [ 3 ] a temperature difference exists within the oviduct , and that this temperature difference is established at ovulation in rabbits due to a temperature drop in the oviduct near the junction with the uterus , creating a temperature gradient between the sperm storage site and the fertilization site in the oviduct [ 4 ] (Figure 1), led to investigation whether mammalian spermatozoa can respond to a temperature gradient by thermotaxis. [ 5 ]
Mammalian sperm thermotaxis was, hitherto, demonstrated in three species: humans, rabbits, and mice. [ 5 ] [ 6 ] This was done by two methods. One involved a Zigmond chamber , modified to make the temperature in each well separately controllable and measurable. A linear temperature gradient was established between the wells and the swimming of spermatozoa in this gradient was analyzed. A small fraction of the spermatozoa (at the order of ~10%), shown to be the capacitated cells, biased their swimming direction according to the gradient, moving towards the warmer temperature. [ 5 ] The other method involved two [ 7 ] [ 8 ] - or three [ 6 ] -compartment separation tube placed within a thermoseparation device that maintains a linear temperature gradient. Sperm accumulation at the warmer end of the separation tube was much higher than the accumulation at the same temperature but in the absence of a temperature gradient. [ 6 ] This gradient-dependent sperm accumulation was observed over a wide temperature range (29-41 °C). [ 8 ]
Since temperature affects almost every process, much attention has been devoted to the question of whether the measurements, mentioned just above, truly demonstrate thermotaxis or whether they reflect another temperature-dependent process. The most pronounced effect of temperature in liquid is convection, which raised the concern that the apparent thermotactic response could have been a reflection of a passive drift in the liquid current or a rheotactic response [ 9 ] to the current (rather than to the temperature gradient per se). Another concern was that the temperature could have changed the local pH of the buffer solution in which the spermatozoa are suspended. This could generate a pH gradient along the temperature gradient, and the spermatozoa might have responded to the formed pH gradient by chemotaxis. However, careful experimental examinations of all these possibilities with proper controls demonstrated that the measured responses to temperature are true thermotactic responses and that they are not a reflection of any other temperature-sensitive process, including rheotaxis and chemotaxis. [ 1 ] [ 8 ]
The behavioral mechanism of sperm thermotaxis has been so far only investigated in human spermatozoa. [ 10 ] Like the behavioral mechanisms of bacterial chemotaxis [ 11 ] and human sperm chemotaxis , [ 12 ] the behavioral mechanism of human sperm thermotaxis appears to be stochastic rather than deterministic. Capacitated human spermatozoa swim in rather straight lines interrupted by turns and brief episodes of hyperactivation . Each such episode results in swimming in a new direction. When the spermatozoa sense a decrease in temperature, the frequency of turns and hyperactivation events increases due to increased flagellar-wave amplitude that results in enhanced side-to-side head displacement. With time, this response undergoes partial adaptation. The opposite happens in response to an increase in temperature. This suggests that when capacitated spermatozoa swim up a temperature gradient, turns are repressed and the spermatozoa continue swimming in the gradient direction. When they happen to swim down the gradient, they turn again and again until their swimming direction is again up the gradient.
The response of spermatozoa to temporal temperature changes even when the temperature is kept constant spatially [ 10 ] suggests that, as in the case of human sperm chemotaxis, [ 12 ] [ 13 ] sperm thermotaxis involves temporal gradient sensing. In other words, spermatozoa apparently compare the temperature (or a temperature-dependent function) between consecutive time points. This, however, does not exclude the occurrence of spatial temperature sensing in addition to temporal sensing. Human spermatozoa can respond thermotactically within a wide temperature range (at least 29–41 °C). [ 8 ] Within this range they preferentially accumulate in warmer temperatures rather than at a single specific, preferred temperature. Amazingly, they can sense and thermotactically respond to temperature gradients as low as <0.014 °C/mm. This means that when human spermatozoa swim a distance that equals their body length (~46 μm) they respond to a temperature difference of <0.0006 °C!
The molecular mechanism underlying thermotaxis, in general, and thermosensing with such extreme sensitivity, in particular, is obscure. It is known that, unlike other recognized thermosensors in mammals, the thermosensors for sperm thermotaxis do not seem to be temperature-sensitive ion channels . They are rather opsins , [ 6 ] known to be G-protein-coupled receptors that act as photosensors in vision . The opsins are present in spermatozoa at specific sites, which depend on the species and the opsin type. [ 6 ] They are involved in sperm thermotaxis via two signaling pathways—a phospholipase C signaling pathway and a cyclic-nucleotide pathway. The former was shown by pharmacological means in human spermatozoa to involve the enzyme phospholipase C, an inositol trisphosphate receptor calcium channel located on internal calcium stores, the calcium channel TRPC3 , and intracellular calcium. [ 6 ] [ 7 ] The latter was hitherto shown to involve phosphodiesterase . [ 6 ] Blocking both pathways fully inhibits sperm thermotaxis. [ 6 ] | https://en.wikipedia.org/wiki/Sperm_thermotaxis |
Spermarche , also known as semenarche , is the time at which a male experiences his first ejaculation. [ 1 ] It is considered to be the counterpart of menarche in females. [ 2 ] [ 3 ] Depending on upbringing, cultural differences, and prior sexual knowledge, males may have different reactions to spermarche, ranging from fear to excitement. [ 4 ] Spermarche is one of the first events in the life of a male leading to sexual maturity . It occurs at the time when the secondary sex characteristics are just beginning to develop. [ 5 ] Researchers have had difficulty determining the onset of spermarche because it is reliant on self-reporting. Other methods to determine it have included the examination of urine samples to determine the presence of spermatozoa . The presence of sperm in urine is referred to as spermaturia . [ 3 ]
Research on the subject has varied for the reasons stated above, as well as changes in the average age of pubescence, which has been decreasing at an average rate of three months a decade. [ 6 ] Research from 2010 indicated that the average age for spermarche in the U.S. was 12–16. [ 7 ] In 2015, researchers in China determined that the average age for spermarche in China was 14. [ 8 ] Historical data from countries including Nigeria [ 9 ] and the United States also suggest 14 as an average age. [ 10 ]
Puberty onset before the age of 9 in males is considered medically abnormal, and is defined as precocious puberty ; research on both organic and ideopathic precocious puberty in males has described puberty onset as early as nine months old. [ 11 ] [ 12 ] As semenarche has a wide range of onset within puberty (with some research indicating spermatogenesis in some cases in early pubertal development) [ 13 ] it is difficult to determine a minimum age for spermarche should one exist. [ 11 ] [ 13 ] Research on the subject, though lacking has described ejaculation in males as young as six years old. [ 14 ]
Various studies have examined the circumstances in which the first ejaculation occurred. Most commonly this occurred via a nocturnal emission , with a significant number experiencing semenarche via masturbation , which is very common at that stage. Less commonly, the first ejaculation occurred during sexual intercourse with a partner. [ 15 ] [ 9 ] | https://en.wikipedia.org/wiki/Spermarche |
The spermatheca (pronounced / s p ər m ə ˈ θ iː k ə / pl. : spermathecae / s p ər m ə ˈ θ iː s iː / ), also called receptaculum seminis ( pl. : receptacula seminis ), is an organ of the female reproductive tract in insects , e.g. ants , bees , [ 1 ] some molluscs , Oligochaeta worms and certain other invertebrates and vertebrates . [ 2 ] Its purpose is to receive and store sperm from the male or, in the case of hermaphrodites , the male component of the body. Spermathecae can sometimes be the site of fertilisation when the oocytes are sufficiently developed. [ 3 ]
Some species of animal have multiple spermathecae. For example, certain species of earthworms have four pairs of spermathecae—one pair each in the 6th, 7th, 8th, and 9th segments. The spermathecae receive and store the spermatozoa of another earthworm during copulation. [ 4 ] They are lined with epithelium and are variable in shape: some are thin, heavily coiled tubes, while others are vague outpocketings from the main reproductive tract. It is one of the many variations in sexual reproduction.
The nematode Caenorhabditis elegans has two spermathecae, one at the end of each gonad. [ 5 ] The C. elegans spermatheca is made up of 24 smooth muscle-like cells that form a stretchable tubular structure. [ 6 ] Actin filaments line the spermatheca in a circumferential manner. The C. elegans spermatheca is used as a model to study mechanotransduction . [ 7 ] [ 8 ]
An apiculturist may examine the spermatheca of a dead queen bee to find out whether it had received sperm from a male. [ 9 ] In many species of stingless bees, especially Melipona bicolor , the queen lays her eggs during the provisioning and oviposition process and the spermatheca fertilizes the egg as it passes along the oviduct . The haplo-diploid system of sex determination makes it possible for the queen to choose the sex of the egg. [ 10 ] | https://en.wikipedia.org/wiki/Spermatheca |
Spermicide is a contraceptive substance that destroys sperm , inserted vaginally prior to intercourse to prevent pregnancy . As a contraceptive, spermicide may be used alone. However, the pregnancy rate experienced by couples using only spermicide is higher than that of couples using other methods. Usually, spermicides are combined with contraceptive barrier methods such as diaphragms , condoms , cervical caps , and sponges . Combined methods are believed to result in lower pregnancy rates than either method alone. [ 2 ]
Spermicides are typically unscented, clear, unflavored, non-staining, and lubricative.
The most common active ingredient of spermicides is nonoxynol-9 . Spermicides containing nonoxynol-9 are available in many forms, such as jelly (gel), films, and foams. Used alone, spermicides have a perfect use failure rate of 6% per year when used correctly and consistently, and 16% failure rate per year in typical use. [ 1 ]
This list of examples was provided by the Mayo Clinic: [ 3 ]
Nonoxynol-9 is the primary chemical in spermicides to inhibit sperm motility. Active secondary spermicidal ingredients can include octoxynol-9 , benzalkonium chloride and menfegol . [ 4 ] These secondary ingredients are not mainstream in the United States, where nonoxynol-9 alone is typical. Preventing sperm motility will inhibit the sperm from travelling towards the egg moving down the fallopian tubes to the uterus. The deep proper insertion of spermicide should effectively block the cervix so that sperm cannot make it past the cervix to the uterus or the fallopian tubes. A study observing the distribution of spermicide containing nonoxynol-9 in the vaginal tract showed “After 10 min the gel spread within the vaginal canal providing a contiguous covering of the epithelium of variable thickness.” [ 5 ] The sole goal of spermicide is to prevent fertilization.
Menfegol is a spermicide manufactured as a foaming tablet. [ 6 ] It is available only in Europe.
Octoxynol-9 was previously a common spermicide, but was removed from the U.S. market in 2002 after manufacturers failed to perform new studies required by the FDA. [ 7 ]
The spermicides benzalkonium chloride and sodium cholate are used in some contraceptive sponges . [ 8 ] Benzalkonium chloride might also be available in Canada as a suppository. [ 9 ]
The 2008 Ig Nobel Prize (a parody of the Nobel Prizes ) in Chemistry was awarded to Sheree Umpierre, Joseph Hill, and Deborah Anderson, for discovering that Coca-Cola is an effective spermicide, [ 10 ] and to C.Y. Hong, C.C. Shieh, P. Wu, and B.N. Chiang for proving it is not. [ 11 ] [ 12 ]
Lemon juice solutions have been shown to immobilize sperm in the laboratory, [ 13 ] as has Krest Bitter Lemon drink. [ 14 ] While the authors of the Krest Bitter Lemon study suggested its use as a postcoital douche, this is unlikely to be effective, as sperm begin leaving the ejaculate (out of the reach of any douche) within 1.5 minutes of deposition. No published studies appear to have been done on the effectiveness of lemon juice preparations in preventing pregnancy, though they are advocated by some as 'natural' spermicides. [ 15 ]
Lactic acid preparations have also been shown to have some spermicidal effect, and commercial lactic acid-based spermicides are available. [ 16 ] [ 17 ] A contraceptive containing lactic acid, citric acid, and potassium bitartrate (Phexxi) was approved for use in the United States in May 2020. [ 18 ]
Extractives of the neem plant such as neem oil have also been proposed as spermicides based on laboratory studies. [ 19 ] Animal studies of creams and pessaries derived from neem have shown they have contraceptive effects; [ 20 ] however, trials in humans to determine its effectiveness in preventing pregnancy have not yet been conducted.
Spermicides are believed to increase the contraceptive effectiveness of condoms. [ 2 ]
However, condoms that are spermicidally lubricated by the manufacturer have a shorter shelf life [ 21 ] and may cause urinary tract infections in women. [ 22 ] The World Health Organization says that spermicidally lubricated condoms should no longer be promoted. However, they recommend using a nonoxynol-9 lubricated condom over no condom at all. [ 23 ]
Spermicides used alone are only about 91 percent effective. [ 24 ] When spermicides are used in conjunction with condoms and other barrier methods there is a 97 percent effective rate for pregnancy prevention.
Temporary local skin irritation involving the vulva, vagina, or penis is the most common problem associated with spermicide use. [ 25 ]
Frequent use (two times or more a day) of nonoxynol-9 containing spermicide is inadvisable if STI/HIV exposure is likely, because in this situation there is increased vulvovaginal epithelial disruption and increased risk of HIV acquisition. [ 25 ]
In 2007, the United States Food and Drug Administration (FDA) mandated that labels for nonoxynol-9 over-the-counter (OTC) contraceptive products carry a new warning saying they do not protect against STDs and HIV/AIDS. [ 26 ] [ 27 ]
The first written record of spermicide use is found in the Kahun Papyrus , an Egyptian document dating to 1850 BCE. It described a pessary of crocodile dung and fermented dough. [ 28 ] It is believed that the low pH of the dung may have had a spermicidal effect. [ 29 ]
Further formulations are found in the Ebers Papyrus from approximately 1500 BCE. It recommended mixing seed wool, acacia, dates and honey, and placing the mixture in the vagina. It probably had some effectiveness, in part as a physical barrier due to the thick, sticky consistency, and also because of the lactic acid (a known spermicide) formed from the acacia. [ 29 ]
Writings by Soranus , a 2nd-century Greek physician, contained formulations for a number of acidic concoctions claimed to be spermicidal. His instructions were to soak wool in one of the mixtures, then place near the cervix. [ 28 ]
Laboratory testing of substances to see if they inhibited sperm motility began in the 1800s. Modern spermicides nonoxynol-9 and menfegol were developed from this line of research. [ 28 ] However, many other substances of dubious contraceptive value were also promoted. Especially after the prohibition of contraception in the U.S. by the 1873 Comstock Act , spermicides—the most popular of which was Lysol —were marketed only as "feminine hygiene" products and were not held to any standard of effectiveness. Worse, many manufacturers recommended using the products as a douche after intercourse, too late to affect all the sperm. Medical estimates during the 1930s placed the pregnancy rate of women using many over-the-counter spermicides at seventy percent per year. [ 30 ]
A misconception about spermicides existed in the 1980s and 1990s. A 1988 literature review article noted that in vitro studies of nonoxynol-9 and other spermicides showed inactivation of STI pathogens, including HIV. [ 31 ] But a 2002 systemic review and meta-analysis of nine randomized controlled trials of vaginal nonoxynol-9 for HIV and STI prevention involving more than 5,000 women (predominantly sex workers) found no statistically significant reduction in risk of HIV and STIs, but found a small statistically significant increase in genital lesions among nonoxynol-9 spermicide users. [ 32 ] And in a high-risk population using a nonoxynol-9 vaginal gel more than three applications per day on average, the risk of HIV acquisition was increased. [ 25 ]
Currently available spermicides containing nonoxynol-9 are ineffective as microbicides, in particular as HIV-preventive measures. 17 Thus, spermicides used alone are not recommended to prevent HIV or other STIs. Furthermore, frequent use (more than 2 times a day) of spermicide causes more vulvovovaginal epithelial disruption, 18 which theoretically could increase susceptibility to HIV. In a high-risk population using a vaginal gel with nonoxynol-9 more than three applications per day on average, the risk of HIV acquisition was increased compared with placebo. 19
Disadvantages and cautions Local irritation Temporary skin irritation involving the vulva, vagina, or penis caused by either local toxicity or allergy to the formulation is the most common problem associated with spermicide use... Although vaginal epithelial disruption has been associated with frequent use (twice a day or more) of spermicides containing N-9, this is usually asymptomatic. In a low risk population, long-term use of N-9 containing methods was not associated with epithelial disruption. 22
N-9 spermicides are inadvisable if STI/HIV exposure is likely in situations that would involve frequent use defined as 2 times or more a day. | https://en.wikipedia.org/wiki/Spermicide |
Spermine synthase ( EC 2.5.1.22 , spermidine aminopropyltransferase , spermine synthetase ) is an enzyme that converts spermidine into spermine . [ 1 ] [ 2 ] This enzyme catalyses the following chemical reaction
Spermine synthase is an enzyme involved in polyamine biosynthesis. It is present in all eukaryotes and plays a role in a variety of biological functions in plants [ 3 ] Its structure consists of two identical monomers of 41 kDa with three domains each, creating a homodimer formed via dimerization . The interactions between one of the three domains, the N-terminals of the monomers, is responsible for dimerization as that is where the active site is located; the central terminal consisting of four β- strands structurally forming a lid for the third domain, the C-terminal domain. [ 4 ]
This biochemistry article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Spermine_synthase |
In plant science , the spermosphere is the zone surrounding a seed where soil, microorganisms, and seed germinating interact. [ 1 ] The zone is a small area, typically 1–10 mm from the seed, but varying with seed type, the variety of soil microorganisms , the level of soil moisture, and other factors. [ 2 ]
Within the spermosphere, a range of complex interactions take place among the germinating seed, the soil, and the microbiome . [ 3 ] [ 1 ] Because germination is a brief process, the spermosphere is transient, but the impact of the microbial activity within the spermosphere can have strong and long-lasting effects on the developing plant. [ 3 ] The spermosphere can even have impacts on managing stress during germination, as seen with Bacillus strains and peanut plants. [ 4 ]
Seeds exude various molecules that influence their surrounding microbial communities, either inhibiting or stimulating their growth. [ 1 ] [ 3 ] The composition of the exudates varies according to the plant type and such properties of the soil as its pH and moisture content. Soil type matters much more than seed type, specifically soil with a higher content of organic matter. [ 5 ]
With these biochemical effects, the spermosphere develops both downward—to form the rhizosphere (upon the emergence of the plant's radicle ) [ 3 ] —and upward to form the laimosphere , which is the soil surrounding the growing plant stem , and the phyllosphere , which is the microbial community on the part of the plant above the soil. Specifically, the floral microbiota can play a role in the composition of the spermosphere like in plants such as wheat, [ 6 ] grapevine, [ 7 ] and rice. [ 8 ] As the seed germinates, the function of the microbial community changes rather than its composition. [ 5 ]
The spermosphere also acts as biological control for the germinating seed which means certain beneficial microorganisms can protect the seed from plant pathogens. Many plant pathogens such as Fusarium and Pythium ultimum can colonize a newly germinating seed within the first few hours of planting. [ 9 ] The seed can exudate molecules and nutrients that attract beneficial microorganisms to their spermosphere which then prevent the colonization of pathogens. [ 5 ] There has been specific research in this area with cottonseeds in which they were coated with a beneficial bacteria which prevented the seed from being infected with a Pythium ultimum infection. [ 10 ] | https://en.wikipedia.org/wiki/Spermosphere |
In mathematics , Sperner's lemma is a combinatorial result on colorings of triangulations , analogous to the Brouwer fixed point theorem , which is equivalent to it. [ 1 ] It states that every Sperner coloring (described below) of a triangulation of an n {\displaystyle n} -dimensional simplex contains a cell whose vertices all have different colors.
The initial result of this kind was proved by Emanuel Sperner , in relation with proofs of invariance of domain . Sperner colorings have been used for effective computation of fixed points and in root-finding algorithms , and are applied in fair division (cake cutting) algorithms.
According to the Soviet Mathematical Encyclopaedia (ed. I.M. Vinogradov ), a related 1929 theorem (of Knaster , Borsuk and Mazurkiewicz ) had also become known as the Sperner lemma – this point is discussed in the English translation (ed. M. Hazewinkel). It is now commonly known as the Knaster–Kuratowski–Mazurkiewicz lemma .
In one dimension, Sperner's Lemma can be regarded as a discrete version of the intermediate value theorem . In this case, it essentially says that if a discrete function takes only the values 0 and 1, begins at the value 0 and ends at the value 1, then it must switch values an odd number of times.
The two-dimensional case is the one referred to most frequently. It is stated as follows:
Subdivide a triangle ABC arbitrarily into a triangulation consisting of smaller triangles meeting edge to edge. Then a Sperner coloring of the triangulation is defined as an assignment of three colors to the vertices of the triangulation such that
Then every Sperner coloring of every triangulation has at least one "rainbow triangle", a smaller triangle in the triangulation that has its vertices colored with all three different colors. More precisely, there must be an odd number of rainbow triangles.
In the general case the lemma refers to a n -dimensional simplex :
Consider any triangulation T , a disjoint division of A {\displaystyle {\mathcal {A}}} into smaller n -dimensional simplices, again meeting face-to-face. Denote the coloring function as:
where S is the set of vertices of T . A coloring function defines a Sperner coloring when:
A i 1 A i 2 … A i k + 1 {\displaystyle A_{i_{1}}A_{i_{2}}\ldots A_{i_{k+1}}}
are colored only with the colors
i 1 , i 2 , … , i k + 1 . {\displaystyle i_{1},i_{2},\ldots ,i_{k+1}.}
Then every Sperner coloring of every triangulation of the n -dimensional simplex has an odd number of instances of a rainbow simplex , meaning a simplex whose vertices are colored with all n + 1 colors. In particular, there must be at least one rainbow simplex.
We shall first address the two-dimensional case. Consider a graph G built from the triangulation T as follows:
Note that on the interval AB there is an odd number of borders colored 1-2 (simply because A is colored 1, B is colored 2; and as we move along AB , there must be an odd number of color changes in order to get different colors at the beginning and at the end). On the intervals BC and CA, there are no borders colored 1-2 at all. Therefore, the vertex of G corresponding to the outer area has an odd degree. But it is known (the handshaking lemma ) that in a finite graph there is an even number of vertices with odd degree. Therefore, the remaining graph, excluding the outer area, has an odd number of vertices with odd degree corresponding to members of T .
It can be easily seen that the only possible degree of a triangle from T is 0, 1, or 2, and that the degree 1 corresponds to a triangle colored with the three colors 1, 2, and 3.
Thus we have obtained a slightly stronger conclusion, which says that in a triangulation T there is an odd number (and at least one) of full-colored triangles.
A multidimensional case can be proved by induction on the dimension of a simplex. We apply the same reasoning, as in the two-dimensional case, to conclude that in a n -dimensional triangulation there is an odd number of full-colored simplices.
Here is an elaboration of the proof given previously, for a reader new to graph theory .
This diagram numbers the colors of the vertices of the example given previously. The small triangles whose vertices all have different numbers are shaded in the graph. Each small triangle becomes a node in the new graph derived from the triangulation. The small letters identify the areas, eight inside the figure, and area i designates the space outside of it.
As described previously, those nodes that share an edge whose endpoints are numbered 1 and 2 are joined in the derived graph. For example, node d shares an edge with the outer area i , and its vertices all have different numbers, so it is also shaded. Node b is not shaded because two vertices have the same number, but it is joined to the outer area.
One could add a new full-numbered triangle, say by inserting a node numbered 3 into the edge between 1 and 1 of node a , and joining that node to the other vertex of a . Doing so would have to create a pair of new nodes, like the situation with nodes f and g .
Andrew McLennan and Rabee Tourky presented a different proof, using the volume of a simplex . It proceeds in one step, with no induction. [ 2 ] [ 3 ]
Suppose there is a d -dimensional simplex of side-length N , and it is triangulated into sub-simplices of side-length 1. There is a function that, given any vertex of the triangulation, returns its color. The coloring is guaranteed to satisfy Sperner's boundary condition. How many times do we have to call the function in order to find a rainbow simplex? Obviously, we can go over all the triangulation vertices, whose number is O( N d ), which is polynomial in N when the dimension is fixed. But, can it be done in time O(poly(log N )), which is polynomial in the binary representation of N?
This problem was first studied by Christos Papadimitriou . He introduced a complexity class called PPAD , which contains this as well as related problems (such as finding a Brouwer fixed point ). He proved that finding a Sperner simplex is PPAD-complete even for d =3. Some 15 years later, Chen and Deng proved PPAD-completeness even for d =2. [ 4 ] It is believed that PPAD-hard problems cannot be solved in time O(poly(log N )).
Suppose that each vertex of the triangulation may be labeled with multiple colors, so that the coloring function is F : S → 2 [ n +1] .
For every sub-simplex, the set of labelings on its vertices is a set-family over the set of colors [ n + 1] . This set-family can be seen as a hypergraph .
If, for every vertex v on a face of the simplex, the colors in f ( v ) are a subset of the set of colors on the face endpoints, then there exists a sub-simplex with a balanced labeling – a labeling in which the corresponding hypergraph admits a perfect fractional matching . To illustrate, here are some balanced labeling examples for n = 2 :
This was proved by Shapley in 1973. [ 5 ] It is a combinatorial analogue of the KKMS lemma .
Suppose that we have a d -dimensional polytope P with n vertices. P is triangulated, and each vertex of the triangulation is labeled with a label from {1, …, n }. Every main vertex i is labeled i . A sub-simplex is called fully-labeled if it is d -dimensional, and each of its d + 1 vertices has a different label. If every vertex in a face F of P is labeled with one of the labels on the endpoints of F , then there are at least n – d fully-labeled simplices. Some special cases are:
The general statement was conjectured by Atanassov in 1996, who proved it for the case d = 2 . [ 6 ] The proof of the general case was first given by de Loera, Peterson, and Su in 2002. [ 7 ] They provide two proofs: the first is non-constructive and uses the notion of pebble sets ; the second is constructive and is based on arguments of following paths in graphs .
Meunier [ 8 ] extended the theorem from polytopes to polytopal bodies, which need not be convex or simply-connected. In particular, if P is a polytope, then the set of its faces is a polytopal body. In every Sperner labeling of a polytopal body with vertices v 1 , …, v n , there are at least:
fully-labeled simplices such that any pair of these simplices receives two different labelings. The degree deg B ( P ) ( v i ) is the number of edges of B ( P ) to which v i belongs. Since the degree is at least d , the lower bound is at least n – d . But it can be larger. For example, for the cyclic polytope in 4 dimensions with n vertices, the lower bound is:
Musin [ 9 ] further extended the theorem to d -dimensional piecewise-linear manifolds , with or without a boundary.
Asada, Frick, Pisharody, Polevy, Stoner, Tsang and Wellner [ 10 ] further extended the theorem to pseudomanifolds with boundary, and improved the lower bound on the number of facets with pairwise-distinct labels.
Suppose that, instead of a simplex triangulated into sub-simplices, we have an n -dimensional cube partitioned into smaller n -dimensional cubes.
Harold W. Kuhn [ 11 ] proved the following lemma. Suppose the cube [0, M ] n , for some integer M , is partitioned into M n unit cubes. Suppose each vertex of the partition is labeled with a label from {1, …, n + 1}, such that for every vertex v : (1) if v i = 0 then the label on v is at most i ; (2) if v i = M then the label on v is not i . Then there exists a unit cube with all the labels {1, …, n + 1} (some of them more than once). The special case n = 2 is: suppose a square is partitioned into sub-squares, and each vertex is labeled with a label from {1,2,3}. The left edge is labeled with 1 (= at most 1); the bottom edge is labeled with 1 or 2 (= at most 2); the top edge is labeled with 1 or 3 (= not 2); and the right edge is labeled with 2 or 3 (= not 1). Then there is a square labeled with 1,2,3.
Another variant, related to the Poincaré–Miranda theorem , [ 12 ] is as follows. Suppose the cube [0, M ] n is partitioned into M n unit cubes. Suppose each vertex is labeled with a binary vector of length n , such that for every vertex v : (1) if v i = 0 then the coordinate i of label on v is 0; (2) if v i = M then coordinate i of the label on v is 1; (3) if two vertices are neighbors, then their labels differ by at most one coordinate. Then there exists a unit cube in which all 2 n labels are different. In two dimensions, another way to formulate this theorem is: [ 13 ] in any labeling that satisfies conditions (1) and (2), there is at least one cell in which the sum of labels is 0 [a 1-dimensional cell with (1,1) and (-1,-1) labels, or a 2-dimensional cells with all four different labels].
Wolsey [ 14 ] strengthened these two results by proving that the number of completely-labeled cubes is odd.
Musin [ 13 ] extended these results to general quadrangulations .
Suppose that, instead of a single labeling, we have n different Sperner labelings. We consider pairs (simplex, permutation) such that, the label of each vertex of the simplex is chosen from a different labeling (so for each simplex, there are n ! different pairs). Then there are at least n ! fully labeled pairs. This was proved by Ravindra Bapat [ 15 ] for any triangulation. A simpler proof, which only works for specific triangulations, was presented later by Su. [ 16 ]
Another way to state this lemma is as follows. Suppose there are n people, each of whom produces a different Sperner labeling of the same triangulation. Then, there exists a simplex, and a matching of the people to its vertices, such that each vertex is labeled by its owner differently (one person labels its vertex by 1, another person labels its vertex by 2, etc.). Moreover, there are at least n ! such matchings. This can be used to find an envy-free cake-cutting with connected pieces.
Asada, Frick, Pisharody, Polevy, Stoner, Tsang and Wellner [ 10 ] extended this theorem to pseudomanifolds with boundary.
More generally, suppose we have m different Sperner labelings, where m may be different than n . Then: [ 17 ] : Thm 2.1
Both versions reduce to Sperner's lemma when m = 1 , or when all m labelings are identical.
See [ 18 ] for similar generalizations.
Brown and Cairns [ 19 ] strengthened Sperner's lemma by considering the orientation of simplices. Each sub-simplex has an orientation that can be either +1 or -1 (if it is fully-labeled), or 0 (if it is not fully-labeled). They proved that the sum of all orientations of simplices is +1. In particular, this implies that there is an odd number of fully-labeled simplices.
As an example for n = 3 , suppose a triangle is triangulated and labeled with {1,2,3}. Consider the cyclic sequence of labels on the boundary of the triangle. Define the degree of the labeling as the number of switches from 1 to 2, minus the number of switches from 2 to 1. See examples in the table at the right. Note that the degree is the same if we count switches from 2 to 3 minus 3 to 2, or from 3 to 1 minus 1 to 3.
Musin proved that the number of fully labeled triangles is at least the degree of the labeling . [ 20 ] In particular, if the degree is nonzero, then there exists at least one fully labeled triangle.
If a labeling satisfies the Sperner condition, then its degree is exactly 1: there are 1-2 and 2-1 switches only in the side between vertices 1 and 2, and the number of 1-2 switches must be one more than the number of 2-1 switches (when walking from vertex 1 to vertex 2). Therefore, the original Sperner lemma follows from Musin's theorem.
There is a similar lemma about finite and infinite trees and cycles . [ 21 ]
Mirzakhani and Vondrak [ 22 ] study a weaker variant of a Sperner labeling, in which the only requirement is that label i is not used on the face opposite to vertex i . They call it Sperner-admissible labeling . They show that there are Sperner-admissible labelings in which every cell contains at most 4 labels. They also prove an optimal lower bound on the number of cells that must have at least two different labels in each Sperner-admissible labeling. They also prove that, for any Sperner-admissible partition of the regular simplex, the total area of the boundary between the parts is minimized by the Voronoi partition .
Sperner colorings have been used for effective computation of fixed points . A Sperner coloring can be constructed such that fully labeled simplices correspond to fixed points of a given function. By making a triangulation smaller and smaller, one can show that the limit of the fully labeled simplices is exactly the fixed point. Hence, the technique provides a way to approximate fixed points.
A related application is the numerical detection of periodic orbits and symbolic dynamics . [ 23 ] Sperner's lemma can also be used in root-finding algorithms and fair division algorithms; see Simmons–Su protocols .
Sperner's lemma is one of the key ingredients of the proof of Monsky's theorem , that a square cannot be cut into an odd number of equal-area triangles . [ 24 ]
Sperner's lemma can be used to find a competitive equilibrium in an exchange economy , although there are more efficient ways to find it. [ 25 ] : 67
Fifty years after first publishing it, Sperner presented a survey on the development, influence and applications of his combinatorial lemma. [ 26 ]
There are several fixed-point theorems which come in three equivalent variants: an algebraic topology variant, a combinatorial variant and a set-covering variant. Each variant can be proved separately using totally different arguments, but each variant can also be reduced to the other variants in its row. Additionally, each result in
the top row can be deduced from the one below it in the same column. [ 27 ] | https://en.wikipedia.org/wiki/Sperner's_lemma |
Sperner's theorem , in discrete mathematics , describes the largest possible families of finite sets none of which contain any other sets in the family. It is one of the central results in extremal set theory . It is named after Emanuel Sperner , who published it in 1928.
This result is sometimes called Sperner's lemma, but the name " Sperner's lemma " also refers to an unrelated result on coloring triangulations. To differentiate the two results, the result on the size of a Sperner family is now more commonly known as Sperner's theorem.
A family of sets in which none of the sets is a strict subset of another is called a Sperner family , or an antichain of sets, or a clutter. For example, the family of k -element subsets of an n -element set is a Sperner family. No set in this family can contain any of the others, because a containing set has to be strictly bigger than the set it contains, and in this family all sets have equal size. The value of k that makes this example have as many sets as possible is n /2 if n is even, or either of the nearest integers to n /2 if n is odd. For this choice, the number of sets in the family is ( n ⌊ n / 2 ⌋ ) {\displaystyle {\tbinom {n}{\lfloor n/2\rfloor }}} .
Sperner's theorem states that these examples are the largest possible Sperner families over an n -element set.
Formally, the theorem states that,
Sperner's theorem can also be stated in terms of partial order width . The family of all subsets of an n -element set (its power set ) can be partially ordered by set inclusion; in this partial order, two distinct elements are said to be incomparable when neither of them contains the other. The width of a partial order is the largest number of elements in an antichain , a set of pairwise incomparable elements. Translating this terminology into the language of sets, an antichain is just a Sperner family, and the width of the partial order is the maximum number of sets in a Sperner family.
Thus, another way of stating Sperner's theorem is that the width of the inclusion order on a power set is ( n ⌊ n / 2 ⌋ ) {\displaystyle {\binom {n}{\lfloor n/2\rfloor }}} .
A graded partially ordered set is said to have the Sperner property when one of its largest antichains is formed by a set of elements that all have the same rank. In this terminology, Sperner's theorem states that the partially ordered set of all subsets of a finite set, partially ordered by set inclusion, has the Sperner property.
There are many proofs of Sperner's theorem, each leading to different generalizations (see Anderson (1987) ).
The following proof is due to Lubell (1966) . Let s k denote the number of k -sets in S . For all 0 ≤ k ≤ n ,
and, thus,
Since S is an antichain, we can sum over the above inequality from k = 0 to n and then apply the LYM inequality to obtain
which means
This completes the proof of part 1.
To have equality, all the inequalities in the preceding proof must be equalities. Since
if and only if k = ⌊ n / 2 ⌋ {\displaystyle k=\lfloor {n/2}\rfloor } or ⌈ n / 2 ⌉ , {\displaystyle \lceil {n/2}\rceil ,} we conclude that equality implies that S consists only of sets of sizes ⌊ n / 2 ⌋ {\displaystyle \lfloor {n/2}\rfloor } or ⌈ n / 2 ⌉ . {\displaystyle \lceil {n/2}\rceil .} For even n that concludes the proof of part 2.
For odd n there is more work to do, which we omit here because it is complicated. See Anderson (1987) , pp. 3–4.
There are several generalizations of Sperner's theorem for subsets of P ( E ) , {\displaystyle {\mathcal {P}}(E),} the poset of all subsets of E .
A chain is a subfamily { S 0 , S 1 , … , S r } ⊆ P ( E ) {\displaystyle \{S_{0},S_{1},\dots ,S_{r}\}\subseteq {\mathcal {P}}(E)} that is totally ordered, i.e., S 0 ⊂ S 1 ⊂ ⋯ ⊂ S r {\displaystyle S_{0}\subset S_{1}\subset \dots \subset S_{r}} (possibly after renumbering). The chain has r + 1 members and length r . An r -chain-free family (also called an r -family ) is a family of subsets of E that contains no chain of length r . Erdős (1945) proved that the largest size of an r -chain-free family is the sum of the r largest binomial coefficients ( n i ) {\displaystyle {\binom {n}{i}}} . The case r = 1 is Sperner's theorem.
In the set P ( E ) p {\displaystyle {\mathcal {P}}(E)^{p}} of p -tuples of subsets of E , we say a p -tuple ( S 1 , … , S p ) {\displaystyle (S_{1},\dots ,S_{p})} is ≤ another one, ( T 1 , … , T p ) , {\displaystyle (T_{1},\dots ,T_{p}),} if S i ⊆ T i {\displaystyle S_{i}\subseteq T_{i}} for each i = 1,2,..., p . We call ( S 1 , … , S p ) {\displaystyle (S_{1},\dots ,S_{p})} a p -composition of E if the sets S 1 , … , S p {\displaystyle S_{1},\dots ,S_{p}} form a partition of E . Meshalkin (1963) proved that the maximum size of an antichain of p -compositions is the largest p -multinomial coefficient ( n n 1 n 2 … n p ) , {\displaystyle {\binom {n}{n_{1}\ n_{2}\ \dots \ n_{p}}},} that is, the coefficient in which all n i are as nearly equal as possible (i.e., they differ by at most 1). Meshalkin proved this by proving a generalized LYM inequality.
The case p = 2 is Sperner's theorem, because then S 2 = E ∖ S 1 {\displaystyle S_{2}=E\setminus S_{1}} and the assumptions reduce to the sets S 1 {\displaystyle S_{1}} being a Sperner family.
Beck & Zaslavsky (2002) combined the Erdös and Meshalkin theorems by adapting Meshalkin's proof of his generalized LYM inequality. They showed that the largest size of a family of p -compositions such that the sets in the i -th position of the p -tuples, ignoring duplications, are r -chain-free, for every i = 1 , 2 , … , p − 1 {\displaystyle i=1,2,\dots ,p-1} (but not necessarily for i = p ), is not greater than the sum of the r p − 1 {\displaystyle r^{p-1}} largest p -multinomial coefficients.
In the finite projective geometry PG( d , F q ) of dimension d over a finite field of order q , let L ( p , F q ) {\displaystyle {\mathcal {L}}(p,F_{q})} be the family of all subspaces. When partially ordered by set inclusion, this family is a lattice. Rota & Harper (1971) proved that the largest size of an antichain in L ( p , F q ) {\displaystyle {\mathcal {L}}(p,F_{q})} is the largest Gaussian coefficient [ d + 1 k ] ; {\displaystyle {\begin{bmatrix}d+1\\k\end{bmatrix}};} this is the projective-geometry analog, or q -analog , of Sperner's theorem.
They further proved that the largest size of an r -chain-free family in L ( p , F q ) {\displaystyle {\mathcal {L}}(p,F_{q})} is the sum of the r largest Gaussian coefficients. Their proof is by a projective analog of the LYM inequality.
Beck & Zaslavsky (2003) obtained a Meshalkin-like generalization of the Rota–Harper theorem. In PG( d , F q ), a Meshalkin sequence of length p is a sequence ( A 1 , … , A p ) {\displaystyle (A_{1},\ldots ,A_{p})} of projective subspaces such that no proper subspace of PG( d , F q ) contains them all and their dimensions sum to d − p + 1 {\displaystyle d-p+1} . The theorem is that a family of Meshalkin sequences of length p in PG( d , F q ), such that the subspaces appearing in position i of the sequences contain no chain of length r for each i = 1 , 2 , … , p − 1 , {\displaystyle i=1,2,\dots ,p-1,} is not more than the sum of the largest r p − 1 {\displaystyle r^{p-1}} of the quantities
where [ d + 1 n 1 n 2 … n p ] {\displaystyle {\begin{bmatrix}d+1\\n_{1}\ n_{2}\ \dots \ n_{p}\end{bmatrix}}} (in which we assume that d + 1 = n 1 + ⋯ + n p {\displaystyle d+1=n_{1}+\cdots +n_{p}} ) denotes the p -Gaussian coefficient
and
the second elementary symmetric function of the numbers n 1 , n 2 , … , n p . {\displaystyle n_{1},n_{2},\dots ,n_{p}.} | https://en.wikipedia.org/wiki/Sperner's_theorem |
In order-theoretic mathematics , a graded partially ordered set is said to have the Sperner property (and hence is called a Sperner poset ), if no antichain within it is larger than the largest rank level (one of the sets of elements of the same rank) in the poset. [ 1 ] Since every rank level is itself an antichain, the Sperner property is equivalently the property that some rank level is a maximum antichain. [ 2 ] The Sperner property and Sperner posets are named after Emanuel Sperner , who proved Sperner's theorem stating that the family of all subsets of a finite set (partially ordered by set inclusion) has this property. The lattice of partitions of a finite set typically lacks the Sperner property. [ 3 ]
A k -Sperner poset is a graded poset in which no union of k antichains is larger than the union of the k largest rank levels, [ 1 ] or, equivalently, the poset has a maximum k-family consisting of k rank levels. [ 2 ]
A strict Sperner poset is a graded poset in which all maximum antichains are rank levels. [ 2 ]
A strongly Sperner poset is a graded poset which is k-Sperner for all values of k up to the largest rank value. [ 2 ]
This combinatorics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Sperner_property_of_a_partially_ordered_set |
Sphaerobacter is a genus of bacteria. When originally described it was placed in its own subclass (Spahaerobacteridae) within the class Actinomycetota . Subsequently, phylogenetic studies have now placed it in its own order Sphaerobacterales within the phylum Thermomicrobiota . [ 1 ] [ 2 ] Up to now there is only one species of this genus known ( Sphaerobacter thermophilus ). [ 3 ] The closest related cultivated organism to S. thermophilus is Thermomicrobium roseum with an 87% sequence similarity which indicates that S. thermophilus is one of the most isolated bacterial species.[4]
4. Pati, A., Labutti, K., Pukall, R., Nolan, M., Glavina Del Rio, T., Tice, H., … Lapidus, A. (2010). Complete genome sequence of Sphaerobacter thermophilus type strain (S 6022). Standards in genomic sciences , 2(1), 49–56. doi:10.4056/sigs.601105
This bacteria -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Sphaerobacter |
Sphaerotilus natans is an aquatic periphyton bacterial organism associated with polluted water. These tightly sheathed filamentous bacteria colonies are commonly but inaccurately known as " sewage fungus " [ 1 ]
Straight or smoothly curved filaments 1.5 μm in diameter and 100 to more than 500 μm in length are formed by rod-shaped cells with clear septa growing within a long, tubular sheath. An adhesive basal element at one end of the filament can aid attachment to solid surfaces. [ 2 ] The sheath offers some protection from predators, and the ability to anchor in flowing water allows access to a passing stream of food and nutrients. [ 3 ] Individual mature cells swarm out of the protective tube to colonize new sites. [ 4 ] Each motile mature cell has an intertwined bundle of flagella appearing as a single flagellum consisting of a long filament with a short hook and a basal body complex, but it is distinguishable by electron microscope as 10 to 30 strands with diameters of 12.5 to 16 nm each. S. natans stores reserves of poly- beta -hydroxybutyrate as internal bioplastic globules making up 30 to 40% of the dry weight of a colony. [ 3 ] Gram and Neisser staining reactions are negative. [ 5 ]
S. natans requires dissolved simple sugars or organic acids as a food supply, but needs less phosphorus than many competing organisms and can tolerate low oxygen concentrations. [ 5 ] Capability to deposit elemental sulfur intracellularly in the presence of hydrogen sulfide is believed to be a detoxifying mechanism. S. natans requires either cobalamin or methionine as a trace nutrient. [ 3 ] S. natans filaments can aid development of a periphyton biofilm trapping suspended particles and stabilizing colonies of other organisms including Klebsiella and Pseudomonas . [ 2 ]
S. natans is described as a key taxon in sewage fungus , a polymicrobial biofilm that proliferates in rivers with a high organic loading [ 6 ] [ 7 ] [ 8 ] such as from sewage discharges, industrial effluents or runoff from airport de-icing. [ 9 ] It is also implicated in active sludge bulking [ 10 ]
Sphaerotilus natans is often associated with a buoyant floc (or "bulking sludge") causing poor solids separation in activated sludge clarifiers of secondary sewage treatment . [ 4 ] Metal surfaces covered with S. natans may experience accelerated corrosion if the slime creates a barrier causing differential oxygen concentrations. [ 11 ] S. natans slimes may reduce quality of paper produced by paper mills that use recycled water. [ 2 ] | https://en.wikipedia.org/wiki/Sphaerotilus_natans |
A sphere (from Greek σφαῖρα , sphaîra ) [ 1 ] is a surface analogous to the circle , a curve . In solid geometry , a sphere is the set of points that are all at the same distance r from a given point in three-dimensional space . [ 2 ] That given point is the center of the sphere, and the distance r is the sphere's radius . The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians .
The sphere is a fundamental surface in many fields of mathematics . Spheres and nearly-spherical shapes also appear in nature and industry. Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is often approximated as a sphere in geography , and the celestial sphere is an important concept in astronomy . Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres roll smoothly in any direction, so most balls used in sports and toys are spherical, as are ball bearings .
As mentioned earlier r is the sphere's radius; any line from the center to a point on the sphere is also called a radius. 'Radius' is used in two senses: as a line segment and also as its length. [ 3 ]
If a radius is extended through the center to the opposite side of the sphere, it creates a diameter . Like the radius, the length of a diameter is also called the diameter, and denoted d . Diameters are the longest line segments that can be drawn between two points on the sphere: their length is twice the radius, d = 2 r . Two points on the sphere connected by a diameter are antipodal points of each other. [ 3 ]
A unit sphere is a sphere with unit radius ( r = 1 ). For convenience, spheres are often taken to have their center at the origin of the coordinate system , and spheres in this article have their center at the origin unless a center is mentioned.
A great circle on the sphere has the same center and radius as the sphere, and divides it into two equal hemispheres .
Although the figure of Earth is not perfectly spherical, terms borrowed from geography are convenient to apply to the sphere.
A particular line passing through its center defines an axis (as in Earth's axis of rotation ).
The sphere-axis intersection defines two antipodal poles ( north pole and south pole ). The great circle equidistant to the poles is called the equator . Great circles through the poles are called lines of longitude or meridians . Small circles on the sphere that are parallel to the equator are circles of latitude (or parallels ). In geometry unrelated to astronomical bodies, geocentric terminology should be used only for illustration and noted as such, unless there is no chance of misunderstanding. [ 3 ]
Mathematicians consider a sphere to be a two-dimensional closed surface embedded in three-dimensional Euclidean space . They draw a distinction between a sphere and a ball , which is a solid figure , a three-dimensional manifold with boundary that includes the volume contained by the sphere. An open ball excludes the sphere itself, while a closed ball includes the sphere: a closed ball is the union of the open ball and the sphere, and a sphere is the boundary of a (closed or open) ball. The distinction between ball and sphere has not always been maintained and especially older mathematical references talk about a sphere as a solid. The distinction between " circle " and " disk " in the plane is similar.
Small spheres or balls are sometimes called spherules (e.g., in Martian spherules ).
In analytic geometry , a sphere with center ( x 0 , y 0 , z 0 ) and radius r is the locus of all points ( x , y , z ) such that
Since it can be expressed as a quadratic polynomial, a sphere is a quadric surface , a type of algebraic surface . [ 3 ]
Let a, b, c, d, e be real numbers with a ≠ 0 and put
Then the equation
has no real points as solutions if ρ < 0 {\displaystyle \rho <0} and is called the equation of an imaginary sphere . If ρ = 0 {\displaystyle \rho =0} , the only solution of f ( x , y , z ) = 0 {\displaystyle f(x,y,z)=0} is the point P 0 = ( x 0 , y 0 , z 0 ) {\displaystyle P_{0}=(x_{0},y_{0},z_{0})} and the equation is said to be the equation of a point sphere . Finally, in the case ρ > 0 {\displaystyle \rho >0} , f ( x , y , z ) = 0 {\displaystyle f(x,y,z)=0} is an equation of a sphere whose center is P 0 {\displaystyle P_{0}} and whose radius is ρ {\displaystyle {\sqrt {\rho }}} . [ 2 ]
If a in the above equation is zero then f ( x , y , z ) = 0 is the equation of a plane. Thus, a plane may be thought of as a sphere of infinite radius whose center is a point at infinity . [ 4 ]
A parametric equation for the sphere with radius r > 0 {\displaystyle r>0} and center ( x 0 , y 0 , z 0 ) {\displaystyle (x_{0},y_{0},z_{0})} can be parameterized using trigonometric functions .
The symbols used here are the same as those used in spherical coordinates . r is constant, while θ varies from 0 to π and φ {\displaystyle \varphi } varies from 0 to 2 π .
In three dimensions, the volume inside a sphere (that is, the volume of a ball , but classically referred to as the volume of a sphere) is
where r is the radius and d is the diameter of the sphere. Archimedes first derived this formula ( On the Sphere and Cylinder c. 225 BCE) by showing that the volume inside a sphere is twice the volume between the sphere and the circumscribed cylinder of that sphere (having the height and diameter equal to the diameter of the sphere). [ 6 ] This may be proved by inscribing a cone upside down into semi-sphere, noting that the area of a cross section of the cone plus the area of a cross section of the sphere is the same as the area of the cross section of the circumscribing cylinder, and applying Cavalieri's principle . [ 7 ] This formula can also be derived using integral calculus (i.e., disk integration ) to sum the volumes of an infinite number of circular disks of infinitesimally small thickness stacked side by side and centered along the x -axis from x = − r to x = r , assuming the sphere of radius r is centered at the origin.
At any given x , the incremental volume ( δV ) equals the product of the cross-sectional area of the disk at x and its thickness ( δx ):
The total volume is the summation of all incremental volumes:
In the limit as δx approaches zero, [ 8 ] this equation becomes:
At any given x , a right-angled triangle connects x , y and r to the origin; hence, applying the Pythagorean theorem yields:
Using this substitution gives
which can be evaluated to give the result
An alternative formula is found using spherical coordinates , with volume element
so
For most practical purposes, the volume inside a sphere inscribed in a cube can be approximated as 52.4% of the volume of the cube, since V = π / 6 d 3 , where d is the diameter of the sphere and also the length of a side of the cube and π / 6 ≈ 0.5236. For example, a sphere with diameter 1 m has 52.4% the volume of a cube with edge length 1 m, or about 0.524 m 3 .
The surface area of a sphere of radius r is:
Archimedes first derived this formula [ 9 ] from the fact that the projection to the lateral surface of a circumscribed cylinder is area-preserving. [ 10 ] Another approach to obtaining the formula comes from the fact that it equals the derivative of the formula for the volume with respect to r because the total volume inside a sphere of radius r can be thought of as the summation of the surface area of an infinite number of spherical shells of infinitesimal thickness concentrically stacked inside one another from radius 0 to radius r . At infinitesimal thickness the discrepancy between the inner and outer surface area of any given shell is infinitesimal, and the elemental volume at radius r is simply the product of the surface area at radius r and the infinitesimal thickness.
At any given radius r , [ note 1 ] the incremental volume ( δV ) equals the product of the surface area at radius r ( A ( r ) ) and the thickness of a shell ( δr ):
The total volume is the summation of all shell volumes:
In the limit as δr approaches zero [ 8 ] this equation becomes:
Substitute V :
Differentiating both sides of this equation with respect to r yields A as a function of r :
This is generally abbreviated as:
where r is now considered to be the fixed radius of the sphere.
Alternatively, the area element on the sphere is given in spherical coordinates by dA = r 2 sin θ dθ dφ . The total area can thus be obtained by integration :
The sphere has the smallest surface area of all surfaces that enclose a given volume, and it encloses the largest volume among all closed surfaces with a given surface area. [ 11 ] The sphere therefore appears in nature: for example, bubbles and small water drops are roughly spherical because the surface tension locally minimizes surface area.
The surface area relative to the mass of a ball is called the specific surface area and can be expressed from the above stated equations as
where ρ is the density (the ratio of mass to volume).
A sphere can be constructed as the surface formed by rotating a circle one half revolution about any of its diameters ; this is very similar to the traditional definition of a sphere as given in Euclid's Elements . Since a circle is a special type of ellipse , a sphere is a special type of ellipsoid of revolution . Replacing the circle with an ellipse rotated about its major axis , the shape becomes a prolate spheroid ; rotated about the minor axis, an oblate spheroid. [ 12 ]
A sphere is uniquely determined by four points that are not coplanar . More generally, a sphere is uniquely determined by four conditions such as passing through a point, being tangent to a plane, etc. [ 13 ] This property is analogous to the property that three non-collinear points determine a unique circle in a plane.
Consequently, a sphere is uniquely determined by (that is, passes through) a circle and a point not in the plane of that circle.
By examining the common solutions of the equations of two spheres , it can be seen that two spheres intersect in a circle and the plane containing that circle is called the radical plane of the intersecting spheres. [ 14 ] Although the radical plane is a real plane, the circle may be imaginary (the spheres have no real point in common) or consist of a single point (the spheres are tangent at that point). [ 15 ]
The angle between two spheres at a real point of intersection is the dihedral angle determined by the tangent planes to the spheres at that point. Two spheres intersect at the same angle at all points of their circle of intersection. [ 16 ] They intersect at right angles (are orthogonal ) if and only if the square of the distance between their centers is equal to the sum of the squares of their radii. [ 4 ]
If f ( x , y , z ) = 0 and g ( x , y , z ) = 0 are the equations of two distinct spheres then
is also the equation of a sphere for arbitrary values of the parameters s and t . The set of all spheres satisfying this equation is called a pencil of spheres determined by the original two spheres. In this definition a sphere is allowed to be a plane (infinite radius, center at infinity) and if both the original spheres are planes then all the spheres of the pencil are planes, otherwise there is only one plane (the radical plane) in the pencil. [ 4 ]
In their book Geometry and the Imagination , David Hilbert and Stephan Cohn-Vossen describe eleven properties of the sphere and discuss whether these properties uniquely determine the sphere. [ 17 ] Several properties hold for the plane , which can be thought of as a sphere with infinite radius. These properties are:
The basic elements of Euclidean plane geometry are points and lines . On the sphere, points are defined in the usual sense. The analogue of the "line" is the geodesic , which is a great circle ; the defining characteristic of a great circle is that the plane containing all its points also passes through the center of the sphere. Measuring by arc length shows that the shortest path between two points lying on the sphere is the shorter segment of the great circle that includes the points.
Many theorems from classical geometry hold true for spherical geometry as well, but not all do because the sphere fails to satisfy some of classical geometry's postulates , including the parallel postulate . In spherical trigonometry , angles are defined between great circles. Spherical trigonometry differs from ordinary trigonometry in many respects. For example, the sum of the interior angles of a spherical triangle always exceeds 180 degrees. Also, any two similar spherical triangles are congruent.
Any pair of points on a sphere that lie on a straight line through the sphere's center (i.e., the diameter) are called antipodal points – on the sphere, the distance between them is exactly half the length of the circumference. [ note 2 ] Any other (i.e., not antipodal) pair of distinct points on a sphere
Spherical geometry is a form of elliptic geometry , which together with hyperbolic geometry makes up non-Euclidean geometry .
The sphere is a smooth surface with constant Gaussian curvature at each point equal to 1/ r 2 . [ 9 ] As per Gauss's Theorema Egregium , this curvature is independent of the sphere's embedding in 3-dimensional space. Also following from Gauss, a sphere cannot be mapped to a plane while maintaining both areas and angles. Therefore, any map projection introduces some form of distortion.
A sphere of radius r has area element d A = r 2 sin θ d θ d φ {\displaystyle dA=r^{2}\sin \theta \,d\theta \,d\varphi } . This can be found from the volume element in spherical coordinates with r held constant. [ 9 ]
A sphere of any radius centered at zero is an integral surface of the following differential form :
This equation reflects that the position vector and tangent plane at a point are always orthogonal to each other. Furthermore, the outward-facing normal vector is equal to the position vector scaled by 1/r .
In Riemannian geometry , the filling area conjecture states that the hemisphere is the optimal (least area) isometric filling of the Riemannian circle .
Remarkably, it is possible to turn an ordinary sphere inside out in a three-dimensional space with possible self-intersections but without creating any creases, in a process called sphere eversion .
The antipodal quotient of the sphere is the surface called the real projective plane , which can also be thought of as the Northern Hemisphere with antipodal points of the equator identified.
Circles on the sphere are, like circles in the plane, made up of all points a certain distance from a fixed point on the sphere. The intersection of a sphere and a plane is a circle, a point, or empty. [ 18 ] Great circles are the intersection of the sphere with a plane passing through the center of a sphere: others are called small circles.
More complicated surfaces may intersect a sphere in circles, too: the intersection of a sphere with a surface of revolution whose axis contains the center of the sphere (are coaxial ) consists of circles and/or points if not empty. For example, the diagram to the right shows the intersection of a sphere and a cylinder, which consists of two circles. If the cylinder radius were that of the sphere, the intersection would be a single circle. If the cylinder radius were larger than that of the sphere, the intersection would be empty.
In navigation , a loxodrome or rhumb line is a path whose bearing , the angle between its tangent and due North, is constant. Loxodromes project to straight lines under the Mercator projection . Two special cases are the meridians which are aligned directly North–South and parallels which are aligned directly East–West. For any other bearing, a loxodrome spirals infinitely around each pole. For the Earth modeled as a sphere, or for a general sphere given a spherical coordinate system , such a loxodrome is a kind of spherical spiral . [ 19 ]
Another kind of spherical spiral is the Clelia curve, for which the longitude (or azimuth) φ {\displaystyle \varphi } and the colatitude (or polar angle) θ {\displaystyle \theta } are in a linear relationship, φ = c θ {\displaystyle \varphi =c\theta } . Clelia curves project to straight lines under the equirectangular projection . Viviani's curve ( c = 1 {\displaystyle c=1} ) is a special case. Clelia curves approximate the ground track of satellites in polar orbit .
The analog of a conic section on the sphere is a spherical conic , a quartic curve which can be defined in several equivalent ways.
Many theorems relating to planar conic sections also extend to spherical conics.
If a sphere is intersected by another surface, there may be more complicated spherical curves.
The intersection of the sphere with equation x 2 + y 2 + z 2 = r 2 {\displaystyle \;x^{2}+y^{2}+z^{2}=r^{2}\;} and the cylinder with equation ( y − y 0 ) 2 + z 2 = a 2 , y 0 ≠ 0 {\displaystyle \;(y-y_{0})^{2}+z^{2}=a^{2},\;y_{0}\neq 0\;} is not just one or two circles. It is the solution of the non-linear system of equations
(see implicit curve and the diagram)
An ellipsoid is a sphere that has been stretched or compressed in one or more directions. More exactly, it is the image of a sphere under an affine transformation . An ellipsoid bears the same relationship to the sphere that an ellipse does to a circle.
Spheres can be generalized to spaces of any number of dimensions . For any natural number n , an n -sphere, often denoted S n , is the set of points in ( n + 1 )-dimensional Euclidean space that are at a fixed distance r from a central point of that space, where r is, as before, a positive real number. In particular:
Spheres for n > 2 are sometimes called hyperspheres .
The n -sphere of unit radius centered at the origin is denoted S n and is often referred to as "the" n -sphere. The ordinary sphere is a 2-sphere, because it is a 2-dimensional surface which is embedded in 3-dimensional space.
In topology , the n -sphere is an example of a compact topological manifold without boundary . A topological sphere need not be smooth ; if it is smooth, it need not be diffeomorphic to the Euclidean sphere (an exotic sphere ).
The sphere is the inverse image of a one-point set under the continuous function ‖ x ‖ , so it is closed; S n is also bounded, so it is compact by the Heine–Borel theorem .
More generally, in a metric space ( E , d ) , the sphere of center x and radius r > 0 is the set of points y such that d ( x , y ) = r .
If the center is a distinguished point that is considered to be the origin of E , as in a normed space, it is not mentioned in the definition and notation. The same applies for the radius if it is taken to equal one, as in the case of a unit sphere .
Unlike a ball , even a large sphere may be an empty set. For example, in Z n with Euclidean metric , a sphere of radius r is nonempty only if r 2 can be written as sum of n squares of integers .
An octahedron is a sphere in taxicab geometry , and a cube is a sphere in geometry using the Chebyshev distance .
The geometry of the sphere was studied by the Greeks. Euclid's Elements defines the sphere in book XI, discusses various properties of the sphere in book XII, and shows how to inscribe the five regular polyhedra within a sphere in book XIII. Euclid does not include the area and volume of a sphere, only a theorem that the volume of a sphere varies as the third power of its diameter, probably due to Eudoxus of Cnidus . The volume and area formulas were first determined in Archimedes 's On the Sphere and Cylinder by the method of exhaustion . Zenodorus was the first to state that, for a given surface area, the sphere is the solid of maximum volume. [ 3 ]
Archimedes wrote about the problem of dividing a sphere into segments whose volumes are in a given ratio, but did not solve it. A solution by means of the parabola and hyperbola was given by Dionysodorus . [ 20 ] A similar problem – to construct a segment equal in volume to a given segment, and in surface to another segment – was solved later by al-Quhi . [ 3 ] | https://en.wikipedia.org/wiki/Sphere |
In differential topology , sphere eversion is a theoretical process of turning a sphere inside out in a three-dimensional space (the word eversion means "turning inside out"). It is possible to smoothly and continuously turn a sphere inside out in this way (allowing self-intersections of the sphere's surface) without cutting or tearing it or creating any crease . This is surprising, both to non-mathematicians and to those who understand regular homotopy , and can be regarded as a veridical paradox ; that is something that, while being true, on first glance seems false.
More precisely, let
be the standard embedding ; then there is a regular homotopy of immersions
such that ƒ 0 = ƒ and ƒ 1 = − ƒ .
An existence proof for crease-free sphere eversion was first created by Stephen Smale ( 1958 ).
It is difficult to visualize a particular example of such a turning, although some digital animations have been produced that make it somewhat easier. The first example was exhibited through the efforts of several mathematicians, including Arnold S. Shapiro and Bernard Morin , who was blind. On the other hand, it is much easier to prove that such a "turning" exists, and that is what Smale did.
Smale's graduate adviser Raoul Bott at first told Smale that the result was obviously wrong ( Levy 1995 ).
His reasoning was that the degree of the Gauss map must be preserved in such "turning"—in particular it follows that there is no such turning of S 1 in R 2 . But the degrees of the Gauss map for the embeddings f and − f in R 3 are both equal to 1, and do not have opposite sign as one might incorrectly guess. The degree of the Gauss map of all immersions of S 2 in R 3 is 1, so there is no obstacle. The term "veridical paradox" applies perhaps more appropriately at this level: until Smale's work, there was no documented attempt to argue for or against the eversion of S 2 , and later efforts are in hindsight, so there never was a historical paradox associated with sphere eversion, only an appreciation of the subtleties in visualizing it by those confronting the idea for the first time.
See h -principle for further generalizations.
Smale's original proof was indirect: he identified (regular homotopy) classes of immersions of spheres with a homotopy group of the Stiefel manifold . Since the homotopy group that corresponds to immersions of S 2 {\displaystyle S^{2}} in R 3 {\displaystyle \mathbb {R} ^{3}} vanishes, the standard embedding and the inside-out one must be regular homotopic. In principle the proof can be unwound to produce an explicit regular homotopy, but this is not easy to do.
There are several ways of producing explicit examples and mathematical visualization : | https://en.wikipedia.org/wiki/Sphere_eversion |
In geometry, sphere packing in a cube is a three-dimensional sphere packing problem with the objective of packing spheres inside a cube . It is the three-dimensional equivalent of the circle packing in a square problem in two dimensions. The problem consists of determining the optimal packing of a given number of spheres inside the cube.
Gensane [ 1 ] traces the origin of the problem to work of J. Schaer in the mid-1960s. [ 2 ] Reviewing Schaer's work, H. S. M. Coxeter writes that he "proves that the arrangements for k = 2 , 3 , 4 , 8 , 9 {\displaystyle k=2,3,4,8,9} are what anyone would have guessed". [ 3 ] The cases k = 7 {\displaystyle k=7} and k = 10 {\displaystyle k=10} were resolved in later work of Schaer, [ 4 ] and a packing for k = 14 {\displaystyle k=14} was proven optimal by Joós. [ 5 ] For larger numbers of spheres, all results so far are conjectural. [ 1 ] In a 1971 paper, Goldberg found many non-optimal packings for other values of k {\displaystyle k} and three that may still be optimal. [ 6 ] Gensane improved the rest of Goldberg's packings and found good packings for up to 32 spheres. [ 1 ]
Goldberg also conjectured that for numbers of spheres of the form k = ⌊ p 3 / 2 ⌋ {\displaystyle k=\lfloor p^{3}/2\rfloor } , the optimal packing of spheres in a cube is a form of cubic close-packing . However, omitting as few as two spheres from this number allows a different and tighter packing. [ 7 ]
This geometry-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Sphere_packing_in_a_cube |
Sphere packing in a cylinder is a three-dimensional packing problem with the objective of packing a given number of identical spheres inside a cylinder of specified diameter and length. For cylinders with diameters on the same order of magnitude as the spheres, such packings result in what are called columnar structures .
These problems are studied extensively in the context of biology , nanoscience , materials science , and so forth due to the analogous assembly of small particles (like cells and atoms ) into cylindrical crystalline structures .
The book "Columnar Structures of Spheres: Fundamentals and Applications" [ 1 ] serves as a notable contributions to this field of study. Authored by Winkelmann and Chan, the book reviews theoretical foundations and practical applications of densely packed spheres within cylindrical confinements.
Columnar structures appear in various research fields on a broad range of length scales from metres down to the nanoscale. On the largest scale, such structures can be found in botany where seeds of a plant assemble around the stem. On a smaller scale bubbles of equal size crystallise to columnar foam structures when confined in a glass tube. In nanoscience such structures can be found in man-made objects which are on length scales from a micron to the nanoscale.
Columnar structures were first studied in botany due to their diverse appearances in plants. [ 2 ] D'Arcy Thompson analysed such arrangement of plant parts around the stem in his book " On Growth and Form " (1917). But they are also of interest in other biological areas, including bacteria, [ 3 ] viruses, [ 4 ] microtubules , [ 5 ] and the notochord of the zebra fish . [ 6 ]
One of the largest flowers where the berries arrange in a regular cylindrical form is the titan arum . This flower can be up to 3m in height and is natively solely found in western Sumatra and western Java.
On smaller length scales, the berries of the Arum maculatum form a columnar structure in autumn. Its berries are similar to that of the corpse flower, since the titan arum is its larger relative. However, the cuckoo-pint is much smaller in height (height ≈ 20 cm). The berry arrangement varies with the stem to berry size.
Another plant that can be found in many gardens of residential areas is the Australian bottlebrush . It assembles its seed capsules around a branch of the plant. The structure depends on the seed capsule size to branch size.
A further occurrence of ordered columnar arrangement on the macroscale are foam structures confined inside a glass tube. They can be realised experimentally with equal-sized soap bubbles inside a glass tube, produced by blowing air of constant gas flow through a needle dipped in a surfactant solution. [ 7 ] By putting the resulting foam column under forced drainage (feeding it with surfactant solution from the top), the foam can be adjusted to either a dry (bubbles shaped as polyhedrons ) or wet (spherical bubbles) structure. [ 8 ]
Due to this simple experimental set-up, many columnar structures have been discovered and investigated in the context of foams with experiments as well as simulation. Many simulations have been carried out using the Surface Evolver to investigate dry structure or the hard sphere model for the wet limit where the bubbles are spherical.
In the zigzag structure the bubbles are stacked on top of each other in a continuous w-shape. For this particular structure a moving interface with increasing liquid fraction was reported by Hutzler et al. in 1997. [ 9 ] This included an unexpected 180° twist interface, whose explanation is still lacking.
The first experimental observation of a line-slip structure was discovered by Winkelmann et al. in a system of bubbles. [ 10 ]
Further discovered structures include complex structures with internal spheres/foam cells. Some dry foam structures with interior cells were found to consist of a chain of pentagonal dodecahedra or Kelvin cells in the centre of the tube. [ 11 ] For many more arrangements of this type, it was observed that the outside bubble layer is ordered, with each internal layer resembling a different, simpler columnar structure by using X-ray tomography . [ 7 ]
Columnar structures have also been studied intensively in the context of nanotubes . Their physical or chemical properties can be altered by trapping identical particles inside them. [ 12 ] [ 13 ] [ 14 ] These are usually done by self-assembling fullerenes such as C60 , C70, or C78 into carbon nanotubes, [ 12 ] but also boron nitride nanotubes [ 15 ]
Such structures also assemble when particles are coated on the surface of a spherocylinder as in the context of pharmaceutical research. Lazáro et al. examined the morphologies of virus capsid proteins self-assembled around metal nanorods. [ 16 ] Drug particles were coated as densely as possible on a spherocylinder to provide the best medical treatment.
Wu et al. built rods of the size of several microns. These microrods are created by densely packing silica colloidal particles inside cylindrical pores. By solidifying the assembled structures the microrods were imaged and examined using scanning electron microscopy (SEM). [ 17 ]
Columnar arrangements are also investigated as a possible candidate of optical metamaterials (i.e. materials with a negative refractive index) which find applications in super lenses [ 18 ] or optical cloaking. [ 19 ] Tanjeem et al. are constructing such a resonator by self-assembling nanospheres on the surface of the cylinder. [ 20 ] [ 21 ] The nanospheres are suspended in an SDS solution together with a cylinder of diameter D {\textstyle D} , much larger than the diameter of the nanospheres d {\displaystyle d} ( D / d ≈ 3 to 5 {\textstyle D/d\approx 3{\text{ to }}5} ). The nanospheres then stick to the surface of the cylinders by a depletion force .
The most common way of classifying ordered columnar structures uses the phyllotactic notation , adopted from botany. It is used to describe arrangements of leaves of a plant, pine cones, or pineapples, but also planar patterns of florets in a sunflower head. While the arrangement in the former are cylindrical, the spirals in the latter are arranged on a disk. For columnar structures phyllotaxis in the context of cylindrical structures is adopted.
The phyllotactic notation describes such structures by a triplet of positive integers ( l = m + n , m , n ) {\displaystyle (l=m+n,m,n)} with l ≥ m ≥ n {\textstyle l\geq m\geq n} . Each number l {\textstyle l} , m {\displaystyle m} , and n {\textstyle n} describes a family of spirals in the 3-dimensional packing. They count the number of spirals in each direction until the spiral repeats. This notation, however, only applies to triangular lattices and is therefore restricted to the ordered structures without internal spheres.
Ordered columnar structures without internal spheres are categorised into two separate classes: uniform and line-slip structures. For each structure that can be identified with the triplet ( l , m , n ) {\textstyle (l,m,n)} , there exist a uniform structure and at least one line slip.
A uniform structure is identified by each sphere having the same number of contacting neighbours. [ 22 ] [ 1 ] This gives each sphere an identical neighbourhood. In the example image on the side each sphere has six neighbouring contacts.
The number of contacts is best visualised in the rolled-out contact network. It is created by rolling out the contact network into a plane of height z {\textstyle z} and azimuthal angle θ {\textstyle \theta } of each sphere. For a uniform structure such as the one in the example image, this leads to a regular hexagonal lattice . Each dot in this pattern represents a sphere of the packing and each line a contact between adjacent spheres.
For all uniform structures above a diameter ratio of D / d > 2.0 {\displaystyle D/d>2.0} , the regular hexagonal lattice is its characterising feature since this lattice type has the maximum number of contacts. [ 22 ] [ 1 ] For different uniform structures ( l , m , n ) {\displaystyle (l,m,n)} the rolled-out contact pattern only varies by a rotation in the z - θ {\textstyle z{\text{-}}\theta } plane. Each uniform structure is thus distinguished by its periodicity vector V {\textstyle V} , which is defined by the phyllotactic triplet ( l , m , n ) {\displaystyle (l,m,n)} .
For each uniform structure, there also exists a related but different structure, called a line-slip arrangement. [ 22 ] [ 1 ]
The differences between uniform and line-slip structures are marginal and difficult to spot from images of the sphere packings. However, by comparing their rolled-out contact networks, one can spot that certain lines (which represent contacts) are missing.
All spheres in a uniform structure have the same number of contacts, but the number of contacts for spheres in a line slip may differ from sphere to sphere. For the example line slip in the image on the right side, some spheres count five and others six contacts. Thus a line slip structure is characterised by these gaps or loss of contacts.
Such a structure is termed line slip because the losses of contacts occur along a line in the rolled-out contact network. It was first identified by Picket et al. , but not termed line slip. [ 23 ]
The direction, in which the loss of contacts occur can be denoted in the phyllotactic notation ( l , m , n ) {\textstyle (l,m,n)} , since each number represents one of the lattice vectors in the hexagonal lattice. [ 22 ] [ 1 ] This is usually indicated by a bold number.
By shearing the row of spheres below the loss of contact against a row above the loss of contact, one can regenerate two uniform structures related to this line slip. Thus, each line slip is related to two adjacent uniform structures, one at a higher and one at a lower diameter ratio D / d {\textstyle D/d} . [ 22 ] [ 1 ] [ 24 ]
Winkelmann et al. were the first to experimentally realise such a structure using soap bubbles in a system of deformable spheres. [ 10 ]
Columnar structures arise naturally in the context of dense hard sphere packings inside a cylinder. Mughal et al. studied such packings using simulated annealing up to the diameter ratio of D / d = 2.873 {\textstyle D/d=2.873} for cylinder diameter D {\textstyle D} to sphere diameter d {\textstyle d} . [ 24 ] This includes some structures with internal spheres that are not in contact with the cylinder wall.
They calculated the packing fraction for all these structures as a function of the diameter ratio. At the peaks of this curve lie the uniform structures. In-between these discrete diameter ratios are the line slips at a lower packing density. Their packing fraction is significantly smaller than that of an unconfined lattice packing such as fcc , bcc, or hcp due to the free volume left by the cylindrical confinement.
The rich variety of such ordered structures can also be obtained by sequential depositioning the spheres into the cylinder. [ 25 ] Chan reproduced all dense sphere packings up to D / d < 2.7013 {\textstyle D/d<2.7013} using an algorithm, in which the spheres are placed sequentially dropped inside the cylinder.
Mughal et al. also discovered that such structures can be related to disk packings on a surface of a cylinder. [ 24 ] The contact network of both packings are identical. For both packing types, it was found that different uniform structures are connected with each other by line slips. [ 24 ]
Fu et al. extended this work to higher diameter ratios D / d < 4.0 {\textstyle D/d<4.0} using linear programming and discovered 17 new dense structures with internal spheres that are not in contact with the cylinder wall. [ 26 ]
A similar variety of dense crystalline structures have also been discovered for columnar packings of spheroids through Monte Carlo simulations . [ 27 ] Such packings include achiral structures with specific spheroid orientations and chiral helical structures with rotating spheroid orientations.
A further dynamic method to assemble such structures was introduced by Lee et al . [ 28 ] Here, polymeric beads are placed together with a fluid of higher density inside a rotating lathe .
When the lathe is static, the beads float on top of the liquid. With increasing rotational speed, the centripetal force then pushes the fluid outwards and the beads toward the central axis. Hence, the beads are essentially confined by a potential given by the rotational energy E rot = 1 2 m R 2 ω 2 , {\displaystyle E_{\text{rot}}={\frac {1}{2}}mR^{2}\omega ^{2},} where m {\textstyle m} is the mass of the beads, R {\textstyle R} the distance from the central axis, and ω {\textstyle \omega } the rotational speed. Due to the R 2 {\textstyle R^{2}} proportionality, the confining potential resembles that of a cylindrical harmonic oscillator .
Depending on number of spheres and rotational speed, a variety of ordered structures that are comparable to the dense sphere packings were discovered.
A comprehensive theory to this experiment was developed by Winkelmann et al. [ 29 ] It is based on analytic energy calculations using a generic sphere model and predicts peritectoid structure transitions. | https://en.wikipedia.org/wiki/Sphere_packing_in_a_cylinder |
In Riemannian geometry , the sphere theorem , also known as the quarter-pinched sphere theorem , strongly restricts the topology of manifolds admitting metrics with a particular curvature bound. The precise statement of the theorem is as follows. If M {\displaystyle M} is a complete , simply-connected , n -dimensional Riemannian manifold with sectional curvature taking values in the interval ( 1 , 4 ] {\displaystyle (1,4]} then M {\displaystyle M} is homeomorphic to the n -sphere . (To be precise, we mean the sectional curvature of every tangent 2-plane at each point must lie in ( 1 , 4 ] {\displaystyle (1,4]} .) Another way of stating the result is that if M {\displaystyle M} is not homeomorphic to the sphere, then it is impossible to put a metric on M {\displaystyle M} with quarter-pinched curvature.
Note that the conclusion is false if the sectional curvatures are allowed to take values in the closed interval [ 1 , 4 ] {\displaystyle [1,4]} . The standard counterexample is complex projective space with the Fubini–Study metric ; sectional curvatures of this metric take on values between 1 {\displaystyle 1} and 4 {\displaystyle 4} , with endpoints included. Other counterexamples may be found among the rank one symmetric spaces .
The original proof of the sphere theorem did not conclude that M {\displaystyle M} was necessarily diffeomorphic to the n -sphere. This complication is because spheres in higher dimensions admit smooth structures that are not diffeomorphic. (For more information, see the article on exotic spheres .) However, in 2007 Simon Brendle and Richard Schoen utilized Ricci flow to prove that with the above hypotheses, M {\displaystyle M} is necessarily diffeomorphic to the n -sphere with its standard smooth structure. Moreover, the proof of Brendle and Schoen only uses the weaker assumption of pointwise rather than global pinching. This result is known as the differentiable sphere theorem .
Heinz Hopf conjectured that a simply connected manifold with pinched sectional curvature is a sphere. [ 1 ] In 1951, Harry Rauch showed that a simply connected manifold with curvature in [ 3 / 4 , 1 ] {\displaystyle [3/4,1]} is homeomorphic to a sphere. [ 2 ] In 1960, both Marcel Berger and Wilhelm Klingenberg proved the topological version of the sphere theorem with the optimal pinching constant. [ 3 ] [ 4 ] Berger discusses the history of the theorem in his book A Panoramic View of Riemannian Geometry , originally published in 2003. [ 5 ] | https://en.wikipedia.org/wiki/Sphere_theorem |
In mathematics, in the topology of 3-manifolds , the sphere theorem of Christos Papakyriakopoulos ( 1957 ) gives conditions for elements of the second homotopy group of a 3-manifold to be represented by embedded spheres.
One example is the following:
Let M {\displaystyle M} be an orientable 3-manifold such that π 2 ( M ) {\displaystyle \pi _{2}(M)} is not the trivial group. Then there exists a non-zero element of π 2 ( M ) {\displaystyle \pi _{2}(M)} having a representative that is an embedding S 2 → M {\displaystyle S^{2}\to M} .
The proof of this version of the theorem can be based on transversality methods, see Jean-Loïc Batude ( 1971 ).
Another more general version (also called the projective plane theorem, and due to David B. A. Epstein ) is:
Let M {\displaystyle M} be any 3-manifold and N {\displaystyle N} a π 1 ( M ) {\displaystyle \pi _{1}(M)} - invariant subgroup of π 2 ( M ) {\displaystyle \pi _{2}(M)} . If f : S 2 → M {\displaystyle f\colon S^{2}\to M} is a general position map such that [ f ] ∉ N {\displaystyle [f]\notin N} and U {\displaystyle U} is any neighborhood of the singular set Σ ( f ) {\displaystyle \Sigma (f)} , then there is a map g : S 2 → M {\displaystyle g\colon S^{2}\to M} satisfying
quoted in ( Hempel 1976 , p. 54). | https://en.wikipedia.org/wiki/Sphere_theorem_(3-manifolds) |
The spherical Bernstein's problem is a possible generalization of the original Bernstein's problem in the field of global differential geometry , first proposed by Shiing-Shen Chern in 1969, and then later in 1970, during his plenary address at the International Congress of Mathematicians in Nice .
Are the equators in S n + 1 {\displaystyle \mathbb {S} ^{n+1}} the only smooth embedded minimal hypersurfaces which are topological n {\displaystyle n} -dimensional spheres?
Additionally, the spherical Bernstein's problem , while itself a generalization of the original Bernstein's problem, can, too, be generalized further by replacing the ambient space S n + 1 {\displaystyle \mathbb {S} ^{n+1}} by a simply-connected, compact symmetric space. Some results in this direction are due to Wu-Chung Hsiang and Wu-Yi Hsiang work.
Below are two alternative ways to express the problem:
Let the ( n − 1) sphere be embedded as a minimal hypersurface in S n {\displaystyle S^{n}} (1). Is it necessarily an equator?
By the Almgren – Calabi theorem, it's true when n = 3 (or n = 2 for the 1st formulation).
Wu-Chung Hsiang proved it for n ∈ {4, 5, 6, 7, 8, 10, 12, 14} (or n ∈ {3, 4, 5, 6, 7, 9, 11, 13}, respectively)
In 1987, Per Tomter proved it for all even n (or all odd n , respectively).
Thus, it only remains unknown for all odd n ≥ 9 (or all even n ≥ 8, respectively)
Is it true that an embedded, minimal hypersphere inside the Euclidean n {\displaystyle n} -sphere is
necessarily an equator?
Geometrically, the problem is analogous to the following problem:
Is the local topology at an isolated singular point of a minimal hypersurface necessarily different from that of a disc?
For example, the affirmative answer for spherical Bernstein problem when n = 3 is equivalent to the fact that the local topology at an isolated singular point of any minimal hypersurface in an arbitrary Riemannian 4-manifold must be different from that of a disc. | https://en.wikipedia.org/wiki/Spherical_Bernstein's_problem |
In organic chemistry , spherical aromaticity is formally used to describe an unusually stable nature of some spherical compounds such as fullerenes and polyhedral boranes .
In 2000, Andreas Hirsch and coworkers in Erlangen , Germany , formulated a rule to determine when a spherical compound would be aromatic . They found that those with 2( n +1) 2 π- electrons could display aromatic properties, as spherical molecular orbitals are filled when there are 2( n +1) 2 π-electrons for some positive integer n . For example, in buckminsterfullerene (C 60 ) this happens for the species C 60 10+ , which has 50 π-electrons: 50/2 = 25, which is a perfect square . [ 1 ]
In 2011, Jordi Poater and Miquel Solà expanded Hirsch's rule to open-shell spherical compounds, which have unfilled outer shells but are still aromatic. They found that spherical compounds with 2 n 2 +2 n +1 π- electrons with spin S = (n + 1/2) would also display aromatic properties, sometimes more aromatic than comparable closed-shell species. This corresponds to the outer shell being half-filled, [ 2 ] and is similar to Baird's rule . For example buckminsterfullerene with one additional electron, (C 60 1– ) is aromatic, with S = 11/2 and a bond-length alternation of 0.2 pm . | https://en.wikipedia.org/wiki/Spherical_aromaticity |
In pure and applied mathematics , particularly quantum mechanics and computer graphics and their applications, a spherical basis is the basis used to express spherical tensors . [ definition needed ] The spherical basis closely relates to the description of angular momentum in quantum mechanics and spherical harmonic functions.
While spherical polar coordinates are one orthogonal coordinate system for expressing vectors and tensors using polar and azimuthal angles and radial distance, the spherical basis are constructed from the standard basis and use complex numbers .
A vector A in 3D Euclidean space R 3 can be expressed in the familiar Cartesian coordinate system in the standard basis e x , e y , e z , and coordinates A x , A y , A z :
or any other coordinate system with associated basis set of vectors. From this extend the scalars to allow multiplication by complex numbers, so that we are now working in C 3 {\displaystyle \mathbb {C} ^{3}} rather than R 3 {\displaystyle \mathbb {R} ^{3}} .
In the spherical bases denoted e + , e − , e 0 , and associated coordinates with respect to this basis, denoted A + , A − , A 0 , the vector A is:
where the spherical basis vectors can be defined in terms of the Cartesian basis using complex -valued coefficients in the xy plane: [ 1 ]
in which i {\displaystyle i} denotes the imaginary unit , and one normal to the plane in the z direction:
The inverse relations are:
While giving a basis in a 3-dimensional space is a valid definition for a spherical tensor, it only covers the case for when the rank k {\displaystyle k} is 1. For higher ranks, one may use either the commutator, or rotation definition of a spherical tensor. The commutator definition is given below, any operator T q ( k ) {\displaystyle T_{q}^{(k)}} that satisfies the following relations is a spherical tensor: [ J ± , T q ( k ) ] = ℏ ( k ∓ q ) ( k ± q + 1 ) T q ± 1 ( k ) {\displaystyle [J_{\pm },T_{q}^{(k)}]=\hbar {\sqrt {(k\mp q)(k\pm q+1)}}T_{q\pm 1}^{(k)}} [ J z , T q ( k ) ] = ℏ q T q ( k ) {\displaystyle [J_{z},T_{q}^{(k)}]=\hbar qT_{q}^{(k)}}
Analogously to how the spherical harmonics transform under a rotation, a general spherical tensor transforms as follows, when the states transform under the unitary Wigner D-matrix D ( R ) {\displaystyle {\mathcal {D}}(R)} , where R is a (3×3 rotation) group element in SO(3) . That is, these matrices represent the rotation group elements. With the help of its Lie algebra , one can show these two definitions are equivalent.
For the spherical basis, the coordinates are complex-valued numbers A + , A 0 , A − , and can be found by substitution of ( 3B ) into ( 1 ), or directly calculated from the inner product ⟨, ⟩ ( 5 ):
with inverse relations:
In general, for two vectors with complex coefficients in the same real-valued orthonormal basis e i , with the property e i · e j = δ ij , the inner product is:
where · is the usual dot product and the complex conjugate * must be used to keep the magnitude (or "norm") of the vector positive definite .
The spherical basis is an orthonormal basis , since the inner product ⟨, ⟩ ( 5 ) of every pair vanishes meaning the basis vectors are all mutually orthogonal :
and each basis vector is a unit vector :
hence the need for the normalizing factors of 1 / 2 {\displaystyle 1/\!{\sqrt {2}}} .
The defining relations ( 3A ) can be summarized by a transformation matrix U :
with inverse:
It can be seen that U is a unitary matrix , in other words its Hermitian conjugate U † ( complex conjugate and matrix transpose ) is also the inverse matrix U −1 .
For the coordinates:
and inverse:
Taking cross products of the spherical basis vectors, we find an obvious relation:
where q is a placeholder for +, −, 0, and two less obvious relations:
The inner product between two vectors A and B in the spherical basis follows from the above definition of the inner product: | https://en.wikipedia.org/wiki/Spherical_basis |
In spherical geometry , a spherical circle (often shortened to circle ) is the locus of points on a sphere at constant spherical distance (the spherical radius ) from a given point on the sphere (the pole or spherical center ). It is a curve of constant geodesic curvature relative to the sphere, analogous to a line or circle in the Euclidean plane ; the curves analogous to straight lines are called great circles , and the curves analogous to planar circles are called small circles or lesser circles . If the sphere is embedded in three-dimensional Euclidean space , its circles are the intersections of the sphere with planes , and the great circles are intersections with planes passing through the center of the sphere.
A spherical circle with zero geodesic curvature is called a great circle , and is a geodesic analogous to a straight line in the plane. A great circle separates the sphere into two equal hemispheres , each with the great circle as its boundary. If a great circle passes through a point on the sphere, it also passes through the antipodal point (the unique furthest other point on the sphere). For any pair of distinct non-antipodal points, a unique great circle passes through both. Any two points on a great circle separate it into two arcs analogous to line segments in the plane; the shorter is called the minor arc and is the shortest path between the points, and the longer is called the major arc .
A circle with non-zero geodesic curvature is called a small circle , and is analogous to a circle in the plane. A small circle separates the sphere into two spherical disks or spherical caps , each with the circle as its boundary. For any triple of distinct non-antipodal points a unique small circle passes through all three. Any two points on the small circle separate it into two arcs , analogous to circular arcs in the plane.
Every circle has two antipodal poles (or centers) intrinsic to the sphere. A great circle is equidistant to its poles, while a small circle is closer to one pole than the other. Concentric circles are sometimes called parallels , because they each have constant distance to each-other, and in particular to their concentric great circle, and are in that sense analogous to parallel lines in the plane.
If the sphere is isometrically embedded in Euclidean space , the sphere's intersection with a plane is a circle, which can be interpreted extrinsically to the sphere as a Euclidean circle: a locus of points in the plane at a constant Euclidean distance (the extrinsic radius ) from a point in the plane (the extrinsic center ). A great circle lies on a plane passing through the center of the sphere, so its extrinsic radius is equal to the radius of the sphere itself, and its extrinsic center is the sphere's center. A small circle lies on a plane not passing through the sphere's center, so its extrinsic radius is smaller than that of the sphere and its extrinsic center is an arbitrary point in the interior of the sphere. Parallel planes cut the sphere into parallel (concentric) small circles; the pair of parallel planes tangent to the sphere are tangent at the poles of these circles, and the diameter through these poles, passing through the sphere's center and perpendicular to the parallel planes, is called the axis of the parallel circles.
The sphere's intersection with a second sphere is also a circle, and the sphere's intersection with a concentric right circular cylinder or right circular cone is a pair of antipodal circles.
In the geographic coordinate system on a globe, the parallels of latitude are small circles, with the Equator the only great circle. By contrast, all meridians of longitude , paired with their opposite meridian in the other hemisphere , form great circles.
This geometry-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Spherical_circle |
In geometry and coding theory , a spherical code with parameters ( n , N , t ) is a set of N points on the unit hypersphere in n dimensions for which the dot product of unit vectors from the origin to any two points is less than or equal to t . The kissing number problem may be stated as the problem of finding the maximal N for a given n for which a spherical code with parameters ( n , N ,1/2) exists. The Tammes problem may be stated as the problem of finding a spherical code with minimal t for given n and N .
This geometry-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Spherical_code |
The spherical cow is a humorous metaphor for highly simplified scientific models of complex phenomena. [ 1 ] [ 2 ] [ 3 ] [ 4 ] Originating in theoretical physics , the metaphor refers to some scientific tendencies to develop toy models that reduce a problem to the simplest form imaginable, making calculations more feasible, even if the simplification hinders the model's application to reality.
The phrase comes from a joke that spoofs the simplifying assumptions sometimes used in theoretical physics. [ 5 ]
Milk production at a dairy farm was low, so the farmer wrote to the local university, asking for help from academia. A multidisciplinary team of professors was assembled, headed by a theoretical physicist, and two weeks of intensive on-site investigation took place. The scholars then returned to the university, notebooks crammed with data, where the task of writing the report was left to the team leader. Shortly thereafter the physicist returned to the farm, saying to the farmer, "I have the solution, but it works only in the case of spherical cows in a vacuum."
John Harte , who received his Ph.D. from the University of Wisconsin in 1965, [ 6 ] reported that he first heard the joke as a graduate student . [ 7 ] One of the earliest published references is in a 1970 article by Arthur O. Williams Jr. of Brown University , who described it as "a professional joke that circulated among scientists a few years ago". [ 8 ]
The story is told in many variants, [ 9 ] including a joke about a physicist who said he could predict the winner of any race provided it involved spherical horses moving through a vacuum. [ 10 ] [ 11 ] A 1973 letter to the editor in the journal Science describes the "famous story" about a physicist whose solution to a poultry farm's egg-production problems began with "Postulate a spherical chicken". [ 12 ]
The concept is familiar enough that the phrase is sometimes used as shorthand for the entire issue of proper modeling. For example, Consider a Spherical Cow is a 1985 book about problem solving using simplified models. [ 7 ] A 2015 paper on the systemic errors introduced by simplifying assumptions about spherical symmetries in galactic dark-matter haloes was titled "Milking the spherical cow – on aspherical dynamics in spherical coordinates". [ 13 ]
References to the joke appear even outside the field of scientific modeling. "Spherical Cow" was chosen as the code name for the Fedora 18 Linux distribution . [ 14 ] In the sitcom The Big Bang Theory , a joke is told by Dr. Leonard Hofstadter with the punchline mentioning "spherical chickens in a vacuum", in " The Cooper-Hofstadter Polarization " episode. [ 15 ] In the space gravity simulator educational video game Universe Sandbox , a spherical cow was added as a user-placeable object in March 2023. [ 16 ] | https://en.wikipedia.org/wiki/Spherical_cow |
A spherical roller bearing is a rolling-element bearing that permits rotation with low friction, and permits angular misalignment. Typically these bearings support a rotating shaft in the bore of the inner ring that may be misaligned in respect to the outer ring. The misalignment is possible due to the spherical internal shape of the outer ring and spherical rollers. [ 1 ] Despite what their name may imply, spherical roller bearings are not truly spherical in shape. The rolling elements of spherical roller bearings are mainly cylindrical in shape, but have a (barrel like) profile that makes them appear like cylinders that have been slightly over-inflated [ 2 ] (i.e. like a barrel).
Spherical roller bearings consist of an inner ring with two raceways inclined at an angle to the bearing axis, an outer ring with a common spherical raceway, spherical rollers, cages and, in certain designs, also internal guide rings or center rings. These bearings can also be sealed.
The spherical roller bearing was invented by engineer Arvid Palmgren [ 3 ] and was introduced on the market 1919 by SKF . [ 4 ] The design of the bearing that Arvid Palmgren invented is similar to the design that is still in use in modern machines.
Most spherical roller bearings are designed with two rows of rollers, allowing them to take very heavy radial loads and heavy axial loads. There are also designs with one row of rollers, suitable for lower radial loads and virtually no axial load. These are also called "barrel roller bearings" or "Tonnenlager" and are typically available in the 202- and 203-series. [ 5 ]
The internal design of the bearing is not standardised by ISO, so it varies between different manufacturers and different series. Some features that may or may not exist in different bearings are:
External dimensions of spherical roller bearings are standardised by ISO in the standard ISO 15:1998. [ 6 ] Some of the common series of spherical roller bearings are: 213, 222, 223, 230, 231, 232, 238, 239, 240, 241, 248, 249. [ 7 ]
Bearing rings and rolling elements can be made of a number of different materials, but the most common is "chrome steel", (high carbon chromium) a material with approximately 1.5% chrome content. Such "chrome steel" has been standardized by a number of authorities, and there are therefore a number of similar materials, such as: AISI 52100 (USA), 100CR6 (Germany), SUJ2 (Japan) and GCR15 (China). [ 8 ]
Some common materials for bearing cages: [ 9 ]
The choice of material is mainly done by the manufacturing volume and method. For large-volume bearings, cages are often of stamped sheet-metal or injection molded polyamide, whereas low volume manufacturers or low volume series often have cages of machined brass or machined steel. For some specific applications, special material for coating (e.g. PTFE coated cylindrical bore for vibratory applications) is adopted.
Some manufacturers of spherical roller bearings are SKF , Schaeffler , Timken Company , NSK Ltd. , NTN Corporation and JTEKT . [ citation needed ]
Since SKF introduced the spherical roller bearing in 1919, spherical roller bearings have purposefully been refined through the decades to improve carrying capacity and to reduce operational friction. This has been possible by playing with a palette of parameters such as materials, internal geometry, tolerance and lubricant. Nowadays, spherical roller bearing manufacturers are striving to refine the bearing knowledge towards more environmentally-friendly and energy-efficient solutions.
Spherical bearings are used in countless industrial applications where there are heavy loads, moderate speeds and possibly misalignment. Some common application areas are: [ 4 ] [ 10 ] | https://en.wikipedia.org/wiki/Spherical_roller_bearing |
A spherical roller thrust bearing is a rolling-element bearing of thrust type that permits rotation with low friction , and permits angular misalignment. The bearing is designed to take radial loads, and heavy axial loads in one direction. Typically these bearings support a rotating shaft in the bore of the shaft washer that may be misaligned in respect to the housing washer. The misalignment is possible due to the spherical internal shape of the house washer. [ 1 ]
Spherical roller thrust bearings consist of a shaft washer (for radial bearings often called "inner ring"), a housing washer (for radial bearings often called "outer ring"), asymmetrical rollers and a cage. [ 2 ] There are also bearing units available that can take axial loads in two directions.
The spherical roller thrust bearing was introduced by SKF in 1939. [ 3 ] The design of the early bearings is similar to the design that is still in use in modern machines.
The internal design of the bearing is not standardised by ISO, so it varies between different manufacturers and different series. Some of the design parameters are:
The spherical roller thrust bearings have the highest load rating density of all thrust bearings. [ 4 ]
External dimensions of spherical roller bearings are standardised by ISO in the standard ISO 104:2015. [ 5 ] Some common series of spherical roller bearings are:
Bearing rings and rolling elements can be made of a number of different materials, but the most common is "chrome steel", a material with approximately 1.5% chrome content. Such "chrome steel" has been standardized by a number of authorities, and there are therefore a number of similar materials, such as: AISI 52100 (USA), 100CR6 (Germany), SUJ2 (Japan) and GCR15 (China). [ 6 ]
Some common materials for bearing cages: [ 7 ]
The choice of material is mainly done by the manufacturing volume and method. For large-volume bearings, cages are often of stamped sheet-metal, whereas low volume series often have cages of machined brass or machined steel.
Some manufacturers of spherical roller bearings are SKF , Schaeffler , Timken Company and NSK Ltd.
Spherical roller thrust bearings are used in industrial applications, where there are heavy axial loads, moderate speeds and possibly misalignment.
Some common application areas are: [ 4 ] | https://en.wikipedia.org/wiki/Spherical_roller_thrust_bearing |
In geometry , a spherical sector , [ 1 ] also known as a spherical cone , [ 2 ] is a portion of a sphere or of a ball defined by a conical boundary with apex at the center of the sphere. It can be described as the union of a spherical cap and the cone formed by the center of the sphere and the base of the cap. It is the three-dimensional analogue of the sector of a circle .
If the radius of the sphere is denoted by r and the height of the cap by h , the volume of the spherical sector is V = 2 π r 2 h 3 . {\displaystyle V={\frac {2\pi r^{2}h}{3}}\,.}
This may also be written as V = 2 π r 3 3 ( 1 − cos φ ) , {\displaystyle V={\frac {2\pi r^{3}}{3}}(1-\cos \varphi )\,,} where φ is half the cone aperture angle, i.e., φ is the angle between the rim of the cap and the axis direction to the middle of the cap as seen from the sphere center. The limiting case is for φ approaching 180 degrees, which then describes a complete sphere.
The height, h is given by h = r ( 1 − cos φ ) . {\displaystyle h=r(1-\cos \varphi )\,.}
The volume V of the sector is related to the area A of the cap by: V = r A 3 . {\displaystyle V={\frac {rA}{3}}\,.}
The curved surface area of the spherical sector (on the surface of the sphere, excluding the cone surface) is A = 2 π r h . {\displaystyle A=2\pi rh\,.}
It is also A = Ω r 2 {\displaystyle A=\Omega r^{2}} where Ω is the solid angle of the spherical sector in steradians , the SI unit of solid angle. One steradian is defined as the solid angle subtended by a cap area of A = r 2 .
The volume can be calculated by integrating the differential volume element d V = ρ 2 sin ϕ d ρ d ϕ d θ {\displaystyle dV=\rho ^{2}\sin \phi \,d\rho \,d\phi \,d\theta } over the volume of the spherical sector, V = ∫ 0 2 π ∫ 0 φ ∫ 0 r ρ 2 sin ϕ d ρ d ϕ d θ = ∫ 0 2 π d θ ∫ 0 φ sin ϕ d ϕ ∫ 0 r ρ 2 d ρ = 2 π r 3 3 ( 1 − cos φ ) , {\displaystyle V=\int _{0}^{2\pi }\int _{0}^{\varphi }\int _{0}^{r}\rho ^{2}\sin \phi \,d\rho \,d\phi \,d\theta =\int _{0}^{2\pi }d\theta \int _{0}^{\varphi }\sin \phi \,d\phi \int _{0}^{r}\rho ^{2}d\rho ={\frac {2\pi r^{3}}{3}}(1-\cos \varphi )\,,} where the integrals have been separated, because the integrand can be separated into a product of functions each with one dummy variable.
The area can be similarly calculated by integrating the differential spherical area element d A = r 2 sin ϕ d ϕ d θ {\displaystyle dA=r^{2}\sin \phi \,d\phi \,d\theta } over the spherical sector, giving A = ∫ 0 2 π ∫ 0 φ r 2 sin ϕ d ϕ d θ = r 2 ∫ 0 2 π d θ ∫ 0 φ sin ϕ d ϕ = 2 π r 2 ( 1 − cos φ ) , {\displaystyle A=\int _{0}^{2\pi }\int _{0}^{\varphi }r^{2}\sin \phi \,d\phi \,d\theta =r^{2}\int _{0}^{2\pi }d\theta \int _{0}^{\varphi }\sin \phi \,d\phi =2\pi r^{2}(1-\cos \varphi )\,,} where φ is inclination (or elevation) and θ is azimuth (right). Notice r is a constant. Again, the integrals can be separated.
This geometry-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Spherical_sector |
In geometry , a spherical segment is the solid defined by cutting a sphere or a ball with a pair of parallel planes .
It can be thought of as a spherical cap with the top truncated, and so it corresponds to a spherical frustum .
The surface of the spherical segment (excluding the bases) is called spherical zone .
If the radius of the sphere is called R , the radii of the spherical segment bases are a and b , and the height of the segment (the distance from one parallel plane to the other) called h , then the volume of the spherical segment is
For the special case of the top plane being tangent to the sphere, we have b = 0 {\displaystyle b=0} and the solid reduces to a spherical cap . [ 1 ]
The equation above for volume of the spherical segment can be arranged to
Thus, the segment volume equals the sum of three volumes: two right circular cylinders one of radius a and the second of radius b (both of height h / 2 {\displaystyle h/2} ) and a sphere of radius h / 2 {\displaystyle h/2} .
The curved surface area of the spherical zone—which excludes the top and bottom bases—is given by
Thus the surface area of the segment depends only on the distance between the cutting planes, and not their absolute heights.
This geometry-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Spherical_segment |
In geometric topology , the spherical space form conjecture (now a theorem) states that a finite group acting on the 3-sphere is conjugate to a group of isometries of the 3-sphere.
The conjecture was posed by Heinz Hopf in 1926 after determining the fundamental groups of three-dimensional spherical space forms as a generalization of the Poincaré conjecture to the non-simply connected case. [ 1 ] [ 2 ]
The conjecture is implied by Thurston 's geometrization conjecture , which was proven by Grigori Perelman in 2003. The conjecture was independently proven for groups whose actions have fixed points —this special case is known as the Smith conjecture . It is also proven for various groups acting without fixed points, such as cyclic groups whose orders are a power of two (George Livesay, Robert Myers) and cyclic groups of order 3 ( J. Hyam Rubinstein ). [ 3 ]
This topology-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Spherical_space_form_conjecture |
Spherical surface acoustic wave sensors use a type of surface acoustic wave (SAW) that travels along the surface of a medium exhibiting elasticity with exponentially decaying amplitude along depth. MEMS-IDT technology allows the use of SAW waves to sense various gases. Sensitivity up to 10 ppm of hydrogen using a spherical Ball SAW device is obtained. [ 1 ]
Conventional planar SAW sensors are based on principle that the parameters such as amplitude, speed and phase of Surface acoustic wave changes on adsorption of gas molecules. Limitation of planar SAW based sensors is that the change in above mentioned parameters is very small due to limited path offered to Surface acoustic wave by planar sensor. In case of Spherical sensors surface acoustic wave make several round trips along the equator of a ball as shown in fig, which offer longer paths to Surface acoustic wave hence even smaller change in parameters is amplified with multiple turns, which increases the sensitivity of the sensor considerably.
This acoustics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Spherical_surface_acoustic_wave_(SAW)_sensor |
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles , traditionally expressed using trigonometric functions . On the sphere , geodesics are great circles . Spherical trigonometry is of great importance for calculations in astronomy , geodesy , and navigation .
The origins of spherical trigonometry in Greek mathematics and the major developments in Islamic mathematics are discussed fully in History of trigonometry and Mathematics in medieval Islam . The subject came to fruition in Early Modern times with important developments by John Napier , Delambre and others, and attained an essentially complete form by the end of the nineteenth century with the publication of Todhunter's textbook Spherical trigonometry for the use of colleges and Schools . [ 1 ] Since then, significant developments have been the application of vector methods, quaternion methods, and the use of numerical methods.
A spherical polygon is a polygon on the surface of the sphere. Its sides are arcs of great circles —the spherical geometry equivalent of line segments in plane geometry .
Such polygons may have any number of sides greater than 1. Two-sided spherical polygons— lunes , also called digons or bi-angles —are bounded by two great-circle arcs: a familiar example is the curved outward-facing surface of a segment of an orange. Three arcs serve to define a spherical triangle, the principal subject of this article. Polygons with higher numbers of sides (4-sided spherical quadrilaterals, 5-sided spherical pentagons, etc.) are defined in similar manner. Analogously to their plane counterparts, spherical polygons with more than 3 sides can always be treated as the composition of spherical triangles.
One spherical polygon with interesting properties is the pentagramma mirificum , a 5-sided spherical star polygon with a right angle at every vertex.
From this point in the article, discussion will be restricted to spherical triangles, referred to simply as triangles .
In particular, the sum of the angles of a spherical triangle is strictly greater than the sum of the angles of a triangle defined on the Euclidean plane, which is always exactly π radians.
The polar triangle associated with a triangle △ ABC is defined as follows. Consider the great circle that contains the side BC . This great circle is defined by the intersection of a diametral plane with the surface. Draw the normal to that plane at the centre: it intersects the surface at two points and the point that is on the same side of the plane as A is (conventionally) termed the pole of A and it is denoted by A' . The points B' and C' are defined similarly.
The triangle △ A'B'C' is the polar triangle corresponding to triangle △ ABC . The angles and sides of the polar triangle are
given by (Todhunter, [ 1 ] Art.27) A ′ = π − a , B ′ = π − b , C ′ = π − c , a ′ = π − A , b ′ = π − B , c ′ = π − C . {\displaystyle {\begin{alignedat}{3}A'&=\pi -a,&\qquad B'&=\pi -b,&\qquad C'&=\pi -c,\\a'&=\pi -A,&b'&=\pi -B,&c'&=\pi -C.\end{alignedat}}} Therefore, if any identity is proved for △ ABC then we can immediately derive a second identity by applying the first identity to the polar triangle by making the above substitutions. This is how the supplemental cosine equations are derived from the cosine equations. Similarly, the identities for a quadrantal triangle can be derived from those for a right-angled triangle. The polar triangle of a polar triangle is the original triangle.
If the 3 × 3 matrix M has the positions A , B , and C as its columns then the rows of the matrix inverse M −1 , if normalized to unit length, are the positions A′ , B′ , and C′ . In particular, when △ A′B′C′ is the polar triangle of △ ABC then △ ABC is the polar triangle of △ A′B′C′ .
The cosine rule is the fundamental identity of spherical trigonometry: all other identities, including the sine rule, may be derived from the cosine rule: cos a = cos b cos c + sin b sin c cos A , cos b = cos c cos a + sin c sin a cos B , cos c = cos a cos b + sin a sin b cos C . {\displaystyle {\begin{aligned}\cos a&=\cos b\cos c+\sin b\sin c\cos A,\\[2pt]\cos b&=\cos c\cos a+\sin c\sin a\cos B,\\[2pt]\cos c&=\cos a\cos b+\sin a\sin b\cos C.\end{aligned}}}
These identities generalize the cosine rule of plane trigonometry , to which they are asymptotically equivalent
in the limit of small interior angles. (On the unit sphere, if a , b , c → 0 {\displaystyle a,b,c\rightarrow 0} set sin a ≈ a {\displaystyle \sin a\approx a} and cos a ≈ 1 − a 2 2 {\displaystyle \cos a\approx 1-{\frac {a^{2}}{2}}} etc.; see Spherical law of cosines .)
The spherical law of sines is given by the formula sin A sin a = sin B sin b = sin C sin c . {\displaystyle {\frac {\sin A}{\sin a}}={\frac {\sin B}{\sin b}}={\frac {\sin C}{\sin c}}.} These identities approximate the sine rule of plane trigonometry when the sides are much smaller than the radius of the sphere.
The spherical cosine formulae were originally proved by elementary geometry and the planar cosine rule (Todhunter, [ 1 ] Art.37). He also gives a derivation using simple coordinate geometry and the planar cosine rule (Art.60). The approach outlined here uses simpler vector methods. (These methods are also discussed at Spherical law of cosines .)
Consider three unit vectors OA → , OB → , OC → drawn from the origin to the vertices of the triangle (on the unit sphere). The arc BC subtends an angle of magnitude a at the centre and therefore OB → · OC → = cos a . Introduce a Cartesian basis with OA → along the z -axis and OB → in the xz -plane making an angle c with the z -axis. The vector OC → projects to ON in the xy -plane and the angle between ON and the x -axis is A . Therefore, the three vectors have components:
O A → : ( 0 , 0 , 1 ) O B → : ( sin c , 0 , cos c ) O C → : ( sin b cos A , sin b sin A , cos b ) . {\displaystyle {\begin{aligned}{\vec {OA}}:&\quad (0,\,0,\,1)\\{\vec {OB}}:&\quad (\sin c,\,0,\,\cos c)\\{\vec {OC}}:&\quad (\sin b\cos A,\,\sin b\sin A,\,\cos b).\end{aligned}}}
The scalar product OB → · OC → in terms of the components is O B → ⋅ O C → = sin c sin b cos A + cos c cos b . {\displaystyle {\vec {OB}}\cdot {\vec {OC}}=\sin c\sin b\cos A+\cos c\cos b.} Equating the two expressions for the scalar product gives cos a = cos b cos c + sin b sin c cos A . {\displaystyle \cos a=\cos b\cos c+\sin b\sin c\cos A.} This equation can be re-arranged to give explicit expressions for the angle in terms of the sides: cos A = cos a − cos b cos c sin b sin c . {\displaystyle \cos A={\frac {\cos a-\cos b\cos c}{\sin b\sin c}}.}
The other cosine rules are obtained by cyclic permutations.
This derivation is given in Todhunter, [ 1 ] (Art.40). From the identity sin 2 A = 1 − cos 2 A {\displaystyle \sin ^{2}A=1-\cos ^{2}A} and the explicit expression for cos A given immediately above sin 2 A = 1 − ( cos a − cos b cos c sin b sin c ) 2 = ( 1 − cos 2 b ) ( 1 − cos 2 c ) − ( cos a − cos b cos c ) 2 sin 2 b sin 2 c sin A sin a = 1 − cos 2 a − cos 2 b − cos 2 c + 2 cos a cos b cos c sin a sin b sin c . {\displaystyle {\begin{aligned}\sin ^{2}A&=1-\left({\frac {\cos a-\cos b\cos c}{\sin b\sin c}}\right)^{2}\\[5pt]&={\frac {(1-\cos ^{2}b)(1-\cos ^{2}c)-(\cos a-\cos b\cos c)^{2}}{\sin ^{2}\!b\,\sin ^{2}\!c}}\\[5pt]{\frac {\sin A}{\sin a}}&={\frac {\sqrt {1-\cos ^{2}\!a-\cos ^{2}\!b-\cos ^{2}\!c+2\cos a\cos b\cos c}}{\sin a\sin b\sin c}}.\end{aligned}}} Since the right hand side is invariant under a cyclic permutation of a , b , and c the spherical sine rule follows immediately.
There are many ways of deriving the fundamental cosine and sine rules and the other rules developed in the following sections. For example, Todhunter [ 1 ] gives two proofs of the cosine rule (Articles 37 and 60) and two proofs of the sine rule (Articles 40 and 42). The page on Spherical law of cosines gives four different proofs of the cosine rule. Text books on geodesy [ 2 ] and spherical astronomy [ 3 ] give different proofs and the online resources of MathWorld provide yet more. [ 4 ] There are even more exotic derivations, such as that of Banerjee [ 5 ] who derives the formulae using the linear algebra of projection matrices and also quotes methods in differential geometry and the group theory of rotations.
The derivation of the cosine rule presented above has the merits of simplicity and directness and the derivation of the sine rule emphasises the fact that no separate proof is required other than the cosine rule. However, the above geometry may be used to give an independent proof of the sine rule. The scalar triple product , OA → · ( OB → × OC → ) evaluates to sin b sin c sin A in the basis shown. Similarly, in a basis oriented with the z -axis along OB → , the triple product OB → · ( OC → × OA → ) , evaluates to sin c sin a sin B . Therefore, the invariance of the triple product under cyclic permutations gives sin b sin A = sin a sin B which is the first of the sine rules. See curved variations of the law of sines to see details of this derivation.
When any three of the differentials da , db , dc , dA , dB , dC are known, the following equations, which are found by differentiating the cosine rule and using the sine rule, can be used to calculate the other three by elimination: [ 6 ]
d a = cos C d b + cos B d c + sin b sin C d A , d b = cos A d c + cos C d a + sin c sin A d B , d c = cos B d a + cos A d b + sin a sin B d C . {\displaystyle {\begin{aligned}da=\cos C\ db+\cos B\ dc+\sin b\ \sin C\ dA,\\db=\cos A\ dc+\cos C\ da+\sin c\ \sin A\ dB,\\dc=\cos B\ da+\cos A\ db+\sin a\ \sin B\ dC.\\\end{aligned}}}
Applying the cosine rules to the polar triangle gives (Todhunter, [ 1 ] Art.47), i.e. replacing A by π – a , a by π – A etc., cos A = − cos B cos C + sin B sin C cos a , cos B = − cos C cos A + sin C sin A cos b , cos C = − cos A cos B + sin A sin B cos c . {\displaystyle {\begin{aligned}\cos A&=-\cos B\,\cos C+\sin B\,\sin C\,\cos a,\\\cos B&=-\cos C\,\cos A+\sin C\,\sin A\,\cos b,\\\cos C&=-\cos A\,\cos B+\sin A\,\sin B\,\cos c.\end{aligned}}}
The six parts of a triangle may be written in cyclic order as ( aCbAcB ). The cotangent, or four-part, formulae relate two sides and two angles forming four consecutive parts around the triangle, for example ( aCbA ) or BaCb ). In such a set there are inner and outer parts: for example in the set ( BaCb ) the inner angle is C , the inner side is a , the outer angle is B , the outer side is b . The cotangent rule may be written as (Todhunter, [ 1 ] Art.44) cos ( inner side ) cos ( inner angle ) = cot ( outer side ) sin ( inner side ) − cot ( outer angle ) sin ( inner angle ) , {\displaystyle \cos \!{\Bigl (}{\begin{smallmatrix}{\text{inner}}\\{\text{side}}\end{smallmatrix}}{\Bigr )}\cos \!{\Bigl (}{\begin{smallmatrix}{\text{inner}}\\{\text{angle}}\end{smallmatrix}}{\Bigr )}=\cot \!{\Bigl (}{\begin{smallmatrix}{\text{outer}}\\{\text{side}}\end{smallmatrix}}{\Bigr )}\sin \!{\Bigl (}{\begin{smallmatrix}{\text{inner}}\\{\text{side}}\end{smallmatrix}}{\Bigr )}-\cot \!{\Bigl (}{\begin{smallmatrix}{\text{outer}}\\{\text{angle}}\end{smallmatrix}}{\Bigr )}\sin \!{\Bigl (}{\begin{smallmatrix}{\text{inner}}\\{\text{angle}}\end{smallmatrix}}{\Bigr )},} and the six possible equations are (with the relevant set shown at right): (CT1) cos b cos C = cot a sin b − cot A sin C ( a C b A ) (CT2) cos b cos A = cot c sin b − cot C sin A ( C b A c ) (CT3) cos c cos A = cot b sin c − cot B sin A ( b A c B ) (CT4) cos c cos B = cot a sin c − cot A sin B ( A c B a ) (CT5) cos a cos B = cot c sin a − cot C sin B ( c B a C ) (CT6) cos a cos C = cot b sin a − cot B sin C ( B a C b ) {\displaystyle {\begin{alignedat}{5}{\text{(CT1)}}&&\qquad \cos b\,\cos C&=\cot a\,\sin b-\cot A\,\sin C\qquad &&(aCbA)\\[0ex]{\text{(CT2)}}&&\cos b\,\cos A&=\cot c\,\sin b-\cot C\,\sin A&&(CbAc)\\[0ex]{\text{(CT3)}}&&\cos c\,\cos A&=\cot b\,\sin c-\cot B\,\sin A&&(bAcB)\\[0ex]{\text{(CT4)}}&&\cos c\,\cos B&=\cot a\,\sin c-\cot A\,\sin B&&(AcBa)\\[0ex]{\text{(CT5)}}&&\cos a\,\cos B&=\cot c\,\sin a-\cot C\,\sin B&&(cBaC)\\[0ex]{\text{(CT6)}}&&\cos a\,\cos C&=\cot b\,\sin a-\cot B\,\sin C&&(BaCb)\end{alignedat}}} To prove the first formula start from the first cosine rule and on the right-hand side substitute for cos c from the third cosine rule: cos a = cos b cos c + sin b sin c cos A = cos b ( cos a cos b + sin a sin b cos C ) + sin b sin C sin a cot A cos a sin 2 b = cos b sin a sin b cos C + sin b sin C sin a cot A . {\displaystyle {\begin{aligned}\cos a&=\cos b\cos c+\sin b\sin c\cos A\\&=\cos b\ (\cos a\cos b+\sin a\sin b\cos C)+\sin b\sin C\sin a\cot A\\\cos a\sin ^{2}b&=\cos b\sin a\sin b\cos C+\sin b\sin C\sin a\cot A.\end{aligned}}} The result follows on dividing by sin a sin b . Similar techniques
with the other two cosine rules give CT3 and CT5. The other three equations follow by applying rules 1, 3 and 5 to the polar triangle.
With 2 s = ( a + b + c ) {\displaystyle 2s=(a+b+c)} and 2 S = ( A + B + C ) , {\displaystyle 2S=(A+B+C),} sin 1 2 A = sin ( s − b ) sin ( s − c ) sin b sin c sin 1 2 a = − cos S cos ( S − A ) sin B sin C cos 1 2 A = sin s sin ( s − a ) sin b sin c cos 1 2 a = cos ( S − B ) cos ( S − C ) sin B sin C tan 1 2 A = sin ( s − b ) sin ( s − c ) sin s sin ( s − a ) tan 1 2 a = − cos S cos ( S − A ) cos ( S − B ) cos ( S − C ) {\displaystyle {\begin{alignedat}{5}\sin {\tfrac {1}{2}}A&={\sqrt {\frac {\sin(s-b)\sin(s-c)}{\sin b\sin c}}}&\qquad \qquad \sin {\tfrac {1}{2}}a&={\sqrt {\frac {-\cos S\cos(S-A)}{\sin B\sin C}}}\\[2ex]\cos {\tfrac {1}{2}}A&={\sqrt {\frac {\sin s\sin(s-a)}{\sin b\sin c}}}&\cos {\tfrac {1}{2}}a&={\sqrt {\frac {\cos(S-B)\cos(S-C)}{\sin B\sin C}}}\\[2ex]\tan {\tfrac {1}{2}}A&={\sqrt {\frac {\sin(s-b)\sin(s-c)}{\sin s\sin(s-a)}}}&\tan {\tfrac {1}{2}}a&={\sqrt {\frac {-\cos S\cos(S-A)}{\cos(S-B)\cos(S-C)}}}\end{alignedat}}}
Another twelve identities follow by cyclic permutation.
The proof (Todhunter, [ 1 ] Art.49) of the first formula starts from the identity 2 sin 2 A 2 = 1 − cos A , {\displaystyle 2\sin ^{2}\!{\tfrac {A}{2}}=1-\cos A,} using the cosine rule to express A in terms of the sides and replacing the sum of two cosines by a product. (See sum-to-product identities .) The second formula starts from the identity 2 cos 2 A 2 = 1 + cos A , {\displaystyle 2\cos ^{2}\!{\tfrac {A}{2}}=1+\cos A,} the third is a quotient and the remainder follow by applying the results to the polar triangle.
The Delambre analogies (also called Gauss analogies) were published independently by Delambre, Gauss, and Mollweide in 1807–1809. [ 7 ]
sin 1 2 ( A + B ) cos 1 2 C = cos 1 2 ( a − b ) cos 1 2 c sin 1 2 ( A − B ) cos 1 2 C = sin 1 2 ( a − b ) sin 1 2 c cos 1 2 ( A + B ) sin 1 2 C = cos 1 2 ( a + b ) cos 1 2 c cos 1 2 ( A − B ) sin 1 2 C = sin 1 2 ( a + b ) sin 1 2 c {\displaystyle {\begin{aligned}{\frac {\sin {\tfrac {1}{2}}(A+B)}{\cos {\tfrac {1}{2}}C}}={\frac {\cos {\tfrac {1}{2}}(a-b)}{\cos {\tfrac {1}{2}}c}}&\qquad \qquad &{\frac {\sin {\tfrac {1}{2}}(A-B)}{\cos {\tfrac {1}{2}}C}}={\frac {\sin {\tfrac {1}{2}}(a-b)}{\sin {\tfrac {1}{2}}c}}\\[2ex]{\frac {\cos {\tfrac {1}{2}}(A+B)}{\sin {\tfrac {1}{2}}C}}={\frac {\cos {\tfrac {1}{2}}(a+b)}{\cos {\tfrac {1}{2}}c}}&\qquad &{\frac {\cos {\tfrac {1}{2}}(A-B)}{\sin {\tfrac {1}{2}}C}}={\frac {\sin {\tfrac {1}{2}}(a+b)}{\sin {\tfrac {1}{2}}c}}\end{aligned}}} Another eight identities follow by cyclic permutation.
Proved by expanding the numerators and using the half angle formulae. (Todhunter, [ 1 ] Art.54 and Delambre [ 8 ] )
tan 1 2 ( A + B ) = cos 1 2 ( a − b ) cos 1 2 ( a + b ) cot 1 2 C tan 1 2 ( a + b ) = cos 1 2 ( A − B ) cos 1 2 ( A + B ) tan 1 2 c tan 1 2 ( A − B ) = sin 1 2 ( a − b ) sin 1 2 ( a + b ) cot 1 2 C tan 1 2 ( a − b ) = sin 1 2 ( A − B ) sin 1 2 ( A + B ) tan 1 2 c {\displaystyle {\begin{aligned}\tan {\tfrac {1}{2}}(A+B)={\frac {\cos {\tfrac {1}{2}}(a-b)}{\cos {\tfrac {1}{2}}(a+b)}}\cot {\tfrac {1}{2}}C&\qquad &\tan {\tfrac {1}{2}}(a+b)={\frac {\cos {\tfrac {1}{2}}(A-B)}{\cos {\tfrac {1}{2}}(A+B)}}\tan {\tfrac {1}{2}}c\\[2ex]\tan {\tfrac {1}{2}}(A-B)={\frac {\sin {\tfrac {1}{2}}(a-b)}{\sin {\tfrac {1}{2}}(a+b)}}\cot {\tfrac {1}{2}}C&\qquad &\tan {\tfrac {1}{2}}(a-b)={\frac {\sin {\tfrac {1}{2}}(A-B)}{\sin {\tfrac {1}{2}}(A+B)}}\tan {\tfrac {1}{2}}c\end{aligned}}}
Another eight identities follow by cyclic permutation.
These identities follow by division of the Delambre formulae. (Todhunter, [ 1 ] Art.52)
Taking quotients of these yields the law of tangents , first stated by Persian mathematician Nasir al-Din al-Tusi (1201–1274),
tan 1 2 ( A − B ) tan 1 2 ( A + B ) = tan 1 2 ( a − b ) tan 1 2 ( a + b ) {\displaystyle {\frac {\tan {\tfrac {1}{2}}(A-B)}{\tan {\tfrac {1}{2}}(A+B)}}={\frac {\tan {\tfrac {1}{2}}(a-b)}{\tan {\tfrac {1}{2}}(a+b)}}}
When one of the angles, say C , of a spherical triangle is equal to π /2 the various identities given above are considerably simplified. There are ten identities relating three elements chosen from the set a , b , c , A , and B .
Napier [ 9 ] provided an elegant mnemonic aid for the ten independent equations: the mnemonic is called Napier's circle or Napier's pentagon (when the circle in the above figure, right, is replaced by a pentagon).
First, write the six parts of the triangle (three vertex angles, three arc angles for the sides) in the order they occur around any circuit of the triangle: for the triangle shown above left, going clockwise starting with a gives aCbAcB . Next replace the parts that are not adjacent to C (that is A , c , and B ) by their complements and then delete the angle C from the list. The remaining parts can then be drawn as five ordered, equal slices of a pentagram, or circle, as shown in the above figure (right). For any choice of three contiguous parts, one (the middle part) will be adjacent to two parts and opposite the other two parts. The ten Napier's Rules are given by
The key for remembering which trigonometric function goes with which part is to look at the first vowel of the kind of part: middle parts take the sine, adjacent parts take the tangent, and opposite parts take the cosine.
For an example, starting with the sector containing a we have: sin a = tan ( π 2 − B ) tan b = cos ( π 2 − c ) cos ( π 2 − A ) = cot B tan b = sin c sin A . {\displaystyle {\begin{aligned}\sin a&=\tan({\tfrac {\pi }{2}}-B)\,\tan b\\[2pt]&=\cos({\tfrac {\pi }{2}}-c)\,\cos({\tfrac {\pi }{2}}-A)\\[2pt]&=\cot B\,\tan b\\[4pt]&=\sin c\,\sin A.\end{aligned}}} The full set of rules for the right spherical triangle is (Todhunter, [ 1 ] Art.62) (R1) cos c = cos a cos b , (R6) tan b = cos A tan c , (R2) sin a = sin A sin c , (R7) tan a = cos B tan c , (R3) sin b = sin B sin c , (R8) cos A = sin B cos a , (R4) tan a = tan A sin b , (R9) cos B = sin A cos b , (R5) tan b = tan B sin a , (R10) cos c = cot A cot B . {\displaystyle {\begin{alignedat}{4}&{\text{(R1)}}&\qquad \cos c&=\cos a\,\cos b,&\qquad \qquad &{\text{(R6)}}&\qquad \tan b&=\cos A\,\tan c,\\&{\text{(R2)}}&\sin a&=\sin A\,\sin c,&&{\text{(R7)}}&\tan a&=\cos B\,\tan c,\\&{\text{(R3)}}&\sin b&=\sin B\,\sin c,&&{\text{(R8)}}&\cos A&=\sin B\,\cos a,\\&{\text{(R4)}}&\tan a&=\tan A\,\sin b,&&{\text{(R9)}}&\cos B&=\sin A\,\cos b,\\&{\text{(R5)}}&\tan b&=\tan B\,\sin a,&&{\text{(R10)}}&\cos c&=\cot A\,\cot B.\end{alignedat}}}
A quadrantal spherical triangle is defined to be a spherical triangle in which one of the sides subtends an angle of π /2 radians at the centre of the sphere: on the unit sphere the side has length π /2. In the case that the side c has length π /2 on the unit sphere the equations governing the remaining sides and angles may be obtained by applying the rules for the right spherical triangle of the previous section to the polar triangle △ A'B'C' with sides a', b', c' such that A' = π − a , a' = π − A etc. The results are: (Q1) cos C = − cos A cos B , (Q6) tan B = − cos a tan C , (Q2) sin A = sin a sin C , (Q7) tan A = − cos b tan C , (Q3) sin B = sin b sin C , (Q8) cos a = sin b cos A , (Q4) tan A = tan a sin B , (Q9) cos b = sin a cos B , (Q5) tan B = tan b sin A , (Q10) cos C = − cot a cot b . {\displaystyle {\begin{alignedat}{4}&{\text{(Q1)}}&\qquad \cos C&=-\cos A\,\cos B,&\qquad \qquad &{\text{(Q6)}}&\qquad \tan B&=-\cos a\,\tan C,\\&{\text{(Q2)}}&\sin A&=\sin a\,\sin C,&&{\text{(Q7)}}&\tan A&=-\cos b\,\tan C,\\&{\text{(Q3)}}&\sin B&=\sin b\,\sin C,&&{\text{(Q8)}}&\cos a&=\sin b\,\cos A,\\&{\text{(Q4)}}&\tan A&=\tan a\,\sin B,&&{\text{(Q9)}}&\cos b&=\sin a\,\cos B,\\&{\text{(Q5)}}&\tan B&=\tan b\,\sin A,&&{\text{(Q10)}}&\cos C&=-\cot a\,\cot b.\end{alignedat}}}
Substituting the second cosine rule into the first and simplifying gives: cos a = ( cos a cos c + sin a sin c cos B ) cos c + sin b sin c cos A cos a sin 2 c = sin a cos c sin c cos B + sin b sin c cos A {\displaystyle {\begin{aligned}\cos a&=(\cos a\,\cos c+\sin a\,\sin c\,\cos B)\cos c+\sin b\,\sin c\,\cos A\\[4pt]\cos a\,\sin ^{2}c&=\sin a\,\cos c\,\sin c\,\cos B+\sin b\,\sin c\,\cos A\end{aligned}}} Cancelling the factor of sin c gives cos a sin c = sin a cos c cos B + sin b cos A {\displaystyle \cos a\sin c=\sin a\,\cos c\,\cos B+\sin b\,\cos A}
Similar substitutions in the other cosine and supplementary cosine formulae give a large variety of 5-part rules. They are rarely used.
Multiplying the first cosine rule by cos A gives cos a cos A = cos b cos c cos A + sin b sin c − sin b sin c sin 2 A . {\displaystyle \cos a\cos A=\cos b\,\cos c\,\cos A+\sin b\,\sin c-\sin b\,\sin c\,\sin ^{2}A.} Similarly multiplying the first supplementary cosine rule by cos a yields cos a cos A = − cos B cos C cos a + sin B sin C − sin B sin C sin 2 a . {\displaystyle \cos a\cos A=-\cos B\,\cos C\,\cos a+\sin B\,\sin C-\sin B\,\sin C\,\sin ^{2}a.} Subtracting the two and noting that it follows from the sine rules that sin b sin c sin 2 A = sin B sin C sin 2 a {\displaystyle \sin b\,\sin c\,\sin ^{2}A=\sin B\,\sin C\,\sin ^{2}a} produces Cagnoli's equation sin b sin c + cos b cos c cos A = sin B sin C − cos B cos C cos a {\displaystyle \sin b\,\sin c+\cos b\,\cos c\,\cos A=\sin B\,\sin C-\cos B\,\cos C\,\cos a} which is a relation between the six parts of the spherical triangle. [ 10 ]
The solution of triangles is the principal purpose of spherical trigonometry: given three, four or five elements of the triangle, determine the others. The case of five given elements is trivial, requiring only a single application of the sine rule. For four given elements there is one non-trivial case, which is discussed below. For three given elements there are six cases: three sides, two sides and an included or opposite angle, two angles and an included or opposite side, or three angles. (The last case has no analogue in planar trigonometry.) No single method solves all cases. The figure below shows the seven non-trivial cases: in each case the given sides are marked with a cross-bar and the given angles with an arc. (The given elements are also listed below the triangle). In the summary notation here such as ASA, A refers to a given angle and S refers to a given side, and the sequence of A's and S's in the notation refers to the corresponding sequence in the triangle.
The solution methods listed here are not the only possible choices: many others are possible. In general it is better to choose methods that avoid taking an inverse sine because of the possible ambiguity between an angle and its supplement. The use of half-angle formulae is often advisable because half-angles will be less than π /2 and therefore free from ambiguity. There is a full discussion in Todhunter. The article Solution of triangles#Solving spherical triangles presents variants on these methods with a slightly different notation.
There is a full discussion of the solution of oblique triangles in Todhunter. [ 1 ] : Chap. VI See also the discussion in Ross. [ 11 ] Nasir al-Din al-Tusi was the first to list the six distinct cases (2–7 in the diagram) of a right triangle in spherical trigonometry. [ 12 ]
Another approach is to split the triangle into two right-angled triangles. For example, take the Case 3 example where b , c , and B are given. Construct the great circle from A that is normal to the side BC at the point D . Use Napier's rules to solve the triangle △ ABD : use c and B to find the sides AD and BD and the angle ∠ BAD . Then use Napier's rules to solve the triangle △ ACD : that is use AD and b to find the side DC and the angles C and ∠ DAC . The angle A and side a follow by addition.
Not all of the rules obtained are numerically robust in extreme examples, for example when an angle approaches zero or π . Problems and solutions may have to be examined carefully, particularly when writing code to solve an arbitrary triangle.
Consider an N -sided spherical polygon and let A n denote the n -th interior angle. The area of such a polygon is given by (Todhunter, [ 1 ] Art.99) Area of polygon (on the unit sphere) ≡ E N = ( ∑ n = 1 N A n ) − ( N − 2 ) π . {\displaystyle {{\text{Area of polygon}} \atop {\text{(on the unit sphere)}}}\equiv E_{N}=\left(\sum _{n=1}^{N}A_{n}\right)-(N-2)\pi .}
For the case of a spherical triangle with angles A , B , and C this reduces to Girard's theorem Area of triangle (on the unit sphere) ≡ E = E 3 = A + B + C − π , {\displaystyle {{\text{Area of triangle}} \atop {\text{(on the unit sphere)}}}\equiv E=E_{3}=A+B+C-\pi ,} where E is the amount by which the sum of the angles exceeds π radians, called the spherical excess of the triangle. This theorem is named after its author, Albert Girard . [ 13 ] An earlier proof was derived, but not published, by the English mathematician Thomas Harriot . On a sphere of radius R both of the above area expressions are multiplied by R 2 . The definition of the excess is independent of the radius of the sphere.
The converse result may be written as
A + B + C = π + 4 π × Area of triangle Area of the sphere . {\displaystyle A+B+C=\pi +{\frac {4\pi \times {\text{Area of triangle}}}{\text{Area of the sphere}}}.}
Since the area of a triangle cannot be negative the spherical excess is always positive. It is not necessarily small, because the sum of the angles may attain 5 π (3 π for proper angles). For example,
an octant of a sphere is a spherical triangle with three right angles, so that the excess is π /2. In practical applications it is often small: for example the triangles of geodetic survey typically have a spherical excess much less than 1' of arc. [ 14 ] On the Earth the excess of an equilateral triangle with sides 21.3 km (and area 393 km 2 ) is approximately 1 arc second.
There are many formulae for the excess. For example, Todhunter, [ 1 ] (Art.101—103) gives ten examples including that of L'Huilier : tan 1 4 E = tan 1 2 s tan 1 2 ( s − a ) tan 1 2 ( s − b ) tan 1 2 ( s − c ) {\displaystyle \tan {\tfrac {1}{4}}E={\sqrt {\tan {\tfrac {1}{2}}s\,\tan {\tfrac {1}{2}}(s-a)\,\tan {\tfrac {1}{2}}(s-b)\,\tan {\tfrac {1}{2}}(s-c)}}} where s = 1 2 ( a + b + c ) {\displaystyle s={\tfrac {1}{2}}(a+b+c)} . This formula is reminiscent of Heron's formula for planar triangles.
Because some triangles are badly characterized by
their edges (e.g., if a = b ≈ 1 2 c {\textstyle a=b\approx {\frac {1}{2}}c} ), it is often better to use
the formula for the excess in terms of two edges and their included angle tan 1 2 E = tan 1 2 a tan 1 2 b sin C 1 + tan 1 2 a tan 1 2 b cos C . {\displaystyle \tan {\tfrac {1}{2}}E={\frac {\tan {\frac {1}{2}}a\tan {\frac {1}{2}}b\sin C}{1+\tan {\frac {1}{2}}a\tan {\frac {1}{2}}b\cos C}}.}
When triangle △ ABC is a right triangle with right angle at C , then cos C = 0 and sin C = 1 , so this reduces to tan 1 2 E = tan 1 2 a tan 1 2 b . {\displaystyle \tan {\tfrac {1}{2}}E=\tan {\tfrac {1}{2}}a\tan {\tfrac {1}{2}}b.}
Angle deficit is defined similarly for hyperbolic geometry .
The spherical excess of a spherical quadrangle bounded by the equator, the two meridians of longitudes λ 1 {\displaystyle \lambda _{1}} and λ 2 , {\displaystyle \lambda _{2},} and the great-circle arc between two points with longitude and latitude ( λ 1 , φ 1 ) {\displaystyle (\lambda _{1},\varphi _{1})} and ( λ 2 , φ 2 ) {\displaystyle (\lambda _{2},\varphi _{2})} is tan 1 2 E 4 = sin 1 2 ( φ 2 + φ 1 ) cos 1 2 ( φ 2 − φ 1 ) tan 1 2 ( λ 2 − λ 1 ) . {\displaystyle \tan {\tfrac {1}{2}}E_{4}={\frac {\sin {\tfrac {1}{2}}(\varphi _{2}+\varphi _{1})}{\cos {\tfrac {1}{2}}(\varphi _{2}-\varphi _{1})}}\tan {\tfrac {1}{2}}(\lambda _{2}-\lambda _{1}).}
This result is obtained from one of Napier's analogies. In the limit where φ 1 , φ 2 , λ 2 − λ 1 {\displaystyle \varphi _{1},\varphi _{2},\lambda _{2}-\lambda _{1}} are all small, this reduces to the familiar trapezoidal area, E 4 ≈ 1 2 ( φ 2 + φ 1 ) ( λ 2 − λ 1 ) {\textstyle E_{4}\approx {\frac {1}{2}}(\varphi _{2}+\varphi _{1})(\lambda _{2}-\lambda _{1})} .
The area of a polygon can be calculated from individual quadrangles of the above type, from (analogously) individual triangle bounded by a segment of the polygon and two meridians, [ 15 ] by a line integral with Green's theorem , [ 16 ] or via an equal-area projection as commonly done in GIS. The other algorithms can still be used with the side lengths calculated using a great-circle distance formula. | https://en.wikipedia.org/wiki/Spherical_trigonometry |
Spherical wave transformations leave the form of spherical waves as well as the laws of optics and electrodynamics invariant in all inertial frames . They were defined between 1908 and 1909 by Harry Bateman and Ebenezer Cunningham , with Bateman giving the transformation its name. [ M 1 ] They correspond to the conformal group of "transformations by reciprocal radii" in relation to the framework of Lie sphere geometry , which were already known in the 19th century. Time is used as fourth dimension as in Minkowski space , so spherical wave transformations are connected to the Lorentz transformation of special relativity , and it turns out that the conformal group of spacetime includes the Lorentz group and the Poincaré group as subgroups. However, only the Lorentz/Poincaré groups represent symmetries of all laws of nature including mechanics, whereas the conformal group is related to certain areas such as electrodynamics. [ 1 ] [ 2 ] [ 3 ] In addition, it can be shown that the conformal group of the plane (corresponding to the Möbius group of the extended complex plane ) is isomorphic to the Lorentz group. [ 4 ]
A special case of Lie sphere geometry is the transformation by reciprocal directions or Laguerre inversion, being a generator of the Laguerre group . It transforms not only spheres into spheres but also planes into planes. [ 5 ] [ 6 ] [ 7 ] If time is used as fourth dimension, a close analogy to the Lorentz transformation as well as isomorphism to the Lorentz group was pointed out by several authors such as Bateman, Cartan or Poincaré . [ M 2 ] [ 8 ] [ M 3 ] [ 9 ] [ 10 ] [ 11 ] [ 12 ] [ 13 ]
Inversions preserving angles between circles were first discussed by Durrande (1820), with Quetelet (1827) and Plücker (1828) writing down the corresponding transformation formula, k {\displaystyle k} being the radius of inversion: [ 14 ]
These inversions were later called "transformations by reciprocal radii", and became better known when Thomson (1845, 1847) applied them on spheres with coordinates x , y , z {\displaystyle x,y,z} in the course of developing the method of inversion in electrostatics . [ 15 ] Joseph Liouville (1847) demonstrated its mathematical meaning by showing that it belongs to the conformal transformations producing the following quadratic form : [ M 4 ]
Liouville himself [ M 5 ] and more extensively Sophus Lie (1871) [ M 6 ] showed that the related conformal group can be differentiated ( Liouville's theorem ): For instance, λ = 1 {\displaystyle \lambda =1} includes the Euclidean group of ordinary motions; λ ≠ 1 {\displaystyle \lambda \neq 1} scale or similarity transformations in which the coordinates of the previous transformations are multiplied by λ {\displaystyle {\sqrt {\lambda }}} ; and λ = k 4 / ( x 2 + y 2 + z 2 ) 2 {\displaystyle \lambda =k^{4}/\left(x^{2}+y^{2}+z^{2}\right)^{2}} gives Thomson's transformation by reciprocal radii (inversions): [ M 5 ]
Subsequently, Liouville's theorem was extended to n {\displaystyle n} dimensions by Lie (1871) [ M 6 ] and others such as Darboux (1878): [ M 7 ]
This group of conformal transformations by reciprocal radii preserves angles and transforms spheres into spheres or hyperspheres (see Möbius transformation , conformal symmetry , special conformal transformation ). It is a 6-parameter group in the plane R 2 which corresponds to the Möbius group of the extended complex plane , [ 16 ] [ 4 ] a 10-parameter group in space R 3 , and a 15-parameter group in R 4 . In R 2 it represents only a small subset of all conformal transformations therein, whereas in R 2+n it is identical to the group of all conformal transformations (corresponding to the Möbius transformations in higher dimensions) therein, in accordance with Liouville's theorem. [ 16 ] Conformal transformations in R 3 were often applied to what Darboux (1873) called "pentaspherical coordinates" by relating the points to homogeneous coordinates based on five spheres. [ 17 ] [ 18 ]
Another method for solving such sphere problems was to write down the coordinates together with the sphere's radius. [ 19 ] This was employed by Lie (1871) in the context of Lie sphere geometry which represents a general framework of sphere-transformations (being a special case of contact transformations ) conserving lines of curvature and transforming spheres into spheres. [ M 8 ] The previously mentioned 10-parameter group in R 3 related to pentaspherical coordinates is extended to the 15-parameter group of Lie sphere transformations related to "hexaspherical coordinates" (named by Klein in 1893) by adding a sixth homogeneous coordinate related to the radius. [ M 9 ] [ 17 ] [ 20 ] Since the radius of a sphere can have a positive or negative sign, one sphere always corresponds to two transformed spheres. It is advantageous to remove this ambiguity by attributing a definite sign to the radius, consequently giving the spheres a definite orientation too, so that one oriented sphere corresponds to one transformed oriented sphere. [ 21 ] This method was occasionally and implicitly employed by Lie (1871) [ M 6 ] himself and explicitly introduced by Laguerre (1880). [ M 10 ] In addition, Darboux (1887) brought the transformations by reciprocal radii into a form by which the radius r of a sphere can be determined if the radius of the other one is known: [ M 11 ]
Using coordinates together with the radius was often connected to a method called "minimal projection" by Klein (1893), [ M 12 ] which was later called "isotropy projection" by Blaschke (1926) emphasizing the relation to oriented circles and spheres. [ 22 ] For instance, a circle with rectangular coordinates x , y {\displaystyle x,y} and radius r {\displaystyle r} in R 2 corresponds to a point in R 3 with coordinates x , y , z {\displaystyle x,y,z} . This method was known for some time in circle geometry (though without using the concept of orientation) and can be further differentiated depending on whether the additional coordinate is treated as imaginary or real: z = i r {\displaystyle z=ir} was used by Chasles (1852), Möbius (1857), Cayley (1867), and Darboux (1872); [ M 13 ] z = r {\displaystyle z=r} was used by Cousinery (1826), Druckenmüller (1842), and in the "cyclography" of Fiedler (1882), therefore the latter method was also called "cyclographic projection" – see E. Müller (1910) for a summary. [ 23 ] This method was also applied to spheres [ M 14 ] by Darboux (1872), [ M 15 ] Lie (1871), [ M 6 ] or Klein (1893). [ M 12 ] Let x , y , z , r {\displaystyle x,y,z,r} and x ′ , y ′ , z ′ , r ′ {\displaystyle x',y',z',r'} be the center coordinates and radii of two spheres in three-dimensional space R 3 . If the spheres are touching each other with same orientation, their equation is given
Setting t = i r {\displaystyle t=ir} , these coordinates correspond to rectangular coordinates in four-dimensional space R 4 : [ M 15 ] [ M 12 ]
In general, Lie (1871) showed that the conformal point transformations in R n (composed of motions, similarities, and transformations by reciprocal radii) correspond in R n-1 to those sphere transformations which are contact transformations. [ M 16 ] [ 24 ] Klein (1893) pointed out that by using minimal projection on hexaspherical coordinates, the 15-parameter Lie sphere transformations in R 3 are simply the projections of the 15-parameter conformal point transformations in R 4 , whereas the points in R 4 can be seen as the stereographic projection of the points of a sphere in R 5 . [ M 9 ] [ 25 ]
Harry Bateman and Ebenezer Cunningham (1909) [ M 1 ] showed that the electromagnetic equations are not only Lorentz invariant, but also scale and conformal invariant. [ 26 ] They are invariant under the 15-parameter group of conformal transformations G 15 {\displaystyle G_{15}} (transformations by reciprocal radii) in R 4 producing the relation
where u = i c t {\displaystyle u=ict} includes t {\displaystyle t} as time component and c {\displaystyle c} as the speed of light . Bateman (1909) also noticed the equivalence to the previously mentioned Lie sphere transformations in R 3 , because the radius r {\displaystyle r} used in them can be interpreted as the radius c t {\displaystyle ct} of a spherical wave contracting or expanding with c {\displaystyle c} , therefore he called them "spherical wave transformations". [ M 17 ] He wrote: [ M 18 ]
When we use Darboux's representation of a point in S 4 {\displaystyle S_{4}} by a spherical wave in S 3 {\displaystyle S_{3}} , the group G 15 {\displaystyle G_{15}} becomes the group of spherical wave transformations which transform a spherical wave into a spherical wave. This group of transformations has been discussed by S. Lie; it is the group of transformations which transform lines of curvature on a surface enveloped by spherical waves into lines of curvature on the surface enveloped by the corresponding spherical waves.
Depending on λ {\displaystyle \lambda } they can be differentiated into subgroups: [ 27 ]
(a) λ = 1 {\displaystyle \lambda =1} correspond to mappings which transform not only spheres into spheres but also planes into planes. These are called Laguerre transformations/inversions forming the Laguerre group, which in physics correspond to the Lorentz transformations forming the 6-parameter Lorentz group or 10-parameter Poincaré group with translations. [ 28 ]
(b) λ ≠ 1 {\displaystyle \lambda \neq 1} represents scale or similarity transformations by multiplication of the space-time variables of the Lorentz transformations by a constant factor depending on λ {\displaystyle \lambda } . [ 29 ] For instance, if l = λ {\displaystyle l={\sqrt {\lambda }}} is used, then the transformation given by Poincaré in 1905 follows: [ M 19 ]
However, it was shown by Poincaré and Einstein that only l = 1 {\displaystyle l=1} produces a group that is a symmetry of all laws of nature as required by the principle of relativity (the Lorentz group), while the group of scale transformations is only a symmetry of optics and electrodynamics.
(c) Setting λ = r 4 / ( x 2 + y 2 + z 2 + u 2 ) 2 {\displaystyle \lambda =r^{4}/\left(x^{2}+y^{2}+z^{2}+u^{2}\right)^{2}} particularly relates to the wide conformal group of transformations by reciprocal radii. It consists of elementary transformations that represent a generalized inversion into a four-dimensional hypersphere : [ 30 ]
which become real spherical wave transformations in terms of Lie sphere geometry if the real radius c t {\displaystyle ct} is used instead of u = i c t {\displaystyle u=ict} , thus x 2 + y 2 + z 2 − c 2 t 2 {\displaystyle x^{2}+y^{2}+z^{2}-c^{2}t^{2}} is given in the denominator. [ M 1 ]
Felix Klein (1921) pointed out the similarity of these relations to Lie's and his own researches of 1871, adding that the conformal group doesn't have the same meaning as the Lorentz group, because the former applies to electrodynamics whereas the latter is a symmetry of all laws of nature including mechanics. [ M 20 ] The possibility was discussed for some time, whether conformal transformations allow for the transformation into uniformly accelerated frames. [ 31 ] Later, conformal invariance became important again in certain areas such as conformal field theory . [ 32 ]
It turns out that also the 6-parameter conformal group of R 2 (i.e. the Möbius group composed of automorphisms of the Riemann sphere ), [ 4 ] which in turn is isomorphic to the 6-parameter group of hyperbolic motions (i.e. isometric automorphisms of a hyperbolic space ) in R 3 , [ 33 ] can be physically interpreted: It is isomorphic to the Lorentz group.
For instance, Fricke and Klein (1897) started by defining an "absolute" Cayley metric in terms of a one-part curvilinear surface of second degree, which can be represented by a sphere whose interior represents hyperbolic space with the equation [ 34 ]
where z 1 , z 2 , z 3 , z 4 {\displaystyle z_{1},\ z_{2},\ z_{3},\ z_{4}} are homogeneous coordinates. They pointed out that motions of hyperbolic space into itself also transform this sphere into itself. They developed the corresponding transformation by defining a complex parameter ξ {\displaystyle \xi } of the sphere [ 35 ]
which is connected to another parameter ξ ′ {\displaystyle \xi '} by the substitution
where α , β , γ , δ {\displaystyle \alpha ,\beta ,\gamma ,\delta } are complex coefficients. They furthermore showed that by setting z 1 : z 2 : z 3 : z 4 = X : Y : Z : 1 {\displaystyle z_{1}:z_{2}:z_{3}:z_{4}=X:Y:Z:1} , the above relations assume the form in terms of the unit sphere in R 3 : [ 36 ]
which is identical to the stereographic projection of the ξ {\displaystyle \xi } -plane on a spherical surface already given by Klein in 1884. [ M 21 ] Since the substitutions ξ , ξ ′ {\displaystyle \xi ,\xi '} are Möbius transformations ( German : Kreisverwandtschaften ) in the ξ {\displaystyle \xi } -plane or upon the ξ {\displaystyle \xi } -sphere, they concluded that by carrying out an arbitrary motion of hyperbolic space in itself, the ξ {\displaystyle \xi } -sphere undergoes a Möbius transformation, that the entire group of hyperbolic motions gives all direct Möbius transformations, and finally that any direct Möbius transformation corresponds to a motion of hyperbolic space. [ 37 ]
Based on the work of Fricke & Klein, the isomorphism of that group of hyperbolic motions (and consequently of the Möbius group) to the Lorentz group was demonstrated by Gustav Herglotz (1909). [ M 22 ] Namely, the Minkowski metric corresponds to the above Cayley metric (based on a real conic section), if the spacetime coordinates are identified with the above homogeneous coordinates
by which the above parameter become
Herglotz concluded, that any such substitution corresponds to a Lorentz transformation, establishing a one-to-one correspondence to hyperbolic motions in R 3 . The relation between the Lorentz group and the Cayley metric in hyperbolic space was also pointed out by Klein (1910) [ M 23 ] as well as Pauli (1921). [ 38 ] The corresponding isomorphism of the Möbius group to the Lorentz group was employed, among others, by Roger Penrose .
Above, the connection of conformal transformations with coordinates including the radius of spheres within Lie sphere geometry was mentioned. The special case λ = 1 {\displaystyle \lambda =1} corresponds to a sphere transformation given by Edmond Laguerre (1880–1885), who called it the "transformation by reciprocal directions" and who laid down the foundation of a geometry of oriented spheres and planes . [ M 10 ] [ 5 ] [ 6 ] According to Darboux [ M 24 ] and Bateman, [ M 2 ] similar relations were discussed before by Albert Ribaucour (1870) [ M 25 ] and by Lie himself (1871). [ M 6 ] Stephanos (1881) pointed out that Laguerre's geometry is indeed a special case of Lie's sphere geometry. [ M 26 ] He also represented Laguerre's oriented spheres by quaternions (1883). [ M 27 ]
Lines, circles, planes, or spheres with radii of certain orientation are called by Laguerre half-lines, half-circles (cycles), half-planes, half-spheres, etc. A tangent is a half-line cutting a cycle at a point where both have the same direction. The transformation by reciprocal directions transforms oriented spheres into oriented spheres and oriented planes into oriented planes, leaving invariant the "tangential distance" of two cycles (the distance between the points of each one of their common tangents), and also conserves the lines of curvature . [ 39 ] Laguerre (1882) applied the transformation to two cycles under the following conditions: Their radical axis is the axis of transformation, and their common tangents are parallel to two fixed directions of the half-lines that are transformed into themselves (Laguerre called this specific method the "transformation by reciprocal half-lines", which was later called "Laguerre inversion" [ 40 ] [ 41 ] ). Setting R {\displaystyle R} and R ′ {\displaystyle R'} as the radii of the cycles, and D {\displaystyle D} and D ′ {\displaystyle D'} as the distances of their centers to the axis, he obtained: [ M 28 ]
with the transformation: [ M 29 ]
Darboux (1887) obtained the same formulas in different notation (with z = D {\displaystyle z=D} and k = α {\displaystyle k=\alpha } ) in his treatment of the "transformation by reciprocal directions", though he included the x {\displaystyle x} and y {\displaystyle y} coordinates as well: [ M 30 ]
with
consequently he obtained the relation
As mentioned above, oriented spheres in R 3 can be represented by points of four-dimensional space R 4 using minimal (isotropy) projection, which became particularly important in Laguerre's geometry. [ 5 ] For instance, E. Müller (1898) based his discussion of oriented spheres on the fact that they can be mapped upon the points of a plane manifold of four dimensions (which he likened to Fiedler's "cyclography" from 1882). He systematically compared the transformations by reciprocal radii (calling it "inversion at a sphere") with the transformations by reciprocal directions (calling it "inversion at a plane sphere complex"). [ M 31 ] Following Müller's paper, Smith (1900) discussed Laguerre's transformation and the related "group of the geometry of reciprocal directions". Alluding to Klein's (1893) treatment of minimal projection, he pointed out that this group "is simply isomorphic with the group of all displacements and symmetry transformations in space of four dimensions". [ M 32 ] Smith obtained the same transformation as Laguerre and Darboux in different notation, calling it "inversion into a spherical complex": [ M 33 ]
with the relations
In 1905 both Poincaré and Einstein pointed out that the Lorentz transformation of special relativity (setting c = 1 {\displaystyle c=1} )
leaves the relation x 2 + y 2 + z 2 − t 2 {\displaystyle x^{2}+y^{2}+z^{2}-t^{2}} invariant. [ 2 ] Einstein stressed the point that by this transformation a spherical light wave in one frame is transformed into a spherical light wave in another one. [ 42 ] Poincaré showed that the Lorentz transformation can be seen as a rotation in four-dimensional space with time as fourth coordinate, with Minkowski deepening this insight much further (see History of special relativity ).
As shown above, also Laguerre's transformation by reciprocal directions or half-lines – later called Laguerre inversion [ 40 ] [ 41 ] – in the form given by Darboux (1887) leaves the expression x 2 + y 2 + z 2 − R 2 {\displaystyle x^{2}+y^{2}+z^{2}-R^{2}} invariant. Subsequently, the relation to the Lorentz transformation was noted by several authors. For instance, Bateman (1910) argued that this transformation (which he attributed to Ribaucour) is "identical" to the Lorentz transformation. [ M 2 ] In particular, he argued (1912) that the variant given by Darboux (1887) corresponds to the Lorentz transformation in z {\displaystyle z} direction, if R = c t {\displaystyle R=ct} , R ′ = c t ′ {\displaystyle R'=ct'} , and the k {\displaystyle k} terms are replaced by velocities. [ M 34 ] Bateman (1910) also sketched geometric representations of relativistic light spheres using such spherical systems. [ M 35 ] [ 43 ] However, Kubota (1925) responded to Bateman by arguing that the Laguerre inversion is involutory whereas the Lorentz transformation is not. He concluded that in order to make them equivalent, the Laguerre inversion has to be combined with a reversal of direction of the cycles. [ M 36 ]
The specific relation between the Lorentz transformation and the Laguerre inversion can also be demonstrated as follows (see H.R. Müller (1948) [ M 37 ] for analogous formulas in different notation). Laguerre's inversion formulas from 1882 (equivalent to those of Darboux in 1887) read:
by setting
it follows
finally by setting D = x , D ′ = x ′ , R = t , R ′ = t ′ {\displaystyle D=x,D'=x',R=t,R'=t'} the Laguerre inversion becomes very similar to the Lorentz transformation except that the expression t − v x {\displaystyle t-vx} is reversed into w x − t {\displaystyle wx-t} :
According to Müller, the Lorentz transformation can be seen as the product of an even number of such Laguerre inversions that change the sign. First an inversion is conducted into plane π 1 {\displaystyle \pi _{1}} which is inclined with respect to plane π {\displaystyle \pi } under a certain angle, followed by another inversion back to π {\displaystyle \pi } . [ M 37 ] See section #Laguerre group isomorphic to Lorentz group for more details of the connection between the Laguerre inversion to other variants of Laguerre transformations.
Timerding (1911) [ M 38 ] used Laguerre's concept of oriented spheres in order to represent and derive the Lorentz transformation. Given a sphere of radius r {\displaystyle r} , with x {\displaystyle x} as the distance between its center and the central plane, he obtained the relations to a corresponding sphere
resulting in the transformation
By setting λ = v / c {\displaystyle \lambda =v/c} and r = c t {\displaystyle r=ct} , it becomes the Lorentz transformation.
Following Timerding and Bateman, Ogura (1913) analyzed a Laguerre transformation of the form [ M 39 ]
which become the Lorentz transformation with
He stated that "the Laguerre transformation in sphere manifoldness is equivalent to the Lorentz transformation in spacetime manifoldness".
As shown above, the group of conformal point transformations in R n (composed of motions, similarities, and inversions) can be related by minimal projection to the group of contact transformations in R n-1 transforming circles or spheres into other circles or spheres. In addition, Lie (1871, 1896) pointed out that in R 3 there is a 7-parameter subgroup of point transformations composed of motions and similarities, which by using minimal projection corresponds to a 7-parameter subgroup of contact transformations in R 2 transforming circles into circles. [ M 40 ] These relations were further studied by Smith (1900), [ M 32 ] Blaschke (1910), [ M 41 ] Coolidge (1916) [ 44 ] and others, who pointed out the connection to Laguerre's geometry of reciprocal directions related to oriented lines, circles, planes and spheres. Therefore, Smith (1900) called it the "group of the geometry of reciprocal directions", [ M 32 ] and Blaschke (1910) used the expression "Laguerre group". [ M 41 ] The "extended Laguerre group" consists of motions and similarities, having 7 parameters in R 2 transforming oriented lines and circles, or 11 parameters in R 3 transforming oriented planes and spheres. If similarities are excluded, it becomes the "restricted Laguerre group" having 6 parameters in R 2 and 10 parameters in R 3 , consisting of orientation-preserving or orientation-reversing motions, and preserving the tangential distance between oriented circles or spheres. [ M 42 ] [ 45 ] Subsequently, it became common that the term Laguerre group only refers to the restricted Laguerre group. [ 45 ] [ 46 ] It was also noted that the Laguerre group is part of a wider group conserving tangential distances, called the "equilong group" by Scheffers (1905). [ M 43 ] [ 47 ]
In R 2 the Laguerre group leaves invariant the relation d x 2 + d y 2 − d r 2 {\displaystyle dx^{2}+dy^{2}-dr^{2}} , which can be extended to arbitrary R n as well. [ 48 ] For instance, in R 3 it leaves invariant the relation d x 2 + d y 2 + d z 2 − d r 2 {\displaystyle dx^{2}+dy^{2}+dz^{2}-dr^{2}} . [ 49 ] This is equivalent to relation d x 2 + d y 2 + d z 2 + d r 2 {\displaystyle dx^{2}+dy^{2}+dz^{2}+dr^{2}} in R 4 by using minimal (isotropy) projection with imaginary radius coordinate, or cyclographic projection (in descriptive geometry ) with real radius coordinate. [ 9 ] The transformations forming the Laguerre group can be further differentiated into "direct Laguerre transformations" which are related to motions preserving both the tangential distance as well as the sign; or "indirect Laguerre transformations" which are related to orientation-reversing motions, preserving the tangential distance with the sign reversed. [ M 43 ] [ 50 ] The Laguerre inversion first given by Laguerre in 1882 is involutory , thus it belongs to the indirect Laguerre transformations. Laguerre himself did not discuss the group related to his inversion, but it turned out that every Laguerre transformation can be generated by at most four Laguerre inversions and every direct Laguerre transformation is the product of two involutory transformations, thus Laguerre inversions are of special importance because they are generating operators of the entire Laguerre group. [ M 44 ] [ 51 ]
It was noted that the Laguerre group is indeed isomorphic to the Lorentz group (or the Poincaré group if translations are included), as both groups leave invariant the form d x 1 2 + d x 2 2 + d x 3 2 − d x 4 2 {\displaystyle dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}-dx_{4}^{2}} . After the first comparison of the Lorentz transformation and the Laguerre inversion by Bateman (1910) as mentioned above , the equivalence of both groups was pointed out by Cartan in 1912 [ M 45 ] and 1914, [ M 46 ] and he expanded upon it in 1915 (published 1955) in the French version of Klein's encyclopedia . [ 8 ] Also Poincaré (1912, published 1921) wrote: [ M 3 ] [ 52 ]
Mr. Cartan has recently given a curious example. We know the importance in mathematical physics of what has been called the Lorentz group; it is this group upon which our new ideas on the principle of relativity and the dynamics of the electron are based. On the other hand, Laguerre once introduced into geometry a group of transformations that change the spheres into spheres. These two groups are isomorphic, so that mathematically these two theories, one physical, the other one geometric, show no essential difference. [ M 47 ]
Others who noticed this connection include Coolidge (1916), [ 9 ] Klein & Blaschke (1926), [ 10 ] Blaschke (1929), [ 11 ] H.R. Müller , [ M 48 ] Kunle & Fladt (1970), [ 12 ] Benz (1992). [ 13 ] It was recently pointed out:
A Laguerre transformation (L-transform) is a mapping which is bijective on the sets of oriented planes and oriented spheres, respectively, and preserves tangency between plane and sphere. L-transforms are more easily understood if we use the so-called cyclographic model of Laguerre geometry. There, an oriented sphere S {\displaystyle S} is represented as point S := ( m , R ) ∈ R 4 {\displaystyle \mathbf {S} \operatorname {\text{:=}} (\mathbf {m} ,R)\in \mathbb {R} ^{4}} . An oriented plane P {\displaystyle P} in E 3 {\displaystyle E^{3}} may be interpreted as the set of all oriented spheres which are tangent to P {\displaystyle P} . Mapping P {\displaystyle P} via this set of spheres into R 4 {\displaystyle \mathbb {R} ^{4}} , one finds a hyperplane in R 4 {\displaystyle \mathbb {R} ^{4}} which is parallel to a tangent hyperplane of the cone x 1 2 + x 2 2 + x 3 2 − x 4 2 = 0 {\displaystyle x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2}=0} . In the cyclographic model, an L-transform is seen as a special affine map (Lorentz transformation),... | https://en.wikipedia.org/wiki/Spherical_wave_transformation |
In mathematics, a field K with an absolute value is called spherically complete if the intersection of every decreasing sequence of balls (in the sense of the metric induced by the absolute value) is nonempty: [ 1 ]
The definition can be adapted also to a field K with a valuation v taking values in an arbitrary ordered abelian group: ( K , v ) is spherically complete if every collection of balls that is totally ordered by inclusion has a nonempty intersection.
Spherically complete fields are important in nonarchimedean functional analysis , since many results analogous to theorems of classical functional analysis require the base field to be spherically complete. [ 2 ]
This mathematical analysis –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Spherically_complete_field |
Sphericity is a measure of how closely the shape of a physical object resembles that of a perfect sphere . For example, the sphericity of the balls inside a ball bearing determines the quality of the bearing, such as the load it can bear or the speed at which it can turn without failing. Sphericity is a specific example of a compactness measure of a shape.
Sphericity applies in three dimensions ; its analogue in two dimensions , such as the cross sectional circles along a cylindrical object such as a shaft , is called roundness .
Defined by Wadell in 1935, [ 1 ] the sphericity, Ψ {\displaystyle \Psi } , of an object is the ratio of the surface area of a sphere with the same volume to the object's surface area:
where V p {\displaystyle V_{p}} is volume of the object and A p {\displaystyle A_{p}} is the surface area. The sphericity of a sphere is unity by definition and, by the isoperimetric inequality , any shape which is not a sphere will have sphericity less than 1.
The sphericity, Ψ {\displaystyle \Psi } , of an oblate spheroid (similar to the shape of the planet Earth ) is:
where a and b are the semi-major and semi-minor axes respectively.
Hakon Wadell defined sphericity as the surface area of a sphere of the same volume as the object divided by the actual surface area of the object.
First we need to write surface area of the sphere, A s {\displaystyle A_{s}} in terms of the volume of the object being measured, V p {\displaystyle V_{p}}
therefore
hence we define Ψ {\displaystyle \Psi } as: | https://en.wikipedia.org/wiki/Sphericity |
In graph theory , the sphericity of a graph is a graph invariant defined to be the smallest dimension of Euclidean space required to realize the graph as an intersection graph of unit spheres . The sphericity of a graph is a generalization of the boxicity and cubicity invariants defined by F.S. Roberts in the late 1960s. [ 1 ] [ 2 ] The concept of sphericity was first introduced by Hiroshi Maehara in the early 1980s. [ 3 ]
Let G {\displaystyle G} be a graph. Then the sphericity of G {\displaystyle G} , denoted by sph ( G ) {\displaystyle \operatorname {sph} (G)} , is the smallest integer n {\displaystyle n} such that G {\displaystyle G} can be realized as an intersection graph of unit spheres in n {\displaystyle n} -dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} . [ 4 ]
Sphericity can also be defined using the language of space graphs as follows. For a finite set of points in some n {\displaystyle n} -dimensional Euclidean space, a space graph is built by connecting pairs of points with a line segment when their Euclidean distance is less than some specified constant. Then the sphericity of a graph G {\displaystyle G} is the minimum n {\displaystyle n} such that G {\displaystyle G} is isomorphic to a space graph in R n {\displaystyle \mathbb {R} ^{n}} . [ 3 ]
Graphs of sphericity 1 are known as interval graphs or indifference graphs . Graphs of sphericity 2 are known as unit disk graphs .
The sphericity of certain graph classes can be computed exactly. The following sphericities were given by Maehara on page 56 of his original paper on the topic.
The most general known upper bound on sphericity is as follows. Assuming the graph is not complete , then sph ( G ) ≤ | G | − ω ( G ) {\displaystyle \operatorname {sph} (G)\leq |G|-\omega (G)} where ω ( G ) {\displaystyle \omega (G)} is the clique number of G {\displaystyle G} and | G | {\displaystyle |G|} denotes the number of vertices of G . {\displaystyle G.} [ 3 ] | https://en.wikipedia.org/wiki/Sphericity_(graph_theory) |
Spherics (sometimes spelled sphaerics or sphaerica ) is a term used in the history of mathematics for historical works on spherical geometry , [ 1 ] [ 2 ] exemplified by the Spherics ( Ancient Greek : τὰ σφαιρικά tá sphairiká ), a treatise by the Hellenistic mathematician Theodosius (2nd or early 1st century BC), [ 3 ] and another treatise of the same title by Menelaus of Alexandria ( c. 100 AD ). [ 4 ]
This geometry-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Spherics |
In four-dimensional geometry , the spherinder , or spherical cylinder or spherical prism , is a geometric object, defined as the Cartesian product of a 3- ball (or solid 2- sphere ) of radius r 1 and a line segment of length 2 r 2 :
Like the duocylinder , it is also analogous to a cylinder in 3-space, which is the Cartesian product of a disk with a line segment . It is a rotatope and a toratope.
It can be seen in 3-dimensional space by stereographic projection as two concentric spheres, in a similar way that a tesseract (cubic prism) can be projected as two concentric cubes, and how a circular cylinder can be projected into 2-dimensional space as two concentric circles.
One can define a "spherindrical" coordinate system ( r , θ , φ , w ) , consisting of spherical coordinates with an extra coordinate w . This is analogous to how cylindrical coordinates are defined: r and φ being polar coordinates with an elevation coordinate z . Spherindrical coordinates can be converted to Cartesian coordinates using the formulas x = r cos φ sin θ y = r sin φ sin θ z = r cos θ w = w {\displaystyle {\begin{aligned}x&=r\cos \varphi \sin \theta \\y&=r\sin \varphi \sin \theta \\z&=r\cos \theta \\w&=w\end{aligned}}} where r is the radius, θ is the zenith angle, φ is the azimuthal angle, and w is the height. Cartesian coordinates can be converted to spherindrical coordinates using the formulas r = x 2 + y 2 + z 2 φ = arctan y x θ = arccot z x 2 + y 2 w = w {\displaystyle {\begin{aligned}r&={\sqrt {x^{2}+y^{2}+z^{2}}}\\\varphi &=\arctan {\frac {y}{x}}\\\theta &=\operatorname {arccot} {\frac {z}{\sqrt {x^{2}+y^{2}}}}\\w&=w\end{aligned}}} The hypervolume element for spherindrical coordinates is d H = r 2 sin θ d r d θ d φ d w , {\displaystyle \mathrm {d} H=r^{2}\sin {\theta }\,\mathrm {d} r\,\mathrm {d} \theta \,\mathrm {d} \varphi \,\mathrm {d} w,} which can be derived by computing the Jacobian .
Given a spherinder with a spherical base of radius r and a height h , the hypervolume of the spherinder is given by H = 4 3 π r 3 h {\displaystyle H={\frac {4}{3}}\pi r^{3}h}
The surface volume of a spherinder, like the surface area of a cylinder, is made up of three parts:
Therefore, the total surface volume is
S V = 8 3 π r 3 + 4 π r 2 h {\displaystyle SV={\frac {8}{3}}\pi r^{3}+4\pi r^{2}h}
The above formulas for hypervolume and surface volume can be proven using integration. The hypervolume of an arbitrary 4D region is given by the quadruple integral H = ⨌ D d H {\displaystyle H=\iiiint \limits _{D}\mathrm {d} H}
The hypervolume of the spherinder can be integrated over spherindrical coordinates. H s p h e r i n d e r = ⨌ D d H = ∫ 0 h ∫ 0 2 π ∫ 0 π ∫ 0 R r 2 sin θ d r d θ d φ d w = 4 3 π R 3 h {\displaystyle H_{\mathrm {spherinder} }=\iiiint \limits _{D}\mathrm {d} H=\int _{0}^{h}\int _{0}^{2\pi }\int _{0}^{\pi }\int _{0}^{R}r^{2}\sin {\theta }\,\mathrm {d} r\,\mathrm {d} \theta \,\mathrm {d} \varphi \,\mathrm {d} w={\frac {4}{3}}\pi R^{3}h}
The spherinder is related to the uniform prismatic polychora , which are Cartesian products of a regular or semiregular polyhedron and a line segment . There are eighteen convex uniform prisms based on the Platonic and Archimedean solids ( tetrahedral prism , truncated tetrahedral prism , cubic prism , cuboctahedral prism , octahedral prism , rhombicuboctahedral prism , truncated cubic prism , truncated octahedral prism , truncated cuboctahedral prism , snub cubic prism , dodecahedral prism , icosidodecahedral prism , icosahedral prism , truncated dodecahedral prism , rhombicosidodecahedral prism , truncated icosahedral prism , truncated icosidodecahedral prism , snub dodecahedral prism ), plus an infinite family based on antiprisms , and another infinite family of uniform duoprisms , which are products of two regular polygons . | https://en.wikipedia.org/wiki/Spherinder |
The " spherium " model consists of two electrons trapped on the surface of a sphere of radius R {\displaystyle R} . It has been used by Berry and collaborators [ 1 ] to understand both weakly and strongly correlated systems and to suggest an "alternating" version of Hund's rule . Seidl studies this system in the context of density functional theory (DFT) to develop new correlation functionals within the adiabatic connection . [ 2 ]
The electronic Hamiltonian in atomic units is
where u {\displaystyle u} is the interelectronic distance.
For the singlet S states, it can be then shown [ 3 ] that the wave function S ( u ) {\displaystyle S(u)} satisfies the Schrödinger equation
By introducing the dimensionless variable x = u / 2 R {\displaystyle x=u/2R} , this becomes a Heun equation with singular points at x = − 1 , 0 , + 1 {\displaystyle x=-1,0,+1} . Based on the known solutions of the Heun equation, we seek wave functions of the form
and substitution into the previous equation yields the recurrence relation
with the starting values s 0 = s 1 = 1 {\displaystyle s_{0}=s_{1}=1} . Thus, the Kato cusp condition is
The wave function reduces to the polynomial
(where m {\displaystyle m} the number of roots between 0 {\displaystyle 0} and 2 R {\displaystyle 2R} ) if, and only if, s n + 1 = s n + 2 = 0 {\displaystyle s_{n+1}=s_{n+2}=0} . Thus, the energy E n , m {\displaystyle E_{n,m}} is a root of the polynomial equation s n + 1 = 0 {\displaystyle s_{n+1}=0} (where deg s n + 1 = ⌊ ( n + 1 ) / 2 ⌋ {\displaystyle \deg s_{n+1}=\lfloor (n+1)/2\rfloor } ) and the corresponding radius R n , m {\displaystyle R_{n,m}} is found from the previous equation which yields
S n , m ( u ) {\displaystyle S_{n,m}(u)} is the exact wave function of the m {\displaystyle m} -th excited state of singlet S symmetry for the radius R n , m {\displaystyle R_{n,m}} .
We know from the work of Loos and Gill [ 3 ] that the HF energy of the lowest singlet S state is E H F = 1 / R {\displaystyle E_{\rm {HF}}=1/R} . It follows that the exact correlation energy for R = 3 / 2 {\displaystyle R={\sqrt {3}}/2} is E c o r r = 1 − 2 / 3 ≈ − 0.1547 {\displaystyle E_{\rm {corr}}=1-2/{\sqrt {3}}\approx -0.1547} which is much larger than the limiting correlation energies of the helium-like ions ( − 0.0467 {\displaystyle -0.0467} ) or Hooke's atoms ( − 0.0497 {\displaystyle -0.0497} ). This confirms the view that electron correlation on the surface of a sphere is qualitatively different from that in three-dimensional physical space.
Loos and Gill [ 4 ] considered the case of two electrons confined to a 3-sphere repelling Coulombically. They report a ground state energy of ( − .0476 {\displaystyle -.0476} ). | https://en.wikipedia.org/wiki/Spherium |
A spheroplast (or sphaeroplast in British usage) is a microbial cell from which the cell wall has been almost completely removed, as by the action of penicillin or lysozyme. According to some definitions, the term is used to describe Gram-negative bacteria . [ 3 ] [ 4 ] According to other definitions, the term also encompasses yeasts . [ 5 ] [ 6 ] The name spheroplast stems from the fact that after the microbe's cell wall is digested, membrane tension causes the cell to acquire a characteristic spherical shape. [ 4 ] Spheroplasts are osmotically fragile, and will lyse if transferred to a hypotonic solution. [ 5 ]
When used to describe Gram-negative bacteria, the term spheroplast refers to cells from which the peptidoglycan component but not the outer membrane component of the cell wall has been removed. [ 2 ] [ 5 ]
Various antibiotics convert Gram-negative bacteria into spheroplasts. These include peptidoglycan synthesis inhibitors such as fosfomycin , vancomycin , moenomycin, lactivicin and the β-lactam antibiotics . [ 1 ] [ 2 ] Antibiotics that inhibit biochemical pathways directly upstream of peptidoglycan synthesis induce spheroplasts too (e.g. fosmidomycin , phosphoenolpyruvate ). [ 1 ] [ 2 ]
In addition to the above antibiotics, inhibitors of protein synthesis (e.g. chloramphenicol , oxytetracycline , several aminoglycosides ) and inhibitors of folic acid synthesis (e.g. trimethoprim , sulfamethoxazole ) also cause Gram-negative bacteria to form spheroplasts. [ 2 ]
The enzyme lysozyme causes Gram-negative bacteria to form spheroplasts, but only if a membrane permeabilizer such as lactoferrin or ethylenediaminetetraacetate (EDTA) is used to ease the enzyme's passage through the outer membrane . [ 2 ] [ 7 ] EDTA acts as a permeabilizer by binding to divalent ions such as Ca 2+ and removing them from the outer membrane. [ 8 ]
The yeast Candida albicans can be converted to spheroplasts using the enzymes lyticase , chitinase and β-glucuronidase . [ 9 ]
From the 1960s into the 1990s, Merck and Co. used a spheroplast screen as a primary method for discovery of antibiotics that inhibit cell wall biosynthesis. In this screen devised by Eugene Dulaney, growing bacteria were exposed to test substances under hypertonic conditions. Inhibitors of cell wall synthesis caused growing bacteria to form spheroplasts. This screen enabled the discovery of fosfomycin, cephamycin C , thienamycin and several carbapenems . [ 1 ]
Specially prepared giant spheroplasts of Gram-negative bacteria can be used to study the function of bacterial ion channels through a technique called patch clamp , which was originally designed for characterizing the behavior of neurons and other excitable cells. To prepare giant spheroplasts, bacteria are treated with a septation inhibitor (e.g. cephalexin ). This causes the bacteria to form filaments , elongated cells that lack internal cross-walls. [ 10 ] After a period of time, the cell walls of the filaments are digested, and the bacteria collapse into very large spheres surrounded by just their cytoplasmic and outer membranes. The membranes can then be analyzed on a patch clamp apparatus to determine the phenotype of the ion channels embedded in it. It is also common to overexpress a particular channel to amplify its effect and make it easier to characterize.
The technique of patch clamping giant E. coli spheroplasts has been used to study the native mechanosensitive channels (MscL, MscS, and MscM) of E. coli . [ 11 ] [ 12 ] It has been extended to study other heterologously expressed ion channels and it has been shown that the giant E. coli spheroplast can be used as an ion-channel expression system comparable to the Xenopus oocyte . [ 13 ] [ 14 ] [ 15 ] [ 16 ]
Yeast cells are normally protected by a thick cell wall which makes extraction of cellular proteins difficult. [ citation needed ] Enzymatic digestion of the cell wall with zymolyase, creating spheroplasts, renders the cells vulnerable to easy lysis with detergents or rapid osmolar pressure changes. [ 9 ]
Bacterial spheroplasts, with suitable recombinant DNA inserted into them, can be used to transfect animal cells. Spheroplasts with recombinant DNA are introduced into the media containing animal cells and are fused by polyethylene glycol (PEG). With this method, nearly 100% of the animal cells may take up the foreign DNA. [ 17 ] Upon conducting experiments following a modified Hanahan protocol using calcium chloride in E. coli , it was determined that spheroplasts may be able to transform at 4.9x10 −4 . [ 18 ] | https://en.wikipedia.org/wiki/Spheroplast |
In polymer physics , spherulites (from Greek sphaira = ball and lithos = stone) are spherical semicrystalline regions inside non- branched linear polymers. Their formation is associated with crystallization of polymers from the melt and is controlled by several parameters such as the number of nucleation sites, structure of the polymer molecules, cooling rate, etc. Depending on those parameters, spherulite diameter may vary in a wide range from a few micrometers to millimeters. Spherulites are composed of highly ordered lamellae , which result in higher density, hardness, but also brittleness when compared to disordered regions in a polymer. The lamellae are connected by amorphous regions which provide elasticity and impact resistance. Alignment of the polymer molecules within the lamellae results in birefringence producing a variety of colored patterns, including a Maltese cross , when spherulites are viewed between crossed polarizers in an optical microscope .
If a molten linear polymer (such as polyethylene ) is cooled down rapidly, then the orientation of its molecules, which are randomly aligned, curved and entangled remain frozen and the solid has disordered structure. However, upon slow cooling, some polymer chains take on a certain orderly configuration : they align themselves in plates called crystalline lamellae . [ 2 ]
Growth from the melt would follow the temperature gradient (see figure). For example, if the gradient is directed normal to the direction of molecular alignment then the lamella growth sideward into a planar crystallite. However, in absence of thermal gradient, growth occurs radially, in all directions resulting in spherical aggregates, that is spherulites. The largest surfaces of the lamellae are terminated by molecular bends and kinks, and growth in this direction results in disordered regions. Therefore, spherulites have semicrystalline structure where highly ordered lamellae plates are interrupted by amorphous regions. [ 2 ] [ 3 ]
The size of spherulites varies in a wide range, from micrometers up to 8 centimeter [ 4 ] and is controlled by the nucleation. Strong supercooling or intentional addition of crystallization seeds results in relatively large number of nucleation sites; then spherulites are numerous and small and interact with each other upon growth. In case of fewer nucleation sites and slow cooling, a few larger spherulites are created. [ 5 ] [ 6 ]
The seeds can be induced by impurities, plasticizers, fillers, dyes and other substances added to improve other properties of the polymer. This effect is poorly understood and irregular, so that the same additive can promote nucleation in one polymer, but not in another. Many of the good nucleating agents are metal salts of organic acids, which themselves are crystalline at the solidification temperature of the polymer solidification. [ 1 ]
Formation of spherulites affects many properties of the polymer material; in particular, crystallinity, density , tensile strength and Young's modulus of polymers increase during spherulization. This increase is due to the lamellae fraction within the spherulites, where the molecules are more densely packed than in the amorphous phase. Stronger intermolecular interaction within the lamellae accounts for increased hardness, but also for higher brittleness. On the other hand, the amorphous regions between the lamellae within the spherulites give the material certain elasticity and impact resistance. [ 2 ]
Changes in mechanical properties of polymers upon formation of spherulites however strongly depend on the size and density of the spherulites. A representative example is shown in the figure demonstrating that the strain at failure rapidly decreases with the increase in the spherulite size and thus with the decrease in their number in isotactic polypropylene . Similar trends are observed for tensile strength, yield stress and toughness. [ 7 ] Increase in the total volume of the spherulites results in their interaction as well as shrinkage of the polymer, which becomes brittle and easily cracks under load along the boundaries between the spherulites. [ 7 ]
Alignment of the polymer molecules within the lamellae results in birefringence producing a variety of colored patterns when spherulites are viewed between crossed polarizers in an optical microscope . In particular, the so-called " Maltese cross " is often present which consists of four dark perpendicular cones diverging from the origin (see right picture), sometimes with a bright center (front picture). Its formation can be explained as follows. Linear polymer chains can be regarded as a linear polarizers . If their direction coincides with that of one of the crossed polarizers then little light is transmitted; the transmission is increased when the chains make a non-zero angle with both polarizers, and the induced transmittance is dependent on the wavelength, partly because of the absorption properties of the polymer. [ 8 ] [ 9 ]
This effect results in the dark perpendicular cones ( Maltese cross ) and colored brighter regions in between them in the front and right pictures. It reveals that the molecular axis of the polymer molecules in the spherules is either normal or perpendicular to the radius vector , i.e. molecular orientation is uniform when going along a line from the spherulite center to its edge along its radius. However, this orientation changes with rotation angle. [ 8 ] [ 9 ] The pattern may be different (bright or dark) for the center of the spherulites indicating misorientation of the molecules in the nucleation seeds of individual spherulites. Any dark or light spots are dependent on the angle made with the polarizer, which results in a symmetrical image due to the spherical shape.
When spherulites were rotated in their plane, the corresponding Maltese cross patterns did not change, indicating that the molecular arrangement is homogeneous versus the polar angle. From the birefringence point of view, spherulites can be positive or negative. This distinction depends not on the orientation of the molecules (parallel or perpendicular to the radial direction) but to the orientation of the major refractive index of the molecule relative to the radial vector. The spherulite polarity depends on the constituent molecules, but it can also change with temperature. [ 4 ] | https://en.wikipedia.org/wiki/Spherulite_(polymer_physics) |
The sphincter of Boyden (also known as the choledochal sphincter ) is a sphincter located in the common bile duct before it joins with the pancreatic duct to form the ampulla of vater . This sphincter controls the flow of bile into the pancreatic duct and it helps in filling up of the gallbladder with bile .
The sphincter of Boyden is a smooth muscle sphincter surrounding the common bile duct (ductus choledocus). [ 1 ] [ 2 ] It occurs just before the junction with the pancreatic duct , where the ampulla of Vater is formed. [ 1 ] Occasionally, some fibres also surround the pancreatic duct. [ 2 ]
It is subdivided into two parts - pars superior and pars inferior. The pars inferior is the strongest component of the sphincter of Oddi complex . [ 3 ] [ 4 ]
The sphincter of Boyden controls the flow of bile from the common bile duct into the pancreatic duct . [ 1 ] This helps with filling of the gallbladder with bile .
Its contractions regulate the passage of bile into the gall bladder or the duodenum . [ 4 ] [ 3 ]
This is named after the American anatomist Edward Allen Boyden (1886-1976), who served as the 32nd president of the American Association of Anatomists from 1956 to 1957. | https://en.wikipedia.org/wiki/Sphincter_of_Boyden |
Sphingolipids are a class of lipids containing a backbone of sphingoid bases, which are a set of aliphatic amino alcohols that includes sphingosine . They were discovered in brain extracts in the 1870s and were named after the mythological sphinx because of their enigmatic nature. [ 1 ] [ 2 ] These compounds play important roles in signal transduction and cell recognition . [ 3 ] Sphingolipidoses , or disorders of sphingolipid metabolism, have particular impact on neural tissue . A sphingolipid with a terminal hydroxyl group is a ceramide . Other common groups bonded to the terminal oxygen atom include phosphocholine , yielding a sphingomyelin , and various sugar monomers or dimers , yielding cerebrosides and globosides , respectively. Cerebrosides and globosides are collectively known as glycosphingolipids .
The long-chain bases, sometimes simply known as sphingoid bases, are the first non-transient products of de novo sphingolipid synthesis in both yeast and mammals. These compounds, specifically known as phytosphingosine and dihydrosphingosine (also known as sphinganine, [ 4 ] although this term is less common), are mainly C 18 compounds, with somewhat lower levels of C 20 bases. [ 5 ] Ceramides and glycosphingolipids are N -acyl derivatives of these compounds. [ 6 ]
The sphingosine backbone is O-linked to a (usually) charged head group such as ethanolamine , serine , or choline . [ citation needed ]
The backbone is also amide-linked to an acyl group , such as a fatty acid . [ citation needed ]
Simple sphingolipids, which include the sphingoid bases and ceramides, make up the early products of the sphingolipid synthetic pathways.
Complex sphingolipids may be formed by addition of head groups to ceramide or phytoceramide:
De novo sphingolipid synthesis begins with formation of 3-keto-dihydrosphingosine by serine palmitoyltransferase . [ 9 ] The preferred substrates for this reaction are palmitoyl-CoA and serine . However, studies have demonstrated that serine palmitoyltransferase has some activity toward other species of fatty acyl-CoA [ 10 ] and alternative amino acids , [ 11 ] and the diversity of sphingoid bases has recently been reviewed. [ 12 ] Next, 3-keto-dihydrosphingosine is reduced to form dihydrosphingosine. Dihydrosphingosine is acylated by one of six (dihydro)-ceramide synthase, CerS - originally termed LASS - to form dihydroceramide. [ 13 ] The six CerS enzymes have different specificity for acyl-CoA substrates, resulting in the generation of dihydroceramides with differing chain lengths (ranging from C14-C26). Dihydroceramides are then desaturated to form ceramide. [ 14 ]
De novo generated ceramide is the central hub of the sphingolipid network and subsequently has several fates. It may be phosphorylated by ceramide kinase to form ceramide-1-phosphate. Alternatively, it may be glycosylated by glucosylceramide synthase or galactosylceramide synthase . Additionally, it can be converted to sphingomyelin by the addition of a phosphorylcholine headgroup by sphingomyelin synthase . Diacylglycerol is generated by this process. Finally, ceramide may be broken down by a ceramidase to form sphingosine . Sphingosine may be phosphorylated to form sphingosine-1-phosphate. This may be dephosphorylated to reform sphingosine. [ 15 ]
Breakdown pathways allow the reversion of these metabolites to ceramide. The complex glycosphingolipids are hydrolyzed to glucosylceramide and galactosylceramide. These lipids are then hydrolyzed by beta-glucosidases and beta-galactosidases to regenerate ceramide. Similarly, sphingomyelin may be broken down by sphingomyelinase to form ceramide. [ citation needed ]
The only route by which sphingolipids are converted to non-sphingolipids is through sphingosine-1-phosphate lyase. This forms ethanolamine phosphate and hexadecenal. [ 16 ]
Sphingolipids are commonly believed to protect the cell surface against harmful environmental factors by forming a mechanically stable and chemically resistant outer leaflet of the plasma membrane lipid bilayer . Certain complex glycosphingolipids were found to be involved in specific functions, such as cell recognition and signaling . Cell recognition depends mainly on the physical properties of the sphingolipids, whereas signaling involves specific interactions of the glycan structures of glycosphingolipids with similar lipids present on neighboring cells or with proteins . [ citation needed ]
Recently, simple sphingolipid metabolites , such as ceramide and sphingosine-1-phosphate , have been shown to be important mediators in the signaling cascades involved in apoptosis , proliferation , stress responses, necrosis , inflammation , autophagy , senescence , and differentiation . [ 17 ] [ 18 ] [ 19 ] [ 20 ] [ 21 ] Ceramide-based lipids self-aggregate in cell membranes and form separate phases less fluid than the bulk phospholipids. These sphingolipid-based microdomains, or " lipid rafts " were originally proposed to sort membrane proteins along the cellular pathways of membrane transport. At present, most research focuses on the organizing function during signal transduction. [ 22 ]
Sphingolipids are synthesized in a pathway that begins in the ER and is completed in the Golgi apparatus , but these lipids are enriched in the plasma membrane and in endosomes , where they perform many of their functions. [ 23 ] Transport occurs via vesicles and monomeric transport in the cytosol . Sphingolipids are virtually absent from mitochondria and the ER , but constitute a 20-35 molar fraction of plasma membrane lipids. [ 24 ]
In experimental animals, feeding sphingolipids inhibits colon carcinogenesis , reduces LDL cholesterol and elevates HDL cholesterol . [ 25 ]
Sphingolipids are universal in eukaryotes but are rare in bacteria and archaea , meaning that they are evolutionally very old. Bacteria that do produce sphingolipids are found in some members of the superphylum FCB group ( Sphingobacteria ), particularly family Sphingomonadaceae , some members of the Bdellovibrionota , and some members of the Myxococcota . [ 26 ]
Because of the incredible complexity of mammalian systems, yeast are often used as a model organism for working out new pathways. These single-celled organisms are often more genetically tractable than mammalian cells, and strain libraries are available to supply strains harboring almost any non-lethal open reading frame single deletion. The two most commonly used yeasts are Saccharomyces cerevisiae and Schizosaccharomyces pombe , although research is also done in the pathogenic yeast Candida albicans . [ citation needed ]
In addition to the important structural functions of complex sphingolipids (inositol phosphorylceramide and its mannosylated derivatives), the sphingoid bases phytosphingosine and dihydrosphingosine (sphinganine) play vital signaling roles in S. cerevisiae . These effects include regulation of endocytosis , ubiquitin-dependent proteolysis (and, thus, regulation of nutrient uptake [ 27 ] ), cytoskeletal dynamics, the cell cycle , translation , posttranslational protein modification, and the heat stress response. [ 28 ] Additionally, modulation of sphingolipid metabolism by phosphatidylinositol (4,5)-bisphosphate signaling via Slm1p and Slm2p and calcineurin has recently been described. [ 29 ] Additionally, a substrate-level interaction has been shown between complex sphingolipid synthesis and cycling of phosphatidylinositol 4-phosphate by the phosphatidylinositol kinase Stt4p and the lipid phosphatase Sac1p. [ 30 ]
Higher plants contain a wider variety of sphingolipids than animals and fungi. [ citation needed ]
There are several disorders of sphingolipid metabolism, known as sphingolipidoses . The main members of this group are Niemann-Pick disease , Fabry disease , Krabbe disease , Gaucher disease , Tay–Sachs disease and Metachromatic leukodystrophy . They are generally inherited in an autosomal recessive fashion, but notably Fabry disease is X-linked . Taken together, sphingolipidoses have an incidence of approximately 1 in 10,000, but substantially more in certain populations such as Ashkenazi Jews . Enzyme replacement therapy is available to treat mainly Fabry disease and Gaucher disease , and people with these types of sphingolipidoses may live well into adulthood. The other types are generally fatal by age 1 to 5 years for infantile forms, but progression may be mild for juvenile- or adult-onset forms. [ citation needed ]
Sphingolipids have also been implicated with the frataxin protein (Fxn), the deficiency of which is associated with Friedreich's ataxia (FRDA). Loss of Fxn in the nervous system in mice also activates an iron/sphingolipid/PDK1/Mef2 pathway, indicating that the mechanism is evolutionarily conserved. Furthermore, sphingolipid levels and PDK1 activity are also increased in hearts of FRDA patients, suggesting that a similar pathway is affected in FRDA. [ 31 ] Other research has demonstrated that iron accumulation in the nervous systems of flies enhances the synthesis of sphingolipids, which in turn activates 3-phosphoinositide dependent protein kinase-1 (Pdk1) and myocyte enhancer factor-2 (Mef2) to trigger neurodegeneration of adult photoreceptors. [ 32 ]
Sphingolipids play a key role in neuronal survival in Parkinson's Disease (PD) and their catabolic pathway alteration in the brain is partly represented in cerebrospinal fluid and blood tissues (Table1) and have the diagnostic potential. [ 33 ] | https://en.wikipedia.org/wiki/Sphingolipid |
Sphingosine (2-amino-4-trans-octadecene-1,3-diol) is an 18-carbon amino alcohol with an unsaturated hydrocarbon chain, which forms a primary part of sphingolipids , a class of cell membrane lipids that include sphingomyelin , an important phospholipid .
Sphingosine can be phosphorylated in vivo via two kinases , sphingosine kinase type 1 and sphingosine kinase type 2 . This leads to the formation of sphingosine-1-phosphate , a potent signaling lipid .
Sphingolipid metabolites, such as ceramides , sphingosine and sphingosine-1-phosphate , are lipid signaling molecules involved in diverse cellular processes.
Sphingosine is synthesized from palmitoyl CoA and serine in a condensation required to yield dihydrosphingosine.
Dehydrosphingosine is then reduced by NADPH to dihydrosphingosine (sphinganine), acylated to dihydroceramide and finally oxidized by FAD to ceramide. Sphingosine is then solely formed via degradation of sphingolipid in the lysosome. | https://en.wikipedia.org/wiki/Sphingosine |
Sphingosyl phosphatide refers to a lipid containing phosphorus and a long-chain base.
This biochemistry article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Sphingosyl_phosphatide |
The Sphinx ( Russian : СФИНКС , romanized : SFINKS ) was an experimental Soviet project for a home automation system, commissioned by the State Committee for Science and Technology and designed by Dmitry Azrikan , in collaboration with A. Kolotushkin and V. Goessen, in 1987. [ 1 ] Sphinx, an acronym for Super Functional Integrated Communication System ( Russian : С упер ф ункциональная ин тегрированная к оммуникативная с истема ), was intended to be an ensemble of modules that would allow consumers to easily interact with information systems.
The home environment, as described in a 1987 issue of Soviet magazine Technical Aesthetics ( Russian : Техническая эстетика ), would be composed of "spherical speakers, a detachable monitor, headphones, a handheld remote control with a removable display, a diskette drive , a processor with three memory blocks and more". The modules were designed to be used collectively, or individually by family members, and the number of memory blocks was supposed to be possibly increased endlessly according to the needs of the household so different family members could activate different programs simultaneously.
According to Sergey Moiseyev, Head the VNIITE (Russian design research institute):
The SPHINX equipment was designed to have everything integrated into one single system, and it was not just about creating a smart house : it also had a lot to do with solving some of the more important problems facing Soviet men and women. Say, for instance, that someone wanted to increase the functionality of his or her tape recorder . Back in the day, they would most certainly have come face to face with a number of difficulties with compatibility . Ergonomics too had its share of issues, since quite often even the appearance of the TV and the recorder had little, if anything, in common.
The configuration of the Sphinx station, with detachable monitors and speakers, prefigured the environment of computer stations with peripheral touch pads and accessories that characterises informatics systems in the beginnings of the 21st century. [ 3 ] | https://en.wikipedia.org/wiki/Sphinx_(home_automation_system) |
Spice , spiciness , or spicity , symbol τ , is a term in oceanography referring to variations in the temperature and salinity of seawater over space or time, whose combined effects leave the water's density unchanged. For a given spice, any change in temperature is offset by a change in salinity to maintain unchanged density. An increase in temperature decreases density, but an increase in salinity increases density. Such density-compensated thermohaline variability is ubiquitous in the upper ocean. Warmer, saltier water is more spicy while cooler, less salty water is more minty. [ 1 ] For a density ratio of 1, all the thermohaline variability is spice, and there are no density fluctuations.
The density of seawater controls much of the movement of water, or the thermohaline flow, in the ocean. The density of seawater is primarily determined by the temperature and salinity of that water. Changes in these two main parameters, potential temperature Θ and salinity S, are multiplied with their thermal expansion α {\displaystyle \alpha } or haline contraction coefficient β {\displaystyle \beta } equal to each other; α d Θ {\displaystyle \alpha d\Theta } and β d S {\displaystyle \beta dS} are both proportional to a change in density and are both terms of the linearized equation of state of the ocean ( TEOS-10 ). This similarity is supposed to be relevant for understanding the consequences of sea water mixing. [ 2 ]
α d Θ = β d S {\displaystyle \alpha d\Theta =\beta dS}
The density ρ {\displaystyle \rho } doesn't change over an isopycnal . However, by mixing a change in temperature and salinity can occur. Therefore spiciness τ {\displaystyle \tau } is introduced as variable that is proportional to thermal expansion and haline contraction. Integration of this variable along an isopycnal leads to the following equation.
∫ ρ d τ = ∫ ρ α d Θ = ∫ ρ β d S {\displaystyle \int _{\rho }d\tau =\int _{\rho }\alpha d\Theta =\int _{\rho }\beta dS}
Spiciness could be described as the isothermal gradient of the density that equals the isohaline gradient of the density.
τ = 2 ∫ d ρ d S | Θ d S = − 2 ∫ d ρ d Θ | S d Θ {\displaystyle \tau =2\int {\frac {d\rho }{dS}}|_{\Theta }\quad dS=-2\int {\frac {d\rho }{d\Theta }}|_{S}\,d\Theta }
The isopycnal gradient of spiciness should equal to the isopycnal gradient of temperature and salinity by multiplication with the derivative in the other variable of the density.
d τ | ρ = 2 ρ S d S | ρ = − 2 ρ Θ d Θ | ρ {\displaystyle d\tau |_{\rho }=2\rho _{S}dS|_{\rho }=-2\rho _{\Theta }d\Theta |_{\rho }}
Another mathematical implication for the existence of a spiciness influence manifests itself in a S , Θ {\displaystyle S,\Theta } -diagram, where the negative slope of the isopleths equals the ratio between the temperature- and salinity derivative of the spiciness. [ 3 ]
d S d Θ | τ = − τ Θ τ S {\displaystyle {\frac {dS}{d\Theta }}|_{\tau }=-{\frac {\tau _{\Theta }}{\tau _{S}}}}
A purpose for introducing spiciness is to decrease the amount of state variables needed; the density at constant depth is a function of potential temperature and salinity and of using both, spiciness can be used. If the goal is to only quantify the variation of water parcels along isopycnals, the variation in absolute salinity or temperature can be used instead because it gives the same information with the same amount of variables. [ 3 ]
Another purpose is to examine how the stability ratio R ρ {\displaystyle R_{\rho }} varies vertically on a water column. The stability ratio is a number determining the involvement of temperature changes relative to the involvement salinity changes in a vertical profile, which yields relevant information about the stability of the water column:
R ρ = ( − ρ Θ Θ z ) / ( ρ S S z ) {\displaystyle R_{\rho }=(-\rho _{\Theta }\Theta _{z})/(\rho _{S}S_{z})}
The vertical variation of this number is often shown in a spiciness-potential density diagram and/or plot, where the angle shows the stability. [ 5 ]
The spiciness can be calculated in several programming languages with the Gibbs SeaWater (GSW) toolbox. [ 6 ] It is used to derive thermodynamic seawater properties and is adopted by the Intergovernmental Oceanographic Commission (IOC), International Association for the Physical Sciences of the Oceans (IAPSO) and the Scientific Committee on Oceanic Research (SCOR). They use the definition of spiciness (gsw_spiciness0(), gsw_spiciness1(), gsw_spiciness2() at respectively 0, 1000 and 2000 dbar) provided by. [ 7 ] These isobars are chosen because they correspond to commonly used potential density surfaces. Areas with constant density but different spiciness have a net water flow of heat and salinity due to diffusion.
The exact definition of spiciness is debated. Specifically, the orthogonality of the density with spiciness and the used scaling factor of potential temperature and salinity. McDougall [ 3 ] claims that orthogonality should not be imposed because:
McDougall [ 7 ] is adopted by the Intergovernmental Oceanographic Commission (IOC), International Association for the Physical Sciences of the Oceans (IAPSO) and the Scientific Committee on Oceanic Research (SCOR) due to their implementation of spiciness in the TEOS-10. [ 6 ]
Huang [ 8 ] claims that the orthogonal system is superior to the non orthogonal system because the coordinates can be regarded as independent and distances between points can be calculated more easily.
McDougall [ 3 ] recommended that the spiciness should not be used. Instead, they recommend that the variation of salinity should be used to differentiate between isopycnal water parcels and the stability ratio R ρ {\displaystyle R_{\rho }} on vertical water columns for stability. | https://en.wikipedia.org/wiki/Spice_(oceanography) |
SpiderCloud Wireless [ 1 ] was founded in November 2006 as Evoke Networks by Peter Wexler, Allan Baw, and Mark Gallagher. The trio incubated the company as Copivia Inc. and hired Mike Gallagher as CEO in October 2007. After closing the Series-A funding in January 2008, the company soon changed its name to SpiderCloud Wireless. The company is now headquartered in Milpitas, California . The company is backed by investors Charles River Ventures , Matrix Partners , Opus Capital, and Shasta Ventures. It has raised around US$125 million in venture capital and is generating revenue from customers such as Vodafone UK , Vodafone Netherlands , Verizon Wireless , Warid Telecom , and more. The company helps mobile operators improve service quality for enterprise customers. [ 2 ]
SpiderCloud was acquired by Corning Inc. on July 19, 2017. [ 3 ] | https://en.wikipedia.org/wiki/SpiderCloud_Wireless |
Spider is a balloon-borne experiment designed to search for primordial gravitational waves imprinted on the cosmic microwave background (CMB). Measuring the strength of this signal puts limits on inflationary theory .
The Spider instrument consists of six degree-resolution telescopes cooled to liquid Helium temperature (4 K ) which observe at frequencies of 100 GHz, 150 GHz, and 280 GHz (corresponding to wavelengths of 3 mm, 2 mm, and 1.1 mm). Each telescope is coupled to a polarisation -sensitive transition-edge bolometer (TES) array cooled to 300 mK . Spider was the first instrument to successfully demonstrate time-domain multiplexed TES detectors in a space-like environment. At the time of the first flight over Antarctica in 2015, Spider was the most sensitive microwave instrument ever made. [ 1 ] [ 2 ]
The primary science goals include:
The first balloon flight of the experiment launched in January 2015 from McMurdo Station , Antarctica, with support from NASA's Columbia Scientific Balloon Facility . This Long Duration Balloon flight lasted for about 17 days, mapping about 10% of the full sky. The data from this flight produced high signal-to-noise images of the intensity and linear polarization of the Cosmic Microwave Background, with noise levels 3—5 times lower than the Planck spacecraft in the same region of the sky, resulting in precise measurements of the CMB and Galactic foreground radiation, as well as a robust limit on the cosmological tensor-to-scalar ratio. Further flights planned for successive seasons enable upgrades and changes to the modular telescope, increased frequency coverage and depth. | https://en.wikipedia.org/wiki/Spider_(polarimeter) |
Spider silk is a protein fibre or silk spun by spiders . Spiders use silk to make webs or other structures that function as adhesive traps to catch prey, to entangle and restrain prey before biting, to transmit tactile information, or as nests or cocoons to protect their offspring. They can use the silk to suspend themselves from height, to float through the air , or to glide away from predators. Most spiders vary the thickness and adhesiveness of their silk according to its use.
In some cases, spiders may use silk as a food source. [ 1 ] While methods have been developed to collect silk from a spider by force, [ 2 ] gathering silk from many spiders is more difficult than from silk-spinning organisms such as silkworms .
All spiders produce silk, although some spiders do not make webs. Silk is tied to courtship and mating . Silk produced by females provides a transmission channel for male vibratory courtship signals, while webs and draglines provide a substrate for female sex pheromones . Observations of male spiders producing silk during sexual interactions are common across widespread taxa. The function of male-produced silk in mating has received little study. [ 3 ]
Silks have a hierarchical structure. The primary structure is the amino acid sequence of its proteins ( spidroin ), mainly consisting of highly repetitive glycine and alanine blocks, [ 4 ] [ 5 ] which is why silks are often referred to as a block co-polymer . On a secondary level, the short side-chained alanine is mainly found in the crystalline domains ( beta sheets ) of the nanofibril. Glycine is mostly found in the so-called amorphous matrix consisting of helical and beta turn structures. [ 5 ] [ 6 ] The interplay between the hard crystalline segments and the strained elastic semi-amorphous regions gives spider silk its extraordinary properties. [ 7 ] [ 8 ] Various compounds other than protein are used to enhance the fibre's properties. Pyrrolidine has hygroscopic properties that keep the silk moist while warding off ant invasion. It occurs in high concentration in glue threads. Potassium hydrogen phosphate releases hydrogen ions in aqueous solution, resulting in a pH of about 4, making the silk acidic and thus protecting it from fungi and bacteria that would otherwise digest the protein. Potassium nitrate is believed to prevent the protein from denaturing in the acidic milieu. [ 9 ]
Termonia introduced this first basic model of silk in 1994. [ 10 ] He suggested crystallites embedded in an amorphous matrix interlinked with hydrogen bonds . Refinements to this model include: semi-crystalline regions were found [ 5 ] as well as a fibrillar skin core model suggested for spider silk, [ 11 ] later visualised by AFM and TEM . [ 12 ] Sizes of the nanofibrillar structure and the crystalline and semi-crystalline regions were revealed by neutron scattering . [ 13 ]
The fibres' microstructural information and macroscopic mechanical properties are related. [ 14 ] Ordered regions (i) mainly reorient by deformation for low-stretched fibres and (ii) the fraction of ordered regions increases progressively for higher fibre stretching.
Each spider and each type of silk has a set of mechanical properties optimised for their biological function.
Most silks, in particular dragline silk, have exceptional mechanical properties. They exhibit a unique combination of high tensile strength and extensibility ( ductility ). This enables a silk fibre to absorb a large amount of energy before breaking ( toughness , the area under a stress-strain curve).
Strength and toughness are distinct quantities. Weight for weight, silk is stronger than steel, but not as strong as Kevlar . Spider silk is, however, tougher than both.
The variability of spider silk fibre mechanical properties is related to their degree of molecular alignment. [ 16 ] Mechanical properties also depend on ambient conditions, i.e. humidity and temperature. [ 17 ]
Young's modulus is the resistance to deformation elastically along the tensile force direction. Unlike steel or Kevlar which are stiff, spider silk is ductile and elastic, having lower Young's modulus. According to Spider Silkome Database, Ariadna lateralis silk has the highest Young's modulus with 37 GPa, [ 18 ] compared to 208 GPa for steel [ 19 ] and 112 GPa for Kevlar. [ 20 ]
A dragline silk's tensile strength is comparable to that of high-grade alloy steel (450−2000 MPa), [ 21 ] [ 22 ] and about half as strong as aramid filaments, such as Twaron or Kevlar (3000 MPa). [ 23 ] According to Spider Silkome Database, Clubiona vigil silk has the highest tensile strength. [ 18 ]
Consisting of mainly protein, silks are about a sixth of the density of steel (1.3 g/cm 3 ). As a result, a strand long enough to circle the Earth would weigh about 2 kilograms (4.4 lb). (Spider dragline silk has a tensile strength of roughly 1.3 GPa . The tensile strength listed for steel might be slightly higher – e.g. 1.65 GPa, [ 24 ] [ 25 ] but spider silk is a much less dense material, so that a given weight of spider silk is five times as strong as the same weight of steel.)
The energy density of dragline spider silk is roughly 1.2 × 10 8 J/m 3 . [ 26 ]
Silks are ductile , with some able to stretch up to five times their relaxed length without breaking.
The combination of strength and ductility gives dragline silks a high toughness (or work to fracture), which "equals that of commercial polyaramid (aromatic nylon) filaments, which themselves are benchmarks of modern polymer fibre technology". [ 27 ] [ 28 ] According to Spider Silkome Database, Araneus ishisawai silk is the toughest. [ 18 ]
Elongation at break compares initial object length to final length at break. According to Spider Silkome Database, Caerostris darwini silk has the highest strain at break for any spider silk, breaking at 65% extension. [ 18 ]
While unlikely to be relevant in nature, dragline silks can hold their strength below -40 °C (-40 °F) and up to 220 °C (428 °F). [ 29 ] As occurs in many materials, spider silk fibres undergo a glass transition . The glass-transition temperature depends on humidity, as water is a plasticiser for spider silk. [ 17 ]
When exposed to water, dragline silks undergo supercontraction, shrinking up to 50% in length and behaving like a weak rubber under tension. [ 17 ] Many hypotheses have attempted to explain its use in nature, most popularly to re-tension webs built in the night using the morning dew. [ citation needed ]
The toughest known spider silk is produced by the species Darwin's bark spider ( Caerostris darwini ): "The toughness of forcibly silked fibers averages 350 MJ/m 3 , with some samples reaching 520 MJ/m 3 . Thus, C. darwini silk is more than twice as tough as any previously described silk and over 10 times tougher than Kevlar". [ 30 ]
Silk fibre is a two-compound pyriform secretion, spun into patterns (called "attachment discs") using a minimum of silk substrate. [ 31 ] The pyriform threads polymerise under ambient conditions, become functional immediately, and are usable indefinitely, remaining biodegradable, versatile and compatible with other materials in the environment. [ 31 ] The adhesive and durability properties of the attachment disc are controlled by functions within the spinnerets. [ 32 ] Some adhesive properties of the silk resemble glue , consisting of microfibrils and lipid enclosures. [ 31 ]
All spiders produce silks, and a single spider can produce up to seven different types of silk for different uses. [ 33 ] This is in contrast to insect silks, where an individual usually only produces a single type. [ 34 ] Spiders use silks in many ways, in accord with the silk's properties. As spiders have evolved, so has their silks' complexity and uses, for example from primitive tube webs 300–400 million years ago to complex orb webs 110 million years ago. [ 35 ]
Meeting the specification for all these ecological uses requires different types of silk presenting different properties, as either a fibre, a structure of fibres, or a globule. These types include glues and fibres. Some types of fibres are used for structural support, others for protective structures. Some can absorb energy effectively, whereas others transmit vibration efficiently. These silk types are produced in different glands; so the silk from a particular gland can be linked to its use.
Many species have different glands to produce silk with different properties for different purposes, including housing, web construction, defence, capturing and detaining prey , egg protection, and mobility (fine "gossamer" thread for ballooning , or for a strand allowing the spider to drop down as silk is extruded). [ 39 ] [ 40 ]
Silk production differs in an important aspect from that of most other fibrous biomaterials. It is pulled on demand from a precursor out of specialised glands, [ 41 ] rather than continuously grown like plant cell walls. [ 26 ]
The spinning process occurs when a fibre is pulled away from the body of a spider, whether by the spider's legs, by the spider's falling under its own weight, or by any other method. The term "spinning" is misleading because no rotation occurs. It comes from analogy to the textile spinning wheels . Silk production is a pultrusion , [ 42 ] similar to extrusion, with the subtlety that the force is induced by pulling at the finished fibre rather than squeezing it out of a reservoir. The fibre is pulled through (possibly multiple) silk glands of multiple types. [ 41 ]
The gland's visible, or external, part is termed the spinneret . Depending on the complexity of the species, spiders have two to eight spinnerets, usually in pairs. Species have varying specialised glands, ranging from a sac with an opening at one end, to the complex, multiple-section ampullate glands of the golden silk orb-weavers . [ 56 ]
Behind each spinneret on the surface of the spider lies a gland, a generalised form of which is shown in the figure.
Throughout the process the silk appears to have a nematic texture, [ 63 ] in a manner similar to a liquid crystal , arising in part due to the high protein concentration of silk dope (around 30% in terms of weight per volume). [ 64 ] This allows the silk to flow through the duct as a liquid while maintaining molecular order.
As an example of a complex spinning field, the spinneret apparatus of an adult Araneus diadematus (garden cross spider) consists of many glands shown below. [ 9 ] A similar gland architecture appears in the black widow spider. [ 65 ]
To artificially synthesise spider silk into fibres, two broad tasks are required. These are synthesis of the feedstock (the unspun silk dope in spiders), and synthesis of the production conditions (the funnel, valve, tapering duct, and spigot). Few strategies have produced silk that can efficiently be synthesised into fibres.
The molecular structure of unspun silk is both complex and long. Though this endows the fibres with desirable properties, it also complicates replication. Various organisms have been used as a basis for attempts to replicate necessary protein components. These proteins must then be extracted, purified, and then spun before their properties can be tested.
Spider silks with comparatively simple molecular structure need complex ducts to be able to form an effective fibre. Approaches:
Feedstock is forced through a hollow needle using a syringe. [ 74 ] [ 75 ]
Although cheap and easy to produce, gland shape and conditions are loosely approximated. Fibres created using this method may need encouragement to solidify by removing water from the fibre with chemicals such as (environmentally undesirable) methanol [ 76 ] or acetone , [ 75 ] and also may require later stretching of the fibre to achieve desirable properties. [ 77 ] [ 74 ]
Placing a solution of spider silk on a superhydrophobic surface can generate sheets, particles, and nanowires of spider silk. [ 78 ] [ 79 ]
Self-assembly of silk at standing liquid-gas interphases of a solution tough and strong sheets. These sheets are now explored for mimicking the basal membrane in tissue modeling. [ 80 ] [ 81 ]
Microfluidics have the advantage of being controllable and able to test spin small volumes of unspun fibre, [ 82 ] [ 83 ] but setup and development costs are high. A patent has been granted and continuously spun fibres have achieved commercial use. [ 84 ]
Electrospinning is an old technique whereby a fluid is held in a container such that it flows out through capillary action. A conducting substrate is positioned below, and a difference in electrical potential is applied between the fluid and the substrate. The fluid is attracted to the substrate, and tiny fibres jump from their point of emission, the Taylor cone , to the substrate, drying as they travel. This method creates nano-scale fibres from silk dissected from organisms and regenerated silk fibroin. [ citation needed ]
Silk can be formed into other shapes and sizes such as spherical capsules for drug delivery, cell scaffolds and wound healing, textiles, cosmetics, coatings, and many others. [ 85 ] [ 86 ] Spider silk proteins can self-assemble on superhydrophobic surfaces into nanowires, as well as micron-sized circular sheets. [ 86 ] Recombinant spider silk proteins can self-assemble at the liquid-air interface of a standing solution to form protein-permeable, strong and flexible nanomembranes that support cell proliferation. Potential applications include skin transplants, and supportive membranes in organ-on-a-chip. [ 87 ] These nanomembranes have been used to create a static in-vitro model of a blood vessel. [ 88 ]
Replicating the complex conditions required to produce comparable fibres has challenged research and early-stage manufacturing. Through genetic engineering , E. coli bacteria, yeasts, plants, silkworms, and animals other than silkworms have been used to produce spider silk-like proteins, which have different characteristics than those from a spider. [ 89 ] Extrusion of protein fibres in an aqueous environment is known as "wet-spinning". This process has produced silk fibres of diameters ranging from 10 to 60 μm, compared to diameters of 2.5–4 μm for natural spider silk. Artificial spider silks have fewer and simpler proteins than natural dragline silk, and consequently offer half the diameter, strength, and flexibility of natural dragline silk. [ 89 ]
The earliest recorded attempt to weave fabric from spider silk was in 1709 by François Xavier Bon who, using a process similar to creating silkworm silk, wove silk derived spider's egg cocoons into stockings and gloves. Fifty years later Jesuit missionary Ramón M. Termeyer [ pl ] invented a reeling device for harvesting spider silk directly from spiders, allowing it to be spun into threads. Neither Bon nor Termeyer were successful in producing commercially viable quantities. [ 121 ]
The development of methods to mass-produce spider silk led to the manufacturing of military, medical, and consumer goods, such as ballistic armour , athletic footwear, personal care products, breast implant and catheter coatings, mechanical insulin pumps, fashion clothing, and outerwear . [ 89 ] However, due to the difficulties in extracting and processing, the largest known piece of cloth made of spider silk is an 11-by-4-foot (3.4 by 1.2 m) textile with a golden tint made in Madagascar in 2009. [ 122 ] Eighty-two people worked for four years to collect over one million golden orb spiders and extract silk from them. [ 123 ] In 2012, spider silk fibres were used to create a set of violin strings. [ 124 ]
Peasants in the southern Carpathian Mountains used to cut up tubes built by Atypus and cover wounds with the inner lining. It reportedly facilitated healing, and connected with the skin. This is believed to be due to the silk's antiseptic properties, [ 125 ] and because silk is rich in vitamin K , which can aid in clotting blood. [ 126 ] [ verify ] N. clavipes silk was used in research concerning mammalian neuronal regeneration. [ 127 ]
Spider silk has been used as a thread for crosshairs in optical instruments such as telescopes, microscopes, [ 128 ] and telescopic rifle sights . [ 129 ] In 2011, silk fibres were used to generate fine diffraction patterns over N-slit interferometric signals used in optical communications. [ 130 ] Silk has been used to create biolenses that could be used in conjunction with lasers to create high-resolution images of the inside of the human body. [ 131 ]
Silk has been used to suspend inertial confinement fusion targets during laser ignition, as it remains considerably elastic and has a high energy to break at temperatures as low as 10–20 K. In addition, it is made from "light" atomic number elements that emit no x-rays during irradiation that could preheat the target, limiting the pressure differential required for fusion. [ 132 ] | https://en.wikipedia.org/wiki/Spider_silk |
Spike-in controls or spike-ins are known quantities of molecules—such as oligonucleotide sequences (RNA, DNA), proteins , or metabolites —added to a biological sample for more accurate quantitative estimation of the molecule of interest across samples and batches. [ 1 ] Spike-ins are particularly used in high-throughput sequencing assays, [ 2 ] where they act as an internal reference to monitor and normalize technical and biological biases introduced during sample processing such as library preparation , handling, and measurement. [ 3 ] [ 4 ] [ 5 ]
Spike-ins can adjust for specific technical biases and enable accurate estimation of the endogenous molecules of interest, resulting in improved data quality and standardization across different samples or experiments. Spike-ins can be synthetic or exogenous material (not originally part of the sample). In sequencing-based assays, exogenous material is typically derived from the genome of a different species such as Drosophila melanogaster or Arabidopsis thaliana . [ 6 ]
Spike-ins are subjected to the same experimental steps and potential biases as the native molecules within a sample after they have been added. They are added early in the experimental workflow, often during or immediately after sample lysis or extraction and prior to sequencing. [ 7 ] As such, the suitability of spike-ins, their design, and subsequently analysis should allow accounting for as many sources of experimental variation as possible. Ideally, the spike-ins closely resemble the input material containing epitopes of interest but allow clear differentiation from the native molecules. [ 1 ] Since the initial amount of each spike-in molecule is known, its measured quantity at the end of the experiment reflects the cumulative effects of technical factors, such as extraction efficiency, enzymatic reaction efficiencies (e.g., reverse transcription , ligation , amplification ), sample loss, and measurement sensitivity.
In sequencing assays, spike-ins can further be combined with unique molecular identifiers to increase sensitivity and specificity. [ 8 ] [ 9 ]
The information obtained from spike-ins is typically leveraged after initial bioinformatics analyses have been carried out — with the final output of such analyses being absolute counts of different spike-in controls for each library. Various spike-in normalization or calibration methods then utilize this information as baseline to adjust the primary signal of interest.
The choice of a normalization method can significantly influence the post-normalization conclusions drawn from an experiment. [ 10 ] The first spike-in normalization method, known as reference-adjusted reads per million (RRPM) used a scaling factor determined from the number of reads aligned to the exogenous genome. [ 11 ] Subsequent methods modified RRPM to consider the counts of spike-in reads derived from the input sample. [ 12 ] A common approach involves determining the ratio between the observed spike-in read counts and the expected counts, or simply calculating the total spike-in reads per sample. These values are then used to derive sample-specific scaling factors. For instance, if a sample yields fewer spike-in reads than expected or fewer than another sample normalized to the same input, its endogenous gene counts are scaled upwards, under the assumption that the lower spike-in recovery reflects a global technical loss for that sample.
More sophisticated methods may use regression analysis [ 13 ] or factor analysis [ 14 ] across multiple spike-ins added at various concentrations to model the relationship between input amount and sequencing output, aiming for a more robust estimate of technical bias.
Several types of spike-in controls are used depending on the application:
Other less used spike-ins may include peptide or metabolite spike-ins. In proteomics and metabolomics , often stable isotope -labeled synthetic peptides (e.g., AQUA peptides) or metabolites, purified proteins or endogenous metabolites or non-endogenous small molecules are added in known amounts for quantification and normalization. [ 17 ] [ 18 ] | https://en.wikipedia.org/wiki/Spike-in_controls |
The spike response model (SRM) [ 1 ] is a spiking neuron model in which spikes are generated by either a deterministic [ 2 ] or a stochastic [ 1 ] threshold process. In the SRM, the membrane voltage V is described as a linear sum of the postsynaptic potentials (PSPs) caused by spike arrivals to which the effects of refractoriness and adaptation are added. The threshold is either fixed or dynamic. In the latter case it increases after each spike. The SRM is flexible enough to account for a variety of neuronal firing pattern in response to step current input. [ 2 ] The SRM has also been used in the theory of computation to quantify the capacity of spiking neural networks; [ 3 ] and in the neurosciences to predict the subthreshold voltage and the firing times of cortical neurons during stimulation with a time-dependent current stimulation. [ 4 ] The name Spike Response Model points to the property that the two important filters ε {\displaystyle \varepsilon } and η {\displaystyle \eta } of the model can be interpreted as the response of the membrane potential to an incoming spike (response kernel ε {\displaystyle \varepsilon } , the PSP) and to an outgoing spike (response kernel η {\displaystyle \eta } , also called refractory kernel). The SRM has been formulated in continuous time and in discrete time. [ 5 ] The SRM can be viewed as a generalized linear model (GLM) [ 6 ] [ 7 ] or as an (integrated version of) a generalized integrate-and-fire model with adaptation. [ 2 ] [ 8 ] [ 9 ]
In the SRM, at each moment in time t, a spike can be generated stochastically with instantaneous stochastic intensity or 'escape function' [ 8 ] [ 9 ] [ 5 ]
that depends on the momentary difference between the membrane voltage V (t) and the dynamic threshold ϑ ( t ) {\displaystyle \vartheta (t)} .
The membrane voltage V (t) at time t is given by [ 8 ] [ 5 ]
where t f is the firing time of spike number f of the neuron, V rest is the resting voltage in the absence of input, I(t-s) is the input current at time t − s and κ ( s ) {\displaystyle \kappa (s)} is a linear filter (also called kernel) that describes the contribution of an input current pulse at time t − s to the voltage at time t . The contributions to the voltage caused by a spike at time t f {\displaystyle t^{f}} are described by the refractory kernel η ( t − t f ) {\displaystyle \eta (t-t^{f})} . In particular,
η ( t − t f ) {\displaystyle \eta (t-t^{f})} describes the time course of the action potential starting at time t f {\displaystyle t^{f}} as well as the spike-afterpotential.
The dynamic threshold ϑ ( t ) {\displaystyle \vartheta (t)} is given by [ 8 ] [ 4 ]
where ϑ 0 {\displaystyle \vartheta _{0}} is the firing threshold of an inactive neuron and θ 1 ( t − t f ) {\displaystyle \theta _{1}(t-t^{f})} describes the increase of the threshold after a spike at time t f {\displaystyle t^{f}} . In case of a fixed threshold [i.e., θ 1 ( t − t f ) {\displaystyle \theta _{1}(t-t^{f})} =0], the refractory kernel η ( t − t f ) {\displaystyle \eta (t-t^{f})} should include only the spike-afterpotential, but not the shape of the spike itself.
A common choice [ 8 ] [ 9 ] [ 5 ] for the 'escape rate' f {\displaystyle f} (that is consistent with biological data [ 4 ] ) is
where τ 0 {\displaystyle \tau _{0}} is a time constant that describes how quickly a spike is fired once the membrane potential reaches the threshold and β {\displaystyle \beta } is a sharpness parameter. For β → ∞ {\displaystyle \beta \to \infty } the threshold becomes sharp and spike firing occurs deterministically at the moment when the membrane potential hits the threshold from below. The sharpness value found in experiments is 1 / β ≈ 4 m V {\displaystyle 1/\beta \approx 4mV} which that neuronal firing becomes non-neglibable as soon the membrane potential is a few mV below the formal firing threshold. The escape rate process via a soft threshold is reviewed in Chapter 9 of the textbook Neuronal Dynamics. [ 8 ]
In a network of N SRM neurons 1 ≤ i ≤ N {\displaystyle 1\leq i\leq N} , the membrane voltage of neuron i {\displaystyle i} is given by [ 5 ]
where t j f ′ {\displaystyle t_{j}^{f'}} are the firing times of neuron j (i.e., its spike train), and η i ( t − t i f ) {\displaystyle \eta _{i}(t-t_{i}^{f})} describes the time course of the spike and the spike after-potential for neuron i, w i j {\displaystyle w_{ij}} and ε i j ( t − t j f ′ ) {\displaystyle \varepsilon _{ij}(t-t_{j}^{f'})} describe the amplitude and time course of an excitatory or inhibitory postsynaptic potential (PSP) caused by the spike t j f ′ {\displaystyle t_{j}^{f'}} of the presynaptic neuron j. The time course ε i j ( s ) {\displaystyle \varepsilon _{ij}(s)} of the PSP results from the convolution of the postsynaptic current I ( t ) {\displaystyle I(t)} caused by the arrival of a presynaptic spike from neuron j.
For simulations, the SRM is usually implemented in discrete time. [ 5 ] [ 10 ] In time step t n {\displaystyle t_{n}} of duration Δ t {\displaystyle \Delta t} , a spike is generated with probability
that depends on the momentary difference between the membrane voltage V and the dynamic threshold ϑ {\displaystyle \vartheta } . The function F is often taken as a standard sigmoidal F ( x ) = 0.5 [ 1 + tanh ( γ x ) ] {\displaystyle F(x)=0.5[1+\tanh(\gamma x)]} [ 11 ] with steepness parameter γ {\displaystyle \gamma } . But the functional form of F can also be calculated from the stochastic intensity f {\displaystyle f} in continuous time as F ( y n ) ≈ 1 − exp [ y n Δ t ] {\displaystyle F(y_{n})\approx 1-\exp[y_{n}\,\Delta t]} where y n = V ( t n ) − ϑ ( t n ) {\displaystyle y_{n}=V(t_{n})-\vartheta (t_{n})} is the distance to threshold. [ 8 ] [ 5 ]
The membrane voltage V ( t n ) {\displaystyle V(t_{n})} in discrete time is given by
where t f is the discretized firing time of the neuron, V rest is the resting voltage in the absence of input, and I ( t k ) {\displaystyle I(t_{k})} is the input current at time t k {\displaystyle t_{k}} (integrated over one time step). The input filter κ ( s ) {\displaystyle \kappa (s)} and the spike-afterpotential η ( s ) {\displaystyle \eta (s)} are defined as in the case of the SRM in continuous time.
For networks of SRM neurons in discrete time we define the spike train of neuron j as a sequence of zeros and ones, { X j ( t m ) ∈ { 0 , 1 } ; m = 1 , 2 , 3 , … } {\displaystyle \{X_{j}(t_{m})\in \{0,1\};m=1,2,3,\dots \}} and rewrite the membrane potential as [ 5 ]
In this notation, the refractory kernel κ ( s ) {\displaystyle \kappa (s)} and the PSP shape ε i j ( s ) {\displaystyle \varepsilon _{ij}(s)} can be interpreted as linear response filters applied to the binary spike trains X j {\displaystyle X_{j}} .
Since the formulation as SRM provides an explicit expression for the membrane voltage (without the detour via a differential equations), SRMs have been the dominant mathematical model in a formal theory of computation with spiking neurons. [ 12 ] [ 13 ] [ 3 ]
The SRM with dynamic threshold has been used to predict the firing time of cortical neurons with a precision of a few milliseconds. [ 4 ] Neurons were stimulated, via current injection, with time-dependent currents of different means and variance while the membrane voltage was recorded. The reliability of predicted spikes was close to the intrinsic reliability when the same time-dependent current was repeated several times. Moreover, extracting the shape of the filters κ ( s ) {\displaystyle \kappa (s)} and η ( s ) {\displaystyle \eta (s)} directly from the experimental data revealed that adaptation extends over time scales from tens of milliseconds to tens of seconds. [ 14 ] [ 15 ] [ 16 ] Thanks to the convexity properties of the likelihood in Generalized Linear Models, [ 6 ] [ 7 ] parameter extraction is efficient. [ 17 ]
SRM0 neurons have been used to construct an associative memory in a network of spiking neurons. [ 5 ] [ 11 ] The SRM network [ 5 ] which stored a finite number of stationary patterns as attractors using a Hopfield-type connectivity matrix [ 18 ] was one of the first examples of attractor networks with spiking neurons. [ 19 ] [ 20 ] [ 21 ]
For SRM neurons, an important variable characterizing the internal state of the neuron is the time since the last spike (or 'age' of the neuron) which enters into the refractory kernel η ( s ) {\displaystyle \eta (s)} . The population activity equations for SRM neurons can be formulated alternatively either as integral equations, [ 5 ] [ 10 ] [ 22 ] or as partial differential equations for the 'refractory density'. [ 5 ] [ 10 ] Because the refractory kernel may include a time scale slower than that of the membrane potential, the population equations for SRM neurons provide powerful alternatives [ 23 ] [ 22 ] [ 24 ] to the more broadly used partial differential equations for the 'membrane potential density'. [ 20 ] [ 25 ] [ 26 ] Reviews of the population activity equation based on refractory densities can be found in [ 24 ] as well in Chapter 14 of the textbook Neuronal Dynamics. [ 8 ]
SRMs are useful to understand theories of neural coding . A network SRM neurons has stored attractors that form reliable spatio-temporal spike patterns [ 1 ] (also known as synfire chains [ 27 ] ) example of temporal coding for stationary inputs. Moreover, the population activity equations for SRM exhibit temporally precise transients after a stimulus switch, indicating reliable spike firing. [ 10 ]
The Spike Response Model has been introduced in a series of papers between 1991 [ 11 ] and 2000. [ 2 ] [ 5 ] [ 10 ] [ 28 ] The name Spike Response Model probably appeared for the first time in 1993. [ 1 ] Some papers used exclusively the deterministic limit with a hard threshold [ 2 ] others the soft threshold with escape noise. [ 5 ] Precursors of the Spike Response Model are the integrate-and-fire model introduced by Lapicque in 1907 as well as models used in auditory neuroscience. [ 29 ] [ 30 ] [ 31 ] [ 32 ]
An important variant of the model is SRM0 [ 10 ] which is related to time-dependent nonlinear renewal theory . The main difference to the voltage equation of the SRM introduced above is that in the term containing the refractory kernel η ( s ) {\displaystyle \eta (s)} there is no summation sign over past spikes: only the most recent spike matters. The model SRM0 is closely related to the inhomogeneous Markov interval process [ 33 ] and to age-dependent models of refractoriness. [ 30 ] [ 31 ] [ 32 ]
The equations of the SRM as introduced above are equivalent to Generalized Linear Models in neuroscience (GLM) . [ 6 ] [ 7 ] In the neuroscience, GLMs have been introduced as an extension of the Linear-Nonlinear-Poisson model (LNP) by adding self-interaction of an output spike with the internal state of the neuron [ 6 ] [ 7 ] (therefore also called 'Recursive LNP'). The self-interaction is equivalent to the kernel η ( s ) {\displaystyle \eta (s)} of the SRM. The GLM framework enables to formulate a maximum likelihood approach [ 34 ] applied to the likelihood of an observed spike train under the assumption that an SRM could have generated the spike train. [ 8 ] Despite the mathematical equivalence there is a conceptual difference in interpretation: in the SRM the variable V is interpreted as membrane voltage whereas in the recursive LNP it is a 'hidden' variable to which no meaning is assigned. The SRM interpretation is useful if measurements of subthreshold voltage are available [ 4 ] [ 14 ] [ 15 ] whereas the recursive LNP is useful in systems neuroscience where spikes (in response to sensory stimulation) are recorded extracellulary without access to the subthreshold voltage. [ 6 ] [ 7 ]
A leaky integrate-and-fire neuron with spike-triggered adaptation has a subthreshold membrane potential generated by the following differential equations
where τ m {\displaystyle \tau _{m}} is the membrane time constant and w k is an adaptation current number, with index k, E rest is the resting potential and t f is the firing time of the neuron and the Greek delta denotes the Dirac delta function. Whenever the voltage reaches the firing threshold the voltage is reset to a value V r below the firing threshold. Integration of the linear differential equations gives a formula identical to the voltage equation of the SRM. [ 2 ] However, in this case, the refractory kernel η ( s ) {\displaystyle \eta (s)} does not include the spike shape but only the spike-afterpotential. In the absence of adaptation currents, we retrieve the standard LIF model which is equivalent to a refractory kernel η ( s ) {\displaystyle \eta (s)} that decays exponentially with the membrane time constant τ m {\displaystyle \tau _{m}} . | https://en.wikipedia.org/wiki/Spike_response_model |
A spike strip (also referred to as a spike belt , road spikes , traffic spikes , tire shredders , stingers , stop sticks , by the trademark Stinger or formally known as a Tire Deflation Device or TDD ) is a device or incident weapon used to impede or stop the movement of wheeled vehicles by puncturing their tires .
Generally, the strip is composed of a collection of 35-to-75-millimetre-long ( 1 + 1 ⁄ 2 to 3 in) metal barbs , teeth or spikes pointing upward. The spikes are designed to puncture and flatten tires when a vehicle is driven over them; they may be portable, as a police weapon, or strongly secured to the ground, as those found at security checkpoint entrances in certain facilities. (These particular models, however, retract and do not cause damage when a vehicle drives over them from the proper direction.) They also may be detachable, with new spikes fitted to the strip after use. The spikes may be hollow or solid; hollow ones are designed to detach and become embedded in the tires, allowing air to escape at a steady rate to reduce the risk of the driver losing control and crashing. [ 1 ] They are historically a development of the caltrop , with anti-cavalry and anti-personnel versions being used as early as 331 BC by Darius III against Alexander the Great at the Battle of Gaugamela in Persia . [ citation needed ]
In the United States, five officers were killed deploying spike strips in 2011, having been struck by fleeing vehicles. Dallas , Texas police are among those banned from using them, in response to the hazards. [ 2 ]
Remotely deployable spike strips have been invented to reduce the danger to police officers deploying them. [ 3 ]
Private possession of spike strips was banned in New South Wales , Australia in 2003 after a strip cheaply constructed from a steel pipe studded with nails was used against a police vehicle. John Watkins , a member of New South Wales Legislative Assembly, stated they would be added to the New South Wales prohibited weapons list. [ 4 ]
Following the rise in terrorist vehicle attacks whereby a vehicle is driven at speed into pedestrians, a net with steel spikes that can be deployed by two people in less than a minute, reported able to stop a vehicle of up to 17 tonnes, was developed for preventive use at public events in the UK, with the name "Talon". It has steel spikes to puncture tires, and becomes entangled around the front wheels, halting the vehicle. It is designed to reduce risk to crowds by making the vehicle skid in a straight line without veering unpredictably. It was first deployed to protect a parade on 11 September 2017. [ 5 ] | https://en.wikipedia.org/wiki/Spike_strip |
Spiling is a traditional technique used in temperate regions of the world for the prevention of erosion to river and stream banks.
Willow spiling is currently used in the United Kingdom; live willow rods are woven between live willow uprights and the area behind is filled with soil for the willow to root into. [ 1 ]
Kipling 's poem The Land mentions it: "They spiled along the water-course with trunks of willow-trees, And planks of elms behind 'em and immortal oaken knees." [ 2 ]
The species of willow used are riparian (associated with rivers); the posts, 10 centimetres (4 in) in diameter, are usually Salix alba or S. fragilis , and S. viminalis varieties are used for the interwoven rods. The living willow posts are driven into the bank, to a depth of 30 centimetres (1 ft) or more, at 60-centimetre (2 ft) intervals and the thinner rods are woven in between, the rods are best woven at an angle slightly above horizontal to ensure good survival rates. A row of stones, gabions or wooden planks held by posts can be added to the bottom of each "spile" to prevent undercutting when the willow is establishing itself. All works should be done during the dormant period, winter in temperate zones. A layer of seeded coir matting can be pegged onto the soil on top of the spiles to prevent the soil being washed out during flood events. This method is an example of soft engineering , techniques which tend to be less expensive and more sustainable than others. [ 3 ] | https://en.wikipedia.org/wiki/Spiling |
Spill kits are typically single-use kits containing chemicals used for absorption of ( hazardous ) wastes. Ready-for-use kits typically contain personal protective equipment , decontamination or neutralizing agents, disposal containers, and signage markers. Sometimes, spill containment material will also be included.
Specialized kits are available for specific spills: [ 1 ] [ 2 ]
This chemistry -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Spill_kit |
A spill metric is a heuristic metric used by register allocators to decide which registers to spill. Popular spill metrics are:
Where cost is the estimated cost of spilling a value from registers into memory.
This computing article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Spill_metric |
Spillover infection , also known as pathogen spillover and spillover event , occurs when a reservoir population with a high pathogen prevalence comes into contact with a novel host population. The pathogen is transmitted from the reservoir population and may or may not be transmitted within the host population. [ 1 ] Due to climate change and land use expansion, the risk of viral spillover is predicted to significantly increase. [ 2 ] [ 3 ]
Spillover is a common event; in fact, more than two-thirds of human viruses are zoonotic . [ 4 ] [ 5 ] Most spillover events result in self-limited cases with no further human-to-human transmission, as occurs, for example, with rabies, anthrax, histoplasmosis or hydatidosis. Other zoonotic pathogens are able to be transmitted by humans to produce secondary cases and even to establish limited chains of transmission. Some examples are the Ebola and Marburg filoviruses, the MERS and SARS coronaviruses and some avian flu viruses. Finally, some spillover events can result in the final adaptation of the microbe to humans, who can become a new stable reservoir, as occurred with the HIV virus resulting in the AIDS epidemic and with SARS-CoV-2 resulting in the COVID-19 pandemic . [ 5 ]
If the history of mutual adaptation is long enough, permanent host-microbe associations can be established resulting in co-evolution, and even permanent integration of the microbe genome with the human genome, as is the case of endogenous viruses. [ 6 ] The closer the two target host species are in phylogenetic terms, the easier it is for microbes to overcome the biological barrier to produce successful spillovers. [ 1 ] For this reason, other mammals are the main source of zoonotic agents for humans. For example, in the case of the Ebola virus, fruit bats are the hypothesized zoonotic agent. [ 7 ]
During the late 20th century, zoonotic spillover increased as the environmental impact of agriculture promoted increased land use and deforestation , changing wildlife habitat . As species shift their geographic range in response to climate change , the risk of zoonotic spillover is predicted to substantially increase, particularly in tropical regions that are experiencing rapid warming. [ 8 ] As forested areas of land are cleared for human use, there is increased proximity and interaction between wild animals and humans thereby increasing the potential for exposure. [ 9 ]
Commercially bred bumblebees used to pollinate greenhouses can be reservoirs for several pollinator parasites including the protozoans Crithidia bombi , and Apicystis bombi , [ 10 ] the microsporidians Nosema bombi and Nosema ceranae , [ 10 ] [ 11 ] plus viruses such as Deformed wing virus and the tracheal mites Locustacarus buchneri . [ 11 ] Commercial bees that escape the greenhouse environment may then infect wild bee populations. Infection may be via direct interactions between managed and wild bees or via shared flower use and contamination. [ 12 ] [ 13 ] One study found that half of all wild bees found near greenhouses were infected with C. bombi . Rates and incidence of infection decline dramatically the further away from the greenhouses where the wild bees are located. [ 14 ] [ 15 ] Instances of spillover between bumblebees are well documented across the world, particularly in Japan, North America, and the United Kingdom. [ 16 ] [ 17 ]
Zoonotic spillover is a relatively uncommon but incredibly dangerous natural phenomenon—as is evidenced by the Ebola epidemic and Coronavirus pandemic. For zoonotic spillover to occur, several important factors have to occur in tandem. [ 1 ] Such factors include altered ecological niches, epidemiological susceptibility, and the natural behavior of pathogens and novel host or spillover host species. [ 29 ] By suggesting that the natural behavior of pathogens and host species impacts zoonotic spillover, simple Darwinian theories are being referenced. As with all species, a pathogen's main goal is to survive. When a stressor puts pressure on the survival of the pathogenic species, it will have to adapt to said stressor in order to survive. [ 30 ] For example, the ecological niche of the novel host may be subject to a lack of food which leads to a decrease in the novel host population. In order for a virus to replicate, it must invade a eukaryotic organism. [ 31 ] When the novel eukaryotic organism is not available for the virus to infect, it must jump to another host. [ 30 ] In order for the virus to make the jump to the spillover host, the spillover host must be epidemiologically susceptible to this virus. Although it is not well understood what makes one spillover host "better" than another host, it is known that the susceptibility has to do with the shedding rate of the virus, how well the virus survives and moves while not within a host, the genotypic similarities between the novel and spillover hosts, and the behavior of the spillover host that leads to contact with a high dose of the virus. [ 1 ] | https://en.wikipedia.org/wiki/Spillover_infection |
In quantum mechanics , spin-exchange is an interaction process between two particles mediated by an exchange interaction . [ 1 ] It preserves total angular momentum of the system but may allow other aspects of the system to change. When two spin-polarized atoms in their ground state experience a spin-exchange collision, the total spin of the atoms is preserved yet the orientation of the individual spins may change. For example, if atoms A {\displaystyle A} and B {\displaystyle B} are oppositely polarized , a spin-exchange collision reverses the spins: [ 2 ]
In a typical vapor of alkali metal atoms, spin-exchange collisions are the dominant type of interaction between atoms. The collisions happen so rapidly that they only alter the state of the electron spins and do not significantly affect the nuclear spins. Thus, spin-exchange collisions between alkali metal atoms can change the hyperfine state of the atoms while preserving total angular momentum of the colliding pair. As a result, spin-exchange collisions cause decoherence in ensembles of polarized atoms precessing in the presence of a magnetic field .
The time between spin-exchange collisions for a vapor of alkali metal atoms is
where the spin exchange cross section for alkali metals such as K, Rb, and Cs is [ 3 ]
and where n {\displaystyle n} is the vapor density and v ¯ r e l {\displaystyle {\bar {v}}_{rel}} is the average relative velocity given by the Maxwell–Boltzmann distribution :
where R {\displaystyle R} is the ideal gas constant , T {\displaystyle T} is the temperature, and m {\displaystyle m} is the molar mass of the atoms.
This quantum mechanics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Spin-exchange |
In chemistry , reactions that involve a change in spin state are known as spin-forbidden reactions Such reactions show increased activation energy when compared to a similar reaction in which the spin states of the reactant and product are isomorphic. As a result of this increased activation energy, a decreased rate of reaction is observed. A famous example of spin-forbidden reaction is the very slow reaction of O 2 with hydrocarbons.
The dissociation of nitrous oxide is a well-studied process: [ 1 ]
O atoms have a triplet ground state.
Methoxy cation has a triplet ground state. In a mass spectrometer, it dissociates into singlet products (formyl cation and H 2 ):
Numerous spin-forbidden reactions are encountered in transition metal chemistry since many metal ions can adopt multiple spin states. For example, ferrous porphyrin complexes containing one axial donor are high spin ferrous. These complexes, which are represented by myoglobin and hemoglobin, bind CO to give singlet products:
Cobalt(I) dicarbonyl complexes of a trispyrazolylborate are diamagnetic. The corresponding monocarbonyls have triplet ground states.
The addition of CO to Fe(CO) 4 is an example showing the slowing effect of spin-forbidden reaction takes place when Fe(CO) x is placed under CO pressure. [ 3 ]
When a reaction converts a metal from a singlet to triplet state (or vice versa ):
Strong spin-orbital coupling can satisfy the 2nd condition. Parameter 1, however, can lead to very slow reactions due to large disparities between the metal complex's potential energy surfaces , which only cross at high energy leading to a substantial activation barrier . [ 4 ]
Spin-forbidden reactions formally fall into the category of electronically non-adiabatic reactions . [ 5 ] In general, potential energy surfaces fall into either the adiabatic and diabatic classification. Potential Energy Surfaces that are adiabatic rely on the use of the full electronic Hamiltonian , which includes the spin-orbit term. Those that are diabatic are likewise derived by solving the eigenvalues of the Schrödinger equation , but in this case one or more terms are omitted. [ 1 ]
Once a minimum energy crossing point is reached and parameter 1 above is satisfied, the system needs to hop from one diabatic surface to the other, as stated above by parameter 2. At a given energy ( E ), the rate coefficient [ k(E) ] of a spin-forbidden reaction can be calculated using the density of rovibrational states of the reactant [ ρ(E) ] and the effective integrated density of states in the crossing seam between the two surfaces [ N er (E) ].
where
The probability of hopping ( p sh ) is calculated from Landau-Zener theory giving
where
in which the spin-orbit coupling derived off the diagonal Hamiltonian matrix element between two electronic states ( H 12 ), the relative slope of the two surfaces at the crossing seam [ F(Δ) ], the reduced mass of the system through its movement along the hopping coordinate ( μ ), and the kinetic energy of the system passing through the crossing point ( E ) are used.
It is useful to note that when E h < E c (when below the minimum energy crossing point) the probability of hopping between spin states is null. [ 6 ]
Insertion into C-H bonds, known as C-H activation , is an integral first step in C-H functionalization. [ 7 ] For some metal complexes with identical ligands, C-H activation is rapid when one metal is used and slow when other metals are used, often first row transition metals, due to the spin allowed nature of the former case and the spin-forbidden nature of the latter case. The difference in rates of C-H activation of methane for CoCp(CO), RhCp(CO), and IrCp(CO) readily demonstrate this property. CoCp(CO), the starting material in a C-H activation, exists in a triplet spin state while RhCp(CO) exists in a singlet state, with the triplet state only 5.9 kcal/mol away. IrCp(CO) is unique among these complexes in that its starting state is essentially degenerate between the triplet and singlet states. The given product of C-H insertion, CpMH(CO)(CH 3 ), where M = Co, Rh, Ir, is in a singlet state meaning that the C-H activation with CoCp(CO) must reach the minimum energy crossing point for the reactant and product's potential energy surfaces, thus requiring relatively high energies to proceed. [ 8 ]
Metal-oxo species, due to their small spatial extent of metal-centered d orbitals leading to weak bonding, often have similar energies for both the low spin ( M = O {\displaystyle {\ce {M=O}}} ) and high spin configuration ( ⋅ M − O ⋅ {\displaystyle {\ce {*M-O*}}} ). [ 9 ] This similarity in energy between the low- and high spin configurations of oxo-species lends itself to the study of spin-forbidden reactions, such as Mn(salen)-catalyzed epoxidation. The Mn(salen)-oxo species can exist in either a triplet or quintet state. While the product of the quintet lies at a lower energy, both the triplet and quintet products can be observed. [ 10 ] | https://en.wikipedia.org/wiki/Spin-forbidden_reactions |
Spin-polarized electron energy loss spectroscopy or SPEELS is a technique mainly used to measure the dispersion relation of the collective excitations, over the whole Brillouin zone .
Spin waves are collective perturbations in a magnetic solid. Their properties depend on their wavelength (or wave vector ). For long wavelength (short wave vector) spin waves, the resulting spin precession has a very low frequency and the spin waves can be treated classically . Ferromagnetic resonance (FMR) and Brillouin light scattering (BLS) experiments explain the long wavelength spin waves in ultrathin magnetic films and nanostructures . If the wavelength is comparable to the lattice constant , the spin waves are governed by the microscopic exchange coupling and a quantum mechanical description is needed. Therefore, experimental information on these short wavelength (large wave vector) spin waves in ultrathin films is highly desired and may lead to fundamentally new insights into the spin dynamics in reduced dimensions in the future.
SPEELS is one of the few techniques that can be used to measure the dispersion of such short wavelength spin waves in ultrathin films and nanostructures. [ citation needed ]
For the first time Kirschner's group [ 1 ] in Max Planck institute of Microstructure Physics showed that the signature of the large wave vector spin waves can be detected by spin polarized electron energy loss spectroscopy (SPEELS). [ 2 ] [ 3 ] Later, with a better momentum resolution, the spin wave dispersion was fully measured in 8 monolayer (ML) fcc cobalt film on Cu (001) [ 4 ] and 8 ML hcp cobalt on W (110), [ 5 ] respectively. Those spin waves were obtained up to the surface Brillouin zone (SBZ) at the energy range about few hundreds of meV. Another recent example is the investigation of 1 and 2 monolayer iron films grown on W(110) measured at 120 K and 300 K, respectively. [ 6 ] [ 7 ]
This spectroscopy -related article is a stub . You can help Wikipedia by expanding it .
This scattering –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Spin-polarized_electron_energy_loss_spectroscopy |
Spin is an intrinsic form of angular momentum carried by elementary particles , and thus by composite particles such as hadrons , atomic nuclei , and atoms. [ 1 ] [ 2 ] : 183 –184 Spin is quantized, and accurate models for the interaction with spin require relativistic quantum mechanics or quantum field theory .
The existence of electron spin angular momentum is inferred from experiments, such as the Stern–Gerlach experiment , in which silver atoms were observed to possess two possible discrete angular momenta despite having no orbital angular momentum. [ 3 ] The relativistic spin–statistics theorem connects electron spin quantization to the Pauli exclusion principle : observations of exclusion imply half-integer spin, and observations of half-integer spin imply exclusion.
Spin is described mathematically as a vector for some particles such as photons, and as a spinor or bispinor for other particles such as electrons. Spinors and bispinors behave similarly to vectors : they have definite magnitudes and change under rotations; however, they use an unconventional "direction". All elementary particles of a given kind have the same magnitude of spin angular momentum, though its direction may change. These are indicated by assigning the particle a spin quantum number . [ 2 ] : 183–184
The SI units of spin are the same as classical angular momentum (i.e., N · m · s , J ·s, or kg ·m 2 ·s −1 ). In quantum mechanics, angular momentum and spin angular momentum take discrete values proportional to the Planck constant . In practice, spin is usually given as a dimensionless spin quantum number by dividing the spin angular momentum by the reduced Planck constant ħ . Often, the "spin quantum number" is simply called "spin".
The earliest models for electron spin imagined a rotating charged mass, but this model fails when examined in detail: the required space distribution does not match limits on the electron radius : the required rotation speed exceeds the speed of light. [ 4 ] In the Standard Model , the fundamental particles are all considered "point-like": they have their effects through the field that surrounds them. [ 5 ] Any model for spin based on mass rotation would need to be consistent with that model.
Wolfgang Pauli , a central figure in the history of quantum spin, initially rejected any idea that the "degree of freedom" he introduced to explain experimental observations was related to rotation. He called it "classically non-describable two-valuedness". Later, he allowed that it is related to angular momentum, but insisted on considering spin an abstract property. [ 6 ] This approach allowed Pauli to develop a proof of his fundamental Pauli exclusion principle , a proof now called the spin-statistics theorem . [ 7 ] In retrospect, this insistence and the style of his proof initiated the modern particle-physics era, where abstract quantum properties derived from symmetry properties dominate. Concrete interpretation became secondary and optional. [ 6 ]
The first classical model for spin proposed a small rigid particle rotating about an axis, as ordinary use of the word may suggest. Angular momentum can be computed from a classical field as well. [ 8 ] [ 9 ] : 63 By applying Frederik Belinfante 's approach to calculating the angular momentum of a field, Hans C. Ohanian showed that "spin is essentially a wave property ... generated by a circulating flow of charge in the wave field of the electron". [ 10 ] This same concept of spin can be applied to gravity waves in water: "spin is generated by subwavelength circular motion of water particles". [ 11 ]
Unlike classical wavefield circulation, which allows continuous values of angular momentum, quantum wavefields allow only discrete values. [ 10 ] Consequently, energy transfer to or from spin states always occurs in fixed quantum steps. Only a few steps are allowed: for many qualitative purposes, the complexity of the spin quantum wavefields can be ignored and the system properties can be discussed in terms of "integer" or "half-integer" spin models as discussed in quantum numbers below.
Spin can be understood differently depending on the interpretations of quantum mechanics . In the de Broglie–Bohm interpretation , particles have definitive trajectories but their motion is driven by the wave function or pilot wave. In this interpretation, the spin is a property of the pilot wave and not of the particle themselves. [ 12 ]
Quantitative calculations of spin properties for electrons requires the Dirac relativistic wave equation . [ 7 ]
As the name suggests, spin was originally conceived as the rotation of a particle around some axis. Historically orbital angular momentum related to particle orbits. [ 13 ] : 131 While the names based on mechanical models have survived, the physical explanation has not. Quantization fundamentally alters the character of both spin and orbital angular momentum.
Since elementary particles are point-like, self-rotation is not well-defined for them. However, spin implies that the phase of the particle depends on the angle as e i S θ , {\displaystyle e^{iS\theta }\ ,} for rotation of angle θ around the axis parallel to the spin S . This is equivalent to the quantum-mechanical interpretation of momentum as phase dependence in the position, and of orbital angular momentum as phase dependence in the angular position.
For fermions, the picture is less clear: From the Ehrenfest theorem , the angular velocity is equal to the derivative of the Hamiltonian to its conjugate momentum , which is the total angular momentum operator J = L + S . Therefore, if the Hamiltonian H has any dependence on the spin S , then ∂ H / ∂ S must be non-zero; consequently, for classical mechanics , the existence of spin in the Hamiltonian will produce an actual angular velocity, and hence an actual physical rotation – that is, a change in the phase-angle, θ , over time. However, whether this holds true for free electrons is ambiguous, since for an electron, | S | ² is a constant 1 / 2 ℏ , and one might decide that since it cannot change, no partial ( ∂ ) can exist. Therefore it is a matter of interpretation whether the Hamiltonian must include such a term, and whether this aspect of classical mechanics extends into quantum mechanics (any particle's intrinsic spin angular momentum, S , is a quantum number arising from a " spinor " in the mathematical solution to the Dirac equation , rather than being a more nearly physical quantity, like orbital angular momentum L ). Nevertheless, spin appears in the Dirac equation , and thus the relativistic Hamiltonian of the electron, treated as a Dirac field , can be interpreted as including a dependence in the spin S . [ 9 ]
Spin obeys the mathematical laws of angular momentum quantization . The specific properties of spin angular momenta include:
The conventional definition of the spin quantum number is s = n / 2 , where n can be any non-negative integer . Hence the allowed values of s are 0, 1 / 2 , 1, 3 / 2 , 2, etc. The value of s for an elementary particle depends only on the type of particle and cannot be altered in any known way (in contrast to the spin direction described below). The spin angular momentum S of any physical system is quantized . The allowed values of S are S = ℏ s ( s + 1 ) = h 2 π n 2 ( n + 2 ) 2 = h 4 π n ( n + 2 ) , {\displaystyle S=\hbar \,{\sqrt {s(s+1)}}={\frac {h}{2\pi }}\,{\sqrt {{\frac {n}{2}}{\frac {(n+2)}{2}}}}={\frac {h}{4\pi }}\,{\sqrt {n(n+2)}},} where h is the Planck constant , and ℏ = h 2 π {\textstyle \hbar ={\frac {h}{2\pi }}} is the reduced Planck constant. In contrast, orbital angular momentum can only take on integer values of s ; i.e., even-numbered values of n .
Those particles with half-integer spins, such as 1 / 2 , 3 / 2 , 5 / 2 , are known as fermions , while those particles with integer spins, such as 0, 1, 2, are known as bosons . The two families of particles obey different rules and broadly have different roles in the world around us. A key distinction between the two families is that fermions obey the Pauli exclusion principle : that is, there cannot be two identical fermions simultaneously having the same quantum numbers (meaning, roughly, having the same position, velocity and spin direction). Fermions obey the rules of Fermi–Dirac statistics . In contrast, bosons obey the rules of Bose–Einstein statistics and have no such restriction, so they may "bunch together" in identical states. Also, composite particles can have spins different from their component particles. For example, a helium-4 atom in the ground state has spin 0 and behaves like a boson, even though the quarks and electrons which make it up are all fermions.
This has some profound consequences:
The spin–statistics theorem splits particles into two groups: bosons and fermions , where bosons obey Bose–Einstein statistics , and fermions obey Fermi–Dirac statistics (and therefore the Pauli exclusion principle ). Specifically, the theorem requires that particles with half-integer spins obey the Pauli exclusion principle while particles with integer spin do not. As an example, electrons have half-integer spin and are fermions that obey the Pauli exclusion principle, while photons have integer spin and do not. The theorem was derived by Wolfgang Pauli in 1940; it relies on both quantum mechanics and the theory of special relativity . Pauli described this connection between spin and statistics as "one of the most important applications of the special relativity theory". [ 15 ]
Particles with spin can possess a magnetic dipole moment , just like a rotating electrically charged body in classical electrodynamics . These magnetic moments can be experimentally observed in several ways, e.g. by the deflection of particles by inhomogeneous magnetic fields in a Stern–Gerlach experiment , or by measuring the magnetic fields generated by the particles themselves.
The intrinsic magnetic moment μ of a spin- 1 / 2 particle with charge q , mass m , and spin angular momentum S is [ 16 ]
where the dimensionless quantity g s is called the spin g -factor . For exclusively orbital rotations, it would be 1 (assuming that the mass and the charge occupy spheres of equal radius).
The electron, being a charged elementary particle, possesses a nonzero magnetic moment . One of the triumphs of the theory of quantum electrodynamics is its accurate prediction of the electron g -factor , which has been experimentally determined to have the value −2.002 319 304 360 92 (36) , with the digits in parentheses denoting measurement uncertainty in the last two digits at one standard deviation . [ 17 ] The value of 2 arises from the Dirac equation , a fundamental equation connecting the electron's spin with its electromagnetic properties; and the deviation from −2 arises from the electron's interaction with the surrounding quantum fields, including its own electromagnetic field and virtual particles . [ 18 ]
Composite particles also possess magnetic moments associated with their spin. In particular, the neutron possesses a non-zero magnetic moment despite being electrically neutral. This fact was an early indication that the neutron is not an elementary particle. In fact, it is made up of quarks , which are electrically charged particles. The magnetic moment of the neutron comes from the spins of the individual quarks and their orbital motions.
Neutrinos are both elementary and electrically neutral. The minimally extended Standard Model that takes into account non-zero neutrino masses predicts neutrino magnetic moments of: [ 19 ] [ 20 ] [ 21 ]
where the μ ν are the neutrino magnetic moments, m ν are the neutrino masses, and μ B is the Bohr magneton . New physics above the electroweak scale could, however, lead to significantly higher neutrino magnetic moments. It can be shown in a model-independent way that neutrino magnetic moments larger than about 10 −14 μ B are "unnatural" because they would also lead to large radiative contributions to the neutrino mass. Since the neutrino masses are known to be at most about 1 eV/ c 2 , fine-tuning would be necessary in order to prevent large contributions to the neutrino mass via radiative corrections. [ 22 ] The measurement of neutrino magnetic moments is an active area of research. Experimental results have put the neutrino magnetic moment at less than 1.2 × 10 −10 times the electron's magnetic moment.
On the other hand, elementary particles with spin but without electric charge, such as the photon and Z boson , do not have a magnetic moment.
In classical mechanics, the angular momentum of a particle possesses not only a magnitude (how fast the body is rotating), but also a direction (either up or down on the axis of rotation of the particle). Quantum-mechanical spin also contains information about direction, but in a more subtle form. Quantum mechanics states that the component of angular momentum for a spin- s particle measured along any direction can only take on the values [ 23 ]
where S i is the spin component along the i -th axis (either x , y , or z ), s i is the spin projection quantum number along the i -th axis, and s is the principal spin quantum number (discussed in the previous section). Conventionally the direction chosen is the z axis:
where S z is the spin component along the z axis, s z is the spin projection quantum number along the z axis.
One can see that there are 2 s + 1 possible values of s z . The number " 2 s + 1 " is the multiplicity of the spin system. For example, there are only two possible values for a spin- 1 / 2 particle: s z = + 1 / 2 and s z = − 1 / 2 . These correspond to quantum states in which the spin component is pointing in the + z or − z directions respectively, and are often referred to as "spin up" and "spin down". For a spin- 3 / 2 particle, like a delta baryon , the possible values are + 3 / 2 , + 1 / 2 , − 1 / 2 , − 3 / 2 .
For a given quantum state , one could think of a spin vector ⟨ S ⟩ {\textstyle \langle S\rangle } whose components are the expectation values of the spin components along each axis, i.e., ⟨ S ⟩ = [ ⟨ S x ⟩ , ⟨ S y ⟩ , ⟨ S z ⟩ ] {\textstyle \langle S\rangle =[\langle S_{x}\rangle ,\langle S_{y}\rangle ,\langle S_{z}\rangle ]} . This vector then would describe the "direction" in which the spin is pointing, corresponding to the classical concept of the axis of rotation . It turns out that the spin vector is not very useful in actual quantum-mechanical calculations, because it cannot be measured directly: s x , s y and s z cannot possess simultaneous definite values, because of a quantum uncertainty relation between them. However, for statistically large collections of particles that have been placed in the same pure quantum state, such as through the use of a Stern–Gerlach apparatus , the spin vector does have a well-defined experimental meaning: It specifies the direction in ordinary space in which a subsequent detector must be oriented in order to achieve the maximum possible probability (100%) of detecting every particle in the collection. For spin- 1 / 2 particles, this probability drops off smoothly as the angle between the spin vector and the detector increases, until at an angle of 180°—that is, for detectors oriented in the opposite direction to the spin vector—the expectation of detecting particles from the collection reaches a minimum of 0%.
As a qualitative concept, the spin vector is often handy because it is easy to picture classically. For instance, quantum-mechanical spin can exhibit phenomena analogous to classical gyroscopic effects . For example, one can exert a kind of " torque " on an electron by putting it in a magnetic field (the field acts upon the electron's intrinsic magnetic dipole moment —see the following section). The result is that the spin vector undergoes precession , just like a classical gyroscope. This phenomenon is known as electron spin resonance (ESR). The equivalent behaviour of protons in atomic nuclei is used in nuclear magnetic resonance (NMR) spectroscopy and imaging.
Mathematically, quantum-mechanical spin states are described by vector-like objects known as spinors . There are subtle differences between the behavior of spinors and vectors under coordinate rotations . For example, rotating a spin- 1 / 2 particle by 360° does not bring it back to the same quantum state, but to the state with the opposite quantum phase ; this is detectable, in principle, with interference experiments. To return the particle to its exact original state, one needs a 720° rotation. (The plate trick and Möbius strip give non-quantum analogies.) A spin-zero particle can only have a single quantum state, even after torque is applied. Rotating a spin-2 particle 180° can bring it back to the same quantum state, and a spin-4 particle should be rotated 90° to bring it back to the same quantum state. The spin-2 particle can be analogous to a straight stick that looks the same even after it is rotated 180°, and a spin-0 particle can be imagined as sphere, which looks the same after whatever angle it is turned through.
Spin obeys commutation relations [ 24 ] analogous to those of the orbital angular momentum : [ S ^ j , S ^ k ] = i ℏ ε j k l S ^ l , {\displaystyle \left[{\hat {S}}_{j},{\hat {S}}_{k}\right]=i\hbar \varepsilon _{jkl}{\hat {S}}_{l},} where ε jkl is the Levi-Civita symbol . It follows (as with angular momentum ) that the eigenvectors of S ^ 2 {\displaystyle {\hat {S}}^{2}} and S ^ z {\displaystyle {\hat {S}}_{z}} (expressed as kets in the total S basis ) are [ 2 ] : 166 S ^ 2 | s , m s ⟩ = ℏ 2 s ( s + 1 ) | s , m s ⟩ , S ^ z | s , m s ⟩ = ℏ m s | s , m s ⟩ . {\displaystyle {\begin{aligned}{\hat {S}}^{2}|s,m_{s}\rangle &=\hbar ^{2}s(s+1)|s,m_{s}\rangle ,\\{\hat {S}}_{z}|s,m_{s}\rangle &=\hbar m_{s}|s,m_{s}\rangle .\end{aligned}}}
The spin raising and lowering operators acting on these eigenvectors give S ^ ± | s , m s ⟩ = ℏ s ( s + 1 ) − m s ( m s ± 1 ) | s , m s ± 1 ⟩ , {\displaystyle {\hat {S}}_{\pm }|s,m_{s}\rangle =\hbar {\sqrt {s(s+1)-m_{s}(m_{s}\pm 1)}}|s,m_{s}\pm 1\rangle ,} where S ^ ± = S ^ x ± i S ^ y {\displaystyle {\hat {S}}_{\pm }={\hat {S}}_{x}\pm i{\hat {S}}_{y}} . [ 2 ] : 166
But unlike orbital angular momentum, the eigenvectors are not spherical harmonics . They are not functions of θ and φ . There is also no reason to exclude half-integer values of s and m s .
All quantum-mechanical particles possess an intrinsic spin s {\displaystyle s} (though this value may be equal to zero). The projection of the spin s {\displaystyle s} on any axis is quantized in units of the reduced Planck constant , such that the state function of the particle is, say, not ψ = ψ ( r ) {\displaystyle \psi =\psi (\mathbf {r} )} , but ψ = ψ ( r , s z ) {\displaystyle \psi =\psi (\mathbf {r} ,s_{z})} , where s z {\displaystyle s_{z}} can take only the values of the following discrete set: s z ∈ { − s ℏ , − ( s − 1 ) ℏ , … , + ( s − 1 ) ℏ , + s ℏ } . {\displaystyle s_{z}\in \{-s\hbar ,-(s-1)\hbar ,\dots ,+(s-1)\hbar ,+s\hbar \}.}
One distinguishes bosons (integer spin) and fermions (half-integer spin). The total angular momentum conserved in interaction processes is then the sum of the orbital angular momentum and the spin.
The quantum-mechanical operators associated with spin- 1 / 2 observables are S ^ = ℏ 2 σ , {\displaystyle {\hat {\mathbf {S} }}={\frac {\hbar }{2}}{\boldsymbol {\sigma }},} where in Cartesian components S x = ℏ 2 σ x , S y = ℏ 2 σ y , S z = ℏ 2 σ z . {\displaystyle S_{x}={\frac {\hbar }{2}}\sigma _{x},\quad S_{y}={\frac {\hbar }{2}}\sigma _{y},\quad S_{z}={\frac {\hbar }{2}}\sigma _{z}.}
For the special case of spin- 1 / 2 particles, σ x , σ y and σ z are the three Pauli matrices : σ x = ( 0 1 1 0 ) , σ y = ( 0 − i i 0 ) , σ z = ( 1 0 0 − 1 ) . {\displaystyle \sigma _{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\quad \sigma _{y}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}},\quad \sigma _{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}.}
The Pauli exclusion principle states that the wavefunction ψ ( r 1 , σ 1 , … , r N , σ N ) {\displaystyle \psi (\mathbf {r} _{1},\sigma _{1},\dots ,\mathbf {r} _{N},\sigma _{N})} for a system of N identical particles having spin s must change upon interchanges of any two of the N particles as ψ ( … , r i , σ i , … , r j , σ j , … ) = ( − 1 ) 2 s ψ ( … , r j , σ j , … , r i , σ i , … ) . {\displaystyle \psi (\dots ,\mathbf {r} _{i},\sigma _{i},\dots ,\mathbf {r} _{j},\sigma _{j},\dots )=(-1)^{2s}\psi (\dots ,\mathbf {r} _{j},\sigma _{j},\dots ,\mathbf {r} _{i},\sigma _{i},\dots ).}
Thus, for bosons the prefactor (−1) 2 s will reduce to +1, for fermions to −1.
This permutation postulate for N -particle state functions has most important consequences in daily life, e.g. the periodic table of the chemical elements.
As described above, quantum mechanics states that components of angular momentum measured along any direction can only take a number of discrete values. The most convenient quantum-mechanical description of particle's spin is therefore with a set of complex numbers corresponding to amplitudes of finding a given value of projection of its intrinsic angular momentum on a given axis. For instance, for a spin- 1 / 2 particle, we would need two numbers a ±1/2 , giving amplitudes of finding it with projection of angular momentum equal to + ħ / 2 and − ħ / 2 , satisfying the requirement | a + 1 / 2 | 2 + | a − 1 / 2 | 2 = 1. {\displaystyle |a_{+1/2}|^{2}+|a_{-1/2}|^{2}=1.}
For a generic particle with spin s , we would need 2 s + 1 such parameters. Since these numbers depend on the choice of the axis, they transform into each other non-trivially when this axis is rotated. It is clear that the transformation law must be linear, so we can represent it by associating a matrix with each rotation, and the product of two transformation matrices corresponding to rotations A and B must be equal (up to phase) to the matrix representing rotation AB. Further, rotations preserve the quantum-mechanical inner product, and so should our transformation matrices: ∑ m = − j j a m ∗ b m = ∑ m = − j j ( ∑ n = − j j U n m a n ) ∗ ( ∑ k = − j j U k m b k ) , {\displaystyle \sum _{m=-j}^{j}a_{m}^{*}b_{m}=\sum _{m=-j}^{j}\left(\sum _{n=-j}^{j}U_{nm}a_{n}\right)^{*}\left(\sum _{k=-j}^{j}U_{km}b_{k}\right),} ∑ n = − j j ∑ k = − j j U n p ∗ U k q = δ p q . {\displaystyle \sum _{n=-j}^{j}\sum _{k=-j}^{j}U_{np}^{*}U_{kq}=\delta _{pq}.}
Mathematically speaking, these matrices furnish a unitary projective representation of the rotation group SO(3) . Each such representation corresponds to a representation of the covering group of SO(3), which is SU(2) . [ 25 ] There is one n -dimensional irreducible representation of SU(2) for each dimension, though this representation is n -dimensional real for odd n and n -dimensional complex for even n (hence of real dimension 2 n ). For a rotation by angle θ in the plane with normal vector θ ^ {\textstyle {\hat {\boldsymbol {\theta }}}} , U = e − i ℏ θ ⋅ S , {\displaystyle U=e^{-{\frac {i}{\hbar }}{\boldsymbol {\theta }}\cdot \mathbf {S} },} where θ = θ θ ^ {\textstyle {\boldsymbol {\theta }}=\theta {\hat {\boldsymbol {\theta }}}} , and S is the vector of spin operators .
Working in the coordinate system where θ ^ = z ^ {\textstyle {\hat {\theta }}={\hat {z}}} , we would like to show that S x and S y are rotated into each other by the angle θ . Starting with S x . Using units where ħ = 1 : S x → U † S x U = e i θ S z S x e − i θ S z = S x + ( i θ ) [ S z , S x ] + ( 1 2 ! ) ( i θ ) 2 [ S z , [ S z , S x ] ] + ( 1 3 ! ) ( i θ ) 3 [ S z , [ S z , [ S z , S x ] ] ] + ⋯ {\displaystyle {\begin{aligned}S_{x}\rightarrow U^{\dagger }S_{x}U&=e^{i\theta S_{z}}S_{x}e^{-i\theta S_{z}}\\&=S_{x}+(i\theta )\left[S_{z},S_{x}\right]+\left({\frac {1}{2!}}\right)(i\theta )^{2}\left[S_{z},\left[S_{z},S_{x}\right]\right]+\left({\frac {1}{3!}}\right)(i\theta )^{3}\left[S_{z},\left[S_{z},\left[S_{z},S_{x}\right]\right]\right]+\cdots \end{aligned}}}
Using the spin operator commutation relations , we see that the commutators evaluate to i S y for the odd terms in the series, and to S x for all of the even terms. Thus: U † S x U = S x [ 1 − θ 2 2 ! + ⋯ ] − S y [ θ − θ 3 3 ! ⋯ ] = S x cos θ − S y sin θ , {\displaystyle {\begin{aligned}U^{\dagger }S_{x}U&=S_{x}\left[1-{\frac {\theta ^{2}}{2!}}+\cdots \right]-S_{y}\left[\theta -{\frac {\theta ^{3}}{3!}}\cdots \right]\\&=S_{x}\cos \theta -S_{y}\sin \theta ,\end{aligned}}} as expected. Note that since we only relied on the spin operator commutation relations, this proof holds for any dimension (i.e., for any principal spin quantum number s ) [ 26 ] : 164
A generic rotation in 3-dimensional space can be built by compounding operators of this type using Euler angles : R ( α , β , γ ) = e − i α S x e − i β S y e − i γ S z . {\displaystyle {\mathcal {R}}(\alpha ,\beta ,\gamma )=e^{-i\alpha S_{x}}e^{-i\beta S_{y}}e^{-i\gamma S_{z}}.}
An irreducible representation of this group of operators is furnished by the Wigner D-matrix : D m ′ m s ( α , β , γ ) ≡ ⟨ s m ′ | R ( α , β , γ ) | s m ⟩ = e − i m ′ α d m ′ m s ( β ) e − i m γ , {\displaystyle D_{m'm}^{s}(\alpha ,\beta ,\gamma )\equiv \langle sm'|{\mathcal {R}}(\alpha ,\beta ,\gamma )|sm\rangle =e^{-im'\alpha }d_{m'm}^{s}(\beta )e^{-im\gamma },} where d m ′ m s ( β ) = ⟨ s m ′ | e − i β s y | s m ⟩ {\displaystyle d_{m'm}^{s}(\beta )=\langle sm'|e^{-i\beta s_{y}}|sm\rangle } is Wigner's small d-matrix . Note that for γ = 2π and α = β = 0 ; i.e., a full rotation about the z axis, the Wigner D-matrix elements become D m ′ m s ( 0 , 0 , 2 π ) = d m ′ m s ( 0 ) e − i m 2 π = δ m ′ m ( − 1 ) 2 m . {\displaystyle D_{m'm}^{s}(0,0,2\pi )=d_{m'm}^{s}(0)e^{-im2\pi }=\delta _{m'm}(-1)^{2m}.}
Recalling that a generic spin state can be written as a superposition of states with definite m , we see that if s is an integer, the values of m are all integers, and this matrix corresponds to the identity operator. However, if s is a half-integer, the values of m are also all half-integers, giving (−1) 2 m = −1 for all m , and hence upon rotation by 2 π the state picks up a minus sign. This fact is a crucial element of the proof of the spin–statistics theorem .
We could try the same approach to determine the behavior of spin under general Lorentz transformations , but we would immediately discover a major obstacle. Unlike SO(3), the group of Lorentz transformations SO(3,1) is non-compact and therefore does not have any faithful, unitary, finite-dimensional representations.
In case of spin- 1 / 2 particles, it is possible to find a construction that includes both a finite-dimensional representation and a scalar product that is preserved by this representation. We associate a 4-component Dirac spinor ψ with each particle. These spinors transform under Lorentz transformations according to the law ψ ′ = exp ( 1 8 ω μ ν [ γ μ , γ ν ] ) ψ , {\displaystyle \psi '=\exp {\left({\tfrac {1}{8}}\omega _{\mu \nu }[\gamma _{\mu },\gamma _{\nu }]\right)}\psi ,} where γ ν are gamma matrices , and ω μν is an antisymmetric 4 × 4 matrix parametrizing the transformation. It can be shown that the scalar product ⟨ ψ | ϕ ⟩ = ψ ¯ ϕ = ψ † γ 0 ϕ {\displaystyle \langle \psi |\phi \rangle ={\bar {\psi }}\phi =\psi ^{\dagger }\gamma _{0}\phi } is preserved. It is not, however, positive-definite, so the representation is not unitary.
Each of the ( Hermitian ) Pauli matrices of spin- 1 / 2 particles has two eigenvalues , +1 and −1. The corresponding normalized eigenvectors are ψ x + = | 1 2 , + 1 2 ⟩ x = 1 2 ( 1 1 ) , ψ x − = | 1 2 , − 1 2 ⟩ x = 1 2 ( 1 − 1 ) , ψ y + = | 1 2 , + 1 2 ⟩ y = 1 2 ( 1 i ) , ψ y − = | 1 2 , − 1 2 ⟩ y = 1 2 ( 1 − i ) , ψ z + = | 1 2 , + 1 2 ⟩ z = ( 1 0 ) , ψ z − = | 1 2 , − 1 2 ⟩ z = ( 0 1 ) . {\displaystyle {\begin{array}{lclc}\psi _{x+}=\left|{\frac {1}{2}},{\frac {+1}{2}}\right\rangle _{x}=\displaystyle {\frac {1}{\sqrt {2}}}\!\!\!\!\!&{\begin{pmatrix}{1}\\{1}\end{pmatrix}},&\psi _{x-}=\left|{\frac {1}{2}},{\frac {-1}{2}}\right\rangle _{x}=\displaystyle {\frac {1}{\sqrt {2}}}\!\!\!\!\!&{\begin{pmatrix}{1}\\{-1}\end{pmatrix}},\\\psi _{y+}=\left|{\frac {1}{2}},{\frac {+1}{2}}\right\rangle _{y}=\displaystyle {\frac {1}{\sqrt {2}}}\!\!\!\!\!&{\begin{pmatrix}{1}\\{i}\end{pmatrix}},&\psi _{y-}=\left|{\frac {1}{2}},{\frac {-1}{2}}\right\rangle _{y}=\displaystyle {\frac {1}{\sqrt {2}}}\!\!\!\!\!&{\begin{pmatrix}{1}\\{-i}\end{pmatrix}},\\\psi _{z+}=\left|{\frac {1}{2}},{\frac {+1}{2}}\right\rangle _{z}=&{\begin{pmatrix}1\\0\end{pmatrix}},&\psi _{z-}=\left|{\frac {1}{2}},{\frac {-1}{2}}\right\rangle _{z}=&{\begin{pmatrix}0\\1\end{pmatrix}}.\end{array}}}
(Because any eigenvector multiplied by a constant is still an eigenvector, there is ambiguity about the overall sign. In this article, the convention is chosen to make the first element imaginary and negative if there is a sign ambiguity. The present convention is used by software such as SymPy ; while many physics textbooks, such as Sakurai and Griffiths, prefer to make it real and positive.)
By the postulates of quantum mechanics , an experiment designed to measure the electron spin on the x , y , or z axis can only yield an eigenvalue of the corresponding spin operator ( S x , S y or S z ) on that axis, i.e. ħ / 2 or − ħ / 2 . The quantum state of a particle (with respect to spin), can be represented by a two-component spinor : ψ = ( a + b i c + d i ) . {\displaystyle \psi ={\begin{pmatrix}a+bi\\c+di\end{pmatrix}}.}
When the spin of this particle is measured with respect to a given axis (in this example, the x axis), the probability that its spin will be measured as ħ / 2 is just | ⟨ ψ x + | ψ ⟩ | 2 {\displaystyle {\big |}\langle \psi _{x+}|\psi \rangle {\big |}^{2}} . Correspondingly, the probability that its spin will be measured as − ħ / 2 is just | ⟨ ψ x − | ψ ⟩ | 2 {\displaystyle {\big |}\langle \psi _{x-}|\psi \rangle {\big |}^{2}} . Following the measurement, the spin state of the particle collapses into the corresponding eigenstate. As a result, if the particle's spin along a given axis has been measured to have a given eigenvalue, all measurements will yield the same eigenvalue (since | ⟨ ψ x + | ψ x + ⟩ | 2 = 1 {\displaystyle {\big |}\langle \psi _{x+}|\psi _{x+}\rangle {\big |}^{2}=1} , etc.), provided that no measurements of the spin are made along other axes.
The operator to measure spin along an arbitrary axis direction is easily obtained from the Pauli spin matrices. Let u = ( u x , u y , u z ) be an arbitrary unit vector. Then the operator for spin in this direction is simply S u = ℏ 2 ( u x σ x + u y σ y + u z σ z ) . {\displaystyle S_{u}={\frac {\hbar }{2}}(u_{x}\sigma _{x}+u_{y}\sigma _{y}+u_{z}\sigma _{z}).}
The operator S u has eigenvalues of ± ħ / 2 , just like the usual spin matrices. This method of finding the operator for spin in an arbitrary direction generalizes to higher spin states, one takes the dot product of the direction with a vector of the three operators for the three x -, y -, z -axis directions.
A normalized spinor for spin- 1 / 2 in the ( u x , u y , u z ) direction (which works for all spin states except spin down, where it will give 0 / 0 ) is 1 2 + 2 u z ( 1 + u z u x + i u y ) . {\displaystyle {\frac {1}{\sqrt {2+2u_{z}}}}{\begin{pmatrix}1+u_{z}\\u_{x}+iu_{y}\end{pmatrix}}.}
The above spinor is obtained in the usual way by diagonalizing the σ u matrix and finding the eigenstates corresponding to the eigenvalues. In quantum mechanics, vectors are termed "normalized" when multiplied by a normalizing factor, which results in the vector having a length of unity.
Since the Pauli matrices do not commute , measurements of spin along the different axes are incompatible. This means that if, for example, we know the spin along the x axis, and we then measure the spin along the y axis, we have invalidated our previous knowledge of the x axis spin. This can be seen from the property of the eigenvectors (i.e. eigenstates) of the Pauli matrices that | ⟨ ψ x ± | ψ y ± ⟩ | 2 = | ⟨ ψ x ± | ψ z ± ⟩ | 2 = | ⟨ ψ y ± | ψ z ± ⟩ | 2 = 1 2 . {\displaystyle {\big |}\langle \psi _{x\pm }|\psi _{y\pm }\rangle {\big |}^{2}={\big |}\langle \psi _{x\pm }|\psi _{z\pm }\rangle {\big |}^{2}={\big |}\langle \psi _{y\pm }|\psi _{z\pm }\rangle {\big |}^{2}={\tfrac {1}{2}}.}
So when physicists measure the spin of a particle along the x axis as, for example, ħ / 2 , the particle's spin state collapses into the eigenstate | ψ x + ⟩ {\displaystyle |\psi _{x+}\rangle } . When we then subsequently measure the particle's spin along the y axis, the spin state will now collapse into either | ψ y + ⟩ {\displaystyle |\psi _{y+}\rangle } or | ψ y − ⟩ {\displaystyle |\psi _{y-}\rangle } , each with probability 1 / 2 . Let us say, in our example, that we measure − ħ / 2 . When we now return to measure the particle's spin along the x axis again, the probabilities that we will measure ħ / 2 or − ħ / 2 are each 1 / 2 (i.e. they are | ⟨ ψ x + | ψ y − ⟩ | 2 {\displaystyle {\big |}\langle \psi _{x+}|\psi _{y-}\rangle {\big |}^{2}} and | ⟨ ψ x − | ψ y − ⟩ | 2 {\displaystyle {\big |}\langle \psi _{x-}|\psi _{y-}\rangle {\big |}^{2}} respectively). This implies that the original measurement of the spin along the x axis is no longer valid, since the spin along the x axis will now be measured to have either eigenvalue with equal probability.
The spin- 1 / 2 operator S = ħ / 2 σ forms the fundamental representation of SU(2) . By taking Kronecker products of this representation with itself repeatedly, one may construct all higher irreducible representations. That is, the resulting spin operators for higher-spin systems in three spatial dimensions can be calculated for arbitrarily large s using this spin operator and ladder operators . For example, taking the Kronecker product of two spin- 1 / 2 yields a four-dimensional representation, which is separable into a 3-dimensional spin-1 ( triplet states ) and a 1-dimensional spin-0 representation ( singlet state ).
The resulting irreducible representations yield the following spin matrices and eigenvalues in the z-basis:
Also useful in the quantum mechanics of multiparticle systems, the general Pauli group G n is defined to consist of all n -fold tensor products of Pauli matrices.
The analog formula of Euler's formula in terms of the Pauli matrices R ^ ( θ , n ^ ) = e i θ 2 n ^ ⋅ σ = I cos θ 2 + i ( n ^ ⋅ σ ) sin θ 2 {\displaystyle {\hat {R}}(\theta ,{\hat {\mathbf {n} }})=e^{i{\frac {\theta }{2}}{\hat {\mathbf {n} }}\cdot {\boldsymbol {\sigma }}}=I\cos {\frac {\theta }{2}}+i\left({\hat {\mathbf {n} }}\cdot {\boldsymbol {\sigma }}\right)\sin {\frac {\theta }{2}}} for higher spins is tractable, but less simple. [ 27 ]
In tables of the spin quantum number s for nuclei or particles, the spin is often followed by a "+" or "−". [ citation needed ] This refers to the parity with "+" for even parity (wave function unchanged by spatial inversion) and "−" for odd parity (wave function negated by spatial inversion). For example, see the isotopes of bismuth , in which the list of isotopes includes the column nuclear spin and parity. For Bi-209, the longest-lived isotope, the entry 9/2− means that the nuclear spin is 9/2 and the parity is odd.
The nuclear spin of atoms can be determined by sophisticated improvements to the original Stern-Gerlach experiment . [ 28 ] A single-energy (monochromatic) molecular beam of atoms in an inhomogeneous magnetic field will split into beams representing each possible spin quantum state. For an atom with electronic spin S and nuclear spin I , there are (2 S + 1)(2 I + 1) spin states. For example, neutral Na atoms, which have S = 1/2 , were passed through a series of inhomogeneous magnetic fields that selected one of the two electronic spin states and separated the nuclear spin states, from which four beams were observed. Thus, the nuclear spin for 23 Na atoms was found to be I = 3/2 . [ 29 ] [ 30 ]
The spin of pions , a type of elementary particle, was determined by the principle of detailed balance applied to those collisions of protons that produced charged pions and deuterium . p + p → π + + d {\displaystyle p+p\rightarrow \pi ^{+}+d} The known spin values for protons and deuterium allows analysis of the collision cross-section to show that π + {\displaystyle \pi ^{+}} has spin s π = 0 {\displaystyle s_{\pi }=0} . A different approach is needed for neutral pions. In that case the decay produced two gamma ray photons with spin one: π 0 → 2 γ {\displaystyle \pi ^{0}\rightarrow 2\gamma } This result supplemented with additional analysis leads to the conclusion that the neutral pion also has spin zero. [ 31 ] : 66
Spin has important theoretical implications and practical applications. Well-established direct applications of spin include:
Electron spin plays an important role in magnetism , with applications for instance in computer memories. The manipulation of nuclear spin by radio-frequency waves ( nuclear magnetic resonance ) is important in chemical spectroscopy and medical imaging.
Spin–orbit coupling leads to the fine structure of atomic spectra, which is used in atomic clocks and in the modern definition of the second . Precise measurements of the g -factor of the electron have played an important role in the development and verification of quantum electrodynamics . Photon spin is associated with the polarization of light ( photon polarization ).
An emerging application of spin is as a binary information carrier in spin transistors . The original concept, proposed in 1990, is known as Datta–Das spin transistor . [ 32 ] Electronics based on spin transistors are referred to as spintronics . The manipulation of spin in dilute magnetic semiconductor materials , such as metal-doped ZnO or TiO 2 imparts a further degree of freedom and has the potential to facilitate the fabrication of more efficient electronics. [ 33 ]
There are many indirect applications and manifestations of spin and the associated Pauli exclusion principle , starting with the periodic table of chemistry.
Spin was first discovered in the context of the emission spectrum of alkali metals . Starting around 1910, many experiments on different atoms produced a collection of relationships involving quantum numbers for atomic energy levels partially summarized in Bohr's model for the atom [ 34 ] : 106 Transitions between levels obeyed selection rules and the rules were known to be correlated with even or odd atomic number . Additional information was known from changes to atomic spectra observed in strong magnetic fields, known as the Zeeman effect . In 1924, Wolfgang Pauli used this large collection of empirical observations to propose a new degree of freedom, [ 7 ] introducing what he called a "two-valuedness not describable classically" [ 35 ] associated with the electron in the outermost shell .
The physical interpretation of Pauli's "degree of freedom" was initially unknown. Ralph Kronig , one of Alfred Landé 's assistants, suggested in early 1925 that it was produced by the self-rotation of the electron. When Pauli heard about the idea, he criticized it severely, noting that the electron's hypothetical surface would have to be moving faster than the speed of light in order for it to rotate quickly enough to produce the necessary angular momentum. This would violate the theory of relativity . Largely due to Pauli's criticism, Kronig decided not to publish his idea. [ 36 ]
In the autumn of 1925, the same thought came to Dutch physicists George Uhlenbeck and Samuel Goudsmit at Leiden University . Under the advice of Paul Ehrenfest , they published their results. [ 37 ] The young physicists immediately regretted the publication: Hendrik Lorentz and Werner Heisenberg both pointed out problems with the concept of a spinning electron. [ 38 ]
Pauli was especially unconvinced and continued to pursue his two-valued degree of freedom. This allowed him to formulate the Pauli exclusion principle , stating that no two electrons can have the same quantum state in the same quantum system.
Fortunately, by February 1926, Llewellyn Thomas managed to resolve a factor-of-two discrepancy between experimental results for the fine structure in the hydrogen spectrum and calculations based on Uhlenbeck and Goudsmit's (and Kronig's unpublished) model. [ 2 ] : 385 This discrepancy was due to a relativistic effect, the difference between the electron's rotating rest frame and the nuclear rest frame; the effect is now known as Thomas precession . [ 7 ] Thomas' result convinced Pauli that electron spin was the correct interpretation of his two-valued degree of freedom, while he continued to insist that the classical rotating charge model is invalid. [ 35 ] [ 6 ]
In 1927, Pauli formalized the theory of spin using the theory of quantum mechanics invented by Erwin Schrödinger and Werner Heisenberg . He pioneered the use of Pauli matrices as a representation of the spin operators and introduced a two-component spinor wave-function.
Pauli's theory of spin was non-relativistic. In 1928, Paul Dirac published his relativistic electron equation, using a four-component spinor (known as a " Dirac spinor ") for the electron wave-function. In 1940, Pauli proved the spin–statistics theorem , which states that fermions have half-integer spin, and bosons have integer spin. [ 7 ]
In retrospect, the first direct experimental evidence of the electron spin was the Stern–Gerlach experiment of 1922. However, the correct explanation of this experiment was only given in 1927. [ 39 ] The original interpretation assumed the two spots observed in the experiment were due to quantized orbital angular momentum . However, in 1927 Ronald Fraser showed that Sodium atoms are isotropic with no orbital angular momentum and suggested that the observed magnetic properties were due to electron spin. [ 40 ] In the same year, Phipps and Taylor applied the Stern-Gerlach technique to hydrogen atoms; the ground state of hydrogen has zero angular momentum but the measurements again showed two peaks. [ 41 ] Once the quantum theory became established, it became clear that the original interpretation could not have been correct:
the possible values of orbital angular momentum along one axis is always an odd number, unlike the observations. Hydrogen atoms have a single electron with two spin states giving the two spots observed; silver atoms have closed shells which do not contribute to the magnetic moment and only the unmatched outer electron's spin responds to the field. | https://en.wikipedia.org/wiki/Spin_(physics) |
The spin Hall effect (SHE) is a transport phenomenon predicted by Russian physicists Mikhail I. Dyakonov and Vladimir I. Perel in 1971. [ 1 ] [ 2 ] It consists of the appearance of spin accumulation on the lateral surfaces of an electric current -carrying sample, the signs of the spin directions being opposite on the opposing boundaries. In a cylindrical wire, the current-induced surface spins will wind around the wire. When the current direction is reversed, the directions of spin orientation is also reversed.
The spin Hall effect is a transport phenomenon consisting of the appearance of spin accumulation on the lateral surfaces of a sample carrying electric current. The opposing surface boundaries will have spins of opposite sign. It is analogous to the classical Hall effect , where charges of opposite sign appear on the opposing lateral surfaces in an electric-current carrying sample in a magnetic field . In the case of the classical Hall effect the charge build up at the boundaries is in compensation for the Lorentz force acting on the charge carriers in the sample due to the magnetic field. No magnetic field is needed for the spin Hall effect which is a purely spin -based phenomenon. The spin Hall effect belongs to the same family as the anomalous Hall effect , known for a long time in ferromagnets , which also originates from spin–orbit interaction .
The spin Hall effect (direct and inverse) was predicted by Russian physicists Mikhail I. Dyakonov and Vladimir I. Perel in 1971. [ 1 ] [ 2 ] They also introduced for the first time the notion of spin current .
In 1983 Averkiev and Dyakonov [ 3 ] proposed a way to measure the inverse spin Hall effect under optical spin orientation in semiconductors. The first experimental demonstration of the inverse spin Hall effect, based on this idea, was performed by Bakun et al. in 1984 [ 4 ]
The term "spin Hall effect" was introduced by Hirsch [ 5 ] who re-predicted this effect in 1999.
Experimentally, the (direct) spin Hall effect was observed in semiconductors [ 6 ] [ 7 ] more than 30 years after the original prediction.
Two possible mechanisms give origin to the spin Hall effect, in which an electric current (composed of moving charges) transforms into a spin current (a current of moving spins without charge flow). The original (extrinsic) mechanism devised by Dyakonov and Perel consisted of spin-dependent Mott scattering , where carriers with opposite spin diffuse in opposite directions when colliding with impurities in the material. The second mechanism is due to intrinsic properties of the material, where the carrier's trajectories are distorted due to spin–orbit interaction as a consequence of the asymmetries in the material. [ 8 ]
One can intuitively picture the intrinsic effect by using the classical analogy between an electron and a spinning tennis ball. The tennis ball deviates from its straight path in air in a direction depending on the sense of rotation, also known as the Magnus effect . In a solid, the air is replaced by an effective electric field due to asymmetries in the material, the relative motion between the magnetic moment (associated to the spin) and the electric field creates a coupling that distorts the motion of the electrons.
Similar to the standard Hall effect, both the extrinsic and the intrinsic mechanisms lead to an accumulation of spins of opposite signs on opposing lateral boundaries.
The spin current is described [ 1 ] [ 2 ] by a second-rank tensor q ij , where the first index refers to the direction of flow, and the second one to the spin component that is flowing. Thus q xy denotes the flow density of the y -component of spin in the x -direction. Introduce also the vector q i of charge flow density (which is related to the normal current density j = e q ), where e is the elementary charge. The coupling between spin and charge currents is due to spin-orbit interaction. It may be described in a very simple way [ 9 ] by introducing a single dimensionless coupling parameter ʏ .
No magnetic field is needed for spin Hall effect. However, if a strong enough magnetic field is applied in the direction perpendicular to the orientation of the spins at the surfaces, spins will precess around the direction of the magnetic field and the spin Hall effect will disappear. Thus in the presence of magnetic field, the combined action of the direct and inverse spin Hall effect leads to a change of the sample resistance, an effect that is of second order in spin-orbit interaction. This was noted by Dyakonov and Perel already in 1971 [ 2 ] and later elaborated in more detail by Dyakonov. [ 9 ] In recent years, the spin Hall magnetoresistance was extensively studied experimentally both in magnetic and non-magnetic materials (heavy metals, such as Pt, Ta, Pd, where the spin-orbit interaction is strong).
A transformation of spin currents consisting in interchanging ( swapping ) of the spin and flow directions ( q ij → q ji ) was predicted by Lifshits and Dyakonov. [ 10 ] Thus a flow in the x -direction of spins polarized along y is transformed to a flow in the y -direction of spins polarized along x . This prediction has yet not been confirmed experimentally.
The direct and inverse spin Hall effect can be monitored by optical means. The spin accumulation induces circular polarization of the emitted light , as well as the Faraday (or Kerr ) polarization rotation of the transmitted (or reflected) light. Observing the polarization of emitted light allows the spin Hall effect to be observed.
More recently, the existence of both direct and inverse effects was demonstrated not only in semiconductors , [ 11 ] but also in metals . [ 12 ] [ 13 ] [ 14 ]
The spin Hall effect can be used to manipulate electron spins electrically. For example, in combination with the electric stirring effect, the spin Hall effect leads to spin polarization in a localized conducting region. [ 15 ]
For a review of spin Hall effect, see for example: | https://en.wikipedia.org/wiki/Spin_Hall_effect |
The spin Nernst effect is a phenomenon of spin current generation caused by the thermal flow of electrons or magnons in condensed matter. Under a thermal drive such as temperature gradient or chemical potential gradient, spin-up and spin-down carriers can flow perpendicularly to the thermal current and towards opposite directions without the application of a magnetic field . This effect is similar to the spin Hall effect , where a pure spin current is induced by an electrical current . The spin Nernst effect can be detected by the spatial separation of opposite spin species, typically in the form of spin polarization (imbalanced spin accumulation) on the transverse boundaries of a material.
The spin Nernst effect of electrons was first experimentally observed in 2016 and published by two independent groups in 2017. [ 1 ] [ 2 ]
The spin Nernst effect of magnons (quanta of spin wave excitations) was theoretically proposed in 2016 [ 3 ] [ 4 ] in collinear antiferromagnetic materials , but its experimental confirmation remains elusive. In 2017, around the same time when its electronic counterpart was experimentally observed, the spin Nernst effect of magnons was first claimed in transition metal trichalcogenide MnPS 3 . [ 5 ] However, the experiment involved ambiguities that cannot convincingly verify the spin Nernst effect of magnons, awaiting further experimental studies. With a more accurate description accounting for real device geometry, it was believed that optical detection should be more reliable than electronic detection. [ 6 ] At present, optical detection of the spin Nernst effect of magnons has not been reported. | https://en.wikipedia.org/wiki/Spin_Nernst_Effect |
The spin angular momentum of light ( SAM ) is the component of angular momentum of light that is associated with the quantum spin and the rotation between the polarization degrees of freedom of the photon.
Spin is the fundamental property that distinguishes the two types of elementary particles: fermions , with half-integer spins; and bosons , with integer spins. Photons , which are the quanta of light , have been long recognized as spin-1 gauge bosons . The polarization of the light is commonly accepted as its “intrinsic” spin degree of freedom . However, in free space, only two transverse polarizations are allowed. Thus, the photon spin is always only connected to the two circular polarizations. To construct the full quantum spin operator of light, longitudinal polarized photon modes have to be introduced.
An electromagnetic wave is said to have circular polarization when its electric and magnetic fields rotate continuously around the beam axis during propagation. The circular polarization is left ( L {\displaystyle \mathrm {L} } ) or right ( R {\displaystyle \mathrm {R} } ) depending on the field rotation direction and, according to the convention used: either from the point of view of the source, or the receiver. Both conventions are used in science, depending on the context.
When a light beam is circularly polarized, each of its photons carries a spin angular momentum (SAM) of ± ℏ {\displaystyle \pm \hbar } , where ℏ {\displaystyle \hbar } is the reduced Planck constant and the ± {\displaystyle \pm } sign is positive for left and negative for right circular polarizations (this is adopting the convention from the point of view of the receiver most commonly used in optics ). This SAM is directed along the beam axis (parallel if positive, antiparallel if negative).
The above figure shows the instantaneous structure of the electric field of left ( L {\displaystyle \mathrm {L} } ) and right ( R {\displaystyle \mathrm {R} } ) circularly polarized light in space. The green arrows indicate the propagation direction.
The mathematical expressions reported under the figures give the three electric-field components of a circularly polarized plane wave propagating in the z {\displaystyle z} direction, in complex notation.
The general expression for the spin angular momentum is [ 1 ]
S = 1 c ∫ d 3 x π × A , {\displaystyle \mathbf {S} ={\frac {1}{c}}\int d^{3}x\mathbf {\pi } \times \mathbf {A} ,}
where c {\displaystyle c} is the speed of light in free space and π {\displaystyle \mathbf {\pi } } is the conjugate canonical momentum of the vector potential A {\displaystyle \mathbf {A} } . The general expression for the orbital angular momentum of light is L = 1 c ∫ d 3 x π μ x × ∇ A μ , {\displaystyle \mathbf {L} ={\frac {1}{c}}\int d^{3}x\pi ^{\mu }\mathbf {x} \times \mathbf {\nabla } A_{\mu },} where μ = { 0 , 1 , 2 , 3 } {\displaystyle \mu =\{0,1,2,3\}} denotes four indices of the spacetime and Einstein's summation convention has been applied.
To quantize light, the basic equal-time commutation relations have to be postulated, [ 2 ] [ A μ ( x , t ) , π ν ( x ′ , t ) ] = i ℏ c g μ ν δ 3 ( x − x ′ ) , {\displaystyle [A^{\mu }(\mathbf {x} ,t),\pi ^{\nu }(\mathbf {x} ',t)]=i\hbar cg^{\mu \nu }\delta ^{3}(\mathbf {x} -\mathbf {x} '),} [ A μ ( x , t ) , A ν ( x ′ , t ) ] = [ π μ ( x , t ) , π ν ( x ′ , t ) ] = 0 , {\displaystyle [A^{\mu }(\mathbf {x} ,t),A^{\nu }(\mathbf {x} ',t)]=[\pi ^{\mu }(\mathbf {x} ,t),\pi ^{\nu }(\mathbf {x} ',t)]=0,} where ℏ {\displaystyle \hbar } is the reduced Planck constant and g μ ν = d i a g { 1 , − 1 , − 1 , − 1 } {\displaystyle g^{\mu \nu }={\rm {{diag}\{1,-1,-1,-1\}}}} is the metric tensor of the Minkowski space .
Then, one can verify that both S {\displaystyle \mathbf {S} } and L {\displaystyle \mathbf {L} } satisfy the canonical angular momentum commutation relations [ S i , S j ] = i ℏ ϵ i j k S k , {\displaystyle [S_{i},S_{j}]=i\hbar \epsilon _{ijk}S_{k},} [ L i , L j ] = i ℏ ϵ i j k L k , {\displaystyle [L_{i},L_{j}]=i\hbar \epsilon _{ijk}L_{k},} and they commute with each other [ S i , L j ] = 0 {\displaystyle [S_{i},L_{j}]=0} .
After the plane-wave expansion , the photon spin can be re-expressed in a simple and intuitive form in the wave-vector space S = ℏ ∫ d 3 k ϕ ^ k † s ^ ϕ ^ k {\displaystyle \mathbf {S} =\hbar \int d^{3}k{\hat {\phi }}_{\mathbf {k} }^{\dagger }\mathbf {\hat {s}} {\hat {\phi }}_{\mathbf {k} }} where the vector ϕ ^ k = ( a ^ k , 1 , a ^ k , 2 , a ^ k , 3 ) {\displaystyle {\hat {\phi }}_{\mathbf {k} }=({\hat {a}}_{\mathbf {k} ,1},{\hat {a}}_{\mathbf {k} ,2},{\hat {a}}_{\mathbf {k} ,3})} is the field operator of the photon in wave-vector space and the 3 × 3 {\displaystyle 3\times 3} matrix s ^ = ∑ λ = 1 3 s ^ λ ϵ ( k , λ ) {\displaystyle \mathbf {\hat {s}} =\sum _{\lambda =1}^{3}{\hat {s}}_{\lambda }\mathbf {\epsilon } (\mathbf {k} ,\lambda )} is the spin-1 operator of the photon with the SO(3) rotation generators s ^ 1 = [ 0 0 0 0 0 − i 0 i 0 ] , s ^ 2 = [ 0 0 i 0 0 0 − i 0 0 ] , s ^ 3 = [ 0 − i 0 i 0 0 0 0 0 ] , {\displaystyle {\hat {s}}_{1}={\begin{bmatrix}0&0&0\\0&0&-i\\0&i&0\end{bmatrix}},\qquad {\hat {s}}_{2}={\begin{bmatrix}0&0&i\\0&0&0\\-i&0&0\end{bmatrix}},\qquad {\hat {s}}_{3}={\begin{bmatrix}0&-i&0\\i&0&0\\0&0&0\end{bmatrix}},} and the two unit vectors ϵ ( k , 1 ) ⋅ k = ϵ ( k , 2 ) ⋅ k = 0 {\displaystyle {\boldsymbol {\epsilon }}(\mathbf {k} ,1)\cdot \mathbf {k} ={\boldsymbol {\epsilon }}(\mathbf {k} ,2)\cdot \mathbf {k} =0} denote the two transverse polarizations of light in free space and unit vector ϵ ( k , 3 ) = k / | k | {\displaystyle {\boldsymbol {\epsilon }}(\mathbf {k} ,3)=\mathbf {k} /|\mathbf {k} |} denotes the longitudinal polarization.
Due to the longitudinal polarized photon and scalar photon have been involved, both S {\displaystyle \mathbf {S} } and L {\displaystyle \mathbf {L} } are not gauge invariant. To incorporate the gauge invariance into the photon angular momenta, a re-decomposition of the total QED angular momentum and the Lorenz gauge condition have to be enforced. Finally, the direct observable part of spin and orbital angular momenta of light are given by S o b s = i ℏ ∫ d 3 k ( a ^ k , 2 † a ^ k , 1 − a ^ k , 1 † a ^ k , 2 ) k | k | = ε 0 ∫ d 3 x E ⊥ × A ⊥ , {\displaystyle \mathbf {S} ^{\rm {obs}}=i\hbar \int d^{3}k({\hat {a}}_{\mathbf {k} ,2}^{\dagger }{\hat {a}}_{\mathbf {k} ,1}-{\hat {a}}_{\mathbf {k} ,1}^{\dagger }{\hat {a}}_{\mathbf {k} ,2}){\frac {\mathbf {k} }{|\mathbf {k} |}}=\varepsilon _{0}\int d^{3}x\mathbf {E} _{\perp }\times \mathbf {A} _{\perp },} and L M o b s = ε 0 ∫ d 3 x E ⊥ j x × ∇ A ⊥ j {\displaystyle \mathbf {L} _{M}^{\rm {obs}}=\varepsilon _{0}\int d^{3}xE_{\perp }^{j}\mathbf {x} \times \mathbf {\nabla } A_{\perp }^{j}} which recover the angular momenta of classical transverse light. [ 3 ] Here, E ⊥ {\displaystyle \mathbf {E} _{\perp }} ( A ⊥ {\displaystyle \mathbf {A} _{\perp }} ) is the transverse part of the electric field ( vector potential ), ε 0 {\displaystyle \varepsilon _{0}} is the vacuum permittivity , and we are using SI units .
We can define the annihilation operators for circularly polarized transverse photons: a ^ k , L = 1 2 ( a ^ k , 1 − i a ^ k , 2 ) , {\displaystyle {\hat {a}}_{\mathbf {k} ,\mathrm {L} }={\frac {1}{\sqrt {2}}}\left({\hat {a}}_{\mathbf {k} ,1}-i{\hat {a}}_{\mathbf {k} ,2}\right),} a ^ k , R = 1 2 ( a ^ k , 1 + i a ^ k , 2 ) , {\displaystyle {\hat {a}}_{\mathbf {k} ,\mathrm {R} }={\frac {1}{\sqrt {2}}}\left({\hat {a}}_{\mathbf {k} ,1}+i{\hat {a}}_{\mathbf {k} ,2}\right),} with polarization unit vectors e ( k , L ) = 1 2 [ e ( k , 1 ) + i e ( k , 2 ) ] , {\displaystyle \mathbf {e} (\mathbf {k} ,\mathrm {L} )={\frac {1}{\sqrt {2}}}\left[\mathbf {e} (\mathbf {k} ,1)+i\mathbf {e} (\mathbf {k} ,2)\right],} e ( k , R ) = 1 2 [ e ( k , 1 ) − i e ( k , 2 ) ] . {\displaystyle \mathbf {e} (\mathbf {k} ,\mathrm {R} )={\frac {1}{\sqrt {2}}}\left[\mathbf {e} (\mathbf {k} ,1)-i\mathbf {e} (\mathbf {k} ,2)\right].}
Then, the transverse-field photon spin can be re-expressed as S o b s = ∫ d 3 k ℏ ( a ^ k , L † a ^ k , L − a ^ k , R † a ^ k , R ) k | k | , {\displaystyle \mathbf {S} ^{\rm {obs}}=\int d^{3}k\hbar \left({\hat {a}}_{\mathbf {k} ,L}^{\dagger }{\hat {a}}_{\mathbf {k} ,L}-{\hat {a}}_{\mathbf {k} ,R}^{\dagger }{\hat {a}}_{\mathbf {k} ,R}\right){\frac {\mathbf {k} }{|\mathbf {k} |}},}
For a single plane-wave photon , the spin can only have two values ± ℏ {\displaystyle \pm \hbar } , which are eigenvalues of the spin operator s ^ 3 {\displaystyle {\hat {s}}_{3}} . The corresponding eigenfunctions describing photons with well defined values of SAM are described as circularly polarized waves: | ± ⟩ = ( 1 ± i 0 ) . {\displaystyle |\pm \rangle ={\begin{pmatrix}1\\\pm i\\0\end{pmatrix}}.} | https://en.wikipedia.org/wiki/Spin_angular_momentum_of_light |
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