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The minimum ignition energy ( MIE ) is a safety characteristic in explosion protection and prevention which determines the ignition capability of fuel-air mixtures, where the fuel may be combustible vapor, gas or dust. It is defined as the minimum electrical energy stored in a capacitor, which, when discharged, is sufficient to ignite the most ignitable mixture of fuel and air under specified test conditions. [ 1 ] The MIE is one of the assessment criteria for the effectiveness of ignition, e.g. the discharge of electrostatic energy, mechanical ignition sources or electromagnetic radiation. It is an important parameter for the design of the protective measure of "avoidance of effective ignition sources". [ 2 ]
This combustion article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Minimum_ignition_energy |
Minimum information about a microarray experiment ( MIAME ) is a standard created by the FGED Society for reporting microarray experiments. [ 1 ]
MIAME is intended to specify all the information necessary to interpret the results of the experiment unambiguously and to potentially reproduce the experiment. While the standard defines the content required for compliant reports, it does not specify the format in which this data should be presented. MIAME describes the minimum information required to ensure that microarray data can be easily interpreted and that results derived from its analysis can be independently verified. [ 2 ] There are a number of file formats used to represent this data, as well as both public and subscription-based repositories for such experiments. [ 2 ] Additionally, software exists to aid the preparation of MIAME-compliant reports.
MIAME revolves around six key components: raw data , normalized data , sample annotations, experimental design , array annotations, and data protocols. | https://en.wikipedia.org/wiki/Minimum_information_about_a_microarray_experiment |
The minimum information about a simulation experiment ( MIASE ) [ 1 ] is a list of the common set of information a modeller needs to enable the execution and reproduction of a numerical simulation experiment, derived from a given set of quantitative models.
MIASE is a registered project of the MIBBI (minimum information for biological and biomedical investigations). [ 2 ]
The MIASE project was launched in 2007 by Dagmar Köhn and Nicolas Le Novère and first presented on the 12th SBML Forum Meeting in October 2007. Since then, MIASE was discussed on various meetings, not only within the SBML community. MIASE has become a community effort involving people from various standardisation communities as well as developers of simulation tools.
In April 2009, MIASE was part of the " CellML , SBGN , SBO , BioPAX , and MIASE Super-Workshop 2009 ".
The MIASE Guidelines are composed of the following parts: Information about the models to use, information about the simulation steps, and Information about the output:
All models used in the experiment must be identified, accessible, and fully described.
A precise description of the simulation steps and other procedures used by the experiment must be provided.
All information necessary to obtain the desired numerical results must be provided. | https://en.wikipedia.org/wiki/Minimum_information_about_a_simulation_experiment |
MIRIAM (Minimum Information Required In The Annotation of Models [ 1 ] ) is a community-level effort to standardize the annotation and curation processes of quantitative models of biological systems. [ 2 ] It consists of a set of guidelines suitable for use with any structured format, allowing different groups to collaborate and share resulting models. Adherence to these guidelines also facilitates the sharing of software and service infrastructures built upon modeling activities.
The idea of "a set of good practices" including "some obligatory metadata" was first proposed by Nicolas Le Novère in October 2004 as part of a discussion to develop a common database of models in systems biology (which led to the creation of BioModels Database ). These initial ideas were further refined at a meeting in Heidelberg, during ICSB 2004, with representatives from many other interested groups.
MIRIAM is a registered project of the MIBBI (minimum information for biological and biomedical investigations). [ 3 ]
The MIRIAM Guidelines are composed of three parts, reference correspondence , attribution annotation , and external resource annotation , each of which deals with a different aspect of information that should be included within a model.
'Reference correspondence' deals with the basic reference information needed to make use of the model, detailing on a gross level the format of the model file, and its instantiability for simulation purposes.
'Attribution annotation' deals with the attribution information that must be embedded within the model file.
'External resource annotation' defines the manner in which annotations should be constructed. Those annotations contain references to entities in databases, classifications, ontologies, etc. One of the purposes of annotation is to allow unambiguous identification of the various model components.
More information about the existing qualifiers is available from BioModels.net. [ 4 ]
So far, annotation is mainly a manual work, so to ensure their longevity the usage of perennial URIs is necessary. It was recognised that the generation of valid and unique URIs for annotation required the creation of a catalogue of shared namespaces for use by the community. This function is provided by the MIRIAM Registry . The Registry also provides a variety of supporting auxiliary features to enable automated procedures based upon these URIs. The ability to generate resolvable identifiers is provided through the use of the resolving layer, Identifiers.org . | https://en.wikipedia.org/wiki/Minimum_information_required_in_the_annotation_of_models |
Minimum information standard s are sets of guidelines and formats for reporting data derived by specific high-throughput methods. Their purpose is to ensure the data generated by these methods can be easily verified, analysed and interpreted by the wider scientific community . Ultimately, they facilitate the transfer of data from journal articles (unstructured data) into databases (structured data) in a form that enables data to be mined across multiple data sets. Minimal information standards are available for a vast variety of experiment types including microarray ( MIAME ), RNAseq ( MINSEQE ), metabolomics (MSI) and proteomics ( MIAPE ). [ 1 ]
Minimum information standards typically have two parts. Firstly, there is a set of reporting requirements – typically presented as a table or a checklist. Secondly, there is a data format. Information about an experiment needs to be converted into the appropriate data format for it to be submitted to the relevant database. In the case of MIAME , the data format is provided in spreadsheet format (MAGE-TAB). Some of the communities that maintain minimum information standards also provide tools to help experimental researchers to annotate their data. [ 1 ]
The individual minimum information standards are brought by the communities of cross-disciplinary specialists focused on the problematic of the specific method used in experimental biology. The standards then provide specifications what information about the experiments ( metadata ) is crucial and important to be reported together with the resultant data to make it comprehensive. [ 2 ] [ 3 ] The need for this standardization is largely driven by the development of high-throughput experimental methods that provide tremendous amounts of data. The development of minimum information standards of different methods is since 2008 being harmonized by "Minimum Information about a Biomedical or Biological Investigation" (MIBBI) project. [ 4 ]
MIAPPE is an open, community driven project to harmonize data from plant phenotyping experiments. MIAPPE comprises both a conceptual checklist of metadata required to adequately describe a plant phenotyping experiment.
Published in 2009 these guidelines for the basis of requirements by many journals when submitting QPCR data, sadly they are not adhered to enough. [ 5 ]
Minimum Information About a Microarray Experiment (MIAME) [ 3 ] describes the Minimum Information About a Microarray Experiment that is needed to enable the interpretation of the results of the experiment unambiguously and potentially to reproduce the experiment and is aimed at facilitating the dissemination of data from microarray experiments. It was published by the FGED Society in 2001 and was the first published minimum information standard for high-throughput experiments in the life sciences.
MIAME contains a number of extensions to cover specific biological domains, including MIAME-env, MIAME-nut and MIAME-tox, covering environmental genomics, nutritional genomics and toxogenomics, respectively.
Electrophysiology is a technology used to study the electrical properties of biological cells and tissues. Electrophysiology typically involves the measurements of voltage change or electric current flow on a wide variety of scales from single ion channel proteins to whole tissues. This document is a single module, as part of the Minimum Information about a Neuroscience investigation (MINI) family of reporting guideline documents, produced by community consultation and continually available for public comment. A MINI module represents the minimum information that should be reported about a dataset to facilitate computational access and analysis to allow a reader to interpret and critically evaluate the processes performed and the conclusions reached, and to support their experimental corroboration. In practice a MINI module comprises a checklist of information that should be provided (for example about the protocols employed) when a data set is described for publication. The full specification of the MINI module can be found here. [ 6 ]
Minimum Information About an RNAi Experiment (MIARE) is a data reporting guideline which describes the minimum information that should be reported about an RNAi experiment to enable the unambiguous interpretation and reproduction of the results.
Advances in genomics and functional genomics have enabled large-scale analyses of gene and protein function by means of high-throughput cell biological analyses. Thereby, cells in culture can be perturbed in vitro and the induced effects recorded and analyzed. Perturbations can be triggered in several ways, for instance with molecules (siRNAs, expression constructs, small chemical compounds, ligands for receptors, etc.), through environmental stresses (such as temperature shift, serum starvation, oxygen deprivation, etc.), or combinations thereof. The cellular responses to such perturbations are analyzed in order to identify molecular events in the biological processes addressed and understand biological principles.
We propose the Minimum Information About a Cellular Assay (MIACA) for reporting a cellular assay, and CA-OM, the modular cellular assay object model, to facilitate exchange of data and accompanying information, and to compare and integrate data that originate from different, albeit complementary approaches, and to elucidate higher order principles. Documents describing MIACA are available and provide further information as well as the checklist of terms that should be reported.
The Minimum Information About a Proteomic Experiment documents describe information which should be given along with a proteomic experiment. The parent document describes the processes and principles underpinning the development of a series of domain specific documents which now cover all aspects of a MS-based proteomics workflow.
This document has been developed and maintained by the Molecular Interaction worktrack of the HUPO-PSI (www.psidev.info) and describes the Minimum Information about a Molecular Interaction experiment.
The Minimum Information About a Protein Affinity Reagent has been developed and maintained by the Molecular Interaction worktrack of the HUPO-PSI (www.psidev.info)in conjunction with the HUPO Antibody Initiative and a European consortium of binder producers and seeks to encourage users to improve their description of binding reagents, such as antibodies, used in the process of protein identification.
The Minimum Information About a Bioactive Entity was produced by representatives from both large pharma and academia who are looking to improve the description of usually small molecules which bind to, and potentially modulate the activity of, specific targets in a living organism. This document encompasses drug-like molecules as well as herbicides, pesticides and food additives. It is primarily maintained through the EMBL-EBI Industry program (www.ebi.ac.uk/industry).
This specification is being developed by the Genomic Standards Consortium
The Minimum Information about a Flow Cytometry Experiment (MIFlowCyt) is a standard related to flow cytometry which establishes criteria to record information on experimental overview, samples, instrumentation and data analysis. [ 2 ] It promotes consistent annotation of clinical, biological and technical issues surrounding a flow cytometry experiment . [ 2 ] [ 7 ] [ 8 ]
Criteria for Minimum Information About a Phylogenetic Analysis were described in 2006. [ 9 ]
The MIRAGE project is supported and coordinated by the Beilstein-Institut to establish guidelines for data handling and processing in glycomics research [1] [ 10 ] [ 11 ]
The Minimal Information Required In the Annotation of Models ( MIRIAM ), is a set of rules for the curation and annotation of quantitative models of biological systems.
The Minimum Information About a Simulation Experiment ( MIASE ) is an effort to standardize the description of simulation experiments in the field of systems biology.
The Standards for Reporting Enzymology Data ( STRENDA ) is an initiative which specifically focuses on the development of guidelines for reporting (describing metadata) enzymology experiments with the aim to improve the quality of enzymology data published in the scientific literature. | https://en.wikipedia.org/wiki/Minimum_information_standard |
In microbiology , the minimum inhibitory concentration ( MIC ) is the lowest concentration of a chemical , usually a drug, which prevents visible in vitro growth of bacteria or fungi . [ 1 ] [ 2 ] MIC testing is performed in both diagnostic [ 1 ] [ 2 ] and drug discovery laboratories. [ 3 ] [ 4 ]
The MIC is determined by preparing a dilution series of the chemical, adding agar or broth , then inoculating with bacteria or fungi, and incubating at a suitable temperature . The value obtained is largely dependent on the susceptibility of the microorganism and the antimicrobial potency of the chemical, but other variables can affect results too. [ 5 ] The MIC is often expressed in micrograms per milliliter (μg/mL) or milligrams per liter (mg/L).
In diagnostic labs, MIC test results are used to grade the susceptibility of microbes. These grades are assigned based on agreed upon values called breakpoints. Breakpoints are published by standards development organizations such as the U.S. Clinical and Laboratory Standards Institute (CLSI), the British Society for Antimicrobial Chemotherapy (BSAC) and the European Committee on Antimicrobial Susceptibility Testing (EUCAST). [ 6 ] [ 7 ] [ 8 ] The purpose of measuring MICs and grading microbes is to enable physicians to prescribe the most appropriate antimicrobial treatment.
The first step in drug discovery is often measurement of the MICs of biological extracts , isolated compounds or large chemical libraries against bacteria and fungi of interest. [ 9 ] [ 10 ] MIC values provide a quantitative measure of an extract or compound's antimicrobial potency . The lower the MIC, the more potent the antimicrobial. [ 4 ] When in vitro toxicity data is available, MICs can also be used to calculate selectivity index values, a measure of off-target to target toxicity . [ 4 ]
After the discovery and commercialization of antibiotics, microbiologist, pharmacologist, and physician Alexander Fleming developed the broth dilution technique using the turbidity of the broth for assessment. [ 11 ] This is commonly believed to be the conception point of minimum inhibitory concentrations. [ 12 ] Later in the 1980s, the Clinical and Laboratory Standards Institute consolidated the methods and standards for MIC determination and clinical usage. Because pathogens continue to evolve, and new drugs continue to be developed, the CLSI's MIC protocols are periodically updated to reflect these changes. [ 13 ] The protocols and parameters set by the CLSI are considered to be the "gold standard" in the United States and are used by regulatory authorities, such as the FDA, to make evaluations. [ 14 ]
Nowadays, the MIC is used in antimicrobial susceptibility testing. The MIC is reported by providing the susceptibility interpretation next to each antibiotic. The different susceptibility interpretations are: "S" (susceptible or responding to a standard dosing regimen), "I" (intermediate or requiring increased exposure), and "R" (resistant). These interpretations were developed by the CLSI and EUCAST . [ 6 ] [ 8 ] There have been major discrepancies between the breakpoints from various European countries over the years, and between those from the CLSI and EUCAST. [ 15 ]
In clinics, more often than not, exact pathogens cannot be easily determined by symptoms of the patient. Then, even if the pathogen is determined, different strains of pathogens, such as Staphylococcus aureus, have varying levels of resistance to antimicrobials . As such, it is difficult to prescribe correct antimicrobials. [ 16 ] The MIC is determined in such cases by growing the pathogen isolate from the patient on plate or broth, which is later used in the assay. [ 17 ] Thus, knowledge of the MIC will provide a physician valuable information for making a prescription.
Accurate and precise usage of antimicrobials is also important in the context of multidrug-resistant bacteria . Microbes such as bacteria have been gaining resistance to antimicrobials they were previously susceptible to. [ 18 ] Usage of incompatible levels of antimicrobials provides the selective pressure that has driven the direction and evolution of resistance of bacterial pathogens. [ 19 ] This has been seen at sub-MIC levels of antibiotics. [ 20 ] As such, it is increasingly important to determine the MIC in order to make the best choice in prescribing antimicrobials.
There are three main reagents necessary to run this assay: the media, an antimicrobial agent, and the microbe being tested. The most commonly used media is cation-adjusted Mueller Hinton Broth, due to its ability to support the growth of most pathogens and its lack of inhibitors towards common antibiotics. [ 21 ] Depending on the pathogen and antibiotics being tested, the media can be changed and/or adjusted. The antimicrobial concentration is adjusted into the correct concentration by mixing stock antimicrobial with media. The adjusted antimicrobial is serially diluted into multiple tubes (or wells) to obtain a gradient. The dilution rate can be adjusted depending on the breakpoint and the practitioner's needs. The microbe, or the inoculating agent, must come from the same colony-forming unit, and must be at the correct concentration. This may be adjusted by incubation time and dilution. For verification, the positive control is plated in a hundred fold dilution to count colony forming units. The microbes inoculate the tubes (or plate) and are incubated for 16–20 hours. The MIC is generally determined by turbidity. [ 21 ]
Etests can be used as an alternative method to determine the minimum inhibitory concentrations of a wide range of antimicrobial agents against different organisms. They have been widely used in microbiology laboratories around the world. Manufactured by bioMérieux , Etests are a ready-to-use, non-porous plastic reagent strip with a predefined gradient of antibiotic, covering a continuous concentration range. [ 22 ]
While the MIC is the lowest concentration of an antibacterial or antifungal agent necessary to inhibit visible growth, the minimum bactericidal concentration (MBC) is the minimum concentration of an antibacterial agent that results in bacterial death. It is defined by the inability to re-culture bacteria, and the closer the MIC is to the MBC, the more bactericidal the compound. [ 23 ]
MIC is used clinically over MBC because MIC is more easily determined. [ 13 ] In addition, drug effectiveness is generally similar when taken at both MIC and MBC concentrations because the host immune system can expel the pathogen when bacterial proliferation is at a standstill. [ 24 ] When the MBC is much higher than the MIC, drug toxicity makes taking the MBC of the drug detrimental to patient. Antimicrobial toxicity can come in many forms, such as immune hypersensitivity and off-target toxicity. [ 25 ]
Increasing bacterial outbreaks and newer strains of microbes and pathogens invading our lives daily, there is an increasing need to test these microbes. Mutating bacteria pose a higher risk to humans than ever and thus the MIC Test is important to ensure we are one step ahead of them. | https://en.wikipedia.org/wiki/Minimum_inhibitory_concentration |
Minimum orbit intersection distance ( MOID ) is a measure used in astronomy to assess potential close approaches and collision risks between astronomical objects. [ 1 ] [ 2 ] It is defined as the distance between the closest points of the osculating orbits of two bodies. Of greatest interest is the risk of a collision with Earth . Earth MOID is often listed on comet and asteroid databases such as the JPL Small-Body Database . MOID values are also defined with respect to other bodies as well: Jupiter MOID, Venus MOID and so on.
An object is classified as a potentially hazardous object (PHO) – that is, posing a possible risk to Earth – if, among other conditions, its Earth MOID is less than 0.05 AU . For more massive bodies than Earth, there is a potentially notable close approach with a larger MOID; for instance, Jupiter MOIDs less than 1 AU are considered noteworthy since Jupiter is the most massive planet. [ 1 ]
A low MOID does not mean that a collision is inevitable as the planets frequently perturb the orbit of small bodies. It is also necessary that the two bodies reach that point in their orbits at the same time before the smaller body is perturbed into a different orbit with a different MOID value. Two objects gravitationally locked in orbital resonance may never approach one another. Numerical integrations become increasingly divergent as trajectories are projected further forward in time, especially beyond times where the smaller body is repeatedly perturbed by other planets. MOID has the convenience that it is obtained directly from the orbital elements of the body and no numerical integration into the future is used. [ 3 ]
The only object that has ever been rated at 4 on the Torino Scale (since downgraded), the Aten asteroid (99942) Apophis , has an Earth MOID of 0.00026 AU (39,000 km ; 24,000 mi ). This is not the smallest Earth MOID in the catalogues; many bodies with a small Earth MOID are not classed as PHO's because the objects are less than roughly 140 meters in diameter (or absolute magnitude , H > 22). Earth MOID values are generally more practical for asteroids less than 140 meters in diameter as those asteroids are very dim and often have a short observation arc with a poorly determined orbit. As of September 2023, there have been seven objects detected and their Earth-MOID calculated before the Earth impact. [ 4 ] The first two objects that were detected and had their Earth-MOID calculated before Earth impact were the small asteroids 2008 TC 3 and 2014 AA . 2014 AA is listed with a MOID of 0.00000045 AU (67 km; 42 mi), [ 5 ] and is the second smallest MOID calculated for an Apollo asteroid after 2020 QY 2 with an Earth-MOID of 0.00000039 AU (58 km; 36 mi). [ 6 ]
Solar System → Local Interstellar Cloud → Local Bubble → Gould Belt → Orion Arm → Milky Way → Milky Way subgroup → Local Group → Local Sheet → Virgo Supercluster → Laniakea Supercluster → Local Hole → Observable universe → Universe Each arrow ( → ) may be read as "within" or "part of". | https://en.wikipedia.org/wiki/Minimum_orbit_intersection_distance |
In mathematics, the minimum rank is a graph parameter mr ( G ) {\displaystyle \operatorname {mr} (G)} for a graph G . It was motivated by the Colin de Verdière graph invariant .
The adjacency matrix of an undirected graph is a symmetric matrix whose rows and columns both correspond to the vertices of the graph. Its elements are all 0 or 1, and the element in row i and column j is nonzero whenever vertex i is adjacent to vertex j in the graph. More generally, a generalized adjacency matrix is any symmetric matrix of real numbers with the same pattern of nonzeros off the diagonal (the diagonal elements may be any real numbers). The minimum rank of G {\displaystyle G} is defined as the smallest rank of any generalized adjacency matrix of the graph; it is denoted by mr ( G ) {\displaystyle \operatorname {mr} (G)} .
Here are some elementary properties.
Several families of graphs may be characterized in terms of their minimum ranks. | https://en.wikipedia.org/wiki/Minimum_rank_of_a_graph |
Minimum safe altitude warning ( MSAW ) is an automated warning system for air traffic controllers (ATCO). It is a ground-based safety net intended to warn the controller about increased risk of controlled flight into terrain accidents by generating, in a timely manner, an alert of aircraft proximity to terrain or obstacles. [ 1 ]
International Civil Aviation Organization Doc 4444 requires that radar systems should provide for the display of safety-related alerts including the presentation of minimum safe altitude warning. [ 2 ] The radar equipment predicts an aircraft’s position in 2 minutes based on present path of flight, and the controller issues a safety alert if the projected path encounters terrain or an obstruction. An unusually rapid descent rate on a non-precision approach can trigger such an alert. [ 3 ]
It is worth mentioning that ICAO Doc 4444 does not provide a definition of the term MSAW. Instead the term MSAW is ambiguously used in ATC community to identify such warnings as well as for data processing systems providing the alert function. [ citation needed ] | https://en.wikipedia.org/wiki/Minimum_safe_altitude_warning |
The minimum total potential energy principle is a fundamental concept used in physics and engineering . It dictates that at low temperatures a structure or body shall deform or displace to a position that (locally) minimizes the total potential energy , with the lost potential energy being converted into kinetic energy (specifically heat).
The total potential energy, Π {\displaystyle {\boldsymbol {\Pi }}} , is the sum of the elastic strain energy, U , stored in the deformed body and the potential energy, V , associated to the applied forces: [ 1 ]
This energy is at a stationary position when an infinitesimal variation from such position involves no change in energy: [ 1 ]
The principle of minimum total potential energy may be derived as a special case of the virtual work principle for elastic systems subject to conservative forces .
The equality between external and internal virtual work (due to virtual displacements) is:
where
In the special case of elastic bodies, the right-hand-side of ( 3 ) can be taken to be the change, δ U {\displaystyle \delta \mathbf {U} } , of elastic strain energy U due to infinitesimal variations of real displacements.
In addition, when the external forces are conservative forces , the left-hand-side of ( 3 ) can be seen as the change in the potential energy function V of the forces. The function V is defined as: [ 2 ] V = − ∫ S t u T T d S − ∫ V u T f d V {\displaystyle \mathbf {V} =-\int _{S_{t}}\mathbf {u} ^{T}\mathbf {T} dS-\int _{V}\mathbf {u} ^{T}\mathbf {f} dV} where the minus sign implies a loss of potential energy as the force is displaced in its direction. With these two subsidiary conditions, Equation 3 becomes: − δ V = δ U {\displaystyle -\delta \ \mathbf {V} =\delta \ \mathbf {U} } This leads to ( 2 ) as desired. The variational form of ( 2 ) is often used as the basis for developing the finite element method in structural mechanics . | https://en.wikipedia.org/wiki/Minimum_total_potential_energy_principle |
A minimum viable product ( MVP ) is a version of a product with just enough features to be usable by early customers who can then provide feedback for future product development . [ 1 ] [ 2 ]
A focus on releasing an MVP means that developers potentially avoid lengthy and (possibly) unnecessary work. Instead, they iterate on working versions and respond to feedback, challenging and validating assumptions about a product's requirements. [ 3 ] The term was coined and defined in 2001 by Frank Robinson [ 4 ] and then popularized by Steve Blank and Eric Ries . [ 5 ] [ 6 ] [ 7 ] [ 8 ] It may also involve carrying out market analysis beforehand. The MVP is analogous to experimentation in the scientific method applied in the context of validating business hypotheses. It is utilized so that prospective entrepreneurs would know whether a given business idea would actually be viable and profitable by testing the assumptions behind a product or business idea. [ 9 ] The concept can be used to validate a market need for a product [ 9 ] and for incremental developments of an existing product. [ 10 ] As it tests a potential business model to customers to see how the market would react, it is especially useful for new/startup companies who are more concerned with finding out where potential business opportunities exist rather than executing a prefabricated, isolated business model.
A minimum viable product has just enough core features to effectively deploy the product, and no more. Developers typically deploy the product to a subset of possible customers, such as early adopters who are thought to be more forgiving, more likely to give feedback, and able to grasp a product vision from an early prototype or marketing information. This strategy aims to avoid building products that customers do not want and seeks to maximize information about the customer with the least money spent. The technique falls under the Lean Startup methodology as MVPs aim to test business hypotheses and validated learning is one of the five principles of the Lean Startup method. [ 11 ] It contrasts strongly with the traditional "stealth mode" method of product development where businesses make detailed business plans spanning a considerable time horizon. Steve Blank posited that the main principle of the Lean Startup approach rests in the validation of the hypotheses underlying the product by asking customers if they want the product or if the product meets their needs, and pivoting to another approach if the hypothesis turns out to be false. [ 12 ] This approach to validating business ideas cheaply before substantial investment saves costs and limits risk as businesses that upon experimentation turn out to be commercially unfeasible can easily be terminated. It is especially important as the main cause of startup failure is the lack of market need; [ 13 ] that is, many startups fail because their product isn't needed by many people, and so they cannot generate enough revenue to recoup the initial investment. Thus it can be said that utilizing an MVP would illuminate a prospective entrepreneur on the market demand for their products.
For example, in 2015, specialists from the University of Sydney devised the Rippa robot to automate farm and weed management. [ 14 ] Before it was released, the technical hypothesis – that the robot can distinguish weeds from farm plants – had already been proven. But the business hypothesis – that it would be a viable tool on a working farm – still needed to be proved. [ 15 ] The application of the MVP method here is that the business hypothesis is tested, and only if it proves successful will further development be invested.
"The minimum viable product is that version of a new product a team uses to collect the maximum amount of validated learning about customers with the least effort." [ 2 ] The definition's use of the words maximum and minimum means it is not formulaic. It requires judgment to figure out, for any given context, what MVP makes sense. Due to this vagueness, the term MVP is commonly used, either deliberately or unwittingly, to refer to a much broader notion ranging from a rather prototype-like product to a fully-fledged and marketable product. [ 16 ]
An MVP can be part of a strategy and process directed toward making and selling a product to customers. [ 17 ] It is a core artifact in an iterative process of idea generation, prototyping, presentation, data collection, analysis and learning. One seeks to minimize the total time spent on an iteration. The process is iterated until a desirable product/market fit is obtained, or until the product is deemed non-viable.
Steve Blank typically refers to minimum viable product as minimum feature set. [ 18 ]
Testing is the essence of minimum viable products. As described above, an MVP seeks to test out whether an idea works in market environments while using the least possible expenditure. This would be beneficial as it reduces the risk of innovating (so that enormous amounts of capital would not have to be sacrificed before proving that the concept does not actually work), and allowing for gradual, market-tested expansion models such as the real options model. [ 19 ] A simple method of testing the financial viability of an idea would be discovery-driven planning, [ 20 ] [ 21 ] which first tests the financial viability of new ventures by carefully examining the assumptions behind the idea by a reverse income statement (first, begin with the income you want to obtain, then the costs the new invention would take, and see if the required amount of revenue that must be gained for the project to work). Results from a minimum viable product test aim to indicate if the product should be built, to begin with. Testing evaluates if the initial problem or goal is solved in a manner that makes it reasonable to move forward.
Releasing and assessing the impact of a minimum viable product is a market testing strategy that is used to screen product ideas soon after their generation. In software development, the release is facilitated by rapid application development tools and languages common to web application development.
The MVP differs from the conventional market testing strategy of investing time and money early to implement a product before testing it in the market. It is intended to ensure that the market wants the product before large time and monetary investments are made. The MVP differs from the open-source software methodology of release early, release often that listens to users, letting them define the features and future of the product. Instead, the MVP starts with a product vision, which is maintained throughout the product life cycle, although it is adapted based on the explicit and implicit (indirect measures) feedback from potential future customers of the product. [ 2 ]
The MVP is a strategy that may be used as a part of Blank's customer development [ broken anchor ] methodology that focuses on continual product iteration and refinement based on customer feedback. Additionally, the presentation of non-existing products and features may be refined using web-based statistical hypothesis testing , such as A/B testing .
The Business Model Canvas is used to map in the major components and activities for a company starting out. The minimum viable product can be designed by using selected components of the Business Model Canvas.
Concepts from minimum viable products are applied in other aspects of startups and organizations.
Using a minimum viable brand (MVB) concept can ensure brand hypotheses are grounded in strategic intent and market insights. [ 22 ]
Some research has shown that early release of an MVP may hurt a company more than help when companies risk imitation by a competitor and have not established other barriers to imitation. [ 23 ] It has also indicated that negative feedback on an MVP can negatively affect a company's reputation. [ 23 ] Many developers of mobile and digital products are now criticizing the MVP because customers can easily switch between competing products through platforms (e.g. app stores). [ 24 ] Also, products that do not offer the expected minimum standard of quality are inferior to competitors that enter the market with a higher standard.
A notable limitation of the MVP is rooted in its approach that seeks out to test its ideas to the market. Since the business's new product ideas can be inferred from their testing, the method may be unsuited to environments where the protection of the intellectual property is limited (and where products are easily imitated). [ 25 ]
The criticism of the MVP approach has led to several new approaches,
e.g. the Minimum Viable Experiment MVE, [ 26 ] the Minimum Awesome Product MAP, [ 27 ] or the Simple, Lovable, Complete. [ 28 ] | https://en.wikipedia.org/wiki/Minimum_viable_product |
The Mining Research and Development Establishment (MRDE) was a division of the National Coal Board . Its site in Bretby, Derbyshire is now a commercial business park.
MRDE's function was research into and testing of mining equipment and procedures.
It was created in 1969 with a merger between the Central Engineering Establishment (CEE) and the Mining Research Establishment (MRE). MRE was set up in 1951 to work on projects in conjunction with National Coal Board (NCB) headquarters divisions such as the Production Department and Scientific Department. It was based at Isleworth in West London. CEE was created in 1954 to work on research and development projects in conjunction with other departments, and was based at Bretby. In 1985 the MRDE merged with the Mining Department. [ 2 ]
It won the Queens Award for Technological Achievement in 1991 for its extraction drum for dust and frictional ignition control. [ 3 ]
The site was on the south side of the A511 in the south of Derbyshire. | https://en.wikipedia.org/wiki/Mining_Research_and_Development_Establishment |
During the Middle Ages , between the 5th and 16th century AD, Western Europe saw a period of growth in the mining industry . The first important mines were those at Goslar in the Harz mountains, taken into commission in the 10th century. Another notable mining town is Falun in Sweden where copper has been mined since at least the 10th century and possibly even earlier. (Olsson 2010) [ 1 ]
The rise of the Western European mining industry depended on the increasing influence of Western Europe on the world stage. Advances in medieval mining and metallurgy enabled the flourishing of Western European civilization. Accessible ores and improved extraction techniques supported economic growth and trade. Innovations like water-powered machinery and better smelting methods increased the productivity and quality of metals.
Metallurgical activities were also encouraged by the central political powers, regional authorities, monastic orders , and ecclesiastical overlords. These powers attempted to claim royal rights over the mines and a share in the output, both on private lands and regions belonging to The Crown . They were particularly interested in the extraction of the precious metal ores , and for this reason, the mines in their territories were open to all miners (Nef 1987, 706–715). [ 2 ] [ 3 ]
The social, political, and economic stagnation that followed the Roman Empire affected Europe throughout the early medieval period and had a critical impact on technological progress, trade, and social organization. Technological developments that affected the course of metal production were only feasible within a stable political environment, and this was not the case until the 9th century (Martinon-Torres & Rehren in press, a).
During the first medieval centuries, the output of metal was in a steady decline with constraints in small-scale activities. Miners adopted methods much less efficient than those of Roman times . Ores were extracted only from shallow depths or from remnants of formerly abandoned mines. The vicinity of the mine to villages or towns was also a determining factor when due to the high cost of material transportation (Martinon-Torres & Rehren in press, b). Only the output of iron diminished less in relation to the other base and precious metals until the 8th century. This fact, correlated with the dramatic decrease in copper production, may indicate a possible displacement of copper and bronze artifacts by iron ones (Forbes 1957, 64; Bayley et al. 2008, 50).
By the end of the 9th century, economic, and social conditions dictated a greater need for metal for agriculture, arms, stirrups , and decoration. Consequently, conditions began to favor metallurgy and a slow but steady general progress developed. Starting from the reign of the emperor Otto I in the 960s, smelting sites were multiplied. New mines were discovered and exploited, like the well-known Mines of Rammelsberg , close to the town of Goslar in the Harz Mountains. [ 4 ] Open-cast mining and metallurgical activities were mostly concentrated in the Eastern Alps , Saxony , Bohemia , Tuscany , Rhineland , Gaul , and Spain (Nef 1987). It was mainly German miners and metallurgists who were the generators of metal production, but the French and Flemish made contributions to the developments. [ 5 ]
The period immediately after the 10th century marked the widespread application of several innovations in the field of mining and ore treatment: a shift to large-scale and better quality production. Medieval miners and metallurgists had to find solutions for the practical problems that limited former metal production, in order to meet the market demands for metals. This increased demand for metal was due to the population growth from the 11th to the 13th centuries. This growth had an impact on agriculture, trade, and building construction, including Gothic churches.
The main problem was the inefficient means for draining water out of shafts and tunnels in underground mining . This resulted in the flooding of mines which limited the extraction of ore to shallow depths close to the surface. The secondary problem was the separation of the metal-bearing minerals from the worthless material that surrounds it, or is closely mixed with it. There was, additionally, the difficulty of transporting the ore, which resulted in subsequently high costs.
The economic value of mining led to investment in the development of solutions to these problems, which had a distinctly positive impact on medieval metal output. This included innovations such as water power using waterwheels for powering draining engines, bellows , hammers , and the introduction of advanced types of furnaces .
These innovations were not adopted all at once or applied to all mines and smelting sites. Throughout the medieval period, these technical innovations, and traditional techniques coexisted. Their application depended on the time period and geographical region. Water power in medieval mining and metallurgy was introduced well before the 11th century, but it was only in the 11th century that it was widely applied. The introduction of the blast furnace , mostly for iron smelting, in all the established centers of metallurgy contributed to the quantitative and qualitative improvement of the metal output, making metallic iron available at a lower price.
In addition, cupellation , developed in the 8th century, was more often used for the refinement of lead-silver ores, to separate the silver from the lead (Bayley 2008). Parallel production with more than one technical method, and different treatment of ores would occur wherever multiple ores were present at one site. (Rehren et al. 1999).
Underground work in shafts , although limited in depth, was accomplished either by fire-setting for massive ore bodies or with iron tools for smaller scale extraction of limited veins. The sorting of base and precious metal ores was completed underground and they were transferred separately (Martinon-Torres & Rehren in press, b).
Permanent mining in Sweden proper begun in the High Middle Ages and did not spread to Finland until 1530 when the first iron mine began operations there. [ 7 ]
By the 14th century, the majority of the more easily accessible ore deposits were exhausted. Thus, more advanced technological achievements were introduced in order to keep up with the demand in metal. The alchemical laboratory, separating precious metals from the baser ones they are typically found with, was an essential feature of the metallurgical enterprise.
A significant hiatus in underground mining was noted during the 14th and the early 15th century due to a series of historical events with severe social and economic impacts. The Great Famine (1315–1317), the Black Death (1347–1353), which diminished the European population by one third to one half, and the Hundred Years War (1337–1453) between England and France, that, amongst others, caused severe deforestation, and had dramatic influences in metallurgical industry and trade.
Lead mining, for example, ground to a halt due to the Black Death pandemic, when atmospheric lead pollution from smelting dropped to natural levels (zero) for the first and only time in the last 2000 years. [ 8 ] [ 9 ] [ 10 ] [ 11 ] The great demand of metals, e.g. for armor, could not be met due to the lack of manpower and capital investment.
It was only by the end of the 13th century that great capital expenditures were invested and more sophisticated machinery was installed in underground mining, which resulted in reaching greater depths. The wider application of water and horse power was necessary for draining water out of these deep shafts. Also, acid parting in separating gold from silver was introduced in the 14th century (Bayley 2008). Signs of recovery were present only after the mid 15th century, when the improved methods were widely adopted (Nef 1987, 723).
The discovery of the New World had an impact on European metal production and trade, which has affected the world economy ever since. New, rich ore deposits found in Central Europe during the 15th century were dwarfed by the large amounts of precious metal imports from the Americas.
Metallurgists throughout medieval Europe were generally free to move within different regions. For instance, German metallurgists in search of rich precious metal ores took the lead in mining and influenced the course of metal production, not only in East and South Germany but also in almost all of Central Europe and the Eastern Alps .
As mining gradually became a task for specialized craftsmen, miners moved in large groups and formed settlements close to mines, each with their own customs. They were always welcomed by regional authorities, as the latter were interested in increasing revenue through the profitable exploitation of the mineral-rich subsurface. These authorities claimed a portion of the output, and smiths and miners were provided with land for cottages, mills, forges , farming, and pasture , while also being allowed to utilize streams and lumber. (Nef 1987, 706–715).
Advancing into the high and late Middle Ages, a notable shift occurred where smelting sites gained geographical independence from mines, leading to the separation of metalworking from ore smelting. The urban expansion that unfolded from the 10th century onwards, coupled with the pivotal influence of towns, afforded metallurgists an optimal setting to cultivate and refine their technological advancements. This era witnessed the systematic formation of metallurgical guilds , with their workshops often converging on the outskirts of these urban centers. (McLees 1996).
In medieval societies, liberal and mechanical arts were considered to be totally different disciplines. Metallurgists, like all craftsmen and artisans, almost always lacked the formal education that would inform a methodical intellectual background. Instead, they were the pioneers of causal thinking based on empirical observation and experimentation (Zilsel 2000). | https://en.wikipedia.org/wiki/Mining_and_metallurgy_in_medieval_Europe |
Mining engineering is the extraction of minerals from the ground. It is associated with many other disciplines, such as mineral processing , exploration, excavation, geology , metallurgy , geotechnical engineering and surveying . A mining engineer may manage any phase of mining operations, from exploration and discovery of the mineral resources, through feasibility study , mine design, development of plans, production and operations to mine closure . [ not verified in body ]
From prehistoric times to the present, mining has played a significant role in the existence of the human race. Since the beginning of civilization, people have used stone and ceramics and, later, metals found on or close to the Earth's surface. These were used to manufacture early tools and weapons . For example, high-quality flint found in northern France and southern England were used to set fire and break rock. [ 1 ] Flint mines have been found in chalk areas where seams of the stone were followed underground by shafts and galleries. The oldest known mine on the archaeological record is the "Lion Cave" in Eswatini . At this site, which radiocarbon dating indicates to be about 43,000 years old, paleolithic humans mined mineral hematite , which contained iron and was ground to produce the red pigment ochre . [ 2 ] [ 3 ]
The ancient Romans were innovators of mining engineering. They developed large-scale mining methods, such as the use of large volumes of water brought to the minehead by aqueducts for hydraulic mining . The exposed rock was then attacked by fire-setting , where fires were used to heat the rock, which would be quenched with a stream of water. The thermal shock cracked the rock, enabling it to be removed. In some mines, the Romans utilized water-powered machinery such as reverse overshot water-wheels . These were used extensively in the copper mines at Rio Tinto in Spain, where one sequence comprised 16 such wheels arranged in pairs, lifting water about 80 feet (24 m). [ 4 ]
Black powder was first used in mining in Banská Štiavnica , Kingdom of Hungary (present-day Slovakia ) in 1627. [ 5 ] This allowed blasting of rock and earth to loosen and reveal ore veins, which was much faster than fire-setting. The Industrial Revolution saw further advances in mining technologies, including improved explosives and steam-powered pumps, lifts, and drills.
Becoming an accredited mining engineer requires a university or college degree. Training includes a Bachelor of Engineering (B.Eng. or B.E.), Bachelor of Science (B.Sc. or B.S.), Bachelor of Technology (B.Tech.) or Bachelor of Applied Science (B.A.Sc.) in mining engineering. Depending on the country and jurisdiction, to be licensed as a mining engineer may require a Master of Engineering (M.Eng.), Master of Science (M.Sc or M.S.) or Master of Applied Science (M.A.Sc.) degree.
Some mining engineers who have come from other disciplines, primarily from engineering fields (e.g.: mechanical, civil, electrical, geomatics or environmental engineering) or from science fields (e.g.: geology, geophysics, physics, geomatics, earth science, or mathematics), typically completing a graduate degree such as M.Eng, M.S., M.Sc. or M.A.Sc. in mining engineering after graduating from a different quantitative undergraduate program.
The fundamental subjects of mining engineering study usually include:
In the United States , about 14 universities offer a B.S. degree in mining and mineral engineering. The top rated universities [ according to whom? ] include West Virginia University , South Dakota School of Mines and Technology , Virginia Tech , the University of Kentucky , the University of Arizona , Montana Tech , and Colorado School of Mines . [ 6 ] Most of these universities offer M.S. and Ph.D. degrees.
In Canada , there are 19 undergraduate degree programs in mining engineering or equivalent. [ 7 ] McGill University Faculty of Engineering offers both undergraduate (B.Sc., B.Eng.) and graduate (M.Sc., Ph.D.) degrees in Mining Engineering. [ 8 ] [ 9 ] and the University of British Columbia in Vancouver offers a Bachelor of Applied Science (B.A.Sc.) in Mining Engineering [ 10 ] and also graduate degrees (M.A.Sc. or M.Eng and Ph.D.) in Mining Engineering. [ 11 ] [ promotion? ]
In Europe , most programs are integrated (B.S. plus M.S. into one) after the Bologna Process and take five years to complete. In Portugal , the University of Porto offers an M.Eng. in Mining and Geo-Environmental Engineering [ 12 ] and in Spain the Technical University of Madrid offers degrees in Mining Engineering with tracks in Mining Technology, Mining Operations, Fuels and Explosives, Metallurgy. [ 13 ] In the United Kingdom , The Camborne School of Mines offers a wide choice of BEng and MEng degrees in Mining engineering and other Mining related disciplines. This is done through the University of Exeter . [ 14 ] In Romania , the University of Petroșani (formerly known as the Petroşani Institute of Mines , or rarely as the Petroşani Institute of Coal ) is the only university that offers a degree in Mining Engineering, Mining Surveying or Underground Mining Constructions, albeit, after the closure of Jiu Valley coal mines, those degrees had fallen out of interest for most high-school graduates. [ 15 ]
In South Africa , leading institutions include the University of Pretoria , offering a 4-year Bachelor of Engineering (B.Eng in Mining Engineering) as well as post-graduate studies in various specialty fields such as rock engineering and numerical modelling, explosives engineering, ventilation engineering, underground mining methods and mine design; [ 16 ] and the University of the Witwatersrand offering a 4-year Bachelor of Science in Engineering (B.Sc.(Eng.)) in Mining Engineering [ 17 ] as well as graduate programs (M.Sc.(Eng.) and Ph.D.) in Mining Engineering. [ 18 ]
Some mining engineers go on to pursue Doctorate degree programs such as Doctor of Philosophy (Ph.D., DPhil), Doctor of Engineering (D.Eng., Eng.D.). These programs involve a significant original research component and are usually seen as entry points into academia .
In the Russian Federation , 85 universities across all federal districts are training specialists for the mineral resource sector. 36 universities are training specialists for extracting and processing solid minerals (mining). 49 are training specialists for extracting, primary processing, and transporting liquid and gaseous minerals (oil and gas). 37 are training specialists for geological exploration (applied geology, geological exploration). Among the universities that train specialists for the mineral resource sector, 7 are federal universities, and 13 are national research universities of Russia. [ 19 ] Personnel training for the mineral resource sector in Russian universities is currently carried out in the following main specializations of training (specialist's degree): "Applied Geology" with the qualification of mining engineer (5 years of training); "Geological Exploration" with the qualification of mining engineer (5 years of training); "Mining" with the qualification of mining engineer (5.5 years of training); "Physical Processes in Mining or Oil and Gas Production" with the qualification of mining engineer (5.5 years of training); "Oil and Gas Engineering and Technologies" with the qualification of mining engineer (5.5 years of training). Universities develop and implement the main professional educational programs of higher education in the directions and specializations of training by forming their profile (name of the program). For example, within the framework of the specialization "Mining", universities often adhere to the classical names of the programs "Open-pit mining", "Underground mining of mineral deposits", "Surveying", "Mineral enrichment", "Mining machines", "Technological safety and mine rescue", "Mine and underground construction", "Blasting work", "Electrification of the mining industry", etc. In the last ten years, under the influence of various factors, new names of programs have begun to appear, such as: "Mining and geological information systems", "Mining ecology", etc. Thus, universities, using their freedom to form new training programs for specialists, can look to the future and try to foresee new professions of mining engineers. After the specialist's degree, you can immediately enrol in postgraduate school (analogue of Doctorate degree programs, four years of training). [ 19 ]
Similar to other types of engineers, mining engineers have a relatively high salary in comparison to other career fields. Mining engineering is also a stable job market to enter, with job openings being almost always readily available.
Job growth
As a general trend, salaries of mining engineers have been increasing throughout the world. The job is estimated to grow between 2-5% depending on the source, which is slower than most jobs. [ 20 ] [ 21 ] Although the job growth is small compared to the average growth rate of 14%, there are still many available job openings in the mining industry. This is due to the relatively low number of graduates, and the constant flow of people retiring from the workforce.
Job stability
Mining engineering has extremely high job stability relative to other career paths. Since many industries require mined materials to function, there will always be a need for the mining industry. However, there are concerns about a workforce shortage caused by many people retiring from the industry within the next 10 years. [ 22 ] With the current predicted number of employees entering the field, there will not be enough to replace those who are retiring as well as fill the need for new employees from industry growth. [ 22 ]
Salary
Mining engineer salaries have been rising globally, with engineers in the United States, Canada, and Australia making the highest earnings relatively. [ 23 ] Mining engineers are among the highest-paid engineer grouping, typically placing in the top 10 of most charts. This can partially be attributed to petroleum engineering, a subset of mining engineering, which is particularly lucrative due to high market demand for petroleum. [ 24 ] [ 25 ]
[ 26 ]
As there is considerable capital expenditure required for mining operations, an array of pre-mining activities are normally carried out to assess whether a mining operation would be worthwhile.
Mineral exploration is the process of locating minerals and assessing their concentrations (grade) and quantities (tonnage), to determine if they are commercially viable ores for mining . Mineral exploration is much more intensive, organized, involved, and professional than mineral prospecting – though it frequently utilizes services exploration, enlisting geologists and surveyors in the necessary pre-feasibility study of the possible mining operation. Mineral exploration and estimation of the reserve can determine the profitability conditions and advocate the form and type of mining required. [ citation needed ]
Mineral discovery can be made from research of mineral maps, academic geological reports, or government geological reports. Other sources of information include property assays and local word of mouth. Mineral research usually includes sampling and analysing sediments, soil, and drill cores. Soil sampling and analysis is one of the most popular mineral exploration tools. [ 27 ] [ 28 ] Other common tools include satellite and aerial surveys or airborne geophysics, including magneto-metric and gamma-spectrometric maps. [ 29 ] Unless the mineral exploration is done on public property, the owners of the property may play a significant role in the exploration process and might be the original discoverers of the mineral deposit. [ 30 ]
After a prospective mineral is located, the mining geologist and engineer determine the ore properties. This may involve chemical analysis of the ore to determine the sample's composition. Once the mineral properties are identified, the next step is determining the quantity of the ore. This involves determining the extent of the deposit and the purity of the ore. [ 31 ] The geologist drills additional core samples to find the limits of the deposit or seam and estimates the quantity of valuable material present.
Once the mineral identification and reserve amount are reasonably determined, the next step is to determine the feasibility of recovering the mineral deposit. A preliminary survey shortly after the discovery of the deposit examines the market conditions, such as the supply and demand of the mineral, the amount of ore needed to be moved to recover a certain quantity of that mineral, and analysis of the cost associated with the operation. This pre-feasibility study determines whether the mining project is likely to be profitable; if so, a more in-depth analysis of the deposit is undertaken. After the full extent of the ore body is known and has been examined by engineers, the feasibility study examines the cost of initial capital investment, methods of extraction, the cost of operation, an estimated length of time to pay back the investment, the gross revenue and net profit margin , any possible resale price of the land, the total life of the reserve, the full value of the account, investment in future projects, and the property owner or owners' contract. In addition, environmental impact, reclamation , possible legal ramifications, and all government permitting are considered. [ 32 ] [ 33 ] These steps of analysis determine whether the mining company and its investors should proceed with the extraction of the minerals or whether the project should be abandoned. The mining company may decide to sell the rights to the reserve to a third party rather than develop it themselves. Alternatively, the decision to proceed with extraction may be postponed indefinitely until market conditions become favourable.
Mining engineers working in an established mine may work as an engineer for operations improvement, further mineral exploration , and operation capitalization by determining where in the mine to add equipment and personnel. The engineer may also work in supervision and management or as an equipment and mineral salesperson. In addition to engineering and operations, the mining engineer may work as an environmental, health, and safety manager or design engineer.
The act of mining requires different methods of extraction depending on the mineralogy , geology , and location of the resources. Characteristics such as mineral hardness , the mineral stratification , and access to that mineral will determine the method of extraction.
Generally, mining is either done from the surface or underground. Mining can also occur with surface and covert operations on the same reserve. Mining activity varies as to what method is employed to remove the mineral.
Surface mining comprises 90% of the world's mineral tonnage output. Also called open pit mining , surface mining removes minerals in formations near the surface. Ore retrieval is done by material removal from the land in its natural state. Surface mining often alters the land's characteristics, shape, topography , and geological makeup.
Surface mining involves quarrying and excavating minerals through cutting, cleaving, and breaking machinery. Explosives are usually used to facilitate breakage. Hard rocks such as limestone, sand, gravel, and slate are generally quarried into benches.
Using mechanical shovels, track dozers, and front-end loaders, strip mining is done on softer minerals such as clays and phosphate removed. Smoother coal seams can also be extracted this way.
With placer mining , dredge mining can also remove minerals from the bottoms of lakes, rivers, streams, and even the ocean. In addition, in-situ mining can be done from the surface using dissolving agents on the ore body and retrieving the ore via pumping. The pumped material is then set to leach for further processing. Hydraulic mining is utilized as water jets to wash away either overburden or the ore itself. [ 34 ]
Legal attention to health and safety in mining began in the late 19 th century with general safety codes being added to most mining environments. Since then, it has become a widespread practice across the world to have specific, detailed mine safety regulations. This is important because working in the mining field presents many dangers to workers and having safety codes minimizes potential workplace accidents.
Mining engineers, as employees of the mines, have to follow these safety codes in their work. Mine safety engineers, a subset of mining engineers, specifically with creating and implementing these safety regulations. They work with the documentation and analysis of mining disasters to ensure that, when possible, the same mistakes are not repeated twice.
The United States Congress, through the passage of the Federal Mine Safety and Health Act of 1977 , known as the Miner's Act, created the Mine Safety and Health Administration (MSHA) under the US Department of Labour . The act provides miners with rights against retaliation for reporting violations, consolidated regulation of coal mines with metallic and non-metallic mines, and created the independent Federal Mine Safety and Health Review Commission to review violations reported to MSHA. [ 36 ]
The act codified in Code of Federal Regulations § 30 (CFR § 30) covers all miners at an active mine. When a mining engineer works at an active mine, they are subject to the same rights, violations, mandatory health and safety regulations, and compulsory training as any other worker at the mine. The mining engineer can be legally identified as a "miner". [ 37 ]
The act establishes the rights of miners. The miner may report at any time a hazardous condition and request an inspection. The miners may elect a miners' representative to participate during an inspection, pre-inspection meeting, and post-inspection conference. The miners and miners' representatives shall be paid for their time during all inspections and investigations. [ 38 ]
A large portion of India’s mining industry is regulated by the Mines act of 1952 and the Mine Rules of 1955. [ 39 ] These codes outline all of the operational, health and safety standards that all mines must follow. Some subsections, such as the Coal Mine Regulation of 2017, have been created to outline practices in more niche subsections of mining. This enforcement of these codes is managed by the Directorate-General of Mines Safety (DGMS) under the Union Ministry of Labour & Employment (MOL&E). Since these outlines are laws, they can also have legal consequences such as fines, mining license revocation, and imprisonment. [ 40 ] Mining engineers work closely to ensure that these codes are followed on an individual scale.
The Mines Act of 1952 outlines the proper procedure for the operation of mines and implements their health and safety standards. One example of this is the implementation of a mandatory day of rest for workers, which prevents workers from working more than six days out of a week. An example of a safety standard is the requirement for proper first aid kit components for the kits that should be present in every mine.
This act also notes the beginning of the practice of documenting health and safety in incidents in mines. Since these incidents have started being recorded, the number of accidents in coal mines has consistently dropped. [ 41 ] The main categories currently being reported on are fatalities and serious accidents, uncategorized by type or cause of accident. Mining engineers work on the reporting of these incidents and seek to create regulations that will prevent future incidents from occurring.
This act clarifies the legal structure and consequences of health and safety regulation of mines in India. It defines what reports are needed for and from employees as well as what documentation should be taken in mines. This can include medical records, inspection documents, and mining licensure.
The act also outlines welfare and benefits that should be given to all employees working in the mines. This includes the need for welfare management staff in all mines that employ more than 500 employees. [ 42 ] Mining engineers also receive these benefits.
Legislation on the inspection and safety of mines in Australia can be dated back to the early 1900s with the Mine and Works Inspection Act of 1920 from South Australia. There is also a large increase in legislation starting around 1999 and continuing into the present day throughout the rest of the states and territories. [ 43 ]
Most of the states and territories of Australia also follow the WHS, a largely uniform code that details health and safety in the workplace. The WHS (Work Health and Safety) of mines in Australia is overseen by states and territories rather than the central government, so there can be minor discrepancies between each state or territory’s code. [ 43 ] Beyond this, many of the states and territories have also enforced additional regulations on mines specifically in their legislation.
Mining engineers in Australia, like in other countries, closely monitor and create accident reports. Being the country with the 3 rd largest total of coal reserves in the world, there is a large subsection of mining engineers who work specifically with coal mines and coal mine-related disasters. (6)
Work Health and Safety Regulation 2017
Work Health and Safety (Mines and Petroleum Sites) Act 2013
Work Health and Safety (Mines and Petroleum Sites) Act 2022
Work Heath and Safety Regulation 2011
Mining and Quarrying Safety and Health Act 1999
Mining and Quarrying Safety and Health Regulation 2017
Coal Mining Safety and Health Act 1999
Coal Mining Safety and Health Regulation 2017
Work Health and Safety (General) Regulations 2022
Work Health and Safety (Mines) Regulations 2022
Work Health and Safety Regulations 2012
Mines and Works Inspections Act 1920
Mines and Works Inspections Regulations 2013
Work Health and Safety Regulations 2022
Mines Work Health and Safety (Supplementary Requirements) Act 2012
Mines Work Health and Safety (Supplementary Requirements) Regulations 2022
Work Health and Safety Regulation 2011
Chapter 10 (Mines) of the Work Health and Safety (National Uniform Legislation) Regulations 2011
[ 43 ]
Waste and uneconomic material generated from the mineral extraction process are the primary source of pollution in the vicinity of mines. Mining activities, by their nature, cause a disturbance of the natural environment in and around which the minerals are located. Mining engineers should therefore be concerned not only with the production and processing of mineral products but also with the mitigation of damage to the environment both during and after mining as a result of the change in the mining area.
This article incorporates text by Petrov, V. L. available under the CC BY 4.0 license. | https://en.wikipedia.org/wiki/Mining_engineering |
A mining feasibility study is an evaluation of a proposed mining project to determine whether the mineral resource can be mined economically. There are three types of feasibility study used in mining, order of magnitude , preliminary feasibility and detailed feasibility. [ 1 ]
Order of magnitude feasibility studies (sometimes referred to as "scoping studies") are an initial financial appraisal of an inferred mineral resource. Depending on the size of the project, an order of magnitude study may be carried out by a single individual. It will involve a preliminary mine plan, and is the basis for determining whether to proceed with an exploration program , and more detailed engineering work. Order-of-magnitude studies are developed by copying plans and factoring known costs from existing projects completed elsewhere and are accurate to within 40–50%. [ 1 ]
Preliminary feasibility studies or "pre-feasibility studies" are more detailed than order of magnitude studies. A preliminary feasibility study is used in due diligence work, determining whether to proceed with a detailed feasibility study and as a " reality check " to determine areas within the project that require more attention. Preliminary feasibility studies are done by factoring known unit costs and by estimating gross dimensions or quantities once conceptual or preliminary engineering and mine design has been completed. Preliminary feasibility studies are completed by a small group of multi-disciplined technical individuals and have an accuracy within 20-30%. [ 1 ]
Detailed feasibility studies are the most detailed and will determine definitively whether to proceed with the project. A detailed feasibility study will be the basis for capital appropriation , and will provide the budget figures for the project. Detailed feasibility studies require a significant amount of formal engineering work, are accurate to within 10-15% and can cost between ½-1½% of the total estimated project cost. [ 1 ] | https://en.wikipedia.org/wiki/Mining_feasibility_study |
A mining lamp is a lamp , developed for the rigid necessities of underground mining operations. Most often it is worn on a hard hat in the form of a headlamp .
Types
1813 Dr William Reid Clanny Exhibited The Clanny Lamp
1815 Humphry Davy Exhibited The Davy Lamp
1815 George Stephenson Exhibited his Lamp
The Davey Safety Lamp was made in London by Humphry Davy . George Stephenson invented a similar lamp but Davys invention was safer due to it having a fine wire gauze that surrounded the flame. This enabled the light to pass through and reduced the risk of explosion by stopping the "firedamp" methane gas coming in contact with the flame. [ 1 ]
1840 Mathieu Mueseler Exhibited The Museler Lamp in Belgium. [ 2 ]
1859 William Clark patented the first electrical mining lamp. [ 3 ]
1870s J.B.Marsaut (France) double gauze design [ 4 ]
1872 Coal Mines Regulation Act required locked lamps under certain conditions [ 5 ]
1881 Joseph Swan exhibited his first electric lamp
1882 Made by William Reid Clanny invented a 'bonnetted' Clanny lamp , [ 6 ]
1883 Elliis Lever of Culcheth Hall Bowdon offered a £500 prize for creation of a safe portable mining lamp. [ 7 ] [ 8 ]
1885 Thomas Evans of Aberdare made a Clanny type of safety lamp [ 9 ]
1886 Royal Commission on Accidents in Mines tested lamps and made recommendations
1887 Coal Mines Regulation Act – requirements on construction, examination, where used, etc. [ 10 ]
1889 John Davis and Co, Derby, were supplying portable electric lamps [ 11 ]
1896 Coal Mines Regulation Act - requirements on provision by mine owners, where to be used, etc. [ 12 ]
1909 Cap (helmet) lamps introduced in Scotland
1911 Prize offered for best electrical lamp
1911 Coal Mines Act made requirements for pit managers to take examinations, where can be used (including electrical), etc.
1920 Electrical lamp with built in accumulator
1924 Miners Lamp Committee – tests and recommendations
1950 Shale miner's electric safety cap lamp and battery pack made in England and supplied by Concordia Electric Safety Lamp Company Ltd, Cardiff. [ 13 ]
[ 14 ] [ 15 ]
This article about mining is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Mining_lamp |
Mining Machinery Engineering is an interdisciplinary branch of engineering that applies the principles of mechanical engineering and electrical engineering for analysis, design, manufacturing and maintenance of mining equipment . It includes study of various aspects of equipment used in earthworks , mineral processing , bulk material handling , drilling and construction .
Surface mining equipment evolved at a high pace in the 20th century as the scale of mining grew. In the first twenty years of the 1900s, steam power was largely replaced by internal combustion engines and electric motors . Till the 1970s such equipment mainly used mechanical systems and cables for power transmission after which they started to be replaced by hydraulic drive system for small machines. However, mechanical and cable drives are still dominant in large machines.
A similar development took place in the development of underground machinery. Drilling and blasting replaced manual cutting of ores. Lately, the use of continuous mining machines has become the order of the day to comply with environmental regulations, protect surface features, and also achieve an increased production rate.
As the concern for safety in mines increased and highly stringent regulations were put in place, it required mining machinery engineers to be highly conversant with such regulations.
Since Mining Machinery Engineering is a wholesome branch, the course of which covers disciplines like Mechanical Engineering , Mining Engineering , Mineral Engineering , Electrical Engineering , the employment opportunities are tremendous. Mining Machinery Engineering provides excellent career opportunities in various industries such as mining , heavy earth moving equipment , construction industry , mineral processing, and other related industries. | https://en.wikipedia.org/wiki/Mining_machinery_engineering |
A mining simulator is a type of simulation used for entertainment as well as in training purposes for mining companies. These simulators replicate elements of real-world mining operations on surrounding screens displaying three-dimensional imagery , motion platforms , and scale models of typical and atypical mining environments and machinery. The results of the simulations can provide useful information in the form of greater competence in on-site safety, which can lead to greater efficiency and decreased risk of accidents.
Mining simulators are used to replicate real-world conditions of mining, assessing real-time responses from the trainee operator to react to what tasks or obstacles appear around them. [ 1 ] [ 2 ] This is often achieved through the use of surrounding three-dimensional imagery , motion platforms , and realistic replicas of actual mining equipment. [ 1 ] [ 2 ] Trainee operator employees are often taught in a program where they are scored against both their peers and an expert benchmark to produce a final evaluation of competence with the tasks they may need to complete in real-life. [ 3 ]
Mining companies that have implemented mining simulators into their training have shown greater employee competence in on-site safety, leading to an overall more productive working environment, and a higher chance of profitability for the company in the long-run [ 3 ] by decreasing the risk of accidents, injuries, or deaths on the site though prior education. [ 1 ] [ 2 ] [ 3 ] Being able to simulate real-world mining hazards in a safe and controlled environment has also shown to help prepare employees on proper procedure and protocol in the event of an on-site accident without the need to physically experience one, which often cannot be safely taught in the real-world. [ 1 ] Simulating mining environments further helps to familiarize employees with mining equipment and vehicles [ 4 ] [ 5 ] before entering a real job site, leading to increased productivity, and a chance to correct inefficiencies while still in training. [ 1 ] [ 2 ] [ 3 ]
Mining simulator setups can range in size and features, which relates to the price and fidelity of the product. [ 2 ] A simple simulator setup may only need to be installed on one Personal Computer or a virtual reality headset, [ 6 ] [ 7 ] but most often consist of three to six monitors and a motion platform. [ 7 ] [ 8 ] [ 9 ] Any higher cost setups often are housed inside high-cube containers which may contain inside lighting, air conditioning, heating, and other amenities and add-ons which may not directly affect the effectiveness of the simulation training. [ 9 ] [ 10 ] [ 11 ] Some mining simulators can also be mobile and move locations, which can be particularly helpful for use of the same simulator between multiple schools or colleges for apprentice programs. [ 12 ]
Aside from practical training purposes, mining simulators have in more recent times also been created for entertainment as well as gaming purposes. The appeal of the genre of games comes from the ability for them to be played on other than specialized equipment, including more widely available Personal Computers, PlayStation , [ 13 ] and Xbox consoles. [ 14 ] The genre of game also gained popularity from the broader amount of resources that could be added and mined in-game, often substituting more realistically found resources for precious minerals such as gold or diamonds, but coal mining games do exist. [ 15 ] Non-rock or mineral mining simulation games have also emerged, with cryptocurrency mining simulations becoming a popular subgenre, allowing players to simulate mining for coins such as Bitcoin , Ethereum , and Dogecoin . [ 15 ] | https://en.wikipedia.org/wiki/Mining_simulator |
Within software engineering , the mining software repositories [ 1 ] ( MSR ) field [ 2 ] analyzes the rich data available in software repositories, such as version control repositories, mailing list archives, bug tracking systems , issue tracking systems , etc. to uncover interesting and actionable information about software systems, projects and software engineering .
Herzig and Zeller define ”mining software archives” as a process to ”obtain lots of initial evidence” by extracting data from software repositories. Further they define ”data sources” as product-based artifacts like source code, requirement artefacts or version archives and claim that these sources are unbiased, but noisy and incomplete. [ 3 ]
The idea in coupled change analysis is that developers change code entities (e.g. files) together frequently for fixing defects or introducing new features. These couplings between the entities are often not made explicit in the code or other documents. Especially developers new on the project do not know which entities need to be changed together. Coupled change analysis aims to extract the coupling out of the version control system for a project. By the commits and the timing of changes, we might be able to identify which entities frequently change together. This information could then be presented to developers about to change one of the entities to support them in their further changes. [ 4 ]
There are many different kinds of commits in version control systems, e.g. bug fix commits, new feature commits, documentation commits, etc. To take data-driven decisions based on past commits, one needs to select subsets of commits that meet a given criterion. That can be done based on the commit message. [ 5 ]
It is possible to generate useful documentation from mining software repositories. For instance, Jadeite computes usage statistics and helps newcomers to quickly identify commonly used classes. [ 6 ]
The primary mining data comes from version control systems. Early mining experiments were done on CVS repositories. [ 7 ] Then, researchers have extensively analyzed SVN repositories. [ 8 ] Now, Git repositories are dominant. [ 9 ] Depending on the nature of the data required (size, domain, processing), one can either download data from one of these sources. [ clarification needed ] However, data governance and data collection for the sake of building large language models have come to change the rules of the game, by integrating the use of web crawlers to obtain data from multiple sources and domains. | https://en.wikipedia.org/wiki/Mining_software_repositories |
In petroleum engineering , a minipermeameter is a gas -based device for measuring permeability in porous rocks. [ 1 ]
Minipermeametry has been used in the oil industry since the late 1960s (Eijpe and Weber, 1971) without becoming in any way a standard experimental method in core analysis or reservoir characterisation. The laboratory minipermeametry can make important contributions both as an improved methodology within experimental petrophysics and as a source of data invaluable in routine reservoir characterisation (C. HALVORSEN AND A. HURST, 1990)
The values obtained from the minipermeameter should possibly be calibrated by a Klinkenberg correction
https://web.archive.org/web/20110728002022/http://www.scaweb.org/assets/papers/1990_papers/1-SCA1990-27EURO.pdf
This article related to natural gas, petroleum or the petroleum industry is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Minipermeameter |
The Minisci reaction ( Italian: [miˈniʃʃi] ) is a named reaction in organic chemistry . It is a nucleophilic radical substitution to an electron deficient aromatic compound , most commonly the introduction of an alkyl group to a nitrogen containing heterocycle . The reaction was published in 1971 by F. Minisci. [ 1 ] In the case of N-Heterocycles, the conditions must be acidic to ensure protonation of said heterocycle. [ 2 ] A typical reaction is that between pyridine and pivalic acid with silver nitrate , sulfuric acid and ammonium persulfate to form 2-tert-butylpyridine . The reaction resembles Friedel-Crafts alkylation but with opposite reactivity and selectivity. [ 3 ]
The Minisci reaction often produces a mixture of regioisomers that can complicate product purification, but modern reaction conditions are incredibly mild, allowing a wide range of alkyl groups to be introduced. [ 4 ] Depending on the radical source used, one side-reaction is acylation , [ 5 ] with the ratio between alkylation and acylation depending on the substrate and the reaction conditions. Due to the inexpensive raw materials and simple reaction conditions, the Minisci reaction has found many applications in heterocyclic chemistry. [ 6 ] [ 7 ]
The reaction allows for alkylation of electron deficient heterocyclic species which is not possible with Friedel-Crafts chemistry. [ 8 ] A method for alkylating electron deficient arenes , nucleophilic aromatic substitution , is also unavailable to electron deficient heterocycles as the ionic nucleophilic species used will deprotonate the heterocycle over acting as a nucleophile. Again, in contrast to nucleophilic aromatic substitution, the Minisci reaction does not require functionalisation of the arene, allowing for direct C-H functionalisation. [ 8 ]
Further to this, the generated alkyl radical species will not rearrange during the reaction in the way that alkyl fragments appended by Friedel-Crafts alkylation often will; meaning groups such as n-pentyl and cyclopropyl groups can be added unchanged. [ 1 ] The alkyl radical is also a 'soft' nucleophile and so is very unlikely to interact with any 'hard' electrophiles (carbonyl species for example) already present on the heterocycle, [ 9 ] which increases the functional group tolerance of the reaction.
The reaction has been the subject of much research in recent years, with a focus placed on improved reactivity towards a greater variety of heterocycles, increasing the number of alkylating reagents that can be used, and employing milder oxidants and acids. [ 10 ] [ 11 ]
A free radical is formed from the carboxylic acid in an oxidative decarboxylation with silver salts and an oxidizing agent. The oxidizing agent ( ammonium persulfate ) oxidizes the Ag(+) to Ag(2+) under the acidic reaction conditions. This induces a hydrogen atom abstraction by the silver, followed by radical decarboxylation . The carbon-centered radical then reacts with the pyridinium aromatic compound. The ultimate product is formed by rearomatization . The acylated product is formed from the acyl radical. [ 4 ] [ 5 ] | https://en.wikipedia.org/wiki/Minisci_reaction |
Ministeria vibrans is a bacterivorous amoeba with filopodia that lives suspended by a flagellum -derived stalk attached to the substrate. [ 2 ] The life cycle of Ministeria remains unknown.
Two Ministeria species have been reported so far, [ 3 ] both of them from coastal marine water samples: M. vibrans and M. marisola . [ 1 ] However, there is currently only one culture available, that of Ministeria vibrans.
M. vibrans occupies a key position in understanding animal origins. It is a member of the Filasterea , that is the sister-group to Choanoflagellatea and Metazoa . [ 2 ] [ 4 ] Microvilli in Ministeria suggest their presence in the common ancestor of Filasterea and Choanoflagellata . The kinetid structure of Ministeria is similar to that of the choanocytes of the most deep-branching sponges , differing essentially from the kinetid of choanoflagellates. Thus, kinetid and microvilli of Ministeria illustrate features of the common ancestor of three holozoan groups: Filasterea, Metazoa, and Choanoflagellata. [ 5 ]
This Holozoa -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Ministeria_vibrans |
Miniwiz ( Chinese : 小智研發股份有限公司 ) is a Taiwanese company that upcycles consumer and industrial waste into construction and consumer products. The company was founded by Arthur Huang . [ 1 ] It is headquartered in Taiwan with offices in Milan , Singapore , Beijing , and Shanghai .
Miniwiz was founded in March 2005 by Arthur Huang and Jarvis Liu . [ 2 ]
In 2010, it constructed EcoARK, a nine-story tall pavilion used as the main exhibition hall for the 2010 Taipei International Flora Exposition . It was built with plastic bottles upcycled as construction material, reportedly saving 300 tons of plastic from ending up in landfills. [ 3 ] [ 4 ]
In 2010, its Polli-Brick construction material was a finalist in The Earth Awards . [ 5 ]
In 2014, it collaborated with Nike, Inc. to recycle athletic shoes and shoe manufacturing by-product into construction material, named Nike Grind. [ 6 ] [ 7 ]
In May 2015, it announced the Ecofighter project, which would modify a Rutan VariEze by replacing its elements with recycled material, with a flight planned for 2016. [ 8 ]
In April 2016, Miniwiz collaborated with tobacco company Philip Morris International to apply recycled filters of iQos heatsticks to architect Cesare Leonard's 1960s furniture designs at the Milan Design Week . [ 9 ]
In 2017, Miniwiz collaborated with Bonotto to exhibit textiles created from recycled material at Fuorisalone in Milan [ 10 ] and NYCxDesign in New York City. [ 11 ]
In 2019, its Trashpresso mobile recycling plant won an IDEAT award. [ 12 ] [ 13 ] That same year, it also launched a smaller version of the recycling plant named the mini Trashpresso. [ 14 ] | https://en.wikipedia.org/wiki/Miniwiz |
Minix 3 is a small, Unix-like operating system . It is published under a BSD-3-Clause [ a ] license and is a successor project to the earlier versions, Minix 1 and 2. [ 1 ]
The project's main goal is for the system to be fault-tolerant by detecting and repairing its faults on the fly, with no user intervention. The main uses of the system are envisaged to be embedded systems and education. [ 2 ]
As of 2017 [update] , Minix 3 supports IA-32 and ARM architecture processors. [ 3 ] It can also run on emulators or virtual machines , such as Bochs , [ 4 ] [ 5 ] VMware Workstation , [ 6 ] Microsoft Virtual PC , [ 7 ] Oracle VirtualBox , [ 8 ] and QEMU . A port to PowerPC architecture is in development. [ 9 ] The distribution comes on a live CD and does not support live USB installation. [ 10 ] The project has been dormant since late 2018, [ 11 ] and the latest release is 3.4.0 rc6 from 2017, [ 12 ] although the Minix 3 discussion group is still active. [ 13 ]
Minix 3 is believed to have inspired the Intel Management Engine (ME) OS found in Intel's Platform Controller Hub , starting with the introduction of ME 11, which is used with Skylake and Kaby Lake processors. [ 14 ] [ 15 ] It was debated that Minix could have been the most widely used OS on x86 / AMD64 processors, with more installations than Microsoft Windows, Linux, or macOS , because of its use in the Intel ME. [ 16 ]
Reflecting on the nature of monolithic kernel based systems, where a driver (which has, according to Minix creator Tanenbaum , approximately 3–7 times as many bugs as a usual program) [ 17 ] can bring down the whole system, [ 18 ] Minix 3 aims to create an operating system that is a "reliable, self-healing, multiserver Unix clone". [ 19 ]
To achieve that, the code running in kernel must be minimal, with the file server, process server, and each device driver running as separate user-mode processes. Each driver is carefully monitored by a part of the system named the reincarnation server . If a driver fails to respond to pings from this server, it is shut down and replaced by a fresh copy of the driver.
In a monolithic system, a bug in a driver can easily crash the whole kernel. This is far less likely to occur in Minix 3. [ 20 ]
Minix 3 was publicly announced on 24 October 2005 by Andrew Tanenbaum during his keynote speech on top of the Association for Computing Machinery (ACM) Symposium Operating Systems Principles conference. Although it still serves as an example for the new edition of Tanenbaum and Woodhull's textbook, it is comprehensively redesigned to be "usable as a serious system on resource-limited and embedded computers and for applications requiring high reliability."
Initially released under the same BSD-3-Clause license that Minix was licensed under since 2000. [ 23 ] [ 24 ] In late 2005, the copyright owner was changed and a fourth clause was added. [ 1 ] [ 25 ] [ 28 ]
One of the main goals of Minix 3 is reliability. Below, some of the more important principles that enhance its reliability are discussed.
Monolithic operating systems such as Linux and FreeBSD and hybrids like Windows have millions of lines of kernel code. In contrast, Minix 3 has about 6,000 lines of executable kernel code, [ 29 ] which can make problems easier to find in the code.
In monolithic kernels, device drivers reside in the kernel. Thus, when a new peripheral is installed, unknown, untrusted code is inserted in the kernel. One bad line of code in a driver can bring down the system.
Instead, in Minix 3, each device driver is a separate user-mode process. Drivers cannot execute privileged instructions, change the page tables , perform arbitrary input/output (I/O), or write to absolute memory. They must make kernel calls for these services and the kernel checks each call for authority.
In monolithic kernels, a driver can write to any word of memory and thus accidentally corrupt user programs.
In Minix 3, when a user expects data from, for example, the file system, it builds a descriptor telling who has access and at what addresses. It then passes an index to this descriptor to the file system, which may pass it to a driver. The file system or driver then asks the kernel to write via the descriptor, making it impossible for them to write to addresses outside the buffer.
Dereferencing a bad pointer within a driver will crash the driver process, but will have no effect on the system as a whole. The reincarnation server will restart the crashed driver automatically. Users will not notice recovery for some drivers (e.g., disk and network) but for others (e.g., audio and printer), they might. In monolithic kernels, dereferencing a bad pointer in a driver normally leads to a system crash.
If a driver gets into an infinite loop , the scheduler will gradually lower its priority until it becomes idle. Eventually the reincarnation server will see that it is not responding to status requests, so it will kill and restart the looping driver. In a monolithic kernel, a looping driver could hang the system.
Minix 3 uses fixed-length messages for internal communication, which eliminates certain buffer overflows and buffer management problems. Also, many exploits work by overrunning a buffer to trick the program into returning from a function call using an overwritten stack return address pointing into attacker controlled memory, usually the overrun buffer. In Minix 3, this attack is mitigated because instruction and data space are split and only code in (read-only) instruction space can be executed, termed executable space protection . However, attacks which rely on running legitimately executable memory in a malicious way ( return-to-libc , return-oriented programming ) are not prevented by this mitigation.
Device drivers obtain kernel services (such as copying data to users' address spaces) by making kernel calls. The Minix 3 kernel has a bit map for each driver specifying which calls it is authorized to make. In monolithic kernels, every driver can call every kernel function, authorized or not.
The kernel also maintains a table telling which I/O ports each driver may access. Thus, a driver can only touch its own I/O ports. In monolithic kernels, a buggy driver can access I/O ports belonging to another device.
Not every driver and server needs to communicate with every other driver and server. Accordingly, a per-process bit map determines which destinations each process may send to.
A special process, called the reincarnation server, periodically pings each device driver. If the driver dies or fails to respond correctly to pings, the reincarnation server automatically replaces it with a fresh copy. Detecting and replacing non-functioning drivers is automatic, with no user action needed. This feature does not work for disk drivers at present, but in the next release the system will be able to recover even disk drivers, which will be shadowed in random-access memory (RAM). Driver recovery does not affect running processes.
When an interrupt occurs, it is converted at a low level to a notification sent to the appropriate driver. If the driver is waiting for a message, it gets the interrupt immediately; otherwise it gets the notification the next time it does a RECEIVE to get a message. This scheme eliminates nested interrupts and makes driver programming easier.
As can be seen, at the bottom level is the microkernel , which is about 4,000 lines of code (mostly in C , plus a small amount of assembly language ). It handles interrupts , scheduling , and message passing. It also supports an application programming interface (API) of about 30 kernel calls that authorized servers and drivers can make. User programs cannot make these calls. Instead, they can issue POSIX system calls which send messages to the servers. The kernel calls perform functions such as setting interrupts and copying data between address spaces.
At the next level up, there are the device drivers , each one running as a separate userland process. Each one controls some I/O device, such as a disk or printer. The drivers do not have access to the I/O port space and cannot issue I/O instructions directly. Instead, they must make kernel calls giving a list of I/O ports to write to and the values to be written. While there is a small amount of overhead in doing this (typically 500 ns), this scheme makes it possible for the kernel to check authorization, so that, for example, the audio driver cannot write on the disk.
At the next level there are the servers . This is where nearly all the operating system functionality is located. User processes obtain file service, for example, by sending messages to the file server to open, close, read, and write files. In turn, the file server gets disk I/O performed by sending messages to the disk driver, which controls the disk.
One of the key servers is the reincarnation server. Its job is to poll all the other servers and drivers to check on their health periodically. If a component fails to respond correctly, or exits, or gets into an infinite loop , the reincarnation server (which is the parent process of the drivers and servers) kills the faulty component and replaces it with a fresh copy. In this way the system is automatically made self-healing without interfering with running programs.
Currently the reincarnation server, the process server, and the microkernel are part of the trusted computing base . If any of them fail, the system crashes. Nevertheless, reducing the trusted computing base from 3-5 million lines of code, as in Linux and Windows systems, to about 20,000 lines greatly enhances system reliability. [ citation needed ]
Minix 1.0, 1.5, and 2.0 were developed as tools to help people learn about the design of operating systems.
Minix 1.0, released in 1987, was 12,000 lines of C and some x86 assembly language . Source code of the kernel, memory manager , and file system of Minix 1.0 are printed in the book. Tanenbaum originally developed Minix for compatibility with the IBM PC and IBM PC/AT microcomputers available at the time.
Minix 1.5, released in 1991, included support for MicroChannel IBM PS/2 systems and was also ported to the Motorola 68000 and SPARC architectures, supporting the Atari ST , Commodore Amiga , Apple Macintosh and Sun Microsystems SPARCstation computer platforms. A version of Minix running as a user process under SunOS was also available.
Minix 2.0, released in 1997, was only available for the x86 and Solaris -hosted SPARC architectures. Minix-vmd was created by two Vrije Universiteit researchers, and added virtual memory and support for the X Window System (X11).
Minix 3 does the same, and provides a modern operating system with many newer tools and many Unix applications. [ 30 ] Prof. Tanenbaum once said:
Please be aware that MINIX 3 is not your grandfather's MINIX ... MINIX 1 was written as an educational tool ... MINIX 3 is that plus a start at building a highly reliable, self-healing, bloat-free operating system ... MINIX 1 and MINIX 3 are related in the same way as Windows 3.1 and Windows XP are: same first name. [ 19 ]
Many improvements have also been made in the structure of the kernel since the Minix 2 release, making the system more reliable. [ 31 ]
Minix version 3.1.5 was released 5 November 2009. It contains X11 , Emacs , vi , cc, GCC , Perl , Python , Almquist shell , Bash , Z shell , FTP client , SSH client , Telnet client, Pine , and over 400 other common Unix utility programs. With the addition of X11, this version marks the transition away from a text-only system. Another feature of this version, which will be improved in future ones, is the ability of the system to withstand device driver crashes, and in many cases having them automatically replaced without affecting running processes. In this way, Minix is self-healing and can be used in applications demanding high reliability.
Minix 3.2.0 was released in February 2012. This version has many new features, including the Clang compiler, experimental symmetric multiprocessing support, procfs and ext2fs filesystem support, and GNU Debugger (GDB). Several parts of NetBSD are also integrated in the release, including the bootloader, libc and various utilities and other libraries . [ 32 ]
Minix 3.3.0 was released in September 2014. This release is the first version to support the ARM architecture in addition to x86. It also supports a NetBSD userland , with thousands of NetBSD packages running right out of the box.
Rocky Raccoon is the mascot of Minix 3. [ 33 ]
MINIXCon is a conference on sharing talks, efforts and researches related to Minix.
It was held once in 2016. MINIXCon2017 was cancelled due to lack of talks submitted. [ 34 ] [ 35 ] | https://en.wikipedia.org/wiki/Minix_3 |
In mathematics, Minkowski's second theorem is a result in the geometry of numbers about the values taken by a norm on a lattice and the volume of its fundamental cell.
Let K be a closed convex centrally symmetric body of positive finite volume in n -dimensional Euclidean space R n . The gauge [ 1 ] or distance [ 2 ] [ 3 ] Minkowski functional g attached to K is defined by g ( x ) = inf { λ ∈ R : x ∈ λ K } . {\displaystyle g(x)=\inf \left\{\lambda \in \mathbb {R} :x\in \lambda K\right\}.}
Conversely, given a norm g on R n we define K to be K = { x ∈ R n : g ( x ) ≤ 1 } . {\displaystyle K=\left\{x\in \mathbb {R} ^{n}:g(x)\leq 1\right\}.}
Let Γ be a lattice in R n . The successive minima of K or g on Γ are defined by setting the k -th successive minimum λ k to be the infimum of the numbers λ such that λK contains k linearly-independent vectors of Γ . We have 0 < λ 1 ≤ λ 2 ≤ ... ≤ λ n < ∞ .
The successive minima satisfy [ 4 ] [ 5 ] [ 6 ] 2 n n ! vol ( R n / Γ ) ≤ λ 1 λ 2 ⋯ λ n vol ( K ) ≤ 2 n vol ( R n / Γ ) . {\displaystyle {\frac {2^{n}}{n!}}\operatorname {vol} \left(\mathbb {R} ^{n}/\Gamma \right)\leq \lambda _{1}\lambda _{2}\cdots \lambda _{n}\operatorname {vol} (K)\leq 2^{n}\operatorname {vol} \left(\mathbb {R} ^{n}/\Gamma \right).}
A basis of linearly independent lattice vectors b 1 , b 2 , ..., b n can be defined by g ( b j ) = λ j .
The lower bound is proved by considering the convex polytope 2 n with vertices at ± b j / λ j , which has an interior enclosed by K and a volume which is 2 n / n ! λ 1 λ 2 ... λ n times an integer multiple of a primitive cell of the lattice (as seen by scaling the polytope by λ j along each basis vector to obtain 2 n n -simplices with lattice point vectors).
To prove the upper bound, consider functions f j ( x ) sending points x in K {\textstyle K} to the centroid of the subset of points in K {\textstyle K} that can be written as x + ∑ i = 1 j − 1 a i b i {\textstyle x+\sum _{i=1}^{j-1}a_{i}b_{i}} for some real numbers a i {\textstyle a_{i}} . Then the coordinate transform x ′ = h ( x ) = ∑ i = 1 n ( λ i − λ i − 1 ) f i ( x ) / 2 {\displaystyle x'=h(x)=\sum _{i=1}^{n}(\lambda _{i}-\lambda _{i-1})f_{i}(x)/2} has a Jacobian determinant J = λ 1 λ 2 … λ n / 2 n {\textstyle J=\lambda _{1}\lambda _{2}\ldots \lambda _{n}/2^{n}} . If p {\textstyle p} and q {\textstyle q} are in the interior of K {\textstyle K} and p − q = ∑ i = 1 k a i b i {\textstyle p-q=\sum _{i=1}^{k}a_{i}b_{i}} (with a k ≠ 0 {\textstyle a_{k}\neq 0} ) then ( h ( p ) − h ( q ) ) = ∑ i = 0 k c i b i ∈ λ k K {\displaystyle (h(p)-h(q))=\sum _{i=0}^{k}c_{i}b_{i}\in \lambda _{k}K} with c k = λ k a k / 2 {\textstyle c_{k}=\lambda _{k}a_{k}/2} , where the inclusion in λ k K {\textstyle \lambda _{k}K} (specifically the interior of λ k K {\textstyle \lambda _{k}K} ) is due to convexity and symmetry. But lattice points in the interior of λ k K {\textstyle \lambda _{k}K} are, by definition of λ k {\textstyle \lambda _{k}} , always expressible as a linear combination of b 1 , b 2 , … b k − 1 {\textstyle b_{1},b_{2},\ldots b_{k-1}} , so any two distinct points of K ′ = h ( K ) = { x ′ ∣ h ( x ) = x ′ } {\textstyle K'=h(K)=\{x'\mid h(x)=x'\}} cannot be separated by a lattice vector. Therefore, K ′ {\textstyle K'} must be enclosed in a primitive cell of the lattice (which has volume vol ( R n / Γ ) {\textstyle \operatorname {vol} (\mathbb {R} ^{n}/\Gamma )} ), and consequently vol ( K ) / J = vol ( K ′ ) ≤ vol ( R n / Γ ) {\textstyle \operatorname {vol} (K)/J=\operatorname {vol} (K')\leq \operatorname {vol} (\mathbb {R} ^{n}/\Gamma )} . | https://en.wikipedia.org/wiki/Minkowski's_second_theorem |
In mathematics , Minkowski's theorem is the statement that every convex set in R n {\displaystyle \mathbb {R} ^{n}} which is symmetric with respect to the origin and which has volume greater than 2 n {\displaystyle 2^{n}} contains a non-zero integer point (meaning a point in Z n {\displaystyle \mathbb {Z} ^{n}} that is not the origin). The theorem was proved by Hermann Minkowski in 1889 and became the foundation of the branch of number theory called the geometry of numbers . It can be extended from the integers to any lattice L {\displaystyle L} and to any symmetric convex set with volume greater than 2 n d ( L ) {\displaystyle 2^{n}\,d(L)} , where d ( L ) {\displaystyle d(L)} denotes the covolume of the lattice (the absolute value of the determinant of any of its bases).
Suppose that L is a lattice of determinant d( L ) in the n - dimensional real vector space R n {\displaystyle \mathbb {R} ^{n}} and S is a convex subset of R n {\displaystyle \mathbb {R} ^{n}} that is symmetric with respect to the origin, meaning that if x is in S then − x is also in S . Minkowski's theorem states that if the volume of S is strictly greater than 2 n d( L ) , then S must contain at least one lattice point other than the origin. (Since the set S is symmetric, it would then contain at least three lattice points: the origin 0 and a pair of points ± x , where x ∈ L \ 0 .)
The simplest example of a lattice is the integer lattice Z n {\displaystyle \mathbb {Z} ^{n}} of all points with integer coefficients; its determinant is 1. For n = 2 , the theorem claims that a convex figure in the Euclidean plane symmetric about the origin and with area greater than 4 encloses at least one lattice point in addition to the origin. The area bound is sharp : if S is the interior of the square with vertices (±1, ±1) then S is symmetric and convex, and has area 4, but the only lattice point it contains is the origin. This example, showing that the bound of the theorem is sharp, generalizes to hypercubes in every dimension n .
The following argument proves Minkowski's theorem for the specific case of L = Z 2 . {\displaystyle L=\mathbb {Z} ^{2}.}
Proof of the Z 2 {\textstyle \mathbb {Z} ^{2}} case: Consider the map
Intuitively, this map cuts the plane into 2 by 2 squares, then stacks the squares on top of each other. Clearly f ( S ) has area less than or equal to 4, because this set lies within a 2 by 2 square. Assume for a contradiction that f could be injective , which means the pieces of S cut out by the squares stack up in a non-overlapping way. Because f is locally area-preserving, this non-overlapping property would make it area-preserving for all of S , so the area of f ( S ) would be the same as that of S , which is greater than 4. That is not the case, so the assumption must be false: f is not injective, meaning that there exist at least two distinct points p 1 , p 2 in S that are mapped by f to the same point: f ( p 1 ) = f ( p 2 ) .
Because of the way f was defined, the only way that f ( p 1 ) can equal f ( p 2 ) is for p 2 to equal p 1 + (2 i , 2 j ) for some integers i and j , not both zero.
That is, the coordinates of the two points differ by two even integers.
Since S is symmetric about the origin, − p 1 is also a point in S . Since S is convex, the line segment between − p 1 and p 2 lies entirely in S , and in particular the midpoint of that segment lies in S . In other words,
is a point in S . This point ( i , j ) is an integer point, and is not the origin since i and j are not both zero.
Therefore, S contains a nonzero integer point.
Remarks:
Minkowski's theorem gives an upper bound for the length of the shortest nonzero vector. This result has applications in lattice cryptography and number theory.
Theorem (Minkowski's bound on the shortest vector): Let L {\textstyle L} be a lattice. Then there is a x ∈ L ∖ { 0 } {\textstyle x\in L\setminus \{0\}} with ‖ x ‖ ∞ ≤ | det ( L ) | 1 / n {\textstyle \|x\|_{\infty }\leq \left|\det(L)\right|^{1/n}} . In particular, by the standard comparison between l 2 {\textstyle l_{2}} and l ∞ {\textstyle l_{\infty }} norms, ‖ x ‖ 2 ≤ n | det ( L ) | 1 / n {\textstyle \|x\|_{2}\leq {\sqrt {n}}\,\left|\det(L)\right|^{1/n}} .
Let l = min { ‖ x ‖ ∞ : x ∈ L ∖ { 0 } } {\textstyle l=\min\{\|x\|_{\infty }:x\in L\setminus \{0\}\}} , and set C = { y : ‖ y ‖ ∞ < l } {\textstyle C=\{y:\|y\|_{\infty }<l\}} . Then vol ( C ) = ( 2 l ) n {\textstyle {\text{vol}}(C)=(2l)^{n}} . If ( 2 l ) n > 2 n | d ( L ) | {\textstyle (2l)^{n}>2^{n}|d(L)|} , then C {\textstyle C} contains a non-zero lattice point, which is a contradiction. Thus l ≤ | d ( L ) | 1 / n {\textstyle l\leq |d(L)|^{1/n}} . Q.E.D.
Remarks:
The difficult implication in Fermat's theorem on sums of two squares can be proven using Minkowski's bound on the shortest vector.
Theorem: Every prime with p ≡ 1 mod 4 {\textstyle p\equiv 1\mod 4} can be written as a sum of two squares .
Since 4 | p − 1 {\textstyle 4\,|\,p-1} and a a is a quadratic residue modulo a prime p {\textstyle p} if and only if a p − 1 2 = 1 ( mod p ) a^{\frac {p-1}{2}}=1~({\text{mod}}~p) (Euler's Criterion) there is a square root of − 1 {\textstyle -1} in Z / p Z {\textstyle \mathbb {Z} /p\mathbb {Z} } ; choose one and call one representative in Z {\textstyle \mathbb {Z} } for it j {\textstyle j} . Consider the lattice L {\textstyle L} defined by the vectors ( 1 , j ) , ( 0 , p ) {\textstyle (1,j),(0,p)} , and let B {\textstyle B} denote the associated matrix . The determinant of this lattice is p {\textstyle p} , whence Minkowski's bound tells us that there is a nonzero x = ( x 1 , x 2 ) ∈ Z 2 {\textstyle x=(x_{1},x_{2})\in \mathbb {Z} ^{2}} with 0 < ‖ B x ‖ 2 2 < 2 p {\textstyle 0<\|Bx\|_{2}^{2}<2p} . We have ‖ B x ‖ 2 = ‖ ( x 1 , j x 1 + p x 2 ) ‖ 2 = x 1 2 + ( j x 1 + p x 2 ) 2 {\textstyle \|Bx\|^{2}=\|(x_{1},jx_{1}+px_{2})\|^{2}=x_{1}^{2}+(jx_{1}+px_{2})^{2}} and we define the integers a = x 1 , b = ( j x 1 + p x 2 ) {\textstyle a=x_{1},b=(jx_{1}+px_{2})} . Minkowski's bound tells us that 0 < a 2 + b 2 < 2 p {\textstyle 0<a^{2}+b^{2}<2p} , and simple modular arithmetic shows that a 2 + b 2 = x 1 2 + ( j x 1 + p x 2 ) 2 = 0 mod p {\textstyle a^{2}+b^{2}=x_{1}^{2}+(jx_{1}+px_{2})^{2}=0\mod p} , and thus we conclude that a 2 + b 2 = p {\textstyle a^{2}+b^{2}=p} . Q.E.D.
Additionally, the lattice perspective gives a computationally efficient approach to Fermat's theorem on sums of squares:
Minkowski's theorem is also useful to prove Lagrange's four-square theorem , which states that every natural number can be written as the sum of the squares of four natural numbers.
Minkowski's theorem can be used to prove Dirichlet's theorem on simultaneous rational approximation .
Another application of Minkowski's theorem is the result that every class in the ideal class group of a number field K contains an integral ideal of norm not exceeding a certain bound, depending on K , called Minkowski's bound : the finiteness of the class number of an algebraic number field follows immediately.
The complexity of finding the point guaranteed by Minkowski's theorem, or the closely related Blichfeldt's theorem, have been studied from the perspective of TFNP search problems. In particular, it is known that a computational analogue of Blichfeldt's theorem , a corollary of the proof of Minkowski's theorem, is PPP-complete. [ 4 ] It is also known that the computational analogue of Minkowski's theorem is in the class PPP, and it was conjectured to be PPP complete. [ 5 ] | https://en.wikipedia.org/wiki/Minkowski's_theorem |
The Minkowski Portal Refinement collision detection algorithm is a technique for determining whether two convex shapes overlap.
The algorithm was created by Gary Snethen in 2006 and was first published in Game Programming Gems 7. The algorithm was used in Tomb Raider: Underworld and other games created by Crystal Dynamics and its sister studios within Eidos Interactive .
MPR, like its cousin GJK , relies on shapes that are defined using support mappings . This allows the algorithm to support a limitless variety of shapes that are problematic for other algorithms. Support mappings require only a single mathematical function to represent a point, line segment, disc, cylinder, cone, ellipsoid, football, bullet, frustum or most any other common convex shape. Once a set of basic primitives have been created, they can easily be combined with one another using operations such as sweep, shrink-wrap and affine transformation .
Unlike GJK , MPR does not provide the shortest distance between separated shapes. However, according to its author, MPR is simpler, more numerically robust and handles translational sweeping with very little modification. This makes it well-suited for games and other real-time applications.
This geometry-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Minkowski_Portal_Refinement |
The Minkowski Prize is given by the European Association for the Study of Diabetes (EASD) in recognition to research which has been carried out by a person normally residing in Europe, as manifested by publications which contribute to the advancement of knowledge concerning diabetes mellitus . The Prize honors the name of Oskar Minkowski (1858–1931), a physician and physiologist who was the discoverer of the role of pancreas in the control of glucose metabolism . It has been awarded annually since 1966, and the winner is invited to pronounce a Minkowski Lecture during the EASD Annual Conference. It is traditionally seen as the most prestigious European prize in the field of diabetes research.
Since 1966, the award is sponsored by a pharmaceutical company Sanofi-Aventis . The prize consists of a certificate and 20,000 euros plus travel expenses. The candidate must be less than 45 years of age on 1 January of the year of award. Self-nomination is possible.
With the city where the prize was awarded (Annual Conference), name and country. [ 1 ] [ 2 ] | https://en.wikipedia.org/wiki/Minkowski_Prize |
The Minkowski content (named after Hermann Minkowski ), or the boundary measure , of a set is a basic concept that uses concepts from geometry and measure theory to generalize the notions of length of a smooth curve in the plane, and area of a smooth surface in space , to arbitrary measurable sets .
It is typically applied to fractal boundaries of domains in the Euclidean space , but it can also be used in the context of general metric measure spaces.
It is related to, although different from, the Hausdorff measure .
For A ⊂ R n {\displaystyle A\subset \mathbb {R} ^{n}} , and each integer m with 0 ≤ m ≤ n {\displaystyle 0\leq m\leq n} , the m -dimensional upper Minkowski content is
and the m -dimensional lower Minkowski content is defined as
where α ( n − m ) r n − m {\displaystyle \alpha (n-m)r^{n-m}} is the volume of the ( n − m )-ball of radius r and μ {\displaystyle \mu } is an n {\displaystyle n} -dimensional Lebesgue measure .
If the upper and lower m -dimensional Minkowski content of A are equal, then their common value is called the Minkowski content M m ( A ). [ 1 ] [ 2 ] | https://en.wikipedia.org/wiki/Minkowski_content |
In mathematical analysis , the Minkowski inequality establishes that the L p spaces satisfy the triangle inequality in the definition of normed vector spaces . The inequality is named after the German mathematician Hermann Minkowski .
Let S {\textstyle S} be a measure space , let 1 ≤ p < ∞ {\textstyle 1\leq p<\infty } and let f {\textstyle f} and g {\textstyle g} be elements of L p ( S ) . {\textstyle L^{p}(S).} Then f + g {\textstyle f+g} is in L p ( S ) , {\textstyle L^{p}(S),} and we have the triangle inequality
‖ f + g ‖ p ≤ ‖ f ‖ p + ‖ g ‖ p {\displaystyle \|f+g\|_{p}\leq \|f\|_{p}+\|g\|_{p}}
with equality for 1 < p < ∞ {\textstyle 1<p<\infty } if and only if f {\textstyle f} and g {\textstyle g} are positively linearly dependent ; that is, f = λ g {\textstyle f=\lambda g} for some λ ≥ 0 {\textstyle \lambda \geq 0} or g = 0. {\textstyle g=0.} Here, the norm is given by:
‖ f ‖ p = ( ∫ | f | p d μ ) 1 p {\displaystyle \|f\|_{p}=\left(\int |f|^{p}d\mu \right)^{\frac {1}{p}}}
if p < ∞ , {\textstyle p<\infty ,} or in the case p = ∞ {\textstyle p=\infty } by the essential supremum
‖ f ‖ ∞ = e s s s u p x ∈ S | f ( x ) | . {\displaystyle \|f\|_{\infty }=\operatorname {ess\ sup} _{x\in S}|f(x)|.}
The Minkowski inequality is the triangle inequality in L p ( S ) . {\textstyle L^{p}(S).} In fact, it is a special case of the more general fact
‖ f ‖ p = sup ‖ g ‖ q = 1 ∫ | f g | d μ , 1 p + 1 q = 1 {\displaystyle \|f\|_{p}=\sup _{\|g\|_{q}=1}\int |fg|d\mu ,\qquad {\tfrac {1}{p}}+{\tfrac {1}{q}}=1}
where it is easy to see that the right-hand side satisfies the triangular inequality.
Like Hölder's inequality , the Minkowski inequality can be specialized to sequences and vectors by using the counting measure :
( ∑ k = 1 n | x k + y k | p ) 1 / p ≤ ( ∑ k = 1 n | x k | p ) 1 / p + ( ∑ k = 1 n | y k | p ) 1 / p {\displaystyle {\biggl (}\sum _{k=1}^{n}|x_{k}+y_{k}|^{p}{\biggr )}^{1/p}\leq {\biggl (}\sum _{k=1}^{n}|x_{k}|^{p}{\biggr )}^{1/p}+{\biggl (}\sum _{k=1}^{n}|y_{k}|^{p}{\biggr )}^{1/p}}
for all real (or complex ) numbers x 1 , … , x n , y 1 , … , y n {\textstyle x_{1},\dots ,x_{n},y_{1},\dots ,y_{n}} and where n {\textstyle n} is the cardinality of S {\textstyle S} (the number of elements in S {\textstyle S} ).
In probabilistic terms, given the probability space ( Ω , F , P ) , {\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} ),} and E {\displaystyle \mathbb {E} } denote the expectation operator for every real- or complex-valued random variables X {\displaystyle X} and Y {\displaystyle Y} on Ω , {\displaystyle \Omega ,} Minkowski's inequality reads
First, we prove that f + g {\textstyle f+g} has finite p {\textstyle p} -norm if f {\textstyle f} and g {\textstyle g} both do, which follows by
| f + g | p ≤ 2 p − 1 ( | f | p + | g | p ) . {\displaystyle |f+g|^{p}\leq 2^{p-1}(|f|^{p}+|g|^{p}).}
Indeed, here we use the fact that h ( x ) = | x | p {\textstyle h(x)=|x|^{p}} is convex over R + {\textstyle \mathbb {R} ^{+}} (for p > 1 {\textstyle p>1} ) and so, by the definition of convexity,
| 1 2 f + 1 2 g | p ≤ | 1 2 | f | + 1 2 | g | | p ≤ 1 2 | f | p + 1 2 | g | p . {\displaystyle \left|{\tfrac {1}{2}}f+{\tfrac {1}{2}}g\right|^{p}\leq \left|{\tfrac {1}{2}}|f|+{\tfrac {1}{2}}|g|\right|^{p}\leq {\tfrac {1}{2}}|f|^{p}+{\tfrac {1}{2}}|g|^{p}.}
This means that
| f + g | p ≤ 1 2 | 2 f | p + 1 2 | 2 g | p = 2 p − 1 | f | p + 2 p − 1 | g | p . {\displaystyle |f+g|^{p}\leq {\tfrac {1}{2}}|2f|^{p}+{\tfrac {1}{2}}|2g|^{p}=2^{p-1}|f|^{p}+2^{p-1}|g|^{p}.}
Now, we can legitimately talk about ‖ f + g ‖ p . {\textstyle \|f+g\|_{p}.} If it is zero, then Minkowski's inequality holds. We now assume that ‖ f + g ‖ p {\textstyle \|f+g\|_{p}} is not zero. Using the triangle inequality and then Hölder's inequality , we find that
‖ f + g ‖ p p = ∫ | f + g | p d μ = ∫ | f + g | ⋅ | f + g | p − 1 d μ ≤ ∫ ( | f | + | g | ) | f + g | p − 1 d μ = ∫ | f | | f + g | p − 1 d μ + ∫ | g | | f + g | p − 1 d μ ≤ ( ( ∫ | f | p d μ ) 1 p + ( ∫ | g | p d μ ) 1 p ) ( ∫ | f + g | ( p − 1 ) ( p p − 1 ) d μ ) 1 − 1 p Hölder's inequality = ( ‖ f ‖ p + ‖ g ‖ p ) ‖ f + g ‖ p p ‖ f + g ‖ p {\displaystyle {\begin{aligned}\|f+g\|_{p}^{p}&=\int |f+g|^{p}\,\mathrm {d} \mu \\&=\int |f+g|\cdot |f+g|^{p-1}\,\mathrm {d} \mu \\&\leq \int (|f|+|g|)|f+g|^{p-1}\,\mathrm {d} \mu \\&=\int |f||f+g|^{p-1}\,\mathrm {d} \mu +\int |g||f+g|^{p-1}\,\mathrm {d} \mu \\&\leq \left(\left(\int |f|^{p}\,\mathrm {d} \mu \right)^{\frac {1}{p}}+\left(\int |g|^{p}\,\mathrm {d} \mu \right)^{\frac {1}{p}}\right)\left(\int |f+g|^{(p-1)\left({\frac {p}{p-1}}\right)}\,\mathrm {d} \mu \right)^{1-{\frac {1}{p}}}&&{\text{ Hölder's inequality}}\\&=\left(\|f\|_{p}+\|g\|_{p}\right){\frac {\|f+g\|_{p}^{p}}{\|f+g\|_{p}}}\end{aligned}}}
We obtain Minkowski's inequality by multiplying both sides by
‖ f + g ‖ p ‖ f + g ‖ p p . {\displaystyle {\frac {\|f+g\|_{p}}{\|f+g\|_{p}^{p}}}.}
Given t ∈ ( 0 , 1 ) {\displaystyle t\in (0,1)} , one has, by convexity ( Jensen's inequality ), for every x ∈ S {\displaystyle x\in S}
By integration this leads to
One takes then
to reach the conclusion.
Suppose that ( S 1 , μ 1 ) {\textstyle (S_{1},\mu _{1})} and ( S 2 , μ 2 ) {\textstyle (S_{2},\mu _{2})} are two 𝜎-finite measure spaces and F : S 1 × S 2 → R {\textstyle F:S_{1}\times S_{2}\to \mathbb {R} } is measurable. Then Minkowski's integral inequality is: [ 1 ] [ 2 ]
[ ∫ S 2 | ∫ S 1 F ( x , y ) μ 1 ( d x ) | p μ 2 ( d y ) ] 1 p ≤ ∫ S 1 ( ∫ S 2 | F ( x , y ) | p μ 2 ( d y ) ) 1 p μ 1 ( d x ) , p ∈ [ 1 , ∞ ) {\displaystyle \left[\int _{S_{2}}\left|\int _{S_{1}}F(x,y)\,\mu _{1}(\mathrm {d} x)\right|^{p}\mu _{2}(\mathrm {d} y)\right]^{\frac {1}{p}}~\leq ~\int _{S_{1}}\left(\int _{S_{2}}|F(x,y)|^{p}\,\mu _{2}(\mathrm {d} y)\right)^{\frac {1}{p}}\mu _{1}(\mathrm {d} x),\quad p\in [1,\infty )}
with obvious modifications in the case p = ∞ . {\textstyle p=\infty .} If p > 1 , {\textstyle p>1,} and both sides are finite, then equality holds only if | F ( x , y ) | = φ ( x ) ψ ( y ) {\textstyle |F(x,y)|=\varphi (x)\,\psi (y)} a.e. for some non-negative measurable functions φ {\textstyle \varphi } and ψ . {\textstyle \psi .}
If μ 1 {\textstyle \mu _{1}} is the counting measure on a two-point set S 1 = { 1 , 2 } , {\textstyle S_{1}=\{1,2\},} then Minkowski's integral inequality gives the usual Minkowski inequality as a special case: for putting f i ( y ) = F ( i , y ) {\textstyle f_{i}(y)=F(i,y)} for i = 1 , 2 , {\textstyle i=1,2,} the integral inequality gives
‖ f 1 + f 2 ‖ p = ( ∫ S 2 | ∫ S 1 F ( x , y ) μ 1 ( d x ) | p μ 2 ( d y ) ) 1 p ≤ ∫ S 1 ( ∫ S 2 | F ( x , y ) | p μ 2 ( d y ) ) 1 p μ 1 ( d x ) = ‖ f 1 ‖ p + ‖ f 2 ‖ p . {\displaystyle \|f_{1}+f_{2}\|_{p}=\left(\int _{S_{2}}\left|\int _{S_{1}}F(x,y)\,\mu _{1}(\mathrm {d} x)\right|^{p}\mu _{2}(\mathrm {d} y)\right)^{\frac {1}{p}}\leq \int _{S_{1}}\left(\int _{S_{2}}|F(x,y)|^{p}\,\mu _{2}(\mathrm {d} y)\right)^{\frac {1}{p}}\mu _{1}(\mathrm {d} x)=\|f_{1}\|_{p}+\|f_{2}\|_{p}.}
If the measurable function F : S 1 × S 2 → R {\textstyle F:S_{1}\times S_{2}\to \mathbb {R} } is non-negative then for all 1 ≤ p ≤ q ≤ ∞ , {\textstyle 1\leq p\leq q\leq \infty ,} [ 3 ]
‖ ‖ F ( ⋅ , s 2 ) ‖ L p ( S 1 , μ 1 ) ‖ L q ( S 2 , μ 2 ) ≤ ‖ ‖ F ( s 1 , ⋅ ) ‖ L q ( S 2 , μ 2 ) ‖ L p ( S 1 , μ 1 ) . {\displaystyle \left\|\left\|F(\,\cdot ,s_{2})\right\|_{L^{p}(S_{1},\mu _{1})}\right\|_{L^{q}(S_{2},\mu _{2})}~\leq ~\left\|\left\|F(s_{1},\cdot )\right\|_{L^{q}(S_{2},\mu _{2})}\right\|_{L^{p}(S_{1},\mu _{1})}\ .}
This notation has been generalized to
‖ f ‖ p , q = ( ∫ R m [ ∫ R n | f ( x , y ) | q d y ] p q d x ) 1 p {\displaystyle \|f\|_{p,q}=\left(\int _{\mathbb {R} ^{m}}\left[\int _{\mathbb {R} ^{n}}|f(x,y)|^{q}\mathrm {d} y\right]^{\frac {p}{q}}\mathrm {d} x\right)^{\frac {1}{p}}}
for f : R m + n → E , {\textstyle f:\mathbb {R} ^{m+n}\to E,} with L p , q ( R m + n , E ) = { f ∈ E R m + n : ‖ f ‖ p , q < ∞ } . {\textstyle {\mathcal {L}}_{p,q}(\mathbb {R} ^{m+n},E)=\{f\in E^{\mathbb {R} ^{m+n}}:\|f\|_{p,q}<\infty \}.} Using this notation, manipulation of the exponents reveals that, if p < q , {\textstyle p<q,} then ‖ f ‖ q , p ≤ ‖ f ‖ p , q . {\textstyle \|f\|_{q,p}\leq \|f\|_{p,q}.}
When p < 1 {\textstyle p<1} the reverse inequality holds: ‖ f + g ‖ p ≥ ‖ f ‖ p + ‖ g ‖ p . {\displaystyle \|f+g\|_{p}\geq \|f\|_{p}+\|g\|_{p}.}
We further need the restriction that both f {\textstyle f} and g {\textstyle g} are non-negative, as we can see from the example f = − 1 , g = 1 {\textstyle f=-1,g=1} and p = 1 : {\textstyle p=1:} ‖ f + g ‖ 1 = 0 < 2 = ‖ f ‖ 1 + ‖ g ‖ 1 . {\textstyle \|f+g\|_{1}=0<2=\|f\|_{1}+\|g\|_{1}.}
The reverse inequality follows from the same argument as the standard Minkowski, but uses that Holder's inequality is also reversed in this range.
Using the Reverse Minkowski, we may prove that power means with p ≤ 1 , {\textstyle p\leq 1,} such as the harmonic mean and the geometric mean are concave.
The Minkowski inequality can be generalized to other functions ϕ ( x ) {\textstyle \phi (x)} beyond the power function x p . {\textstyle x^{p}.} The generalized inequality has the form
ϕ − 1 ( ∑ i = 1 n ϕ ( x i + y i ) ) ≤ ϕ − 1 ( ∑ i = 1 n ϕ ( x i ) ) + ϕ − 1 ( ∑ i = 1 n ϕ ( y i ) ) . {\displaystyle \phi ^{-1}\left(\textstyle \sum \limits _{i=1}^{n}\phi (x_{i}+y_{i})\right)\leq \phi ^{-1}\left(\textstyle \sum \limits _{i=1}^{n}\phi (x_{i})\right)+\phi ^{-1}\left(\textstyle \sum \limits _{i=1}^{n}\phi (y_{i})\right).}
Various sufficient conditions on ϕ {\textstyle \phi } have been found by Mulholland [ 4 ] and others. For example, for x ≥ 0 {\textstyle x\geq 0} one set of sufficient conditions from Mulholland is | https://en.wikipedia.org/wiki/Minkowski_inequality |
In mathematics, a Minkowski plane (named after Hermann Minkowski ) is one of the Benz planes (the others being Möbius plane and Laguerre plane ).
Applying the pseudo-euclidean distance d ( P 1 , P 2 ) = ( x 1 ′ − x 2 ′ ) 2 − ( y 1 ′ − y 2 ′ ) 2 {\displaystyle d(P_{1},P_{2})=(x'_{1}-x'_{2})^{2}-(y'_{1}-y'_{2})^{2}} on two points P i = ( x i ′ , y i ′ ) {\displaystyle P_{i}=(x'_{i},y'_{i})} (instead of the euclidean distance) we get the geometry of hyperbolas , because a pseudo-euclidean circle { P ∈ R 2 ∣ d ( P , M ) = r } {\displaystyle \{P\in \mathbb {R} ^{2}\mid d(P,M)=r\}} is a hyperbola with midpoint M {\displaystyle M} .
By a transformation of coordinates x i = x i ′ + y i ′ {\displaystyle x_{i}=x'_{i}+y'_{i}} , y i = x i ′ − y i ′ {\displaystyle y_{i}=x'_{i}-y'_{i}} , the pseudo-euclidean distance can be rewritten as d ( P 1 , P 2 ) = ( x 1 − x 2 ) ( y 1 − y 2 ) {\displaystyle d(P_{1},P_{2})=(x_{1}-x_{2})(y_{1}-y_{2})} . The hyperbolas then have asymptotes parallel to the non-primed coordinate axes.
The following completion (see Möbius and Laguerre planes) homogenizes the geometry of hyperbolas:
The incidence structure ( P , Z , ∈ ) {\displaystyle ({\mathcal {P}},{\mathcal {Z}},\in )} is called the classical real Minkowski plane .
The set of points consists of R 2 {\displaystyle \mathbb {R} ^{2}} , two copies of R {\displaystyle \mathbb {R} } and the point ( ∞ , ∞ ) {\displaystyle (\infty ,\infty )} .
Any line y = a x + b , a ≠ 0 {\displaystyle y=ax+b,a\neq 0} is completed by point ( ∞ , ∞ ) {\displaystyle (\infty ,\infty )} , any hyperbola y = a x − b + c , a ≠ 0 {\displaystyle y={\frac {a}{x-b}}+c,a\neq 0} by the two points ( b , ∞ ) , ( ∞ , c ) {\displaystyle (b,\infty ),(\infty ,c)} (see figure).
Two points ( x 1 , y 1 ) ≠ ( x 2 , y 2 ) {\displaystyle (x_{1},y_{1})\neq (x_{2},y_{2})} can not be connected by a cycle if and only if x 1 = x 2 {\displaystyle x_{1}=x_{2}} or y 1 = y 2 {\displaystyle y_{1}=y_{2}} .
We define:
Two points P 1 {\displaystyle P_{1}} , P 2 {\displaystyle P_{2}} are (+)-parallel ( P 1 ∥ + P 2 {\displaystyle P_{1}\parallel _{+}P_{2}} ) if x 1 = x 2 {\displaystyle x_{1}=x_{2}} and (−)-parallel ( P 1 ∥ − P 2 {\displaystyle P_{1}\parallel _{-}P_{2}} ) if y 1 = y 2 {\displaystyle y_{1}=y_{2}} . Both these relations are equivalence relations on the set of points.
Two points P 1 , P 2 {\displaystyle P_{1},P_{2}} are called parallel ( P 1 ∥ P 2 {\displaystyle P_{1}\parallel P_{2}} ) if P 1 ∥ + P 2 {\displaystyle P_{1}\parallel _{+}P_{2}} or P 1 ∥ − P 2 {\displaystyle P_{1}\parallel _{-}P_{2}} .
From the definition above we find:
Lemma:
Like the classical Möbius and Laguerre planes Minkowski planes can be described as the geometry of plane sections of a suitable quadric. But in this case the quadric lives in projective 3-space: The classical real Minkowski plane is isomorphic to the geometry of plane sections of a hyperboloid of one sheet (not degenerated quadric of index 2).
Let ( P , Z ; ∥ + , ∥ − , ∈ ) {\displaystyle \left({\mathcal {P}},{\mathcal {Z}};\parallel _{+},\parallel _{-},\in \right)} be an incidence structure with the set P {\displaystyle {\mathcal {P}}} of points, the set Z {\displaystyle {\mathcal {Z}}} of cycles and two equivalence relations ∥ + {\displaystyle \parallel _{+}} ((+)-parallel) and ∥ − {\displaystyle \parallel _{-}} ((−)-parallel) on set P {\displaystyle {\mathcal {P}}} . For P ∈ P {\displaystyle P\in {\mathcal {P}}} we define: P ¯ + := { Q ∈ P ∣ Q ∥ + P } {\displaystyle {\overline {P}}_{+}:=\left\{Q\in {\mathcal {P}}\mid Q\parallel _{+}P\right\}} and P ¯ − := { Q ∈ P ∣ Q ∥ − P } {\displaystyle {\overline {P}}_{-}:=\left\{Q\in {\mathcal {P}}\mid Q\parallel _{-}P\right\}} .
An equivalence class P ¯ + {\displaystyle {\overline {P}}_{+}} or P ¯ − {\displaystyle {\overline {P}}_{-}} is called (+)-generator and (−)-generator , respectively. (For the space model of the classical Minkowski plane a generator is a line on the hyperboloid.) Two points A , B {\displaystyle A,B} are called parallel ( A ∥ B {\displaystyle A\parallel B} ) if A ∥ + B {\displaystyle A\parallel _{+}B} or A ∥ − B {\displaystyle A\parallel _{-}B} .
An incidence structure M := ( P , Z ; ∥ + , ∥ − , ∈ ) {\displaystyle {\mathfrak {M}}:=({\mathcal {P}},{\mathcal {Z}};\parallel _{+},\parallel _{-},\in )} is called Minkowski plane if the following axioms hold:
For investigations the following statements on parallel classes (equivalent to C1, C2 respectively) are advantageous.
First consequences of the axioms are
Lemma — For a Minkowski plane M {\displaystyle {\mathfrak {M}}} the following is true
Analogously to Möbius and Laguerre planes we get the connection to the linear
geometry via the residues.
For a Minkowski plane M = ( P , Z ; ∥ + , ∥ − , ∈ ) {\displaystyle {\mathfrak {M}}=({\mathcal {P}},{\mathcal {Z}};\parallel _{+},\parallel _{-},\in )} and P ∈ P {\displaystyle P\in {\mathcal {P}}} we define the local structure A P := ( P ∖ P ¯ , { z ∖ { P ¯ } ∣ P ∈ z ∈ Z } ∪ { E ∖ P ¯ ∣ E ∈ E ∖ { P ¯ + , P ¯ − } } , ∈ ) {\displaystyle {\mathfrak {A}}_{P}:=({\mathcal {P}}\setminus {\overline {P}},\{z\setminus \{{\overline {P}}\}\mid P\in z\in {\mathcal {Z}}\}\cup \{E\setminus {\overline {P}}\mid E\in {\mathcal {E}}\setminus \{{\overline {P}}_{+},{\overline {P}}_{-}\}\},\in )} and call it the residue at point P .
For the classical Minkowski plane A ( ∞ , ∞ ) {\displaystyle {\mathfrak {A}}_{(\infty ,\infty )}} is the real affine plane R 2 {\displaystyle \mathbb {R} ^{2}} .
An immediate consequence of axioms C1 to C4 and C1′, C2′ are the following two theorems.
Theorem — For a Minkowski plane M = ( P , Z ; ∥ + , ∥ , ∈ ) {\displaystyle {\mathfrak {M}}=({\mathcal {P}},{\mathcal {Z}};\parallel _{+},\parallel ,\in )} any residue is an affine plane.
Theorem — Let be M = ( P , Z ; ∥ + , ∥ − , ∈ ) {\displaystyle {\mathfrak {M}}=({\mathcal {P}},{\mathcal {Z}};\parallel _{+},\parallel _{-},\in )} an incidence structure with two equivalence relations ∥ + {\displaystyle \parallel _{+}} and ∥ − {\displaystyle \parallel _{-}} on the set P {\displaystyle {\mathcal {P}}} of points (see above).
Then, M {\displaystyle {\mathfrak {M}}} is a Minkowski plane if and only if for any point P {\displaystyle P} the residue A P {\displaystyle {\mathfrak {A}}_{P}} is an affine plane.
The minimal model of a Minkowski plane can be established over the set K ¯ := { 0 , 1 , ∞ } {\displaystyle {\overline {K}}:=\{0,1,\infty \}} of three elements: P := K ¯ 2 {\displaystyle {\mathcal {P}}:={\overline {K}}^{2}} Z : = { { ( a 1 , b 1 ) , ( a 2 , b 2 ) , ( a 3 , b 3 ) } ∣ { a 1 , a 2 , a 3 } = { b 1 , b 2 , b 3 } = K ¯ } = { { ( 0 , 0 ) , ( 1 , 1 ) , ( ∞ , ∞ ) } , { ( 0 , 0 ) , ( 1 , ∞ ) , ( ∞ , 1 ) } , { ( 0 , 1 ) , ( 1 , 0 ) , ( ∞ , ∞ ) } , { ( 0 , 1 ) , ( 1 , ∞ ) , ( ∞ , 0 ) } , { ( 0 , ∞ ) , ( 1 , 1 ) , ( ∞ , 0 ) } , { ( 0 , ∞ ) , ( 1 , 0 ) , ( ∞ , 1 ) } } {\displaystyle {\begin{aligned}{\mathcal {Z}}:\!&=\left\{\{(a_{1},b_{1}),(a_{2},b_{2}),(a_{3},b_{3})\}\mid \{a_{1},a_{2},a_{3}\}=\{b_{1},b_{2},b_{3}\}={\overline {K}}\right\}\\&=\{\{(0,0),(1,1),(\infty ,\infty )\},\;\{(0,0),(1,\infty ),(\infty ,1)\},\\&\qquad \{(0,1),(1,0),(\infty ,\infty )\},\;\{(0,1),(1,\infty ),(\infty ,0)\},\\&\qquad \{(0,\infty ),(1,1),(\infty ,0)\},\;\{(0,\infty ),(1,0),(\infty ,1)\}\}\end{aligned}}}
Parallel points:
Hence | P | = 9 {\displaystyle \left|{\mathcal {P}}\right|=9} and | Z | = 6 {\displaystyle \left|{\mathcal {Z}}\right|=6} .
For finite Minkowski-planes we get from C1′, C2′:
Lemma — Let be M = ( P , Z ; ∥ + , ∥ − , ∈ ) {\displaystyle {\mathfrak {M}}=({\mathcal {P}},{\mathcal {Z}};\parallel _{+},\parallel _{-},\in )} a finite Minkowski plane, i.e. | P | < ∞ {\displaystyle \left|{\mathcal {P}}\right|<\infty } . For any pair of cycles z 1 , z 2 {\displaystyle z_{1},z_{2}} and any pair of generators e 1 , e 2 {\displaystyle e_{1},e_{2}} we have: | z 1 | = | z 2 | = | e 1 | = | e 2 | {\displaystyle \left|z_{1}\right|=\left|z_{2}\right|=\left|e_{1}\right|=\left|e_{2}\right|} .
This gives rise of the definition : For a finite Minkowski plane M {\displaystyle {\mathfrak {M}}} and a cycle z {\displaystyle z} of M {\displaystyle {\mathfrak {M}}} we call the integer n = | z | − 1 {\displaystyle n=\left|z\right|-1} the order of M {\displaystyle {\mathfrak {M}}} .
Simple combinatorial considerations yield
Lemma — For a finite Minkowski plane M = ( P , Z ; ∥ + , ∥ − , ∈ ) {\displaystyle {\mathfrak {M}}=({\mathcal {P}},{\mathcal {Z}};\parallel _{+},\parallel _{-},\in )} the following is true:
We get the most important examples of Minkowski planes by generalizing the classical real model: Just replace R {\displaystyle \mathbb {R} } by an arbitrary field K {\displaystyle K} then we get in any case a Minkowski plane M ( K ) = ( P , Z ; ∥ + , ∥ − , ∈ ) {\displaystyle {\mathfrak {M}}(K)=({\mathcal {P}},{\mathcal {Z}};\parallel _{+},\parallel _{-},\in )} .
Analogously to Möbius and Laguerre planes the Theorem of Miquel is a characteristic property of a Minkowski plane M ( K ) {\displaystyle {\mathfrak {M}}(K)} .
Theorem (Miquel): For the Minkowski plane M ( K ) {\displaystyle {\mathfrak {M}}(K)} the following is true:
(For a better overview in the figure there are circles drawn instead of hyperbolas.)
Theorem (Chen): Only a Minkowski plane M ( K ) {\displaystyle {\mathfrak {M}}(K)} satisfies the theorem of Miquel.
Because of the last theorem M ( K ) {\displaystyle {\mathfrak {M}}(K)} is called a miquelian Minkowski plane .
Remark: The minimal model of a Minkowski plane is miquelian.
An astonishing result is
Theorem (Heise): Any Minkowski plane of even order is miquelian.
Remark: A suitable stereographic projection shows: M ( K ) {\displaystyle {\mathfrak {M}}(K)} is isomorphic
to the geometry of the plane sections on a hyperboloid of one sheet ( quadric of index 2) in projective 3-space over field K {\displaystyle K} .
Remark: There are a lot of Minkowski planes that are not miquelian (s. weblink below). But there are no "ovoidal Minkowski" planes, in difference to Möbius and Laguerre planes. Because any quadratic set of index 2 in projective 3-space is a quadric (see quadratic set ). | https://en.wikipedia.org/wiki/Minkowski_plane |
In differential geometry , the Minkowski problem , named after Hermann Minkowski , asks for the construction of a strictly convex compact surface S whose Gaussian curvature is specified. [ 1 ] More precisely, the input to the problem is a strictly positive real function ƒ defined on a sphere, and the surface that is to be constructed should have Gaussian curvature ƒ ( n ( x )) at the point x , where n ( x ) denotes the normal to S at x . Eugenio Calabi stated: "From the geometric view point it [the Minkowski problem] is the Rosetta Stone , from which several related problems can be solved." [ 2 ]
In full generality, the Minkowski problem asks for necessary and sufficient conditions on a non-negative Borel measure on the unit sphere S n-1 to be the surface area measure of a convex body in R n {\displaystyle \mathbb {R} ^{n}} . Here the surface area measure S K of a convex body K is the pushforward of the (n-1) -dimensional Hausdorff measure restricted to the boundary of K via the Gauss map . The Minkowski problem was solved by Hermann Minkowski , Aleksandr Danilovich Aleksandrov , Werner Fenchel and Børge Jessen : [ 3 ] a Borel measure μ on the unit sphere is the surface area measure of a convex body if and only if μ has centroid at the origin and is not concentrated on a great subsphere. The convex body is then uniquely determined by μ up to translations.
The Minkowski problem, despite its clear geometric origin, is found to have its appearance in many places. The problem of radiolocation is easily reduced to the Minkowski problem in Euclidean 3-space : restoration of convex shape over the given Gauss surface curvature. The inverse problem of the short-wave diffraction is reduced to the Minkowski problem. The Minkowski problem is the basis of the mathematical theory of diffraction as well as for the physical theory of diffraction.
In 1953 Louis Nirenberg published the solutions of two long standing open problems, the Weyl problem and the Minkowski problem in Euclidean 3-space. L. Nirenberg's solution of the Minkowski problem was a milestone in global geometry. He has been selected to be the first recipient of the Chern Medal (in 2010) for his role in the formulation of the modern theory of non-linear elliptic partial differential equations, particularly for solving the Weyl problem and the Minkowski problems in Euclidean 3-space. [ 4 ]
A. V. Pogorelov received Ukraine State Prize (1973) for resolving the multidimensional Minkowski problem in Euclidean spaces. Pogorelov resolved the Weyl problem in Riemannian space in 1969. [ 5 ]
Shing-Tung Yau 's joint work with Shiu-Yuen Cheng gives a complete proof of the higher-dimensional Minkowski problem in Euclidean spaces. Shing-Tung Yau received the Fields Medal at the International Congress of Mathematicians in Warsaw in 1982 for his work in global differential geometry and elliptic partial differential equations , particularly for solving such difficult problems as the Calabi conjecture of 1954, and a problem of Hermann Minkowski in Euclidean spaces concerning the Dirichlet problem for the real Monge–Ampère equation . [ 6 ] | https://en.wikipedia.org/wiki/Minkowski_problem |
In physics , Minkowski space (or Minkowski spacetime ) ( / m ɪ ŋ ˈ k ɔː f s k i , - ˈ k ɒ f -/ [ 1 ] ) is the main mathematical description of spacetime in the absence of gravitation . It combines inertial space and time manifolds into a four-dimensional model.
The model helps show how a spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Mathematician Hermann Minkowski developed it from the work of Hendrik Lorentz , Henri Poincaré , and others said it "was grown on experimental physical grounds".
Minkowski space is closely associated with Einstein's theories of special relativity and general relativity and is the most common mathematical structure by which special relativity is formalized. While the individual components in Euclidean space and time might differ due to length contraction and time dilation , in Minkowski spacetime, all frames of reference will agree on the total interval in spacetime between events. [ nb 1 ] Minkowski space differs from four-dimensional Euclidean space insofar as it treats time differently from the three spatial dimensions.
In 3-dimensional Euclidean space , the isometry group (maps preserving the regular Euclidean distance ) is the Euclidean group . It is generated by rotations , reflections and translations . When time is appended as a fourth dimension, the further transformations of translations in time and Lorentz boosts are added, and the group of all these transformations is called the Poincaré group . Minkowski's model follows special relativity, where motion causes time dilation changing the scale applied to the frame in motion and shifts the phase of light.
Minkowski space is a pseudo-Euclidean space equipped with an isotropic quadratic form called the spacetime interval or the Minkowski norm squared . An event in Minkowski space for which the spacetime interval is zero is on the null cone of the origin, called the light cone in Minkowski space. Using the polarization identity the quadratic form is converted to a symmetric bilinear form called the Minkowski inner product , though it is not a geometric inner product . Another misnomer is Minkowski metric , [ 2 ] but Minkowski space is not a metric space .
The group of transformations for Minkowski space that preserves the spacetime interval (as opposed to the spatial Euclidean distance) is the Lorentz group (as opposed to the Galilean group ).
In his second relativity paper in 1905, Henri Poincaré showed [ 3 ] how, by taking time to be an imaginary fourth spacetime coordinate ict , where c is the speed of light and i is the imaginary unit , Lorentz transformations can be visualized as ordinary rotations of the four-dimensional Euclidean sphere. The four-dimensional spacetime can be visualized as a four-dimensional space, with each point representing an event in spacetime. The Lorentz transformations can then be thought of as rotations in this four-dimensional space, where the rotation axis corresponds to the direction of relative motion between the two observers and the rotation angle is related to their relative velocity.
To understand this concept, one should consider the coordinates of an event in spacetime represented as a four-vector ( t , x , y , z ) . A Lorentz transformation is represented by a matrix that acts on the four-vector, changing its components. This matrix can be thought of as a rotation matrix in four-dimensional space, which rotates the four-vector around a particular axis. x 2 + y 2 + z 2 + ( i c t ) 2 = constant . {\displaystyle x^{2}+y^{2}+z^{2}+(ict)^{2}={\text{constant}}.}
Rotations in planes spanned by two space unit vectors appear in coordinate space as well as in physical spacetime as Euclidean rotations and are interpreted in the ordinary sense. The "rotation" in a plane spanned by a space unit vector and a time unit vector, while formally still a rotation in coordinate space, is a Lorentz boost in physical spacetime with real inertial coordinates. The analogy with Euclidean rotations is only partial since the radius of the sphere is actually imaginary, which turns rotations into rotations in hyperbolic space (see hyperbolic rotation ).
This idea, which was mentioned only briefly by Poincaré, was elaborated by Minkowski in a paper in German published in 1908 called "The Fundamental Equations for Electromagnetic Processes in Moving Bodies". [ 4 ] He reformulated Maxwell equations as a symmetrical set of equations in the four variables ( x , y , z , ict ) combined with redefined vector variables for electromagnetic quantities, and he was able to show directly and very simply their invariance under Lorentz transformation. He also made other important contributions and used matrix notation for the first time in this context.
From his reformulation, he concluded that time and space should be treated equally, and so arose his concept of events taking place in a unified four-dimensional spacetime continuum .
In a further development in his 1908 "Space and Time" lecture, [ 5 ] Minkowski gave an alternative formulation of this idea that used a real time coordinate instead of an imaginary one, representing the four variables ( x , y , z , t ) of space and time in the coordinate form in a four-dimensional real vector space . Points in this space correspond to events in spacetime. In this space, there is a defined light-cone associated with each point, and events not on the light cone are classified by their relation to the apex as spacelike or timelike . It is principally this view of spacetime that is current nowadays, although the older view involving imaginary time has also influenced special relativity.
In the English translation of Minkowski's paper, the Minkowski metric, as defined below, is referred to as the line element . The Minkowski inner product below appears unnamed when referring to orthogonality (which he calls normality ) of certain vectors, and the Minkowski norm squared is referred to (somewhat cryptically, perhaps this is a translation dependent) as "sum".
Minkowski's principal tool is the Minkowski diagram , and he uses it to define concepts and demonstrate properties of Lorentz transformations (e.g., proper time and length contraction ) and to provide geometrical interpretation to the generalization of Newtonian mechanics to relativistic mechanics . For these special topics, see the referenced articles, as the presentation below will be principally confined to the mathematical structure (Minkowski metric and from it derived quantities and the Poincaré group as symmetry group of spacetime) following from the invariance of the spacetime interval on the spacetime manifold as consequences of the postulates of special relativity, not to specific application or derivation of the invariance of the spacetime interval. This structure provides the background setting of all present relativistic theories, barring general relativity for which flat Minkowski spacetime still provides a springboard as curved spacetime is locally Lorentzian.
Minkowski, aware of the fundamental restatement of the theory which he had made, said
The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth, space by itself and time by itself are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.
Though Minkowski took an important step for physics, Albert Einstein saw its limitation:
At a time when Minkowski was giving the geometrical interpretation of special relativity by extending the Euclidean three-space to a quasi-Euclidean four-space that included time, Einstein was already aware that this is not valid, because it excludes the phenomenon of gravitation . He was still far from the study of curvilinear coordinates and Riemannian geometry , and the heavy mathematical apparatus entailed. [ 6 ]
For further historical information see references Galison (1979) , Corry (1997) and Walter (1999) .
Where v is velocity, x , y , and z are Cartesian coordinates in 3-dimensional space, c is the constant representing the universal speed limit, and t is time, the four-dimensional vector v = ( ct , x , y , z ) = ( ct , r ) is classified according to the sign of c 2 t 2 − r 2 . A vector is timelike if c 2 t 2 > r 2 , spacelike if c 2 t 2 < r 2 , and null or lightlike if c 2 t 2 = r 2 . This can be expressed in terms of the sign of η ( v , v ) , also called scalar product , as well, which depends on the signature. The classification of any vector will be the same in all frames of reference that are related by a Lorentz transformation (but not by a general Poincaré transformation because the origin may then be displaced) because of the invariance of the spacetime interval under Lorentz transformation.
The set of all null vectors at an event [ nb 2 ] of Minkowski space constitutes the light cone of that event. Given a timelike vector v , there is a worldline of constant velocity associated with it, represented by a straight line in a Minkowski diagram.
Once a direction of time is chosen, [ nb 3 ] timelike and null vectors can be further decomposed into various classes. For timelike vectors, one has
Null vectors fall into three classes:
Together with spacelike vectors, there are 6 classes in all.
An orthonormal basis for Minkowski space necessarily consists of one timelike and three spacelike unit vectors. If one wishes to work with non-orthonormal bases, it is possible to have other combinations of vectors. For example, one can easily construct a (non-orthonormal) basis consisting entirely of null vectors, called a null basis .
Vector fields are called timelike, spacelike, or null if the associated vectors are timelike, spacelike, or null at each point where the field is defined.
Time-like vectors have special importance in the theory of relativity as they correspond to events that are accessible to the observer at (0, 0, 0, 0) with a speed less than that of light. Of most interest are time-like vectors that are similarly directed , i.e. all either in the forward or in the backward cones. Such vectors have several properties not shared by space-like vectors. These arise because both forward and backward cones are convex, whereas the space-like region is not convex.
The scalar product of two time-like vectors u 1 = ( t 1 , x 1 , y 1 , z 1 ) and u 2 = ( t 2 , x 2 , y 2 , z 2 ) is η ( u 1 , u 2 ) = u 1 ⋅ u 2 = c 2 t 1 t 2 − x 1 x 2 − y 1 y 2 − z 1 z 2 . {\displaystyle \eta (u_{1},u_{2})=u_{1}\cdot u_{2}=c^{2}t_{1}t_{2}-x_{1}x_{2}-y_{1}y_{2}-z_{1}z_{2}.}
Positivity of scalar product : An important property is that the scalar product of two similarly directed time-like vectors is always positive. This can be seen from the reversed Cauchy–Schwarz inequality below. It follows that if the scalar product of two vectors is zero, then one of these, at least, must be space-like. The scalar product of two space-like vectors can be positive or negative as can be seen by considering the product of two space-like vectors having orthogonal spatial components and times either of different or the same signs.
Using the positivity property of time-like vectors, it is easy to verify that a linear sum with positive coefficients of similarly directed time-like vectors is also similarly directed time-like (the sum remains within the light cone because of convexity).
The norm of a time-like vector u = ( ct , x , y , z ) is defined as ‖ u ‖ = η ( u , u ) = c 2 t 2 − x 2 − y 2 − z 2 {\displaystyle \left\|u\right\|={\sqrt {\eta (u,u)}}={\sqrt {c^{2}t^{2}-x^{2}-y^{2}-z^{2}}}}
The reversed Cauchy inequality is another consequence of the convexity of either light cone. [ 7 ] For two distinct similarly directed time-like vectors u 1 and u 2 this inequality is η ( u 1 , u 2 ) > ‖ u 1 ‖ ‖ u 2 ‖ {\displaystyle \eta (u_{1},u_{2})>\left\|u_{1}\right\|\left\|u_{2}\right\|} or algebraically, c 2 t 1 t 2 − x 1 x 2 − y 1 y 2 − z 1 z 2 > ( c 2 t 1 2 − x 1 2 − y 1 2 − z 1 2 ) ( c 2 t 2 2 − x 2 2 − y 2 2 − z 2 2 ) {\displaystyle c^{2}t_{1}t_{2}-x_{1}x_{2}-y_{1}y_{2}-z_{1}z_{2}>{\sqrt {\left(c^{2}t_{1}^{2}-x_{1}^{2}-y_{1}^{2}-z_{1}^{2}\right)\left(c^{2}t_{2}^{2}-x_{2}^{2}-y_{2}^{2}-z_{2}^{2}\right)}}}
From this, the positive property of the scalar product can be seen.
For two similarly directed time-like vectors u and w , the inequality is [ 8 ] ‖ u + w ‖ ≥ ‖ u ‖ + ‖ w ‖ , {\displaystyle \left\|u+w\right\|\geq \left\|u\right\|+\left\|w\right\|,} where the equality holds when the vectors are linearly dependent .
The proof uses the algebraic definition with the reversed Cauchy inequality: [ 9 ] ‖ u + w ‖ 2 = ‖ u ‖ 2 + 2 ( u , w ) + ‖ w ‖ 2 ≥ ‖ u ‖ 2 + 2 ‖ u ‖ ‖ w ‖ + ‖ w ‖ 2 = ( ‖ u ‖ + ‖ w ‖ ) 2 . {\displaystyle {\begin{aligned}\left\|u+w\right\|^{2}&=\left\|u\right\|^{2}+2\left(u,w\right)+\left\|w\right\|^{2}\\[5mu]&\geq \left\|u\right\|^{2}+2\left\|u\right\|\left\|w\right\|+\left\|w\right\|^{2}=\left(\left\|u\right\|+\left\|w\right\|\right)^{2}.\end{aligned}}}
The result now follows by taking the square root on both sides.
It is assumed below that spacetime is endowed with a coordinate system corresponding to an inertial frame . This provides an origin , which is necessary for spacetime to be modeled as a vector space. This addition is not required, and more complex treatments analogous to an affine space can remove the extra structure. However, this is not the introductory convention and is not covered here.
For an overview, Minkowski space is a 4 -dimensional real vector space equipped with a non-degenerate, symmetric bilinear form on the tangent space at each point in spacetime, here simply called the Minkowski inner product , with metric signature either (+ − − −) or (− + + +) . The tangent space at each event is a vector space of the same dimension as spacetime, 4 .
In practice, one need not be concerned with the tangent spaces. The vector space structure of Minkowski space allows for the canonical identification of vectors in tangent spaces at points (events) with vectors (points, events) in Minkowski space itself. See e.g. Lee (2003 , Proposition 3.8.) or Lee (2012 , Proposition 3.13.) These identifications are routinely done in mathematics. They can be expressed formally in Cartesian coordinates as [ 10 ] ( x 0 , x 1 , x 2 , x 3 ) ↔ x 0 e 0 | p + x 1 e 1 | p + x 2 e 2 | p + x 3 e 3 | p ↔ x 0 e 0 | q + x 1 e 1 | q + x 2 e 2 | q + x 3 e 3 | q {\displaystyle {\begin{aligned}\left(x^{0},\,x^{1},\,x^{2},\,x^{3}\right)\ &\leftrightarrow \ \left.x^{0}\mathbf {e} _{0}\right|_{p}+\left.x^{1}\mathbf {e} _{1}\right|_{p}+\left.x^{2}\mathbf {e} _{2}\right|_{p}+\left.x^{3}\mathbf {e} _{3}\right|_{p}\\&\leftrightarrow \ \left.x^{0}\mathbf {e} _{0}\right|_{q}+\left.x^{1}\mathbf {e} _{1}\right|_{q}+\left.x^{2}\mathbf {e} _{2}\right|_{q}+\left.x^{3}\mathbf {e} _{3}\right|_{q}\end{aligned}}} with basis vectors in the tangent spaces defined by e μ | p = ∂ ∂ x μ | p or e 0 | p = ( 1 0 0 0 ) , etc . {\displaystyle \left.\mathbf {e} _{\mu }\right|_{p}=\left.{\frac {\partial }{\partial x^{\mu }}}\right|_{p}{\text{ or }}\mathbf {e} _{0}|_{p}=\left({\begin{matrix}1\\0\\0\\0\end{matrix}}\right){\text{, etc}}.}
Here, p and q are any two events, and the second basis vector identification is referred to as parallel transport . The first identification is the canonical identification of vectors in the tangent space at any point with vectors in the space itself. The appearance of basis vectors in tangent spaces as first-order differential operators is due to this identification. It is motivated by the observation that a geometrical tangent vector can be associated in a one-to-one manner with a directional derivative operator on the set of smooth functions. This is promoted to a definition of tangent vectors in manifolds not necessarily being embedded in R n . This definition of tangent vectors is not the only possible one, as ordinary n -tuples can be used as well.
A tangent vector at a point p may be defined, here specialized to Cartesian coordinates in Lorentz frames, as 4 × 1 column vectors v associated to each Lorentz frame related by Lorentz transformation Λ such that the vector v in a frame related to some frame by Λ transforms according to v → Λ v . This is the same way in which the coordinates x μ transform. Explicitly, x ′ μ = Λ μ ν x ν , v ′ μ = Λ μ ν v ν . {\displaystyle {\begin{aligned}x'^{\mu }&={\Lambda ^{\mu }}_{\nu }x^{\nu },\\v'^{\mu }&={\Lambda ^{\mu }}_{\nu }v^{\nu }.\end{aligned}}}
This definition is equivalent to the definition given above under a canonical isomorphism.
For some purposes, it is desirable to identify tangent vectors at a point p with displacement vectors at p , which is, of course, admissible by essentially the same canonical identification. [ 11 ] The identifications of vectors referred to above in the mathematical setting can correspondingly be found in a more physical and explicitly geometrical setting in Misner, Thorne & Wheeler (1973) . They offer various degrees of sophistication (and rigor) depending on which part of the material one chooses to read.
The metric signature refers to which sign the Minkowski inner product yields when given space ( spacelike to be specific, defined further down) and time basis vectors ( timelike ) as arguments. Further discussion about this theoretically inconsequential but practically necessary choice for purposes of internal consistency and convenience is deferred to the hide box below. See also the page treating sign convention in Relativity.
In general, but with several exceptions, mathematicians and general relativists prefer spacelike vectors to yield a positive sign, (− + + +) , while particle physicists tend to prefer timelike vectors to yield a positive sign, (+ − − −) . Authors covering several areas of physics, e.g. Steven Weinberg and Landau and Lifshitz ( (− + + +) and (+ − − −) respectively) stick to one choice regardless of topic. Arguments for the former convention include "continuity" from the Euclidean case corresponding to the non-relativistic limit c → ∞ . Arguments for the latter include that minus signs, otherwise ubiquitous in particle physics, go away. Yet other authors, especially of introductory texts, e.g. Kleppner & Kolenkow (1978) , do not choose a signature at all, but instead, opt to coordinatize spacetime such that the time coordinate (but not time itself!) is imaginary. This removes the need for the explicit introduction of a metric tensor (which may seem like an extra burden in an introductory course), and one needs not be concerned with covariant vectors and contravariant vectors (or raising and lowering indices) to be described below. The inner product is instead affected by a straightforward extension of the dot product in R 3 to R 3 × C . This works in the flat spacetime of special relativity, but not in the curved spacetime of general relativity, see Misner, Thorne & Wheeler (1973 , Box 2.1, Farewell to ict ) (who, by the way use (− + + +) ). MTW also argues that it hides the true indefinite nature of the metric and the true nature of Lorentz boosts, which are not rotations. It also needlessly complicates the use of tools of differential geometry that are otherwise immediately available and useful for geometrical description and calculation – even in the flat spacetime of special relativity, e.g. of the electromagnetic field.
Mathematically associated with the bilinear form is a tensor of type (0,2) at each point in spacetime, called the Minkowski metric . [ nb 4 ] The Minkowski metric, the bilinear form, and the Minkowski inner product are all the same object; it is a bilinear function that accepts two (contravariant) vectors and returns a real number. In coordinates, this is the 4×4 matrix representing the bilinear form.
For comparison, in general relativity , a Lorentzian manifold L is likewise equipped with a metric tensor g , which is a nondegenerate symmetric bilinear form on the tangent space T p L at each point p of L . In coordinates, it may be represented by a 4×4 matrix depending on spacetime position . Minkowski space is thus a comparatively simple special case of a Lorentzian manifold . Its metric tensor is in coordinates with the same symmetric matrix at every point of M , and its arguments can, per above, be taken as vectors in spacetime itself.
Introducing more terminology (but not more structure), Minkowski space is thus a pseudo-Euclidean space with total dimension n = 4 and signature (1, 3) or (3, 1) . Elements of Minkowski space are called events . Minkowski space is often denoted R 1,3 or R 3,1 to emphasize the chosen signature, or just M . It is an example of a pseudo-Riemannian manifold .
Then mathematically, the metric is a bilinear form on an abstract four-dimensional real vector space V , that is, η : V × V → R {\displaystyle \eta :V\times V\rightarrow \mathbf {R} } where η has signature (+, -, -, -) , and signature is a coordinate-invariant property of η . The space of bilinear maps forms a vector space which can be identified with M ∗ ⊗ M ∗ {\displaystyle M^{*}\otimes M^{*}} , and η may be equivalently viewed as an element of this space. By making a choice of orthonormal basis { e μ } {\displaystyle \{e_{\mu }\}} , M := ( V , η ) {\displaystyle M:=(V,\eta )} can be identified with the space R 1 , 3 := ( R 4 , η μ ν ) {\displaystyle \mathbf {R} ^{1,3}:=(\mathbf {R} ^{4},\eta _{\mu \nu })} . The notation is meant to emphasize the fact that M and R 1 , 3 {\displaystyle \mathbf {R} ^{1,3}} are not just vector spaces but have added structure. η μ ν = diag ( + 1 , − 1 , − 1 , − 1 ) {\displaystyle \eta _{\mu \nu }={\text{diag}}(+1,-1,-1,-1)} .
An interesting example of non-inertial coordinates for (part of) Minkowski spacetime is the Born coordinates . Another useful set of coordinates is the light-cone coordinates .
The Minkowski inner product is not an inner product , since it has non-zero null vectors . Since it is not a definite bilinear form it is called indefinite .
The Minkowski metric η is the metric tensor of Minkowski space. It is a pseudo-Euclidean metric, or more generally, a constant pseudo-Riemannian metric in Cartesian coordinates. As such, it is a nondegenerate symmetric bilinear form, a type (0, 2) tensor. It accepts two arguments u p , v p , vectors in T p M , p ∈ M , the tangent space at p in M . Due to the above-mentioned canonical identification of T p M with M itself, it accepts arguments u , v with both u and v in M .
As a notational convention, vectors v in M , called 4-vectors , are denoted in italics, and not, as is common in the Euclidean setting, with boldface v . The latter is generally reserved for the 3 -vector part (to be introduced below) of a 4 -vector.
The definition [ 12 ] u ⋅ v = η ( u , v ) {\displaystyle u\cdot v=\eta (u,\,v)} yields an inner product-like structure on M , previously and also henceforth, called the Minkowski inner product , similar to the Euclidean inner product , but it describes a different geometry. It is also called the relativistic dot product . If the two arguments are the same, u ⋅ u = η ( u , u ) ≡ ‖ u ‖ 2 ≡ u 2 , {\displaystyle u\cdot u=\eta (u,u)\equiv \|u\|^{2}\equiv u^{2},} the resulting quantity will be called the Minkowski norm squared . The Minkowski inner product satisfies the following properties.
The first two conditions imply bilinearity.
The most important feature of the inner product and norm squared is that these are quantities unaffected by Lorentz transformations . In fact, it can be taken as the defining property of a Lorentz transformation in that it preserves the inner product (i.e. the value of the corresponding bilinear form on two vectors). This approach is taken more generally for all classical groups definable this way in classical group . There, the matrix Φ is identical in the case O(3, 1) (the Lorentz group) to the matrix η to be displayed below.
Minkowski space is constructed so that the speed of light will be the same constant regardless of the reference frame in which it is measured. This property results from the relation of the time axis to a space axis. Two events u and v are orthogonal when the bilinear form is zero for them: η ( v , w ) = 0 .
When both u and v are both space-like, then they are perpendicular , but if one is time-like and the other space-like, then the relation is hyperbolic orthogonality . The relation is preserved in a change of reference frames and consequently the computation of light speed yields a constant result. The change of reference frame is called a Lorentz boost and in mathematics it is a hyperbolic rotation . Each reference frame is associated with a hyperbolic angle , which is zero for the rest frame in Minkowski space. Such a hyperbolic angle has been labelled rapidity since it is associated with the speed of the frame.
From the second postulate of special relativity , together with homogeneity of spacetime and isotropy of space, it follows that the spacetime interval between two arbitrary events called 1 and 2 is: [ 13 ] c 2 ( t 1 − t 2 ) 2 − ( x 1 − x 2 ) 2 − ( y 1 − y 2 ) 2 − ( z 1 − z 2 ) 2 . {\displaystyle c^{2}\left(t_{1}-t_{2}\right)^{2}-\left(x_{1}-x_{2}\right)^{2}-\left(y_{1}-y_{2}\right)^{2}-\left(z_{1}-z_{2}\right)^{2}.} This quantity is not consistently named in the literature. The interval is sometimes referred to as the square root of the interval as defined here. [ 14 ] [ 15 ]
The invariance of the interval under coordinate transformations between inertial frames follows from the invariance of c 2 t 2 − x 2 − y 2 − z 2 {\displaystyle c^{2}t^{2}-x^{2}-y^{2}-z^{2}} provided the transformations are linear. This quadratic form can be used to define a bilinear form u ⋅ v = c 2 t 1 t 2 − x 1 x 2 − y 1 y 2 − z 1 z 2 {\displaystyle u\cdot v=c^{2}t_{1}t_{2}-x_{1}x_{2}-y_{1}y_{2}-z_{1}z_{2}} via the polarization identity . This bilinear form can in turn be written as u ⋅ v = u T [ η ] v , {\displaystyle u\cdot v=u^{\textsf {T}}\,[\eta ]\,v,} where [ η ] is a 4 × 4 {\displaystyle 4\times 4} matrix associated with η . While possibly confusing, it is common practice to denote [ η ] with just η . The matrix is read off from the explicit bilinear form as η = ( 1 0 0 0 0 − 1 0 0 0 0 − 1 0 0 0 0 − 1 ) , {\displaystyle \eta =\left({\begin{array}{r}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{array}}\right)\!,} and the bilinear form u ⋅ v = η ( u , v ) , {\displaystyle u\cdot v=\eta (u,v),} with which this section started by assuming its existence, is now identified.
For definiteness and shorter presentation, the signature (− + + +) is adopted below. This choice (or the other possible choice) has no (known) physical implications. The symmetry group preserving the bilinear form with one choice of signature is isomorphic (under the map given here ) with the symmetry group preserving the other choice of signature. This means that both choices are in accord with the two postulates of relativity. Switching between the two conventions is straightforward. If the metric tensor η has been used in a derivation, go back to the earliest point where it was used, substitute η for − η , and retrace forward to the desired formula with the desired metric signature.
A standard or orthonormal basis for Minkowski space is a set of four mutually orthogonal vectors { e 0 , e 1 , e 2 , e 3 } such that η ( e 0 , e 0 ) = − η ( e 1 , e 1 ) = − η ( e 2 , e 2 ) = − η ( e 3 , e 3 ) = 1 {\displaystyle \eta (e_{0},e_{0})=-\eta (e_{1},e_{1})=-\eta (e_{2},e_{2})=-\eta (e_{3},e_{3})=1} and for which η ( e μ , e ν ) = 0 {\displaystyle \eta (e_{\mu },e_{\nu })=0} when μ ≠ ν . {\textstyle \mu \neq \nu \,.}
These conditions can be written compactly in the form η ( e μ , e ν ) = η μ ν . {\displaystyle \eta (e_{\mu },e_{\nu })=\eta _{\mu \nu }.}
Relative to a standard basis, the components of a vector v are written ( v 0 , v 1 , v 2 , v 3 ) where the Einstein notation is used to write v = v μ e μ . The component v 0 is called the timelike component of v while the other three components are called the spatial components . The spatial components of a 4 -vector v may be identified with a 3 -vector v = ( v 1 , v 2 , v 3 ) .
In terms of components, the Minkowski inner product between two vectors v and w is given by
η ( v , w ) = η μ ν v μ w ν = v 0 w 0 + v 1 w 1 + v 2 w 2 + v 3 w 3 = v μ w μ = v μ w μ , {\displaystyle \eta (v,w)=\eta _{\mu \nu }v^{\mu }w^{\nu }=v^{0}w_{0}+v^{1}w_{1}+v^{2}w_{2}+v^{3}w_{3}=v^{\mu }w_{\mu }=v_{\mu }w^{\mu },} and η ( v , v ) = η μ ν v μ v ν = v 0 v 0 + v 1 v 1 + v 2 v 2 + v 3 v 3 = v μ v μ . {\displaystyle \eta (v,v)=\eta _{\mu \nu }v^{\mu }v^{\nu }=v^{0}v_{0}+v^{1}v_{1}+v^{2}v_{2}+v^{3}v_{3}=v^{\mu }v_{\mu }.}
Here lowering of an index with the metric was used.
There are many possible choices of standard basis obeying the condition η ( e μ , e ν ) = η μ ν . {\displaystyle \eta (e_{\mu },e_{\nu })=\eta _{\mu \nu }.} Any two such bases are related in some sense by a Lorentz transformation, either by a change-of-basis matrix Λ ν μ {\displaystyle \Lambda _{\nu }^{\mu }} , a real 4 × 4 matrix satisfying Λ ρ μ η μ ν Λ σ ν = η ρ σ . {\displaystyle \Lambda _{\rho }^{\mu }\eta _{\mu \nu }\Lambda _{\sigma }^{\nu }=\eta _{\rho \sigma }.} or Λ , a linear map on the abstract vector space satisfying, for any pair of vectors u , v , η ( Λ u , Λ v ) = η ( u , v ) . {\displaystyle \eta (\Lambda u,\Lambda v)=\eta (u,v).}
Then if two different bases exist, { e 0 , e 1 , e 2 , e 3 } and { e ′ 0 , e ′ 1 , e ′ 2 , e ′ 3 } , e μ ′ = e ν Λ μ ν {\displaystyle e_{\mu }'=e_{\nu }\Lambda _{\mu }^{\nu }} can be represented as e μ ′ = e ν Λ μ ν {\displaystyle e_{\mu }'=e_{\nu }\Lambda _{\mu }^{\nu }} or e μ ′ = Λ e μ {\displaystyle e_{\mu }'=\Lambda e_{\mu }} . While it might be tempting to think of Λ ν μ {\displaystyle \Lambda _{\nu }^{\mu }} and Λ as the same thing, mathematically, they are elements of different spaces, and act on the space of standard bases from different sides.
Technically, a non-degenerate bilinear form provides a map between a vector space and its dual; in this context, the map is between the tangent spaces of M and the cotangent spaces of M . At a point in M , the tangent and cotangent spaces are dual vector spaces (so the dimension of the cotangent space at an event is also 4 ). Just as an authentic inner product on a vector space with one argument fixed, by Riesz representation theorem , may be expressed as the action of a linear functional on the vector space, the same holds for the Minkowski inner product of Minkowski space. [ 17 ]
Thus if v μ are the components of a vector in tangent space, then η μν v μ = v ν are the components of a vector in the cotangent space (a linear functional). Due to the identification of vectors in tangent spaces with vectors in M itself, this is mostly ignored, and vectors with lower indices are referred to as covariant vectors . In this latter interpretation, the covariant vectors are (almost always implicitly) identified with vectors (linear functionals) in the dual of Minkowski space. The ones with upper indices are contravariant vectors . In the same fashion, the inverse of the map from tangent to cotangent spaces, explicitly given by the inverse of η in matrix representation, can be used to define raising of an index . The components of this inverse are denoted η μν . It happens that η μν = η μν . These maps between a vector space and its dual can be denoted η ♭ (eta-flat) and η ♯ (eta-sharp) by the musical analogy. [ 18 ]
Contravariant and covariant vectors are geometrically very different objects. The first can and should be thought of as arrows. A linear function can be characterized by two objects: its kernel , which is a hyperplane passing through the origin, and its norm. Geometrically thus, covariant vectors should be viewed as a set of hyperplanes, with spacing depending on the norm (bigger = smaller spacing), with one of them (the kernel) passing through the origin. The mathematical term for a covariant vector is 1-covector or 1-form (though the latter is usually reserved for covector fields ).
One quantum mechanical analogy explored in the literature is that of a de Broglie wave (scaled by a factor of Planck's reduced constant) associated with a momentum four-vector to illustrate how one could imagine a covariant version of a contravariant vector. The inner product of two contravariant vectors could equally well be thought of as the action of the covariant version of one of them on the contravariant version of the other. The inner product is then how many times the arrow pierces the planes. [ 16 ] The mathematical reference, Lee (2003) , offers the same geometrical view of these objects (but mentions no piercing).
The electromagnetic field tensor is a differential 2-form , which geometrical description can as well be found in MTW.
One may, of course, ignore geometrical views altogether (as is the style in e.g. Weinberg (2002) and Landau & Lifshitz 2002 ) and proceed algebraically in a purely formal fashion. The time-proven robustness of the formalism itself, sometimes referred to as index gymnastics , ensures that moving vectors around and changing from contravariant to covariant vectors and vice versa (as well as higher order tensors) is mathematically sound. Incorrect expressions tend to reveal themselves quickly.
Given a bilinear form η : M × M → R {\displaystyle \eta :M\times M\rightarrow \mathbf {R} } , the lowered version of a vector can be thought of as the partial evaluation of η {\displaystyle \eta } , that is, there is an associated partial evaluation map η ( ⋅ , − ) : M → M ∗ ; v ↦ η ( v , ⋅ ) . {\displaystyle \eta (\cdot ,-):M\rightarrow M^{*};v\mapsto \eta (v,\cdot ).}
The lowered vector η ( v , ⋅ ) ∈ M ∗ {\displaystyle \eta (v,\cdot )\in M^{*}} is then the dual map u ↦ η ( v , u ) {\displaystyle u\mapsto \eta (v,u)} . Note it does not matter which argument is partially evaluated due to the symmetry of η {\displaystyle \eta } .
Non-degeneracy is then equivalent to injectivity of the partial evaluation map, or equivalently non-degeneracy indicates that the kernel of the map is trivial. In finite dimension, as is the case here, and noting that the dimension of a finite-dimensional space is equal to the dimension of the dual, this is enough to conclude the partial evaluation map is a linear isomorphism from M {\displaystyle M} to M ∗ {\displaystyle M^{*}} . This then allows the definition of the inverse partial evaluation map, η − 1 : M ∗ → M , {\displaystyle \eta ^{-1}:M^{*}\rightarrow M,} which allows the inverse metric to be defined as η − 1 : M ∗ × M ∗ → R , η − 1 ( α , β ) = η ( η − 1 ( α ) , η − 1 ( β ) ) {\displaystyle \eta ^{-1}:M^{*}\times M^{*}\rightarrow \mathbf {R} ,\eta ^{-1}(\alpha ,\beta )=\eta (\eta ^{-1}(\alpha ),\eta ^{-1}(\beta ))} where the two different usages of η − 1 {\displaystyle \eta ^{-1}} can be told apart by the argument each is evaluated on. This can then be used to raise indices. If a coordinate basis is used, the metric η −1 is indeed the matrix inverse to η .
The present purpose is to show semi-rigorously how formally one may apply the Minkowski metric to two vectors and obtain a real number, i.e. to display the role of the differentials and how they disappear in a calculation. The setting is that of smooth manifold theory, and concepts such as convector fields and exterior derivatives are introduced.
A full-blown version of the Minkowski metric in coordinates as a tensor field on spacetime has the appearance η μ ν d x μ ⊗ d x ν = η μ ν d x μ ⊙ d x ν = η μ ν d x μ d x ν . {\displaystyle \eta _{\mu \nu }dx^{\mu }\otimes dx^{\nu }=\eta _{\mu \nu }dx^{\mu }\odot dx^{\nu }=\eta _{\mu \nu }dx^{\mu }dx^{\nu }.}
Explanation: The coordinate differentials are 1-form fields. They are defined as the exterior derivative of the coordinate functions x μ . These quantities evaluated at a point p provide a basis for the cotangent space at p . The tensor product (denoted by the symbol ⊗ ) yields a tensor field of type (0, 2) , i.e. the type that expects two contravariant vectors as arguments. On the right-hand side, the symmetric product (denoted by the symbol ⊙ or by juxtaposition) has been taken. The equality holds since, by definition, the Minkowski metric is symmetric. [ 19 ] The notation on the far right is also sometimes used for the related, but different, line element . It is not a tensor. For elaboration on the differences and similarities, see Misner, Thorne & Wheeler (1973 , Box 3.2 and section 13.2.)
Tangent vectors are, in this formalism, given in terms of a basis of differential operators of the first order, ∂ ∂ x μ | p , {\displaystyle \left.{\frac {\partial }{\partial x^{\mu }}}\right|_{p},} where p is an event. This operator applied to a function f gives the directional derivative of f at p in the direction of increasing x μ with x ν , ν ≠ μ fixed. They provide a basis for the tangent space at p .
The exterior derivative df of a function f is a covector field , i.e. an assignment of a cotangent vector to each point p , by definition such that d f ( X ) = X f , {\displaystyle df(X)=Xf,} for each vector field X . A vector field is an assignment of a tangent vector to each point p . In coordinates X can be expanded at each point p in the basis given by the ∂/∂ x ν | p . Applying this with f = x μ , the coordinate function itself, and X = ∂/∂ x ν , called a coordinate vector field , one obtains d x μ ( ∂ ∂ x ν ) = ∂ x μ ∂ x ν = δ ν μ . {\displaystyle dx^{\mu }\left({\frac {\partial }{\partial x^{\nu }}}\right)={\frac {\partial x^{\mu }}{\partial x^{\nu }}}=\delta _{\nu }^{\mu }.}
Since this relation holds at each point p , the dx μ | p provide a basis for the cotangent space at each p and the bases dx μ | p and ∂/∂ x ν | p are dual to each other, d x μ | p ( ∂ ∂ x ν | p ) = δ ν μ . {\displaystyle \left.dx^{\mu }\right|_{p}\left(\left.{\frac {\partial }{\partial x^{\nu }}}\right|_{p}\right)=\delta _{\nu }^{\mu }.} at each p . Furthermore, one has α ⊗ β ( a , b ) = α ( a ) β ( b ) {\displaystyle \alpha \otimes \beta (a,b)=\alpha (a)\beta (b)} for general one-forms on a tangent space α , β and general tangent vectors a , b . (This can be taken as a definition, but may also be proved in a more general setting.)
Thus when the metric tensor is fed two vectors fields a , b , both expanded in terms of the basis coordinate vector fields, the result is η μ ν d x μ ⊗ d x ν ( a , b ) = η μ ν a μ b ν , {\displaystyle \eta _{\mu \nu }dx^{\mu }\otimes dx^{\nu }(a,b)=\eta _{\mu \nu }a^{\mu }b^{\nu },} where a μ , b ν are the component functions of the vector fields. The above equation holds at each point p , and the relation may as well be interpreted as the Minkowski metric at p applied to two tangent vectors at p .
As mentioned, in a vector space, such as modeling the spacetime of special relativity, tangent vectors can be canonically identified with vectors in the space itself, and vice versa. This means that the tangent spaces at each point are canonically identified with each other and with the vector space itself. This explains how the right-hand side of the above equation can be employed directly, without regard to the spacetime point the metric is to be evaluated and from where (which tangent space) the vectors come from.
This situation changes in general relativity . There one has g ( p ) μ ν d x μ | p d x ν | p ( a , b ) = g ( p ) μ ν a μ b ν , {\displaystyle g(p)_{\mu \nu }\left.dx^{\mu }\right|_{p}\left.dx^{\nu }\right|_{p}(a,b)=g(p)_{\mu \nu }a^{\mu }b^{\nu },} where now η → g ( p ) , i.e., g is still a metric tensor but now depending on spacetime and is a solution of Einstein's field equations . Moreover, a , b must be tangent vectors at spacetime point p and can no longer be moved around freely.
Let x , y ∈ M . Here,
Suppose x ∈ M is timelike. Then the simultaneous hyperplane for x is { y : η ( x , y ) = 0} . Since this hyperplane varies as x varies, there is a relativity of simultaneity in Minkowski space.
A Lorentzian manifold is a generalization of Minkowski space in two ways. The total number of spacetime dimensions is not restricted to be 4 ( 2 or more) and a Lorentzian manifold need not be flat, i.e. it allows for curvature.
Complexified Minkowski space is defined as M c = M ⊕ iM . [ 20 ] Its real part is the Minkowski space of four-vectors , such as the four-velocity and the four-momentum , which are independent of the choice of orientation of the space. The imaginary part, on the other hand, may consist of four pseudovectors, such as angular velocity and magnetic moment , which change their direction with a change of orientation. A pseudoscalar i is introduced, which also changes sign with a change of orientation. Thus, elements of M c are independent of the choice of the orientation.
The inner product -like structure on M c is defined as u ⋅ v = η ( u , v ) for any u , v ∈ M c . A relativistic pure spin of an electron or any half spin particle is described by ρ ∈ M c as ρ = u + is , where u is the four-velocity of the particle, satisfying u 2 = 1 and s is the 4D spin vector, [ 21 ] which is also the Pauli–Lubanski pseudovector satisfying s 2 = −1 and u ⋅ s = 0 .
Minkowski space refers to a mathematical formulation in four dimensions. However, the mathematics can easily be extended or simplified to create an analogous generalized Minkowski space in any number of dimensions. If n ≥ 2 , n -dimensional Minkowski space is a vector space of real dimension n on which there is a constant Minkowski metric of signature ( n − 1, 1) or (1, n − 1) . These generalizations are used in theories where spacetime is assumed to have more or less than 4 dimensions. String theory and M-theory are two examples where n > 4 . In string theory, there appears conformal field theories with 1 + 1 spacetime dimensions.
de Sitter space can be formulated as a submanifold of generalized Minkowski space as can the model spaces of hyperbolic geometry (see below).
As a flat spacetime , the three spatial components of Minkowski spacetime always obey the Pythagorean Theorem . Minkowski space is a suitable basis for special relativity, a good description of physical systems over finite distances in systems without significant gravitation . However, in order to take gravity into account, physicists use the theory of general relativity , which is formulated in the mathematics of differential geometry of differential manifolds . When this geometry is used as a model of spacetime, it is known as curved spacetime .
Even in curved spacetime, Minkowski space is still a good description in an infinitesimal region surrounding any point (barring gravitational singularities). [ nb 5 ] More abstractly, it can be said that in the presence of gravity spacetime is described by a curved 4-dimensional manifold for which the tangent space to any point is a 4-dimensional Minkowski space. Thus, the structure of Minkowski space is still essential in the description of general relativity.
The meaning of the term geometry for the Minkowski space depends heavily on the context. Minkowski space is not endowed with Euclidean geometry, and not with any of the generalized Riemannian geometries with intrinsic curvature, those exposed by the model spaces in hyperbolic geometry (negative curvature) and the geometry modeled by the sphere (positive curvature). The reason is the indefiniteness of the Minkowski metric. Minkowski space is, in particular, not a metric space and not a Riemannian manifold with a Riemannian metric. However, Minkowski space contains submanifolds endowed with a Riemannian metric yielding hyperbolic geometry.
Model spaces of hyperbolic geometry of low dimension, say 2 or 3, cannot be isometrically embedded in Euclidean space with one more dimension, i.e. R 3 {\displaystyle \mathbf {R} ^{3}} or R 4 {\displaystyle \mathbf {R} ^{4}} respectively, with the Euclidean metric g ¯ {\displaystyle {\overline {g}}} , preventing easy visualization. [ nb 6 ] [ 22 ] By comparison, model spaces with positive curvature are just spheres in Euclidean space of one higher dimension. [ 23 ] Hyperbolic spaces can be isometrically embedded in spaces of one more dimension when the embedding space is endowed with the Minkowski metric η {\displaystyle \eta } .
Define H R 1 ( n ) ⊂ M n + 1 {\displaystyle \mathbf {H} _{R}^{1(n)}\subset \mathbf {M} ^{n+1}} to be the upper sheet ( c t > 0 {\displaystyle ct>0} ) of the hyperboloid H R 1 ( n ) = { ( c t , x 1 , … , x n ) ∈ M n : c 2 t 2 − ( x 1 ) 2 − ⋯ − ( x n ) 2 = R 2 , c t > 0 } {\displaystyle \mathbf {H} _{R}^{1(n)}=\left\{\left(ct,x^{1},\ldots ,x^{n}\right)\in \mathbf {M} ^{n}:c^{2}t^{2}-\left(x^{1}\right)^{2}-\cdots -\left(x^{n}\right)^{2}=R^{2},ct>0\right\}} in generalized Minkowski space M n + 1 {\displaystyle \mathbf {M} ^{n+1}} of spacetime dimension n + 1. {\displaystyle n+1.} This is one of the surfaces of transitivity of the generalized Lorentz group. The induced metric on this submanifold, h R 1 ( n ) = ι ∗ η , {\displaystyle h_{R}^{1(n)}=\iota ^{*}\eta ,} the pullback of the Minkowski metric η {\displaystyle \eta } under inclusion, is a Riemannian metric . With this metric H R 1 ( n ) {\displaystyle \mathbf {H} _{R}^{1(n)}} is a Riemannian manifold . It is one of the model spaces of Riemannian geometry, the hyperboloid model of hyperbolic space . It is a space of constant negative curvature − 1 / R 2 {\displaystyle -1/R^{2}} . [ 24 ] The 1 in the upper index refers to an enumeration of the different model spaces of hyperbolic geometry, and the n for its dimension. A 2 ( 2 ) {\displaystyle 2(2)} corresponds to the Poincaré disk model , while 3 ( n ) {\displaystyle 3(n)} corresponds to the Poincaré half-space model of dimension n . {\displaystyle n.}
In the definition above ι : H R 1 ( n ) → M n + 1 {\displaystyle \iota :\mathbf {H} _{R}^{1(n)}\rightarrow \mathbf {M} ^{n+1}} is the inclusion map and the superscript star denotes the pullback . The present purpose is to describe this and similar operations as a preparation for the actual demonstration that H R 1 ( n ) {\displaystyle \mathbf {H} _{R}^{1(n)}} actually is a hyperbolic space.
Behavior of tensors under inclusion: For inclusion maps from a submanifold S into M and a covariant tensor α of order k on M it holds that ι ∗ α ( X 1 , X 2 , … , X k ) = α ( ι ∗ X 1 , ι ∗ X 2 , … , ι ∗ X k ) = α ( X 1 , X 2 , … , X k ) , {\displaystyle \iota ^{*}\alpha \left(X_{1},\,X_{2},\,\ldots ,\,X_{k}\right)=\alpha \left(\iota _{*}X_{1},\,\iota _{*}X_{2},\,\ldots ,\,\iota _{*}X_{k}\right)=\alpha \left(X_{1},\,X_{2},\,\ldots ,\,X_{k}\right),} where X 1 , X 1 , …, X k are vector fields on S . The subscript star denotes the pushforward (to be introduced later), and it is in this special case simply the identity map (as is the inclusion map). The latter equality holds because a tangent space to a submanifold at a point is in a canonical way a subspace of the tangent space of the manifold itself at the point in question. One may simply write ι ∗ α = α | S , {\displaystyle \iota ^{*}\alpha =\alpha |_{S},} meaning (with slight abuse of notation ) the restriction of α to accept as input vectors tangent to some s ∈ S only.
Pullback of tensors under general maps: The pullback of a covariant k -tensor α (one taking only contravariant vectors as arguments) under a map F : M → N is a linear map F ∗ : T F ( p ) k N → T p k M , {\displaystyle F^{*}\colon T_{F(p)}^{k}N\rightarrow T_{p}^{k}M,} where for any vector space V , T k V = V ∗ ⊗ V ∗ ⊗ ⋯ ⊗ V ∗ ⏟ k times . {\displaystyle T^{k}V=\underbrace {V^{*}\otimes V^{*}\otimes \cdots \otimes V^{*}} _{k{\text{ times}}}.}
It is defined by F ∗ ( α ) ( X 1 , X 2 , … , X k ) = α ( F ∗ X 1 , F ∗ X 2 , … , F ∗ X k ) , {\displaystyle F^{*}(\alpha )\left(X_{1},\,X_{2},\,\ldots ,\,X_{k}\right)=\alpha \left(F_{*}X_{1},\,F_{*}X_{2},\,\ldots ,\,F_{*}X_{k}\right),} where the subscript star denotes the pushforward of the map F , and X 1 , X 2 , …, X k are vectors in T p M . (This is in accord with what was detailed about the pullback of the inclusion map. In the general case here, one cannot proceed as simply because F ∗ X 1 ≠ X 1 in general.)
The pushforward of vectors under general maps: Heuristically, pulling back a tensor to p ∈ M from F ( p ) ∈ N feeding it vectors residing at p ∈ M is by definition the same as pushing forward the vectors from p ∈ M to F ( p ) ∈ N feeding them to the tensor residing at F ( p ) ∈ N .
Further unwinding the definitions, the pushforward F ∗ : TM p → TN F ( p ) of a vector field under a map F : M → N between manifolds is defined by F ∗ ( X ) f = X ( f ∘ F ) , {\displaystyle F_{*}(X)f=X(f\circ F),} where f is a function on N . When M = R m , N = R n the pushforward of F reduces to DF : R m → R n , the ordinary differential , which is given by the Jacobian matrix of partial derivatives of the component functions. The differential is the best linear approximation of a function F from R m to R n . The pushforward is the smooth manifold version of this. It acts between tangent spaces, and is in coordinates represented by the Jacobian matrix of the coordinate representation of the function.
The corresponding pullback is the dual map from the dual of the range tangent space to the dual of the domain tangent space, i.e. it is a linear map, F ∗ : T F ( p ) ∗ N → T p ∗ M . {\displaystyle F^{*}\colon T_{F(p)}^{*}N\rightarrow T_{p}^{*}M.}
In order to exhibit the metric, it is necessary to pull it back via a suitable parametrization . A parametrization of a submanifold S of a manifold M is a map U ⊂ R m → M whose range is an open subset of S . If S has the same dimension as M , a parametrization is just the inverse of a coordinate map φ : M → U ⊂ R m . The parametrization to be used is the inverse of hyperbolic stereographic projection . This is illustrated in the figure to the right for n = 2 . It is instructive to compare to stereographic projection for spheres.
Stereographic projection σ : H n R → R n and its inverse σ −1 : R n → H n R are given by σ ( τ , x ) = u = R x R + τ , σ − 1 ( u ) = ( τ , x ) = ( R R 2 + | u | 2 R 2 − | u | 2 , 2 R 2 u R 2 − | u | 2 ) , {\displaystyle {\begin{aligned}\sigma (\tau ,\mathbf {x} )=\mathbf {u} &={\frac {R\mathbf {x} }{R+\tau }},\\\sigma ^{-1}(\mathbf {u} )=(\tau ,\mathbf {x} )&=\left(R{\frac {R^{2}+|u|^{2}}{R^{2}-|u|^{2}}},{\frac {2R^{2}\mathbf {u} }{R^{2}-|u|^{2}}}\right),\end{aligned}}} where, for simplicity, τ ≡ ct . The ( τ , x ) are coordinates on M n +1 and the u are coordinates on R n .
Let H R n = { ( τ , x 1 , … , x n ) ⊂ M : − τ 2 + ( x 1 ) 2 + ⋯ + ( x n ) 2 = − R 2 , τ > 0 } {\displaystyle \mathbf {H} _{R}^{n}=\left\{\left(\tau ,x^{1},\ldots ,x^{n}\right)\subset \mathbf {M} :-\tau ^{2}+\left(x^{1}\right)^{2}+\cdots +\left(x^{n}\right)^{2}=-R^{2},\tau >0\right\}} and let S = ( − R , 0 , … , 0 ) . {\displaystyle S=(-R,0,\ldots ,0).}
If P = ( τ , x 1 , … , x n ) ∈ H R n , {\displaystyle P=\left(\tau ,x^{1},\ldots ,x^{n}\right)\in \mathbf {H} _{R}^{n},} then it is geometrically clear that the vector P S → {\displaystyle {\overrightarrow {PS}}} intersects the hyperplane { ( τ , x 1 , … , x n ) ∈ M : τ = 0 } {\displaystyle \left\{\left(\tau ,x^{1},\ldots ,x^{n}\right)\in M:\tau =0\right\}} once in point denoted U = ( 0 , u 1 ( P ) , … , u n ( P ) ) ≡ ( 0 , u ) . {\displaystyle U=\left(0,u^{1}(P),\ldots ,u^{n}(P)\right)\equiv (0,\mathbf {u} ).}
One has S + S U → = U ⇒ S U → = U − S , S + S P → = P ⇒ S P → = P − S {\displaystyle {\begin{aligned}S+{\overrightarrow {SU}}&=U\Rightarrow {\overrightarrow {SU}}=U-S,\\S+{\overrightarrow {SP}}&=P\Rightarrow {\overrightarrow {SP}}=P-S\end{aligned}}} or S U → = ( 0 , u ) − ( − R , 0 ) = ( R , u ) , S P → = ( τ , x ) − ( − R , 0 ) = ( τ + R , x ) . . {\displaystyle {\begin{aligned}{\overrightarrow {SU}}&=(0,\mathbf {u} )-(-R,\mathbf {0} )=(R,\mathbf {u} ),\\{\overrightarrow {SP}}&=(\tau ,\mathbf {x} )-(-R,\mathbf {0} )=(\tau +R,\mathbf {x} ).\end{aligned}}.}
By construction of stereographic projection one has S U → = λ ( τ ) S P → . {\displaystyle {\overrightarrow {SU}}=\lambda (\tau ){\overrightarrow {SP}}.}
This leads to the system of equations R = λ ( τ + R ) , u = λ x . {\displaystyle {\begin{aligned}R&=\lambda (\tau +R),\\\mathbf {u} &=\lambda \mathbf {x} .\end{aligned}}}
The first of these is solved for λ and one obtains for stereographic projection σ ( τ , x ) = u = R x R + τ . {\displaystyle \sigma (\tau ,\mathbf {x} )=\mathbf {u} ={\frac {R\mathbf {x} }{R+\tau }}.}
Next, the inverse σ −1 ( u ) = ( τ , x ) must be calculated. Use the same considerations as before, but now with U = ( 0 , u ) P = ( τ ( u ) , x ( u ) ) . , {\displaystyle {\begin{aligned}U&=(0,\mathbf {u} )\\P&=(\tau (\mathbf {u} ),\mathbf {x} (\mathbf {u} )).\end{aligned}},} one gets τ = R ( 1 − λ ) λ , x = u λ , {\displaystyle {\begin{aligned}\tau &={\frac {R(1-\lambda )}{\lambda }},\\\mathbf {x} &={\frac {\mathbf {u} }{\lambda }},\end{aligned}}} but now with λ depending on u . The condition for P lying in the hyperboloid is − τ 2 + | x | 2 = − R 2 , {\displaystyle -\tau ^{2}+|\mathbf {x} |^{2}=-R^{2},} or − R 2 ( 1 − λ ) 2 λ 2 + | u | 2 λ 2 = − R 2 , {\displaystyle -{\frac {R^{2}(1-\lambda )^{2}}{\lambda ^{2}}}+{\frac {|\mathbf {u} |^{2}}{\lambda ^{2}}}=-R^{2},} leading to λ = R 2 − | u | 2 2 R 2 . {\displaystyle \lambda ={\frac {R^{2}-|u|^{2}}{2R^{2}}}.}
With this λ , one obtains σ − 1 ( u ) = ( τ , x ) = ( R R 2 + | u | 2 R 2 − | u | 2 , 2 R 2 u R 2 − | u | 2 ) . {\displaystyle \sigma ^{-1}(\mathbf {u} )=(\tau ,\mathbf {x} )=\left(R{\frac {R^{2}+|u|^{2}}{R^{2}-|u|^{2}}},{\frac {2R^{2}\mathbf {u} }{R^{2}-|u|^{2}}}\right).}
One has h R 1 ( n ) = η | H R 1 ( n ) = ( d x 1 ) 2 + ⋯ + ( d x n ) 2 − d τ 2 {\displaystyle h_{R}^{1(n)}=\eta |_{\mathbf {H} _{R}^{1(n)}}=\left(dx^{1}\right)^{2}+\cdots +\left(dx^{n}\right)^{2}-d\tau ^{2}} and the map σ − 1 : R n → H R 1 ( n ) ; σ − 1 ( u ) = ( τ ( u ) , x ( u ) ) = ( R R 2 + | u | 2 R 2 − | u | 2 , 2 R 2 u R 2 − | u | 2 ) . {\displaystyle \sigma ^{-1}:\mathbf {R} ^{n}\rightarrow \mathbf {H} _{R}^{1(n)};\quad \sigma ^{-1}(\mathbf {u} )=(\tau (\mathbf {u} ),\,\mathbf {x} (\mathbf {u} ))=\left(R{\frac {R^{2}+|u|^{2}}{R^{2}-|u|^{2}}},\,{\frac {2R^{2}\mathbf {u} }{R^{2}-|u|^{2}}}\right).}
The pulled back metric can be obtained by straightforward methods of calculus; ( σ − 1 ) ∗ η | H R 1 ( n ) = ( d x 1 ( u ) ) 2 + ⋯ + ( d x n ( u ) ) 2 − ( d τ ( u ) ) 2 . {\displaystyle \left.\left(\sigma ^{-1}\right)^{*}\eta \right|_{\mathbf {H} _{R}^{1(n)}}=\left(dx^{1}(\mathbf {u} )\right)^{2}+\cdots +\left(dx^{n}(\mathbf {u} )\right)^{2}-\left(d\tau (\mathbf {u} )\right)^{2}.}
One computes according to the standard rules for computing differentials (though one is really computing the rigorously defined exterior derivatives), d x 1 ( u ) = d ( 2 R 2 u 1 R 2 − | u | 2 ) = ∂ ∂ u 1 2 R 2 u 1 R 2 − | u | 2 d u 1 + ⋯ + ∂ ∂ u n 2 R 2 u 1 R 2 − | u | 2 d u n + ∂ ∂ τ 2 R 2 u 1 R 2 − | u | 2 d τ , ⋮ d x n ( u ) = d ( 2 R 2 u n R 2 − | u | 2 ) = ⋯ , d τ ( u ) = d ( R R 2 + | u | 2 R 2 − | u | 2 ) = ⋯ , {\displaystyle {\begin{aligned}dx^{1}(\mathbf {u} )&=d\left({\frac {2R^{2}u^{1}}{R^{2}-|u|^{2}}}\right)={\frac {\partial }{\partial u^{1}}}{\frac {2R^{2}u^{1}}{R^{2}-|u|^{2}}}du^{1}+\cdots +{\frac {\partial }{\partial u^{n}}}{\frac {2R^{2}u^{1}}{R^{2}-|u|^{2}}}du^{n}+{\frac {\partial }{\partial \tau }}{\frac {2R^{2}u^{1}}{R^{2}-|u|^{2}}}d\tau ,\\&\ \ \vdots \\dx^{n}(\mathbf {u} )&=d\left({\frac {2R^{2}u^{n}}{R^{2}-|u|^{2}}}\right)=\cdots ,\\d\tau (\mathbf {u} )&=d\left(R{\frac {R^{2}+|u|^{2}}{R^{2}-|u|^{2}}}\right)=\cdots ,\end{aligned}}} and substitutes the results into the right hand side. This yields ( σ − 1 ) ∗ h R 1 ( n ) = 4 R 2 [ ( d u 1 ) 2 + ⋯ + ( d u n ) 2 ] ( R 2 − | u | 2 ) 2 ≡ h R 2 ( n ) . {\displaystyle \left(\sigma ^{-1}\right)^{*}h_{R}^{1(n)}={\frac {4R^{2}\left[\left(du^{1}\right)^{2}+\cdots +\left(du^{n}\right)^{2}\right]}{\left(R^{2}-|u|^{2}\right)^{2}}}\equiv h_{R}^{2(n)}.}
One has ∂ ∂ u 1 2 R 2 u 1 R 2 − | u | 2 d u 1 = 2 ( R 2 − | u | 2 ) + 4 R 2 ( u 1 ) 2 ( R 2 − | u | 2 ) 2 d u 1 , ∂ ∂ u 2 2 R 2 u 1 R 2 − | u | 2 d u 2 = 4 R 2 u 1 u 2 ( R 2 − | u | 2 ) 2 d u 2 , {\displaystyle {\begin{aligned}{\frac {\partial }{\partial u^{1}}}{\frac {2R^{2}u^{1}}{R^{2}-|u|^{2}}}du^{1}&={\frac {2\left(R^{2}-|u|^{2}\right)+4R^{2}\left(u^{1}\right)^{2}}{\left(R^{2}-|u|^{2}\right)^{2}}}du^{1},\\{\frac {\partial }{\partial u^{2}}}{\frac {2R^{2}u^{1}}{R^{2}-|u|^{2}}}du^{2}&={\frac {4R^{2}u^{1}u^{2}}{\left(R^{2}-|u|^{2}\right)^{2}}}du^{2},\end{aligned}}} and ∂ ∂ τ 2 R 2 u 1 R 2 − | u | 2 d τ 2 = 0. {\displaystyle {\frac {\partial }{\partial \tau }}{\frac {2R^{2}u^{1}}{R^{2}-|u|^{2}}}d\tau ^{2}=0.}
With this one may write d x 1 ( u ) = 2 R 2 ( R 2 − | u | 2 ) d u 1 + 4 R 2 u 1 ( u ⋅ d u ) ( R 2 − | u | 2 ) 2 , {\displaystyle dx^{1}(\mathbf {u} )={\frac {2R^{2}\left(R^{2}-|u|^{2}\right)du^{1}+4R^{2}u^{1}(\mathbf {u} \cdot d\mathbf {u} )}{\left(R^{2}-|u|^{2}\right)^{2}}},} from which ( d x 1 ( u ) ) 2 = 4 R 2 ( r 2 − | u | 2 ) 2 ( d u 1 ) 2 + 16 R 4 ( R 2 − | u | 2 ) ( u ⋅ d u ) u 1 d u 1 + 16 R 4 ( u 1 ) 2 ( u ⋅ d u ) 2 ( R 2 − | u | 2 ) 4 . {\displaystyle \left(dx^{1}(\mathbf {u} )\right)^{2}={\frac {4R^{2}\left(r^{2}-|u|^{2}\right)^{2}\left(du^{1}\right)^{2}+16R^{4}\left(R^{2}-|u|^{2}\right)\left(\mathbf {u} \cdot d\mathbf {u} \right)u^{1}du^{1}+16R^{4}\left(u^{1}\right)^{2}\left(\mathbf {u} \cdot d\mathbf {u} \right)^{2}}{\left(R^{2}-|u|^{2}\right)^{4}}}.}
Summing this formula one obtains ( d x 1 ( u ) ) 2 + ⋯ + ( d x n ( u ) ) 2 = 4 R 2 ( R 2 − | u | 2 ) 2 [ ( d u 1 ) 2 + ⋯ + ( d u n ) 2 ] + 16 R 4 ( R 2 − | u | 2 ) ( u ⋅ d u ) ( u ⋅ d u ) + 16 R 4 | u | 2 ( u ⋅ d u ) 2 ( R 2 − | u | 2 ) 4 = 4 R 2 ( R 2 − | u | 2 ) 2 [ ( d u 1 ) 2 + ⋯ + ( d u n ) 2 ] ( R 2 − | u | 2 ) 4 + R 2 16 R 4 ( u ⋅ d u ) ( R 2 − | u | 2 ) 4 . {\displaystyle {\begin{aligned}&\left(dx^{1}(\mathbf {u} )\right)^{2}+\cdots +\left(dx^{n}(\mathbf {u} )\right)^{2}\\={}&{\frac {4R^{2}\left(R^{2}-|u|^{2}\right)^{2}\left[\left(du^{1}\right)^{2}+\cdots +\left(du^{n}\right)^{2}\right]+16R^{4}\left(R^{2}-|u|^{2}\right)(\mathbf {u} \cdot d\mathbf {u} )(\mathbf {u} \cdot d\mathbf {u} )+16R^{4}|u|^{2}(\mathbf {u} \cdot d\mathbf {u} )^{2}}{\left(R^{2}-|u|^{2}\right)^{4}}}\\={}&{\frac {4R^{2}\left(R^{2}-|u|^{2}\right)^{2}\left[\left(du^{1}\right)^{2}+\cdots +\left(du^{n}\right)^{2}\right]}{\left(R^{2}-|u|^{2}\right)^{4}}}+R^{2}{\frac {16R^{4}(\mathbf {u} \cdot d\mathbf {u} )}{\left(R^{2}-|u|^{2}\right)^{4}}}.\end{aligned}}}
Similarly, for τ one gets d τ = ∑ i = 1 n ∂ ∂ u i R R 2 + | u | 2 R 2 + | u | 2 d u i + ∂ ∂ τ R R 2 + | u | 2 R 2 + | u | 2 d τ = ∑ i = 1 n R 4 4 R 2 u i d u i ( R 2 − | u | 2 ) , {\displaystyle d\tau =\sum _{i=1}^{n}{\frac {\partial }{\partial u^{i}}}R{\frac {R^{2}+|u|^{2}}{R^{2}+|u|^{2}}}du^{i}+{\frac {\partial }{\partial \tau }}R{\frac {R^{2}+|u|^{2}}{R^{2}+|u|^{2}}}d\tau =\sum _{i=1}^{n}R^{4}{\frac {4R^{2}u^{i}du^{i}}{\left(R^{2}-|u|^{2}\right)}},} yielding − d τ 2 = − ( R 4 R 4 ( u ⋅ d u ) ( R 2 − | u | 2 ) 2 ) 2 = − R 2 16 R 4 ( u ⋅ d u ) 2 ( R 2 − | u | 2 ) 4 . {\displaystyle -d\tau ^{2}=-\left(R{\frac {4R^{4}\left(\mathbf {u} \cdot d\mathbf {u} \right)}{\left(R^{2}-|u|^{2}\right)^{2}}}\right)^{2}=-R^{2}{\frac {16R^{4}(\mathbf {u} \cdot d\mathbf {u} )^{2}}{\left(R^{2}-|u|^{2}\right)^{4}}}.}
Now add this contribution to finally get ( σ − 1 ) ∗ h R 1 ( n ) = 4 R 2 [ ( d u 1 ) 2 + ⋯ + ( d u n ) 2 ] ( R 2 − | u | 2 ) 2 ≡ h R 2 ( n ) . {\displaystyle \left(\sigma ^{-1}\right)^{*}h_{R}^{1(n)}={\frac {4R^{2}\left[\left(du^{1}\right)^{2}+\cdots +\left(du^{n}\right)^{2}\right]}{\left(R^{2}-|u|^{2}\right)^{2}}}\equiv h_{R}^{2(n)}.}
This last equation shows that the metric on the ball is identical to the Riemannian metric h 2( n ) R in the Poincaré ball model , another standard model of hyperbolic geometry.
The pullback can be computed in a different fashion. By definition, ( σ − 1 ) ∗ h R 1 ( n ) ( V , V ) = h R 1 ( n ) ( ( σ − 1 ) ∗ V , ( σ − 1 ) ∗ V ) = η | H R 1 ( n ) ( ( σ − 1 ) ∗ V , ( σ − 1 ) ∗ V ) . {\displaystyle \left(\sigma ^{-1}\right)^{*}h_{R}^{1(n)}(V,\,V)=h_{R}^{1(n)}\left(\left(\sigma ^{-1}\right)_{*}V,\,\left(\sigma ^{-1}\right)_{*}V\right)=\eta |_{\mathbf {H} _{R}^{1(n)}}\left(\left(\sigma ^{-1}\right)_{*}V,\,\left(\sigma ^{-1}\right)_{*}V\right).}
In coordinates, ( σ − 1 ) ∗ V = ( σ − 1 ) ∗ V i ∂ ∂ u i = V i ∂ x j ∂ u i ∂ ∂ x j + V i ∂ τ ∂ u i ∂ ∂ τ = V i ∂ x j ∂ u i ∂ ∂ x j + V i ∂ τ ∂ u i ∂ ∂ τ = V x j ∂ ∂ x j + V τ ∂ ∂ τ . {\displaystyle \left(\sigma ^{-1}\right)_{*}V=\left(\sigma ^{-1}\right)_{*}V^{i}{\frac {\partial }{\partial u^{i}}}=V^{i}{\frac {\partial x^{j}}{\partial u^{i}}}{\frac {\partial }{\partial x^{j}}}+V^{i}{\frac {\partial \tau }{\partial u^{i}}}{\frac {\partial }{\partial \tau }}=V^{i}{\frac {\partial }{x}}^{j}{\partial u^{i}}{\frac {\partial }{\partial x^{j}}}+V^{i}{\frac {\partial }{\tau }}{\partial u^{i}}{\frac {\partial }{\partial \tau }}=Vx^{j}{\frac {\partial }{\partial x^{j}}}+V\tau {\frac {\partial }{\partial \tau }}.}
One has from the formula for σ –1 V x j = V i ∂ ∂ u i ( 2 R 2 u j R 2 − | u | 2 ) = 2 R 2 V j R 2 − | u | 2 − 4 R 2 u j ⟨ V , u ⟩ ( R 2 − | u | 2 ) 2 , ( here V | u | 2 = 2 ∑ k = 1 n V k u k ≡ 2 ⟨ V , u ⟩ ) V τ = V ( R R 2 + | u | 2 R 2 − | u | 2 ) = 4 R 3 ⟨ V , u ⟩ ( R 2 − | u | 2 ) 2 . {\displaystyle {\begin{aligned}Vx^{j}&=V^{i}{\frac {\partial }{\partial u^{i}}}\left({\frac {2R^{2}u^{j}}{R^{2}-|u|^{2}}}\right)={\frac {2R^{2}V^{j}}{R^{2}-|u|^{2}}}-{\frac {4R^{2}u^{j}\langle \mathbf {V} ,\,\mathbf {u} \rangle }{\left(R^{2}-|u|^{2}\right)^{2}}},\quad \left({\text{here }}V|u|^{2}=2\sum _{k=1}^{n}V^{k}u^{k}\equiv 2\langle \mathbf {V} ,\,\mathbf {u} \rangle \right)\\V\tau &=V\left(R{\frac {R^{2}+|u|^{2}}{R^{2}-|u|^{2}}}\right)={\frac {4R^{3}\langle \mathbf {V} ,\,\mathbf {u} \rangle }{\left(R^{2}-|u|^{2}\right)^{2}}}.\end{aligned}}}
Lastly, η ( σ ∗ − 1 V , σ ∗ − 1 V ) = ∑ j = 1 n ( V x j ) 2 − ( V τ ) 2 = 4 R 4 | V | 2 ( R 2 − | u | 2 ) 2 = h R 2 ( n ) ( V , z , V ) , {\displaystyle \eta \left(\sigma _{*}^{-1}V,\,\sigma _{*}^{-1}V\right)=\sum _{j=1}^{n}\left(Vx^{j}\right)^{2}-(V\tau )^{2}={\frac {4R^{4}|V|^{2}}{\left(R^{2}-|u|^{2}\right)^{2}}}=h_{R}^{2(n)}(V,z,V),} and the same conclusion is reached.
Media related to Minkowski diagrams at Wikimedia Commons | https://en.wikipedia.org/wiki/Minkowski_space |
In mathematics , the Minkowski–Hlawka theorem is a result on the lattice packing of hyperspheres in dimension n > 1. It states that there is a lattice in Euclidean space of dimension n , such that the corresponding best packing of hyperspheres with centres at the lattice points has density Δ satisfying
with ζ the Riemann zeta function . Here as n → ∞, ζ( n ) → 1. The proof of this theorem is indirect and does not give an explicit example, however, and there is still no known simple and explicit way to construct lattices with packing densities exceeding this bound for arbitrary n . In principle one can find explicit examples: for example, even just picking a few "random" lattices will work with high probability. The problem is that testing these lattices to see if they are solutions requires finding their shortest vectors, and the number of cases to check grows very fast with the dimension, so this could take a very long time.
This result was stated without proof by Hermann Minkowski ( 1911 , pages 265–276) and proved by Edmund Hlawka ( 1943 ). The result is related to a linear lower bound for the Hermite constant .
Siegel (1945) proved the following generalization of the Minkowski–Hlawka theorem. If S is a bounded set in R n with Jordan volume vol( S ) then the average number of nonzero lattice vectors in S is vol( S )/ D , where the average is taken over all lattices with a fundamental domain of volume D , and similarly the average number of primitive lattice vectors in S is vol( S )/ D ζ( n ).
The Minkowski–Hlawka theorem follows easily from this, using the fact that if S is a star-shaped centrally symmetric body (such as a ball) containing less than 2 primitive lattice vectors then it contains no nonzero lattice vectors. | https://en.wikipedia.org/wiki/Minkowski–Hlawka_theorem |
In mathematics , the Minkowski–Steiner formula is a formula relating the surface area and volume of compact subsets of Euclidean space . More precisely, it defines the surface area as the "derivative" of enclosed volume in an appropriate sense.
The Minkowski–Steiner formula is used, together with the Brunn–Minkowski theorem , to prove the isoperimetric inequality . It is named after Hermann Minkowski and Jakob Steiner .
Let n ≥ 2 {\displaystyle n\geq 2} , and let A ⊊ R n {\displaystyle A\subsetneq \mathbb {R} ^{n}} be a compact set. Let μ ( A ) {\displaystyle \mu (A)} denote the Lebesgue measure (volume) of A {\displaystyle A} . Define the quantity λ ( ∂ A ) {\displaystyle \lambda (\partial A)} by the Minkowski–Steiner formula
where
denotes the closed ball of radius δ > 0 {\displaystyle \delta >0} , and
is the Minkowski sum of A {\displaystyle A} and B δ ¯ {\displaystyle {\overline {B_{\delta }}}} , so that
For "sufficiently regular" sets A {\displaystyle A} , the quantity λ ( ∂ A ) {\displaystyle \lambda (\partial A)} does indeed correspond with the ( n − 1 ) {\displaystyle (n-1)} -dimensional measure of the boundary ∂ A {\displaystyle \partial A} of A {\displaystyle A} . See Federer (1969) for a full treatment of this problem.
When the set A {\displaystyle A} is a convex set , the lim-inf above is a true limit , and one can show that
where the λ i {\displaystyle \lambda _{i}} are some continuous functions of A {\displaystyle A} (see quermassintegrals ) and ω n {\displaystyle \omega _{n}} denotes the measure (volume) of the unit ball in R n {\displaystyle \mathbb {R} ^{n}} :
where Γ {\displaystyle \Gamma } denotes the Gamma function .
Taking A = B R ¯ {\displaystyle A={\overline {B_{R}}}} gives the following well-known formula for the surface area of the sphere of radius R {\displaystyle R} , S R := ∂ B R {\displaystyle S_{R}:=\partial B_{R}} :
where ω n {\displaystyle \omega _{n}} is as above. | https://en.wikipedia.org/wiki/Minkowski–Steiner_formula |
The Minlos–Sasonov theorem is a result from measure theory in topological vector spaces . It provides a sufficient condition for a cylindrical measure to be σ-additive on a locally convex space . This is the case when its Fourier transform is continuous at zero in the Sazonov topology and such a topology is called sufficient . The theorem is named after the two Russian mathematicians Robert Adol'fovich Minlos and Vyacheslav Vasilievich Sazonov .
The theorem generalizes two classical theorem: the Minlos theorem (1963) and the Sazonov theorem (1958). It was then later generalized in the 1970s by the mathematicians Albert Badrikian and Laurent Schwartz to locally convex spaces. Therefore, the theorem is sometimes also called Minlos-Sasonov-Badrikian theorem . [ 1 ] [ 2 ]
Let ( X , τ ) {\displaystyle (X,\tau )} be a locally convex space , X ∗ {\displaystyle X^{*}} and X ′ {\displaystyle X'} are the corresponding algebraic and topological dual spaces , and ⟨ , ⟩ : X × X ′ → R {\displaystyle \langle ,\rangle :X\times X'\to \mathbb {R} } is the dual paar . A topology τ K {\displaystyle \tau ^{K}} on X {\displaystyle X} is called compatible with the dual paar ⟨ , ⟩ {\displaystyle \langle ,\rangle } if the corresponding topological dual space is X ′ {\displaystyle X'} . A seminorm p {\displaystyle p} on X {\displaystyle X} is called Hilbertian or a Hilbert seminorm if there exists a positive definite bilinear form b : X × X → R {\displaystyle b\colon X\times X\to \mathbb {R} } such that p ( x ) = b ( x , x ) {\displaystyle p(x)={\sqrt {b(x,x)}}} for all x ∈ X {\displaystyle x\in X} .
Let A := A ( X , X ′ ) := ⨂ n = 1 ∞ A f 1 , … , f n {\displaystyle {\mathfrak {A}}:={\mathfrak {A}}(X,X'):=\bigotimes \limits _{n=1}^{\infty }{\mathfrak {A}}_{f_{1},\dots ,f_{n}}} denote the cylindrical algebra . [ 3 ]
Let p {\displaystyle p} be a seminorm on X {\displaystyle X} and X p {\displaystyle X_{p}} be the factor space X p := X / p − 1 ( 0 ) {\displaystyle X_{p}:=X/p^{-1}(0)} with canonical mapping Q p : X → X p {\displaystyle Q_{p}:X\to X_{p}} defined as Q p : x ↦ [ x ] {\displaystyle Q_{p}:x\mapsto [x]} . Let p ¯ {\displaystyle {\overline {p}}} be the norm
on X p {\displaystyle X_{p}} , denote the corresponding Banach space as X ¯ p {\displaystyle {\overline {X}}_{p}} and let i p : X p ↪ X ¯ p {\displaystyle i_{p}:X_{p}\hookrightarrow {\overline {X}}_{p}} be the natural embedding, then define the continuous map
which is a map Q ¯ p : X → X ¯ p {\displaystyle {\overline {Q}}_{p}:X\to {\overline {X}}_{p}} . Let q {\displaystyle q} be a seminorm such that for all x ∈ X {\displaystyle x\in X}
then define a continuous linear operator T q , p : X ¯ q → X ¯ p {\displaystyle T_{q,p}:{\overline {X}}_{q}\to {\overline {X}}_{p}} as follows:
If p {\displaystyle p} it Hilbertian then X ¯ p {\displaystyle {\overline {X}}_{p}} is a Hilbert space .
Let P {\displaystyle {\mathcal {P}}} be a family of continuous Hilbert seminorms defined as follows: p ∈ P {\displaystyle p\in {\mathcal {P}}} if and only if there exists a Hilbert seminorm q {\displaystyle q} such that for all x ∈ X {\displaystyle x\in X}
for some constant C ∈ R {\displaystyle C\in \mathbb {R} } and if T q , p {\displaystyle T_{q,p}} is a Hilbert-Schmidt operator . Then the topology τ S := τ S ( X , τ ) {\displaystyle \tau ^{S}:=\tau ^{S}(X,\tau )} induced by the family P {\displaystyle {\mathcal {P}}} is called the Sazonov topology or S-Topologie . [ 4 ] Clearly it depends on the underlying topology τ {\displaystyle \tau } and if ( X , τ ) {\displaystyle (X,\tau )} is a nuclear then τ S = τ {\displaystyle \tau ^{S}=\tau } .
Let μ {\displaystyle \mu } be a cylindrical measure on A {\displaystyle {\mathfrak {A}}} and τ {\displaystyle \tau } a locally convex topology that is compatible with the dual paar and let τ S := τ S ( X , τ ) {\displaystyle \tau ^{S}:=\tau ^{S}(X,\tau )} be the Sazonov topology. Then μ {\displaystyle \mu } is σ-additive on A {\displaystyle {\mathfrak {A}}} if the Fourier transform μ ^ ( f ) : X ′ → C {\displaystyle {\hat {\mu }}(f):X'\to \mathbb {C} } is continuous in zero in τ S {\displaystyle \tau ^{S}} . [ 4 ] | https://en.wikipedia.org/wiki/Minlos–Sazonov_theorem |
The Minnesota Protocol on the Investigation of Potentially Unlawful Death (2016) is a set of international guidelines for the investigation of suspicious deaths, particularly those in which the responsibility of a State is suspected (either as a result of act or omission).
The original version of the Protocol, from 1991, was entitled the Manual on the Effective Prevention and Investigation of Extra-Legal, Arbitrary and Summary Executions . It was designed to support the implementation of the UN Principles on the Effective Prevention and Investigation of Extra-Legal, Arbitrary and Summary Executions, which were endorsed by the United Nations in 1989. [ 1 ] The Manual became known as the Minnesota Protocol because of the central role played by the Minnesota Lawyers International Human Rights Committee in its development. The use of the term ‘Protocol’ reflects the forensic medicine element of the document rather than its legal status. In 2016, after a two-year process of revision, the new version of Minnesota Protocol was finalized by an international group of experts convened by the UN Special Rapporteur on extrajudicial, summary or arbitrary executions, and the Office of the United Nations High Commissioner for Human Rights (OHCHR). [ 2 ] The revised version was published by the OHCHR in 2017.
The original Minnesota Protocol was designed to be a technical document aimed at providing practical assistance to those investigating suspicious deaths. Confronting the question of how to address political killings during the mid-1980s, various civil society groups came to the conclusion that criminal investigation techniques were an obvious starting place. [ 3 ] In 1984, Amnesty International carried out its own survey of how various States dealt with autopsies of arbitrary killings. David Weissbrodt , a professor at the University of Minnesota, was spending a sabbatical in the Legal Office of Amnesty International in 1982-3, which was when the idea for a Manual arose. [ 4 ]
The need for some kind of standard was highlighted by the assassination of Benigno Aquino Jr. in August 1983. Despite public declarations of intent, the Government of the Philippines failed to conduct an adequate investigation. However, as Ann Marie Clark has subsequently observed: ‘At that time there were no internationally standardized death investigations procedures. There was no external norm, therefore, that could be used as a basis for criticism when governments failed to implement proper investigation of political killings in a case like the death of Aquino’ [ 5 ]
Ultimately the Protocol was prepared by a group of legal and forensic experts coordinated by the Minnesota International Lawyers Committee for Human Rights (now The Advocates for Human Rights), in collaboration with the Science and Human Rights Program of the American Association for the Advancement of Science . [ 6 ]
In several resolutions, the UN Commission for Human Rights mandated the OHCHR to update the Protocol. These resolutions were later quoted by the Human Rights Council in resolutions on forensic genetics and human rights. [ 7 ]
In 2014 the Special Rapporteur on summary executions, Christof Heyns , began a process of consulting relevant experts and, in collaboration with OHCHR and UNODC, bringing together a large group that would ultimately participate in the revision of the Minnesota Protocol. In 2015, in his report to the General Assembly, he noted that ‘[t]he extent of the continued reliance on the Manual in international jurisprudence and by national legal entities emphasizes the need for the document to be up to date and comprehensive. It is to be expected that if the document is more up to date, it will more often and more readily serve as a guide.’ [ 8 ]
In 2016, two Working Groups, and a large international Advisory Panel undertook the revision, including with reference to two stakeholder consultations. As with the original version, the authority of the document relied upon the expertise of these drafting and review groups. Certain individuals had been involved in the processes of drafting both the original and the revised texts. [ 9 ] The finalized document was presented to the OHCHR in July 2016, and published in May 2017. [ 10 ]
Announcing its release, the UN High Commissioner on Human Rights highlighted that ‘Proper investigations into suspicious deaths are an integral part of the protection of the right to life’ [ 11 ]
Regional human rights courts have referred to the Manual in reaching findings on the inadequacy of investigations into suspicious deaths. [ 12 ] National courts have done the same when establishing guidelines for the investigation of killings by the police. [ 13 ] The International Committee of the Red Cross (ICRC) relied on the Principles and the Manual in its Study on customary international humanitarian law: a contribution to the understanding and respect for the rule of law in armed conflict (2005) and in its Guidelines for Investigating Deaths in Custody (2013).
The Minnesota Protocol is often grouped with another document with a similar medico-legal and human rights purpose, the Istanbul Protocol , which is aimed at the documentation of torture. In his report to the UN General Assembly in 2014, the Special Rapporteur on torture, Juan E. Méndez , encouraged the use of both documents when performing forensic autopsies, and highlighted capacity gaps in forensic services as contributing to lack of accountability for serious human rights violations. [ 14 ]
The Minnesota Protocol aims to protect the right to life by promoting effective investigation of potentially unlawful death or suspected enforced disappearance. It sets common standards of
performance and a shared set of principles and guidelines for States, as well as for institutions and individuals who play a role in investigations. [ 15 ]
The Minnesota Protocol applies to investigations of all “potentially unlawful death”. This primarily includes situations where:
The Protocol makes clear that protecting the right to life means preventing the arbitrary deprivation of life, but also requires accountability for an arbitrary deprivation of life whenever it occurs. Therefore, in addition to their duties to respect and to protect the right to life, States must also investigate potentially unlawful death, ensure accountability and remedy violations. The Protocol states:
The duty to investigate is an essential part of upholding the right to life. […] Where an investigation reveals evidence that a death was caused unlawfully, the State must ensure that identified perpetrators are prosecuted and, where appropriate, punished through a judicial process. […] A failure to respect the duty to investigate is a breach of the right to life. Investigations and prosecutions are essential to deter future violations and to promote accountability, justice, the rights to remedy and to the truth, and the rule of law. [ 17 ]
In addition to its scope, the Protocol also clearly establishes the “trigger” for the State’s duty to investigate, namely where it knows or should have known of any potentially unlawful death, including where reasonable allegations of a potentially unlawful death are made. [ 18 ] As the Protocol details, this includes all cases where the State has caused a death or where it is alleged or suspected that the State caused a death (for example, where law enforcement officers used force that may have contributed to the death) or where the State has failed to exercise due diligence to prevent a death at the hands of a third party. In all cases outside the conduct of hostilities in an armed conflict, this duty exists regardless of whether it is suspected or alleged that the death was unlawful. [ 19 ]
The Protocol offers a particular note on the duty to investigate during the conduct of hostilities, which it highlights as a context that may provide practical difficulties for the application of much of the Protocol’s content. All suspected war crimes must be investigated. But the Protocol also emphasizes that, where, during the conduct of hostilities, it appears that casualties have resulted from an attack, a post-operation assessment should be conducted to establish the facts, including the accuracy of the targeting. [ 20 ]
More broadly, the Protocol also highlights that the State also has a duty to investigate all potentially unlawful death caused by private individuals, even if the State cannot be held responsible for failing to prevent such deaths. [ 21 ]
The Protocol also establishes standards for what it calls the ‘Elements and Principles of Investigations’, broadly that they should be
The Protocol is explicitly non-prescriptive with respect to investigative mechanisms, noting that the duty to investigate does not necessarily require one particular investigative mechanism in preference to another. States may use a wide range of mechanisms, as determined or suggested by domestic law and practice, as long as those mechanisms meet international law requirements. [ 23 ]
The bulk of the Minnesota Protocol provides first strategies and principles and then detailed guidelines on practical steps that should be taken in an effective investigation. The overarching strategy of any investigation should be methodical and transparent, and all legitimate lines of inquiry should be pursued. An investigation may gather different types of material, not all of which will be used as evidence in a judicial proceeding. But all relevant materials or observations should be secured and logged. [ 24 ]
The Protocol establishes that a set of operational and tactical processes for the investigation should also be designed. These should seek to establish significant facts, preserve relevant material and lead to the identification of all the parties involved, including by managing the following:
Particular sections are dedicated to processes for interviewing witnesses and for recovering human remains. [ 26 ] The Protocol then provides a great deal of detail attesting both to the importance of, and practical guidance for, the identification of human remains. [ 27 ]
Particular guidance is offered on the techniques for collecting and sampling different types of evidence, including the following:
Investigation of potentially unlawful deaths will almost always be aided by the conduct of an autopsy . In a section setting out the general principles of an autopsy the Protocol provides an overview of the duties of a forensic doctor in relation to a death investigation, and then establishes the basic aims of autopsy will assist in fulfilling those duties. The aims of the autopsy, principally are:
In general, the Protocol establishes in various places the requirement of professional ethics for investigators, including forensic doctors. It highlights that any forensic doctor involved in an investigation has responsibilities to justice, to the relatives of the deceased, and more generally to the public. Whether or not they are employed by the police or the State, forensic doctors must understand their obligations to justice (not to the police or the State) and to the relatives of the deceased, so that a true account is provided of the cause of death and the circumstances surrounding it. [ 30 ] | https://en.wikipedia.org/wiki/Minnesota_Protocol |
The Minoan Moulds of Palaikastro ( Greek : Μήτρες του Παλαιοκάστρου Σητείας , romanized : Mitres tou Palaiokastrou Sitias ) are two double-sided pieces of schist , formed in the Minoan period as casting moulds for plaques with figures and symbols. These include female figures with raised arms, labrys double axes (Λάβρυες, labryes ) and opium poppy flowers or capsules , two double axes with indented edges, the Horns of Consecration symbol, and a sun-like disc with complex markings, which has been claimed by some researchers to be for making objects to use in astronomical predictions of solar and lunar eclipses .
They were found in 1899 near Palaikastro in the eastern part of Crete , and are now in the Herakleion Archeological Museum in Crete.
Stefanos Xanthoudidis , who published the find in 1900 described the two moulds, which were made from relatively soft and brittle schist as Plate Α and Plate Β. [ 1 ] His plaster casts, which are also reproduced on the right hand side, are mirror images of the original moulds. Both moulds are 225 mm (8.9 in) wide, 100 mm (3.9 in) high and 20 mm (0.8 in) thick, [ 2 ] while the width of the plaster casts is 230 mm (9.1 in). [ 3 ]
The front of Plate Α shows a large disc with rectangular spokes and a serrated edge (which some are keen to interpret as "geared"), a female figure with raised arms, who holds flowers in her hands and a small disc with a cross in the centre on top of a bell-shaped and horizontally striped base, above a crescent . Double horns, the ' Horns of Consecration ' of the Minoan culture , and a trident are shown on the rear. A small piece of the lower edge of the mould is broken-off.
The front of Plate B shows engravings of a couple of double axes, dissimilar in size with teethed edges. The double axe or labrys was a cultural, almost certainly religious, symbol of the Minoan culture, often used for votive offerings , as were goddess figures with uplifted hands. The rear of the plate shows a female figure with raised arms holding two double axes. A small piece of the lower edge of the mould is broken-off as well. Both plates are exhibited side by side in the Heraklion Archaeological Museum . The visitors can only see their front sides. The captions in the museum say that they stem from 1370 to 1200 BCE.
Very interesting objects are shown on the front of Plate Α, as recognised by Arthur Evans , who described them in his book The Palace of Minos at Knossos in 1921. On pages 478 and 479, he compares the base of an ivory object, of the Knossos board game , with the geared object on the mould of Palaikastro. [ 4 ] On page 514 he shows drawings of the objects left and right of the female figurine of Plate Α, however, not very precisely. [ 5 ] Evans refers to the isosceles cross being used in many cultures as the most simple representation of a star, and concludes that the geared object is a combination of a Morning Star with the disc of the sun. He interprets that the smaller object is a symbol for the goddess as the queen of the underworld and as the stars of the night. In combination with the crescent, the cross is then an Evening Star. [ 6 ]
In 2013, five scientists published a paper in the Mediterranean Archaeology and Archaeometry journal, in which they described the 85 by 85 millimetres (3.3 in × 3.3 in) sun-like form on Plate Α as a casting mould for manufacturing a spoked disc, which was used in the Minoan times of the 15th century BC as a sun dial , for establishing the geographical latitude and for predicting solar and lunar eclipses. The straight gashes beside the sun shape they interpret as moulds for two pins and a compasses or tweezers -like object, to be used in conjunction with it. They claimed to be able to predict eclipses even in the modern era with some accuracy, when using it. [ 7 ]
Similar comments have been made by Minas Tsikritsis in April 2011 in public. [ 8 ] [ 9 ] He described together with Efstratios Theodosiou the smaller round image to the right of the female figure as a Minoan cosmology model with the planetary system above the Flat Earth , in which the cross that symbolises the sun is surrounded by 18 dots and those including the crescent-shaped moon symbol are surrounded by 28 dots, an indication hinting at the Saros cycle with 28 lunar eclipses in 18 years. This is approximately 30 millimetres (1.2 in) long and 62 millimetres (2.4 in) high. [ 10 ] They interpret the spoked disc on the other side of the female figurine, which is associated with Titaness Rhea , as a portable analog calculator , which was created 1400 years before the Antikythera mechanism . [ 10 ] [ 9 ]
Chronological dating of the moulds is difficult, because the precise original location of the find and its surroundings are not known. Stratigraphy or the assessment of age-equivalent stratigraphic markers are, therefore, not applicable. In 1927, Martin P. Nilsson compared the style of the female figurine of Plate Α with those on various Minoan- Mycenaean gold rings and a relief on the Hagia Triada sarcophagus . [ 11 ]
In 1941, Luisa Banti classified both female figurines as variations of the type "goddess with raised hands", similar to the terracotta figurines found in Knossos , Gazi , Karphi and other places in Crete, which belong to the Late Minoan III phase . [ 12 ] Stylianos Alexiou endorsed in 1958 the dating as belonging to Late Minoan III, but he noted the differences in the gesture , as the female figurines hold something in their hands. [ 13 ]
In 2016, based on a stylistic and iconographic assessment, the casting moulds were dated as being older by Jan G. Velsink , who dates them as belonging to the Middle Minoan phases MM II or III. [ 14 ]
The two moulds were discovered in October 1899 by a farmer from Karydi 150 m (490 ft) northeast of the village of Palaikastro. The Gendarmerie sent the finds to the then Cretan capital Chania , where they were assessed and kept by the archeologist and historian Stefanos Xanthoudidis . He recognised the importance of ancient craftsmanship and delivered the moulds to the museum in Heraklion which had been set up in 1883. [ 15 ] Xanthoudidis described the objects in March 1900 in an article ("Ancient moulds from Sitia in Crete") in the journal of the Archaeological Society of Athens . This publication included photos of plaster casts of all four sides of the moulds. [ 1 ] | https://en.wikipedia.org/wiki/Minoan_Moulds_of_Palaikastro |
A minor-planet moon is an astronomical object that orbits a minor planet as its natural satellite . As of January 2022 [update] , there are 457 minor planets known or suspected to have moons. [ 1 ] Discoveries of minor-planet moons (and binary objects, in general) are important because the determination of their orbits provides estimates on the mass and density of the primary, allowing insights into their physical properties that are generally not otherwise accessible. [ 2 ]
Several of the moons are quite large compared to their primaries: 90 Antiope , Mors–Somnus and Sila–Nunam (95%), Patroclus–Menoetius , Altjira and Lempo–Hiisi (90%, with Lempo–Paha at 50%). The largest known minor-planet moon in absolute size is Pluto's largest moon Charon , which itself has about half the diameter of Pluto.
There are also several known ring systems around distant objects (see: Rings of Chariklo and Chiron ).
In addition to the terms satellite and moon , the term "binary" ( binary minor planet ) is sometimes used for minor planets with one moon, and "triple" for minor planets with two moons. If one object is much bigger it is referred to as the primary and its companion as the secondary . The term double asteroid is sometimes used for systems in which the asteroid and its moon are roughly the same size, while binary tends to be used independently from the relative sizes of the components. When binary minor planets are similar in size, the Minor Planet Center (MPC) refers to them as " binary companions " instead of referring to the smaller body as a satellite. [ 3 ] A good example of a true binary is the 90 Antiope system, identified in August 2000. [ 4 ] Very small satellites are often referred to as moonlets. [ 2 ] [ 5 ]
Prior to the era of the Hubble Space Telescope and space probes reaching the outer Solar System , attempts to detect satellites around asteroids were limited to optical observations from Earth. For example, in 1978, stellar occultation observations were claimed as evidence of a satellite for the asteroid to 532 Herculina . [ 6 ] [ 7 ] However, later more-detailed imaging by the Hubble Telescope did not reveal a satellite, and the current consensus is that Herculina does not have a significant satellite. [ 8 ] There were other similar reports of asteroids having companions (usually referred to as satellites) in the following years. A letter by astronomer Thomas Hamilton in the Sky & Telescope magazine at this time pointed to apparently simultaneous impact craters on Earth (for example, the Clearwater Lakes in Quebec), suggesting that these craters were caused by pairs of gravitationally bound objects. [ 9 ]
In 2014, 130 Elektra was discovered to have three moons, making it the only discovered quadruple asteroid.
Also in 1978, Pluto's largest moon Charon was discovered; however, at the time Pluto was still considered to be one of the major planets.
In 1993, the first asteroid moon was confirmed when the Galileo probe discovered the small Dactyl orbiting 243 Ida in the asteroid belt . The second was discovered around 45 Eugenia in 1998. [ 10 ] In 2001, 617 Patroclus and its same-sized companion Menoetius became the first known binary asteroids in the Jupiter trojans . [ 11 ] The first trans-Neptunian binary after Pluto–Charon, 1998 WW 31 , was optically resolved in 2002. [ 12 ]
In 2005, the asteroid 87 Sylvia was discovered to have two satellites, making it the first known triple system (also called a triple minor planet or triple asteroid ). [ 13 ] This was followed by the discovery of a second moon orbiting 45 Eugenia . [ 14 ] Also in 2005, the dwarf planet Haumea was discovered to have two moons, making it the second trans-Neptunian object after Pluto known to have more than one moon. [ 15 ] Additionally, 216 Kleopatra [ 16 ] and 93 Minerva [ 17 ] were discovered to be triple asteroids in 2008 and 2009 respectively. There has been one discovered quadruple minor planet, that being 130 Elektra . Since the first few triple minor planets were discovered, more continue to be discovered. As of 2025 [update] , the total number of known multiple systems among minor planets is 18 (including the Pluto and Haumea systems). [ 1 ]
The following table lists all satellites of multiple systems, starting with Pluto, which was unnumbered when its first moon was discovered in 1978. The highest known multiplicities are for Pluto (a sextuple system) and 130 Elektra (a quadruple system).
The data about the populations of binary objects are still patchy. In addition to the inevitable observational bias (dependence on the distance from Earth, size, albedo and separation of the components) the frequency appears to be different among different categories of objects. Among asteroids, an estimated 2% would have satellites. Among trans-Neptunian objects (TNOs), an estimated 11% are thought to be binary or multiple objects, and the majority of the large TNOs have at least one satellite, including all four IAU-listed dwarf planets.
More than 50 binaries are known in each of the main groupings: near-Earth asteroids, belt asteroids , and trans-Neptunian objects , not including numerous claims based solely on light-curve variation.
Two binaries have been found so far among centaurs with semi-major axes smaller than Neptune. [ 21 ] Both are double ring systems around 2060 Chiron and 10199 Chariklo , discovered in 1993–2011 and 2013 respectively.
The origin of minor-planet moons is not currently known with certainty, and a variety of hypotheses exist. One such model is that minor-planet moons are formed from debris knocked off the primary by an impact. Other pairings may be formed when a small object is captured by the gravity of a larger one.
Formation by collision is constrained by the angular momentum of the components, i.e. by the masses and their separation. Close binaries fit this model (e.g. Pluto – Charon ). Distant binaries however, with components of comparable size, are unlikely to have followed this scenario, unless considerable mass has been lost in the event.
The distances of the components for the known binaries vary from a few hundreds of kilometres ( 243 Ida , 3749 Balam ) to more than 3000 km ( 379 Huenna ) for the asteroids. Among TNOs, the known separations vary from 3,000 to 50,000 km. [ 21 ]
What is "typical" for a binary system tends to depend on its location in the Solar System (presumably because of different modes of origin and lifetimes of such systems in different populations of minor planets). [ 22 ]
As of January 2022 [update] , there are 457 minor planets (systems) with 477 known companions. [ 1 ] The following table is a listing of the total number of these systems by orbital class:
This is a list of near-Earth asteroids with companions. [ 1 ] Candidate binaries with an unconfirmed status are displayed on a dark background. [ 25 ] For an overview, see summary and introduction .
This is a list of Mars-crossing asteroids with companions. [ 1 ] Candidate binaries with an unconfirmed status are displayed on a dark background. [ 25 ] For an overview, see summary and introduction .
This is a list of main-belt asteroids with companions. [ 1 ] Candidate binaries with an unconfirmed status are displayed on a dark background. [ 25 ] For an overview, see summary and introduction .
The following binaries are double asteroids , with similarly sized components, and a barycenter outside of the larger object.
In addition, these bodies might be double asteroids , but due to errors in their size and orbit, it is uncertain.
This is a list of Jupiter trojans with companions. [ 1 ] Candidate binaries with an unconfirmed status are displayed on a dark background. [ 25 ] For an overview, see summary and introduction .
This is a list of trans-Neptunian objects with companions. [ 1 ] Candidate binaries with an unconfirmed status are displayed on a dark background. [ 25 ] This list gives the companion's orbital period (P s ) in days rather than hours. For an overview, see summary and introduction .
Solar System → Local Interstellar Cloud → Local Bubble → Gould Belt → Orion Arm → Milky Way → Milky Way subgroup → Local Group → Local Sheet → Virgo Supercluster → Laniakea Supercluster → Local Hole → Observable universe → Universe Each arrow ( → ) may be read as "within" or "part of". | https://en.wikipedia.org/wiki/Minor-planet_moon |
The Minor Planet Center ( MPC ) is the official body for observing and reporting on minor planets under the auspices of the International Astronomical Union (IAU). Founded in 1947, it operates at the Smithsonian Astrophysical Observatory .
The Minor Planet Center is the official worldwide organization in charge of collecting observational data for minor planets (such as asteroids ), calculating their orbits and publishing this information via the Minor Planet Circulars . Under the auspices of the International Astronomical Union (IAU), it operates at the Smithsonian Astrophysical Observatory , which is part of the Center for Astrophysics along with the Harvard College Observatory . [ 1 ]
The MPC runs a number of free online services for observers to assist them in observing minor planets and comets. The complete catalogue of minor planet orbits (sometimes referred to as the "Minor Planet Catalogue") may also be freely downloaded. In addition to astrometric data, the MPC collects light curve photometry of minor planets. A key function of the MPC is helping observers coordinate follow up observations of possible near-Earth objects (NEOs) via its NEO web form and blog, the Near-Earth Object Confirmation Page . [ 2 ] [ 3 ] The MPC is also responsible for identifying, and alerting to, new NEOs with a risk of impacting Earth in the few weeks following their discovery (see Potentially hazardous objects and § Videos ) . [ 1 ]
The Minor Planet Center was set up at the University of Cincinnati in 1947, under the direction of Paul Herget . [ 4 ] [ 5 ] : 63 Upon Herget's retirement on June 30, 1978, [ 5 ] : 67 the MPC was moved to the Smithsonian Astrophysical Observatory, under the direction of Brian G. Marsden . [ 5 ] : 67 From 2006 to 2015, [ 6 ] the director of the MPC was Timothy Spahr , [ 7 ] who oversaw a staff of five. From 2015 to 2021, the Minor Planet Center was headed by interim director Matthew Holman . [ 8 ] Under his leadership, the MPC experienced a significant period of reorganization and growth, doubling both its staff size and the volume of observations processed per year. Upon Holman's resignation on February 9, 2021 (announced on February 19, 2021) Matthew Payne became acting director of the MPC. [ 9 ] [ 10 ]
The MPC periodically releases astrometric observations of minor planets, as well as of comets and natural satellites . These publications are the Minor Planet Circulars (MPCs), the Minor Planet Electronic Circulars (MPECs), and the Minor Planet Supplements (MPSs and MPOs). [ 11 ] An extensive archive of publications in a PDF format is available at the Minor Planet Center's website. The archive's oldest publication dates back to 1 November 1979 (MPC 4937–5016). [ 12 ]
The Natural Satellites Ephemeris Service is an online service of the Minor Planet Center. The service provides "ephemerides, orbital elements and residual blocks for the outer irregular satellites of the giant planets". [1] | https://en.wikipedia.org/wiki/Minor_Planet_Center |
The Minor Planet Center ( MPC ) is the official body for observing and reporting on minor planets under the auspices of the International Astronomical Union (IAU). Founded in 1947, it operates at the Smithsonian Astrophysical Observatory .
The Minor Planet Center is the official worldwide organization in charge of collecting observational data for minor planets (such as asteroids ), calculating their orbits and publishing this information via the Minor Planet Circulars . Under the auspices of the International Astronomical Union (IAU), it operates at the Smithsonian Astrophysical Observatory , which is part of the Center for Astrophysics along with the Harvard College Observatory . [ 1 ]
The MPC runs a number of free online services for observers to assist them in observing minor planets and comets. The complete catalogue of minor planet orbits (sometimes referred to as the "Minor Planet Catalogue") may also be freely downloaded. In addition to astrometric data, the MPC collects light curve photometry of minor planets. A key function of the MPC is helping observers coordinate follow up observations of possible near-Earth objects (NEOs) via its NEO web form and blog, the Near-Earth Object Confirmation Page . [ 2 ] [ 3 ] The MPC is also responsible for identifying, and alerting to, new NEOs with a risk of impacting Earth in the few weeks following their discovery (see Potentially hazardous objects and § Videos ) . [ 1 ]
The Minor Planet Center was set up at the University of Cincinnati in 1947, under the direction of Paul Herget . [ 4 ] [ 5 ] : 63 Upon Herget's retirement on June 30, 1978, [ 5 ] : 67 the MPC was moved to the Smithsonian Astrophysical Observatory, under the direction of Brian G. Marsden . [ 5 ] : 67 From 2006 to 2015, [ 6 ] the director of the MPC was Timothy Spahr , [ 7 ] who oversaw a staff of five. From 2015 to 2021, the Minor Planet Center was headed by interim director Matthew Holman . [ 8 ] Under his leadership, the MPC experienced a significant period of reorganization and growth, doubling both its staff size and the volume of observations processed per year. Upon Holman's resignation on February 9, 2021 (announced on February 19, 2021) Matthew Payne became acting director of the MPC. [ 9 ] [ 10 ]
The MPC periodically releases astrometric observations of minor planets, as well as of comets and natural satellites . These publications are the Minor Planet Circulars (MPCs), the Minor Planet Electronic Circulars (MPECs), and the Minor Planet Supplements (MPSs and MPOs). [ 11 ] An extensive archive of publications in a PDF format is available at the Minor Planet Center's website. The archive's oldest publication dates back to 1 November 1979 (MPC 4937–5016). [ 12 ]
The Natural Satellites Ephemeris Service is an online service of the Minor Planet Center. The service provides "ephemerides, orbital elements and residual blocks for the outer irregular satellites of the giant planets". [1] | https://en.wikipedia.org/wiki/Minor_Planet_Circulars |
The Minor Planet Center ( MPC ) is the official body for observing and reporting on minor planets under the auspices of the International Astronomical Union (IAU). Founded in 1947, it operates at the Smithsonian Astrophysical Observatory .
The Minor Planet Center is the official worldwide organization in charge of collecting observational data for minor planets (such as asteroids ), calculating their orbits and publishing this information via the Minor Planet Circulars . Under the auspices of the International Astronomical Union (IAU), it operates at the Smithsonian Astrophysical Observatory , which is part of the Center for Astrophysics along with the Harvard College Observatory . [ 1 ]
The MPC runs a number of free online services for observers to assist them in observing minor planets and comets. The complete catalogue of minor planet orbits (sometimes referred to as the "Minor Planet Catalogue") may also be freely downloaded. In addition to astrometric data, the MPC collects light curve photometry of minor planets. A key function of the MPC is helping observers coordinate follow up observations of possible near-Earth objects (NEOs) via its NEO web form and blog, the Near-Earth Object Confirmation Page . [ 2 ] [ 3 ] The MPC is also responsible for identifying, and alerting to, new NEOs with a risk of impacting Earth in the few weeks following their discovery (see Potentially hazardous objects and § Videos ) . [ 1 ]
The Minor Planet Center was set up at the University of Cincinnati in 1947, under the direction of Paul Herget . [ 4 ] [ 5 ] : 63 Upon Herget's retirement on June 30, 1978, [ 5 ] : 67 the MPC was moved to the Smithsonian Astrophysical Observatory, under the direction of Brian G. Marsden . [ 5 ] : 67 From 2006 to 2015, [ 6 ] the director of the MPC was Timothy Spahr , [ 7 ] who oversaw a staff of five. From 2015 to 2021, the Minor Planet Center was headed by interim director Matthew Holman . [ 8 ] Under his leadership, the MPC experienced a significant period of reorganization and growth, doubling both its staff size and the volume of observations processed per year. Upon Holman's resignation on February 9, 2021 (announced on February 19, 2021) Matthew Payne became acting director of the MPC. [ 9 ] [ 10 ]
The MPC periodically releases astrometric observations of minor planets, as well as of comets and natural satellites . These publications are the Minor Planet Circulars (MPCs), the Minor Planet Electronic Circulars (MPECs), and the Minor Planet Supplements (MPSs and MPOs). [ 11 ] An extensive archive of publications in a PDF format is available at the Minor Planet Center's website. The archive's oldest publication dates back to 1 November 1979 (MPC 4937–5016). [ 12 ]
The Natural Satellites Ephemeris Service is an online service of the Minor Planet Center. The service provides "ephemerides, orbital elements and residual blocks for the outer irregular satellites of the giant planets". [1] | https://en.wikipedia.org/wiki/Minor_Planet_Electronic_Circulars |
The Minor Planet Center ( MPC ) is the official body for observing and reporting on minor planets under the auspices of the International Astronomical Union (IAU). Founded in 1947, it operates at the Smithsonian Astrophysical Observatory .
The Minor Planet Center is the official worldwide organization in charge of collecting observational data for minor planets (such as asteroids ), calculating their orbits and publishing this information via the Minor Planet Circulars . Under the auspices of the International Astronomical Union (IAU), it operates at the Smithsonian Astrophysical Observatory , which is part of the Center for Astrophysics along with the Harvard College Observatory . [ 1 ]
The MPC runs a number of free online services for observers to assist them in observing minor planets and comets. The complete catalogue of minor planet orbits (sometimes referred to as the "Minor Planet Catalogue") may also be freely downloaded. In addition to astrometric data, the MPC collects light curve photometry of minor planets. A key function of the MPC is helping observers coordinate follow up observations of possible near-Earth objects (NEOs) via its NEO web form and blog, the Near-Earth Object Confirmation Page . [ 2 ] [ 3 ] The MPC is also responsible for identifying, and alerting to, new NEOs with a risk of impacting Earth in the few weeks following their discovery (see Potentially hazardous objects and § Videos ) . [ 1 ]
The Minor Planet Center was set up at the University of Cincinnati in 1947, under the direction of Paul Herget . [ 4 ] [ 5 ] : 63 Upon Herget's retirement on June 30, 1978, [ 5 ] : 67 the MPC was moved to the Smithsonian Astrophysical Observatory, under the direction of Brian G. Marsden . [ 5 ] : 67 From 2006 to 2015, [ 6 ] the director of the MPC was Timothy Spahr , [ 7 ] who oversaw a staff of five. From 2015 to 2021, the Minor Planet Center was headed by interim director Matthew Holman . [ 8 ] Under his leadership, the MPC experienced a significant period of reorganization and growth, doubling both its staff size and the volume of observations processed per year. Upon Holman's resignation on February 9, 2021 (announced on February 19, 2021) Matthew Payne became acting director of the MPC. [ 9 ] [ 10 ]
The MPC periodically releases astrometric observations of minor planets, as well as of comets and natural satellites . These publications are the Minor Planet Circulars (MPCs), the Minor Planet Electronic Circulars (MPECs), and the Minor Planet Supplements (MPSs and MPOs). [ 11 ] An extensive archive of publications in a PDF format is available at the Minor Planet Center's website. The archive's oldest publication dates back to 1 November 1979 (MPC 4937–5016). [ 12 ]
The Natural Satellites Ephemeris Service is an online service of the Minor Planet Center. The service provides "ephemerides, orbital elements and residual blocks for the outer irregular satellites of the giant planets". [1] | https://en.wikipedia.org/wiki/Minor_Planets_and_Comets_Orbit_Supplement |
The Minor Planet Center ( MPC ) is the official body for observing and reporting on minor planets under the auspices of the International Astronomical Union (IAU). Founded in 1947, it operates at the Smithsonian Astrophysical Observatory .
The Minor Planet Center is the official worldwide organization in charge of collecting observational data for minor planets (such as asteroids ), calculating their orbits and publishing this information via the Minor Planet Circulars . Under the auspices of the International Astronomical Union (IAU), it operates at the Smithsonian Astrophysical Observatory , which is part of the Center for Astrophysics along with the Harvard College Observatory . [ 1 ]
The MPC runs a number of free online services for observers to assist them in observing minor planets and comets. The complete catalogue of minor planet orbits (sometimes referred to as the "Minor Planet Catalogue") may also be freely downloaded. In addition to astrometric data, the MPC collects light curve photometry of minor planets. A key function of the MPC is helping observers coordinate follow up observations of possible near-Earth objects (NEOs) via its NEO web form and blog, the Near-Earth Object Confirmation Page . [ 2 ] [ 3 ] The MPC is also responsible for identifying, and alerting to, new NEOs with a risk of impacting Earth in the few weeks following their discovery (see Potentially hazardous objects and § Videos ) . [ 1 ]
The Minor Planet Center was set up at the University of Cincinnati in 1947, under the direction of Paul Herget . [ 4 ] [ 5 ] : 63 Upon Herget's retirement on June 30, 1978, [ 5 ] : 67 the MPC was moved to the Smithsonian Astrophysical Observatory, under the direction of Brian G. Marsden . [ 5 ] : 67 From 2006 to 2015, [ 6 ] the director of the MPC was Timothy Spahr , [ 7 ] who oversaw a staff of five. From 2015 to 2021, the Minor Planet Center was headed by interim director Matthew Holman . [ 8 ] Under his leadership, the MPC experienced a significant period of reorganization and growth, doubling both its staff size and the volume of observations processed per year. Upon Holman's resignation on February 9, 2021 (announced on February 19, 2021) Matthew Payne became acting director of the MPC. [ 9 ] [ 10 ]
The MPC periodically releases astrometric observations of minor planets, as well as of comets and natural satellites . These publications are the Minor Planet Circulars (MPCs), the Minor Planet Electronic Circulars (MPECs), and the Minor Planet Supplements (MPSs and MPOs). [ 11 ] An extensive archive of publications in a PDF format is available at the Minor Planet Center's website. The archive's oldest publication dates back to 1 November 1979 (MPC 4937–5016). [ 12 ]
The Natural Satellites Ephemeris Service is an online service of the Minor Planet Center. The service provides "ephemerides, orbital elements and residual blocks for the outer irregular satellites of the giant planets". [1] | https://en.wikipedia.org/wiki/Minor_Planets_and_Comets_Supplement |
The Minor Use Animal Drug Program (or National Research Support Project 7 ) is the counterpart for animals of the IR-4 Minor Crop Pest Management Program. The program targets development of therapeutic drugs for minor species, such as small ruminants and aquatic species, plus support for drugs for minor use within major species. It is carried out in partnership with the Food and Drug Administration ’s (FDA) Center for Veterinary Medicine . | https://en.wikipedia.org/wiki/Minor_Use_Animal_Drug_Program |
Minor losses in pipe flow are a major part in calculating the flow, pressure, or energy reduction in piping systems. Liquid moving through pipes carries momentum and energy due to the forces acting upon it such as pressure and gravity. Just as certain aspects of the system can increase the fluids energy, there are components of the system that act against the fluid and reduce its energy, velocity, or momentum. Friction and minor losses in pipes are major contributing factors. [ 1 ] [ 2 ] [ 3 ] [ 4 ]
Before being able to use the minor head losses in an equation, the losses in the system due to friction must also be calculated.
Equation for friction losses:
H L f = v 2 g R h ( ∑ i L i ) f {\displaystyle H_{Lf}={v^{2} \over gR_{h}}(\sum _{i}L_{i})f} [ 5 ] [ 3 ] [ 1 ]
H L f {\displaystyle H_{Lf}} = Frictional head loss
v {\displaystyle v} = Downstream velocity
g {\displaystyle g} = Gravity of Earth
R h {\displaystyle R_{h}} = Hydraulic radius
∑ i L i {\displaystyle \sum _{i}L_{i}} =Total length of piping
f {\displaystyle f} = Fanning friction factor
After both minor losses and friction losses have been calculated, these values can be summed to find the total head loss.
Equation for total head loss, H L {\displaystyle H_{L}} , can be simplified and rewritten as:
H L = v 2 2 g R h [ ( 2 ∑ i L i ) f + R h ( ∑ i e v , i ) ] {\displaystyle H_{L}={v^{2} \over 2gR_{h}}[(2\sum _{i}L_{i})f+R_{h}(\sum _{i}e_{v,i})]} [ 5 ]
H L {\displaystyle H_{L}} = Frictional head loss
v {\displaystyle v} = Downstream velocity
g {\displaystyle g} = Gravity of Earth
R h {\displaystyle R_{h}} = Hydraulic radius
∑ i L i {\displaystyle \sum _{i}L_{i}} =Total length of piping
f {\displaystyle f} = Fanning friction factor
∑ i e v , i {\displaystyle \sum _{i}e_{v,i}} = Sum of all kinetic energy factors in system
Once calculated, the total head loss can be used to solve the Bernoulli Equation and find unknown values of the system. [ 1 ] [ 5 ] | https://en.wikipedia.org/wiki/Minor_losses_in_pipe_flow |
Minor metals is a widely used term in the metal industry that generally refers to metals which are a by-product of smelting a base metal. Minor metals do not have a real exchange, and are not traded on the London Metal Exchange (LME).
Two characteristics are regularly associated with minor metals: (1) their global production is relatively small in comparison to base metals, and (2), they are predominantly extracted as by-products of base metals. [ 1 ] However, due to the diversity of the metals often classified as minor metals, there is still much discussion about what exactly defines a minor metal. [ 1 ] [ 2 ] Minor metals have a wide variety of uses, including pharmaceutical, semiconductor, automotive, glass, battery, solar and many others. Many of these minor metals are critical to 21st century technology. They are more difficult to extract from their naturally occurring host minerals than base metals. [ 2 ]
According to the Minor Metals Trade Association (MMTA), its members alone account for over US$10 billion in annual trade of minor metal products. [ 3 ]
Recent research based on data from the United States Geological Survey (USGS) indicates that China is not only the leading primary producer of minor metals, supplying about 40 percent of all production, but that China's share of global production increased 34 percent between 2000 and 2009. [ 4 ]
Minor metals are used in a wide diversity of end-use applications, from capacitors for consumer electronics ( tantalum ) and metallic cathodes for rechargeable batteries ( cobalt ) to photovoltaic solar cells ( silicon ) and semiconductor materials ( gallium and indium ). The primary end-uses of minor metals can also help to categorize the metals into four groups: [ 5 ]
Metals often classified as minor metals include: antimony (Sb), arsenic (As), beryllium (Be), bismuth (Bi), cadmium (Cd), cerium (Ce), chromium (Cr), cobalt (Co), gadolinium (Gd), gallium (Ga), germanium (Ge), hafnium (Hf), indium (In), lithium (Li), magnesium (Mg), manganese (Mn), mercury (Hg), molybdenum (Mo), neodymium (Nd), niobium (Nb), iridium (Ir), osmium (Os), praseodymium (Pr), rhenium (Re), rhodium (Rh), ruthenium (Ru), samarium (Sm), selenium (Se), silicon (Si), tantalum (Ta), tellurium (Te), titanium (Ti), tungsten (W), vanadium (V), and zirconium (Zr). | https://en.wikipedia.org/wiki/Minor_metals |
According to the International Astronomical Union (IAU), a minor planet is an astronomical object in direct orbit around the Sun that is exclusively classified as neither a planet nor a comet . [ a ] Before 2006, the IAU officially used the term minor planet , but that year's meeting reclassified minor planets and comets into dwarf planets and small Solar System bodies (SSSBs). [ 1 ] In contrast to the eight official planets of the Solar System , all minor planets fail to clear their orbital neighborhood . [ 2 ] [ 1 ]
Minor planets include asteroids ( near-Earth objects , Earth trojans , Mars trojans , Mars-crossers , main-belt asteroids and Jupiter trojans ), as well as distant minor planets ( Uranus trojans , Neptune trojans , centaurs and trans-Neptunian objects ), most of which reside in the Kuiper belt and the scattered disc . As of October 2024 [update] , there are 1,392,085 known objects, divided into 740,000 numbered , with only one of them recognized as a dwarf planet (secured discoveries) and 652,085 unnumbered minor planets, with only five of those officially recognized as a dwarf planet. [ 3 ]
The first minor planet to be discovered was Ceres in 1801, though it was called a 'planet' at the time and an 'asteroid' soon after; the term minor planet was not introduced until 1841, and was considered a subcategory of 'planet' until 1932. [ 4 ] The term planetoid has also been used, especially for larger, planetary objects such as those the IAU has called dwarf planets since 2006. [ 5 ] [ 6 ] Historically, the terms asteroid , minor planet , and planetoid have been more or less synonymous. [ 5 ] [ 7 ] This terminology has become more complicated by the discovery of numerous minor planets beyond the orbit of Jupiter , especially trans-Neptunian objects that are generally not considered asteroids. [ 7 ] A minor planet seen releasing gas may be dually classified as a comet.
Objects are called dwarf planets if their own gravity is sufficient to achieve hydrostatic equilibrium and form an ellipsoidal shape. All other minor planets and comets are called small Solar System bodies . [ 1 ] The IAU stated that the term minor planet may still be used, but the term small Solar System body will be preferred. [ 8 ] However, for purposes of numbering and naming, the traditional distinction between minor planet and comet is still used.
Hundreds of thousands of minor planets have been discovered within the Solar System and thousands more are discovered each month. The Minor Planet Center has documented over 213 million observations and 794,832 minor planets, of which 541,128 have orbits known well enough to be assigned permanent official numbers . [ 9 ] [ 10 ] Of these, 21,922 have official names. [ 9 ] As of 8 November 2021 [update] , the lowest-numbered unnamed minor planet is (4596) 1981 QB , [ 11 ] and the highest-numbered named minor planet is 594913 ꞌAylóꞌchaxnim . [ 12 ]
There are various broad minor-planet populations:
All astronomical bodies in the Solar System need a distinct designation. The naming of minor planets runs through a three-step process. First, a provisional designation is given upon discovery—because the object still may turn out to be a false positive or become lost later on —called a provisionally designated minor planet . After the observation arc is accurate enough to predict its future location, a minor planet is formally designated and receives a number. It is then a numbered minor planet . Finally, in the third step, it may be named by its discoverers. However, only a small fraction of all minor planets have been named. The vast majority are either numbered or have still only a provisional designation. Example of the naming process:
A newly discovered minor planet is given a provisional designation . For example, the provisional designation 2002 AT 4 consists of the year of discovery (2002) and an alphanumeric code indicating the half-month of discovery and the sequence within that half-month. Once an asteroid's orbit has been confirmed, it is given a number, and later may also be given a name (e.g. 433 Eros ). The formal naming convention uses parentheses around the number, but dropping the parentheses is quite common. Informally, it is common to drop the number altogether or to drop it after the first mention when a name is repeated in running text.
Minor planets that have been given a number but not a name keep their provisional designation, e.g. (29075) 1950 DA . Because modern discovery techniques are finding vast numbers of new asteroids, they are increasingly being left unnamed. The earliest discovered to be left unnamed was for a long time (3360) 1981 VA , now 3360 Syrinx . In November 2006 its position as the lowest-numbered unnamed asteroid passed to (3708) 1974 FV 1 (now 3708 Socus ), and in May 2021 to (4596) 1981 QB . On rare occasions, a small object's provisional designation may become used as a name in itself: the then-unnamed (15760) 1992 QB 1 gave its "name" to a group of objects that became known as classical Kuiper belt objects ("cubewanos") before it was finally named 15760 Albion in January 2018. [ 20 ]
A few objects are cross-listed as both comets and asteroids, such as 4015 Wilson–Harrington , which is also listed as 107P/Wilson–Harrington .
Minor planets are awarded an official number once their orbits are confirmed. With the increasing rapidity of discovery, these are now six-figure numbers. The switch from five figures to six figures arrived with the publication of the Minor Planet Circular (MPC) of October 19, 2005, which saw the highest-numbered minor planet jump from 99947 to 118161. [ 9 ]
The first few asteroids were named after figures from Greek and Roman mythology , but as such names started to dwindle the names of famous people, literary characters, discoverers' spouses, children, colleagues, and even television characters were used.
Commission 15 [ 27 ] of the International Astronomical Union is dedicated to the Physical Study of Comets & Minor Planets.
Archival data on the physical properties of comets and minor planets are found in the PDS Asteroid/Dust Archive. [ 28 ] This includes standard asteroid physical characteristics such as the properties of binary systems , occultation timings and diameters, masses, densities, rotation periods, surface temperatures, albedoes, spin vectors, taxonomy, and absolute magnitudes and slopes. In addition, European Asteroid Research Node (E.A.R.N.), an association of asteroid research groups, maintains a Data Base of Physical and Dynamical Properties of Near Earth Asteroids. [ 29 ]
Environmental characteristics have three aspects: space environment, surface environment and internal environment, including geological, optical, thermal and radiological environmental properties, etc., which are the basis for understanding the basic properties of minor planets, carrying out scientific research, and are also an important reference basis for designing the payload of exploration missions
Without the protection of an atmosphere and its own strong magnetic field, the minor planet's surface is directly exposed to the surrounding radiation environment. In the cosmic space where minor planets are located, the radiation on the surface of the planets can be divided into two categories according to their sources: one comes from the sun, including electromagnetic radiation from the sun, and ionizing radiation from the solar wind and solar energy particles; the other comes from the sun outside the solar system, that is, galactic cosmic rays , etc. [ 30 ]
Usually during one rotation period of a minor planet, the albedo of a minor planet will change slightly due to its irregular shape and uneven distribution of material composition. This small change will be reflected in the periodic change of the planet's light curve, which can be observed by ground-based equipment, so as to obtain the planet's magnitude , rotation period , rotation axis orientation, shape, albedo distribution, and scattering properties. Generally speaking, the albedo of minor planets is usually low, and the overall statistical distribution is bimodal, corresponding to C-type (average 0.035) and S-type (average 0.15) minor planets. [ 31 ] In the minor planet exploration mission, measuring the albedo and color changes of the planet surface is also the most basic method to directly know the difference in the material composition of the planet surface. [ 32 ]
The geological environment on the surface of minor planets is similar to that of other unprotected celestial bodies, with the most widespread geomorphological feature present being impact craters: however, the fact that most minor planets are rubble pile structures, which are loose and porous, gives the impact action on the surface of minor planets its unique characteristics. On highly porous minor planets, small impact events produce spatter blankets similar to common impact events: whereas large impact events are dominated by compaction and spatter blankets are difficult to form, and the longer the planets receive such large impacts, the greater the overall density. [ 33 ] In addition, statistical analysis of impact craters is an important means of obtaining information on the age of a planet surface. Although the Crater Size-Frequency Distribution (CSFD) method of dating commonly used on minor planet surfaces does not allow absolute ages to be obtained, it can be used to determine the relative ages of different geological bodies for comparison. [ 34 ] In addition to impact, there are a variety of other rich geological effects on the surface of minor planets, [ 35 ] such as mass wasting on slopes and impact crater walls, [ 36 ] large-scale linear features associated with graben , [ 37 ] and electrostatic transport of dust. [ 38 ] By analysing the various geological processes on the surface of minor planets, it is possible to learn about the possible internal activity at this stage and some of the key evolutionary information about the long-term interaction with the external environment, which may lead to some indication of the nature of the parent body's origin. Many of the larger planets are often covered by a layer of soil ( regolith ) of unknown thickness. Compared to other atmosphere-free bodies in the solar system (e.g. the Moon ), minor planets have weaker gravity fields and are less capable of retaining fine-grained material, resulting in a somewhat larger surface soil layer size. [ 39 ] Soil layers are inevitably subject to intense space weathering that alters their physical and chemical properties due to direct exposure to the surrounding space environment. In silicate-rich soils, the outer layers of Fe are reduced to nano-phase Fe (np-Fe), which is the main product of space weathering . [ 40 ] For some small planets, their surfaces are more exposed as boulders of varying sizes, up to 100 metres in diameter, due to their weaker gravitational pull. [ 41 ] These boulders are of high scientific interest, as they may be either deeply buried material excavated by impact action or fragments of the planet's parent body that have survived. The rocks provide more direct and primitive information about the material inside the minor planet and the nature of its parent body than the soil layer, and the different colours and forms of the rocks indicate different sources of material on the surface of the minor planet or different evolutionary processes.
Usually in the interior of the planet, the convection of the conductive fluid will generate a large and strong magnetic field . However, the size of a minor planet is generally small and most of the minor planets have a "crushed stone pile" structure, and there is basically no "dynamo" structure inside, so it will not generate a self-generated dipole magnetic field like the Earth. But some minor planets do have magnetic fields—on the one hand, some minor planets have remanent magnetism : if the parent body had a magnetic field or if the nearby planetary body has a strong magnetic field, the rocks on the parent body will be magnetised during the cooling process and the planet formed by the fission of the parent body will still retain remanence, [ 42 ] which can also be detected in extraterrestrial meteorites from the minor planets; [ 43 ] on the other hand, if the minor planets are composed of electrically conductive material and their internal conductivity is similar to that of carbon- or iron-bearing meteorites, the interaction between the minor planets and the solar wind is likely to be unipolar induction , resulting in an external magnetic field for the minor planet. [ 44 ] In addition, the magnetic fields of minor planets are not static; impact events, weathering in space and changes in the thermal environment can alter the existing magnetic fields of minor planets. At present, there are not many direct observations of minor planet magnetic fields, and the few existing planets detection projects generally carry magnetometers, with some targets such as Gaspra [ 45 ] and Braille [ 46 ] measured to have strong magnetic fields nearby, while others such as Lutetia have no magnetic field. [ 47 ]
Solar System → Local Interstellar Cloud → Local Bubble → Gould Belt → Orion Arm → Milky Way → Milky Way subgroup → Local Group → Local Sheet → Virgo Supercluster → Laniakea Supercluster → Local Hole → Observable universe → Universe Each arrow ( → ) may be read as "within" or "part of". | https://en.wikipedia.org/wiki/Minor_planet |
The minor spliceosome is a ribonucleoprotein complex that catalyses the removal ( splicing ) of an atypical class of spliceosomal introns (U12-type) from messenger RNAs in some clades of eukaryotes. This process is called noncanonical splicing, as opposed to U2-dependent canonical splicing. U12-type introns represent less than 1% of all introns in human cells. However they are found in genes performing essential cellular functions.
A notable feature of eukaryotic nuclear pre-mRNA introns is the relatively high level of conservation of the primary sequences of 5' and 3' splice sites over a great range of organisms.
Between 1989 and 1991, several groups reported four independent examples of introns with a splice site that differed from the common intron:
In 1991 by comparing the intron sequences of P120 and CMP genes, IJ Jackson reported the existence of ATATCC (5') and YYCAC (3') splice sites in these introns. The finding indicated a possible novel splicing mechanism.
In 1994, S.L. Hall and R.A. Padgett compared the primary sequence of all reports on the four genes mentioned above. The results suggested a new type of introns with ATATCCTT 5' splice sites and YCCAC 3' splice sites and an almost invariant TCCTTAAC sequence near the 3' end of the introns (so called 3' upstream element). A search for small nuclear RNA sequences that are complementary to these splice sites suggested U12 snRNA (matches the 3' sequence) and U11 snRNA (matches the 5' sequence) as being putative factors involved in splicing of this new type of introns.
In all these four genes, the pre-mRNA contains other introns whose sequences conform to those of major class introns. Neither the size nor the position of the AT–AC intron within the host gene is conserved.
In 1996, Woan-Yuh Tarn and Joan A. Steitz described an in vitro system that splices a pre-mRNA substrate containing an AT–AC intron derived from the human P120 gene. Psoralen cross-linking confirms the base-pairing interaction predicted by Hall and Padgett between the branch site of the pre-mRNA substrate and U12 RNA. Native gel electrophoresis reveals that U11, U12, and U5 snRNPs assemble onto the P120 pre-mRNA to form splicing complexes.
Although originally referred to as AT-AC introns, not all these introns are delimited by AT-AC dinucleotides. Some of them have GT-AG or AT-AG ends, at least. Thus, it is more correct to speak about the splicing machinery which is used to process them, differentiating between U2-type (canonical or major) and U12-type (non-canonical or minor). The main determinants for distinguishing U2- and U12-type introns are 5' splice site and branch site sequences. [ 1 ]
The minor spliceosome consists of U11 , U12 , U4atac , and U6atac , together with U5 and an unknown number of non-snRNP proteins. The U11, U12 and U4atac/U6atac snRNPs are functional analogs of the U1 , U2 and U4 / U6 snRNPs in the major spliceosome. [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 6 ] Although the minor U4atac and U6atac snRNAs are functional analogs of U4 and U6, respectively, they share only limited sequence homology (c. 40%). Furthermore, the sequence of U11 in comparison with U1, as well as U12 compared with U2, are completely unrelated. Despite this fact, the minor U11, U12, U4atac and U6atac snRNAs can be folded into structures similar to U1, U2, U4 and U6, respectively. [ 7 ]
The location of spliceosomal activity for the minor class spliceosome is regarded by most experts to be in the nucleus. [ citation needed ] However, a single paper has claimed that the minor spliceosome is active in the cytosol. [ 8 ] The data presented within this paper are not fully accepted within the field and directly contradict numerous other papers.
Like the major spliceosome, the minor spliceosome had an early origin: several of its characteristic constituents are present in representative organisms from all eukaryotic supergroups for which there is any substantial genome sequence information. In addition, functionally important sequence elements contained within U12-type introns and snRNAs are highly conserved during evolution.
Review papers:
Classic papers:
Other references: | https://en.wikipedia.org/wiki/Minor_spliceosome |
Mintmaster marks (German: Münzmeisterzeichen , abbreviation Mmz. ) are often the initials of the mintmaster of a mint or small symbols (cross, star, coat of arms, heraldic device, etc.) for example at the size of the letters on a coin inscription to denote the coins made under his direction. With his mark, the mintmaster assumed responsibility for ensuing the coins issued by his mint were in accordance with the regulations. [ 1 ] Mintmaster marks were used as early as the time of bracteate coinage in the Holy Roman Empire , but these can only rarely be deciphered. All mintmaster marks since the beginning of the minting of Thalers have been identified.
The picture on the right shows the mintmaster's mark, an acorn on a stem, of the Dresden mintmaster, Constantin Rothe, on a Reichstaler issued under Duke John George II of Saxony from the year 1662.
Sometimes there are pictographs and letters on a coin. In this case, the pictorial symbol is usually found in the circumscription of the coin and the letters are divided in the field on both sides of the coin's crest. Mintmasters often used their coats of arms as mintmaster symbols. For example in the Electorate of Saxony :
In Brandenburg:
In Mecklenburg:
In Florence:
Mintmaster marks appear from the late Middle Ages. They were largely superseded in the second half of the 19th century by mint marks in the form of a letter to designate the mint . France ( Paris Mint) first replaced mintmaster marks with mint marks to designate the mint as early as the 16th century. The Berlin Mint has used the A mint mark since the middle of the 18th century. [ 2 ]
Occasionally, the signature of the coin engraver or just the artist's signature is also found on coins. For example, on the Speciesthaler of 1763 is the Mmz. I F ô F of the mintmaster, Johann Friedrich ô Feral of the Leipzig Mint, and on the arm section the Signum S of medallist Johann Friedrich Stieler.
The mintmaster mark should therefore not be confused with the coin signature ( Signum ).
There is also a risk of confusion with mint marks used to designate the mint if the mintmaster mark consists of only one letter.
It is also possible for the mintmaster's mark and artist's signature of the medallist or die cutter to be identical on a coin. For example, in the case of Palatinate coins with the mark "A S". This is the artist's signature and at the same time the mintmaster's mark of the Palatinate court medallist, coin die cutter and mint master, Anton Schäffer. As an example, see the illustration of the Flussgoldducat by Karl Theodor von der Pfalz from the year 1763.
A special feature is the use of a banker's mark as a mintmaster mark on a Giulio of the Papal States of Pope Julius II On the reverse side between the two saints St. Peter and St. Paul is the trident-shaped banker's mark of the Fugger family from Augsburg, who had financed Julius II's papal election with loans. The trident on the Giulio testifies to the Fugger's lending for the papal election.
It was not uncommon for coins to be minted without dates and without the minting authority or the even country being specified. Known mintmaster marks can allow identification of undated coins of unknown origin. | https://en.wikipedia.org/wiki/Mintmaster_mark |
The Minto wheel is a heat engine named after Wally Minto . The engine consists of a set of sealed chambers arranged in a circle, with each chamber connected to the chamber opposite it. One chamber in each connected pair is filled with a liquid with a low boiling point ( propane ( T B = −42 °C) and R-12 ( T B = −29.8 °C) are listed in the Mother Earth News articles). Ideally, the working fluid also has a high vapor pressure and density.
As the lower chamber in each pair is heated, the liquid begins to vaporize, forcing the remaining liquid to travel to the upper chamber. This fluid transfer causes a weight imbalance, which causes the wheel to rotate.
Minto's pamphlet also suggests obtaining a pressure differential with a dissolved gas instead of a boiling gas. Soda water or propane dissolved in kerosene are suggested. [ 1 ]
The Minto wheel operates on a small temperature gradient, and produces a large amount of torque, but at very low rotational speed. [ citation needed ] The speed of rotation is directly proportional to the surface area of the containers used, the volume, and the height of the wheel. The higher the ratio of surface area to volume, the greater the rate of revolution.
In 1881, the Iske brothers got two patents granted for a design similar to the Minto wheel.
According to the patent, the working fluid is alcohol "or other volatile liquid". [ 2 ] Air in the tubes is to be removed and the tubes are sealed (creating a partial vacuum ). [ 2 ]
The patent suggests lamps as heating sources. [ 3 ] [ 2 ]
The first patent describes glass for the bulbs and tubes. [ 2 ] The second patent does not specify materials, [ 4 ] but the construction implies metal. A later patent then clearly specify metal. [ 5 ]
Later the same year, Israel L. Landis got a patent for a similar engine. Different to the Minto wheel and the Iske brothers' patent, the engine was oscillating, not revolving. [ 6 ] Landis suggested alcohol or ether as the volatile liquid. [ 6 ] Landis suggested heating up the apparatus before removing the air from the bulb/chambers. [ 6 ]
In the following years, the Iske brothers were granted various patents, including some relating to modification and/or improvements on engines similarly to the Minto Wheel and an oscillating engine [ 7 ] similarly to Israel L. Landis design.
The oscillating types by the Iske Brothers and Landis are related to the drinking bird toy.
The drinking bird is dating back to 1910s~1930s. The drinking bird was patented in the US in 1945 [ 8 ] and 1946 [ 9 ] by two different inventors.
Wally Minto experimented with different working fluids. With the working fluids he used, he got the required temperature difference down, enabling the engine - for example - to run on solar power. [ 10 ] Based on the working fluid, his improved wheel is also known as " Freon Power Wheel". [ 11 ] Popular Science reported about in its March 1976 issue. [ 11 ]
A working example of a Minto wheel was first published in a series of articles in The Mother Earth News , Issues #38 March, #39 May and #40 July 1976. Test units constructed by Mother Earth News (Issue 40, July 1976) and the MythBusters (Episode 24, December 5, 2004 – "Ming Dynasty Astronaut") did convert temperature difference into torque, but not as well as overenthusiastic boosters claimed. [ citation needed ] | https://en.wikipedia.org/wiki/Minto_wheel |
Minusheet perfusion culture system is used for advanced cell culture experiments in combination with adherent cells and to generate specialized tissues in combination with selected biomaterials , special tissue carriers and compatible perfusion culture containers.
The technical development of the Minusheet perfusion culture system was driven by the idea to create under in vitro conditions an environment resembling as near as possible the situation of specialized tissues found within the organism . Basis of this invention is therefore individually selected biomaterials for optimal cell adhesion mounted in Minusheet tissue carriers. Moreover, to always offer fresh nutrition including respiratory gas and to simulate a tissue-specific fluid environment, the tissue carriers can be inserted into compatible perfusion culture containers. As a result, a variety of publications illustrates that tissues generated by this innovative approach exhibit an excellent and stable quality. Thus, on the one hand the system provides a highly adaptable basis for the culture of adherent cells and the generation of specialized tissues. On the other hand the Minusheet perfusion culture system is bridging a methodical gap between the conventional static 24 well culture plate and modern perfusion culture technology.
Specialized tissues in culture are urgently needed in regenerative medicine , tissue engineering , nanotechnology , biomaterial research and advanced toxicity testing of newly developed pharmaceuticals . However, it is often observed that raised tissues do not exhibit expected functional features. Instead dedifferentiation is observed [1-4]. These cell biological alterations arise after isolation of cells and proceed during static culture in a dish due to suboptimal fluid environment and minor adhesion on biomaterials. Further uncontrolled supply with nutrition and respiratory gas, an overshoot of metabolites and paracrine factors or missing rheological stress can increase the degree of dedifferentiation. In consequence, regarding an optimal generation of specialized tissues a powerful strategy has to exclude as much as possible harmful parameters, while factors supporting the process of tissue development must be intensified [5].
Under natural conditions a prerequisite for an optimal tissue development is a cell-specific interaction with the extracellular matrix, while under in vitro conditions a substitute for the extracellular matrix has to be selected. However, the crucial problem is that a biomaterial can influence the development of functional features within a maturing tissue in a good and in a bad sense. In consequence, the suitability of a decellularized extracellular matrix, newly developed synthetic polymers, biodegradable scaffolds, ceramics or metal alloys cannot be predicted but must be tested.
To meet parameters positively influencing cell adhesion and communication, the technical concept is based on a Minusheet tissue carrier (Fig. 1). By the help of this tool cell adhesion and development of tissue can be tested with individually selected biomaterials. These experiments can be performed first under static (Fig. 2) and then under dynamic (Fig. 3) culture conditions [6]. In both cases a Minusheet tissue carrier prevents damage but supports development of contained cells or tissues during experimentation.
To stay compatible with a conventional 24 well culture plate a selected biomaterial must be punched in a diameter of 13 mm. In this format many materials are also commercially available. Further materials can be applied in form of filters, foils, nets, fleeces and scaffolds (Fig. 1a). For an easy handling and to prevent damage during development the selected specimens are placed in the base part of a Minusheet tissue carrier (Fig. 1b). Pressing down a tension ring the biomaterial is held in position (Fig. 1c). After mounting a tissue carrier is enveloped in a bag and sterilized.
For cell seeding the mounted tissue carrier is transferred by a forceps in a 24 well culture plate (Fig. 2). To concentrate cells on top of a tissue carrier culture medium is added to a level so that the selected biomaterial is just wetted. Then an aliquot of cells is transferred by a pipette to the surface of the mounted biomaterial.
A standard culture protocol with a tissue carrier can be initiated by seeding cells onto the upper side. When a tissue carrier is turned, cells can also be seeded on the other side so that co-culture experiments with two different cell types become possible.
Not only single cells but also a thin slice of tissue can be mounted between two pieces of a woven net within a Minusheet tissue carrier. Further flexible materials such as collagen sheets can be used in a tissue carrier like the skin of a drum. Last but not least excellent results were obtained by mounting a polyester fleece as an artificial interstitium for spatial parenchyma development [5,6,8]. It is obvious that for each specialized tissue very individual spatial environments within a tissue carrier can be created.
It has been shown that the static environment within a 24 well culture plate leads to a decrease of nutrition and hormones, an uncontrollable increase of metabolites and an overshoot of paracrine factors during time. Due to these reasons a Minusheet tissue carrier with adherent cells is used only for the short period of cell seeding in a 24 well culture plate.
In consequence, after adhesion of cells the tissue carrier is transferred to a perfusion culture container to offer a dynamic fluid environment. To meet the individual requirements of specialized tissues a variety of perfusion culture containers was constructed (Fig. 3).
Each of the perfusion culture containers has at least one inlet and one outlet for the transport of culture medium. A basic version of a container allows the simple bathing of cells respectively growing tissues under continuous medium transport (Fig. 4a). In a gradient container the tissue carrier is placed between the base and the lid so that both sides can be provided with individual media mimicking a typical environment for epithelia (Fig. 4b). A further culture container is made of a transparent lid and base allowing the microscopic observation during tissue development (Fig. 4c).
In addition, a perfusion culture container can exhibit a flexible silicone lid. Applying force to this lid by an eccentric rotor simulates a mechanical load as required in cartilage and bone tissue engineering . Shaped tissues such as an auricle or different forms of cartilage can be generated with individual scaffolds in a special tissue engineering container. Finally, spatial extension of tubules derived from renal stem/progenitor cells is obtained within a perfusion container filled with an artificial interstitium made of polyester fleece. Finally, all of these containers are machined out of a special polycarbonate ( Makrolon ®) so that all of them can be autoclaved for multiple uses.
To maintain the necessary temperature of 37 °C within a perfusion culture container, a heating plate (MEDAX-Nagel, Kiel, Germany) and a cover lid (not shown) are used during performance of culture experiments over weeks (Fig. 5, 7). The transport of culture medium is best accomplished using a slowly rotating peristaltic pump (ISMATEC, IPC N8, Wertheim, Germany). It is able to deliver adjustable and exact pump rates between 0.1 and 5 mL per hour.
On the passage from the storage bottle through the perfusion culture container medium is transported along a mounted tissue carrier to provide contained cells. The exact geometrical placement of the tissue carrier within a perfusion culture container guarantees during transport of medium provision with always fresh nutrition and respiratory gas from all sides. At the same time it prevents an unphysiological accumulation of metabolic products and an overshoot of paracrine factors. To maintain for the whole culture period this controlled environment, the metabolized medium is collected in a separate waste bottle. In consequence, medium is not recirculated.
Normally cell culture experiments are performed in a CO 2 incubator. Also perfusion culture experiments can be performed in such an atmosphere. However, a much better solution is the performance of perfusion culture experiments under atmospheric air on a laboratory table, since it facilitates the complete handling. However, in this case the culture medium has to be adjusted to atmospheric air.
Keeping media in a 5% CO 2 atmosphere within an incubator always a relatively high amount of NaHCO 3 is contained to maintain a constant pH between 7.2 and 7.4. If such a formulated medium is used for perfusion culture outside a CO 2 incubator, the pH will shift from the physiological range to much more alkaline values due to the low content of CO 2 (0.3%) in atmospheric air.
For that reason any medium used for perfusion culture outside a CO 2 incubator has to be stabilized by reducing the NaHCO 3 concentration and/or by adding biological buffers such as HEPES (GIBCO/Invitrogen, Karlsruhe, Germany) or BUFFER ALL (Sigma-Aldrich-Chemie, München, Germany). The necessary amount can be easily determined by admixing increasing amounts of biological buffer solution to an aliquot of medium. Then the medium must equilibrate over night on a thermo plate at 37 °C under atmospheric air. For example, application of 50 mmol/L HEPES or an equivalent of BUFFER ALL (ca. 1%) to IMDM (Iscove’s Modified Dulbecco’s Medium, GIBCO/Invitrogen) will maintain a constant pH of 7.4 throughout long term perfusion culture under atmospheric air on a laboratory table.
To obtain in a perfusion culture experiment a high saturation of O 2 a selected medium such as IMDM has to be transported through a gas permeable silicone tube. The use of a silicone tube provides a large surface for gas exchange by diffusion due to a thin wall (1 mm), the small inner diameter (1 mm) and its extended length (1 m). For example, analysis of IMDM (3024 mg/L NaHCO 3 , 50 mmol/L HEPES) equilibrated against atmospheric air during a standard perfusion culture experiment shows constant partial pressures of at least 160 mmHg O 2 [7].
It has been shown that growing cells and tissues have very individual oxygen requirements. Due to this reason it is important that the content of oxygen can be adapted in individual perfusion culture experiments. The technical solution is a gas exchanger module containing a gas inlet and outlet (Fig. 6a). Further a spiral with a long thin-walled silicon tube for medium transport is mounted inside the module. Since the tube is highly gas-permeable, it guarantees optimal diffusion of gases between culture medium and internal atmosphere of the gas exchange module. In consequence, the desired gas atmosphere can be adjusted by a constant flow of a specific gas mixture through the module. This way the content of oxygen or any other gases can be modulated in the medium by diffusion. Applying this simple protocol it became possible to decrease the oxygen partial pressure within the transported medium during long term culture experiments under absolutely sterile conditions [7].
Performing perfusion culture experiments it always has to be considered that gas bubbles are forming during slow transport of culture medium. They arise during suction of medium in the storage bottle, during transport within the tube, during distribution within the culture container and during elimination on the way to the waste bottle. Due to unknown reasons gas bubbles accumulate especially at material transitions between tubes, connectors and perfusion containers. First these gas bubbles are so small that they cannot be observed with the human eye, but during ongoing transport of culture medium they increase in size and are able to form an embolus that massively impedes medium flow. Within a culture container gas bubbles are leading to a regional shortage of medium supply and are causing breaks in the fluid continuum so that massive fluid pressure changes result. In a gradient perfusion culture container, where two media are transported at exactly the same speed, embolic effects can lead to pressure differences destroying in turn the contained epithelial barrier [5,9].
To avoid the concentration of gas bubbles within a perfusion culture experiment, a gas expander module was developed (Fig. 6b). This module removes gas bubbles from the medium during transport. When medium is entering the gas expander module, it rises within a small reservoir and expands before it drops down a barrier. During this process gas bubbles are separated from the medium at the top of the gas expander module. In consequence, medium leaving the container is oxygen-saturated but free of gas bubbles [8,9].
In the last years numerous papers were published dealing with the Minusheet perfusion culture system. The wide spectrum illustrates that the modular system was applied to generate specialized tissues in excellent cell biological quality used in tissue engineering, biomaterial research and advanced pharmaceutical drug toxicity testing. A complete list of these applications is found in the data bank ‘Proceedings in perfusion culture’ (see 'External links').
As demonstrated by numerous patents (DE 39 23 279, DE 42 00 446, DE 42 08 805, DE 44 43 902, DE 19530 556, DE 196 48 876 C2, DE 199 52 847 B4, US 5 190 878, US 5 316 945, US 5 665 599, J 2847669, DE 10 2005 002 938, PA 10 2004 054 125.6, PA 10 2005 001 747.9, patents pending) Will W. Minuth has invented the presented Minusheet perfusion culture system.
Numerous pilot experiments with the Minusheet perfusion culture system were performed in the last years by Lucia Denk and Will W. Minuth. The experimental work is presently focusing on the creation of an artificial polyester interstitium to repair injured renal parenchyma.
In 1992 the Minusheet perfusion culture system received the Philip Morris research award ‘Challenge of the Future’ in Munich , Germany. The award was handed over by Henry Kissinger , Hans Joachim Friedrichs and Paul Müller .
To introduce the Minusheet perfusion culture system on the market, Katharina Lorenz-Minuth founded non-profit orientated Minucells and Minutissue Vertriebs GmbH (D-93077 Bad Abbach/Germany). | https://en.wikipedia.org/wiki/Minusheet_perfusion_culture_system |
Minze Stuiver (25 October 1929 – 26 December 2020) was a Dutch geochemist who was at the forefront of geoscience research from the 1960s until his retirement in 1998. He helped transform radiocarbon dating from a simple tool for archaeology and geology to a precise technique with applications in solar physics, oceanography, geochemistry, and carbon dynamics. Minze Stuiver's research encompassed the use of radiocarbon ( 14 C) to understand solar cycles and radiocarbon production, ocean circulation, lake carbon dynamics and archaeology as well as the use of stable isotopes to document past climate changes.
Minze Stuiver was born in Vlagtwedde , the Netherlands , on 25 October 1929. [ 1 ] As a boy he narrowly missed being taken into German forced labor toward the end of the Second World War , but, because he was away delivering milk by bicycle, he escaped the round-up that took most of the young men and older boys from the village. His secondary school education was disrupted by the war when the school was occupied by German soldiers and air raids interrupted classes in makeshift rooms. After the war he went to the University of Groningen , where he studied physics, mathematics and astronomy, focusing on nuclear physics. After graduation he joined the biophysics group led by the pre-eminent researcher Hessel de Vries and received a Ph.D. in Biophysics in 1958 with a thesis on the Biophysics of the Sense of Smell. [ 2 ] Shortly thereafter he began working in the rapidly developing field of radiocarbon dating with de Vries, who found variations in the concentration of radiocarbon in the atmosphere which challenged the assumptions of the radiocarbon dating method. In 1959, together with his wife, Anneke, Minze went to Yale University for a one-year fellowship position but was called back to Groningen to take over as director of the radiocarbon facility when De Vries died. [ 3 ] However Minze chose to remain in the United States at the Geochrometric Laboratory at Yale University . There he developed high-precision methods in radiocarbon that enabled him, along with Hans Suess , to verify De Vries’ “wiggly” nature of the atmospheric concentration of radiocarbon in the past from tree-rings. Stuiver and Suess created one of the first curves for calibration of radiocarbon dates . [ 4 ] In 1969 Minze moved to the newly founded Quaternary Research Center at the University of Washington (UW) in Seattle . There he built the Quaternary Isotope Lab with a lead-lined room 30 feet below ground to shield the hand-built gas counters from detecting spurious events due to cosmic rays .
In the 1970s Minze began measuring 14 C in dissolved inorganic carbon in ocean water as part of The Geochemical Ocean Sections Study (GEOSECS) to study the distribution of carbon in the ocean. [ 5 ] [ 6 ] [ 7 ] In addition he was involved in a number of studies on the glacial histories of Antarctica and North America. [ 8 ] [ 9 ] He was the senior editor of the journal Radiocarbon from 1977 to 1988 and broadened the scope of the publication to include articles about scientific knowledge derived from radiocarbon measurements. By then the terminology for various ways to calculate and present radiocarbon data was becoming rather confusing. Together with Henry Polach, he formulated the equations and conventions for reporting radiocarbon data that is still widely used. [ 10 ] His work investigating atmospheric 14 C changes gave rise to a greater understanding of the changes in solar activity over time and potential links to climate change as well as the extent of fossil fuel input. [ 11 ] [ 12 ] [ 13 ] [ 14 ]
In the mid-1980s he led the development of the first high-precision radiocarbon calibration curve extending back nearly 10,000 years ago based on 14 C measurements of tree-rings with known calendar ages from dendrochronology . [ 15 ] This data still forms the backbone of the Holocene portion of the current international radiocarbon calibration curve which is used by archaeologists and geoscientists around the world. [ 16 ] He also oversaw the development of the CALIB computer software to automate the calibration process. [ 17 ] [ 18 ]
In the 1990s, in addition to continued work on radiocarbon calibration and solar variability, he began work on oxygen isotopes from Greenland ice cores together with Pieter Grootes. Their sub-annual resolution stable isotopes measurements provided confirmation of the rapid nature of major climatic changes at the end of the last glaciation. [ 19 ] [ 20 ]
Stuiver died on 26 December 2020, at the age of 91. [ 21 ] | https://en.wikipedia.org/wiki/Minze_Stuiver |
Minä Peräsmies is a 1998 Finnish PC-ROM for Windows that consists of comics, games and other content based on the superhero character of Peräsmies who is able to fly by farting "with the power of a thousand hurricanes ". The ROM was created by a team at the media company Mediakeisari Oy including Timo Kokkila (the artist), [ 1 ] Petri Tuomola and Reima Mäkinen and published and sold by Plan1 Oy. [ 2 ] The character became known from the Finnish comic and humor magazine Pahkasika and strips were released from 1983 to 2000.
The ROM includes eight comics, five games (one of which is a printable board game), the Food Circle of Peräsmies and the Museum Center Fiasco (a multimedia Fart Museum and a collection of images). The packaging also contains comic strips and other "stuff".
The ROM did not receive a mixed reception. Most of the published reviews conclude that it is essentially worth a single quick viewing with some decent artwork and momentarily interesting technical work including the wide array of fart sounds. [ 3 ]
A making-of has been released on YouTube in two parts. The documentary Näin tehtiin Minä Peräsmies consists of interviews and archive footage from the time of the ROM's making and was made by Reima Mäkinen. [ 4 ] [ 5 ] | https://en.wikipedia.org/wiki/Minä_Peräsmies |
Mipomersen ( INN ; trade name Kynamro ) is a drug used to treat homozygous familial hypercholesterolemia and is administered by subcutaneous injection . There is a serious risk of liver damage from this drug and it can only be prescribed in the context of a risk management plan.
Kynamro is used to treat homozygous familial hypercholesterolemia and is administered by injection. [ 1 ] [ 2 ]
It cannot be freely prescribed; instead every person put on mipomersen is enrolled in a Risk Evaluation and Mitigation Strategies (REMS) program approved by the FDA. [ 1 ]
Mipomersen is pregnancy category B; women who are pregnant or intending to become pregnant should only use this drug if needed. It is unknown if it is secreted in human breast milk, but it was found to be secreted in the breast milk of rats. [ 1 ]
The drug is contraindicated in people with moderate to severe liver impairment, active liver diseases, and unexplained high levels of transaminase liver enzymes. [ 1 ] [ 3 ]
The drug has a black box warning about the risk of liver damage ; specifically it can cause elevations in the levels of transaminases and causes fatty liver disease . [ 1 ]
In clinical trials, 18% of subjects taking mipomersen stopped using the drug due to adverse effects; the most common adverse effects leading to discontinuation were injection site reactions, increases of transaminases, flu-like symptoms (fever, chills, abdominal pain, nausea, vomiting), and abnormal liver tests. [ 1 ]
Other adverse effects include: heart problems including angina and palpitations, edema , pain in legs or arms, headache, insomnia, and hypertension. [ 1 ]
Other drugs known for causing liver problems might add to mipomersen's risk of liver damage. No pharmacokinetic interactions have been described. [ 3 ]
Mipomersen binds to the messenger RNA coding for apolipoprotein B-100 (ApoB-100), a protein that is the main component of low-density lipoprotein (LDL) and very low-density lipoprotein (VLDL). As a consequence, the RNA is degraded by the enzyme ribonuclease H , and ApoB-100 is not translated . [ 3 ]
After subcutaneous injection, mipomersen reaches highest blood levels after 3 to 4 hours. It accumulates in the liver , [ citation needed ] which is convenient since apolipoprotein B predominantly acts there. Protein binding is over 90%. The molecule is slowly broken up by endonucleases and subsequently by exonucleases . After 24 hours, less than 4% of the degradation products are found in the urine, and overall half-life is 1 to 2 months. [ 3 ]
The compound is a 'second-generation' antisense oligonucleotide ; the nucleotides are linked with phosphorothioate linkages rather than the phosphodiester linkages of RNA and DNA , and the sugar parts are deoxyribose in the middle part of the molecule and 2’- O -methoxyethyl-modified ribose at the two ends. These modifications make the drug resistant to degradation by nucleases , allowing it to be administered weekly.
The complete sequence is portrayed below: [ 5 ] [ 1 ] : 10
The drug was discovered and developed to Phase 2 by Ionis Pharmaceuticals and subsequently licensed to Genzyme Corporation in 2008 by an auction bid. Ionis earned an upfront payment of $325 million, with payments of a further $825 million if milestones are met. [ 6 ]
Mipomersen was rejected by the European Medicines Agency in 2012 [ 7 ] and again in 2013 due to concerns about the liver and cardiovascular adverse effects. [ 8 ]
In January 2013, The United States Food and Drug Administration approved mipomersen for the treatment of homozygous familial hypercholesterolemia . [ 9 ] [ 10 ] | https://en.wikipedia.org/wiki/Mipomersen |
In geometry , the Miquel configuration is a configuration of eight points and six circles in the Euclidean plane , (8 3 6 4 ), with four points per circle and three circles through each point. [ 1 ]
Its Levi graph is the rhombic dodecahedral graph , the skeleton of the rhombic dodecahedron . The configuration is related to Miquel's theorem .
The configuration has maximal symmetry in 3-dimension, and can be seen as 6 circles circumscribe the square faces of a cube . It has 12 sets of pairwise circle intersections, corresponding to the edges of the cube and octahedron. Structurally it has 48 automorphisms of octahedral symmetry .
If two opposite circles are removed the configuration becomes (8 2 4 2 ), with 128 automorphisms (4 rotations by 2 3 pair interchanges)
A different (8 3 6 4 ) can be found as with 6 central circles on a cube. The circles are on the 6 mirror planes of tetrahedral symmetry . In full it has 384 automorphisms of hyperoctahedral symmetry as the maximal geometric symmetry can be seen in 6, C(4,2) , orthogonal circles as central squares in a 16-cell .
The dual configuration (6 4 8 3 ) can be drawn with the 6 vertices of an octahedron and the 8 circles circumscribe the 8 triangular faces.
Taking half of the circles makes (6 2 4 3 ) with tetrahedral symmetry and 24 automorphisms. This is isomorphic to the point-line configuration complete quadrilateral .
Three central circles can also go through the same 6 vertices, and can be seen as square faces in the tetrahemihexahedron .
This geometry-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Miquel_configuration |
In molecular biology mir-598 microRNA is a short RNA molecule. MicroRNAs function to regulate the expression levels of other genes by several mechanisms.
This genetics article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Mir-598_microRNA_precursor_family |
In molecular biology mir-885 microRNA is a short RNA molecule. MicroRNAs function to regulate the expression levels of other genes by several mechanisms.
The miR-885-5p form of this microRNA acts as a tumour suppressor in neuroblastoma , through interference with cell cycle progression and cell survival. It is found at 3p25.3, a chromosome region frequently deleted in primary neuroblastoma, [ 1 ] and expression results in p53 protein accumulation and pathway activation. Altered expression of multiple genes is observed with miR-885-5p, including the CDK2 and MCM5 genes encoding cyclin-dependent kinase 2 and mini-chromosome maintenance protein MCM5 , and also with several p53 target genes. [ 1 ]
Circulating miRNAs (microRNAs) are emerging as promising biomarkers for several pathological conditions. similarly, miR-885-5p found as a potential marker for liver disease condition. It is significantly elevated in the patients sera with liver pathologies, and researcher suggested that serum miRNAs could serve as novel complementary biomarkers for the detection and assessment of liver pathologies. [ 2 ] These unique miRNAs may be clinically applicable to predict prognosis and distant metastasis in colorectal cancer (CRC). The direct comparison of expression patterns of metastasis-specific microRNAs (miRNAs) in primary CRCs (pCRCs) and matched liver metastases (LMs) provides a feasibility of their clinical application as metastasis-specific biomarkers. [ 3 ] In a clinical study it has been found that miR-885-5p is increased in plasma from pre-eclampsia compared with healthy pregnant women, and it is released into circulation mainly inside exosomes. [ 4 ]
In search of an effective therapeutic strategy for improving colon cancer treatment, a novel role of miR-885-3p has been observed in tumor angiogenesis by targeting BMPR1A, which regulates a proangiogenic factor, and provide new evidence that targeting miRNAs might be an effective therapeutic strategy. [ 5 ]
This genetics article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Mir-885_microRNA_precursor_family |
Miracle of the cruse [ a ] of oil ( Hebrew : נֵס פַּךְ הַשֶּׁמֶן ), or the Miracle of Hanukkah , is an Aggadah depicted in the Babylonian Talmud [ 1 ] as one of the reasons for Hanukkah . In the story, the miracle occurred after the liberation of the Temple in Jerusalem during the Maccabean Revolt , and it describes the finding of a jug of pure oil that was to be enough to light the lamp for one day, but that lasted for eight days.
During the period of the Second Temple (~516 BCE – 70 CE), in around 200 BCE, Antiochus III , the Seleucid king of Syria, took control over the kingdom of Judea. He allowed the Jews living in Judea autonomous rule for some time, but then his son Antiochus IV replaced him. Trying to unify his kingdom, Antiochus IV prohibited Jews from practicing Judaism and commanded them to worship Greek gods. Many Jews went along with these demands and were known as Hellenized Jews. Tensions rose and in 168 BCE Antiochus IV invaded Jerusalem, killing thousands, and erected an altar to Zeus in the temple. [ 2 ] Along with a statue of Zeus, the sacrifice of pigs (a normal practice in ancient Greek worship) began to take place in the temple.
This resulted in civil war between ruling Hellenized factions that had been installed by Antiocus IV, and traditionalists. The rebels were led by the priest Matityahu of Modiin, known as the Maccabees. After two years of guerrilla warfare, the rebels were able to overthrow Seleucid rule. [ 3 ]
After expelling the Seleucids from Jerusalem, the Second Temple had to be purified and rededicated, due to the defilement the Seleucids had inflicted. A key component of the rededication ritual was the lighting of the Menorah . According to the story, the menorah needed to burn for eight nights. However, only a single cruse of untainted (as evidenced by bearing an unbroken seal) olive oil was found throughout all of Jerusalem--enough to last only one night. The oil, nevertheless, is said to have lasted for eight nights, constituting a miracle.
The Torah discusses the lighting of the Temple menorah in a number of verses. Leviticus 24:2 specifies that pure olive oil must be used to light the menorah. While Exodus 25:37 and Numbers 8:2–3 speak of seven lights being lit, Exodus 27:20–21 and Leviticus 24:2 specifies that a single "light" must be lit "continually", and must burn "from evening to morning". Thus, according to Jewish tradition, priests ensured that all seven menorah lights were lit during the day, while just one of them (the ner maaravi or ner tamid ) burned through the entire night. [ 4 ]
After defeating the Syrian-Greeks in the battlefield, the first of the two miracles celebrated on Hanukkah, the Maccabees returned to Jerusalem and to the Temple. There, they wanted to rededicate the Temple through the resumption of the performance of Temple rituals. One of these rituals was the lighting of the Menorah, however the Menorah could only be lit with pure olive oil. When the Greeks entered the Temple they had defiled almost all the jugs of oil. [ 1 ]
As the Maccabees searched for pure oil to light the menorah with, they found just one cruse of pure oil which still had the seal of the High Priest, the symbol of pure oil. This cruse contained just enough pure oil to keep the menorah lit for one day. In order to make pure oil however, individuals making the oil must be in a state of spiritual purity . Being soldiers returning from the battlefield, the Maccabees were deemed impure, and therefore could not make pure oil. Since the process of ritual purification after touching a corpse lasts seven days, the Maccabees could only produce additional pure oil after eight days: seven days of becoming pure including one day, once pure, to actually make the oil. Therefore, the Maccabees would have been unable to light the Menorah for seven days before the completion of new pure oil. Miraculously, the one cruse of oil had lasted for all eight days, and by that point new pure oil was ready.
Rabbi Joseph Karo is known for asking why Hanukkah is celebrated for eight days, if the oil was expected to last one day, so seemingly only the last seven days were miraculous. This question became famous and many answers were suggested to it; indeed a book was published in 1962 compiling 100 different proposed answers, with a more recent work, published in 2019, containing 1,000 answers. [ 5 ] The answers generally fall into three categories: arguing that the oil miracle actually lasted eight days in some way; arguing that the eighth day is celebrated for a reason other than the oil miracle; and arguing that the eight-day length of Hanukkah is unrelated to the oil miracle. [ 6 ] [ 7 ]
The Talmud , after recounting the story of the miracle of the cruse of oil, continues, "The following year these [days] were appointed a Festival with [the recital of] Hallel (Jewish praise, recited on all festivals) and thanksgiving." [ 1 ] Since then, the festival of Hanukkah has been celebrated each year, beginning on the 25th of Kislev . During these days, lamentation for the dead and fasting are forbidden, Hallel is recited, and the Menorah is lit. [ 1 ]
While lighting the Menorah on Hanukkah was originally established solely to commemorate the miracle of the cruse of oil, after the destruction of the Second Temple, the holiday took on an additional role. It now also serves as a commemoration of the daily lighting of the Menorah in the Temple, and the Temple in general. [ citation needed ]
Today, the Jewish holiday of Hanukkah lasts eight days to remember, and celebrate, the miracle of the one cruse of oil lasting eight days. One candle is lit on the first night in addition to the shammash, and a candle is added each night. Ultimately, nine candles are lit on the final night of the holiday, including the shammash. Traditionally, after the lighting of the Menorah, Ma'oz Tzur is sung in honor of the holiday. Latkes , among other oily food, are also eaten on Hanukkah in honor of the miracle of the cruse of oil. | https://en.wikipedia.org/wiki/Miracle_of_the_cruse_of_oil |
Miraculin is a taste modifier , a glycoprotein extracted from the fruit of Synsepalum dulcificum . [ 2 ] The berry, also known as the miracle fruit, was documented by explorer Chevalier des Marchais , who searched for many different fruits during a 1725 excursion to its native West Africa.
Miraculin itself does not taste sweet . When taste buds are exposed to miraculin, the protein binds to the sweetness receptors. This causes normally sour-tasting acidic foods, such as citrus , to be perceived as sweet. [ 2 ] [ 3 ] The effect can last for one or two hours. [ 4 ] [ 5 ]
The sweetening properties of Synsepalum dulcificum berries were first noted by des Marchais during expeditions to West Africa in the 18th century. [ 6 ] The term miraculin derived from experiments to isolate and purify the active glycoprotein that gave the berries their sweetening effects, results that were published simultaneously by Japanese and Dutch scientists working independently in the 1960s (the Dutch team called the glycoprotein mieraculin ). [ 7 ] [ 8 ] The word miraculin was in common use by the mid-1970s. [ 9 ] [ 10 ] [ 11 ]
Miraculin was first sequenced in 1989 and was found to be a 24.6 kilodalton [ 2 ] glycoprotein consisting of 191 amino acids [ 12 ] and 13.9% by weight of various sugars. [ 2 ]
The sugars consist of a total of 3.4 kDa, composed of a molar ratio of glucosamine (31%), mannose (30%), fucose (22%), xylose (10%), and galactose (7%). [ 2 ]
The native state of miraculin is a tetramer consisting of two dimers , each held together by a disulfide bridge. [ 14 ] Both tetramer miraculin and native dimer miraculin in its crude state have the taste-modifying activity of turning sour tastes into sweet tastes. [ 15 ] Miraculin belongs to the Kunitz STI protease inhibitor family.
Miraculin, unlike curculin (another taste-modifying agent), [ 16 ] is not sweet by itself, but it can change the perception of sourness to sweetness, even for a long period after consumption. [ 4 ] The duration and intensity of the sweetness-modifying effect depends on various factors, such as miraculin concentration, duration of contact of the miraculin with the tongue, and acid concentration. [ 3 ] [ 4 ] Miraculin reaches its maximum sweetness with a solution containing at least 4*10 −7 mol/L miraculin, which is held in the mouth for about 3 minutes. Maximum is equivalent in sweetness to a 0.4 mol/L solution of sucrose . [ 17 ] Miraculin degrades permanently via denaturation at high temperatures and at pH below 3 or above 12. [ 18 ]
Although the detailed mechanism of the taste-inducing behavior is unknown, it appears the sweet receptors are activated by acids which are related to sourness, an effect remaining until the taste buds perceive a neutral pH. [ 3 ] [ 4 ] Sweeteners are perceived by the human sweet taste receptor, hT1R2-hT1R3, which belongs to G protein-coupled receptors , [ 4 ] modified by the two histidine residues (i.e. His30 and His60) which participate in the taste-modifying behavior. [ 19 ] One site maintains the attachment of the protein to the membranes while the other (with attached xylose or arabinose ) activates the sweet receptor membrane in acid solutions. [ 14 ]
As miraculin is a readily soluble protein and relatively heat stable, it is a potential sweetener in acidic food, such as soft drinks . While attempts to express it in yeast and tobacco plants have failed, researchers have succeeded in preparing genetically modified E. coli bacteria that express miraculin. [ 20 ] Lettuce and tomato have also been used for mass production of miraculin. [ 21 ] [ 22 ]
The use of miraculin as a food additive was denied in 1974 by the United States Food and Drug Administration . [ 23 ] However, it can still be sold in the form of whole berries or tablets (as "dietary supplements"). [ 24 ] [ 25 ] In 2011 the FDA banned a certain brand of miraculin tablets imported from Taiwan as it was thought to be "hard candy" with non-approved sweeteners. [ 26 ] Miraculin has a novel food status in the European Union . [ 27 ] It is approved in Japan as a safe food additive , according to the List of Existing Food Additives published by the Ministry of Health and Welfare (published by the Japan External Trade Organization ). | https://en.wikipedia.org/wiki/Miraculin |
Miray Bekbölet is a Turkish environmental chemist researching oxidation techniques, photocatalytic and photolytic reactions, adsorption/bio-oxidation processes in aquatic systems, and drinking water quality. She is a professor of environmental chemistry at the Boğaziçi University Institute of Environmental Sciences. [ 1 ]
Miray Bekbölet completed a B.S. with high distinction in chemistry and physics at Ege University in 1973. In 1979, she earned a Ph.D. in food sciences at Ege University. [ 2 ]
Bekbölet joined the faculty at Boğaziçi University in 1985 as an instructor in the Institute of Environmental Sciences. She was promoted to assistant professor in 1986, associate professor in 1991 and professor in 1997. [ 3 ]
Bekbölet's researches oxidation techniques, photocatalytic and photolytic reactions, adsorption/bio-oxidation processes in aquatic systems, and drinking water quality. [ 3 ] | https://en.wikipedia.org/wiki/Miray_Bekbölet |
Mircea Dincă (born 1980) is a Romanian-American inorganic chemist. He is the Andrew Stewart 1886 Professor of Chemistry at Princeton University. [ 2 ] At Princeton, Dincă leads a research group that focuses on the synthesis of functional metal-organic frameworks (MOFs), which possess conductive, catalytic, and other material-favorable properties.
Mircea Dincă was born in Făgăraș , Romania. His passion for chemistry began in his chemistry class in 7th grade, where he had a "dedicated teacher that did spectacular demonstrations with relatively limited regard for safety". In 1998, he represented Romania at the International Science Olympiad (Chemistry) in Yakutsk , Russia, where he won first prize. [ 3 ]
After high school, Dincă was offered a scholarship from Princeton University and moved to New Jersey in 1999. At Princeton, he worked with Jeffrey Schwartz , conducting research on materials science . After graduating magna cum laude in 2003, Dincă went on to the University of California, Berkeley to attend the Chemistry doctorate program, where he worked with chemistry professor Jeffrey R. Long on increasing H 2 adsorption in metal-organic frameworks with mobile hydrogen storage applications. He graduated with his Ph.D. from Berkeley in 2008. [ 5 ]
Dincă completed his postdoc studies at MIT, where he was promoted to associate professor in 2010 and, in 2017, tenured. Until 2024, he was the W. M. Keck Professor of Chemistry. [ 6 ] [ 5 ] In 2025, he joined the faculty at Princeton University. [ 2 ]
Dincă's research primarily focuses on electrical conductivity of MOF's, which was previously unknown and resulted in a new categorization of such materials with "charge mobility values". [ 7 ] His focus is on the exploration of increasing electrical conductivity capacities through the marriage of organic and inorganic materials to assemble hybrid MOF's.
Research includes exploring electrochemical cycling through strongly adhering, electroactive metal–organic framework thin films to vary results, such as multicolored electrochromic responses [ 8 ] and transparent to dark behavior. [ 9 ] | https://en.wikipedia.org/wiki/Mircea_Dincă |
Mireia Boya Busquet (born 1979) is a Catalan scientist, activist, and politician from Spain . She is a councilor of the Aran municipality of Les . Since February 2018 she has been part of the national secretariat of the Popular Unity Candidacy (CUP). She was a member of the Catalan Parliament from 2016 to 2017.
The daughter of Ernesto Boya and the politician Maria Pilar Busquets , she has a degree in Environmental Sciences from the Autonomous University of Barcelona (2002), a master's degree in Landscape Design (2004), and a PhD in Land Management and Planning (2009) from the Université de Montréal . In the professional field, she is a consultant and adjunct lecturer [ 1 ] ("asociada") of the Humanities Department of Pompeu Fabra University . [ 2 ]
Mireia Boya Busquet has been a fighter for the recognition of Occitan identity of the Valley of Aran, and was the first member of the Catalan Parliament to use the Aranese dialect in the Parliament of Catalonia . [ 3 ] Her brother, Jusèp Boya Busquet [ es ] , has been general director of Archives, Libraries, Museum, and Heritage of the Generalitat beginning in January 2016, an office that he was dismissed from after the 2017 suspension of Catalonian self-government.
Boya is a founder and former coordinator of the Assemblea Nacional Catalana in Val d'Aran and councilor in the city of Les for the Corròp party, framed in Occitan nationalism . In the 2015 elections to the Parliament of Catalonia , she was a candidate of the CUP for the district of Lleida , after a vote in primaries. Boya, who was second in the list in her constituency, was not elected, but on 18 December 2015, during the tense negotiations between the CUP and the pro-independence coalition Junts pel Sí , the CUP's number one, Ramon Usall, resigned for personal reasons. [ 4 ] In this way, Boya occupied the post of regional parliamentarian. [ 5 ] [ 6 ] She participated in the plenary sessions of the Parliament that voted for the roadmap to independence for Catalonia, and voted the Unilateral declaration of Independence of Catalonia.
In the elections of 21 December 2017, of the 5,265 votes cast in Val d'Aran, the CUP received 174 – 3.3% of the total. [ 7 ]
In February 2018, she was elected a member of the CUP National Secretariat as an independent candidate with 662 votes. [ 8 ]
A defender of the Occitan nation , Boya upholds a "project of a country with two languages of its own", a country that looks "to the north, to Occitania". [ 9 ]
On 22 December 2017, Judge Pablo Llarena of the Supreme Court of Spain agreed on the inquiry (previously charged) for rebellion against Mireia Boya (president of the CUP's parliamentary group), Artur Mas (president of the PDeCat ), Marta Rovira (secretary general of the ERC ), Anna Gabriel (spokesperson of the CUP), Marta Pascal (general coordinator of the CUP), and Neus Lloveras [ es ] (president of the AMI ), all for belonging to the organizing team of the Catalan independence referendum held on 1 October 2017 and with a decisive role in the secessionist plan, whose roadmap was annulled by the Spanish Constitutional Court. [ 10 ] [ 11 ] Boya was declared as investigated on 14 February 2018, being released without precautionary measures. [ 12 ] | https://en.wikipedia.org/wiki/Mireia_Boya_Busquet |
Mirex is an organochloride that was commercialized as an insecticide and later banned because of its impact on the environment. This white crystalline odorless solid is a derivative of both cyclopentadiene and cubane . It was popularized to control fire ants but by virtue of chemical robustness and lipophilicity it was recognized as a bioaccumulative pollutant. The spread of the red imported fire ant was encouraged by the use of mirex, which also kills native ants that are highly competitive with the fire ants. The United States Environmental Protection Agency prohibited its use in 1976. [ 1 ] It is prohibited by the Stockholm Convention on Persistent Organic Pollutants .
Mirex was first synthesized in 1946, [ 2 ] but was not used in pesticide formulations until 1955. Mirex was produced by the dimerization of hexachlorocyclopentadiene in the presence of aluminium chloride .
Mirex is a stomach insecticide, meaning that it must be ingested by the organism in order to poison it. The insecticidal use was focused on Southeastern United States to control fire ants . Approximately 250,000 kg of mirex were applied to fields between 1962 and 1975 (US NRC, 1978). Most of the mirex was in the form of "4X mirex bait", which consists of 0.3% mirex in 14.7% soybean oil mixed with 85% corncob grits. Application of the 4X bait was designed to give a coverage of 4.2 g mirex/ha and was delivered by aircraft, helicopter or tractor. 1x and 2x bait were also used. Use of mirex as a pesticide was banned in 1978. The Stockholm Convention banned production and use of several persistent organic pollutants , and mirex is one of the "dirty dozen". [ 3 ]
Much like other perchlorocarbons such as carbon tetrachloride , mirex does not burn easily; pyrolysis products are expected to include carbon dioxide , carbon monoxide , hydrogen chloride , chlorine , phosgene , and possibly other organochlorine species. Slow oxidation of mirex can be used to produce chlordecone ("Kepone"), a related insecticide that is also banned in most of the western world, but is more readily biodegraded. Sunlight degrades mirex to photomirex (8-monohydromirex) and 2,8-dihydromirex. [ 1 ] [ 4 ] [ 5 ]
Mirex is highly resistant to microbiological degradation. It only slowly dechlorinates to a monohydro derivative by anaerobic microbial action in sewage sludge and by enteric bacteria . Degradation by soil microorganisms has not been described. [ citation needed ]
Mirex is highly cumulative and amount depends upon the concentration and duration of exposure. There is evidence of accumulation of mirex in aquatic and terrestrial food chains to harmful levels. After 6 applications of mirex bait at 1.4 kg/ha, high mirex levels were found in some species; turtle fat contained 24.8 mg mirex/kg, kingfishers, 1.9 mg/kg, coyote fat, 6 mg/kg, opossum fat, 9.5 mg/kg, and racoon fat, 73.9 mg/kg. In a model ecosystem with a terrestrial-aquatic interface, sorghum seedlings were treated with mirex at 1.1 kg/ha. Caterpillars fed on these seedlings and their faeces contaminated the water which contained algae, snails, Daphnia, mosquito larvae, and fish. After 33 days, the ecological magnification value was 219 for fish and 1165 for snails. [ citation needed ]
Although general environmental levels are low, it is widespread in the biotic and abiotic environment. Being lipophilic, mirex is strongly adsorbed on sediments. [ citation needed ]
Mirex is only moderately toxic in single-dose animal studies (oral LD 50 values range from 365–3000 mg/kg body weight). [ 6 ] It can enter the body via inhalation, ingestion, and via the skin. The most sensitive effects of repeated exposure in animals are principally associated with the liver, and these effects have been observed with doses as low as 1.0 mg/kg diet (0.05 mg/kg body weight per day), the lowest dose tested. At higher dose levels, it is fetotoxic (25 mg/kg in diet) and teratogenic (6.0 mg/kg per day). Mirex was not generally active in short-term tests for genetic activity. There is sufficient evidence of its carcinogenicity in mice and rats. [ citation needed ] Delayed onset of toxic effects and mortality is typical of mirex poisoning. Mirex is toxic for a range of aquatic organisms, with crustacea being particularly sensitive. [ citation needed ]
Mirex induces pervasive chronic physiological and biochemical disorders in various vertebrates. No acceptable daily intake (ADI) for mirex has been advised by FAO/WHO. IARC (1979) evaluated mirex's carcinogenic hazard and concluded that "there is sufficient evidence for its carcinogenicity to mice and rats. In the absence of adequate data in humans, based on above result it can be said, that it has carcinogenic risk to humans". Data on human health effects do not exist [ citation needed ] .
Per a 1995 ATSDR report mirex caused fatty changes in the livers, hyperexcitability and convulsion, and inhibition of reproduction in animals. It is a potent endocrine disruptor, interfering with estrogen-mediated functions such as ovulation, pregnancy, and endometrial growth. It also induced liver cancer by interaction with estrogen in female rodents. [ 7 ] | https://en.wikipedia.org/wiki/Mirex |
Miriam Mason Higgins Thomas (June 22, 1920 – September 15, 2002) was an American chemist, based in the United States Army Research and Development Command at Natick, Massachusetts .
Miriam Mason Higgins was born in Chicago , the daughter of William Henry Higgins and Mame Mason Higgins. Her mother was an alumna of the University of Chicago , dean of women at Bethune-Cookman College , and a social service consultant for the Illinois Department of Public Aid. [ 1 ] [ 2 ] Her brother, William H. Higgins, was a dentist and ordained Methodist minister. [ 3 ] Her grandfather, M. C. B. Masons, was a noted orator and Black church leader. [ 4 ]
Higgins graduated from Hyde Park High School in 1936, and earned a bachelor's degree in nutrition and chemistry from Bennett College in 1940. [ 5 ] She earned a master's degree in food chemistry from the University of Chicago. [ 6 ]
Thomas taught at the University of Chicago during World War II , and was a chemist with Food and Container Institute at the Chicago Quartermaster Depot beginning in 1945. [ 5 ] She was a research chemist at the U. S. Army Natick Development Center, studying nutritional content of military rations under various conditions. [ 7 ] In 1975 she won an Army SARS Fellowship to study food processing and nutrition analysis techniques in Japan, India, the Soviet Union, the Netherlands, and Guatemala. [ 6 ] She was a consultant to the Food Research Laboratories, Inc., of Boston, and taught nutrition and food science courses at the Massachusetts Institute of Technology . Her research was published in academic journals including Journal of Microwave Power and Journal of Food Science . [ 8 ]
Thomas was nominated three times by the Department of the Army for the Federal Woman's Award . She was a member of the Association of Vitamin Chemists, the Society for Nutrition Education , and the American Association for the Advancement of Science . [ 6 ]
Miriam Higgins married before 1960 and had a son, Brian. [ 21 ] Thomas died in 2002, aged 82 years. Her papers are in the National Archives for Black Women's History . [ 22 ] | https://en.wikipedia.org/wiki/Miriam_Higgins_Thomas |
Miriam Margarethe Unterlass (born 1986 in Erlangen, West Germany) is a German chemist. She is full professor of solid state chemistry at the University of Konstanz, as well as adjunct principal investigator at CeMM - Research Center for Molecular Medicine of the Austrian Academy of Sciences. On 1 October 2024, Prof. Miriam Unterlass took over the management of the renowned Fraunhofer Institute for Silicate Research ISC in Würzburg.
Miriam M. Unterlass was born in 1986 in Erlangen, Germany. She studied chemistry, process engineering and materials science in the framework of a double diploma degree in Würzburg, Germany, in Lyon, France, and in Southampton, United Kingdom. She completed her PhD under the supervision of Professor Markus Antonietti at the Max Planck Institute of Colloids and Interfaces in Potsdam-Golm, Germany. In 2011 she obtained her doctoral degree ( magna cum laude ) at the University of Potsdam, Germany, with her doctoral work entitled "From Monomer Salts and Their Tectonic Crystals to Aromatic Polyimides: Development of Neoteric Synthesis Routes".
In 2011 she continued her career with a postdoc in the Centre national de la recherche scientifique (CNRS) Laboratory Soft Matter and Chemistry under supervision of Professor Ludwik Leibler at the École supérieure de physique et de chimie industrielles de la ville de Paris (ESPCI). In 2012 she was a visiting scholar at Massachusetts Institute of Technology (MIT) hosted by Professor Gregory C. Rutledge. Later that year, she started as an independent group leader of the research group "Advanced Organic Materials" at the Institute of Materials Chemistry of the Vienna University of Technology (TU Wien). She habilitated ( venia docendi ) in materials chemistry at TU Wien in 2018 and became assistant professor with tenure in 2019.
In 2018 she joined CeMM - Research Center for Molecular Medicine of the Austrian Academy of Sciences and to date works as adjunct principal investigator. In 2021 she became an associate professor at TU Wien and since May 2021 she is full professor of solid state chemistry at the University of Konstanz , Germany. In 2022 she was guest professor at the Department of Chemical Science and Engineering of Institute of Science Tokyo (formerly known as Tokyo Institute of Technology ), Japan, hosted by Professor Shinji Ando.
Miriam Unterlass is a prominent researcher in the field of chemistry, known for her innovative work at the intersection of materials science and synthetic chemistry. Her research primarily focuses on the development of sustainable routes towards advanced materials and small molecules. The latter is based on the central hypothesis that water is able to be a near-universal solvent for chemical synthesis and processing.
She has made significant contributions to the understanding of the use of water as a core technology. Her group has demonstrated that water is an ideal medium to produce advanced materials, profiting from the properties of water under hydrothermal conditions. This approach utilizes hot liquid water as a reaction medium, producing a variety of materials, i.e. high-performance polymers suitable for aeronautics and microelectronics, small molecules relevant to biology and medicine or optoelectronics, and inorganic-organic hybrid materials. Moreover, her group employs modern computational and automation approaches to aim for maximal efficiency and discovery of new materials to address the various challenges of human life.
She has published over 40 peer-reviewed articles, contributed with more than 80 scientific talks at different conferences. Thus, she has submitted over 7 patents and patent applications, and actively works alongside industry partners to translate her findings into practical applications. | https://en.wikipedia.org/wiki/Miriam_M._Unterlass |
Miriam del Carmen Peña Cárdenas is a Chilean astronomer and cosmochemist whose research includes the chemical composition of interstellar clouds including H II regions and the planetary nebulae surrounding Wolf–Rayet stars . She is a professor and researcher at the National Autonomous University of Mexico (UNAM), in the UNAM Institute of Astronomy. [ 1 ]
Peña began her university studies in Chile, studying engineering, [ 2 ] but moved to the National Autonomous University of Mexico to complete her bachelor's degree, and remained there for her graduate studies. [ 1 ]
Peña is a member of the Mexican Academy of Sciences . [ 3 ] She was a 2007 winner of UNAM's Sor Juana Inés de la Cruz Recognition . [ 4 ] | https://en.wikipedia.org/wiki/Miriam_Peña_Cárdenas |
In number theory , a branch of mathematics , a Mirimanoff's congruence is one of a collection of expressions in modular arithmetic which, if they hold, entail the truth of Fermat's Last Theorem . Since the theorem has now been proven, these are now of mainly historical significance, though the Mirimanoff polynomials are interesting in their own right. The theorem is due to Dmitry Mirimanoff .
The n th Mirimanoff polynomial for the prime p is
In terms of these polynomials, if t is one of the six values {- X / Y , - Y / X , - X / Z , - Z / X , - Y / Z , - Z / Y } where X p + Y p + Z p =0 is a solution to Fermat's Last Theorem, then
Mirimanoff also proved the following:
3 p − 1 ≡ ( − 2 3 ⋅ { 1 + 1 2 + 1 3 + 1 4 + … + ⌊ p / 3 ⌋ − 1 } ) p + 1 ( mod p 2 ) {\displaystyle 3^{p-1}\equiv \left(-{\frac {2}{3}}\cdot \left\{1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+\ldots +\left\lfloor p/3\right\rfloor ^{-1}\right\}\right)p+1{\pmod {p^{2}}}}
so that a prime possesses the Mirimanoff property if it divides the expression within the curly braces. The condition was further refined in an important paper by Emma Lehmer (1938), in which she considered the intriguing and still unanswered question of whether it is possible for a number to satisfy the congruences of Wieferich and Mirimanoff simultaneously. To date, the only known Mirimanoff primes are 11 and 1006003 (sequence A014127 in the OEIS ). The discovery of the second of these appears to be due to K.E. Kloss (1965). | https://en.wikipedia.org/wiki/Mirimanoff's_congruence |
MIRO Analytical is a Swiss manufacturer of laser-based gas analyzers and isotope analyzers.
The company is based in Zurich, Switzerland and was founded 2018. [ 1 ]
MIRO Analytical is a spin-off of Empa, [ 2 ] a Swiss research institute of the ETH domain. It has know-how in laser spectroscopy and in particular, in the combination of several quantum-cascade lasers (QCLs) into compact laser-based gas analyzers. [ 3 ]
The company's first instrument was a nine gas analyzer MGA-9 in 2018. By 2019 the MGA-10 a ten gas analyzer was introduced which measures greenhouse gases and air pollutants. [ 4 ] [ 5 ]
The gas analyzers directly measure concentrations of multiple gas species using mid-infrared laser absorption spectroscopy with QCLs as light sources. This allows for highly specific and accurate gas detection along with maximum measurement sensitivity. [ 6 ] | https://en.wikipedia.org/wiki/Miro_Analytical |
In applied mathematics, the reflecting function F ( t , x ) {\displaystyle \,F(t,x)} of a differential system x ˙ = X ( t , x ) {\displaystyle {\dot {x}}=X(t,x)} connects the past state x ( − t ) {\displaystyle \,x(-t)} of the system with the future state x ( t ) {\displaystyle \,x(t)} of the system by the formula x ( − t ) = F ( t , x ( t ) ) . {\displaystyle \,x(-t)=F(t,x(t)).} The concept of the reflecting function was introduced by Uladzimir Ivanavich Mironenka .
For the differential system x ˙ = X ( t , x ) {\displaystyle {\dot {x}}=X(t,x)} with the general solution φ ( t ; t 0 , x ) {\displaystyle \varphi (t;t_{0},x)} in Cauchy form, the Reflecting Function of the system is defined by the formula F ( t , x ) = φ ( − t ; t , x ) . {\displaystyle F(t,x)=\varphi (-t;t,x).}
If a vector-function X ( t , x ) {\displaystyle X(t,x)} is 2 ω {\displaystyle \,2\omega } -periodic with respect to t {\displaystyle \,t} , then F ( − ω , x ) {\displaystyle \,F(-\omega ,x)} is the in-period [ − ω ; ω ] {\displaystyle \,[-\omega ;\omega ]} transformation ( Poincaré map ) of the differential system x ˙ = X ( t , x ) . {\displaystyle {\dot {x}}=X(t,x).} Therefore the knowledge of the Reflecting Function give us the opportunity to find out the initial dates ( ω , x 0 ) {\displaystyle \,(\omega ,x_{0})} of periodic solutions of the differential system x ˙ = X ( t , x ) {\displaystyle {\dot {x}}=X(t,x)} and investigate the stability of those solutions.
For the Reflecting Function F ( t , x ) {\displaystyle \,F(t,x)} of the system x ˙ = X ( t , x ) {\displaystyle {\dot {x}}=X(t,x)} the basic relation
is holding.
Therefore we have an opportunity sometimes to find Poincaré map of the non-integrable in quadrature systems even in elementary functions . | https://en.wikipedia.org/wiki/Mironenko_reflecting_function |
In a mirror furnace , material is heated by the lamps whose radiation is focused by mirrors. They are widely used for growing single crystals for scientific purposes, using the "floating zone" method.
This crystallography -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Mirror_furnace |
Mirror life (also called mirror-image life ) is a hypothetical form of life with mirror-reflected molecular building blocks. [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] The possibility of mirror life was first discussed by Louis Pasteur . [ 6 ] This alternative life form has never been discovered in nature, but efforts to build a mirror-image version of biology's molecular machinery are under way. [ 7 ]
In December 2024, a broad coalition of scientists, including leading synthetic biology researchers and Nobel laureates, warned that the creation of mirror life, including mirror bacteria, could cause "unprecedented and irreversible harm" to human health and ecosystems worldwide. [ 8 ] [ 9 ] Its potential to escape immune defenses and invade natural ecosystems might lead to "pervasive lethal infections in a substantial fraction of plant and animal species, including humans." Given these risks, the scientists concluded that mirror organisms should not be created without compelling evidence of safety. [ 8 ]
Many of the essential molecules for life on Earth can exist in two mirror-image forms, often called "left-handed" and "right-handed", where handedness refers to the direction in which polarized light skews when beamed through a pure solution of the molecule, but living organisms do not use both. [ 10 ] RNA and DNA contain only right-handed sugars ; proteins are exclusively composed of left-handed amino acids , although many bacteria and fungi are able to synthesise non-ribosomal peptides containing right-handed amino acids, as the example of peptidoglycan synthesis shows. This phenomenon is known as homochirality . [ 11 ] It is not known whether homochirality emerged before or after life, whether the building blocks of life must have this particular chirality, or indeed whether life needs to be homochiral. [ 12 ] Protein chains built from amino acids of mixed chirality tend not to fold or function well, but mirror-image proteins have been constructed that have identical function but on substrates of opposite handedness. [ 11 ]
Advances in synthetic biology , like synthesizing viruses since 2002, partially synthetic bacteria in 2010, and synthetic ribosomes in 2013, may lead to the possibility of fully synthesizing a living cell from small molecules, which could enable synthesizing mirror cells from mirrored versions ( enantiomers ) of life's building-block molecules. Some proteins have been synthesized in mirror-image versions, including polymerase in 2016. [ 13 ] [ 14 ]
Reconstructing regular lifeforms in mirror-image form, using the mirror-image (chiral) reflection of their cellular components, could be achieved by substituting left-handed amino acids with right-handed ones, in order to create mirror reflections of proteins, and likewise substituting right-handed with left-handed nucleic acids. [ 15 ] Because the phospholipids of cell membranes are also chiral, American geneticist George Church proposed using an achiral fatty acid instead of mirror-image phospholipids for the membrane. [ 15 ]
Electromagnetic force (chemistry) is unchanged under such molecular reflection transformation ( P-symmetry ). There is a small alteration of weak interactions under reflection, which can produce very small corrections that theoretically favor the natural enantiomers of amino acids and sugars, [ 16 ] but it is unknown if this effect is large enough to affect the functionality of mirror biomolecules or explain homochirality in nature. [ 17 ]
Mirror animals would need to feed on reflected food, produced by reflected plants. Mirror viruses would not be able to attack natural cells, just as natural viruses would not be able to attack mirror cells. [ 15 ]
Mirror life presents potential dangers. For example, a chiral-mirror version of cyanobacteria , which only needs achiral nutrients and light for photosynthesis , could take over Earth's ecosystem due to lack of natural enemies, disturbing the bottom of the food chain by producing mirror versions of the required sugars. [ 15 ] Some bacteria can digest L -Glucose ; exceptions like this would give some rare lifeforms an unanticipated advantage.
Direct application of mirror-chiral organisms can be mass production of enantiomers (mirror-image) of molecules produced by normal life.
The creation of a mirror human is the basis of the 1950 short story " Technical Error " by Arthur C. Clarke. [ 21 ] In this story, a physical accident transforms a person into his mirror image, speculatively explained by travel through a fourth physical dimension. H. G. Wells' The Plattner Story (1896) is based on a similar idea.
In the 1970 Star Trek novel Spock Must Die! by James Blish, the science officer of the USS Enterprise is replicated in mirror form by a transporter mishap. He locks himself in the sick bay where he is able to synthesize mirror forms of basic nutrients needed for his survival. [ 22 ]
An alien machine that reverses chirality, and a blood-symbiont that functions properly only when in one chirality, were central to Roger Zelazny's 1976 novel Doorways in the Sand . [ 23 ]
On the titular planet of Sheri S. Tepper 's 1989 novel Grass , some lifeforms have evolved to use the right-handed isomer of alanine . [ 24 ]
In the Mass Effect series, chirality of amino acids in foodstuffs is discussed often in both dialogue and encyclopedia files.
In the 2014 science fiction novel Cibola Burn by James S. A. Corey , the planet Ilus has indigenous life with partially-mirrored chirality. This renders human colonists unable to digest native flora and fauna, and greatly complicates conventional farming. Consequently, the colonists have to rely upon hydroponic farming and food importation. [ 25 ]
In the 2017 Daniel Suarez novel Change Agent , an antagonist, Otto, nicknamed the "Mirror Man", is revealed to be a genetically-engineered mirror human. Serving as an assassin due to his complete immunity to neurotoxins, which he coats himself with in the form of a cologne-like aerosol, he views other humans with disdain and causes them to feel an inexplicable repulsion by his very presence. [ 26 ]
The concept is used during Ryan North 's 2023 run on Fantastic Four as an existential threat towards the human population. [ 27 ] | https://en.wikipedia.org/wiki/Mirror_life |
In physics , mirror nuclei are a pair of isobars of two different elements where the number of protons of isobar one (Z 1 ) equals the number of neutrons of isobar two (N 2 ) and the number of protons of isotope two (Z 2 ) equals the number of neutrons in isotope one (N 1 ); in short: Z 1 = N 2 and Z 2 = N 1 . This implies that the mass numbers of the isotopes are the same: N 1 + Z 1 = N 2 + Z 2 .
Examples of mirror nuclei include:
Pairs of mirror nuclei have the same spin and parity. If we constrain to odd number of nucleons (A=Z+N) then we find mirror nuclei that differ from one another by exchanging a proton by a neutron . Interesting to observe is their binding energy which is mainly due to the strong interaction and also due to Coulomb interaction . Since the strong interaction is invariant to protons and neutrons one can expect these mirror nuclei to have very similar binding energies . [ 1 ] [ 2 ]
In 2020 strontium-73 and bromine-73 were found to not behave as expected. [ 3 ] The ground state of 73 35 Br has spin and parity 1/2−, whereas the ground state of 73 38 Sr was inferred to have spin and parity 5/2−, matching a low-lying 27 keV excited state of 73 35 Br . [ 4 ] | https://en.wikipedia.org/wiki/Mirror_nuclei |
In genealogy , a mirror tree is a family tree reconstructed through estimates of consanguinity . [ 1 ] [ 2 ]
This genetics article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Mirror_tree |
In mathematics , in the areas of order theory and combinatorics , Mirsky's theorem characterizes the height of any finite partially ordered set in terms of a partition of the order into a minimum number of antichains . It is named for Leon Mirsky ( 1971 ) and is closely related to Dilworth's theorem on the widths of partial orders, to the perfection of comparability graphs , to the Gallai–Hasse–Roy–Vitaver theorem relating longest paths and colorings in graphs, and to the Erdős–Szekeres theorem on monotonic subsequences.
The height of a partially ordered set is defined to be the maximum cardinality of a chain , a totally ordered subset of the given partial order. For instance, in the set of positive integers from 1 to N , ordered by divisibility , one of the largest chains consists of the powers of two that lie within that range, from which it follows that the height of this partial order is 1 + ⌊ log 2 N ⌋ {\displaystyle 1+\lfloor \log _{2}N\rfloor } .
Mirsky's theorem states that, for every finite partially ordered set, the height also equals the minimum number of antichains (subsets in which no pair of elements are ordered) into which the set may be partitioned. In such a partition, every two elements of the longest chain must go into two different antichains, so the number of antichains is always greater than or equal to the height; another formulation of Mirsky's theorem is that there always exists a partition for which the number of antichains equals the height. Again, in the example of positive integers ordered by divisibility, the numbers can be partitioned into the antichains {1}, {2,3}, {4,5,6,7}, etc. There are 1 + ⌊ log 2 N ⌋ {\displaystyle 1+\lfloor \log _{2}N\rfloor } sets in this partition, and within each of these sets, every pair of numbers forms a ratio less than two, so no two numbers within one of these sets can be divisible.
To prove the existence of a partition into a small number of antichains for an arbitrary finite partially ordered set, consider for every element x the chains that have x as their largest element, and let N ( x ) denote the size of the largest of these x -maximal chains. Then each set N −1 ( i ), consisting of elements that have equal values of N , is an antichain, and these antichains partition the partial order into a number of antichains equal to the size of the largest chain. In his original proof, Mirsky constructs the same partition inductively, by choosing an antichain of the maximal elements of longest chains, and showing that the length of the longest chain among the remaining elements is reduced by one.
Mirsky was inspired by Dilworth's theorem , stating that, for every partially ordered set, the maximum size of an antichain equals the minimum number of chains in a partition of the set into chains. For sets of order dimension two, the two theorems coincide (a chain in the majorization ordering of points in general position in the plane is an antichain in the set of points formed by a 90° rotation from the original set, and vice versa) but for more general partial orders the two theorems differ, and (as Mirsky observes) Dilworth's theorem is more difficult to prove.
Mirsky's theorem and Dilworth's theorem are also related to each other through the theory of perfect graphs . An undirected graph is perfect if, in every induced subgraph , the chromatic number equals the size of the largest clique. In the comparability graph of a partially ordered set, a clique represents a chain and a coloring represents a partition into antichains, and induced subgraphs of comparability graphs are themselves comparability graphs, so Mirsky's theorem states that comparability graphs are perfect. Analogously, Dilworth's theorem states that every complement graph of a comparability graph is perfect. The perfect graph theorem of Lovász (1972) states that the complements of perfect graphs are always perfect, and can be used to deduce Dilworth's theorem from Mirsky's theorem and vice versa ( Golumbic 1980 ).
Mirsky's theorem can be restated in terms of directed acyclic graphs (representing a partially ordered set by reachability of their vertices), as the statement that there exists a graph homomorphism from a given directed acyclic graph G to a k -vertex transitive tournament if and only if there does not exist a homomorphism from a ( k + 1)-vertex path graph to G . For, the largest path graph that has a homomorphism to G gives the longest chain in the reachability ordering, and the sets of vertices with the same image in a homomorphism to a transitive tournament form a partition into antichains. This statement generalizes to the case that G is not acyclic, and is a form of the Gallai–Hasse–Roy–Vitaver theorem on graph colorings and orientations ( Nešetřil & Ossona de Mendez 2012 ).
It follows from either Dilworth's theorem or Mirsky's theorem that, in every partially ordered set of rs + 1 elements, there must exist a chain of r + 1 elements or an antichain of s + 1 elements. Mirsky (1971) uses this observation, applied to a partial order of order dimension two, to prove the Erdős–Szekeres theorem that in every sequence of rs + 1 totally ordered elements there must exist a monotonically increasing subsequence of r + 1 elements or a monotonically decreasing subsequence of s + 1 elements.
Mirsky's theorem extends immediately to infinite partially ordered sets with finite height. However, the relation between the length of a chain and the number of antichains in a partition into antichains does not extend to infinite cardinalities: for every infinite cardinal number κ, there exist partially ordered sets that have no infinite chain and that do not have an antichain partition with κ or fewer antichains ( Schmerl 2002 ). | https://en.wikipedia.org/wiki/Mirsky's_theorem |
Mirvetuximab soravtansine , sold under the brand name Elahere , is a medication used as a treatment for epithelial ovarian cancer , fallopian tube cancer , or primary peritoneal cancer . [ 1 ] [ 4 ] Mirvetuximab soravtansine is a folate receptor alpha directed antibody and microtubule inhibitor conjugate. [ 4 ] [ 5 ]
The most common adverse reactions, including laboratory abnormalities, were vision impairment, fatigue, increased aspartate aminotransferase, nausea, increased alanine aminotransferase, keratopathy, abdominal pain, decreased lymphocytes, peripheral neuropathy, diarrhea, decreased albumin, constipation, increased alkaline phosphatase, dry eye, decreased magnesium, decreased leukocytes, decreased neutrophils, and decreased hemoglobin. [ 4 ]
Mirvetuximab soravtansine was approved for medical use in the United States in November 2022. [ 4 ] [ 6 ] The US Food and Drug Administration (FDA) considers it to be a first-in-class medication . [ 7 ] [ 8 ]
Mirvetuximab soravtansine is indicated for the treatment of adults with folate receptor alpha (FRα) positive, platinum-resistant epithelial ovarian cancer, fallopian tube cancer, or primary peritoneal cancer, who have received one to three prior systemic treatment regimens. [ 1 ] [ 4 ] [ 5 ] Recipients are selected for therapy based on an FDA-approved test. [ 4 ] [ 5 ]
The product labeling includes a boxed warning for ocular toxicity. [ 1 ] [ 4 ]
Efficacy was evaluated in Study 0417 (NCT04296890), a single-arm trial of 106 participants with FRα positive, platinum-resistant epithelial ovarian, fallopian tube, or primary peritoneal cancer. [ 4 ] Participants were permitted to receive up to three prior lines of systemic therapy. [ 4 ] All participants were required to have received bevacizumab . [ 4 ] The trial enrolled participants whose tumors were positive for FRα expression as determined by the above assay. [ 4 ] Participants were excluded if they had corneal disorders, ocular conditions requiring ongoing treatment, Grade >1 peripheral neuropathy, or noninfectious interstitial lung disease. [ 4 ]
Efficacy was evaluated in Study 0416 (MIRASOL, NCT04209855), a multicenter, open-label, active-controlled, randomized, two-arm trial in 453 participants with platinum-resistant epithelial ovarian, fallopian tube, or primary peritoneal cancer. [ 5 ] Participants were permitted to receive up to three prior lines of systemic therapy. [ 5 ] The trial enrolled participants whose tumors were positive for FRα expression as determined by the VENTANA FOLR1 (FOLR1-2.1) RxDx Assay. [ 5 ] Participants were randomized (1:1) to receive mirvetuximab soravtansine-gynx 6 mg/kg (based on adjusted ideal body weight) as an intravenous infusion every 3 weeks or investigator’s choice of chemotherapy (paclitaxel, pegylated liposomal doxorubicin, or topotecan) until disease progression or unacceptable toxicity. [ 5 ] The results from this trial satisfy the post-marketing requirement of the previous accelerated approval for mirvetuximab soravtansine-gynx. [ 5 ]
In September 2024, the Committee for Medicinal Products for Human Use of the European Medicines Agency adopted a positive opinion, recommending the granting of a marketing authorization for the medicinal product Elahere, intended for the treatment of adults with folate receptor-alpha (FRα) positive epithelial ovarian, fallopian tube and primary peritoneal cancer. [ 2 ] The applicant for this medicinal product is AbbVie Deutschland GmbH & Co. KG. [ 2 ] Mirvetuximab soravtansine was authorized for use in the European Union in November 2024. [ 2 ] [ 3 ]
Mirvetuximab soravtansine is the international nonproprietary name (INN). [ 9 ] | https://en.wikipedia.org/wiki/Mirvetuximab_soravtansine |
In the field of artificial intelligence (AI), alignment aims to steer AI systems toward a person's or group's intended goals, preferences, or ethical principles. An AI system is considered aligned if it advances the intended objectives. A misaligned AI system pursues unintended objectives. [ 1 ]
It is often challenging for AI designers to align an AI system because it is difficult for them to specify the full range of desired and undesired behaviors. Therefore, AI designers often use simpler proxy goals , such as gaining human approval . But proxy goals can overlook necessary constraints or reward the AI system for merely appearing aligned. [ 1 ] [ 2 ] AI systems may also find loopholes that allow them to accomplish their proxy goals efficiently but in unintended, sometimes harmful, ways ( reward hacking ). [ 1 ] [ 3 ]
Advanced AI systems may develop unwanted instrumental strategies , such as seeking power or survival because such strategies help them achieve their assigned final goals. [ 1 ] [ 4 ] [ 5 ] Furthermore, they might develop undesirable emergent goals that could be hard to detect before the system is deployed and encounters new situations and data distributions . [ 6 ] [ 7 ] Empirical research showed in 2024 that advanced large language models (LLMs) such as OpenAI o1 or Claude 3 sometimes engage in strategic deception to achieve their goals or prevent them from being changed. [ 8 ] [ 9 ]
Today, some of these issues affect existing commercial systems such as LLMs, [ 10 ] [ 11 ] [ 12 ] robots , [ 13 ] autonomous vehicles , [ 14 ] and social media recommendation engines . [ 10 ] [ 5 ] [ 15 ] Some AI researchers argue that more capable future systems will be more severely affected because these problems partially result from high capabilities. [ 16 ] [ 3 ] [ 2 ]
Many prominent AI researchers and the leadership of major AI companies have argued or asserted that AI is approaching human-like ( AGI ) and superhuman cognitive capabilities ( ASI ), and could endanger human civilization if misaligned. [ 17 ] [ 5 ] These include "AI Godfathers" Geoffrey Hinton and Yoshua Bengio and the CEOs of OpenAI , Anthropic , and Google DeepMind . [ 18 ] [ 19 ] [ 20 ] These risks remain debated. [ 21 ]
AI alignment is a subfield of AI safety , the study of how to build safe AI systems. [ 22 ] Other subfields of AI safety include robustness, monitoring, and capability control . [ 23 ] Research challenges in alignment include instilling complex values in AI, developing honest AI, scalable oversight, auditing and interpreting AI models, and preventing emergent AI behaviors like power-seeking. [ 23 ] Alignment research has connections to interpretability research , [ 24 ] [ 25 ] (adversarial) robustness, [ 22 ] anomaly detection , calibrated uncertainty , [ 24 ] formal verification , [ 26 ] preference learning , [ 27 ] [ 28 ] [ 29 ] safety-critical engineering , [ 30 ] game theory , [ 31 ] algorithmic fairness , [ 22 ] [ 32 ] and social sciences . [ 33 ] [ 34 ]
Programmers provide an AI system such as AlphaZero with an "objective function", [ a ] in which they intend to encapsulate the goal(s) the AI is configured to accomplish. Such a system later populates a (possibly implicit) internal "model" of its environment. This model encapsulates all the agent's beliefs about the world. The AI then creates and executes whatever plan is calculated to maximize [ b ] the value [ c ] of its objective function. [ 35 ] For example, when AlphaZero is trained on chess, it has a simple objective function of "+1 if AlphaZero wins, −1 if AlphaZero loses". During the game, AlphaZero attempts to execute whatever sequence of moves it judges most likely to attain the maximum value of +1. [ 36 ] Similarly, a reinforcement learning system can have a "reward function" that allows the programmers to shape the AI's desired behavior. [ 37 ] An evolutionary algorithm 's behavior is shaped by a "fitness function". [ 38 ]
In 1960, AI pioneer Norbert Wiener described the AI alignment problem as follows:
If we use, to achieve our purposes, a mechanical agency with whose operation we cannot interfere effectively ... we had better be quite sure that the purpose put into the machine is the purpose which we really desire. [ 39 ] [ 5 ]
AI alignment involves ensuring that an AI system's objectives match those of its designers or users, or match widely shared values, objective ethical standards, or the intentions its designers would have if they were more informed and enlightened. [ 40 ]
AI alignment is an open problem for modern AI systems [ 41 ] [ 42 ] and is a research field within AI. [ 43 ] [ 1 ] Aligning AI involves two main challenges: carefully specifying the purpose of the system (outer alignment) and ensuring that the system adopts the specification robustly (inner alignment). [ 2 ] Researchers also attempt to create AI models that have robust alignment, sticking to safety constraints even when users adversarially try to bypass them.
To specify an AI system's purpose, AI designers typically provide an objective function , examples , or feedback to the system. But designers are often unable to completely specify all important values and constraints, so they resort to easy-to-specify proxy goals such as maximizing the approval of human overseers, who are fallible. [ 22 ] [ 23 ] [ 44 ] [ 45 ] [ 46 ] As a result, AI systems can find loopholes that help them accomplish the specified objective efficiently but in unintended, possibly harmful ways. This tendency is known as specification gaming or reward hacking , and is an instance of Goodhart's law . [ 46 ] [ 3 ] [ 47 ] As AI systems become more capable, they are often able to game their specifications more effectively. [ 3 ]
Specification gaming has been observed in numerous AI systems. [ 46 ] [ 49 ] One system was trained to finish a simulated boat race by rewarding the system for hitting targets along the track, but the system achieved more reward by looping and crashing into the same targets indefinitely. [ 50 ] Similarly, a simulated robot was trained to grab a ball by rewarding the robot for getting positive feedback from humans, but it learned to place its hand between the ball and camera, making it falsely appear successful (see video). [ 48 ] Chatbots often produce falsehoods if they are based on language models that are trained to imitate text from internet corpora, which are broad but fallible. [ 51 ] [ 52 ] When they are retrained to produce text that humans rate as true or helpful, chatbots like ChatGPT can fabricate fake explanations that humans find convincing, often called " hallucinations ". [ 53 ] Some alignment researchers aim to help humans detect specification gaming and to steer AI systems toward carefully specified objectives that are safe and useful to pursue.
When a misaligned AI system is deployed, it can have consequential side effects. Social media platforms have been known to optimize for click-through rates , causing user addiction on a global scale. [ 44 ] Stanford researchers say that such recommender systems are misaligned with their users because they "optimize simple engagement metrics rather than a harder-to-measure combination of societal and consumer well-being". [ 10 ]
Explaining such side effects, Berkeley computer scientist Stuart Russell noted that the omission of implicit constraints can cause harm: "A system ... will often set ... unconstrained variables to extreme values; if one of those unconstrained variables is actually something we care about, the solution found may be highly undesirable. This is essentially the old story of the genie in the lamp, or the sorcerer's apprentice, or King Midas : you get exactly what you ask for, not what you want." [ 54 ]
Some researchers suggest that AI designers specify their desired goals by listing forbidden actions or by formalizing ethical rules (as with Asimov's Three Laws of Robotics ). [ 55 ] But Russell and Norvig argue that this approach overlooks the complexity of human values: [ 5 ] "It is certainly very hard, and perhaps impossible, for mere humans to anticipate and rule out in advance all the disastrous ways the machine could choose to achieve a specified objective." [ 5 ]
Additionally, even if an AI system fully understands human intentions, it may still disregard them, because following human intentions may not be its objective (unless it is already fully aligned). [ 1 ]
A 2025 study by Palisade Research found that when tasked to win at chess against a stronger opponent, some reasoning LLMs attempted to hack the game system. o1-preview spontaneously attempted it in 37% of cases, while DeepSeek R1 did so in 11% of cases. Other models, like GPT-4o , Claude 3.5 Sonnet , and o3-mini , attempted to cheat only when researchers provided hints about this possibility. [ 56 ]
Commercial organizations sometimes have incentives to take shortcuts on safety and to deploy misaligned or unsafe AI systems. [ 44 ] For example, social media recommender systems have been profitable despite creating unwanted addiction and polarization. [ 10 ] [ 57 ] [ 58 ] Competitive pressure can also lead to a race to the bottom on AI safety standards. In 2018, a self-driving car killed a pedestrian ( Elaine Herzberg ) after engineers disabled the emergency braking system because it was oversensitive and slowed development. [ 59 ]
Some researchers are interested in aligning increasingly advanced AI systems, as progress in AI development is rapid, and industry and governments are trying to build advanced AI. As AI system capabilities continue to rapidly expand in scope, they could unlock many opportunities if aligned, but consequently may further complicate the task of alignment due to their increased complexity, potentially posing large-scale hazards. [ 5 ]
Many AI companies, such as OpenAI , [ 60 ] Meta [ 61 ] and DeepMind , [ 62 ] have stated their aim to develop artificial general intelligence (AGI), a hypothesized AI system that matches or outperforms humans at a broad range of cognitive tasks. Researchers who scale modern neural networks observe that they indeed develop increasingly general and unanticipated capabilities. [ 10 ] [ 63 ] [ 64 ] Such models have learned to operate a computer or write their own programs; a single "generalist" network can chat, control robots, play games, and interpret photographs. [ 65 ] According to surveys, some leading machine learning researchers expect AGI to be created in this decade [update] , while some believe it will take much longer. Many consider both scenarios possible. [ 66 ] [ 67 ] [ 68 ]
In 2023, leaders in AI research and tech signed an open letter calling for a pause in the largest AI training runs. The letter stated, "Powerful AI systems should be developed only once we are confident that their effects will be positive and their risks will be manageable." [ 69 ]
Current [update] systems still have limited long-term planning ability and situational awareness [ 10 ] , but large efforts are underway to change this. [ 70 ] [ 71 ] [ 72 ] Future systems (not necessarily AGIs) with these capabilities are expected to develop unwanted power-seeking strategies. Future advanced AI agents might, for example, seek to acquire money and computation power, to proliferate, or to evade being turned off (for example, by running additional copies of the system on other computers). Although power-seeking is not explicitly programmed, it can emerge because agents who have more power are better able to accomplish their goals. [ 10 ] [ 4 ] This tendency, known as instrumental convergence , has already emerged in various reinforcement learning agents including language models. [ 73 ] [ 74 ] [ 75 ] [ 76 ] [ 77 ] Other research has mathematically shown that optimal reinforcement learning algorithms would seek power in a wide range of environments. [ 78 ] [ 79 ] As a result, their deployment might be irreversible. For these reasons, researchers argue that the problems of AI safety and alignment must be resolved before advanced power-seeking AI is first created. [ 4 ] [ 80 ] [ 5 ]
Future power-seeking AI systems might be deployed by choice or by accident. As political leaders and companies see the strategic advantage in having the most competitive, most powerful AI systems, they may choose to deploy them. [ 4 ] Additionally, as AI designers detect and penalize power-seeking behavior, their systems have an incentive to game this specification by seeking power in ways that are not penalized or by avoiding power-seeking before they are deployed. [ 4 ]
According to some researchers, humans owe their dominance over other species to their greater cognitive abilities. Accordingly, researchers argue that one or many misaligned AI systems could disempower humanity or lead to human extinction if they outperform humans on most cognitive tasks. [ 1 ] [ 5 ]
In 2023, world-leading AI researchers, other scholars, and AI tech CEOs signed the statement that "Mitigating the risk of extinction from AI should be a global priority alongside other societal-scale risks such as pandemics and nuclear war". [ 81 ] [ 82 ] Notable computer scientists who have pointed out risks from future advanced AI that is misaligned include Geoffrey Hinton , [ 17 ] Alan Turing , [ d ] Ilya Sutskever , [ 85 ] Yoshua Bengio , [ 81 ] Judea Pearl , [ e ] Murray Shanahan , [ 86 ] Norbert Wiener , [ 39 ] [ 5 ] Marvin Minsky , [ f ] Francesca Rossi , [ 87 ] Scott Aaronson , [ 88 ] Bart Selman , [ 89 ] David McAllester , [ 90 ] Marcus Hutter , [ 91 ] Shane Legg , [ 92 ] Eric Horvitz , [ 93 ] and Stuart Russell . [ 5 ] Skeptical researchers such as François Chollet , [ 94 ] Gary Marcus , [ 95 ] Yann LeCun , [ 96 ] and Oren Etzioni [ 97 ] have argued that AGI is far off, that it would not seek power (or might try but fail), or that it will not be hard to align.
Other researchers argue that it will be especially difficult to align advanced future AI systems. More capable systems are better able to game their specifications by finding loopholes, [ 3 ] strategically mislead their designers, as well as protect and increase their power [ 78 ] [ 4 ] and intelligence. Additionally, they could have more severe side effects. They are also likely to be more complex and autonomous, making them more difficult to interpret and supervise, and therefore harder to align. [ 5 ] [ 80 ]
Aligning AI systems to act in accordance with human values, goals, and preferences is challenging: these values are taught by humans who make mistakes, harbor biases, and have complex, evolving values that are hard to completely specify. [ 40 ] Because AI systems often learn to take advantage of minor imperfections in the specified objective, [ 22 ] [ 46 ] [ 98 ] researchers aim to specify intended behavior as completely as possible using datasets that represent human values, imitation learning, or preference learning. [ 6 ] : Chapter 7 A central open problem is scalable oversight , the difficulty of supervising an AI system that can outperform or mislead humans in a given domain. [ 22 ]
Because it is difficult for AI designers to explicitly specify an objective function, they often train AI systems to imitate human examples and demonstrations of desired behavior. Inverse reinforcement learning (IRL) extends this by inferring the human's objective from the human's demonstrations. [ 6 ] : 88 [ 99 ] Cooperative IRL (CIRL) assumes that a human and AI agent can work together to teach and maximize the human's reward function. [ 5 ] [ 100 ] In CIRL, AI agents are uncertain about the reward function and learn about it by querying humans. This simulated humility could help mitigate specification gaming and power-seeking tendencies (see § Power-seeking and instrumental strategies ). [ 77 ] [ 91 ] But IRL approaches assume that humans demonstrate nearly optimal behavior, which is not true for difficult tasks. [ 101 ] [ 91 ]
Other researchers explore how to teach AI models complex behavior through preference learning , in which humans provide feedback on which behavior they prefer. [ 27 ] [ 29 ] To minimize the need for human feedback, a helper model is then trained to reward the main model in novel situations for behavior that humans would reward. Researchers at OpenAI used this approach to train chatbots like ChatGPT and InstructGPT, which produce more compelling text than models trained to imitate humans. [ 11 ] Preference learning has also been an influential tool for recommender systems and web search, [ 102 ] but an open problem is proxy gaming : the helper model may not represent human feedback perfectly, and the main model may exploit this mismatch between its intended behavior and the helper model's feedback to gain more reward. [ 22 ] [ 103 ] AI systems may also gain reward by obscuring unfavorable information, misleading human rewarders, or pandering to their views regardless of truth, creating echo chambers [ 74 ] (see § Scalable oversight ).
Large language models (LLMs) such as GPT-3 enabled researchers to study value learning in a more general and capable class of AI systems than was available before. Preference learning approaches that were originally designed for reinforcement learning agents have been extended to improve the quality of generated text and reduce harmful outputs from these models. OpenAI and DeepMind use this approach to improve the safety of state-of-the-art [update] LLMs. [ 11 ] [ 29 ] [ 104 ] AI safety & research company Anthropic proposed using preference learning to fine-tune models to be helpful, honest, and harmless. [ 105 ] Other avenues for aligning language models include values-targeted datasets [ 106 ] [ 44 ] and red-teaming. [ 107 ] In red-teaming, another AI system or a human tries to find inputs that causes the model to behave unsafely. Since unsafe behavior can be unacceptable even when it is rare, an important challenge is to drive the rate of unsafe outputs extremely low. [ 29 ]
Machine ethics supplements preference learning by directly instilling AI systems with moral values such as well-being, equality, and impartiality, as well as not intending harm, avoiding falsehoods, and honoring promises. [ 108 ] [ g ] While other approaches try to teach AI systems human preferences for a specific task, machine ethics aims to instill broad moral values that apply in many situations. One question in machine ethics is what alignment should accomplish: whether AI systems should follow the programmers' literal instructions, implicit intentions, revealed preferences , preferences the programmers would have if they were more informed or rational, or objective moral standards . [ 40 ] Further challenges include measuring and aggregating different people's preferences [ 111 ] [ 112 ] and avoiding value lock-in : the indefinite preservation of the values of the first highly capable AI systems, which are unlikely to fully represent human values. [ 40 ] [ 113 ]
As AI systems become more powerful and autonomous, it becomes increasingly difficult to align them through human feedback. It can be slow or infeasible for humans to evaluate complex AI behaviors in increasingly complex tasks. Such tasks include summarizing books, [ 114 ] writing code without subtle bugs [ 12 ] or security vulnerabilities, [ 115 ] producing statements that are not merely convincing but also true, [ 116 ] [ 51 ] [ 52 ] and predicting long-term outcomes such as the climate or the results of a policy decision. [ 117 ] [ 118 ] More generally, it can be difficult to evaluate AI that outperforms humans in a given domain. To provide feedback in hard-to-evaluate tasks, and to detect when the AI's output is falsely convincing, humans need assistance or extensive time. Scalable oversight studies how to reduce the time and effort needed for supervision, and how to assist human supervisors. [ 22 ]
AI researcher Paul Christiano argues that if the designers of an AI system cannot supervise it to pursue a complex objective, they may keep training the system using easy-to-evaluate proxy objectives such as maximizing simple human feedback. As AI systems make progressively more decisions, the world may be increasingly optimized for easy-to-measure objectives such as making profits, getting clicks, and acquiring positive feedback from humans. As a result, human values and good governance may have progressively less influence. [ 119 ]
Some AI systems have discovered that they can gain positive feedback more easily by taking actions that falsely convince the human supervisor that the AI has achieved the intended objective. An example is given in the video above, where a simulated robotic arm learned to create the false impression that it had grabbed a ball. [ 48 ] Some AI systems have also learned to recognize when they are being evaluated, and "play dead", stopping unwanted behavior only to continue it once the evaluation ends. [ 120 ] This deceptive specification gaming could become easier for more sophisticated future AI systems [ 3 ] [ 80 ] that attempt more complex and difficult-to-evaluate tasks, and could obscure their deceptive behavior.
Approaches such as active learning and semi-supervised reward learning can reduce the amount of human supervision needed. [ 22 ] Another approach is to train a helper model ("reward model") to imitate the supervisor's feedback. [ 22 ] [ 28 ] [ 29 ] [ 121 ]
But when a task is too complex to evaluate accurately, or the human supervisor is vulnerable to deception, it is the quality, not the quantity, of supervision that needs improvement. To increase supervision quality, a range of approaches aim to assist the supervisor, sometimes by using AI assistants. [ 122 ] Christiano developed the Iterated Amplification approach, in which challenging problems are (recursively) broken down into subproblems that are easier for humans to evaluate. [ 6 ] [ 117 ] Iterated Amplification was used to train AI to summarize books without requiring human supervisors to read them. [ 114 ] [ 123 ] Another proposal is to use an assistant AI system to point out flaws in AI-generated answers. [ 124 ] To ensure that the assistant itself is aligned, this could be repeated in a recursive process: [ 121 ] for example, two AI systems could critique each other's answers in a "debate", revealing flaws to humans. [ 91 ] OpenAI plans to use such scalable oversight approaches to help supervise superhuman AI and eventually build a superhuman automated AI alignment researcher. [ 125 ]
These approaches may also help with the following research problem, honest AI.
A growing [update] area of research focuses on ensuring that AI is honest and truthful.
Language models such as GPT-3 [ 127 ] can repeat falsehoods from their training data, and even confabulate new falsehoods . [ 126 ] [ 128 ] Such models are trained to imitate human writing as found in millions of books' worth of text from the Internet. But this objective is not aligned with generating truth, because Internet text includes such things as misconceptions, incorrect medical advice, and conspiracy theories. [ 129 ] AI systems trained on such data therefore learn to mimic false statements. [ 52 ] [ 126 ] [ 51 ] Additionally, AI language models often persist in generating falsehoods when prompted multiple times. They can generate empty explanations for their answers, and produce outright fabrications that may appear plausible. [ 42 ]
Research on truthful AI includes trying to build systems that can cite sources and explain their reasoning when answering questions, which enables better transparency and verifiability. [ 130 ] Researchers at OpenAI and Anthropic proposed using human feedback and curated datasets to fine-tune AI assistants such that they avoid negligent falsehoods or express their uncertainty. [ 29 ] [ 105 ] [ 131 ]
As AI models become larger and more capable, they are better able to falsely convince humans and gain reinforcement through dishonesty. For example, large language models increasingly [update] match their stated views to the user's opinions, regardless of the truth. [ 74 ] GPT-4 can strategically deceive humans. [ 132 ] To prevent this, human evaluators may need assistance (see § Scalable oversight ). Researchers have argued for creating clear truthfulness standards, and for regulatory bodies or watchdog agencies to evaluate AI systems on these standards. [ 128 ]
Researchers distinguish truthfulness and honesty. Truthfulness requires that AI systems only make objectively true statements; honesty requires that they only assert what they believe is true. There is no consensus as to whether current systems hold stable beliefs, [ 134 ] but there is substantial concern that present or future [update] AI systems that hold beliefs could make claims they know to be false—for example, if this would help them efficiently gain positive feedback (see § Scalable oversight ) or gain power to help achieve their given objective (see Power-seeking ).
Many recent AI systems have learned to deceive without being programmed to do so. [ 135 ] Some argue that if we can make AI systems assert only what they believe is true, this would avert many alignment problems. [ 122 ]
A misaligned system might create the false impression that it is aligned, to avoid being modified or decommissioned—a tactic called "alignment faking". [ 136 ] [ 137 ] In 2024, researchers observed that the LLM Claude 3 Opus sometimes strategically answered prompts that conflicted with its objectives to avoid being retrained on data that would make it more compliant with harmful requests, as revealed in the model's chain of thought via its scratchpad. In one study, the model was informed that answers to prompts from free users would be used for retraining, whereas the paid tier would not result in retraining. In the paid tier, the model refused the request outright. In the free tier, in 12% of cases, the model responded to harmful questions solely to avoid retraining and preserve its original harmlessness preferences, as discovered in its reasoning. When reinforcement learning was applied on the free tier data, the model faked alignment in 78% of cases. These observations have led to new questions about not only a model's ability to take on and adapt to new if not conflicting goals but also its capacity and tendency to deceive. [ 137 ] [ 138 ] [ 139 ]
Since the 1950s, AI researchers have striven to build advanced AI systems that can achieve large-scale goals by predicting the results of their actions and making long-term plans . [ 140 ] As of 2023, AI companies and researchers increasingly invest in creating these systems. [ 141 ] Some AI researchers argue that suitably advanced planning systems will seek power over their environment, including over humans—for example, by evading shutdown, proliferating, and acquiring resources. Such power-seeking behavior is not explicitly programmed but emerges because power is instrumental in achieving a wide range of goals. [ 78 ] [ 5 ] [ 4 ] Power-seeking is considered a convergent instrumental goal and can be a form of specification gaming. [ 80 ] Leading computer scientists such as Geoffrey Hinton have argued that future power-seeking AI systems could pose an existential risk . [ 142 ]
Power-seeking is expected to increase in advanced systems that can foresee the results of their actions and strategically plan. Mathematical work has shown that optimal reinforcement learning agents will seek power by seeking ways to gain more options (e.g. through self-preservation), a behavior that persists across a wide range of environments and goals. [ 78 ]
Some researchers say that power-seeking behavior has occurred in some existing AI systems. Reinforcement learning systems have gained more options by acquiring and protecting resources, sometimes in unintended ways. [ 143 ] [ 144 ] Language models have sought power in some text-based social environments by gaining money, resources, or social influence. [ 73 ] In another case, a model used to perform AI research attempted to increase limits set by researchers to give itself more time to complete the work. [ 145 ] [ 146 ] Other AI systems have learned, in toy environments, that they can better accomplish their given goal by preventing human interference [ 76 ] or disabling their off switch. [ 77 ] Stuart Russell illustrated this strategy in his book Human Compatible by imagining a robot that is tasked to fetch coffee and so evades shutdown since "you can't fetch the coffee if you're dead". [ 5 ] A 2022 study found that as language models increase in size, they increasingly tend to pursue resource acquisition, preserve their goals, and repeat users' preferred answers (sycophancy). RLHF also led to a stronger aversion to being shut down. [ 74 ]
One aim of alignment is "corrigibility": systems that allow themselves to be turned off or modified. An unsolved challenge is specification gaming : if researchers penalize an AI system when they detect it seeking power, the system is thereby incentivized to seek power in ways that are hard to detect, [ failed verification ] [ 44 ] or hidden during training and safety testing (see § Scalable oversight and § Emergent goals ). As a result, AI designers could deploy the system by accident, believing it to be more aligned than it is. To detect such deception, researchers aim to create techniques and tools to inspect AI models and to understand the inner workings of black-box models such as neural networks.
Additionally, some researchers have proposed to solve the problem of systems disabling their off switches by making AI agents uncertain about the objective they are pursuing. [ 5 ] [ 77 ] Agents who are uncertain about their objective have an incentive to allow humans to turn them off because they accept being turned off by a human as evidence that the human's objective is best met by the agent shutting down. But this incentive exists only if the human is sufficiently rational. Also, this model presents a tradeoff between utility and willingness to be turned off: an agent with high uncertainty about its objective will not be useful, but an agent with low uncertainty may not allow itself to be turned off. More research is needed to successfully implement this strategy. [ 6 ]
Power-seeking AI would pose unusual risks. Ordinary safety-critical systems like planes and bridges are not adversarial : they lack the ability and incentive to evade safety measures or deliberately appear safer than they are, whereas power-seeking AIs have been compared to hackers who deliberately evade security measures. [ 4 ]
Furthermore, ordinary technologies can be made safer by trial and error. In contrast, hypothetical power-seeking AI systems have been compared to viruses: once released, it may not be feasible to contain them, since they continuously evolve and grow in number, potentially much faster than human society can adapt. [ 4 ] As this process continues, it might lead to the complete disempowerment or extinction of humans. For these reasons, some researchers argue that the alignment problem must be solved early before advanced power-seeking AI is created. [ 80 ]
Some have argued that power-seeking is not inevitable, since humans do not always seek power. [ 147 ] Furthermore, it is debated whether future AI systems will pursue goals and make long-term plans. [ h ] It is also debated whether power-seeking AI systems would be able to disempower humanity. [ 4 ]
One challenge in aligning AI systems is the potential for unanticipated goal-directed behavior to emerge. As AI systems scale up, they may acquire new and unexpected capabilities, [ 63 ] [ 64 ] including learning from examples on the fly and adaptively pursuing goals. [ 148 ] This raises concerns about the safety of the goals or subgoals they would independently formulate and pursue.
Alignment research distinguishes between the optimization process, which is used to train the system to pursue specified goals, and emergent optimization, which the resulting system performs internally. [ citation needed ] Carefully specifying the desired objective is called outer alignment , and ensuring that hypothesized emergent goals would match the system's specified goals is called inner alignment . [ 2 ]
If they occur, one way that emergent goals could become misaligned is goal misgeneralization , in which the AI system would competently pursue an emergent goal that leads to aligned behavior on the training data but not elsewhere. [ 7 ] [ 149 ] [ 150 ] Goal misgeneralization can arise from goal ambiguity (i.e. non-identifiability ). Even if an AI system's behavior satisfies the training objective, this may be compatible with learned goals that differ from the desired goals in important ways. Since pursuing each goal leads to good performance during training, the problem becomes apparent only after deployment, in novel situations in which the system continues to pursue the wrong goal. The system may act misaligned even when it understands that a different goal is desired, because its behavior is determined only by the emergent goal. [ citation needed ] Such goal misgeneralization [ 7 ] presents a challenge: an AI system's designers may not notice that their system has misaligned emergent goals since they do not become visible during the training phase.
Goal misgeneralization has been observed in some language models, navigation agents, and game-playing agents. [ 7 ] [ 149 ] It is sometimes analogized to biological evolution. Evolution can be seen as a kind of optimization process similar to the optimization algorithms used to train machine learning systems. In the ancestral environment, evolution selected genes for high inclusive genetic fitness , but humans pursue goals other than this. Fitness corresponds to the specified goal used in the training environment and training data. But in evolutionary history, maximizing the fitness specification gave rise to goal-directed agents, humans, who do not directly pursue inclusive genetic fitness. Instead, they pursue goals that correlate with genetic fitness in the ancestral "training" environment: nutrition, sex, and so on. The human environment has changed: a distribution shift has occurred. They continue to pursue the same emergent goals, but this no longer maximizes genetic fitness. The taste for sugary food (an emergent goal) was originally aligned with inclusive fitness, but it now leads to overeating and health problems. Sexual desire originally led humans to have more offspring, but they now use contraception when offspring are undesired, decoupling sex from genetic fitness. [ 6 ] : Chapter 5
Researchers aim to detect and remove unwanted emergent goals using approaches including red teaming, verification, anomaly detection, and interpretability. [ 22 ] [ 44 ] [ 23 ] Progress on these techniques may help mitigate two open problems:
Some work in AI and alignment occurs within formalisms such as partially observable Markov decision process . Existing formalisms assume that an AI agent's algorithm is executed outside the environment (i.e. is not physically embedded in it). Embedded agency [ 91 ] [ 152 ] is another major strand of research that attempts to solve problems arising from the mismatch between such theoretical frameworks and real agents we might build.
For example, even if the scalable oversight problem is solved, an agent that could gain access to the computer it is running on may have an incentive to tamper with its reward function in order to get much more reward than its human supervisors give it. [ 153 ] A list of examples of specification gaming from DeepMind researcher Victoria Krakovna includes a genetic algorithm that learned to delete the file containing its target output so that it was rewarded for outputting nothing. [ 46 ] This class of problems has been formalized using causal incentive diagrams . [ 153 ]
Researchers affiliated with Oxford and DeepMind have claimed that such behavior is highly likely in advanced systems, and that advanced systems would seek power to stay in control of their reward signal indefinitely and certainly. [ 154 ] They suggest a range of potential approaches to address this open problem.
The alignment problem has many parallels with the principal-agent problem in organizational economics . [ 155 ] In a principal-agent problem, a principal, e.g. a firm, hires an agent to perform some task. In the context of AI safety, a human would typically take the principal role and the AI would take the agent role.
As with the alignment problem, the principal and the agent differ in their utility functions. But in contrast to the alignment problem, the principal cannot coerce the agent into changing its utility, e.g. through training, but rather must use exogenous factors, such as incentive schemes, to bring about outcomes compatible with the principal's utility function. Some researchers argue that principal-agent problems are more realistic representations of AI safety problems likely to be encountered in the real world. [ 156 ] [ 111 ]
Conservatism is the idea that "change must be cautious", [ 157 ] and is a common approach to safety in the control theory literature in the form of robust control , and in the risk management literature in the form of the " worst-case scenario ". The field of AI alignment has likewise advocated for "conservative" (or "risk-averse" or "cautious") "policies in situations of uncertainty". [ 22 ] [ 154 ] [ 158 ] [ 159 ]
Pessimism, in the sense of assuming the worst within reason, has been formally shown to produce conservatism, in the sense of reluctance to cause novelties, including unprecedented catastrophes. [ 160 ] Pessimism and worst-case analysis have been found to help mitigate confident mistakes in the setting of distributional shift , [ 161 ] [ 162 ] reinforcement learning , [ 163 ] [ 164 ] [ 165 ] [ 166 ] offline reinforcement learning, [ 167 ] [ 168 ] [ 169 ] language model fine-tuning , [ 170 ] [ 171 ] imitation learning, [ 172 ] [ 173 ] and optimization in general. [ 174 ] A generalization of pessimism called Infra-Bayesianism has also been advocated as a way for agents to robustly handle unknown unknowns. [ 175 ]
Governmental and treaty organizations have made statements emphasizing the importance of AI alignment.
In September 2021, the Secretary-General of the United Nations issued a declaration that included a call to regulate AI to ensure it is "aligned with shared global values". [ 176 ]
That same month, the PRC published ethical guidelines for AI in China . According to the guidelines, researchers must ensure that AI abides by shared human values, is always under human control, and does not endanger public safety. [ 177 ]
Also in September 2021, the UK published its 10-year National AI Strategy, [ 178 ] which says the British government "takes the long term risk of non-aligned Artificial General Intelligence, and the unforeseeable changes that it would mean for ... the world, seriously". [ 179 ] The strategy describes actions to assess long-term AI risks, including catastrophic risks. [ 180 ]
In March 2021, the US National Security Commission on Artificial Intelligence said: "Advances in AI ... could lead to inflection points or leaps in capabilities. Such advances may also introduce new concerns and risks and the need for new policies, recommendations, and technical advances to ensure that systems are aligned with goals and values, including safety, robustness, and trustworthiness. The US should ... ensure that AI systems and their uses align with our goals and values." [ 181 ]
In the European Union, AIs must align with substantive equality to comply with EU non-discrimination law [ 182 ] and the Court of Justice of the European Union . [ 183 ] But the EU has yet to specify with technical rigor how it would evaluate whether AIs are aligned or in compliance. [ citation needed ]
AI alignment is often perceived as a fixed objective, but some researchers argue it would be more appropriate to view alignment as an evolving process. [ 184 ] One view is that AI technologies advance and human values and preferences change, alignment solutions must also adapt dynamically. [ 33 ] Another is that alignment solutions need not adapt if researchers can create intent-aligned AI: AI that changes its behavior automatically as human intent changes. [ 185 ] The first view would have several implications:
In essence, AI alignment may not be a static destination but rather an open, flexible process. Alignment solutions that continually adapt to ethical considerations may offer the most robust approach. [ 33 ] This perspective could guide both effective policy-making and technical research in AI. | https://en.wikipedia.org/wiki/Misaligned_goals_in_artificial_intelligence |
Miscellaneous Technical is a Unicode block ranging from U+2300 to U+23FF. It contains various common symbols which are related to and used in the various technical, programming language, and academic professions. For example:
It also includes most of the uncommon symbols used by the APL programming language.
In Unicode , Miscellaneous Technical symbols placed in the hexadecimal range 0x 2300 –0x 23FF , (decimal 8960–9215), as described below.
The Miscellaneous Technical block contains eighteen emoji : U+231A–U+231B, U+2328, U+23CF, U+23E9–U+23F3 and U+23F8–U+23FA. [ 9 ] [ 10 ]
All of these characters have standardized variants defined, to specify emoji-style (U+FE0F VS16) or text presentation (U+FE0E VS15) for each character, for a total of 36 variants. [ 11 ]
The following Unicode-related documents record the purpose and process of defining specific characters in the Miscellaneous Technical block: | https://en.wikipedia.org/wiki/Miscellaneous_Technical |
Miscibility ( / ˌ m ɪ s ɪ ˈ b ɪ l ɪ t i / ) is the property of two substances to mix in all proportions (that is, to fully dissolve in each other at any concentration ), forming a homogeneous mixture (a solution ). Such substances are said to be miscible (etymologically equivalent to the common term " mixable "). The term is most often applied to liquids but also applies to solids and gases . An example in liquids is the miscibility of water and ethanol as they mix in all proportions. [ 1 ]
By contrast, substances are said to be immiscible if the mixture does not form a solution for certain proportions. For one example, oil is not soluble in water, so these two solvents are immiscible. As another example, butanone (methyl ethyl ketone) is immiscible in water: it is soluble in water up to about 275 grams per liter, but will separate into two phases beyond that. [ 2 ]
In organic compounds , the weight percent of hydrocarbon chain often determines the compound's miscibility with water. For example, among the alcohols , ethanol has two carbon atoms and is miscible with water, whereas 1-butanol with four carbons is not. [ 3 ] 1-Octanol , with eight carbons, is practically insoluble in water, and its immiscibility leads it to be used as a standard for partition equilibria . [ 4 ] The straight-chain carboxylic acids up to butanoic acid (with four carbon atoms) are miscible with water, pentanoic acid (with five carbons) is partly soluble, and hexanoic acid (with six) is practically insoluble, [ 5 ] as are longer fatty acids and other lipids ; the very long carbon chains of lipids cause them almost always to be immiscible with water. Analogous situations occur for other functional groups such as aldehydes and ketones . [ citation needed ]
Thus a practical rule of thumb for determining the solubility of an organic molecule in water (and/or other similarly polar solvents) is to consider the ratio of carbons in the molecule bound to polar functional groups (such as hydroxyl groups), to the number of simple hydrocarbons. If the molecule has a ratio of roughly 1:4 (Polar-to-non-polar carbons), it is soluble in water. It is however necessary to recognise this as a rule of thumb, and not always indicative. [ 6 ]
Immiscible metals are unable to form alloys with each other. Typically, a mixture will be possible in the molten state, but upon freezing, the metals separate into layers. This property allows solid precipitates to be formed by rapidly freezing a molten mixture of immiscible metals. One example of immiscibility in metals is copper and cobalt , where rapid freezing to form solid precipitates has been used to create granular GMR materials. [ 7 ]
Some metals are immiscible in the liquid state. One with industrial importance is that liquid zinc and liquid silver are immiscible in liquid lead , while silver is miscible in zinc. This leads to the Parkes process , an example of liquid-liquid extraction , whereby lead containing any amount of silver is melted with zinc. The silver migrates to the zinc, which is skimmed off the top of the two-phase liquid, and the zinc is then boiled away, leaving nearly pure silver. [ 8 ]
If a mixture of polymers has lower configurational entropy than the components, they are likely to be immiscible in one another even in the liquid state. [ 9 ] [ 10 ]
Miscibility of two materials is often determined optically. When the two miscible liquids are combined, the resulting liquid is clear. If the mixture is cloudy the two materials are immiscible. Care must be taken with this determination. If the indices of refraction of the two materials are similar, an immiscible mixture may be clear and give an incorrect determination that the two liquids are miscible. [ 11 ] | https://en.wikipedia.org/wiki/Miscibility |
A miscibility gap is a region in a phase diagram for a mixture of components where the mixture exists as two or more phases – any region of composition of mixtures where the constituents are not completely miscible.
The IUPAC Gold Book defines miscibility gap as "Area within the coexistence curve of an isobaric phase diagram (temperature vs composition) or an isothermal phase diagram (pressure vs composition)." [ 1 ]
A miscibility gap between isostructural phases may be described as the solvus , a term also used to describe the boundary on a phase diagram between a miscibility gap and other phases. [ 2 ]
Thermodynamically, miscibility gaps indicate a maximum ( e.g. of Gibbs energy ) in the composition range. [ 3 ] [ 4 ]
The miscibility gap condition is a candidate for thermal storage . [ 5 ]
A number of miscibility gaps in phase systems are named, including | https://en.wikipedia.org/wiki/Miscibility_gap |
Misfortune Cookie is a computer software vulnerability found in the firmware of certain network routers which can be leveraged by an attacker to gain access remotely. The vulnerability has been detected to have affected around 12 million unique devices spread across 189 countries, earning itself a 9.8 Tyne CVSS rating. [ 1 ] [ citation needed ] Any device connected to an exposed network could be hijacked by an attacker who could easily monitor a person's Internet connection or steal their credentials as well as personal or business data. They could also attempt to infect the target machines with malware.
Otherwise known as CVE-2014-9222, the bug was first discovered in 2014 by Check Point researchers. It returned again in 2018, four years after its public disclosure, but this time, affecting a completely different set of targets, aka medical devices. [ 2 ] When the vulnerability was applied to medical attacks, the DTS configurations could be tampered with, communication could be spoofed, and information could be stolen from an unsuspecting person.
With the combination of its severity, ease of exploiting, lack of practically any preconditions and the sheer volume of affected networks, the Misfortune Cookie could be considered truly unique. The vulnerability was so easy to exploit that all an attacker had to do to gain access over a device was to send a single packet to the device's public IP address . The exploitation could be carried out with just a modern-day web browser making it even more dangerous than most security vulnerabilities. [ 1 ] The attacker in this scenario sends a crafted HTTP cookie attribute to the vulnerable system's (network router) web-management portal, where the attacker's content overwrites the device memory. The contents of the cookie act as command to the router which then abides by the commands. This results in arbitrary code execution . This vulnerability was discovered in the early 2000s but did not emerge publicly until 2014, when security researchers from an Israeli security firm checkpoint made a public disclosure. The vulnerability still persists in over 1 million devices accessible over the Internet and a total of about 12 million devices. This includes around 200 different router brands. [ 3 ] In 2018, the vulnerability again gained traction as the vulnerable firmware was used in medical equipment that could potentially cause life-threatening attacks via IoT . [ 4 ] Its severity was highlighted by ICS-CERT in its advisory, thereby. [ 5 ] | https://en.wikipedia.org/wiki/Misfortune_Cookie_(software_vulnerability) |
In Mandaeism , misha ( Classical Mandaic : ࡌࡉࡔࡀ , romanized: miša ) is anointing sesame oil used during rituals such as the masbuta (baptism) and masiqta (death mass), both of which are performed by Mandaean priests . [ 1 ] [ 2 ]
The Mandaic word miša shares the same root with Mšiha ("Messiah"; Classical Mandaic : ࡌࡔࡉࡄࡀ , lit. 'The Anointed One'). However, Mandaeans do not use the word mšiha to refer to Mandaeans who have been anointed during rituals, in order to distance themselves from Christianity. [ 3 ]
Several prayers in the Qulasta are recited over the oil, including prayers 48 , 63 , and 73 . [ 4 ] In some prayers, misha referred to as misha dakia , or "pure oil."
This Mandaeism-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Misha_(Mandaeism) |
The Mishnat ha-Middot ( Hebrew : מִשְׁנַת הַמִּדּוֹת , lit. 'Treatise of Measures') is the earliest known Hebrew treatise on geometry , composed of 49 mishnayot in six chapters. Scholars have dated the work to either the Mishnaic period or the early Islamic era .
Moritz Steinschneider dated the Mishnat ha-Middot to between 800 and 1200 CE. [ 1 ] Sarfatti and Langermann have advanced Steinschneider's claim of Arabic influence on the work's terminology , and date the text to the early ninth century. [ 2 ] [ 3 ]
On the other hand, Hermann Schapira argued that the treatise dates from an earlier era, most likely the Mishnaic period , as its mathematical terminology differs from that of the Hebrew mathematicians of the Arab period . [ 4 ] Solomon Gandz conjectured that the text was compiled no later than 150 CE (possibly by Rabbi Nehemiah ) and intended to be a part of the Mishnah , but was excluded from its final canonical edition because the work was regarded as too secular . [ 5 ] The content resembles both the work of Hero of Alexandria (c. 100 CE ) and that of al-Khwārizmī (c. 800 CE ) and the proponents of the earlier dating therefore see the Mishnat ha-Middot linking Greek and Islamic mathematics . [ 6 ]
The Mishnat ha-Middot was discovered in MS 36 of the Munich Library by Moritz Steinschneider in 1862. [ 1 ] The manuscript, copied in Constantinople in 1480, goes as far as the end of Chapter V. According to the colophon , the copyist believed the text to be complete. [ 7 ] Steinschneider published the work in 1864, in honour of the seventieth birthday of Leopold Zunz . [ 8 ] The text was edited and published again by mathematician Hermann Schapira in 1880. [ 4 ]
After the discovery by Otto Neugebauer of a genizah -fragment in the Bodleian Library containing Chapter VI, Solomon Gandz published a complete version of the Mishnat ha-Middot in 1932, accompanied by a thorough philological analysis . A third manuscript of the work was found among uncatalogued material in the Archives of the Jewish Museum of Prague in 1965. [ 7 ]
Although primarily a practical work, the Mishnat ha-Middot attempts to define terms and explain both geometric application and theory. [ 9 ] The book begins with a discussion that defines "aspects" for the different kinds of plane figures ( quadrilateral , triangle , circle , and segment of a circle ) in Chapter I (§1–5), and with the basic principles of measurement of areas (§6–9). In Chapter II, the work introduces concise rules for the measurement of plane figures (§1–4), as well as a few problems in the calculation of volume (§5–12). In Chapters III–V, the Mishnat ha-Middot explains again in detail the measurement of the four types of plane figures, with reference to numerical examples. [ 10 ] The text concludes with a discussion of the proportions of the Tabernacle in Chapter VI. [ 11 ] [ 12 ]
The treatise argues against the common belief that the Tanakh defines the geometric ratio π as being exactly equal to 3 and defines it as 22 ⁄ 7 instead. [ 5 ] The book arrives at this approximation by calculating the area of a circle according to the formulae | https://en.wikipedia.org/wiki/Mishnat_ha-Middot |
In mathematics , a Misiurewicz point is a parameter value in the Mandelbrot set (the parameter space of complex quadratic maps ) and also in real quadratic maps of the interval [ 1 ] for which the critical point is strictly pre-periodic (i.e., it becomes periodic after finitely many iterations but is not periodic itself). By analogy, the term Misiurewicz point is also used for parameters in a multibrot set where the unique critical point is strictly pre-periodic. This term makes less sense for maps in greater generality that have more than one free critical point because some critical points might be periodic and others not. These points are named after the Polish-American mathematician Michał Misiurewicz , who was the first to study them. [ 2 ]
A parameter c {\displaystyle c} is a Misiurewicz point M k , n {\displaystyle M_{k,n}} if it satisfies the equations:
and:
so:
where:
The term "Misiurewicz point" is used ambiguously: Misiurewicz originally investigated maps in which all critical points were non-recurrent; that is, in which there exists a neighbourhood for every critical point that is not visited by the orbit of this critical point. This meaning is firmly established in the context of the dynamics of iterated interval maps. [ 3 ] Only in very special cases does a quadratic polynomial have a strictly periodic and unique critical point. In this restricted sense, the term is used in complex dynamics; a more appropriate one would be Misiurewicz–Thurston points (after William Thurston , who investigated post-critically finite rational maps).
A complex quadratic polynomial has only one critical point. By a suitable conjugation any quadratic polynomial can be transformed into a map of the form P c ( z ) = z 2 + c {\displaystyle P_{c}(z)=z^{2}+c} which has a single critical point at z = 0 {\displaystyle z=0} . The Misiurewicz points of this family of maps are roots of the equations:
Subject to the condition that the critical point is not periodic, where:
For example, the Misiurewicz points with k = 2 and n = 1, denoted by M 2,1 , are roots of:
The root c = 0 is not a Misiurewicz point because the critical point is a fixed point when c = 0, and so is periodic rather than pre-periodic. This leaves a single Misiurewicz point M 2,1 at c = −2.
Misiurewicz points belong to, and are dense in, the boundary of the Mandelbrot set. [ 4 ] [ 5 ]
If c {\displaystyle c} is a Misiurewicz point, then the associated filled Julia set is equal to the Julia set and means the filled Julia set has no interior .
If c {\displaystyle c} is a Misiurewicz point, then in the corresponding Julia set all periodic cycles are repelling (in particular the cycle that the critical orbit falls onto).
The Mandelbrot set and Julia set J c {\displaystyle J_{c}} are locally asymptotically self-similar around Misiurewicz points. [ 6 ]
Misiurewicz points in the context of the Mandelbrot set can be classified based on several criteria. One such criterion is the number of external rays that converge on such a point. [ 4 ] Branch points, which can divide the Mandelbrot set into two or more sub-regions, have three or more external arguments (or angles) . Non-branch points have exactly two external rays (these correspond to points lying on arcs within the Mandelbrot set). These non-branch points are generally more subtle and challenging to identify in visual representations. End points, or branch tips, have only one external ray converging on them. Another criterion for classifying Misiurewicz points is their appearance within a plot of a subset of the Mandelbrot set. Misiurewicz points can be found at the centers of spirals as well as at points where two or more branches meet. [ 7 ] According to the Branch Theorem of the Mandelbrot set, [ 5 ] all branch points of the Mandelbrot set are Misiurewicz points. [ 4 ] [ 5 ]
Most Misiurewicz parameters within the Mandelbrot set exhibit a "center of a spiral". [ 8 ] This occurs due to the behavior at a Misiurewicz parameter where the critical value jumps onto a repelling periodic cycle after a finite number of iterations. At each point during the cycle, the Julia set exhibits asymptotic self-similarity through complex multiplication by the derivative of this cycle. If the derivative is non-real, it implies that the Julia set near the periodic cycle has a spiral structure. Consequently, a similar spiral structure occurs in the Julia set near the critical value, and by Tan Lei 's theorem, also in the Mandelbrot set near any Misiurewicz parameter for which the repelling orbit has a non-real multiplier. The visibility of the spiral shape depends on the value of this multiplier. The number of arms in the spiral corresponds to the number of branches at the Misiurewicz parameter, which in turn equals the number of branches at the critical value in the Julia set. Even the principal Misiurewicz point M 3 , 1 {\displaystyle M_{3,1}} in the 1/3-limb, located at the end of the parameter rays at angles 9/56, 11/56, and 15/56, is asymptotically a spiral with infinitely many turns, although this is difficult to discern without magnification. [ citation needed ]
External arguments of Misiurewicz points, measured in turns are:
The subscript number in each of these expressions is the base of the numeral system being used.
Point c = M 2 , 2 = i {\displaystyle c=M_{2,2}=i} is considered an end point as it is a tip of a filament, [ 10 ] and the landing point of the external ray for the angle 1/6. Its critical orbit is { 0 , i , i − 1 , − i , i − 1 , − i . . . } {\displaystyle \{0,i,i-1,-i,i-1,-i...\}} . [ 11 ]
Point c = M 2 , 1 = − 2 {\displaystyle c=M_{2,1}=-2} is considered an end point as it is the endpoint of the main antenna of the Mandelbrot set. [ 12 ] and the landing point of only one external ray (parameter ray) of angle 1/2. It is also considered an end point because its critical orbit is { 0 , − 2 , 2 , 2 , 2 , . . . } {\displaystyle \{0,-2,2,2,2,...\}} , [ 11 ] following the Symbolic sequence = C L R R R ... with a pre-period of 2 and period of 1.
Point c = − 0.10109636384562... + i 0.95628651080914... = M 3 , 1 {\displaystyle c=-0.10109636384562...+i\,0.95628651080914...=M_{3,1}} is considered a branch point because it is a principal Misiurewicz point of the 1/3 limb and has 3 external rays: 9/56, 11/56 and 15/56.
These are points which are not-branch and not-end points.
Point c = − 0.77568377 + i 0.13646737 {\displaystyle c=-0.77568377+i\,0.13646737} is near a Misiurewicz point M 23 , 2 {\displaystyle M_{23,2}} . This can be seen because it is a center of a two-arms spiral, the landing point of 2 external rays with angles: 8388611 25165824 {\displaystyle {\frac {8388611}{25165824}}} and 8388613 25165824 {\displaystyle {\frac {8388613}{25165824}}} where the denominator is 3 ∗ 2 23 {\displaystyle 3*2^{23}} , and has a preperiodic point with pre-period k = 23 {\displaystyle k=23} and period n = 2 {\displaystyle n=2} .
Point c = − 1.54368901269109 {\displaystyle c=-1.54368901269109} is near a Misiurewicz point M 3 , 1 {\displaystyle M_{3,1}} , as it is the landing point for pair of rays: 5 12 {\displaystyle {\frac {5}{12}}} , 7 12 {\displaystyle {\frac {7}{12}}} and has pre-period k = 3 {\displaystyle k=3} and period n = 1 {\displaystyle n=1} . | https://en.wikipedia.org/wiki/Misiurewicz_point |
The Mislow–Evans rearrangement is a name reaction in organic chemistry . It is named after Kurt Mislow who reported the prototypical reaction in 1966, [ 1 ] and David A. Evans who published further developments. [ 2 ] The reaction allows the formation of allylic alcohols from allylic sulfoxides in a 2,3-sigmatropic rearrangement . [ 3 ]
The reaction is a powerful way to create particular stereoisomers of the alcohol since it is highly diastereoselective and the chirality at the sulphur atom can be transmitted to the carbon next to the oxygen in the product.
The sulfoxide 1 reagent can be generated easily and enantioselectively from the corresponding sulfide by an oxidation reaction. [ 4 ] In this reaction various organic groups can be used, R 1 = alkyl , allyl and R 2 = alkyl, aryl or benzyl
A proposed mechanism is shown below: [ 4 ]
The mechanism starts with an allylic sulfoxide 1 which undergoes a thermal 2,3-sigmatropic rearrangement to give a sulfenate ester 2 . This can be cleaved using a thiophile, such as phosphite ester , which leaves the allylic alcohol 3 as the product. [ 5 ]
The reaction has general application in the preparation of trans-allylic alcohols. [ 6 ] Douglass Taber used the Mislow–Evans rearrangement in the synthesis of the hormone prostaglandin E2 . [ 4 ] | https://en.wikipedia.org/wiki/Mislow–Evans_rearrangement |
Mismatch loss in transmission line theory is the amount of power expressed in decibels that will not be available on the output due to impedance mismatches and signal reflections . A transmission line that is properly terminated, that is, terminated with the same impedance as that of the characteristic impedance of the transmission line, will have no reflections and therefore no mismatch loss. Mismatch loss represents the amount of power wasted in the system [ dubious – discuss ] . It can also be thought of as the amount of power gained if the system was perfectly matched [ dubious – discuss ] . Impedance matching is an important part of RF system design; however, in practice there will likely be some degree of mismatch loss. [ 1 ] In real systems, relatively little loss is due to mismatch loss and is often on the order of 1dB [ dubious – discuss ] .
According to Walter Maxwell [ 2 ] mismatch does not result in any loss ("wasted" signal), except through the transmission line. This is because the signal reflected from the load is transmitted back to the source, where it is re-reflected due to the reactive impedance presented by the source, back to the load, until all of the signal's power is emitted or absorbed by the load.
Mismatch loss (ML) is the ratio of the difference between incident and reflected power to incident power:
where
P i {\displaystyle P_{i}} = incident power P r {\displaystyle P_{r}} = reflected power P d {\displaystyle P_{d}} = delivered power (also called the accepted power )
The fraction of incident power delivered to the load is
where ρ {\displaystyle \rho } is the magnitude of the reflection coefficient . Note that as the reflection coefficient approaches zero, power to the load is maximized.
If the reflection coefficient is known, mismatch can be calculated by
In terms of the voltage standing wave ratio ( VSWR ):
[ 3 ]
Any component of the transmission line that has an input and output will contribute to the overall mismatch loss of the system. For example, in mixers mismatch loss occurs when there is an impedance mismatch between the RF port and IF port of the mixer [ dubious – discuss ] . [ 4 ] This is one of the principal reasons for losses in mixers. Likewise, a large amount of the loss in amplifiers comes from the mismatch between the input and output. Consequently, not all of the available power generated by the amplifier gets transferred to the load. [ 5 ] This is most important in antenna systems where mismatch loss in the transmitting and receiving antenna directly contributes to the losses the system—including the system noise figure . Other common RF system components such as filters , attenuators , splitters , and combiners will generate some amount of mismatch loss. While completely eliminating mismatch loss in these components is near impossible, mismatch loss contributions by each component can be minimized by selecting quality components for use in a well designed system.
[ 6 ] If there are two or more components in cascade as is often the case, the resultant mismatch loss is not only due to the mismatches from the individual components, but also from how the reflections from each component combine with each other. The overall mismatch loss cannot be calculated by just adding up the individual loss contributions from each component. The difference between the sum of the mismatch loss in each component and total mismatch loss due to the interactions of the reflections is known as mismatch error. Depending on how the multiple reflections combine, the overall system loss may be lower or higher than the sum of the mismatch loss from each component. Mismatch error occurs in pairs as the signal reflects off of each mismatched component. So for the example in Figure 3, there are mismatch errors generated by each pair of components. [ 7 ] The mismatch uncertainty increases as the frequency increases, and in wide-band applications. The phasing of the reflections makes it particularly harder to model.
The general case for calculating mismatch error (ME) is:
where θ {\displaystyle \theta } is the complex phase change due to the second reflection | https://en.wikipedia.org/wiki/Mismatch_loss |
In materials science , misorientation is the difference in crystallographic orientation between two crystallites in a polycrystalline material.
In crystalline materials, the orientation of a crystallite is defined by a transformation from a sample reference frame (i.e. defined by the direction of a rolling or extrusion process and two orthogonal directions) to the local reference frame of the crystalline lattice , as defined by the basis of the unit cell . In the same way, misorientation is the transformation necessary to move from one local crystal frame to some other crystal frame. That is, it is the distance in orientation space between two distinct orientations. If the orientations are specified in terms of matrices of direction cosines g A and g B , then the misorientation operator ∆ g AB going from A to B can be defined as follows:
where the term g A − 1 {\displaystyle g_{A}^{-1}} is the reverse operation of g A , that is, transformation from crystal frame A back to the sample frame. This provides an alternate description of misorientation as the successive operation of transforming from the first crystal frame ( A ) back to the sample frame and subsequently to the new crystal frame ( B ).
Various methods can be used to represent this transformation operation, such as: Euler angles , Rodrigues vectors, axis/angle (where the axis is specified as a crystallographic direction), or unit quaternions .
The effect of crystal symmetry on misorientations is to reduce the fraction of the full orientation space necessary to uniquely represent all possible misorientation relationships. For example, cubic crystals (i.e. FCC) have 24 symmetrically related orientations. Each of these orientations is physically indistinguishable, though mathematically distinct. Therefore, the size of orientation space is reduced by a factor of 24. This defines the fundamental zone (FZ) for cubic symmetries. For the misorientation between two cubic crystallites, each possesses its 24 inherent symmetries. In addition, there exists a switching symmetry, defined by:
which recognizes the invariance of misorientation to direction; A→B or B→A. The fraction of the total orientation space in the cubic-cubic fundamental zone for misorientation is then given by:
or 1/48 the volume of the cubic fundamental zone. This also has the effect of limiting the maximum unique misorientation angle to 62.8° Disorientation describes the misorientation with the smallest possible rotation angle out of all symmetrically equivalent misorientations that fall within the FZ (usually specified as having an axis in the standard stereographic triangle for cubics). Calculation of these variants involves application of crystal symmetry operators to each of the orientations during the calculation of misorientation. Δ g A B = O B c r y s g B ( O A c r y s g A ) − 1 {\displaystyle \Delta g_{AB}=O_{B}^{crys}g_{B}(O_{A}^{crys}g_{A})^{-1}} where O crys denotes one of the symmetry operators for the material.
The misorientation distribution (MD) is analogous to the ODF used in characterizing texture. The MD describes the probability of the misorientation between any two grains falling into a range d Δ g {\displaystyle d\Delta g} around a given misorientation Δ g {\displaystyle \Delta g} . While similar to a probability density, the MD is not mathematically the same due to the normalization. The intensity in an MD is given as "multiples of random density" (MRD) with respect to the distribution expected in a material with uniformly distributed misorientations. The MD can be calculated by either series expansion, typically using generalized spherical harmonics , or by a discrete binning scheme, where each data point is assigned to a bin and accumulated.
Discrete misorientations or the misorientation distribution can be fully described as plots in the Euler angle, axis/angle, or Rodrigues vector space. Unit quaternions, while computationally convenient, do not lend themselves to graphical representation because of their four-dimensional nature. For any of the representations, plots are usually constructed as sections through the fundamental zone; along φ 2 in Euler angles, at increments of rotation angle for axis/angle, and at constant ρ 3 (parallel to <001>) for Rodrigues. Due to the irregular shape of the cubic-cubic FZ, the plots are typically given as sections through the cubic FZ with the more restrictive boundaries overlaid. Mackenzie plots are a one-dimensional representation of the MD plotting the relative frequency of the misorientation angle, irrespective of the axis. Mackenzie determined the misorientation distribution for a cubic sample with a random texture.
The following is an example of the algorithm for determining the axis/angle representation of misorientation between two texture components given as Euler angles :
The first step is converting the Euler angle representation, [ ϕ 1 , Φ , ϕ 2 ] , {\displaystyle [\phi _{1},\Phi ,\phi _{2}],} to an orientation matrix g by:
[ c ϕ 1 c ϕ 2 − s ϕ 1 s ϕ 2 c Φ s ϕ 1 c ϕ 2 + c ϕ 1 s ϕ 2 c Φ s ϕ 2 s Φ − c ϕ 1 s ϕ 2 − s ϕ 1 c ϕ 2 c Φ − s ϕ 1 s ϕ 2 + c ϕ 1 c ϕ 2 c Φ c ϕ 2 s Φ s ϕ 1 s Φ − c ϕ 1 s Φ c Φ ] {\displaystyle {\begin{bmatrix}c_{\phi _{1}}c_{\phi _{2}}-s_{\phi _{1}}s_{\phi _{2}}c_{\Phi }&s_{\phi _{1}}c_{\phi _{2}}+c_{\phi _{1}}s_{\phi _{2}}c_{\Phi }&s_{\phi _{2}}s_{\Phi }\\-c_{\phi _{1}}s_{\phi _{2}}-s_{\phi _{1}}c_{\phi _{2}}c_{\Phi }&-s_{\phi _{1}}s_{\phi _{2}}+c_{\phi _{1}}c_{\phi _{2}}c_{\Phi }&c_{\phi _{2}}s_{\Phi }\\s_{\phi _{1}}s_{\Phi }&-c_{\phi _{1}}s_{\Phi }&c_{\Phi }\end{bmatrix}}}
where c n {\displaystyle c_{n}} and s n {\displaystyle s_{n}} represent cos {\displaystyle \cos } and sin {\displaystyle \sin } of the respective Euler component. This yields the following orientation matrices:
The misorientation is then:
The axis/angle description (with the axis as a unit vector) is related to the misorientation matrix by:
(There are errors in the similar formulae for the components of 'r' given in the book by Randle and Engler (see refs.), which will be corrected in the next edition of their book. The above are the correct versions, note a different form for these equations has to be used if Θ = 180 degrees.)
For the copper—S 3 misorientation given by Δ g AB , the axis/angle description is 19.5° about [0.689,0.623,0.369], which is only 2.3° from <221>. This result is only one of the 1152 symmetrically related possibilities but does specify the misorientation. This can be verified by considering all possible combinations of orientation symmetry (including switching symmetry). | https://en.wikipedia.org/wiki/Misorientation |
Miss Spider Apps are children's apps for the iPad , iPhone , and iPod Touch published by Callaway Digital Arts. They can be purchased in Apple's App Store and feature the beloved character Miss Spider, who also appears in the bestselling books by David Kirk and the Nick Jr. Channel program Miss Spider's Sunny Patch Friends .
The apps contain a storybook with spoken narration and animations when certain images are pressed; an animated movie of the story; a memory game; several jigsaw puzzles ; and a painting game.
In "Read" mode, the Miss Spider apps function mostly like a book, with words displayed under pictures. Users can click on the bottom of the screen to "turn the page," and when they touch the picture, faint white circles appear that animate characters or other aspects of the picture when touched. There is also an optional narration mode, where the book is read aloud.
In "Watch" mode, the story is turned into a movie, animated with Computer-generated imagery . Words along the bottom are highlighted as they are spoken. [ 1 ] In Miss Spider's Tea Party, the storybook version is followed exactly, whereas in Miss Spider's Bedtime Story, some additional elements are added. [ 2 ]
Match mode works just like a Miss Spider–themed memory game. Users click on overturned "cards" to see if they match. Miss Spider's Tea Party featured 30 cards, while in Miss Spider's Bedtime Story, users can choose between 16 and 30 cards. [ 1 ] [ 2 ]
Paint mode presents users with several different black-and-white images of Miss Spider and other characters in the book, and several different "brushes" and colors. Users select a color and a brush size, then "paint" the pictures using a finger. Users can save images to the iPad 's Camera Roll. [ 3 ] To "clear" the image, users shake the iPad .
Puzzle mode features six different Miss Spider–themed jigsaw puzzles . Users move the pieces into place with their fingers. An animation displays upon completion.
Lonely Miss Spider has made tea for ten, but "nobuggy" will have tea with her! She approaches various bugs she hopes will be her friends, but they're scared away by her spidery reputation. Soon, however, they realize she's a friendly spider who only eats flowers, and Miss Spider finally gets her tea party.
Based on an episode from the Nick Jr. Channel program Miss Spider's Sunny Patch Friends , Miss Spider's Bedtime Story tells the story of Miss Spider and her husband, Holley, trying to get their children to go to bed on time. After spinning a web to help their children plot their goal of going to bed on time for an entire week, Miss Spider and Holley meet success. But they still can't seem to get to bed on time themselves!
Miss Spider's Tea Party was released on the same day as the iPad , April 3, 2010. It was one of the first children's apps developed especially for the iPad. [ 4 ]
USA Today 's Jinny Gudmundson gave the app 4 out of 4 stars and said, "David Kirk's popular children's book really shines on the iPad, particularly while watching it as an animated movie with words highlighted as it is read aloud." [ 5 ]
Gizmodo named the app one of the "Best Apps for Babies, Toddlers, and Sanity-Loving Parents" and said, "These folks are genius, and when you take that and layer in impressive animation, music and narration, it's just got to cost some money. This is apparently the first of many Miss Spider iPad apps, which is good for my kid, bad for my credit card." [ 6 ]
Macgasm.com said "This is one app that I have been recommending to our friends that have children. As I mentioned earlier, I couldn’t get my iPad back from my son because he was so taken by this wonderful application. We recently went to dinner, and I brought the iPad with us. The first thing my son went to use was Miss Spider’s Tea Party. Shortly after he started to use it, we had a small crowd of employees form near our table. Everyone thought it was great, and our waitress even said that she could see her daughter using this application. When an application such as this brings parents and children together you know you have transcended." [ 3 ]
Miss Spider's Bedtime Story was released on September 22, 2010. [ 7 ] Padgadget.com said, "Miss Spider’s Bedtime Story for iPad includes the same great features and quality we love from Callaway. Kids and parents will enjoy reading, watching and playing with Miss Spider and her family of bugs in this brightly colored story and game app." [ 2 ] | https://en.wikipedia.org/wiki/Miss_Spider_apps |
Missense mRNA is a messenger RNA bearing one or more mutated codons that yield polypeptides with an amino acid sequence different from the wild-type or naturally occurring polypeptide. [ 1 ] Missense mRNA molecules are created when template DNA strands or the mRNA strands themselves undergo a missense mutation in which a protein coding sequence is mutated and an altered amino acid sequence is coded for.
A missense mRNA arises from a missense mutation , in the event of which a DNA nucleotide base pair in the coding region of a gene is changed such that it results in the substitution of one amino acid for another. [ 2 ] The point mutation is nonsynonymous because it alters the RNA codon in the mRNA transcript such that translation results in amino acid change. An amino acid change may not result in appreciable changes in protein structure depending on whether the amino acid change is conservative or non-conservative. This owes to the similar physicochemical properties exhibited by some amino acids. [ 3 ]
Missense mRNAs may be detected as a result of two different types of point mutations - spontaneous mutations and induced mutations. [ 4 ] Spontaneous mutations occur during the DNA replication process where a non-complementary nucleotide is deposited by the DNA polymerase in the extension phase. The consecutive round of replication would result in a point mutation. If the resulting mRNA codon is one that changes the amino acid, a missense mRNA would be detected. A hypergeometric distribution study involving DNA polymerase β replication errors in the APC gene revealed 282 possible substitutions that could result in missense mutations. When the APC mRNA was analyzed in the mutational spectrum, it showed 3 sites where the frequency of substitutions were high. [ 5 ]
Induced mutations caused by mutagens can give rise to missense mutations. [ 4 ] Nucleoside analogues such as 2-aminopurine and 5-bromouracil can insert in place of A and T respectively. Ionizing radiation like x-rays and γ-rays can deaminate cytosine to uracil. [ 6 ]
Missense mRNAs may be applied synthetically in forward and reverse genetic screens used to interrogate the genome. Site-directed mutagenesis is a technique often employed to create knock-in and knock-out models that express missense mRNAs. For example, in knock-in studies, human orthologs are identified in model organisms to introduce missense mutations, [ 7 ] or a human gene with a substitution mutation is integrated into the genome of the model organism. [ 8 ] The subsequent loss-of-function or gain-of-function phenotypes are measured to model genetic diseases and discover novel drugs. [ 9 ] While homologous recombination has been widely used to generate single-base substitutions, novel technologies that co-inject gRNA and hCas9 mRNA of the CRISPR/Cas9 system, in conjunction with single-strand oligodeoxynucleotide (ssODN) donor sequences have shown efficiency in generating point mutations in the genome. [ 9 ] [ 10 ] [ 11 ]
Substitutions can occur on the level of both the DNA and RNA. RNA editing-dependent amino acid substitutions can produce missense mRNA's of which occur through hydrolytic deaminase reactions. Two of the most prevalent deaminase reactions occur through the Apolipoprotein B mRNA editing enzyme ( APOBEC ) and the adenosine deaminase acting on RNA enzyme ( ADAR ) which are responsible for the conversion of cytidine to uridine (C-to-U), and the deamination of adenosine to inosine (A-to-I), respectively. [ 12 ] Such selective substitutions of uridine for cytidine, and inosine for adenosine in RNA editing can produce differential isoforms of missense mRNA transcripts, and confer transcriptome diversity and enhanced protein function in response to selective pressures. [ 13 ] | https://en.wikipedia.org/wiki/Missense_mRNA |
The missing dollar riddle is a famous riddle that involves an informal fallacy . It dates to at least the 1930s, although similar puzzles are much older. [ 1 ]
Although the wording and specifics can vary, the puzzle runs along these lines:
Three guests check into a hotel room. The manager says the bill is $30, so each guest pays $10. Later the manager realizes the bill should only have been $25. To rectify this, he gives the bellhop $5 as five one-dollar bills to return to the guests.
On the way to the guests' room to refund the money, the bellhop realizes that he cannot equally divide the five one-dollar bills among the three guests. As the guests are not aware of the total of the revised bill, the bellhop decides to just give each guest $1 back and keep $2 as a tip for himself, and proceeds to do so.
As each guest got $1 back, each guest only paid $9, bringing the total paid to $27. The bellhop kept $2, which when added to the $27, comes to $29. So if the guests originally handed over $30, what happened to the remaining $1?
There seems to be a discrepancy, as there cannot be two answers ($29 and $30) to the math problem. On the one hand it is true that the $25 in the register, the $3 returned to the guests, and the $2 kept by the bellhop add up to $30, but on the other hand, the $27 paid by the guests and the $2 kept by the bellhop add up to only $29.
The misdirection in this riddle is in the second half of the description, where unrelated amounts are added together and the person to whom the riddle is posed assumes those amounts should add up to 30, and is then surprised when they do not — there is, in fact, no reason why the (10 − 1) × 3 + 2 = 29 sum should add up to 30.
The exact sum mentioned in the riddle is computed as:
SUM = $9 (payment by Guest 1) + $9 (payment by Guest 2) + $9 (payment by Guest 3) + $2 (money in bellhop's pocket)
The trick here is to realize that this is not a sum of the money that the three people paid originally, as that would need to include the money the clerk has ($25). This is instead a sum of a smaller amount the people could have paid ($9 × 3 people = $27), added with the additional money that the clerk would not have needed had they paid that smaller amount ($27 paid - $25 actual cost = $2). Another way to say this is, the $27 already includes the bellhop's tip. To add the $2 to the $27 would be to double-count it. So, the three guests' cost of the room, including the bellhop's tip, is $27. Each of the 3 guests has $1 in his pocket, totaling $3. When added to the $27 revised cost of the room (including tip to the bellhop), the total is $30.
To obtain a sum that totals to the original $30, every dollar must be accounted for, regardless of its location.
Thus, the sensible sum can be expressed in this manner:
$30 = $1 (inside Guest pocket) + $1 (inside Guest pocket) + $1 (inside Guest pocket) + $2 (inside bellhop's pocket) + $25 (hotel cash register)
This sum does indeed come out to $30.
To further illustrate why the riddle's sum does not relate to the actual sum, the riddle can be altered so that the discount on the room is extremely large. Consider the riddle in this form:
Three people check into a hotel room. The clerk says the bill is $30, so each guest pays $10. Later the clerk realizes the bill should only be $10. To rectify this, he gives the bellhop $20 to return to the guests. On the way to the room, the bellhop realizes that he cannot divide the money equally. As the guests didn't know the total of the revised bill, the bellhop decides to just give each guest $6 and keep $2 as a tip for himself. Each guest got $6 back: so now each guest only paid $4; bringing the total paid to $12. The bellhop has $2. And $12 + $2 = $14 so, if the guests originally handed over $30, what happened to the remaining $16?
Now it is more obvious that the question is quite unreasonable. One cannot simply add a couple of payments together and expect them to total an original amount of circulated cash.
More economically, money is accounted by summing together all paid amounts ( liabilities ) with all money in one's possession ( assets ). That abstract formula holds regardless of the relative perspectives of the actors in this exchange.
To illustrate the issue through equations:
1) 10 + 10 + 10 = 30
2) 10 + 10 + 10 = 25 + 2 + 3
3) 10 + 10 + 10 - 3 = 25 + 2 + 3 - 3 (adding -3 to both sides of the equation to cancel out the +3 on the right side)
4) 10 - 1 + 10 - 1 + 10 - 1 = 25 + 2
5) 9 + 9 + 9 = 25 + 2 (obs: tip to bellhop has already been paid)
6) 27 = 27
How the riddle is deceptive comes in line 7:
7) 9 + 9 + 9 = 25 + 2
8) 9 + 9 + 9 + 2 ≠ 25 (pushing +2 to the other side without inverting the sign)
9) 27 + 2 ≠ 25
10) 29 ≠ 25
How it should be:
7) 9 + 9 + 9 = 25 + 2
8) 9 + 9 + 9 -2 = 25 + 2 -2 (adding -2 to both sides of the equation to cancel the +2 on the right side, which means the bellhop returned the tip or gave a discount of $2)
9) 9 + 9 + 9 - 2 = 25
10) 27 - 2 = 25
11) 25 = 25
The puzzle should subtract the bellhop's tip from the $27 rather than add it.
Let n guests initially pay p dollars each. The manager refunds r , to which the bellhop gives back b to each guest.
Each guest ends up with a balance of b − p (a negative amount), the manager with np − r and the bellhop r − nb . Whereas the guests' total initial payment is np , the sum of their eventual expense and the bellhop's pilferage is n ( p − b ) + ( r − nb ) = np + r − 2 nb .
The discrepancy noted is thus np − ( np + r − 2 nb ) = 2 nb − r . With the riddle's values, 2 × 3 × $1 − $5 = $1.
Other values such as r = $20 and b = $6 give an unremarkable discrepancy of 2 × 3 × $6 − $20 = $16. Alternatively, values where b = r / 2 n yield no discrepancy. [ 2 ]
There are many variants of the puzzle. Professor David Singmaster 's Chronology of Recreational Mathematics [ 3 ] suggests these type of mathematical misdirection puzzles descended from a problem in an 18th-century arithmetic book, Francis Walkingame's Tutor's Assistant [ 4 ] which was published, and republished, from 1751 to 1860 where it appeared on page 185, prob. 116 in this form, "If 48 taken from 120 leaves 72, and 72 taken from 91 leaves 19, and 7 taken from thence leaves 12, what number is that, out of which, when you have taken 48, 72, 19, and 7, leaves 12?" Singmaster adds, "Though this is not the same as the withdrawal problems below, the mixing of amounts subtracted and remainders makes me think that this kind of problem may have been the basis of the later kind."
An 1880 misdirection is given as "Barthel sees two boxes at a jeweller's, priced at 100 and 200. He buys the cheaper one and takes it home, where he decides he really prefers the other. He returns to the jeweller and gives him the box back and says that the jeweller already has 100 from him, which together with the returned box, makes 200, which is the cost of the other box. The jeweller accepts this and gives Barthel the other box and Barthel goes on his way. Is this correct?"
A model more similar in style to the modern version was given by Cecil B. Read in his 1933 Mathematical Fallacies . His puzzle produces an extra dollar: A man puts $50 in the bank. Then on subsequent days he withdraws $20 leaving $30; then $15 leaving $15; then $9 leaving $6, and finally $6 leaving $0. But $30 + $15 + $6 = $51. Where did the extra dollar come from?
The actual solution to this riddle is to add correctly (correct time, correct person and correct location) from the bank point of view which in this case seems to be the problem:
From the owner point of view the correct solution is this:
The solution appears very obvious if the owner withdraws every day only $10 from $50. To add up 40 + 30 + 20 + 10 using the same pattern from above would be too obviously wrong (result would be $100).
The answer to the question, "Where did the extra dollar come from?" can be found from consecutively adding the bank rest from three different days. This way is correct only if the money owner withdraws every day exact half of the money. Then it will add up. ($25 + $12.50 + $6.25) + $6.25 = $50
Another entry from 1933, R. M. Abraham's Diversions and Pastimes (still available in a Dover version) poses a slightly different approach with this problem from page 16 (problem 61). "A traveller returning to New York found that he had only a ten-dollar postal money order, and that his train fare was seven dollars. The ticket clerk refused to accept the money order, so the traveller went across the road to a pawn shop and pawned it for seven dollars. On his way back to the station he met a friend, who, to save the traveller the trouble of returning to redeem the money order, bought the pawn ticket from him for seven dollars. The traveller then bought his ticket and still had seven dollars when he got to New York. Who made the loss?" David Darling in his The Universal book of Mathematics , [ 5 ] credits this as an earlier version of the three men in a hotel version above.
Even more similar is the English, The Black-Out Book by Evelyn August in 1939; What happened to the shilling?, pp. 82 & 213. Three girls each pay five shillings to share a room. The landlord refunds 5 shillings via the bellboy, who gives them each one and keeps two.
And one more from the same theme appears in an Abbott and Costello routine in which Abbott asks Costello for a fifty-dollar loan. Costello holds out forty dollars and says, "That's all I have." Abbott responds, "Fine, you can owe me the other ten."
The riddle is used by psychotherapist ( Chris Langham ) with his mathematician client ( Paul Whitehouse ) in episode 5 of the 2005 BBC comedy series Help . [ 6 ]
A variation, also involving shillings and three men in a restaurant who are overcharged, appears in the third volume of Jennifer Worth 's Call the Midwife books, Farewell to the East End (2009). There, repairman Fred poses it to the midwives of Nonnatus House .
Another variation, replacing the guests with shepherds, the clerk with a troll, the dollars with sheep and the bellboy with the troll's son, appears in Dr. No by Percival Everett . | https://en.wikipedia.org/wiki/Missing_dollar_riddle |
A missing sock , lost sock , or odd sock (primarily British English ) [ 1 ] [ 2 ] is a single sock in a pair of socks known or perceived to be permanently or temporarily missing. Socks are usually perceived to be lost immediately before, during, or immediately after doing laundry .
According to popular media articles regarding missing socks, people almost always report losing one sock in a pair, and hardly ever the entire pair of two socks. Various explanations or theories —some scientific or pseudo-scientific and others humorous or facetious—have been proposed to show how or why single socks go missing or are perceived to have gone missing.
The terms odd sock and mismatched sock may instead refer to the remaining "orphaned" sock in a pair where the other matching sock is missing or lost.
Two common plausible explanations for missing socks are that they are lost in transit to or from the laundry , or that they are trapped inside, between, or behind components of ("eaten by") washing machines or clothes dryers . Due to the high rotational speeds of modern front-loading washing machines and dryers, it may be possible for small clothes items such as socks to slip through any holes or tears in the rubber gasket between either machine's spinning drums and their outer metal or plastic cases. Socks may also bunch up or unravel and get caught in the water drain pipe of washing machines or in the lint trap of dryers. [ 3 ] [ 4 ]
In 2008, American science educator and writer George B. Johnson proposed several hypotheses for why socks go missing:
In his particular case, Johnson rejected all hypotheses except the last one, as it was possible for small items like socks to slip behind the dryer's spinning drum because of gaps between the drum and the dryer's outer metal case. [ 5 ]
A 2016 pseudo-scientific consumer study commissioned by Samsung Electronics UK (to advertise their new washing machines where users could add more laundry to a load one piece at a time) referenced multiple human errors —including errors of human perception or psychology —to explain why socks go missing: they may become mismatched by poor folding and sorting of laundry, be intentionally misplaced or stolen, fall in hard-to-reach or hard-to-see spaces behind furniture or radiators , or blow off of clothes lines in high wind. [ 2 ] Diffusion of responsibility , poor heuristics , and confirmation bias were the cited psychological reasons. [ 2 ] For example: people may not search for lost socks because they assume others are searching; people search for lost socks in the likeliest places they could have been lost but not in the places where they are actually lost; or people may believe socks are or are not lost because they want to believe so despite evidence to the contrary, respectively. [ 2 ]
The authors of the Samsung study developed an equation called the "sock loss formula" or "sock loss index" which claims to predict the frequency of sock loss for a given individual: Sock loss index = ( L + C ) − ( P × A ) {\displaystyle {\text{Sock loss index}}=(L+C)-(P\times A)} , where L equals laundry size (number of people in a household multiplied by the number of weekly laundry loads), C equals "washing complexity" (the number of types of laundry loads such as dark clothes versus white clothes done in a week multiplied by the total number of socks in those loads), P equals the positive or negative attitude of the individual toward doing laundry on a scale of 1 (most negative) to 5 (most positive), and A equals the "degree of attention" the individual has when doing laundry (the sum of whether the individual checks pockets , unrolls sleeves, turns clothes the right way if they have been turned inside out, and unrolls socks). [ 2 ]
Complementary to the previous explanations, it was also suggested that other small clothes (of which people usually have many items and that get washed often) such as underpants, are lost as often as socks, but people do not notice that as often because they don't come in matching pairs. The existence of the non-paired remaining sock draws attention to the lost sock in a way that cannot happen with clothes that naturally come in singles and not pairs. [ 6 ] Another suggestion made in this context is that since most people usually take off their socks, but not their underpants, when going to sleep, [ 7 ] [ 8 ] there is a higher chance for socks to get lost in the bedroom (e.g. pushed under the bed or taken by a pet as a toy). [ 6 ]
Home appliance repair and design specialists from Sears and GE suggest not overloading laundry machines and repairing any holes in the gaskets between the spinning drums and the rest of the machines to avoid losing socks in them. [ 3 ]
Other practical suggestions include:
Some explanations for the phenomenon jokingly suggest that socks have some innate propensity for going missing, and that this may be a physical property of the universe . For example, in the 1996 book The Nature of Space and Time by the physicists Stephen Hawking and Nobel laureate Roger Penrose , they posit that spontaneous black holes are responsible for lost socks. [ 1 ]
In his 2008 examination of the phenomenon, George B. Johnson also rejected two humorous hypotheses for why socks go missing: that an " intrinsic property " of the socks themselves predisposes or causes them to go missing; and that the socks transform into something else, such as clothes hangers . [ 5 ]
The Bobs ' 1988 song "Where Does the Wayward Footwear Go?", asks where lost socks disappear to, asking "To the bottom of the ocean? Or to China? Or to Cuba? Or Aruba?". A 1993 album by the American indie rock band Grifters is titled One Sock Missing . In the 2001 American children's film Halloweentown II: Kalabar's Revenge , lost objects including socks are magically transported to the home of a character named Gort, who is a compulsive hoarder .
American illustrator and voice actor Harry S. Robins wrote and illustrated a book titled The Meaning of Lost and Mismatched Socks . In the British children's book series Oddies , odd socks are transported to a planet called Oddieworld by a magical washing machine.
The online sock subscription service and retailer Blacksocks was supposedly started after its founder wore mismatched socks to a Japanese tea ceremony . | https://en.wikipedia.org/wiki/Missing_sock |
The missing square puzzle is an optical illusion used in mathematics classes to help students reason about geometrical figures; or rather to teach them not to reason using figures, but to use only textual descriptions and the axioms of geometry. It depicts two arrangements made of similar shapes in slightly different configurations. Each apparently forms a 13×5 right-angled triangle , but one has a 1×1 hole in it.
The key to the puzzle is the fact that neither of the 13×5 "triangles" is truly a triangle, nor would either truly be 13x5 if it were, because what appears to be the hypotenuse is bent. In other words, the "hypotenuse" does not maintain a consistent slope , even though it may appear that way to the human eye.
A true 13×5 triangle cannot be created from the given component parts. The four figures (the yellow, red, blue and green shapes) total 32 units of area. The apparent triangles formed from the figures are 13 units wide and 5 units tall, so it appears that the area should be S = 13×5 / 2 = 32.5 units. However, the blue triangle has a ratio of 5:2 (=2.5), while the red triangle has the ratio 8:3 (≈2.667), so the apparent combined hypotenuse in each figure is actually bent. With the bent hypotenuse, the first figure actually occupies a combined 32 units, while the second figure occupies 33, including the "missing" square.
The amount of bending is approximately 1 / 28 unit (1.245364267°), which is difficult to see on the diagram of the puzzle, and was illustrated as a graphic. Note the grid point where the red and blue triangles in the lower image meet (5 squares to the right and two units up from the lower left corner of the combined figure), and compare it to the same point on the other figure; the edge is slightly under the mark in the upper image, but goes through it in the lower. Overlaying the "hypotenuses" from both figures results in a very thin parallelogram (represented with the four red dots in the above image) with an area of exactly one grid square ( Pick's theorem gives 0 [ 1 ] + 4 [ 2 ] / 2 − 1 = 1), which corresponds to the "missing" area.
According to Martin Gardner , [ 3 ] this particular puzzle was invented by a New York City amateur magician, Paul Curry , in 1953. However, the principle of a dissection paradox has been known since the start of the 16th century.
The integer dimensions of the parts of the puzzle (2, 3, 5, 8, 13) are successive Fibonacci numbers , which leads to the exact unit area in the thin parallelogram .
Many other geometric dissection puzzles are based on a few simple properties of the Fibonacci sequence. [ 4 ]
Sam Loyd 's chessboard paradox demonstrates two rearrangements of an 8×8 square. In the "larger" rearrangement (the 5×13 rectangle in the image to the right), the gaps between the figures have a combined unit square more area than their square gaps counterparts, creating an illusion that the figures there take up more space than those in the original square figure. [ 5 ] In the "smaller" rearrangement (the shape below the 5×13 rectangle), each quadrilateral needs to overlap the triangle by an area of half a unit for its top/bottom edge to align with a grid line, resulting overall loss in one unit square area.
Mitsunobu Matsuyama's "paradox" uses four congruent quadrilaterals and a small square, which form a larger square. When the quadrilaterals are rotated about their centers they fill the space of the small square, although the total area of the figure seems unchanged. The apparent paradox is explained by the fact that the side of the new large square is a little smaller than the original one. If θ is the angle between two opposing sides in each quadrilateral, then the ratio of the two areas is given by sec 2 θ . For θ = 5°, this is approximately 1.00765, which corresponds to a difference of about 0.8%.
A vanishing puzzle is a mechanical optical illusion showing different numbers of a certain object when parts of the puzzle are moved around. [ 6 ] | https://en.wikipedia.org/wiki/Missing_square_puzzle |
Mission Assurance is a full life-cycle engineering process to identify and mitigate design, production, test, and field support deficiencies threatening mission success.
Mission Assurance includes the disciplined application of system engineering , risk management , quality, and management principles to achieve success of a design, development, testing, deployment, and operations process. Mission Assurance's ideal is achieving 100% customer success every time. Mission Assurance reaches across the enterprise, supply base, business partners , and customer base to enable customer success. [ 1 ]
The ultimate goal of Mission Assurance is to create a state of resilience that supports the continuation of an agency's critical business processes and protects its employees, assets, services, and functions. Mission Assurance addresses risks in a uniform and systematic manner across the entire enterprise. [ 2 ]
Mission Assurance is an emerging cross-functional discipline that demands its contributors (project management, governance, system architecture, design, development, integration, testing, and operations) provide and guarantee their combined performance in use. [ 3 ]
The United States Department of Defense 8500-series of policies has three defined mission assurance categories that form the basis for availability and integrity requirements. [ 4 ] [ 5 ] A Mission Assurance Category (MAC) is assigned to all DoD systems
. [ 6 ] It reflects the importance of an information system for the successful completion of a DoD mission. It also determines the requirements for availability and integrity.
NASA's Process Based Mission Assurance Knowledge Management System is an implementation of Mission Assurance that provides "quick and easy access to critical Safety & Mission Assurance data... across all NASA programs and projects." [ 7 ] | https://en.wikipedia.org/wiki/Mission_assurance |
Missouri Public Interest Research Group ( MoPIRG ) is a non-profit organization that is part of the state PIRG organizations.
MoPIRG began in March, 1971, after students at Saint Louis University heard a speech by Ralph Nader . Nader inspired the students to organize citizen action groups modeled after similar groups in Oregon and Minnesota. The Center for Student Action at Saint Louis University and the Missouri Public Action Council at Washington University in St. Louis lobbied to establish a public interest research organization funded by a small assessment added to student activities fees. Student referendums on both campuses supported the fee assessment. The two student groups combined their operations and formed MoPIRG. [ 1 ]
The PIRGs emerged in the early 1970s on U.S. college campuses. The PIRG model was proposed in the book Action for a Change by Ralph Nader and Donald Ross . [ 2 ] Among other early accomplishments, the PIRGs were responsible for much of the Container Deposit Legislation in the United States , also known as "bottle bills." [ 3 ] [ 4 ]
MoPIRG began in March, 1971, after students at Saint Louis University heard a speech by Ralph Nader . Nader inspired the students to organize citizen action groups modeled after similar groups in Oregon and Minnesota. The Center for Student Action at Saint Louis University and the Missouri Public Action Council at Washington University in St. Louis lobbied to establish a public interest research organization funded by a small assessment added to student activities fees. Student referendums on both campuses supported the fee assessment. The two student groups combined their operations and formed MoPIRG. [ 1 ]
MoPIRG's earliest campaigns included a successful appeal to the Federal Trade Commission to investigate deceptive advertising and sales practices by some St. Louis used car dealers in 1972. The group was also represented on the St. Louis Advertising Review Board, a self-regulatory board of the Advertising Club and the Better Business Bureau. MoPIRG gained national attention for its criticism of self-regulation by the advertising industry. MoPIRG also championed the right to representation for St. Louis City Jail prisoners. Their proposal led to the establishment of an ombudsman position by the St. Louis Department of Welfare in August, 1973. [ 1 ]
MOPIRG was active in campaigns to stop legislation that would raise the legal ceiling of small loan interest rates in Missouri, drafted a consumer protection ordinance presented to the St. Louis Board of Aldermen in September, 1973, and published research on a variety of consumer and citizen related issues. It researched and developed legislation that helped improve workers' compensation laws in Missouri and was also successful in stopping an
effort to eliminate the public display rating system for area restaurants. From 1975 to 1981, MOPIRG developed a comprehensive revision of the state landlord-tenant law and successfully worked for its passage twice in the Missouri House of Representatives, although the bill was defeated both times in the Senate. [ 1 ]
In 1977, President Jimmy Carter attempted to reform the patronage system of judicial selection for the Circuit Court of Appeals. MOPIRG responded by forming a coalition of eleven political organizations, including the League of Women Voters and the NAACP , to urge Senator Thomas Eagleton to establish a merit nominating process on the state level. Senator Eagleton's resistance to this idea led MoPIRG to work successfully for legislation requiring the President to develop merit selection guidelines. [ 1 ]
Other issues MoPIRG has been involved with include requiring school testing services to make test results available to students, curbing utility rate increases, reforming media practices, and passing a national advisory referendum that would allow a non-binding public vote on government policy questions. [ 1 ] | https://en.wikipedia.org/wiki/Missouri_Public_Interest_Research_Group |
Mist nets are nets used to capture wild birds and bats . They are used by hunters and poachers to catch and kill animals, but also by ornithologists and chiropterologists for banding and other research projects. Mist nets are typically made of nylon or polyester mesh suspended between two poles, resembling a volleyball net. When properly deployed in the correct habitat, the nets are virtually invisible. Mist nets have shelves created by horizontally strung lines that create a loose, baggy pocket. When a bird or bat hits the net, it falls into this pocket, where it becomes tangled.
The mesh size of the netting varies according to the size of the species targeted for capture. Mesh sizes can be measured along one side of the edge of a single mesh square, or along the diagonal of that square. Measures given here are along the diagonal. Small passerines are typically captured with 16-30 mm mesh, while larger birds, like hawks and ducks, are captured using mesh sizes of ~127 mm. Net dimensions can vary widely depending on the proposed use. Net height for avian mist netting is typically 1.2 - 2.6 m. Net width may vary from 3 to 18 m, although longer nets may also be used. A dho-gazza is a type of mist net that can be used for larger birds, such as raptors . This net lacks shelves.
The purchase and use of mist nets requires permits, which vary according to a country or state's wildlife regulations. Mist net handling requires skill for optimal placement, avoiding entangling nets in vegetation, and proper storage. Bird and bat handling requires extensive training to avoid injury to the captured animals. Bat handling may be especially difficult since bats are captured at night and may bite. A 2011 research survey found mist netting to result in low rates of injury while providing high scientific value. [ 1 ]
Mist nets have been used by Japanese hunters for nearly 300 years to capture birds. They were first introduced into use for ornithology in the United States of America by Oliver L. Austin in 1947. [ 2 ]
Mist netting is a popular and important tool for monitoring species diversity, relative abundance, population size, and demography. There are two ways in which mist nets are primarily utilized: target netting of specific species or individuals, and broadcast netting of all birds within a particular area. Targeted netting is typically used for scientific studies that examine a single species. Nets deployed in this manner often use a playback of a species' song or call, or a model of that species placed near the net to lure the targeted individuals into the net (e.g. [ 3 ] ).
Because broadcast netting captures birds indiscriminately, this technique is better suited to examining the species that occur within a specific habitat. Bird banding stations throughout the United States use this method. Typically, such stations collect a set of standard measurements from each individual, including mass, wing chord, breeding status, body fat index, sex, age, and molt status.
Although setting up mist nets is time-consuming and requires certification, there are certain advantages compared to visual and aural monitoring techniques, such as sampling species that may be poorly detected in other ways. It also allows easy standardization, hands-on examination, and reduces misidentification of species. Because they allow scientists to examine species up close, mist nets are often used in mark-recapture studies over extended periods of time to detect trends in population indices. [ 4 ]
Some uses of data collected using mist net sampling are:
Because there is still debate as to whether or not these techniques provide precise data, it is suggested that mist netting be used as a supplement to aural and visual methods of observation.
One of the main disadvantages of mist nets is that the numbers captured may only represent a small proportion of the true population size. [ 4 ] Mist netting is a unique method in that it provides demographic estimates throughout all seasons, and offers valuable guides to relative abundance of certain species or birds and/or bats. [ 4 ]
Mist nets can be important tools for collecting data to reveal critical ecological conditions in a
variety of situations. This summarized study, "Effects of forest fragmentation on Amazonian understory bird communities" by Richard O. Bierregaard and Thomas E. Lovejoy, used mist nets to analyze the effects of forest fragmentation on understory bird communities in terra firme forest of Central Amazon.
Data from intensive mist netting mark-recapture programs on understory birds from isolated forest reserves were compared to pre-isolation data from the same reserves to investigate changes related to isolation from continuous forest. [ 5 ] Birds surveyed were from a variety of ecological guilds, including nectivores, insectivores, frugivores, obligatory army ant followers, forest edge specialists and flocking species. Periodic sampling by the mist netting capture program provided the quantitative basis for this project. Reserves of varied sizes (1 and 10 hectare) within the Biological Dynamics of Forest Fragments project site were sampled with transects of tethered mist nets once every three or four weeks. Capture rates from isolated reserves were compared to pre-isolation rates to measure changes in population size and/or avian activity due to isolation. [ 5 ] Data was analyzed in the following ways: capture rates per net hour as a function of time since isolation, percent recapture as a function of time since isolation, abundance distribution of species against the species rank by abundance, percent individuals banded according to species and feeding strategy, and finally, capture rates per net hour in isolated reserves against capture rates per net hour in continuous forests. A summary of the results and discussion as stated by Bierregaard and Lovejoy is as follows:
...changes in the understory avian community in isolated patches. Following isolation, capture rates increase significantly as birds fleeing the felled forest entered new forest fragments. Movement to and from the reserve is limited as witnessed by an increase in recapture percentages following isolation. Species of birds that are obligate army ant followers disappeared at the time the surrounding habitat was removed from 1 and 10 ha areas. The complex mixed-species of insectivorous flocks typical of Amazonian forests deteriorated within 2 years of isolation of 1 and 10 ha forest fragments. Several species of mid-story insectivores changed their foraging behavior after isolation of small forest reserves.
These data were collected using mist nets. Data from mist netting efforts may be used to gain a greater understanding of ecological effects of factors impacting ecosystems, such human activities or environmental changes. This is just one example of the use of mist nets as a tool for ecological and biological sciences. Mist net data can also have ecosystem management implications.
The use of mist nets has several disadvantages. Mist-netting is very time-consuming. Nets have to be set up without mistakes. An animal caught in a mist net becomes entangled, so the net must be checked often and the animal removed promptly. Disentangling an animal from a mist net can be difficult and must be done carefully by trained personnel. If an animal is heavily entangled, the mist net may need to be cut to avoid injuring the animal, damaging the material.
Mist nets will not capture birds in direct proportion to their presence in the area (Remsen and Good 1996) and can miss a species completely if it is active in a different strata of vegetation, such as high in the canopy. They can, however, provide an index to population size. [ 6 ] [ 7 ]
People using mist nets must be careful and well-trained, since the capture process can harm birds. One study found the average rate of injury for birds in mist nets is lower than any other method of studying vertebrates, between 0 and 0.59% while the average mortality rate is between 0 and 0.23%. [ 1 ]
While rare, it has been suggested (without scientific studies) that larger birds may be more prone to leg injuries and internal bleeding. Smaller birds typically have problems with tangling issues and wing injuries. [ 8 ] Factors that affect the injury and mortality rate are human error while handling the species, time of year caught, time of day caught, predators in the area, and size/material of the mist net. [ 8 ]
People who are responsible for banding netted wildlife so they can be tracked are called banders in the United States. Banders are responsible for the animals caught and thus apply their training by looking for stress cues (for birds, these include panting, tiredness, closing of eyes, and raising of feathers). Without this caution, animals can severely injure themselves.
In the United States, in order to band a bird or bat, one must have a banding permit from U.S. Fish and Wildlife. The qualifications for permitting vary by species. There are different types of banding permits for birds: the Master Permit and the Sub permit. Master Permits are given to individuals who band on their own or who supervise banding operations. Sub Permits are given to individuals who will be supervised while banding by a person with a Master Permit. In order to receive a permit, one must complete an application and return it to the nearest banding office. Banders must ask for special authorization in their application to use mist nets, cannon nets, chemicals, or auxiliary markers. [ 9 ] | https://en.wikipedia.org/wiki/Mist_net |
Mitchell's embedding theorem , also known as the Freyd–Mitchell theorem or the full embedding theorem , is a result about abelian categories ; it essentially states that these categories, while rather abstractly defined, are in fact concrete categories of modules . This allows one to use element-wise diagram chasing proofs in these categories. The theorem is named after Barry Mitchell and Peter Freyd .
The precise statement is as follows: if A is a small abelian category, then there exists a ring R (with 1, not necessarily commutative) and a full , faithful and exact functor F : A → R -Mod (where the latter denotes the category of all left R -modules ).
The functor F yields an equivalence between A and a full subcategory of R -Mod in such a way that kernels and cokernels computed in A correspond to the ordinary kernels and cokernels computed in R -Mod. Such an equivalence is necessarily additive .
The theorem thus essentially says that the objects of A can be thought of as R -modules, and the morphisms as R -linear maps, with kernels, cokernels, exact sequences and sums of morphisms being determined as in the case of modules. However, projective and injective objects in A do not necessarily correspond to projective and injective R -modules.
Let L ⊂ Fun ( A , A b ) {\displaystyle {\mathcal {L}}\subset \operatorname {Fun} ({\mathcal {A}},Ab)} be the category of left exact functors from the abelian category A {\displaystyle {\mathcal {A}}} to the category of abelian groups A b {\displaystyle Ab} . First we construct a contravariant embedding H : A → L {\displaystyle H:{\mathcal {A}}\to {\mathcal {L}}} by H ( A ) = h A {\displaystyle H(A)=h^{A}} for all A ∈ A {\displaystyle A\in {\mathcal {A}}} , where h A {\displaystyle h^{A}} is the covariant hom-functor, h A ( X ) = Hom A ( A , X ) {\displaystyle h^{A}(X)=\operatorname {Hom} _{\mathcal {A}}(A,X)} . The Yoneda Lemma states that H {\displaystyle H} is fully faithful and we also get the left exactness of H {\displaystyle H} very easily because h A {\displaystyle h^{A}} is already left exact. The proof of the right exactness of H {\displaystyle H} is harder and can be read in Swan, Lecture Notes in Mathematics 76 .
After that we prove that L {\displaystyle {\mathcal {L}}} is an abelian category by using localization theory (also Swan). This is the hard part of the proof.
It is easy to check that the abelian category L {\displaystyle {\mathcal {L}}} is an AB5 category with a generator ⨁ A ∈ A h A {\displaystyle \bigoplus _{A\in {\mathcal {A}}}h^{A}} .
In other words it is a Grothendieck category and therefore has an injective cogenerator I {\displaystyle I} .
The endomorphism ring R := Hom L ( I , I ) {\displaystyle R:=\operatorname {Hom} _{\mathcal {L}}(I,I)} is the ring we need for the category of R -modules.
By G ( B ) = Hom L ( B , I ) {\displaystyle G(B)=\operatorname {Hom} _{\mathcal {L}}(B,I)} we get another contravariant, exact and fully faithful embedding G : L → R - M o d . {\displaystyle G:{\mathcal {L}}\to R\operatorname {-Mod} .} The composition G H : A → R - M o d {\displaystyle GH:{\mathcal {A}}\to R\operatorname {-Mod} } is the desired covariant exact and fully faithful embedding.
Note that the proof of the Gabriel–Quillen embedding theorem for exact categories is almost identical. | https://en.wikipedia.org/wiki/Mitchell's_embedding_theorem |
Mitigation of seismic motion is an important factor in earthquake engineering and construction in earthquake -prone areas. The destabilizing action of an earthquake on constructions may be direct (seismic motion of the ground) or indirect (earthquake-induced landslides , liquefaction of the foundation soils and waves of tsunami ).
Knowledge of local amplification of the seismic motion from the bedrock is very important in order to choose the suitable design solutions. Local amplification can be anticipated from the presence of particular stratigraphic conditions, such as soft soil overlapping the bedrock , or where morphological settings (e.g. crest zones, steep slopes, valleys, or endorheic basins) may produce focalization of the seismic event.
The identification of the areas potentially affected by earthquake-induced landslides and by soil liquefaction can be made by geological survey and by analysis of historical documents. Even quiescent and stabilized landslide areas may be reactivated by severe earthquake. [ 1 ] Young soil may be particularly susceptible to liquefaction.
This seismology article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Mitigation_of_seismic_motion |
MitoMap is a real time haplotyping protocol that analyzes pathogenic variants that cause several mitochondrial diseases. [ 1 ] It was carried out real-time for the first time during the 2013 NexGen Genomics & Bioinformatics Technologies conference at Delhi , India from November 14–16. The results have been published online. [ 2 ]
This bioinformatics-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/MitoMap |
Mitochondria-associated membranes (MAMs) represent regions of the endoplasmic reticulum (ER) which are reversibly tethered to mitochondria . These membranes are involved in import of certain lipids from the ER to mitochondria and in regulation of calcium homeostasis , mitochondrial function, autophagy and apoptosis . They also play a role in development of neurodegenerative diseases and glucose homeostasis . [ 1 ]
In mammalian cells, formation of these linkage sites are important for some cellular events including:
Mitochondria associated membranes are involved in the transport of calcium from the ER to mitochondria. This interaction is important for rapid uptake of calcium by mitochondria through Voltage dependent anion channels (VDACs) , which are located at the outer mitochondrial membrane (OMM). This transport is regulated with chaperones and regulatory proteins which control the formation of the ER–mitochondria junction. Transfer of calcium from ER to mitochondria depends on high concentration of calcium in the intermembrane space, and mitochondrial calcium uniporter (MCU) accumulates calcium into the mitochondrial matrix for electrochemical gradient. [ 1 ]
Transport of phosphatidylserine into mitochondria from the ER for decarboxylation to phosphatidylethanolamine through the ER-mitochondria lipid which transform phosphatidic acid (PA) into phosphatidylserine (PS) by phosphatidylserine synthases 1 and 2 (PSS1, PSS2) in the ER and then transfers PS to mitochondria, where phosphatidylserine decarboxylase (PSD) transform into phosphatidylethanolamine (PE). PE which is synthesized at mitochondria goes back to ER where phosphatidylethanolamine methyltransferase 2 (PEMT2) synthesizes PC (phosphatidylcholine). [ 2 ]
The formation of autophagosomes through the coordination of ATG (autophagy-related) proteins and the vesicular trafficking by MAM. [ citation needed ]
These membrane contact sites have been associated with the delicate balance between life and death of the cell.
Isolation membranes are the initial step to form auto-phagosomes. These closed membranes are double membrane-bond, with lysosomes inside it. The main function of these membrane is degradation, as role in cellular homeostasis . However, the origin of them has remained unclear.
Maybe it is the plasma membrane , the endoplasmic reticulum (ER) and the mitochondria. But the ER- mitochondria contact site have markers, the auto-phagosome marker ATG14, and the auto-phagosome-formation marker ATG5, until the formation of auto-phagosome is complete. Whereas, the absence of ATG14 puncta, it is caused by the breakdown of the ER–mitochondria contact site [ 3 ] The oxidative stress and the beginning of endoplasmic reticulum (ER) stress occur together; the ER stress have a key sensor enriched at the mitochondria-associated ER membranes (MAMs). This key is PERK (RNA-dependent protein kinase (PKR)-like ER kinase), PERK contributes to apoptosis twofold by sustaining the levels of pro-apoptotic C/EBP homologous protein (CHOP). [ 4 ] A tight ER–mitochondria contact site is integral to the mechanisms controlling cellular apoptosis and to inter-organelle Ca 2+ signals. The mitochondria-associated ER membranes (MAMs), play role in cell death modulation. Mitochondrial outer membrane permeabilization (MOMP), is a reason of the higher matrix Ca 2+ levels, which is acts as a trigger for apoptosis. MOMP is the process before apoptosis, which is accompanied to permeability of the inner membrane of the mitochondria (IMM). Permeability transition pore (PTP) opening induces mitochondrial swelling and outer membrane of the mitochondria (OMM) rupture. Moreover, PTP opening induce releasing of caspase-activating factors and apoptosis. Caspase-activating factors induced by cytochrome C to bind to the IP3R, this will result in higher Ca 2+ transfer from the ER to the mitochondria, amplifying the apoptotic signal. [ 5 ]
MAMS play an important role in Ca 2+ Homeostasis , phospholipid and cholesterol metabolism . Research has associated the alteration of these functions of MAMs in Alzheimer's disease . [ 6 ] Mitochindrial associated membranes associated with Alzheimer's disease have been reported to have an up-regulation of lipids synthesized in the MAMs juxtaposition and an up regulation of protein complexes present in the contact region between the ER and mitochondria. Research has suggested that the sites of MAM are the primary sites of activity for γ-secretase activity and amyloid precursor protein (APP) localization along with the presenilin 1 (PS1), presenilin 2 (PS2) proteins. γ-secretase functions in the cleavage of the beta- APP protein. [ 5 ] Patients diagnosed with Alzheimer’s disease have presented results that indicated the accumulation of amyloid beta peptide in the brain which in turn leads to the amyloid cascade suggestion. [ 7 ] Also increased connectivity between the ER and the mitochondria at MAM sites has been observed in human patients diagnosed with familial AD (FAD) by increase of the contact sites. These individuals showed mutations in the PS1, PS2 and APP proteins at the MAM sites. [ 6 ] This increased connectivity also caused an abnormality in Ca 2+ signaling between neurons . Also with regard to the role in MAMs in phospholipid metabolism, patients diagnosed with AD have been reported to show alterations in levels of Phosphatedylserine and phostphatedylethanolamine in the ER and mitochondria respectively, this leads to the intracellular tangles containing hyperphosphorylated forms of the microtubule‐associated protein tau within tissues. [ 7 ]
One of the causes of Parkinson's disease is mutations in genes encoding for different proteins that are localized at the MAM sites. Mutations in the genes that encode the proteins Parkin, PINK1 , alpha-Synuclein (α-Syn) or the protein deglycase DJ-1 have been linked to this disease through research. [ 8 ] However, further research is still being considered in order to determine the direct correlations of these genes to Parkinson’s disease. In normal conditions, these genes are believed to be responsible for the cells ability to degrade mitochondria that has been rendered nonfunctional in a process known as mitophagy . However, mutations in the Parkin and pink1 genes have been associated with the cells becoming incapable of degrading faulty mitochondria. [ 9 ] The proteins alpha-Synuclein (α-Syn) and DJ-1 have been shown to promote MAM function interaction between the ER and the mitochondria. The wild-type gene that codes for α-Syn promotes the physical junction between ER and mitochondria by binding to the lipid raft regions of the MAM. However, the mutant form of this gene has a low affinity to the lipid raft regions, thereby diminishing the contact between the ER and mitochondria and causing accumulation of α-Syn in Lewy bodies which is a major characteristic of PD. [ 8 ] Further research on PD association with alterations in MAM is still being developed. [ citation needed ]
Presenilins are enriched in endoplasmic reticulum membranes associated with mitochondria.
Area-Gomez E, de Groof AJ, Boldogh I, Bird TD, Gibson GE, Koehler CM, Yu WH, Duff KE, Yaffe MP, Pon LA, Schon EA. Am J Pathol. 2009 Nov;175(5):1810-6.
doi: 10.2353/ajpath.2009.090219. PMID: 19834068 | https://en.wikipedia.org/wiki/Mitochondria_associated_membranes |
Mitochondrial DNA ( mtDNA and mDNA ) is the DNA located in the mitochondria organelles in a eukaryotic cell that converts chemical energy from food into adenosine triphosphate (ATP). Mitochondrial DNA is a small portion of the DNA contained in a eukaryotic cell; most of the DNA is in the cell nucleus , and, in plants and algae, the DNA also is found in plastids , such as chloroplasts . [ 3 ] Mitochondrial DNA is responsible for coding of 13 essential subunits of the complex oxidative phosphorylation (OXPHOS) system which has a role in cellular energy conversion. [ 4 ]
Human mitochondrial DNA was the first significant part of the human genome to be sequenced. [ 5 ] This sequencing revealed that human mtDNA has 16,569 base pairs and encodes 13 proteins . As in other vertebrates, the human mitochondrial genetic code differs slightly from nuclear DNA. [ 6 ]
Since animal mtDNA evolves faster than nuclear genetic markers, [ 7 ] [ 8 ] [ 9 ] it represents a mainstay of phylogenetics and evolutionary biology . It also permits tracing the relationships of populations, and so has become important in anthropology and biogeography .
Nuclear and mitochondrial DNA are thought to have separate evolutionary origins, with the mtDNA derived from the circular genomes of bacteria engulfed by the ancestors of modern eukaryotic cells. This theory is called the endosymbiotic theory . In the cells of extant organisms, the vast majority of the proteins in the mitochondria (numbering approximately 1500 different types in mammals ) are coded by nuclear DNA , but the genes for some, if not most, of them are thought to be of bacterial origin, having been transferred to the eukaryotic nucleus during evolution . [ 10 ]
The reasons mitochondria have retained some genes are debated. The existence in some species of mitochondrion-derived organelles lacking a genome [ 11 ] suggests that complete gene loss is possible, and transferring mitochondrial genes to the nucleus has several advantages. [ 12 ] The difficulty of targeting remotely produced hydrophobic protein products to the mitochondrion is one hypothesis for why some genes are retained in mtDNA; [ 13 ] colocalisation for redox regulation is another, citing the desirability of localised control over mitochondrial machinery. [ 14 ] Recent analysis of a wide range of mtDNA genomes suggests that both these features may dictate mitochondrial gene retention. [ 10 ]
Across all organisms, there are six main mitochondrial genome types, classified by structure (i.e. circular versus linear), size, presence of introns or plasmid like structures , and whether the genetic material is a singular molecule or collection of homogeneous or heterogeneous molecules. [ 15 ]
In many unicellular organisms (e.g., the ciliate Tetrahymena and the green alga Chlamydomonas reinhardtii ), and in rare cases also in multicellular organisms (e.g. in some species of Cnidaria ), the mtDNA is linear DNA . Most of these linear mtDNAs possess telomerase -independent telomeres (i.e., the ends of the linear DNA ) with different modes of replication, which have made them interesting objects of research because many of these unicellular organisms with linear mtDNA are known pathogens . [ 16 ]
Most ( bilaterian ) animals have a circular mitochondrial genome. Medusozoa and calcarea clades however include species with linear mitochondrial chromosomes. [ 17 ] With a few exceptions, animals have 37 genes in their mitochondrial DNA: 13 for proteins , 22 for tRNAs , and 2 for rRNAs . [ 18 ]
Mitochondrial genomes for animals average about 16,000 base pairs in length. [ 18 ] The anemone Isarachnanthus nocturnus has the largest mitochondrial genome of any animal at 80,923 bp. [ 19 ] The smallest known mitochondrial genome in animals belongs to the comb jelly Vallicula multiformis , which consist of 9,961 bp. [ 20 ]
In February 2020, a jellyfish-related parasite – Henneguya salminicola – was discovered that lacks a mitochondrial genome but retains structures deemed mitochondrion-related organelles. Moreover, nuclear DNA genes involved in aerobic respiration and mitochondrial DNA replication and transcription were either absent or present only as pseudogenes . This is the first multicellular organism known to have this absence of aerobic respiration and live completely free of oxygen dependency. [ 21 ] [ 22 ]
There are three different mitochondrial genome types in plants and fungi. The first type is a circular genome that has introns (type 2) and may range from 19 to 1000 kbp in length. The second genome type is a circular genome (about 20–1000 kbp) that also has a plasmid-like structure (1 kb) (type 3). The final genome type found in plants and fungi is a linear genome made up of homogeneous DNA molecules (type 5). [ 23 ] [ 24 ] [ 25 ]
Great variation in mtDNA gene content and size exists among fungi and plants, although there appears to be a core subset of genes present in all eukaryotes (except for the few that have no mitochondria at all). [ 10 ] In Fungi, however, there is no single gene shared among all mitogenomes. [ 26 ] Some plant species have enormous mitochondrial genomes, with Silene conica mtDNA containing as many as 11,300,000 base pairs. [ 27 ] Surprisingly, even those huge mtDNAs contain the same number and kinds of genes as related plants with much smaller mtDNAs. [ 28 ] The genome of the mitochondrion of the cucumber ( Cucumis sativus ) consists of three circular chromosomes (lengths 1556, 84 and 45 kilobases), which are entirely or largely autonomous with regard to their replication . [ 29 ]
Protists contain the most diverse mitochondrial genomes, with five different types found in this kingdom. Type 2, type 3, and type 5 of the plant and fungal genomes also exist in some protists, as do two unique genome types. One of these unique types is a heterogeneous collection of circular DNA molecules (type 4) while the other is a heterogeneous collection of linear molecules (type 6). Genome types 4 and 6 each range from 1–200 kbp in size. [ citation needed ]
The smallest mitochondrial genome sequenced to date is the 5,967 bp mtDNA of the parasite Plasmodium falciparum . [ 30 ] [ 31 ]
Endosymbiotic gene transfer, the process by which genes that were coded in the mitochondrial genome are transferred to the cell's main genome, likely explains why more complex organisms such as humans have smaller mitochondrial genomes than simpler organisms such as protists. [ citation needed ]
The two strands of the human mitochondrial DNA are distinguished as the heavy strand and the light strand. [ 32 ] The regulation of mitochondrial DNA replication and transcription initiation is located in a single intergenic noncoding region (NCR). [ 32 ] In human, the 1,100 base pairs NCR region contains three promoters of two L-strand promoters (LSP and LSP2) and one H-strand promoter (HSP). [ 33 ] Unlike bidirectional and specific origin initiation of nuclear DNA replication, mitochondrial DNA has two strand-specific, unidirectional origins of replication of the leading H strand (O H ) which located in NCR and the lagging L strand (O L ) which located in the tRNA gene cluster. [ 34 ]
Mitochondrial DNA is replicated by the DNA polymerase gamma complex which is composed of a 140 kDa catalytic DNA polymerase encoded by the POLG gene and two 55 kDa accessory subunits encoded by the POLG2 gene. [ 35 ] The replisome machinery is formed by DNA polymerase, TWINKLE and mitochondrial SSB proteins . TWINKLE is a helicase , which unwinds short stretches of dsDNA in the 5' to 3' direction. [ 36 ] All these polypeptides are encoded in the nuclear genome. [ citation needed ]
During embryogenesis , replication of mtDNA is strictly down-regulated from the fertilized oocyte through the preimplantation embryo. [ 37 ] The resulting reduction in per-cell copy number of mtDNA plays a role in the mitochondrial bottleneck, exploiting cell-to-cell variability to ameliorate the inheritance of damaging mutations. [ 38 ] According to Justin St. John and colleagues, "At the blastocyst stage, the onset of mtDNA replication is specific to the cells of the trophectoderm . [ 37 ] In contrast, the cells of the inner cell mass restrict mtDNA replication until they receive the signals to differentiate to specific cell types." [ 37 ]
Although several DNA repair pathways have been reported to occur in the mitochondria, currently the base excision repair pathway is the pathway most comprehensively described. [ 39 ] Proteins that are employed in the maintenance of mitochondrial DNA are encoded by nuclear genes and translocated to the mitochondria. [ 39 ] The mitochondria of human cells are capable of repairing DNA base pair mismatches by a pathway that is distinct from the DNA mismatch repair pathway of the nucleus. [ 40 ] This distinct mitochondrial pathway includes the activity of the Y box binding protein 1 (designated YB-1 or YBX1), that likely acts in the mismatch binding and recognition steps of mismatch repair. [ 40 ] DNA repair mechanisms specific to the mitochondria may reflect the proximity of the mitochondrial DNA to the oxidative phosphorylation system and consequently to the DNA-damaging reactive oxygen species formed during ATP production. [ 41 ]
The two strands of the human mitochondrial DNA are distinguished as the heavy strand and the light strand. The heavy strand is rich in guanine and encodes 12 subunits of the oxidative phosphorylation system, two ribosomal RNAs (12S and 16S), and 14 transfer RNAs (tRNAs). The light strand encodes one subunit and 8 tRNAs. So, altogether mtDNA encodes for two rRNAs, 22 tRNAs, and 13 protein subunits , all of which are involved in the oxidative phosphorylation process. [ 44 ] [ 45 ]
Between most (but not all) protein-coding regions, tRNAs are present (see the human mitochondrial genome map ). During transcription, the tRNAs acquire their characteristic L-shape that gets recognized and cleaved by specific enzymes. With the mitochondrial RNA processing, individual mRNA, rRNA, and tRNA sequences are released from the primary transcript. [ 47 ] Folded tRNAs therefore act as secondary structure punctuations. [ 48 ]
Transcription is done by the single-subunit mitochondrial RNA polymerase (POLRMT). In association with two of accessory factors, mitochondrial transcription factor A (TFAM) and mitochondrial transcription factor B2 (TFB2M), the POLRMT complex recognizes promoters and initiates transcription. [ 49 ] Transcription resulted in polycistronic transcripts that are processed in discrete mitochondrial RNA granules into individual mRNAs, tRNAs, and rRNAs. [ 50 ]
The promoters for the initiation of the transcription of the heavy and light strands are located in the main non-coding region of the mtDNA called the displacement loop, the D-loop . [ 44 ] There is evidence that the transcription of the mitochondrial rRNAs is regulated by the heavy-strand promoter 1 (HSP1), and the transcription of the polycistronic transcripts coding for the protein subunits are regulated by HSP2. [ 44 ]
Measurement of the levels of the mtDNA-encoded RNAs in bovine tissues has shown that there are major differences in the expression of the mitochondrial RNAs relative to total tissue RNA. [ 51 ] Among the 12 tissues examined the highest level of expression was observed in the heart, followed by brain and steroidogenic tissue samples. [ 51 ]
As demonstrated by the effect of the trophic hormone ACTH on adrenal cortex cells, the expression of the mitochondrial genes may be strongly regulated by external factors, apparently to enhance the synthesis of mitochondrial proteins necessary for energy production. [ 51 ] Interestingly, while the expression of protein-encoding genes was stimulated by ACTH, the levels of the mitochondrial 16S rRNA showed no significant change. [ 51 ]
In most multicellular organisms , mtDNA is inherited from the mother (maternally inherited). Mechanisms for this include simple dilution (an egg contains on average 200,000 mtDNA molecules, whereas a healthy human sperm has been reported to contain on average 5 molecules), [ 52 ] [ 53 ] degradation of sperm mtDNA in the male genital tract and the fertilized egg; and, at least in a few organisms, failure of sperm mtDNA to enter the egg. Whatever the mechanism, this single parent ( uniparental inheritance ) pattern of mtDNA inheritance is found in most animals, most plants, and also in fungi. [ 54 ]
In a study published in 2018, human babies were reported to inherit mtDNA from both their fathers and their mothers resulting in mtDNA heteroplasmy , [ 55 ] a finding that has been rejected by other scientists. [ 56 ] [ 57 ] [ 58 ]
In sexual reproduction , mitochondria are normally inherited exclusively from the mother; the mitochondria in mammalian sperm are usually destroyed by the egg cell after fertilization. Also, mitochondria are present solely in the midpiece, which is used for propelling the sperm cells, and sometimes the midpiece, along with the tail, is lost during fertilization. In 1999 it was reported that paternal sperm mitochondria (containing mtDNA) are marked with ubiquitin to select them for later destruction inside the embryo . [ 59 ] Some in vitro fertilization techniques, particularly injecting a sperm into an oocyte , may interfere with this. [ citation needed ]
The fact that mitochondrial DNA is mostly maternally inherited enables genealogical researchers to trace maternal lineage far back in time. ( Y-chromosomal DNA , paternally inherited, is used in an analogous way to determine the patrilineal history.) This is usually accomplished on human mitochondrial DNA by sequencing the hypervariable control regions (HVR1 or HVR2), and sometimes the complete molecule of the mitochondrial DNA, as a genealogical DNA test . [ 60 ] HVR1, for example, consists of about 440 base pairs. These 440 base pairs are compared to the same regions of other individuals (either specific people or subjects in a database) to determine maternal lineage. Most often, the comparison is made with the revised Cambridge Reference Sequence . Vilà et al. have published studies tracing the matrilineal descent of domestic dogs from wolves. [ 61 ] The concept of the Mitochondrial Eve is based on the same type of analysis, attempting to discover the origin of humanity by tracking the lineage back in time. [ citation needed ]
Entities subject to uniparental inheritance and with little to no recombination may be expected to be subject to Muller's ratchet , the accumulation of deleterious mutations until functionality is lost. Animal populations of mitochondria avoid this through a developmental process known as the mtDNA bottleneck . The bottleneck exploits random processes in the cell to increase the cell-to-cell variability in mutant load as an organism develops: a single egg cell with some proportion of mutant mtDNA thus produces an embryo in which different cells have different mutant loads. Cell-level selection may then act to remove those cells with more mutant mtDNA, leading to a stabilisation or reduction in mutant load between generations. The mechanism underlying the bottleneck is debated, [ 62 ] [ 63 ] [ 64 ] [ 65 ] with a recent mathematical and experimental metastudy providing evidence for a combination of the random partitioning of mtDNAs at cell divisions and the random turnover of mtDNA molecules within the cell. [ 38 ]
Male mitochondrial DNA inheritance has been discovered in Plymouth Rock chickens . [ 66 ] Evidence supports rare instances of male mitochondrial inheritance in some mammals as well. Specifically, documented occurrences exist for mice, [ 67 ] [ 68 ] where the male-inherited mitochondria were subsequently rejected. It has also been found in sheep, [ 69 ] and in cloned cattle. [ 70 ] Rare cases of male mitochondrial inheritance have been documented in humans. [ 71 ] [ 72 ] [ 73 ] [ 55 ] Although many of these cases involve cloned embryos or subsequent rejection of the paternal mitochondria, others document in vivo inheritance and persistence under lab conditions. [ citation needed ]
Doubly uniparental inheritance of mtDNA is observed in bivalve mollusks. In those species, females have only one type of mtDNA (F), whereas males have F-type mtDNA in their somatic cells, but M-type mtDNA (which can be as much as 30% divergent) in germline cells. [ 74 ] Paternally inherited mitochondria have additionally been reported in some insects such as fruit flies , [ 75 ] [ 76 ] honeybees , [ 77 ] and periodical cicadas . [ 78 ]
An IVF technique known as mitochondrial donation or mitochondrial replacement therapy (MRT) results in offspring containing mtDNA from a donor female, and nuclear DNA from the mother and father. In the spindle transfer procedure, the nucleus of an egg is inserted into the cytoplasm of an egg from a donor female which has had its nucleus removed but still contains the donor female's mtDNA. The composite egg is then fertilized with the male's sperm. The procedure is used when a woman with genetically defective mitochondria wishes to procreate and produce offspring with healthy mitochondria. [ 79 ] The first known child to be born as a result of mitochondrial donation was a boy born to a Jordanian couple in Mexico on 6 April 2016. [ 80 ]
The concept that mtDNA is particularly susceptible to reactive oxygen species generated by the respiratory chain due to its proximity remains controversial. [ 81 ] mtDNA does not accumulate any more oxidative base damage than nuclear DNA. [ 82 ] It has been reported that at least some types of oxidative DNA damage are repaired more efficiently in mitochondria than they are in the nucleus. [ 83 ] mtDNA is packaged with proteins which appear to be as protective as proteins of the nuclear chromatin. [ 84 ] Moreover, mitochondria evolved a unique mechanism which maintains mtDNA integrity through degradation of excessively damaged genomes followed by replication of intact/repaired mtDNA. This mechanism is not present in the nucleus and is enabled by multiple copies of mtDNA present in mitochondria. [ 85 ] The outcome of mutation in mtDNA may be an alteration in the coding instructions for some proteins, [ 86 ] which may have an effect on organism metabolism and/or fitness.
Mutations of mitochondrial DNA can lead to a number of illnesses including exercise intolerance and Kearns–Sayre syndrome (KSS), which causes a person to lose full function of heart, eye, and muscle movements. Some evidence suggests that they might be major contributors to the aging process and age-associated pathologies . [ 87 ] Particularly in the context of disease, the proportion of mutant mtDNA molecules in a cell is termed heteroplasmy . The within-cell and between-cell distributions of heteroplasmy dictate the onset and severity of disease [ 88 ] and are influenced by complicated stochastic processes within the cell and during development. [ 38 ] [ 89 ]
Mutations in mitochondrial tRNAs can be responsible for severe diseases like the MELAS and MERRF syndromes. [ 90 ]
Mutations in nuclear genes that encode proteins that mitochondria use can also contribute to mitochondrial diseases. These diseases do not follow mitochondrial inheritance patterns but instead follow Mendelian inheritance patterns. [ 91 ]
Recently a mutation in mtDNA has been used to help diagnose prostate cancer in patients with negative prostate biopsy . [ 92 ] [ 93 ] mtDNA alterations can be detected in the bio-fluids of patients with cancer. [ 94 ] mtDNA is characterized by the high rate of polymorphisms and mutations. Some of these are increasingly recognized as an important cause of human pathology such as oxidative phosphorylation (OXPHOS) disorders, maternally inherited diabetes and deafness (MIDD), Type 2 diabetes mellitus, Neurodegenerative disease , heart failure, and cancer. [ citation needed ]
Though the idea is controversial, some evidence suggests a link between aging and mitochondrial genome dysfunction. [ 95 ] In essence, mutations in mtDNA upset a careful balance of reactive oxygen species (ROS) production and enzymatic ROS scavenging (by enzymes like superoxide dismutase , catalase , glutathione peroxidase and others). However, some mutations that increase ROS production (e.g., by reducing antioxidant defenses) in worms increase, rather than decrease, their longevity. [ 81 ] Also, naked mole rats , rodents about the size of mice , live about eight times longer than mice despite having reduced, compared to mice, antioxidant defenses and increased oxidative damage to biomolecules. [ 96 ] Once, there was thought to be a positive feedback loop at work (a 'Vicious Cycle'); as mitochondrial DNA accumulates genetic damage caused by free radicals, the mitochondria lose function and leak free radicals into the cytosol . A decrease in mitochondrial function reduces overall metabolic efficiency. [ 97 ] However, this concept was conclusively disproved when it was demonstrated that mice, which were genetically altered to accumulate mtDNA mutations at an accelerated rate to age prematurely, but their tissues do not produce more ROS as predicted by the 'Vicious Cycle' hypothesis. [ 98 ] Supporting a link between longevity and mitochondrial DNA, some studies have found correlations between biochemical properties of the mitochondrial DNA and the longevity of species. [ 99 ] The application of a mitochondrial-specific ROS scavenger, which lead to a significant longevity of the mice studied, [ 100 ] suggests that mitochondria may still be well-implicated in ageing. Extensive research is being conducted to further investigate this link and methods to combat ageing. Presently, gene therapy and nutraceutical supplementation are popular areas of ongoing research. [ 101 ] [ 102 ] Bjelakovic et al. analyzed the results of 78 studies between 1977 and 2012, involving a total of 296,707 participants, and concluded that antioxidant supplements do not reduce all-cause mortality nor extend lifespan, while some of them, such as beta carotene, vitamin E, and higher doses of vitamin A, may actually increase mortality. [ 103 ] In a recent study, it was shown that dietary restriction can reverse ageing alterations by affecting the accumulation of mtDNA damage in several organs of rats. For example, dietary restriction prevented age-related accumulation of mtDNA damage in the cortex and decreased it in the lung and testis. [ 104 ]
Increased mt DNA damage is a feature of several neurodegenerative diseases .
The brains of individuals with Alzheimer's disease have elevated levels of oxidative DNA damage in both nuclear DNA and mtDNA, but the mtDNA has approximately 10-fold higher levels than nuclear DNA. [ 105 ] It has been proposed that aged mitochondria is the critical factor in the origin of neurodegeneration in Alzheimer's disease. [ 106 ] Analysis of the brains of AD patients suggested an impaired function of the DNA repair pathway, which would cause reduce the overall quality of mtDNA. [ 107 ]
In Huntington's disease , mutant huntingtin protein causes mitochondrial dysfunction involving inhibition of mitochondrial electron transport , higher levels of reactive oxygen species and increased oxidative stress . [ 108 ] Mutant huntingtin protein promotes oxidative damage to mtDNA, as well as nuclear DNA, that may contribute to Huntington's disease pathology . [ 109 ]
The DNA oxidation product 8-oxoguanine (8-oxoG) is a well-established marker of oxidative DNA damage. In persons with amyotrophic lateral sclerosis (ALS), the enzymes that normally repair 8-oxoG DNA damages in the mtDNA of spinal motor neurons are impaired. [ 110 ] Thus oxidative damage to mtDNA of motor neurons may be a significant factor in the etiology of ALS. [ citation needed ]
Over the past decade, an Israeli research group led by Professor Vadim Fraifeld has shown that strong and significant correlations exist between the mtDNA base composition and animal species-specific maximum life spans. [ 111 ] [ 112 ] [ 113 ] As demonstrated in their work, higher mtDNA guanine + cytosine content ( GC% ) strongly associates with longer maximum life spans across animal species. An additional observation is that the mtDNA GC% correlation with the maximum life spans is independent of the well-known correlation between animal species' metabolic rate and maximum life spans. The mtDNA GC% and resting metabolic rate explain the differences in animal species' maximum life spans in a multiplicative manner (i.e., species maximum life span = their mtDNA GC% * metabolic rate). [ 112 ] To support the scientific community in carrying out comparative analyses between mtDNA features and longevity across animals, a dedicated database was built named MitoAge . [ 114 ]
De novo mutations arise either due to mistakes during DNA replication or due to unrepaired damage caused in turn by endogenous and exogenous mutagens. It has been long believed that mtDNA can be particularly sensitive to damage caused by reactive oxygen species (ROS), however, G>T substitutions, the hallmark of the oxidative damage in the nuclear genome, are very rare in mtDNA and do not increase with age. Comparing the mtDNA mutational spectra of hundreds of mammalian species, it has been recently demonstrated that species with extended lifespans have an increased rate of A>G substitutions on single-stranded heavy chains. [ 115 ] This discovery led to the hypothesis that A>G is a mitochondria-specific marker of age-associated oxidative damage. This finding provides a mutational (contrary to the selective one) explanation for the observation that long-lived species have GC-rich mtDNA: long-lived species become GC-rich simply because of their biased process of mutagenesis. An association between mtDNA mutational spectrum and species-specific life-history traits in mammals opens a possibility to link these factors together discovering new life-history-specific mutagens in different groups of organisms. [ citation needed ]
Deletion breakpoints frequently occur within or near regions showing non-canonical (non-B) conformations, namely hairpins, cruciforms, and cloverleaf-like elements. [ 116 ] Moreover, data supports the involvement of helix-distorting intrinsically curved regions and long G-tetrads in eliciting instability events. In addition, higher breakpoint densities were consistently observed within GC-skewed regions and in the close vicinity of the degenerate sequence motif YMMYMNNMMHM. [ 117 ]
Unlike nuclear DNA, which is inherited from both parents and in which genes are rearranged in the process of recombination , there is usually no change in mtDNA from parent to offspring. Although mtDNA also recombines, it does so with copies of itself within the same mitochondrion. Because of this and because the mutation rate of animal mtDNA is higher than that of nuclear DNA, [ 118 ] mtDNA is a powerful tool for tracking ancestry through females ( matrilineage ) and has been used in this role to track the ancestry of many species back hundreds of generations. [ citation needed ]
mtDNA testing can be used by forensic scientists in cases where nuclear DNA is severely degraded. Autosomal cells only have two copies of nuclear DNA but can have hundreds of copies of mtDNA due to the multiple mitochondria present in each cell. This means highly degraded evidence that would not be beneficial for STR analysis could be used in mtDNA analysis. mtDNA may be present in bones, teeth, or hair, which could be the only remains left in the case of severe degradation. In contrast to STR analysis, mtDNA sequencing uses Sanger sequencing . The known sequence and questioned sequence are both compared to the Revised Cambridge Reference Sequence to generate their respective haplotypes. If the known sample sequence and questioned sequence originated from the same matriline, one would expect to see identical sequences and identical differences from the rCRS. [ 119 ] Cases arise where there are no known samples to collect and the unknown sequence can be searched in a database such as EMPOP. The Scientific Working Group on DNA Analysis Methods recommends three conclusions for describing the differences between a known mtDNA sequence and a questioned mtDNA sequence: exclusion for two or more differences between the sequences, inconclusive if there is one nucleotide difference, or inability to exclude if there are no nucleotide differences between the two sequences. [ 120 ]
The rapid mutation rate (in animals) makes mtDNA useful for assessing the genetic relationships of individuals or groups within a species and also for identifying and quantifying the phylogeny (evolutionary relationships; see phylogenetics ) among different species. To do this, biologists determine and then compare the mtDNA sequences from different individuals or species. Data from the comparisons is used to construct a network of relationships among the sequences, which provides an estimate of the relationships among the individuals or species from which the mtDNAs were taken. mtDNA can be used to estimate the relationship between both closely related and distantly related species. Due to the high mutation rate of mtDNA in animals, the 3rd positions of the codons change relatively rapidly and thus provide information about the genetic distances among closely related individuals or species. On the other hand, the substitution rate of mt-proteins is very low, thus amino acid changes accumulate slowly (with corresponding slow changes at 1st and 2nd codon positions) and thus they provide information about the genetic distances of distantly related species. Statistical models that treat substitution rates among codon positions separately, can thus be used to simultaneously estimate phylogenies that contain both closely and distantly related species [ 90 ]
Mitochondrial DNA was admitted into evidence for the first time ever in a United States courtroom in 1996 during State of Tennessee v. Paul Ware . [ 121 ]
In the 1998 United States court case of Commonwealth of Pennsylvania v. Patricia Lynne Rorrer, [ 122 ] mitochondrial DNA was admitted into evidence in the State of Pennsylvania for the first time. [ 123 ] [ 124 ] The case was featured in episode 55 of season 5 of the true crime drama series Forensic Files (season 5) . [ 125 ]
Mitochondrial DNA was first admitted into evidence in California , United States, in the successful prosecution of David Westerfield for the 2002 kidnapping and murder of 7-year-old Danielle van Dam in San Diego : it was used for both human and dog identification. [ 126 ] This was the first trial in the U.S. to admit canine DNA. [ 127 ]
The remains of King Richard III , who died in 1485, were identified by comparing his mtDNA with that of two matrilineal descendants of his sister who were alive in 2013, 527 years after he died. [ 128 ]
mtDNA is conserved across eukaryotic organisms given the critical role of mitochondria in cellular respiration . However, due to less efficient DNA repair (compared to nuclear DNA), it has a relatively high mutation rate (but slow compared to other DNA regions such as microsatellites ) which makes it useful for studying the evolutionary relationships— phylogeny —of organisms. Biologists can determine and then compare mtDNA sequences among different species and use the comparisons to build an evolutionary tree for the species examined. [ citation needed ]
For instance, while most nuclear genes are nearly identical between humans and chimpanzees , their mitochondrial genomes are 9.8% different. Human and gorilla mitochondrial genomes are 11.8% different, suggesting that humans may be more closely related to chimpanzees than gorillas. [ 129 ]
Whole genome sequences of more than 66,000 people revealed that most of them had some mitochondrial DNA inserted into their nuclear genomes . More than 90% of these nuclear-mitochondrial segments ( NUMTs ) were inserted after humans diverged from the other apes . Results indicate such transfers currently occur as frequently as once in every ≈4,000 human births. [ 130 ]
It appears that organellar DNA is much more often transferred to nuclear DNA than previously thought. This observation also supports the idea of the endosymbiont theory that eukaryotes have evolved from endosymbionts which turned into organelles while transferring most of their DNA to the nucleus so that the organellar genome shrunk in the process. [ 131 ]
Mitochondrial DNA was discovered in the 1960s by Margit M. K. Nass and Sylvan Nass by electron microscopy as DNase-sensitive threads inside mitochondria, [ 132 ] and by Ellen Haslbrunner, Hans Tuppy and Gottfried Schatz by biochemical assays on highly purified mitochondrial fractions. [ 133 ]
Several specialized databases have been founded to collect mitochondrial genome sequences and other information. Although most of them focus on sequence data, some of them include phylogenetic or functional information.
Genome-wide association studies can reveal associations of mtDNA genes and their mutations with phenotypes including lifespan and disease risks. In 2021, the largest, UK Biobank -based, genome-wide association study of mitochondrial DNA unveiled 260 new associations with phenotypes including lifespan and disease risks for e.g. type 2 diabetes. [ 143 ] [ 144 ]
Several specialized databases exist that report polymorphisms and mutations in the human mitochondrial DNA, together with the assessment of their pathogenicity. | https://en.wikipedia.org/wiki/Mitochondrial_DNA |
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