text stringlengths 11 320k | source stringlengths 26 161 |
|---|---|
Material handling equipment ( MHE ) is mechanical equipment used for the movement, storage, control, and protection of materials, goods and products throughout the process of manufacturing, distribution, consumption, and disposal. [ 1 ] The different types of equipment can be classified into four major categories: [ 2 ] transport equipment, positioning equipment, unit load formation equipment, and storage equipment.
Transport equipment is used to move material from one location to another (e.g., between workplaces, between a loading dock and a storage area, etc.), while positioning equipment is used to manipulate material at a single location. [ 3 ] The major subcategories of transport equipment are conveyors, cranes, and industrial trucks. Material can also be transported manually using no equipment.
Conveyors are used when material is to be moved frequently between specific points over a fixed path and when there is a sufficient flow volume to justify the fixed conveyor investment. [ 4 ] Different types of conveyors can be characterized by the type of product being handled: unit load or bulk load ; the conveyor's location: in-floor , on-floor , or overhead , and whether or not loads can accumulate on the conveyor. Accumulation allows intermittent movement of each unit of material transported along the conveyor, while all units move simultaneously on conveyors without accumulation capability. [ 5 ] For example, while both the roller and flat-belt are unit-load on-floor conveyors, the roller provides accumulation capability while the flat-belt does not; similarly, both the power-and-free and trolley are unit-load overhead conveyors, with the power-and-free designed to include an extra track in order to provide the accumulation capability lacking in the trolley conveyor. Examples of bulk-handling conveyors include the magnetic-belt, troughed-belt, bucket, and screw conveyors. A sortation conveyor system is used for merging, identifying, inducting, and separating products to be conveyed to specific destinations, and typically consists of flat-belt, roller, and chute conveyor segments together with various moveable arms and/or pop-up wheels and chains that deflect, push, or pull products to different destinations. [ 6 ]
Cranes are used to transport loads over variable (horizontal and vertical) paths within a restricted area and when there is insufficient (or intermittent) flow volume such that the use of a conveyor cannot be justified. Cranes provide more flexibility in movement than conveyors because the loads handled can be more varied with respect to their shape and weight. Cranes provide less flexibility in movement than industrial trucks because they only can operate within a restricted area, though some can operate on a portable base. Most cranes utilize trolley-and-tracks for horizontal movement and hoists for vertical movement, although manipulators can be used if precise positioning of the load is required. The most common cranes include the jib, bridge, gantry, and stacker cranes.
Industrial trucks are trucks that are not licensed to travel on public roads ( commercial trucks are licensed to travel on public roads [ 7 ] ). Industrial trucks are used to move materials over variable paths and when there is insufficient (or intermittent) flow volume such that the use of a conveyor cannot be justified. They provide more flexibility in movement than conveyors and cranes because there are no restrictions on the area covered, and they provide vertical movement if the truck has lifting capabilities. Different types of industrial trucks can be characterized by whether or not they have forks for handling pallets , provide powered or require manual lifting and travel capabilities, allow the operator to ride on the truck or require that the operator walk with the truck during travel, provide load stacking capability, and whether or not they can operate in narrow aisles .
Hand trucks (including carts and dollies), the simplest type of industrial truck, cannot transport or stack pallets, is non-powered, and requires the operator to walk. A pallet jack , which cannot stack a pallet, uses front wheels mounted inside the end of forks that extend to the floor as the pallet is only lifted enough to clear the floor for subsequent travel. [ 8 ] A counterbalanced lift truck (sometimes referred to as a forklift truck , but other attachments besides forks can be used) can transport and stack pallets and allows the operator to ride on the truck. The weight of the vehicle (and operator) behind the front wheels of truck counterbalances weight of the load (and weight of vehicle beyond front wheels); the front wheels act as a fulcrum or pivot point. Narrow-aisle trucks usually require that the operator stand-up while riding in order to reduce the truck's turning radius. Reach mechanisms and outrigger arms that straddle and support a load can be used in addition to the just the counterbalance of the truck. On a turret truck, the forks rotate during stacking, eliminating the need for the truck itself to turn in narrow aisles. An order picker allows the operator to be lifted with the load to allow for less-than-pallet-load picking. Automated guided vehicles (AGVs) are industrial trucks that can transport loads without requiring a human operator.
Rail or wheel steered transfer carts are preferred for areas that do not have favourable conditions for the operation of forklifts. Rail transfer carts are carts that can move on the rail line. Wheel steered transfer carts can move independently of the route with battery powered energy systems.
An electric tug is a small battery powered and pedestrian operated machine capable of either pushing or pulling a significantly heavier load than itself.
Commonly used to assist in moving smaller loads where larger equipment would struggle, manual handling equipment such as pallet trucks, trolleys , and sack trucks can be an essential part of any material handling.
A yard ramp, sometimes called a mobile yard ramp, is a movable metal ramp for loading and unloading of vehicles. A yard ramp is placed at the back of a vehicle to provide access for forklifts to ascend the ramp. Using a yard ramp for vehicle loading or unloading allows the work to be carried out by a forklift. [ 10 ]
Positioning equipment is used to handle material at a single location. It can be used at a workplace to feed, orient, load/unload, or otherwise manipulate materials so that are in the correct position for subsequent handling, machining, transport, or storage. As compared to manual handling, the use of positioning equipment can raise the productivity of each worker when the frequency of handling is high, improve product quality and limit damage to materials and equipment when the item handled is heavy or awkward to hold and damage is likely through human error or inattention, and can reduce fatigue and injuries when the environment is hazardous or inaccessible. [ 11 ] In many cases, positioning equipment is required for and can be justified by the ergonomic requirements of a task. Examples of positioning equipment include lift/tilt/turn tables, hoists, balancers, manipulators, and industrial robots . Manipulators act as “muscle multipliers” by counterbalancing the weight of a load so that an operator lifts only a small portion (1%) of the load's weight, and they fill the gap between hoists and industrial robots: they can be used for a wider range of positioning tasks than hoists and are more flexible than industrial robots due to their use of manual control. [ 12 ] They can be powered manually, electrically, or pneumatically, and a manipulator's end-effector can be equipped with mechanical grippers, vacuum grippers, electromechanical grippers, or other tooling.
Unit load formation equipment is used to restrict materials so that they maintain their integrity when handled a single load during transport and for storage. If materials are self-restraining (e.g., a single part or interlocking parts), then they can be formed into a unit load with no equipment. Examples of unit load formation equipment include pallets, skids, slipsheets, tote pans, bins/baskets, cartons, bags, and crates. A pallet is a platform made of wood (the most common), paper, plastic, rubber, or metal with enough clearance beneath its top surface (or face) to enable the insertion of forks for subsequent lifting purposes. [ 13 ] A slipsheet is a thick piece of paper, corrugated fiber, or plastic upon which a load is placed and has tabs that can be grabbed by special push/pull lift truck attachments. They are used in place of a pallet to reduce weight and volume, but loading/unloading is slower.
Storage equipment is used for holding or buffering materials over a period of time. The design of each type of storage equipment, along with its use in warehouse design, represents a trade-off between minimizing handling costs, by making material easily accessible, and maximizing the utilization of space (or cube). [ 14 ] If materials are stacked directly on the floor, then no storage equipment is required, but, on average, each different item in storage will have a stack only half full; to increase cube utilization, storage racks can be used to allow multiple stacks of different items to occupy the same floor space at different levels. The use of racks becomes preferable to floor storage as the number of units per item requiring storage decreases. Similarly, the depth at which units of an item are stored affects cube utilization in proportion to the number of units per item requiring storage.
Pallets can be stored using single- and double-deep racks when the number of units per item is small, while pallet-flow and push-back racks are used when the units per item are mid-range, and floor-storage or drive-in racks are used when the number of units per item is large, with drive-in providing support for pallet loads that cannot be stacked on top of each other. Individual cartons can either be picked from pallet loads or can be stored in carton-flow racks, which are designed to allow first-in, first-out (FIFO) carton access. For individual piece storage, bin shelving, storage drawers, carousels, and A-frames can be used.
Engineered systems are automated solutions designed to streamline and optimize material handling processes. [ 15 ] An automatic storage/retrieval system (AS/RS) is an integrated computer-controlled storage system that combines storage medium, transport mechanism, and controls with various levels of automation for fast and accurate random storage of products and materials. [ 16 ]
Equipment used to collect and communicate the information that is used to coordinate the flow of materials within a facility and between a facility and its suppliers and customers. The identification of materials and associated control can be performed manually with no specialized equipment. [ 17 ] | https://en.wikipedia.org/wiki/Material-handling_equipment |
Material efficiency is a description or metric (( M p) (the ratio of material used to the supplied material)) which refers to decreasing the amount of a particular material needed to produce a specific product. [ 1 ] Making a usable item out of thinner stock than a prior version increases the material efficiency of the manufacturing process. Material efficiency is associated with Green building and Energy conservation , as well as other ways of incorporating Renewable resources in the building process from start to finish.
The impacts can include material efficiency include reducing energy demand, reducing Greenhouse gas emissions , and other environmental impacts such as land use , water scarcity , air pollution , water pollution , and waste management . [ 2 ] A growing population with increasing wealth can increase demand for material extraction, and therefore processing may double in the next 40 years. [ 3 ]
Increasing Material efficiency can reduce the impacts of material consumption. [ 4 ] Some forms of Material Efficiency include increasing the life of existing products, using them more in entirety, re-using components to avoid waste, or reducing the amount of material through a lightweight product design. [ 3 ]
Material efficiency in manufacturing refers to Increasing the efficiency of raw materials to manufactured product, generating less waste per product, and improving waste management. [ 5 ] Using building materials such as steel, reinforced concrete, and aluminum release CO 2 during production. [ 6 ] In 2015, materials manufacturing for building construction were responsible for 11% of global energy-related CO 2 emissions. [ 7 ] The largest market for aluminum is the transportation sector, smaller applications of aluminum include building, construction, and packaging. [ 8 ]
The potential in manufacturing can also refer to improving waste segregation (e.g., separating plastics from combustibles). Recycling and reusing components allow for remanufacturing during the process improvement in creating the product, increasing the material's durability, technology development, and correct component/material purchasing. [ 9 ]
Material efficiency can contribute to a circular economy and capturing value in the industry. [ 10 ] Some companies have applied the circular economy theory to design strategies and business models to close material loops. [ 11 ]
Since 1971, global steel demand has increased by three times, cement by slightly under seven times, primary aluminum by almost six times, and plastics by over ten times. [ 12 ] Significant materials, such as iron and steel, aluminum, cement, chemical products, and pulp and paper, impact the building process. However, employing more efficient strategies to produce these materials will reduce energy and cost without ignoring the reduction of carbon emissions. [ 13 ]
One process is using recycled steel saves room in landfills that the steel would otherwise occupy, saves 75% of the energy required to produce steel in the production process, and saves trees from being cut down to build structures. The recycled steel can be fashioned in the exact dimensions needed for the building and can be made into "customized steel beams and panels to fit each specific design." [ 14 ]
During the manufacturing process, each stage can increase material efficiency, from design and fabrication, through use, and finally to the end of life. [ 12 ]
Some strategies are:
Recycling can allow for lower-emission second purposes to new materials like steel, aluminum, and other metals. [ 12 ] Incorporating recycled materials into the manufacturing process of new goods is a necessary change. Recycling is standard for most materials and is found in every country and economy. [ 1 ] Some materials that can be recycled are:
Aluminum cans from recycled material requiring as little as 4% of the energy needed to make the same cans from bauxite ore . Metals don't degrade as they're recycled in the same way plastics and paper do, fibers shortening every cycle, so many metals are prime candidates for recycling, especially considering their value per ton compared to other recyclables. [ 16 ] Aluminum is a highly desirable metal for recycling because it retains the same properties and quality, no matter how many times the aluminum can be recycled. After all, once it's melted, the structure doesn't change. [ 8 ]
Approximately 36% of all plastic produced is used to create packaging, 85% of which ends up in landfills. [ 17 ] Plastic waste is a mixture of different types of plastics. [ 18 ] Plastic recycling has several challenges. Plastic cannot be recycled several times without quickly degrading in quality; The total bottle recycling rate for 2020 was 27.2%, down from 28.7% in 2019. Every hour, 2.5 million plastic bottles are thrown away in the U.S. Currently, around 75 and 199 million tons of plastic are in our oceans, without considering microplastics . [ 17 ]
Paper (particularly newspaper) have lower energy savings than other materials, with recycled products costing 45% and 21% less energy, respectively. Recycled paper has a large market in China. However, work still needs to be done to facilitate mixed paper recycling instead of newspaper. [ 16 ] Utilizing these recycling methods would permit spending less energy and resources on extracting new resources to use in manufacturing. Despite significant progress in recycling over the last decades, the paper sector is a substantial contributor to global greenhouse gas emissions. [ 19 ] The pulp and paper industries produce 50% of their energy from biomass, which still requires vast energy. [ 8 ]
Public policies help to discuss and provide a market incentive for more efficient use of materials. Impediments to material efficiency improvement include hesitation to invest, a lack of available and accessible information, and economic disincentives. [ 20 ] However, a wide range of policy strategies and innovations have been created in some countries to achieve the mentioned goals. [ 20 ] These include regulation and guidelines; economic incentives; voluntary agreements and actions; information, education, and training; and funding for research, development, and demonstration. [ 21 ]
In 2022, the United States released "The Critical Material Innovation, Efficiency, And Alternatives" program. It will be to study, develop, demonstrate, and trade with the primary goal of creating new alternatives to critical material, promoting efficient manufacturing and use. [ 22 ] In addition, The U.S. Department of Energy released a new "Energy Efficiency Materials Pilot Program for Nonprofits" program to provide nonprofit organizations with funding to upgrade building materials to improve energy efficiency , lower utility costs, and reduce carbon emissions. | https://en.wikipedia.org/wiki/Material_efficiency |
Material failure theory is an interdisciplinary field of materials science and solid mechanics which attempts to predict the conditions under which solid materials fail under the action of external loads . The failure of a material is usually classified into brittle failure ( fracture ) or ductile failure ( yield ). Depending on the conditions (such as temperature , state of stress , loading rate) most materials can fail in a brittle or ductile manner or both. However, for most practical situations, a material may be classified as either brittle or ductile.
In mathematical terms, failure theory is expressed in the form of various failure criteria which are valid for specific materials. Failure criteria are functions in stress or strain space which separate "failed" states from "unfailed" states. A precise physical definition of a "failed" state is not easily quantified and several working definitions are in use in the engineering community. Quite often, phenomenological failure criteria of the same form are used to predict brittle failure and ductile yields.
In materials science , material failure is the loss of load carrying capacity of a material unit. This definition introduces to the fact that material failure can be examined in different scales, from microscopic , to macroscopic . In structural problems, where the structural response may be beyond the initiation of nonlinear material behaviour, material failure is of profound importance for the determination of the integrity of the structure. On the other hand, due to the lack of globally accepted fracture criteria, the determination of the structure's damage, due to material failure, is still under intensive research.
Material failure can be distinguished in two broader categories depending on the scale in which the material is examined:
Microscopic material failure is defined in terms of crack initiation and propagation. Such methodologies are useful for gaining insight in the cracking of specimens and simple structures under well defined global load distributions. Microscopic failure considers the initiation and propagation of a crack. Failure criteria in this case are related to microscopic fracture. Some of the most popular failure models in this area are the micromechanical failure models, which combine the advantages of continuum mechanics and classical fracture mechanics . [ 1 ] Such models are based on the concept that during plastic deformation , microvoids nucleate and grow until a local plastic neck or fracture of the intervoid matrix occurs, which causes the coalescence of neighbouring voids. Such a model, proposed by Gurson and extended by Tvergaard and Needleman , is known as GTN. Another approach, proposed by Rousselier, is based on continuum damage mechanics (CDM) and thermodynamics . Both models form a modification of the von Mises yield potential by introducing a scalar damage quantity, which represents the void volume fraction of cavities, the porosity f .
Macroscopic material failure is defined in terms of load carrying capacity or energy storage capacity, equivalently. Li [ 2 ] presents a classification of macroscopic failure criteria in four categories:
Five general levels are considered, at which the meaning of deformation and failure is interpreted differently: the structural element scale, the macroscopic scale where macroscopic stress and strain are defined, the mesoscale which is represented by a typical void, the microscale and the atomic scale. The material behavior at one level is considered as a collective of its behavior at a sub-level. An efficient deformation and failure model should be consistent at every level.
Failure of brittle materials can be determined using several approaches:
The failure criteria that were developed for brittle solids were the maximum stress / strain criteria. The maximum stress criterion assumes that a material fails when the maximum principal stress σ 1 {\displaystyle \sigma _{1}} in a material element exceeds the uniaxial tensile strength of the material. Alternatively, the material will fail if the minimum principal stress σ 3 {\displaystyle \sigma _{3}} is less than the uniaxial compressive strength of the material. If the uniaxial tensile strength of the material is σ t {\displaystyle \sigma _{t}} and the uniaxial compressive strength is σ c {\displaystyle \sigma _{c}} , then the safe region for the material is assumed to be
Note that the convention that tension is positive has been used in the above expression.
The maximum strain criterion has a similar form except that the principal strains are compared with experimentally determined uniaxial strains at failure, i.e.,
The maximum principal stress and strain criteria continue to be widely used in spite of severe shortcomings.
Numerous other phenomenological failure criteria can be found in the engineering literature. The degree of success of these criteria in predicting failure has been limited. Some popular failure criteria for various type of materials are:
The approach taken in linear elastic fracture mechanics is to estimate the amount of energy needed to grow a preexisting crack in a brittle material. The earliest fracture mechanics approach for unstable crack growth is Griffiths' theory. [ 3 ] When applied to the mode I opening of a crack, Griffiths' theory predicts that the critical stress ( σ {\displaystyle \sigma } ) needed to propagate the crack is given by
where E {\displaystyle E} is the Young's modulus of the material, γ {\displaystyle \gamma } is the surface energy per unit area of the crack, and a {\displaystyle a} is the crack length for edge cracks or 2 a {\displaystyle 2a} is the crack length for plane cracks. The quantity σ π a {\displaystyle \sigma {\sqrt {\pi a}}} is postulated as a material parameter called the fracture toughness . The mode I fracture toughness for plane strain is defined as
where σ c {\displaystyle \sigma _{c}} is a critical value of the far field stress and Y {\displaystyle Y} is a dimensionless factor that depends on the geometry, material properties, and loading condition. The quantity K I c {\displaystyle K_{\rm {Ic}}} is related to the stress intensity factor and is determined experimentally. Similar quantities K I I c {\displaystyle K_{\rm {IIc}}} and K I I I c {\displaystyle K_{\rm {IIIc}}} can be determined for mode II and model III loading conditions.
The state of stress around cracks of various shapes can be expressed in terms of their stress intensity factors . Linear elastic fracture mechanics predicts that a crack will extend when the stress intensity factor at the crack tip is greater than the fracture toughness of the material. Therefore, the critical applied stress can also be determined once the stress intensity factor at a crack tip is known.
The linear elastic fracture mechanics method is difficult to apply for anisotropic materials (such as composites ) or for situations where the loading or the geometry are complex. The strain energy release rate approach has proved quite useful for such situations. The strain energy release rate for a mode I crack which runs through the thickness of a plate is defined as
where P {\displaystyle P} is the applied load, t {\displaystyle t} is the thickness of the plate, u {\displaystyle u} is the displacement at the point of application of the load due to crack growth, and a {\displaystyle a} is the crack length for edge cracks or 2 a {\displaystyle 2a} is the crack length for plane cracks. The crack is expected to propagate when the strain energy release rate exceeds a critical value G I c {\displaystyle G_{\rm {Ic}}} - called the critical strain energy release rate .
The fracture toughness and the critical strain energy release rate for plane stress are related by
where E {\displaystyle E} is the Young's modulus. If an initial crack size is known, then a critical stress can be determined using the strain energy release rate criterion.
A yield criterion often expressed as yield surface, or yield locus, is a hypothesis concerning the limit of elasticity under any combination of stresses. There are two interpretations of yield criterion: one is purely mathematical in taking a statistical approach while other models attempt to provide a justification based on established physical principles. Since stress and strain are tensor qualities they can be described on the basis of three principal directions, in the case of stress these are denoted by σ 1 {\displaystyle \sigma _{1}\,\!} , σ 2 {\displaystyle \sigma _{2}\,\!} , and σ 3 {\displaystyle \sigma _{3}\,\!} .
The following represent the most common yield criterion as applied to an isotropic material (uniform properties in all directions). Other equations have been proposed or are used in specialist situations.
Maximum principal stress theory – by William Rankine (1850). Yield occurs when the largest principal stress exceeds the uniaxial tensile yield strength. Although this criterion allows for a quick and easy comparison with experimental data it is rarely suitable for design purposes. This theory gives good predictions for brittle materials.
Maximum principal strain theory – by St.Venant. Yield occurs when the maximum principal strain reaches the strain corresponding to the yield point during a simple tensile test. In terms of the principal stresses this is determined by the equation:
Maximum shear stress theory – Also known as the Tresca yield criterion , after the French scientist Henri Tresca . This assumes that yield occurs when the shear stress τ {\displaystyle \tau \!} exceeds the shear yield strength τ y {\displaystyle \tau _{y}\!} :
Total strain energy theory – This theory assumes that the stored energy associated with elastic deformation at the point of yield is independent of the specific stress tensor. Thus yield occurs when the strain energy per unit volume is greater than the strain energy at the elastic limit in simple tension. For a 3-dimensional stress state this is given by:
Maximum distortion energy theory ( von Mises yield criterion ) also referred to as octahedral shear stress theory . [ 4 ] – This theory proposes that the total strain energy can be separated into two components: the volumetric ( hydrostatic ) strain energy and the shape (distortion or shear ) strain energy. It is proposed that yield occurs when the distortion component exceeds that at the yield point for a simple tensile test. This theory is also known as the von Mises yield criterion .
The yield surfaces corresponding to these criteria have a range of forms. However, most isotropic yield criteria correspond to convex yield surfaces.
When a metal is subjected to large plastic deformations the grain sizes and orientations change in the direction of deformation. As a result, the plastic yield behavior of the material shows directional dependency. Under such circumstances, the isotropic yield criteria such as the von Mises yield criterion are unable to predict the yield behavior accurately. Several anisotropic yield criteria have been developed to deal with such situations.
Some of the more popular anisotropic yield criteria are:
The yield surface of a ductile material usually changes as the material experiences increased deformation . Models for the evolution of the yield surface with increasing strain, temperature, and strain rate are used in conjunction with the above failure criteria for isotropic hardening , kinematic hardening , and viscoplasticity . Some such models are:
There is another important aspect to ductile materials - the prediction of the ultimate failure strength of a ductile material. Several models for predicting the ultimate strength have been used by the engineering community with varying levels of success. For metals, such failure criteria are usually expressed in terms of a combination of porosity and strain to failure or in terms of a damage parameter. | https://en.wikipedia.org/wiki/Material_failure_theory |
Material flow (or "materials flow") is the description of the transportation of raw materials , pre-fabricates, parts, components, integrated objects and final products as a flow of entities. [ 1 ] The term applies mainly to advanced modeling of supply chain management and its use has been largely subsumed under this heading. [ 2 ]
As industrial material flow can easily become very complex, several different specialized simulation tools have been developed for complex systems. Typical tools include:
This engineering-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Material_flow |
In logic , inference is the process of deriving logical conclusions from premises known or assumed to be true. In checking a logical inference for formal and material validity, the meaning of only its logical vocabulary and of both its logical and extra-logical vocabulary [ clarification needed ] is considered, respectively.
For example, the inference " Socrates is a human, and each human must eventually die, therefore Socrates must eventually die " is a formally valid inference; it remains valid if the nonlogical vocabulary " Socrates ", " is human ", and " must eventually die " is arbitrarily, but consistently replaced. [ note 1 ]
In contrast, the inference " Montreal is north of New York, therefore New York is south of Montreal " is materially valid only; its validity relies on the extra-logical relations " is north of " and " is south of " being converse to each other. [ note 2 ]
Classical formal logic considers the above "north/south" inference as an enthymeme , that is, as an incomplete inference; it can be made formally valid by supplementing the tacitly used conversity relationship explicitly: " Montreal is north of New York, and whenever a location x is north of a location y, then y is south of x; therefore New York is south of Montreal ".
In contrast, the notion of a material inference has been developed by Wilfrid Sellars [ 1 ] in order to emphasize his view that such supplements are not necessary to obtain a correct argument.
Robert Brandom adopted Sellars' view, [ 2 ] arguing that everyday (practical) reasoning is usually non-monotonic , i.e. additional premises can turn a practically valid inference into an invalid one, e.g.
Therefore, practically valid inference is different from formally valid inference (which is monotonic - the above argument that Socrates must eventually die cannot be challenged by whatever additional information), and should better be modelled by materially valid inference. While a classical logician could add a ceteris paribus clause to 1. to make it usable in formally valid inferences:
However, Brandom doubts that the meaning of such a clause can be made explicit, and prefers to consider it as a hint to non-monotony rather than a miracle drug to establish monotony.
Moreover, the "match" example shows that a typical everyday inference can hardly be ever made formally complete. In a similar way, Lewis Carroll 's dialogue " What the Tortoise Said to Achilles " demonstrates that the attempt to make every inference fully complete can lead to an infinite regression. [ 3 ]
Material inference should not be confused with the following concepts, which refer to formal , not material validity: | https://en.wikipedia.org/wiki/Material_inference |
A material passport is a digital document listing all the materials that are included in a product or construction during its life cycle in order to facilitate strategizing circularity decisions in supply chain management. [ 1 ] Passports generally consists of a set of data describing defined characteristics of materials in products, which enables the identification of value for recovery , recycling and re-use . [ 2 ] These passports have been adopted as a best practice for business process analysis and improvement in the widely applied supply chain operation reference (SCOR) by the association for supply chain management . [ 3 ]
The core idea behind the concept is that a material passport will contribute to a more circular economy , in which materials are being recovered, recycled and/or re-used in an open-traded material market. The concept of the 'material passport’ is currently being developed by multiple parties in primarily European countries . Such a passport could make possible second-hand material markets or material banks in the future.
Similar types of passports for the circular economy are being developed by several parties under a variety of terminology. [ 1 ] Other names for the material passport are:
Closely related concepts, which share some of the life cycle registrations that passports also support, are the bill of materials , product life cycle management , digital twin , and ecolabels . The key difference in these concepts is that a passport provides an identity of a single identifiable object and acts as a certified interface to all life-cycle registrations a product is concerned with. [ 1 ]
"According to United Nations estimates, construction accounts for some 50 percent of raw material consumption in Europe and 60 percent of waste." [ 7 ]
Assuming that the earth is a closed system, this situation is objectively untenable. There is an urgent need to deal with raw materials in a more sophisticated manner. A shift in the building sector would greatly benefit movement towards needing less material, and using material more effectively, e.g., by ensuring a much longer and more useful life cycle. Proponents of the material passport argue that it is a step in this direction.
The material passport gives material an identity. By acknowledging that the material exists in a given form in a specific building, it ensures that the material receives and keeps a value, e.g., through a possible re-use after the deconstruction of a building for example.
Like a personal passport , the material passport allows the material to ‘travel,' or identifies the most useful future destination after it has served in a building (or other project/product). This could be in another building or in another product altogether.
By recognizing the individual materials in buildings (or other products), new ownership structures could be facilitated that would enable more functions to be offered as a service. As lighting can be provided as a service, functions such as "shelter from elements" could be a service instead of owning a roof.
In general, material passports create incentives for suppliers to produce and developers / managers / renovators to choose healthy, sustainable and circular materials/building products. They fit into a broader and growing movement that aims at developing circular building business models .
The material passport can be applied to every product or construction. There are different levels in which a product/construct can be discomposed:
For a building, a material passport could be a complete description of all products (staircase, window, furnace, ...), components (iron beam, glass panel, ...), and raw materials (wood, steel, ...), that are present in the building. Ideally, this database would be created during construction and continuously updated. In case an existing building does not yet have a material passport, it can be created through various methods (e.g., plan analysis, digital 3D scanning ).
A material passport allows the owner of a product/construction to know exactly what it is made of. This is of importance at the end of its useful life, to enable the most effective re-use of the materials. It allows the owner to view a product/construct as a depot, inventory of valuable materials.
Furthermore, the process of creating a material passport also shapes the design of the building. The easier the materials can be extracted and re-used on deconstruction of the building, the better. This will lead to an increase of ‘recoverable’ or ‘reversible’ buildings, buildings that can be dis-assembled as easily as they were assembled.
Another possibility is that a material passport can enable the owner to get better insight into the value of the product/construction. Besides the value of the location and of the space, it could also improve the valuation of the materials used. A higher, or more accurate, valuation of product/construction could be made possible.
The first scientific publication about a material passport (2012) was written by Maayke Damen and is called " A resources passport for a circular economy ". It provides a comprehensive overview of the advantages and disadvantages of a material passport for every actor in the supply chain. It includes an outline for the content of a material passport. | https://en.wikipedia.org/wiki/Material_passport |
The material point method ( MPM ) is a numerical technique used to simulate the behavior of solids , liquids , gases , and any other continuum material. Especially, it is a robust spatial discretization method for simulating multi-phase (solid-fluid-gas) interactions. In the MPM, a continuum body is described by a number of small Lagrangian elements referred to as 'material points'. These material points are surrounded by a background mesh/grid that is used to calculate terms such as the deformation gradient. Unlike other mesh-based methods like the finite element method , finite volume method or finite difference method , the MPM is not a mesh based method and is instead categorized as a meshless/meshfree or continuum-based particle method, examples of which are smoothed particle hydrodynamics and peridynamics . Despite the presence of a background mesh, the MPM does not encounter the drawbacks of mesh-based methods (high deformation tangling, advection errors etc.) which makes it a promising and powerful tool in computational mechanics .
The MPM was originally proposed, as an extension of a similar method known as FLIP (a further extension of a method called PIC ) to computational solid dynamics, in the early 1990 by Professors Deborah L. Sulsky , Zhen Chen and Howard L. Schreyer at University of New Mexico. After this initial development, the MPM has been further developed both in the national labs as well as the University of New Mexico , Oregon State University , University of Utah and more across the US and the world. Recently the number of institutions researching the MPM has been growing with added popularity and awareness coming from various sources such as the MPM's use in the Disney film Frozen .
An MPM simulation consists of the following stages:
(Prior to the time integration phase)
(During the time integration phase - explicit formulation )
The PIC was originally conceived to solve problems in fluid dynamics, and developed by Harlow at Los Alamos National Laboratory in 1957. [ 1 ] One of the first PIC codes was the Fluid-Implicit Particle (FLIP) program, which was created by Brackbill in 1986 [ 2 ] and has been constantly in development ever since. Until the 1990s, the PIC method was used principally in fluid dynamics.
Motivated by the need for better simulating penetration problems in solid dynamics, Sulsky, Chen and Schreyer started in 1993 to reformulate the PIC and develop the MPM, with funding from Sandia National Laboratories. [ 3 ] The original MPM was then further extended by Bardenhagen et al. . to include frictional contact, [ 4 ] which enabled the simulation of granular flow, [ 5 ] and by Nairn to include explicit cracks [ 6 ] and crack propagation (known as CRAMP).
Recently, an MPM implementation based on a micro-polar Cosserat continuum [ 7 ] has been used to simulate high-shear granular flow, such as silo discharge. MPM's uses were further extended into Geotechnical engineering with the recent development of a quasi-static, implicit MPM solver which provides numerically stable analyses of large-deformation problems in Soil mechanics . [ 8 ]
Annual workshops on the use of MPM are held at various locations in the United States. The Fifth MPM Workshop was held at Oregon State University , in Corvallis, OR , on April 2 and 3, 2009.
The uses of the PIC or MPM method can be divided into two broad categories: firstly, there are many applications involving fluid dynamics, plasma physics, magnetohydrodynamics, and multiphase applications. The second category of applications comprises problems in solid mechanics.
The PIC method has been used to simulate a wide range of fluid-solid interactions, including sea ice dynamics, [ 9 ] penetration of biological soft tissues, [ 10 ] fragmentation of gas-filled canisters, [ 11 ] dispersion of atmospheric pollutants, [ 12 ] multiscale simulations coupling molecular dynamics with MPM, [ 13 ] [ 14 ] and fluid-membrane interactions. [ 15 ] In addition, the PIC-based FLIP code has been applied in magnetohydrodynamics and plasma processing tools, and simulations in astrophysics and free-surface flow. [ 16 ]
As a result of a joint effort between UCLA's mathematics department and Walt Disney Animation Studios , MPM was successfully used to simulate snow in the 2013 animated film Frozen . [ 17 ] [ 18 ] [ 19 ]
MPM has also been used extensively in solid mechanics, to simulate impact, penetration, collision and rebound, as well as crack propagation. [ 20 ] [ 21 ] MPM has also become a widely used method within the field of soil mechanics: it has been used to simulate granular flow, quickness test of sensitive clays, [ 22 ] landslides, [ 23 ] [ 24 ] [ 25 ] silo discharge, pile driving, fall-cone test, [ 26 ] [ 27 ] [ 28 ] [ 29 ] bucket filling, and material failure; and to model soil stress distribution, [ 30 ] compaction, and hardening. It is now being used in wood mechanics problems such as simulations of transverse compression on the cellular level including cell wall contact. [ 31 ] The work also received the George Marra Award for paper of the year from the Society of Wood Science and Technology. [ 32 ]
One subset of numerical methods are Meshfree methods , which are defined as methods for which "a predefined mesh is not necessary, at least in field variable interpolation". Ideally, a meshfree method does not make use of a mesh "throughout the process of solving the problem governed by partial differential equations, on a given arbitrary domain, subject to all kinds of boundary conditions," although existing methods are not ideal and fail in at least one of these respects. Meshless methods, which are also sometimes called particle methods, share a "common feature that the history of state variables is traced at points (particles) which are not connected with any element mesh, the distortion of which is a source of numerical difficulties." As can be seen by these varying interpretations, some scientists consider MPM to be a meshless method, while others do not. All agree, however, that MPM is a particle method.
The Arbitrary Lagrangian Eulerian (ALE) methods form another subset of numerical methods which includes MPM. Purely Lagrangian methods employ a framework in which a space is discretised into initial subvolumes, whose flowpaths are then charted over time. Purely Eulerian methods, on the other hand, employ a framework in which the motion of material is described relative to a mesh that remains fixed in space throughout the calculation. As the name indicates, ALE methods combine Lagrangian and Eulerian frames of reference.
PIC methods may be based on either the strong form collocation or a weak form discretisation of the underlying partial differential equation (PDE). Those based on the strong form are properly referred to as finite-volume PIC methods. Those based on the weak form discretisation of PDEs may be called either PIC or MPM.
MPM solvers can model problems in one, two, or three spatial dimensions, and can also model axisymmetric problems. MPM can be implemented to solve either quasi-static or dynamic equations of motion , depending on the type of problem that is to be modeled. Several versions of MPM include Generalized Interpolation Material Point Method [ 33 ] ;Convected Particle Domain Interpolation Method; [ 34 ] Convected Particle Least Squares Interpolation Method. [ 35 ]
The time-integration used for MPM may be either explicit or implicit . The advantage to implicit integration is guaranteed stability, even for large timesteps. On the other hand, explicit integration runs much faster and is easier to implement.
Unlike FEM , MPM does not require periodical remeshing steps and remapping of state variables, and is therefore better suited to the modeling of large material deformations. In MPM, particles and not the mesh points store all the information on the state of the calculation. Therefore, no numerical error results from the mesh returning to its original position after each calculation cycle, and no remeshing algorithm is required.
The particle basis of MPM allows it to treat crack propagation and other discontinuities better than FEM, which is known to impose the mesh orientation on crack propagation in a material. Also, particle methods are better at handling history-dependent constitutive models.
Because in MPM nodes remain fixed on a regular grid, the calculation of gradients is trivial.
In simulations with two or more phases it is rather easy to detect contact between entities, as particles can interact via the grid with other particles in the same body, with other solid bodies, and with fluids.
MPM is more expensive in terms of storage than other methods, as MPM makes use of mesh as well as particle data. MPM is more computationally expensive than FEM, as the grid must be reset at the end of each MPM calculation step and reinitialised at the beginning of the following step. Spurious oscillation may occur as particles cross the boundaries of the mesh in MPM, although this effect can be minimized by using generalized interpolation methods (GIMP). In MPM as in FEM, the size and orientation of the mesh can impact the results of a calculation: for example, in MPM, strain localisation is known to be particularly sensitive to mesh refinement.
One stability problem in MPM that does not occur in FEM is the cell-crossing errors and null-space errors [ 36 ] because the number of integration points (material points) does not remain constant in a cell. | https://en.wikipedia.org/wiki/Material_point_method |
The thermodynamic properties of materials are intensive thermodynamic parameters which are specific to a given material. Each is directly related to a second order differential of a thermodynamic potential . Examples for a simple 1-component system are:
where P is pressure , V is volume , T is temperature , S is entropy , and N is the number of particles .
For a single component system, only three second derivatives are needed in order to derive all others, and so only three material properties are needed to derive all others. For a single component system, the "standard" three parameters are the isothermal compressibility κ T {\displaystyle \kappa _{T}} , the specific heat at constant pressure c P {\displaystyle c_{P}} , and the coefficient of thermal expansion α {\displaystyle \alpha } .
For example, the following equations are true:
The three "standard" properties are in fact the three possible second derivatives of the Gibbs free energy with respect to temperature and pressure. Moreover, considering derivatives such as ∂ 3 G ∂ P ∂ T 2 {\displaystyle {\frac {\partial ^{3}G}{\partial P\partial T^{2}}}} and the related Schwartz relations, shows that the properties triplet is not independent. In fact, one property function can be given as an expression of the two others, up to a reference state value. [ 1 ]
The second principle of thermodynamics has implications on the sign of some thermodynamic properties such isothermal compressibility. [ 1 ] [ 2 ] | https://en.wikipedia.org/wiki/Material_properties_(thermodynamics) |
Material selection is a step in the process of designing any physical object. In the context of product design , the main goal of material selection is to minimize cost while meeting product performance goals. [ 1 ] Systematic selection of the best material for a given application begins with properties and costs of candidate materials. Material selection is often benefited by the use of material index or performance index relevant to the desired material properties. [ 2 ] For example, a thermal blanket must have poor thermal conductivity in order to minimize heat transfer for a given temperature difference. It is essential that a designer should have a thorough knowledge of the properties of the materials and their behavior under working conditions. Some of the important characteristics of materials are : strength, durability, flexibility, weight, resistance to heat and corrosion, ability to cast, welded or hardened, machinability, electrical conductivity, etc. [ 3 ] In contemporary design, sustainability is a key consideration in material selection. [ 4 ] Growing environmental consciousness prompts professionals to prioritize factors such as ecological impact, recyclability, and life cycle analysis in their decision-making process.
Systematic selection for applications requiring multiple criteria is more complex. For example, when the material should be both stiff and light, for a rod a combination of high Young's modulus and low density indicates the best material, whereas for a plate the cube root of stiffness divided by density E 3 / ρ {\displaystyle {\sqrt[{3}]{E}}/\rho } is the best indicator, since a plate's bending stiffness scales by its thickness cubed. Similarly, again considering both stiffness and lightness, for a rod that will be pulled in tension the specific modulus , or modulus divided by density E / ρ {\displaystyle E/\rho } should be considered, whereas for a beam that will be subject to bending, the material index E 2 / ρ {\displaystyle {\sqrt[{2}]{E}}/\rho } is the best indicator.
Reality often presents limitations, and the utilitarian factor must be taken in consideration. The cost of the ideal material, depending on shape, size and composition, may be prohibitive, and the demand, the commonality of frequently utilized and known items, its characteristics and even the region of the market dictate its availability.
An Ashby plot, named for Michael Ashby of Cambridge University , is a scatter plot which displays two or more properties of many materials or classes of materials. [ 5 ] These plots are useful to compare the ratio between different properties. For the example of the stiff/light part discussed above would have Young's modulus on one axis and density on the other axis, with one data point on the graph for each candidate material. On such a plot, it is easy to find not only the material with the highest stiffness, or that with the lowest density, but that with the best ratio E / ρ {\displaystyle E/\rho } . Using a log scale on both axes facilitates selection of the material with the best plate stiffness E 3 / ρ {\displaystyle {\sqrt[{3}]{E}}/\rho } .
The first plot on the right shows density and Young's modulus, in a linear scale. The second plot shows the same materials attributes in a log-log scale. Materials families (polymers, foams, metals, etc.) are identified by colors.
Cost of materials plays a very significant role in their selection. The most straightforward way to weight cost against properties is to develop a monetary metric for properties of parts. For example, life cycle assessment can show that the net present value of reducing the weight of a car by 1 kg averages around $5, so material substitution which reduces the weight of a car can cost up to $5 per kilogram of weight reduction more than the original material. [ citation needed ] However, the geography- and time-dependence of energy, maintenance and other operating costs, and variation in discount rates and usage patterns (distance driven per year in this example) between individuals, means that there is no single correct number for this. For commercial aircraft, this number is closer to $450/kg, and for spacecraft, launch costs around $20,000/kg dominate selection decisions. [ 6 ]
Thus as energy prices have increased and technology has improved, automobiles have substituted increasing amounts of lightweight magnesium and aluminium alloys for steel , aircraft are substituting carbon fiber reinforced plastic and titanium alloys for aluminium, and satellites have long been made out of exotic composite materials .
Of course, cost per kg is not the only important factor in material selection. An important concept is 'cost per unit of function'. For example, if the key design objective was the stiffness of a plate of the material, as described in the introductory paragraph above, then the designer would need a material with the optimal combination of density, Young's modulus, and price. Optimizing complex combinations of technical and price properties is a hard process to achieve manually, so rational material selection software is an important tool.
Utilizing an "Ashby chart" is a common method for choosing the appropriate material. First, three different sets of variables are identified:
Next, an equation for the performance index is derived. This equation numerically quantifies how desirable the material will be for a specific situation. By convention, a higher performance index denotes a better material. Lastly, the performance index is plotted on the Ashby chart. Visual inspection reveals the most desirable material.
In this example, the material will be subject to both tension and bending . Therefore, the optimal material will perform well under both circumstances.
In the first situation the beam experiences two forces: the weight of gravity w {\displaystyle w} and tension P {\displaystyle P} . The material variables are density ρ {\displaystyle \rho } and strength σ {\displaystyle \sigma } . Assume that the length L {\displaystyle L} and tension P {\displaystyle P} are fixed, making them design variables. Lastly the cross sectional area A {\displaystyle A} is a free variable. The objective in this situation is to minimize the weight w {\displaystyle w} by choosing a material with the best combination of material variables ρ , σ {\displaystyle \rho ,\sigma } . Figure 1 illustrates this loading.
The stress in the beam is measured as P / A {\displaystyle P/A} whereas weight is described by w = ρ A L {\displaystyle w=\rho AL} . Deriving a performance index requires that all free variables are removed, leaving only design variables and material variables. In this case that means that A {\displaystyle A} must be removed. The axial stress equation can be rearranged to give A = P / σ {\displaystyle A=P/\sigma } . Substituting this into the weight equation gives w = ρ ( P / σ ) L = ρ L P / σ {\displaystyle w=\rho (P/\sigma )L=\rho LP/\sigma } . Next, the material variables and design variables are grouped separately, giving w = ( ρ / σ ) L P {\displaystyle w=(\rho /\sigma )LP} .
Since both L {\displaystyle L} and P {\displaystyle P} are fixed, and since the goal is to minimize w {\displaystyle w} , then the ratio ρ / σ {\displaystyle \rho /\sigma } should be minimized. By convention, however, the performance index is always a quantity which should be maximized. Therefore, the resulting equation is Performance index = P c r = σ / ρ {\displaystyle {\text{Performance index}}=P_{cr}=\sigma /\rho }
Next, suppose that the material is also subjected to bending forces. The max tensile stress equation of bending is σ = ( − M y ) / I {\displaystyle \sigma =(-My)/I} , where M {\displaystyle M} is the bending moment , y {\displaystyle y} is the distance from the neutral axis, and I {\displaystyle I} is the moment of inertia. This is shown in Figure 2. Using the weight equation above and solving for the free variables, the solution arrived at is w = 6 M b L 2 ( ρ / σ ) {\displaystyle w={\sqrt {6MbL^{2}}}(\rho /{\sqrt {\sigma }})} , where L {\displaystyle L} is the length and b {\displaystyle b} is the height of the beam. Assuming that b {\displaystyle b} , L {\displaystyle L} , and M {\displaystyle M} are fixed design variables, the performance index for bending becomes P C R = σ / ρ {\displaystyle P_{CR}={\sqrt {\sigma }}/\rho } .
At this point two performance indices that have been derived: for tension σ / ρ {\displaystyle \sigma /\rho } and for bending σ / ρ {\displaystyle {\sqrt {\sigma }}/\rho } . The first step is to create a log-log plot and add all known materials in the appropriate locations. However, the performance index equations must be modified before being plotted on the log-log graph.
For the tension performance equation P C R = σ / ρ {\displaystyle P_{CR}=\sigma /\rho } , the first step is to take the log of both sides. The resulting equation can be rearranged to give log ( σ ) = log ( ρ ) + log ( P C R ) {\displaystyle \log(\sigma )=\log(\rho )+\log(P_{CR})} . Note that this follows the format of y = x + b {\displaystyle y=x+b} , making it linear on a log-log graph. Similarly, the y-intercept is the log of P C R {\displaystyle P_{CR}} . Thus, the fixed value of P C R {\displaystyle P_{CR}} for tension in Figure 3 is 0.1.
The bending performance equation P C R = σ / ρ {\displaystyle P_{CR}={\sqrt {\sigma }}/\rho } can be treated similarly. Using the power property of logarithms it can be derived that log ( σ ) = 2 × ( log ( ρ ) + log ( P C R ) ) {\displaystyle \log(\sigma )=2\times (\log(\rho )+\log(P_{CR}))} . The value for P C R {\displaystyle P_{CR}} for bending is ≈ 0.0316 in Figure 3. Finally, both lines are plotted on the Ashby chart.
First, the best bending materials can be found by examining which regions are higher on the graph than the σ / ρ {\displaystyle {\sqrt {\sigma }}/\rho } bending line. In this case, some of the foams (blue) and technical ceramics (pink) are higher than the line. Therefore those would be the best bending materials. In contrast, materials which are far below the line (like metals in the bottom-right of the gray region) would be the worst materials.
Lastly, the σ / ρ {\displaystyle \sigma /\rho } tension line can be used to "break the tie" between foams and technical ceramics. Since technical ceramics are the only material which is located higher than the tension line, then the best-performing tension materials are technical ceramics. Therefore, the overall best material is a technical ceramics in the top-left of the pink region such as boron carbide .
The performance index can then be plotted on the Ashby chart by converting the equation to a log scale. This is done by taking the log of both sides, and plotting it similar to a line with P c r {\displaystyle P_{cr}} being the y-axis intercept. This means that the higher the intercept, the higher the performance of the material. By moving the line up the Ashby chart, the performance index gets higher. Each materials the line passes through, has the performance index listed on the y-axis. So, moving to the top of the chart while still touching a region of material is where the highest performance will be.
As seen from figure 3 the two lines intercept near the top of the graph at Technical ceramics and Composites. This will give a performance index of 120 for tensile loading and 15 for bending. When taking into consideration the cost of the engineering ceramics, especially because the intercept is around the Boron carbide, this would not be the optimal case. A better case with lower performance index but more cost effective solutions is around the Engineering Composites near CFRP. | https://en.wikipedia.org/wiki/Material_selection |
Material take off ( MTO ) is a term used in engineering and construction , and refers to a list of materials with quantities and types (such as specific grades of steel ) that are required to build a designed structure or item. This list is generated by analysis of a blueprint or other design document. The list of required materials for construction is sometimes referred to as the material take off list (MTOL).
Material take off is not limited to the amount of required material, but also the weight of the items taken off. This is important when dealing with larger structures, allowing the company that does the take off to determine total weight of the item and how best to move the item (if necessary) when construction is completed.
A material take off (MTO) is the process of analyzing the drawings and determining all the materials required to accomplish the design. Thereafter, the material take off is used to create a bill of materials (BOM). Procurement and requisition are activities that occur after the bill of materials is complete, distinct from Inspection.
The final stages of the MTO is as instrumentally visible in the GAD (General Arrangement Drawing) for specific equipment. The MTO sheet is such an important document in projects as it presents a huge detail such as list of all Materials, quantities, weights, material types, material codes etc. | https://en.wikipedia.org/wiki/Material_take_off |
Material unaccounted for ( MUF ), in the context of nuclear material , refers to any discrepancy between a nuclear-weapons state 's physical inventory of nuclear material, and the book inventory. [ 1 ] The difference can be either a positive discrepancy (an apparent gain of material) or a negative discrepancy (an apparent loss of material). Nuclear accounting discrepancies are commonplace and inevitable due to the problem of accurately measuring nuclear materials. This problem of inaccurate measurement provides a potential loophole for diversion of nuclear materials for weapons production. In a large plant , even a tiny percentage of the annual through-put of nuclear material will suffice to build one or more nuclear weapons . [ 2 ]
MUF is a term used within nuclear material monitoring , the organisational and physical tests used in the monitoring of fissile material and the detection of any impermissible removal. [ 3 ] An associated term is limit of error for the material unaccounted for ( LEMUF ), meaning the associated statistical limits of error possible for the MUF. [ 4 ] In a civilian context, MUF is also sometimes referred to as the inventory difference ( ID ). [ 5 ] [ 6 ]
A 2014 report by the United States Army War College 's Strategic Studies Institute states that although the quantity of MUF globally is unknown, it is "significant." They add that "U.S. nuclear weapons MUF alone is pegged at nearly six tons—i.e., enough to fashion at least 800 low-tech, multi- kiloton bombs," with Russian MUF numbers assumed to be as large. "As for Chinese, Indian, Pakistani, Israeli, and North Korean MUF figures, though, we have only a general idea of what they might be […] The civilian production of nuclear weapons-usable plutonium in the United States, United Kingdom (UK), Japan, France, and India also is a worry. We know that specific accounting losses in the case of civilian plutonium reprocessing and fuel making in the UK and Japan have been significant—measured in scores of bombs worth. What they might be elsewhere, again, is unknown." [ 5 ]
The International Atomic Energy Agency define MUF as the "difference between the book inventory and the physical inventory. This definition may be with respect to either the element or isotope weight." A more exact definition is represented by the following equation:
MUF = I - 0 + B - E
where I designates inputs, 0 designates outputs (which are sometimes subdivided into product and waste streams), B refers to beginning inventory, and E to ending inventory. The three terms in (eq. 3.4.1), I, 0, and B, collectively represent the book inventory, while E represents the physical inventory. Note that the physical inventory for one accounting period becomes a part of the book inventory for the subsequent period.
The IAEA also notes that the "definition of MUF implicitly assumes that the material balance is based completely on measured data. The use of by-difference accounting results in a meaningless MUF. For example, if the contents of waste streams are calculated as the differences between the measured amounts entering a process step and those exiting the step, it is clear that the calculated MUF would be zero over that particular material balance area, i.e., it would be meaningless as a performance index." [ 7 ]
The Center for Public Integrity (CPI) reported that during the Cold War nuclear weapons production was so frantic that approximately six tons of nuclear material, enough to fuel "hundreds of nuclear explosives", has been declared as MUF by the government. They add that "most of it [is] presumed to have been trapped in factory pipes, filters, and machines, or improperly logged in paperwork." [ 8 ] A Defense Nuclear Facilities Safety Board report highlights, “On at least one occasion, in trying to determine the cause of a MUF in excess of 40 kg in 1969, all of the vessels, sumps, and catchbasins were flushed and inspected: total plutonium yield was less than 10% of the MUF." [ 9 ]
Charles D. Ferguson , former president of the Federation of American Scientists , [ 10 ] writes that this "was due to an emphasis […] on fast production rather than accurate accounting." He adds that "Given the history of U.S. production of enriched uranium […] one has to realize that tens of thousands of tons of uranium hexafluoride gas were pumped through these plants to produce the approximately 750 metric tons of HEU for military purposes. It is not surprising then that a few metric tons are considered MUF." Ferguson's assessment of the whereabouts differs from that of the CPI's, stating "discharges of nuclear material in waste streams and the environment were most likely the major reasons why the MUF values were 2.4 metric tons for plutonium and 3.2 metric tons for HEU." [ 9 ] Elmer B. Staats stated, in a 1978 report, "For the most part, MUF is attributed by DOE and NRC to such things as inaccurate measurements and difficult to measure material held up in pipes, filters and machines used in processing special nuclear material." The other main reason given was " clerical errors." [ 5 ]
In 1974, Karen Silkwood , a lab technician at Kerr-McGee , revealed to the Atomic Energy Commission that among many others irregularities, 40 pounds of plutonium was missing from the company's inventory. [ 11 ] In 1977, The New York Times reported that 8,000 pounds of HEU and plutonium was unaccounted for across nuclear plants in the U.S. [ 12 ]
Soviet figures are unknown, as, unlike Western producers of plutonium, Soviet Russia did not use the 'MUF' accounting system (or an equivalent) to track its physical inventory of nuclear materials. Instead, it simply relied on the physical security of its plants. [ 13 ] However, as stated above, Russian figures are presumed to be as large as the United States counterpart.
Worries that weapons-grade nuclear material could perhaps leak onto the black market following the collapse of the Soviet Union at the end of 1991 grabbed headlines at the time. However, several reported deals involving the sale of nuclear material turned out to be hoaxes. Nonetheless, of the approximately 20 known seizures of nuclear weapons materials since the dissolution of the Soviet Union, all have been made in former Soviet states. Thomas Cochran, formerly of the Natural Resources Defense Council stated that "The Russian problem is by far the most serious. People have an incentive to make money. There is clear evidence that stuff is leaking out." The CPI also notes that "although roughly two dozen countries have enough nuclear explosives to make a bomb, Russia's materials have long been the chief Western concern." The main threat about a potential nuclear explosion on Western soil post-cold war, as perceived by the United States, "[has] always been centered around the risk that explosive materials — more than a bomb's mechanical workings — could fall into the wrong hands." In 2005, Porter Goss warned that "There is sufficient material unaccounted for so that it would be possible for those with know-how to construct a nuclear weapon." [ 13 ] [ 14 ]
The CPI also reports that the United States government has spent "$4 billion over the past 25 years to help [Russia] tighten control of the weapons-usable materials inside its vast nuclear complex." In the same report, a US intelligence official alleges that former Russian military and intelligence personnel have been suspects in Nuclear trafficking. However, Russia has dismissed allegations of a leakage of nuclear materials as a smear campaign. Under Vladimir Putin , who first came to power in 1999, Russia has slowly reduced its nuclear security cooperation with the United States, stating that it has no further need of financial or technical assistance from Washington. Michael McFaul states it became a "tertiary issue" under Putin. In his 2007 memoir, George Tenet stated that upon hearing Al-Qaeda were attempting to purchase Russian nuclear devices in 2003, a Department of Energy intelligence official was sent to Moscow seeking information about "reports we had received of missing material," but the Russian government refused to provide details. [ 15 ]
In 2018, the CPI also reported that two United States Department of Energy nuclear specialists "drove to San Antonio to pick up nuclear material from a research lab and transport it to an Idaho lab." However, "before they were able to complete the mission, radioactive material that they brought with them to calibrate radiation detectors was stolen from their vehicle while they stayed at a hotel." Over a year later, the CPI found that the nuclear materials in question, plutonium and cesium , have not been located, "and are now among an unknown amount of military-grade nuclear materials that have gone missing over the years." The same report also found "gaps" in the amount of plutonium manufactured by weapons companies and the amount that the government can account for. [ 16 ]
Charles D. Ferguson writes that the "biggest concern is statistically significant positive MUF values bigger than the LEMUF, because this could indicate diversion, loss, or theft. The inventory difference also has to reconcile losses of uranium and plutonium through radioactive decay and transmutations of an element to a different element in a nuclear reactor or accelerator. Other losses or consumptions of nuclear material occur in nuclear explosives or reactors via fission. The Department of Energy (DOE) has tried to take into account these natural and manmade losses and consumptions in its historical assessment of uranium and plutonium stockpiles." [ 9 ]
As stated by the Office for Nuclear Regulation (ONR), however, the primary cause of MUF is uncertainties "inherent in measurement systems." They elaborate, "[a] finished product such as Plutonium Oxide , can be more accurately weighed and the uncertainty will be smaller." For materials that cannot be easily handled due to dangerous levels of radioactivity, or material in large quantities, "measurements may be more difficult and the associated uncertainties higher. Such measurement uncertainties are a major cause of [MUF] figures, and their existence does not mean that material has been found or lost." [ 17 ] [ 18 ] For example, according to a DOE report dated September 1996, "Most of the HEU in waste has been removed from the U.S. inventory as ‘normal operating losses’ because it is technically too difficult or uneconomical to recover." [ 5 ] The ONR also note that "The magnitude of [the MUF] due to measurement uncertainty will depend strongly on the throughput of material at the plant concerned […] This is particularly the case in bulk processes such as reprocessing, where large volumes of material (hundreds of Tonnes per annum) pass through the plant, often in liquid solution form." [ 18 ]
Residual holdup, which refers to the nuclear material remaining in and around the process equipment and handling areas after operation, is also a problem. The NRC write that "Uranium accumulates in cracks, pores, and zones of poor circulation within and around process equipment. The walls of process vessels and associated plumbing often become coated with uranium during processing of solutions. Uranium also accumulates in air filters and
associated ductwork. The absolute amounts of uranium holdup must be small for efficient processing and proper hazards control. However, the total amount of uranium holdup may be significant in the context of the plant MUF." [ 19 ]
The typical LEMUF by Western standards allows 3% of production to go missing. [ 13 ] For example, in 2005, 29.6 kilograms of plutonium went unaccounted for at Sellafield in the United Kingdom . Although some, including Irish Green Party leader Trevor Sargent , expressed concerns it could fall into the hands of terrorists via the black market , Britain's Department of Trade and Industry (DTI) announced that it was an auditing mistake and involved no actual nuclear material. The Atomic Energy Authority said "The MUF figures for 2003/04 were all within international standards of expected measurement accuracies for closing a nuclear material balance at the type of facility concerned […] There is no evidence to suggest that any of the apparent losses reported were real losses of nuclear material." [ 20 ]
Stringent monitoring of nuclear material stockpiles is common practice amongst the nuclear powers, and can help limit the potential for MUF. [ 21 ] Most have some form of regulations in place to provide for this. As stated by the ONR, however, "Though procedures for nuclear materials accountancy are well developed they cannot be mathematically precise. The presence of positive inventory differences does not mean that material not in existence has somehow been found just as a negative figure does not imply a real loss of material." [ 18 ] Accounting for materials in nuclear waste is a problematic task. Despite efforts to minimise the amount of plutonium or uranium that goes into waste, one cannot eliminate it entirely. MUF, therefore, presents an on-going challenge for nuclear facility operators and the term is instead one amongst a number used in the field of nuclear material monitoring . [ 22 ]
To help prevent residual holdup, defined above, the NRC advises that "When the limit of error of uranium holdup is compatible with the plant LEMUF, the material balance can be computed using the measured contents of uranium holdup. Additional cleanout and recovery for accountability will then not be necessary." However, "when the limit of error of uranium holdup is not compatible with the plant LEMUF, the information obtained in the holdup survey can be used to locate principal uranium accumulations. Once located, substantial accumulations can be recovered, transforming the uranium to a more accurately measurable inventory component. Having reduced the amount of uranium holdup, the limit of error on the remeasurement of the remaining holdup may be sufficiently reduced to be compatible with overall plant LEMUF requirements." [ 19 ]
Theft is still taken seriously. Security at nuclear facilities to detect and prevent any impermissible removal of nuclear material is high. The Strategic Studies Institute (SSI) notes that "Because of the layout and design of fuel cycle facilities, these MUFs can grow over time and may only be resolved by dismantlement and careful clean-out. Unless and until the source of the MUF can be identified, it is impossible to rule out the possibility of diversion or theft." [ 5 ] The Federation of American Scientists comment that "As part of the inventory difference (ID) evaluation, other security events are reviewed to ensure that IDs are not linked to breaches of physical security or insider acts. If there is no evidence of security breaches, then IDs are less likely to be caused by malevolent acts, since integrated security and safeguards work to provide defense-in-depth ." [ 6 ]
Diversion of nuclear materials from civil nuclear power to the production of nuclear weapons by non-nuclear weapons states remains a cause of concern. Elaborating on the methods of detection used, the Arms Control Association write; "The IAEA typically is alerted to the diversion of declared nuclear material in a number of ways. It can detect the removal or alteration of objects containing nuclear materials that the IAEA has sealed or placed under video surveillance, or it can employ accountancy methods to detect shipper-receiver differences and material unaccounted for that exceed limits set by measurement uncertainties." [ 23 ] However, the SSI also highlight that "Despite technological advances in monitoring and accounting systems since 1990, large MUFs have occurred repeatedly at facilities with IAEA-quality safeguards […] These failures have arisen both in non-nuclear weapons states, subject to IAEA safeguards , and in nuclear weapons states subject to analogous domestic regulations." [ 5 ]
The Treaty on the Non-Proliferation of Nuclear Weapons (NPT) was signed in 1968 and became effective in 1970. It was signed by 191 parties, with the notable exceptions of India, Israel, North Korea and Pakistan. [ 24 ] The signatories all agree to International Atomic Energy Agency (IAEA) Safeguards , which requires them to report to the IAEA all information on the amount of nuclear material held in their inventories, and by extension, to allow the IAEA to dispatch inspectors to confirm the authenticity of the report. Nuclear-weapons states , however, are not subject to IAEA safeguards and instead regulate their own respective nuclear industries. [ 25 ]
The Convention on the Physical Protection of Nuclear Material (CPPNM) became effective in 1987. The United States Department of State write that "The CPPNM provides for certain levels of physical protection during international transport of nuclear material. It also establishes a general framework for cooperation among states in the protection, recovery, and return of stolen nuclear material. Further, the Convention lists certain serious offenses involving nuclear material which state parties are to make punishable and for which offenders shall be subject to a system of extradition or submission for prosecution." [ 26 ] There are currently 157 signatories to the convention plus the European Atomic Energy Community . [ 27 ]
The following countries are either recognised nuclear-weapons states (US, UK, Russia, China, and France) or non-signatories to the NPT (North Korea, Israel, India, and Pakistan). As such they are regulated by their own domestic legislation and not subject to International Atomic Energy Agency (IAEA) safeguards . France, however, voluntarily chooses to be subject to IAEA safeguards, in conjunction with its own domestic authorities. Iran is also included, due to its unique circumstance pertaining to the Joint Comprehensive Plan of Action .
The regulations laid out by the United States ' Nuclear Regulatory Commission (NRC) require that each licensee maintain an official material control and accounting programme (known as MC&A) that tracks all special nuclear material (SNM) on site. All licensees are required to maintain accounts showing the receipt, inventory, acquisition, transfer, and disposal of all SNM in its possession regardless of its origin or method of acquisition. [ 28 ] Physical inventories are inspected yearly. Within 60 days of taking the physical inventory, an SNM Physical Inventory Summary Report must be written, outlining any discrepancies between the physical and book inventory. If discrepancies are found, a further report is required to identify and resolve said discrepancies. This report must be sent within 30 days. [ 29 ] [ 30 ] Additionally, the Office of Nuclear Material Safety and Safeguards (NMSS), a branch of the NRC, is responsible for ensuring that security at nuclear facilities remains satisfactory, as well as providing security for the transport of nuclear material. [ 31 ]
Prior to 1 January 2021, the United Kingdom abided by the laws and regulations as outlined by the European Atomic Energy Community (Euratom). However, following the UK's withdrawal from Euratom , the Office for Nuclear Regulation (ONR) assumed its safeguards and nuclear material accountancy duties. The Nuclear Safeguards (EU Exit) Regulations 2019 , which became British law in 2021— stipulate that for each nuclear material balance area, reports must be sent to the ONR showing:
"(i)the beginning physical inventory;
(ii)inventory changes (first increases, then decreases);
(iii)ending book inventory;
(iv)ending physical inventory; and
(v)material unaccounted for".
The regulations also demand that these reports be made "at the latest within the period of 15 days beginning with the day on which the physical inventory was taken." The physical inventory itself "must be taken every calendar year and the period between two successive physical inventory takings must not exceed 14 months." These regulations replaced existing Euratom law. Additionally, the levels of security at nuclear sites are very high. All sites are required to comply with a security plan approved by the Civil Nuclear Security Division of ONR and the measures taken exceed international recommendations in this area. [ 18 ] [ 32 ]
Historically, Russia had no regulations in place pertaining to the monitoring of its nuclear material inventories. Damon Moglen, Director for the Climate and Energy Project stated in 2011 that "The Russians would not know even if there was anything missing." Instead, it simply relied on its physical security to prevent the loss of material. [ 33 ] [ 13 ] However, in 2012 new nuclear security regulations, known as the Federal Rules and Regulations Regarding the Use of Atomic Energy- NP-030-12 - "Basic Nuclear Material Control and Accounting Rules" — were adopted by the Federal Environmental, Industrial, and Nuclear Supervision Service of Russia and made Russian Law. [ 34 ] Shortly thereafter, a non-military overseer group was founded to ensure their implementation. Additionally, Russian officials have stated progress has been made in improving training for security guards, installing new barriers at nuclear facilities, and upgrading sensor technology. [ 35 ]
A report by the University of Maryland School of Public Policy comments that "Specific improvements in the regulations that are worth noting, include: requiring that each site establish a designated MC&A (material control and accounting) organization; requiring that the book inventory be adjusted on the basis of a physical inventory taking; requiring the application of seals with unique identifiers to the most attractive categories of nuclear materials; and requiring the adoption of a two-person rule when accessing and working with nuclear material in certain situations." However, the same report emphasises that while the regulations do "establish the general requirement that statistical analysis be used at the completion of each physical inventory-taking […] it does not establish specific requirements, goals, or criteria. Several Russian facilities are developing relevant analytical methods, but it will take several years for these methods to be fully developed and tested, for personnel to be trained in using them, and for facilities to acquire the necessary technological capabilities to conduct them." [ 36 ]
China has placed particular emphasis on nuclear security. The IAEA comment that the "Chinese government has been continuously strengthening and improving its nuclear security capacity. China has kept an excellent record on nuclear security during the past 60 years." [ 37 ] Licensees in China are required to implement a strict physical inventory inspection procedure, with inventories required at least once per year. For nuclear materials such as plutonium-239 or uranium-233 , inspections are required at least twice per year. For nuclear materials that are inaccessible or cannot be handled due to dangerous levels of radioactivity, inventory inspections rely on operational records and calculations. The National Nuclear Safety Administration is the government agency responsible for enforcing these regulations. China is believed to have been the first to implement a computer-based accounting system in 1996 to monitor its nuclear materials. [ 17 ]
In addition, in 2016, China opened the State Nuclear Security Technology Centre (SNSTC), a state-of-the-art facility specialising in the use of technology to improve the security of nuclear material. Zhenhua Xu, the SNSTC's Deputy Director General stated that "Protecting nuclear or other radioactive material from falling into the hands of terrorists is of growing importance in a country like China, which is expanding its nuclear power programme." China cooperates actively with the IAEA to improve nuclear security, both domestically and globally. [ 38 ] The Chinese government has recently asserted that "For more than 50 years, China has not lost a single gram or single piece of important nuclear material." [ 39 ] Though Reuters contradict this claim. [ 40 ]
Article 67 of the 1978 agreement between the IAEA and France stipulates that the "Material balance reports shall include the following entries unless
otherwise agreed in the Subsidiary Arrangements:
(a) Beginning physical inventory;
(b) Inventory changes (first increases, then decreases);
(c) Ending book inventory;
(d) Shipper/receiver differences;
(e) Adjusted ending book inventory;
(f) Ending physical inventory; and
(g) Material unaccounted for."
Article 72 allows the IAEA to "verify information on the possible causes of material unaccounted for, shipper/receiver differences and uncertainties in the book inventory." For each
inventory change, the date of the inventory change and the originating material balance area and the receiving material balance area or the recipient must be indicated. This agreement became French Law in 1981. [ 41 ] The Autorité de sûreté nucléaire (ASN) is responsible for ensuring these regulations are implemented, as well as for overseeing security. [ 42 ] The IAEA comment that "The nuclear security regime in France is robust and well-established, and incorporates the fundamental principles of the amended CPPNM ." [ 43 ]
North Korea became a member of the IAEA in 1974 and ratified the NPT in 1985. In 1990, North Korea reported to the IAEA that after a "hot test" at a nuclear facility— the purpose being to ultimately extract 60 grams of plutonium— it lost almost 30% of the plutonium in waste streams. [ 44 ] [ 45 ] North Korea did not include the required safeguards agreement with the IAEA until an agreement was signed in 1992. [ 46 ] Article 67 of the 1992 agreement stipulated that ( verbatim with the 1978 French agreement) "Material balance reports shall include the following entries, unless otherwise agreed by the Democratic People's Republic of Korea and the Agency:
(a) beginning physical inventory;
(b) inventory changes (first increases, then decreases);
(c) ending book inventory;
(d) shipper/receiver differences;
(e) adjusted ending book inventory;
(f) ending physical inventory; and
(g) material unaccounted for." Articles 72 and 81 of the agreement also gave provision for "routine inspections" to be carried out by the IAEA to determine compliance in that regard. [ 47 ] However, the agreement only lasted until North Korea's withdrawal from the IAEA in 1994. [ 48 ] North Korea was also suspended from the NPT in June 1993, before later withdrawing altogether in 2011. [ 49 ]
A North Korean official stated in 2004 that the country's annual throughput of spent fuel was 110 tons. There is little information known beyond what North Korea reveals as the country repeatedly refuses international inspectors access to any of its nuclear facilities. An issue facing North Korea is whether the country's frequent power failures allow their nuclear reprocessing facilities to operate continuously, as shutdowns can lead to plutonium losses. [ 45 ]
Iran signed the NPT in 1968 and as such is subject to the standard IAEA safeguards . [ 24 ] The 2015 Joint Comprehensive Plan of Action (JCPOA) between Iran and the P5+1 does not specifically mention material unaccounted for. However, the agreement does stipulate that "Iran will provide the IAEA with all necessary information […] to verify the production of uranium ore concentrate and the inventory of uranium ore concentrate produced in Iran or obtained from any other source for 25 years." [ 50 ] During the period of negotiation for the JCPOA, Iran concealed the existence of what the Middle East Forum describe as a "secret nuclear weapons archive." Once exposed, Iran allowed the IAEA access to its "undeclared nuclear sites" where tests performed by the IAEA proved the existence of MUF. However, "by that time the containers had been moved and the area 'sanitized'." The IAEA still have an unresolved, open investigation into this matter. [ 51 ] In 2020 Iran refused to cooperate with inspectors sent by the IAEA to investigate material unaccounted for. [ 52 ] The investigation was prompted by the IAEA's discovery of a spike in Iran's nuclear-fuel stockpile, far above the levels permitted under the JCPOA. [ 53 ] The British, French and German governments jointly expressed "concerns over possible undeclared and unaccounted for nuclear material". [ 54 ]
Israel has a long-standing policy of deliberate ambiguity with regards to its nuclear program, and as such, has not signed the NPT. On 18 September 2009 the General Conference of the IAEA called on Israel to open its nuclear facilities to IAEA inspection and adhere to the NPT as part of a resolution on "Israeli nuclear capabilities," which passed by a narrow margin of 49–45 with 16 abstentions. The chief Israeli delegate stated that "Israel will not co-operate in any matter with this resolution." [ 55 ]
India has criticised the NPT because it "discriminate[s] against states not possessing nuclear weapons on 1 January 1967," and has said it will only sign the NPT if it is allowed to join as a nuclear-weapons state. But this is seen as unlikely. [ 56 ] The Atomic Energy Regulatory Board is the government organisation responsible carrying out certain regulatory and safety functions under India's Atomic Energy Act, 1962 . However, India's regulatory structure has been criticised by the Nuclear Threat Initiative for failures in its security and accounting practices. A report from 2014 states that key provisions "on security; and, in some cases, security measures are recommended, but not required. Weaknesses are particularly apparent in the areas of transport security, material control and accounting, and measures to protect against insider threat, such as personnel vetting and mandatory reporting of suspicious behaviour." [ 57 ] India did agree in 2005, however, to the India–United States Civil Nuclear Agreement , classifying 14 of its 22 nuclear power plants as being for civilian use and to place them under IAEA safeguards , subject to inspections. [ 58 ] Nonetheless, there is little information available from India's government in regards to nuclear matters except at the most general level, especially in regards to non-civilian use. In relation to the size of India's stockpiles of fissile materials (and related MUF figures), unofficial estimates have considerable uncertainties. [ 59 ]
Pakistan has not signed the NPT, arguing, like India, that it is discriminatory. Foreign Secretary Aizaz Ahmad Chaudhry has said "It is a discriminatory treaty. Pakistan has the right to defend itself, so Pakistan will not sign the NPT. Why should we?" [ 60 ] Pakistan had always asserted that it would sign the NPT if India did so. However, in 2010, Pakistan abandoned this position, stating that, like India, it would only join the NPT as a recognised nuclear-weapon state. [ 61 ] Pakistan did notably sign a safeguards agreement with the IAEA in 1977 for the import of uranium concentrate from Niger . By extension, Pakistan's civilian nuclear component, which was built with foreign assistance, is under IAEA safeguards. Since 1974, however, Pakistan's nuclear complex has also had a significant military component, of which, there is no official quantitative information available in regards to fissile-material production or losses of said material. [ 59 ] | https://en.wikipedia.org/wiki/Material_unaccounted_for |
Materials & Design is a peer-reviewed open access scientific journal published by Elsevier . It covers research on the practical applications of engineering materials including materials processing. Article formats are regular, express, and review articles (typically commissioned by the editors). The editor-in-chief is Alexander M. Korsunsky ( Trinity College, Oxford ). The journal was established in 1978 as the International Journal of Materials in Engineering Applications and obtained its current title in 1980.
The journal is abstracted and indexed by: [ 1 ]
According to the Journal Citation Reports , the journal has a 2022 impact factor of 8.4. [ 2 ] | https://en.wikipedia.org/wiki/Materials_&_Design |
Materials is a semi-monthly peer-reviewed open access scientific journal covering materials science and engineering . It was established in 2008 and is published by MDPI . The editor-in-chief is Maryam Tabrizian ( McGill University ). The journal publishes reviews, regular research papers, short communications, and book reviews . There are currently hundreds of calls for submissions to special issues, [ 1 ] a fact that has led to serious concerns. [ 2 ]
The journal is abstracted and indexed in:
According to the Journal Citation Reports , the journal has a 2021 impact factor of 3.748. [ 10 ] | https://en.wikipedia.org/wiki/Materials_(journal) |
Materials Chemistry and Physics (including Materials Science Communications ) is a peer-reviewed scientific journal published 18 times per year by Elsevier . The focus of the journal is interrelationships among structure, properties, processing and performance of materials. It covers conventional and advanced materials. Publishing formats are short communications, full-length papers and feature articles. The editor-in-chief is Jenq-Gong Duh ( National Tsing Hua University ). [ 1 ]
According to the Journal Citation Reports , the journal has a 2022 impact factor of 4.6, ranking it 57th out of 423 in the category of Condensed Matter Physics . [ 2 ]
This journal is abstracted and indexed by: | https://en.wikipedia.org/wiki/Materials_Chemistry_and_Physics |
Materials Evaluation is a monthly peer-reviewed scientific journal covering nondestructive testing, evaluation, and inspection published by the American Society for Nondestructive Testing . The journal was established in 1942.
This article about a materials science journal is a stub . You can help Wikipedia by expanding it .
See tips for writing articles about academic journals . Further suggestions might be found on the article's talk page . | https://en.wikipedia.org/wiki/Materials_Evaluation |
Materials Horizons is a bimonthly peer-reviewed scientific journal that covers research across the breadth of materials science at the interface between chemistry , physics , biology and engineering . The current editor-in-chief is Martina Stenzel . [ 1 ] The journal was established in 2014. [ 2 ] [ 3 ] A sister journal Nanoscale Horizons was launched in 2016. [ 4 ]
The journal publishes "communications" (articles for rapid publication), "reviews" (state-of-the-art accounts of a research field), "mini-reviews" (research highlights in an emerging area of materials science, usually from the past 2–3 years) and "focus articles" (educational articles providing an overview of a concept in materials science). [ 5 ]
The journal is indexed in the Science Citation Index . [ 6 ] Selective content is also indexed in Polymer Library , Inspec , Biotechnology and Bioengineering Abstracts, METADEX , Mechanical Engineering Abstracts, Solid State and Superconductivity Abstracts, Metal Abstracts and CSA Technology Research Database, and CABI . [ 7 ] | https://en.wikipedia.org/wiki/Materials_Horizons |
Materials Letters is an interdisciplinary, peer-reviewed journal published by Elsevier which according to its website "is dedicated to publishing novel, cutting edge reports of broad interest to the materials community." [ 1 ]
The journal is abstracted and indexed in: [ 2 ]
According to the Journal Citation Reports , the journal has a 2023 impact factor of 2.7. [ 3 ]
This article about a materials science journal is a stub . You can help Wikipedia by expanding it .
See tips for writing articles about academic journals . Further suggestions might be found on the article's talk page . | https://en.wikipedia.org/wiki/Materials_Letters |
The Materials Project is an open-access database offering material properties [ 2 ] to accelerate the development of technology by predicting how new materials–both real and hypothetical–can be used. [ 3 ] The project was established in 2011 with an emphasis on battery research, [ 4 ] but includes property calculations for many areas of clean energy systems such as photovoltaics , thermoelectric materials, and catalysts . [ 5 ] Most of the known 35,000 molecules and over 130,000 inorganic compounds are included in the database. [ 6 ] [ 7 ]
Dr. Kristin Persson of Lawrence Berkeley National Laboratory founded and leads the initiative, which uses supercomputers at Berkeley, among other institutions, to run calculations using Density Functional Theory (DFT). Commonly computed values include enthalpy of formation, crystal structure, and band gap. The assembled databases of computed structures and properties is freely available to anyone under a CC 4.0 license and was developed with ease of use in mind. The data have been used to predict new materials that should be synthesizable, [ 8 ] and screen existing materials for useful properties. [ 9 ]
The project can be traced back to Persson's postdoc research at MIT in 2004, during which she was given access to a supercomputer to do DFT calculations. [ 1 ] After joining Berkeley Lab in 2008, Persson received the necessary funding to make the data from her research freely available. [ 1 ]
This article about materials science is a stub . You can help Wikipedia by expanding it .
This website-related article is a stub . You can help Wikipedia by expanding it .
This database -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Materials_Project |
Materials Research Bulletin is a peer-reviewed , scientific journal that covers the study of materials science and engineering. The journal is published by Elsevier and was established in 1966. [ 1 ] The Editor-in-Chief is Rick Ubic. [ 2 ]
The journal focuses on the development and understanding of materials, including their properties, structure, and processing, and the application of these materials in various fields. The scope of the journal includes the following areas: ceramics, metals, polymers, composites, electronic and optical materials, and biomaterials. [ 3 ]
Materials Research Bulletin features original research articles, review articles, and short communications.
The journal is abstracted and indexed for example in: [ 4 ]
According to the Journal Citation Reports , the journal has a 2021 impact factor of 5.6. [ 7 ]
This article about a materials science journal is a stub . You can help Wikipedia by expanding it .
See tips for writing articles about academic journals . Further suggestions might be found on the article's talk page . | https://en.wikipedia.org/wiki/Materials_Research_Bulletin |
Materials Research Innovations is a scientific journal published by Maney Publishing . It covers all areas of Materials Research. [ 1 ]
The journal is indexed in :
This article about a materials science journal is a stub . You can help Wikipedia by expanding it .
See tips for writing articles about academic journals . Further suggestions might be found on the article's talk page . | https://en.wikipedia.org/wiki/Materials_Research_Innovations |
Materials Research Letters is an open-access, peer-reviewed scientific journal, targeted to be a high impact, fast communication letters journal for the materials research community. It was established in 2013. According to the Journal Citation Reports , the journal has a 2020 impact factor of 7.323. [ 1 ]
This article about a materials science journal is a stub . You can help Wikipedia by expanding it .
See tips for writing articles about academic journals . Further suggestions might be found on the article's talk page . | https://en.wikipedia.org/wiki/Materials_Research_Letters |
The Materials Research Society (MRS) is a non-profit, professional organization for materials researchers, scientists and engineers. Established in 1973, MRS is a member-driven organization of approximately 13,000 materials researchers from academia, industry and government.
Headquartered in Warrendale , Pennsylvania, MRS membership spans over 90 countries, with approximately 48% of MRS members residing outside the United States.
MRS members work in all areas of materials science and research, including physics , chemistry , biology , mathematics and engineering . MRS provides a collaborative environment for idea exchange across all disciplines of materials science through its meetings, publications and other programs designed to foster networking and cooperation. [ 1 ]
The Society’s mission is to promote communication for the advancement of interdisciplinary materials research to improve the quality of life. [ 2 ]
MRS is governed by a board of directors which is composed of the Society's officers and 12 to 21 Directors, the exact number determined by resolution of the board. Directors are elected by the membership. Up to 25% of the Directors, however, may be appointed by the Board MRS Officers include a President, Vice President, Secretary, Treasurer, and Immediate Past President. [ 3 ]
MRS hosts two annual meetings for its members and the materials community to network, exchange technical information, and contribute to the advancement of research. These meetings are held in Boston, Massachusetts , every fall, and in different cities (on the west coast) every spring. [ 4 ] Each meeting incorporates more than 50 technical symposia as well as many “broader impact” sessions that include professional development, government policies and funding opportunities, student activities, award talks and special events. [ 5 ] Each of these meetings is attended by approximately 5,000–6,000 materials scientists, researchers and engineers. [ 6 ]
MRS also partners with other materials organizations to develop meetings such as the International Materials Research Congress (IMRC), held annually in Cancun, Mexico. [ 7 ]
In addition, MRS offers meeting expertise and logistical/operational infrastructure to other scientific communities in need of conference support via the Conference Services Program. [ 8 ]
In partnership with Springer Nature , MRS publishes the following periodicals for the materials community: [ 9 ]
Through the MRS Publishing program, MRS publishes materials-related monographs, handbooks and textbooks, including:
MRS, through its Government Affairs Committee, advocates for sustainable funding of science, provides forums for public-policy discussions, offers itself as a scientific resource for policymakers, and delivers timely information on emerging public policy issues, federal programs and other activities of importance to its members and the materials community.
MRS advocacy efforts include:
The Materials Research Society Foundation was founded in 2012 to support the MRS mission and to ensure and enrich MRS’s education, outreach and peer-recognition programs. Foundation programs include: [ 11 ] | https://en.wikipedia.org/wiki/Materials_Research_Society |
The Materials Science Citation Index is a citation index , established in 1992, by Thomson ISI ( Thomson Reuters ). Its overall focus is cited reference searching of the notable and significant journal literature in materials science . The database makes accessible the various properties , behaviors, and materials in the materials science discipline. This then encompasses applied physics , ceramics , composite materials , metals and metallurgy , polymer engineering , semiconductors , thin films , biomaterials , dental technology , as well as optics . The database indexes relevant materials science information from over 6,000 scientific journals that are part of the ISI database which is multidisciplinary . Author abstracts are searchable, which links articles sharing one or more bibliographic references. The database also allows a researcher to use an appropriate (or related to research) article as a base to search forward in time to discover more recently published articles that cite it. [ 1 ]
Materials Science Citation Index lists 625 high-impact journals, and is accessible via the Science Citation Index Expanded collection of databases. [ 2 ]
Coverage of Materials science is accomplished with the following editions: [ 3 ] [ 4 ]
This article about a materials science journal is a stub . You can help Wikipedia by expanding it .
See tips for writing articles about academic journals . Further suggestions might be found on the article's talk page . | https://en.wikipedia.org/wiki/Materials_Science_Citation_Index |
The Materials Science Laboratory (MSL) of the European Space Agency is a payload on board the International Space Station for materials science experiments in low gravity.
It is installed in NASA's first Materials Science Research Rack which is placed in the Destiny laboratory on board the ISS. Its purpose is to process material samples in different ways: directional solidification of metals and alloys, crystal growth of semi-conducting materials, thermo-physical properties and diffusion experiments of alloys and glass-forming materials, and investigations on polymers and ceramics at the liquid-solid phase transition . [ 1 ]
MSL was built for ESA by EADS Astrium in Friedrichshafen, Germany. It is operated and monitored by the Microgravity User Support Center (MUSC) of the German Aerospace Center (DLR) in Cologne, Germany.
MSL was launched with Space Shuttle Discovery on its STS-128 mission at the end of August 2009. It was transferred from the Multi-Purpose Logistics Module to the Destiny Laboratory shortly after the shuttle docked at the International Space Station some two days after launch.
After that the commissioning activities started to check out first the functionality of the Materials Science Research Rack and MSL inside MSRR. The commissioning included the processing of the first two samples which took place at the beginning of November. After bringing those two samples back to ground for analysis by the scientists the rest of the samples from batch 1 will be processed in early 2010.
The Materials Science Laboratory (MSL) facility is the contribution of the European Space Agency to NASA's MSRR-1. It occupies one half of an International Standard Payload Rack .
The MSL consists of a Core Facility , together with associated support sub-systems. The Core Facility consists mainly of a vacuum-tight stainless steel cylinder ( Process Chamber ) capable of accommodating different individual Furnace Inserts (FIs), within which sample processing is carried out. The processing chamber provides an accurately controlled processing environment and measurement of microgravity levels. It can house several different Furnace Inserts . During the first batch of experiments the Low Gradient Furnace (LGF) is installed. Another furnace, the Solidification and Quenching Furnace (SQF) is already produced and waiting on ground for future operations. The FI can be moved with a dedicated drive mechanism, to process each sample according to requirements from the scientists. Processing normally takes place under vacuum.
The Core Facility supports FIs with up to eight heating elements, and provides the mechanical, thermal and electrical infrastructure necessary to handle the FIs, the Sample Cartridge Assembly (SCA) , together with any associated experiment-dedicated electronics that may be required.
A FI is an arrangement of heating elements, isolating zones and cooling zones contained in a thermal insulation assembly. On the outer envelope of this assembly is a water-cooled metal jacket forming the mechanical interface to the Core Facility .
The major characteristics of the two produced Furnace Inserts are:
The LGF is designed mainly for Bridgman crystal growth of semiconductor materials. It consists of two heated cavities separated by an adiabatic zone. This assembly can establish low and precisely controlled gradients between two very stable temperature levels.
The SQF is designed mainly for metallurgical research, with the option of quenching the solidification interface at the end of processing by quickly displacing the cooling zone. It consists of a heated cavity and a water-cooled cooling zone separated by an adiabatic zone. It can establish medium to steep temperature gradients along the experiment sample. For creating large gradients, a Liquid Metal Ring enhances the thermal coupling between the SCA and the cooling zone. [ 2 ]
The samples to be processed are contained in experiment cartridges, the SCAs, that consist of a leak-tight tube, crucible, sensors for process control, sample probe and cartridge foot (i.e. the mechanical and electrical interface to the process chamber). The MSL safety concept requires that experiment samples containing toxic compounds are contained in SCAs that support the detection of potential leaks. The volume between the experiment sample and the cartridge tube is filled with a pre-defined quantity of krypton, allowing leak detection by mass spectrometry. However the first batch of experiments does not contain any toxic substances.
Up to 12 scientific thermocouples provide the sample's temperature profile and allow differential thermal analysis. [ 2 ]
Materials Science Laboratory - Columnar-to-Equiaxed Transition in Solidification Processing (CETSOL) and Microstructure Formation in Casting of Technical Alloys under Diffusive and Magnetically Controlled Convective Conditions (MICAST) are two investigations which will examine different growth patterns and evolution of microstructures during crystallization of metallic alloys in microgravity.
MICAST studies microstructure formation during casting of technical alloys under diffusive and magnetically controlled convective conditions. The experimental results together with parametric studies using numerical simulations, will be used to optimize industrial casting processes. MICAST identifies and controls experimentally the fluid-flow patterns that affect microstructure evolution during casting processes, and to develop analytical and advanced numerical models. The microgravity environment of the International Space Station is of special importance to this project because only there are all gravity-induced convections eliminated and well-defined conditions for solidification prevail that can be disturbed by artificial fluid flow being under full control of the experimenters. Design solutions that make it possible to improve casting processes and especially aluminium alloys with well-defined properties will be provided. MICAST studies the influence of pure diffusive and convective conditions on aluminium-silicon (AlSi) and aluminium-silicon-iron (AlSiFe) cast alloys on the microstructure evolution during directional solidification with and without rotating magnetic field.
The major objective of CETSOL is to improve and validate the modelling of Columnar-Equiaxed Transition (CET) and of the grain microstructure in solidification processing. This aims to give industry confidence in the reliability of the numerical tools introduced in their integrated numerical models of casting, and their relationship. To achieve this goal, intensive deepening of the quantitative characterization of the basic physical phenomena that, from the microscopic to the macroscopic scales, govern microstructure
formation and CET will be pursued.
CET occurs during columnar growth when new grains grow ahead of the columnar front in the undercooled liquid. Under certain conditions, these grains can stop the columnar growth and then the solidification microstructure becomes equiaxed. Experiments have to take place on the ISS due to the long-duration required to solidify samples with the objective to study the CET. Indeed, the length scale of the grain structure when columnar growth takes place is of the order of the casting scale rather than the microstructure scale. This is due to the fact that, to a first approximation, it is the heat flow that controls the transition rather than the solute flow. Experimental programs are being carried out on aluminium-nickel and aluminium-silicon alloys. [ 3 ]
Scientific research on the ISS | https://en.wikipedia.org/wiki/Materials_Science_Laboratory |
Materials Science and Engineering may refer to several journals in the field of materials science and engineering : | https://en.wikipedia.org/wiki/Materials_Science_and_Engineering |
Materials Science and Engineering: A — Structural Materials: Properties, Microstructure and Processing is a peer-reviewed scientific journal . It is the section of Materials Science and Engineering dedicated to "theoretical and experimental studies related to the load-bearing capacity of materials as influenced by their basic properties, processing history, microstructure and operating environment" [ 1 ] and is published monthly by Elsevier . The current editor-in-chiefs are H. W. Hahn ( University of Oklahoma ), E. J. Lavernia ( Texas A&M University ), and B. B. Wei ( Northwestern Polytechnical University ). [ 2 ]
The journal is indexed and abstracted in the following bibliographic databases: [ 3 ]
According to the Journal Citation Reports , the journal has a 2022 impact factor of 6.4, ranking 9th out of 79 in the category 'Metallurgy & Metallurgical Engineering'. [ 4 ]
This article about a physics journal is a stub . You can help Wikipedia by expanding it .
See tips for writing articles about academic journals . Further suggestions might be found on the article's talk page . | https://en.wikipedia.org/wiki/Materials_Science_and_Engineering_A |
Materials Science and Engineering: B — Advanced Functional Solid-State Materials is a peer-reviewed scientific journal . It is the section of Materials Science and Engineering dedicated to "calculation, synthesis, processing, characterization, and understanding of advanced quantum materials" [ 1 ] and is published monthly by Elsevier . It aims at providing a leading international forum for material researchers across the disciplines of theory, experiment, and device applications. The current editor-in-chief is Jing Xia ( University of California Irvine ). [ 2 ]
According to the Journal Citation Reports , the journal has a 2021 impact factor of 3.407. [ 3 ]
This article about a physics journal is a stub . You can help Wikipedia by expanding it .
See tips for writing articles about academic journals . Further suggestions might be found on the article's talk page . | https://en.wikipedia.org/wiki/Materials_Science_and_Engineering_B |
Materials Science and Engineering: C was a peer-reviewed scientific journal that has since been renamed to Biomaterials Advances .
According to the Journal Citation Reports , the journal had a 2020 impact factor of 7.328. [ 1 ]
This article about a materials science journal is a stub . You can help Wikipedia by expanding it .
See tips for writing articles about academic journals . Further suggestions might be found on the article's talk page . | https://en.wikipedia.org/wiki/Materials_Science_and_Engineering_C |
Materials Science and Engineering R: Reports is a monthly peer-reviewed scientific journal . It is the review section of Materials Science and Engineering and is published by Elsevier . It was established in 1993, when the journal Materials Science Reports was split into Materials Science and Engineering C and Materials Science and Engineering R: Reports .
According to the Journal Citation Reports , Materials Science and Engineering R: Reports has a 2020 impact factor of 36.214, ranking it 3rd out of 160 in the category Physics, Applied . [ 1 ]
This article about a materials science journal is a stub . You can help Wikipedia by expanding it .
See tips for writing articles about academic journals . Further suggestions might be found on the article's talk page . | https://en.wikipedia.org/wiki/Materials_Science_and_Engineering_R:_Reports |
Materials Studio is software for simulating and modeling materials. It is developed and distributed by BIOVIA (formerly Accelrys ), a firm specializing in research software for computational chemistry , bioinformatics , cheminformatics , molecular dynamics simulation, and quantum mechanics . [ 3 ]
This software is used in advanced research of various materials, such as polymers , carbon nanotubes , catalysts , metals , ceramics , and so on, by universities (e.g., North Dakota State University [ 4 ] ), research centers, and high tech companies.
Materials Studio is a client–server model software package with Microsoft Windows -based PC clients and Windows and Linux -based servers running on PCs, Linux IA-64 workstations (including Silicon Graphics (SGI) Altix ) and HP XC clusters. | https://en.wikipedia.org/wiki/Materials_Studio |
Materials Today is a monthly peer-reviewed scientific journal , website, and journal family. The parent journal was established in 1998 and covers all aspects of materials science . It is published by Elsevier and the editors-in-chief are Jun Lou ( Rice University ) and Gleb Yushin ( Georgia Institute of Technology ). [ 1 ] The journal principally publishes invited review articles , but other formats are also included, such as primary research articles, news items, commentaries, and opinion pieces on subjects of interest to the field. The website publishes news, educational webinars , podcasts , and blogs , as well as a jobs and events board. According to the Journal Citation Reports , the journal has a 2020 impact factor of 31.041. [ 2 ]
The journal family includes Applied Materials Today , Materials Today Chemistry , Materials Today Energy , Materials Today Physics , Materials Today Nano, Materials Today Sustainability, Materials Today Communications , Materials Today Advances and Materials Today: Proceedings ; as well as an extended collection of related publications. [ 3 ] [ 4 ]
The journal was established in 1998 as a collaboration between Elsevier and the European Materials Research Society . The founding editor was Phil Mestecky. The journal was distributed free of charge to society members and to anyone else who requested a subscription. The spin-off titles Materials Today Communications , Materials Today: Proceedings , and Applied Materials Today were launched between 2014 and 2015. In October 2016, Materials Today announced plans to further develop the journal and related family: including the appointment of new editors, the inclusion of primary research articles, and the planned launch of an extended family of titles. [ 4 ] The journal transitioned into an open access publication in 2012 but announced the introduction of subscription articles alongside open-access articles from 2017. [ 4 ] | https://en.wikipedia.org/wiki/Materials_Today |
Materials and Structures is a peer-reviewed scientific journal published by Springer Science+Business Media on behalf of RILEM (the International Union of Laboratories and Experts in Construction Materials, Systems and Structures ). It covers research on fundamental properties of building materials, their characterization and processing techniques, modeling, standardization of test methods, and the application of research results in building and civil engineering. Materials and Structures also publishes comprehensive reports prepared by RILEM Technical Committees. The current editor-in-chief is Giovanni Plizzari ( University of Brescia ).
The journal is abstracted and indexed in:
According to the Journal Citation Reports , the journal has a 2020 impact factor of 3.428. [ 1 ] | https://en.wikipedia.org/wiki/Materials_and_Structures |
Materials for use in vacuum are materials that show very low rates of outgassing in vacuum and, where applicable, are tolerant to bake-out temperatures. The requirements grow increasingly stringent with the desired degree of vacuum to be achieved in the vacuum chamber .
The materials can produce gas by several mechanisms. Molecules of gases and water can be adsorbed on the material surface (therefore materials with low affinity to water have to be chosen, which eliminates many plastics). Materials may sublimate in vacuum (this includes some metals and their alloys, most notably cadmium and zinc). Or the gases can be released from porous materials or from cracks and crevices. Traces of lubricants, residues from machining, can be present on the surfaces. A specific risk is outgassing of solvents absorbed in plastics after cleaning.
The gases liberated from the materials not only lower the vacuum quality, but also can be reabsorbed on other surfaces, creating deposits and contaminating the chamber.
Yet another problem is diffusion of gases through the materials themselves. Atmospheric helium can diffuse even through Pyrex glass, even if slowly (and elevated temperatures above room temperature are generally needed); [ 1 ] this however is usually not an issue. Some materials might also expand or increase in size causing problems in delicate equipment.
In addition to the gas-related issues, the materials have to maintain adequate strength through the entire required temperature range (sometimes reaching cryogenic temperatures), maintain their properties (elasticity, plasticity, electrical and thermal conductivity or lack of it, etc.), be machinable, and if possible not be overly expensive. Yet another concern is the thermal expansion coefficient match of adjacent parts.
Materials outgas by three mechanisms: release of absorbed gases (de sorption from the bulk of the material), release of adsorbed gases ( desorption from the surface only), and evaporation of the material itself. The former can be reduced by a bakeout, the latter is an intrinsic property of the material. [ 2 ] Some outgassed materials can deposit on other surfaces, contaminate the vacuum system and be difficult to get rid of.
The most common sources of trouble (out-gassing) in vacuum systems are:
There are also additional physical issues which come with vacuum, including the growth of whiskers from materials such as Tin or Zinc, which can cause physical issues or electrical shorts [ 4 ]
Lubrication of moving parts is a problem for vacuum. Many lubricants have unacceptable outgassing rates, [ 5 ] others (e.g. graphite ) lose lubricating properties.
In addition to the concerns above, materials for use in spacecraft applications have to cope with radiation damage and high-intensity ultraviolet radiation , thermal loads from solar radiation, radiation cooling of the vehicle in other directions, and heat produced within the spacecraft's systems. Another concern, for orbits closer to Earth, is the presence of atomic oxygen , leading to corrosion of exposed surfaces; aluminium is an especially sensitive material [ citation needed ] . Silver, often used for surface-deposited interconnects, forms layer of silver oxide that flakes off and may erode up to a total failure.
Corrosion-sensitive surfaces can be protected by a suitable plating , most often with gold ; a silica layer is also possible. However the coating layer is subject to erosion by micrometeoroids . | https://en.wikipedia.org/wiki/Materials_for_use_in_vacuum |
Materials informatics is a field of study that applies the principles of informatics and data science to materials science and engineering to improve the understanding, use, selection , development, and discovery of materials. The term "materials informatics" is frequently used interchangeably with "data science", "machine learning", and "artificial intelligence" by the community. This is an emerging field, with a goal to achieve high-speed and robust acquisition, management, analysis, and dissemination of diverse materials data with the goal of greatly reducing the time and risk required to develop, produce, and deploy new materials, which generally takes longer than 20 years. [ 1 ] [ 2 ] [ 3 ] This field of endeavor is not limited to some traditional understandings of the relationship between materials and information. Some more narrow interpretations include combinatorial chemistry , process modeling , materials databases, materials data management , and product life cycle management . Materials informatics is at the convergence of these concepts, but also transcends them and has the potential to achieve greater insights and deeper understanding by applying lessons learned from data gathered on one type of material to others. By gathering appropriate meta data, the value of each individual data point can be greatly expanded.
Databases are essential for any informatics research and applications. In material informatics many databases exist containing both empirical data obtained experimentally, and theoretical data obtained computationally. Big data that can be used for machine learning is particularly difficult to obtain for experimental data due to the lack of a standard for reporting data and the variability in the experimental environment. This lack of big data has led to growing effort in developing machine learning techniques that utilize data extremely data sets. On the other hand, large uniform database of theoretical density functional theory (DFT) calculations exists. These databases have proven their utility in high-throughput material screening and discovery.
Some common DFT databases and high throughput tools are listed below:
The concept of materials informatics is addressed by the Materials Research Society . For example, materials informatics was the theme of the December 2006 issue of the MRS Bulletin . The issue was guest-edited by John Rodgers of Innovative Materials, Inc., and David Cebon of Cambridge University , who described the "high payoff for developing methodologies that will accelerate the insertion of materials, thereby saving millions of investment dollars."
The editors focused on the limited definition of materials informatics as primarily focused on computational methods to process and interpret data. They stated that "specialized informatics tools for data capture, management, analysis, and dissemination" and "advances in computing power, coupled with computational modeling and simulation and materials properties databases" will enable such accelerated insertion of materials.
A broader definition of materials informatics goes beyond the use of computational methods to carry out the same experimentation, [ 4 ] viewing materials informatics as a framework in which a measurement or computation is one step in an information-based learning process that uses the power of a collective to achieve greater efficiency in exploration. When properly organized, this framework crosses materials boundaries to uncover fundamental knowledge of the basis of physical, mechanical, and engineering [ 5 ] properties.
While there are many who believe in the future of informatics in the materials development and scaling process, many challenges remain. Hill, et al., write that "Today, the materials community faces serious challenges to bringing about this data-accelerated research paradigm, including diversity of research areas within materials, lack of data standards, and missing incentives for sharing, among others. Nonetheless, the landscape is rapidly changing in ways that should benefit the entire materials research enterprise." [ 6 ] This remaining tension between traditional materials development methodologies and the use of more computationally, machine learning, and analytics approaches will likely exist for some time as the materials industry overcomes some of the cultural barriers necessary to fully embrace such new ways of thinking.
The overarching goals of bioinformatics and systems biology may provide a useful analogy. Andrew Murray of Harvard University expresses the hope that such an approach "will save us from the era of "one graduate student, one gene, one PhD". [ 7 ] Similarly, the goal of materials informatics is to save us from one graduate student, one alloy, one PhD. Such goals will require more sophisticated strategies and research paradigms than applying data-science methods to the same tasks set currently undertaken by students. | https://en.wikipedia.org/wiki/Materials_informatics |
A materials oscilloscope is a time-resolved synchrotron high-energy X-ray technique to study rapid phase composition and microstructural related changes in a polycrystalline sample. [ 1 ] [ better source needed ] Such device has been developed for in-situ studies of specimens undergoing physical thermo-mechanical simulation. [ 2 ] [ 3 ]
Two-dimensional diffraction images of a fine synchrotron beam interacting with the specimen are recorded in time frames, such that reflections stemming from individual crystallites of the polycrystalline material can be distinguished. Data treatment is undertaken in a way that diffraction rings are straightened and presented line by line streaked in time. [ 3 ] The traces, so-called timelines in azimuthal-angle/time plots resemble to traces of an oscilloscope, giving insight on the processes happening in the material, while undergoing plastic deformation, or heating, or both, [ 4 ] [ 5 ] [ 6 ] [ 7 ] [ 8 ] [ 9 ] [ 10 ] These timelines allow to distinguish grain growth or refinement, subgrain formation, slip deformation systems, crystallographic twinning, dynamic recovery, dynamic recrystallization, simultaneously in multiple phases.
The development has been undertaken from a project on modern diffraction methods for the investigation of thermo-mechanical processeses, [ 11 ] [ better source needed ] and started with cold deformation of a copper specimen at the ESRF in 2007, followed by hot deformation of zirconium alloy at APS in 2008. Soon afterwards, a series of other materials has been tested and experience with the timeline traces gained. While ESRF and APS played the major role in experimental facilities, the Japanese high-energy synchrotron in the round, SPring-8 followed in 2013 by performing feasibility studies of this kind. Meanwhile, the new PETRA III synchrotron at DESY built a dedicated beamline for this purpose, opening the Materials Oscilloscope investigations to a larger public. The name materials oscilloscope was introduced in 2013 and used onward upon conferences such as MRS and TMS. [ 12 ] [ 13 ] [ better source needed ]
Besides setups in multi-purpose facilities, the first dedicated end-station has been built at the PETRA-III storage ring, where this technique is routinely applied. | https://en.wikipedia.org/wiki/Materials_oscilloscope |
A materials recovery facility , materials reclamation facility , materials recycling facility or multi re-use facility ( MRF , pronounced "murf") is a specialized waste sorting and recycling system [ 1 ] that receives, separates and prepares recyclable materials for marketing to end-user manufacturers. Generally, the main recyclable materials include ferrous metal, non-ferrous metal, plastics, paper, glass. Organic food waste is used to assist anaerobic digestion or composting. Inorganic inert waste is used to make building materials. Non-recyclable high calorific value waste is used to making refuse-derived fuel (RDF) and solid recovered fuel (SRF).
In the United States, there are over 300 materials recovery facilities. [ 2 ] The total market size is estimated at $6.6B as of 2019. [ 3 ]
As of 2016, the top 75 were headed by Sims Municipal Recycling out of Brooklyn, New York. [ 4 ] Waste Management operated 95 MRF facilities total, with 26 in the top 75. ReCommunity operated 6 in the top 75. Republic Services operated 6 in the top 75. Waste Connections operated 4 in the top 75.
In 2018, a survey in the Northeast United States found that the processing cost per ton was $82, versus a value of around $45 per ton. [ 5 ] Composition of the ton included 28% mixed paper and 24% old corrugated containers (OCC). [ 5 ]
Prices for OCC declined into 2019. [ 6 ] Three paper mill companies have announced initiatives to use more recycled fiber. [ 7 ]
Glass recycling is expensive for these facilities, but a study estimated that costs could be cut significantly by investments in improved glass processing. [ 8 ] In Texas, Austin and Houston have facilities which have invested glass recycling, built and operated by Balcones Recycling and FCC Environment , respectively. [ 9 ]
Robots have spread across the industry, helping with sorting. [ 10 ]
Waste enters a MRF when it is dumped onto the tipping floor by the collection trucks. The materials are then scooped up and placed onto conveyor belts, which transports it to the pre-sorting area. Here, human workers remove some items that are not recyclable, which will either be sent to a landfill or an incinerator. [ 11 ] Between 5 and 45% of "dirty" MRF material is recovered. [ citation needed ] Potential hazards are also removed, such as lithium batteries, propane tanks, and aerosol cans, which can create fires. Materials like plastic bags and hoses, which can entangle the recycling equipment, are also removed. [ 11 ] From there, materials are transported via another conveyer belt to the disk screen, which separates wide and flat materials like flattened cardboard boxes from items like cans, jars, paper, and bottles. Flattened boxes ride across the disk screen to the other side, while all other materials fall below, where paper is separated from the waste stream with a blower. The stream of cardboard and paper is overseen by more human workers, who ensure no plastic, metal, or glass is present. [ 11 ] Newer MRFs or retrofitted ones may use industrial robots instead of humans for pre-sorting and for quality control. [ 11 ] However, complete removal of human labor from the sortation process is unlikely for the foreseeable future, as one needs to replicate the dexterity of the human hand and nervous system for removing every type of contaminant within a material stream. The technical limitations of this involve advanced concepts in mechatronics and computer science, where a robot hand would need to be designed, and a highly flexible algorithm that creates another precise movement algorithm within the time constraints of the system (say, the highly approximate estimate of 30,000 lines of code to do this on a modern processor would trigger too long of a delay to be effective on a sortation line). In other words, one would need to search an encyclopedia of said robotic hand motions for every configuration of waste for every pick, and this may be computationally insurmountable, even with quantum computing, as every conditional would need to be checked every iteration. [ citation needed ]
Metal is separated from plastics and glass first with electromagnets , which removes ferrous metals. Non-ferrous metals like aluminum are then removed with eddy current separators . [ 11 ]
The glass and plastic streams are separated by further disk screens. The glass is crushed into cullet for ease of transportation. The plastics are then separated by polymer type, often using infrared technology ( optical sorting ). Infrared light reflects differently off different polymer types; once identified, a jet of air shoots the plastic into the appropriate bin. MRFs might only collect and recycle a few polymers of plastic, sending the rest to landfills or incinerators. The separated materials are baled and sent to the shipping dock of the facility. [ 11 ]
A clean MRF accepts recyclable materials that have already been separated at the source from municipal solid waste generated by either residential or commercial sources. There are a variety of clean MRFs. The most common are single stream where all recyclable material is mixed, or dual stream MRFs, where source-separated recyclables are delivered in a mixed container stream (typically glass, ferrous metal , aluminum and other non-ferrous metals, PET [No.1] and HDPE [No.2] plastics) and a mixed paper stream including corrugated cardboard boxes, newspapers, magazines, office paper and junk mail. Material is sorted to specifications, then baled, shredded, crushed, compacted, or otherwise prepared for shipment to market.
A mixed-waste processing system, sometimes referred to as a dirty MRF, accepts a mixed solid waste stream and then proceeds to separate out designated recyclable materials through a combination of manual and mechanical sorting. The sorted recyclable materials may undergo further processing required to meet technical specifications established by end-markets while the balance of the mixed waste stream is sent to a disposal facility such as a landfill . Today, MWPFs are attracting renewed interest as a way to address low participation rates for source-separated recycling collection systems and prepare fuel products and/or feedstocks for conversion technologies. MWPFs can give communities the opportunity to recycle at much higher rates than has been demonstrated by curbside or other waste collection systems. Advances in technology make today’s MWPF different and, in many respects better, than older versions. [ 12 ]
Around 2004, new mechanical biological treatment technologies were beginning to utilise wet MRFs . [ 13 ] These combine a dirty MRF with water, which acts to densify, separate and clean the output streams. It also hydrocrushes and dissolves biodegradable organics in solution to make them suitable for anaerobic digestion .
In the United States, modern MRFs began in the 1970s. Peter Karter established Resource Recovery Systems, Inc. in Branford, Connecticut, the "first materials recovery facility (MRF)" in the US. [ 14 ] [ 15 ] [ 16 ] | https://en.wikipedia.org/wiki/Materials_recovery_facility |
Materials science is an interdisciplinary field of researching and discovering materials . Materials engineering is an engineering field of finding uses for materials in other fields and industries.
The intellectual origins of materials science stem from the Age of Enlightenment , when researchers began to use analytical thinking from chemistry , physics , and engineering to understand ancient, phenomenological observations in metallurgy and mineralogy . [ 1 ] [ 2 ] Materials science still incorporates elements of physics, chemistry, and engineering. As such, the field was long considered by academic institutions as a sub-field of these related fields. Beginning in the 1940s, materials science began to be more widely recognized as a specific and distinct field of science and engineering, and major technical universities around the world created dedicated schools for its study.
Materials scientists emphasize understanding how the history of a material ( processing ) influences its structure, and thus the material's properties and performance. The understanding of processing -structure-properties relationships is called the materials paradigm. This paradigm is used to advance understanding in a variety of research areas, including nanotechnology , biomaterials , and metallurgy .
Materials science is also an important part of forensic engineering and failure analysis – investigating materials, products, structures or components, which fail or do not function as intended, causing personal injury or damage to property. Such investigations are key to understanding, for example, the causes of various aviation accidents and incidents .
The material of choice of a given era is often a defining point. Phases such as Stone Age , Bronze Age , Iron Age , and Steel Age are historic, if arbitrary examples. Originally deriving from the manufacture of ceramics and its putative derivative metallurgy, materials science is one of the oldest forms of engineering and applied science. [ 3 ] Modern materials science evolved directly from metallurgy , which itself evolved from the use of fire. A major breakthrough in the understanding of materials occurred in the late 19th century, when the American scientist Josiah Willard Gibbs demonstrated that the thermodynamic properties related to atomic structure in various phases are related to the physical properties of a material. [ 4 ] Important elements of modern materials science were products of the Space Race ; the understanding and engineering of the metallic alloys , and silica and carbon materials, used in building space vehicles enabling the exploration of space. Materials science has driven, and been driven by, the development of revolutionary technologies such as rubbers , plastics , semiconductors , and biomaterials .
Before the 1960s (and in some cases decades after), many eventual materials science departments were metallurgy or ceramics engineering departments, reflecting the 19th and early 20th-century emphasis on metals and ceramics. The growth of material science in the United States was catalyzed in part by the Advanced Research Projects Agency , which funded a series of university-hosted laboratories in the early 1960s, "to expand the national program of basic research and training in the materials sciences." [ 5 ] In comparison with mechanical engineering, the nascent material science field focused on addressing materials from the macro-level and on the approach that materials are designed on the basis of knowledge of behavior at the microscopic level. [ 6 ] Due to the expanded knowledge of the link between atomic and molecular processes as well as the overall properties of materials, the design of materials came to be based on specific desired properties. [ 6 ] The materials science field has since broadened to include every class of materials, including ceramics, polymers , semiconductors, magnetic materials, biomaterials, and nanomaterials , generally classified into three distinct groups: ceramics, metals, and polymers. The prominent change in materials science during the recent decades is active usage of computer simulations to find new materials, predict properties and understand phenomena.
A material is defined as a substance (most often a solid, but other condensed phases can be included) that is intended to be used for certain applications. [ 7 ] There are a myriad of materials around us; they can be found in anything from [ 8 ] new and advanced materials that are being developed include nanomaterials , biomaterials , [ 9 ] and energy materials to name a few. [ 10 ]
The basis of materials science is studying the interplay between the structure of materials, the processing methods to make that material, and the resulting material properties. The complex combination of these produce the performance of a material in a specific application. Many features across many length scales impact material performance, from the constituent chemical elements, its microstructure , and macroscopic features from processing. Together with the laws of thermodynamics and kinetics materials scientists aim to understand and improve materials.
Structure is one of the most important components of the field of materials science. The very definition of the field holds that it is concerned with the investigation of "the relationships that exist between the structures and properties of materials". [ 11 ] Materials science examines the structure of materials from the atomic scale, all the way up to the macro scale. [ 3 ] Characterization is the way materials scientists examine the structure of a material. This involves methods such as diffraction with X-rays , electrons or neutrons , and various forms of spectroscopy and chemical analysis such as Raman spectroscopy , energy-dispersive spectroscopy , chromatography , thermal analysis , electron microscope analysis, etc.
Structure is studied in the following levels.
Atomic structure deals with the atoms of the materials, and how they are arranged to give rise to molecules, crystals, etc. Much of the electrical, magnetic and chemical properties of materials arise from this level of structure. The length scales involved are in angstroms ( Å ). The chemical bonding and atomic arrangement (crystallography) are fundamental to studying the properties and behavior of any material.
To obtain a full understanding of the material structure and how it relates to its properties, the materials scientist must study how the different atoms, ions and molecules are arranged and bonded to each other. This involves the study and use of quantum chemistry or quantum physics . Solid-state physics , solid-state chemistry and physical chemistry are also involved in the study of bonding and structure.
Crystallography is the science that examines the arrangement of atoms in crystalline solids. Crystallography is a useful tool for materials scientists. One of the fundamental concepts regarding the crystal structure of a material includes the unit cell , which is the smallest unit of a crystal lattice (space lattice) that repeats to make up the macroscopic crystal structure. Most common structural materials include parallelpiped and hexagonal lattice types. [ 13 ] In single crystals , the effects of the crystalline arrangement of atoms is often easy to see macroscopically, because the natural shapes of crystals reflect the atomic structure. Further, physical properties are often controlled by crystalline defects. The understanding of crystal structures is an important prerequisite for understanding crystallographic defects . Examples of crystal defects consist of dislocations including edges, screws, vacancies, self inter-stitials, and more that are linear, planar, and three dimensional types of defects. [ 14 ] New and advanced materials that are being developed include nanomaterials , biomaterials . [ 15 ] Mostly, materials do not occur as a single crystal, but in polycrystalline form, as an aggregate of small crystals or grains with different orientations. Because of this, the powder diffraction method , which uses diffraction patterns of polycrystalline samples with a large number of crystals, plays an important role in structural determination. Most materials have a crystalline structure, but some important materials do not exhibit regular crystal structure. [ 16 ] Polymers display varying degrees of crystallinity, and many are completely non-crystalline. Glass , some ceramics, and many natural materials are amorphous , not possessing any long-range order in their atomic arrangements. The study of polymers combines elements of chemical and statistical thermodynamics to give thermodynamic and mechanical descriptions of physical properties.
Materials, which atoms and molecules form constituents in the nanoscale (i.e., they form nanostructures) are called nanomaterials. Nanomaterials are the subject of intense research in the materials science community due to the unique properties that they exhibit.
Nanostructure deals with objects and structures that are in the 1 – 100 nm range. [ 17 ] In many materials, atoms or molecules agglomerate to form objects at the nanoscale. This causes many interesting electrical, magnetic, optical, and mechanical properties.
In describing nanostructures, it is necessary to differentiate between the number of dimensions on the nanoscale .
Nanotextured surfaces have one dimension on the nanoscale, i.e., only the thickness of the surface of an object is between 0.1 and 100 nm.
Nanotubes have two dimensions on the nanoscale, i.e., the diameter of the tube is between 0.1 and 100 nm; its length could be much greater.
Finally, spherical nanoparticles have three dimensions on the nanoscale, i.e., the particle is between 0.1 and 100 nm in each spatial dimension. The terms nanoparticles and ultrafine particles (UFP) often are used synonymously although UFP can reach into the micrometre range. The term 'nanostructure' is often used, when referring to magnetic technology. Nanoscale structure in biology is often called ultrastructure .
Microstructure is defined as the structure of a prepared surface or thin foil of material as revealed by a microscope above 25× magnification. It deals with objects from 100 nm to a few cm. The microstructure of a material (which can be broadly classified into metallic, polymeric, ceramic and composite) can strongly influence physical properties such as strength, toughness, ductility, hardness, corrosion resistance, high/low temperature behavior, wear resistance, and so on. [ 18 ] Most of the traditional materials (such as metals and ceramics) are microstructured.
The manufacture of a perfect crystal of a material is physically impossible. For example, any crystalline material will contain defects such as precipitates , grain boundaries ( Hall–Petch relationship ), vacancies, interstitial atoms or substitutional atoms. [ 19 ] The microstructure of materials reveals these larger defects and advances in simulation have allowed an increased understanding of how defects can be used to enhance material properties.
Macrostructure is the appearance of a material in the scale millimeters to meters, it is the structure of the material as seen with the naked eye.
Materials exhibit myriad properties, including the following.
The properties of a material determine its usability and hence its engineering application.
Synthesis and processing involves the creation of a material with the desired micro-nanostructure. A material cannot be used in industry if no economically viable production method for it has been developed. Therefore, developing processing methods for materials that are reasonably effective and cost-efficient is vital to the field of materials science. Different materials require different processing or synthesis methods. For example, the processing of metals has historically defined eras such as the Bronze Age and Iron Age and is studied under the branch of materials science named physical metallurgy . Chemical and physical methods are also used to synthesize other materials such as polymers , ceramics , semiconductors , and thin films . As of the early 21st century, new methods are being developed to synthesize nanomaterials such as graphene . [ 20 ]
Thermodynamics is concerned with heat and temperature and their relation to energy and work . It defines macroscopic variables, such as internal energy , entropy , and pressure , that partly describe a body of matter or radiation. It states that the behavior of those variables is subject to general constraints common to all materials. These general constraints are expressed in the four laws of thermodynamics. Thermodynamics describes the bulk behavior of the body, not the microscopic behaviors of the very large numbers of its microscopic constituents, such as molecules. The behavior of these microscopic particles is described by, and the laws of thermodynamics are derived from, statistical mechanics .
The study of thermodynamics is fundamental to materials science. It forms the foundation to treat general phenomena in materials science and engineering, including chemical reactions, magnetism, polarizability, and elasticity. [ 21 ] It explains fundamental tools such as phase diagrams and concepts such as phase equilibrium .
Chemical kinetics is the study of the rates at which systems that are out of equilibrium change under the influence of various forces. When applied to materials science, it deals with how a material changes with time (moves from non-equilibrium to equilibrium state) due to application of a certain field. It details the rate of various processes evolving in materials including shape, size, composition and structure. Diffusion is important in the study of kinetics as this is the most common mechanism by which materials undergo change. [ 22 ] Kinetics is essential in processing of materials because, among other things, it details how the microstructure changes with application of heat.
Materials science is a highly active area of research. Together with materials science departments, physics , chemistry , and many engineering departments are involved in materials research. Materials research covers a broad range of topics; the following non-exhaustive list highlights a few important research areas.
Nanomaterials describe, in principle, materials of which a single unit is sized (in at least one dimension) between 1 and 1000 nanometers (10 −9 meter), but is usually 1 nm – 100 nm. Nanomaterials research takes a materials science based approach to nanotechnology , using advances in materials metrology and synthesis, which have been developed in support of microfabrication research. Materials with structure at the nanoscale often have unique optical, electronic, or mechanical properties. The field of nanomaterials is loosely organized, like the traditional field of chemistry, into organic (carbon-based) nanomaterials, such as fullerenes, and inorganic nanomaterials based on other elements, such as silicon. Examples of nanomaterials include fullerenes , carbon nanotubes , nanocrystals, etc.
A biomaterial is any matter, surface, or construct that interacts with biological systems . [ 23 ] Biomaterials science encompasses elements of medicine, biology, chemistry, tissue engineering, and materials science.
Biomaterials can be derived either from nature or synthesized in a laboratory using a variety of chemical approaches using metallic components, polymers , bioceramics , or composite materials . They are often intended or adapted for medical applications, such as biomedical devices which perform, augment, or replace a natural function. Such functions may be benign, like being used for a heart valve , or may be bioactive with a more interactive functionality such as hydroxylapatite -coated hip implants . Biomaterials are also used every day in dental applications, surgery, and drug delivery. For example, a construct with impregnated pharmaceutical products can be placed into the body, which permits the prolonged release of a drug over an extended period of time. A biomaterial may also be an autograft , allograft or xenograft used as an organ transplant material.
Semiconductors, metals, and ceramics are used today to form highly complex systems, such as integrated electronic circuits, optoelectronic devices, and magnetic and optical mass storage media. These materials form the basis of our modern computing world, and hence research into these materials is of vital importance.
Semiconductors are a traditional example of these types of materials. They are materials that have properties that are intermediate between conductors and insulators . Their electrical conductivities are very sensitive to the concentration of impurities, which allows the use of doping to achieve desirable electronic properties. Hence, semiconductors form the basis of the traditional computer.
This field also includes new areas of research such as superconducting materials, spintronics , metamaterials , etc. The study of these materials involves knowledge of materials science and solid-state physics or condensed matter physics .
With continuing increases in computing power, simulating the behavior of materials has become possible. This enables materials scientists to understand behavior and mechanisms, design new materials, and explain properties formerly poorly understood. Efforts surrounding integrated computational materials engineering are now focusing on combining computational methods with experiments to drastically reduce the time and effort to optimize materials properties for a given application. This involves simulating materials at all length scales, using methods such as density functional theory , molecular dynamics , Monte Carlo , dislocation dynamics, phase field , finite element , and many more. [ 26 ]
Radical materials advances can drive the creation of new products or even new industries, but stable industries also employ materials scientists to make incremental improvements and troubleshoot issues with currently used materials. Industrial applications of materials science include materials design, cost-benefit tradeoffs in industrial production of materials, processing methods ( casting , rolling , welding , ion implantation , crystal growth , thin-film deposition , sintering , glassblowing , etc.), and analytic methods (characterization methods such as electron microscopy , X-ray diffraction , calorimetry , nuclear microscopy (HEFIB) , Rutherford backscattering , neutron diffraction , small-angle X-ray scattering (SAXS), etc.).
Besides material characterization, the material scientist or engineer also deals with extracting materials and converting them into useful forms. Thus ingot casting, foundry methods, blast furnace extraction, and electrolytic extraction are all part of the required knowledge of a materials engineer. Often the presence, absence, or variation of minute quantities of secondary elements and compounds in a bulk material will greatly affect the final properties of the materials produced. For example, steels are classified based on 1/10 and 1/100 weight percentages of the carbon and other alloying elements they contain. Thus, the extracting and purifying methods used to extract iron in a blast furnace can affect the quality of steel that is produced.
Solid materials are generally grouped into three basic classifications: ceramics, metals, and polymers. This broad classification is based on the empirical makeup and atomic structure of the solid materials, and most solids fall into one of these broad categories. [ 27 ] An item that is often made from each of these materials types is the beverage container. The material types used for beverage containers accordingly provide different advantages and disadvantages, depending on the material used. Ceramic (glass) containers are optically transparent, impervious to the passage of carbon dioxide, relatively inexpensive, and are easily recycled, but are also heavy and fracture easily. Metal (aluminum alloy) is relatively strong, is a good barrier to the diffusion of carbon dioxide, and is easily recycled. However, the cans are opaque, expensive to produce, and are easily dented and punctured. Polymers (polyethylene plastic) are relatively strong, can be optically transparent, are inexpensive and lightweight, and can be recyclable, but are not as impervious to the passage of carbon dioxide as aluminum and glass.
Another application of materials science is the study of ceramics and glasses , typically the most brittle materials with industrial relevance. Many ceramics and glasses exhibit covalent or ionic-covalent bonding with SiO 2 ( silica ) as a fundamental building block. Ceramics – not to be confused with raw, unfired clay – are usually seen in crystalline form. The vast majority of commercial glasses contain a metal oxide fused with silica. At the high temperatures used to prepare glass, the material is a viscous liquid which solidifies into a disordered state upon cooling. Windowpanes and eyeglasses are important examples. Fibers of glass are also used for long-range telecommunication and optical transmission. Scratch resistant Corning Gorilla Glass is a well-known example of the application of materials science to drastically improve the properties of common components.
Engineering ceramics are known for their stiffness and stability under high temperatures, compression and electrical stress. Alumina, silicon carbide , and tungsten carbide are made from a fine powder of their constituents in a process of sintering with a binder. Hot pressing provides higher density material. Chemical vapor deposition can place a film of a ceramic on another material. Cermets are ceramic particles containing some metals. The wear resistance of tools is derived from cemented carbides with the metal phase of cobalt and nickel typically added to modify properties.
Ceramics can be significantly strengthened for engineering applications using the principle of crack deflection . [ 28 ] This process involves the strategic addition of second-phase particles within a ceramic matrix, optimizing their shape, size, and distribution to direct and control crack propagation. This approach enhances fracture toughness, paving the way for the creation of advanced, high-performance ceramics in various industries. [ 29 ]
Another application of materials science in industry is making composite materials . These are structured materials composed of two or more macroscopic phases.
Applications range from structural elements such as steel-reinforced concrete, to the thermal insulating tiles, which play a key and integral role in NASA's Space Shuttle thermal protection system , which is used to protect the surface of the shuttle from the heat of re-entry into the Earth's atmosphere. One example is reinforced Carbon-Carbon (RCC), the light gray material, which withstands re-entry temperatures up to 1,510 °C (2,750 °F) and protects the Space Shuttle's wing leading edges and nose cap. [ 30 ] RCC is a laminated composite material made from graphite rayon cloth and impregnated with a phenolic resin . After curing at high temperature in an autoclave , the laminate is pyrolized to convert the resin to carbon, impregnated with furfuryl alcohol in a vacuum chamber, and cured-pyrolized to convert the furfuryl alcohol to carbon. To provide oxidation resistance for reusability, the outer layers of the RCC are converted to silicon carbide .
Other examples can be seen in the "plastic" casings of television sets, cell-phones and so on. These plastic casings are usually a composite material made up of a thermoplastic matrix such as acrylonitrile butadiene styrene (ABS) in which calcium carbonate chalk, talc , glass fibers or carbon fibers have been added for added strength, bulk, or electrostatic dispersion . These additions may be termed reinforcing fibers, or dispersants, depending on their purpose.
Polymers are chemical compounds made up of a large number of identical components linked together like chains. [ 31 ] Polymers are the raw materials (the resins) used to make what are commonly called plastics and rubber . Plastics and rubber are the final product, created after one or more polymers or additives have been added to a resin during processing, which is then shaped into a final form. Plastics in former and in current widespread use include polyethylene , polypropylene , polyvinyl chloride (PVC), polystyrene , nylons , polyesters , acrylics , polyurethanes , and polycarbonates . Rubbers include natural rubber, styrene-butadiene rubber, chloroprene , and butadiene rubber . Plastics are generally classified as commodity , specialty and engineering plastics .
Polyvinyl chloride (PVC) is widely used, inexpensive, and annual production quantities are large. It lends itself to a vast array of applications, from artificial leather to electrical insulation and cabling, packaging , and containers . Its fabrication and processing are simple and well-established. The versatility of PVC is due to the wide range of plasticisers and other additives that it accepts. [ 32 ] The term "additives" in polymer science refers to the chemicals and compounds added to the polymer base to modify its material properties.
Polycarbonate would be normally considered an engineering plastic (other examples include PEEK , ABS). Such plastics are valued for their superior strengths and other special material properties. They are usually not used for disposable applications, unlike commodity plastics.
Specialty plastics are materials with unique characteristics, such as ultra-high strength, electrical conductivity, electro-fluorescence, high thermal stability, etc.
The dividing lines between the various types of plastics is not based on material but rather on their properties and applications. For example, polyethylene (PE) is a cheap, low friction polymer commonly used to make disposable bags for shopping and trash, and is considered a commodity plastic, whereas medium-density polyethylene (MDPE) is used for underground gas and water pipes, and another variety called ultra-high-molecular-weight polyethylene (UHMWPE) is an engineering plastic which is used extensively as the glide rails for industrial equipment and the low-friction socket in implanted hip joints .
The alloys of iron ( steel , stainless steel , cast iron , tool steel , alloy steels ) make up the largest proportion of metals today both by quantity and commercial value.
Iron alloyed with various proportions of carbon gives low , mid and high carbon steels . An iron-carbon alloy is only considered steel if the carbon level is between 0.01% and 2.00% by weight. For steels, the hardness and tensile strength of the steel is related to the amount of carbon present, with increasing carbon levels also leading to lower ductility and toughness. Heat treatment processes such as quenching and tempering can significantly change these properties, however. In contrast, certain metal alloys exhibit unique properties where their size and density remain unchanged across a range of temperatures. [ 33 ] Cast iron is defined as an iron–carbon alloy with more than 2.00%, but less than 6.67% carbon. Stainless steel is defined as a regular steel alloy with greater than 10% by weight alloying content of chromium . Nickel and molybdenum are typically also added in stainless steels.
Other significant metallic alloys are those of aluminium , titanium , copper and magnesium . Copper alloys have been known for a long time (since the Bronze Age ), while the alloys of the other three metals have been relatively recently developed. Due to the chemical reactivity of these metals, the electrolytic extraction processes required were only developed relatively recently. The alloys of aluminium, titanium and magnesium are also known and valued for their high strength to weight ratios and, in the case of magnesium, their ability to provide electromagnetic shielding. [ 34 ] These materials are ideal for situations where high strength to weight ratios are more important than bulk cost, such as in the aerospace industry and certain automotive engineering applications.
A semiconductor is a material that has a resistivity between a conductor and insulator . Modern day electronics run on semiconductors, and the industry had an estimated US$530 billion market in 2021. [ 35 ] Its electronic properties can be greatly altered through intentionally introducing impurities in a process referred to as doping. Semiconductor materials are used to build diodes , transistors , light-emitting diodes (LEDs), and analog and digital electric circuits , among their many uses. Semiconductor devices have replaced thermionic devices like vacuum tubes in most applications. Semiconductor devices are manufactured both as single discrete devices and as integrated circuits (ICs), which consist of a number—from a few to millions—of devices manufactured and interconnected on a single semiconductor substrate . [ 36 ]
Of all the semiconductors in use today, silicon makes up the largest portion both by quantity and commercial value. Monocrystalline silicon is used to produce wafers used in the semiconductor and electronics industry . Gallium arsenide (GaAs) is the second most popular semiconductor used. Due to its higher electron mobility and saturation velocity compared to silicon, it is a material of choice for high-speed electronics applications. These superior properties are compelling reasons to use GaAs circuitry in mobile phones, satellite communications, microwave point-to-point links and higher frequency radar systems. Other semiconductor materials include germanium , silicon carbide , and gallium nitride and have various applications.
Materials science evolved, starting from the 1950s because it was recognized that to create, discover and design new materials, one had to approach it in a unified manner. Thus, materials science and engineering emerged in many ways: renaming and/or combining existing metallurgy and ceramics engineering departments; splitting from existing solid state physics research (itself growing into condensed matter physics ); pulling in relatively new polymer engineering and polymer science ; recombining from the previous, as well as chemistry , chemical engineering , mechanical engineering , and electrical engineering ; and more.
The field of materials science and engineering is important both from a scientific perspective, as well as for applications field. Materials are of the utmost importance for engineers (or other applied fields) because usage of the appropriate materials is crucial when designing systems. As a result, materials science is an increasingly important part of an engineer's education.
Materials physics is the use of physics to describe the physical properties of materials. It is a synthesis of physical sciences such as chemistry , solid mechanics , solid state physics , and materials science. Materials physics is considered a subset of condensed matter physics and applies fundamental condensed matter concepts to complex multiphase media, including materials of technological interest. Current fields that materials physicists work in include electronic, optical, and magnetic materials, novel materials and structures, quantum phenomena in materials, nonequilibrium physics, and soft condensed matter physics. New experimental and computational tools are constantly improving how materials systems are modeled and studied and are also fields when materials physicists work in.
The field is inherently interdisciplinary , and the materials scientists or engineers must be aware and make use of the methods of the physicist, chemist and engineer. Conversely, fields such as life sciences and archaeology can inspire the development of new materials and processes, in bioinspired and paleoinspired approaches. Thus, there remain close relationships with these fields. Conversely, many physicists, chemists and engineers find themselves working in materials science due to the significant overlaps between the fields.
early uses [ 37 ]
early uses [ 38 ] [ 39 ]
early uses [ 44 ] [ 45 ]
The main branches of materials science stem from the four main classes of materials: ceramics, metals, polymers and composites.
There are additionally broadly applicable, materials independent, endeavors.
There are also relatively broad focuses across materials on specific phenomena and techniques. | https://en.wikipedia.org/wiki/Materials_science |
Materials science in science fiction is the study of how materials science is portrayed in works of science fiction . The accuracy of the materials science portrayed spans a wide range – sometimes it is an extrapolation of existing technology, sometimes it is a physically realistic portrayal of a far-out technology, and sometimes it is simply a plot device that looks scientific, but has no basis in science. Examples are:
Critical analysis of materials science in science fiction falls into the same general categories. The predictive aspects are emphasized, for example, in the motto of the Georgia Tech 's department of materials science and engineering – Materials scientists lead the way in turning yesterday's science fiction into tomorrow's reality . This is also the theme of many technical articles, such as Material By Design: Future Science or Science Fiction? , [ 3 ] found in IEEE Spectrum , the flagship magazine of the Institute of Electrical and Electronics Engineers .
On the other hand, there is criticism of the unrealistic materials science used in science fiction. In the professional materials science journal JOM , for example, there are articles such as The (Mostly Improbable) Materials Science and Engineering of the Star Wars Universe [ 4 ] and Personification: The Materials Science and Engineering of Humanoid Robots . [ 5 ]
In many cases, the materials science aspect of a fictional work was interesting enough that someone other than the author has remarked on it. Here are some examples, and their relationship to real world materials science usage, if any.
In real life, scientists have announced a ceramic, aluminum oxynitride (trade name Alon) which is as strong as steel, but transparent. [ 6 ]
A duralumin briefcase was featured in the game Resident Evil: Code Veronica . Two feature in Danganronpa 2: Goodbye Despair .
Duralumin is one of many metals used for magic in the Mistborn series [ 12 ]
The behavior of ice-nine in the book probably comes from the fact that pure water can be supercooled , that is, cooled below the freezing temperature, while remaining liquid, until an impurity or seed crystal is introduced, which causes the water to solidify. It was not inspired by the real Ice-IX, which was only discovered 5 years after the book was published. [ 21 ]
Daxamites are highly susceptible to lead poisoning.
Lead poisoning is a very real effect.
In the X-COM series, in reference to this kind of UFO theory, "element 115" is known as elerium-115 or just elerium .
A stable isotope of "element 115" occurs in the game Dark Reign .
A stable isotope of "Element 115" powers the "Back Step" time machine system in the American television series Seven Days . [ 25 ] An accidental environmental contamination once caused a large number of congenital disorders .
Element 115 is featured in the Call of Duty: Black Ops subseries in the 'Zombies' PVE-style game mode, where it is called Divinium . In the game, Divinium is used for multiple purposes, such as powering weapons, teleporters, liquid drinks known as "Perk-a-Colas", special gumballs known as "Gobblegum", and even creating the zombies themselves.
In Tomb Raider III , "Element 115" is one of the four pieces of meteorite rock acquired by Lara Croft during the course of the game. The element can shoot powerful turquoise blasts, and can also be used to speed up and personally alter evolution, even evolving an already developed life form.
In the 2016 tenth season of the television show The X-Files , the episode " My Struggle " features a triangular, levitating aircraft built from alien technology. When Fox Mulder asks a scientist how the aircraft could turn invisible, the scientist states "Element 115: Ununpentium," apparently obtained from the alien spacecraft crash site at Roswell, New Mexico in 1947.
The protagonists of the film Evolution use hundreds of gallons of Head & Shoulders shampoo (which they say contains selenium) to defeat the titular alien menace. Critics have noted the method of picking selenium as a poison is less than scientific. [ 38 ]
In the book I, Robot , in the story " Runaround ", selenium is used on Mercury to generate power, and to protect Gregory Powell and Michael Donovan from the heat of the Mercurian sun .
In the Lexx episode "Twilight," Stanley Tweedle becomes ill due to a selenium deficiency. He is eventually cured with a dose of dandruff shampoo .
Head & Shoulders shampoo actually uses a zinc-based active ingredient , while Selsun Blue , Extra Strength Head & Shoulders, and many other brands of anti-dandruff shampoo do contain selenium sulfide .
Photo-sensitivity of selenium was discovered in the 19th century. It is really used in some types of photocells, but many alternatives are available today.
The bounty hunting mutants of Strontium Dog attribute their deformities and freakish powers to strontium-90 contained in the fallout of atomic wars.
In the video game Fallout 3 , one of the consumable items is called the "Nuka-Cola Quantum", which supposedly gets its unique properties from the addition of strontium-90 in its formula.
In Island of Terror , bone-eating monsters appear to be unstoppable until doctors discover that strontium-90 is deadly to them.
YouTube content creator "raxdflipnote" included it in their video "swimming pool" , where an unsuspecting stickman is informed that the pool he's in is filled with strontium-90.
Thorium is also used as a highly explosive material in the game Star Wars: Knights of the Old Republic II .
A Soviet doomsday device in Stanley Kubrick 's film Dr. Strangelove employs "Cobalt Thorium G".
In the game World of Warcraft , thorium is a workable metal mined from rock deposits that are greenish in color.
The DSiWare game "Thorium Wars" envisions a future "era of peace and prosperity" powered by thorium which is shattered when "Thorions—a super species of Thorium-based machines" turn against mankind. [ 41 ]
In the movie Up, Up and Away , tin foil acts as kryptonite for the superheroes .
In current times, the material known as tin foil is made of aluminium , not tin . This matters little for the intended use since both are conductive and ductile metals. | https://en.wikipedia.org/wiki/Materials_science_in_science_fiction |
In continuum physics , materials with memory , also referred as materials with hereditary effects are a class of materials whose constitutive equations contains a dependence upon the past history of thermodynamic , kinetic , electromagnetic or other kind of state variables .
The study of these materials arises from the pioneering articles of Ludwig Boltzmann [ 1 ] [ 2 ] and Vito Volterra , [ 3 ] [ 4 ] in which they sought an extension of the concept of an elastic material . [ 5 ] The key assumption of their theory was that the local stress value at a time t depends upon the history of the local deformation up to t . In general, in materials with memory the local value of some constitutive quantity (stress, heat flux, electric current, polarization and magnetization, etc.) at a time t depends upon the history of the state variables (deformation, temperature, electric and magnetic fields, etc.). The hypothesis that the remote history of a variable has less influence than its values in the recent past, was stated in modern continuum mechanics as the fading memory principle by Bernard Coleman and Walter Noll .
This assumption was implicit in the pioneer works: when restricted to cyclic hystories, it traces back to the closed cycle principle stated by Volterra, [ 4 ] which leads to a constitutive relation of integral convolution type.
In the linear case, this relation takes the form of a Volterra equation.
In the linear case, this relation takes the form of a Volterra equation : | https://en.wikipedia.org/wiki/Materials_with_memory |
Materiomics is the holistic study of material systems. Materiomics examines links between physicochemical material properties and material characteristics and function. The focus of materiomics is system functionality and behavior, rather than a piecewise collection of properties, a paradigm similar to systems biology . While typically applied to complex biological systems and biomaterials, materiomics is equally applicable to non-biological systems. Materiomics investigates the material properties of natural and synthetic materials by examining fundamental links between processes, structures and properties at multiple scales, from nano to macro, by using systematic experimental, theoretical or computational methods.
The term has been independently proposed with slightly different definitions in 2004 by T. Akita et al. (AIST/Japan [ 1 ] ), in 2008 by Markus J. Buehler (MIT/USA [ 2 ] [ 3 ] ), and Clemens van Blitterswijk , Jan de Boer and Hemant Unadkat (University of Twente/The Netherlands [ 4 ] ) in analogy to genomics , the study of an organism's entire genome . Similarly, materiomics refers to the study of the processes, structures and properties of materials from a fundamental, systematic perspective by incorporating all relevant scales, from nano to macro, in the synthesis and function of materials and structures. The integrated view of these interactions at all scales is referred to as a material's materiome. [ 5 ]
[ 6 ] [ 7 ]
New techniques for evaluating materials at the tissue level, such as reference point indentation (RPI) and raman spectroscopy are lending insight into the nature of these highly complex, functional relationships.
Materiomics is related to proteomics , where the difference is the focus on material properties, stability, failure and mechanistic insight into multi-scale phenomena.
In higher education, innovative study programmes in materiomics are emerging. [ 8 ] | https://en.wikipedia.org/wiki/Materiomics |
A maternal effect is a situation where the phenotype of an organism is determined not only by the environment it experiences and its genotype , but also by the environment and genotype of its mother. In genetics , maternal effects occur when an organism shows the phenotype expected from the genotype of the mother, irrespective of its own genotype, often due to the mother supplying messenger RNA or proteins to the egg. Maternal effects can also be caused by the maternal environment independent of genotype, sometimes controlling the size, sex, or behaviour of the offspring . These adaptive maternal effects lead to phenotypes of offspring that increase their fitness. Further, it introduces the concept of phenotypic plasticity , an important evolutionary concept. It has been proposed that maternal effects are important for the evolution of adaptive responses to environmental heterogeneity .
In genetics , a maternal effect occurs when the phenotype of an organism is determined by the genotype of its mother. [ 1 ] For example, if a mutation is maternal effect recessive , then a female homozygous for the mutation may appear phenotypically normal, however her offspring will show the mutant phenotype, even if they are heterozygous for the mutation.
Maternal effects often occur because the mother supplies a particular mRNA or protein to the oocyte, hence the maternal genome determines whether the molecule is functional. Maternal supply of mRNAs to the early embryo is important, as in many organisms the embryo is initially transcriptionally inactive. [ 2 ] Because of the inheritance pattern of maternal effect mutations, special genetic screens are required to identify them. These typically involve examining the phenotype of the organisms one generation later than in a conventional ( zygotic ) screen, as their mothers will be potentially homozygous for maternal effect mutations that arise. [ 3 ] [ 4 ]
A Drosophila melanogaster oocyte develops in an egg chamber in close association with a set of cells called nurse cells . Both the oocyte and the nurse cells are descended from a single germline stem cell , however cytokinesis is incomplete in these cell divisions , and the cytoplasm of the nurse cells and the oocyte is connected by structures known as ring canals . [ 5 ] Only the oocyte undergoes meiosis and contributes DNA to the next generation.
Many maternal effect Drosophila mutants have been found that affect the early steps in embryogenesis such as axis determination , including bicoid , dorsal , gurken and oskar . [ 6 ] [ 7 ] [ 8 ] For example, embryos from homozygous bicoid mothers fail to produce head and thorax structures.
Once the gene that is disrupted in the bicoid mutant was identified, it was shown that bicoid mRNA is transcribed in the nurse cells and then relocalized to the oocyte. [ 9 ] Other maternal effect mutants either affect products that are similarly produced in the nurse cells and act in the oocyte, or parts of the transportation machinery that are required for this relocalization. [ 10 ] Since these genes are expressed in the (maternal) nurse cells and not in the oocyte or fertilised embryo, the maternal genotype determines whether they can function.
Maternal effect genes [ 11 ] are expressed during oogenesis by the mother (expressed prior to fertilization) and develop the anterior-posterior and dorsal ventral polarity of the egg. The anterior end of the egg becomes the head; posterior end becomes the tail. the dorsal side is on the top; the ventral side is in underneath. The products of maternal effect genes called maternal mRNAs are produced by nurse cell and follicle cells and deposited in the egg cells (oocytes). At the start of development process, mRNA gradients are formed in oocytes along anterior-posterior and dorsal ventral axes.
About thirty maternal genes are involved in pattern formation have been identified. In particular, products of four maternal effect genes are critical to the formation of anterior-posterior axis. The product of two maternal effect gene, bicoid and hunchback, regulates formation of anterior structure while another pair nanos and caudal, specifies protein that regulates formation of posterior part of embryo.
The transcript of all four genes-bicoid, hunchback, caudal, nanos are synthesized by nurse and follicle cells and transported into the oocytes.
In birds, mothers may pass down hormones in their eggs that affect an offspring's growth and behavior. Experiments in domestic canaries have shown that eggs that contain more yolk androgens develop into chicks that display more social dominance. Similar variation in yolk androgen levels has been seen in bird species like the American coot , though the mechanism of effect has yet to be established. [ 12 ]
In 2015, obesity theorist Edward Archer published "The Childhood Obesity Epidemic as a Result of Nongenetic Evolution: The Maternal Resources Hypothesis" and a series of works on maternal effects in human obesity and health. [ 13 ] [ 14 ] [ 15 ] [ 16 ] In this body of work, Archer argued that accumulative maternal effects via the non-genetic evolution of matrilineal nutrient metabolism is responsible for the increased global prevalence of obesity and diabetes mellitus type 2 . Archer posited that decrements in maternal metabolic control altered fetal pancreatic beta cell , adipocyte (fat cell) and myocyte (muscle cell) development thereby inducing an enduring competitive advantage of adipocytes in the acquisition and sequestering on nutrient energy.
The environmental cues such as light, temperature, soil moisture and nutrients that the mother plant encounters can cause variations in seed quality, even within the same genotype. Thus, the mother plant greatly influences seed traits such as seed size, germination rate, and viability. [ 17 ]
The environment or condition of the mother can also in some situations influence the phenotype of her offspring, independent of the offspring's genotype.
In contrast, a paternal effect is when a phenotype results from the genotype of the father, rather than the genotype of the individual. [ 18 ] The genes responsible for these effects are components of sperm that are involved in fertilization and early development. [ 19 ] An example of a paternal-effect gene is the ms(3)sneaky in Drosophila . Males with a mutant allele of this gene produce sperm that are able to fertilize an egg, but the sneaky-inseminated eggs do not develop normally. However, females with this mutation produce eggs that undergo normal development when fertilized. [ 20 ]
Adaptive maternal effects induce phenotypic changes in offspring that result in an increase in fitness. [ 21 ] These changes arise from mothers sensing environmental cues that work to reduce offspring fitness, and then responding to them in a way that then “prepares” offspring for their future environments. A key characteristic of “adaptive maternal effects” phenotypes is their plasticity. Phenotypic plasticity gives organisms the ability to respond to different environments by altering their phenotype. With these “altered” phenotypes increasing fitness it becomes important to look at the likelihood that adaptive maternal effects will evolve and become a significant phenotypic adaptation to an environment.
When traits are influenced by either the maternal environment or the maternal phenotype, it is said to be influenced by maternal effects. Maternal effects work to alter the phenotypes of the offspring through pathways other than DNA. [ 22 ] Adaptive maternal effects are when these maternal influences lead to a phenotypic change that increases the fitness of the offspring. [ 23 ] In general, adaptive maternal effects are a mechanism to cope with factors that work to reduce offspring fitness; [ 24 ] they are also environment specific.
It can sometimes be difficult to differentiate between maternal and adaptive maternal effects. Consider the following: Gypsy moths reared on foliage of black oak, rather than chestnut oak, had offspring that developed faster. [ 25 ] This is a maternal, not an adaptive maternal effect. In order to be an adaptive maternal effect, the mother's environment would have to have led to a change in the eating habits or behavior of the offspring. [ 25 ] The key difference between the two therefore, is that adaptive maternal effects are environment specific. The phenotypes that arise are in response to the mother sensing an environment that would reduce the fitness of her offspring. By accounting for this environment she is then able to alter the phenotypes to actually increase the offspring's fitness. Maternal effects are not in response to an environmental cue, and further they have the potential to increase offspring fitness, but they may not.
When looking at the likelihood of these “altered” phenotypes evolving there are many factors and cues involved. Adaptive maternal effects evolve only when offspring can face many potential environments; when a mother can “predict” the environment into which her offspring will be born; and when a mother can influence her offspring's phenotype, thereby increasing their fitness. [ 25 ] The summation of all of these factors can then lead to these “altered” traits becoming favorable for evolution.
The phenotypic changes that arise from adaptive maternal effects are a result of the mother sensing that a certain aspect of the environment may decrease the survival of her offspring. When sensing a cue the mother “relays” information to the developing offspring and therefore induces adaptive maternal effects. This tends to then cause the offspring to have a higher fitness because they are “prepared” for the environment they are likely to experience. [ 24 ] These cues can include responses to predators, habitat, high population density, and food availability [ 26 ] [ 27 ] [ 28 ]
The increase in size of Northern American red squirrels is a great example of an adaptive maternal effect producing a phenotype that resulted in an increased fitness. The adaptive maternal effect was induced by the mothers sensing the high population density and correlating it to low food availability per individual. Her offspring were on average larger than other squirrels of the same species; they also grew faster. Ultimately, the squirrels born during this period of high population density showed an increased survival rate (and therefore fitness) during their first winter. [ 26 ]
When analyzing the types of changes that can occur to a phenotype, we can see changes that are behavioral, morphological, or physiological. A characteristic of the phenotype that arises through adaptive maternal effects, is the plasticity of this phenotype. Phenotypic plasticity allows organisms to adjust their phenotype to various environments, thereby enhancing their fitness to changing environmental conditions. [ 24 ] Ultimately it is a key attribute to an organism's, and a population's, ability to adapt to short term environmental change. [ 29 ] [ 30 ]
Phenotypic plasticity can be seen in many organisms, one species that exemplifies this concept is the seed beetle Stator limbatus . This seed beetle reproduces on different host plants, two of the more common ones being Cercidium floridum and Acacia greggii . When C. floridum is the host plant, there is selection for a large egg size; when A. greggii is the host plant, there is a selection for a smaller egg size. In an experiment it was seen that when a beetle who usually laid eggs on A. greggii was put onto C. floridum , the survivorship of the laid eggs was lower compared to those eggs produced by a beetle that was conditioned and remained on the C. florium host plant. Ultimately these experiments showed the plasticity of egg size production in the beetle, as well as the influence of the maternal environment on the survivorship of the offspring. [ 27 ]
In many insects:
Related to adaptive maternal effects are epigenetic effects. Epigenetics is the study of long lasting changes in gene expression that are produced by modifications to chromatin instead of changes in DNA sequence, as is seen in DNA mutation. This "change" refers to DNA methylation , histone acetylation , or the interaction of non-coding RNAs with DNA. DNA methylation is the addition of methyl groups to the DNA. When DNA is methylated in mammals, the transcription of the gene at that location is turned down or turned off entirely. The induction of DNA methylation is highly influenced by the maternal environment. Some maternal environments can lead to a higher methylation of an offspring's DNA, while others lower methylation.[22] [ citation needed ] The fact that methylation can be influenced by the maternal environment, makes it similar to adaptive maternal effects. Further similarities are seen by the fact that methylation can often increase the fitness of the offspring. Additionally, epigenetics can refer to histone modifications or non-coding RNAs that create a sort of cellular memory . Cellular memory refers to a cell's ability to pass nongenetic information to its daughter cell during replication. For example, after differentiation, a liver cell performs different functions than a brain cell; cellular memory allows these cells to "remember" what functions they are supposed to perform after replication. Some of these epigenetic changes can be passed down to future generations, while others are reversible within a particular individual's lifetime. This can explain why individuals with identical DNA can differ in their susceptibility to certain chronic diseases.
Currently, researchers are examining the correlations between maternal diet during pregnancy and its effect on the offspring's susceptibility for chronic diseases later in life. The fetal programming hypothesis highlights the idea that environmental stimuli during critical periods of fetal development can have lifelong effects on body structure and health and in a sense they prepare offspring for the environment they will be born into. Many of these variations are thought to be due to epigenetic mechanisms brought on by maternal environment such as stress, diet, gestational diabetes , and exposure to tobacco and alcohol. These factors are thought to be contributing factors to obesity and cardiovascular disease, neural tube defects, cancer, diabetes, etc. [ 32 ] Studies to determine these epigenetic mechanisms are usually performed through laboratory studies of rodents and epidemiological studies of humans.
Knowledge of maternal diet induced epigenetic changes is important not only for scientists, but for the general public. Perhaps the most obvious place of importance for maternal dietary effects is within the medical field. In the United States and worldwide, many non-communicable diseases, such as cancer, obesity, and heart disease, have reached epidemic proportions. The medical field is working on methods to detect these diseases, some of which have been discovered to be heavily driven by epigenetic alterations due to maternal dietary effects. Once the genomic markers for these diseases are identified, research can begin to be implemented to identify the early onset of these diseases and possibly reverse the epigenetic effects of maternal diet in later life stages. The reversal of epigenetic effects will utilize the pharmaceutical field in an attempt to create drugs which target the specific genes and genomic alterations. The creation of drugs to cure these non-communicable diseases could be used to treat individuals who already have these illnesses. General knowledge of the mechanisms behind maternal dietary epigenetic effects is also beneficial in terms of awareness. The general public can be aware of the risks of certain dietary behaviors during pregnancy in an attempt to curb the negative consequences which may arise in offspring later in their lives. Epigenetic knowledge can lead to an overall healthier lifestyle for the billions of people worldwide.
The effect of maternal diet in species other than humans is also relevant. Many of the long term effects of global climate change are unknown. Knowledge of epigenetic mechanisms can help scientists better predict the impacts of changing community structures on species which are ecologically, economically, and/or culturally important around the world. Since many ecosystems will see changes in species structures, the nutrient availability will also be altered, ultimately affecting the available food choices for reproducing females. Maternal dietary effects may also be used to improve agricultural and aquaculture practices. Breeders may be able to utilize scientific data to create more sustainable practices, saving money for themselves, as well as the consumers.
Hyperglycemia during pregnancy is thought to cause epigenetic changes in the leptin gene of newborns leading to a potential increased risk for obesity and heart disease. Leptin is sometimes known as the “satiety hormone” because it is released by fat cells to inhibit hunger. By studying both animal models and human observational studies, it has been suggested that a leptin surge in the perinatal period plays a critical role in contributing to long-term risk of obesity. The perinatal period begins at 22 weeks gestation and ends a week after birth.[34] DNA methylation near the leptin locus has been examined to determine if there was a correlation between maternal glycemia and neonatal leptin levels. Results showed that glycemia was inversely associated with the methylation states of LEP gene, which controls the production of the leptin hormone. Therefore, higher glycemic levels in mothers corresponded to lower methylation states in LEP gene in their children. With this lower methylation state, the LEP gene is transcribed more often, thereby inducing higher blood leptin levels. [ 33 ] These higher blood leptin levels during the perinatal period were linked to obesity in adulthood, perhaps due to the fact that a higher “normal” level of leptin was set during gestation. Because obesity is a large contributor to heart disease, this leptin surge is not only correlated with obesity but also heart disease.
High fat diets in utero are believed to cause metabolic syndrome. Metabolic syndrome is a set of symptoms including obesity and insulin resistance that appear to be related. This syndrome is often associated with type II diabetes as well as hypertension and atherosclerosis. Using mice models, researchers have shown that high fat diets in utero cause modifications to the adiponectin and leptin genes that alter gene expression; these changes contribute to metabolic syndrome. The adiponectin genes regulate glucose metabolism as well as fatty acid breakdown; however, the exact mechanisms are not entirely understood. In both human and mice models, adiponectin has been shown to add insulin-sensitizing and anti-inflammatory properties to different types of tissue, specifically muscle and liver tissue. Adiponectin has also been shown to increase the rate of fatty acid transport and oxidation in mice, which causes an increase in fatty acid metabolism. [ 34 ] With a high fat diet during gestation, there was an increase in methylation in the promoter of the adiponectin gene accompanied by a decrease in acetylation. These changes likely inhibit the transcription of the adiponectin genes because increases in methylation and decreases in acetylation usually repress transcription. Additionally, there was an increase in methylation of the leptin promoter, which turns down the production of the leptin gene. Therefore, there was less adiponectin to help cells take up glucose and break down fat, as well as less leptin to cause a feeling of satiety. The decrease in these hormones caused fat mass gain, glucose intolerance, hypertriglyceridemia, abnormal adiponectin and leptin levels, and hypertension throughout the animal's lifetime. However, the effect was abolished after three subsequent generations with normal diets. This study highlights the fact that these epigenetic marks can be altered in as many as one generation and can even be completely eliminated over time. [ 35 ] This study highlighted the connection between high fat diets to the adiponectin and leptin in mice. In contrast, few studies have been done in humans to show the specific effects of high fat diets in utero on humans. However, it has been shown that decreased adiponectin levels are associated with obesity, insulin resistance, type II diabetes, and coronary artery disease in humans. It is postulated that a similar mechanism as the one described in mice may also contribute to metabolic syndrome in humans. [ 34 ]
In addition, high fat diets cause chronic low-grade inflammation in the placenta, adipose, liver, brain, and vascular system. Inflammation is an important aspect of the bodies’ natural defense system after injury, trauma, or disease. During an inflammatory response, a series of physiological reactions, such as increased blood flow, increased cellular metabolism, and vasodilation, occur in order to help treat the wounded or infected area. However, chronic low-grade inflammation has been linked to long-term consequences such as cardiovascular disease, renal failure, aging, diabetes, etc. This chronic low-grade inflammation is commonly seen in obese individuals on high fat diets. In a mice model, excessive cytokines were detected in mice fed on a high fat diet. Cytokines aid in cell signaling during immune responses, specifically sending cells towards sites of inflammation, infection, or trauma. The mRNA of proinflammatory cytokines was induced in the placenta of mothers on high fat diets. The high fat diets also caused changes in microbiotic composition, which led to hyperinflammatory colonic responses in offspring. This hyperinflammatory response can lead to inflammatory bowel diseases such as Crohn's disease or ulcerative colitis .[35] As previously mentioned, high fat diets in utero contribute to obesity; however, some proinflammatory factors, like IL-6 and MCP-1, are also linked to body fat deposition. It has been suggested that histone acetylation is closely associated with inflammation because the addition of histone deacetylase inhibitors has been shown to reduce the expression of proinflammatory mediators in glial cells . This reduction in inflammation resulted in improved neural cell function and survival. This inflammation is also often associated with obesity, cardiovascular disease, fatty liver , brain damage, as well as preeclampsia and preterm birth. Although it has been shown that high fat diets induce inflammation, which contribute to all these chronic diseases; it is unclear as to how this inflammation acts as a mediator between diet and chronic disease. [ 36 ]
A study done after the Dutch Hunger Winter of 1944-1945 showed that undernutrition during the early stages of pregnancy are associated with hypomethylation of the insulin-like growth factor II (IGF2) gene even after six decades. These individuals had significantly lower methylation rates as compared to their same sex sibling who had not been conceived during the famine. A comparison was done with children conceived prior to the famine so that their mothers were nutrient deprived during the later stages of gestation; these children had normal methylation patterns. The IGF2 stands for insulin-like growth factor II; this gene is a key contributor in human growth and development. IGF2 gene is also maternally imprinted meaning that the mother's gene is silenced. The mother's gene is typically methylated at the differentially methylated region (DMR); however, when hypomethylated, the gene is bi-allelically expressed. Thus, individuals with lower methylation states likely lost some of the imprinting effect. Similar results have been demonstrated in the Nr3c1 and Ppara genes of the offspring of rats fed on an isocaloric protein-deficient diet before starting pregnancy. This further implies that the undernutrition was the cause of the epigenetic changes. Surprisingly, there was not a correlation between methylation states and birth weight. This displayed that birth weight may not be an adequate way to determine nutritional status during gestation. This study stressed that epigenetic effects vary depending on the timing of exposure and that early stages of mammalian development are crucial periods for establishing epigenetic marks. Those exposed earlier in gestation had decreased methylation while those who were exposed at the end of gestation had relatively normal methylation levels. [ 37 ] The offspring and descendants of mothers with hypomethylation were more likely to develop cardiovascular disease. Epigenetic alterations that occur during embryogenesis and early fetal development have greater physiologic and metabolic effects because they are transmitted over more mitotic divisions. In other words, the epigenetic changes that occur earlier are more likely to persist in more cells. [ 37 ]
In another study, researchers discovered that perinatal nutrient restriction resulting in intrauterine growth restriction (IUGR) contributes to diabetes mellitus type 2 (DM2). IUGR refers to the poor growth of the baby in utero. In the pancreas, IUGR caused a reduction in the expression of the promoter of the gene encoding a critical transcription factor for beta cell function and development. Pancreatic beta cells are responsible for making insulin; decreased beta cell activity is associated with DM2 in adulthood. In skeletal muscle, IUGR caused a decrease in expression of the Glut-4 gene. The Glut-4 gene controls the production of the Glut-4 transporter; this transporter is specifically sensitive to insulin. Thus, when insulin levels rise, more glut-4 transporters are brought to the cell membrane to increase the uptake of glucose into the cell. This change is caused by histone modifications in the cells of skeletal muscle that decrease the effectiveness of the glucose transport system into the muscle. Because the main glucose transporters are not operating at optimal capacity, these individuals are more likely to develop insulin resistance with energy rich diets later in life, contributing to DM2. [ 38 ]
Further studies have examined the epigenetic changes resulting from a high protein/low carbohydrate diet during pregnancy. This diet caused epigenetic changes that were associated with higher blood pressure, higher cortisol levels, and a heightened Hypothalamic-pituitary-adrenal (HPA) axis response to stress. Increased methylation in the 11β-hydroxysteroid dehydrogenase type 2 (HSD2), glucocorticoid receptor (GR) , and H19 ICR were positively correlated with adiposity and blood pressure in adulthood. Glucocorticoids play a vital role in tissue development and maturation as well as having effects on metabolism. Glucocorticoids’ access to GR is regulated by HSD1 and HSD2. H19 is an imprinted gene for a long coding RNA (lncRNA) , which has limiting effects on body weight and cell proliferation. Therefore, higher methylation rates in H19 ICR repress transcription and prevent the lncRNA from regulating body weight. Mothers who reported higher meat/fish and vegetable intake and lower bread/potato intake in late pregnancy had a higher average methylation in GR and HSD2. However, one common challenge of these types of studies is that many epigenetic modifications have tissue and cell-type specificity DNA methylation patterns. Thus, epigenetic modification patterns of accessible tissues, like peripheral blood, may not represent the epigenetic patterns of the tissue involved in a particular disease. [ 39 ]
Strong evidence in rats supports the conclusion that neonatal estrogen exposure plays a role in the development of prostate cancer . Using a human fetal prostate xenograft model, researchers studied the effects of early exposure to estrogen with and without secondary estrogen and testosterone treatment. A xenograft model is a graft of tissue transplanted between organisms of different species. In this case, human tissue was transplanted into rats; therefore, there was no need to extrapolate from rodents to humans. Histopathological lesions, proliferation, and serum hormone levels were measured at various time-points after xenografting. At day 200, the xenograft that had been exposed to two treatments of estrogen showed the most severe changes. Additionally, researchers looked at key genes involved in prostatic glandular and stromal growth, cell-cycle progression, apoptosis, hormone receptors, and tumor suppressors using a custom PCR array. Analysis of DNA methylation showed methylation differences in CpG sites of the stromal compartment after estrogen treatment. These variations in methylation are likely a contributing cause to the changes in the cellular events in the KEGG prostate cancer pathway that inhibit apoptosis and increase cell cycle progression that contribute to the development of cancer. [ 40 ]
In utero or neonatal exposure to bisphenol A (BPA) , a chemical used in manufacturing polycarbonate plastic, is correlated with higher body weight, breast cancer, prostate cancer, and an altered reproductive function. In a mice model, the mice fed on a BPA diet were more likely to have a yellow coat corresponding to their lower methylation state in the promoter regions of the retrotransposon upstream of the Agouti gene. The Agouti gene is responsible for determining whether an animal's coat will be banded (agouti) or solid (non-agouti). However, supplementation with methyl donors like folic acid or phytoestrogen abolished the hypomethylating effect. This demonstrates that the epigenetic changes can be reversed through diet and supplementation. [ 41 ]
Maternal dietary effects are not just seen in humans, but throughout many taxa in the animal kingdom. These maternal dietary effects can result in ecological changes on a larger scale throughout populations and from generation to generation. The plasticity involved in these epigenetic changes due to maternal diet represents the environment into which the offspring will be born. Many times, epigenetic effects on offspring from the maternal diet during development will genetically prepare the offspring to be better adapted for the environment in which they will first encounter. The epigenetic effects of maternal diet can be seen in many species, utilizing different ecological cues and epigenetic mechanisms to provide an adaptive advantage to future generations.
Within the field of ecology, there are many examples of maternal dietary effects. Unfortunately, the epigenetic mechanisms underlying these phenotypic changes are rarely investigated. In the future, it would be beneficial for ecological scientists as well as epigenetic and genomic scientists to work together to fill the holes within the ecology field to produce a complete picture of environmental cues and epigenetic alterations producing phenotypic diversity.
A pyralid moth species , Plodia interpunctella , commonly found in food storage areas, exhibits maternal dietary effects, as well as paternal dietary effects, on its offspring. Epigenetic changes in moth offspring affect the production of phenoloxidase, an enzyme involved with melanization and correlated with resistance of certain pathogens in many invertebrate species. In this study, parent moths were housed in food rich or food poor environments during their reproductive period. Moths who were housed in food poor environments produced offspring with less phenoloxidase, and thus had a weaker immune system, than moths who reproduced in food rich environments. This is believed to be adaptive because the offspring develop while receiving cues of scarce nutritional opportunities. These cues allow the moth to allocate energy differentially, decreasing energy allocated for the immune system and devoting more energy towards growth and reproduction to increase fitness and insure future generations. One explanation for this effect may be imprinting, the expression of only one parental gene over the other, but further research has yet to be done. [ 42 ]
Parental-mediated dietary epigenetic effects on immunity has a broader significance on wild organisms. Changes in immunity throughout an entire population may make the population more susceptible to an environmental disturbance, such as the introduction of a pathogen. Therefore, these transgenerational epigenetic effects can influence the population dynamics by decreasing the stability of populations who inhabit environments different from the parental environment that offspring are epigenetically modified for.
Food availability also influences the epigenetic mechanisms driving growth rate in the mouthbrooding cichlid , Simochromis pleurospilus . When nutrient availability is high, reproducing females will produce many small eggs, versus fewer, larger eggs in nutrient poor environments. Egg size often correlates with fish larvae body size at hatching: smaller larvae hatch from smaller eggs. In the case of the cichlid, small larvae grow at a faster rate than their larger egg counterparts. This is due to the increased expression of GHR, the growth hormone receptor. Increased transcription levels of GHR genes increase the receptors available to bind with growth hormone , GH, leading to an increased growth rate in smaller fish. Fish of larger size are less likely to be eaten by predators, therefore it is advantageous to grow quickly in early life stages to insure survival. The mechanism by which GHR transcription is regulated is unknown, but it may be due to hormones within the yolk produced by the mother, or just by the yolk quantity itself. This may lead to DNA methylation or histone modifications which control genic transcription levels. [ 43 ]
Ecologically, this is an example of the mother utilizing her environment and determining the best method to maximize offspring survival, without actually making a conscious effort to do so. Ecology is generally driven by the ability of an organism to compete to obtain nutrients and successfully reproduce. If a mother is able to gather a plentiful amount of resources, she will have a higher fecundity and produce offspring who are able to grow quickly to avoid predation. Mothers who are unable to obtain as many nutrients will produce fewer offspring, but the offspring will be larger in hopes that their large size will help insure survival into sexual maturation. Unlike the moth example, the maternal effects provided to the cichlid offspring do not prepare the cichlids for the environment that they will be born into; this is because mouth brooding cichlids provide parental care to their offspring, providing a stable environment for the offspring to develop. Offspring who have a greater growth rate can become independent more quickly than slow growing counterparts, therefore decreasing the amount of energy spent by the parents during the parental care period.
A similar phenomenon occurs in the sea urchin , Strongylocentrotus droebachiensis . Urchin mothers in nutrient rich environments produce a large number of small eggs. Offspring from these small eggs grow at a faster rate than their large egg counterparts from nutrient poor mothers. Again, it is beneficial for sea urchin larvae, known as planula , to grow quickly to decrease the duration of their larval phase and metamorphose into a juvenile to decrease predation risks. Sea urchin larvae have the ability to develop into one of two phenotypes, based on their maternal and larval nutrition. Larvae who grow at a fast rate from high nutrition, are able to devote more of their energy towards development into the juvenile phenotype. Larvae who grow at a slower rate with low nutrition, devote more energy towards growing spine-like appendages to protect themselves from predators in an attempt to increase survival into the juvenile phase. The determination of these phenotypes is based on both the maternal and the juvenile nutrition. The epigenetic mechanisms behind these phenotypic changes is unknown, but it is believed that there may be a nutritional threshold that triggers epigenetic changes affecting development and, ultimately, the larval phenotype. [ 44 ] | https://en.wikipedia.org/wiki/Maternal_effect |
Maternal effect dominant embryonic arrest ( Medea ) is a selfish gene composed of a toxin and an antidote. A mother carrying Medea will express the toxin in her germline, killing her progeny. If the children also carry Medea , they produce copies of the antidote, saving their lives. Therefore, if a mother has one Medea allele and one non- Medea allele, half of her children will inherit Medea and survive while the other half will inherit the non- Medea allele and die (unless they receive Medea from their father).
Medea ' s selfish behavior gives it a selective advantage over normal genes. If introduced into a population at sufficiently high levels, the Medea gene will spread, replacing entire populations of normal beetles with beetles carrying Medea . [ 1 ] Because of this, Medea has been proposed as a way of genetically modifying insect populations. By linking the Medea construct to a gene of interest – for instance, a gene conferring resistance to malaria – Medea ' s unique dynamics could be exploited to drive both genes into a population. These findings have dramatic implications for the control of insect-borne diseases such as malaria and dengue fever .
Medea , which has been found in nature only in flour beetles, is an example of a selfish gene that has been simulated in the lab and tested in the fruit fly Drosophila melanogaster . The toxin was a microRNA that blocked the expression of myd88 , a gene vital for embryonic development in insects . The antidote was an extra copy of myd88 . The offspring receiving the extra copy of myd88 survived and hatched, while those without the extra copy died. In lab trials where 25% of the original members were homozygous for Medea , the gene spread to the entire population within 10 to 12 generations. [ 2 ]
Medea was named for the Greek mythological figure of Medea , who killed her children when her husband left her for another woman. | https://en.wikipedia.org/wiki/Maternal_effect_dominant_embryonic_arrest |
Maternal to zygotic transition ( MZT ), also known as embryonic genome activation , is the stage in embryonic development during which development comes under the exclusive control of the zygotic genome rather than the maternal (egg) genome. The egg contains stored maternal genetic material mRNA which controls embryo development until the onset of MZT. After MZT the diploid embryo takes over genetic control. [ 1 ] [ 2 ] This requires both zygotic genome activation (ZGA) and degradation of maternal products. This process is important because it is the first time that the new embryonic genome is utilized and the paternal and maternal genomes are used in combination (ie. different alleles will be expressed ). The zygotic genome now drives embryo development.
MZT is often thought to be synonymous with midblastula transition (MBT), but these processes are, in fact, distinct. [ 3 ] However, the MBT roughly coincides with ZGA in many metazoans , [ 4 ] and thus may share some common regulatory features. For example, both processes are proposed to be regulated by the nucleocytoplasmic ratio . [ 5 ] [ 6 ] MBT strictly refers to changes in the cell cycle and cell motility that occur just prior to gastrulation . [ 3 ] [ 4 ] In the early cleavage stages of embryogenesis , rapid divisions occur synchronously and there are no "gap" stages in the cell cycle . [ 3 ] During these stages, there is also little to no transcription of mRNA from the zygotic genome , [ 5 ] but zygotic transcription is not required for MBT to occur. [ 3 ] Cellular functions during early cleavage are carried out primarily by maternal products – proteins and mRNAs contributed to the egg during oogenesis .
To begin transcription of zygotic genes , the embryo must first overcome the silencing that has been established. The cause of this silencing could be due to several factors: chromatin modifications leading to repression, lack of adequate transcription machinery , or lack of time in which significant transcription can occur due to the shortened cell cycles. [ 7 ] Evidence for the first method was provided by Newport and Kirschner's experiments showing that nucleocytoplasmic ratio plays a role in activating zygotic transcription . [ 5 ] [ 8 ] They suggest that a defined amount of repressor is packaged into the egg, and that the exponential amplification of DNA at each cell cycle results in titration of the repressor at the appropriate time. Indeed, in Xenopus embryos in which excess DNA is introduced, transcription begins earlier. [ 5 ] [ 8 ] More recently, evidence has been shown that transcription of a subset of genes in Drosophila is delayed by one cell cycle in haploid embryos . [ 9 ] The second mechanism of repression has also been addressed experimentally. Prioleau et al. show that by introducing TATA binding protein (TBP) into Xenopus oocytes, the block in transcription can be partially overcome. [ 10 ] The hypothesis that shortened cell cycles can cause repression of transcription is supported by the observation that mitosis causes transcription to cease. [ 11 ] The generally accepted mechanism for the initiation of embryonic gene regulatory networks in mammals is the occurrence of multiple waves of MZT. In mice, the first of these occurs in the zygote, where expression of a few pioneering transcription factors gradually increases the expression of target genes downstream. This induction of genes leads to a second major MZT event. [ 12 ]
To eliminate the contribution of maternal gene products to development, maternally-supplied mRNAs must be degraded in the embryo . Studies in Drosophila have shown that sequences in the 3' UTR of maternal transcripts mediate their degradation [ 13 ] These sequences are recognized by regulatory proteins that cause destabilization or degradation of the transcripts. Recent studies in both zebrafish and Xenopus have found evidence of a role for microRNAs in degradation of maternal transcripts. In zebrafish , the microRNA miR-430 is expressed at the onset of zygotic transcription and targets several hundred mRNAs for deadenylation and degradation. Many of these targets are genes that are expressed maternally. [ 14 ] Similarly, in Xenopus , the miR-430 ortholog miR-427 has been shown to target maternal mRNAs for deadenylation. Specifically, miR-427 targets include cell cycle regulators such as Cyclin A1 and Cyclin B2 . [ 15 ] | https://en.wikipedia.org/wiki/Maternal_to_zygotic_transition |
Matgrounds are strong surface layers of seabed-hardening bacterial fauna preserved in the Proterozoic and lower Cambrian . Wrinkled matgrounds are informally named "elephant skin" because of its wrinkled surface in the fossil record. Matgrounds supported themselves until early burrowing worms were ubiquitous enough to unharden them. [ 1 ] [ 2 ] [ 3 ] [ 4 ] Burrowing animals broke down the hardy mats to further penetrate the underlying sediment for protection and feeding. [ 5 ] Once matgrounds disappeared, exceptional preservation of lagerstätten such as the Burgess Shale or Ediacara Hills also did so too. [ 5 ] Trace fossils such as Treptichnus are evidence for soft-bodied burrowers more anatomically complex than the Ediacaran biota that also caused the matgrounds disappearance. [ 6 ] [ 7 ]
This paleontology article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Matground |
Mathematical Markup Language ( MathML ) is a pair of mathematical markup languages , an application of XML for describing mathematical notations and capturing both its structure and content. Its aim is to natively integrate mathematical formulae into World Wide Web pages and other documents. It is part of HTML5 and standardised by ISO /IEC since 2015. [ 1 ]
Following some experiments in the Arena browser based on proposals for mathematical markup in HTML, [ 4 ] MathML 1 was released as a W3C recommendation in April 1998 as the first XML language to be recommended by the W3C . Version 1.01 of the format was released in July 1999 and version 2.0 appeared in February 2001. Implementations of the specification appeared in Amaya 1.1 , Mozilla 1.0 and Opera 9.5 . [ 5 ] [ 6 ] In October 2003, the second edition of MathML Version 2.0 was published as the final release by the W3C Math Working Group .
MathML was originally designed before the finalization of XML namespaces . However, it was assigned a namespace immediately after the Namespace Recommendation was completed, and for XML use, the elements should be in the namespace with namespace URL http://www.w3.org/1998/Math/MathML . When MathML is used in HTML (as opposed to XML) this namespace is automatically inferred by the HTML parser and need not be specified in the document. [ 7 ]
Version 3 of the MathML specification was released as a W3C recommendation on 20 October 2010. A recommendation of A MathML for CSS Profile was later released on 7 June 2011; [ 8 ] this is a subset of MathML suitable for CSS formatting. Another subset, Strict Content MathML , provides a subset of content MathML with a uniform structure and is designed to be compatible with OpenMath . Other content elements are defined in terms of a transformation to the strict subset. New content elements include <bind> which associates bound variables ( <bvar> ) to expressions, for example a summation index. The new <share> element allows structure sharing. [ 9 ]
The development of MathML 3.0 went through a number of stages. In June 2006, the W3C rechartered the MathML Working Group to produce a MathML 3 Recommendation until February 2008, and in November 2008 extended the charter to April 2010. A sixth Working Draft of the MathML 3 revision was published in June 2009. On 10 August 2010 version 3 graduated to become a "Proposed Recommendation" rather than a draft. [ 9 ] An implementation of MathML 2 landed in WebKit around this same time, [ 10 ] with a Chromium implementation following a couple of years later, [ 11 ] although that implementation was removed from Chromium after less than a year. [ 12 ]
The Second Edition of MathML 3.0 was published as a W3C Recommendation on 10 April 2014. [ 2 ] The specification was approved as an ISO/IEC international standard 40314:2015 on 23 June 2015. [ 13 ] Also in 2015, the MathML Association was founded to support the adoption of the MathML standard. [ 14 ] At that time, according to a member of the MathJax team, none of the major browser makers paid any of their developers for any MathML-rendering work; whatever support existed was overwhelmingly the result of unpaid volunteer time/work. [ 15 ]
In August 2021, a new specification called MathML Core was published, described as the “core subset of Mathematical Markup Language, or MathML, that is suitable for browser implementation.” [ 16 ] MathML Core set itself apart from MathML 3.0 by including detailed rendering rules and integration with CSS , automated browser support testing resources, and focusing on a fundamental subset of MathML. An implementation was added to Chromium at the beginning of 2023. [ 17 ]
MathML deals not only with the presentation but also the meaning of formula components (the latter part of MathML is known as "Content MathML"). Because the meaning of the equation is preserved separate from the presentation, how the content is communicated can be left up to the user. For example, web pages with MathML embedded in them can be viewed as normal web pages with many browsers, but visually impaired users can also have the same MathML read to them through the use of screen readers (e.g. using the VoiceOver in Safari ). JAWS from version 16 onward supports MathML voicing as well as braille output. [ 20 ]
The quality of rendering of MathML in a browser depends on the installed fonts. The STIX Fonts project have released a comprehensive set of mathematical fonts under an open license. The Cambria Math font supplied with Microsoft Windows had slightly more limited support. [ 21 ]
A valid MathML document typically consists of the XML declaration, DOCTYPE declaration, and document element. The document body then contains MathML expressions which appear in < math > elements as needed in the document. Often, MathML will be embedded in more general documents, such as HTML , DocBook , or other XML -based formats.
Presentation MathML focuses on the display of an equation, and has about 30 elements. The elements' names all begin with m . A Presentation MathML expression is built up out of tokens that are combined using higher-level elements, which control their layout. Finer details of presentation are affected by close to 50 attributes.
Token elements generally only contain characters (not other elements). They include:
Note, however, that these token elements may be used as extension points, allowing markup in host languages.
MathML in HTML5 allows most inline HTML markup in mtext, and <mtext><b> non </b> zero </mtext> is conforming, with the HTML markup being used within the MathML to mark up the embedded text (making the first word bold in this example).
These are combined using layout elements, that generally contain only elements. They include:
As usual in HTML and XML, many entities are available for specifying special symbols by name, such as π and → . An interesting feature of MathML is that entities also exist to express normally-invisible operators, such as ⁢ (or the shorthand ⁢ ) for implicit multiplication. They are:
The full specification of MathML entities [ 22 ] is closely coordinated with the corresponding specifications for use with HTML and XML in general. [ 23 ]
Thus, the expression a x 2 + b x + c {\displaystyle ax^{2}+bx+c} requires two layout elements: one to create the overall horizontal row and one for the superscripted exponent. However, the individual tokens also have to be identified as identifiers ( <mi> ), operators ( <mo> ), or numbers ( <mn> ). Adding the token markup, the full form ends up as
A complete document that consists of just the MathML example above, is shown here:
Content MathML focuses on the semantics, or meaning, of the expression rather than its layout. Central to Content MathML is the <apply> element that represents function application. The function being applied is the first child element under <apply> , and its operands or parameters are the remaining child elements. Content MathML uses only a few attributes.
Tokens such as identifiers and numbers are individually marked up, much as for Presentation MathML, but with elements such as <ci> and <cn> . Rather than being merely another type of token, operators are represented by specific elements, whose mathematical semantics are known to MathML: <times> , <power> , etc. There are over a hundred different elements for different functions and operators. [ 24 ]
For example, <apply><sin/><ci> x </ci></apply> represents sin ( x ) {\displaystyle \sin(x)} and <apply><plus/><ci> x </ci><cn> 5 </cn></apply> represents x + 5 {\displaystyle x+5} . The elements representing operators and functions are empty elements, because their operands are the other elements under the containing <apply> .
The expression a x 2 + b x + c {\displaystyle ax^{2}+bx+c} could be represented as
Content MathML is nearly isomorphic to expressions in a functional language such as Scheme and other dialects of Lisp . <apply> ... </apply> amounts to Scheme's ( ... ) , and the many operator and function elements amount to Scheme functions. With this trivial literal transformation, plus un-tagging the individual tokens, the example above becomes:
This reflects the long-known close relationship between XML element structures, and LISP or Scheme S-expressions . [ 25 ] [ 26 ]
According to the OM Society, [ 27 ] OpenMath Content Dictionaries can be employed as collections of symbols and identifiers with declarations of their semantics – names, descriptions and rules. A 2018 paper presented at the SIGIR conference [ 28 ] proposed that the semantic knowledge base Wikidata could be used as an OpenMath Content Dictionary to link semantic elements of a mathematical formula to unique and language-independent Wikidata items.
The well-known quadratic formula could be represented in Presentation MathML as an expression tree made up from layout elements like <mfrac> or <msqrt> :
This example uses the <annotation> element, which can be used to embed a semantic annotation in non-XML format, for example to store the formula in the format used by an equation editor such as StarMath or the markup using LaTeX syntax. The encoding field is usually a MIME type , although most of the equation encodings don't have such a registration; freeform text may be used in such cases.
Although less compact than other formats, the XML structuring of MathML makes its content widely usable and accessible, allows near-instant display in applications such as web browsers , and facilitates an interpretation of its meaning in mathematical software products. MathML is not intended to be written or edited directly by humans. [ 29 ]
MathML, being XML, can be embedded inside other XML files such as XHTML files using XML namespaces.
Inline MathML is also supported in HTML5 files. There is no need to specify namespaces as there was in XHTML .
Another standard called OpenMath that has been more specifically designed (largely by the same people who devised Content MathML) for storing formulae semantically can be used to complement MathML. OpenMath data can be embedded in MathML using the <annotation-xml encoding= "OpenMath" > element. OpenMath content dictionaries can be used to define the meaning of <csymbol> elements. The following would define P 1 ( x ) to be the first Legendre polynomial :
The OMDoc format has been created for markup of larger mathematical structures than formulae, from statements like definitions, theorems, proofs, and examples, to complete theories and even entire text books. Formulae in OMDoc documents can either be written in Content MathML or in OpenMath; for presentation, they are converted to Presentation MathML.
The ISO / IEC standard Office Open XML (OOXML) defines a different XML math syntax, derived from Microsoft Office products. However, it is partially compatible [ 30 ] through XSL Transformations . | https://en.wikipedia.org/wiki/MathML |
MathMagic is a mathematical WYSIWYG equation editor.
In June 2012, " MathMagic Lite Edition " was introduced for macOS platforms, with some limited features. [ 2 ]
In 2013, Adobe bundled a custom version of MathMagic to Adobe Captivate 7 for both macOS and Windows. [ 3 ]
In September 2014, " MathMagic Lite for Windows " was released. [ 4 ]
In 2022, the 64-bit versions of MathMagic for macOS were released in Universal binary format for both Intel Macs and M1 Apple silicon Macs. [ citation needed ]
MathMagic supports MathML , LaTeX , Plain TeX , SVG , MathType equations, and others.
MathMagic does not support computation.
Its website supports the HTTP protocol, not the more secure HTTPS. | https://en.wikipedia.org/wiki/MathMagic |
Mathcad is computer software for the verification, validation, documentation and re-use of mathematical calculations in engineering and science, notably mechanical, chemical, electrical, and civil engineering. [ 2 ] Released in 1986 on DOS , it introduced live editing ( WYSIWYG ) of typeset mathematical notation in an interactive notebook , combined with automatic computations. It was originally developed by Mathsoft , and since 2006 has been a product of Parametric Technology Corporation .
Mathcad was conceived and developed by Allen Razdow and Josh Bernoff at Mathsoft founded by David Blohm and Razdow. It was released in 1986. It was the first system to support WYSIWYG editing and recalculation of mathematical calculations mixed with text. [ 3 ] It was also the first to check the consistency of engineering units through the full calculation. Other equation solving systems existed at the time, but did not provide a notebook interface: Software Arts ' TK Solver was released in 1982, and Borland 's Eureka: The Solver was released in 1987. [ 4 ]
Mathcad was acquired by Parametric Technology in April 2006. [ 5 ]
Mathcad was named "Best of '87" and "Best of '88" by PC Magazine ' s editors. [ 6 ]
Mathcad's central interface is an interactive notebook in which equations and expressions are created and manipulated in the same graphical format in which they are presented (WYSIWYG). This approach was adopted by systems such as Mathematica , Maple , Macsyma , MATLAB , and Jupyter .
Mathcad today includes some of the capabilities of a computer algebra system , but remains oriented towards ease of use and documentation of numerical engineering applications.
Mathcad is part of a broader product development system developed by PTC, addressing analytical steps in systems engineering. It integrates with PTC's Creo Elements/Pro , Windchill , and Creo Elements/View . Its live feature-level integration with Creo Elements/Pro enables Mathcad analytical models to be directly used in driving CAD geometry, and its structural awareness within Windchill allows live calculations to be re-used and re-applied toward multiple design models.
The Mathcad interface allows users to combine a variety of different elements (mathematics, descriptive text, and supporting imagery) into a worksheet, in which dependent calculations are dynamically recalculated as inputs change. This allows for simple manipulation of input variables, assumptions, and expressions. Mathcad's functionality includes:
Although Mathcad is mostly oriented to non-programmers, it is also used in more complex projects to visualize results of mathematical modeling by using distributed computing and coupling with programs written using more traditional languages such as C++ .
As of 2025, the latest release from PTC is Mathcad Prime 11.0.0.0. This release is a freemium variant: if the software is not activated after a Mathcad Prime 30-day trial, it is possible to continue using PTC Mathcad Express for an unlimited time as "PTC Mathcad Express Free-for-Life Engineering Calculations Software". This freemium pilot is a new marketing approach for PTC. Review and markup of engineering notes can now be done directly by team members without them all requiring a full Mathcad Prime license. [ 8 ]
The last release of the traditional (pre "Prime") product line, Mathcad 15.0, came out in June 2010 and shares the same worksheet file structure as Mathcad 14.0. The last service release, Mathcad 15.0 M050, which added support for Windows 10, was released in 2017. Mathcad 15.0 is no longer actively developed but in "sustained support".
Mathcad only runs on Microsoft Windows . Mathcad Prime 6.0 requires a 64-bit version of Windows 7 , Windows 8.1 or Windows 10 . Until 1998, Mathcad also supported Mac OS . [ 9 ]
Starting in 2011 (Mathcad 15.0) the first year of maintenance and support has been included in the purchase or upgrade price. | https://en.wikipedia.org/wiki/Mathcad |
MatheAss (former Math-Assist ) is a computer program for numerical solutions in school mathematics and functions in some points similar to Microsoft Mathematics . [ 1 ] MatheAss is common in math classes throughout Germany, and schools in the German federal state of Hessen possess a state license which allows all secondary schools to use MatheAss. [ 2 ] [ 3 ]
Its functionality is limited compared to other numerical programs, for example, MatheAss has no script language and does no symbolic computation . On the other side it is easy to use and offers the user fully worked out solutions, in which only the necessary quantities need to be entered. MatheAss covers the topics algebra, geometry, analysis, stochastics, and linear algebra. [ 4 ]
After a precursor for the home computers , usual around 1980, MatheAss appeared in 1983 as a shareware version for the PC, so it was one of the first shareware programs on the German market. MatheAss is available on the manufacturer's website for download for various versions of the Windows operating system. [ 5 ]
Since version 8.2 (released in February 2011) MatheAss again offers a context-sensitive help, which was supplemented in many places by showing mathematical examples and background information. [ 6 ] The MatheAss help file can also be viewed online. [ 7 ]
This computer-programming -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Matheass |
Mathemalchemy (French: MathémAlchimie ) is a traveling art installation dedicated to a celebration of the intersection of art and mathematics . It is a collaborative work led by Duke University mathematician Ingrid Daubechies [ 6 ] and fiber artist Dominique Ehrmann. [ 7 ] The cross-disciplinary team of 24 people, who collectively built the installation during the calendar years 2020 and 2021, includes artists, mathematicians, and craftspeople who employed a wide variety of materials to illustrate, amuse, and educate the public on the wonders, mystery, and beauty of mathematics. [ 4 ] Including the core team of 24, about 70 people contributed in some way to the realization of Mathemalchemy . [ 5 ]
The art installation occupies a footprint approximately 20 by 10.5 feet (6.1 by 3.2 m), which extends up to 9.5 feet (2.9 m) in height (in addition, small custom-fabricated tables are arranged around the periphery to protect the more fragile elements). A map shows the 14 or so different zones or regions within the exhibit, which is filled with hundreds of detailed mathematical artifacts, some smaller than 0.5 inches (13 mm); the entire exhibit comprises more than 1,000 parts which must be packed for shipment. Versions of some of the complex mathematical objects can be purchased through an associated "Mathemalchemy Boutique" website. [ 8 ]
The art installation contains puns (such as " Pi " in a bakery) and Easter eggs , such as a miniature model of the Antikythera mechanism hidden on the bottom of "Knotilus Bay". Mathematically sophisticated visitors may enjoy puzzling out and decoding the many mathematical allusions symbolized in the exhibit, while viewers of all levels are invited to enjoy the self-guided tours, detailed explanations, and videos available on the accompanying official website [1] . [ 9 ]
A downloadable comic book was created to explore some of the themes of the exhibition, using an independent narrative set in the world of Mathemalchemy . [ 10 ]
The installation features or illustrates mathematical concepts at many different levels. [ 2 ] All of the participants regard " recreational mathematics "—especially when it has a strong visual component—as having an important role in education and in culture in general. Jessica Sklar maintains that "mathematics is, at heart, a human endeavor" and feels compelled to make it accessible to those who don't regard themselves as "math people". [ 2 ] Bronna Butler talks about the heritage of JH Conway, whose lectures were "almost magical in quality" because they used what looked like curios and tricks but in the end arrived at answers to "fundamental questions of mathematics". [ 3 ]
Henry Segerman , who wrote the book Visualizing Mathematics With 3D Printing, [ 11 ] contributed 3D pieces that explore stereographic projection and polyhedra . According to Susan Goldstine , "The interplay between mathematics and fiber arts is endlessly fascinating [and] allows for a deeper understanding ways that these crafts can illuminate complex concepts in mathematics". [ 12 ] Edmund Harriss says, "You don’t need a background in math to appreciate the installation, just like you can enjoy a concert without being a musician". [ 13 ]
The creators had the goal of illustrating as much of mathematics as possible. Thus the various exhibits touch on number theory , fractals , tessellations , probability theory , Zeno's paradoxes , Venn diagrams , knot theory , calculus , chaos theory , topology , hyperbolic geometry , symbolic logic —and much else—all in a setting that is beautiful and fun. Mathematicians explicitly mentioned or alluded to include Vladimir Arnold , John H. Conway , Felix Klein , Sofya Kovalevskaya , Henri Lebesgue , Ada Lovelace , Benoit Mandelbrot , Maryam Mirzakhani , August Möbius , Emmy Noether , Marjorie Rice , Bernhard Riemann , Caroline Series , Wacław Sierpiński , Alicia Boole Stott , William Thurston , Helge von Koch , Gladys West , Zeno , and many others. [ 13 ] [ 14 ] [ 2 ] [ 15 ]
Twenty of the "mathemalchemists" are women, and the facility especially celebrates the contributions of women in mathematics, from amateur Marjorie Rice , who found new kinds of pentagon tilings , [ 15 ] to Maryam Mirzakhani , the first woman to ever garner a Fields Medal . [ 15 ]
Daubechies and Ehrmann presented the project in a special session at the 2020 Joint Mathematics Meetings (JMM) in Denver, Colorado . [ 14 ] [ 9 ] They soon had a core group of more than a dozen interested mathematicians and artists who in turn suggested other people not at JMM. Eventually the group would grow to 24 people. [ 16 ] [ 9 ]
Originally, the intent was to collectively design and fabricate in a series of workshops to be held at Duke University in Durham, North Carolina , starting in March 2020. [ 7 ] The COVID-19 pandemic disrupted these plans. [ 14 ] Working instead over Zoom , under the guidance of Dominique Ehrmann and various "team leaders" for different parts of the installation, the 16-by-12-by-10-foot (4.9 by 3.7 by 3.0 m) installation was collectively designed and discussed.
In July 2021 the team could finally get together at Duke for the first in-person meeting, where the components that had been fabricated in various locations in the US and Canada were assembled for the first time, leading to the first complete full-scale construction. [ 16 ] The 24 members of the team employed ceramics , knitting , crocheting , quilting , beadwork , 3D printing , welding , woodworking , textile embellishment, origami , metal-folding, water-sculpted brick, and temari balls [ 15 ] to create the room-sized installation. [ 13 ]
The finished installation was originally displayed at Duke University, then moving to the National Academy of Sciences (NAS) building in Washington DC , where it was on display from December 4, 2021, until June 12, 2022. The installation next showed at Juniata College in Huntingdon, Pennsylvania [ 17 ] before moving to Boston University from January to March 2023, partially overlapping with the 2023 Joint Mathematics Meetings in Boston. [ 9 ] The exhibit then moved to Beaty Biodiversity Museum in Vancouver, British Columbia [ 18 ] and then in November of that year it went to Northern Kentucky University where it remained until February 2024. [ 19 ] From May 22 to October 27, 2024 Mathemalchemy was at the National Museum of Mathematics (MoMath) in New York City. From November 6, 2024 to May 2, 2025, the University of Quebec in Montreal (UQAM) hosts the exhibition. As of November 2024 [update] , fundraising is underway to mount the exhibition at the Navajo Nation Museum in Window Rock, Arizona . [ 20 ]
The exhibit is planned to ultimately reside in the Duke University mathematics building, on permanent display. [ 14 ] [ failed verification ] | https://en.wikipedia.org/wiki/Mathemalchemy |
Theresa Marie Korn (née McLaughlin, November 5, 1926 – April 9, 2020) was an American engineer, radio enthusiast, and airplane pilot. The first woman to earn an engineering degree from what is now Carnegie Mellon University , [ 1 ] [ 2 ] she was the author of multiple books on engineering and mathematics.
A fictionalized version of Korn is one of the characters in the novel Kay Everett Calls CQ by Amelia Lobsenz (Vanguard Press, 1951), describing a girls' summer road trip adventure in the 1940s with ham radio and flying components. [ 3 ]
Theresa McLaughlin was born in St. Louis, Missouri , on November 5, 1926, the daughter of a civil engineer . [ 4 ] When she was one year old, a storm damaged her family home, breaking her nose, [ 1 ] and the family moved to Greensburg, Pennsylvania , where she grew up. [ 4 ] As a high school student, she became a ham radio operator in 1941, [ 5 ] and flew Atlantic reconnaissance patrols as an airplane pilot for the Civil Air Patrol , becoming the youngest pilot and radio operator in the country. She became a member of the Ninety-Nines society of female pilots, [ 1 ] and graduated as the valedictorian of Greensburg High School in 1943, winning the Bausch and Lomb Science Award and a Carnegie Scholarship to the Carnegie Institute of Technology , which later became Carnegie Mellon University . [ 4 ]
Since its founding in 1903, the Carnegie Institute had admitted women as students, but only through its Margaret Morrison Carnegie College for women, not through its engineering school, and her scholarship was to this college, through which McLaughlin could take engineering classes but would be barred from earning an engineering degree. By refusing her scholarship and instead accepting money from her pilot friends to pay for her tuition, McLaughlin was able to gain admission to the engineering school instead of to the women's college, becoming the first female student at the school. [ 1 ] [ 4 ] While studying, she earned a radio license and began working for WHGB , a local radio station, [ 4 ] but quit over being paid less than the station's male employees, and took another job working on the electrical systems of arcade games . [ 1 ] Despite opposition to teaching her from some male faculty members, she graduated with a bachelor's degree in electrical engineering in 1947, and was nominated for membership in Eta Kappa Nu , the international honor society of the IEEE . The society refused her nomination because she was a woman, instead giving her a certificate as the best student in her class. [ 4 ]
She became a junior engineer for Curtiss-Wright , working in the restricted research section on missile development. In 1948 she married Granino Arthur Korn, [ 4 ] a German-born physicist, the son of physicist and inventor Arthur Korn . [ 1 ] Granino was head of analysis at Curtiss-Wright, and because of the anti- nepotism rules then in place at Curtiss-Wright, this marriage caused her to lose her position there. A few years later, they both moved to Boeing in Seattle and she returned to work, on airplane engineering. [ 4 ] [ 1 ] The Korns co-founded an engineering consulting company in 1952, and Theresa Korn earned a master's degree in 1954 from the University of California, Los Angeles . [ 4 ] In 1957, her husband became a professor of computer and electrical engineering at the University of Arizona , while Theresa Korn managed the consulting business and became active in Tucson society. [ 4 ] After Granino Korn retired in 1983, the Korns moved to Wenatchee, Washington . Granino died on December 17, 2013, [ 6 ] and Theresa Korn died from COVID-19 on April 9, 2020, in Wenatchee during the COVID-19 pandemic in Washington (state) . [ 1 ]
Korn was the author of: | https://en.wikipedia.org/wiki/Mathematical_Handbook_for_Scientists_and_Engineers |
Mathematical Magick (complete title: Mathematical Magick, or, The wonders that may by performed by mechanical geometry .) is a treatise by the English clergyman , natural philosopher , polymath and author John Wilkins (1614 – 1672). It was first published in 1648 in London, [ 1 ] another edition was printed in 1680 [ 2 ] and further editions were published in 1691 and 1707.
Wilkins dedicated his work to His Highness the Prince Elector Palatine ( Charles I Louis ) who was in London at the time. It is divided into two books, one headed Archimedes , because he was the chiefest in discovering of Mechanical powers , the other was called Daedalus because he was one of the first and most famous amongst the Ancients for his skill in making Automata. [ 3 ] Wilkins sets out and explains the principles of mechanics in the first book and gives an outlook in the second book on future technical developments like flying which he anticipates as certain if only sufficient exercise, research and development would be directed to these topics. The treatise is an example of his general intention to disseminate scientific knowledge and method and of his attempts to persuade his readers to pursue further scientific studies. [ 4 ]
In the 20 chapters of the first book, traditional mechanical devices are discussed such as the balance , the lever , the wheel or pulley and the block and tackle , the wedge , and the screw . The powers acting on them are compared to those acting in the human body. The book deals with the phrase attributed to Archimedes saying that if he did but know where to stand and fasten his instrument, he could move the world and shows the effect of a series of gear transmissions one linked to the other. It shows the importance of various speeds and the theoretical possibility to increase speed beyond the speed of the earth at the equator . Finally, siege engines like catapults are compared with the cost and effect of then-modern guns.
In the 15 chapters of the second book, various devices are examined which move independently of human interference like clocks and watches, water mills and wind mills. Wilkins explains devices being driven by the motion of air in a chimney or by pressurized air. A land yacht is proposed driven by two sails on two masts, and a wagon powered by a vertical axis wind turbine . A number of independently moving small artificial figures representing men and animals are described. The possibilities are considered to improve the type of submarine designed and built by Cornelis Drebbel . The tales about various flying devices are related and doubts as to their truth are dissipated. Wilkins explains that it should be possible for a man, too, to fly by himself [ 5 ] if a frame were built where the person could sit and if this frame was sufficiently pushed in the air.
In chapter VII, Wilkins discusses various methods how a man could fly, namely by the help of spirits and good or evil angels (as related on various occasions in the Bible), by the help of fowls, by wings fastened immediately to the body or by a flying chariot. The whole of this chapter (and of the following one) concern the possibilities of flying. In a single preliminary phrase, he refers to previous reports of flight attempts:
Tis related of a certain English Monk called Elmerus [probably Eilmer of Malmesbury ], about the Confessors time, that he did by such wings fly from a Tower above a furlong; and so another from Saint Marks steeple in Venice; another at Norinberge; and Busbequius speaks of a Turk in Constantinople, who attempted something this way. Mt. Burton mentioning this quotation, doth believe that some new-fangled wit ('tis his Cynical phrase) will some time or other find out this art. Though the truth is, most of these Artists did unfortunately miscarry by falling down and breaking their arms or legs, yet that may be imputed to their want of experience ...
He writes that sufficient practise should enable a man to fly, most probably by "a flying chariot, which may be so contrived as to carry a man within it" [ 6 ] and equipped with a sort of engine, or else big enough to carry several people, each successively working to fly it. He used the next chapter to dissipate any doubts there may be as to the possibility of such a flying chariot, should a number of particular items be developed and tested.
In Chapters IX to XV, extensive discussions and deliberations are set out why a perpetual motion should be feasible, why the stories about lamps burning for hundreds of years were true and how such lamps could be made and perpetual motions created. | https://en.wikipedia.org/wiki/Mathematical_Magick |
Mathematical Methods of Classical Mechanics is a textbook by mathematician Vladimir I. Arnold . It was originally written in Russian, and later translated into English by A. Weinstein and K. Vogtmann . [ 1 ] It is aimed at graduate students.
The original Russian first edition Математические методы классической механики was published in 1974 by Наука . A second edition was published in 1979, and a third in 1989. The book has since been translated into a number of other languages, including French, German, Japanese and Mandarin.
The Bulletin of the American Mathematical Society said, "The [book] under review [...] written by a distinguished mathematician [...is one of] the first textbooks [to] successfully to present to students of mathematics and physics, [sic] classical mechanics in a modern setting." [ 2 ]
A book review in the journal Celestial Mechanics said, "In summary, the author has succeeded in producing a mathematical synthesis of the science of dynamics. The book is well presented and beautifully translated [...] Arnold's book is pure poetry; one does not simply read it, one enjoys it." [ 3 ] | https://en.wikipedia.org/wiki/Mathematical_Methods_of_Classical_Mechanics |
Mathematical Models is a book on the construction of physical models of mathematical objects for educational purposes. It was written by Martyn Cundy and A. P. Rollett, and published by the Clarendon Press in 1951, [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 6 ] with a second edition in 1961. [ 2 ] [ 7 ] Tarquin Publications published a third edition in 1981. [ 8 ]
The vertex configuration of a uniform polyhedron , a generalization of the Schläfli symbol that describes the pattern of polygons surrounding each vertex , was devised in this book as a way to name the Archimedean solids , and has sometimes been called the Cundy–Rollett symbol as a nod to this origin. [ 9 ]
The first edition of the book had five chapters, including its introduction which discusses model-making in general and the different media and tools with which one can construct models. [ 5 ] The media used for the constructions described in the book include "paper, cardboard, plywood, plastics, wire, string, and sheet metal". [ 1 ]
The second chapter concerns plane geometry, and includes material on the golden ratio , [ 5 ] the Pythagorean theorem , [ 6 ] dissection problems , the mathematics of paper folding , tessellations , and plane curves , which are constructed by stitching, by graphical methods, and by mechanical devices. [ 1 ]
The third chapter, and the largest part of the book, concerns polyhedron models , [ 1 ] made from cardboard or plexiglass. [ 6 ] It includes information about the Platonic solids , Archimedean solids , their stellations and duals , uniform polyhedron compounds , and deltahedra . [ 1 ]
The fourth chapter is on additional topics in solid geometry [ 5 ] and curved surfaces , particularly quadrics [ 1 ] but also including topological manifolds such as the torus , Möbius strip and Klein bottle , and physical models helping to visualize the map coloring problem on these surfaces . [ 1 ] [ 3 ] Also included are sphere packings . [ 4 ] The models in this chapter are constructed as the boundaries of solid objects, via two-dimensional paper cross-sections, and by string figures . [ 1 ]
The fifth chapter, and the final one of the first edition, includes mechanical apparatus including harmonographs and mechanical linkages , [ 1 ] the bean machine and its demonstration of the central limit theorem , and analogue computation using hydrostatics . [ 3 ] The second edition expands this chapter, and adds another chapter on computational devices such as the differential analyser of Vannevar Bush . [ 7 ]
Much of the material on polytopes was based on the book Regular Polytopes by H. S. M. Coxeter , and some of the other material has been drawn from resources previously published in 1945 by the National Council of Teachers of Mathematics . [ 1 ]
At the time they wrote the book, Cundy and Rollett were sixth form teachers in the UK, [ 1 ] [ 4 ] and they intended the book to be used by mathematics students and teachers for educational activities at that level. [ 1 ] [ 6 ] However, it may also be enjoyed by a general audience of mathematics enthusiasts. [ 3 ]
Reviewer Michael Goldberg notes some minor errors in the book's historical credits and its notation, and writes that for American audiences some of the British terminology may be unfamiliar, but concludes that it could still be valuable for students and teachers. Stanley Ogilvy complains about the inconsistent level of rigor of the mathematical descriptions, with some proofs given and others omitted, for no clear reason, but calls this issue minor and in general calls the book's presentation excellent. Dirk ter Haar is more enthusiastic, recommending it to anyone interested in mathematics, and suggesting that it should be required for mathematics classrooms. [ 3 ] Similarly, B. J. F. Dorrington recommends it to all mathematical libraries, [ 5 ] and The Basic Library List Committee of the Mathematical Association of America has given it their strong recommendation for inclusion in undergraduate mathematics libraries. [ 8 ] By the time of its second edition, H. S. M. Coxeter states that Mathematical Models had become "well known". [ 7 ] | https://en.wikipedia.org/wiki/Mathematical_Models_(Cundy_and_Rollett) |
Mathematical Models: From the Collections of Universities and Museums – Photograph Volume and Commentary is a book on the physical models of concepts in mathematics that were constructed in the 19th century and early 20th century and kept as instructional aids at universities. It credits Gerd Fischer as editor, but its photographs of models are also by Fischer. [ 1 ] It was originally published by Vieweg+Teubner Verlag for their bicentennial in 1986, both in German (titled Mathematische Modelle. Aus den Sammlungen von Universitäten und Museen. Mit 132 Fotografien. Bildband und Kommentarband ) [ 2 ] and (separately) in English translation, [ 3 ] [ 4 ] in each case as a two-volume set with one volume of photographs and a second volume of mathematical commentary. [ 2 ] [ 3 ] [ 4 ] Springer Spektrum reprinted it in a second edition in 2017, as a single dual-language volume. [ 1 ]
The work consists of 132 full-page photographs of mathematical models, [ 4 ] divided into seven categories, and seven chapters of mathematical commentary written by experts in the topic area of each category. [ 1 ]
These categories are:
This book can be viewed as a supplement to Mathematical Models by Martyn Cundy and A. P. Rollett (1950), on instructions for making mathematical models, which according to reviewer Tony Gardiner "should be in every classroom and on every lecturer's shelf" but in fact sold very slowly. Gardiner writes that the photographs may be useful in undergraduate mathematics lectures, while the commentary is best aimed at mathematics professionals in giving them an understanding of what each model depicts. Gardiner also suggests using the book as a source of inspiration for undergraduate research projects that use its models as starting points and build on the mathematics they depict. Although Gardiner finds the commentary at times overly telegraphic and difficult to understand, [ 4 ] reviewer O. Giering, writing about the German-language version of the same commentary, calls it detailed, easy-to-read, and stimulating. [ 2 ]
By the time of the publication of the second edition, in 2017, reviewer Hans-Peter Schröcker evaluates the visualizations in the book as "anachronistic", superseded by the ability to visualize the same phenomena more easily with modern computer graphics, and he writes that some of the commentary is also "slightly outdated". Nevertheless, he writes that the photos are "beautiful and aesthetically pleasing", writing approvingly that they use color sparingly and aim to let the models speak for themselves rather than dazzling with many color images. And despite the fading strength of its original purpose, he finds the book valuable both for its historical interest and for what it still has to say about visualizing mathematics in a way that is both beautiful and informative. [ 1 ] | https://en.wikipedia.org/wiki/Mathematical_Models_(Fischer) |
Mathematical Operators is a Unicode block containing characters for mathematical, logical, and set notation.
Notably absent are the plus sign (+), greater than sign (>) and less than sign (<), due to them already appearing in the Basic Latin Unicode block, and the plus-or-minus sign (±), multiplication sign (×) and obelus (÷), due to them already appearing in the Latin-1 Supplement block, although a distinct minus sign (−) is included, semantically different from the Basic Latin hyphen-minus (-).
The Mathematical Operators block has sixteen variation sequences defined for standardized variants . [ 3 ] [ 4 ] They use U+FE00 VARIATION SELECTOR-1 (VS01) to denote variant symbols (depending on the font):
The following Unicode-related documents record the purpose and process of defining specific characters in the Mathematical Operators block: | https://en.wikipedia.org/wiki/Mathematical_Operators_(Unicode_block) |
Mathematical Platonism is the form of realism that suggests that mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging. This is often claimed to be the view most people have of numbers.
The term Platonism is used because such a view is seen to parallel Plato 's Theory of Forms and a "World of Ideas" (Greek: eidos (εἶδος)) described in Plato's allegory of the cave : the everyday world can only imperfectly approximate an unchanging, ultimate reality. Both Plato's cave and Platonism have meaningful, not just superficial connections, because Plato's ideas were preceded and probably influenced by the hugely popular Pythagoreans of ancient Greece, who believed that the world was, quite literally, generated by numbers .
A major question considered in mathematical Platonism is: Precisely where and how do the mathematical entities exist, and how do we know about them? Is there a world, completely separate from our physical one, that is occupied by the mathematical entities? How can we gain access to this separate world and discover truths about the entities? One proposed answer is the Ultimate Ensemble , a theory that postulates that all structures that exist mathematically also exist physically in their own universe.
Kurt Gödel 's Platonism [ 1 ] postulates a special kind of mathematical intuition that lets us perceive mathematical objects directly. (This view bears resemblances to many things Edmund Husserl said about mathematics, and supports Immanuel Kant 's idea that mathematics is synthetic a priori .) Philip J. Davis and Reuben Hersh have suggested in their 1999 book The Mathematical Experience that most mathematicians act as though they are Platonists, even though, if pressed to defend the position carefully, they may retreat to formalism .
Full-blooded Platonism is a modern variation of Platonism, which is in reaction to the fact that different sets of mathematical entities can be proven to exist depending on the axioms and inference rules employed (for instance, the law of the excluded middle , and the axiom of choice ). It holds that all mathematical entities exist. They may be provable, even if they cannot all be derived from a single consistent set of axioms. [ 2 ]
Set-theoretic realism (also set-theoretic Platonism ) [ 3 ] a position defended by Penelope Maddy , is the view that set theory is about a single universe of sets. [ 4 ] This position (which is also known as naturalized Platonism because it is a naturalized version of mathematical Platonism) has been criticized by Mark Balaguer on the basis of Paul Benacerraf 's epistemological problem . [ 5 ]
A similar view, termed Platonized naturalism , was later defended by the Stanford–Edmonton School : according to this view, a more traditional kind of Platonism is consistent with naturalism ; the more traditional kind of Platonism they defend is distinguished by general principles that assert the existence of abstract objects . [ 6 ] | https://en.wikipedia.org/wiki/Mathematical_Platonism |
The Mathematical Tables Project [ 1 ] [ 2 ] was one of the largest and most sophisticated computing organizations that operated prior to the invention of the digital electronic computer. Begun in the United States in 1938 as a project of the Works Progress Administration (WPA), it employed 450 unemployed clerks to tabulate higher mathematical functions , such as exponential functions , logarithms , and trigonometric functions . These tables were eventually published in a 28-volume set by Columbia University Press .
The group was led by a group of mathematicians and physicists, most of whom had been unable to find professional work during the Great Depression . The mathematical leader was Gertrude Blanch , who had just finished her doctorate in mathematics at Cornell University . She had been unable to find a university position and was working at a photographic company before joining the project.
The administrative director was Arnold Lowan , who had a degree in physics from Columbia University and had spent a year at the Institute for Advanced Study in Princeton University before returning to New York without a job. Perhaps the most accomplished mathematician to be associated with the group was Cornelius Lanczos , who had once served as an assistant to Albert Einstein . He spent a year with the project and organized seminars on computation and applied mathematics at the project's office in Lower Manhattan .
In addition to computing tables of mathematical functions, the project did large computations for sciences, including the physicist Hans Bethe , and did calculations for a variety of war projects, including tables for the LORAN navigation system, tables for microwave radar, bombing tables, and shock wave propagation tables.
The Mathematical Tables Project survived the termination of the WPA in 1943 and continued to operate in New York until 1948. At that point, roughly 25 members of the group moved to Washington, D.C., to become the Computation Laboratory of the National Bureau of Standards, now the National Institute of Standards and Technology . Blanch moved to Los Angeles to lead the computing office of the Institute for Numerical Analysis at UCLA and Arnold Lowan joined the faculty of Yeshiva University in New York. The greatest legacy of the project is the Handbook of Mathematical Functions , [ 3 ] which was published 16 years after the group disbanded. Edited by two veterans of the project, Milton Abramowitz and Irene Stegun , it became a widely circulated mathematical and scientific reference. | https://en.wikipedia.org/wiki/Mathematical_Tables_Project |
Analysis is the branch of mathematics dealing with continuous functions , limits , and related theories, such as differentiation , integration , measure , infinite sequences , series , and analytic functions . [ 1 ] [ 2 ]
These theories are usually studied in the context of real and complex numbers and functions . Analysis evolved from calculus , which involves the elementary concepts and techniques of analysis.
Analysis may be distinguished from geometry ; however, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space ) or specific distances between objects (a metric space ).
Mathematical analysis formally developed in the 17th century during the Scientific Revolution , [ 3 ] but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were implicitly present in the early days of ancient Greek mathematics . For instance, an infinite geometric sum is implicit in Zeno's paradox of the dichotomy . [ 4 ] (Strictly speaking, the point of the paradox is to deny that the infinite sum exists.) Later, Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids. [ 5 ] The explicit use of infinitesimals appears in Archimedes' The Method of Mechanical Theorems , a work rediscovered in the 20th century. [ 6 ] In Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century CE to find the area of a circle. [ 7 ] From Jain literature, it appears that Hindus were in possession of the formulae for the sum of the arithmetic and geometric series as early as the 4th century BCE. [ 8 ] Ācārya Bhadrabāhu uses the sum of a geometric series in his Kalpasūtra in 433 BCE . [ 9 ]
Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere in the 5th century. [ 10 ] In the 12th century, the Indian mathematician Bhāskara II used infinitesimal and used what is now known as Rolle's theorem . [ 11 ]
In the 14th century, Madhava of Sangamagrama developed infinite series expansions, now called Taylor series , of functions such as sine , cosine , tangent and arctangent . [ 12 ] Alongside his development of Taylor series of trigonometric functions , he also estimated the magnitude of the error terms resulting of truncating these series, and gave a rational approximation of some infinite series. His followers at the Kerala School of Astronomy and Mathematics further expanded his works, up to the 16th century.
The modern foundations of mathematical analysis were established in 17th century Europe. [ 3 ] This began when Fermat and Descartes developed analytic geometry , which is the precursor to modern calculus. Fermat's method of adequality allowed him to determine the maxima and minima of functions and the tangents of curves. [ 13 ] Descartes's publication of La Géométrie in 1637, which introduced the Cartesian coordinate system , is considered to be the establishment of mathematical analysis. It would be a few decades later that Newton and Leibniz independently developed infinitesimal calculus , which grew, with the stimulus of applied work that continued through the 18th century, into analysis topics such as the calculus of variations , ordinary and partial differential equations , Fourier analysis , and generating functions . During this period, calculus techniques were applied to approximate discrete problems by continuous ones.
In the 18th century, Euler introduced the notion of a mathematical function . [ 14 ] Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the modern definition of continuity in 1816, [ 15 ] but Bolzano's work did not become widely known until the 1870s. In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra widely used in earlier work, particularly by Euler. Instead, Cauchy formulated calculus in terms of geometric ideas and infinitesimals . Thus, his definition of continuity required an infinitesimal change in x to correspond to an infinitesimal change in y . He also introduced the concept of the Cauchy sequence , and started the formal theory of complex analysis . Poisson , Liouville , Fourier and others studied partial differential equations and harmonic analysis . The contributions of these mathematicians and others, such as Weierstrass , developed the (ε, δ)-definition of limit approach, thus founding the modern field of mathematical analysis. Around the same time, Riemann introduced his theory of integration , and made significant advances in complex analysis.
Towards the end of the 19th century, mathematicians started worrying that they were assuming the existence of a continuum of real numbers without proof. Dedekind then constructed the real numbers by Dedekind cuts , in which irrational numbers are formally defined, which serve to fill the "gaps" between rational numbers, thereby creating a complete set: the continuum of real numbers, which had already been developed by Simon Stevin in terms of decimal expansions . Around that time, the attempts to refine the theorems of Riemann integration led to the study of the "size" of the set of discontinuities of real functions.
Also, various pathological objects , (such as nowhere continuous functions , continuous but nowhere differentiable functions , and space-filling curves ), commonly known as "monsters", began to be investigated. In this context, Jordan developed his theory of measure , Cantor developed what is now called naive set theory , and Baire proved the Baire category theorem . In the early 20th century, calculus was formalized using an axiomatic set theory . Lebesgue greatly improved measure theory, and introduced his own theory of integration, now known as Lebesgue integration , which proved to be a big improvement over Riemann's. Hilbert introduced Hilbert spaces to solve integral equations . The idea of normed vector space was in the air, and in the 1920s Banach created functional analysis .
In mathematics , a metric space is a set where a notion of distance (called a metric ) between elements of the set is defined.
Much of analysis happens in some metric space; the most commonly used are the real line , the complex plane , Euclidean space , other vector spaces , and the integers . Examples of analysis without a metric include measure theory (which describes size rather than distance) and functional analysis (which studies topological vector spaces that need not have any sense of distance).
Formally, a metric space is an ordered pair ( M , d ) {\displaystyle (M,d)} where M {\displaystyle M} is a set and d {\displaystyle d} is a metric on M {\displaystyle M} , i.e., a function
such that for any x , y , z ∈ M {\displaystyle x,y,z\in M} , the following holds:
By taking the third property and letting z = x {\displaystyle z=x} , it can be shown that d ( x , y ) ≥ 0 {\displaystyle d(x,y)\geq 0} ( non-negative ).
A sequence is an ordered list. Like a set , it contains members (also called elements , or terms ). Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Most precisely, a sequence can be defined as a function whose domain is a countable totally ordered set, such as the natural numbers .
One of the most important properties of a sequence is convergence . Informally, a sequence converges if it has a limit . Continuing informally, a ( singly-infinite ) sequence has a limit if it approaches some point x , called the limit, as n becomes very large. That is, for an abstract sequence ( a n ) (with n running from 1 to infinity understood) the distance between a n and x approaches 0 as n → ∞, denoted
Real analysis (traditionally, the "theory of functions of a real variable") is a branch of mathematical analysis dealing with the real numbers and real-valued functions of a real variable. [ 16 ] [ 17 ] In particular, it deals with the analytic properties of real functions and sequences , including convergence and limits of sequences of real numbers, the calculus of the real numbers, and continuity , smoothness and related properties of real-valued functions.
Complex analysis (traditionally known as the "theory of functions of a complex variable") is the branch of mathematical analysis that investigates functions of complex numbers . [ 18 ] It is useful in many branches of mathematics, including algebraic geometry , number theory , applied mathematics ; as well as in physics , including hydrodynamics , thermodynamics , mechanical engineering , electrical engineering , and particularly, quantum field theory .
Complex analysis is particularly concerned with the analytic functions of complex variables (or, more generally, meromorphic functions ). Because the separate real and imaginary parts of any analytic function must satisfy Laplace's equation , complex analysis is widely applicable to two-dimensional problems in physics .
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product , norm , topology , etc.) and the linear operators acting upon these spaces and respecting these structures in a suitable sense. [ 19 ] [ 20 ] The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous , unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations .
Harmonic analysis is a branch of mathematical analysis concerned with the representation of functions and signals as the superposition of basic waves . This includes the study of the notions of Fourier series and Fourier transforms ( Fourier analysis ), and of their generalizations. Harmonic analysis has applications in areas as diverse as music theory , number theory , representation theory , signal processing , quantum mechanics , tidal analysis , and neuroscience .
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders . [ 21 ] [ 22 ] [ 23 ] Differential equations play a prominent role in engineering , physics , economics , biology , and other disciplines.
Differential equations arise in many areas of science and technology, specifically whenever a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space or time (expressed as derivatives) is known or postulated. This is illustrated in classical mechanics , where the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow one (given the position, velocity, acceleration and various forces acting on the body) to express these variables dynamically as a differential equation for the unknown position of the body as a function of time. In some cases, this differential equation (called an equation of motion ) may be solved explicitly.
A measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. [ 24 ] In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space , which assigns the conventional length , area , and volume of Euclidean geometry to suitable subsets of the n {\displaystyle n} -dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} . For instance, the Lebesgue measure of the interval [ 0 , 1 ] {\displaystyle \left[0,1\right]} in the real numbers is its length in the everyday sense of the word – specifically, 1.
Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X {\displaystyle X} . It must assign 0 to the empty set and be ( countably ) additive: the measure of a 'large' subset that can be decomposed into a finite (or countable) number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller" subsets. In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure . This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are required to form a σ {\displaystyle \sigma } -algebra . This means that the empty set, countable unions , countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement. Indeed, their existence is a non-trivial consequence of the axiom of choice .
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations ) for the problems of mathematical analysis (as distinguished from discrete mathematics ). [ 25 ]
Modern numerical analysis does not seek exact answers, because exact answers are often impossible to obtain in practice. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors.
Numerical analysis naturally finds applications in all fields of engineering and the physical sciences, but in the 21st century, the life sciences and even the arts have adopted elements of scientific computations. Ordinary differential equations appear in celestial mechanics (planets, stars and galaxies); numerical linear algebra is important for data analysis; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology.
Vector analysis , also called vector calculus , is a branch of mathematical analysis dealing with vector-valued functions . [ 26 ]
Scalar analysis is a branch of mathematical analysis dealing with values related to scale as opposed to direction. Values such as temperature are scalar because they describe the magnitude of a value without regard to direction, force, or displacement that value may or may not have.
Techniques from analysis are also found in other areas such as:
The vast majority of classical mechanics , relativity , and quantum mechanics is based on applied analysis, and differential equations in particular. Examples of important differential equations include Newton's second law , the Schrödinger equation , and the Einstein field equations .
Functional analysis is also a major factor in quantum mechanics .
When processing signals, such as audio , radio waves , light waves, seismic waves , and even images, Fourier analysis can isolate individual components of a compound waveform, concentrating them for easier detection or removal. A large family of signal processing techniques consist of Fourier-transforming a signal, manipulating the Fourier-transformed data in a simple way, and reversing the transformation. [ 27 ]
Techniques from analysis are used in many areas of mathematics, including: | https://en.wikipedia.org/wiki/Mathematical_analysis |
Mathematical anxiety , also known as math phobia , is a feeling of tension and anxiety that interferes with the manipulation of numbers and the solving of mathematical problems in daily life and academic situations. [ 1 ]
Mark H. Ashcraft defines math anxiety as "a feeling of tension, apprehension, or fear that interferes with math performance" (2002, p. 1). [ 2 ] It is a phenomenon that is often considered when examining students' problems in mathematics. According to the American Psychological Association, mathematical anxiety is often linked to testing anxiety. This anxiety can cause distress and likely causes a dislike and avoidance of all math-related tasks. The academic study of math anxiety originates as early as the 1950s, when Mary Fides Gough introduced the term mathemaphobia to describe the phobia-like feelings of many towards mathematics. [ 3 ] The first math anxiety measurement scale was developed by Richardson and Suinn in 1972. [ 4 ] Since this development, several researchers have examined math anxiety in empirical studies . [ 2 ] Hembree [ 5 ] (1990) conducted a meta-analysis of 151 studies concerning math anxiety. The study determined that math anxiety is related to poor math performance on math achievement tests and to negative attitudes concerning math. Hembree also suggests that math anxiety is directly connected with math avoidance.
In contrast, a study by the University of Cambridge [ 6 ] found that 77% of children with high maths anxiety were normal to high achievers on curriculum maths tests. Maths Anxiety has also been linked to perfectionism. [ 7 ]
Ashcraft [ 2 ] (2002) suggests that highly anxious math students will avoid situations in which they have to perform mathematical tasks. Unfortunately, math avoidance results in less competency, exposure and math practice, leaving students more anxious and mathematically unprepared to achieve. In college and university, anxious math students take fewer math courses and tend to feel negative toward the subject. In fact, Ashcraft found that the correlation between math anxiety and variables such as self- confidence and motivation in math is strongly negative .
According to Schar, [ 8 ] because math anxiety can cause math avoidance, an empirical dilemma arises. For instance, when a highly math-anxious student performs disappointingly on a math question, it could be due to math anxiety or the lack of competency in math because of math avoidance. Ashcraft determined that by administering a test that becomes increasingly more mathematically challenging, he noticed that even highly math-anxious individuals do well on the first portion of the test measuring performance. However, on the latter and more difficult portion of the test, there was a stronger negative relationship between accuracy and math anxiety.
According to the research found at the University of Chicago by Sian Beilock and her group, math anxiety is not simply about being bad at math. After using brain scans, scholars confirmed that the anticipation or the thought of solving math actually causes math anxiety. The brain scans showed that the area of the brain that is triggered when someone has math anxiety overlaps the same area of the brain where bodily harm is registered. [ 9 ] And Trezise and Reeve [ 10 ] [ 11 ] show that students' math anxiety can fluctuate throughout the duration of a math class.
The impact of mathematics anxiety on mathematics performance has been studied in more recent literature. An individual with math anxiety does not necessarily lack ability in mathematics, rather, they cannot perform to their full potential due to the interfering symptoms of their anxiety. [ 12 ] Math anxiety manifests itself in a variety of ways, including physical, psychological, and behavioral symptoms, that can all disrupt a student's mathematical performance. [ 13 ] The strong negative correlation between high math anxiety and low achievement is often thought to be due to the impact of math anxiety on working memory. Working memory has a limited capacity. A large portion of this capacity is dedicated to problem-solving when solving mathematical tasks. However, in individuals with math anxiety, much of this space is taken up by anxious thoughts, thus compromising the individual's ability to perform. [ 14 ] In addition, a frequent reliance in schools on high-stakes and timed testing, where students tend to feel the most anxiety, can lead to lower achievement for math-anxious individuals. [ 15 ] Programme for International Student Assessment (PISA) results demonstrate that students experiencing high math anxiety demonstrate mathematics scores that are 34 points lower than students who do not have math anxiety, equivalent to one full year of school. [ 16 ] Besides, researchers Elisa Cargnelutti et al show that the influence of mathematical anxiety on math-related performance increases over time due to the accumulation of passive experience in the subject or other factors like more requirements on mathematics as children grow up. [ 17 ] These findings demonstrate the clear link between math anxiety and reduced levels of achievement, suggesting that alleviating math anxiety may lead to a marked improvement in student achievement.
A rating scale for mathematics anxiety was developed in 1972 by Richardson and Suinn. [ 18 ] Richardson and Suinn defined mathematical anxiety as "feelings of apprehension and tension concerning manipulation of numbers and completion of mathematical problems in various contexts". [ 19 ] Richardson and Suinn introduced the MARS (Mathematics Anxiety Rating Scale) in 1972. Elevated scores on the MARS test translate to high math anxiety. The authors presented the normative data, including a mean score of 215.38 with a standard deviation of 65.29, collected from 397 students that replied to an advertisement for behavior therapy treatment for math anxiety. [ 20 ] For test-retest reliability, the Pearson product-moment coefficient was used and a score of 0.85 was calculated, which was favorable and comparable to scores found on other anxiety tests. Richardson and Suinn validated the construct of this test by sharing previous results from three other studies that were very similar to the results achieved in this study. They also administered the Differential Aptitude Test, a 10-minute math test including simple to complex problems.
Calculation of the Pearson product-moment correlation coefficient between the MARS test and Differential Aptitude Test scores was −0.64 (p < .01), indicating that higher MARS scores relate to lower math test scores and "since high anxiety interferes with performance, and poor performance produces anxiety, this result provides evidence that the MARS does measure mathematics anxiety". [ 21 ] This test was intended for use in diagnosing math anxiety, testing the efficacy of different math anxiety treatment approaches and possibly designing an anxiety hierarchy to be used in desensitization treatments. [ 20 ] The MARS test is of interest to those in counseling psychology [ 22 ] and the test is used profusely in math anxiety research. It is available in several versions of varying lengths [ 23 ] and is considered psychometrically sound. [ 24 ] Other tests are often given to measure different dimensionalities of math anxiety, such as Elizabeth Fennema and Julia Sherman's Fennema-Sherman Mathematics Attitudes Scales (FSMAS). The FSMAS evaluates nine specific domains using Likert-type scales: attitude toward success, mathematics as a male domain, mother's attitude, father's attitude, teacher's attitude, confidence in learning mathematics, mathematics anxiety, affectance motivation and mathematics usefulness. [ 25 ] Despite the introduction of newer instrumentation, the use of the MARS test appears to be the educational standard for measuring math anxiety due to its specificity and prolific use. [ 26 ] [ 27 ]
While there are overarching similarities concerning the acquisition of math skills, researchers have shown that children's mathematical abilities differ across countries. In Canada, students score substantially lower in math problem-solving and operations than students in Korea, India and Singapore. Researchers [ who? ] have conducted thorough comparisons between countries and determined that in some areas, such as Taiwan and Japan, parents place more emphasis on effort rather than one's innate intellectual ability in school success. By placing a higher emphasis on effort rather than one's innate intellectual ability, they are helping their child develop a growth mindset . [ 28 ] People who develop a growth mindset believe that everyone has the ability to grow their intellectual ability, learn from their mistakes, and become more resilient learners. Rather than getting stuck on a problem and giving up, students with a growth mindset try other strategies to solve the problem. A growth mindset can benefit everyone, not just people trying to solve math computations. Moreover, parents in these countries tend to set higher expectations and standards for their children. In turn, students spend more time on homework and value homework more than American children. [ 29 ]
In addition, researchers Jennifer L. Brown et al. shows that difference in level of mathematical anxiety among different countries may result from varying degrees of the courses. In the same culture, there is little difference in anxiety scale that is associated with gender, while the anxiety is more related with its type. Samples show greater degree of anxiety at subscale.
MEA (Mathematical Evaluation Anxiety) compared with LMA (Learning Mathematical Anxiety). [ 30 ]
Another difference in mathematic abilities often explored in research concerns gender disparities. There has been research examining gender differences in performance on standardized tests across various countries. Beller and Gafni's have shown that children at approximately nine years of age do not show consistent gender differences in relation to math skills. However, in 17 out of the 20 countries examined in this study, 13-year-old boys tended to score higher than girls. Moreover, mathematics is often labeled as a masculine ability; as a result, girls often have low confidence in their math capabilities. [ 31 ] These gender stereotypes can reinforce low confidence in girls and can cause math anxiety as research has shown that performance on standardized math tests is affected by one's confidence. [ 32 ] As a result, educators have been trying to abolish this stereotype by fostering confidence in math in all students in order to avoid math anxiety. [ 33 ]
While on the other hand, results obtained by Monika Szczygiel show that girls have a higher level of anxiety on testing and in total, although there is no gender difference in general learning math anxiety. Therefore, the gender gap in math anxiety may result from the type of anxiety. Tests triggers greater anxiety in girls compared with boys, but they feel same level of anxiety learning math. [ 34 ]
The principles of mathematics are generally understood at an early age; preschoolers can comprehend the majority of principles underlying counting. By kindergarten, it is common for children to use counting in a more sophisticated manner by adding and subtracting numbers. While kindergarteners tend to use their fingers to count, this habit is soon abandoned and replaced with a more refined and efficient strategy; children begin to perform addition and subtraction mentally at approximately six years of age. When children reach approximately eight years of age, they can retrieve answers to mathematical equations from memory. With proper instruction, most children acquire these basic mathematical skills and are able to solve more complex mathematical problems with sophisticated training. [ 33 ]
High-risk teaching styles are often explored to gain a better understanding of math anxiety. Goulding, Rowland, and Barber [ 35 ] suggest that there are linkages between a teacher's lack of subject knowledge and the ability to plan teaching material effectively. These findings suggest that teachers who do not have a sufficient background in mathematics may struggle with the development of comprehensive lesson plans for their students. Similarly, Laturner's research [ 36 ] shows that teachers with certification in math are more likely to be passionate and committed to teaching math than those without certification. However, those without certification vary in their commitment to the profession depending on coursework preparation.
A study conducted by Kawakami, Steele, Cifa, Phills, and Dovidio [ 37 ] examined attitudes towards math and behavior during math examinations. The study examined the effect of extensive training in teaching women how to approach math. The results showed that women who were trained to approach rather than avoid math showed a positive implicit attitude towards math. These findings were only consistent with women low in initial identification with math. This study was replicated with women who were either encouraged to approach math or who received neutral training. Results were consistent and demonstrated that women taught to approach math had an implicit positive attitude and completed more math problems than women taught to approach math in a neutral manner.
Johns, Schmader, and Martens [ 38 ] conducted a study in which they examined the effect of teaching stereotype threat as a means of improving women's math performance. The researchers concluded that women tended to perform worse than men when problems were described as math equations. However, women did not differ from men when the test sequence was described as problem-solving or in a condition in which they learned about stereotype threats. This research has practical implications. The results suggested that teaching students about stereotype threat could offer a practical means of reducing its detrimental effects and lead to an improvement in a girl's performance and mathematical ability, leading the researchers to conclude that educating female teachers about stereotype threat can reduce its negative effects in the classroom.
According to Margaret Murray, female mathematicians in the United States have almost always been a minority. Although the exact difference fluctuates with the times, as she has explored in her book Women Becoming Mathematicians: Creating a Professional Identity in Post-World War II America , "Since 1980, women have earned over 17 percent of the mathematics doctorates.... [In The United States]". [ 39 ] The trends in gender are by no means clear, but perhaps parity is still a way to go. Since 1995, studies have shown that the gender gap favored males in most mathematical standardized testing as boys outperformed girls in 15 out of 28 countries. However, as of 2015 the gender gap has almost been reversed, showing an increase in female presence. This is being caused by women's steadily increasing performance on math and science testing and enrollment, but also by males' losing ground at the same time. This role reversal can largely be associated with the gender normative stereotypes that are found in the Science, technology, engineering, and mathematics (STEM) field, deeming "who math is for" and "who STEM careers are for". These stereotypes can fuel mathematical anxiety that is already present among young female populations. [ 40 ] Thus parity will take more work to overcome mathematical anxiety and this is one reason why women in mathematics are role models for younger women.
According to John Taylor Gatto , as expounded in several lengthy books, [ 41 ] [ page needed ] modern Western schools were deliberately [ dubious – discuss ] designed during the late 19th century to create an environment which is ideal for fostering fear and anxiety, and for preventing or delaying learning. Many who are sympathetic to Gatto's thesis regard his position as unnecessarily extreme. [ 42 ] Diane Ravitch , former assistant secretary of education during the George H. W. Bush administration, agrees with Gatto up to a point, conceding that there is an element of social engineering (i.e. the manufacture of the compliant citizenry) in the construction of the American education system, [ 42 ] which prioritizes conformance over learning.
The role of attachment has been suggested as having an impact in the development of the anxiety. [ 43 ] Children with an insecure attachment style were more likely to demonstrate the anxiety.
Math used to be taught as a right and wrong subject and as if getting the right answer were paramount. In contrast to most subjects, mathematics problems almost always have a right answer but there are many ways to obtain the answer. Previously, the subject was often taught as if there were a right way to solve the problem and any other approaches would be wrong, even if students got the right answer. Thankfully, mathematics has evolved and so has teaching it. Students used to have higher anxiety because of the way math was taught. "Teachers benefit children most when they encourage them to share their thinking process and justify their answers out loud or in writing as they perform math operations. ... With less of an emphasis on right or wrong and more of an emphasis on process, teachers can help alleviate students' anxiety about math". [ 44 ]
There have been many studies that show parent involvement in developing a child's educational processes is essential. A student's success in school is increased if their parents are involved in their education both at home and school. [ 45 ] As a result, one of the easiest ways to reduce math anxiety is for the parent to be more involved in their child's education. In addition, research has shown that a parent's perception on mathematics influences their child's perception and achievement in mathematics. [ 46 ]
Furthermore, studies by Herbert P. Ginsburg , Columbia University, show the influence of parents' and teachers' attitudes on " 'the child's expectations in that area of learning.'... It is less the actual teaching and more the attitude and expectations of the teacher or parents that count". This is further supported by a survey of Montgomery County, Maryland students who "pointed to their parents as the primary force behind the interest in mathematics". [ 47 ]
Claudia Zaslavsky [ 47 ] contends that math has two components. The first component is to calculate the answer. This component also has two subcomponents, namely the answer and the process or method used to determine the answer. Focusing more on the process or method enables students to make mistakes, but not "fail at math". The second component is to understand the mathematical concepts that underlay the problem being studied. "... and in this respect studying mathematics is much more like studying, say, music or painting than it is like studying history or biology."
Amongst others supporting this viewpoint is the work of Eugene Geist . [ 48 ] Geist's recommendations include focusing on the concepts rather than the right answer and letting students work on their own and discuss their solutions before the answer is given.
National Council of Teachers of Mathematics (NCTM) (1989, 1995b) suggestions for teachers seeking to prevent math anxiety include:
Hackworth [ 49 ] suggests that the following activities can help students in reducing and mitigating mathematical anxiety:
B R Alimin and D B Widjajanti [ 50 ] recommend teachers:
Several studies have shown that relaxation techniques, including controlled breathing, can be used to help alleviate anxiety related to mathematics. In her workbook Conquering Math Anxiety , Cynthia Arem offers specific strategies to reduce math avoidance and anxiety. One strategy she advocates for is relaxation exercises and indicates that by practicing relaxation techniques on a regular basis for 10–20 minutes students can significantly reduce their anxiety. [ 51 ]
Dr. Edmundo Jacobson's Progressive Muscle Relaxation taken from the book Mental Toughness Training for Sports, Loehr (1986) can be used in a modified form to reduce anxiety as posted on the website HypnoGenesis. [ 52 ]
According to Mina Bazargan and Mehdi Amiri, Modular Cognitive Behavior Therapy (MCBT) can reduce the level of mathematical anxiety and increase students' self-esteem. [ 53 ]
Visualization has also been used effectively to help reduce math anxiety. Arem has a chapter that deals with reducing test anxiety and advocates the use of visualization. In her chapter titled Conquer Test Anxiety (Chapter 9) she has specific exercises devoted to visualization techniques to help the student feel calm and confident during testing. [ 54 ]
Studies have shown students learn best when they are active rather than passive learners. [ 55 ]
The theory of multiple intelligences suggests that there is a need for addressing different learning styles. Math lessons can be tailored for visual / spatial , logical/mathematics, musical, auditory , body/kinesthetic , interpersonal and intrapersonal and verbal/linguistic learning styles. This theory of learning styles has never been demonstrated to be true in controlled trials. Studies show no evidence to support tailoring lessons to an individual students learning style to be beneficial. [ 56 ]
New concepts can be taught through play acting, cooperative groups, visual aids, hands on activities or information technology. [ 57 ] To help with learning statistics, there are many applets found on the Internet that help students learn about many things from probability distributions to linear regression. These applets are commonly used in introductory statistics classes, as many students benefit from using them. [ original research? ] [ who? ]
Active learners ask critical questions, such as: Why do we do it this way, and not that way ? Some teachers may find these questions annoying or difficult to answer, and indeed may have been trained to respond to such questions with hostility and contempt, designed to instill fear. Better teachers respond eagerly to these questions, and use them to help the students deepen their understanding by examining alternative methods so the students can choose for themselves which method they prefer. This process can result in meaningful class discussions. Talking is the way in which students increase their understanding and command of math. [ 58 ] Teachers can give students insight as to why they learn certain content by asking students questions such as "what purpose is served by solving this problem?" and "why are we being asked to learn this?" [ 59 ]
Reflective journals help students develop metacognitive skills by having them think about their understanding. According to Pugalee, [ 60 ] writing helps students organize their thinking which helps them better understand mathematics. Moreover, writing in mathematics classes helps students problem solve and improve mathematical reasoning. When students know how to use mathematical reasoning, they are less anxious about solving problems.
Children learn best when math is taught in a way that is relevant to their everyday lives. Children enjoy experimenting. To learn mathematics in any depth, students should be engaged in exploring, conjecturing, and thinking, as well as in rote learning of rules and procedures. [ 61 ] | https://en.wikipedia.org/wiki/Mathematical_anxiety |
Mathematical beauty is the aesthetic pleasure derived from the abstractness, purity, simplicity, depth or orderliness of mathematics . Mathematicians may express this pleasure by describing mathematics (or, at least, some aspect of mathematics) as beautiful or describe mathematics as an art form, (a position taken by G. H. Hardy [ 1 ] ) or, at a minimum, as a creative activity .
Comparisons are made with music and poetry .
Mathematicians commonly describe an especially pleasing method of proof as elegant . [ 2 ] Depending on context, this may mean:
In the search for an elegant proof, mathematicians may search for multiple independent ways to prove a result, as the first proof that is found can often be improved. The theorem for which the greatest number of different proofs have been discovered is possibly the Pythagorean theorem , with hundreds of proofs being published up to date. [ 3 ] Another theorem that has been proved in many different ways is the theorem of quadratic reciprocity . In fact, Carl Friedrich Gauss alone had eight different proofs of this theorem, six of which he published. [ 4 ]
Conversely, results that are logically correct but involve laborious calculations, over-elaborate methods, highly conventional approaches or a large number of powerful axioms or previous results are usually not considered to be elegant, and may be even referred to as ugly or clumsy .
Some mathematicians see beauty in mathematical results that establish connections between two areas of mathematics that at first sight appear to be unrelated. [ 5 ] These results are often described as deep . While it is difficult to find universal agreement on whether a result is deep, some examples are more commonly cited than others. One such example is Euler's identity : [ 6 ]
e i π + 1 = 0 . {\displaystyle \displaystyle e^{i\pi }+1=0\,.}
This elegant expression ties together arguably the five most important mathematical constants ( e , i , π {\displaystyle \pi } , 1, and 0) with the two most common mathematical symbols (+, =). Euler's identity is a special case of Euler's formula , which the physicist Richard Feynman called "our jewel" and "the most remarkable formula in mathematics". [ 7 ] Modern examples include the modularity theorem , which establishes an important connection between elliptic curves and modular forms (work on which led to the awarding of the Wolf Prize to Andrew Wiles and Robert Langlands ), and " monstrous moonshine ", which connects the Monster group to modular functions via string theory (for which Richard Borcherds was awarded the Fields Medal ).
Other examples of deep results include unexpected insights into mathematical structures. For example, Gauss's Theorema Egregium is a deep theorem that states that the gaussian curvature is invariant under isometry of the surface. Another example is the fundamental theorem of calculus [ 8 ] (and its vector versions including Green's theorem and Stokes' theorem ).
The opposite of deep is trivial . A trivial theorem may be a result that can be derived in an obvious and straightforward way from other known results, or which applies only to a specific set of particular objects such as the empty set . In some occasions, a statement of a theorem can be original enough to be considered deep, though its proof is fairly obvious.
In his 1940 essay A Mathematician's Apology , G. H. Hardy suggested that a beautiful proof or result possesses "inevitability", "unexpectedness", and "economy". [ 9 ]
In 1997, Gian-Carlo Rota , disagreed with unexpectedness as a sufficient condition for beauty and proposed a counterexample:
A great many theorems of mathematics, when first published, appear to be surprising; thus for example some twenty years ago [from 1977] the proof of the existence of non-equivalent differentiable structures on spheres of high dimension was thought to be surprising, but it did not occur to anyone to call such a fact beautiful, then or now. [ 10 ]
In contrast, Monastyrsky wrote in 2001:
It is very difficult to find an analogous invention in the past to Milnor 's beautiful construction of the different differential structures on the seven-dimensional sphere... The original proof of Milnor was not very constructive, but later E. Brieskorn showed that these differential structures can be described in an extremely explicit and beautiful form. [ 11 ]
This disagreement illustrates both the subjective nature of mathematical beauty and its connection with mathematical results: in this case, not only the existence of exotic spheres, but also a particular realization of them.
Interest in pure mathematics that is separate from empirical study has been part of the experience of various civilizations , including that of the ancient Greeks , who "did mathematics for the beauty of it". [ 12 ] The aesthetic pleasure that mathematical physicists tend to experience in Einstein's theory of general relativity has been attributed (by Paul Dirac , among others) to its "great mathematical beauty". [ 13 ] The beauty of mathematics is experienced when the physical reality of objects are represented by mathematical models . Group theory , developed in the early 1800s for the sole purpose of solving polynomial equations, became a fruitful way of categorizing elementary particles —the building blocks of matter. Similarly, the study of knots provides important insights into string theory and loop quantum gravity . [ citation needed ]
Some [ who? ] believe that in order to appreciate mathematics, one must engage in doing mathematics. [ 14 ]
For example, Math Circles are after-school enrichment programs where students engage with mathematics through lectures and activities; there are also some teachers who encourage student engagement by teaching mathematics in kinesthetic learning . In a general Math Circle lesson, students use pattern finding, observation, and exploration to make their own mathematical discoveries. For example, mathematical beauty arises in a Math Circle activity on symmetry designed for 2nd and 3rd graders, where students create their own snowflakes by folding a square piece of paper and cutting out designs of their choice along the edges of the folded paper. When the paper is unfolded, a symmetrical design reveals itself. In a day to day elementary school mathematics class, symmetry can be presented as such in an artistic manner where students see aesthetically pleasing results in mathematics. [ citation needed ]
Some [ who? ] teachers prefer to use mathematical manipulatives to present mathematics in an aesthetically pleasing way. Examples of a manipulative include algebra tiles , cuisenaire rods , and pattern blocks . For example, one can teach the method of completing the square by using algebra tiles. Cuisenaire rods can be used to teach fractions, and pattern blocks can be used to teach geometry. Using mathematical manipulatives helps students gain a conceptual understanding that might not be seen immediately in written mathematical formulas. [ 15 ]
Another example of beauty in experience involves the use of origami . Origami, the art of paper folding, has aesthetic qualities and many mathematical connections. One can study the mathematics of paper folding by observing the crease pattern on unfolded origami pieces. [ 16 ]
Combinatorics , the study of counting, has artistic representations which some [ who? ] find mathematically beautiful. There are many visual examples which illustrate combinatorial concepts. Some of the topics and objects seen in combinatorics courses with visual representations include, among others Four color theorem , Young tableau , Permutohedron , Graph theory , Partition of a set . [ 17 ]
Brain imaging experiments conducted by Semir Zeki and his colleagues [ 18 ] show that the experience of mathematical beauty has, as a neural correlate, activity in field A1 of the medial orbito-frontal cortex (mOFC) of the brain and that this activity is parametrically related to the declared intensity of beauty. The location of the activity is similar to the location of the activity that correlates with the experience of beauty from other sources, such as music or joy or sorrow. Moreover, mathematicians seem resistant to revising their judgment of the beauty of a mathematical formula in light of contradictory opinion given by their peers. [ 19 ]
Some [ who? ] mathematicians are of the opinion that the doing of mathematics is closer to discovery than invention, for example:
There is no scientific discoverer, no poet, no painter, no musician, who will not tell you that he found ready made his discovery or poem or picture—that it came to him from outside, and that he did not consciously create it from within.
These mathematicians believe that the detailed and precise results of mathematics may be reasonably taken to be true without any dependence on the universe in which we live. For example, they would argue that the theory of the natural numbers is fundamentally valid, in a way that does not require any specific context. Some mathematicians have extrapolated this viewpoint that mathematical beauty is truth further, in some cases becoming mysticism .
In Plato 's philosophy there were two worlds, the physical one in which we live and another abstract world which contained unchanging truth, including mathematics. He believed that the physical world was a mere reflection of the more perfect abstract world. [ 20 ]
Hungarian mathematician Paul Erdős [ 21 ] spoke of an imaginary book, in which God has written down all the most beautiful mathematical proofs. When Erdős wanted to express particular appreciation of a proof, he would exclaim "This one's from The Book!"
Twentieth-century French philosopher Alain Badiou claimed that ontology is mathematics. [ 22 ] Badiou also believes in deep connections between mathematics, poetry and philosophy.
In many cases, natural philosophers and other scientists who have made extensive use of mathematics have made leaps of inference between beauty and physical truth in ways that turned out to be erroneous. For example, at one stage in his life, Johannes Kepler believed that the proportions of the orbits of the then-known planets in the Solar System have been arranged by God to correspond to a concentric arrangement of the five Platonic solids , each orbit lying on the circumsphere of one polyhedron and the insphere of another. As there are exactly five Platonic solids, Kepler's hypothesis could only accommodate six planetary orbits and was disproved by the subsequent discovery of Uranus .
G. H. Hardy [ 23 ] analysed the beauty of mathematical proofs into these six dimensions: general, serious, deep, unexpected, inevitable, economical (simple). Paul Ernest [ 24 ] proposes seven dimensions for any mathematical objects, including concepts, theorems, proofs and theories. These are
1. Economy, simplicity, brevity, succinctness, elegance;
2. Generality, abstraction, power;
3. Surprise, ingenuity, cleverness;
4. Pattern, structure, symmetry, regularity, visual design;
5. Logicality, rigour, tight reasoning and deduction, pure thought;
6. Interconnectedness, links, unification;
7. Applicability, modelling power, empirical generality.
He argues that individual mathematicians and communities of mathematicians will have preferred choices from this list. Some, like Hardy, will reject some (Hardy claimed that applied mathematics is ugly). However, Rentuya Sa and colleagues [ 25 ] compared the views of British mathematicians and undergraduates and Chinese mathematicians on the beauty of 20 well known equations and found a strong measure of agreement between their views.
In the 1970s, Abraham Moles and Frieder Nake analyzed links between beauty, information processing , and information theory . [ 26 ] [ 27 ] In the 1990s, Jürgen Schmidhuber formulated a mathematical theory of observer-dependent subjective beauty based on algorithmic information theory : the most beautiful objects among subjectively comparable objects have short algorithmic descriptions (i.e., Kolmogorov complexity ) relative to what the observer already knows. [ 28 ] [ 29 ] [ 30 ] Schmidhuber explicitly distinguishes between beautiful and interesting. The latter corresponds to the first derivative of subjectively perceived beauty: the observer continually tries to improve the predictability and compressibility of the observations by discovering regularities such as repetitions and symmetries and fractal self-similarity . Whenever the observer's learning process (possibly a predictive artificial neural network ) leads to improved data compression such that the observation sequence can be described by fewer bits than before, the temporary interesting-ness of the data corresponds to the compression progress, and is proportional to the observer's internal curiosity reward. [ 31 ] [ 32 ]
Examples of the use of mathematics in music include the stochastic music of Iannis Xenakis , the Fibonacci sequence in Tool 's Lateralus , counterpoint of Johann Sebastian Bach , polyrhythmic structures (as in Igor Stravinsky 's The Rite of Spring ), the Metric modulation of Elliott Carter , permutation theory in serialism beginning with Arnold Schoenberg , and application of Shepard tones in Karlheinz Stockhausen 's Hymnen . They also include the application of Group theory to transformations in music in the theoretical writings of David Lewin .
Examples of the use of mathematics in the visual arts include applications of chaos theory and fractal geometry to computer-generated art , symmetry studies of Leonardo da Vinci , projective geometries in development of the perspective theory of Renaissance art, grids in Op art , optical geometry in the camera obscura of Giambattista della Porta , and multiple perspective in analytic cubism and futurism .
Sacred geometry is a field of its own, giving rise to countless art forms including some of the best known mystic symbols and religious motifs, and has a particularly rich history in Islamic architecture . It also provides a means of meditation and comtemplation, for example the study of the Kaballah Sefirot (Tree Of Life) and Metatron's Cube ; and also the act of drawing itself.
The Dutch graphic designer M. C. Escher created mathematically inspired woodcuts , lithographs , and mezzotints . These feature impossible constructions, explorations of infinity , architecture, visual paradoxes and tessellations .
Some painters and sculptors create work distorted with the mathematical principles of anamorphosis , including South African sculptor Jonty Hurwitz .
British constructionist artist John Ernest created reliefs and paintings inspired by group theory. [ 33 ] A number of other British artists of the constructionist and systems schools of thought also draw on mathematics models and structures as a source of inspiration, including Anthony Hill and Peter Lowe . [ 34 ] Computer-generated art is based on mathematical algorithms .
Bertrand Russell expressed his sense of mathematical beauty in these words:
Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry. [ 35 ]
Paul Erdős expressed his views on the ineffability of mathematics when he said, "Why are numbers beautiful? It's like asking why is Beethoven's Ninth Symphony beautiful. If you don't see why, someone can't tell you. I know numbers are beautiful. If they aren't beautiful, nothing is". [ 36 ] | https://en.wikipedia.org/wiki/Mathematical_beauty |
Mathematical chemistry [ 1 ] is the area of research engaged in novel applications of mathematics to chemistry ; it concerns itself principally with the mathematical modeling of chemical phenomena. [ 2 ] Mathematical chemistry has also sometimes been called computer chemistry , but should not be confused with computational chemistry .
Major areas of research in mathematical chemistry include chemical graph theory , which deals with topology such as the mathematical study of isomerism and the development of topological descriptors or indices which find application in quantitative structure-property relationships ; and chemical aspects of group theory , which finds applications in stereochemistry and quantum chemistry . Another important area is molecular knot theory and circuit topology that describe the topology of folded linear molecules such as proteins and nucleic acids.
The history of the approach may be traced back to the 19th century. Georg Helm published a treatise titled "The Principles of Mathematical Chemistry: The Energetics of Chemical Phenomena" in 1894. [ 3 ] Some of the more contemporary periodical publications specializing in the field are MATCH Communications in Mathematical and in Computer Chemistry, first published in 1975, and the Journal of Mathematical Chemistry, first published in 1987. In 1986 a series of annual conferences MATH/CHEM/COMP taking place in Dubrovnik was initiated by the late Ante Graovac .
The basic models for mathematical chemistry are molecular graph and topological index .
In 2005 the International Academy of Mathematical Chemistry (IAMC) was founded in Dubrovnik (Croatia) by Milan Randić . The Academy has 82 members (2009) from all over the world, including six scientists awarded with a Nobel Prize. | https://en.wikipedia.org/wiki/Mathematical_chemistry |
A mathematical coincidence is said to occur when two expressions with no direct relationship show a near-equality which has no apparent theoretical explanation.
For example, there is a near-equality close to the round number 1000 between powers of 2 and powers of 10:
Some mathematical coincidences are used in engineering when one expression is taken as an approximation of another.
A mathematical coincidence often involves an integer , and the surprising feature is the fact that a real number arising in some context is considered by some standard as a "close" approximation to a small integer or to a multiple or power of ten, or more generally, to a rational number with a small denominator . Other kinds of mathematical coincidences, such as integers simultaneously satisfying multiple seemingly unrelated criteria or coincidences regarding units of measurement, may also be considered. In the class of those coincidences that are of a purely mathematical sort, some simply result from sometimes very deep mathematical facts, while others appear to come 'out of the blue'.
Given the countably infinite number of ways of forming mathematical expressions using a finite number of symbols, the number of symbols used and the precision of approximate equality might be the most obvious way to assess mathematical coincidences; but there is no standard, and the strong law of small numbers is the sort of thing one has to appeal to with no formal opposing mathematical guidance. [ citation needed ] Beyond this, some sense of mathematical aesthetics could be invoked to adjudicate the value of a mathematical coincidence, and there are in fact exceptional cases of true mathematical significance (see Ramanujan's constant below, which made it into print some years ago as a scientific April Fools' joke [ 1 ] ). All in all, though, they are generally to be considered for their curiosity value, or perhaps to encourage new mathematical learners at an elementary level.
Sometimes simple rational approximations are exceptionally close to interesting irrational values. These are explainable in terms of large terms in the continued fraction representation of the irrational value, but further insight into why such improbably large terms occur is often not available.
Rational approximants (convergents of continued fractions) to ratios of logs of different numbers are often invoked as well, making coincidences between the powers of those numbers. [ 2 ]
Many other coincidences are combinations of numbers that put them into the form that such rational approximants provide close relationships.
In music, the distances between notes (intervals) are measured as ratios of their frequencies, with near-rational ratios often sounding harmonious. In western twelve-tone equal temperament , the ratio between consecutive note frequencies is 2 12 {\displaystyle {\sqrt[{12}]{2}}} .
with the last accurate to 14 or 15 decimal places.
The speed of light is (by definition) exactly 299 792 458 m/s , extremely close to 3.0 × 10 8 m/s ( 300 000 000 m/s ). This is a pure coincidence, as the metre was originally defined as 1 / 10 000 000 of the distance between the Earth's pole and equator along the surface at sea level, and the Earth's circumference just happens to be about 2/15 of a light-second. [ 40 ] It is also roughly equal to one foot per nanosecond (the actual number is 0.9836 ft/ns).
As seen from Earth, the angular diameter of the Sun varies between 31′27″ and 32′32″, while that of the Moon is between 29′20″ and 34′6″. The fact that the intervals overlap (the former interval is contained in the latter) is a coincidence, and has implications for the types of solar eclipses that can be observed from Earth.
While not constant but varying depending on latitude and altitude , the numerical value of the acceleration caused by Earth's gravity on the surface lies between 9.74 and 9.87 m/s 2 , which is quite close to 10. This means that as a result of Newton's second law , the weight of a kilogram of mass on Earth's surface corresponds roughly to 10 newtons of force exerted on an object. [ 41 ]
This is related to the aforementioned coincidence that the square of pi is close to 10. One of the early definitions of the metre was the length of a pendulum whose half swing had a period equal to one second. Since the period of the full swing of a pendulum is approximated by the equation below, algebra shows that if this definition was maintained, gravitational acceleration measured in metres per second per second would be exactly equal to π 2 . [ 42 ]
The upper limit of gravity on Earth's surface (9.87 m/s 2 ) is equal to π 2 m/s 2 to four significant figures. It is approximately 0.6% greater than standard gravity (9.80665 m/s 2 ).
The Rydberg constant , when multiplied by the speed of light and expressed as a frequency, is close to π 2 3 × 10 15 Hz {\displaystyle {\frac {\pi ^{2}}{3}}\times 10^{15}\ {\text{Hz}}} : [ 40 ]
This is also approximately the number of feet in one meter:
As discovered by Randall Munroe , a cubic mile is close to 4 3 π {\displaystyle {\frac {4}{3}}\pi } cubic kilometres (within 0.5%). This means that a sphere with radius n kilometres has almost exactly the same volume as a cube with side length n miles. [ 44 ] [ 45 ]
The ratio of a mile to a kilometre is approximately the Golden ratio . As a consequence, a Fibonacci number of miles is approximately the next Fibonacci number of kilometres.
The ratio of a mile to a kilometre is also very close to ln ( 5 ) {\displaystyle \ln(5)} (within 0.006%). That is, 5 m ≈ e k {\displaystyle 5^{m}\approx e^{k}} where m is the number of miles, k is the number of kilometres and e is Euler's number .
A density of one ounce per cubic foot is very close to one kilogram per cubic metre: 1 oz/ft 3 = 1 oz × 0.028349523125 kg/oz / (1 ft × 0.3048 m/ft) 3 ≈ 1.0012 kg/m 3 .
The ratio between one troy ounce and one gram is approximately 10 π − π 10 = 99 10 π {\displaystyle 10\pi -{\frac {\pi }{10}}={\frac {99}{10}}\pi } .
The fine-structure constant α {\displaystyle \alpha } is close to, and was once conjectured to be precisely equal to 1 / 137 . [ 46 ] Its CODATA recommended value is
α {\displaystyle \alpha } is a dimensionless physical constant , so this coincidence is not an artifact of the system of units being used.
The number of seconds in one year, based on the Gregorian calendar , can be calculated by: 365.2425 ( days year ) × 24 ( hours day ) × 60 ( minutes hour ) × 60 ( seconds minute ) = 31 , 556 , 952 ( seconds year ) {\displaystyle 365.2425\left({\frac {\text{days}}{\text{year}}}\right)\times 24\left({\frac {\text{hours}}{\text{day}}}\right)\times 60\left({\frac {\text{minutes}}{\text{hour}}}\right)\times 60\left({\frac {\text{seconds}}{\text{minute}}}\right)=31,556,952\left({\frac {\text{seconds}}{\text{year}}}\right)}
This value can be approximated by π × 10 7 {\displaystyle \pi \times 10^{7}} or 31,415,926.54 with less than one percent of an error: [ 1 − ( 31 , 415 , 926.54 31 , 556 , 952 ) ] × 100 = 0.4489 % {\displaystyle \left[1-\left({\frac {31,415,926.54}{31,556,952}}\right)\right]\times 100=0.4489\%} | https://en.wikipedia.org/wiki/Mathematical_coincidence |
A mathematical constant is a number whose value is fixed by an unambiguous definition, often referred to by a special symbol (e.g., an alphabet letter ), or by mathematicians' names to facilitate using it across multiple mathematical problems . [ 1 ] Constants arise in many areas of mathematics , with constants such as e and π occurring in such diverse contexts as geometry , number theory , statistics , and calculus .
Some constants arise naturally by a fundamental principle or intrinsic property, such as the ratio between the circumference and diameter of a circle ( π ). Other constants are notable more for historical reasons than for their mathematical properties. The more popular constants have been studied throughout the ages and computed to many decimal places.
All named mathematical constants are definable numbers , and usually are also computable numbers ( Chaitin's constant being a significant exception).
These are constants which one is likely to encounter during pre-college education in many countries.
The square root of 2 , often known as root 2 or Pythagoras' constant , and written as √ 2 , is the unique positive real number that, when multiplied by itself, gives the number 2 . It is more precisely called the principal square root of 2 , to distinguish it from the negative number with the same property.
Geometrically the square root of 2 is the length of a diagonal across a square with sides of one unit of length ; this follows from the Pythagorean theorem . It is an irrational number, possibly the first number to be known as such, and an algebraic number . Its numerical value truncated to 50 decimal places is:
Alternatively, the quick approximation 99/70 (≈ 1.41429) for the square root of two was frequently used before the common use of electronic calculators and computers . Despite having a denominator of only 70, it differs from the correct value by less than 1/10,000 (approx. 7.2 × 10 −5 ).
Its simple continued fraction is periodic and given by:
2 = 1 + 1 2 + 1 2 + 1 2 + 1 ⋱ {\displaystyle {\sqrt {2}}=1+{\frac {1}{2+{\frac {1}{2+{\frac {1}{2+{\frac {1}{\ddots }}}}}}}}}
The constant π (pi) has a natural definition in Euclidean geometry as the ratio between the circumference and diameter of a circle. It may be found in many other places in mathematics: for example, the Gaussian integral , the complex roots of unity , and Cauchy distributions in probability . However, its ubiquity is not limited to pure mathematics. It appears in many formulas in physics, and several physical constants are most naturally defined with π or its reciprocal factored out. For example, the ground state wave function of the hydrogen atom is
where a 0 {\displaystyle a_{0}} is the Bohr radius .
π is an irrational number , transcendental number and an algebraic period .
The numeric value of π is approximately:
Unusually good approximations are given by the fractions 22/7 and 355/113 .
Memorizing as well as computing increasingly more digits of π is a world record pursuit.
Euler's number e , also known as the exponential growth constant, appears in many areas of mathematics, and one possible definition of it is the value of the following expression:
The constant e is intrinsically related to the exponential function x ↦ e x {\displaystyle x\mapsto e^{x}} .
The Swiss mathematician Jacob Bernoulli discovered that e arises in compound interest : If an account starts at $1, and yields interest at annual rate R , then as the number of compounding periods per year tends to infinity (a situation known as continuous compounding ), the amount of money at the end of the year will approach e R dollars.
The constant e also has applications to probability theory , where it arises in a way not obviously related to exponential growth. As an example, suppose that a slot machine with a one in n probability of winning is played n times, then for large n (e.g., one million), the probability that nothing will be won will tend to 1/ e as n tends to infinity.
Another application of e , discovered in part by Jacob Bernoulli along with French mathematician Pierre Raymond de Montmort , is in the problem of derangements , also known as the hat check problem . [ 2 ] Here, n guests are invited to a party, and at the door each guest checks his hat with the butler, who then places them into labelled boxes. The butler does not know the name of the guests, and hence must put them into boxes selected at random. The problem of de Montmort is: what is the probability that none of the hats gets put into the right box. The answer is
which, as n tends to infinity, approaches 1/ e .
e is an irrational number and a transcendental number.
The numeric value of e is approximately:
The imaginary unit or unit imaginary number , denoted as i , is a mathematical concept which extends the real number system R {\displaystyle \mathbb {R} } to the complex number system C . {\displaystyle \mathbb {C} .} The imaginary unit's core property is that i 2 = −1 . The term " imaginary " was coined because there is no ( real ) number having a negative square .
There are in fact two complex square roots of −1, namely i and − i , just as there are two complex square roots of every other real number (except zero , which has one double square root).
In contexts where the symbol i is ambiguous or problematic, j or the Greek iota ( ι ) is sometimes used. This is in particular the case in electrical engineering and control systems engineering , where the imaginary unit is often denoted by j , because i is commonly used to denote electric current .
The number φ , also called the golden ratio , turns up frequently in geometry , particularly in figures with pentagonal symmetry . Indeed, the length of a regular pentagon 's diagonal is φ times its side. The vertices of a regular icosahedron are those of three mutually orthogonal golden rectangles . Also, it is related to the Fibonacci sequence , related to growth by recursion . [ 3 ] Kepler proved that it is the limit of the ratio of consecutive Fibonacci numbers. [ 4 ] The golden ratio has the slowest convergence of any irrational number. [ 5 ] It is, for that reason, one of the worst cases of Lagrange's approximation theorem and it is an extremal case of the Hurwitz inequality for diophantine approximations . This may be why angles close to the golden ratio often show up in phyllotaxis (the growth of plants). [ 6 ] It is approximately equal to:
or, more precisely 1 + 5 2 . {\displaystyle {\frac {1+{\sqrt {5}}}{2}}.}
These are constants which are encountered frequently in higher mathematics .
Euler's constant or the Euler–Mascheroni constant is defined as the limiting difference between the harmonic series and the natural logarithm :
It appears frequently in mathematics, especially in number theoretical contexts such as Mertens' third theorem or the growth rate of the divisor function . It has relations to the gamma function and its derivatives as well as the zeta function and there exist many different integrals and series involving γ {\displaystyle \gamma } .
Despite the ubiquity of the Euler-Mascheroni constant, many of its properties remain unknown. That includes the major open questions of whether it is a rational or irrational number and whether it is algebraic or transcendental. In fact, γ {\displaystyle \gamma } has been described as a mathematical constant "shadowed only π {\displaystyle \pi } and e {\displaystyle e} in importance." [ 7 ]
The numeric value of γ {\displaystyle \gamma } is approximately:
Apery's constant is defined as the sum of the reciprocals of the cubes of the natural numbers: ζ ( 3 ) = ∑ n = 1 ∞ 1 n 3 = 1 + 1 2 3 + 1 3 3 + 1 4 3 + 1 5 3 ⋯ {\displaystyle \zeta (3)=\sum _{n=1}^{\infty }{\frac {1}{n^{3}}}=1+{\frac {1}{2^{3}}}+{\frac {1}{3^{3}}}+{\frac {1}{4^{3}}}+{\frac {1}{5^{3}}}\cdots } It is the special value of the Riemann zeta function ζ ( s ) {\displaystyle \zeta (s)} at s = 3 {\displaystyle s=3} . The quest to find an exact value for this constant in terms of other known constants and elementary functions originated when Euler famously solved the Basel problem by giving ζ ( 2 ) = 1 6 π 2 {\displaystyle \zeta (2)={\frac {1}{6}}\pi ^{2}} . To date no such value has been found and it is conjectured that there is none. [ 8 ] However, there exist many representations of ζ ( 3 ) {\displaystyle \zeta (3)} in terms of infinite series.
Apéry's constant arises naturally in a number of physical problems, including in the second- and third-order terms of the electron 's gyromagnetic ratio , computed using quantum electrodynamics . [ 9 ]
ζ ( 3 ) {\displaystyle \zeta (3)} is known to be an irrational number which was proven by the French mathematician Roger Apéry in 1979. It is however not known whether it is algebraic or transcendental.
The numeric value of Apéry's constant is approximately:
Catalan's constant is defined by the alternating sum of the reciprocals of the odd square numbers :
It is the special value of the Dirichlet beta function β ( s ) {\displaystyle \beta (s)} at s = 2 {\displaystyle s=2} . Catalan's constant appears frequently in combinatorics and number theory and also outside mathematics such as in the calculation of the mass distribution of spiral galaxies . [ 10 ]
Questions about the arithmetic nature of this constant also remain unanswered, G {\displaystyle G} having been called "arguably the most basic constant whose irrationality and transcendence (though strongly suspected) remain unproven." [ 11 ] There exist many integral and series representations of Catalan's constant.
It is named after the French and Belgian mathematician Charles Eugène Catalan .
The numeric value of G {\displaystyle G} is approximately:
Iterations of continuous maps serve as the simplest examples of models for dynamical systems . [ 12 ] Named after mathematical physicist Mitchell Feigenbaum , the two Feigenbaum constants appear in such iterative processes: they are mathematical invariants of logistic maps with quadratic maximum points [ 7 ] and their bifurcation diagrams . Specifically, the constant α is the ratio between the width of a tine and the width of one of its two subtines, and the constant δ is the limiting ratio of each bifurcation interval to the next between every period-doubling bifurcation .
The logistic map is a polynomial mapping, often cited as an archetypal example of how chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popularized in a seminal 1976 paper by the Australian biologist Robert May , [ 13 ] in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst . The difference equation is intended to capture the two effects of reproduction and starvation.
The Feigenbaum constants in bifurcation theory are analogous to π in geometry and e in calculus . Neither of them is known to be irrational or even transcendental. However proofs of their universality exist. [ 14 ]
The respective approximate numeric values of δ and α are:
Some constants, such as the square root of 2 , Liouville's constant and Champernowne constant :
are not important mathematical invariants but retain interest being simple representatives of special sets of numbers, the irrational numbers , [ 16 ] the transcendental numbers [ 17 ] and the normal numbers (in base 10) [ 18 ] respectively. The discovery of the irrational numbers is usually attributed to the Pythagorean Hippasus of Metapontum who proved, most likely geometrically, the irrationality of the square root of 2. As for Liouville's constant, named after French mathematician Joseph Liouville , it was the first number to be proven transcendental. [ 19 ]
In the computer science subfield of algorithmic information theory , Chaitin's constant is the real number representing the probability that a randomly chosen Turing machine will halt, formed from a construction due to Argentine - American mathematician and computer scientist Gregory Chaitin . Chaitin's constant, though not being computable , has been proven to be transcendental and normal . Chaitin's constant is not universal, depending heavily on the numerical encoding used for Turing machines; however, its interesting properties are independent of the encoding.
It is common to express the numerical value of a constant by giving its decimal representation (or just the first few digits of it). For two reasons this representation may cause problems. First, even though rational numbers all have a finite or ever-repeating decimal expansion, irrational numbers don't have such an expression making them impossible to completely describe in this manner. Also, the decimal expansion of a number is not necessarily unique. For example, the two representations 0.999... and 1 are equivalent [ 20 ] [ 21 ] in the sense that they represent the same number.
Calculating digits of the decimal expansion of constants has been a common enterprise for many centuries. For example, German mathematician Ludolph van Ceulen of the 16th century spent a major part of his life calculating the first 35 digits of pi. [ 22 ] Using computers and supercomputers , some of the mathematical constants, including π, e , and the square root of 2, have been computed to more than one hundred billion digits. Fast algorithms have been developed, some of which — as for Apéry's constant — are unexpectedly fast.
Some constants differ so much from the usual kind that a new notation has been invented to represent them reasonably. Graham's number illustrates this as Knuth's up-arrow notation is used. [ 23 ] [ 24 ]
It may be of interest to represent them using continued fractions to perform various studies, including statistical analysis. Many mathematical constants have an analytic form , that is they can be constructed using well-known operations that lend themselves readily to calculation. Not all constants have known analytic forms, though; Grossman's constant [ 25 ] and Foias' constant [ 26 ] are examples.
Symbolizing constants with letters is a frequent means of making the notation more concise. A common convention , instigated by René Descartes in the 17th century and Leonhard Euler in the 18th century, is to use lower case letters from the beginning of the Latin alphabet a , b , c , … {\displaystyle a,b,c,\dots } or the Greek alphabet α , β , γ , … {\displaystyle \alpha ,\beta ,\,\gamma ,\dots } when dealing with constants in general.
However, for more important constants, the symbols may be more complex and have an extra letter, an asterisk , a number, a lemniscate or use different alphabets such as Hebrew , Cyrillic or Gothic . [ 24 ]
Sometimes, the symbol representing a constant is a whole word. For example, American mathematician Edward Kasner 's 9-year-old nephew coined the names googol and googolplex . [ 24 ] [ 27 ]
Other names are either related to the meaning of the constant ( universal parabolic constant , twin prime constant , ...) or to a specific person ( Sierpiński's constant , Josephson constant , and so on).
Abbreviations used: | https://en.wikipedia.org/wiki/Mathematical_constant |
In mathematics , certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept called mathematical fallacy . There is a distinction between a simple mistake and a mathematical fallacy in a proof, in that a mistake in a proof leads to an invalid proof while in the best-known examples of mathematical fallacies there is some element of concealment or deception in the presentation of the proof.
For example, the reason why validity fails may be attributed to a division by zero that is hidden by algebraic notation. There is a certain quality of the mathematical fallacy: as typically presented, it leads not only to an absurd result, but does so in a crafty or clever way. [ 1 ] Therefore, these fallacies, for pedagogic reasons, usually take the form of spurious proofs of obvious contradictions . Although the proofs are flawed, the errors, usually by design, are comparatively subtle, or designed to show that certain steps are conditional, and are not applicable in the cases that are the exceptions to the rules.
The traditional way of presenting a mathematical fallacy is to give an invalid step of deduction mixed in with valid steps, so that the meaning of fallacy is here slightly different from the logical fallacy . The latter usually applies to a form of argument that does not comply with the valid inference rules of logic, whereas the problematic mathematical step is typically a correct rule applied with a tacit wrong assumption. Beyond pedagogy, the resolution of a fallacy can lead to deeper insights into a subject (e.g., the introduction of Pasch's axiom of Euclidean geometry , [ 2 ] the five colour theorem of graph theory ). Pseudaria , an ancient lost book of false proofs, is attributed to Euclid . [ 3 ]
Mathematical fallacies exist in many branches of mathematics. In elementary algebra , typical examples may involve a step where division by zero is performed, where a root is incorrectly extracted or, more generally, where different values of a multiple valued function are equated. Well-known fallacies also exist in elementary Euclidean geometry and calculus . [ 4 ] [ 5 ]
Examples exist of mathematically correct results derived by incorrect lines of reasoning. Such an argument, however true the conclusion appears to be, is mathematically invalid and is commonly known as a howler . The following is an example of a howler involving anomalous cancellation : 16 64 = 1 6 6 4 = 1 4 . {\displaystyle {\frac {16}{64}}={\frac {1{\cancel {6}}}{{\cancel {6}}4}}={\frac {1}{4}}.}
Here, although the conclusion 16 / 64 = 1 / 4 is correct, there is a fallacious, invalid cancellation in the middle step. [ note 1 ] Another classical example of a howler is proving the Cayley–Hamilton theorem by simply substituting the scalar variables of the characteristic polynomial with the matrix.
Bogus proofs, calculations, or derivations constructed to produce a correct result in spite of incorrect logic or operations were termed "howlers" by Edwin Maxwell . [ 2 ] Outside the field of mathematics the term howler has various meanings, generally less specific.
The division-by-zero fallacy has many variants. The following example uses a disguised division by zero to "prove" that 2 = 1, but can be modified to prove that any number equals any other number.
The fallacy is in line 5: the progression from line 4 to line 5 involves division by a − b , which is zero since a = b . Since division by zero is undefined, the argument is invalid.
Mathematical analysis as the mathematical study of change and limits can lead to mathematical fallacies — if the properties of integrals and differentials are ignored. For instance, a naïve use of integration by parts can be used to give a false proof that 0 = 1. [ 7 ] Letting u = 1 / log x and dv = dx / x ,
after which the antiderivatives may be cancelled yielding 0 = 1. The problem is that antiderivatives are only defined up to a constant and shifting them by 1 or indeed any number is allowed. The error really comes to light when we introduce arbitrary integration limits a and b .
Since the difference between two values of a constant function vanishes, the same definite integral appears on both sides of the equation.
Many functions do not have a unique inverse . For instance, while squaring a number gives a unique value, there are two possible square roots of a positive number. The square root is multivalued . One value can be chosen by convention as the principal value ; in the case of the square root the non-negative value is the principal value, but there is no guarantee that the square root given as the principal value of the square of a number will be equal to the original number (e.g. the principal square root of the square of −2 is 2). This remains true for nth roots .
Care must be taken when taking the square root of both sides of an equality . Failing to do so results in a "proof" of [ 8 ] 5 = 4.
Proof:
The fallacy is in the second to last line, where the square root of both sides is taken: a 2 = b 2 only implies a = b if a and b have the same sign, which is not the case here. In this case, it implies that a = – b , so the equation should read
which, by adding 9 / 2 on both sides, correctly reduces to 5 = 5.
Another example illustrating the danger of taking the square root of both sides of an equation involves the following fundamental identity [ 9 ]
which holds as a consequence of the Pythagorean theorem . Then, by taking a square root,
Evaluating this when x = π , we get that
or
which is incorrect.
The error in each of these examples fundamentally lies in the fact that any equation of the form
where a ≠ 0 {\displaystyle a\neq 0} , has two solutions:
and it is essential to check which of these solutions is relevant to the problem at hand. [ 10 ] In the above fallacy, the square root that allowed the second equation to be deduced from the first is valid only when cos x is positive. In particular, when x is set to π , the second equation is rendered invalid.
Invalid proofs utilizing powers and roots are often of the following kind:
The fallacy is that the rule x y = x y {\displaystyle {\sqrt {xy}}={\sqrt {x}}{\sqrt {y}}} is generally valid only if at least one of x {\displaystyle x} and y {\displaystyle y} is non-negative (when dealing with real numbers), which is not the case here. [ 11 ]
Alternatively, imaginary roots are obfuscated in the following:
The error here lies in the incorrect usage of multiple-valued functions. ( − 1 ) 1 2 {\displaystyle (-1)^{\frac {1}{2}}} has two values i {\displaystyle i} and − i {\displaystyle -i} without a prior choice of branch, while − 1 {\displaystyle {\sqrt {-1}}} only denotes the principal value i {\displaystyle i} . [ 12 ] Similarly, 1 1 4 {\displaystyle 1^{\frac {1}{4}}} has four different values 1 {\displaystyle 1} , i {\displaystyle i} , − 1 {\displaystyle -1} , and − i {\displaystyle -i} , of which only i {\displaystyle i} is equal to the left side of the first equality.
When a number is raised to a complex power, the result is not uniquely defined (see Exponentiation § Failure of power and logarithm identities ). If this property is not recognized, then errors such as the following can result:
The error here is that the rule of multiplying exponents as when going to the third line does not apply unmodified with complex exponents, even if when putting both sides to the power i only the principal value is chosen. When treated as multivalued functions , both sides produce the same set of values, being { e 2 π n | n ∈ Z } . {\displaystyle \{e^{2\pi n}|n\in \mathbb {Z} \}.}
Many mathematical fallacies in geometry arise from using an additive equality involving oriented quantities (such as adding vectors along a given line or adding oriented angles in the plane) to a valid identity, but which fixes only the absolute value of (one of) these quantities. This quantity is then incorporated into the equation with the wrong orientation, so as to produce an absurd conclusion. This wrong orientation is usually suggested implicitly by supplying an imprecise diagram of the situation, where relative positions of points or lines are chosen in a way that is actually impossible under the hypotheses of the argument, but non-obviously so.
In general, such a fallacy is easy to expose by drawing a precise picture of the situation, in which some relative positions will be different from those in the provided diagram. In order to avoid such fallacies, a correct geometric argument using addition or subtraction of distances or angles should always prove that quantities are being incorporated with their correct orientation.
The fallacy of the isosceles triangle, from ( Maxwell 1959 , Chapter II, § 1), purports to show that every triangle is isosceles , meaning that two sides of the triangle are congruent . This fallacy was known to Lewis Carroll and may have been discovered by him. It was published in 1899. [ 13 ] [ 14 ]
Given a triangle △ABC, prove that AB = AC:
Q.E.D.
As a corollary, one can show that all triangles are equilateral, by showing that AB = BC and AC = BC in the same way.
The error in the proof is the assumption in the diagram that the point O is inside the triangle. In fact, O always lies on the circumcircle of the △ABC (except for isosceles and equilateral triangles where AO and OD coincide). Furthermore, it can be shown that, if AB is longer than AC, then R will lie within AB, while Q will lie outside of AC, and vice versa (in fact, any diagram drawn with sufficiently accurate instruments will verify the above two facts). Because of this, AB is still AR + RB, but AC is actually AQ − QC; and thus the lengths are not necessarily the same.
There exist several fallacious proofs by induction in which one of the components, basis case or inductive step, is incorrect. Intuitively, proofs by induction work by arguing that if a statement is true in one case, it is true in the next case, and hence by repeatedly applying this, it can be shown to be true for all cases. The following "proof" shows that all horses are the same colour . [ 15 ] [ note 3 ]
The fallacy in this proof arises in line 3. For N = 1, the two groups of horses have N − 1 = 0 horses in common, and thus are not necessarily the same colour as each other, so the group of N + 1 = 2 horses is not necessarily all of the same colour. The implication "every N horses are of the same colour, then N + 1 horses are of the same colour" works for any N > 1, but fails to be true when N = 1. The basis case is correct, but the induction step has a fundamental flaw. | https://en.wikipedia.org/wiki/Mathematical_fallacy |
Mathematical fiction is a genre of creative fictional work in which mathematics and mathematicians play important roles. The form and the medium of the works are not important. The genre may include poems, short stories, novels or plays; comic books; films, videos, or audios. One of the earliest, and much studied, work of this genre is Flatland: A Romance of Many Dimensions , an 1884 satirical novella by the English schoolmaster Edwin Abbott Abbott . Mathematical fiction may have existed since ancient times, but it was recently rediscovered as a genre of literature; since then there has been a growing body of literature in this genre, and the genre has attracted a growing body of readers. [ 1 ] [ 2 ] For example, Abbott's Flatland spawned a sequel in the 21st century: a novel titled Flatterland , authored by Ian Stewart and published in 2001. [ 3 ]
Alex Kasman, a professor of mathematics at the College of Charleston , who maintains a database of works that could possibly be included in this genre, has a broader definition for the genre: Any work "containing mathematics or mathematicians" has been treated as mathematical fiction. Accordingly, Gulliver's Travels by Jonathan Swift , War and Peace by Lev Tolstoy , Mrs. Warren's Profession by George Bernard Shaw , and several similar literary works appear in Kasman's database because these works contain references to mathematics or mathematicians, even though mathematics and mathematicians are not important in their plots. According to this broader approach, the oldest extant work of mathematical fiction is The Birds , a comedy by the ancient Greek playwright Aristophanes performed in 414 BCE. Kasman's database has a list of more than one thousand items of diverse categories like literature, comic books and films. [ 4 ] [ 5 ] [ 6 ] | https://en.wikipedia.org/wiki/Mathematical_fiction |
In common mathematical parlance, a mathematical result is called folklore if it is an unpublished result with no clear originator, but which is well-circulated and believed to be true among the specialists. More specifically, folk mathematics , or mathematical folklore , is the body of theorems, definitions, proofs, facts or techniques that circulate among mathematicians by word of mouth, but have not yet appeared in print, either in books or in scholarly journals. [ 1 ]
Quite important at times for researchers are folk theorems , which are results known, at least to experts in a field, and are considered to have established status, though not published in complete form. [ 1 ] Sometimes, these are only alluded to in the public literature.
An example is a book of exercises, described on the back cover:
This book contains almost 350 exercises in the basics of ring theory . The problems form the "folklore" of ring theory, and the solutions are given in as much detail as possible. [ 2 ]
Another distinct category is well-knowable mathematics, a term introduced by John Conway . [ 3 ] These mathematical matters are known and factual, but not in active circulation in relation with current research (i.e., untrendy). Both of these concepts are attempts to describe the actual context in which research work is done.
Some people, in particular non-mathematicians, use the term folk mathematics to refer to the informal mathematics studied in many ethno-cultural studies of mathematics. [ citation needed ] Although the term "mathematical folklore" can also be used within the mathematics circle to describe the various aspects of their esoteric culture and practices (e.g., slang, proverb, limerick, joke). [ 4 ]
Mathematical folklore can also refer to the unusual (and possibly apocryphal) stories or jokes involving mathematicians or mathematics that are told verbally in mathematics departments. Compilations include tales collected in G. H. Hardy 's A Mathematician's Apology and ( Krantz 2002 ); examples include: | https://en.wikipedia.org/wiki/Mathematical_folklore |
The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics . This mathematical formalism uses mainly a part of functional analysis , especially Hilbert spaces , which are a kind of linear space . Such are distinguished from mathematical formalisms for physics theories developed prior to the early 1900s by the use of abstract mathematical structures, such as infinite-dimensional Hilbert spaces ( L 2 space mainly), and operators on these spaces. In brief, values of physical observables such as energy and momentum were no longer considered as values of functions on phase space , but as eigenvalues ; more precisely as spectral values of linear operators in Hilbert space. [ 1 ]
These formulations of quantum mechanics continue to be used today. At the heart of the description are ideas of quantum state and quantum observables , which are radically different from those used in previous models of physical reality. While the mathematics permits calculation of many quantities that can be measured experimentally, there is a definite theoretical limit to values that can be simultaneously measured. This limitation was first elucidated by Heisenberg through a thought experiment , and is represented mathematically in the new formalism by the non-commutativity of operators representing quantum observables.
Prior to the development of quantum mechanics as a separate theory , the mathematics used in physics consisted mainly of formal mathematical analysis , beginning with calculus , and increasing in complexity up to differential geometry and partial differential equations . Probability theory was used in statistical mechanics . Geometric intuition played a strong role in the first two and, accordingly, theories of relativity were formulated entirely in terms of differential geometric concepts. The phenomenology of quantum physics arose roughly between 1895 and 1915, and for the 10 to 15 years before the development of quantum mechanics (around 1925) physicists continued to think of quantum theory within the confines of what is now called classical physics , and in particular within the same mathematical structures. The most sophisticated example of this is the Sommerfeld–Wilson–Ishiwara quantization rule, which was formulated entirely on the classical phase space .
In the 1890s, Planck was able to derive the blackbody spectrum , which was later used to avoid the classical ultraviolet catastrophe by making the unorthodox assumption that, in the interaction of electromagnetic radiation with matter , energy could only be exchanged in discrete units which he called quanta . Planck postulated a direct proportionality between the frequency of radiation and the quantum of energy at that frequency. The proportionality constant, h , is now called the Planck constant in his honor.
In 1905, Einstein explained certain features of the photoelectric effect by assuming that Planck's energy quanta were actual particles, which were later dubbed photons .
All of these developments were phenomenological and challenged the theoretical physics of the time. Bohr and Sommerfeld went on to modify classical mechanics in an attempt to deduce the Bohr model from first principles. They proposed that, of all closed classical orbits traced by a mechanical system in its phase space, only the ones that enclosed an area which was a multiple of the Planck constant were actually allowed. The most sophisticated version of this formalism was the so-called Sommerfeld–Wilson–Ishiwara quantization . Although the Bohr model of the hydrogen atom could be explained in this way, the spectrum of the helium atom (classically an unsolvable 3-body problem ) could not be predicted. The mathematical status of quantum theory remained uncertain for some time.
In 1923, de Broglie proposed that wave–particle duality applied not only to photons but to electrons and every other physical system.
The situation changed rapidly in the years 1925–1930, when working mathematical foundations were found through the groundbreaking work of Erwin Schrödinger , Werner Heisenberg , Max Born , Pascual Jordan , and the foundational work of John von Neumann , Hermann Weyl and Paul Dirac , and it became possible to unify several different approaches in terms of a fresh set of ideas. The physical interpretation of the theory was also clarified in these years after Werner Heisenberg discovered the uncertainty relations and Niels Bohr introduced the idea of complementarity .
Werner Heisenberg's matrix mechanics was the first successful attempt at replicating the observed quantization of atomic spectra . Later in the same year, Schrödinger created his wave mechanics . Schrödinger's formalism was considered easier to understand, visualize and calculate as it led to differential equations , which physicists were already familiar with solving. Within a year, it was shown that the two theories were equivalent.
Schrödinger himself initially did not understand the fundamental probabilistic nature of quantum mechanics, as he thought that the absolute square of the wave function of an electron should be interpreted as the charge density of an object smeared out over an extended, possibly infinite, volume of space. It was Max Born who introduced the interpretation of the absolute square of the wave function as the probability distribution of the position of a pointlike object . Born's idea was soon taken over by Niels Bohr in Copenhagen who then became the "father" of the Copenhagen interpretation of quantum mechanics. Schrödinger's wave function can be seen to be closely related to the classical Hamilton–Jacobi equation . The correspondence to classical mechanics was even more explicit, although somewhat more formal, in Heisenberg's matrix mechanics. In his PhD thesis project, Paul Dirac [ 2 ] discovered that the equation for the operators in the Heisenberg representation , as it is now called, closely translates to classical equations for the dynamics of certain quantities in the Hamiltonian formalism of classical mechanics, when one expresses them through Poisson brackets , a procedure now known as canonical quantization .
Already before Schrödinger, the young postdoctoral fellow Werner Heisenberg invented his matrix mechanics , which was the first correct quantum mechanics – the essential breakthrough. Heisenberg's matrix mechanics formulation was based on algebras of infinite matrices, a very radical formulation in light of the mathematics of classical physics, although he started from the index-terminology of the experimentalists of that time, not even aware that his "index-schemes" were matrices, as Born soon pointed out to him. In fact, in these early years, linear algebra was not generally popular with physicists in its present form.
Although Schrödinger himself after a year proved the equivalence of his wave-mechanics and Heisenberg's matrix mechanics, the reconciliation of the two approaches and their modern abstraction as motions in Hilbert space is generally attributed to Paul Dirac, who wrote a lucid account in his 1930 classic The Principles of Quantum Mechanics . He is the third, and possibly most important, pillar of that field (he soon was the only one to have discovered a relativistic generalization of the theory). In his above-mentioned account, he introduced the bra–ket notation , together with an abstract formulation in terms of the Hilbert space used in functional analysis; he showed that Schrödinger's and Heisenberg's approaches were two different representations of the same theory, and found a third, most general one, which represented the dynamics of the system. His work was particularly fruitful in many types of generalizations of the field.
The first complete mathematical formulation of this approach, known as the Dirac–von Neumann axioms , is generally credited to John von Neumann 's 1932 book Mathematical Foundations of Quantum Mechanics , although Hermann Weyl had already referred to Hilbert spaces (which he called unitary spaces ) in his 1927 classic paper and book. It was developed in parallel with a new approach to the mathematical spectral theory based on linear operators rather than the quadratic forms that were David Hilbert 's approach a generation earlier. Though theories of quantum mechanics continue to evolve to this day, there is a basic framework for the mathematical formulation of quantum mechanics which underlies most approaches and can be traced back to the mathematical work of John von Neumann. In other words, discussions about interpretation of the theory , and extensions to it, are now mostly conducted on the basis of shared assumptions about the mathematical foundations.
The application of the new quantum theory to electromagnetism resulted in quantum field theory , which was developed starting around 1930. Quantum field theory has driven the development of more sophisticated formulations of quantum mechanics, of which the ones presented here are simple special cases.
A related topic is the relationship to classical mechanics. Any new physical theory is supposed to reduce to successful old theories in some approximation. For quantum mechanics, this translates into the need to study the so-called classical limit of quantum mechanics . Also, as Bohr emphasized, human cognitive abilities and language are inextricably linked to the classical realm, and so classical descriptions are intuitively more accessible than quantum ones. In particular, quantization , namely the construction of a quantum theory whose classical limit is a given and known classical theory, becomes an important area of quantum physics in itself.
Finally, some of the originators of quantum theory (notably Einstein and Schrödinger) were unhappy with what they thought were the philosophical implications of quantum mechanics. In particular, Einstein took the position that quantum mechanics must be incomplete, which motivated research into so-called hidden-variable theories . The issue of hidden variables has become in part an experimental issue with the help of quantum optics .
A physical system is generally described by three basic ingredients: states ; observables ; and dynamics (or law of time evolution ) or, more generally, a group of physical symmetries . A classical description can be given in a fairly direct way by a phase space model of mechanics: states are points in a phase space formulated by symplectic manifold , observables are real-valued functions on it, time evolution is given by a one-parameter group of symplectic transformations of the phase space, and physical symmetries are realized by symplectic transformations. A quantum description normally consists of a Hilbert space of states, observables are self-adjoint operators on the space of states, time evolution is given by a one-parameter group of unitary transformations on the Hilbert space of states, and physical symmetries are realized by unitary transformations . (It is possible, to map this Hilbert-space picture to a phase space formulation , invertibly. See below.)
The following summary of the mathematical framework of quantum mechanics can be partly traced back to the Dirac–von Neumann axioms . [ 3 ]
Each isolated physical system is associated with a (topologically) separable complex Hilbert space H with inner product ⟨ φ | ψ ⟩ .
The state of an isolated physical system is represented, at a fixed time t {\displaystyle t} , by a state vector | ψ ⟩ {\displaystyle |\psi \rangle } belonging to a Hilbert space H {\displaystyle {\mathcal {H}}} called the state space .
Separability is a mathematically convenient hypothesis, with the physical interpretation that the state is uniquely determined by countably many observations. Quantum states can be identified with equivalence classes in H , where two vectors (of length 1) represent the same state if they differ only by a phase factor : [ 4 ] [ 5 ] | ψ k ⟩ ∼ | ψ l ⟩ ⇔ | ψ k ⟩ = e i α | ψ l ⟩ , α ∈ R . {\displaystyle |\psi _{k}\rangle \sim |\psi _{l}\rangle \;\;\Leftrightarrow \;\;|\psi _{k}\rangle =e^{i\alpha }|\psi _{l}\rangle ,\quad \ \alpha \in \mathbb {R} .} As such, a quantum state forms a ray in projective Hilbert space , not a vector . [ 6 ]
Accompanying Postulate I is the composite system postulate: [ 7 ]
The Hilbert space of a composite system is the Hilbert space tensor product of the state spaces associated with the component systems. For a non-relativistic system consisting of a finite number of distinguishable particles, the component systems are the individual particles.
In the presence of quantum entanglement , the quantum state of the composite system cannot be factored as a tensor product of states of its local constituents; Instead, it is expressed as a sum, or superposition , of tensor products of states of component subsystems. A subsystem in an entangled composite system generally cannot be described by a state vector (or a ray), but instead is described by a density operator ; Such quantum state is known as a mixed state . The density operator of a mixed state is a trace class , nonnegative ( positive semi-definite ) self-adjoint operator ρ normalized to be of trace 1. In turn, any density operator of a mixed state can be represented as a subsystem of a larger composite system in a pure state (see purification theorem ).
In the absence of quantum entanglement, the quantum state of the composite system is called a separable state . The density matrix of a bipartite system in a separable state can be expressed as ρ = ∑ k p k ρ 1 k ⊗ ρ 2 k {\displaystyle \rho =\sum _{k}p_{k}\rho _{1}^{k}\otimes \rho _{2}^{k}} , where ∑ k p k = 1 {\displaystyle \;\sum _{k}p_{k}=1} . If there is only a single non-zero p k {\displaystyle p_{k}} , then the state can be expressed just as ρ = ρ 1 ⊗ ρ 2 , {\textstyle \rho =\rho _{1}\otimes \rho _{2},} and is called simply separable or product state.
Physical observables are represented by Hermitian matrices on H . Since these operators are Hermitian, their eigenvalues are always real, and represent the possible outcomes/results from measuring the corresponding observable. If the spectrum of the observable is discrete , then the possible results are quantized .
Every measurable physical quantity A {\displaystyle {\mathcal {A}}} is described by a Hermitian operator A {\displaystyle A} acting in the state space H {\displaystyle {\mathcal {H}}} . This operator is an observable, meaning that its eigenvectors form a basis for H {\displaystyle {\mathcal {H}}} . The result of measuring a physical quantity A {\displaystyle {\mathcal {A}}} must be one of the eigenvalues of the corresponding observable A {\displaystyle A} .
By spectral theory, we can associate a probability measure to the values of A in any state ψ . We can also show that the possible values of the observable A in any state must belong to the spectrum of A . The expectation value (in the sense of probability theory) of the observable A for the system in state represented by the unit vector ψ ∈ H is ⟨ ψ | A | ψ ⟩ {\displaystyle \langle \psi |A|\psi \rangle } . If we represent the state ψ in the basis formed by the eigenvectors of A , then the square of the modulus of the component attached to a given eigenvector is the probability of observing its corresponding eigenvalue.
When the physical quantity A {\displaystyle {\mathcal {A}}} is measured on a system in a normalized state | ψ ⟩ {\displaystyle |\psi \rangle } , the probability of obtaining an eigenvalue (denoted a n {\displaystyle a_{n}} for discrete spectra and α {\displaystyle \alpha } for continuous spectra) of the corresponding observable A {\displaystyle A} is given by the amplitude squared of the appropriate wave function (projection onto corresponding eigenvector).
P ( a n ) = | ⟨ a n | ψ ⟩ | 2 (Discrete, nondegenerate spectrum) P ( a n ) = ∑ i g n | ⟨ a n i | ψ ⟩ | 2 (Discrete, degenerate spectrum) d P ( α ) = | ⟨ α | ψ ⟩ | 2 d α (Continuous, nondegenerate spectrum) {\displaystyle {\begin{alignedat}{3}\mathbb {P} (a_{n})&=|\langle a_{n}|\psi \rangle |^{2}&&\,\,{\text{(Discrete, nondegenerate spectrum)}}\\\mathbb {P} (a_{n})&=\sum _{i}^{g_{n}}|\langle a_{n}^{i}|\psi \rangle |^{2}&&\,\,{\text{(Discrete, degenerate spectrum)}}\\d\mathbb {P} (\alpha )&=|\langle \alpha |\psi \rangle |^{2}d\alpha &&\,\,{\text{(Continuous, nondegenerate spectrum)}}\end{alignedat}}}
For a mixed state ρ , the expected value of A in the state ρ is tr ( A ρ ) {\displaystyle \operatorname {tr} (A\rho )} , and the probability of obtaining an eigenvalue a n {\displaystyle a_{n}} in a discrete, nondegenerate spectrum of the corresponding observable A {\displaystyle A} is given by P ( a n ) = tr ( | a n ⟩ ⟨ a n | ρ ) = ⟨ a n | ρ | a n ⟩ {\displaystyle \mathbb {P} (a_{n})=\operatorname {tr} (|a_{n}\rangle \langle a_{n}|\rho )=\langle a_{n}|\rho |a_{n}\rangle } .
If the eigenvalue a n {\displaystyle a_{n}} has degenerate , orthonormal eigenvectors { | a n 1 ⟩ , | a n 2 ⟩ , … , | a n m ⟩ } {\displaystyle \{|a_{n1}\rangle ,|a_{n2}\rangle ,\dots ,|a_{nm}\rangle \}} , then the projection operator onto the eigensubspace can be defined as the identity operator in the eigensubspace: P n = | a n 1 ⟩ ⟨ a n 1 | + | a n 2 ⟩ ⟨ a n 2 | + ⋯ + | a n m ⟩ ⟨ a n m | , {\displaystyle P_{n}=|a_{n1}\rangle \langle a_{n1}|+|a_{n2}\rangle \langle a_{n2}|+\dots +|a_{nm}\rangle \langle a_{nm}|,} and then P ( a n ) = tr ( P n ρ ) {\displaystyle \mathbb {P} (a_{n})=\operatorname {tr} (P_{n}\rho )} .
Postulates II.a and II.b are collectively known as the Born rule of quantum mechanics.
When a measurement is performed, only one result is obtained (according to some interpretations of quantum mechanics ). This is modeled mathematically as the processing of additional information from the measurement, confining the probabilities of an immediate second measurement of the same observable. In the case of a discrete, non-degenerate spectrum, two sequential measurements of the same observable will always give the same value assuming the second immediately follows the first. Therefore, the state vector must change as a result of measurement, and collapse onto the eigensubspace associated with the eigenvalue measured.
If the measurement of the physical quantity A {\displaystyle {\mathcal {A}}} on the system in the state | ψ ⟩ {\displaystyle |\psi \rangle } gives the result a n {\displaystyle a_{n}} , then the state of the system immediately after the measurement is the normalized projection of | ψ ⟩ {\displaystyle |\psi \rangle } onto the eigensubspace associated with a n {\displaystyle a_{n}} | ψ ⟩ ⟹ a n P n | ψ ⟩ ⟨ ψ | P n | ψ ⟩ {\displaystyle |\psi \rangle \quad {\overset {a_{n}}{\Longrightarrow }}\quad {\frac {P_{n}|\psi \rangle }{\sqrt {\langle \psi |P_{n}|\psi \rangle }}}}
For a mixed state ρ , after obtaining an eigenvalue a n {\displaystyle a_{n}} in a discrete, nondegenerate spectrum of the corresponding observable A {\displaystyle A} , the updated state is given by ρ ′ = P n ρ P n † tr ( P n ρ P n † ) {\textstyle \rho '={\frac {P_{n}\rho P_{n}^{\dagger }}{\operatorname {tr} (P_{n}\rho P_{n}^{\dagger })}}} . If the eigenvalue a n {\displaystyle a_{n}} has degenerate, orthonormal eigenvectors { | a n 1 ⟩ , | a n 2 ⟩ , … , | a n m ⟩ } {\displaystyle \{|a_{n1}\rangle ,|a_{n2}\rangle ,\dots ,|a_{nm}\rangle \}} , then the projection operator onto the eigensubspace is P n = | a n 1 ⟩ ⟨ a n 1 | + | a n 2 ⟩ ⟨ a n 2 | + ⋯ + | a n m ⟩ ⟨ a n m | {\displaystyle P_{n}=|a_{n1}\rangle \langle a_{n1}|+|a_{n2}\rangle \langle a_{n2}|+\dots +|a_{nm}\rangle \langle a_{nm}|} .
Postulates II.c is sometimes called the "state update rule" or "collapse rule"; Together with the Born rule (Postulates II.a and II.b), they form a complete representation of measurements , and are sometimes collectively called the measurement postulate(s).
Note that the projection-valued measures (PVM) described in the measurement postulate(s) can be generalized to positive operator-valued measures (POVM), which is the most general kind of measurement in quantum mechanics. A POVM can be understood as the effect on a component subsystem when a PVM is performed on a larger, composite system (see Naimark's dilation theorem ).
Though it is possible to derive the Schrödinger equation, which describes how a state vector evolves in time, most texts assert the equation as a postulate. Common derivations include using the de Broglie hypothesis or path integrals .
The time evolution of the state vector | ψ ( t ) ⟩ {\displaystyle |\psi (t)\rangle } is governed by the Schrödinger equation, where H ( t ) {\displaystyle H(t)} is the observable associated with the total energy of the system (called the Hamiltonian ) i ℏ d d t | ψ ( t ) ⟩ = H ( t ) | ψ ( t ) ⟩ {\displaystyle i\hbar {\frac {d}{dt}}|\psi (t)\rangle =H(t)|\psi (t)\rangle }
Equivalently, the time evolution postulate can be stated as:
The time evolution of a closed system is described by a unitary transformation on the initial state. | ψ ( t ) ⟩ = U ( t ; t 0 ) | ψ ( t 0 ) ⟩ {\displaystyle |\psi (t)\rangle =U(t;t_{0})|\psi (t_{0})\rangle }
For a closed system in a mixed state ρ , the time evolution is ρ ( t ) = U ( t ; t 0 ) ρ ( t 0 ) U † ( t ; t 0 ) {\displaystyle \rho (t)=U(t;t_{0})\rho (t_{0})U^{\dagger }(t;t_{0})} .
The evolution of an open quantum system can be described by quantum operations (in an operator sum formalism) and quantum instruments , and generally does not have to be unitary.
Furthermore, to the postulates of quantum mechanics one should also add basic statements on the properties of spin and Pauli's exclusion principle , see below.
In addition to their other properties, all particles possess a quantity called spin , an intrinsic angular momentum . Despite the name, particles do not literally spin around an axis, and quantum mechanical spin has no correspondence in classical physics. In the position representation, a spinless wavefunction has position r and time t as continuous variables, ψ = ψ ( r , t ) . For spin wavefunctions the spin is an additional discrete variable: ψ = ψ ( r , t , σ ) , where σ takes the values; σ = − S ℏ , − ( S − 1 ) ℏ , … , 0 , … , + ( S − 1 ) ℏ , + S ℏ . {\displaystyle \sigma =-S\hbar ,-(S-1)\hbar ,\dots ,0,\dots ,+(S-1)\hbar ,+S\hbar \,.}
That is, the state of a single particle with spin S is represented by a (2 S + 1) -component spinor of complex-valued wave functions.
Two classes of particles with very different behaviour are bosons which have integer spin ( S = 0, 1, 2, ... ), and fermions possessing half-integer spin ( S = 1 ⁄ 2 , 3 ⁄ 2 , 5 ⁄ 2 , ... ).
In quantum mechanics, two particles can be distinguished from one another using two methods. By performing a measurement of intrinsic properties of each particle, particles of different types can be distinguished. Otherwise, if the particles are identical, their trajectories can be tracked which distinguishes the particles based on the locality of each particle. While the second method is permitted in classical mechanics, (i.e. all classical particles are treated with distinguishability), the same cannot be said for quantum mechanical particles since the process is infeasible due to the fundamental uncertainty principles that govern small scales. Hence the requirement of indistinguishability of quantum particles is presented by the symmetrization postulate. The postulate is applicable to a system of bosons or fermions, for example, in predicting the spectra of helium atom . The postulate, explained in the following sections, can be stated as follows:
The wavefunction of a system of N identical particles (in 3D) is either totally symmetric (Bosons) or totally antisymmetric (Fermions) under interchange of any pair of particles.
Exceptions can occur when the particles are constrained to two spatial dimensions where existence of particles known as anyons are possible which are said to have a continuum of statistical properties spanning the range between fermions and bosons. [ 10 ] The connection between behaviour of identical particles and their spin is given by spin statistics theorem .
It can be shown that two particles localized in different regions of space can still be represented using a symmetrized/antisymmetrized wavefunction and that independent treatment of these wavefunctions gives the same result. [ 11 ] Hence the symmetrization postulate is applicable in the general case of a system of identical particles.
In a system of identical particles, let P be known as exchange operator that acts on the wavefunction as:
If a physical system of identical particles is given, wavefunction of all particles can be well known from observation but these cannot be labelled to each particle. Thus, the above exchanged wavefunction represents the same physical state as the original state which implies that the wavefunction is not unique. This is known as exchange degeneracy. [ 12 ]
More generally, consider a linear combination of such states, | Ψ ⟩ {\displaystyle |\Psi \rangle } . For the best representation of the physical system, we expect this to be an eigenvector of P since exchange operator is not excepted to give completely different vectors in projective Hilbert space. Since P 2 = 1 {\displaystyle P^{2}=1} , the possible eigenvalues of P are +1 and −1. The | Ψ ⟩ {\displaystyle |\Psi \rangle } states for identical particle system are represented as symmetric for +1 eigenvalue or antisymmetric for -1 eigenvalue as follows:
The explicit symmetric/antisymmetric form of | Ψ ⟩ {\displaystyle |\Psi \rangle } is constructed using a symmetrizer or antisymmetrizer operator. Particles that form symmetric states are called bosons and those that form antisymmetric states are called as fermions. The relation of spin with this classification is given from spin statistics theorem which shows that integer spin particles are bosons and half integer spin particles are fermions.
The property of spin relates to another basic property concerning systems of N identical particles: the Pauli exclusion principle , which is a consequence of the following permutation behaviour of an N -particle wave function; again in the position representation one must postulate that for the transposition of any two of the N particles one always should have
ψ ( … , r i , σ i , … , r j , σ j , … ) = ( − 1 ) 2 S ⋅ ψ ( … , r j , σ j , … , r i , σ i , … ) {\displaystyle \psi (\dots ,\,\mathbf {r} _{i},\sigma _{i},\,\dots ,\,\mathbf {r} _{j},\sigma _{j},\,\dots )=(-1)^{2S}\cdot \psi (\dots ,\,\mathbf {r} _{j},\sigma _{j},\,\dots ,\mathbf {r} _{i},\sigma _{i},\,\dots )}
i.e., on transposition of the arguments of any two particles the wavefunction should reproduce , apart from a prefactor (−1) 2 S which is +1 for bosons, but ( −1 ) for fermions .
Electrons are fermions with S = 1/2 ; quanta of light are bosons with S = 1 .
Due to the form of anti-symmetrized wavefunction:
if the wavefunction of each particle is completely determined by a set of quantum number, then two fermions cannot share the same set of quantum numbers since the resulting function cannot be anti-symmetrized (i.e. above formula gives zero). The same cannot be said of Bosons since their wavefunction is:
where n j {\displaystyle n_{j}} is the number of particles with same wavefunction.
In nonrelativistic quantum mechanics all particles are either bosons or fermions ; in relativistic quantum theories also "supersymmetric" theories exist, where a particle is a linear combination of a bosonic and a fermionic part. Only in dimension d = 2 can one construct entities where (−1) 2 S is replaced by an arbitrary complex number with magnitude 1, called anyons . In relativistic quantum mechanics, spin statistic theorem can prove that under certain set of assumptions that the integer spins particles are classified as bosons and half spin particles are classified as fermions . Anyons which form neither symmetric nor antisymmetric states are said to have fractional spin.
Although spin and the Pauli principle can only be derived from relativistic generalizations of quantum mechanics, the properties mentioned in the last two paragraphs belong to the basic postulates already in the non-relativistic limit. Especially, many important properties in natural science, e.g. the periodic system of chemistry, are consequences of the two properties.
The time evolution of the state is given by a differentiable function from the real numbers R , representing instants of time, to the Hilbert space of system states. This map is characterized by a differential equation as follows:
If | ψ ( t )⟩ denotes the state of the system at any one time t , the following Schrödinger equation holds:
i ℏ d d t | ψ ( t ) ⟩ = H | ψ ( t ) ⟩ {\displaystyle i\hbar {\frac {d}{dt}}\left|\psi (t)\right\rangle =H\left|\psi (t)\right\rangle }
where H is a densely defined self-adjoint operator, called the system Hamiltonian , i is the imaginary unit and ħ is the reduced Planck constant . As an observable, H corresponds to the total energy of the system.
Alternatively, by Stone's theorem one can state that there is a strongly continuous one-parameter unitary map U ( t ) : H → H such that | ψ ( t + s ) ⟩ = U ( t ) | ψ ( s ) ⟩ {\displaystyle \left|\psi (t+s)\right\rangle =U(t)\left|\psi (s)\right\rangle } for all times s , t . The existence of a self-adjoint Hamiltonian H such that U ( t ) = e − ( i / ℏ ) t H {\displaystyle U(t)=e^{-(i/\hbar )tH}} is a consequence of Stone's theorem on one-parameter unitary groups. It is assumed that H does not depend on time and that the perturbation starts at t 0 = 0 ; otherwise one must use the Dyson series , formally written as U ( t ) = T [ exp ( − i ℏ ∫ t 0 t d t ′ H ( t ′ ) ) ] , {\displaystyle U(t)={\mathcal {T}}\left[\exp \left(-{\frac {i}{\hbar }}\int _{t_{0}}^{t}dt'\,H(t')\right)\right],} where T {\displaystyle {\mathcal {T}}} is Dyson's time-ordering symbol.
(This symbol permutes a product of noncommuting operators of the form B 1 ( t 1 ) ⋅ B 2 ( t 2 ) ⋅ ⋯ ⋅ B n ( t n ) {\displaystyle B_{1}(t_{1})\cdot B_{2}(t_{2})\cdot \dots \cdot B_{n}(t_{n})} into the uniquely determined re-ordered expression B i 1 ( t i 1 ) ⋅ B i 2 ( t i 2 ) ⋅ ⋯ ⋅ B i n ( t i n ) {\displaystyle B_{i_{1}}(t_{i_{1}})\cdot B_{i_{2}}(t_{i_{2}})\cdot \dots \cdot B_{i_{n}}(t_{i_{n}})} with t i 1 ≥ t i 2 ≥ ⋯ ≥ t i n . {\displaystyle t_{i_{1}}\geq t_{i_{2}}\geq \dots \geq t_{i_{n}}\,.}
d d t A ( t ) = i ℏ [ H , A ( t ) ] + ∂ A ( t ) ∂ t , {\displaystyle {\frac {d}{dt}}A(t)={\frac {i}{\hbar }}[H,A(t)]+{\frac {\partial A(t)}{\partial t}},}
i ℏ d d t | ψ ( t ) ⟩ = H i n t ( t ) | ψ ( t ) ⟩ {\displaystyle i\hbar {\frac {d}{dt}}\left|\psi (t)\right\rangle ={H}_{\rm {int}}(t)\left|\psi (t)\right\rangle }
i ℏ d d t A ( t ) = [ A ( t ) , H 0 ] . {\displaystyle i\hbar {\frac {d}{dt}}A(t)=[A(t),H_{0}].}
The interaction picture does not always exist, though. In interacting quantum field theories, Haag's theorem states that the interaction picture does not exist. This is because the Hamiltonian cannot be split into a free and an interacting part within a superselection sector . Moreover, even if in the Schrödinger picture the Hamiltonian does not depend on time, e.g. H = H 0 + V , in the interaction picture it does, at least, if V does not commute with H 0 , since H i n t ( t ) ≡ e ( i / ℏ ) t H 0 V e ( − i / ℏ ) t H 0 . {\displaystyle H_{\rm {int}}(t)\equiv e^{{(i/\hbar })tH_{0}}\,V\,e^{{(-i/\hbar })tH_{0}}.}
So the above-mentioned Dyson-series has to be used anyhow.
The Heisenberg picture is the closest to classical Hamiltonian mechanics (for example, the commutators appearing in the above equations directly translate into the classical Poisson brackets ); but this is already rather "high-browed", and the Schrödinger picture is considered easiest to visualize and understand by most people, to judge from pedagogical accounts of quantum mechanics. The Dirac picture is the one used in perturbation theory , and is specially associated to quantum field theory and many-body physics .
Summary :
The original form of the Schrödinger equation depends on choosing a particular representation of Heisenberg's canonical commutation relations . The Stone–von Neumann theorem dictates that all irreducible representations of the finite-dimensional Heisenberg commutation relations are unitarily equivalent. A systematic understanding of its consequences has led to the phase space formulation of quantum mechanics, which works in full phase space instead of Hilbert space , so then with a more intuitive link to the classical limit thereof. This picture also simplifies considerations
of quantization , the deformation extension from classical to quantum mechanics.
The quantum harmonic oscillator is an exactly solvable system where the different representations are easily compared. There, apart from the Heisenberg, or Schrödinger (position or momentum), or phase-space representations, one also encounters the Fock (number) representation and the Segal–Bargmann (Fock-space or coherent state) representation (named after Irving Segal and Valentine Bargmann ). All four are unitarily equivalent.
The framework presented so far singles out time as the parameter that everything depends on. It is possible to formulate mechanics in such a way that time becomes itself an observable associated with a self-adjoint operator. At the classical level, it is possible to arbitrarily parameterize the trajectories of particles in terms of an unphysical parameter s , and in that case the time t becomes an additional generalized coordinate of the physical system. At the quantum level, translations in s would be generated by a "Hamiltonian" H − E , where E is the energy operator and H is the "ordinary" Hamiltonian. However, since s is an unphysical parameter, physical states must be left invariant by " s -evolution", and so the physical state space is the kernel of H − E (this requires the use of a rigged Hilbert space and a renormalization of the norm).
This is related to the quantization of constrained systems and quantization of gauge theories . It
is also possible to formulate a quantum theory of "events" where time becomes an observable. [ 13 ]
The picture given in the preceding paragraphs is sufficient for description of a completely isolated system. However, it fails to account for one of the main differences between quantum mechanics and classical mechanics, that is, the effects of measurement . [ 14 ] The von Neumann description of quantum measurement of an observable A , when the system is prepared in a pure state ψ is the following (note, however, that von Neumann's description dates back to the 1930s and is based on experiments as performed during that time – more specifically the Compton–Simon experiment ; it is not applicable to most present-day measurements within the quantum domain):
For example, suppose the state space is the n -dimensional complex Hilbert space C n and A is a Hermitian matrix with eigenvalues λ i , with corresponding eigenvectors ψ i . The projection-valued measure associated with A , E A , is then E A ( B ) = | ψ i ⟩ ⟨ ψ i | , {\displaystyle \operatorname {E} _{A}(B)=|\psi _{i}\rangle \langle \psi _{i}|,} where B is a Borel set containing only the single eigenvalue λ i . If the system is prepared in state | ψ ⟩ {\displaystyle |\psi \rangle } Then the probability of a measurement returning the value λ i can be calculated by integrating the spectral measure ⟨ ψ ∣ E A ψ ⟩ {\displaystyle \langle \psi \mid \operatorname {E} _{A}\psi \rangle } over B i . This gives trivially ⟨ ψ | ψ i ⟩ ⟨ ψ i ∣ ψ ⟩ = | ⟨ ψ ∣ ψ i ⟩ | 2 . {\displaystyle \langle \psi |\psi _{i}\rangle \langle \psi _{i}\mid \psi \rangle =|\langle \psi \mid \psi _{i}\rangle |^{2}.}
The characteristic property of the von Neumann measurement scheme is that repeating the same measurement will give the same results. This is also called the projection postulate .
A more general formulation replaces the projection-valued measure with a positive-operator valued measure (POVM). To illustrate, take again the finite-dimensional case. Here we would replace the rank-1 projections | ψ i ⟩ ⟨ ψ i | {\displaystyle |\psi _{i}\rangle \langle \psi _{i}|} by a finite set of positive operators F i F i ∗ {\displaystyle F_{i}F_{i}^{*}} whose sum is still the identity operator as before (the resolution of identity). Just as a set of possible outcomes { λ 1 ... λ n } is associated to a projection-valued measure, the same can be said for a POVM. Suppose the measurement outcome is λ i . Instead of collapsing to the (unnormalized) state | ψ i ⟩ ⟨ ψ i | ψ ⟩ {\displaystyle |\psi _{i}\rangle \langle \psi _{i}|\psi \rangle } after the measurement, the system now will be in the state F i | ψ ⟩ . {\displaystyle F_{i}|\psi \rangle .}
Since the F i F i * operators need not be mutually orthogonal projections, the projection postulate of von Neumann no longer holds.
The same formulation applies to general mixed states .
In von Neumann's approach, the state transformation due to measurement is distinct from that due to time evolution in several ways. For example, time evolution is deterministic and unitary whereas measurement is non-deterministic and non-unitary. However, since both types of state transformation take one quantum state to another, this difference was viewed by many as unsatisfactory. The POVM formalism views measurement as one among many other quantum operations , which are described by completely positive maps which do not increase the trace.
In any case it seems that the above-mentioned problems can only be resolved if the time evolution included not only the quantum system, but also, and essentially, the classical measurement apparatus (see above).
Part of the folklore of the subject concerns the mathematical physics textbook Methods of Mathematical Physics put together by Richard Courant from David Hilbert 's Göttingen University courses. The story is told (by mathematicians) that physicists had dismissed the material as not interesting in the current research areas, until the advent of Schrödinger's equation. At that point it was realised that the mathematics of the new quantum mechanics was already laid out in it. It is also said that Heisenberg had consulted Hilbert about his matrix mechanics, and Hilbert observed that his own experience with infinite-dimensional matrices had derived from differential equations, advice which Heisenberg ignored, missing the opportunity to unify the theory as Weyl and Dirac did a few years later. Whatever the basis of the anecdotes, the mathematics of the theory was conventional at the time, whereas the physics was radically new.
The main tools include: | https://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics |
A mathematical instrument is a tool or device used in the study or practice of mathematics . In geometry , construction of various proofs was done using only a compass and straightedge ; arguments in these proofs relied only on idealized properties of these instruments and literal construction was regarded as only an approximation. In applied mathematics , mathematical instruments were used for measuring angles and distances, in astronomy , navigation , surveying and in the measurement of time. [ 1 ]
Instruments such as the astrolabe , the quadrant , and others were used to measure and accurately record the relative positions and movements of planets and other celestial objects. The sextant and other related instruments were essential for navigation at sea.
Most instruments are used within the field of geometry , including the ruler , dividers , protractor , set square , compass, ellipsograph , T-square and opisometer . Others are used in arithmetic (for example the abacus , slide rule and calculator ) or in algebra (the integraph ). In astronomy, many [ by whom? ] have said the pyramids (along with Stonehenge) were actually instruments used for tracking the stars over long periods or for the annual planting seasons.
The Oxford Set of Mathematical Instruments is a set of instruments used by generations of school children in the United Kingdom and around the world in mathematics and geometry lessons. It includes two set squares, a 180° protractor, a 15 cm ruler, a metal compass, a metal divider, a 9 cm pencil, a pencil sharpener, an eraser and a 10mm stencil.
This mathematics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Mathematical_instrument |
Mathematical linguistics is the application of mathematics to model phenomena and solve problems in general linguistics and theoretical linguistics . Mathematical linguistics has a significant amount of overlap with computational linguistics .
Discrete mathematics is used in language modeling, including formal grammars, language representation, and historical linguistic trends.
Semantic classes , word classes , natural classes , and the allophonic variations of each phoneme in a language are all examples of applied set theory . Set theory and concatenation theory are used extensively in phonetics and phonology.
In phonotactics , combinatorics is useful for determining which sequences of phonemes are permissible in a given language, and for calculating the total number of possible syllables or words, based on a given set of phonological constraints. Combinatorics on words can reveal patterns within words, morphemes, and sentences.
Context-sensitive rewriting rules of the form a → b / c _ d , used in linguistics to model phonological rules and sound change , are computationally equivalent to finite-state transducers , provided that application is nonrecursive, i.e. the rule is not allowed to rewrite the same substring twice. [ 1 ]
Weighted FSTs found applications in natural language processing , including machine translation , and in machine learning . [ 2 ] [ 3 ] An implementation for part-of-speech tagging can be found as one component of the OpenGrm [ 4 ] library.
Optimality theory (OT) and maximum entropy (Maxent) phonotactics use algorithmic approaches when evaluating candidate forms (phoneme strings) for determining the phonotactic constraints of a language. [ 5 ]
Trees have several applications in linguistics, including:
Other graphs that are used in linguistics include:
Formal linguistics is the branch of linguistics which uses formal languages , formal grammars and first-order logical expressions for the analysis of natural languages . Since the 1980s, the term is often used to refer to Chomskyan linguistics . [ 6 ] Generative models of formal linguistics, such as head-driven phrase structure grammar , have also been used in natural language processing.
Logic is used to model syntax , formal semantics , and pragmatics . Modal logic can model syntax that employs different grammatical moods . [ 7 ] Most linguistic universals (e.g. Greenberg's linguistic universals ) employ propositional logic . Lexical relations between words can be determined based on whether a pair of words satisfies conditional propositions. [ 8 ]
Methods of formal linguistics were introduced by semioticians such as Charles Sanders Peirce and Louis Hjelmslev . Building on the work of David Hilbert and Rudolf Carnap , Hjelmslev proposed the use of formal grammars to analyse, generate and explain language in his 1943 book Prolegomena to a Theory of Language . [ 9 ] [ 10 ] In this view, language is regarded as arising from a mathematical relationship between meaning and form.
The formal description of language was further developed by linguists including J. R. Firth and Simon Dik , giving rise to modern grammatical frameworks such as systemic functional linguistics and functional discourse grammar . Computational methods have been developed by the framework functional generative description among others.
Dependency grammar , created by French structuralist Lucien Tesnière , [ 11 ] has been used widely in natural language processing .
The Fast Fourier Transform , Kalman filters , and autoencoding are all used in signal processing (advanced phonetics, speech recognition).
In linguistics, statistical methods are necessary to describe and validate research results, as well as to understand observations and trends within an area of study.
Student's t -test can be used to determine whether the occurrence of a collocation in a corpus is statistically significant. [ 12 ] For a bigram w 1 w 2 {\displaystyle w_{1}w_{2}} , let P ( w 1 ) = # w 1 N {\displaystyle P(w_{1})={\frac {\#w_{1}}{N}}} be the unconditional probability of occurrence of w 1 {\displaystyle w_{1}} in a corpus with size N {\displaystyle N} , and let P ( w 2 ) = # w 2 N {\displaystyle P(w_{2})={\frac {\#w_{2}}{N}}} be the unconditional probability of occurrence of w 2 {\displaystyle w_{2}} in the corpus. The t-score for the bigram w 1 w 2 {\displaystyle w_{1}w_{2}} is calculated as:
where x ¯ = # w i w j N {\displaystyle {\bar {x}}={\frac {\#w_{i}w_{j}}{N}}} is the sample mean of the occurrence of w 1 w 2 {\displaystyle w_{1}w_{2}} , # w 1 w 2 {\displaystyle \#w_{1}w_{2}} is the number of occurrences of w 1 w 2 {\displaystyle w_{1}w_{2}} , μ = P ( w i ) P ( w j ) {\displaystyle \mu =P(w_{i})P(w_{j})} is the probability of w 1 w 2 {\displaystyle w_{1}w_{2}} under the null-hypothesis that w 1 {\displaystyle w_{1}} and w 2 {\displaystyle w_{2}} appear independently in the text, and s 2 = x ¯ ( 1 − x ¯ ) ≈ x ¯ {\displaystyle s^{2}={\bar {x}}(1-{\bar {x}})\approx {\bar {x}}} is the sample variance. With a large N {\displaystyle N} , the t -test is equivalent to a Z -test .
Lexicostatistics can model the lexical similarities between languages that share a language family, sprachbund , language contact , or other historical connections.
Quantitative linguistics (QL) deals with language learning, language change, and application as well as structure of natural languages. QL investigates languages using statistical methods; its most demanding objective is the formulation of language laws and, ultimately, of a general theory of language in the sense of a set of interrelated languages laws. [ 13 ] Synergetic linguistics was from its very beginning specifically designed for this purpose. [ 14 ] QL is empirically based on the results of language statistics, a field which can be interpreted as statistics of languages or as statistics of any linguistic object. This field is not necessarily connected to substantial theoretical ambitions. Corpus linguistics and computational linguistics are other fields which contribute important empirical evidence .
Quantitative comparative linguistics is a subfield of quantitative linguistics which applies quantitative analysis to comparative linguistics . It makes use of lexicostatistics and glottochronology , and the borrowing of phylogenetics from biology. | https://en.wikipedia.org/wiki/Mathematical_linguistics |
A mathematical markup language is a computer notation for representing mathematical formulae , based on mathematical notation . Specialized markup languages are necessary because computers normally deal with linear text and more limited character sets (although increasing support for Unicode is obsoleting very simple uses). A formally standardized syntax also allows a computer to interpret otherwise ambiguous content, for rendering or even evaluating. For computer-interpretable syntaxes, the most popular are TeX / LaTeX , MathML (Mathematical Markup Language), OpenMath and OMDoc .
Popular languages for input by humans and interpretation by computers include TeX [ 1 ] / LaTeX [ 2 ] and eqn . [ 3 ]
Computer algebra systems such as Macsyma , Mathematica ( Wolfram Language ), Maple , and MATLAB each have their own syntax.
When the purpose is informal communication with other humans, syntax is often ad hoc, sometimes called "ASCII math notation". Academics sometimes use syntax based on TeX due to familiarity with it from writing papers. Those used to programming languages may also use shorthands like "!" for ¬ {\displaystyle \neg } . Web pages may also use a limited amount of HTML to mark up a small subset, for example superscripting . [ 4 ] Ad hoc syntax requires context to interpret ambiguous syntax, for example "<=" could be "is implied by" or "less than or equal to", and "dy/dx" is likely to denote a derivative , but strictly speaking could also mean a finite quantity dy divided by dx .
Unicode improves the support for mathematics, compared to ASCII only. [ 5 ] [ 6 ]
Markup languages optimized for computer-to-computer communication include MathML , [ 7 ] OpenMath , and OMDoc . These are designed for clarity, parseability and to minimize ambiguity, at the price of verbosity. However, the verbosity makes them clumsier for humans to type directly. [ 7 ]
Many input, rendering, and conversion tools exist.
Microsoft Word included Equation Editor , a limited version of MathType , until 2007. These allow entering formulae using a graphical user interface , and converting to standard markup languages such as MathML. With Microsoft's release of Microsoft Office 2007 and the Office Open XML file formats , they introduced a new equation editor which uses a new format, "Office Math Markup Language" (OMML). The lack of compatibility led some prestigious scientific journals to refuse to accept manuscripts which had been produced using Microsoft Office 2007 . [ 8 ] [ 9 ]
SciWriter is another GUI that can generate MathML and LaTeX. [ 10 ]
ASCIIMathML , a JavaScript program, can convert ad hoc ASCII notation to MathML. [ 11 ] | https://en.wikipedia.org/wiki/Mathematical_markup_language |
Mathematical methods are integral to the study of electronics.
Mathematical Methods in Electronics Engineering involves applying mathematical principles to analyze, design, and optimize electronic circuits and systems. Key areas include: [ 1 ] [ 2 ]
These methods are integral to systematically analyzing and improving the performance and functionality of electronic devices and systems.
A number of fundamental electrical laws and theorems apply to all electrical networks. These include: [ 3 ]
In addition to the foundational principles and theorems, several analytical methods are integral to the study of electronics: [ 4 ] [ 5 ]
These methods build on the foundational laws and theorems provide insights and tools for the analysis and design of complex electronic systems. | https://en.wikipedia.org/wiki/Mathematical_methods_in_electronics |
Mathematical models can project how infectious diseases progress to show the likely outcome of an epidemic (including in plants ) and help inform public health and plant health interventions. Models use basic assumptions or collected statistics along with mathematics to find parameters for various infectious diseases and use those parameters to calculate the effects of different interventions, like mass vaccination programs. The modelling can help decide which intervention(s) to avoid and which to trial, or can predict future growth patterns, etc.
The modelling of infectious diseases is a tool that has been used to study the mechanisms by which diseases spread, to predict the future course of an outbreak and to evaluate strategies to control an epidemic. [ 1 ]
The first scientist who systematically tried to quantify causes of death was John Graunt in his book Natural and Political Observations made upon the Bills of Mortality , in 1662. The bills he studied were listings of numbers and causes of deaths published weekly. Graunt's analysis of causes of death is considered the beginning of the "theory of competing risks" which according to Daley and Gani [ 1 ] is "a theory that is now well established among modern epidemiologists".
The earliest account of mathematical modelling of spread of disease was carried out in 1760 by Daniel Bernoulli . Trained as a physician, Bernoulli created a mathematical model to defend the practice of inoculating against smallpox . [ 2 ] The calculations from this model showed that universal inoculation against smallpox would increase the life expectancy from 26 years 7 months to 29 years 9 months. [ 3 ] Daniel Bernoulli's work preceded the modern understanding of germ theory . [ 4 ]
In the early 20th century, William Hamer [ 5 ] and Ronald Ross [ 6 ] applied the law of mass action to explain epidemic behaviour.
The 1920s saw the emergence of compartmental models. The Kermack–McKendrick epidemic model (1927) and the Reed–Frost epidemic model (1928) both describe the relationship between susceptible , infected and immune individuals in a population. The Kermack–McKendrick epidemic model was successful in predicting the behavior of outbreaks very similar to that observed in many recorded epidemics. [ 7 ]
Recently, agent-based models (ABMs) have been used in exchange for simpler compartmental models . [ 8 ] For example, epidemiological ABMs have been used to inform public health (nonpharmaceutical) interventions against the spread of SARS-CoV-2 . [ 9 ] Epidemiological ABMs, in spite of their complexity and requiring high computational power, have been criticized for simplifying and unrealistic assumptions. [ 10 ] [ 11 ] Still, they can be useful in informing decisions regarding mitigation and suppression measures in cases when ABMs are accurately calibrated. [ 12 ]
Models are only as good as the assumptions on which they are based. If a model makes predictions that are out of line with observed results and the mathematics is correct, the initial assumptions must change to make the model useful. [ 13 ]
"Stochastic" means being or having a random variable. A stochastic model is a tool for estimating probability distributions of potential outcomes by allowing for random variation in one or more inputs over time. Stochastic models depend on the chance variations in risk of exposure, disease and other illness dynamics. Statistical agent-level disease dissemination in small or large populations can be determined by stochastic methods. [ 14 ] [ 15 ] [ 16 ]
When dealing with large populations, as in the case of tuberculosis, deterministic or compartmental mathematical models are often used. In a deterministic model, individuals in the population are assigned to different subgroups or compartments, each representing a specific stage of the epidemic. [ 17 ]
The transition rates from one class to another are mathematically expressed as derivatives, hence the model is formulated using differential equations. While building such models, it must be assumed that the population size in a compartment is differentiable with respect to time and that the epidemic process is deterministic. In other words, the changes in population of a compartment can be calculated using only the history that was used to develop the model. [ 7 ]
Formally, these models belong to the class of deterministic models; however, they incorporate heterogeneous social features into the dynamics, such as individuals' levels of sociality, opinion, wealth, geographic location, which profoundly influence disease propagation. These models are typically represented by partial differential equations, in contrast to classical models described as systems of ordinary differential equations. Following the derivation principles of kinetic theory, they provide a more rigorous description of epidemic dynamics by starting from agent-based interactions. [ 18 ]
A common explanation for the growth of epidemics holds that 1 person infects 2, those 2 infect 4 and so on and so on with the number of infected doubling every generation.
It is analogous to a game of tag where 1 person tags 2, those 2 tag 4 others who've never been tagged and so on. As this game progresses it becomes increasing frenetic as the tagged run past the previously tagged to hunt down those who have never been tagged.
Thus this model of an epidemic leads to a curve that grows exponentially until it crashes to zero as all the population have been infected. i.e. no herd immunity and no peak and gradual decline as seen in reality. [ 19 ]
Epidemics can be modeled as diseases spreading over networks of contact between people. Such a network can be represented mathematically with a graph and is called the contact network. [ 20 ] Every node in a contact network is a representation of an individual and each link (edge) between a pair of nodes represents the contact between them. Links in the contact networks may be used to transmit the disease between the individuals and each disease has its own dynamics on top of its contact network. The combination of disease dynamics under the influence of interventions, if any, on a contact network may be modeled with another network, known as a transmission network. In a transmission network, all the links are responsible for transmitting the disease. If such a network is a locally tree-like network, meaning that any local neighborhood in such a network takes the form of a tree , then the basic reproduction can be written in terms of the average excess degree of the transmission network such that:
R 0 = ⟨ k 2 ⟩ ⟨ k ⟩ − 1 , {\displaystyle R_{0}={\frac {\langle k^{2}\rangle }{\langle k\rangle }}-1,}
where ⟨ k ⟩ {\displaystyle {\langle k\rangle }} is the mean-degree (average degree) of the network and ⟨ k 2 ⟩ {\displaystyle {\langle k^{2}\rangle }} is the second moment of the transmission network degree distribution . It is, however, not always straightforward to find the transmission network out of the contact network and the disease dynamics. [ 21 ] For example, if a contact network can be approximated with an Erdős–Rényi graph with a Poissonian degree distribution , and the disease spreading parameters are as defined in the example above, such that β {\displaystyle \beta } is the transmission rate per person and the disease has a mean infectious period of 1 γ {\displaystyle {\dfrac {1}{\gamma }}} , then the basic reproduction number is R 0 = β γ ⟨ k ⟩ {\displaystyle R_{0}={\dfrac {\beta }{\gamma }}{\langle k\rangle }} [ 22 ] [ 23 ] since ⟨ k 2 ⟩ − ⟨ k ⟩ 2 = ⟨ k ⟩ {\displaystyle {\langle k^{2}\rangle }-{\langle k\rangle }^{2}={\langle k\rangle }} for a Poisson distribution.
The basic reproduction number (denoted by R 0 ) is a measure of how transferable a disease is. It is the average number of people that a single infectious person will infect over the course of their infection. This quantity determines whether the infection will increase sub-exponentially , die out, or remain constant: if R 0 > 1, then each person on average infects more than one other person so the disease will spread; if R 0 < 1, then each person infects fewer than one person on average so the disease will die out; and if R 0 = 1, then each person will infect on average exactly one other person, so the disease will become endemic: it will move throughout the population but not increase or decrease. [ 24 ]
An infectious disease is said to be endemic when it can be sustained in a population without the need for external inputs. This means that, on average, each infected person is infecting exactly one other person (any more and the number of people infected will grow sub-exponentially and there will be an epidemic , any less and the disease will die out). In mathematical terms, that is:
The basic reproduction number ( R 0 ) of the disease, assuming everyone is susceptible, multiplied by the proportion of the population that is actually susceptible ( S ) must be one (since those who are not susceptible do not feature in our calculations as they cannot contract the disease). Notice that this relation means that for a disease to be in the endemic steady state , the higher the basic reproduction number, the lower the proportion of the population susceptible must be, and vice versa. This expression has limitations concerning the susceptibility proportion, e.g. the R 0 equals 0.5 implicates S has to be 2, however this proportion exceeds the population size. [ citation needed ]
Assume the rectangular stationary age distribution and let also the ages of infection have the same distribution for each birth year. Let the average age of infection be A , for instance when individuals younger than A are susceptible and those older than A are immune (or infectious). Then it can be shown by an easy argument that the proportion of the population that is susceptible is given by:
We reiterate that L is the age at which in this model every individual is assumed to die. But the mathematical definition of the endemic steady state can be rearranged to give:
Therefore, due to the transitive property :
This provides a simple way to estimate the parameter R 0 using easily available data.
For a population with an exponential age distribution ,
This allows for the basic reproduction number of a disease given A and L in either type of population distribution.
Compartmental models are formulated as Markov chains . [ 25 ] A classic compartmental model in epidemiology is the SIR model, which may be used as a simple model for modelling epidemics. Multiple other types of compartmental models are also employed.
In 1927, W. O. Kermack and A. G. McKendrick created a model in which they considered a fixed population with only three compartments: susceptible, S ( t ) {\displaystyle S(t)} ; infected, I ( t ) {\displaystyle I(t)} ; and recovered, R ( t ) {\displaystyle R(t)} . The compartments used for this model consist of three classes: [ 26 ]
There are many modifications of the SIR model, including those that include births and deaths, where upon recovery there is no immunity (SIS model), where immunity lasts only for a short period of time (SIRS), where there is a latent period of the disease where the person is not infectious ( SEIS and SEIR ), and where infants can be born with immunity (MSIR). [ citation needed ]
Mathematical models need to integrate the increasing volume of data being generated on host - pathogen interactions. Many theoretical studies of the population dynamics , structure and evolution of infectious diseases of plants and animals, including humans, are concerned with this problem. [ 27 ]
Research topics include:
If the proportion of the population that is immune exceeds the herd immunity level for the disease, then the disease can no longer persist in the population and its transmission dies out. [ 28 ] Thus, a disease can be eliminated from a population if enough individuals are immune due to either vaccination or recovery from prior exposure to disease. For example, smallpox eradication , with the last wild case in 1977, and certification of the eradication of indigenous transmission of 2 of the 3 types of wild poliovirus (type 2 in 2015, after the last reported case in 1999, and type 3 in 2019, after the last reported case in 2012). [ 29 ]
The herd immunity level will be denoted q . Recall that, for a stable state: [ citation needed ]
In turn,
which is approximately: [ citation needed ]
S will be (1 − q ), since q is the proportion of the population that is immune and q + S must equal one (since in this simplified model, everyone is either susceptible or immune). Then:
Remember that this is the threshold level. Die out of transmission will only occur if the proportion of immune individuals exceeds this level due to a mass vaccination programme.
We have just calculated the critical immunization threshold (denoted q c ). It is the minimum proportion of the population that must be immunized at birth (or close to birth) in order for the infection to die out in the population.
Because the fraction of the final size of the population p that is never infected can be defined as:
Hence,
Solving for R 0 {\displaystyle R_{0}} , we obtain:
If the vaccine used is insufficiently effective or the required coverage cannot be reached, the program may fail to exceed q c . Such a program will protect vaccinated individuals from disease, but may change the dynamics of transmission. [ citation needed ]
Suppose that a proportion of the population q (where q < q c ) is immunised at birth against an infection with R 0 > 1. The vaccination programme changes R 0 to R q where
This change occurs simply because there are now fewer susceptibles in the population who can be infected. R q is simply R 0 minus those that would normally be infected but that cannot be now since they are immune.
As a consequence of this lower basic reproduction number , the average age of infection A will also change to some new value A q in those who have been left unvaccinated.
Recall the relation that linked R 0 , A and L . Assuming that life expectancy has not changed, now: [ citation needed ]
But R 0 = L / A so:
Thus, the vaccination program may raise the average age of infection, and unvaccinated individuals will experience a reduced force of infection due to the presence of the vaccinated group. For a disease that leads to greater clinical severity in older populations, the unvaccinated proportion of the population may experience the disease relatively later in life than would occur in the absence of vaccine.
If a vaccination program causes the proportion of immune individuals in a population to exceed the critical threshold for a significant length of time, transmission of the infectious disease in that population will stop. If elimination occurs everywhere at the same time, then this can lead to eradication . [ citation needed ]
Models have the advantage of examining multiple outcomes simultaneously, rather than making a single forecast. Models have shown broad degrees of reliability in past pandemics, such as SARS , SARS-CoV-2 , [ 30 ] Swine flu , MERS and Ebola . [ 31 ] | https://en.wikipedia.org/wiki/Mathematical_modelling_of_infectious_diseases |
Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship to other areas of philosophy, particularly epistemology and metaphysics . Central questions posed include whether or not mathematical objects are purely abstract entities or are in some way concrete, and in what the relationship such objects have with physical reality consists. [ 1 ]
Major themes that are dealt with in philosophy of mathematics include:
The connection between mathematics and material reality has led to philosophical debates since at least the time of Pythagoras . The ancient philosopher Plato argued that abstractions that reflect material reality have themselves a reality that exists outside space and time. As a result, the philosophical view that mathematical objects somehow exist on their own in abstraction is often referred to as Platonism . Independently of their possible philosophical opinions, modern mathematicians may be generally considered as Platonists, since they think of and talk of their objects of study as real objects. [ 2 ]
Armand Borel summarized this view of mathematics reality as follows, and provided quotations of G. H. Hardy , Charles Hermite , Henri Poincaré and Albert Einstein that support his views. [ 3 ]
Something becomes objective (as opposed to "subjective") as soon as we are convinced that it exists in the minds of others in the same form as it does in ours and that we can think about it and discuss it together. [ 4 ] Because the language of mathematics is so precise, it is ideally suited to defining concepts for which such a consensus exists. In my opinion, that is sufficient to provide us with a feeling of an objective existence, of a reality of mathematics ...
Mathematical reasoning requires rigor . This means that the definitions must be absolutely unambiguous and the proofs must be reducible to a succession of applications of syllogisms or inference rules , [ a ] without any use of empirical evidence and intuition . [ b ] [ 6 ]
The rules of rigorous reasoning have been established by the ancient Greek philosophers under the name of logic . Logic is not specific to mathematics, but, in mathematics, the standard of rigor is much higher than elsewhere.
For many centuries, logic, although used for mathematical proofs, belonged to philosophy and was not specifically studied by mathematicians. [ 7 ] Circa the end of the 19th century, several paradoxes made questionable the logical foundation of mathematics, and consequently the validity of the whole of mathematics. This has been called the foundational crisis of mathematics . Some of these paradoxes consist of results that seem to contradict the common intuition, such as the possibility to construct valid non-Euclidean geometries in which the parallel postulate is wrong, the Weierstrass function that is continuous but nowhere differentiable , and the study by Georg Cantor of infinite sets , which led to consider several sizes of infinity (infinite cardinals ). Even more striking, Russell's paradox shows that the phrase "the set of all sets" is self contradictory.
Several methods have been proposed to solve the problem by changing of logical framework, such as constructive mathematics and intuitionistic logic . Roughly speaking, the first one consists of requiring that every existence theorem must provide an explicit example, and the second one excludes from mathematical reasoning the law of excluded middle and double negation elimination .
These logics have less inference rules than classical logic. On the other hand classical logic was a first-order logic , which means roughly that quantifiers cannot be applied to infinite sets. This means, for example that the sentence "every set of natural numbers has a least element" is nonsensical in any formalization of classical logic. This led to the introduction of higher-order logics , which are presently used commonly in mathematics.
The problems of foundation of mathematics has been eventually resolved with the rise of mathematical logic as a new area of mathematics. In this framework, a mathematical or logical theory consists of a formal language that defines the well-formed of assertions , a set of basic assertions called axioms and a set of inference rules that allow producing new assertions from one or several known assertions. A theorem of such a theory is either an axiom or an assertion that can be obtained from previously known theorems by the application of an inference rule. The Zermelo–Fraenkel set theory with the axiom of choice , generally called ZFC , is a higher-order logic in which all mathematics have been restated; it is used implicitely in all mathematics texts that do not specify explicitly on which foundations they are based. Moreover, the other proposed foundations can be modeled and studied inside ZFC.
It results that "rigor" is no more a relevant concept in mathematics, as a proof is either correct or erroneous, and a "rigorous proof" is simply a pleonasm . Where a special concept of rigor comes into play is in the socialized aspects of a proof. In particular, proofs are rarely written in full details, and some steps of a proof are generally considered as trivial , easy , or straightforward , and therefore left to the reader. As most proof errors occur in these skipped steps, a new proof requires to be verified by other specialists of the subject, and can be considered as reliable only after having been accepted by the community of the specialists, which may need several years. [ 8 ]
Also, the concept of "rigor" may remain useful for teaching to beginners what is a mathematical proof. [ 9 ]
Mathematics is used in most sciences for modeling phenomena, which then allows predictions to be made from experimental laws. [ 10 ] The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model. [ 11 ] Inaccurate predictions, rather than being caused by invalid mathematical concepts, imply the need to change the mathematical model used. [ 12 ] For example, the perihelion precession of Mercury could only be explained after the emergence of Einstein 's general relativity , which replaced Newton's law of gravitation as a better mathematical model. [ 13 ]
There is still a philosophical debate whether mathematics is a science. However, in practice, mathematicians are typically grouped with scientists, and mathematics shares much in common with the physical sciences. Like them, it is falsifiable , which means in mathematics that if a result or a theory is wrong, this can be proved by providing a counterexample . Similarly as in science, theories and results (theorems) are often obtained from experimentation . [ 14 ] In mathematics, the experimentation may consist of computation on selected examples or of the study of figures or other representations of mathematical objects (often mind representations without physical support). For example, when asked how he came about his theorems, Gauss once replied "durch planmässiges Tattonieren" (through systematic experimentation). [ 15 ] However, some authors emphasize that mathematics differs from the modern notion of science by not relying on empirical evidence. [ 16 ] [ 17 ] [ 18 ] [ 19 ]
The unreasonable effectiveness of mathematics is a phenomenon that was named and first made explicit by physicist Eugene Wigner . [ 20 ] It is the fact that many mathematical theories (even the "purest") have applications outside their initial object. These applications may be completely outside their initial area of mathematics, and may concern physical phenomena that were completely unknown when the mathematical theory was introduced. [ 21 ] Examples of unexpected applications of mathematical theories can be found in many areas of mathematics.
A notable example is the prime factorization of natural numbers that was discovered more than 2,000 years before its common use for secure internet communications through the RSA cryptosystem . [ 22 ] A second historical example is the theory of ellipses . They were studied by the ancient Greek mathematicians as conic sections (that is, intersections of cones with planes). It was almost 2,000 years later that Johannes Kepler discovered that the trajectories of the planets are ellipses. [ 23 ]
In the 19th century, the internal development of geometry (pure mathematics) led to definition and study of non-Euclidean geometries, spaces of dimension higher than three and manifolds . At this time, these concepts seemed totally disconnected from the physical reality, but at the beginning of the 20th century, Albert Einstein developed the theory of relativity that uses fundamentally these concepts. In particular, spacetime of special relativity is a non-Euclidean space of dimension four, and spacetime of general relativity is a (curved) manifold of dimension four. [ 24 ] [ 25 ]
A striking aspect of the interaction between mathematics and physics is when mathematics drives research in physics. This is illustrated by the discoveries of the positron and the baryon Ω − . {\displaystyle \Omega ^{-}.} In both cases, the equations of the theories had unexplained solutions, which led to conjecture of the existence of an unknown particle , and the search for these particles. In both cases, these particles were discovered a few years later by specific experiments. [ 26 ] [ 27 ] [ 28 ]
The origin of mathematics is of arguments and disagreements. Whether the birth of mathematics was by chance or induced by necessity during the development of similar subjects, such as physics, remains an area of contention. [ 29 ] [ 30 ]
Many thinkers have contributed their ideas concerning the nature of mathematics. Today, some [ who? ] philosophers of mathematics aim to give accounts of this form of inquiry and its products as they stand, while others emphasize a role for themselves that goes beyond simple interpretation to critical analysis. There are traditions of mathematical philosophy in both Western philosophy and Eastern philosophy . Western philosophies of mathematics go as far back as Pythagoras , who described the theory "everything is mathematics" ( mathematicism ), Plato , who paraphrased Pythagoras, and studied the ontological status of mathematical objects, and Aristotle , who studied logic and issues related to infinity (actual versus potential).
Greek philosophy on mathematics was strongly influenced by their study of geometry . For example, at one time, the Greeks held the opinion that 1 (one) was not a number , but rather a unit of arbitrary length. A number was defined as a multitude. Therefore, 3, for example, represented a certain multitude of units, and was thus "truly" a number. At another point, a similar argument was made that 2 was not a number but a fundamental notion of a pair. These views come from the heavily geometric straight-edge-and-compass viewpoint of the Greeks: just as lines drawn in a geometric problem are measured in proportion to the first arbitrarily drawn line, so too are the numbers on a number line measured in proportion to the arbitrary first "number" or "one". [ citation needed ]
These earlier Greek ideas of numbers were later upended by the discovery of the irrationality of the square root of two. Hippasus , a disciple of Pythagoras , showed that the diagonal of a unit square was incommensurable with its (unit-length) edge: in other words he proved there was no existing (rational) number that accurately depicts the proportion of the diagonal of the unit square to its edge. This caused a significant re-evaluation of Greek philosophy of mathematics. According to legend, fellow Pythagoreans were so traumatized by this discovery that they murdered Hippasus to stop him from spreading his heretical idea. [ 31 ] Simon Stevin was one of the first in Europe to challenge Greek ideas in the 16th century. Beginning with Leibniz , the focus shifted strongly to the relationship between mathematics and logic. This perspective dominated the philosophy of mathematics through the time of Boole , Frege and Russell , but was brought into question by developments in the late 19th and early 20th centuries.
A perennial issue in the philosophy of mathematics concerns the relationship between logic and mathematics at their joint foundations. While 20th-century philosophers continued to ask the questions mentioned at the outset of this article, the philosophy of mathematics in the 20th century was characterized by a predominant interest in formal logic , set theory (both naive set theory and axiomatic set theory ), and foundational issues.
It is a profound puzzle that on the one hand mathematical truths seem to have a compelling inevitability, but on the other hand the source of their "truthfulness" remains elusive. Investigations into this issue are known as the foundations of mathematics program.
At the start of the 20th century, philosophers of mathematics were already beginning to divide into various schools of thought about all these questions, broadly distinguished by their pictures of mathematical epistemology and ontology . Three schools, formalism , intuitionism , and logicism , emerged at this time, partly in response to the increasingly widespread worry that mathematics as it stood, and analysis in particular, did not live up to the standards of certainty and rigor that had been taken for granted. Each school addressed the issues that came to the fore at that time, either attempting to resolve them or claiming that mathematics is not entitled to its status as our most trusted knowledge.
Surprising and counter-intuitive developments in formal logic and set theory early in the 20th century led to new questions concerning what was traditionally called the foundations of mathematics . As the century unfolded, the initial focus of concern expanded to an open exploration of the fundamental axioms of mathematics, the axiomatic approach having been taken for granted since the time of Euclid around 300 BCE as the natural basis for mathematics. Notions of axiom , proposition and proof , as well as the notion of a proposition being true of a mathematical object (see Assignment ) , were formalized, allowing them to be treated mathematically. The Zermelo–Fraenkel axioms for set theory were formulated which provided a conceptual framework in which much mathematical discourse would be interpreted. In mathematics, as in physics, new and unexpected ideas had arisen and significant changes were coming. With Gödel numbering , propositions could be interpreted as referring to themselves or other propositions, enabling inquiry into the consistency of mathematical theories. This reflective critique in which the theory under review "becomes itself the object of a mathematical study" led Hilbert to call such study metamathematics or proof theory . [ 32 ]
At the middle of the century, a new mathematical theory was created by Samuel Eilenberg and Saunders Mac Lane , known as category theory , and it became a new contender for the natural language of mathematical thinking. [ 33 ] As the 20th century progressed, however, philosophical opinions diverged as to just how well-founded were the questions about foundations that were raised at the century's beginning. Hilary Putnam summed up one common view of the situation in the last third of the century by saying:
When philosophy discovers something wrong with science, sometimes science has to be changed— Russell's paradox comes to mind, as does Berkeley 's attack on the actual infinitesimal —but more often it is philosophy that has to be changed. I do not think that the difficulties that philosophy finds with classical mathematics today are genuine difficulties; and I think that the philosophical interpretations of mathematics that we are being offered on every hand are wrong, and that "philosophical interpretation" is just what mathematics doesn't need. [ 34 ] : 169–170
Philosophy of mathematics today proceeds along several different lines of inquiry, by philosophers of mathematics, logicians, and mathematicians, and there are many schools of thought on the subject. The schools are addressed separately in the next section, and their assumptions explained.
Contemporary schools of thought in the philosophy of mathematics include: artistic, Platonism, mathematicism, logicism, formalism, conventionalism, intuitionism, constructivism, finitism, structuralism, embodied mind theories (Aristotelian realism, psychologism, empiricism), fictionalism, social constructivism, and non-traditional schools.
However, many of these schools of thought are mutually compatible. For example, most living mathematicians are together Platonists and formalists, give a great importance to aesthetic , and consider that axioms should be chosen for the results they produce, not for their coherence with human intuition of reality (conventionalism). [ 26 ]
The view that claims that mathematics is the aesthetic combination of assumptions, and then also claims that mathematics is an art . A famous mathematician who claims that is the British G. H. Hardy . [ 35 ] For Hardy, in his book, A Mathematician's Apology , the definition of mathematics was more like the aesthetic combination of concepts. [ 36 ]
Max Tegmark 's mathematical universe hypothesis (or mathematicism ) goes further than Platonism in asserting that not only do all mathematical objects exist, but nothing else does. Tegmark's sole postulate is: All structures that exist mathematically also exist physically . That is, in the sense that "in those [worlds] complex enough to contain self-aware substructures [they] will subjectively perceive themselves as existing in a physically 'real' world". [ 37 ] [ 38 ]
Logicism is the thesis that mathematics is reducible to logic, and hence nothing but a part of logic. [ 39 ] : 41 Logicists hold that mathematics can be known a priori , but suggest that our knowledge of mathematics is just part of our knowledge of logic in general, and is thus analytic , not requiring any special faculty of mathematical intuition. In this view, logic is the proper foundation of mathematics, and all mathematical statements are necessary logical truths .
Rudolf Carnap (1931) presents the logicist thesis in two parts: [ 39 ]
Gottlob Frege was the founder of logicism. In his seminal Die Grundgesetze der Arithmetik ( Basic Laws of Arithmetic ) he built up arithmetic from a system of logic with a general principle of comprehension, which he called "Basic Law V" (for concepts F and G , the extension of F equals the extension of G if and only if for all objects a , Fa equals Ga ), a principle that he took to be acceptable as part of logic.
Frege's construction was flawed. Bertrand Russell discovered that Basic Law V is inconsistent (this is Russell's paradox ). Frege abandoned his logicist program soon after this, but it was continued by Russell and Whitehead . They attributed the paradox to "vicious circularity" and built up what they called ramified type theory to deal with it. In this system, they were eventually able to build up much of modern mathematics but in an altered, and excessively complex form (for example, there were different natural numbers in each type, and there were infinitely many types). They also had to make several compromises in order to develop much of mathematics, such as the " axiom of reducibility ". Even Russell said that this axiom did not really belong to logic.
Modern logicists (like Bob Hale , Crispin Wright , and perhaps others) have returned to a program closer to Frege's. They have abandoned Basic Law V in favor of abstraction principles such as Hume's principle (the number of objects falling under the concept F equals the number of objects falling under the concept G if and only if the extension of F and the extension of G can be put into one-to-one correspondence ). Frege required Basic Law V to be able to give an explicit definition of the numbers, but all the properties of numbers can be derived from Hume's principle. This would not have been enough for Frege because (to paraphrase him) it does not exclude the possibility that the number 3 is in fact Julius Caesar. In addition, many of the weakened principles that they have had to adopt to replace Basic Law V no longer seem so obviously analytic, and thus purely logical.
Formalism holds that mathematical statements may be thought of as statements about the consequences of certain string manipulation rules. For example, in the "game" of Euclidean geometry (which is seen as consisting of some strings called "axioms", and some "rules of inference" to generate new strings from given ones), one can prove that the Pythagorean theorem holds (that is, one can generate the string corresponding to the Pythagorean theorem). According to formalism, mathematical truths are not about numbers and sets and triangles and the like—in fact, they are not "about" anything at all.
Another version of formalism is known as deductivism . [ 40 ] In deductivism, the Pythagorean theorem is not an absolute truth, but a relative one, if it follows deductively from the appropriate axioms. The same is held to be true for all other mathematical statements.
Formalism need not mean that mathematics is nothing more than a meaningless symbolic game. It is usually hoped that there exists some interpretation in which the rules of the game hold. (Compare this position to structuralism .) But it does allow the working mathematician to continue in his or her work and leave such problems to the philosopher or scientist. Many formalists would say that in practice, the axiom systems to be studied will be suggested by the demands of science or other areas of mathematics.
A major early proponent of formalism was David Hilbert , whose program was intended to be a complete and consistent axiomatization of all of mathematics. [ 41 ] Hilbert aimed to show the consistency of mathematical systems from the assumption that the "finitary arithmetic" (a subsystem of the usual arithmetic of the positive integers , chosen to be philosophically uncontroversial) was consistent. Hilbert's goals of creating a system of mathematics that is both complete and consistent were seriously undermined by the second of Gödel's incompleteness theorems , which states that sufficiently expressive consistent axiom systems can never prove their own consistency. Since any such axiom system would contain the finitary arithmetic as a subsystem, Gödel's theorem implied that it would be impossible to prove the system's consistency relative to that (since it would then prove its own consistency, which Gödel had shown was impossible). Thus, in order to show that any axiomatic system of mathematics is in fact consistent, one needs to first assume the consistency of a system of mathematics that is in a sense stronger than the system to be proven consistent.
Hilbert was initially a deductivist, but, as may be clear from above, he considered certain metamathematical methods to yield intrinsically meaningful results and was a realist with respect to the finitary arithmetic. Later, he held the opinion that there was no other meaningful mathematics whatsoever, regardless of interpretation.
Other formalists, such as Rudolf Carnap , Alfred Tarski , and Haskell Curry , considered mathematics to be the investigation of formal axiom systems . Mathematical logicians study formal systems but are just as often realists as they are formalists.
Formalists are relatively tolerant and inviting to new approaches to logic, non-standard number systems, new set theories, etc. The more games we study, the better. However, in all three of these examples, motivation is drawn from existing mathematical or philosophical concerns. The "games" are usually not arbitrary.
The main critique of formalism is that the actual mathematical ideas that occupy mathematicians are far removed from the string manipulation games mentioned above. Formalism is thus silent on the question of which axiom systems ought to be studied, as none is more meaningful than another from a formalistic point of view.
Recently, some [ who? ] formalist mathematicians have proposed that all of our formal mathematical knowledge should be systematically encoded in computer-readable formats, so as to facilitate automated proof checking of mathematical proofs and the use of interactive theorem proving in the development of mathematical theories and computer software. Because of their close connection with computer science , this idea is also advocated by mathematical intuitionists and constructivists in the "computability" tradition— see QED project for a general overview .
The French mathematician Henri Poincaré was among the first to articulate a conventionalist view. Poincaré's use of non-Euclidean geometries in his work on differential equations convinced him that Euclidean geometry should not be regarded as a priori truth. He held that axioms in geometry should be chosen for the results they produce, not for their apparent coherence with human intuitions about the physical world.
In mathematics, intuitionism is a program of methodological reform whose motto is that "there are no non-experienced mathematical truths" ( L. E. J. Brouwer ). From this springboard, intuitionists seek to reconstruct what they consider to be the corrigible portion of mathematics in accordance with Kantian concepts of being, becoming, intuition, and knowledge. Brouwer, the founder of the movement, held that mathematical objects arise from the a priori forms of the volitions that inform the perception of empirical objects. [ 42 ]
A major force behind intuitionism was L. E. J. Brouwer , who rejected the usefulness of formalized logic of any sort for mathematics. His student Arend Heyting postulated an intuitionistic logic , different from the classical Aristotelian logic ; this logic does not contain the law of the excluded middle and therefore frowns upon proofs by contradiction . The axiom of choice is also rejected in most intuitionistic set theories, though in some versions it is accepted.
In intuitionism, the term "explicit construction" is not cleanly defined, and that has led to criticisms. Attempts have been made to use the concepts of Turing machine or computable function to fill this gap, leading to the claim that only questions regarding the behavior of finite algorithms are meaningful and should be investigated in mathematics. This has led to the study of the computable numbers , first introduced by Alan Turing . Not surprisingly, then, this approach to mathematics is sometimes associated with theoretical computer science .
Like intuitionism, constructivism involves the regulative principle that only mathematical entities which can be explicitly constructed in a certain sense should be admitted to mathematical discourse. In this view, mathematics is an exercise of the human intuition, not a game played with meaningless symbols. Instead, it is about entities that we can create directly through mental activity. In addition, some adherents of these schools reject non-constructive proofs, such as using proof by contradiction when showing the existence of an object or when trying to establish the truth of some proposition. Important work was done by Errett Bishop , who managed to prove versions of the most important theorems in real analysis as constructive analysis in his 1967 Foundations of Constructive Analysis. [ 43 ]
Finitism is an extreme form of constructivism , according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps. In her book Philosophy of Set Theory , Mary Tiles characterized those who allow countably infinite objects as classical finitists, and those who deny even countably infinite objects as strict finitists.
The most famous proponent of finitism was Leopold Kronecker , [ 44 ] who said:
God created the natural numbers, all else is the work of man.
Ultrafinitism is an even more extreme version of finitism, which rejects not only infinities but finite quantities that cannot feasibly be constructed with available resources. Another variant of finitism is Euclidean arithmetic, a system developed by John Penn Mayberry in his book The Foundations of Mathematics in the Theory of Sets . [ 45 ] Mayberry's system is Aristotelian in general inspiration and, despite his strong rejection of any role for operationalism or feasibility in the foundations of mathematics, comes to somewhat similar conclusions, such as, for instance, that super-exponentiation is not a legitimate finitary function.
Structuralism is a position holding that mathematical theories describe structures, and that mathematical objects are exhaustively defined by their places in such structures, consequently having no intrinsic properties . For instance, it would maintain that all that needs to be known about the number 1 is that it is the first whole number after 0. Likewise all the other whole numbers are defined by their places in a structure, the number line . Other examples of mathematical objects might include lines and planes in geometry, or elements and operations in abstract algebra .
Structuralism is an epistemologically realistic view in that it holds that mathematical statements have an objective truth value. However, its central claim only relates to what kind of entity a mathematical object is, not to what kind of existence mathematical objects or structures have (not, in other words, to their ontology ). The kind of existence mathematical objects have would clearly be dependent on that of the structures in which they are embedded; different sub-varieties of structuralism make different ontological claims in this regard. [ 46 ]
The ante rem structuralism ("before the thing") has a similar ontology to Platonism . Structures are held to have a real but abstract and immaterial existence. As such, it faces the standard epistemological problem of explaining the interaction between such abstract structures and flesh-and-blood mathematicians (see Benacerraf's identification problem ) .
The in re structuralism ("in the thing") is the equivalent of Aristotelian realism . Structures are held to exist inasmuch as some concrete system exemplifies them. This incurs the usual issues that some perfectly legitimate structures might accidentally happen not to exist, and that a finite physical world might not be "big" enough to accommodate some otherwise legitimate structures.
The post rem structuralism ("after the thing") is anti-realist about structures in a way that parallels nominalism . Like nominalism, the post rem approach denies the existence of abstract mathematical objects with properties other than their place in a relational structure. According to this view mathematical systems exist, and have structural features in common. If something is true of a structure, it will be true of all systems exemplifying the structure. However, it is merely instrumental to talk of structures being "held in common" between systems: they in fact have no independent existence.
Embodied mind theories hold that mathematical thought is a natural outgrowth of the human cognitive apparatus which finds itself in our physical universe. For example, the abstract concept of number springs from the experience of counting discrete objects (requiring the human senses such as sight for detecting the objects, touch; and signalling from the brain). It is held that mathematics is not universal and does not exist in any real sense, other than in human brains. Humans construct, but do not discover, mathematics.
The cognitive processes of pattern-finding and distinguishing objects are also subject to neuroscience ; if mathematics is considered to be relevant to a natural world (such as from realism or a degree of it, as opposed to pure solipsism ).
Its actual relevance to reality, while accepted to be a trustworthy approximation (it is also suggested the evolution of perceptions, the body, and the senses may have been necessary for survival) is not necessarily accurate to a full realism (and is still subject to flaws such as illusion , assumptions (consequently; the foundations and axioms in which mathematics have been formed by humans), generalisations, deception, and hallucinations ). As such, this may also raise questions for the modern scientific method for its compatibility with general mathematics; as while relatively reliable, it is still limited by what can be measured by empiricism which may not be as reliable as previously assumed (see also: 'counterintuitive' concepts in such as quantum nonlocality , and action at a distance ).
Another issue is that one numeral system may not necessarily be applicable to problem solving. Subjects such as complex numbers or imaginary numbers require specific changes to more commonly used axioms of mathematics; otherwise they cannot be adequately understood.
Alternatively, computer programmers may use hexadecimal for its 'human-friendly' representation of binary-coded values, rather than decimal (convenient for counting because humans have ten fingers). The axioms or logical rules behind mathematics also vary through time (such as the adaption and invention of zero ).
As perceptions from the human brain are subject to illusions , assumptions, deceptions, (induced) hallucinations , cognitive errors or assumptions in a general context, it can be questioned whether they are accurate or strictly indicative of truth (see also: philosophy of being ), and the nature of empiricism itself in relation to the universe and whether it is independent to the senses and the universe.
The human mind has no special claim on reality or approaches to it built out of math. If such constructs as Euler's identity are true then they are true as a map of the human mind and cognition .
Embodied mind theorists thus explain the effectiveness of mathematics—mathematics was constructed by the brain in order to be effective in this universe.
The most accessible, famous, and infamous treatment of this perspective is Where Mathematics Comes From , by George Lakoff and Rafael E. Núñez . In addition, mathematician Keith Devlin has investigated similar concepts with his book The Math Instinct , as has neuroscientist Stanislas Dehaene with his book The Number Sense . For more on the philosophical ideas that inspired this perspective, see cognitive science of mathematics .
Aristotelian realism holds that mathematics studies properties such as symmetry, continuity and order that can be literally realized in the physical world (or in any other world there might be). It contrasts with Platonism in holding that the objects of mathematics, such as numbers, do not exist in an "abstract" world but can be physically realized. For example, the number 4 is realized in the relation between a heap of parrots and the universal "being a parrot" that divides the heap into so many parrots. [ 47 ] [ 48 ] Aristotelian realism is defended by James Franklin and the Sydney School in the philosophy of mathematics and is close to the view of Penelope Maddy that when an egg carton is opened, a set of three eggs is perceived (that is, a mathematical entity realized in the physical world). [ 49 ] A problem for Aristotelian realism is what account to give of higher infinities, which may not be realizable in the physical world.
The Euclidean arithmetic developed by John Penn Mayberry in his book The Foundations of Mathematics in the Theory of Sets [ 45 ] also falls into the Aristotelian realist tradition. Mayberry, following Euclid, considers numbers to be simply "definite multitudes of units" realized in nature—such as "the members of the London Symphony Orchestra" or "the trees in Birnam wood". Whether or not there are definite multitudes of units for which Euclid's Common Notion 5 (the whole is greater than the part) fails and which would consequently be reckoned as infinite is for Mayberry essentially a question about Nature and does not entail any transcendental suppositions.
Psychologism in the philosophy of mathematics is the position that mathematical concepts and/or truths are grounded in, derived from or explained by psychological facts (or laws).
John Stuart Mill seems to have been an advocate of a type of logical psychologism, as were many 19th-century German logicians such as Sigwart and Erdmann as well as a number of psychologists , past and present: for example, Gustave Le Bon . Psychologism was famously criticized by Frege in his The Foundations of Arithmetic , and many of his works and essays, including his review of Husserl 's Philosophy of Arithmetic . Edmund Husserl, in the first volume of his Logical Investigations , called "The Prolegomena of Pure Logic", criticized psychologism thoroughly and sought to distance himself from it. The "Prolegomena" is considered [ by whom? ] a more concise, fair, and thorough refutation of psychologism than the criticisms made by Frege, and also it is considered today by many [ by whom? ] as being a memorable refutation for its decisive blow to psychologism. Psychologism was also criticized by Charles Sanders Peirce and Maurice Merleau-Ponty .
Mathematical empiricism is a form of realism that denies that mathematics can be known a priori at all. It says that we discover mathematical facts by empirical research , just like facts in any of the other sciences. It is not one of the classical three positions advocated in the early 20th century, but primarily arose in the middle of the century. However, an important early proponent of a view like this was John Stuart Mill . Mill's view was widely criticized, because, according to critics, such as A.J. Ayer, [ 50 ] it makes statements like "2 + 2 = 4" come out as uncertain, contingent truths, which we can only learn by observing instances of two pairs coming together and forming a quartet.
Karl Popper was another philosopher to point out empirical aspects of mathematics, observing that "most mathematical theories are, like those of physics and biology, hypothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently." [ 51 ] Popper also noted he would "admit a system as empirical or scientific only if it is capable of being tested by experience." [ 52 ]
Contemporary mathematical empiricism, formulated by W. V. O. Quine and Hilary Putnam , is primarily supported by the indispensability argument : mathematics is indispensable to all empirical sciences, and if we want to believe in the reality of the phenomena described by the sciences, we ought also believe in the reality of those entities required for this description. That is, since physics needs to talk about electrons to say why light bulbs behave as they do, then electrons must exist . Since physics needs to talk about numbers in offering any of its explanations, then numbers must exist. In keeping with Quine and Putnam's overall philosophies, this is a naturalistic argument. It argues for the existence of mathematical entities as the best explanation for experience, thus stripping mathematics of being distinct from the other sciences.
Putnam strongly rejected the term " Platonist " as implying an over-specific ontology that was not necessary to mathematical practice in any real sense. He advocated a form of "pure realism" that rejected mystical notions of truth and accepted much quasi-empiricism in mathematics . This grew from the increasingly popular assertion in the late 20th century that no one foundation of mathematics could be ever proven to exist. It is also sometimes called "postmodernism in mathematics" although that term is considered overloaded by some and insulting by others. Quasi-empiricism argues that in doing their research, mathematicians test hypotheses as well as prove theorems. A mathematical argument can transmit falsity from the conclusion to the premises just as well as it can transmit truth from the premises to the conclusion. Putnam has argued that any theory of mathematical realism would include quasi-empirical methods. He proposed that an alien species doing mathematics might well rely on quasi-empirical methods primarily, being willing often to forgo rigorous and axiomatic proofs, and still be doing mathematics—at perhaps a somewhat greater risk of failure of their calculations. He gave a detailed argument for this in New Directions . [ 53 ] Quasi-empiricism was also developed by Imre Lakatos .
The most important criticism of empirical views of mathematics is approximately the same as that raised against Mill. If mathematics is just as empirical as the other sciences, then this suggests that its results are just as fallible as theirs, and just as contingent. In Mill's case the empirical justification comes directly, while in Quine's case it comes indirectly, through the coherence of our scientific theory as a whole, i.e. consilience after E.O. Wilson . Quine suggests that mathematics seems completely certain because the role it plays in our web of belief is extraordinarily central, and that it would be extremely difficult for us to revise it, though not impossible.
For a philosophy of mathematics that attempts to overcome some of the shortcomings of Quine and Gödel's approaches by taking aspects of each see Penelope Maddy 's Realism in Mathematics . Another example of a realist theory is the embodied mind theory .
For experimental evidence suggesting that human infants can do elementary arithmetic, see Brian Butterworth .
Mathematical fictionalism was brought to fame in 1980 when Hartry Field published Science Without Numbers , [ 54 ] which rejected and in fact reversed Quine's indispensability argument. Where Quine suggested that mathematics was indispensable for our best scientific theories, and therefore should be accepted as a body of truths talking about independently existing entities, Field suggested that mathematics was dispensable, and therefore should be considered as a body of falsehoods not talking about anything real. He did this by giving a complete axiomatization of Newtonian mechanics with no reference to numbers or functions at all. He started with the "betweenness" of Hilbert's axioms to characterize space without coordinatizing it, and then added extra relations between points to do the work formerly done by vector fields . Hilbert's geometry is mathematical, because it talks about abstract points, but in Field's theory, these points are the concrete points of physical space, so no special mathematical objects at all are needed.
Having shown how to do science without using numbers, Field proceeded to rehabilitate mathematics as a kind of useful fiction . He showed that mathematical physics is a conservative extension of his non-mathematical physics (that is, every physical fact provable in mathematical physics is already provable from Field's system), so that mathematics is a reliable process whose physical applications are all true, even though its own statements are false. Thus, when doing mathematics, we can see ourselves as telling a sort of story, talking as if numbers existed. For Field, a statement like "2 + 2 = 4" is just as fictitious as " Sherlock Holmes lived at 221B Baker Street"—but both are true according to the relevant fictions.
Another fictionalist, Mary Leng , expresses the perspective succinctly by dismissing any seeming connection between mathematics and the physical world as "a happy coincidence". This rejection separates fictionalism from other forms of anti-realism, which see mathematics itself as artificial but still bounded or fitted to reality in some way. [ 55 ]
By this account, there are no metaphysical or epistemological problems special to mathematics. The only worries left are the general worries about non-mathematical physics, and about fiction in general. Field's approach has been very influential, but is widely rejected. This is in part because of the requirement of strong fragments of second-order logic to carry out his reduction, and because the statement of conservativity seems to require quantification over abstract models or deductions. [ citation needed ]
Social constructivism sees mathematics primarily as a social construct , as a product of culture, subject to correction and change. Like the other sciences, mathematics is viewed as an empirical endeavor whose results are constantly evaluated and may be discarded. However, while on an empiricist view the evaluation is some sort of comparison with "reality", social constructivists emphasize that the direction of mathematical research is dictated by the fashions of the social group performing it or by the needs of the society financing it. However, although such external forces may change the direction of some mathematical research, there are strong internal constraints—the mathematical traditions, methods, problems, meanings and values into which mathematicians are enculturated—that work to conserve the historically defined discipline.
This runs counter to the traditional beliefs of working mathematicians, that mathematics is somehow pure or objective. But social constructivists argue that mathematics is in fact grounded by much uncertainty: as mathematical practice evolves, the status of previous mathematics is cast into doubt, and is corrected to the degree it is required or desired by the current mathematical community. This can be seen in the development of analysis from reexamination of the calculus of Leibniz and Newton. They argue further that finished mathematics is often accorded too much status, and folk mathematics not enough, due to an overemphasis on axiomatic proof and peer review as practices.
The social nature of mathematics is highlighted in its subcultures . Major discoveries can be made in one branch of mathematics and be relevant to another, yet the relationship goes undiscovered for lack of social contact between mathematicians. Social constructivists argue each speciality forms its own epistemic community and often has great difficulty communicating, or motivating the investigation of unifying conjectures that might relate different areas of mathematics. Social constructivists see the process of "doing mathematics" as actually creating the meaning, while social realists see a deficiency either of human capacity to abstractify, or of human's cognitive bias , or of mathematicians' collective intelligence as preventing the comprehension of a real universe of mathematical objects. Social constructivists sometimes reject the search for foundations of mathematics as bound to fail, as pointless or even meaningless.
Contributions to this school have been made by Imre Lakatos and Thomas Tymoczko , although it is not clear that either would endorse the title. [ clarification needed ] More recently Paul Ernest has explicitly formulated a social constructivist philosophy of mathematics. [ 56 ] Some consider the work of Paul Erdős as a whole to have advanced this view (although he personally rejected it) because of his uniquely broad collaborations, which prompted others to see and study "mathematics as a social activity", e.g., via the Erdős number . Reuben Hersh has also promoted the social view of mathematics, calling it a "humanistic" approach, [ 57 ] similar to but not quite the same as that associated with Alvin White; [ 58 ] one of Hersh's co-authors, Philip J. Davis , has expressed sympathy for the social view as well.
Rather than focus on narrow debates about the true nature of mathematical truth , or even on practices unique to mathematicians such as the proof , a growing movement from the 1960s to the 1990s began to question the idea of seeking foundations or finding any one right answer to why mathematics works. The starting point for this was Eugene Wigner 's famous 1960 paper " The Unreasonable Effectiveness of Mathematics in the Natural Sciences ", in which he argued that the happy coincidence of mathematics and physics being so well matched seemed to be unreasonable and hard to explain.
Realist and constructivist theories are normally taken to be contraries. However, Karl Popper [ 59 ] argued that a number statement such as "2 apples + 2 apples = 4 apples" can be taken in two senses. In one sense it is irrefutable and logically true. In the second sense it is factually true and falsifiable. Another way of putting this is to say that a single number statement can express two propositions: one of which can be explained on constructivist lines; the other on realist lines. [ 60 ]
Innovations in the philosophy of language during the 20th century renewed interest in whether mathematics is, as is often said, [ citation needed ] the language of science. Although some [ who? ] mathematicians and philosophers would accept the statement "mathematics is a language" (most consider that the language of mathematics is a part of mathematics to which mathematics cannot be reduced), [ citation needed ] linguists [ who? ] believe that the implications of such a statement must be considered. For example, the tools of linguistics are not generally applied to the symbol systems of mathematics, that is, mathematics is studied in a markedly different way from other languages. If mathematics is a language, it is a different type of language from natural languages . Indeed, because of the need for clarity and specificity, the language of mathematics is far more constrained than natural languages studied by linguists. However, the methods developed by Frege and Tarski for the study of mathematical language have been extended greatly by Tarski's student Richard Montague and other linguists working in formal semantics to show that the distinction between mathematical language and natural language may not be as great as it seems.
Mohan Ganesalingam has analysed mathematical language using tools from formal linguistics. [ 61 ] Ganesalingam notes that some features of natural language are not necessary when analysing mathematical language (such as tense ), but many of the same analytical tools can be used (such as context-free grammars ). One important difference is that mathematical objects have clearly defined types , which can be explicitly defined in a text: "Effectively, we are allowed to introduce a word in one part of a sentence, and declare its part of speech in another; and this operation has no analogue in natural language." [ 61 ] : 251
This argument, associated with Willard Quine and Hilary Putnam , is considered by Stephen Yablo to be one of the most challenging arguments in favor of the acceptance of the existence of abstract mathematical entities, such as numbers and sets. [ 62 ] The form of the argument is as follows.
The justification for the first premise is the most controversial. Both Putnam and Quine invoke naturalism to justify the exclusion of all non-scientific entities, and hence to defend the "only" part of "all and only". The assertion that "all" entities postulated in scientific theories, including numbers, should be accepted as real is justified by confirmation holism . Since theories are not confirmed in a piecemeal fashion, but as a whole, there is no justification for excluding any of the entities referred to in well-confirmed theories. This puts the nominalist who wishes to exclude the existence of sets and non-Euclidean geometry , but to include the existence of quarks and other undetectable entities of physics, for example, in a difficult position. [ 63 ]
The anti-realist " epistemic argument" against Platonism has been made by Paul Benacerraf and Hartry Field . Platonism posits that mathematical objects are abstract entities. By general agreement, abstract entities cannot interact causally with concrete, physical entities ("the truth-values of our mathematical assertions depend on facts involving Platonic entities that reside in a realm outside of space-time" [ 64 ] ). Whilst our knowledge of concrete, physical objects is based on our ability to perceive them, and therefore to causally interact with them, there is no parallel account of how mathematicians come to have knowledge of abstract objects. [ 65 ] [ 66 ] [ 67 ] Another way of making the point is that if the Platonic world were to disappear, it would make no difference to the ability of mathematicians to generate proofs , etc., which is already fully accountable in terms of physical processes in their brains.
Field developed his views into fictionalism . Benacerraf also developed the philosophy of mathematical structuralism , according to which there are no mathematical objects. Nonetheless, some versions of structuralism are compatible with some versions of realism.
The argument hinges on the idea that a satisfactory naturalistic account of thought processes in terms of brain processes can be given for mathematical reasoning along with everything else. One line of defense is to maintain that this is false, so that mathematical reasoning uses some special intuition that involves contact with the Platonic realm. A modern form of this argument is given by Sir Roger Penrose . [ 68 ]
Another line of defense is to maintain that abstract objects are relevant to mathematical reasoning in a way that is non-causal, and not analogous to perception. This argument is developed by Jerrold Katz in his 2000 book Realistic Rationalism .
A more radical defense is denial of physical reality, i.e. the mathematical universe hypothesis . In that case, a mathematician's knowledge of mathematics is one mathematical object making contact with another.
Many practicing mathematicians have been drawn to their subject because of a sense of beauty they perceive in it. One sometimes hears the sentiment that mathematicians would like to leave philosophy to the philosophers and get back to mathematics—where, presumably, the beauty lies.
In his work on the divine proportion , H.E. Huntley relates the feeling of reading and understanding someone else's proof of a theorem of mathematics to that of a viewer of a masterpiece of art—the reader of a proof has a similar sense of exhilaration at understanding as the original author of the proof, much as, he argues, the viewer of a masterpiece has a sense of exhilaration similar to the original painter or sculptor. Indeed, one can study mathematical and scientific writings as literature .
Philip J. Davis and Reuben Hersh have commented that the sense of mathematical beauty is universal amongst practicing mathematicians. By way of example, they provide two proofs of the irrationality of √ 2 . The first is the traditional proof by contradiction , ascribed to Euclid ; the second is a more direct proof involving the fundamental theorem of arithmetic that, they argue, gets to the heart of the issue. Davis and Hersh argue that mathematicians find the second proof more aesthetically appealing because it gets closer to the nature of the problem.
Paul Erdős was well known for his notion of a hypothetical "Book" containing the most elegant or beautiful mathematical proofs. There is not universal agreement that a result has one "most elegant" proof; Gregory Chaitin has argued against this idea.
Philosophers have sometimes criticized mathematicians' sense of beauty or elegance as being, at best, vaguely stated. By the same token, however, philosophers of mathematics have sought to characterize what makes one proof more desirable than another when both are logically sound.
Another aspect of aesthetics concerning mathematics is mathematicians' views towards the possible uses of mathematics for purposes deemed unethical or inappropriate. The best-known exposition of this view occurs in G. H. Hardy 's book A Mathematician's Apology , in which Hardy argues that pure mathematics is superior in beauty to applied mathematics precisely because it cannot be used for war and similar ends. | https://en.wikipedia.org/wiki/Mathematical_monism |
A mathematical object is an abstract concept arising in mathematics . [ 1 ] Typically, a mathematical object can be a value that can be assigned to a symbol , and therefore can be involved in formulas . Commonly encountered mathematical objects include numbers , expressions , shapes , functions , and sets . Mathematical objects can be very complex; for example, theorems , proofs , and even formal theories are considered as mathematical objects in proof theory .
In Philosophy of mathematics , the concept of "mathematical objects" touches on topics of existence , identity , and the nature of reality . [ 2 ] In metaphysics , objects are often considered entities that possess properties and can stand in various relations to one another. [ 3 ] Philosophers debate whether mathematical objects have an independent existence outside of human thought ( realism ), or if their existence is dependent on mental constructs or language ( idealism and nominalism ). Objects can range from the concrete : such as physical objects usually studied in applied mathematics , to the abstract , studied in pure mathematics . What constitutes an "object" is foundational to many areas of philosophy, from ontology (the study of being) to epistemology (the study of knowledge). In mathematics, objects are often seen as entities that exist independently of the physical world , raising questions about their ontological status. [ 4 ] [ 5 ] There are varying schools of thought which offer different perspectives on the matter, and many famous mathematicians and philosophers each have differing opinions on which is more correct. [ 6 ]
Quine-Putnam indispensability is an argument for the existence of mathematical objects based on their unreasonable effectiveness in the natural sciences . Every branch of science relies largely on large and often vastly different areas of mathematics. From physics' use of Hilbert spaces in quantum mechanics and differential geometry in general relativity to biology 's use of chaos theory and combinatorics (see mathematical biology ), not only does mathematics help with predictions , it allows these areas to have an elegant language to express these ideas. Moreover, it is hard to imagine how areas like quantum mechanics and general relativity could have developed without their assistance from mathematics, and therefore, one could argue that mathematics is indispensable to these theories. It is because of this unreasonable effectiveness and indispensability of mathematics that philosophers Willard Quine and Hilary Putnam argue that we should believe the mathematical objects for which these theories depend actually exist, that is, we ought to have an ontological commitment to them. The argument is described by the following syllogism : [ 7 ]
( Premise 1) We ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories.
(Premise 2) Mathematical entities are indispensable to our best scientific theories.
( Conclusion ) We ought to have ontological commitment to mathematical entities
This argument resonates with a philosophy in applied mathematics called Naturalism [ 8 ] (or sometimes Predicativism) [ 9 ] which states that the only authoritative standards on existence are those of science .
Platonism asserts that mathematical objects are seen as real, abstract entities that exist independently of human thought , often in some Platonic realm . Just as physical objects like electrons and planets exist, so do numbers and sets. And just as statements about electrons and planets are true or false as these objects contain perfectly objective properties , so are statements about numbers and sets. Mathematicians discover these objects rather than invent them. [ 10 ] [ 11 ] (See also: Mathematical Platonism )
Some some notable platonists include:
Nominalism denies the independent existence of mathematical objects. Instead, it suggests that they are merely convenient fictions or shorthand for describing relationships and structures within our language and theories. Under this view, mathematical objects do not have an existence beyond the symbols and concepts we use. [ 13 ] [ 14 ]
Some notable nominalists include:
Logicism asserts that all mathematical truths can be reduced to logical truths , and a ll objects forming the subject matter of those branches of mathematics are logical objects. In other words, mathematics is fundamentally a branch of logic , and all mathematical concepts, theorems , and truths can be derived from purely logical principles and definitions. Logicism faced challenges, particularly with the Russillian axioms, the Multiplicative axiom (now called the Axiom of Choice ) and his Axiom of Infinity , and later with the discovery of Gödel's incompleteness theorems , which showed that any sufficiently powerful formal system (like those used to express arithmetic ) cannot be both complete and consistent . This meant that not all mathematical truths could be derived purely from a logical system, undermining the logicist program. [ 16 ]
Some notable logicists include:
Mathematical formalism treats objects as symbols within a formal system . The focus is on the manipulation of these symbols according to specified rules, rather than on the objects themselves. One common understanding of formalism takes mathematics as not a body of propositions representing an abstract piece of reality but much more akin to a game, bringing with it no more ontological commitment of objects or properties than playing ludo or chess . In this view, mathematics is about the consistency of formal systems rather than the discovery of pre-existing objects. Some philosophers consider logicism to be a type of formalism. [ 19 ]
Some notable formalists include:
Mathematical constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove the existence of a mathematical object without "finding" that object explicitly, by assuming its non-existence and then deriving a contradiction from that assumption. Such a proof by contradiction might be called non-constructive, and a constructivist might reject it. The constructive viewpoint involves a verificational interpretation of the existential quantifier , which is at odds with its classical interpretation. [ 23 ] There are many forms of constructivism. [ 24 ] These include Brouwer 's program of intutionism , the finitism of Hilbert and Bernays , the constructive recursive mathematics of mathematicians Shanin and Markov , and Bishop 's program of constructive analysis . [ 25 ] Constructivism also includes the study of constructive set theories such as Constructive Zermelo–Fraenkel and the study of philosophy.
Some notable constructivists include:
Structuralism suggests that mathematical objects are defined by their place within a structure or system. The nature of a number, for example, is not tied to any particular thing, but to its role within the system of arithmetic . In a sense, the thesis is that mathematical objects (if there are such objects) simply have no intrinsic nature. [ 26 ] [ 27 ]
Some notable structuralists include:
Frege famously distinguished between functions and objects . [ 30 ] According to his view, a function is a kind of ‘incomplete’ entity that maps arguments to values, and is denoted by an incomplete expression, whereas an object is a ‘complete’ entity and can be denoted by a singular term. Frege reduced properties and relations to functions and so these entities are not included among the objects. Some authors make use of Frege's notion of ‘object’ when discussing abstract objects. [ 31 ] But though Frege's sense of ‘object’ is important, it is not the only way to use the term. Other philosophers include properties and relations among the abstract objects. And when the background context for discussing objects is type theory , properties and relations of higher type (e.g., properties of properties, and properties of relations) may be all be considered ‘objects’. This latter use of ‘object’ is interchangeable with ‘entity.’ It is this more broad interpretation that mathematicians mean when they use the term 'object'. [ 32 ]
Citations
Further reading | https://en.wikipedia.org/wiki/Mathematical_object |
The Unicode Standard encodes almost all standard characters used in mathematics. [ 1 ] Unicode Technical Report #25 provides comprehensive information about the character repertoire, their properties, and guidelines for implementation. [ 1 ] Mathematical operators and symbols are in multiple Unicode blocks . Some of these blocks are dedicated to, or primarily contain, mathematical characters while others are a mix of mathematical and non-mathematical characters. This article covers all Unicode characters with a derived property of "Math". [ 2 ] [ 3 ]
The Mathematical Operators block (U+2200–U+22FF) contains characters for mathematical, logical, and set notation.
The Supplemental Mathematical Operators block (U+2A00–U+2AFF) contains various mathematical symbols, including N-ary operators, summations and integrals, intersections and unions, logical and relational operators, and subset/superset relations.
The Mathematical Alphanumeric Symbols block (U+1D400–U+1D7FF) contains Latin and Greek letters and decimal digits that enable mathematicians to denote different notions with different letter styles. The reserved code points (the "holes") in the alphabetic ranges up to U+1D551 duplicate characters in the Letterlike Symbols block . In order, these are ℎ / ℬ ℰ ℱ ℋ ℐ ℒ ℳ ℛ / ℯ ℊ ℴ / ℭ ℌ ℑ ℜ ℨ / ℂ ℍ ℕ ℙ ℚ ℝ ℤ.
The Letterlike Symbols block (U+2100–U+214F) includes variables. Most alphabetic math symbols are in the Mathematical Alphanumeric Symbols block shown above .
The math subset of this block is U+2102, U+2107, U+210A–U+2113, U+2115, U+2118–U+211D, U+2124, U+2128–U+2129, U+212C–U+212D, U+212F–U+2131, U+2133–U+2138, U+213C–U+2149, and U+214B. [ 4 ]
The Miscellaneous Mathematical Symbols-A block (U+27C0–U+27EF) contains characters for mathematical, logical, and database notation.
The Miscellaneous Mathematical Symbols-B block (U+2980–U+29FF) contains miscellaneous mathematical symbols, including brackets, angles, and circle symbols.
The Miscellaneous Technical block (U+2300–U+23FF) includes braces and operators.
The math subset of this block is U+2308–U+230B, U+2320–U+2321, U+237C, U+239B–U+23B5, 23B7, U+23D0, and U+23DC–U+23E2.
The Geometric Shapes block (U+25A0–U+25FF) contains geometric shape symbols.
The math subset of this block is U+25A0–25A1, U+25AE–25B7, U+25BC–25C1, U+25C6–25C7, U+25CA–25CB, U+25CF–25D3, U+25E2, U+25E4, U+25E7–25EC, and U+25F8–25FF.
The Arrows block (U+2190–U+21FF) contains line, curve, and semicircle arrows and arrow-like operators.
The math subset of this block is U+2190–U+21A7, U+21A9–U+21AE, U+21B0–U+21B1, U+21B6–U+21B7, U+21BC–U+21DB, U+21DD, U+21E4–U+21E5, U+21F4–U+21FF. [ 5 ]
The Supplemental Arrows-A block (U+27F0–U+27FF) contains arrows and arrow-like operators.
The Supplemental Arrows-B block (U+2900–U+297F) contains arrows and arrow-like operators (arrow tails, crossing arrows, curved arrows, and harpoons).
The Miscellaneous Symbols and Arrows block (U+2B00–U+2BFF Arrows) contains arrows and geometric shapes with various fills.
The math subset of this block is U+2B30–U+2B44, U+2B47–U+2B4C. [ 6 ]
The Combining Diacritical Marks for Symbols block contains arrows, dots, enclosures, and overlays for modifying symbol characters.
The math subset of this block is U+20D0–U+20DC, U+20E1, U+20E5–U+20E6, and U+20EB–U+20EF.
The Arabic Mathematical Alphabetic Symbols block (U+1EE00–U+1EEFF) contains characters used in Arabic mathematical expressions.
Mathematical characters also appear in other blocks. Below is a list of these characters as of Unicode version 16.0:
Note: non-marking character | https://en.wikipedia.org/wiki/Mathematical_operators_and_symbols_in_Unicode |
Mathematical physiology is an interdisciplinary science . Primarily, it investigates ways in which mathematics may be used to give insight into physiological questions. In turn, it also describes how physiological questions can lead to new mathematical problems. The field may be broadly grouped into two physiological application areas: cell physiology – including mathematical treatments of biochemical reactions , ionic flow and regulation of function – and systems physiology – including electrocardiology , circulation and digestion . [ 1 ]
This applied mathematics –related article is a stub . You can help Wikipedia by expanding it .
This biology article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Mathematical_physiology |
Mathematical practice comprises the working practices of professional mathematicians : selecting theorems to prove, using informal notations to persuade themselves and others that various steps in the final proof are convincing, and seeking peer review and publication , as opposed to the end result of proven and published theorems .
Philip Kitcher has proposed a more formal definition of a mathematical practice, as a quintuple. His intention was primarily to document mathematical practice through its historical changes. [ 2 ]
The evolution of mathematical practice was slow, and some contributors to modern mathematics did not follow even the practice of their time. For example, Pierre de Fermat was infamous for withholding his proofs, but nonetheless had a vast reputation for correct assertions of results.
One motivation to study mathematical practice is that, despite much work in the 20th century, some still feel that the foundations of mathematics remain unclear and ambiguous. One proposed remedy is to shift focus to some degree onto 'what is meant by a proof', and other such questions of method.
If mathematics has been informally used throughout history, in numerous cultures and continents, then it could be argued that "mathematical practice" is the practice, or use, of mathematics in everyday life. One definition of mathematical practice, as described above, is the "working practices of professional mathematicians". However, another definition, more in keeping with the predominant usage of mathematics, is that mathematical practice is the everyday practice, or use, of math. Whether one is estimating the total cost of their groceries, calculating miles per gallon, or figuring out how many minutes on the treadmill that chocolate éclair will require, math in everyday life relies on practicality (i.e., does it answer the question?) rather than formal proof.
Mathematical teaching usually requires the use of several important teaching pedagogies or components. Most GCSE , A-Level and undergraduate mathematics require the following components: | https://en.wikipedia.org/wiki/Mathematical_practice |
A mathematical problem is a problem that can be represented , analyzed, and possibly solved, with the methods of mathematics . This can be a real-world problem, such as computing the orbits of the planets in the Solar System , or a problem of a more abstract nature, such as Hilbert's problems . It can also be a problem referring to the nature of mathematics itself, such as Russell's Paradox .
Informal "real-world" mathematical problems are questions related to a concrete setting, such as "Adam has five apples and gives John three. How many has he left?". Such questions are usually more difficult to solve than regular mathematical exercises like "5 − 3", even if one knows the mathematics required to solve the problem. Known as word problems , they are used in mathematics education to teach students to connect real-world situations to the abstract language of mathematics.
In general, to use mathematics for solving a real-world problem, the first step is to construct a mathematical model of the problem. This involves abstraction from the details of the problem, and the modeller has to be careful not to lose essential aspects in translating the original problem into a mathematical one. After the problem has been solved in the world of mathematics, the solution must be translated back into the context of the original problem.
Abstract mathematical problems arise in all fields of mathematics. While mathematicians usually study them for their own sake, by doing so, results may be obtained that find application outside the realm of mathematics. Theoretical physics has historically been a rich source of inspiration .
Some abstract problems have been rigorously proved to be unsolvable, such as squaring the circle and trisecting the angle using only the compass and straightedge constructions of classical geometry, and solving the general quintic equation algebraically. Also provably unsolvable are so-called undecidable problems , such as the halting problem for Turing machines .
Some well-known difficult abstract problems that have been solved relatively recently are the four-colour theorem , Fermat's Last Theorem , and the Poincaré conjecture .
Computers do not need to have a sense of the motivations of mathematicians in order to do what they do. [ 1 ] Formal definitions and computer-checkable deductions are absolutely central to mathematical science .
Mathematics educators using problem solving for evaluation have an issue phrased by Alan H. Schoenfeld :
The same issue was faced by Sylvestre Lacroix almost two centuries earlier:
Such degradation of problems into exercises is characteristic of mathematics in history. For example, describing the preparations for the Cambridge Mathematical Tripos in the 19th century, Andrew Warwick wrote: | https://en.wikipedia.org/wiki/Mathematical_problem |
A mathematical proof is a deductive argument for a mathematical statement , showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems ; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms , [ 2 ] [ 3 ] [ 4 ] along with the accepted rules of inference . Proofs are examples of exhaustive deductive reasoning that establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning that establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture , or a hypothesis if frequently used as an assumption for further mathematical work.
Proofs employ logic expressed in mathematical symbols, along with natural language that usually admits some ambiguity. In most mathematical literature, proofs are written in terms of rigorous informal logic . Purely formal proofs , written fully in symbolic language without the involvement of natural language, are considered in proof theory . The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice , quasi-empiricism in mathematics , and so-called folk mathematics , oral traditions in the mainstream mathematical community or in other cultures. The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language .
The word proof derives from the Latin probare 'to test'; related words include English probe , probation , and probability , as well as Spanish probar 'to taste' (sometimes 'to touch' or 'to test'), [ 5 ] Italian provare 'to try', and German probieren 'to try'. The legal term probity means authority or credibility, the power of testimony to prove facts when given by persons of reputation or status. [ 6 ]
Plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof. [ 7 ] It is likely that the idea of demonstrating a conclusion first arose in connection with geometry , which originated in practical problems of land measurement. [ 8 ] The development of mathematical proof is primarily the product of ancient Greek mathematics , and one of its greatest achievements. [ 9 ] Thales (624–546 BCE) and Hippocrates of Chios (c. 470–410 BCE) gave some of the first known proofs of theorems in geometry. Eudoxus (408–355 BCE) and Theaetetus (417–369 BCE) formulated theorems but did not prove them. Aristotle (384–322 BCE) said definitions should describe the concept being defined in terms of other concepts already known.
Mathematical proof was revolutionized by Euclid (300 BCE), who introduced the axiomatic method still in use today. It starts with undefined terms and axioms , propositions concerning the undefined terms which are assumed to be self-evidently true (from Greek axios 'something worthy'). From this basis, the method proves theorems using deductive logic . Euclid's Elements was read by anyone who was considered educated in the West until the middle of the 20th century. [ 10 ] In addition to theorems of geometry, such as the Pythagorean theorem , the Elements also covers number theory , including a proof that the square root of two is irrational and a proof that there are infinitely many prime numbers .
Further advances also took place in medieval Islamic mathematics . In the 10th century, the Iraqi mathematician Al-Hashimi worked with numbers as such, called "lines" but not necessarily considered as measurements of geometric objects, to prove algebraic propositions concerning multiplication, division, etc., including the existence of irrational numbers . [ 11 ] An inductive proof for arithmetic progressions was introduced in the Al-Fakhri (1000) by Al-Karaji , who used it to prove the binomial theorem and properties of Pascal's triangle .
Modern proof theory treats proofs as inductively defined data structures , not requiring an assumption that axioms are "true" in any sense. This allows parallel mathematical theories as formal models of a given intuitive concept, based on alternate sets of axioms, for example axiomatic set theory and non-Euclidean geometry .
As practiced, a proof is expressed in natural language and is a rigorous argument intended to convince the audience of the truth of a statement. The standard of rigor is not absolute and has varied throughout history. A proof can be presented differently depending on the intended audience. To gain acceptance, a proof has to meet communal standards of rigor; an argument considered vague or incomplete may be rejected.
The concept of proof is formalized in the field of mathematical logic . [ 12 ] A formal proof is written in a formal language instead of natural language. A formal proof is a sequence of formulas in a formal language, starting with an assumption, and with each subsequent formula a logical consequence of the preceding ones. This definition makes the concept of proof amenable to study. Indeed, the field of proof theory studies formal proofs and their properties, the most famous and surprising being that almost all axiomatic systems can generate certain undecidable statements not provable within the system.
The definition of a formal proof is intended to capture the concept of proofs as written in the practice of mathematics. The soundness of this definition amounts to the belief that a published proof can, in principle, be converted into a formal proof. However, outside the field of automated proof assistants , this is rarely done in practice. A classic question in philosophy asks whether mathematical proofs are analytic or synthetic . Kant , who introduced the analytic–synthetic distinction , believed mathematical proofs are synthetic, whereas Quine argued in his 1951 " Two Dogmas of Empiricism " that such a distinction is untenable. [ 13 ]
Proofs may be admired for their mathematical beauty . The mathematician Paul Erdős was known for describing proofs which he found to be particularly elegant as coming from "The Book", a hypothetical tome containing the most beautiful method(s) of proving each theorem. The book Proofs from THE BOOK , published in 2003, is devoted to presenting 32 proofs its editors find particularly pleasing.
In direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier theorems. [ 14 ] For example, direct proof can be used to prove that the sum of two even integers is always even:
This proof uses the definition of even integers, the integer properties of closure under addition and multiplication, and the distributive property .
Despite its name, mathematical induction is a method of deduction , not a form of inductive reasoning . In proof by mathematical induction, a single "base case" is proved, and an "induction rule" is proved that establishes that any arbitrary case implies the next case. Since in principle the induction rule can be applied repeatedly (starting from the proved base case), it follows that all (usually infinitely many) cases are provable. [ 15 ] This avoids having to prove each case individually. A variant of mathematical induction is proof by infinite descent , which can be used, for example, to prove the irrationality of the square root of two .
A common application of proof by mathematical induction is to prove that a property known to hold for one number holds for all natural numbers : [ 16 ] Let N = {1, 2, 3, 4, ... } be the set of natural numbers, and let P ( n ) be a mathematical statement involving the natural number n belonging to N such that
For example, we can prove by induction that all positive integers of the form 2 n − 1 are odd . Let P ( n ) represent " 2 n − 1 is odd":
The shorter phrase "proof by induction" is often used instead of "proof by mathematical induction". [ 17 ]
Proof by contraposition infers the statement "if p then q " by establishing the logically equivalent contrapositive statement : "if not q then not p ".
For example, contraposition can be used to establish that, given an integer x {\displaystyle x} , if x 2 {\displaystyle x^{2}} is even, then x {\displaystyle x} is even:
In proof by contradiction, also known by the Latin phrase reductio ad absurdum (by reduction to the absurd), it is shown that if some statement is assumed true, a logical contradiction occurs, hence the statement must be false. A famous example involves the proof that 2 {\displaystyle {\sqrt {2}}} is an irrational number :
To paraphrase: if one could write 2 {\displaystyle {\sqrt {2}}} as a fraction , this fraction could never be written in lowest terms, since 2 could always be factored from numerator and denominator.
Proof by construction, or proof by example, is the construction of a concrete example with a property to show that something having that property exists. Joseph Liouville , for instance, proved the existence of transcendental numbers by constructing an explicit example . It can also be used to construct a counterexample to disprove a proposition that all elements have a certain property.
In proof by exhaustion, the conclusion is established by dividing it into a finite number of cases and proving each one separately. The number of cases sometimes can become very large. For example, the first proof of the four color theorem was a proof by exhaustion with 1,936 cases. This proof was controversial because the majority of the cases were checked by a computer program, not by hand. [ 18 ]
A closed chain inference shows that a collection of statements are pairwise equivalent.
In order to prove that the statements φ 1 , … , φ n {\displaystyle \varphi _{1},\ldots ,\varphi _{n}} are each pairwise equivalent, proofs are given for the implications φ 1 ⇒ φ 2 {\displaystyle \varphi _{1}\Rightarrow \varphi _{2}} , φ 2 ⇒ φ 3 {\displaystyle \varphi _{2}\Rightarrow \varphi _{3}} , … {\displaystyle \dots } , φ n − 1 ⇒ φ n {\displaystyle \varphi _{n-1}\Rightarrow \varphi _{n}} and φ n ⇒ φ 1 {\displaystyle \varphi _{n}\Rightarrow \varphi _{1}} . [ 19 ] [ 20 ]
The pairwise equivalence of the statements then results from the transitivity of the material conditional .
A probabilistic proof is one in which an example is shown to exist, with certainty, by using methods of probability theory . Probabilistic proof, like proof by construction, is one of many ways to prove existence theorems .
In the probabilistic method, one seeks an object having a given property, starting with a large set of candidates. One assigns a certain probability for each candidate to be chosen, and then proves that there is a non-zero probability that a chosen candidate will have the desired property. This does not specify which candidates have the property, but the probability could not be positive without at least one.
A probabilistic proof is not to be confused with an argument that a theorem is 'probably' true, a 'plausibility argument'. The work toward the Collatz conjecture shows how far plausibility is from genuine proof, as does the disproof of the Mertens conjecture . While most mathematicians do not think that probabilistic evidence for the properties of a given object counts as a genuine mathematical proof, a few mathematicians and philosophers have argued that at least some types of probabilistic evidence (such as Rabin's probabilistic algorithm for testing primality ) are as good as genuine mathematical proofs. [ 21 ] [ 22 ]
A combinatorial proof establishes the equivalence of different expressions by showing that they count the same object in different ways. Often a bijection between two sets is used to show that the expressions for their two sizes are equal. Alternatively, a double counting argument provides two different expressions for the size of a single set, again showing that the two expressions are equal.
A nonconstructive proof establishes that a mathematical object with a certain property exists—without explaining how such an object can be found. Often, this takes the form of a proof by contradiction in which the nonexistence of the object is proved to be impossible. In contrast, a constructive proof establishes that a particular object exists by providing a method of finding it. The following famous example of a nonconstructive proof shows that there exist two irrational numbers a and b such that a b {\displaystyle a^{b}} is a rational number . This proof uses that 2 {\displaystyle {\sqrt {2}}} is irrational (an easy proof is known since Euclid ), but not that 2 2 {\displaystyle {\sqrt {2}}^{\sqrt {2}}} is irrational (this is true, but the proof is not elementary).
The expression "statistical proof" may be used technically or colloquially in areas of pure mathematics , such as involving cryptography , chaotic series , and probabilistic number theory or analytic number theory . [ 23 ] [ 24 ] [ 25 ] It is less commonly used to refer to a mathematical proof in the branch of mathematics known as mathematical statistics . See also the " Statistical proof using data " section below.
Until the twentieth century it was assumed that any proof could, in principle, be checked by a competent mathematician to confirm its validity. [ 7 ] However, computers are now used both to prove theorems and to carry out calculations that are too long for any human or team of humans to check; the first proof of the four color theorem is an example of a computer-assisted proof. Some mathematicians are concerned that the possibility of an error in a computer program or a run-time error in its calculations calls the validity of such computer-assisted proofs into question. In practice, the chances of an error invalidating a computer-assisted proof can be reduced by incorporating redundancy and self-checks into calculations, and by developing multiple independent approaches and programs. Errors can never be completely ruled out in case of verification of a proof by humans either, especially if the proof contains natural language and requires deep mathematical insight to uncover the potential hidden assumptions and fallacies involved.
A statement that is neither provable nor disprovable from a set of axioms is called undecidable (from those axioms). One example is the parallel postulate , which is neither provable nor refutable from the remaining axioms of Euclidean geometry .
Mathematicians have shown there are many statements that are neither provable nor disprovable in Zermelo–Fraenkel set theory with the axiom of choice (ZFC), the standard system of set theory in mathematics (assuming that ZFC is consistent); see List of statements undecidable in ZFC .
Gödel's (first) incompleteness theorem shows that many axiom systems of mathematical interest will have undecidable statements.
While early mathematicians such as Eudoxus of Cnidus did not use proofs, from Euclid to the foundational mathematics developments of the late 19th and 20th centuries, proofs were an essential part of mathematics. [ 26 ] With the increase in computing power in the 1960s, significant work began to be done investigating mathematical objects beyond the proof-theorem framework, [ 27 ] in experimental mathematics . Early pioneers of these methods intended the work ultimately to be resolved into a classical proof-theorem framework, e.g. the early development of fractal geometry , [ 28 ] which was ultimately so resolved.
Although not a formal proof, a visual demonstration of a mathematical theorem is sometimes called a " proof without words ". The left-hand picture below is an example of a historic visual proof of the Pythagorean theorem in the case of the (3,4,5) triangle .
Some illusory visual proofs, such as the missing square puzzle , can be constructed in a way which appear to prove a supposed mathematical fact but only do so by neglecting tiny errors (for example, supposedly straight lines which actually bend slightly) which are unnoticeable until the entire picture is closely examined, with lengths and angles precisely measured or calculated.
An elementary proof is a proof which only uses basic techniques. More specifically, the term is used in number theory to refer to proofs that make no use of complex analysis . For some time it was thought that certain theorems, like the prime number theorem , could only be proved using "higher" mathematics. However, over time, many of these results have been reproved using only elementary techniques.
A particular way of organising a proof using two parallel columns is often used as a mathematical exercise in elementary geometry classes in the United States. [ 29 ] The proof is written as a series of lines in two columns. In each line, the left-hand column contains a proposition, while the right-hand column contains a brief explanation of how the corresponding proposition in the left-hand column is either an axiom, a hypothesis, or can be logically derived from previous propositions. The left-hand column is typically headed "Statements" and the right-hand column is typically headed "Reasons". [ 30 ]
The expression "mathematical proof" is used by lay people to refer to using mathematical methods or arguing with mathematical objects , such as numbers, to demonstrate something about everyday life, or when data used in an argument is numerical. It is sometimes also used to mean a "statistical proof" (below), especially when used to argue from data.
"Statistical proof" from data refers to the application of statistics, data analysis , or Bayesian analysis to infer propositions regarding the probability of data. While using mathematical proof to establish theorems in statistics, it is usually not a mathematical proof in that the assumptions from which probability statements are derived require empirical evidence from outside mathematics to verify. In physics, in addition to statistical methods, "statistical proof" can refer to the specialized mathematical methods of physics applied to analyze data in a particle physics experiment or observational study in physical cosmology . "Statistical proof" may also refer to raw data or a convincing diagram involving data, such as scatter plots , when the data or diagram is adequately convincing without further analysis.
Proofs using inductive logic , while considered mathematical in nature, seek to establish propositions with a degree of certainty, which acts in a similar manner to probability , and may be less than full certainty . Inductive logic should not be confused with mathematical induction .
Bayesian analysis uses Bayes' theorem to update a person's assessment of likelihoods of hypotheses when new evidence or information is acquired.
Psychologism views mathematical proofs as psychological or mental objects. Mathematician philosophers, such as Leibniz , Frege , and Carnap have variously criticized this view and attempted to develop a semantics for what they considered to be the language of thought , whereby standards of mathematical proof might be applied to empirical science .
Philosopher-mathematicians such as Spinoza have attempted to formulate philosophical arguments in an axiomatic manner, whereby mathematical proof standards could be applied to argumentation in general philosophy. Other mathematician-philosophers have tried to use standards of mathematical proof and reason, without empiricism, to arrive at statements outside of mathematics, but having the certainty of propositions deduced in a mathematical proof, such as Descartes ' cogito argument.
Sometimes, the abbreviation "Q.E.D." is written to indicate the end of a proof. This abbreviation stands for "quod erat demonstrandum" , which is Latin for "that which was to be demonstrated" . A more common alternative is to use a square or a rectangle, such as □ or ∎, known as a " tombstone " or "halmos" after its eponym Paul Halmos . Often, "which was to be shown" is verbally stated when writing "QED", "□", or "∎" during an oral presentation. Unicode explicitly provides the "end of proof" character, U+220E (∎) (220E(hex) = 8718(dec)) . | https://en.wikipedia.org/wiki/Mathematical_proof |
Mathematical psychology is an approach to psychological research that is based on mathematical modeling of perceptual, thought , cognitive and motor processes, and on the establishment of law-like rules that relate quantifiable stimulus characteristics with quantifiable behavior (in practice often constituted by task performance). The mathematical approach is used with the goal of deriving hypotheses that are more exact and thus yield stricter empirical validations. There are five major research areas in mathematical psychology: learning and memory , perception and psychophysics , choice and decision-making , language and thinking , and measurement and scaling . [ 1 ]
Although psychology, as an independent subject of science, is a more recent discipline than physics , [ 2 ] the application of mathematics to psychology has been done in the hope of emulating the success of this approach in the physical sciences , which dates back to at least the seventeenth century . [ 3 ] Mathematics in psychology is used extensively roughly in two areas: one is the mathematical modeling of psychological theories and experimental phenomena, which leads to mathematical psychology; the other is the statistical approach of quantitative measurement practices in psychology, which leads to psychometrics . [ 2 ]
As quantification of behavior is fundamental in this endeavor, the theory of measurement is a central topic in mathematical psychology. Mathematical psychology is therefore closely related to psychometrics. However, where psychometrics is concerned with individual differences (or population structure) in mostly static variables, mathematical psychology focuses on process models of perceptual, cognitive and motor processes as inferred from the 'average individual'. Furthermore, where psychometrics investigates the stochastic dependence structure between variables as observed in the population, mathematical psychology almost exclusively focuses on the modeling of data obtained from experimental paradigms and is therefore even more closely related to experimental psychology , cognitive psychology , and psychonomics . Like computational neuroscience and econometrics , mathematical psychology theory often uses statistical optimality as a guiding principle, assuming that the human brain has evolved to solve problems in an optimized way. Central themes from cognitive psychology (e.g., limited vs. unlimited processing capacity, serial vs. parallel processing) and their implications are central in rigorous analysis in mathematical psychology.
Mathematical psychologists are active in many fields of psychology, especially in psychophysics, sensation and perception , problem solving , decision-making , learning , memory , language , and the quantitative analysis of behavior , and contribute to the work of other subareas of psychology such as clinical psychology , social psychology , educational psychology , and psychology of music .
Choice and decision making theory are rooted in the development of statistics theory. In the mid 1600s, Blaise Pascal considered situations in gambling and further extended to Pascal's wager. [ 4 ] In the 18th century, Nicolas Bernoulli proposed the St. Petersburg Paradox in decision making, Daniel Bernoulli gave a solution and Laplace proposed a modification to the solution later on. In 1763, Bayes published the paper " An Essay Towards Solving a Problem in the Doctrine of Chances ", which is the milestone of Bayesian statistics.
Robert Hooke worked on modeling human memory, which is a precursor of the study of memory.
The research developments in Germany and England in the 19th century made psychology a new academic subject. Since the German approach emphasized experiments in the investigation of the psychological processes that all humans share and the English approach was in the measurement of individual differences, the applications of mathematics were also different.
In Germany, Wilhelm Wundt established the first experimental psychology laboratory. The math in German psychology is mainly applied in sensory and psychophysics. Ernst Weber (1795–1878) created the first mathematical law of the mind, Weber's law , based on a variety of experiments. Gustav Fechner (1801–1887) contributed theories in sensations and perceptions and one of them is the Fechner's law , which modifies Weber's law.
Mathematical modeling has a long history in psychology starting in the 19th century with Ernst Weber (1795–1878) and Gustav Fechner (1801–1887) being among the first to apply functional equations to psychological processes. They thereby established the fields of experimental psychology in general, and that of psychophysics in particular.
Researchers in astronomy in the 19th century were mapping distances between stars by denoting the exact time of a star's passing of a cross-hair on a telescope. For lack of the automatic registration instruments of the modern era, these time measurements relied entirely on human response speed. It had been noted that there were small systematic differences in the times measured by different astronomers, and these were first systematically studied by German astronomer Friedrich Bessel (1782–1846). Bessel constructed personal equations from measurements of basic response speed that would cancel out individual differences from the astronomical calculations. Independently, physicist Hermann von Helmholtz measured reaction times to determine nerve conduction speed, developed resonance theory of hearing and the Young-Helmholtz theory of color vision.
These two lines of work came together in the research of Dutch physiologist F. C. Donders and his student J. J. de Jaager , who recognized the potential of reaction times for more or less objectively quantifying the amount of time elementary mental operations required. Donders envisioned the employment of his mental chronometry to scientifically infer the elements of complex cognitive activity by measurement of simple reaction time [ 5 ]
Although there are developments in sensation and perception, Johann Herbart developed a system of mathematical theories in cognitive area to understand the mental process of consciousness.
The origin of English psychology can be traced to the theory of evolution by Darwin. But the emergence of English psychology is because of Francis Galton , who interested in individual differences between humans on psychological variables. The math in English psychology is mainly statistics and the work and methods of Galton is the foundation of psychometrics .
Galton introduced bivariate normal distribution in modeling the traits of the same individual, he also investigated measurement error and built his own model, and he also developed a stochastic branching process to examine the extinction of family names. There is also a tradition of the interest in studying intelligence in English psychology started from Galton. James McKeen Cattell and Alfred Binet developed tests of intelligence.
The first psychological laboratory was established in Germany by Wilhelm Wundt , who amply used Donders' ideas. However, findings that came from the laboratory were hard to replicate and this was soon attributed to the method of introspection that Wundt introduced. Some of the problems resulted from individual differences in response speed found by astronomers. Although Wundt did not seem to take interest in these individual variations and kept his focus on the study of the general human mind , Wundt's U.S. student James McKeen Cattell was fascinated by these differences and started to work on them during his stay in England.
The failure of Wundt's method of introspection led to the rise of different schools of thought. Wundt's laboratory was directed towards conscious human experience, in line with the work of Fechner and Weber on the intensity of stimuli. In the United Kingdom, under the influence of the anthropometric developments led by Francis Galton , interest focussed on individual differences between humans on psychological variables, in line with the work of Bessel. Cattell soon adopted the methods of Galton and helped laying the foundation of psychometrics.
Many statistical methods were developed even before the 20th century: Charles Spearman invented factor analysis which studies individual differences by the variance and covariance. German psychology and English psychology have been combined and taken over by the United States. The statistical methods dominated the field during the beginning of the century. There are two important statistical developments: Structural Equation Modeling (SEM) and analysis of variance (ANOVA). Since factor analysis unable to make causal inferences, the method of structural equation modeling was developed by Sewall Wright to correlational data to infer causality, which is still a major research area today. Those statistical methods formed psychometrics. The Psychometric Society was established in 1935 and the journal Psychometrika was published since 1936.
In the United States, behaviorism arose in opposition to introspectionism and associated reaction-time research, and turned the focus of psychological research entirely to learning theory. [ 5 ] In Europe introspection survived in Gestalt psychology . Behaviorism dominated American psychology until the end of the Second World War , and largely refrained from inference on mental processes. Formal theories were mostly absent (except for vision and hearing ).
During the war, developments in engineering , mathematical logic and computability theory , computer science and mathematics , and the military need to understand human performance and limitations , brought together experimental psychologists, mathematicians, engineers, physicists, and economists. Out of this mix of different disciplines mathematical psychology arose. Especially the developments in signal processing , information theory , linear systems and filter theory , game theory , stochastic processes and mathematical logic gained a large influence on psychological thinking. [ 5 ] [ 6 ]
Two seminal papers on learning theory in Psychological Review helped to establish the field in a world that was still dominated by behaviorists: A paper by Bush and Mosteller instigated the linear operator approach to learning, [ 7 ] and a paper by Estes that started the stimulus sampling tradition in psychological theorizing. [ 8 ] These two papers presented the first detailed formal accounts of data from learning experiments.
Mathematical modeling of learning process were greatly developed in the 1950s as the behavioral learning theory was flourishing. One development is the stimulus sampling theory by Williams K. Estes , the other is linear operator models by Robert R. Bush, and Frederick Mosteller .
Signal processing and detection theory are broadly used in perception, psychophysics and nonsensory area of cognition. Von Neumann 's book The Theory of Games and Economic Behavior establish the importance of game theory and decision making. R. Duncan Luce and Howard Raiffa contributed to the choice and decision making area.
The area of language and thinking comes into the spotlight with the development of computer science and linguistics, especially information theory and computation theory. Chomsky proposed the model of linguistics and computational hierarchy theory. Allen Newell and Herbert Simon proposed the model of human solving problems. The development in artificial intelligence and human computer interface are active areas in both computer science and psychology.
Before the 1950s, psychometricians emphasized the structure of measurement error and the development of high-power statistical methods to the measurement of psychological quantities but little of the psychometric work concerned the structure of the psychological quantities being measured or the cognitive factors behind the response data. Scott and Suppes studied relationship between the structure of data and the structure of numerical systems that represent the data. [ 9 ] Coombs constructed formal cognitive models of the respondent in a measurement situation rather than statistical data processing algorithms, for example the unfolding model. [ 10 ] [ 11 ] Another breakthrough is the development of a new form of the psychophysical scaling function along with new methods of collecting psychophysical data, like Stevens' power law. [ 12 ]
The 1950s saw a surge in mathematical theories of psychological processes, including Luce's theory of choice , Tanner and Swets' introduction of signal detection theory for human stimulus detection, and Miller's approach to information processing. [ 6 ] By the end of the 1950s, the number of mathematical psychologists had increased from a handful by more than a tenfold, not counting psychometricians. Most of these were concentrated at the Indiana University, Michigan, Pennsylvania, and Stanford. [ 6 ] [ 13 ] Some of these were regularly invited by the U.S. Social Science Research Counsel to teach in summer workshops in mathematics for social scientists at Stanford University, promoting collaboration.
To better define the field of mathematical psychology, the mathematical models of the 1950s were brought together in sequence of volumes edited by Luce, Bush, and Galanter: Two readings [ 14 ] and three handbooks. [ 15 ] This series of volumes turned out to be helpful in the development of the field. In the summer of 1963 the need was felt for a journal for theoretical and mathematical studies in all areas in psychology, excluding work that was mainly factor analytical. An initiative led by R. C. Atkinson , R. R. Bush , W. K. Estes , R. D. Luce , and P. Suppes resulted in the appearance of the first issue of the Journal of Mathematical Psychology in January 1964. [ 13 ]
Under the influence of developments in computer science, logic, and language theory, in the 1960s modeling gravitated towards computational mechanisms and devices. Examples of the latter constitute so called cognitive architectures (e.g., production rule systems , ACT-R ) as well as connectionist systems or neural networks . [ citation needed ]
Important mathematical expressions for relations between physical characteristics of stimuli and subjective perception are Weber–Fechner law , Ekman's law , Stevens's power law , Thurstone's law of comparative judgment , the theory of signal detection (borrowed from radar engineering), the matching law , and Rescorla–Wagner rule for classical conditioning. While the first three laws are all deterministic in nature, later established relations are more fundamentally stochastic . This has been a general theme in the evolution in mathematical modeling of psychological processes: from deterministic relations as found in classical physics to inherently stochastic models. [ citation needed ]
Source: [ 16 ]
Developmental psychology is concerned not only with describing the characteristics of psychological change over time but also seeks to explain the principles and internal workings underlying these changes. Psychologists have attempted to better understand these factors by using models . A model must simply account for the means by which a process takes place. This is sometimes done in reference to changes in the brain that may correspond to changes in behavior over the course of the development.
Mathematical modeling is useful in developmental psychology for implementing theory in a precise and easy-to-study manner, allowing generation, explanation, integration, and prediction of diverse phenomena. Several modeling techniques are applied to development: symbolic , connectionist ( neural network ), or dynamical systems models.
Central journals are the Journal of Mathematical Psychology and the British Journal of Mathematical and Statistical Psychology . There are three annual conferences in the field, the annual meeting of the Society for Mathematical Psychology in the U.S, the annual European Mathematical Psychology Group meeting in Europe, and the Australasian Mathematical Psychology conference. | https://en.wikipedia.org/wiki/Mathematical_psychology |
The Mathematical Sciences are a group of areas of study that includes, in addition to mathematics , those academic disciplines that are primarily mathematical in nature but may not be universally considered subfields of mathematics proper.
Statistics , for example, is mathematical in its methods but grew out of bureaucratic and scientific observations , [ 1 ] which merged with inverse probability and then grew through applications in some areas of physics , biometrics , and the social sciences to become its own separate, though closely allied, field. Theoretical astronomy , theoretical physics , theoretical and applied mechanics , continuum mechanics , mathematical chemistry , actuarial science , computer science , computational science , data science , operations research , quantitative biology , control theory , econometrics , geophysics and mathematical geosciences are likewise other fields often considered part of the mathematical sciences.
Some institutions offer degrees in mathematical sciences (e.g. the United States Military Academy , Stanford University , and University of Khartoum ) or applied mathematical sciences (for example, the University of Rhode Island ). | https://en.wikipedia.org/wiki/Mathematical_sciences |
Mathematical tables are lists of numbers showing the results of a calculation with varying arguments. Trigonometric tables were used in ancient Greece and India for applications to astronomy and celestial navigation , and continued to be widely used until electronic calculators became cheap and plentiful in the 1970s, in order to simplify and drastically speed up computation . Tables of logarithms and trigonometric functions were common in math and science textbooks, and specialized tables were published for numerous applications.
The first tables of trigonometric functions known to be made were by Hipparchus (c.190 – c.120 BCE) and Menelaus (c.70–140 CE), but both have been lost. Along with the surviving table of Ptolemy (c. 90 – c.168 CE), they were all tables of chords and not of half-chords, that is, the sine function. [ 1 ] The table produced by the Indian mathematician Āryabhaṭa (476–550 CE) is considered the first sine table ever constructed. [ 1 ] Āryabhaṭa's table remained the standard sine table of ancient India. There were continuous attempts to improve the accuracy of this table, culminating in the discovery of the power series expansions of the sine and cosine functions by Madhava of Sangamagrama (c.1350 – c.1425), and the tabulation of a sine table by Madhava with values accurate to seven or eight decimal places.
Tables of common logarithms were used until the invention of computers and electronic calculators to do rapid multiplications, divisions, and exponentiations, including the extraction of n th roots.
Mechanical special-purpose computers known as difference engines were proposed in the 19th century to tabulate polynomial approximations of logarithmic functions – that is, to compute large logarithmic tables. This was motivated mainly by errors in logarithmic tables made by the human computers of the time. Early digital computers were developed during World War II in part to produce specialized mathematical tables for aiming artillery . From 1972 onwards, with the launch and growing use of scientific calculators , most mathematical tables went out of use.
One of the last major efforts to construct such tables was the Mathematical Tables Project that was started in the United States in 1938 as a project of the Works Progress Administration (WPA), employing 450 out-of-work clerks to tabulate higher mathematical functions. It lasted through World War II. [ 2 ]
Tables of special functions are still used. For example, the use of tables of values of the cumulative distribution function of the normal distribution – so-called standard normal tables – remains commonplace today, especially in schools, although the use of scientific and graphing calculators as well as spreadsheet and dedicated statistical software on personal computers is making such tables redundant.
Creating tables stored in random-access memory is a common code optimization technique in computer programming, where the use of such tables speeds up calculations in those cases where a table lookup is faster than the corresponding calculations (particularly if the computer in question doesn't have a hardware implementation of the calculations). In essence, one trades computing speed for the computer memory space required to store the tables.
Trigonometric calculations played an important role in the early study of astronomy. Early tables were constructed by repeatedly applying trigonometric identities (like the half-angle and angle-sum identities) to compute new values from old ones.
To compute the sine function of 75 degrees, 9 minutes, 50 seconds using a table of trigonometric functions such as the Bernegger table from 1619 illustrated above, one might simply round up to 75 degrees, 10 minutes and then find the 10 minute entry on the 75 degree page, shown above-right, which is 0.9666746.
However, this answer is only accurate to four decimal places. If one wanted greater accuracy, one could interpolate linearly as follows:
From the Bernegger table:
The difference between these values is 0.0000745.
Since there are 60 seconds in a minute of arc, we multiply the difference by 50/60 to get a correction of (50/60)*0.0000745 ≈ 0.0000621; and then add that correction to sin (75° 9′) to get :
A modern calculator gives sin(75° 9′ 50″) = 0.96666219991, so our interpolated answer is accurate to the 7-digit precision of the Bernegger table.
For tables with greater precision (more digits per value), higher order interpolation may be needed to get full accuracy. [ 3 ] In the era before electronic computers, interpolating table data in this manner was the only practical way to get high accuracy values of mathematical functions needed for applications such as navigation, astronomy and surveying.
To understand the importance of accuracy in applications like navigation note that at sea level one minute of arc along the Earth's equator or a meridian (indeed, any great circle ) equals one nautical mile (approximately 1.852 km or 1.151 mi).
Tables containing common logarithms (base-10) were extensively used in computations prior to the advent of electronic calculators and computers because logarithms convert problems of multiplication and division into much easier addition and subtraction problems. Base-10 logarithms have an additional property that is unique and useful: The common logarithm of numbers greater than one that differ only by a factor of a power of ten all have the same fractional part, known as the mantissa . Tables of common logarithms typically included only the mantissas ; the integer part of the logarithm, known as the characteristic , could easily be determined by counting digits in the original number. A similar principle allows for the quick calculation of logarithms of positive numbers less than 1. Thus a single table of common logarithms can be used for the entire range of positive decimal numbers. [ 4 ] See common logarithm for details on the use of characteristics and mantissas.
In 1544, Michael Stifel published Arithmetica integra , which contains a table of integers and powers of 2 that has been considered an early version of a logarithmic table. [ 5 ] [ 6 ] [ 7 ]
The method of logarithms was publicly propounded by John Napier in 1614, in a book entitled Mirifici Logarithmorum Canonis Descriptio ( Description of the Wonderful Rule of Logarithms ). [ 8 ] The book contained fifty-seven pages of explanatory matter and ninety pages of tables related to natural logarithms . The English mathematician Henry Briggs visited Napier in 1615, and proposed a re-scaling of Napier's logarithms to form what is now known as the common or base-10 logarithms. Napier delegated to Briggs the computation of a revised table. In 1617, they published Logarithmorum Chilias Prima ("The First Thousand Logarithms"), which gave a brief account of logarithms and a table for the first 1000 integers calculated to the 14th decimal place. Prior to Napier's invention, there had been other techniques of similar scopes, such as the use of tables of progressions, extensively developed by Jost Bürgi around 1600. [ 9 ] [ 10 ]
The computational advance available via common logarithms, the converse of powered numbers or exponential notation , was such that it made calculations by hand much quicker. | https://en.wikipedia.org/wiki/Mathematical_table |
In physics and cosmology , the mathematical universe hypothesis ( MUH ), also known as the ultimate ensemble theory , is a speculative " theory of everything " (TOE) proposed by cosmologist Max Tegmark . [ 1 ] [ 2 ] According to the hypothesis, the universe is a mathematical object in and of itself. Tegmark extends this idea to hypothesize that all mathematical objects exist, which he describes as a form of Platonism or Modal realism .
The hypothesis has proven controversial. Jürgen Schmidhuber argues that it is not possible to assign an equal weight or probability to all mathematical objects a priori due to there being infinitely many of them. Physicists Piet Hut and Mark Alford have suggested that the idea is incompatible with Gödel's first incompleteness theorem .
Tegmark replies that not only is the universe mathematical, but it is also computable .
In 2014, Tegmark published a popular science book about the topic, titled Our Mathematical Universe .
Tegmark's MUH is the hypothesis that our external physical reality is a mathematical structure. [ 3 ] That is, the physical universe is not merely described by mathematics, but is mathematics — specifically, a mathematical structure . Mathematical existence equals physical existence, and all structures that exist mathematically exist physically as well. Observers, including humans, are "self-aware substructures (SASs)". In any mathematical structure complex enough to contain such substructures, they "will subjectively perceive themselves as existing in a physically 'real' world". [ 4 ]
The theory can be considered a form of Pythagoreanism or Platonism in that it proposes the existence of mathematical entities; a form of mathematicism in that it denies that anything exists except mathematical objects; and a formal expression of ontic structural realism .
Tegmark claims that the hypothesis has no free parameters and is not observationally ruled out. Thus, he reasons, it is preferred over other theories-of-everything by Occam's Razor . Tegmark also considers augmenting the MUH with a second assumption, the computable universe hypothesis ( CUH ), which says that the mathematical structure that is our external physical reality is defined by computable functions . [ 5 ]
The MUH is related to Tegmark's categorization of four levels of the multiverse . [ 6 ] This categorization posits a nested hierarchy of increasing diversity, with worlds corresponding to different sets of initial conditions (level 1), physical constants (level 2), quantum branches (level 3), and altogether different equations or mathematical structures (level 4).
Andreas Albrecht when at Imperial College in London called it a "provocative" solution to one of the central problems facing physics. Although he "wouldn't dare" go so far as to say he believes it, he noted that "it's actually quite difficult to construct a theory where everything we see is all there is". [ 7 ]
Jürgen Schmidhuber [ 8 ] argues that "Although Tegmark suggests that '... all mathematical structures are a priori given equal statistical weight,' there is no way of assigning equal non-vanishing probability to all (infinitely many) mathematical structures." Schmidhuber puts forward a more restricted ensemble which admits only universe representations describable by constructive mathematics , that is, computer programs ; e.g., the Global Digital Mathematics Library and Digital Library of Mathematical Functions , linked open data representations of formalized fundamental theorems intended to serve as building blocks for additional mathematical results. He explicitly includes universe representations describable by non-halting programs whose output bits converge after finite time, although the convergence time itself may not be predictable by a halting program, due to the undecidability of the halting problem . [ 8 ] [ 9 ]
In response, Tegmark notes [ 3 ] : sec. V.E that a constructive mathematics formalized measure of free parameter variations of physical dimensions, constants, and laws over all universes has not yet been constructed for the string theory landscape either, so this should not be regarded as a "show-stopper".
It has also been suggested that the MUH is inconsistent with Gödel's incompleteness theorem . In a three-way debate between Tegmark and fellow physicists Piet Hut and Mark Alford, [ 10 ] the "secularist" (Alford) states that "the methods allowed by formalists cannot prove all the theorems in a sufficiently powerful system... The idea that math is 'out there' is incompatible with the idea that it consists of formal systems."
Tegmark's response [ 10 ] : sec VI.A.1 is to offer a new hypothesis "that only Gödel-complete ( fully decidable ) mathematical structures have physical existence. This drastically shrinks the Level IV multiverse, essentially placing an upper limit on complexity, and may have the attractive side effect of explaining the relative simplicity of our universe." Tegmark goes on to note that although conventional theories in physics are Gödel-undecidable, the actual mathematical structure describing our world could still be Gödel-complete, and "could in principle contain observers capable of thinking about Gödel-incomplete mathematics, just as finite-state digital computers can prove certain theorems about Gödel-incomplete formal systems like Peano arithmetic ." In [ 3 ] : sec. VII he gives a more detailed response, proposing as an alternative to MUH the more restricted "Computable Universe Hypothesis" (CUH) which only includes mathematical structures that are simple enough that Gödel's theorem does not require them to contain any undecidable or uncomputable theorems. Tegmark admits that this approach faces "serious challenges", including (a) it excludes much of the mathematical landscape; (b) the measure on the space of allowed theories may itself be uncomputable; and (c) "virtually all historically successful theories of physics violate the CUH".
Stoeger, Ellis, and Kircher [ 11 ] : sec. 7 note that in a true multiverse theory, "the universes are then completely disjoint and nothing that happens in any one of them is causally linked to what happens in any other one. This lack of any causal connection in such multiverses really places them beyond any scientific support". Ellis [ 12 ] : 29 specifically criticizes the MUH, stating that an infinite ensemble of completely disconnected universes is "completely untestable, despite hopeful remarks sometimes made, see, e.g., Tegmark (1998)." Tegmark maintains that MUH is testable , stating that it predicts (a) that "physics research will uncover mathematical regularities in nature", and (b) by assuming that we occupy a typical member of the multiverse of mathematical structures, one could "start testing multiverse predictions by assessing how typical our universe is". [ 3 ] : sec. VIII.C
The MUH is based on the radical Platonist view that math is an external reality. [ 3 ] : sec V.C However, Jannes [ 13 ] argues that "mathematics is at least in part a human construction", on the basis that if it is an external reality, then it should be found in some other animals as well: "Tegmark argues that, if we want to give a complete description of reality, then we will need a language independent of us humans, understandable for non-human sentient entities, such as aliens and future supercomputers". Brian Greene argues similarly: [ 14 ] : 299 "The deepest description of the universe should not require concepts whose meaning relies on human experience or interpretation. Reality transcends our existence and so shouldn't, in any fundamental way, depend on ideas of our making."
However, there are many non-human entities, plenty of which are intelligent, and many of which can apprehend, memorise, compare and even approximately add numerical quantities. Several animals have also passed the mirror test of self-consciousness . But a few surprising examples of mathematical abstraction notwithstanding (for example, chimpanzees can be trained to carry out symbolic addition with digits, or the report of a parrot understanding a "zero-like concept"), all examples of animal intelligence with respect to mathematics are limited to basic counting abilities. He adds, "non-human intelligent beings should exist that understand the language of advanced mathematics. However, none of the non-human intelligent beings that we know of confirm the status of (advanced) mathematics as an objective language." In the paper "On Math, Matter and Mind" the secularist viewpoint examined argues [ 10 ] : sec. VI.A that math is evolving over time, there is "no reason to think it is converging to a definite structure, with fixed questions and established ways to address them", and also that "The Radical Platonist position is just another metaphysical theory like solipsism... In the end the metaphysics just demands that we use a different language for saying what we already knew." Tegmark responds [ 10 ] : sec VI.A.1 that "The notion of a mathematical structure is rigorously defined in any book on Model Theory ", and that non-human mathematics would only differ from our own "because we are uncovering a different part of what is in fact a consistent and unified picture, so math is converging in this sense." In his 2014 book on the MUH, Tegmark argues that the resolution is not that we invent the language of mathematics, but that we discover the structure of mathematics.
Don Page has argued [ 15 ] : sec 4 that "At the ultimate level, there can be only one world and, if mathematical structures are broad enough to include all possible worlds or at least our own, there must be one unique mathematical structure that describes ultimate reality. So I think it is logical nonsense to talk of Level 4 in the sense of the co-existence of all mathematical structures." This means there can only be one mathematical corpus. Tegmark responds [ 3 ] : sec. V.E that "This is less inconsistent with Level IV than it may sound, since many mathematical structures decompose into unrelated substructures, and separate ones can be unified."
Alexander Vilenkin comments [ 16 ] : Ch. 19, p. 203 that "The number of mathematical structures increases with increasing complexity, suggesting that 'typical' structures should be horrendously large and cumbersome. This seems to be in conflict with the beauty and simplicity of the theories describing our world". He goes on to note [ 16 ] : footnote 8, p. 222 that Tegmark's solution to this problem, the assigning of lower "weights" to the more complex structures [ 6 ] : sec. V.B seems arbitrary ("Who determines the weights?") and may not be logically consistent ("It seems to introduce an additional mathematical structure, but all of them are supposed to be already included in the set").
Tegmark has been criticized as misunderstanding the nature and application of Occam's razor ; Massimo Pigliucci reminds that "Occam's razor is just a useful heuristic , it should never be used as the final arbiter to decide which theory is to be favored". [ 17 ] | https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis |
Mathematical phenomena can be understood and explored via visualization . Classically, this consisted of two-dimensional drawings or building three-dimensional models (particularly plaster models in the 19th and early 20th century). In contrast, today it most frequently consists of using computers to make static two- or three-dimensional drawings, animations, or interactive programs. Writing programs to visualize mathematics is an aspect of computational geometry .
Mathematical visualization is used throughout mathematics, particularly in the fields of geometry and analysis . Notable examples include plane curves , space curves , polyhedra , ordinary differential equations , partial differential equations (particularly numerical solutions, as in fluid dynamics or minimal surfaces such as soap films ), conformal maps , fractals , and chaos .
Geometry can be defined as the study of shapes their size, angles, dimensions and proportions [ 1 ]
In complex analysis , functions of the complex plane are inherently 4-dimensional, but there is no natural geometric projection into lower dimensional visual representations. Instead, colour vision is exploited to capture dimensional information using techniques such as domain coloring .
Many people have a vivid “mind’s eye,” but a team of British scientists has found that tens of millions of people cannot conjure images. The lack of a mental camera is known as aphantasia, and millions more experience extraordinarily strong mental imagery, called hyperphantasia. Researchers are studying how these two conditions arise through changes in the wiring of the brain.
Visualization played an important role at the beginning of topological knot theory, when polyhedral decompositions were used to compute the homology of covering spaces of knots. Extending to 3 dimensions the physically impossible Riemann surfaces used to classify all closed orientable 2-manifolds, Heegaard's 1898 thesis "looked at" similar structures for functions of two complex variables, taking an imaginary 4-dimensional surface in Euclidean 6-space (corresponding to the function f=x^2-y^3) and projecting it stereographically (with multiplicities) onto the 3-sphere. In the 1920s Alexander and Briggs used this technique to compute the homology of cyclic branched covers of knots with 8 or fewer crossings, successfully distinguishing them all from each other (and the unknot). By 1932 Reidemeister extended this to 9 crossings, relying on linking numbers between branch curves of non-cyclic knot covers. The fact that these imaginary objects have no "real" existence does not stand in the way of their usefulness for proving knots distinct. It was the key to Perko's 1973 discovery of the duplicate knot type in Little's 1899 table of 10-crossing knots.
Permutation groups have nice visualizations of their elements that assist in explaining their structure—e.g., the rotated and flipped regular p-gons that comprise the dihedral group of order 2p. They may be used to "see" the relationships among linking numbers between branch curves of dihedral covering spaces of knots and links. [ 3 ]
Stephen Wolfram 's book on cellular automata , A New Kind of Science (2002), is one of the most intensely visual books published in the field of mathematics. It has been criticized for being too heavily visual, with much information conveyed by pictures that do not have formal meaning. [ 5 ]
The cover of the journal The Notices of the American Mathematical Society regularly features a mathematical visualization. | https://en.wikipedia.org/wiki/Mathematical_visualization |
Mathematicism is 'the effort to employ the formal structure and rigorous method of mathematics as a model for the conduct of philosophy', [ 1 ] or the epistemological view that reality is fundamentally mathematical. [ 2 ] The term has been applied to a number of philosophers, including Pythagoras [ 3 ] and René Descartes [ 4 ] although the term was not used by themselves.
The role of mathematics in Western philosophy has grown and expanded from Pythagoras onwards. It is clear that numbers held a particular importance for the Pythagorean school , although it was the later work of Plato that attracts the label of mathematicism from modern philosophers. Furthermore it is René Descartes who provides the first mathematical epistemology which he describes as a mathesis universalis , and which is also referred to as mathematicism.
Although we do not have writings of Pythagoras himself, good evidence that he pioneered the concept of mathematicism is given by Plato, and summed up in the quotation often attributed to him that "everything is mathematics". Aristotle says of the Pythagorean school:
The first to devote themselves to mathematics and to make them progress were the so-called Pythagoreans. They, devoted to this study, believed that the principles of mathematics were also the principles of all things that be. Now, since the principles of mathematics are numbers, and they thought they found in numbers, more than in fire and earth and water, similarities with things that are and that become (they judged, for example, that justice was a particular property of numbers, the soul and mind another, opportunity another, and similarly, so to say, anything else), and since furthermore they saw expressed by numbers the properties and the ratios of harmony, since finally everything in nature appeared to them to be similar to numbers, and numbers appeared to be first among all there is in nature, they thought that the elements of numbers were the elements of all that there is, and that the whole world was harmony and number. And all the properties they could find in numbers and in musical chords, corresponding to properties and parts of the sky, and in general to the whole cosmic order, they gathered and adapted to it. And if something was missing, they made an effort to introduce it, so that their tractation be complete. To clarify with an example: since ten seems to be a perfect number and to contain in itself the whole nature of numbers, they said that the bodies that move in the sky are also ten: and since one can only see nine, they added as tenth the anti-Earth.
Further evidence for the views of Pythagoras and his school, although fragmentary and sometimes contradictory, comes from Alexander Polyhistor. Alexander tells us that central doctrines of the Pythagorieans were the harmony of numbers and the ideal that the mathematical world has primacy over, or can account for the existence of, the physical world. [ 5 ]
According to Aristotle, the Pythagoreans used mathematics for solely mystical reasons, devoid of practical application. [ 6 ] They believed that all things were made of numbers. [ 7 ] [ 8 ] The number one (the monad ) represented the origin of all things [ 9 ] and other numbers similarly had symbolic representations. Nevertheless modern scholars debate whether this numerology was taught by Pythagoras himself or whether it was original to the later philosopher of the Pythagorean school, Philolaus of Croton . [ 10 ]
Walter Burkert argues in his study Lore and Science in Ancient Pythagoreanism , that the only mathematics the Pythagoreans ever actually engaged in was simple, proofless arithmetic , [ 11 ] but that these arithmetic discoveries did contribute significantly to the beginnings of mathematics. [ 12 ]
The Pythagorian school influenced the work of Plato. Mathematical Platonism is the metaphysical view that (a) there are abstract mathematical objects whose existence is independent of us, and (b) there are true mathematical sentences that provide true descriptions of such objects. The independence of the mathematical objects is such that they are non physical and do not exist in space or time. Neither does their existence rely on thought or language. For this reason, mathematical proofs are discovered, not invented. The proof existed before its discovery, and merely became known to the one who discovered it. [ 13 ]
In summary, therefore, Mathematical Platonism can be reduced to three propositions:
It is again not clear the extent to which Plato held to these views himself but they were associated with the Platonist school. Nevertheless, this was a significant progression in the ideas of mathematicism. [ 13 ]
Markus Gabriel refers to Plato in his Fields of Sense: A New Realist Ontology , and in so doing provides a definition for mathematicism. He says:
Ultimately, set-theoretical ontology is a remainder of Platonic mathematicism. Let mathematicism from here on be the view that everything that exists can be studied mathematically either directly or indirectly. It is an instance of theory-reduction, that is, a claim to the effect that every vocabulary can be translated into that of mathematics such that this reduction grounds all derivative vocabulary and helps us understand it significantly better. [ 14 ]
He goes on, however, to show that the term need not be applied merely to the set-theroetical ontology that he takes issue with, but for other mathematical ontologies.
Set-theoretical ontology is just one instance of mathematicism. Depending on one's preferred candidate for the most fundamental theory of quantifiable structure, one can wind up with a graphtheoretical mathematicism, a set-theoretical, category-theoretical, or some other (maybe hybrid) form of mathematicism. However, mathematicism is metaphysics, and metaphysics need not be associated with ontology. [ 14 ]
Although mathematical methods of investigation have been used to establish meaning and analyse the world since Pythagoras, it was Descartes who pioneered the subject as epistemology , setting out Rules for the Direction of the Mind . He proposed that method, rather than intuition, should direct the mind, saying:
So blind is the curiosity with which mortals are possessed that they often direct their minds down untrodden paths, in the groundless hope that they will chance upon what they are seeking, rather like someone who is consumed with such a senseless desire to discover treasure that he continually roams the streets to see if he can find any that a passerby might have dropped [...]
By 'a method' I mean reliable rules which are easy to apply, and such that if one follows them exactly, one will never take what is false to be true or fruitlessly expend one's mental efforts, but will gradually and constantly increase one's knowledge till one arrives at a true understanding of everything within one's capacity
In the discussion of Rule Four , [ 16 ] Descartes' describes what he calls mathesis universalis :
[...] I began my investigation by inquiring what exactly is generally meant by the term 'mathematics' and why it is that, in addition to arithmetic and geometry, sciences such as astronomy, music, optics, mechanics, among others, are called branches of mathematics. [...] This made me realize that there must be a general science which explains all the points that can be raised concerning order and measure irrespective of the subject-matter, and that this science should be termed mathesis universalis — a venerable term with a well-established meaning — for it covers everything that entitles these other sciences to be called branches of mathematics. [...]
The concept of mathesis universalis was, for Descartes, a universal science modeled on mathematics. It is this mathesis universalis that is referred to when writers speak of Descartes' mathematicism. [ 4 ] Following Descartes, Leibniz attempted to derive connections between mathematical logic , algebra , infinitesimal calculus , combinatorics , and universal characteristics in an incomplete treatise titled " Mathesis Universalis ", published in 1695. [ citation needed ] Following on from Leibniz, Benedict de Spinoza and then various 20th century philosophers, including Bertrand Russell , Ludwig Wittgenstein , and Rudolf Carnap have attempted to elaborate and develop Leibniz's work on mathematical logic, syntactic systems and their calculi and to resolve problems in the field of metaphysics.
Leibniz attempted to work out the possible connections between mathematical logic , algebra , infinitesimal calculus , combinatorics , and universal characteristics in an incomplete treatise titled " Mathesis Universalis " in 1695.
In his account of mathesis universalis , Leibniz proposed a dual method of universal synthesis and analysis for the ascertaining truth , described in De Synthesi et Analysi universale seu Arte inveniendi et judicandi (1890). [ 18 ] [ 19 ]
One of the perhaps most prominent critics of the idea of mathesis universalis was Ludwig Wittgenstein and his philosophy of mathematics . [ 20 ] As anthropologist Emily Martin notes: [ 21 ]
Tackling mathematics, the realm of symbolic life perhaps most difficult to regard as contingent on social norms, Wittgenstein commented that people found the idea that numbers rested on conventional social understandings "unbearable".
The Principia Mathematica is a three-volume work on the foundations of mathematics written by the mathematicians Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. According to its introduction, this work had three aims:
There is no doubt that Principia Mathematica is of great importance in the history of mathematics and philosophy: as Irvine has noted, it sparked interest in symbolic logic and advanced the subject by popularizing it; it showcased the powers and capacities of symbolic logic; and it showed how advances in philosophy of mathematics and symbolic logic could go hand-in-hand with tremendous fruitfulness. [ 23 ] Indeed, the work was in part brought about by an interest in logicism , the view on which all mathematical truths are logical truths. It was in part thanks to the advances made in Principia Mathematica that, despite its defects, numerous advances in meta-logic were made, including Gödel's incompleteness theorems .
In The Order of Things , Michel Foucault discuses mathesis as the conjunction point in the ordering of simple natures and algebra, paralleling his concept of taxinomia . Though omitting explicit references to universality, Foucault uses the term to organise and interpret all of human science, as is evident in the full title of his book: " The Order of Things: An Archaeology of the Human Sciences ". [ 24 ]
Tim Maudlin 's mathematical universe hypothesis attempts to construct "a rigorous mathematical structure using primitive terms that give a natural fit with physics" [ citation needed ] and investigating why mathematics should provide such a powerful language for describing the physical world. [ 25 ] According to Maudlin, "the most satisfying possible answer to such a question is: because the physical world literally has a mathematical structure". | https://en.wikipedia.org/wiki/Mathematicism |
Mathematics, Form and Function , a book published in 1986 by Springer-Verlag , is a survey of the whole of mathematics , including its origins and deep structure, by the American mathematician Saunders Mac Lane .
Throughout his book, and especially in chapter I.11, Mac Lane informally discusses how mathematics is grounded in more ordinary concrete and abstract human activities. The following table is adapted from one given on p. 35 of Mac Lane (1986). The rows are very roughly ordered from most to least fundamental. For a bullet list that can be compared and contrasted with this table, see section 3 of Where Mathematics Comes From .
Also see the related diagrams appearing on the following pages of Mac Lane (1986): 149, 184, 306, 408, 416, 422-28.
Mac Lane (1986) cites a related monograph by Lars Gårding (1977).
Mac Lane cofounded category theory with Samuel Eilenberg , which enables a unified treatment of mathematical structures and of the relations among them, at the cost of breaking away from their cognitive grounding . Nevertheless, his views—however informal—are a valuable contribution to the philosophy and anthropology of mathematics. [ 2 ] His views anticipate, in some respects, the more detailed account of the cognitive basis of mathematics given by George Lakoff and Rafael E. Núñez in their Where Mathematics Comes From . Lakoff and Núñez argue that mathematics emerges via conceptual metaphors grounded in the human body , its motion through space and time , and in human sense perceptions. | https://en.wikipedia.org/wiki/Mathematics,_Form_and_Function |
The Mathematics Subject Classification ( MSC ) is an alphanumerical classification scheme that has collaboratively been produced by staff of, and based on the coverage of, the two major mathematical reviewing databases, Mathematical Reviews and Zentralblatt MATH . The MSC is used by many mathematics journals , which ask authors of research papers and expository articles to list subject codes from the Mathematics Subject Classification in their papers. The current version is MSC2020.
The MSC is a hierarchical scheme, with three levels of structure. A classification can be two, three or five digits long, depending on how many levels of the classification scheme are used.
The first level is represented by a two-digit number, the second by a letter, and the third by another two-digit number. For example:
At the top level, 63 mathematical disciplines are labeled with a unique two-digit number. In addition to the typical areas of mathematical research, there are top-level categories for " History and Biography ", " Mathematics Education ", and for the overlap with different sciences. Physics (i.e. mathematical physics) is particularly well represented in the classification scheme with a number of different categories including:
All valid MSC classification codes must have at least the first-level identifier.
The second-level codes are a single letter from the Latin alphabet. These represent specific areas covered by the first-level discipline. The second-level codes vary from discipline to discipline.
For example, for differential geometry, the top-level code is 53 , and the second-level codes are:
In addition, the special second-level code "-" is used for specific kinds of materials. These codes are of the form:
The second and third level of these codes are always the same - only the first level changes. For example, it is not valid to use 53- as a classification. Either 53 on its own or, better yet, a more specific code should be used.
Third-level codes are the most specific, usually corresponding to a specific kind of mathematical object or a well-known problem or research area.
The third-level code 99 exists in every category and means none of the above, but in this section .
The AMS recommends that papers submitted to its journals for publication have one primary classification and one or more optional secondary classifications. A typical MSC subject class line on a research paper looks like
MSC Primary 03C90; Secondary 03-02;
According to the American Mathematical Society (AMS) help page about MSC, [ 1 ] the MSC has been revised a number of times since 1940. Based on a scheme to organize AMS's Mathematical Offprint Service (MOS scheme), the AMS Classification was established for the classification of reviews in Mathematical Reviews in the 1960s. It saw various ad-hoc changes. Despite its shortcomings, Zentralblatt für Mathematik started to use it as well in the 1970s. In the late 1980s, a jointly revised scheme with more formal rules was agreed upon by Mathematical Reviews and Zentralblatt für Mathematik under the new name Mathematics Subject Classification. It saw various revisions as MSC1990 , MSC2000 and MSC2010 . [ 2 ] In July 2016, Mathematical Reviews and zbMATH started collecting input from the mathematical community on the next revision of MSC, [ 3 ] which was released as MSC2020 [ 4 ] in January 2020.
The original classification of older items has not been changed. This can sometimes make it difficult to search for older works dealing with particular topics. Changes at the first level involved the subjects with (present) codes 03, 08, 12-20, 28, 37, 51, 58, 74, 90, 91, 92.
For physics papers the Physics and Astronomy Classification Scheme (PACS) is often used. Due to the large overlap between mathematics and physics research it is quite common to see both PACS and MSC codes on research papers, particularly for multidisciplinary journals and repositories such as the arXiv .
The ACM Computing Classification System (CCS) is a similar hierarchical classification scheme for computer science . There is some overlap between the AMS and ACM classification schemes, in subjects related to both mathematics and computer science, however the two schemes differ in the details of their organization of those topics.
The classification scheme used on the arXiv is chosen to reflect the papers submitted. As arXiv is multidisciplinary its classification scheme does not fit entirely with the MSC, ACM or PACS classification schemes. It is common to see codes from one or more of these schemes on individual papers. | https://en.wikipedia.org/wiki/Mathematics_Subject_Classification |
Mathematics and Plausible Reasoning is a two-volume book by the mathematician George Pólya describing various methods for being a good guesser of new mathematical results. [ 1 ] [ 2 ] In the Preface to Volume 1 of the book Pólya exhorts all interested students of mathematics thus: "Certainly, let us learn proving, but also let us learn guessing." P. R. Halmos reviewing the book summarised the central thesis of the book thus: ". . . a good guess is as important as a good proof." [ 3 ]
Polya begins Volume I with a discussion on induction , not mathematical induction , but as a way of guessing new results. He shows how the chance observations of a few results of the form 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, 10 = 3 + 7, etc., may prompt a sharp mind to formulate the conjecture that every even number greater than 4 can be represented as the sum of two odd prime numbers . This is the well known Goldbach's conjecture . The first problem in the first chapter is to guess the rule according to which the successive terms of the following sequence are chosen: 11, 31, 41, 61, 71, 101, 131, . . . In the next chapter the techniques of generalization, specialization and analogy are presented as possible strategies for plausible reasoning. In the remaining chapters, these ideas are illustrated by discussing the discovery of several results in various fields of mathematics like number theory, geometry, etc. and also in physical sciences.
This volume attempts to formulate certain patterns of plausible reasoning . The relation of these patterns with the calculus of probability are also investigated. Their relation to mathematical invention and instruction are also discussed. The following are
some of the patterns of plausible inference discussed by Polya. | https://en.wikipedia.org/wiki/Mathematics_and_Plausible_Reasoning |
Ideas from mathematics have been used as inspiration for fiber arts including quilt making , knitting , cross-stitch , crochet , embroidery and weaving . A wide range of mathematical concepts have been used as inspiration including topology , graph theory , number theory and algebra . Some techniques such as counted-thread embroidery are naturally geometrical ; other kinds of textile provide a ready means for the colorful physical expression of mathematical concepts .
The IEEE Spectrum has organized a number of competitions on quilt block design, and several books have been published on the subject. Notable quiltmakers include Diana Venters and Elaine Ellison, who have written a book on the subject Mathematical Quilts: No Sewing Required . Examples of mathematical ideas used in the book as the basis of a quilt include the golden rectangle , conic sections , Leonardo da Vinci 's Claw, the Koch curve , the Clifford torus , San Gaku , Mascheroni 's cardioid , Pythagorean triples , spidrons , and the six trigonometric functions . [ 1 ]
Knitted mathematical objects include the Platonic solids , Klein bottles and Boy's surface .
The Lorenz manifold and the hyperbolic plane have been crafted using crochet. [ 2 ] [ 3 ] Knitted and crocheted tori have also been constructed depicting toroidal embeddings of the complete graph K 7 and of the Heawood graph . [ 4 ] The crocheting of hyperbolic planes has been popularized by the Institute For Figuring ; a book by Daina Taimina on the subject, Crocheting Adventures with Hyperbolic Planes , won the 2009 Bookseller/Diagram Prize for Oddest Title of the Year . [ 5 ]
Embroidery techniques such as counted-thread embroidery [ 6 ] including cross-stitch and some canvas work methods such as Bargello make use of the natural pixels of the weave, lending themselves to geometric designs. [ 7 ] [ 8 ]
Ada Dietz (1882 – 1981) was an American weaver best known for her 1949 monograph Algebraic Expressions in Handwoven Textiles , which defines weaving patterns based on the expansion of multivariate polynomials . [ 9 ]
J. C. P. Miller ( 1970 ) used the Rule 90 cellular automaton to design tapestries depicting both trees and abstract patterns of triangles. [ 10 ]
Margaret Greig was a mathematician who articulated the mathematics of worsted spinning . [ 11 ]
The silk scarves from DMCK Designs' 2013 collection are all based on Douglas McKenna's space-filling curve patterns. [ 12 ] The designs are either generalized Peano curves, or based on a new space-filling construction technique. [ 13 ] [ 14 ]
The Issey Miyake Fall-Winter 2010–2011 ready-to-wear collection designs from a collaboration between fashion designer Dai Fujiwara and mathematician William Thurston . The designs were inspired by Thurston's geometrization conjecture , the statement that every 3-manifold can be decomposed into pieces with one of eight different uniform geometries, a proof of which had been sketched in 2003 by Grigori Perelman as part of his proof of the Poincaré conjecture . [ 15 ] | https://en.wikipedia.org/wiki/Mathematics_and_fiber_arts |
Condorcet methods
Positional voting
Cardinal voting
Quota-remainder methods
Approval-based committees
Fractional social choice
Semi-proportional representation
By ballot type
Pathological response
Strategic voting
Paradoxes of majority rule
Positive results
In mathematics and fair division , apportionment problems involve dividing ( apportioning ) a whole number of identical goods fairly across several parties with real -valued entitlements . The original, and best-known, example of an apportionment problem involves distributing seats in a legislature between different federal states or political parties . [ 1 ] However, apportionment methods can be applied to other situations as well, including bankruptcy problems , [ 2 ] inheritance law (e.g. dividing animals ), [ 3 ] [ 4 ] manpower planning (e.g. demographic quotas), [ 5 ] and rounding percentages . [ 6 ]
Mathematically, an apportionment method is just a method of rounding real numbers to natural numbers. Despite the simplicity of this problem, every method of rounding suffers one or more paradoxes , as proven by the Balinski–Young theorem . The mathematical theory of apportionment identifies what properties can be expected from an apportionment method.
The mathematical theory of apportionment was studied as early as 1907 by the mathematician Agner Krarup Erlang . [ citation needed ] It was later developed to a great detail by the mathematician Michel Balinski and the economist Peyton Young . [ 7 ]
The inputs to an apportionment method are:
The output is a vector of integers a 1 , … , a n {\displaystyle a_{1},\ldots ,a_{n}} with ∑ i = 1 n a i = h {\displaystyle \sum _{i=1}^{n}a_{i}=h} , called an apportionment of h {\displaystyle h} , where a i {\displaystyle a_{i}} is the number of items allocated to agent i .
For each party i {\displaystyle i} , the real number q i := t i ⋅ h {\displaystyle q_{i}:=t_{i}\cdot h} is called the entitlement or seat quota for i {\displaystyle i} , and denotes the exact number of items that should be given to i {\displaystyle i} . In general, a "fair" apportionment is one in which each allocation a i {\displaystyle a_{i}} is as close as possible to the quota q i {\displaystyle q_{i}} .
An apportionment method may return a set of apportionment vectors (in other words: it is a multivalued function ). This is required, since in some cases there is no fair way to distinguish between two possible solutions. For example, if h = 101 {\displaystyle h=101} (or any other odd number) and t 1 = t 2 = 1 / 2 {\displaystyle t_{1}=t_{2}=1/2} , then (50,51) and (51,50) are both equally reasonable solutions, and there is no mathematical way to choose one over the other. While such ties are extremely rare in practice, the theory must account for them (in practice, when an apportionment method returns multiple outputs, one of them may be chosen by some external priority rules, or by coin flipping , but this is beyond the scope of the mathematical apportionment theory).
An apportionment method is denoted by a multivalued function M ( t , h ) {\displaystyle M(\mathbf {t} ,h)} ; a particular M {\displaystyle M} -solution is a single-valued function f ( t , h ) {\displaystyle f(\mathbf {t} ,h)} which selects a single apportionment from M ( t , h ) {\displaystyle M(\mathbf {t} ,h)} .
A partial apportionment method is an apportionment method for specific fixed values of n {\displaystyle n} and h {\displaystyle h} ; it is a multivalued function M ∗ ( t ) {\displaystyle M^{*}(\mathbf {t} )} that accepts only n {\displaystyle n} -vectors.
Sometimes, the input also contains a vector of integers r 1 , … , r n {\displaystyle r_{1},\ldots ,r_{n}} representing minimum requirements - r i {\displaystyle r_{i}} represents the smallest number of items that agent i {\displaystyle i} should receive, regardless of its entitlement. So there is an additional requirement on the output: a i ≥ r i {\displaystyle a_{i}\geq r_{i}} for all i {\displaystyle i} .
When the agents are political parties, these numbers are usually 0, so this vector is omitted. But when the agents are states or districts, these numbers are often positive in order to ensure that all are represented. They can be the same for all agents (e.g. 1 for USA states, 2 for France districts), or different (e.g. in Canada or the European parliament).
Sometimes there is also a vector of maximum requirements , but this is less common.
There are basic properties that should be satisfied by any reasonable apportionment method. They were given different names by different authors: the names on the left are from Pukelsheim; [ 8 ] : 75 The names in parentheses on the right are from Balinsky and Young. [ 7 ]
The proportionality of apportionment can be measured by seats-to-votes ratio and Gallagher index . The proportionality of apportionment together with electoral thresholds impact political fragmentation and barrier to entry to the political competition. [ 10 ]
There are many apportionment methods, and they can be classified into several approaches.
The exact quota of agent i {\displaystyle i} is q i = t i ⋅ h {\displaystyle q_{i}=t_{i}\cdot h} . A basic requirement from an apportionment method is that it allocates to each agent i {\displaystyle i} its quota q i {\displaystyle q_{i}} if it is an integer; otherwise, it should allocate it an integer that is near the exact quota, that is, either its lower quota ⌊ q i ⌋ {\displaystyle \lfloor q_{i}\rfloor } or its upper quota ⌈ q i ⌉ {\displaystyle \lceil q_{i}\rceil } . [ 11 ] We say that an apportionment method -
Hamilton's largest-remainder method satisfies both lower quota and upper quota by construction. This does not hold for the divisor methods. [ 7 ] : Prop.6.2, 6.3, 6.4, 6.5
Therefore, no divisor method satisfies both upper quota and lower quota for any number of agents. The uniqueness of Jefferson and Adams holds even in the much larger class of rank-index methods . [ 12 ]
This can be seen as a disadvantage of divisor methods, but it can also be considered a disadvantage of the quota criterion: [ 7 ] : 129
" For example, to give D 26 instead of 25 seats in Table 10.1 would mean taking a seat from one of the smaller states A, B, or C. Such a transfer would penalize the per capita representation of the small state much more - in both absolute and relative terms - than state D is penalized by getting one less than its lower quota. Similar examples can be invented in which some state might reasonably get more than its upper quota. It can be argued that staying within the quota is not really compatible with the idea of proportionality at all, since it allows a much greater variance in the per capita representation of smaller states than it does for larger states ."
In Monte-Carlo simulations, Webster's method satisfies both quotas with a very high probability. Moreover, Webster's method is the only division method that satisfies near quota : [ 7 ] : Thm.6.2 there are no agents i , j {\displaystyle i,j} such that moving a seat from i {\displaystyle i} to j {\displaystyle j} would bring both of them nearer to their quotas:
q i − ( a i − 1 ) < a i − q i and ( a j + 1 ) − q j < q j − a j {\displaystyle q_{i}-(a_{i}-1)~<~a_{i}-q_{i}~~{\text{ and }}~~(a_{j}+1)-q_{j}~<~q_{j}-a_{j}} .
Jefferson's method can be modified to satisfy both quotas, yielding the Quota-Jefferson method. [ 11 ] Moreover, any divisor method can be modified to satisfy both quotas. [ 13 ] This yields the Quota-Webster method, Quota-Hill method, etc. This family of methods is often called the quatatone methods , [ 12 ] as they satisfy both quotas and house-monotonicity .
One way to evaluate apportionment methods is by whether they minimize the amount of inequality between pairs of agents. Clearly, inequality should take into account the different entitlements: if a i / t i = a j / t j {\displaystyle a_{i}/t_{i}=a_{j}/t_{j}} then the agents are treated "equally" (w.r.t. to their entitlements); otherwise, if a i / t i > a j / t j {\displaystyle a_{i}/t_{i}>a_{j}/t_{j}} then agent i {\displaystyle i} is favored, and if a i / t i < a j / t j {\displaystyle a_{i}/t_{i}<a_{j}/t_{j}} then agent j {\displaystyle j} is favored. However, since there are 16 ways to rearrange the equality a i / t i = a j / t j {\displaystyle a_{i}/t_{i}=a_{j}/t_{j}} , there are correspondingly many ways by which inequality can be defined. [ 7 ] : 100–102
This analysis was done by Huntington in the 1920s. [ 14 ] [ 15 ] [ 16 ] Some of the possibilities do not lead to a stable solution. For example, if we define inequality as | a i / a j − t i / t j | {\displaystyle |a_{i}/a_{j}-t_{i}/t_{j}|} , then there are instances in which, for any allocation, moving a seat from one agent to another might decrease their pairwise inequality. There is an example with 3 states with populations (737,534,329) and 16 seats. [ 7 ] : Prop.3.5
The seat bias of an apportionment is the tendency of an apportionment method to systematically favor either large or small parties. Jefferson's method and Droop's method are heavily biased in favor of large states; Adams' method is biased in favor of small states; and the Webster and Huntington–Hill methods are effectively unbiased toward either large or small states.
Consistency properties are properties that characterize an apportionment method , rather than a particular apportionment. Each consistency property compares the outcomes of a particular method on different inputs. Several such properties have been studied.
State-population monotonicity means that, if the entitlement of an agent increases, its apportionment should not decrease. The name comes from the setting where the agents are federal states , whose entitlements are determined by their population. A violation of this property is called the population paradox . There are several variants of this property. One variant - the pairwise PM - is satisfied exclusively by divisor methods. That is, an apportionment method is pairwise PM if-and-only-if it is a divisor method. [ 7 ] : Thm.4.3
When n ≥ 4 {\displaystyle n\geq 4} and h ≥ n + 3 {\displaystyle h\geq n+3} , no partial apportionment method satisfies pairwise-PM, lower quota and upper quota. [ 7 ] : Thm.6.1 Combined with the previous statements, it implies that no divisor method satisfies both quotas.
House monotonicity means that, when the total number of seats h {\displaystyle h} increases, no agent loses a seat. The violation of this property is called the Alabama paradox . It was considered particularly important in the early days of the USA, when the congress size increased every ten years. House-monotonicity is weaker than pairwise-PM. All rank-index methods (hence all divisor methods) are house-monotone - this clearly follows from the iterative procedure. Besides the divisor methods, there are other house-monotone methods, and some of them also satisfy both quotas. For example, the Quota method of Balinsky and Young satisfies house-monotonicity and upper-quota by construction, and it can be proved that it also satisfies lower-quota. [ 11 ] It can be generalized: there is a general algorithm that yields all apportionment methods which are both house-monotone and satisfy both quotas. However, all these quota-based methods (Quota-Jefferson, Quota-Hill, etc.) may violate pairwise-PM: there are examples in which one agent gains in population but loses seats. [ 7 ] : Sec.7
Uniformity (also called coherence [ 17 ] ) means that, if we take some subset of the agents 1 , … , k {\displaystyle 1,\ldots ,k} , and apply the same method to their combined allocation h k = a 1 + ⋯ + a k {\displaystyle h_{k}=a_{1}+\cdots +a_{k}} , then the result is the vector ( a 1 , … , a k ) {\displaystyle (a_{1},\ldots ,a_{k})} . All rank-index methods (hence all divisor methods) are uniform, since they assign seats to agents in a pre-determined method - determined by r ( t , a ) {\displaystyle r(t,a)} , and this order does not depend on the presence or absence of other agents. Moreover, every uniform method that is also anonymous and balanced must be a rank-index method. [ 7 ] : Thm.8.3
Every uniform method that is also anonymous , weakly-exact and concordant (= t i > t j {\displaystyle t_{i}>t_{j}} implies a i ≥ a j {\displaystyle a_{i}\geq a_{j}} ) must be a divisor method. [ 7 ] : Thm.8.4 Moreover, among all anonymous methods: [ 12 ]
When the agents are political parties, they often split or merge. How such splitting/merging affects the apportionment will impact political fragmentation . Suppose a certain apportionment method gives two agents i , j {\displaystyle i,j} some a i , a j {\displaystyle a_{i},a_{j}} seats respectively, and then these two agents form a coalition, and the method is re-activated.
Among the divisor methods: [ 7 ] : Thm.9.1, 9.2, 9.3
Since these are different methods, no divisor method gives every coalition of i , j {\displaystyle i,j} exactly a i + a j {\displaystyle a_{i}+a_{j}} seats. Moreover, this uniqueness can be extended to the much larger class of rank-index methods . [ 12 ]
A weaker property, called "coalitional-stability", is that every coalition of i , j {\displaystyle i,j} should receive between a i + a j − 1 {\displaystyle a_{i}+a_{j}-1} and a i + a j + 1 {\displaystyle a_{i}+a_{j}+1} seats; so a party can gain at most one seat by merging/splitting.
Moreover, every method satisfying both quotas is "almost coalitionally-stable" - it gives every coalition between a i + a j − 2 {\displaystyle a_{i}+a_{j}-2} and a i + a j + 2 {\displaystyle a_{i}+a_{j}+2} seats. [ 12 ]
The following table summarizes uniqueness results for classes of apportionment methods. For example, the top-left cell states that Jefferson's method is the unique divisor method satisfying the lower quota rule. | https://en.wikipedia.org/wiki/Mathematics_of_apportionment |
The mathematics of the Incas (or of the Tawantinsuyu ) was the set of numerical and geometric knowledge and instruments developed and used in the nation of the Incas before the arrival of the Spaniards . It can be mainly characterized by its usefulness in the economic field. The quipus and yupanas are proof of the importance of arithmetic in Inca state administration. This was embodied in a simple but effective arithmetic , for accounting purposes, based on the decimal numeral system ; they too had a concept of zero , [ 1 ] and mastered addition, subtraction, multiplication, and division. The mathematics of the Incas had an eminently applicative character to tasks of management, statistics, and measurement that was far from the Euclidean outline of mathematics as a deductive corpus, since it was suitable and useful for the needs of a centralized administration. [ note 1 ]
On the other hand, the construction of roads, canals and monuments, as well as the layout of cities and fortresses, required the development of practical geometry , which was indispensable for the measurement of lengths and surfaces, in addition to architectural design. At the same time, they developed important measurement systems for length and volume , which took parts of the human body as reference. In addition, they used appropriate objects or actions that allowed to appreciate the result in another way, but relevant and effective.
The prevailing numeral system was the base-ten . [ 2 ] One of the main references confirming this are the chronicles that present a hierarchy of organized authorities, using the decimal numeral system with its arithmometer: Quipu .
of families
It is also possible to confirm the use of the decimal system in the Inca system by the interpretation of the quipus, which are organized in such a way that the knots — according to their location — can represent: units, tens, hundreds, etc. [ 3 ]
However, the main confirmation of the use of this system is expressed in the denomination of the numbers in Quechua , in which the numbers are developed in decimal form. This can be appreciated in the following table: [ note 2 ]
The quipus constituted a mnemonic system based on knotted strings used to record all kinds of quantitative or qualitative information; if they were dealing with the results of mathematical operations, only those previously performed on the "Inca abacuss " or yupanas were cancelled. Although one of its functions is related to mathematics — as it was an instrument capable of accounting — it was also used to store information related to census, product amount, and food kept in state warehouses. [ 4 ] [ 5 ] Quipus are even mentioned as instruments the Incas used to record their traditions and history in a different way than in writing.
Several chroniclers also mention the use of quipus to store historical news. [ note 3 ] However, it has not yet been discovered how this system worked. In the Tahuantinsuyo , it was specialized personnel who handled the strings. They were known as quipucamayoc and they could be in charge of the strings of an entire region or suyu . Although the tradition is being lost, the quipus continue to be used as mnemonic instruments in some indigenous villages where they are used to record the product of the crops and the animals of the communities. [ 5 ]
According to the Jesuit chronicler Bernabé Cobo , the Incas designated to certain specialists the tasks related to accounting. These specialists were called quipo camayos , in whom the Incas placed all their trust. [ 6 ] In his study of the quipu sample VA 42527 ( Museum für Völkerkunde, Berlin ), Sáez-Rodríguez noted that, in order to close the accounting books of the chacras , certain numbers were ordered according to their value in the agricultural calendar , for which the khipukamayuq — the accountant entrusted with the granary — was directly in charge. [ 7 ] [ 8 ]
In the case of numerical information, the mathematical operations were previously carried out on the abacuss or yupanas . These could be made of carved stone or clay, had boxes or compartments that corresponded to the decimal units, and were counted or marked with the help of small stones or grains of corn or quinoa. Units, tens, hundreds, etc. could be indicated according to whether they were implicit in each operation.
Recent research regarding the yupanas suggests that they allowed to calculate considerable numbers based on a probably non-decimal system, [ 9 ] but based in relation to the number 40. If true, it is curious to note the coincidence between the geometric progression achieved in the yupana and the current processing systems; [ 10 ] on the other hand, it is also contradictory that they based their accounting system on the number 40. If the investigations continue and this fact is confirmed, it would be necessary to compare its use with the decimal system, which according to the historical tradition and previous investigations, was the one used by the Incas. [ 11 ]
In October 2010, Peruvian researcher Andrés Chirinos with the support of the Spanish Agency for International Development Cooperation (in Spanish, Agencia Española de Cooperación Internacional para el Desarrollo, AECID), reviewed drawings and ancient descriptions of the indigenous chronicler Guaman Poma de Ayala and finally deciphered the riddle of the yupana — that he calls "pre-Hispanic calculator" — as being capable of adding, subtracting, multiplying, and dividing. This made him hopeful to finally discover how the quipus worked as well. [ 12 ]
There were different units of measurement for magnitudes such as length and volume in pre-Hispanic times. The Andean peoples, as in many other places in the world, took parts of the human body as a reference to establish their units of measurement. There was not a single system of units of obligatory and uniform use throughout the Andean world. Many documents and chronicles have recorded different systems of local origin that remained in use until the 16th century.
Among the units of length measurement, there was the rikra ( fathom ), which is the distance measured between a man's thumbs with arms extended horizontally. [ 13 ] The kukuchu tupu ( kukush tupu ) was equivalent to the Spanish codo ( cubit ) and was the distance measured from the elbow to the end of the fingers of the hand. [ 14 ] There was also the capa ( span ), and the smallest was the yuku or jeme , which was the length between the index finger and the thumb, separating one from the other as much as possible. The distance between two villages would have been evaluated by the number of chasquis required to carry an errand from one village to the other. They would have used direct proportionality between the circumference of a sheepfold and the number of chacra partitions.
The tupu was the unit of measurement of surface area. In general terms it was defined as the plot of land required for the maintenance of a married couple without children. Every hatun runa or "common man" received a plot of land upon marriage and its production had to satisfy the basic needs of food and trade of the spouses. It did not correspond to an exact measurement, since its dimensions varied according to the conditions of each land and from one ethnic group to another. [ 15 ] The quality of the soil was taken into consideration and the necessary rest time was calculated accordingly, which had to be considered after a certain number of agricultural campaigns. After that time, the couple could claim a new tupu from their curaca .
Among the units of measurement of capacity there is the pokcha , which was equivalent to half a fanega or 27.7 liters . Some crops such as corn were measured in containers; liquids were measured in a variety of pitchers and jars. There were boxes of a variety of cántaros and tinajas , and straw or reed boxes in which objects were kept. These boxes were also used in warehouses to store delicate or exquisite products, such as dried fruits. Coca leaves were measured in runcu or large baskets. Other baskets were known as ysanga . Among these measures of capacity there is the poctoy or purash ( almozada ), which is equivalent to the portion of grains or flour that can be kept in the concavity formed with the hands together. [ 16 ] The ancient inhabitants of the Andes knew the scales of saucers and nets as well as the huipe , an instrument similar to steelyards. [ 17 ] Apparently, its presence is associated with the works of jewelry and metallurgy, trades in which it is necessary to know the exact weights to use the right proportions of the alloys.
Especially the volume of their colcas (trojas) and their tambos (state warehouses, located in key points of the Qhapaq Ñan ). They used the runqu (rongos: bales), portable containers or ishanka (baskets) or the capacity of a chacra . They would have handled the proportionality of the volumes of prisms with respect to their heights — without varying the bases. [ 18 ]
To measure time, they used the day (workday), which could include a morning, even an afternoon. Time was also useful, indirectly, to appreciate the distance between two cities; for example, 20 days from Cajamarca to Cusco was the accepted time measurement.
Months, years, and the phases of the moon — much consulted for the tasks of sowing, aporques and harvests and in navigation — were also measured in days. [ 18 ] | https://en.wikipedia.org/wiki/Mathematics_of_the_Incas |
Mathesis universalis (from Greek : μάθησις , mathesis "science or learning", and Latin : universalis "universal") is a hypothetical universal science modelled on mathematics envisaged by Descartes and Leibniz , among a number of other 16th- and 17th-century philosophers and mathematicians. For Leibniz, it would be supported by a calculus ratiocinator . John Wallis invokes the name as title in his Opera Mathematica , a textbook on arithmetic , algebra , and Cartesian geometry .
Descartes' most explicit description of mathesis universalis occurs in Rule Four of the Rules for the Direction of the Mind , written before 1628. [ 1 ] Leibniz attempted to work out the possible connections between mathematical logic , algebra , infinitesimal calculus , combinatorics , and universal characteristics in an incomplete treatise titled " Mathesis Universalis " in 1695.
Predicate logic could be seen as a modern system with some of these universal qualities, at least as far as mathematics and computer science are concerned. More generally, mathesis universalis , along with perhaps François Viète 's algebra , represents one of the earliest attempts to construct a formal system .
One of the perhaps most prominent critics of the idea of mathesis universalis was Ludwig Wittgenstein and his philosophy of mathematics . [ 2 ] As Anthropologist Emily Martin notes: [ 3 ]
Tackling mathematics, the realm of symbolic life perhaps most difficult to regard as contingent on social norms, Wittgenstein commented that people found the idea that numbers rested on conventional social understandings "unbearable".
In Descartes' corpus the term mathesis universalis appears only in the Rules for the Direction of the Mind . [ 1 ] In the discussion of Rule Four , Descartes' provides his clearest description of mathesis universalis :
[...] I began my investigation by inquiring what exactly is generally meant by the term 'mathematics' and why it is that, in addition to arithmetic and geometry, sciences such as astronomy, music, optics, mechanics, among others, are called branches of mathematics. [...] This made me realize that there must be a general science which explains all the points that can be raised concerning order and measure irrespective of the subject-matter, and that this science should be termed mathesis universalis — a venerable term with a well-established meaning — for it covers everything that entitles these other sciences to be called branches of mathematics. [...]
In his account of mathesis universalis , Leibniz proposed a dual method of universal synthesis and analysis for the ascertaining truth , described in De Synthesi et Analysi universale seu Arte inveniendi et judicandi (1890). [ 5 ] [ 6 ]
Ars inveniendi ( Latin for "art of invention") is the constituent part of mathesis universalis corresponding to the method of synthesis. [ 5 ] [ 6 ]
Leibniz also identified synthesis with the ars combinatoria , viewing it in terms of the recombination of symbols or human thoughts. [ 5 ]
Ars judicandi ( Latin for "art of judgement") is the constituent part of mathesis universalis corresponding to the method of analysis. [ 5 ] | https://en.wikipedia.org/wiki/Mathesis_universalis |
A Mathethon is a computational mathematics competition that is primarily focused on computer-based math in contrast to math competitions that use scientific calculators or handwritten only. Mathethons are analogous to hackathons for computer programming competitions. They can very in academic difficulty from elementary competitions, middle school, high school, and college level mathematics. [ 1 ] They can be held in person individually, as a group, or hosted virtually online. [ 2 ] [ 3 ] [ 4 ] | https://en.wikipedia.org/wiki/Mathethon |
The Mathieu transformations make up a subgroup of canonical transformations preserving the differential form
The transformation is named after the French mathematician Émile Léonard Mathieu .
In order to have this invariance , there should exist at least one relation between q i {\displaystyle q_{i}} and Q i {\displaystyle Q_{i}} only (without any p i , P i {\displaystyle p_{i},P_{i}} involved).
where 1 < m ≤ n {\displaystyle 1<m\leq n} . When m = n {\displaystyle m=n} a Mathieu transformation becomes a Lagrange point transformation .
This classical mechanics –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Mathieu_transformation |
In physics , specifically general relativity , the Mathisson–Papapetrou–Dixon equations describe the motion of a massive spinning body moving in a gravitational field . Other equations with similar names and mathematical forms are the Mathisson–Papapetrou equations and Papapetrou–Dixon equations . All three sets of equations describe the same physics.
These equations are named after Myron Mathisson , [ 1 ] William Graham Dixon , [ 2 ] and Achilles Papapetrou , [ 3 ] who worked on them.
Throughout, this article uses the natural units c = G = 1, and tensor index notation .
The Mathisson–Papapetrou–Dixon (MPD) equations for a mass m {\displaystyle m} spinning body are
Here τ {\displaystyle \tau } is the proper time along the trajectory, k ν {\displaystyle k_{\nu }} is the body's four-momentum
the vector V μ {\displaystyle V^{\mu }} is the four-velocity of some reference point X μ {\displaystyle X^{\mu }} in the body, and the skew-symmetric tensor S μ ν {\displaystyle S^{\mu \nu }} is the angular momentum
of the body about this point. In the time-slice integrals we are assuming that the body is compact enough that we can use flat coordinates within the body where the energy-momentum tensor T μ ν {\displaystyle T^{\mu \nu }} is non-zero.
As they stand, there are only ten equations to determine thirteen quantities. These quantities are the six components of S λ μ {\displaystyle S^{\lambda \mu }} , the four components of k ν {\displaystyle k_{\nu }} and the three independent components of V μ {\displaystyle V^{\mu }} . The equations must therefore be supplemented by three additional constraints which serve to determine which point in the body has velocity V μ {\displaystyle V^{\mu }} . Mathison and Pirani originally chose to impose the condition V μ S μ ν = 0 {\displaystyle V^{\mu }S_{\mu \nu }=0} which, although involving four components, contains only three constraints because V μ S μ ν V ν {\displaystyle V^{\mu }S_{\mu \nu }V^{\nu }} is identically zero. This condition, however, does not lead to a unique solution and can give rise to the mysterious "helical motions". [ 4 ] The Tulczyjew–Dixon condition k μ S μ ν = 0 {\displaystyle k_{\mu }S^{\mu \nu }=0} does lead to a unique solution as it selects the reference point X μ {\displaystyle X^{\mu }} to be the body's center of mass in the frame in which its momentum is ( k 0 , k 1 , k 2 , k 3 ) = ( m , 0 , 0 , 0 ) {\displaystyle (k_{0},k_{1},k_{2},k_{3})=(m,0,0,0)} .
Accepting the Tulczyjew–Dixon condition k μ S μ ν = 0 {\displaystyle k_{\mu }S^{\mu \nu }=0} , we can manipulate the second of the MPD equations into the form
This is a form of Fermi–Walker transport of the spin tensor along the trajectory – but one preserving orthogonality to the momentum vector k μ {\displaystyle k^{\mu }} rather than to the tangent vector V μ = d X μ / d τ {\displaystyle V^{\mu }=dX^{\mu }/d\tau } . Dixon calls this M-transport . | https://en.wikipedia.org/wiki/Mathisson–Papapetrou–Dixon_equations |
The mating-type locus is a specialized region in the genomes of some yeast and other fungi , usually organized into heterochromatin and possessing unique histone methylation patterns. The genes in this region regulate the mating type of the organism and therefore determine key events in its life cycle , such as whether it will reproduce sexually or asexually . In fission yeast such as S. pombe , the formation and maintenance of the heterochromatin organization is regulated by RNA-induced transcriptional silencing , a form of RNA interference responsible for genomic maintenance in many organisms. [ 1 ] Mating type regions have also been well studied in budding yeast S. cerevisiae and in the fungus Neurospora crassa . [ 2 ]
In the budding yeast Saccharomyces cerevisiae , mating-type is determined by two non-homologous alleles at the mating-type locus. S. cerevisiae has the capability of undergoing mating-type switching, that is conversion of some haploid cells in a colony from one mating-type to the other. Mating-type switching can occur as frequently as once every generation. Switching involves homologous recombinational repair of a site specific, programmed double-strand break, a highly organized process. [ 3 ] This process replaces one mating type allelic DNA sequence with the sequence encoding the alternative mating-type allele. When two haploid cells of opposite mating type come into contact they can mate to form a diploid cell, a zygote , that may then undergo meiosis . Meiosis tends to occur under nutritionally limiting conditions associated with DNA damage .
This cell biology article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Mating-type_locus |
A mating connection is any method of assembling of two or more component parts with mutually complementing shapes that, with some imagination, resembles the way two animals, male and female , are physically connected during the act of mating . In such connections one of the two components acts as male and the other as female, although more complex relationships exist. [ 1 ] Any electrical connector , bolted joint , and jigsaw puzzle is an example of assembling based on mating connection.
This technology-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Mating_connection |
Mating disruption (MD) is a pest management technique designed to control certain insect pests by introducing artificial stimuli that confuse the individuals and disrupt mate localization and/or courtship, thus preventing mating and blocking the reproductive cycle . It usually involves the use of synthetic sex pheromones , [ 1 ] although other approaches, such as interfering with vibrational communication , are also being developed. [ 2 ]
La confusion sexuelle or mating disruption, was first discussed by the Institut national de la recherche agronomique in 1974 in Bordeaux , France . [ 3 ]
Winemakers in France, Switzerland, Spain, Germany, and Italy were the first to use the method to treat vines against the larvae of the moth genus Cochylis . [ 3 ]
In many insect species of interest to agriculture, such as those in the order Lepidoptera , females emit an airborne trail of a specific chemical blend constituting that species' sex pheromone. This aerial trail is referred to as a pheromone plume. [ 4 ] [ 5 ]
Males of that species use the information contained in the pheromone plume [ 6 ] to locate the emitting female (known as a “calling” female). Mating disruption exploits the male insects' natural response to follow the plume by introducing a synthetic pheromone into the insects’ habitat . The synthetic pheromone is a volatile organic chemical designed to mimic the species-specific sex pheromone produced by the female insect. The general effect of mating disruption is to confuse the male insects by masking the natural pheromone plumes, causing the males to follow “false pheromone trails” at the expense of finding mates, and affecting the males’ ability to respond to “calling" females. Consequently, the male population experiences a reduced probability of successfully locating and mating with females, which leads to the eventual cessation of breeding and collapse of the insect infestation. The California Department of Pesticide Regulation , the California Department of Food and Agriculture , and the United States Environmental Protection Agency consider mating disruption to be among the most environmentally friendly treatments used to eradicate pest infestations. [ 7 ] Mating disruption works best if large areas are treated with pheromones. Ten acres is a good minimum size for a successful MD program, but larger areas are preferable [2]
Pheromone programs are most effective when controlling low to moderate pest population densities. MD has also been identified as a pest control method in which the insect does not become resistant [1] . The scientific community, together with governmental agencies throughout the world, understands the benefits of mating disruption using species-specific sex pheromones, and consider sex-pheromone-based insect control programs among the most environmentally friendly treatments to be used to manage and control insect pest populations. Insect pheromone has been successfully used as an effective tool to slow the spread and to eradicate pests from very large areas in the US; for example to control the spongy moth ( Lymantria dispar ), a devastating forestry pest, and to eradicate the boll weevil and pink bollworm, two of the most damaging pest of cotton. Conventional pesticide based control methods, kill insects directly, whereas mating disruption confuses male insects from accurately locating a mating partner, leading to the eventual collapse of the mating cycle [3] . Mating disruption, due to the specificity of the sex pheromone of the insect species, has the benefit of only affecting the males of that species, while leaving other non target species unaffected [3] . This allows for very targeted pest management, promoting the suppression of a single pest species, leaving the populations of beneficial insects (pollinators and natural enemies) intact. Mating disruption, like most pest management strategies, is a useful technique, but should not be considered a stand-alone treatment program [1] for it targets only a single species in plant production systems that usually have several pests of concern. Mating disruption is a valuable tool that should be used in Integrated Pest Management(IPM) programs.
Pheromone programs have been used for several decades around the globe and to date (2009) there is no documented public health evidence to suggest that agricultural use of synthetic pheromones is harmful to humans or to any other non-target species. However, continuing research is being conducted.
Over the decades that pheromone pest programs have been used several disadvantages have been argued when compared to the use of conventional pesticides. Most pheromones target a single species, so a specific mating disruption formulation controls only the species that uses that pheromone blend; whereas pesticides usually kill indiscriminately a plethora of species, including multiple species with a single application. Some synthetic pheromones have high developmental and production costs, causing the mating disruption technique to be too costly to be adopted by conventional commercial growers. Furthermore most commercial pheromone mating disruption formulations must be applied by hand, which can be an expensive and time consuming. Novel pheromone formulations recently developed to be mechanically applied provide long lasting mating disruption effects (e.g., depending on the target pest a single application of SPLAT [ 8 ] controls the target pest for a complete reproductive cycle, [ 9 ] or for the entire season. [ 10 ] [ 11 ]
Microencapsulated pheromones (MECs) are small droplets of pheromone enclosed within polymer capsules. The capsules control the release rate of the pheromone into the surrounding environment. The capsules are small enough to be applied in the same method as used to spray insecticides . The effective field longevity of the microencapsulated pheromone formulations ranges from a few days to slightly more than a week, depending on climatic conditions, capsule size and chemical properties [1] . Microcapsules in the pheromone formulations are usually kept above a prescribed diameter to avoid the risk of inhalation by humans.
A new, effective and economical concept in pheromone delivery using a flowable formulation to create long lasting monolithic pheromone dispensers has been brought to the market in the past decade. [ 8 ] These novel SPLAT pheromone mating disruption formulations can provide effective season long suppression effect (e.g., depending on the target pest a single application of SPLAT controls the target pest for a complete reproductive cycle, [ 13 ] or for the entire season [ 10 ] [ 14 ] ) and can be manually or mechanically applied. Although mechanical dispersal techniques require specialized off-the-shelf application technology and/or equipment, once the application system is made to work it allows protection of extensive areas using pheromones, one of the most benign and effective pest management techniques available today. A benefit of SPLAT is that the dollop anchors where it lands, avoiding unwanted drift of the formulation once applied in the field, and, depending on the mode of application, the cured dollops are retrievable.
In November 2007, a controversial aerial approach was used to spray microencapsulated LBAM pheromone in urban and rural areas of the counties of Santa Cruz and Monterey California to combat the invasive light brown apple moth . Usually the effect of disruption of orientation of the male moths to females (or monitoring pheromone traps) can be detected by the reduction in moth capture in monitoring pheromone traps. The government campaign using areawide aerial microencapsulated pheromone applications failed to show any sign of mating disruption on the light brown apple moth populations in the treated area. It was found that the first aerial campaign was performed using an incomplete (the wrong) pheromone blend of the light brown apple moth (the wrong blend decreased tremendously the likelihood of success of the mating disruption program), and the LBAM microencapsulated formulation was untested, and finally, microencapsule formulations are notoriously known for their short field life, weak and erratic performance. Furthermore it is possible that the LBAM microencapsulated formulation used in the government campaign was unfit for aerial delivery in urban areas; although pheromone is safe, the formulation used had microcapsules of very small diameter which made it into a possible inhalation hazard that seems to be linked to an increase in allergenic reactions of the population in the target area. This set of LBAM mating disruption aerial applications done by the government has created tremendous dissent of the public in general as well as of several sectors of the scientific community. Now, several years later, the affected communities as well as the nascent US pheromone industry (which provides safer, yet very effective, alternatives to the use of conventional pesticides) are still suffering the ripple effects of these disastrous Bay Area LBAM eradication campaigns.
But there are numerous, successful pest suppression programs that rely on aerial dispersal of pheromone mating disruptants. One of the largest pheromone mating disruption programs in the globe is Slow the Spread. [ 15 ] [ 16 ] Slow the Spread has been implemented across the 1,200-mile (1,900 km) spongy moth frontier from Wisconsin to North Carolina. The program area is located ahead of the advancing front of the spongy moth population. The STS program focuses on early detection and suppression of the low–level populations along this advancing front, disrupting the natural progress of population buildup and spread. Every year hundreds of thousands of acres are aerially sprayed with two pheromone spongy moth pheromone mating disruption formulations, Flakes and SPLAT. A single mating disruption formulation application promotes season-long suppression of spongy moth in the treated areas. With a crew of 8 people it was possible to aerially treat with SPLAT GM over 20,000 acres (81 km 2 ) of forest in a single day. The consortium of Federal and State participants have been able to do the following:
• decrease the new territory invaded by the spongy moth each year from 15,600 square miles (40,000 km 2 ) to 6,000 square miles (16,000 km 2 );
• protect forests, forest–based industries, urban and rural parks, and private property; and
• avoid at least $22 million per year in damage and management costs.
It seems that the tremendous success of the Slow the Spread program is related to extremely well planned campaigns, which involves communication, transparency and clarity of objectives: in advance to an application STS holds meetings that include the area population in general, concerned citizens, public officials, scientists and technical personnel to discuss strategies of management of spongy moths in the areas of concern. There is a movement requesting that new government invasive species eradication campaigns model their pest suppression actions on the existing successful suppression programs like STS, and embrace a more effective policy of communication, transparency and clarity of objectives. With the involvement and education of the public, areawide eradication campaigns will be better planned and more able to deliver decisive end effective pest eradication actions. | https://en.wikipedia.org/wiki/Mating_disruption |
Fungi are a diverse group of organisms that employ a huge variety of reproductive strategies , ranging from fully asexual to almost exclusively sexual species. [ 1 ] Most species can reproduce both sexually and asexually, alternating between haploid and diploid forms. This contrasts with most multicellular eukaryotes, such as mammals, where the adults are usually diploid and produce haploid gametes which combine to form the next generation. In fungi, both haploid and diploid forms can reproduce – haploid individuals can undergo asexual reproduction while diploid forms can produce gametes that combine to give rise to the next generation. [ 2 ]
Mating in fungi is a complex process governed by mating types . Research on fungal mating has focused on several model species with different behaviour. [ 3 ] [ 4 ] Not all fungi reproduce sexually and many that do are isogamous ; thus, for many members of the fungal kingdom, the terms "male" and "female" do not apply. Homothallic species are able to mate with themselves, while in heterothallic species only isolates of opposite mating types can mate.
Mating between isogamous fungi may consist only of a transfer of a nucleus from one cell to another. Vegetative incompatibility within species often prevents a fungal isolate from mating with another isolate. Isolates of the same incompatibility group do not mate or mating does not lead to successful offspring. High variation has been reported including same-chemotype mating, sporophyte to gametophyte mating and biparental transfer of mitochondria.
A zygomycete hypha grows towards a compatible mate and they both form a bridge, called a progametangia , by joining at the hyphal tips via plasmogamy . A pair of septa forms around the merged tips, enclosing nuclei from both isolates. A second pair of septa forms two adjacent cells, one on each side. These adjacent cells, called suspensors provide structural support. The central cell, called the zygosporangium , is destined to become a spore . The zygosporangium is a unique structure to the Zygomycota and is easily recognizable in microscopy due to its characteristic dark color and spiky shape. The nuclei join in a process called karyogamy to form a zygote , which grows into a mature diploid zygomycete. A diploid zygomycete can then undergo meiosis to create spores, which disperse and germinate. The following generations of mycelium can undergo asexual or sexual reproduction. [ 5 ]
The phylum Zygomycota has since been split into two phyla believed to be monophyletic, Mucoromycota and Zoopagomycota (later raised to the subkingdom rank as Mucoromyceta and Zoopagomyceta ). Nevertheless, the two subkingdoms still conform to the behavior described above: "sexual reproduction, if present, via zygospores by gametangial conjugation". [ 6 ]
As it approaches a mate, a haploid sac fungus develops one of two complementary organs, a "female" ascogonium or a "male" antheridium. These organs resemble gametangia except that they contain only nuclei. A bridge, the trichogyne forms, that provides a passage for nuclei to travel from the antheridium to the ascogonium. A dikaryon grows from the ascogonium, and karyogamy occurs in the fruiting body .
Neurospora crassa is a type of red bread mold of the phylum Ascomycota . N. crassa is used as a model organism because it is easy to grow and has a haploid life cycle: this makes genetic analysis simple, since recessive traits will show up in the offspring. Analysis of genetic recombination is facilitated by the ordered arrangement of the products of meiosis within a sac-like structure called an ascus (pl. asci ). In its natural environment, N. crassa lives mainly in tropical and sub-tropical regions. It often can be found growing on dead plant matter after fires.
Neurospora was used by Edward Tatum and George Wells Beadle in the experiments for which they won the Nobel Prize in Physiology or Medicine in 1958. The results of these experiments led directly to the " one gene, one enzyme " hypothesis that specific genes code for specific proteins . This concept launched molecular biology . [ 7 ] Sexual fruiting bodies (perithecia) can only be formed when two cells of different mating type come together (see Figure). Like other Ascomycetes, N. crassa has two mating types that, in this case, are symbolized by A and a . There is no evident morphological difference between the A and a mating type strains. Both can form abundant protoperithecia, the female reproductive structure (see Figure). Protoperithecia are formed most readily in the laboratory when growth occurs on solid (agar) synthetic medium with a relatively low source of nitrogen. [ 8 ] Nitrogen starvation appears to be necessary for expression of genes involved in sexual development. [ 9 ] The protoperithecium consists of an ascogonium, a coiled multicellular hypha that is enclosed in a knot-like aggregation of hyphae. A branched system of slender hyphae, called the trichogyne, extends from the tip of the ascogonium projecting beyond the sheathing hyphae into the air. The sexual cycle is initiated (i.e. fertilization occurs) when a cell, usually a conidium, of opposite mating type contacts a part of the trichogyne (see Figure). Such contact can be followed by cell fusion leading to one or more nuclei from the fertilizing cell migrating down the trichogyne into the ascogonium. Since both A and a strains have the same sexual structures, neither strain can be regarded as exclusively male or female. However, as a recipient, the protoperithecium of both the A and a strains can be thought of as the female structure, and the fertilizing conidium can be thought of as the male participant.
The subsequent steps following fusion of A and a haploid cells have been outlined by Fincham and Day. [ 10 ] and Wagner and Mitchell. [ 11 ] After fusion of the cells, the further fusion of their nuclei is delayed. Instead, a nucleus from the fertilizing cell and a nucleus from the ascogonium become associated and begin to divide synchronously. The products of these nuclear divisions (still in pairs of unlike mating type, i.e. A/a ) migrate into numerous ascogenous hyphae, which then begin to grow out of the ascogonium. Each of these ascogenous hyphae bends to form a hook (or crozier ) at its tip and the A and a pair of haploid nuclei within the crozier divide synchronously. Next, septa form to divide the crozier into three cells. The central cell in the curve of the hook contains one A and one a nucleus (see Figure). This binuclear cell initiates ascus formation and is called an “ascus-initial” cell. Next the two uninucleate cells on either side of the first ascus-forming cell fuse with each other to form a binucleate cell that can grow to form a further crozier that can then form its own ascus-initial cell. This process can then be repeated multiple times.
After formation of the ascus-initial cell, the A and a nuclei fuse with each other to form a diploid nucleus (see Figure). This nucleus is the only diploid nucleus in the entire life cycle of N. crassa . The diploid nucleus has 14 chromosomes formed from the two fused haploid nuclei that had 7 chromosomes each. Formation of the diploid nucleus is immediately followed by meiosis . The two sequential divisions of meiosis lead to four haploid nuclei, two of the A mating type and two of the a mating type. One further mitotic division leads to four A and four a nucleus in each ascus. Meiosis is an essential part of the life cycle of all sexually reproducing organisms, and in its main features, meiosis in N. crassa seems typical of meiosis generally.
As the above events are occurring, the mycelial sheath that had enveloped the ascogonium develops as the wall of the perithecium becomes impregnated with melanin, and blackens. The mature perithecium has a flask-shaped structure.
A mature perithecium may contain as many as 300 asci, each derived from identical fusion diploid nuclei. Ordinarily, in nature, when the perithecia mature the ascospores are ejected rather violently into the air. These ascospores are heat resistant and, in the lab, require heating at 60 °C for 30 minutes to induce germination. For normal strains, the entire sexual cycle takes 10 to 15 days. In a mature ascus containing eight ascospores, pairs of adjacent spores are identical in genetic constitution, since the last division is mitotic, and since the ascospores are contained in the ascus sac that holds them in a definite order determined by the direction of nuclear segregations during meiosis. Since the four primary products are also arranged in sequence, a first division segregation pattern of genetic markers can be distinguished from a second division segregation pattern.
That mating in N. crassa can only occur between strains of different mating type suggests that some degree of outcrossing is favored by natural selection. In haploid multicellular fungi, such as N. crassa , meiosis occurring in the brief diploid stage is one of their most complex processes. The haploid multicellular vegetative stage, although physically much larger than the diploid stage, characteristically has a simple modular construction with little differentiation. In N. crassa , recessive mutations affecting the diploid stage of the life cycle are quite frequent in natural populations. [ 12 ] These mutations, when homozygous in the diploid stage, often cause spores to have maturation defects or to produce barren fruiting bodies with few ascospores (sexual spores). The majority of these homozygous mutations cause abnormal meiosis (e.g. disturbed chromosome pairing or disturbed pachytene or diplotene). [ 13 ] The number of genes affecting the diploid stage was estimated to be at least 435 [ 12 ] (about 4% of the total number of 9,730 genes). Thus, outcrossing, promoted by the necessity for union of opposite mating types, likely provides the benefit of masking recessive mutations that would otherwise be deleterious to sexual spore formation (see Complementation (genetics) ).
Saccharomyces cerevisiae , brewer's and baker's yeast, is in the phylum Ascomycota . During vegetative growth that ordinarily occurs when nutrients are abundant, S. cerevisiae reproduces by mitosis as either haploid or diploid cells. However, when starved, diploid cells undergo meiosis to form haploid spores. [ 14 ] Mating occurs when haploid cells of opposite mating type, MATa and MATα, come into contact. Ruderfer et al. [ 15 ] pointed out that such contacts are frequent between closely related yeast cells for two reasons. The first is that cells of opposite mating type are present together in the same ascus , the sac that contains the tetrad of cells directly produced by a single meiosis, and these cells can mate with each other. The second reason is that haploid cells of one mating type, upon cell division, often produce cells of the opposite mating type with which they may mate.
Katz Ezov et al. [ 16 ] presented evidence that in natural S. cerevisiae populations clonal reproduction and a type of “self-fertilization” (in the form of intratetrad mating) predominate. Ruderfer et al. [ 15 ] analyzed the ancestry of natural S. cerevisiae strains and concluded that outcrossing occurs only about once every 50,000 cell divisions. Thus, although S. cerevisiae is heterothallic, it appears that, in nature, mating is most often between closely related yeast cells. The relative rarity in nature of meiotic events that result from outcrossing suggests that the possible long-term benefits of outcrossing (e.g. generation of genetic diversity ) are unlikely to be sufficient for generally maintaining sex from one generation to the next. [ citation needed ] Instead, a short-term benefit, such as meiotic recombinational repair of DNA damages caused by stressful conditions such as starvation, may be the key to the maintenance of sex in S. cerevisiae . [ 17 ] Alternatively, recessive deleterious mutations accumulate during the diploid expansion phase, and are purged during selfing: this purging has been termed "genome renewal" and provides an advantage of sex that does not depend on outcrossing. [ 18 ] [ 19 ]
Candida albicans is a diploid fungus that grows both as a yeast and as a filament. C. albicans is the most common fungal pathogen in humans. It causes both debilitating mucosal infections and potentially life-threatening systemic infections. C. albicans has maintained an elaborate, but largely hidden, mating apparatus. [ 20 ] Johnson suggested that mating strategies may allow C. albicans to survive in the hostile environment of a mammalian host. In order to mate C. albicans needs to switch from white to opaque cells. The latter are more efficient in mating and referred to as the mating competent cells of C. albicans . Mating in C. albicans is termed a parasexual cycle since meiosis is still not observed in C. albicans . [ 21 ] [ 22 ]
A picture of the mating type mechanism has begun to emerge from studies of particular fungi such as S. cerevisiae . The mating type genes are located in homeobox and encode enzymes for production of pheromones and pheromone receptors . Sexual reproduction thereby depends on pheromones produced from variant alleles of the same gene . Since sexual reproduction takes place in haploid organisms, it cannot proceed until complementary genes are provided by a suitable partner through cell or hyphal fusion. The number of mating types depends on the number of genes and the number of alleles for each.
Depending on the species, sexual reproduction takes place through gametes or hyphal fusion. When a receptor on one haploid detects a pheromone from a complementary mating type, it approaches the source through chemotropic growth or chemotactic movement if it is a gamete.
Some of the species within Basidiomycota have the most complex systems of sexual reproduction known among fungi . In general for fungi there are two main types of sexual reproduction: homothallism , when mating occurs within a single individual, or in other words each individual is self-fertile; and heterothallism , when hyphae from a single individual are self-sterile and need to interact with another compatible individual for mating to take place. Additionally, mating compatibility in the Basidiomycota is further categorized into two types of mating systems: tetrapolar and bipolar.
Heterothallism is the most common mating system in Basidiomycota and in Agaricomycotina (the mushroom-forming fungi) about 90% of the species are heterothallic. [ 23 ] The tetrapolar type of mating system is ruled by two unlinked mating loci termed A and B (in Agaricomycotina) or b and a (in Ustilaginomycotina and Pucciniomycotina ), both of which can be multiallelic. The combination of A and B (or b and a ) alleles, termed mating type , determine the "specificity" or sexual identity of the individual harboring them. Only individuals with different mating types are compatible with each other and therefore able to start the mating event.
A successful mating interaction begins with nuclear exchange and nuclear migration resulting in the formation of dikaryotic hyphae (containing separate haploid nuclei from both initial parents). Dikaryotic hyphae, under the appropriate environmental conditions will give rise to the fruiting body which contains the basidia – specialized cells in which sexual recombination via karyogamy and meiosis occurs. This dikaryotic condition in Basidiomycota is often maintained by a specialized hyphal structure called a clamp connection . The formation of clamp connections is regulated by both mating loci.
Examples of tetrapolar organisms are the smuts Ustilago maydis and U. longissima , [ 24 ] [ 25 ] and the mushrooms Coprinopsis cinerea , Schizophyllum commune , Pleurotus djamor and Laccaria bicolor . [ 26 ]
It is believed that multi-allelic systems favor outcrossing in Basidiomycota. For example, in the case of U. maydis , which bears more than 25 b but only 2 a mating types, an individual has an approximately 50% chance to encounter a compatible mate in nature. [ 27 ] However, species such as C. cinerea , which has more than 240 A and B mating types, each, and S. commune , which has more than 339 A mating types and 64 B mating types, approach close to 100% chance of encountering a compatible partner in nature, due to the huge number of mating types generated by these systems. [ 28 ]
In contrast, bipolar mating systems are ruled by a single allelic mating locus, termed either A or b . In Agaricomycotina, bipolar organisms mostly have multiple alleles for their A mating locus; however, in Ustilaginomycotina and Pucciniomycotina, the b mating locus is predominantly diallelic, which reduces the occurrence of outcrossing within these species . [ 24 ] Bipolarity likely arose via one of two potential routes:
Other bipolar species include the white rot fungus Phanerochaete chrysosporium [ 32 ] and the edible mushroom Pholiota nameko . [ 33 ]
In the B or a locus there are linked genes that code for pheromones and pheromone receptors . The pheromones are short polypeptides with conserved residues [ 28 ] and the pheromone receptors belong to the G protein-coupled family of receptors located in the cell membrane ; they sense different molecules (in this case the pheromones) outside and activate a specific pathway inside of the cell. Pheromone-receptor interaction occurs in a way that the pheromone from one individual interacts with the receptor from the partner and vice versa. The functions of these genes are to regulate reciprocal nuclear exchange, nuclear migration in both mates and ultimately clamp cell fusion. [ 34 ] The first mating pheromone-receptor genes characterized were for U. maydis . [ 35 ]
The A or b mating locus contains genes that code for two types of homeodomain transcription factor proteins, usually tightly linked, that are homologues to the Saccharomyces cerevisiae mating proteins MATα2 and MATa1. In Agaricomycotina the two types of homeodomain transcription factors are termed HD1 and HD2; so the HD1 and HD2 proteins from an individual interacts with the HD2 and HD1 proteins from the other partner, respectively, generating heterodimers able to activate the A transcriptional regulated pathway, which involves formation of clamp cells, coordinated nuclear division and septation . [ 34 ]
Homothallic species may likely have evolved from heterothallic ancestors (Lin and Heitman 2007). In Basidiomycota homothallism is not very common and in Agaricomycotina it is estimated that only 10% of species have homothallic mating behavior. [ 23 ] For example, one subspecies of the ectomycorrhizal Basidiomycete Sistotrema brinkmannii is homothallic, although other subspecies have maintained their ability to outcross. Also, a variety of the edible mushroom Agaricus bisporus , ( A. bisporus var. eurotetrasporus ) produces haploid self-fertile basidiospores. Additionally, in the human pathogen C. neoformans known to outcross under laboratory conditions, both mating types are not normally distributed in natural populations, with the α mating type much more commonly found (>99%), suggesting homothallism is the most prevalent mode of sexual reproduction in ´ C. neoformans in nature. [ 36 ] Finally, the fungus causing witches' broom in cacao, Moniliophthora perniciosa , has a primarily homothallic biology despite having A and B mating type-like genes in its genome . [ 37 ]
Among the 250 known species of aspergilli, about 36% have an identified sexual state [ 38 ] Among those Aspergillus species that exhibit a sexual cycle the overwhelming majority in nature are homothallic (self-fertilizing). [ 38 ] Selfing in the homothallic fungus Aspergillus nidulans involves activation of the same mating pathways characteristic of sex in outcrossing species, i.e. self-fertilization does not bypass required pathways for outcrossing sex but instead requires activation of these pathways within a single individual. [ 39 ] Fusion of haploid nuclei occurs within reproductive structures termed cleistothecia, in which the diploid zygote undergoes meiotic divisions to yield haploid ascospores. | https://en.wikipedia.org/wiki/Mating_in_fungi |
The mating of yeast , also known as yeast sexual reproduction , is a biological process that promotes genetic diversity and adaptation in yeast species. Yeast species, such as Saccharomyces cerevisiae (baker's yeast), are single-celled eukaryotes that can exist as either haploid cells, which contain a single set of chromosomes , or diploid cells, which contain two sets of chromosomes. Haploid yeast cells come in two mating types , a and α, each producing specific pheromones to identify and interact with the opposite type, thus displaying simple sexual differentiation . [ a ] A yeast cell's mating type is determined by a specific genetic locus known as MAT , which governs its mating behaviour. Haploid yeast can switch mating types through a form of genetic recombination , allowing them to change mating type as often as every cell cycle . When two haploid cells of opposite mating types encounter each other, they undergo a complex signaling process that leads to cell fusion and the formation of a diploid cell. Diploid cells can reproduce asexually , but under nutrient-limiting conditions, they undergo meiosis to produce new haploid spores.
The differences between a and α cells, driven by specific gene expression patterns regulated by the MAT locus, are crucial for the mating process. Additionally, the decision to mate involves a highly sensitive and complex signaling pathway that includes pheromone detection and response mechanisms. In nature, yeast mating often occurs between closely related cells, although mating type switching and pheromone signaling allow for occasional outcrossing to enhance genetic variation . Certain yeast species have unique mating behaviors, demonstrating the diversity and adaptability of yeast reproductive strategies.
Yeast cells can stably exist in either a diploid or a haploid form. Both haploid and diploid yeast cells reproduce by mitosis , in which daughter cells bud from mother cells. Haploid cells are capable of mating with other haploid cells of the opposite mating type (an a cell can only mate with an α cell and vice versa) to produce a stable diploid cell. Diploid cells, usually upon facing stressful conditions like nutrient depletion, can undergo meiosis to produce four haploid spores : two a spores and two α spores. [ 1 ] [ 2 ]
a cells produce a -factor, a mating pheromone which signals the presence of an a cell to neighbouring α cells. [ 3 ] a cells respond to α-factor, the α cell mating pheromone, by growing a projection (known as a shmoo, due to its distinctive shape resembling the Al Capp cartoon character Shmoo ) towards the source of α-factor. [ 4 ] Similarly, α cells produce α-factor, and respond to a -factor by growing a projection towards the source of the pheromone. [ 5 ] The selective response of haploid cells to the mating pheromones of the opposite mating type allows mating between a and α cells, but not between cells of the same mating type. [ 6 ]
These phenotypic differences between a and α cells are due to a different set of genes being actively transcribed and repressed in cells of the two mating types. a cells activate genes which produce a -factor and produce a cell surface receptor (Ste2) which binds to α-factor and triggers signaling within the cell. [ 7 ] [ 8 ] a cells also repress the genes associated with being an α cell. Conversely, α cells activate genes which produce α-factor and produce a cell surface receptor (Ste3) which binds and responds to a -factor, and α cells repress the genes associated with being an a cell. [ 9 ]
The different sets of transcriptional repression and activation, which characterize a and α cells, are caused by the presence of one of two alleles for a mating-type locus called MAT : MAT a or MATα located on chromosome III. [ 10 ] The MAT locus is usually divided into five regions (W, X, Y, Z1, and Z2) based on the sequences shared among the two mating types. [ 11 ] The difference lie in the Y region (Y a and Yα), which contains most of the genes and promoters. [ 7 ]
The MAT a allele of MAT encodes a gene called a 1, which directs the a -specific transcriptional program (such as expressing STE2 and repressing STE3 ) that defines an a haploid cell. The MATα allele of MAT encodes the α1 and α2 genes, which directs the α-specific transcriptional program (such as expressing STE3 , repressing STE2, and producing prepro-α-factor ) that defines an α haploid cell. [ 7 ] S. cerevisiae has an a 2 gene with no apparent function that shares much of its sequence with α2; however, other yeast species like Candida albicans do have a functional and distinct MAT a 2 gene. [ 6 ] [ 10 ]
Haploid cells are one of two mating types ( a or α) and respond to the mating pheromone produced by haploid cells of the opposite mating type. [ 4 ] Haploid cells cannot undergo meiosis . [ 12 ] Diploid cells do not produce or respond to either mating pheromone and do not mate, but they can undergo meiosis to produce four haploid cells. [ 13 ]
Like the differences between haploid a and α cells, different patterns of gene repression and activation are responsible for the phenotypic differences between haploid and diploid cells. [ 14 ] In addition to the transcriptional patterns of a and α cells, haploid cells of both mating types share a haploid transcriptional pattern which activates haploid-specific genes (such as HO ) and represses diploid-specific genes (such as IME1 ). [ 15 ] Conversely, diploid cells activate diploid-specific genes and repress haploid-specific genes. [ 16 ]
The different gene expression patterns of haploid and diploid cells are attributable to the MAT locus. Haploid cells only contain one copy of each of the 16 chromosomes and therefore only possess one MAT allele (either MAT a or MATα ), which determines their mating type. [ 17 ] Diploid cells result from the mating of an a cell and an α cell, and they possess 32 chromosomes (in 16 pairs), including one chromosome bearing the MAT a allele and another chromosome bearing the MATα allele. [ 18 ] The combination of the information encoded by the MAT a allele (the a 1 gene) and the MATα allele (the α1 and α2 genes) triggers the diploid transcriptional program. [ 19 ] Conversely, the presence of only one MAT allele, either MAT a or MATα , triggers the haploid transcriptional program. [ 20 ] [ 7 ]
Through genetic engineering , a MAT a allele can be added to a MATα haploid cell, causing it to behave like a diploid cell. [ 21 ] The cell will not produce or respond to mating pheromones, and when starved, the cell will unsuccessfully attempt to undergo meiosis with fatal results. [ 21 ] Similarly, deletion of one copy of the MAT locus in a diploid cell, leaving either a MAT a or MATα allele, will cause a diploid cell to behave like a haploid cell of the associated mating type. [ 22 ] [ 23 ]
α cells with inactivated α1 and α2 genes at the MAT locus will exhibit the mating behavior of a cells. When an a -like faker (alf) cell mates with an α cell, they form a diploid cell lacking an active copy of the a 1 gene. As a result, these diploid cells cannot form the a 1-α2 protein complex needed to repress haploid-specific genes. This diploid cell will act like a haploid α cell, producing α pheromones to mate with an a haploid cell, resulting in aneuploidy . [ 24 ]
Since α cells do not ordinarily mate with each other, the presence of a -like faker cells in a population of α cells can be detected in an a -like faker assay. This test exposes the MATα population, which lacks an active copy of the HIS3 gene, to a tester strain like YPH316 yeast, which lack a HIS1 gene, on YEPD agar . After transferring the pairs of yeast strains onto Sabouraud agar , only those that formed diploid cells by having a -like faker cells mate with the tester strain will be capable of synthesizing the amino acid histidine to survive. The extent of chromosome instability can be inferred from the proportion of surviving pairs since a -like faker cells naturally arise from damage to Chromosome III in yeast cells. [ 25 ]
Mating in yeast is stimulated by a cells' a -factor or α cells' α-factor pheromones binding the Ste3 receptor of α cells or Ste2 receptor of a cells, respectively, activating a heterotrimeric G protein . [ 26 ] [ 27 ] [ 28 ] The dimeric portion of this G-protein recruits Ste5 and its MAPK cascade to the membrane , resulting in the phosphorylation of Fus3 . [ 29 ]
The switching mechanism arises as a result of competition between the Fus3 protein (a MAPK protein) and the phosphatase Ptc1 . [ 30 ] These proteins both attempt to control the four phosphorylation sites of Ste5 , a scaffold protein , with Fus3 attempting to phosphorylate the phosphosites and Ptc1 attempting to dephosphorylate them. [ 31 ]
Presence of α-factor induces recruitment of Ptc1 to Ste5 via a four-amino acid motif located within the Ste5 phosphosites. [ 32 ] Ptc1 then dephosphorylates Ste5, resulting in the dissociation of the Fus3-Ste5 complex. [ 33 ] Fus3 dissociates in a switch-like manner, dependent on the phosphorylation state of the four phosphosites. [ 34 ] All four phosphosites must be dephosphorylated in order for Fus3 to dissociate. [ 35 ] [ 36 ] Fus3's ability to compete with Ptc1 decreases as Ptc1 is recruited, and thus the rate of dephosphorylation increases with the presence of pheromone. [ 37 ]
Kss1, a homologue of Fus3, does not affect shmooing, and does not contribute to the switch-like mating decision. [ 38 ] [ 39 ]
In yeast, mating as well as the production of shmoos occur via an all-or-none, switch-like mechanism. [ 40 ] This switch-like mechanism allows yeast cells to avoid making an unwise commitment to a highly demanding procedure. [ 41 ] The decision to mate must balance being energy-conservative and fast enough to avoid losing the potential mate. [ 42 ]
Yeast maintain an ultra-sensitivity to mating through:
a and α yeast share the same mating response pathway, with the only difference being the type of receptor that each mating type possesses. [ 45 ] Thus, the above description of an a -type yeast stimulated with α-factor resembles the mechanism of an α-type yeast stimulated with a-factor. [ 46 ] [ 47 ]
Wild type haploid yeast are capable of switching mating type between a and α. [ 48 ] Consequently, even if a single haploid cell of a given mating type founds a colony of yeast, mating type switching will cause cells of both a and α mating types to be present in the population. [ 49 ] [ 50 ] Combined with the strong drive for haploid cells to mate with cells of the opposite mating type and form diploids, mating type switching and consequent mating will cause the majority of cells in a colony to be diploid, regardless of whether a haploid or diploid cell founded the colony. [ 51 ] The vast majority of yeast strains studied in laboratories have been altered such that they cannot perform mating type switching (by deletion of the HO gene; see below). This allows the stable propagation of haploid yeast, as haploid cells of the a mating type will remain a cells (and α cells will remain α cells), unable to form diploid cells unless artificially exposed to the other mating type. [ 52 ]
Haploid yeast switch mating type by replacing the information present at the MAT locus. [ 53 ] For example, an a cell will switch to an α cell by replacing the MAT a allele with the MATα allele. [ 54 ] This replacement of one allele of MAT for the other is possible because yeast cells carry an additional silenced copy of both the MAT a and MATα alleles: the HML ( h omothallic m ating l eft) locus typically carries a silenced copy of the MATα allele, and the HMR ( h omothallic m ating r ight) locus typically carries a silenced copy of the MAT a allele. [ 7 ] The silent HML and HMR loci are often referred to as the silent mating cassettes, as the information present there is 'read into' the active MAT locus. [ 55 ]
These additional copies of the mating type information do not interfere with the function of whatever allele is present at the MAT locus because they are not expressed, so a haploid cell with the MAT a allele present at the active MAT locus is still an a cell, despite also having a silenced copy of the MATα allele present at HML . [ 56 ] Only the allele present at the active MAT locus is transcribed, and thus only the allele present at MAT will influence cell behaviour. [ 6 ] Hidden mating type loci are epigenetically silenced by SIR proteins , which form a heterochromatin scaffold that prevents transcription from the silent mating cassettes. [ 57 ]
The process of mating type switching is a gene conversion event initiated by the HO gene. [ 58 ] The HO gene is a tightly regulated haploid-specific gene that is only activated in haploid cells during the G 1 phase of the cell cycle . [ 59 ] The protein encoded by the HO gene is a DNA endonuclease , which physically cleaves DNA, but only at the MAT locus (due to the DNA sequence specificity of the HO endonuclease). [ 60 ]
Once HO cuts the DNA at MAT , exonucleases are attracted to the cut DNA ends and begin to degrade the DNA on both sides of the cut site. [ 61 ] This DNA degradation by exonucleases eliminates the DNA which encoded the MAT allele; however, the resulting gap in the DNA is repaired by copying in the genetic information present at either HML or HMR , filling in a new allele of either the MAT a or MATα gene. Thus, the silenced alleles of MAT a and MATα present at HML and HMR serve as a source of genetic information to repair the HO-induced DNA damage at the active MAT locus. [ 7 ]
The repair of the MAT locus after cutting by the HO endonuclease almost always results in a mating type switch. [ 7 ] [ 60 ] When an a cell cuts the MAT a allele present at the MAT locus, the cut at MAT will almost always be repaired by copying the information present at HML . [ 6 ] This results in MAT being repaired to the MATα allele, switching the mating type of the cell from a to α. [ 62 ] Similarly, an α cell which has its MATα allele cut by the HO endonuclease will almost always repair the damage using the information present at HMR , copying the MAT a gene to the MAT locus and switching the mating type of α cell to a . [ 63 ]
This is the result of a recombination enhancer (RE) located on the left arm of chromosome III. [ 64 ] Normally, a cells have Mcm1 bind to the RE to promote recombination using the HML region. [ 65 ] Deletion of the RE causes a cells to instead repair using HMR, maintaining their status as a cells rather than switching mating types. [ 66 ] In α cells, the α2 factor binds at the RE to repress recombination using the HML region. [ 67 ] Thus, yeast have a predetermined tendency toward DNA repair of the MAT locus using the HMR region. [ 68 ]
In 2006, evolutionary geneticist Leonid Kruglyak found that S. cerevisiae matings only involve out-crossing between different strains roughly once every 50,000 cell divisions. The vast majority of yeast mating instead involves members of the same strain because mating type switching allows a single ascus to produce both mating types from a single haploid cell. [ 69 ] This suggests that yeast primarily maintain their capability to mate through recombinational DNA repair during meiosis, rather than natural selection for fitness among a population with high genetic variability . [ 70 ]
Schizosaccharomyces pombe is a facultative sexual yeast that can undergo mating when nutrients are limited. [ 71 ] Exposure of S. pombe to hydrogen peroxide , which causes oxidative stress to DNA , strongly induces mating, meiosis, and formation of meiotic spores. [ 72 ] Thus, meiosis and meiotic recombination may be an adaptation for repairing DNA damage. [ 73 ] The MAT locus' structure in S. pombe resembles S. cerevisiae . The mating-type switching system is similar but evolved independently. [ 6 ]
Cryptococcus neoformans is a basidiomycetous fungus that grows as a budding yeast in culture and infected hosts. C. neoformans causes life-threatening meningoencephalitis in immunocompromised patients. It undergoes a filamentous transition during the sexual cycle to produce spores, the suspected infectious agent. The vast majority of environmental and clinical isolates of C. neoformans are of mating type α. Filaments ordinarily have haploid nuclei, but these can undergo a process of diploidization (perhaps by endoreduplication or stimulated nuclear fusion) to form diploid cells termed blastospores . [ 74 ]
The diploid nuclei of blastospores can then undergo meiosis, including recombination, to form haploid basidiospores that can then be dispersed. [ 74 ] This process is referred to as monokaryotic fruiting. This process depends on the gene dmc1 , a conserved homologue of the bacterial RecA and eukaryotic RAD51 genes. Dmc1 mediates homologous chromosome pairing during meiosis and repair of double-strand breaks in DNA. [ 75 ] Meiosis in C. neoformans may be performed to promote DNA repair in DNA-damaging environments, such as host-mediated responses to infection. [ 74 ] | https://en.wikipedia.org/wiki/Mating_of_yeast |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.