The University of Pisa (Italian Università di Pisa) is one of the most renowned Italian universities. It is located in Pisa, Tuscany. It was formally founded on the September 3, 1343 by an edict of Pope Clement VI, although there had been lectures on law in Pisa since the 11th century. The University has the oldest Botanical garden (Orto botanico di Pisa) founded in 1543. The University of Pisa is part of the Pisa University System, together with the Scuola Normale Superiore and Sant'Anna School. It offers a wide and wo ...

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• University of Pisa - Organization of the University
• University of Pisa - Notable alumni

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The Lotka-Volterra equations, also known as the predator-prey equations, are a pair of first order, non-linear, differential equations frequently used to describe the dynamics of biological systems in which two species interact, one a predator and one its prey. They were proposed independently by Alfred J. Lotka in 1925 and Vito Volterra in 1926. A classic model using the equations is of the population dynamics of the lynx and the snowshoe hare, popularised due to the extensive data collected on the relative populations of the ...

Including:

• Lotka-Volterra equation - The equations
• Lotka-Volterra equation - Physical meanings of the equations
• Lotka-Volterra equation - Prey
• Lotka-Volterra equation - Predators
• Lotka-Volterra equation - Solutions to the equations
• Lotka-Volterra equation - Dynamics of the system
• Lotka-Volterra equation - Population equilibrium
• Lotka-Volterra equation - Stability of the fixed points
• Lotka-Volterra equation - Bibliography

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Alfred James Lotka (March 2, 1880 - December 5, 1949) was a US mathematician and statistician, most famous for his work in population dynamics. Born in Lemberg, Austria-Hungary (now L'viv, Ukraine) Lotka's parents were US nationals and he was educated internationally, including a degree at the University of Birmingham, England. In 1935, he married Romola Beattie. They had no children. His varied working life included: General Chemical Company US Patent Office National Bureau of Standards ...

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• Alfred J. Lotka - Honors

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Michael was born into a Jewish family in Budapest. His older brother Karl become a famous economist. Their father was an engineer and entrepreneur whose volatile fortunes in railway speculation motivated Polanyi to seek financial stability through a career in medicine. He graduated in 1913, and shortly afterwards served as a physician in the Austro-Hungarian army during World War I, but was hospitalised, and during his convalescence wrote what became a doctorate in physical chem ...

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Michael Polanyi, Michael Polanyi - Early life, Michael Polanyi - Physical chemistry, Michael Polanyi - Philosophy of science, Michael Polanyi - Economics, Michael Polanyi - Honours, Michael Polanyi - Knowledge, Michael Polanyi - Bibliography

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Nowadays the University of Pisa consists of 11 faculties and 56 departments. These faculties offers a notable amount of courses in their related field of studies: Agriculture Arts Economics Engineering Foreign Languages & Literatures Law Mathematical, Physical & Natural Sciences Medicine & Surgery Pharmac ...

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University of Pisa, University of Pisa - Organization of the University, University of Pisa - Notable alumni

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Born in the Obuda district of Budapest, Orowan's father, Berthold, was a mechanical engineer and factory manager, and his mother, Josze Spitzer Ságvári, the daughter of an impoverished land owner. In 1928, Orowan commenced his education at the Technical University of Berlin in mechanical and electrical engineering but soon transfered to physics, completing his doctorate on the fracture of mica in 1932. He seems to have experienced some difficulty in finding immediate employment and spent the next few years living with his mother and ...

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Egon Orowan, Egon Orowan - Life, Egon Orowan - Honours

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Any dislocation can be described by the Burgers vector and the dislocation line. However, an introduction to these and other terms used to describe dislocations can be difficult and it is easer to begin with a simple description of an edge dislocation. Dislocation - Edge dislocations. Edge dislocations can be visualised as being formed by adding an extra half-plane of atoms to a perfect crystal, so that a defect is created in the regular crystal structure along the line where the extra half-plane en ...

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Dislocation, Dislocation - Dislocation geometry, Dislocation - Edge dislocations, Dislocation - Burgers vector, Dislocation - Screw and mixed dislocations, Dislocation - Observation of Dislocations, Dislocation - Dislocations slip and plasticity, Dislocation - Bibliography

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In the model system, the predators thrive when there are plentiful prey but, ultimately, outstrip their food supply and decline. As the predator population is low the prey population will increase again. These dynamics continue in a cycle of growth and decline. Lotka-Volterra equation - Population equilibrium. Population equilibrium occurs in the model when neither of the population levels are changing, i.e. when both of the differential equations are equal to 0. x(α − βy) = 0 See also:

Lotka-Volterra equation, Lotka-Volterra equation - The equations, Lotka-Volterra equation - Physical meanings of the equations, Lotka-Volterra equation - Prey, Lotka-Volterra equation - Predators, Lotka-Volterra equation - Solutions to the equations, Lotka-Volterra equation - Dynamics of the system, Lotka-Volterra equation - Population equilibrium, Lotka-Volterra equation - Stability of the fixed points, Lotka-Volterra equation - Bibliography

Read more here: » Lotka-Volterra equation: Encyclopedia II - Lotka-Volterra equation - Dynamics of the system

In the modern view, functional analysis is seen as the study of complete normed vector spaces over the real or complex numbers. Such spaces are called Banach spaces. An important example is a Hilbert space, where the norm arises from an inner product. These spaces are of fundamental importance in the mathematical formulation of quantum mechanics. More generally, functional analysis includes the study of Fréchet spaces ...

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Functional analysis, Functional analysis - Normed vector spaces, Functional analysis - Hilbert spaces, Functional analysis - Banach spaces, Functional analysis - Major and foundational results, Functional analysis - Foundations of mathematics considerations, Functional analysis - Points of view

Read more here: » Functional analysis: Encyclopedia II - Functional analysis - Normed vector spaces

Until the 1930s, one of the enduring challenges of materials science was to explain plasticity in microscopic terms. A naive attempt to calculate the shear stress at which neighbouring atomic planes slip over each other in a perfect crystal suggests that, for a material with shear modulus G, shear strength τm is given approximately by: As shear modulus in metals is typically within the range 20 000 to 150 000 MPa, this is difficult to reconcile with shear stresses in the range 0.5 t ...

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Dislocation, Dislocation - Dislocation geometry, Dislocation - Edge dislocations, Dislocation - Burgers vector, Dislocation - Screw and mixed dislocations, Dislocation - Observation of Dislocations, Dislocation - Dislocations slip and plasticity, Dislocation - Bibliography

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When a dislocation line intersects the surface of a metallic material, the associated strain field locally increases the relative susceptibility of the material to acidic etching and an etch pit of regular geometrical format results. If the material is strained (deformed) and repeatedly re-etched, a series of etch pits can be produced which effectively trace the movement of the dislocation in question. Transmission electron microscopy can be used to observe dislocations within the microstructure of the material. Thin foils of metallic ...

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Dislocation, Dislocation - Dislocation geometry, Dislocation - Edge dislocations, Dislocation - Burgers vector, Dislocation - Screw and mixed dislocations, Dislocation - Observation of Dislocations, Dislocation - Dislocations slip and plasticity, Dislocation - Bibliography

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These are important results of functional analysis: The uniform boundedness principle is a result on sets of operators with tight bounds. One spectral theorem (there are more of them) gives an integral formula for normal operators on a Hilbert space. It is of central importance in the mathematical formulation of quantum mechanics. The Hahn-Banach theorem is about extending functionals from a subspace to the full space, in a norm-preserving fashion. Another implication is the non-triviality of dual spaces. The open mapping theorem and closed graph theor ...

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Functional analysis, Functional analysis - Normed vector spaces, Functional analysis - Hilbert spaces, Functional analysis - Banach spaces, Functional analysis - Major and foundational results, Functional analysis - Foundations of mathematics considerations, Functional analysis - Points of view

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The usual form of the equations is: where y is the number of some predator (for example, dingoes); x is the number of its prey (for example, wallabies); t represents the growth of the two populations against time; and α, β, γ and δ are parameters representing the interaction of the two species. ...

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Lotka-Volterra equation, Lotka-Volterra equation - The equations, Lotka-Volterra equation - Physical meanings of the equations, Lotka-Volterra equation - Prey, Lotka-Volterra equation - Predators, Lotka-Volterra equation - Solutions to the equations, Lotka-Volterra equation - Dynamics of the system, Lotka-Volterra equation - Population equilibrium, Lotka-Volterra equation - Stability of the fixed points, Lotka-Volterra equation - Bibliography

Read more here: » Lotka-Volterra equation: Encyclopedia II - Lotka-Volterra equation - The equations

Polanyi's scientific interests were diverse, embracing chemical kinetics, x-ray diffraction and the absorption of gases at solid surfaces. In 1934, Polanyi, roughly contemporarily with G. I. Taylor and Egon Orowan realised that the plastic deformation of ductile materials could be explained in terms of the theory of dislocations developed by Vito Volterra in 1905. The insight was critical in developing the modern science of solid mechanics. ...

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Michael Polanyi, Michael Polanyi - Early life, Michael Polanyi - Physical chemistry, Michael Polanyi - Philosophy of science, Michael Polanyi - Economics, Michael Polanyi - Honours, Michael Polanyi - Knowledge, Michael Polanyi - Bibliography

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From the middle years of the Nineteen-Thirties Polanyi began to articulate his opposition to the prevailing positivist account of science, arguing that it failed to recognise the part played by tacit knowledge and the creative role played by the imagination. He viewed positivism as encouraging some to believe that scientific research ought to be directed by the State. He drew attention to what happened to genetics in the Soviet Union, once the doctrines of Trofim Lysenko gained political approval. Polanyi, like Friedrich Hayek, supplied r ...

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Michael Polanyi, Michael Polanyi - Early life, Michael Polanyi - Physical chemistry, Michael Polanyi - Philosophy of science, Michael Polanyi - Economics, Michael Polanyi - Honours, Michael Polanyi - Knowledge, Michael Polanyi - Bibliography

Read more here: » Michael Polanyi: Encyclopedia II - Michael Polanyi - Philosophy of science

When multiplied out, the equations take a form useful for physical interpretation. Lotka-Volterra equation - Prey. The prey equation becomes: The prey are assumed to have an unlimited food supply, and to reproduce exponentially unless subject to predation; this exponential growth is represented in equation above by the term αx. The rate of predation upon the prey is assumed to be proportional to the rate at which the predators and the prey meet; this is represented above by β ...

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Lotka-Volterra equation, Lotka-Volterra equation - The equations, Lotka-Volterra equation - Physical meanings of the equations, Lotka-Volterra equation - Prey, Lotka-Volterra equation - Predators, Lotka-Volterra equation - Solutions to the equations, Lotka-Volterra equation - Dynamics of the system, Lotka-Volterra equation - Population equilibrium, Lotka-Volterra equation - Stability of the fixed points, Lotka-Volterra equation - Bibliography

Read more here: » Lotka-Volterra equation: Encyclopedia II - Lotka-Volterra equation - Physical meanings of the equations