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Non-Euclidean Surface Growth
In the field of surface growth, there are growth processes that result in the surface of an object changing shape over time. As the object grows, its surface may change from flat to curved, or change curvature. Two points on the surface may also change in distance as a result of deformations of the object or accretion of new matter onto the object. The shape of the surface and its changes can be described in terms of non-Euclidean geometry and in particular, Riemannian geometry with a space- and time-dependent curvature. Examples of non-Euclidean surface growth are found in the mechanics of growing gravitational bodies, propagating fronts of phase transitions, epitaxial growth of nanostructures and additive 3D printing, growth of plants, and cell motility Motility is the ability of an organism to move independently using metabolism, metabolic energy. This biological concept encompasses movement at various levels, from whole organisms to cells and subcellular components. Mot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surface Growth
In mathematics and physics, surface growth refers to models used in the dynamical study of the growth of a surface, usually by means of a stochastic differential equation of a field. Examples Popular growth models include: * KPZ equation * Dimer model * Eden growth model * SOS model *Self-avoiding walk *Abelian sandpile model *Kuramoto–Sivashinsky equation (or the flame equation, for studying the surface of a flame front) They are studied for their fractal properties, scaling behavior, critical exponents, universality classes, and relations to chaos theory, dynamical system, non-equilibrium / disordered / complex systems. Popular tools include statistical mechanics, renormalization group, rough path theory, etc. Kinetic Monte Carlo surface growth model Kinetic Monte Carlo (KMC) is a form of computer simulation in which atoms and molecules are allowed to interact at given rate that could be controlled based on known physics. This simulation method is typically used in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curvature
In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or surface is contained in a larger space, curvature can be defined ''extrinsically'' relative to the ambient space. Curvature of Riemannian manifolds of dimension at least two can be defined ''intrinsically'' without reference to a larger space. For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature ''at a point'' of a differentiable curve is the curvature of its osculating circle — that is, the circle that best approximates the curve near this point. The curvature of a straight line is zero. In contrast to the tangent, which is a vector quantity, the curvature at a point is typically a scalar q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deformation (physics)
In physics and continuum mechanics, deformation is the change in the shape or size of an object. It has dimension of length with SI unit of metre (m). It is quantified as the residual displacement of particles in a non-rigid body, from an configuration to a configuration, excluding the body's average translation and rotation (its rigid transformation). A ''configuration'' is a set containing the positions of all particles of the body. A deformation can occur because of external loads, intrinsic activity (e.g. muscle contraction), body forces (such as gravity or electromagnetic forces), or changes in temperature, moisture content, or chemical reactions, etc. In a continuous body, a ''deformation field'' results from a stress field due to applied forces or because of some changes in the conditions of the body. The relation between stress and strain (relative deformation) is expressed by constitutive equations, e.g., Hooke's law for linear elastic materials. Deform ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Accretion (astrophysics)
In astrophysics, accretion is the accumulation of particles into a massive object by gravity, gravitationally attracting more matter, typically gaseous matter, into an accretion disk. Most astronomical objects, such as galaxy, galaxies, stars, and planets, are formed by accretion processes. Overview The accretion model that Earth and the other terrestrial planets formed from meteoric material was proposed in 1944 by Otto Schmidt, followed by the ''protoplanet theory'' of William McCrea (astronomer), William McCrea (1960) and finally the ''capture theory'' of Michael Woolfson. For details of Kant's position, see In 1978, Andrew Prentice resurrected the initial Laplacian ideas about planet formation and developed the ''modern Laplacian theory''. None of these models proved completely successful, and many of the proposed theories were descriptive. The 1944 accretion model by Otto Schmidt was further developed in a quantitative way in 1969 by Viktor Safronov. He calculated, in deta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-Euclidean Geometry
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric requirement. In the former case, one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras, which give rise to kinematic geometries that have also been called non-Euclidean geometry. Principles The essential difference between the metric geometries is the nature of parallel lines. Euclid's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate, which states that, within a two-dimensional plane, for any given line and a point ''A'', which is not on , there is exactly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smoothly from point to point). This gives, in particular, local notions of angle, arc length, length of curves, surface area and volume. From those, some other global quantities can be derived by integral, integrating local contributions. Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture "" ("On the Hypotheses on which Geometry is Based"). It is a very broad and abstract generalization of the differential geometry of surfaces in Three-dimensional space, R3. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gravity
In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force between objects and the Earth. This force is dominated by the combined gravitational interactions of particles but also includes effect of the Earth's rotation. Gravity gives weight to physical objects and is essential to understanding the mechanisms responsible for surface water waves and lunar tides. Gravity also has many important biological functions, helping to guide the growth of plants through the process of gravitropism and influencing the circulation of fluids in multicellular organisms. The gravitational attraction between primordial hydrogen and clumps of dark matter in the early universe caused the hydrogen gas to coalesce, eventually condensing and fusing to form stars. At larger scales this results in galaxies and clust ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phase Transition
In physics, chemistry, and other related fields like biology, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic State of matter, states of matter: solid, liquid, and gas, and in rare cases, plasma (physics), plasma. A phase of a thermodynamic system and the states of matter have uniform physical property, physical properties. During a phase transition of a given medium, certain properties of the medium change as a result of the change of external conditions, such as temperature or pressure. This can be a discontinuous change; for example, a liquid may become gas upon heating to its boiling point, resulting in an abrupt change in volume. The identification of the external conditions at which a transformation occurs defines the phase transition point. Types of phase transition States of matter Phase transitions commonly refer to when a substance tran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Epitaxial Growth
Epitaxy (prefix ''epi-'' means "on top of”) is a type of crystal growth or material deposition in which new crystalline layers are formed with one or more well-defined orientations with respect to the crystalline seed layer. The deposited crystalline film is called an epitaxial film or epitaxial layer. The relative orientation(s) of the epitaxial layer to the seed layer is defined in terms of the orientation of the crystal lattice of each material. For most epitaxial growths, the new layer is usually crystalline and each crystallographic domain of the overlayer must have a well-defined orientation relative to the substrate crystal structure. Epitaxy can involve single-crystal structures, although grain-to-grain epitaxy has been observed in granular films. For most technological applications, single-domain epitaxy, which is the growth of an overlayer crystal with one well-defined orientation with respect to the substrate crystal, is preferred. Epitaxy can also play an important ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nanostructures
A nanostructure is a structure of intermediate size between microscopic and molecular structure A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...s. Nanostructural detail is microstructure at nanoscale. In describing nanostructures, it is necessary to differentiate between the number of dimensions in the volume of an object which are 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 can be far more. Finally, spherical nanoparticles have ''three dimensions'' on the nanoscale, i.e., the particle is between 0.1 and 100 nm in each spat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3D Printing
3D printing, or additive manufacturing, is the construction of a three-dimensional object from a CAD model or a digital 3D model. It can be done in a variety of processes in which material is deposited, joined or solidified under computer control, with the material being added together (such as plastics, liquids or powder grains being fused), typically layer by layer. In the 1980s, 3D printing techniques were considered suitable only for the production of functional or aesthetic prototypes, and a more appropriate term for it at the time was rapid prototyping. , the precision, repeatability, and material range of 3D printing have increased to the point that some 3D printing processes are considered viable as an industrial-production technology; in this context, the term ''additive manufacturing'' can be used synonymously with ''3D printing''. One of the key advantages of 3D printing is the ability to produce very complex shapes or geometries that would be otherwise infeasi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cell Motility
Motility is the ability of an organism to move independently using metabolism, metabolic energy. This biological concept encompasses movement at various levels, from whole organisms to cells and subcellular components. Motility is observed in animals, microorganisms, and even some plant structures, playing crucial roles in activities such as foraging, reproduction, and cellular functions. It is genetically determined but can be influenced by environmental factors. In multicellular organisms, motility is facilitated by systems like the Nervous system, nervous and Human musculoskeletal system, musculoskeletal systems, while at the cellular level, it involves mechanisms such as amoeboid movement and flagellar propulsion. These cellular movements can be directed by external stimuli, a phenomenon known as taxis. Examples include chemotaxis (movement along chemical gradients) and phototaxis (movement in response to light). Motility also includes physiological processes like gastroi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
