1.
Orange (fruit)
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The orange is the fruit of the citrus species Citrus × sinensis in the family Rutaceae. It is also called orange, to distinguish it from the related Citrus × aurantium. The sweet orange reproduces asexually, varieties of sweet orange arise through mutations, the orange is a hybrid between pomelo and mandarin. It has genes that are ~25% pomelo and ~75% mandarin, however, it is not a simple backcrossed BC1 hybrid, the chloroplast genes, and therefore the maternal line, seem to be pomelo. The sweet orange has had its genome sequenced. Earlier estimates of the percentage of pomelo genes varying from ~50% to 6% have been reported, sweet oranges were mentioned in Chinese literature in 314 BC. As of 1987, orange trees were found to be the most cultivated fruit tree in the world, orange trees are widely grown in tropical and subtropical climates for their sweet fruit. The fruit of the tree can be eaten fresh, or processed for its juice or fragrant peel. As of 2012, sweet oranges accounted for approximately 70% of citrus production, in 2014,70.9 million tonnes of oranges were grown worldwide, with Brazil producing 24% of the world total followed by China and India. All citrus trees belong to the single genus Citrus and remain almost entirely interfertile and this includes grapefruits, lemons, limes, oranges, and various other types and hybrids. The fruit of any citrus tree is considered a hesperidium, a kind of modified berry, different names have been given to the many varieties of the genus. Orange applies primarily to the sweet orange – Citrus sinensis Osbeck, the orange tree is an evergreen, flowering tree, with an average height of 9 to 10 m, although some very old specimens can reach 15 m. Its oval leaves, alternately arranged, are 4 to 10 cm long and have crenulate margins, when unripe, the fruit is green. The grainy irregular rind of the fruit can range from bright orange to yellow-orange. Like all other fruits, the sweet orange is non-climacteric. The Citrus sinensis group is subdivided into four classes with distinct characteristics, common oranges, blood or pigmented oranges, navel oranges, other citrus groups also known as oranges are, Bitter orange, also known as Seville orange, sour orange, bigarade orange and marmalade orange. Like the sweet orange, it is a pomelo x mandarin hybrid, bergamot orange, grown mainly in Italy for its peel, producing a primary essence for perfumes, also used to flavor Earl Grey tea. It is a hybrid, probably bitter orange x limetta, trifoliate orange, sometimes included in the genus
2.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
3.
Sphere
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A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball. This distance r is the radius of the ball, and the point is the center of the mathematical ball. The longest straight line through the ball, connecting two points of the sphere, passes through the center and its length is twice the radius. While outside mathematics the terms sphere and ball are used interchangeably. The ball and the share the same radius, diameter. The surface area of a sphere is, A =4 π r 2, at any given radius r, the incremental volume equals the product of the surface area at radius r and the thickness of a shell, δ V ≈ A ⋅ δ r. The total volume is the summation of all volumes, V ≈ ∑ A ⋅ δ r. In the limit as δr approaches zero this equation becomes, V = ∫0 r A d r ′, substitute V,43 π r 3 = ∫0 r A d r ′. Differentiating both sides of equation with respect to r yields A as a function of r,4 π r 2 = A. Which is generally abbreviated as, A =4 π r 2, alternatively, the area element on the sphere is given in spherical coordinates by dA = r2 sin θ dθ dφ. In Cartesian coordinates, the element is d S = r r 2 − ∑ i ≠ k x i 2 ∏ i ≠ k d x i, ∀ k. For more generality, see area element, the total area can thus be obtained by integration, A = ∫02 π ∫0 π r 2 sin θ d θ d φ =4 π r 2. In three dimensions, the volume inside a sphere is derived to be V =43 π r 3 where r is the radius of the sphere, archimedes first derived this formula, which shows that the volume inside a sphere is 2/3 that of a circumscribed cylinder. In modern mathematics, this formula can be derived using integral calculus, at any given x, the incremental volume equals the product of the cross-sectional area of the disk at x and its thickness, δ V ≈ π y 2 ⋅ δ x. The total volume is the summation of all volumes, V ≈ ∑ π y 2 ⋅ δ x. In the limit as δx approaches zero this equation becomes, V = ∫ − r r π y 2 d x. At any given x, a right-angled triangle connects x, y and r to the origin, hence, applying the Pythagorean theorem yields, thus, substituting y with a function of x gives, V = ∫ − r r π d x. Which can now be evaluated as follows, V = π − r r = π − π =43 π r 3
4.
Dimension
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In physics and mathematics, the dimension of a mathematical space is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one only one coordinate is needed to specify a point on it – for example. The inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces, in classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a space but not the one that was found necessary to describe electromagnetism. The four dimensions of spacetime consist of events that are not absolutely defined spatially and temporally, Minkowski space first approximates the universe without gravity, the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. Ten dimensions are used to string theory, and the state-space of quantum mechanics is an infinite-dimensional function space. The concept of dimension is not restricted to physical objects, high-dimensional spaces frequently occur in mathematics and the sciences. They may be parameter spaces or configuration spaces such as in Lagrangian or Hamiltonian mechanics, in mathematics, the dimension of an object is an intrinsic property independent of the space in which the object is embedded. This intrinsic notion of dimension is one of the ways the mathematical notion of dimension differs from its common usages. The dimension of Euclidean n-space En is n, when trying to generalize to other types of spaces, one is faced with the question what makes En n-dimensional. One answer is that to cover a ball in En by small balls of radius ε. This observation leads to the definition of the Minkowski dimension and its more sophisticated variant, the Hausdorff dimension, for example, the boundary of a ball in En looks locally like En-1 and this leads to the notion of the inductive dimension. While these notions agree on En, they turn out to be different when one looks at more general spaces, a tesseract is an example of a four-dimensional object. The rest of this section some of the more important mathematical definitions of the dimensions. A complex number has a real part x and an imaginary part y, a single complex coordinate system may be applied to an object having two real dimensions. For example, an ordinary two-dimensional spherical surface, when given a complex metric, complex dimensions appear in the study of complex manifolds and algebraic varieties. The dimension of a space is the number of vectors in any basis for the space. This notion of dimension is referred to as the Hamel dimension or algebraic dimension to distinguish it from other notions of dimension
5.
Euclidean space
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In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces. It is named after the Ancient Greek mathematician Euclid of Alexandria, the term Euclidean distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions, classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. Geometric constructions are used to define rational numbers. It means that points of the space are specified with collections of real numbers and this approach brings the tools of algebra and calculus to bear on questions of geometry and has the advantage that it generalizes easily to Euclidean spaces of more than three dimensions. From the modern viewpoint, there is only one Euclidean space of each dimension. With Cartesian coordinates it is modelled by the coordinate space of the same dimension. In one dimension, this is the line, in two dimensions, it is the Cartesian plane, and in higher dimensions it is a coordinate space with three or more real number coordinates. One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance, for example, there are two fundamental operations on the plane. One is translation, which means a shifting of the plane so that point is shifted in the same direction. The other is rotation about a point in the plane. In order to all of this mathematically precise, the theory must clearly define the notions of distance, angle, translation. Even when used in theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, measurement instruments. The standard way to such space, as carried out in the remainder of this article, is to define the Euclidean plane as a two-dimensional real vector space equipped with an inner product. The reason for working with vector spaces instead of Rn is that it is often preferable to work in a coordinate-free manner. Once the Euclidean plane has been described in language, it is actually a simple matter to extend its concept to arbitrary dimensions. For the most part, the vocabulary, formulae, and calculations are not made any more difficult by the presence of more dimensions. Intuitively, the distinction says merely that there is no choice of where the origin should go in the space
6.
Packing problems
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Packing problems are a class of optimization problems in mathematics that involve attempting to pack objects together into containers. The goal is to pack a single container as densely as possible or pack all objects using as few containers as possible. Many of these problems can be related to real life packaging, storage, each packing problem has a dual covering problem, which asks how many of the same objects are required to completely cover every region of the container, where objects are allowed to overlap. In a Bin packing problem, you are given, containers A set of some or all of which must be packed into one or more containers. The set may contain different objects with their sizes specified, or an object of a fixed dimension that can be used repeatedly. Usually the packing must be without overlaps between goods and other goods or the container walls, in some variants, the aim is to find the configuration that packs a single container with the maximal density. More commonly, the aim is to all the objects into as few containers as possible. In some variants the overlapping is allowed but should be minimized, many of these problems, when the container size is increased in all directions, become equivalent to the problem of packing objects as densely as possible in infinite Euclidean space. This problem is relevant to a number of disciplines, and has received significant attention. The Kepler conjecture postulated an optimal solution for packing spheres hundreds of years before it was correct by Thomas Callister Hales. Many other shapes have received attention, including ellipsoids, Platonic and Archimedean solids including tetrahedra and these problems are mathematically distinct from the ideas in the circle packing theorem. The related circle packing problem deals with packing circles, possibly of different sizes, on a surface, the counterparts of a circle in other dimensions can never be packed with complete efficiency in dimensions larger than one. That is, there always be unused space if you are only packing circles. The most efficient way of packing circles, hexagonal packing, produces approximately 91% efficiency, in three dimensions, the face-centered cubic lattice offers the best lattice packing of spheres, and is believed to be the optimal of all packings. With simple sphere packings in three dimensions there are nine possible defineable packings, the 8-dimensional E8 lattice and 24-dimensional Leech lattice have also been proven to be optimal in their respective real dimensional space. Cubes can easily be arranged to fill space completely, the most natural packing being the cubic honeycomb. No other Platonic solid can tile space on its own, tetrahedra can achieve a packing of at least 85%. One of the best packings of regular dodecahedra is based on the aforementioned face-centered cubic lattice, tetrahedra and octahedra together can fill all of space in an arrangement known as the tetrahedral-octahedral honeycomb
7.
Circle packing
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This article describes the packing of circles on surfaces. For the related article on circle packing with an intersection graph. In geometry, circle packing is the study of the arrangement of circles on a surface such that no overlapping occurs. The associated packing density, η, of an arrangement is the proportion of the covered by the circles. Generalisations can be made to higher dimensions – this is called sphere packing, while the circle has a relatively low maximum packing density of 0.9069 on the Euclidean plane, it does not have the lowest possible. Packing densities of concave shapes such as star polygons can be arbitrarily small, the density of this arrangement is η h = π23 ≈0.9069. Axel Thue provided the first proof that this was optimal in 1890, showing that the lattice is the densest of all possible circle packings. However, his proof was considered by some to be incomplete, the first rigorous proof is attributed to László Fejes Tóth in 1940. At the other extreme, very low density arrangements of rigidly packed circles have been identified, there are 11 circle packings based on the 11 uniform tilings of the plane. In these packings, every circle can be mapped to every other circle by reflections and rotations, the hexagonal gaps can be filled by one circle and the dodecagonal gaps can be filled with 7 circles, creating 3-uniform packings. The truncated trihexagonal tiling with both types of gaps can be filled as a 4-uniform packing, the snub hexagonal tiling has two mirror-image forms. A related problem is to determine the arrangement of identically interacting points that are constrained to lie within a given surface. The Thomson problem deals with the lowest energy distribution of electric charges on the surface of a sphere. The Tammes problem is a generalisation of this, dealing with maximising the minimum distance between circles on sphere and this is analogous to distributing non-point charges on a sphere. Packing circles in simple bounded shapes is a type of problem in recreational mathematics. The influence of the walls is important, and hexagonal packing is generally not optimal for small numbers of circles. There are also a range of problems which permit the sizes of the circles to be non-uniform, one such extension is to find the maximum possible density of a system with two specific sizes of circle. Only nine particular radius ratios permit compact packing, which is when every pair of circles in contact is in contact with two other circles
8.
Non-Euclidean geometry
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In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry. In the latter case one obtains hyperbolic geometry and elliptic geometry, 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. The essential difference between the geometries is the nature of parallel lines. In hyperbolic geometry, by contrast, there are many lines through A not intersecting ℓ, while in elliptic geometry. In elliptic geometry the lines curve toward each other and intersect, the debate that eventually led to the discovery of the non-Euclidean geometries began almost as soon as Euclids work Elements was written. In the Elements, Euclid began with a number of assumptions. Other mathematicians have devised simpler forms of this property, regardless of the form of the postulate, however, it consistently appears to be more complicated than Euclids other postulates,1. To draw a line from any point to any point. To produce a straight line continuously in a straight line. To describe a circle with any centre and distance and that all right angles are equal to one another. For at least a thousand years, geometers were troubled by the complexity of the fifth postulate. Many attempted to find a proof by contradiction, including Ibn al-Haytham, Omar Khayyám, Nasīr al-Dīn al-Tūsī and these theorems along with their alternative postulates, such as Playfairs axiom, played an important role in the later development of non-Euclidean geometry. These early attempts did, however, provide some early properties of the hyperbolic and elliptic geometries. Another example is al-Tusis son, Sadr al-Din, who wrote a book on the subject in 1298, based on al-Tusis later thoughts and he essentially revised both the Euclidean system of axioms and postulates and the proofs of many propositions from the Elements. His work was published in Rome in 1594 and was studied by European geometers and he finally reached a point where he believed that his results demonstrated the impossibility of hyperbolic geometry. His claim seems to have based on Euclidean presuppositions, because no logical contradiction was present. In this attempt to prove Euclidean geometry he instead unintentionally discovered a new viable geometry, in 1766 Johann Lambert wrote, but did not publish, Theorie der Parallellinien in which he attempted, as Saccheri did, to prove the fifth postulate. He worked with a figure that today we call a Lambert quadrilateral and he quickly eliminated the possibility that the fourth angle is obtuse, as had Saccheri and Khayyam, and then proceeded to prove many theorems under the assumption of an acute angle
9.
Hyperbolic space
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In mathematics, hyperbolic space is a homogeneous space that has a constant negative curvature, where in this case the curvature is the sectional curvature. When embedded to a Euclidean space, every point of a space is a saddle point. Hyperbolic n-space, denoted Hn, is the symmetric, simply connected. Hyperbolic space is a space exhibiting hyperbolic geometry and it is the negative-curvature analogue of the n-sphere. Although hyperbolic space Hn is diffeomorphic to Rn, its negative-curvature metric gives it very different geometric properties, hyperbolic 2-space, H2, is also called the hyperbolic plane. Instead, the postulate is replaced by the following alternative, Given any line L and point P not on L. It is then a theorem that there are many such lines through P. This axiom still does not uniquely characterize the hyperbolic plane up to isometry, there is a constant, the curvature K <0. However, it does uniquely characterize it up to homothety, meaning up to bijections which only change the notion of distance by an overall constant, by choosing an appropriate length scale, one can thus assume, without loss of generality, that K = −1. Models of hyperbolic spaces that can be embedded in a flat spaces may be constructed, in particular, the existence of model spaces implies that the parallel postulate is logically independent of the other axioms of Euclidean geometry. There are several important models of space, the Klein model, the hyperboloid model, the Poincaré ball model. These all model the same geometry in the sense that any two of them can be related by a transformation that preserves all the properties of the space. The hyperboloid model realizes hyperbolic space as a hyperboloid in Rn+1 =, the hyperboloid is the locus Hn of points whose coordinates satisfy x 02 − x 12 − ⋯ − x n 2 =1, x 0 >0. In this model a line is the curve formed by the intersection of Hn with a plane through the origin in Rn+1, the hyperboloid model is closely related to the geometry of Minkowski space. The space Rn+1, equipped with the bilinear form B, is an -dimensional Minkowski space Rn,1, one can associate a distance on the hyperboloid model by defining the distance between two points x and y on H to be d = arcosh B. This function satisfies the axioms of a metric space and it is preserved by the action of the Lorentz group on Rn,1. Hence the Lorentz group acts as a group preserving isometry on Hn. An alternative model of geometry is on a certain domain in projective space
10.
Average
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In colloquial language, an average is the sum of a list of numbers divided by the number of numbers in the list. In mathematics and statistics, this would be called the arithmetic mean, in statistics, mean, median, and mode are all known as measures of central tendency. The most common type of average is the arithmetic mean, one may find that A = /2 =5. Switching the order of 2 and 8 to read 8 and 2 does not change the value obtained for A. The mean 5 is not less than the minimum 2 nor greater than the maximum 8. If we increase the number of terms in the list to 2,8, and 11, one finds that A = /3 =7. Along with the arithmetic mean above, the mean and the harmonic mean are known collectively as the Pythagorean means. The geometric mean of n numbers is obtained by multiplying them all together. See Inequality of arithmetic and geometric means, thus for the above harmonic mean example, AM =50, GM ≈49, and HM =48 km/h. The mode, the median, and the mid-range are often used in addition to the mean as estimates of central tendency in descriptive statistics, the most frequently occurring number in a list is called the mode. For example, the mode of the list is 3 and it may happen that there are two or more numbers which occur equally often and more often than any other number. In this case there is no agreed definition of mode, some authors say they are all modes and some say there is no mode. The median is the number of the group when they are ranked in order. Thus to find the median, order the list according to its elements magnitude, if exactly one value is left, it is the median, if two values, the median is the arithmetic mean of these two. This method takes the list 1,7,3,13, then the 1 and 13 are removed to obtain the list 3,7. Since there are two elements in this remaining list, the median is their arithmetic mean, /2 =5, the table of mathematical symbols explains the symbols used below. Other more sophisticated averages are, trimean, trimedian, and normalized mean, one can create ones own average metric using the generalized f-mean, y = f −1 where f is any invertible function. The harmonic mean is an example of this using f = 1/x, however, this method for generating means is not general enough to capture all averages
11.
Asymptote
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In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, in some contexts, such as algebraic geometry, an asymptote is defined as a line which is tangent to a curve at infinity. The word asymptote is derived from the Greek ἀσύμπτωτος which means not falling together, + σύν together + πτωτ-ός fallen. The term was introduced by Apollonius of Perga in his work on conic sections, there are potentially three kinds of asymptotes, horizontal, vertical and oblique asymptotes. For curves given by the graph of a function y = ƒ, vertical asymptotes are vertical lines near which the function grows without bound. Asymptotes convey information about the behavior of curves in the large, the study of asymptotes of functions, construed in a broad sense, forms a part of the subject of asymptotic analysis. The idea that a curve may come close to a line without actually becoming the same may seem to counter everyday experience. The representations of a line and a curve as marks on a piece of paper or as pixels on a screen have a positive width. So if they were to be extended far enough they would seem to merge, but these are physical representations of the corresponding mathematical entities, the line and the curve are idealized concepts whose width is 0. Therefore, the understanding of the idea of an asymptote requires an effort of reason rather than experience, consider the graph of the function f =1 x shown to the right. The coordinates of the points on the curve are of the form where x is an other than 0. But no matter how large x becomes, its reciprocal 1 x is never 0, so the curve extends farther and farther upward as it comes closer and closer to the y-axis. Thus, both the x and y-axes are asymptotes of the curve and these ideas are part of the basis of concept of a limit in mathematics, and this connection is explained more fully below. The asymptotes most commonly encountered in the study of calculus are of curves of the form y = ƒ and these can be computed using limits and classified into horizontal, vertical and oblique asymptotes depending on its orientation. Horizontal asymptotes are lines that the graph of the function approaches as x tends to +∞ or −∞. As the name indicate they are parallel to the x-axis, vertical asymptotes are vertical lines near which the function grows without bound. Oblique asymptotes are diagonal lines so that the difference between the curve and the line approaches 0 as x tends to +∞ or −∞, more general type of asymptotes can be defined in this case. Only open curves that have some infinite branch, can have an asymptote, no closed curve can have an asymptote
12.
Lattice (group)
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In geometry and group theory, a lattice in R n is a subgroup of R n which is isomorphic to Z n, and which spans the real vector space R n. In other words, for any basis of R n, the subgroup of all linear combinations with integer coefficients of the basis vectors forms a lattice, a lattice may be viewed as a regular tiling of a space by a primitive cell. Lattices have many significant applications in mathematics, particularly in connection to Lie algebras, number theory. More generally, lattice models are studied in physics, often by the techniques of computational physics, a lattice is the symmetry group of discrete translational symmetry in n directions. A pattern with this lattice of translational symmetry cannot have more, as a group a lattice is a finitely-generated free abelian group, and thus isomorphic to Z n. A lattice in the sense of a 3-dimensional array of regularly spaced points coinciding with e. g, a simple example of a lattice in R n is the subgroup Z n. More complicated examples include the E8 lattice, which is a lattice in R8, the period lattice in R2 is central to the study of elliptic functions, developed in nineteenth century mathematics, it generalises to higher dimensions in the theory of abelian functions. Lattices called root lattices are important in the theory of simple Lie algebras, for example, a typical lattice Λ in R n thus has the form Λ = where is a basis for R n. Different bases can generate the lattice, but the absolute value of the determinant of the vectors vi is uniquely determined by Λ. If one thinks of a lattice as dividing the whole of R n into equal polyhedra and this is why d is sometimes called the covolume of the lattice. If this equals 1, the lattice is called unimodular, minkowskis theorem relates the number d and the volume of a symmetric convex set S to the number of lattice points contained in S. The number of lattice points contained in an all of whose vertices are elements of the lattice is described by the polytopes Ehrhart polynomial. Formulas for some of the coefficients of this polynomial involve d as well, Lattice basis reduction is the problem of finding a short and nearly orthogonal lattice basis. The Lenstra-Lenstra-Lovász lattice basis reduction algorithm approximates such a basis in polynomial time, it has found numerous applications. There are five 2D lattice types as given by the crystallographic restriction theorem, below, the wallpaper group of the lattice is given in IUC notation, Orbifold notation, and Coxeter notation, along with a wallpaper diagram showing the symmetry domains. Note that a pattern with this lattice of translational symmetry cannot have more, a full list of subgroups is available. For example below the hexagonal/triangular lattice is given twice, with full 6-fold, if the symmetry group of a pattern contains an n-fold rotation then the lattice has n-fold symmetry for even n and 2n-fold for odd n. For the classification of a lattice, start with one point
13.
Periodic function
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In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the functions, which repeat over intervals of 2π radians. Periodic functions are used throughout science to describe oscillations, waves, any function which is not periodic is called aperiodic. A function f is said to be periodic with period P if we have f = f for all values of x in the domain, geometrically, a periodic function can be defined as a function whose graph exhibits translational symmetry. Specifically, a function f is periodic with period P if the graph of f is invariant under translation in the x-direction by a distance of P and this definition of periodic can be extended to other geometric shapes and patterns, such as periodic tessellations of the plane. A function that is not periodic is called aperiodic, for example, the sine function is periodic with period 2 π, since sin = sin x for all values of x. This function repeats on intervals of length 2 π, everyday examples are seen when the variable is time, for instance the hands of a clock or the phases of the moon show periodic behaviour. Periodic motion is motion in which the position of the system are expressible as periodic functions, for a function on the real numbers or on the integers, that means that the entire graph can be formed from copies of one particular portion, repeated at regular intervals. A simple example of a function is the function f that gives the fractional part of its argument. In particular, f = f = f =, =0.5 The graph of the function f is the sawtooth wave. The trigonometric functions sine and cosine are periodic functions, with period 2π. The subject of Fourier series investigates the idea that a periodic function is a sum of trigonometric functions with matching periods. According to the definition above, some functions, for example the Dirichlet function, are also periodic, in the case of Dirichlet function. For example, f = sin has period 2 π therefore sin will have period 2 π5, a function whose domain is the complex numbers can have two incommensurate periods without being constant. The elliptic functions are such functions, if L is the period of the function then, L =2 π / k One common generalization of periodic functions is that of antiperiodic functions. This is a function f such that f = −f for all x, for example, the sine or cosine function is π-antiperiodic and 2π-periodic. A further generalization appears in the context of Bloch waves and Floquet theory, in this context, the solution is typically a function of the form, f = e i k P f where k is a real or complex number. Functions of this form are sometimes called Bloch-periodic in this context, a periodic function is the special case k =0, and an antiperiodic function is the special case k = π/P
14.
Randomness
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Randomness is the lack of pattern or predictability in events. A random sequence of events, symbols or steps has no order, individual random events are by definition unpredictable, but in many cases the frequency of different outcomes over a large number of events is predictable. For example, when throwing two dice, the outcome of any particular roll is unpredictable, but a sum of 7 will occur twice as often as 4. In this view, randomness is a measure of uncertainty of an outcome, rather than haphazardness, and applies to concepts of chance, probability, the fields of mathematics, probability, and statistics use formal definitions of randomness. In statistics, a variable is an assignment of a numerical value to each possible outcome of an event space. This association facilitates the identification and the calculation of probabilities of the events, Random variables can appear in random sequences. A random process is a sequence of variables whose outcomes do not follow a deterministic pattern. These and other constructs are extremely useful in probability theory and the applications of randomness. Randomness is most often used in statistics to signify well-defined statistical properties, Monte Carlo methods, which rely on random input, are important techniques in science, as, for instance, in computational science. By analogy, quasi-Monte Carlo methods use quasirandom number generators, Random selection is a method of selecting items from a population where the probability of choosing a specific item is the proportion of those items in the population. For example, with a bowl containing just 10 red marbles and 90 blue marbles, note that a random selection mechanism that selected 10 marbles from this bowl would not necessarily result in 1 red and 9 blue. In situations where a population consists of items that are distinguishable and that is, if the selection process is such that each member of a population, of say research subjects, has the same probability of being chosen then we can say the selection process is random. In ancient history, the concepts of chance and randomness were intertwined with that of fate, many ancient peoples threw dice to determine fate, and this later evolved into games of chance. Most ancient cultures used various methods of divination to attempt to circumvent randomness, the Chinese of 3000 years ago were perhaps the earliest people to formalize odds and chance. The Greek philosophers discussed randomness at length, but only in non-quantitative forms and it was only in the 16th century that Italian mathematicians began to formalize the odds associated with various games of chance. The invention of the calculus had a impact on the formal study of randomness. The early part of the 20th century saw a growth in the formal analysis of randomness. In the mid- to late-20th century, ideas of information theory introduced new dimensions to the field via the concept of algorithmic randomness
15.
Symmetry
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Symmetry in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, symmetry has a precise definition, that an object is invariant to any of various transformations. Although these two meanings of symmetry can sometimes be told apart, they are related, so they are discussed together. The opposite of symmetry is asymmetry, a geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion. This means that an object is symmetric if there is a transformation that moves individual pieces of the object, an object has rotational symmetry if the object can be rotated about a fixed point without changing the overall shape. An object has symmetry if it can be translated without changing its overall shape. An object has symmetry if it can be simultaneously translated and rotated in three-dimensional space along a line known as a screw axis. An object has symmetry if it does not change shape when it is expanded or contracted. Fractals also exhibit a form of symmetry, where small portions of the fractal are similar in shape to large portions. Other symmetries include glide reflection symmetry and rotoreflection symmetry, a dyadic relation R is symmetric if and only if, whenever its true that Rab, its true that Rba. Thus, is the age as is symmetrical, for if Paul is the same age as Mary. Symmetric binary logical connectives are and, or, biconditional, nand, xor, the set of operations that preserve a given property of the object form a group. In general, every kind of structure in mathematics will have its own kind of symmetry, examples include even and odd functions in calculus, the symmetric group in abstract algebra, symmetric matrices in linear algebra, and the Galois group in Galois theory. In statistics, it appears as symmetric probability distributions, and as skewness, symmetry in physics has been generalized to mean invariance—that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations. This concept has one of the most powerful tools of theoretical physics. See Noethers theorem, and also, Wigners classification, which says that the symmetries of the laws of physics determine the properties of the found in nature. Important symmetries in physics include continuous symmetries and discrete symmetries of spacetime, internal symmetries of particles, in biology, the notion of symmetry is mostly used explicitly to describe body shapes. Bilateral animals, including humans, are more or less symmetric with respect to the plane which divides the body into left
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Crystal structure
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In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystalline material. Ordered structures occur from the nature of the constituent particles to form symmetric patterns that repeat along the principal directions of three-dimensional space in matter. The smallest group of particles in the material that constitutes the pattern is the unit cell of the structure. The unit cell completely defines the symmetry and structure of the crystal lattice. The repeating patterns are said to be located at the points of the Bravais lattice, the lengths of the principal axes, or edges, of the unit cell and the angles between them are the lattice constants, also called lattice parameters. The symmetry properties of the crystal are described by the concept of space groups, all possible symmetric arrangements of particles in three-dimensional space may be described by the 230 space groups. The crystal structure and symmetry play a role in determining many physical properties, such as cleavage, electronic band structure. The crystal structure of a material can be described in terms of its unit cell, the unit cell is a box containing one or more atoms arranged in three dimensions. The unit cells stacked in three-dimensional space describe the arrangement of atoms of the crystal. Commonly, atomic positions are represented in terms of fractional coordinates, the atom positions within the unit cell can be calculated through application of symmetry operations to the asymmetric unit. The asymmetric unit refers to the smallest possible occupation of space within the unit cell and this does not, however imply that the entirety of the asymmetric unit must lie within the boundaries of the unit cell. Symmetric transformations of atom positions are calculated from the group of the crystal structure. Vectors and planes in a lattice are described by the three-value Miller index notation. It uses the indices ℓ, m, and n as directional parameters, which are separated by 90°, by definition, the syntax denotes a plane that intercepts the three points a1/ℓ, a2/m, and a3/n, or some multiple thereof. That is, the Miller indices are proportional to the inverses of the intercepts of the plane with the unit cell, if one or more of the indices is zero, it means that the planes do not intersect that axis. A plane containing a coordinate axis is translated so that it no longer contains that axis before its Miller indices are determined, the Miller indices for a plane are integers with no common factors. Negative indices are indicated with horizontal bars, as in, in an orthogonal coordinate system for a cubic cell, the Miller indices of a plane are the Cartesian components of a vector normal to the plane. Likewise, the planes are geometric planes linking nodes
17.
Tessellation
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A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries, a periodic tiling has a repeating pattern. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups, a tiling that lacks a repeating pattern is called non-periodic. An aperiodic tiling uses a set of tile shapes that cannot form a repeating pattern. In the geometry of higher dimensions, a space-filling or honeycomb is called a tessellation of space. A real physical tessellation is a made of materials such as cemented ceramic squares or hexagons. Such tilings may be decorative patterns, or may have such as providing durable and water-resistant pavement. Historically, tessellations were used in Ancient Rome and in Islamic art such as in the decorative geometric tiling of the Alhambra palace, in the twentieth century, the work of M. C. Escher often made use of tessellations, both in ordinary Euclidean geometry and in geometry, for artistic effect. Tessellations are sometimes employed for decorative effect in quilting, Tessellations form a class of patterns in nature, for example in the arrays of hexagonal cells found in honeycombs. Tessellations were used by the Sumerians in building wall decorations formed by patterns of clay tiles, decorative mosaic tilings made of small squared blocks called tesserae were widely employed in classical antiquity, sometimes displaying geometric patterns. In 1619 Johannes Kepler made a documented study of tessellations. He wrote about regular and semiregular tessellations in his Harmonices Mundi, he was possibly the first to explore and to explain the structures of honeycomb. Some two hundred years later in 1891, the Russian crystallographer Yevgraf Fyodorov proved that every periodic tiling of the features one of seventeen different groups of isometries. Fyodorovs work marked the beginning of the mathematical study of tessellations. Other prominent contributors include Shubnikov and Belov, and Heinrich Heesch, in Latin, tessella is a small cubical piece of clay, stone or glass used to make mosaics. The word tessella means small square and it corresponds to the everyday term tiling, which refers to applications of tessellations, often made of glazed clay. Tessellation or tiling in two dimensions is a topic in geometry that studies how shapes, known as tiles, can be arranged to fill a plane without any gaps, according to a given set of rules
18.
Close-packing of equal spheres
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In geometry, close-packing of equal spheres is a dense arrangement of congruent spheres in an infinite, regular arrangement. Carl Friedrich Gauss proved that the highest average density – that is, the same packing density can also be achieved by alternate stackings of the same close-packed planes of spheres, including structures that are aperiodic in the stacking direction. The Kepler conjecture states that this is the highest density that can be achieved by any arrangement of spheres and this conjecture was proven by T. C. Highest density is only in case of 1,2,3,8 and 24 dimensions. Many crystal structures are based on a close-packing of a kind of atom. The cubic and hexagonal arrangements are very close to one another in energy, there are two simple regular lattices that achieve this highest average density. They are called face-centered cubic and hexagonal close-packed, based on their symmetry, both are based upon sheets of spheres arranged at the vertices of a triangular tiling, they differ in how the sheets are stacked upon one another. The fcc lattice is known to mathematicians as that generated by the A3 root system. Cannonballs were usually piled in a rectangular or triangular wooden frame, both arrangements produce a face-centered cubic lattice – with different orientation to the ground. Hexagonal close-packing would result in a pyramid with a hexagonal base. In both the fcc and hcp arrangements each sphere has twelve neighbors, for every sphere there is one gap surrounded by six spheres and two smaller gaps surrounded by four spheres. The distances to the centers of these gaps from the centers of the spheres is √ 3⁄2 for the tetrahedral, and √2 for the octahedral. Relative to a layer with positioning A, two more positionings B and C are possible. Every sequence of A, B, and C without immediate repetition of the one is possible. The most regular ones are fcc = ABCABCA hcp = ABABABA, in close-packing, the center-to-center spacing of spheres in the xy plane is a simple honeycomb-like tessellation with a pitch of one sphere diameter. The coordination number of hcp and fcc is 12 and their atomic packing factors are equal to the mentioned above,0.74. The distance between the centers along the shortest path namely that straight line will therefore be r1 + r2 where r1 is the radius of the first sphere, in close packing all of the spheres share a common radius, r. Therefore two centers would simply have a distance 2r, hexagonal close packing of spheres, the coordinate points of the lattice will be the spheres centers
19.
Face centred cubic
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In crystallography, the cubic crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals, there are three main varieties of these crystals, Primitive cubic Body-centered cubic, Face-centered cubic Each is subdivided into other variants listed below. Note that although the cell in these crystals is conventionally taken to be a cube. This is related to the fact that in most cubic crystal systems, a classic isometric crystal has square or pentagonal faces. The three Bravais lattices in the crystal system are, The primitive cubic system consists of one lattice point on each corner of the cube. Each atom at a point is then shared equally between eight adjacent cubes, and the unit cell therefore contains in total one atom. The body-centered cubic system has one point in the center of the unit cell in addition to the eight corner points. It has a net total of 2 lattice points per unit cell, Each sphere in a cF lattice has coordination number 12. The face-centered cubic system is related to the hexagonal close packed system. The plane of a cubic system is a hexagonal grid. Attempting to create a C-centered cubic crystal system would result in a simple tetragonal Bravais lattice, there are a total 36 cubic space groups. Other terms for hexoctahedral are, normal class, holohedral, ditesseral central class, a simple cubic unit cell has a single cubic void in the center. Additionally, there are 24 tetrahedral voids located in a square spacing around each octahedral void and these tetrahedral voids are not local maxima and are not technically voids, but they do occasionally appear in multi-atom unit cells. A face-centered cubic unit cell has eight tetrahedral voids located midway between each corner and the center of the cell, for a total of eight net tetrahedral voids. One important characteristic of a structure is its atomic packing factor. This is calculated by assuming all the atoms are identical spheres. The atomic packing factor is the proportion of space filled by these spheres, assuming one atom per lattice point, in a primitive cubic lattice with cube side length a, the sphere radius would be a⁄2 and the atomic packing factor turns out to be about 0.524. Similarly, in a bcc lattice, the atomic packing factor is 0.680, as a rule, since atoms in a solid attract each other, the more tightly packed arrangements of atoms tend to be more common
20.
Carl Friedrich Gauss
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Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick, in the Duchy of Brunswick-Wolfenbüttel, as the son of poor working-class parents. Gauss later solved this puzzle about his birthdate in the context of finding the date of Easter and he was christened and confirmed in a church near the school he attended as a child. A contested story relates that, when he was eight, he figured out how to add up all the numbers from 1 to 100, there are many other anecdotes about his precocity while a toddler, and he made his first ground-breaking mathematical discoveries while still a teenager. He completed Disquisitiones Arithmeticae, his opus, in 1798 at the age of 21. This work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day, while at university, Gauss independently rediscovered several important theorems. Gauss was so pleased by this result that he requested that a regular heptadecagon be inscribed on his tombstone, the stonemason declined, stating that the difficult construction would essentially look like a circle. The year 1796 was most productive for both Gauss and number theory and he discovered a construction of the heptadecagon on 30 March. He further advanced modular arithmetic, greatly simplifying manipulations in number theory, on 8 April he became the first to prove the quadratic reciprocity law. This remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic, the prime number theorem, conjectured on 31 May, gives a good understanding of how the prime numbers are distributed among the integers. Gauss also discovered that every integer is representable as a sum of at most three triangular numbers on 10 July and then jotted down in his diary the note, ΕΥΡΗΚΑ. On October 1 he published a result on the number of solutions of polynomials with coefficients in finite fields, in 1831 Gauss developed a fruitful collaboration with the physics professor Wilhelm Weber, leading to new knowledge in magnetism and the discovery of Kirchhoffs circuit laws in electricity. It was during this time that he formulated his namesake law and they constructed the first electromechanical telegraph in 1833, which connected the observatory with the institute for physics in Göttingen. In 1840, Gauss published his influential Dioptrische Untersuchungen, in which he gave the first systematic analysis on the formation of images under a paraxial approximation. Among his results, Gauss showed that under a paraxial approximation an optical system can be characterized by its cardinal points and he derived the Gaussian lens formula. In 1845, he became associated member of the Royal Institute of the Netherlands, in 1854, Gauss selected the topic for Bernhard Riemanns Habilitationvortrag, Über die Hypothesen, welche der Geometrie zu Grunde liegen. On the way home from Riemanns lecture, Weber reported that Gauss was full of praise, Gauss died in Göttingen, on 23 February 1855 and is interred in the Albani Cemetery there. Two individuals gave eulogies at his funeral, Gausss son-in-law Heinrich Ewald and Wolfgang Sartorius von Waltershausen and his brain was preserved and was studied by Rudolf Wagner who found its mass to be 1,492 grams and the cerebral area equal to 219,588 square millimeters. Highly developed convolutions were also found, which in the early 20th century were suggested as the explanation of his genius, Gauss was a Lutheran Protestant, a member of the St. Albans Evangelical Lutheran church in Göttingen
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Johannes Kepler
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Johannes Kepler was a German mathematician, astronomer, and astrologer. A key figure in the 17th-century scientific revolution, he is best known for his laws of motion, based on his works Astronomia nova, Harmonices Mundi. These works also provided one of the foundations for Isaac Newtons theory of universal gravitation, Kepler was a mathematics teacher at a seminary school in Graz, where he became an associate of Prince Hans Ulrich von Eggenberg. Later he became an assistant to the astronomer Tycho Brahe in Prague and he was also a mathematics teacher in Linz, and an adviser to General Wallenstein. Kepler lived in an era when there was no distinction between astronomy and astrology, but there was a strong division between astronomy and physics. Kepler was born on December 27, the feast day of St John the Evangelist,1571 and his grandfather, Sebald Kepler, had been Lord Mayor of the city. By the time Johannes was born, he had two brothers and one sister and the Kepler family fortune was in decline and his father, Heinrich Kepler, earned a precarious living as a mercenary, and he left the family when Johannes was five years old. He was believed to have died in the Eighty Years War in the Netherlands and his mother Katharina Guldenmann, an innkeepers daughter, was a healer and herbalist. Born prematurely, Johannes claimed to have weak and sickly as a child. Nevertheless, he often impressed travelers at his grandfathers inn with his phenomenal mathematical faculty and he was introduced to astronomy at an early age, and developed a love for it that would span his entire life. At age six, he observed the Great Comet of 1577, in 1580, at age nine, he observed another astronomical event, a lunar eclipse, recording that he remembered being called outdoors to see it and that the moon appeared quite red. However, childhood smallpox left him with vision and crippled hands. In 1589, after moving through grammar school, Latin school, there, he studied philosophy under Vitus Müller and theology under Jacob Heerbrand, who also taught Michael Maestlin while he was a student, until he became Chancellor at Tübingen in 1590. He proved himself to be a mathematician and earned a reputation as a skilful astrologer. Under the instruction of Michael Maestlin, Tübingens professor of mathematics from 1583 to 1631 and he became a Copernican at that time. In a student disputation, he defended heliocentrism from both a theoretical and theological perspective, maintaining that the Sun was the source of motive power in the universe. Despite his desire to become a minister, near the end of his studies, Kepler was recommended for a position as teacher of mathematics and he accepted the position in April 1594, at the age of 23. Keplers first major work, Mysterium Cosmographicum, was the first published defense of the Copernican system
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Kepler conjecture
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The Kepler conjecture, named after the 17th-century mathematician and astronomer Johannes Kepler, is a mathematical conjecture about sphere packing in three-dimensional Euclidean space. It says that no arrangement of equally sized spheres filling space has an average density than that of the cubic close packing. The density of these arrangements is around 74. 05%, in 1998 Thomas Hales, following an approach suggested by Fejes Tóth, announced that he had a proof of the Kepler conjecture. Hales proof is a proof by exhaustion involving the checking of many cases using complex computer calculations. Referees have said that they are 99% certain of the correctness of Hales proof, in 2014, the Flyspeck project team, headed by Hales, announced the completion of a formal proof of the Kepler conjecture using a combination of the Isabelle and HOL Light proof assistants. Imagine filling a container with small equal-sized spheres. The density of the arrangement is equal to the volume of the spheres divided by the volume of the container. To maximize the number of spheres in the means to create an arrangement with the highest possible density. Experiment shows that dropping the spheres in randomly will achieve a density of around 65%, however, a higher density can be achieved by carefully arranging the spheres as follows. Start with a layer of spheres in a lattice, then put the next layer of spheres in the lowest points you can find above the first layer. The conjecture was first stated by Johannes Kepler in his paper On the six-cornered snowflake and he had started to study arrangements of spheres as a result of his correspondence with the English mathematician and astronomer Thomas Harriot in 1606. Harriot was a friend and assistant of Sir Walter Raleigh, who had set Harriot the problem of determining how best to stack cannonballs on the decks of his ships. Harriot published a study of various stacking patterns in 1591, and this meant that any packing arrangement that disproved the Kepler conjecture would have to be an irregular one. But eliminating all possible arrangements is very difficult, and this is what made the Kepler conjecture so hard to prove. After Gauss, no progress was made towards proving the Kepler conjecture in the nineteenth century. In 1900 David Hilbert included it in his list of twenty three unsolved problems of mathematics—it forms part of Hilberts eighteenth problem, the next step toward a solution was taken by László Fejes Tóth. Fejes Tóth showed that the problem of determining the density of all arrangements could be reduced to a finite number of calculations. This meant that a proof by exhaustion was, in principle, possible, as Fejes Tóth realised, a fast enough computer could turn this theoretical result into a practical approach to the problem
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Thomas Callister Hales
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Thomas Callister Hales is an American mathematician working on the Langlands program. He is known in the area for having worked on the fundamental lemma, many of his ideas were incorporated into the final proof, due to Ngô Bảo Châu. He is also known for his proof of the Kepler conjecture on sphere packing and he received his Ph. D. from Princeton University in 1986, his dissertation was entitled The Subregular Germ of Orbital Integrals. Between 1993 and 2002 he worked at the University of Michigan and he was aided by graduate student Samuel Ferguson. In 1999, Hales proved the conjecture, he also stated that the conjecture may have been present in the minds of mathematicians before Marcus Terentius Varro. After 2002, Hales became the University of Pittsburghs Mellon Professor of mathematics, in 2003, Hales started work on Flyspeck to vindicate his proof of the Kepler conjecture. His proof relied on computer calculation to verify conjectures, the project used two proof assistants, HOL Light and Isabelle. Annals of Mathematics accepted the proof in 2005, but was only 99% sure of the proof, in August 2014, the Flyspeck teams software finally verified the proof to be correct. Hales won the Chauvenet Prize in 2003 and a Lester R. Ford Award in 2008, in 2012 he became a fellow of the American Mathematical Society. The status of the Kepler conjecture, The Mathematical Intelligencer,16, 47–58, historical overview of the Kepler conjecture, Discrete & Computational Geometry,36, 5–20, doi,10. 1007/s00454-005-1210-2, ISSN 0179-5376, MR2229657 Hales, Thomas C. A formulation of the Kepler conjecture, Discrete & Computational Geometry,36, 21–69, doi,10. 1007/s00454-005-1211-1, ISSN 0179-5376, MR2229658 Hales, Thomas C
24.
Laplace transform
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In mathematics the Laplace transform is an integral transform named after its discoverer Pierre-Simon Laplace. It takes a function of a real variable t to a function of a complex variable s. The Laplace transform is similar to the Fourier transform. While the Fourier transform of a function is a function of a real variable. Laplace transforms are usually restricted to functions of t with t >0, a consequence of this restriction is that the Laplace transform of a function is a holomorphic function of the variable s. Unlike the Fourier transform, the Laplace transform of a distribution is generally a well-behaved function, also techniques of complex variables can be used directly to study Laplace transforms. As a holomorphic function, the Laplace transform has a series representation. This power series expresses a function as a superposition of moments of the function. This perspective has applications in probability theory, the Laplace transform is invertible on a large class of functions. The inverse Laplace transform takes a function of a complex variable s, so, for example, Laplace transformation from the time domain to the frequency domain transforms differential equations into algebraic equations and convolution into multiplication. It has many applications in the sciences and technology, the Laplace transform is named after mathematician and astronomer Pierre-Simon Laplace, who used a similar transform in his work on probability theory. The current widespread use of the transform came about during and soon after World War II although it had used in the 19th century by Abel, Lerch, Heaviside. The early history of methods having some similarity to Laplace transform is as follows. From 1744, Leonhard Euler investigated integrals of the form z = ∫ X e a x d x and z = ∫ X x A d x as solutions of differential equations but did not pursue the matter very far. These types of integrals seem first to have attracted Laplaces attention in 1782 where he was following in the spirit of Euler in using the integrals themselves as solutions of equations. He used an integral of the form ∫ x s ϕ d x, akin to a Mellin transform, to transform the whole of a difference equation, in order to look for solutions of the transformed equation. He then went on to apply the Laplace transform in the way and started to derive some of its properties. Laplace also recognised that Joseph Fouriers method of Fourier series for solving the equation could only apply to a limited region of space because those solutions were periodic
25.
Symmetry in biology
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Symmetry in biology is the balanced distribution of duplicate body parts or shapes within the body of an organism. In nature and biology, symmetry is always approximate, for example plant leaves, while considered symmetrical, symmetry creates a class of patterns in nature, where the near-repetition of the pattern element is by reflection or rotation. The body plans of most multicellular organisms exhibit some form of symmetry, whether radial, bilateral, a small minority, notably among the sponges, exhibit no symmetry. Radially symmetric organisms resemble a pie where several cutting planes produce roughly identical pieces, such an organism exhibits no left or right sides. They have a top and a surface only. Most radially symmetric animals are symmetrical about an axis extending from the center of the oral surface, radial symmetry is especially suitable for sessile animals such as the sea anemone, floating animals such as jellyfish, and slow moving organisms such as starfish. Animals in the phyla cnidaria and echinodermata are radially symmetric, although many sea anemones and some corals have bilateral symmetry defined by a single structure, many flowers are radially symmetric or actinomorphic. Roughly identical flower parts – petals, sepals, and stamens occur at intervals around the axis of the flower. Many viruses have radial symmetries, their coats being composed of a small number of protein molecules arranged in a regular pattern to form polyhedrons, spheres. Tetramerism is a variant of radial symmetry found in jellyfish, which have four canals in an otherwise radial body plan, pentamerism, another variant of radial symmetry, means the organism is in five parts around a central axis, 72° apart. Among animals, only the echinoderms such as sea stars, sea urchins, being bilaterian animals, however, they initially develop with mirror symmetry as larvae, then gain pentaradial symmetry later. Flowering plants show fivefold symmetry in many flowers and in various fruits and this is well seen in the arrangement of the five carpels in an apple cut transversely. Hexamerism is found in the corals and sea anemones which are divided into two based on their symmetry. The most common corals in the subclass Hexacorallia have a body plan, their polyps have sixfold internal symmetry. Octamerism is found in corals of the subclass Octocorallia and these have polyps with eight tentacles and octameric radial symmetry. The octopus, however, has symmetry, despite its eight arms. Spherical symmetry occurs in an if it is able to be cut into two identical halves through any cut that runs through the organisms center. Organisms which show approximate spherical symmetry include the green alga Volvox
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Fourier transform
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The Fourier transform decomposes a function of time into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies of its constituent notes. The Fourier transform is called the frequency domain representation of the original signal, the term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time. The Fourier transform is not limited to functions of time, but in order to have a unified language, linear operations performed in one domain have corresponding operations in the other domain, which are sometimes easier to perform. The operation of differentiation in the domain corresponds to multiplication by the frequency. Also, convolution in the domain corresponds to ordinary multiplication in the frequency domain. Concretely, this means that any linear time-invariant system, such as a filter applied to a signal, after performing the desired operations, transformation of the result can be made back to the time domain. Functions that are localized in the domain have Fourier transforms that are spread out across the frequency domain and vice versa. The Fourier transform of a Gaussian function is another Gaussian function, Joseph Fourier introduced the transform in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. The Fourier transform can also be generalized to functions of variables on Euclidean space. In general, functions to which Fourier methods are applicable are complex-valued, the latter is routinely employed to handle periodic functions. The fast Fourier transform is an algorithm for computing the DFT, the Fourier transform of the function f is traditionally denoted by adding a circumflex, f ^. There are several conventions for defining the Fourier transform of an integrable function f, ℝ → ℂ. Here we will use the definition, f ^ = ∫ − ∞ ∞ f e −2 π i x ξ d x. When the independent variable x represents time, the transform variable ξ represents frequency. Under suitable conditions, f is determined by f ^ via the inverse transform, f = ∫ − ∞ ∞ f ^ e 2 π i ξ x d ξ, the functions f and f ^ often are referred to as a Fourier integral pair or Fourier transform pair. For other common conventions and notations, including using the angular frequency ω instead of the frequency ξ, see Other conventions, the Fourier transform on Euclidean space is treated separately, in which the variable x often represents position and ξ momentum. Many other characterizations of the Fourier transform exist, for example, one uses the Stone–von Neumann theorem, the Fourier transform is the unique unitary intertwiner for the symplectic and Euclidean Schrödinger representations of the Heisenberg group. In 1822, Joseph Fourier showed that some functions could be written as an sum of harmonics
27.
Origin (mathematics)
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In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In physical problems, the choice of origin is often arbitrary and this allows one to pick an origin point that makes the mathematics as simple as possible, often by taking advantage of some kind of geometric symmetry. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect, the origin divides each of these axes into two halves, a positive and a negative semiaxis. The coordinates of the origin are all zero, for example in two dimensions and in three. In a polar coordinate system, the origin may also be called the pole, in Euclidean geometry, the origin may be chosen freely as any convenient point of reference. The origin of the plane can be referred as the point where real axis. In other words, it is the number zero
28.
Peter Sarnak
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Peter Clive Sarnak is a South African-born mathematician with dual South-African and American nationalities. He has been Eugene Higgins Professor of Mathematics at Princeton University since 2002, succeeding Andrew Wiles and he is known for his work in analytic number theory. Sarnak is also on the permanent faculty at the School of Mathematics of the Institute for Advanced Study and he also sits on the Board of Adjudicators and the selection committee for the Mathematics award, given under the auspices of the Shaw Prize. Sarnak graduated from the University of the Witwatersrand and Stanford University, sarnak’s highly cited work applied deep results in number theory to Ramanujan graphs, with connections to combinatorics and computer science. Conant Prize in 2003, the Frank Nelson Cole Prize in Number Theory in 2005 and he is the recipient of the 2014 Wolf Prize in Mathematics. The University of the Witwatersrand conferred a doctorate on Professor Peter Sarnak on the 2 July 2014 for his distinguished contribution to the field of mathematics. He was also elected as member of the National Academy of Sciences and he was awarded an honorary doctorate by the Hebrew University of Jerusalem in 2010. He was also awarded a doctorate by the University of Chicago in 2015. Spectral Behavior of Quasi Periodic Potentials, some Applications of Modular Forms,1990 Extremal Riemann Surfaces,1997 Random Matrices, Frobenius Eigenvalues and Monodromy,1998 Peter Sarnak. Some problems in Number Theory, Analysis and Mathematical Physics, arnold, M. Atiyah, P. Lax, B
29.
Particle-size distribution
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Significant energy is usually required to disintegrate soil, etc. particles into the PSD that is then called a grain size distribution. The PSD of a material can be important in understanding its physical and chemical properties and it affects the strength and load-bearing properties of rocks and soils. Particle size distribution can greatly affect the efficiency of any collection device, settling chambers will normally only collect very large particles, those that can be separated using sieve trays. Centrifugal collectors will normally collect particles down to about 20 μm, higher efficiency models can collect particles down to 10 μm. Fabric filters are one of the most efficient and cost effective types of dust collectors available, wet scrubbers that use liquid are commonly known as wet scrubbers. In these systems, the liquid comes into contact with a gas stream containing dust particles. The greater the contact of the gas and liquid streams, the higher the dust removal efficiency, electrostatic precipitators use electrostatic forces to separate dust particles from exhaust gases. They can be efficient at the collection of very fine particles. Filter Press used for filtering liquids by cake filtration mechanism, the PSD plays an important part in the cake formation, cake resistance, and cake characteristics. The filterability of the liquid is determined largely by the size of the particles, ρp, Actual particle density ρg, Gas or sample matrix density r2, Least-squares coefficient of determination. The closer this value is to 1.0, the better the data fit to a hyperplane representing the relationship between the variable and a set of covariate variables. A value equal to 1.0 indicates all data fit perfectly within the hyperplane, λ, Gas mean free path D50, Mass-median-diameter. The log-normal distribution mass median diameter, the MMD is considered to be the average particle diameter by mass. This value is determined mathematically by the equation, σg = D84. 13/D50 = D50/D15.87 The value of σg determines the slope of the least-squares regression curve, α, Relative standard deviation or degree of polydispersity. This value is determined mathematically. For values less than 0.1, the sample can be considered to be monodisperse. α = σg/D50 Re, Particle Reynolds Number, in contrast to the large numerical values noted for flow Reynolds number, particle Reynolds number for fine particles in gaseous mediums is typically less than 0.1. The way PSD is usually defined by the method by which it is determined, the most easily understood method of determination is sieve analysis, where powder is separated on sieves of different sizes
30.
Ionic crystal
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In chemistry, an ionic compound is a chemical compound composed of ions held together by electrostatic forces termed ionic bonding. The compound is overall, but consists of positively charged ions called cations. These can be simple ions such as the sodium and chloride in sodium chloride, or polyatomic species such as the ammonium, Ionic compounds containing hydrogen ions are classified as acids, and those containing basic ions hydroxide or oxide are classified as bases. Ionic compounds without these ions are known as salts and can be formed by acid–base reactions. Ionic compounds typically have high melting and boiling points, and are hard, as solids they are almost always electrically insulating, but when melted or dissolved they become highly conductive, because the ions are mobilized. The word ion is the Greek ἰόν, ion, going and this term was introduced by English physicist and chemist Michael Faraday in 1834 for the then-unknown species that goes from one electrode to the other through an aqueous medium. In 1913 the crystal structure of sodium chloride was determined by William Henry Bragg, many other inorganic compounds were also found to have similar structural features. Principal contributors to the development of a treatment of ionic crystal structures were Max Born, Fritz Haber, Alfred Landé, Erwin Madelung, Paul Peter Ewald. Born predicted crystal energies based on the assumption of ionic constituents, Ionic compounds can be produced from their constituent ions by evaporation, precipitation, or freezing. Reactive metals such as the alkali metals can react directly with the highly electronegative halogen gases to form an ionic product and they can also be synthesized as the product of a high temperature reaction between solids. If the ionic compound is soluble in a solvent, it can be obtained as a compound by evaporating the solvent from this electrolyte solution. This process occurs widely in nature, and is the means of formation of the evaporite minerals, insoluble ionic compounds can be precipitated by mixing two solutions, one with the cation and one with the anion in it. Because all solutions are neutral, the two solutions mixed must also contain counterions of the opposite charges. To ensure that these do not contaminate the precipitated ionic compound, if the two solutions have hydrogen ions and hydroxide ions as the counterions, they will react with one another in what is called an acid–base reaction or a neutralization reaction to form water. Alternately the counterions can be chosen to ensure that even when combined into a solution they will remain soluble as spectator ions. Molten salts will solidify on cooling to below their freezing point and this is sometimes used for the solid-state synthesis of complex ionic compounds from solid reactants, which are first melted together. In other cases the solid reactants do not need to be melted, other synthetic routes use a solid precursor with the correct stoichiometric ratio of non-volatile ions, which is heated to drive off other species. There is also an additional attractive force from van der Waals interactions which contributes only around 1–2% of the cohesive energy for small ions
31.
Stoichiometry
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Stoichiometry /ˌstɔɪkiˈɒmᵻtri/ is the calculation of relative quantities of reactants and products in chemical reactions. This means that if the amounts of the reactants are known. Conversely, if one reactant has a quantity and the quantity of product can be empirically determined. This is illustrated in the image here, where the equation is. Here, one molecule of methane reacts with two molecules of gas to yield one molecule of carbon dioxide and two molecules of water. Stoichiometry measures these quantitative relationships, and is used to determine the amount of products/reactants that are produced/needed in a given reaction, describing the quantitative relationships among substances as they participate in chemical reactions is known as reaction stoichiometry. In the example above, reaction stoichiometry measures the relationship between the methane and oxygen as they react to form carbon dioxide and water. Gas stoichiometry deals with reactions involving gases, where the gases are at a temperature, pressure. For gases, the ratio is ideally the same by the ideal gas law. In practice, due to the existence of isotopes, molar masses are used instead when calculating the mass ratio, the term stoichiometry was first used by Jeremias Benjamin Richter in 1792 when the first volume of Richters Stoichiometry or the Art of Measuring the Chemical Elements was published. The term is derived from the Greek words στοιχεῖον stoicheion element, in patristic Greek, the word Stoichiometria was used by Nicephorus to refer to the number of line counts of the canonical New Testament and some of the Apocrypha. In general, chemical reactions combine in definite ratios of chemicals, since chemical reactions can neither create nor destroy matter, nor transmute one element into another, the amount of each element must be the same throughout the overall reaction. Chemical reactions, as unit operations, consist of simply a very large number of elementary reactions. As the reacting molecules consist of a set of atoms in an integer ratio. A reaction may consume more than one molecule, and the number counts this number, defined as positive for products. Different elements have a different atomic mass, and as collections of atoms, molecules have a definite molar mass. By definition, carbon-12 has a mass of 12 g/mol. The mass ratios can be calculated by dividing each by the total in the whole reaction, Elements in their natural state are mixtures of isotopes of differing mass, thus atomic masses and thus molar masses are not exactly integers
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Electric potential energy
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An object may have electric potential energy by virtue of two key elements, its own electric charge and its relative position to other electrically charged objects. The SI unit of potential energy is the joule. In the CGS system the erg is the unit of energy, also electronvolts may be used,1 eV =1. 602×10−19 J. The following outline of proof states the derivation from the definition of electric potential energy, the electrostatic potential energy UE stored in a system of N charges q1, q2. A common question arises concerning the interaction of a point charge with its own electrostatic potential, since this interaction doesnt act to move the point charge itself, it doesnt contribute to the stored energy of the system. Consider bringing a point charge, q, into its position in the vicinity of a point charge. Some elements in a circuit can convert energy from one form to another, for example, a resistor converts electrical energy to heat. This is known as the Joule effect, a capacitor stores it in its electric field. These latter two expressions are only for cases when the smallest increment of charge is zero such as dielectrics in the presence of metallic electrodes or dielectrics containing many charges
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Ford circle
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In mathematics, a Ford circle is a circle with center at and radius 1 /, where p / q is an irreducible fraction, i. e. p and q are coprime integers. Each Ford circle is tangent to the horizontal axis y =0, Ford circles are a special case of mutually tangent circles, the base line can be thought of as a circle with infinite radius. Systems of mutually tangent circles were studied by Apollonius of Perga, after whom the problem of Apollonius, in the 17th century René Descartes discovered Descartes theorem, a relationship between the reciprocals of the radii of mutually tangent circles. Ford circles also appear in the Sangaku of Japanese mathematics, a typical problem, which is presented on an 1824 tablet in the Gunma Prefecture, covers the relationship of three touching circles with a common tangent. Given the size of the two outer circles, what is the size of the small circle between them. The answer is equivalent to a Ford circle,1 r middle =1 r left +1 r right, Ford circles are named after the American mathematician Lester R. Ford, Sr. who wrote about them in 1938. The Ford circle associated with the fraction p / q is denoted by C or C, there is a Ford circle associated with every rational number. In addition, the line y =1 is counted as a Ford circle – it can be thought of as the Ford circle associated with infinity, two different Ford circles are either disjoint or tangent to one another. No two interiors of Ford circles intersect, even there is a Ford circle tangent to the x-axis at each point on it with rational coordinates. If C and C are two tangent Ford circles, then the circle through and and that is perpendicular to the x -axis also passes through the point where the two circles are tangent to one another, Ford circles can also be thought of as curves in the complex plane. The modular group of transformations of the plane maps Ford circles to other Ford circles. Ford circles are a sub-set of the circles in the Apollonian gasket generated by the lines y =0 and y =1, by interpreting the upper half of the complex plane as a model of the hyperbolic plane Ford circles can be interpreted as horocycles. In hyperbolic geometry any two horocycles are congruent, when these horocycles are circumscribed by apeirogons they tile the hyperbolic plane with an order-3 apeirogonal tiling. There is a link between the area of Ford circles, Eulers totient function φ, the Riemann zeta function ζ, as no two Ford circles intersect, it follows immediately that the total area of the Ford circles is less than 1. In fact the area of these Ford circles is given by a convergent sum. From the definition, the area is A = ∑ q ≥1 ∑ =11 ≤ p < q π2, since ζ = π4 /90, this finally becomes A =452 ζ π3 ≈0.872284041. Fords Touching Circles at cut-the-knot Weisstein, Eric W. Ford Circle
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Infinity
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Infinity is an abstract concept describing something without any bound or larger than any number. In mathematics, infinity is treated as a number but it is not the same sort of number as natural or real numbers. Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th, in the theory he developed, there are infinite sets of different sizes. For example, the set of integers is countably infinite, while the set of real numbers is uncountable. Ancient cultures had various ideas about the nature of infinity, the ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept. The earliest recorded idea of infinity comes from Anaximander, a pre-Socratic Greek philosopher who lived in Miletus and he used the word apeiron which means infinite or limitless. However, the earliest attestable accounts of mathematical infinity come from Zeno of Elea, aristotle called him the inventor of the dialectic. He is best known for his paradoxes, described by Bertrand Russell as immeasurably subtle, however, recent readings of the Archimedes Palimpsest have found that Archimedes had an understanding about actual infinite quantities. The Jain mathematical text Surya Prajnapti classifies all numbers into three sets, enumerable, innumerable, and infinite, on both physical and ontological grounds, a distinction was made between asaṃkhyāta and ananta, between rigidly bounded and loosely bounded infinities. European mathematicians started using numbers in a systematic fashion in the 17th century. John Wallis first used the notation ∞ for such a number, euler used the notation i for an infinite number, and exploited it by applying the binomial formula to the i th power, and infinite products of i factors. In 1699 Isaac Newton wrote about equations with an number of terms in his work De analysi per aequationes numero terminorum infinitas. The infinity symbol ∞ is a symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ infinity and in LaTeX as \infty and it was introduced in 1655 by John Wallis, and, since its introduction, has also been used outside mathematics in modern mysticism and literary symbology. Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers, in real analysis, the symbol ∞, called infinity, is used to denote an unbounded limit. X → ∞ means that x grows without bound, and x → − ∞ means the value of x is decreasing without bound. ∑ i =0 ∞ f = ∞ means that the sum of the series diverges in the specific sense that the partial sums grow without bound. Infinity can be used not only to define a limit but as a value in the real number system
35.
Horosphere
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In hyperbolic geometry, a horosphere is a specific hypersurface in hyperbolic n-space. It is the boundary of a horoball, the limit of a sequence of increasing balls sharing a tangent hyperplane, for n =2 a horosphere is called a horocycle. The concept has its roots in a notion expressed by F. L. Wachter in 1816 in a letter to his teacher Gauss, the term horoball is due to William Thurston, who used it in his work on hyperbolic 3-manifolds. The terms horosphere and horoball are often used in 3-dimensional hyperbolic geometry, in the conformal ball model, a horosphere is represented by a sphere tangent to the horizon sphere. In the upper half-space model, a horosphere can appear either as a tangent to the horizon plane. In the hyperboloid model, a horosphere is represented by a plane whose normal lies in the asymptotic cone, appendix, the theory of space Janos Bolyai,1987, p.143
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Order-6 tetrahedral honeycomb
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In the geometry of hyperbolic 3-space, the order-6 tetrahedral honeycomb is a paracompact regular space-filling tessellation. It is called paracompact because it has infinite vertex figures, with all vertices as points at infinity. It has six tetrahedra around each edge, all vertices are ideal vertices with infinitely many tetrahedra existing around each ideal vertex in an triangular tiling vertex arrangement. A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells and it is an example of the more general mathematical tiling or tessellation in any number of dimensions. Honeycombs are usually constructed in ordinary Euclidean space, like the uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs, any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space. It has a construction as a uniform honeycomb, Schläfli symbol. In Coxeter notation the half symmetry is ↔ or, ↔ and it is similar to the 2-dimensional hyperbolic tiling, infinite-order triangular tiling, having all triangle faces, and all ideal vertices. It is one of 15 regular hyperbolic honeycombs in 3-space,11 of which like this one are paracompact and it is one of 15 uniform paracompact honeycombs in the Coxeter group, along with its dual hexagonal tiling honeycomb. It a part of a sequence of polychora and honeycombs with tetrahedral cells. It a part of a sequence of honeycombs with triangular tiling vertex figures, the rectified order-6 tetrahedral honeycomb, t1 has octahedral and triangular tiling cells connected in a hexagonal prism vertex figure. The cantellated order-6 tetrahedral honeycomb, t0,2 has cuboctahedron, the cantitruncated order-6 tetrahedral honeycomb, t0,1,2 has truncated cuboctahedron and trihexagonal tiling cells connected in an octahedron vertex figure. Convex uniform honeycombs in hyperbolic space List of regular polytopes Coxeter, Regular Polytopes, johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D. Dissertation, University of Toronto,1966 N. W, johnson, Geometries and Transformations, Chapter 13, Hyperbolic Coxeter groups
. Bibcode:2007JAP...102i3511T. doi:10.1063/1.2802184.