1.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times

2.
Manifold
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In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of a manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n. One-dimensional manifolds include lines and circles, but not figure eights, two-dimensional manifolds are also called surfaces. Although a manifold locally resembles Euclidean space, globally it may not, for example, the surface of the sphere is not a Euclidean space, but in a region it can be charted by means of map projections of the region into the Euclidean plane. When a region appears in two neighbouring charts, the two representations do not coincide exactly and a transformation is needed to pass from one to the other, Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. One important class of manifolds is the class of differentiable manifolds and this differentiable structure allows calculus to be done on manifolds. A Riemannian metric on a manifold allows distances and angles to be measured, symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity. After a line, the circle is the simplest example of a topological manifold, Topology ignores bending, so a small piece of a circle is treated exactly the same as a small piece of a line. Consider, for instance, the top part of the circle, x2 + y2 =1. Any point of this arc can be described by its x-coordinate. So, projection onto the first coordinate is a continuous, and invertible, mapping from the arc to the open interval. Such functions along with the regions they map are called charts. Similarly, there are charts for the bottom, left, and right parts of the circle, together, these parts cover the whole circle and the four charts form an atlas for the circle. The top and right charts, χtop and χright respectively, overlap in their domain, Each map this part into the interval, though differently. Let a be any number in, then, T = χ r i g h t = χ r i g h t =1 − a 2 Such a function is called a transition map. The top, bottom, left, and right charts show that the circle is a manifold, charts need not be geometric projections, and the number of charts is a matter of some choice. These two charts provide a second atlas for the circle, with t =1 s Each chart omits a single point, either for s or for t and it can be proved that it is not possible to cover the full circle with a single chart. Viewed using calculus, the transition function T is simply a function between open intervals, which gives a meaning to the statement that T is differentiable

3.
Orientation (geometry)
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In geometry the orientation, angular position, or attitude of an object such as a line, plane or rigid body is part of the description of how it is placed in the space it is in. Namely, it is the rotation that is needed to move the object from a reference placement to its current placement. A rotation may not be enough to reach the current placement and it may be necessary to add an imaginary translation, called the objects location. The location and orientation together fully describe how the object is placed in space, eulers rotation theorem shows that in three dimensions any orientation can be reached with a single rotation around a fixed axis. This gives one way of representing the orientation using an axis–angle representation. Other widely used methods include rotation quaternions, Euler angles, or rotation matrices, more specialist uses include Miller indices in crystallography, strike and dip in geology and grade on maps and signs. Typically, the orientation is given relative to a frame of reference, at least three independent values are needed to describe the orientation of this local frame. Three other values are needed to describe its location, thus, a rigid body free to move in space is said to have six degrees of freedom. All the points of the body change their position during a rotation except for those lying on the rotation axis, if the rigid body has rotational symmetry not all orientations are distinguishable, except by observing how the orientation evolves in time from a known starting orientation. For example, the orientation in space of a line, line segment, another example is the position of a point on the earth, often described using the orientation of a line joining it with the earths center, measured using the two angles of longitude and latitude. Likewise, the orientation of a plane can be described with two values as well, for instance by specifying the orientation of a normal to that plane, or by using the strike. Further details about the methods to represent the orientation of rigid bodies and planes in three dimensions are given in the following sections. In two dimensions the orientation of any object is given by a value, the angle through which it has rotated. There is only one degree of freedom and only one fixed point about which the rotation takes place, several methods to describe orientations of a rigid body in three dimensions have been developed. They are summarized in the following sections, the first attempt to represent an orientation was owed to Leonhard Euler. The values of three rotations are called Euler angles. These are three angles, also known as yaw, pitch and roll, Navigation angles and Cardan angles, in aerospace engineering they are usually referred to as Euler angles. Euler also realized that the composition of two rotations is equivalent to a rotation about a different fixed axis

4.
Mechanism (engineering)
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A mechanism is a device designed to transform input forces and movement into a desired set of output forces and the movement. In this context, his use of machine is generally interpreted to mean mechanism, the combination of force and movement defines power, and a mechanism is designed to manage power in order to achieve a desired set of forces and movement. A mechanism is usually a piece of a process or mechanical system. Sometimes an entire machine may be referred to as a mechanism, examples are the steering mechanism in a car, or the winding mechanism of a wristwatch. From the time of Archimedes through the Renaissance, mechanisms were considered to be constructed from simple machines, such as the lever, pulley, screw, wheel and axle, wedge and inclined plane. It was Reuleaux who focussed on bodies, called links, in order to use geometry to study the movement of a mechanism, its links are modeled as rigid bodies. This means distances between points in a link are assumed to be unchanged as the moves, that is the link does not flex. Thus, the movement between points in two connected links is considered to result from the kinematic pair that joins them. A mechanism is modeled as an assembly of rigid links and kinematic pairs, Reuleaux called the ideal connections between links kinematic pairs. He distinguished between higher pairs which were said to have contact between the two links and lower pairs that have area contact between the links. J. Phillips shows that there are ways to construct pairs that do not fit this simple. Lower pair, A lower pair is a joint that has surface contact between the pair of elements. This imposes five constraints on the movement of the links. This imposes five constraints on the movement of the links. A cylindrical joint requires that a line in the body remain co-linear with a line in the fixed body. It is a combination of a joint and a sliding joint. This joint has two degrees of freedom, a spherical joint, or ball joint, requires that a point in the moving body maintain contact with a point in the fixed body. This joint has three degrees of freedom, a planar joint requires that a plane in the moving body maintain contact with a plane in fixed body

5.
Six degrees of freedom
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Six degrees of freedom refers to the freedom of movement of a rigid body in three-dimensional space. Serial and parallel manipulator systems are designed to position an end-effector with six degrees of freedom. This provides a relationship between actuator positions and the configuration of the manipulator defined by its forward and inverse kinematics. Robot arms are described by their degrees of freedom and this number typically refers to the number of single-axis rotational joints in the arm, where higher number indicates an increased flexibility in positioning a tool. This is a metric, in contrast to the abstract definition of degrees of freedom which measures the aggregate positioning capability of a system. In 2007, Dean Kamen, inventor of the Segway, unveiled a prototype robotic arm with 14 degrees of freedom for DARPA. Humanoid robots typically have 30 or more degrees of freedom, with six degrees of freedom per arm, five or six in each leg, and several more in torso and neck. The term is important in systems, especially biomechanical systems for analyzing and measuring properties of these types of systems that need to account for all six degrees of freedom. Measurement of the six degrees of freedom is accomplished today through both AC and DC magnetic or electromagnetic fields in sensors that transmit positional and angular data to a processing unit. The data is made relevant through software that integrate the data based on the needs, ascension Technology Corporation has recently created a 6DoF device small enough to fit in a biopsy needle, allowing physicians to better research at minute levels. The new sensor passively senses pulsed DC magnetic fields generated by either a transmitter or a flat transmitter and is available for integration. An example of six degree of movement is the motion of a ship at sea. It is described as, Translation, Moving forward and backward on the X-axis, Moving left and right on the Y-axis. Moving up and down on the Z-axis, rotation Tilting side to side on the X-axis. Tilting forward and backward on the Y-axis, turning left and right on the Z-axis. There are three types of envelope in the Six degrees of freedom. 1- Direct type, Involved a degree can be commanded directly without particularly conditions, 2- Semi-direct type, Involved a degree can be commanded when some specific conditions are met. 3- Non-direct type, Involved a degree when is achieved via the interaction with its environment, transitional type also exists in some vehicles

6.
Degrees of freedom (mechanics)
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In physics, the degree of freedom of a mechanical system is the number of independent parameters that define its configuration. The position of a car moving along a track has one degree of freedom because the position of the car is defined by the distance along the track. A train of rigid cars connected by hinges to an engine still has one degree of freedom because the positions of the cars behind the engine are constrained by the shape of the track. An automobile with highly stiff suspension can be considered to be a body traveling on a plane. This body has three independent degrees of freedom consisting of two components of translation and one angle of rotation, skidding or drifting is a good example of an automobiles three independent degrees of freedom. The position and orientation of a body in space is defined by three components of translation and three components of rotation, which means that it has six degrees of freedom. The exact constraint mechanical design method manages the degrees of freedom to neither underconstrain nor overconstrain a device, the number of rotational degrees of freedom comes from the dimension of the rotation group SO. A non-rigid or deformable body may be thought of as a collection of many minute particles, when motion involving large displacements is the main objective of study, a deformable body may be approximated as a rigid body in order to simplify the analysis. The degree of freedom of a system can be viewed as the number of coordinates required to specify a configuration. This reduces the degree of freedom of the system to five, see also Euler angles The trajectory of an airplane in flight has three degrees of freedom and its attitude along the trajectory has three degrees of freedom, for a total of six degrees of freedom. The mobility formula counts the number of parameters that define the configuration of a set of bodies that are constrained by joints connecting these bodies. Consider a system of n rigid bodies moving in space has 6n degrees of freedom measured relative to a fixed frame. In order to count the degrees of freedom of this system, include the frame in the count of bodies. Then the degree-of-freedom of the system of N = n +1 is M =6 n =6. Joints that connect bodies in this system remove degrees of freedom, specifically, hinges and sliders each impose five constraints and therefore remove five degrees of freedom. It is convenient to define the number of constraints c that a joint imposes in terms of the joints freedom f, where c =6 − f. In the case of a hinge or slider, which are one degree of freedom joints, have f =1, the result is that the mobility of a system formed from n moving links and j joints each with freedom fi, i =1. J, is given by M =6 n − ∑ i =1 j =6 + ∑ i =1 j f i Recall that N includes the fixed link, there are two important special cases, a simple open chain, and a simple closed chain

7.
Algebraic variety
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Algebraic varieties are the central objects of study in algebraic geometry. Classically, a variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. For example, some definitions provide that algebraic variety is irreducible, under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility, the concept of an algebraic variety is similar to that of an analytic manifold. An important difference is that a variety may have singular points. Generalizing this result, Hilberts Nullstellensatz provides a correspondence between ideals of polynomial rings and algebraic sets. Using the Nullstellensatz and related results, mathematicians have established a correspondence between questions on algebraic sets and questions of ring theory. This correspondence is the specificity of algebraic geometry, an affine variety over an algebraically closed field is conceptually the easiest type of variety to define, which will be done in this section. Next, one can define projective and quasi-projective varieties in a similar way, the most general definition of a variety is obtained by patching together smaller quasi-projective varieties. It is not obvious that one can construct genuinely new examples of varieties in this way, let k be an algebraically closed field and let An be an affine n-space over k. The polynomials f in the ring k can be viewed as k-valued functions on An by evaluating f at the points in An, i. e. by choosing values in k for each xi. For each set S of polynomials in k, define the zero-locus Z to be the set of points in An on which the functions in S simultaneously vanish, that is to say Z =. This topology is called the Zariski topology.2 Given a subset V of An, let f in k be a homogeneous polynomial of degree d. It is not well-defined to evaluate f on points in Pn in homogeneous coordinates, however, because f is homogeneous, f = λd f , it does make sense to ask whether f vanishes at a point. For each set S of homogeneous polynomials, define the zero-locus of S to be the set of points in Pn on which the functions in S vanish, Given a subset V of Pn, let I be the ideal generated by all homogeneous polynomials vanishing on V. For any projective algebraic set V, the ring of V is the quotient of the polynomial ring by this ideal.10 A quasi-projective variety is a Zariski open subset of a projective variety. Notice that every variety is quasi-projective. In classical algebraic geometry, all varieties were by definition quasiprojective varieties and it might not have an embedding into projective space

8.
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

9.
Degrees of freedom (physics and chemistry)
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In physics, a degree of freedom is an independent physical parameter in the formal description of the state of a physical system. The set of all dimensions of a system is known as a phase space, a degree of freedom of a physical system is an independent parameter that is necessary to characterize the state of a physical system. In general, a degree of freedom may be any property that is not dependent on other variables. The location of a particle in space requires three position coordinates. Similarly, the direction and speed at which a particle moves can be described in terms of three velocity components, each in reference to the three dimensions of space. On the other hand, a system with an object that can rotate or vibrate can have more than six degrees of freedom. In statistical mechanics, a degree of freedom is a scalar number describing the microstate of a system. The specification of all microstates of a system is a point in the phase space. In the 3D ideal chain model in chemistry, two angles are necessary to describe the orientation of each monomer and it is often useful to specify quadratic degrees of freedom. These are degrees of freedom that contribute in a function to the energy of the system. In three-dimensional space, three degrees of freedom are associated with the movement of a particle, a diatomic gas molecule has 7 degrees of freedom. This set may be decomposed in terms of translations, rotations, the center of mass motion of the entire molecule accounts for 3 degrees of freedom. In addition, the molecule has two degrees of motion and two vibrational modes. The rotations occur around the two axes perpendicular to the line between the two atoms, the rotation around the atom–atom bond is not a physical rotation. This yields, for a molecule, a decomposition of. In special cases, such as adsorbed large molecules, the degrees of freedom can be limited to only one. As defined above one can also count degrees of freedom using the number of coordinates required to specify a position. This is done as follows, For a single particle we need 2 coordinates in a 2-D plane to specify its position and 3 coordinates in 3-D space, thus its degree of freedom in a 3-D space is 3

10.
Translation (geometry)
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In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure or a space by the same amount in a given direction. In Euclidean geometry a transformation is a correspondence between two sets of points or a mapping from one plane to another. )A translation can be described as a rigid motion. A translation can also be interpreted as the addition of a constant vector to every point, a translation operator is an operator T δ such that T δ f = f. If v is a vector, then the translation Tv will work as Tv. If T is a translation, then the image of a subset A under the function T is the translate of A by T, the translate of A by Tv is often written A + v. In a Euclidean space, any translation is an isometry, the set of all translations forms the translation group T, which is isomorphic to the space itself, and a normal subgroup of Euclidean group E. The quotient group of E by T is isomorphic to the orthogonal group O, E / T ≅ O, a translation is an affine transformation with no fixed points. Matrix multiplications always have the origin as a fixed point, similarly, the product of translation matrices is given by adding the vectors, T u T v = T u + v. Because addition of vectors is commutative, multiplication of matrices is therefore also commutative. In physics, translation is movement that changes the position of an object, for example, according to Whittaker, A translation is the operation changing the positions of all points of an object according to the formula → where is the same vector for each point of the object. When considering spacetime, a change of time coordinate is considered to be a translation, for example, the Galilean group and the Poincaré group include translations with respect to time