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Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition. [1] : 58.
Examples of the kinds of solutions that are found perturbatively include the solution of the equation of motion ( e.g., the trajectory of a particle), the statistical average of some physical quantity ( e.g., average magnetization), and the ground state energy of a quantum mechanical problem.
Projective geometry can be modeled by the affine plane (or affine space) plus a line (hyperplane) "at infinity" and then treating that line (or hyperplane) as "ordinary". [5] An algebraic model for doing projective geometry in the style of analytic geometry is given by homogeneous coordinates.
In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows: 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that plane.
For planar algebra, non-Euclidean geometry arises in the other cases. When ε 2 = +1, then z is a split-complex number and conventionally j replaces epsilon. Then = (+) = and {z | z z* = 1} is the unit hyperbola. When ε 2 = 0, then z is a dual number.
Esophoria is an eye condition involving inward deviation of the eye, usually due to extra-ocular muscle imbalance. It is a type of heterophoria. Cause. Causes include: Refractive errors; Divergence insufficiency; Convergence excess; this can be due to nerve, muscle, congenital or mechanical anomalies.
In plane geometry, a shear mapping is an affine transformation that displaces each point in a fixed direction by an amount proportional to its signed distance from a given line parallel to that direction.
Strongly based on commutative algebra, scheme theory allows a systematic use of methods of topology and homological algebra. Scheme theory also unifies algebraic geometry with much of number theory, which eventually led to Wiles's proof of Fermat's Last Theorem.
All three implement pre-processing algorithms based on simple-homotopy equivalence and discrete Morse theory to perform homology-preserving reductions of the input cell complexes before resorting to matrix algebra.
In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology. The notion of a submersion is dual to the notion of an immersion .