For two geometric objects P and Q represented by the relations P ( x, y ) and Q ( x, y ) the intersection is the collection of all points ( x, y ) which are in both relations.

For instance, volume and Riemannian curvature are invariants that can distinguish different geometric structures on the same smooth manifold — that is, one can smoothly " flatten out " certain manifolds, but it might require distorting the space and affecting the curvature or volume.

For example, the geometric mean can give a meaningful " average " to compare two companies which are each rated at 0 to 5 for their environmental sustainability, and are rated at 0 to 100 for their financial viability.

For instance, the geometric mean of two numbers, say 2 and 8, is just the square root of their product ; that is.

For all positive data sets containing at least one pair of unequal values, the harmonic mean is always the least of the three means, while the arithmetic mean is always the greatest of the three and the geometric mean is always in between ( see Inequality of arithmetic and geometric means.

For example, the geometric mean of 2 and 8 can be calculated as:

For instance, if one graphs a set of solutions of an equation in some higher dimensional space, he may ask about the geometric properties of the graph.

For example, the geometric mean of five values: 4, 36, 45, 50, 75 is:

For example, a similar geometric system was published in 1987 by Kjell Gustafson, whose method represents a rhythm as a two-dimensional graph.

For a given geometric figure in a given geometric space, consider the following equivalence relation: two automorphisms of space are equivalent if and only if the two images of the figure are the same ( here " the same " does not mean something like e. g. " the same up to translation and rotation ", but it means " exactly the same ").

For instance, an infinite geometric sum is implicit in Zeno's paradox of the dichotomy .< ref name =" Stillwell Infinite Series Early Results "> Later, Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids.

For example, visual hallucinations during occipital lobe seizures are typically visions of brightly colored, geometric shapes that may move across the visual field, multiply, or form concentric rings and generally persist from a few seconds to a few minutes.

: For " outer product " in geometric algebra, see exterior product.

For example, quantum finite automaton, geometric automaton and metric automaton have probabilistic acceptance.

For instance, the substitution x → ax gives the generating function for the geometric sequence 1, a, a < sup > 2 </ sup >, a < sup > 3 </ sup >,... for any constant a:

For example, the primitive equations which weather forecast models solve are more easily expressed in terms of geopotential than geometric height.

For instance in face recognition those geometric constraints can be obtained using principal component analysis ( PCA ) on a database of depth-maps of faces, allowing only surface normals solutions which are found in a normal population.

For example, in his treatise on mineralogy, Kitab al-Jamahir ( Book of Precious Stones ), he is " the most exact of experimental scientists ", while in the introduction to his study of India, he declares that " to execute our project, it has not been possible to follow the geometric method " and develops comparative sociology as a scientific method in the field.

For an infinitesimal deformation the displacements and the displacement gradients are small compared to unity, i. e., and, allowing for the geometric linearisation of the Lagrangian finite strain tensor, and the Eulerian finite strain tensor, i. e. the non-linear or second-order terms of the finite strain tensor can be neglected.

For infinitesimal deformations of a continuum body, in which the displacements and the displacement gradients are small compared to unity, i. e., and, it is possible to perform a geometric linearisation of the Lagrangian finite strain tensor, and the Eulerian finite strain tensor.

For his part Archimedes was the first to calculate the value of π and a geometric series, and also the earliest known mathematical physicist discovering the law of buoyancy, as well as conceiving the irrigation device known as Archimedes ' screw.

For a vector to represent a geometric object, it must be possible to describe how it looks in any other coordinate system.

For example, concerning the parameter of per capita income growth, development economist Ha-Joon Chang writes that considering the record of the last two decades the argument for continuing neo-liberal policy prescriptions are " simply untenable.

For example, if function div stands for the division operation x / y, then div with the parameter x fixed at 1 ( i. e., div 1 ) is another function: the same as the function inv that returns the multiplicative inverse of its argument, defined by inv ( y )

For some distributed parameter systems the vectors may be infinite-dimensional ( typically functions ).

For example, some references, and Wolfram's Mathematica software, define the complete elliptic integral of the first kind in terms of the parameter m, instead of the elliptic modulus k.

For example, consider a model which gives the probability density function of observable random variable X as a function of a parameter θ.

For example, the sample mean () can be used as an estimate of the mean parameter ( μ ) of the population from which the sample was drawn.

For a given choice of those parameters, one could make sense of the theory, but since we can never do infinitely many experiments to fix the values of every parameter, we do not have a meaningful physical theory:

For instance, the sample mean is a statistic which estimates the population mean, which is a parameter.

For example, the parameter may be the average height of 25-year-old men in North America.

For example, the Student's t-test for comparing the means of two groups can be formulated as seeing if an estimated parameter in the model is different from 0.

For example, quantum strings have tension, much like regular strings made of twine ; this tension is considered a fundamental parameter of the theory.

For clarity, the numbers in the second parameter are written as an explicit summation ( that is, with a "+"), rather than presented as a single number ( in this example, 4161 ).

For mach number less than 0. 3, it is the primary parameter that governs the flow characteristics.

For instance, the parameter keyword to apply a text description to any object to be created or changed is spelled the same way for all such commands.

* For programmers who learned Windows programming in C, probably the most memorable examples are the ( word-size parameter ) and ( long-integer parameter ) for the WindowProc () function.

For example: if the null model has 1 free parameter and a log-likelihood of − 8024 and the alternative model has 3 degrees of freedom and a LL of − 8012, then the probability of this difference is that of chi-squared value of + 2 ·( 8024 − 8012 ) = 24 with 3 − 1 = 2 degrees of freedom.

For instance, here is a Caesar cipher using a left rotation of three places ( the shift parameter, here 3, is used as the key ):

For the common case of a steady sinusoid, the s parameter is taken to be, corresponding to a complex sinusoid.

For consistency, this usage is also permitted in non-public settings such as local variables, parameter names, and private members.

For large oscillations of a pendulum, the analysis is complicated by an additional dimensionless parameter, the maximum swing angle.

For systems such as the logistic map, the bifurcation diagram shows a range of parameter values for which apparently the only cycle has period 3.

For this, a phosphor coating is applied to a surface of interest and, usually, the decay time is the emission parameter that indicates temperature.

For constant v on a plane through the Oz axis the parameter u plays the same role for the ellipse of intersection.