Generally these statistics will be scale invariant ( scaling all the numbers by the same factor does not change the output ), to make them independent of population size, which is achieved by using ratios of homogeneous functions, most simply homogeneous linear or homogeneous quadratic functions.
22.
Now if temperature " T " and pressure " P " are held constant, Z = Z ( n _ 1, n _ 2, \ cdots ) is a homogeneous function of degree 1, since doubling the quantities of each component in the mixture will double Z.
23.
For example, a homogeneous function of two variables " x " and " y " is a real-valued function that satisfies the condition f ( \ alpha x, \ alpha y ) = \ alpha ^ k f ( x, y ) for some constant k and all real numbers \ alpha.
24.
The scaling hypothesis is that near the critical point, the free energy f ( t, H ), in d dimensions, can be written as the sum of a slowly varying regular part f _ r and a singular part f _ s, with the singular part being a scaling function, i . e ., a homogeneous function, so that
25.
It follows that if a function is homogeneous of degree then its image under the Legendre transformation is a homogeneous function of degree, where 1 } } . ( Since " x r " / " r " } }, with, implies " p s " / " s " } } . ) Thus, the only monomial whose degree is invariant under Legendre transform is the quadratic.
26.
Showed that a centrally symmetric star-shaped body is an intersection body if and only if the function 1 / | | " x " | | is a positive definite distribution, where | | " x " | | is the homogeneous function of degree 1 that is 1 on the boundary of the body, and used this to show that the unit balls l, 2 " p " norm are intersection bodies for " n " = 4 but are not intersection bodies for " n " e " 5.
27.
The symbol of a differential operator of order " n " on a smooth manifold " X " is defined in much the same way using local coordinate charts, and is a function on the cotangent bundle of " X ", homogeneous of degree " n " on each cotangent space . ( In general, differential operators transform in a rather complicated way under coordinate transforms ( see jet bundle ); however, the highest order terms transform like tensors so we get well defined homogeneous functions on the cotangent spaces that are independent of the choice of local charts . ) More generally, the symbol of a differential operator between two vector bundles " E " and " F " is a section of the pullback of the bundle Hom ( " E ", " F " ) to the cotangent space of " X ".
How to say homogeneous function in Hindi and what is the meaning of homogeneous function in Hindi? homogeneous function Hindi meaning, translation, pronunciation, synonyms and example sentences are provided by Hindlish.com.
