A method of computing the determinant of a square matrixdue to Charles Dodgson (1866) (who is more famous under his pseudonym Lewis Carroll). The method is useful for hand calculations because, for an integer matrix, all entries in submatrices computed along the way must also be integers. The method is also implemented efficiently in a parallel computation. Condensation is also known as the method of contractants (Macmillan 1955, Lotkin 1959).
Given an matrix, condensation successively computes an matrix, an matrix, etc., until arriving at a matrix whose only entry ends up being the determinant of the original matrix. To compute the matrix (), take the connected subdeterminants of the matrix and divide them by the central entries of the matrix, with no divisions performed for . The matrices arrived at in this manner are the matrices of determinants of the connected submatrices of the original matrices.
For example, the first condensation of the matrix
(1)yields the matrix
(2)and the second condensation yields
(3)which is the determinant of the original matrix. Collecting terms gives
(4)of which the nonzero terms correspond to the permutation matrices. In the case, 24 nonzero terms are obtained together with 18 vanishing ones. These 42 terms correspond to the alternating sign matricesfor which any s in a row or column must have a "outside" it (i.e., all s are "bordered" by s).
condensation method and dispersion method
In the preparation of colloids, using the dispersion method, larger particles of the substance are broken down into much smaller parts. Using the condensation method, smaller particles are dispersed and condensed to colloidal size.
Examples of dispersion methods in the preparation of colloids include mechanical methods such as grinding or milling, chemical methods like chemical precipitation, and thermal methods such as vaporization and condensation. These methods help break down larger particles into smaller ones to form stable colloidal suspensions.
evaporation and condensation or just condensation
to reduce to a smaller compaass
Colloids can be destroyed by heating them or adding an electrolyte in order to destroy them. By adding a small amount of a electrolyte, a colloid can be coagulated. Electrophoresis may be another method to destroy a colloid.
Colloids can impact biological systems by altering the distribution and availability of nutrients and contaminants in the environment. They can also affect the transport of pollutants in soil and water systems, leading to potential risks for human health and ecosystem functioning. In terms of the environment, colloids can influence processes such as soil erosion, sedimentation, and water quality by interacting with particles and chemicals in the environment.
After my opinion they are not colloids.
evaporation is a method to condensing
This is part of the water cycle.
evaporation and condensation
none