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This can be concisely written as the matrix inequality , where A is an m×n matrix, x is an n×1 column vector of variables, and b is an m×1 column vector of constants. [citation needed] In the above systems both strict and non-strict inequalities may be used. Not all systems of linear inequalities have solutions.
There are several different notations used to represent different kinds of inequalities: The notation a < b means that a is less than b. The notation a > b means that a is greater than b. In either case, a is not equal to b. These relations are known as strict inequalities, [1] meaning that a is strictly less than or strictly greater than b ...
Theorem 1. e ≤ 3v – 6; Theorem 2. If there are no cycles of length 3, then e ≤ 2v – 4. Theorem 3. f ≤ 2v – 4. In this sense, planar graphs are sparse graphs, in that they have only O(v) edges, asymptotically smaller than the maximum O(v 2). The graph K 3,3, for example, has 6 vertices, 9 edges, and no cycles of length 3. Therefore ...
For every subset F of edges, the dot product 1 E(v) · 1 F represents the number of edges in F that are adjacent to v. Therefore, the following statements are equivalent: A subset F of edges represents a matching in G; For every node v in V: 1 E(v) · 1 F ≤ 1. A G · 1 F ≤ 1 V. The cardinality of a set F of edges is the dot product 1 E · 1 F.
One particularly useful inequality to analyze homomorphism densities is the Cauchy–Schwarz inequality. The effect of applying the Cauchy-Schwarz inequality is "folding" the graph over a line of symmetry to relate it to a smaller graph. This allows for the reduction of densities of large but symmetric graphs to that of smaller graphs.
If X is a random variable with Lorenz curve L X (F), then −X has the Lorenz curve: L − X = 1 − L X (1 − F) The Lorenz curve is changed by translations so that the equality gap F − L(F) changes in proportion to the ratio of the original and translated means.
In mathematics, an inequation is a statement that an inequality holds between two values. [1] [2] It is usually written in the form of a pair of expressions denoting the values in question, with a relational sign between them indicating the specific inequality relation. Some examples of inequations are:
The Grothendieck inequality of a graph is an extension of the Grothendieck inequality because the former inequality is the special case of the latter inequality when is a bipartite graph with two copies of {, …,} as its bipartition classes. Thus,