Polyhedral combinatoricsPolyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes. Research in polyhedral combinatorics falls into two distinct areas. Mathematicians in this area study the combinatorics of polytopes; for instance, they seek inequalities that describe the relations between the numbers of vertices, edges, and faces of higher dimensions in arbitrary polytopes or in certain important subclasses of polytopes, and study other combinatorial properties of polytopes such as their connectivity and diameter (number of steps needed to reach any vertex from any other vertex).
Territorial evolution of the United StatesThe United States of America was formed after thirteen British colonies in North America declared independence from the British Empire on July 4, 1776. In the Lee Resolution, passed by the Second Continental Congress two days prior, the colonies resolved that they were free and independent states. The union was formalized in the Articles of Confederation, which came into force on March 1, 1781, after being ratified by all 13 states. Their independence was recognized by Great Britain in the Treaty of Paris of 1783, which concluded the American Revolutionary War.
Economic planningEconomic planning is a resource allocation mechanism based on a computational procedure for solving a constrained maximization problem with an iterative process for obtaining its solution. Planning is a mechanism for the allocation of resources between and within organizations contrasted with the market mechanism. As an allocation mechanism for socialism, economic planning replaces factor markets with a procedure for direct allocations of resources within an interconnected group of socially owned organizations which together comprise the productive apparatus of the economy.
Analytic hierarchy processIn the theory of decision making, the analytic hierarchy process (AHP), also analytical hierarchy process, is a structured technique for organizing and analyzing complex decisions, based on mathematics and psychology. It was developed by Thomas L. Saaty in the 1970s; Saaty partnered with Ernest Forman to develop Expert Choice software in 1983, and AHP has been extensively studied and refined since then. It represents an accurate approach to quantifying the weights of decision criteria.
Birkhoff polytopeThe Birkhoff polytope Bn (also called the assignment polytope, the polytope of doubly stochastic matrices, or the perfect matching polytope of the complete bipartite graph ) is the convex polytope in RN (where N = n2) whose points are the doubly stochastic matrices, i.e., the n × n matrices whose entries are non-negative real numbers and whose rows and columns each add up to 1. It is named after Garrett Birkhoff. The Birkhoff polytope has n! vertices, one for each permutation on n items.
