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If two numbers b and c have the property that their difference b-c is integrally divisible by a number m (i.e., (b-c)/m is an integer), then b and c are said to be "congruent ...

A connected graph is graph that is connected in the sense of a topological space, i.e., there is a path from any point to any other point in the graph. A graph that is not ...

An n×n complex matrix A is called positive definite if R[x^*Ax]>0 (1) for all nonzero complex vectors x in C^n, where x^* denotes the conjugate transpose of the vector x. In ...

The Zermelo-Fraenkel axioms are the basis for Zermelo-Fraenkel set theory. In the following (Jech 1997, p. 1), exists stands for exists, forall means for all, in stands for ...

Find a way to stack a square of cannonballs laid out on the ground into a square pyramid (i.e., find a square number which is also square pyramidal). This corresponds to ...

Gram-Schmidt orthogonalization, also called the Gram-Schmidt process, is a procedure which takes a nonorthogonal set of linearly independent functions and constructs an ...

There are a number of mathematical curves that produced heart shapes, some of which are illustrated above. A "zeroth" curve is a rotated cardioid (whose name means ...

Let omega_1 and omega_2 be periods of a doubly periodic function, with tau=omega_2/omega_1 the half-period ratio a number with I[tau]!=0. Then Klein's absolute invariant ...

The Knödel graph W_(Delta,n) is a regular bipartite graph of vertex degree Delta on n nodes for even n>=2 and 1<=Delta<=|_log_2n_| with edges defined as follows. Label the ...

A circle that in internally tangent to two sides of a triangle and to the circumcircle is called a mixtilinear incircle. There are three mixtilinear incircles, one ...