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The neighborhood graph of a given graph from a vertex v is the subgraph induced by the neighborhood of a graph from vertex v, most commonly including v itself. Such graphs ...

Let H be a complex Hilbert space, and define a nest as a set N of closed subspaces of H satisfying the conditions: 1. 0,H in N, 2. If N_1,N_2 in N, then either N_1 subset= ...

Beautiful patterns can be created by drawing sets of nested polygons such that the incircle of the nth polygon is the circumcircle of the (n+1)st and successive polygons are ...

The probability that two elements P_1 and P_2 of a symmetric group generate the entire group tends to 3/4 as n->infty (Netto 1964, p. 90). The conjecture was proven by Dixon ...

Polynomials O_n(x) that can be defined by the sum O_n(x)=1/4sum_(k=0)^(|_n/2_|)(n(n-k-1)!)/(k!)(1/2x)^(2k-n-1) (1) for n>=1, where |_x_| is the floor function. They obey the ...

A geometric construction, also called a verging construction, which allows the classical geometric construction rules to be bent in order to permit sliding of a marked ruler. ...

The functions theta_s(u) = (H(u))/(H^'(0)) (1) theta_d(u) = (Theta(u+K))/(Theta(k)) (2) theta_c(u) = (H(u))/(H(K)) (3) theta_n(u) = (Theta(u))/(Theta(0)), (4) where H(u) and ...

A paradox in decision theory. Given two boxes, B1 which contains $1000 and B2 which contains either nothing or a million dollars, you may pick either B2 or both. However, at ...

Samuel Pepys wrote Isaac Newton a long letter asking him to determine the probabilities for a set of dice rolls related to a wager he planned to make. Pepys asked which was ...

Curves with Cartesian equation ay^2=x(x^2-2bx+c) with a>0. The above equation represents the third class of Newton's classification of cubic curves, which Newton divided into ...