We show that every bounded, continuous at infinity game of perfect information has an ε−perfect equilibrium. Our method consists of approximating the payoff function of each player by a sequence of simple functions, and to consider the corresponding sequence of games, each differing form the original game only on the payoff function. In addition, this approach yields a new characterization of p...

A linear (qd, q, t)-perfect hash family of size s in a vector space V of order qd over a field F of order q consists of a set φ1, . . . , φs of linear functionals from V to F with the following property: for all t subsets X ⊆ V there exists i ∈ {1, . . . , s} such that φi is injective when restricted to F . A linear (qd, q, t)-perfect hash family of minimal size d(t− 1) is said to be optimal. I...

We say a number is flat if it can be written as a non-trivial power of 2 times an odd squarefree number. The power is the “exponent” and the number of odd primes the “length”. Let N be flat and 4-perfect with exponent a and length m. If a 6≡ 1 mod 12, then a is even. If a is even and 3 ∤ N then m is also even. If a ≡ 1 mod 12 then 3 | N and m is even. If N is flat and 3-perfect and 3 ∤ N , then...

Super-simple designs are useful in constructing codes and designs such as superimposed codes and perfect hash families. Recently, Gronau et al determined the existence of super-simple (v, 5, 2)-BIBDs with possible exceptions of v ∈ {75, 95, 115, 135, 195, 215, 231, 285, 365, 385, 515}. In this article, we investigate the existence of a super-simple (v, 5, 4)-BIBD and show that such a design exi...

We present two new perfect hashing schemes that can be used for efficient bitboard move generation for sliding pieces in chess-like board games without the need to use rotated bitboards. Moreover, we show that simple variations of these schemes give minimal perfect hashing schemes. The new method is applicable provided N , the number of k-bit spaced positions that may be set to 1, is not more t...

I t is proved that a natural general izat ion of chess to an n x n board is complete in exponential time. This implies that there ex is t chess-positions on an n x n chess-board for which the problem of determining who can win from that posi t ion requires an amount of time which is at least exponential in n .

It is known that necessary and sufficient conditions for the existence of a 2-perfect 6-cycle system of order n are that n = 1 or 9 mod 12 and n > 9 (Lindner, Phelps and Rodger [2]). Hence the smallest possible order of such a system is 13. The existence of a 2-perfect 6-cycle system of order 13 is shown by example in [2]. The example is cyclic. It is obvious that for large n the construction o...