We prove that the bound from the theorem on ‘economic’ maps is best possible. Namely, for m > n + d we construct a map from an n-dimensional simplex to an m-dimensional Euclidean space for which (and for any close map) there exists a d-dimensional plane whose preimage has cardinality not less than the upper bound ⌈(dn + n + 1)/(m − n − d)⌉ + d from the theorem on ‘economic’ maps.
It is shown that every m-by-n matrix that consists of zeros and ones and has Boolean rank equal to n has a column in which at least sqrt(n)/2-1 entries are zero. It is proved that the bound obtained is asymptotically best possible. As an application of the results obtained we show that matrices of full Boolean rank can not have arbitrarily large size provided that those matrices have their tropical or determinantal ranks bounded.
We formulate some term rewriting systems in which the number of computation steps is finite for each output, but this number cannot be bounded by a provably total computable function in Peano arithmetic PA. Thus, the termination of such systems is unprovable in PA. These systems are derived from an independent combinatorial result known as the Worm principle; they can also be viewed as versions of the well-known Hercules-Hydra game introduced by Paris and Kirby.
We study the dynamics of Smale-Vietoris diffeomorphisms
Homology groups of spaces of nonsingular polynomial embeddings R1→Rn of degrees ≤4 are calculated. A general algebraic technique of such calculations for spaces of polynomial knots of arbitrary degrees is described.
We consider the Paley--Wiener spaces of L2 -functions whose Fourier transform has a bounded support. We show that every continuous mapping that generates a superposition operator acting on these spaces is affine and injective.
New lower bounds are obtained for the number N_B(x) of Novák numbers not exceeding the given quantity x. In addition, conditioned on the generalized Riemann Hypothesis, upper bounds are found for the number of prime factors of Novák numbers and a description of the prime factors of Novák numbers N such that 2N is a Novák-Carmichael number is presented.