Let d > 0 be a squarefree integer and a be an integer, which is not 1 nor a square. Let P(a,d)(x) be the number of primes p  x such that p ⌘ 1 mod d and a(p 1)/d ⌘ 1 mod p. Numerical data indicate that the function as approximately equal to a constant multiple of ⇡(x)/(d'(d)) for su ciently large x, where ⇡(x) is the number of primes up to x and '(d) is the Euler-' function. The involved const...

The problem of calculating exact lower bounds for the number k-faces d-polytopes with n vertices, each value k, and characterizing minimizers has recently been solved not exceeding 2d. We establish corresponding result $n=2d+1$; nature minimizing polytopes are quite different in this case. As a byproduct, we also characterize all $d+3$ vertices only one or two edges more than minimum.

Abstract This paper describes a possible approach of designing an unconventional turbocharger using co-simulation method. Our goal is to design turbochargers running optimally during pulsating and transient modes. It means develop tool capable between 3-D 1-D CFD. must be fast precise enough provide reliable result. Influence coarse mesh versus fine was examined. Different time step size applie...