Abstract Let $p=3n+1$ be a prime with $n\in \mathbb {N}=\{0,1,2,\ldots \}$ and let $g\in {Z}$ primitive root modulo p . $0<a_1<\cdots <a_n<p$ all the cubic residues in interval $(0,p)$ Then clearly sequence $a_1 \bmod p,\, a_2 p,\ldots , a_n p$ is permutation of $g^3 p,\,g^6 g^{3n} We determine sign this permutation.

The problem of nding all occurrences of a pattern of length m in a text of length n is considered. It is shown that the Boyer-Moore string matching algorithm performs roughly 3n comparisons and that this bound is tight up to O(n=m); more precisely, an upper bound of 3n n m comparisons is shown, as is a lower bound of 3n(1 o(1)) comparisons, as n m !1 and m!1. While the upper bound is somewhat i...

Using a scale transformation in momentum space a phase equivalent relativistic potential is generated from the nonrelativistic potential. By that transformation a practical method for the relativistic 3N scattering with realistic nucleon-nucleon potentials is introduced. The formalism can be applied to any realistic nonrelativistic potential. We also discuss the locations of the moving logarith...

The three-nucleon photodisintegration of 3 He has been calculated in the whole phase space using consistent Faddeev equations for the three-nucleon bound and scattering states. Modern nucleon-nucleon and 3N forces have been applied as well as different approaches to nuclear currents. Phase space regions are localized where 3N force effects are especially large. In addition semi-exclusive cross ...

We and other investigators have previously shown that membrane-type 1 matrix metalloproteinase (MT1-MMP) is overexpressed in invasive prostate cancer cells. However, the mechanism for this expression is not known. Here, we show that MT1-MMP is minimally expressed in nonmalignant primary prostate cells, moderately expressed in DU-145 cells, and highly expressed in invasive PC-3 and PC-3N cells. ...

Fortuitous sequences of flips of the top of a stack of n burnt pancakes for all n > 24. Summary Burnt pancakes problem was defined by Gates and Papadimitriou in 1979. They proved that g(−f n) ≥ 3n/2 − 1. In 1995 Cohen and Blum conjectured that g(n) = g(−I n) and proved that g(−I n) = g(−f n) + 1 ≥ 3n/2.

In 1995, Hochberg, McDiarmid, and Saks [4] proved that the vertex-bandwidth of the triangular grid Tn is precisely n+1; more recently Balogh, Mubayi, and Pluhár [1] posed the problem of determining the edge-bandwidth of Tn. We show that the edge-bandwidth of Tn is bounded above by 3n− 1 and below by 3n− o(n).