This paper deals with a question regarding tight closure in characteristic zero which we now review. Let R be a commutative ring of prime characteristic p and let I ⊆ R be an ideal. Recall that for e ≥ 0, the e-th Frobenius power of I, denoted I [p e], is the ideal of R generated by all p-th powers of elements in I. We say that f ∈ I∗, the tight closure of I, if there exists a c not in any mini...

Four experiments are reported in which the subjects had to respond to a target that masked a preceding prime via metacontrast masking. In one part of Experiment 1, the subjects discriminated the target's shape (square or diamond) by a motor-choice reaction, and in another part they had to respond with a simple reaction. The prime was neutral (circular) with respect to the target's shape. The da...

“The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors, is known to be one of the most important and useful in arithmetic,” Gauss wrote in his Disquisitiones Arithmeticae in 1801. “The dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated.”...

Let l be a prime. We show that there do not exist any non-zero semi-stable abelian varieties over Q with good reduction outside l if and only if l = 2, 3, 5, 7 or 13. We show that any semi-stable abelian variety over Q with good reduction outside l = 11 is isogenous to a power of the Jacobian variety of the modular curve X0(11). In addition we show that there do not exist any non-zero abelian v...

We show that for the primes l = 2, 3, 5, 7 or 13, there do not exist any non-zero abelian varieties over Q that have good reduction at every prime different from l and are semi-stable at l. We show that any semi-stable abelian variety over Q with good reduction outside l = 11 is isogenous to a power of the Jacobian variety of the modular curve X0(11). In addition we show that for l = 2, 3 and 5...

We prove several basic properties for difference ascending chains including a necessary and sufficient condition for an ascending chain to be the characteristic set of its saturation ideal and a necessary and sufficient condition for an ascending chain to be the characteristic set of a reflexive prime ideal. Based on these properties, we propose an algorithm to decompose the zero set of a finit...

We discuss the existence of zero coefficients in powers determinant polynomial order $n$. D. G. Glynn proved that $m$th power are all nonzero, if $m = p-1$ with a prime $p$. show converse also holds, $n \geq 3$. The proof is quite elementary.

The purpose of this paper is to study topological properties both the set all $k$-prime ideals and congruences for any commutative semiring with zero identity. We first prove that spectrum, i.e. equipped Zariski topology a spectral space, then homeomorphic spectrum respect their topologies.

A combinatorial 2-manifold is called weakly regular if the action of its automorphism group on its vertices is transitive. A combinatorial 2-manifold is called degree-regular if each of its vertices have the same degree. Clearly, a weakly regular combinatorial 2-manifold is degree-regular. In [5], Lutz has classified all the weakly-regular combinatorial 2-manifolds on at most 15 vertices, among...

Treatment with recombinant interleukin 2 and lymphokine-activated killer cells (rIL-2/LAK) has produced a clinical antitumor effect in preliminary human trials. The cytokines gamma-interferon (IFN-gamma), tumor necrosis factor alpha (TNF-alpha), and tumor necrosis factor beta (TNF-beta, lymphotoxin) have potent in vitro antitumor activity and some clinical toxicities similar to interleukin 2 (I...