Let R be a commutative hyperring with identity. We introduceand study prime fuzzy hyperideals of R. We investigate the Zariski topologyon FHspec(R), the spectrum of prime fuzzy hyperideals of R.

The concepts of L-fuzzy ideal generated by a L-fuzzy subset, L-fuzzy prime and completely prime ideal where L is a complete lattice are considered and some results are proved

Let p be a prime k|p−1, t = (p−1)/k and γ(k, p) be the minimal value of s such that every number is a sum of s k-th powers (mod p). We prove Heilbronn’s conjecture that γ(k, p) k1/2 for t > 2. More generally we show that for any positive integer q, γ(k, p) ≤ C(q)k1/q for φ(t) ≥ q. A comparable lower bound is also given. We also establish exact values for γ(k, p) when φ(t) = 2. For instance, whe...

This article deals with the rapid-equilibrium kinetics of the forward and reverse reactions together for the ordered and random enzyme-catalyzed A+B=P+Q and emphasizes the importance of reporting the values of the full set of equilibrium constants. Equilibrium constants that are not in the rate equation can be calculated for random mechanisms using thermodynamic cycles. This treatment is based ...

We give an analysis of the non-constructivity of the following basic result: if X and Y are simplicial sets and Y has the Kan extension property, then Y X also has the Kan extension property. By means of Kripke countermodels we show that even simple consequences of this basic result, such as edge reversal and edge composition, are not constructively provable. We also show that our unprovability...

We investigate the reverse-mathematical status of a version of the Baire Category Theorem known as BCT. In a 1993 paper Brown and Simpson showed that BCT is provable in RCA0. We now show that BCT is equivalent to RCA0 over RCA ∗ 0 .

We prove that if a Π2 sentence is provable in a certain theory of higher order arithmetic without the law of the excluded middle then it is uniformly provable in the weak classical theory RCA0. Applying the contrapositive of this result, we give three examples where results of reverse mathematics can be used to show nonexistence of proofs in certain intuitionistic systems.

We design and analyze a protocol for dividing a state into districts, where parties take turns proposing a division, and freezing a district from the other party’s proposed division. We show that our protocol has predictable and provable guarantees for both the number of districts in which each party has a majority of supporters, and the extent to which either party has the power to pack a spec...

The Nielsen-Schreier Theorem states that every subgroup of a free group is free. To formalize this theorem in weak subsystems of second order arithmetic, one has to choose between defining a subgroup in terms of a set of group elements and defining it in terms of a set of generators. We show that if subgroups are defined by sets, then the Nielsen-Schreier Theorem is provable in RCA0, while if s...