In this article we introduce and study hyperclass-forcing (where the conditions of the forcing notion are themselves classes) in the context of an extension of Morse-Kelley class theory, called MK**. We define this forcing by using a symmetry between MK** models and models of ZFC− plus there exists a strongly inaccessible cardinal (called SetMK**). We develop a coding between β-models ${\cal M}$ of MK** and transitive models M+ of SetMK** which will allow us to go from ${\cal M}$ to M+ and vice versa. So instead of forcing with a hyperclass in MK** we can force over the corresponding SetMK** model with a class of conditions. For class-forcing to work in the context of ZFC− we show that the SetMK** model M+ can be forced to look like LK*[X], where κ* is the height of M+, κ strongly inaccessible in M+ and $X \subseteq \kappa$. Over such a model we can apply definable class forcing and we arrive at an extension of M+ from which we can go back to the corresponding β-model of MK**, which will in turn be an extension of the original ${\cal M}$. Our main result combines hyperclass forcing with coding methods of [3] and [4] to show that every β-model of MK** can be extended to a minimal such model of MK** with the same ordinals. A simpler version of the proof also provides a new and analogous minimality result for models of second-order arithmetic.

Let $(X,B)$ be a projective log canonical pair such that $B$ is a $\mathbb{Q}$-divisor, and that there is a surjective morphism $f: X\to Z$ onto a normal variety $Z$ satisfying $K_X+B\sim _{\mathbb{Q}} f^*M$ for some big $\mathbb{Q}$-divisor $M$, and the augmented base locus ${\mathbf{B}}_+(M)$ does not contain the image of any log canonical centre of $(X,B)$. We will show that $(X,B)$ has a good log minimal model. An interesting special case is when $f$ is the identity morphism.

A recently proposed model for the investigation of diffusivity in planktonic systemscontaining toxin-producing phytoplanktons is here reconsidered. We show the existence ofplanktonic travelling waves. Numerical simulations validate the analytical findings, toelucidate the sensitivity of the results in dependence of the diffusion coefficients.

In this paper, we prove that the log minimal model program in dimension d−1 implies the existence of log minimal models for effective lc pairs (e.g. of non-negative Kodaira dimension) in dimension d. In fact, we prove that the same conclusion follows from a weaker assumption, namely, the log minimal model program with scaling in dimension d−1. This enables us to prove that effective lc pairs in dimension five have log minimal models. We also give new proofs of the existence of log minimal models for effective lc pairs in dimension four and of the Shokurov reduction theorem.

A recent framework of relativized hyperequivalence of programs offers a unifying generalization of strong and uniform equivalence. It seems to be especially well suited for applications in program optimization and modular programming due to its flexibility that allows us to restrict, independently of each other, the head and body alphabets in context programs. We study relativized hyperequivalence for the three semantics of logic programs given by stable, supported, and supported minimal models. For each semantics, we identify four types of contexts, depending on whether the head and body alphabets are given directly or as the complement of a given set. Hyperequivalence relative to contexts where the head and body alphabets are specified directly has been studied before. In this paper, we establish the complexity of deciding relativized hyperequivalence with respect to the three other types of context programs.

We prove that the Iitaka conjecture Cn,m for algebraic fibre spaces holds up to dimension six, that is, when n≤6.

Given an argumentation framework AF, we introduce a mapping function that constructs a disjunctive logic program P, such that the preferred extensions of AF correspond to the stable models of P, after intersecting each stable model with the relevant atoms. The given mapping function is of polynomial size w.r.t. AF.

In particular, we identify that there is a direct relationship between the minimal models of a propositional formula and the preferred extensions of an argumentation framework by working on representing the defeated arguments. Then we show how to infer the preferred extensions of an argumentation framework by using UNSAT algorithms and disjunctive stable model solvers. The relevance of this result is that we define a direct relationship between one of the most satisfactory argumentation semantics and one of the most successful approach of nonmonotonic reasoning i.e., logic programming with the stable model semantics.

We propose and study algorithms to compute minimal models, stable models and answer sets of $t$-CNF theories, and normal and disjunctive $t$-programs. We are especially interested in algorithms with non-trivial worst-case performance bounds. The bulk of the paper is concerned with the classes of 2- and 3-CNF theories, and normal and disjunctive 2- and 3-programs, for which we obtain significantly stronger results than those implied by our general considerations. We show that one can find all minimal models of 2-CNF theories and all answer sets of disjunctive 2-programs in time $O(m1\mbox{.}4422\mbox{..}^n)$. Our main results concern computing stable models of normal 3-programs, minimal models of 3-CNF theories and answer sets of disjunctive 3-programs. We design algorithms that run in time $O(m1\mbox{.}6701\mbox{..}^n)$, in the case of the first problem, and in time $O(mn^2 2\mbox{.}2782\mbox{..}^n)$, in the case of the latter two. All these bounds improve by exponential factors the best algorithms known previously. We also obtain closely related upper bounds on the number of minimal models, stable models and answer sets a $t$-CNF theory, a normal $t$-program or a disjunctive $t$-program may have.

Bertini's theorem on variable singular points may fail in characteristic $p$; there are algebraic fibrations that are pathological in the sense that all fibres are non-smooth though the total space admits only a finite number of singular points. We prove that if an algebraic fibration by plane projective quartic curves in characteristic 7 is pathological then, after an eventual cyclic base extension of degree 3, it is, up to birational equivalence, obtained by a base extension from the pencil of quartic curves cut out by the equations $ zx^3 + xy^3 + t yz^3 = 0$.

More generally, we classify pathological fibrations by canonical curves of arithmetic genus $g = \frac{p - 1}{2}$ in the projective space of dimension $\frac{p - 3}{2}$. We prove that the gap sequences of the smooth Weierstrass points of the fibres have the property that a positive integer $\ell$ is a gap if and only if $2g + 1 - \ell$ is a non-gap. In analogy to the Kodaira–Néron classification of special fibres of minimal fibrations by elliptic curves, we construct minimal proper regular models of pathological fibrations, determine the structure of the bad fibres, and study the global geometry of the total spaces.