Narrow Results

The implication of multivalued dependencies (MVDs) in relational databases has originally and independently been defined in the context of some fixed finite universe by Delobel, Fagin, and Zaniolo. Biskup observed that the original axiomatisation for MVD implication does not reflect the fact that the complementation rule is merely a means to achieve database normalisation. He proposed two alternative ways to overcome this deficiency: i) an axiomatisation that does represent the role of the complementation rule adequately, and ii) a notion of MVD implication in which the underlying universe of attributes is left undetermined together with an axiomatisation of this notion.

In this paper we investigate multivalued dependencies with null values (NMVDs) as defined and axiomatised by Lien. We show that Lien's axiomatisation does not adequately reflect the role of the complementation rule, and extend Biskup's findings for MVDs in total database relations to NMVDs in partial database relations. Moreover, a correspondence between (minimal) axiomatisations in fixed universes that do reflect the property of complementation and (minimal) axiomatisations in undetermined universes is shown.

We study reversible deterministic finite automata (REV-DFAs), that are partial deterministic finite automata whose transition function induces an injective mapping on the state set for every letter of the input alphabet. We give a structural characterization of regular languages that can be accepted by REV-DFAs. This characterization is based on the absence of a forbidden pattern in the (minimal) deterministic state graph. Again with a forbidden pattern approach, we also show that the minimality of REV-DFAs among all equivalent REV-DFAs can be decided. Both forbidden pattern characterizations give rise to $NL$-complete decision algorithms. In fact, our techniques allow us to construct the minimal REV-DFA for a given minimal DFA. These considerations lead to asymptotic upper and lower bounds on the conversion from DFAs to REV-DFAs. Thus, almost all problems that concern uniqueness and the size of minimal REV-DFAs are solved.

Let (M, g) be a Riemannian manifold and T₁ M its unit tangent sphere bundle. Minimality and harmonicity of unit vector fields have been extensively studied by considering on T₁M the Sasaki metric . This metric, and other well-known Riemannian metrics on T₁ M, are particular examples of Riemannian natural metrics. In this paper we equip T₁ M with a Riemannian natural metric and in particular with a natural contact metric structure. Then, we study the minimality for Reeb vector fields of contact metric manifolds and of quasi-umbilical hypersurfaces of a Kähler manifold. Several explicit examples are given. In particular, the Reeb vector field ξ of a K-contact manifold is minimal for any that belongs to a family depending on two parameters of metrics of the Kaluza–Klein type. Next, we show that the Reeb vector field ξ of a K-contact manifold defines a harmonic map for any Riemannian natural metric . Besides this, if the Reeb vector ξ of an almost contact metric manifold is a CR map then the induced almost CR structure on M is strictly pseudoconvex and ξ is a pseudo-Hermitian map; if in addition ξ is geodesic then is a harmonic map. Moreover, the Reeb vector field ξ of a contact metric manifold is a CR map iff ξ is Killing and is a special metric of the Kaluza–Klein type. Finally, in the final section, we obtain that there is a family of strictly pseudoconvex CR structures on T₁S²ⁿ⁺¹ depending on one parameter, for which a Hopf vector field ξ determines a pseudo-harmonic map (in the sense of Barletta–Dragomir–Urakawa [8]) from S²ⁿ⁺¹ to T₁S²ⁿ⁺¹.

In this paper we consider spectral representation of infinite dimensional stationary random fields over an abelian locally compact (or LCA) group, and then extend the results of earlier authors who consider wide sense Markov random fields over a Euclidean group ℝⁿ to the general LCA group context and obtain minimality properties. We also indicate possibilities of some extensions of these results to certain nonstationary classes.

Given a category fibered in groupoids over schemes with a log structure, one produces a category fibered in groupoids over log schemes. We classify the groupoid fibrations over log schemes that arise in this manner in terms of a categorical notion of "minimal" objects. The classification is actually a purely category-theoretic result about groupoid fibrations over fibered categories, though most of the known applications occur in the setting of log geometry, where our categorical framework encompasses many notions of "minimality" previously extant in the literature.

A knot is called minimal if its knot group admits epimorphisms onto the knot groups of only the trivial knot and itself. In this paper, we determine which two-bridge knot $𝔟(p,q)$ is minimal where $q\le 6$ or $p\le 100$.

In [V. O. Manturov, An almost classification of free knots, Dokl. Math.88(2) (2013) 556–558.] the second author constructed an invariant which in some sense generalizes the quantum $\mathrm{\text{sl}}(3)$ link invariant of Kuperberg to the case of free links. In this paper, we generalize this construction to free graph-links. As a result, we obtain an invariant of free graph-links with values in linear combinations of graphs. The main property of this invariant is that under certain conditions on the representative of the free graph-link, we can recover this representative from the value invariant on it. In addition, this invariant allows one to partially classify free graph-links.

Previously we defined an operation $\mu$ that generalizes Turaev’s cobracket for loops on a surface. We showed that, in contrast to the cobracket, this operation gives a formula for the minimum number of self-intersections of a loop in a given free homotopy class. In this paper, we consider the corresponding question for virtual strings, and conjecture that $\mu$ gives a formula for the minimum number of self-intersection points of a virtual string in a given virtual homotopy class. To support the conjecture, we show that $\mu$ gives a bound on the minimal self-intersection number of a virtual string which is stronger than a bound given by Turaev’s virtual string cobracket. We also use Turaev’s based matrices to describe a large set of strings $\alpha$ such that $\mu$ gives a formula for the minimal self-intersection number $\alpha$. Finally, we compare the bound given by $\mu$ to a bound given by Turaev’s based matrix invariant $\rho$, and construct an example that shows the bound on the minimal self-intersection number given by $\mu$ is sometimes stronger than the bound $\rho$.

A virtual$n$-string$(M,\alpha )$ consists of a closed, oriented surface $M$ and a collection $\alpha$ of $n$ oriented, closed curves immersed in $M$. We consider virtual $n$-strings up to virtual homotopy, i.e. stabilizations, destabilizations, stable homeomorphism, and homotopy.

Recently, Cahn proved that any virtual $1$-string can be virtually homotoped to a minimally filling and crossing-minimal representative by monotonically decreasing both genus and the number of self-intersections. We generalize her result to the case of non-parallel virtual $n$-strings.

Cahn also proved that any two crossing-irreducible representatives of a virtual $1$-string are related by isotopy, Type 3 moves, stabilizations, destabilizations, and stable homeomorphism. Kadokami claimed that this held for virtual $n$-strings in general, but Gibson found a counterexample for $5$-strings. We show that Kadokami’s statement holds for non-parallel $n$-strings and exhibit a counterexample for general virtual $3$-strings.

Within the geometrical framework developed in [Geometric constrained variational calculus. I: Piecewise smooth extremals, Int. J. Geom. Methods Mod. Phys.12 (2015) 1550061], the problem of minimality for constrained calculus of variations is analyzed among the class of differentiable curves. A fully covariant representation of the second variation of the action functional, based on a suitable gauge transformation of the Lagrangian, is explicitly worked out. Both necessary and sufficient conditions for minimality are proved, and reinterpreted in terms of Jacobi fields.

The problem of minimality for constrained variational calculus is analyzed within the class of piecewise differentiable extremaloids. A fully covariant representation of the second variation of the action functional based on a family of local gauge transformations of the original Lagrangian is proposed. The necessity of pursuing a local adaptation process, rather than the global one described in [1] is seen to depend on the value of certain scalar attributes of the extremaloid, here called the corners’ strengths. On this basis, both the necessary and the sufficient conditions for minimality are worked out. In the discussion, a crucial role is played by an analysis of the prolongability of the Jacobi fields across the corners. Eventually, in the appendix, an alternative approach to the concept of strength of a corner, more closely related to Pontryagin’s maximum principle, is presented.

By considering the split extension of a free abelian monoid having finite rank by a finite monogenic monoid, the main purposes of this paper are to present examples of efficient monoids and, also, minimal but inefficient monoids.

Although results presented in this paper seem as in the branch of pure mathematics, they are actually related to applications of Combinatorial and Geometric Group-Semigroup Theory, especially computer science, network systems, cryptography and physics etc., which will not be handled here.

We make some attempts to define a general notion of groups and fields of dimension one, and to determine their algebraic properties.

Minimal elements in the family of closed bounded convex sets in a topological vector space X were studied in many papers (see e.g [1], [2], [3], [5], [7]). In this paper we give positive answer to the question asked by R. Urbański concerning the existence of minimal elements in with respect to certain partial orders ≤* and ≤_*.

We are interested in the duality properties of the cohomology of a singular Riemannian foliation. We present here the most simple case where there exists a general duality theorem.