Existence of Optimal Flat Ribbons.

We apply the direct method of the calculus of variations to show that any nonplanar Frenet curve in can be extended to an infinitely narrow flat ribbon having bending energy. We also show that, in general, minimizers are not free of planar points, yet such points must be isolated under the mild condition that the torsion does not vanish.

Planar Pseudo-geodesics and Totally Umbilic Submanifolds.

We study totally umbilic isometric immersions between Riemannian manifolds. First, we provide a novel characterization of the totally umbilic isometric immersions with parallel normalized mean curvature vector, i.e.

Absolutely closed semigroups.

Let be a class of topological semigroups. A semigroup is called if for any homomorphism to a topological semigroup , the image [] is closed in . Let , , and be the classes of , Hausdorff, and Tychonoff zero-dimensional topological semigroups, respectively.

DrForna: visualization of cotranscriptional folding.

Motivation: Understanding RNA folding at the level of secondary structures can give important insights concerning the function of a molecule. We are interested to learn how secondary structures change dynamically during transcription, as well as whether particular secondary structures form already during or only after transcription. While different approaches exist to simulate cotranscriptional folding, the current strategies for visualization are lagging behind.

Total torsion of three-dimensional lines of curvature.

A curve in a Riemannian manifold is if its torsion (signed second curvature function) is well-defined and all higher-order curvatures vanish identically. In particular, when lies on an oriented hypersurface of , we say that is if the curve's principal normal, its torsion vector, and the surface normal are everywhere coplanar. Suppose that is three-dimensional and closed.

Classification of Rank-One Submanifolds.

We study ruled submanifolds of Euclidean space. First, to each (parametrized) ruled submanifold , we associate an integer-valued function, called , measuring the extent to which fails to be cylindrical. In particular, we show that if the degree is constant and equal to , then the singularities of can only occur along an -dimensional "striction" submanifold.

The spectrum of [Formula: see text]-soundness.

We study the [Formula: see text]-soundness spectra of theories. Given a recursively enumerable extension [Formula: see text] of [Formula: see text], [Formula: see text] is defined as the set of all 2-ptykes on which [Formula: see text] is correct about well foundedness. This is a measure of how close [Formula: see text] is to being [Formula: see text]-sound.

How to approximate fuzzy sets: mind-changes and the Ershov Hierarchy.

Computability theorists have introduced multiple hierarchies to measure the complexity of sets of natural numbers. The Kleene Hierarchy classifies sets according to the first-order complexity of their defining formulas. The Ershov Hierarchy classifies limit computable sets with respect to the number of mistakes that are needed to approximate them.

On the critical exponents of generalized ballot sequences in three dimensions and large tandem walks.

We answer some questions on the asymptotics of ballot walks raised in [S. B. Ekhad and D.

Implication in finite posets with pseudocomplemented sections.

It is well-known that relatively pseudocomplemented lattices can serve as an algebraic semantics of intuitionistic logic. To extend the concept of relative pseudocomplementation to non-distributive lattices, the first author introduced so-called sectionally pseudocomplemented lattices, i.e.

Sheffer operation in relational systems.

The concept of a Sheffer operation known for Boolean algebras and orthomodular lattices is extended to arbitrary directed relational systems with involution. It is proved that to every such relational system, there can be assigned a Sheffer groupoid and also, conversely, every Sheffer groupoid induces a directed relational system with involution. Hence, investigations of these relational systems can be transformed to the treatment of special groupoids which form a variety of algebras.

On Boolean posets of numerical events.

With many physical processes in which quantum mechanical phenomena can occur, it is essential to take into account a decision mechanism based on measurement data. This can be achieved by means of so-called numerical events, which are specified as follows: Let be a set of states of a physical system and () the probability of the occurrence of an event when the system is in state . A function is called a numerical event or alternatively, an -probability.

On logicality and natural logic.

In this paper we focus on the logicality of language, i.e. the idea that the language system contains a deductive device to exclude analytic constructions.

Consistent posets.

We introduce so-called consistent posets which are bounded posets with an antitone involution where the lower cones of and of coincide provided that ,  are different from 0, 1 and, moreover, if ,  are different from 0, then their lower cone is different from 0, too. We show that these posets can be represented by means of commutative meet-directoids with an antitone involution satisfying certain identities and implications. In the case of a finite distributive or strongly modular consistent poset, this poset can be converted into a residuated structure and hence it can serve as an algebraic semantics of a certain non-classical logic with unsharp conjunction and implication.

Filters and congruences in sectionally pseudocomplemented lattices and posets.

Together with J. Paseka we introduced so-called sectionally pseudocomplemented lattices and posets and illuminated their role in algebraic constructions. We believe that-similar to relatively pseudocomplemented lattices-these structures can serve as an algebraic semantics of certain intuitionistic logics.

Cichoń's diagram and localisation cardinals.

We reimplement the creature forcing construction used by Fischer et al. (Arch Math Log 56(7-8):1045-1103, 2017. 10.

Discrete curvature and torsion from cross-ratios.

Motivated by a Möbius invariant subdivision scheme for polygons, we study a curvature notion for discrete curves where the cross-ratio plays an important role in all our key definitions. Using a particular Möbius invariant point-insertion-rule, comparable to the classical four-point-scheme, we construct circles along discrete curves. Asymptotic analysis shows that these circles defined on a sampled curve converge to the smooth curvature circles as the sampling density increases.

Classifying equivalence relations in the Ershov hierarchy.

Computably enumerable equivalence relations (ceers) received a lot of attention in the literature. The standard tool to classify ceers is provided by the computable reducibility . This gives rise to a rich degree structure.

The logic induced by effect algebras.

Effect algebras form an algebraic formalization of the logic of quantum mechanics. For lattice effect algebras , we investigate a natural implication and prove that the implication reduct of is term equivalent to . Then, we present a simple axiom system in Gentzen style in order to axiomatize the logic induced by lattice effect algebras.

Ideals and their complements in commutative semirings.

We study conditions under which the lattice of ideals of a given a commutative semiring is complemented. At first we check when the annihilator of a given ideal of is a complement of . Further, we study complements of annihilator ideals.

When does a generalized Boolean quasiring become a Boolean ring?

Generalized Boolean quasirings are ring-like structures used as algebraic models in the foundations of axiomatic quantum mechanics. The quantum mechanical system corresponding to such a quasiring turns out to be a classical one if and only if this quasiring is a Boolean ring with unit. We characterize this situation by a single identity.

Dimensional Lifting through the Generalized Gram-Schmidt Process.

A new way of orthogonalizing ensembles of vectors by "lifting" them to higher dimensions is introduced. This method can potentially be utilized for solving quantum decision and computing problems.

Convex congruences.

For an algebra [Formula: see text] belonging to a quasivariety [Formula: see text], the quotient [Formula: see text] need not belong to [Formula: see text] for every [Formula: see text]. The natural question arises for which [Formula: see text]. We consider algebras [Formula: see text] of type (2, 0) where a partial order relation is determined by the operations [Formula: see text] and 1.