Type-Decomposition of an Effect Algebra

Effect algebras (EAs), play a significant role in quantum logic, are featured in the theory of partially ordered Abelian groups, and generalize orthoalgebras, MV-algebras, orthomodular posets, orthomodular lattices, modular ortholattices, and boolean algebras. We study centrally orthocomplete effect algebras (COEAs), i.e., EAs satisfying the condition that every family of elements that is dominated by an orthogonal family of central elements has a supremum. For COEAs, we introduce a general notion of decomposition into types; prove that a COEA factors uniquely as a direct sum of types I, II, and III; and obtain a generalization for COEAs of Ramsay’s fourfold decomposition of a complete orthomodular lattice.     

Finding small somatic structural variants in exome sequencing data: a machine learning approach

Genetic variation forms the basis for diversity but can as well be harmful and cause diseases, such as tumors. Structural variants (SV) are an example of complex genetic variations that comprise of many nucleotides ranging up to several megabases. Based on recent developments in sequencing technology it has become feasable to elucidate the genetic state of a person’s genes (i.e. the exome) or even the complete genome. Here, a machine learning approach is presented to find small disease-related SVs with the help of sequencing data. The method uses differences in characteristics of mapping patterns between tumor and normal samples at a genomic locus. This way, the method aims to be directly applicable for exome sequencing data to improve detection of SVs since specific SV detection methods are currently lacking. The method has been evaluated based on a simulation study as well as with exome data of patients with acute myeloid leukemia. An implementation of the algorithm is available at https://github.com/lenz99-/svmod.     

Projections in a Synaptic Algebra

A synaptic algebra is an abstract version of the partially ordered Jordan algebra of all bounded Hermitian operators on a Hilbert space. We review the basic features of a synaptic algebra and then focus on the interaction between a synaptic algebra and its orthomodular lattice of projections. Each element in a synaptic algebra determines and is determined by a one-parameter family of projections—its spectral resolution. We observe that a synaptic algebra is commutative if and only if its projection lattice is boolean, and we prove that any commutative synaptic algebra is isomorphic to a subalgebra of the Banach algebra of all continuous functions on the Stone space of its boolean algebra of projections. We study the so-called range-closed elements of a synaptic algebra, prove that (von Neumann) regular elements are range-closed, relate certain range-closed elements to modular pairs of projections, show that the projections in a synaptic algebra form an M-symmetric orthomodular lattice, and give several sufficient conditions for modularity of the projection lattice.     

Properties of Quasi-Hermitian Operators Inherited from Self-Adjoint Operators

We study a generalized effect algebra of unbounded linear operators in an infinite-dimensional complex Hilbert space. This algebra equipped with a certain kind of topology allows us to show that unbounded quasi-Hermitian operators can be expressed as a difference of two infinite sums of bounded quasi-Hermitian operators.     

Energy Estimates and Cavity Interaction for a Critical‐Exponent Cavitation Model

We consider the minimization of in a perforated domain of among maps that are incompressible (det ) and invertible, and satisfy a Dirichlet boundary condition u = g on ?Ω. If the volume enclosed by g (?Ω) is greater than |Ω|, any such deformation u is forced to map the small holes B_ε( a _i) onto macroscopically visible cavities (which do not disappear as ε → 0). We restrict our attention to the critical exponent p = n, where the energy required for cavitation is of the order of and the model is suited, therefore, for an asymptotic analysis (v₁,…, v_M denote the volumes of the cavities). In the spirit of the analysis of vortices in Ginzburg‐Landau theory, we obtain estimates for the “renormalized” energy showing its dependence on the size and the shape of the cavities, on the initial distance between the cavitation points a ₁,…, a _M, and on the distance from these points to the outer boundary ?Ω. Based on those estimates we conclude, for the case of two cavities, that either the cavities prefer to be spherical in shape and well separated, or to be very close to each other and appear as a single equivalent round cavity. This is in agreement with existing numerical simulations and is reminiscent of the interaction between cavities in the mechanism of ductile fracture by void growth and coalescence. © 2012 Wiley Periodicals, Inc.     

Non-Archimedean Probability

We propose an alternative approach to probability theory closely related to the framework of numerosity theory: non-Archimedean probability (NAP). In our approach, unlike in classical probability theory, all subsets of an infinite sample space are measurable and only the empty set gets assigned probability zero (in other words: the probability functions are regular). We use a non-Archimedean field as the range of the probability function. As a result, the property of countable additivity in Kolmogorov’s axiomatization of probability is replaced by a different type of infinite additivity.     

The independence number for polarity graphs of even order planes

In this paper, we use coherent configurations to obtain new upper bounds on the independence number of orthogonal polarity graphs of projective planes of even order. In the case of classical planes of square even order, these bounds differ only by 1 from the size of the largest known independent sets.     

Orthogonal Halogen‐Bonding‐Driven 3D Supramolecular Assembly of Right‐Handed Synthetic Helical Peptides

Peptide‐mediated self‐assembly is a prevalent method for creating highly ordered supramolecular architectures. Herein, we report the first example of orthogonal C?X???X?C/C?X???π halogen bonding and hydrogen bonding driven crystalline architectures based on synthetic helical peptides bearing hybrids of l ‐sulfono‐γ‐AApeptides and natural amino acids. The combination of halogen bonding, intra‐/intermolecular hydrogen bonding, and intermolecular hydrophobic interactions enabled novel 3D supramolecular assembly. The orthogonal halogen bonding in the supramolecular architecture exerts a novel mechanism for the self‐assembly of synthetic peptide foldamers and gives new insights into molecular recognition, supramolecular design, and rational design of biomimetic structures.