Orthomodular lattices that are $$Z_2$$-rich

We study the orthomodular lattices (OMLs) that have an abundance of \(Z_2\)-valued states. We call these OMLs \(Z_2\)-rich. The motivation for the investigation comes from a natural algebraic curiosity that reflects the state of the (orthomodular) art, the consideration also has a certain bearing on the foundation of quantum theories (OMLs are often identified with “quantum logics”) and mathematical logic (\(Z_2\)-states are fundamental in mathematical logic). Before we launch on the subject proper, we observe, for a potential application elsewhere, that there can be a more economic introduction of \(Z_2\)-richness - the \(Z_2\)-richness in the orthocomplemented setup is sufficient to imply orthomodularity. In the further part we review basic examples of OMLs that are \(Z_2\)-rich and that are not. Then we show, as a main result, that the \(Z_2\)-rich OMLs form a large and algebraicly “friendly” class—they form a variety. In the appendix we note that the OMLs that allow for a natural introduction of a symmetric difference provide a source of another type of examples of \(Z_2\)-rich OMLs. We also formulate open questions related to the matter studied.     

Pricing derivatives in a regime switching market with time inhomogenous volatility

This paper studies pricing derivatives in a componentwise semi-Markov (CSM) modulated market. We consider a financial market where the asset price dynamics follows a regime switching geometric Brownian motion model in which the coefficients depend on finitely many age-dependent semi-Markov processes. We further allow the volatility coefficient to depend on time explicitly. Under these market assumptions, we study locally risk minimizing pricing of a class of European options. It is shown that the price function can be obtained by solving a non-local Black–Scholes–Merton-type PDE. We establish existence and uniqueness of a classical solution to the Cauchy problem. We also find another characterization of price function via a system of Volterra integral equation of second kind. This alternative representation leads to computationally efficient methods for finding price and hedging. An explicit expression of quadratic residual risk is also obtained.     

On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator

Extended dynamic mode decomposition (EDMD) (Williams et al. in J Nonlinear Sci 25(6):1307–1346, 2015) is an algorithm that approximates the action of the Koopman operator on an N-dimensional subspace of the space of observables by sampling at M points in the state space. Assuming that the samples are drawn either independently or ergodically from some measure \(\mu \), it was shown in Klus et al. (J Comput Dyn 3(1):51–79, 2016) that, in the limit as \(M\rightarrow \infty \), the EDMD operator \(\mathcal {K}_{N,M}\) converges to \(\mathcal {K}_N\), where \(\mathcal {K}_N\) is the \(L_2(\mu )\)-orthogonal projection of the action of the Koopman operator on the finite-dimensional subspace of observables. We show that, as \(N \rightarrow \infty \), the operator \(\mathcal {K}_N\) converges in the strong operator topology to the Koopman operator. This in particular implies convergence of the predictions of future values of a given observable over any finite time horizon, a fact important for practical applications such as forecasting, estimation and control. In addition, we show that accumulation points of the spectra of \(\mathcal {K}_N\) correspond to the eigenvalues of the Koopman operator with the associated eigenfunctions converging weakly to an eigenfunction of the Koopman operator, provided that the weak limit of the eigenfunctions is nonzero. As a by-product, we propose an analytic version of the EDMD algorithm which, under some assumptions, allows one to construct \(\mathcal {K}_N\) directly, without the use of sampling. Finally, under additional assumptions, we analyze convergence of \(\mathcal {K}_{N,N}\) (i.e., \(M=N\)), proving convergence, along a subsequence, to weak eigenfunctions (or eigendistributions) related to the eigenmeasures of the Perron–Frobenius operator. No assumptions on the observables belonging to a finite-dimensional invariant subspace of the Koopman operator are required throughout.     

A geometric view on learning Bayesian network structures

We recall the basic idea of an algebraic approach to learning Bayesian network (BN) structures, namely to represent every BN structure by a certain (uniquely determined) vector, called a standard imset. The main result of the paper is that the set of standard imsets is the set of vertices (=extreme points) of a certain polytope. Motivated by the geometric view, we introduce the concept of the geometric neighborhood for standard imsets, and, consequently, for BN structures. Then we show that it always includes the inclusion neighborhood, which was introduced earlier in connection with the greedy equivalence search (GES) algorithm. The third result is that the global optimum of an affine function over the polytope coincides with the local optimum relative to the geometric neighborhood.To illustrate the new concept by an example, we describe the geometric neighborhood in the case of three variables and show it differs from the inclusion neighborhood. This leads to a simple example of the failure of the GES algorithm if data are not “generated” from a perfectly Markovian distribution. The point is that one can avoid this failure if the search technique is based on the geometric neighborhood instead. We also found out what is the geometric neighborhood in the case of four and five variables.     

A Raman spectroscopic study of bukovskýite Fe2(AsO4)(SO4)(OH)· 7H2O,a mineral phase with a significant role in arsenic migration

The Raman spectrum of bukovskýite [Fe³⁺₂(OH)(SO₄)(AsO₄)· 7H₂O] has been studied and compared with that of an amorphous gel containing specifically Fe, As and S, which is understood to be an intermediate product in the formation of bukovskýite. The observed bands are assigned to the stretching and bending vibrations of (SO₄)²⁻ and (AsO₄)³⁻ units, stretching and bending vibrations and vibrational modes of hydrogen‐bonded water molecules, stretching and bending vibrations of hydrogen‐bonded (OH)⁻ ions and Fe³⁺ (O,OH) units. The approximate range of O H···O hydrogen bond lengths was inferred from the Raman spectra. Raman spectra of crystalline bukovskýite and of the amorphous gel differ in that the bukovskýite spectrum is more complex, the observed bands are sharp and the degenerate bands of (SO₄)²⁻ and (AsO₄)³⁻ are split and more intense. Lower wavenumbers of δ H₂O bending vibrations in the spectrum of the amorphous gel may indicate the presence of weaker hydrogen bonds compared to those in bukovskýite. Copyright © 2011 John Wiley & Sons, Ltd.     

Polymer Brushes Showing Non‐Fouling in Blood Plasma Challenge the Currently Accepted Design of Protein Resistant Surfaces

Ultra‐low‐fouling poly[N‐(2‐hydroxypropyl) methacrylamide] (poly(HPMA)) brushes have been synthesized for the first time. Similar to the so far only ultra‐low‐fouling surface, poly(carboxybetaine acrylamide), the level of blood plasma fouling was below the detection limit of surface plasmon resonance (SPR, 0.03 ng · cm⁻²) despite being a hydrogen bond donor and displaying a moderate wettability, thus challenging the currently accepted views for the design of antifouling properties. The antifouling properties were preserved even after two years of storage. To demonstrate the potential of poly(HPMA) brushes for the preparation of bioactive ultra‐low fouling surfaces a label‐free SPR immunosensor for detection of G Streptococcus was prepared.

     

Influence of purine on copper behavior in neutral and alkaline sulfate solutions

The effect of purine (concentration range of 1.00 × 10^?6?C1.00 × 10^?2 M) on the behavior of copper in a 0.5 M Na₂SO₄ solution (pH 7 and pH 9) was studied using the open circuit potential measurement, potentiodynamic polarization, and chronoamperometry. Potentiodynamic polarization shows that purine acts as a copper corrosion inhibitor in both alkaline and neutral sulfate solutions. The efficiency of inhibition increases as the purine concentration increases. Chronoamperometric results follow the same trend as the results of potentiodynamic polarization. The inhibition effect can also be observed visually by microscopic examination of the electrode surface. Purine is adsorbed on copper surface according to the Langmuir adsorption isotherm.     

Probing the unconventional superconducting state of LiFeAs by quasiparticle interference

A crucial step in revealing the nature of unconventional superconductivity is to investigate the symmetry of the superconducting order parameter. Scanning tunneling spectroscopy has proven a powerful technique to probe this symmetry by measuring the quasiparticle interference (QPI) which sensitively depends on the superconducting pairing mechanism. A particularly well-suited material to apply this technique is the stoichiometric superconductor LiFeAs as it features clean, charge neutral cleaved surfaces without surface states and a relatively high T(c)～18 K. Our data reveal that in LiFeAs the quasiparticle scattering is governed by a van Hove singularity at the center of the Brillouin zone which is in stark contrast to other pnictide superconductors where nesting is crucial for both scattering and s(±) superconductivity. Indeed, within a minimal model and using the most elementary order parameters, calculations of the QPI suggest a dominating role of the holelike bands for the quasiparticle scattering. Our theoretical findings do not support the elementary singlet pairing symmetries s(++), s(±), and d wave. This brings to mind that the superconducting pairing mechanism in LiFeAs is based on an unusual pairing symmetry such as an elementary p wave (which provides optimal agreement between the experimental data and QPI simulations) or a more complex order parameter (e.g., s+id wave symmetry).     

Removal of lead from aqueous solutions by using the natural and Fe(III)-modified zeolite

In the present study, the sorption of lead by the natural and Fe(III)-modified zeolite (clinoptilolite) is described. The characterization of the natural zeolite-rich rock and the Fe(III)-modified form was performed by chemical analysis, point of the zero charge (pH_pzc), X-ray powder diffraction, applying the Rietveld/RIR method for the quantitative phase analysis, and scanning electron microscopy. The effects of sorbents dose and the initial lead concentrations on its sorption by two sorbents were investigated. For both sorbents, it was determined that at lower initial concentrations of lead, ion exchange of inorganic cations in zeolites with lead, together with uptake of hydrogen dominated, while at higher initial lead concentrations beside these processes, chemisorption of lead occurred. Significantly higher sorption of lead was achieved with Fe(III)-modified zeolite. From sorption isotherms, maximum sorbed amounts of lead, under the applied experimental conditions, were 66 mg/g for the natural and 133 mg/g for Fe(III)-modified zeolite. The best fit of experimental data was achieved with the Freundlich model (R² ≥ 0.94).