Good and Bad Infinitesimals,and States on Pseudo MV-algebras

We study relations among the set of infinitesimal elements of pseudo MV-algebras and the problem of existence of states on them. This is important because in contrast to MV-algebras, it can happen that a pseudo MV-algebra has no states, so no probabilistic evaluation of events on it is possible. We introduce two kinds of radicals, and we deal with their relation. In some cases, they are completely different, which is not the case for MV-algebras. We give many interesting examples describing different situations, and we deal in more details with a subvariety of symmetric pseudo MV-algebras, where both complements coincide. Mathematics Subject Classifications (2000) 06D35, 03B50, 03G12.     

Representable Effect Algebras and Observables

We introduce a class of monotone σ-complete effect algebras, called representable, which are σ-homomorphic images of a class of monotone σ-complete effect algebras of functions taking values in the interval [0, 1] and with effect algebra operations defined by points. We exhibit different types of compatibilities and show their connection to representability. Finally, we study observables and show situations when information of an observable on all intervals of the form (?∞, t) gives full information about the observable.     

MV-Test Spaces Versus MV-Algebras

In analogy with effect algebras, we introduce the test spaces and MV-test spaces. A test corresponds to a hypothesis on the propositional system, or, equivalently, to a partition of unity. We show that there is a close correspondence between MV-algebras and MV-test spaces.     

A new method for complex spectral measurement in millimeter and submillimeter wave range

New method for measuring complex transmission or reflection coefficients has been proposed and demonstrated using a simple setup. High measurement quality was achieved: phase sensitivity 0.005 rad., amplitude reproduceability 1%, dynamic range to 10⁷ ÷ 10⁸ (previously 10⁴), phase shift range to be measured 10⁵. It is effective for BWT-Spectroscopy due to the absence of any mechanical adjustments during electronic frequency scan.     

Postpolymerization effect on molecular weight distribution generated by pulsed laser radical polymerization

The postpolymerization effect on molecular weight distribution (MWDs) and on the Pulsed Laser Polymerization (PLP) technique for evaluation of kinetic constants is investigated. General expressions for moments are derived for a polymerization scheme that contains the reactions of chain initiation, propagation, and termination by recombination or disproportionation, under polymerization initiation by an arbitrary sequence of radiation pulses. The results of calculation of MWDs and of the weight-average degree of polymerization (P _w) for methyl methacrylate are presented. It is shown that the P _w value strongly depends on postpolymerization. A new method for determining the rate constants of chain propagation and chain termination from a single experiment by polymerization with packets of laser pulses is presented.     

Soler's Theorem and Characterization of Inner Product Spaces

Using Soler's result, we show that the existenceof at least one finitely additive probability measure onthe system of all orthogonally closed subspaces of Swhich is concentrated on a one-dimensional subspace of E can imply that E is a real,complex, or quaternionic Hilbert space. In addition,using the concept of test spaces of Foulis and Randalland introducing various systems of subspaces of E , we give some characterizations of inner productspaces which imply that E is a real, complex, orquaternionic Hilbert space.     

Commutative BCK-Algebras and Quantum Structures

We study commutative BCK-algebras with the relative cancellation property, i.e.,if a x, y and x * a = y * a, then x = y. Such algebras generalize Booleanrings as well as Boolean D-posets (= MV-algebras). We show that any suchBCK-algebra X can be embedded into the positive cone of an Abelianlattice-ordered group. Moreover, this group can be chosen to be a universal group forX. We compare BCK-algebras with the relative cancellation property with knownquantum structures as posets with difference, D-posets, orthoalgebras, andquantum MV-algebras, and we show that in many cases we obtain MV-algebras.     

Dark Equations and Their Light Integrability

A relatively new approach to analyzing integrability, based upon differential-algebraic and symplectic techniques, is applied to some “dark equations ”of the type introduced by Boris Kupershmidt. These dark equations have unusual properties and are not particularly well-understood. In particular, dark three-component polynomial Burgers type systems are studied in detail. Their matrix Lax representations are constructed, and the related symmetry recursion operators and infinite hierarchies of integrable nonlinear dynamical systems along with their Lax representations are derived. New linear Lax spectral problems for dark integrable countable hierarchies of dynamical systems are proposed and some special cases are considered as a means of indicating that the approach presented is applicable to a far wider class of dark equations than analyzed here.     

Subdirectly irreducible state-morphism BL-algebras

Recently Flaminio and Montagna (Proceedings of the 5th EUSFLAT Conference, II: 201?C206. Ostrava, 2007), (Inter. J. Approx. Reason. 50:138?C152, 2009) introduced the notion of a state MV-algebra as an MV-algebra with internal state. We have two kinds: state MV-algebras and state-morphism MV-algebras. These notions were also extended for state BL-algebras in (Soft Comput. doi:10.1007/s00500-010-0571-5). In this paper, we completely describe subdirectly irreducible state-morphism BL-algebras and this generalizes an analogous result for state-morphism MV-algebras presented in (Ann. Pure Appl. Logic 161:161?C173, 2009).