Formalization of Ring Theory in PVS

This paper presents a PVS development of relevant results of the theory of rings. The PVS theory includes complete proofs of the three classical isomorphism theorems for rings, and characterizations of principal, prime and maximal ideals. Algebraic concepts and properties are specified and formalized as generally as possible allowing in this manner their application to other algebraic structures. The development provides the required elements to formalize important algebraic theorems. In particular, the paper presents the formalization of the general algebraic-theoretical version of the Chinese remainder theorem (CRT) for the theory of rings, as given in abstract algebra textbooks, proved as a consequence of the first isomorphism theorem. Also, the PVS theory includes a formalization of the number-theoretical version of CRT for the structure of integers, which is the version of CRT found in formalizations. CRT for integers is obtained as a consequence of the general version of CRT for the theory of rings.

     

Matrix Transformation in Boron Containing High-Temperature Co–Re–Cr Alloys

An addition of boron largely increases the ductility in polycrystalline high-temperature Co–Re alloys. Therefore, the effect of boron on the alloy structural characteristics is of high importance for the stability of the matrix at operational temperatures. Volume fractions of ε (hexagonal close-packed—hcp), γ (face-centered cubic—fcc) and σ (Cr₂Re₃ type) phases were measured at ambient and high temperatures (up to 1500 °C) for a boron-containing Co–17Re–23Cr alloy using neutron diffraction. The matrix phase undergoes an allotropic transformation from ε to γ structure at high temperatures, similar to pure cobalt and to the previously investigated, more complex Co–17Re–23Cr–1.2Ta–2.6C alloy. It was determined in this study that the transformation temperature depends on the boron content (0–1000 wt. ppm). Nevertheless, the transformation temperature did not change monotonically with the increase in the boron content but reached a minimum at approximately 200 ppm of boron. A probable reason is the interplay between the amount of boron in the matrix and the amount of σ phase, which binds hcp-stabilizing elements (Cr and Re). Moreover, borides were identified in alloys with high boron content.     

Stimuli-Responsive Nanocomposite Hydrogels for Biomedical Applications

The complex tissue-specific physiology that is orchestrated from the nano- to the macroscale, in conjugation with the dynamic biophysical/biochemical stimuli underlying biological processes, has inspired the design of sophisticated hydrogels and nanoparticle systems exhibiting stimuli-responsive features. Recently, hydrogels and nanoparticles have been combined in advanced nanocomposite hybrid platforms expanding their range of biomedical applications. The ease and flexibility of attaining modular nanocomposite hydrogel constructs by selecting different classes of nanomaterials/hydrogels, or tuning nanoparticle-hydrogel physicochemical interactions widely expands the range of attainable properties to levels beyond those of traditional platforms. This review showcases the intrinsic ability of hybrid constructs to react to external or internal/physiological stimuli in the scope of developing sophisticated and intelligent systems with application-oriented features. Moreover, nanoparticle-hydrogel platforms are overviewed in the context of encoding stimuli-responsive cascades that recapitulate signaling interplays present in native biosystems. Collectively, recent breakthroughs in the design of stimuli-responsive nanocomposite hydrogels improve their potential for operating as advanced systems in different biomedical applications that benefit from tailored single or multi-responsiveness.