Кьороглу і Худий Мемед: парадигма національного героя та авторська свідомість

Стаття присвячена висвітленню загальних тенденцій використання продуктивного фольклоризму відомим турецьким письменником Яшаром Кемалем у його романі “Худий Мемед” та оповіданні “Поява Кьороглу”.Статья посвящена исследованию общих тенденций использования производительного фольклоризма известным турецким писателем Яшаром Кемалем в его романе “Тощий Мемед” и рассказе “Появление Кёроглу”.The following article is devoted to illustrate general tendencies of the usage of productive folklore by the well-known Turkish writer Yashar Kemal in his novel “Memed, My Hawk” and his story “Appearance of Köroğlu”

Constructive pointfree topology eliminates non-constructive representation theorems from Riesz space theory

In Riesz space theory it is good practice to avoid representation theorems which depend on the axiom of choice. Here we present a general methodology to do this using pointfree topology. To illustrate the technique we show that almost f-algebras are commutative. The proof is obtained relatively straightforward from the proof by Buskes and van Rooij by using the pointfree Stone-Yosida representation theorem by Coquand and Spitters

Bohrification of operator algebras and quantum logic

Following Birkhoff and von Neumann, quantum logic has traditionally been based on the lattice of closed linear subspaces of some Hilbert space, or, more generally, on the lattice of projections in a von Neumann algebra A. Unfortunately, the logical interpretation of these lattices is impaired by their nondistributivity and by various other problems. We show that a possible resolution of these difficulties, suggested by the ideas of Bohr, emerges if instead of single projections one considers elementary propositions to be families of projections indexed by a partially ordered set C(A) of appropriate commutative subalgebras of A. In fact, to achieve both maximal generality and ease of use within topos theory, we assume that A is a so-called Rickart C*-algebra and that C(A) consists of all unital commutative Rickart C*-subalgebras of A. Such families of projections form a Heyting algebra in a natural way, so that the associated propositional logic is intuitionistic: distributivity is recovered at the expense of the law of the excluded middle. Subsequently, generalizing an earlier computation for n-by-n matrices, we prove that the Heyting algebra thus associated to A arises as a basis for the internal Gelfand spectrum (in the sense of Banaschewski-Mulvey) of the "Bohrification" of A, which is a commutative Rickart C*-algebra in the topos of functors from C(A) to the category of sets. We explain the relationship of this construction to partial Boolean algebras and Bruns-Lakser completions. Finally, we establish a connection between probability measure on the lattice of projections on a Hilbert space H and probability valuations on the internal Gelfand spectrum of A for A = B(H).Comment: 31 page

Glucose Gradients Influence Zonal Matrix Deposition in 3D Cartilage Constructs

Reproducing the native collagen structure and glycosaminoglycan (GAG) distribution in tissue-engineered cartilage constructs is still a challenge. Articular cartilage has a specific nutrient supply and mechanical environment due to its location and function in the body. Efforts to simulate this native environment have been reported through the use of bioreactor systems. However, few of these devices take into account the existence of gradients over cartilage as a consequence of the nutrient supply by diffusion. We hypothesized that culturing chondrocytes in an environment, in which gradients of nutrients can be mimicked, would induce zonal differentiation. Indeed, we show that glucose gradients facilitating a concentration distribution as low as physiological glucose levels enhanced a zonal chondrogenic capacity similar to the one found in native cartilage. Furthermore, we found that the glucose consumption rates of cultured chondrocytes were higher under physiological glucose concentrations and that GAG production rates were highest in 5 mM glucose. From these findings, we concluded that this condition is better suited for matrix deposition compared to 20 mM glucose standard used in a chondrocyte culture system. Reconsidering the culture conditions in cartilage tissue engineering strategies can lead to cartilaginous constructs that have better mechanical and structural properties, thus holding the potential of further enhancing integration with the host tissue

The Principle of General Tovariance

We tentatively propose two guiding principles for the construction of theories of physics, which should be satisfied by a possible future theory of quantum gravity. These principles are inspired by those that led Einstein to his theory of general relativity, viz. his principle of general covariance and his equivalence principle, as well as by the two mysterious dogmas of Bohr's interpretation of quantum mechanics, i.e. his doctrine of classical concepts and his principle of complementarity. An appropriate mathematical language for combining these ideas is topos theory, a framework earlier proposed for physics by Isham and collaborators.Our principle of general tovariance states that any mathematical structure appearing in the laws of physics must be definable in an arbitrary topos (with natural numbers object) and must be preserved under so-called geometric morphisms. This principle identifies geometric logic as the mathematical language of physics and restricts the constructions and theorems to those valid in intuitionism: neither Aristotle's principle of the excluded third nor Zermelo's Axiom of Choice may be invoked. Subsequently, our equivalence principle states that any algebra of observables (initially defined in the topos Sets) is empirically equivalent to a commutative one in some other topos

The Expectation Monad in Quantum Foundations

The expectation monad is introduced abstractly via two composable adjunctions, but concretely captures measures. It turns out to sit in between known monads: on the one hand the distribution and ultrafilter monad, and on the other hand the continuation monad. This expectation monad is used in two probabilistic analogues of fundamental results of Manes and Gelfand for the ultrafilter monad: algebras of the expectation monad are convex compact Hausdorff spaces, and are dually equivalent to so-called Banach effect algebras. These structures capture states and effects in quantum foundations, and also the duality between them. Moreover, the approach leads to a new re-formulation of Gleason's theorem, expressing that effects on a Hilbert space are free effect modules on projections, obtained via tensoring with the unit interval.Comment: In Proceedings QPL 2011, arXiv:1210.029

A topos for algebraic quantum theory

The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr's idea that the empirical content of quantum physics is accessible only through classical physics, we show how a C*-algebra of observables A induces a topos T(A) in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum S(A) in T(A), which in our approach plays the role of a quantum phase space of the system. Thus we associate a locale (which is the topos-theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*-algebra (which is the noncommutative notion of a space). In this setting, states on A become probability measures (more precisely, valuations) on S(A), and self-adjoint elements of A define continuous functions (more precisely, locale maps) from S(A) to Scott's interval domain. Noting that open subsets of S(A) correspond to propositions about the system, the pairing map that assigns a (generalized) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by A is essentially turned into a classical theory, internal to the topos T(A).Comment: 52 pages, final version, to appear in Communications in Mathematical Physic