The classification of preserved hypothalamic activity in brain death and brainstem death as functional or non-functional has become a subject of debate. While proponents of the neurological criterion claim that these activities lack functional significance (Shemie et al. 2014), Nair-Collins and Joffe (2023) argue for their functional physiological role. However, the interpretation of the term "function" within the medico-legal framework, where death is characterized by the irreversible cessation of all brain functions, remains unclear. -/- My intention here is not (...) to engage in a debate over the validity of different viewpoints regarding the functional significance of preserved activities in the hypothalamus or the heart. Rather, I aim to highlight the critical importance of exploring and defining the concept of function in medicine, which currently lacks clarity. My point is that, whatever our position in the brain death debate, reaching a consensus on this conceptual matter would minimize misunderstandings and establish a robust foundation for further insightful analysis. (shrink)

We study properties of definable sets and functions in power-bounded T -convex fields, proving that the latter have the multidimensional Jacobian property and that the theory of T -convex fields is b -minimal with centers. Through these results and work of I. Halupczok we ensure that a certain kind of geometrical stratifications exist for definable objects in said fields. We then discuss a number of applications of those stratifications, including applications to Archimedean o-minimal geometry.

We characterize elementary equivalences and inclusions between von Neumann regular real closed rings in terms of their boolean algebras of idempotents, and prove that their theories are always decidable. We then show that, under some hypotheses, the map sending an L-structure R to the L-structure of definable functions from R n to R preserves elementary inclusions and equivalences and gives a structure with a decidable theory whenever R is decidable. We briefly consider structures of definable functions (...) satisfying an extra condition such as continuity. (shrink)

We develop a notion of cell decomposition suitable for studying weak p-adic structures definable). As an example, we consider a structure with restricted addition.

We develop a version of Herbrand's theorem for continuous logic and use it to prove that definable functions in infinite-dimensional Hilbert spaces are piecewise approximable by affine functions. We obtain similar results for definable functions in Hilbert spaces expanded by a group of generic unitary operators and Hilbert spaces expanded by a generic subspace. We also show how Herbrand's theorem can be used to characterize definable functions in absolutely ubiquitous structures from classical logic.

Let [Formula: see text] be an expansion of either an ordered field [Formula: see text], or a valued field [Formula: see text]. Given a definable set [Formula: see text] let [Formula: see text] be the ring of continuous definable functions from [Formula: see text] to [Formula: see text]. Under very mild assumptions on the geometry of [Formula: see text] and on the structure [Formula: see text], in particular when [Formula: see text] is [Formula: see text]-minimal or [Formula: (...) see text]-minimal, or an expansion of a local field, we prove that the ring of integers [Formula: see text] is interpretable in [Formula: see text]. If [Formula: see text] is [Formula: see text]-minimal and [Formula: see text] is definably connected of pure dimension [Formula: see text], then [Formula: see text] defines the subring [Formula: see text]. If [Formula: see text] is [Formula: see text]-minimal and [Formula: see text] has no isolated points, then there is a discrete ring [Formula: see text] contained in [Formula: see text] and naturally isomorphic to [Formula: see text], such that the ring of functions [Formula: see text] which take values in [Formula: see text] is definable in [Formula: see text]. (shrink)

The subject of this paper is a characterization of the $\Sigma_1$ -definable set functions of Kripke-Platek set theory with infinity and a uniform version of axiom of choice: $KP\omega+(uniform\;AC)$ . This class of functions is shown to coincide with the collection of set functionals of type 1 primitive recursive in a given choice functional and $x\mapsto\omega$ . This goal is achieved by a Gödel Dialectica-style functional interpretation of $KP\omega+(uniform\;AC)$ and a computability proof for the involved functionals.

Due to several socio-political factors, to many psychiatrists only a strictly objective definition of mental disorder, free of value components, seems really acceptable. In this paper, I will explore a variant of such an objectivist approach to defining metal disorder, natural function objectivism. Proponents of this approach make recourse to the notion of natural function in order to reach a value-free definition of mental disorder. The exploration of Christopher Boorse's 'biostatistical' account of natural function (1) will be followed an investigation (...) of the 'hybrid naturalism' approach to natural functions by Jerome Wakefield (2). In the third part, I will explore two proposals that call into question the whole attempt to define mental disorder (3). I will conclude that while 'natural function objectivism' accounts fail to provide the backdrop for a reliable definition of mental disorder, there is no compelling reason to conclude that a definition cannot be achieved. (shrink)

Using coding devices based on a theorem due to Zsigmondy, Birkhoff and Vandiver, we first define in terms of successor S and coprimeness predicate $\perp$ a full arithmetic over the set of powers of some fixed prime, then we define in the same terms a restriction of the exponentiation. Hence we prove the main result insuring that all arithmetical relations and functions over prime powers and their opposite are $\{S, \perp\}$ -definable over Z. Applications to definability over Z (...) and N are stated as corollaries of the main theorem. (shrink)

A set F of Boolean functions is said to be functionally complete if every Boolean function is definable by combining functions in F. Post clarified when a set of Boolean functions is functionally complete (with respect to classical semantics). In this paper, by extending Post’s theorem, we clarify when a set of Boolean functions is functionally complete with respect to Kripke semantics.

The term functional domain is often used to describe the region containing the cis acting sequences that regulate a gene locus. “Strong” domain models propose that the domain is a spatially isolated entity consisting of a region of extended accessible chromatin bordered by insulators that have evolved to act as functional boundaries. However, the observation that independently regulated loci can overlap partially or completely raises questions about functional requirements for physically isolated domain structures. An alternative model, the “weak” domain model, (...) proposes that domain structure is determined by the distribution of binding sites for positively acting factors, without a requirement for functional boundaries. The domain would effectively be the region that contains these factor-binding sites. Specificity of promoter-enhancer interactions would play a major role in maintaining the functional autonomy of adjacent genes. Sequences that interfere with these interactions (frequently characterised as insulators) would be selected against if they occurred within the domain but not at the edges, or in the interdomain regions. As a result, insulators would often be found near the borders of domains without necessarily being selected to act as boundaries. BioEssays 22:657–665, 2000. © 2000 John Wiley & Sons, Inc. (shrink)

This paper studies the extent to which probability functions are recursively definable. It proves, in particular, that the (absolute) probability of a statement A is recursively definable from a certain point on, to wit: from the (absolute) probabilities of certain atomic components and conjunctions of atomic components of A on, but to no further extent. And it proves that, generally, the probability of a statement A relative to a statement B is recursively definable from a certain (...) point on, to wit: from the probabilities relative to that very B of certain atomic components and conjunctions of atomic components of A, but again to no further extent. These and other results are extended to the less studied case where A and B are compounded from atomic statements by means of `` ∀ '' as well as `` ∼ '' and "&". The absolute probability functions considered are those of Kolmogorov and Carnap, and the relative ones are those of Kolmogorov, Carnap, Renyi, and Popper. (shrink)

The paper builds on a simply typed term system ${\cal PR}^\omega $ providing a notion of partial primitive recursive functional on arbitrary Scott domains $D_\sigma$ that includes a suitable concept of parallelism. Computability on the partial continuous functionals seems to entail that Kleene's schema of higher type simultaneous course-of-values recursion (SCVR) is not reducible to partial primitive recursion. So an extension ${\cal PR}^{\omega e}$ is studied that is closed under SCVR and yet stays within the realm of subrecursiveness. The twist (...) are certain type 1 Gödel recursors ${\cal R}_k$ for simultaneous partial primitive recursion. Formally, the value ${\cal R}_k\vec{g}\vec{H} x i$ is a function $f_i$ of type $\iota \to \iota$ , however, ${\cal R}_k$ is modelled such that $f_i$ is a finite element of $D_{\iota\to\iota}$ or in other words, a partial sequence. It is shown that the Ackermann function is not definable in ${\cal PR}^{\omega e}$. (shrink)

Given any field K, there is a function field F/K in one variable containing definable transcendentals over K, i.e., elements in F \ K first-order definable in the language of fields with parameters from K. Hence, the model-theoretic and the field-theoretic relative algebraic closure of K in F do not coincide. E.g., if K is finite, the model-theoretic algebraic closure of K in the rational function field K(t) is K(t). For the proof, diophantine $\emptyset-definability$ of K in F (...) is established for any function field F/K in one variable, provided K is large, or $K^{x}\,/(K^{x})^n$ is finite for some integer n > 1 coprime to char K. (shrink)

The role played by real-valued functions in functional analysis is fundamental. One often considers metrics, or seminorms, or linear functionals, to mention some important examples. We introduce the notion of definable real-valued function in functional analysis: a real-valued function f defined on a structure of functional analysis is definable if it can be "approximated" by formulas which do not involve f. We characterize definability of real-valued functions in terms of a purely topological condition which does not (...) involve logic. (shrink)

Although the theory of definability had many important antecedents—such as the descriptive set theory initiated by the French semi-intuitionists in the early 1900s—the main ideas were first laid out in precise mathematical terms by Alfred Tarski beginning in 1929. We review here the basic notions of languages, explicit definability, and grammatical complexity, and emphasize common themes in the theories of definability for four important languages underlying, respectively, descriptive set theory, recursive function theory, classical pure logic, and finite-universe logic. We review (...) the history of previous studies of the similarities and differences in the theories of definability of the first three of these four languages. A seminal role leading toward unification of the theories has been played by the separation principles introduced by Nikolai Luzin in 1927. Emphasizing analogies and driving toward further unification embracing finite-universe logic we concentrate on a simple example—the first and second separation principles for existential-universal first-order sentences . Using this as a test case for the fundamental problem of how to “finitize” arguments in classical pure logic to the finite-universe case, we are led to the analogous negative solution by using the theory of certain special graphs: a graph is - special for any positive integers m , n , p , q iff it is bipartite with m red points and n blue points and for every p -tuple of red points there is a blue point to which they are all connected . As an aside we introduce for further study a natural “Ramseyesque” increasing sequence A of positive integers, where A is the least positive integer n for which an -special graph exists. (shrink)

Despite promising early work into the role of immune cells such as eosinophils in adipose tissue (AT) homeostasis, recent findings revealed that elevating the number of eosinophils in AT alone is insufficient for improving metabolic impairments in obese mice. Eosinophils are primarily recognized for their role in allergic immunity and defence against parasitic worms. They have also been detected in AT and appear to contribute to adipose homeostasis and drive energy expenditure, but the underlying mechanisms remain elusive. It has long (...) been recognized that immune cells such as macrophages respond to external signals to regulate adipose homeostasis and energy balance, however, less is known about the relevance of eosinophil activity in AT. As the authors propose in this review, given recent debate over the relative importance of their tissue‐specific abundance, the stage is now set for exploring the functionality and activation states of AT eosinophils. (shrink)

In this paper, we prove a definable version of Kirszbraun's theorem in a non‐Archimedean setting for definable families of functions in one variable. More precisely, we prove that every definable function, where and, that is λ‐Lipschitz in the first variable, extends to a definable function that is λ‐Lipschitz in the first variable.

We evaluated the reliability, validity, and differential item functioning (DIF) of a shorter version of the Defining Issues Test-1 (DIT-1), the behavioral DIT (bDIT), measuring the development of moral reasoning. 353 college students (81 males, 271 females, 1 not reported; age M = 18.64 years, SD = 1.20 years) who were taking introductory psychology classes at a public University in a suburb area in the Southern United States participated in the present study. First, we examined the reliability of the bDIT (...) using Cronbach’s α and its concurrent validity with the original DIT-1 using disattenuated correlation. Second, we compared the test duration between the two measures. Third, we tested the DIF of each question between males and females. Findings reported that first, the bDIT showed acceptable reliability and good concurrent validity. Second, the test duration could be significantly shortened by employing the bDIT. Third, DIF results indicated that the bDIT items did not favour any gender. Practical implications of the present study based on the reported findings are discussed. (shrink)

We prove the existence of a bound to the number of components of an $\forall$-definable set in the reals, using Pfaffian functions, and give some applications.

We prove the existence of a bound to the number of components of an ∀-definable set in the reals, using Pfaffian functions, and give some applications.