The Sometimes Context-Specific Habituation: Theoretical Challenges to Associative Accounts

A substantial corpus of experimental research indicates that in many species, long-term habituation appears to depend on context–stimulus associations. Some authors have recently emphasized that this type of outcome supports Wagner’s priming theory, which affirms that responding is diminished when the eliciting stimulus is predicted by the context where the animal encountered that stimulus in the past. Although we agree with both the empirical reality of the phenomenon as well as the principled adequacy of the theory, we think that the available evidence is more provocative than conclusive and that there are a few nontrivial empirical and theoretical issues that need to be worked out by researchers in the future. In this paper, we comment on these issues within the framework of a quantitative version of priming theory, the SOP model.

Lyapunov exponent of random dynamical systems on the circle

We consider products of an independent and identically distributed sequence in a set $\{f_1,\ldots ,f_m\}$ of orientation-preserving diffeomorphisms of the circle. We can naturally associate a Lyapunov exponent $\lambda $ . Under few assumptions, it is known that $\lambda \leq 0$ and that the equality holds if and only if $f_1,\ldots ,f_m$ are simultaneously conjugated to rotations. In this paper, we state a quantitative version of this fact in the case where $f_1,\ldots ,f_m$ are $C^k$ perturbations of rotations with rotation numbers $\rho (f_1),\ldots ,\rho (f_m)$ satisfying a simultaneous diophantine condition in the sense of Moser [On commuting circle mappings and simultaneous diophantine approximations. Math. Z.205(1) (1990), 105–121]: we give a precise estimate of $\lambda $ (Taylor expansion) and we prove that there exist a diffeomorphism g and rotations $r_i$ such that $\mbox {dist}(gf_ig^{-1},r_i)\ll |\lambda |^{{1}/{2}}$ for $i=1,\ldots , m$ . We also state analogous results for random products of $2\times 2$ matrices, without any diophantine condition.

Sharp Caffarelli–Kohn–Nirenberg inequalities on Riemannian manifolds: the influence of curvature

We first establish a family of sharp Caffarelli–Kohn–Nirenberg type inequalities (shortly, sharp CKN inequalities) on the Euclidean spaces and then extend them to the setting of Cartan–Hadamard manifolds with the same best constant. The quantitative version of these inequalities also is proved by adding a non-negative remainder term in terms of the sectional curvature of manifolds. We next prove several rigidity results for complete Riemannian manifolds supporting the Caffarelli–Kohn–Nirenberg type inequalities with the same sharp constant as in the Euclidean space of the same dimension. Our results illustrate the influence of curvature to the sharp CKN inequalities on the Riemannian manifolds. They extend recent results of Kristály (J. Math. Pures Appl. 119 (2018), 326–346) to a larger class of the sharp CKN inequalities.

Optimal three-ball inequality, quantitative uniqueness for the bi-Laplace equations

In this paper, we prove an optimal three-ball inequality for bi-Laplace equation in some open, connected set. The derivation of such estimate relies on a delicate Carleman estimate for the bi-Laplace equation and some Caccioppoli inequalities to estimate the lower-ters. Based on three -ball inequality, we then derive the vanishing order of solutions, which is a quantitative version of the strong unique continuation property.

Quantitative versions of almost squareness and diameter 2 properties

We introduce a quantitative version (using s ∈ 2 (0; 1]) of almost (local) squareness of Banach spaces. The latter concept (i.e., the s = 1 case) was introduced by Abrahamsen, Langemets, and Lima in 2016. Related diameter 2 properties (local, strong, and symmetric strong) are also relaxed correspondingly. Our note contains some (counter-)examples and results for the s-almost (local) squareness property.

A quantitative Lovász criterion for Property B

A well-known observation of Lovász is that if a hypergraph is not 2-colourable, then at least one pair of its edges intersect at a single vertex. In this short paper we consider the quantitative version of Lovász’s criterion. That is, we ask how many pairs of edges intersecting at a single vertex should belong to a non-2-colourable n-uniform hypergraph. Our main result is an exact answer to this question, which further characterizes all the extremal hypergraphs. The proof combines Bollobás’s two families theorem with Pluhar’s randomized colouring algorithm.

Predicting Regulatory Product Approvals Using a Proposed Quantitative Version of FDA’s Benefit–Risk Framework to Calculate Net-Benefit Score and Benefit–Risk Ratio

Background Approval of regulated medical products in the USA is based upon a rigorous review of the benefits and risks as performed by the US Food and Drug Administration (FDA) staff of scientists and is summarized in a descriptive and qualitative format called the FDA’s Benefit–Risk Framework (BRF). This present method highlights the key factors in regulatory decision-making, but does not clearly define the reason for its final approval. Method This study proposes a quantitative version of FDA’s BRF to calculate a Net-Benefit Score and a Benefit–Risk Ratio as a method to define a single-value summary of the tradeoffs between benefits and risks and allow comparisons among other products. In this retrospective review of five years of new molecular entities and new biologic (N = 185 products) regulatory decision-making, this proposed scoring system codifies and quantitates the information about a product’s benefits, risks, and risk management information in a format that may predict why regulated medical products are approved in the USA. Results Simple calculation of codified benefits, risks, and risk mitigations with numerical limits is proposed to provide a repeatable process and transparency for documenting the net-benefit of regulatory product approval. Conclusion Use of a strict process of collecting, codifying, and analyzing public information to determine a Net-Benefit score and a Benefit–Risk Ratio is possible to anticipate regulatory product approval.

The Kobayashi–Royden metric on punctured spheres

This paper gives an explicit formula of the asymptotic expansion of the Kobayashi–Royden metric on the punctured sphere {\mathbb{CP}^{1}\setminus\{0,1,\infty\}} in terms of the exponential Bell polynomials. We prove a local quantitative version of the Little Picard’s Theorem as an application of the asymptotic expansion. Furthermore, the approach in the paper leads to the interesting consequence that the coefficients in the asymptotic expansion are rational numbers. Furthermore, the explicit formula of the metric and the conclusion regarding the coefficients apply to more general cases of {\mathbb{CP}^{1}\setminus\{a_{1},\ldots,a_{n}\}}, {n\geq 3}, as well, and the metric on {\mathbb{CP}^{1}\setminus\{0,\frac{1}{3},-\frac{1}{6}\pm\frac{\sqrt{3}}{6}i\}} will be given as a concrete example of our results.