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The aim of this paper is to study the families of normal and φ-normal functions on the unit disk 𝔻, and to generalize some normal function criteria of Xu and Qiu [An avoidance criterion for normal functions, C. R. Math 349(2011), 1159-1160] and Yang [A note on the avoidance criterion for normal functions, Anal. Math. Phys. 10, 35(2020)] to the case where derivatives are bounded from above on zero sets.

Stroke is the leading cause of death and long-term disability worldwide, and cognitive impairment and dementia are major complications of ischemic stroke. Cystatin C (CysC) has been found to be a neuroprotective factor in animal studies. However, the relationship between CysC levels and cognitive dysfunction in previous studies has revealed different results. This prospective observational study investigated the correlation between serum CysC levels and post-stroke cognitive dysfunction at 3 months. Data from 638 patients were obtained from the China Antihypertensive Trial in Acute Ischemic Stroke (CATIS). Cognitive dysfunction was assessed using the Mini-Mental State Examination (MMSE) at 3 months after stroke. According to the MMSE score, 308 patients (52.9%) had post-stroke cognitive dysfunction. After adjusting for potential confounding factors, the odds ratio (95% CI) of post-stroke cognitive dysfunction for the highest quartile of serum CysC levels was 0.54 (0.30-0.98), compared with the lowest quartile. The correlation between serum CysC and cognitive dysfunction was modified by renal function status. We observed a negative linear dose-response correlation between CysC and cognitive dysfunction in patients with normal renal function (Plinearity = 0.044), but not in those with abnormal renal function. Elevated serum CysC levels were correlated with a low risk of 3-month cognitive dysfunction in patients with acute ischemic stroke, especially in those with normal renal function. The current results suggest that CysC is a protective factor for post-stroke cognitive dysfunction, and could be used to treat post-stroke cognitive dysfunction. The CATIS study was approved by the Institutional Review Boards at Soochow University from China (approval No. 2012-02) on December 30, 2012, and was registered at ClinicalTrials.gov (identifier No. NCT01840072) on April 25, 2013.

We study the Feynman integral for the three-banana graph defined as the scalar two-point self-energy at three-loop order. The Feynman integral is evaluated for all identical internal masses in two space-time dimensions. Two calculations are given for the Feynman integral: one based on an interpretation of the integral as an inhomogeneous solution of a classical Picard–Fuchs differential equation, and the other using arithmetic algebraic geometry, motivic cohomology, and Eisenstein series. Both methods use the rather special fact that the Feynman integral is a family of regulator periods associated to a family of $K3$ surfaces. We show that the integral is given by a sum of elliptic trilogarithms evaluated at sixth roots of unity. This elliptic trilogarithm value is related to the regulator of a class in the motivic cohomology of the $K3$ family. We prove a conjecture by David Broadhurst which states that at a special kinematical point the Feynman integral is given by a critical value of the Hasse–Weil $L$-function of the $K3$ surface. This result is shown to be a particular case of Deligne’s conjectures relating values of $L$-functions inside the critical strip to periods.

In certain stages of the application of a type-2 fuzzy logic system, it is necessary to perform operations between input or output fuzzy variables in order to compute the union, intersection, aggregation, complement, and so forth. In this context, operators that satisfy the axioms of t-norms and t-conorms are of particular significance, as they are applied to model intersection and union, respectively. Furthermore, the existence of a range of these operators allows for the selection of the t-norm or t-conorm that offers the optimal performance, in accordance with the specific context of the system. In this paper, we obtain new t-norms and t-conorms on some important subfamilies of the set of functions from [0,1] to [0,1]. The structure of these families provides a more solid algebraic foundation for the applications. In particular, we define these new operators on the subsets of the functions that are convex, normal, and normal and convex, as well as the functions taking only the values 0 or 1 and the subset of functions whose support is a finite union of closed intervals. These t-norms and t-conorms are generalized to the type-2 fuzzy set framework.

This article aims at finding sufficient conditions for a family of meromorphic functions to be normal by involving partial sharing of sets with certain differential polynomials. The corresponding results for normal meromorphic functions are also established which improve and generalize many known results. Moreover, sufficient examples are provided to demonstrate sharpness of results.

With the exception of infant growth, there are no well-defined parameters describing normal human lactation. This represents a major gap in the continuum of care that does not exist for other major organs. Biological normality occurs naturally and is characterized by well-integrated function. We have proposed a definition that highlights four key elements that describe parameters for biological normality: comfort, milk supply, infant health, and maternal health. Notwithstanding the current limitations, published data have been collated to provide preliminary markers for the initiation of lactation and to describe objective tests once lactation is established. Reference limits have been calculated for maternal markers of secretory activation, including progesterone in maternal blood and total protein, lactose, sodium, and citrate in maternal milk. Objective measurements for established lactation, including 3-hourly pumping and 24-hour milk production, together with pre-feed to post-feed milk fat changes (a useful indicator of the available milk removed by the infant) have been outlined. Considered together with the parameters describing normal function, this information provides a preliminary objective framework for the assessment of human lactation.

We study normal holomorphic mappings on complex spaces and complex manifolds. Applications are provided.

Cancer biology features the ascription of normal functions to parts of cancers. At least some ascriptions of function in cancer biology track local normality of parts within the global abnormality of the aberration to which those parts belong. That is, cancer biologists identify as functions activities that, in some sense, parts of cancers are supposed to perform, despite cancers themselves having no purpose. The present paper provides a theory to accommodate these normal function ascriptions—I call it the Modeling Account of Normal Function (MA). MA comprises two claims. First, normal functions are activities whose performance by the function-bearing part contributes to the self-maintenance of the whole system and, thereby, results in the continued presence of that part. Second, MA holds that there is a class of models of system-level activities (partly) constitutive of self-maintenance members of which are improved by including a representation of the relevant function-bearing part and by making reference to the activit or/activities which that part performs which contribute(s) to those system-level activities. I contrast MA with two other accounts that seek to explicate the ascription of normal functions in biology, namely, the organizational account and the selected effects account. Both struggle to extend to cancer biology. However, I offer ecumenical readings which allow them to recover some ascriptions of normal function to parts of cancers. So, although I contend that MA excels in this respect, the purpose of this paper is served if it provides materials for bridging the gap between cancer biology, philosophy of cancer, and the literature on function.

In most areas of clinical and preclinical research, the required sample size determines the costs and effort for any project, and thus, optimizing sample size is of primary importance. An experimental design of dose–response studies is determined by the number and choice of dose levels as well as the allocation of sample size to each level. The experimental design of toxicological studies tends to be motivated by convention. Statistical optimal design theory, however, allows the setting of experimental conditions (dose levels, measurement times, etc.) in a way which minimizes the number of required measurements and subjects to obtain the desired precision of the results. While the general theory is well established, the mathematical complexity of the problem so far prevents widespread use of these techniques in practical studies. The paper explains the concepts of statistical optimal design theory with a minimum of mathematical terminology and uses these concepts to generate concrete usable D-optimal experimental designs for dose–response studies on the basis of three common dose–response functions in toxicology: log-logistic, log-normal and Weibull functions with four parameters each. The resulting designs usually require control plus only three dose levels and are quite intuitively plausible. The optimal designs are compared to traditional designs such as the typical setup of cytotoxicity studies for 96-well plates. As the optimal design depends on prior estimates of the dose–response function parameters, it is shown what loss of efficiency occurs if the parameters for design determination are misspecified, and how Bayes optimal designs can improve the situation.

A function (meromorphic or harmonic) from the hyperbolic disk of the complex plane to the Riemann sphere is normal if its dilatation is bounded. In this paper, normal families of meromorphic functions is studied in view of sharing values by their differential monomials. Moreover, several properties of normal harmonic mappings are also studied in view of certain general settings.