New Results on the Aggregation of Norms

It is a natural question if a Cartesian product of objects produces an object of the same type. For example, it is well known that a countable Cartesian product of metrizable topological spaces is metrizable. Related to this question, Borsík and Doboš characterized those functions that allow obtaining a metric in the Cartesian product of metric spaces by means of the aggregation of the metrics of each factor space. This question was also studied for norms by Herburt and Moszyńska. This aggregation procedure can be modified in order to construct a metric or a norm on a certain set by means of a family of metrics or norms, respectively. In this paper, we characterize the functions that allow merging an arbitrary collection of (asymmetric) norms defined over a vector space into a single norm (aggregation on sets). We see that these functions are different from those that allow the construction of a norm in a Cartesian product (aggregation on products). Moreover, we study a related topological problem that was considered in the context of metric spaces by Borsík and Doboš. Concretely, we analyze under which conditions the aggregated norm is compatible with the product topology or the supremum topology in each case.

The urology 2021 curriculum

A new urology curriculum has been approved by the General Medical Council and will be rolled out from August 2021. Major changes include phasing of training with a hard checkpoint at ST5 that facilitates focussed development on core skills during phase 2 (ST3–5) and the development of a special interest during phase 3 (ST6–7). While the syllabus remains unchanged, supervision levels will replace many of the workplace-based assessments currently used in wider clinical practice, learning agreements will transition from placement to placement, and indicative numbers have been replaced with index procedures, in a move from box-ticking to a professional assessment of competency and totality of practice. While change can be daunting, the 2021 urology curriculum can be viewed positively as a move away from the arbitrary collection of records of workplace-based assessments and exposure to indicative numbers, and towards the holistic professional judgements of competency that are more in keeping with lifelong consultant practice. Level of evidence: Not applicable.

Identifiability from a Combination of Observations and Experiments

We study the problem of causal identification from an arbitrary collection of observational and experimental distributions, and substantive knowledge about the phenomenon under investigation, which usually comes in the form of a causal graph. We call this problem g-identifiability, or gID for short. In this paper, we introduce a general strategy to prove non-gID based on thickets and hedgelets, which leads to a necessary and sufficient graphical condition for the corresponding decision problem. We further develop a procedure for systematically computing the target effect, and prove that it is sound and complete for gID instances. In other words, the failure of the algorithm in returning an expression implies that the target effect is not computable from the available distributions. Finally, as a corollary of these results, we show that do-calculus is complete for the task of g-identifiability.

The Meaninglessness of the Mean

The mean (average) or other central tendencies of a set of data is an internal construct that does not necessarily reflect reality. It is possible to determine the central tendency from any arbitrary collection of data as long as they vary on the same dimension. Even if applied to a relevant sample of data, the central tendency may be a poor reflection of data. A virtually infinite number of different collections of data may have the same central tendency and variance. This has very important implications when reasoning from studies reporting means and standard deviations. The same concerns apply to medians as the central tendencies and quartiles as the variability. When translating studies to the individual patient, the cumulative percentage (probability) function may be more helpful. There is a strong inclination to attribute some ontological status (reality) to measures of central tendency that can be misleading.

Philosophical Analysis; Referents of Names

Replacing empirical concepts with unicepts has implications both for philosophical methodology and for some central matters in philosophy of science, plilosophy of language, and philosophy of mind. This chapter gives illustrations that concern the fixing of referents of naming words in a public language, the method of philosophical analysis, referential constancy of names for theoretical objects over theory change, the distinction between so-called “observational concepts” and “theoretical concepts,” and last, so-called “theory of mind.” This is a somewhat arbitrary collection of apparent implications of embracing unicepts, but the discussions of the “observation-theory” distinction and of “theory of mind” will be needed when discussing both perception and the semantics-pragmatics distinction.

An Investigation of a Cyclic Pattern in Mark 4:35–8:21 and Its Theological Significance

This article explores the narrative structure of Mark 4:35–8:21 based on geographical arrangement, literary parallelism, and boat and bread motifs, identifying two cycles (4:35–6:44 and 6:45–8:10) and a conclusion (8:11–21), and presents theological implications of its cyclic structure. The structure of Mark 4–8 seems to be loose in its arrangement of episodes and geographical descriptions. These problems, however, are caused not by Mark's arbitrary collection of sources but by his literary techniques and theological purposes. My proposed structure is based on geographical distinctions; it focuses mainly on tracing Jesus’ geographical movement composed by Mark's literary technique and motifs. In the former cycle, Jesus’ ministry was mostly for the Jews; in the latter cycle, for the Gentiles. A conclusion manifests Jesus as the mighty Savior of God's people. In a Christological view, Jesus’ missional (geographical) movement between west (Jewish territory) and east (Gentile territory) demonstrates his identity as the deliverer, healer, and feeder of both the Jews and the Gentles, regardless of their ethnicity.

A BILINEAR RUBIO DE FRANCIA INEQUALITY FOR ARBITRARY SQUARES

We prove the boundedness of a smooth bilinear Rubio de Francia operator associated with an arbitrary collection of squares (with sides parallel to the axes) in the frequency plane $$\begin{eqnarray}(f,g)\mapsto \biggl(\mathop{\sum }_{\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}}\biggl|\int _{\mathbb{R}^{2}}\hat{f}(\unicode[STIX]{x1D709}){\hat{g}}(\unicode[STIX]{x1D702})\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D714}}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})e^{2\unicode[STIX]{x1D70B}ix(\unicode[STIX]{x1D709}+\unicode[STIX]{x1D702})}\,d\unicode[STIX]{x1D709}\,d\unicode[STIX]{x1D702}\biggr|^{r}\biggr)^{1/r},\end{eqnarray}$$ provided $r>2$. More exactly, we show that the above operator maps $L^{p}\times L^{q}\rightarrow L^{s}$ whenever $p,q,s^{\prime }$ are in the ‘local $L^{r^{\prime }}$’ range, that is, $$\begin{eqnarray}\frac{1}{p}+\frac{1}{q}+\frac{1}{s^{\prime }}=1,\quad 0\leqslant \frac{1}{p},\frac{1}{q}<\frac{1}{r^{\prime }},\quad \text{and}\quad \frac{1}{s^{\prime }}<\frac{1}{r^{\prime }}.\end{eqnarray}$$ Note that we allow for negative values of $s^{\prime }$, which correspond to quasi-Banach spaces $L^{s}$.