One variant of the q-Catalan polynomials is defined in terms of Gaussian polynomials by

$$ \begin{align*}\mathcal{C}_k(q)=\bigg[\begin{matrix}{2k}\\ {k}\end{matrix}\bigg]_q-q \bigg[\begin{matrix}{2k}\\ {k+1} \end{matrix}\bigg]_q.\end{align*} $$

Liu [‘On a congruence involving q-Catalan numbers’, C. R. Math. Acad. Sci. Paris 358 (2020), 211–215] studied congruences of the form $\sum _{k=0}^{n-1} q^k\mathcal {C}_k$ modulo the cyclotomic polynomial $\Phi _n(q)^2$, provided that $n\equiv \pm 1\pmod 3$. Apparently, the case $n\equiv 0\pmod 3$ has been missing from the literature. Our primary purpose is to fill this gap. In addition, we discuss a certain fascinating link to Dirichlet character sum identities.

Vitamin B12, cobalamin, is indispensable for humans owing to its participation in two biochemical reactions: the conversion of l-methylmalonyl coenzyme A to succinyl coenzyme A, and the formation of methionine by methylation of homocysteine. Eukaryotes, encompassing plants, fungi, animals and humans, do not synthesise vitamin B12, in contrast to prokaryotes. Humans must consume it in their diet. The most important sources include meat, milk and dairy products, fish, shellfish and eggs. Due to this, vegetarians are at risk to develop a vitamin B12 deficiency and it is recommended that they consume fortified food. Vitamin B12 behaves differently to most vitamins of the B complex in several aspects, e.g. it is more stable, has a very specific mechanism of absorption and is stored in large amounts in the organism. This review summarises all its biological aspects (including its structure and natural sources as well as its stability in food, pharmacokinetics and physiological function) as well as causes, symptoms, diagnosis (with a summary of analytical methods for its measurement), prevention and treatment of its deficiency, and its pharmacological use and potential toxicity.

The release of ICD-11 has resulted in an expansion of diagnostic entities for trauma- and stress-related disorders. This resulted, at least temporarily, in discrepancies with the DSM-5. This situation is outlined and a look is taken at the potential diagnosis of ‘continuous traumatic stress reaction’.

This article challenges the view that canon law was insignificant in the development of tenth-century English administrative and judicial institutions through a new study of Oda of Canterbury's Constitutiones, an important but neglected episcopal capitulary. Particular attention is paid to Oda's sources, the text's place in the legislative programme of King Edmund and the influence of wider European approaches to episcopal justice. The article shows that Oda's statutes endorsed an emerging system of collaborative justice between secular and ecclesiastical elites, thus demonstrating that tenth-century English governance was informed by a wider range of normative legal traditions than usually thought.

Stegenga (forthcoming) formulates and defends a novel account of scientific progress, according to which science makes progress just in case there is a change in scientific justification. Here, we present several problems for Stegenga’s account, concerning, respectively, (i) obtaining misleading evidence, (ii) losses or destruction of evidence, (iii) oscillations in scientific justification, and (iv) the possibility of scientific regress. We conclude by sketching a substantially different justification-based account of scientific progress that avoids these problems.

After reading these volumes one can hardly conclude otherwise than that the study of urban space in Roman Italy is thriving. These large volumes, collectively exceeding 1,800 pages and comprising 86 chapters/articles (in addition to supplementary materials), include contributions in four languages – though, perhaps surprisingly, none contain contributions in German. The array of contributors suggests that even more modern languages could have been included, if desired. The authors range from early career scholars to renowned professors.

This paper distinguishes news about short-lived events from news about changes in longer term prospects using surveys of expectations. Employing a multivariate GARCH-in-Mean model for the US, the paper illustrates how the different types of news influence business cycle dynamics. The influence of transitory output shocks can be relatively large on impact but gradually diminishes over two to three years. Permanent shocks drive the business cycle, generating immediate stock price reactions and gradually building output effects, although they have more immediate output effects during recessions through the uncertainties they create. Markedly different macroeconomic dynamics are found if these explicitly identified types of news or uncertainty feedbacks are omitted from the analysis.

The English Preposing in PP construction (PiPP; e.g., Happy though/as we were) is extremely rare but displays an intricate set of stable syntactic properties. How do people become proficient with this construction despite such limited evidence? It is tempting to posit innate learning mechanisms, but present-day large language models seem to learn to represent PiPPs as well, even though such models employ only very general learning mechanisms and experience very few instances of the construction during training. This suggests an alternative hypothesis on which knowledge of more frequent constructions helps shape knowledge of PiPPs. I seek to make this idea precise using model-theoretic syntax (MTS). In MTS, a grammar is essentially a set of constraints on forms. In this context, PiPPs can be seen as arising from a mix of construction-specific and general-purpose constraints, all of which seem inferable from general linguistic experience.

Using properties of Ramanujan’s theta functions, we give an elementary proof of Hirschhorn’s conjecture on $2^n$-dissection of Euler’s product $E(q):=(q;q)_\infty $.

The western conifer seed bug, Leptoglossus occidentalis Heidemann (Hemiptera: Coreidae), feeds extensively on the seeds of conifer trees within both its endemic range west of the Rocky Mountains and its expanding introduced range throughout eastern North America, South America, Europe, and Asia. Its damage to conifer seed challenges seed production in natural stands and in orchards that produce seed for forest regeneration and food. In summer 2023, hundreds of L. occidentalis adults were observed feeding on the berries of two cultivated blackberry, Rubus fruticosus Linnaeus (Rosaceae), vines on southern Vancouver Island, British Columbia, Canada. The adults were observed over a period of approximately six weeks, and their stylets were seen penetrating drupelets and causing subsequent damage. Identification was confirmed based on morphological and molecular (CO1 DNA barcoding) methods. To our knowledge, this is the first verifiable report of L. occidentalis feeding on a Rosaceae host, suggesting the insect’s potential for damaging agricultural systems.

Let n be a nonzero integer. A set S of positive integers is a Diophantine tuple with the property $D(n)$ if $ab+n$ is a perfect square for each $a,b \in S$ with $a \neq b$. It is of special interest to estimate the quantity $M_n$, the maximum size of a Diophantine tuple with the property $D(n)$. We show the contribution of intermediate elements is $O(\log \log |n|)$, improving a result by Dujella [‘Bounds for the size of sets with the property $D(n)$’, Glas. Mat. Ser. III 39(59)(2) (2004), 199–205]. As a consequence, we deduce that $M_n\leq (2+o(1))\log |n|$, improving the best known upper bound on $M_n$ by Becker and Murty [‘Diophantine m-tuples with the property $D(n)$’, Glas. Mat. Ser. III 54(74)(1) (2019), 65–75].