In 2007, Andrews introduced Durfee symbols and k-marked Durfee symbols so as to give a combinatorial interpretation for the symmetrized moment function $\eta _{2k}(n)$ of ranks of partitions. He also considered the relations between odd Durfee symbols and the mock theta function $\omega (q)$, and proved that the $2k$th moment function $\eta _{2k}^0(n)$ of odd ranks of odd Durfee symbols counts $(k+1)$-marked odd Durfee symbols of n. In this paper, we first introduce the definition of symmetrized positive odd rank moments $\eta _k^{0+}(n)$ and prove that for all $1\leq i\leq k+1$, $\eta _{2k-1}^{0+}(n)$ is equal to the number of $(k+1)$-marked odd Durfee symbols of n with the ith odd rank equal to zero and $\eta _{2k}^{0+}(n)$ is equal to the number of $(k+1)$-marked Durfee symbols of n with the ith odd rank being positive. Then we calculate the generating functions of $\eta _{k}^{0+}(n)$ and study its asymptotic behavior. Finally, we use Wright’s variant of the Hardy–Ramanujan circle method to obtain an asymptotic formula for $\eta _{k}^{0+}(n)$.

We propose a model of international oligopoly with two countries, two vertically-differentiated goods, and heterogeneous consumers in terms of their willingness to pay for quality. Various sources of pollution are taken into account: consumption, production and the transportation of goods between the two countries. Green persuaded consumers display consumption home bias: they derive additional satisfaction when consuming a domestic good because buying locally abates transportation pollution. We investigate whether consumption home bias effectively curbs global emissions. Finally, we uncover the environmental role played by the globalization of markets.

This article investigates continuities and changes in abbreviation practices from late Middle English to twenty-first-century digital platforms. Adopting a diachronic perspective and lexicological framework, it quantitatively analyses frequency patterns across fifteenth-century memoranda, letters and administrative receipts, seventeenth-century letters and depositions, late nineteenth-century letters, early twentieth-century letters and a subcorpus of WhatsApp instant messages dating from 2018–19. It then presents analyses of the frequencies of various abbreviation forms, such as clippings, and abbreviated lexemes, such as their use for names, over time. The article then provides a qualitative analysis of these lexeme categories over the centuries, with a focus on specific examples. Major changes to overall abbreviation density across time are identified. The forms of abbreviation also go through major change, but the types of lexemes that are abbreviated stay more consistent over time. For example, abbreviations being used for closed-class function words such as the and that are dominant from the earliest data we have looked at to the present day. Overall, the study demonstrates how situating new media abbreviation practices within a historical continuum can enhance our understanding of them.

Theories of vigilantism frequently locate its cause in the ineffectiveness of the state legal system. The state legal system does not work for some, so they take the law into their own hands. Even in communities where vigilantism is prevalent, however, only certain harms are met with vigilantism. Why are some harms met with violence while others are not? To explore this question, I draw on nine months of fieldwork in the eastern Democratic Republic of Congo, a nation with a largely ineffective state legal system and the presence of vigilantism. Yet vigilante remedies are largely reserved for harms attributed to witchcraft, and typically only particularly serious witchcraft harms. In this article I draw on the concept of legal consciousness to understand why people might turn to vigilantism in response to witchcraft. I develop two themes of legal consciousness—a disconnect in harm naming and different logics of justice—that help explain dynamics vigilantism. My research thus contributes to the scholarship on both vigilantism and legal consciousness.

Considerable research has examined Turkey’s discursive governance of the Syrian refugee crisis, identifying the central themes and metaphors in top officials’ refugee-related messages. However, since they tend to rely on qualitative analyses based on convenience or purposive samples, prior studies have failed to assess the relative frequency of these themes and fall short of reliably gauging the shifts and continuities in the official discourse on refugees. Moreover, while several studies have noted the growing emphasis on the repatriation of Syrian refugees in recent years, no research has yet explored how the Turkish government has sought to reconcile this with its pro-refugee posturing. This paper addresses these limitations via a mixed methods analysis of 382 speeches President Erdoğan gave from September 2014 through December 2022. Quantitative findings show that Islamist and neo-Ottomanist themes have played a major role in Erdoğan’s refugee discourse throughout his presidency. However, since 2018, there has been a sharp increase in Erdoğan’s remarks about repatriating Syrian refugees. A critical discourse analysis of these remarks indicates that Erdoğan has appropriated the language of international law and standards on refugee returns so that he can continue to claim the moral high ground while simultaneously advocating mass repatriation of the Syrians.

We prove topological regularity results for isoperimetric sets in PI spaces having a suitable deformation property, which prescribes a control on the increment of the perimeter of sets under perturbations with balls. More precisely, we prove that isoperimetric sets are open, satisfy boundary density estimates and, under a uniform lower bound on the volumes of unit balls, are bounded. Our results apply, in particular, to the class of possibly collapsed $\mathrm {RCD}(K,N)$ spaces. As a consequence, the rigidity in the isoperimetric inequality on possibly collapsed $\mathrm {RCD}(0,N)$ spaces with Euclidean volume growth holds without the additional assumption on the boundedness of isoperimetric sets. Our strategy is of interest even in the Euclidean setting, as it simplifies some classical arguments.

We prove that there exist infinitely many coprime numbers a, b, c with $a+b=c$ and $c>\operatorname {\mathrm {rad}}(abc)\exp (6.563\sqrt {\log c}/\log \log c)$. These are the most extremal examples currently known in the $abc$ conjecture, thereby providing a new lower bound on the tightest possible form of the conjecture. Our work builds on that of van Frankenhuysen (J. Number Theory 82(2000), 91–95) who proved the existence of examples satisfying the above bound with the constant $6.068$ in place of $6.563$. We show that the constant $6.563$ may be replaced by $4\sqrt {2\delta /e}$ where $\delta $ is a constant such that all unimodular lattices of sufficiently large dimension n contain a nonzero vector with $\ell _1$-norm at most $n/\delta $.

Suppose that $f:X\to C$ is a general Jacobian elliptic surface over ${\mathbb {C}}$ of irregularity $q$ and positive geometric genus $h$. Assume that $10 h>12(q-1)$, that $h>0$ and let $\overline {\mathcal {E}\ell \ell }$ denote the stack of generalized elliptic curves. (1) The moduli stack $\mathcal {JE}$ of such surfaces is smooth at the point $X$ and its tangent space $T$ there is naturally a direct sum of lines $(v_a)_{a\in Z}$, where $Z\subset C$ is the ramification locus of the classifying morphism $\phi :C\to \overline {\mathcal {E}\ell \ell }$ that corresponds to $X\to C$. (2) For each $a\in Z$ the map $\overline {\nabla }_{v_a}:H^{2,0}(X)\to H^{1,1}_{\rm prim}(X)$ defined by the derivative $per_*$ of the period map $per$ is of rank one. Its image is a line ${\mathbb {C}}[\eta _a]$ and its kernel is $H^0(X,\Omega ^2_X(-E_a))$, where $E_a=f^{-1}(a)$. (3) The classes $[\eta _a]$ form an orthogonal basis of $H^{1,1}_{\rm prim}(X)$ and $[\eta _a]$ is represented by a meromorphic $2$-form $\eta _a$ in $H^0(X,\Omega ^2_X(2E_a))$ of the second kind. (4) We prove a local Schottky theorem; that is, we give a description of $per_*$ in terms of a certain additional structure on the vector bundles that are involved. Assume further that $8h>10(q-1)$ and that $h\ge q+3$. (5) Given the period point $per(X)$ of $X$ that classifies the Hodge structure on the primitive cohomology $H^2_{\rm prim}(X)$ and the image of $T$ under $per_*$ we recover $Z$ as a subset of ${\mathbb {P}}^{h-1}$ and then, by quadratic interpolation, the curve $C$. (6) We prove a generic Torelli theorem for these surfaces. Everything relies on the construction, via certain kinds of Schiffer variations of curves, of certain variations of $X$ for which $per_*$ can be calculated. (In an earlier version of this paper we used variations constructed by Fay. However, Schiffer variations are slightly more powerful.)

Positive solitude (PS), the choice of being alone to engage in meaningful inner or physical, spiritual, mental, or cognitive activity/ experience, was recently suggested as a stand-alone phenomenon differentiated from loneliness and negative solitude. As loneliness was previously found to have adverse implications for mental health, the present study examined whether the ability to engage in PS can moderate the harmful effect of loneliness on depressive symptoms. The sample consisted of 520 community-dwelling older adults in Israel aged 68–87 (Mage = 72.66). Participants answered an online questionnaire through a survey company (Ipanel) assessing their background characteristics, depressive symptoms, loneliness, and PS. Loneliness was positively associated with depressive symptoms, whereas PS was negatively associated with depressive symptoms. Furthermore, PS moderated the relationship between loneliness and depressive symptoms, such that higher levels of PS weakened this association. The findings indicate that PS may serve as a buffering factor for mental health among older adults by augmenting coping with the adverse outcomes of loneliness. The results provide insight for tailoring future treatment interventions focusing on PS to enhance mental health among older adults.

In 1968, Steinberg [Endomorphisms of Linear Algebraic Groups, Memoirs of the American Mathematical Society, 80 (American Mathematical Society, Providence, RI, 1968)] proved a theorem stating that the exterior powers of an irreducible reflection representation of a Euclidean reflection group are again irreducible and pairwise nonisomorphic. We extend this result to a more general context where the inner product invariant under the group action may not necessarily exist.