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We consider the wave maps problem with domain

This monograph presents a unified framework for determining the congruences on a number of monoids and categories of transformations, diagrams, matrices and braids, and on all their ideals. The key theoretical advances present an iterative process of stacking certain normal subgroup lattices on top of each other to successively build congruence lattices of a chain of ideals. This is applied to several specific categories of: transformations; order/orientation preserving/reversing transformations; partitions; planar/annular partitions; Brauer, Temperley–Lieb and Jones partitions; linear and projective linear transformations; and partial braids. Special considerations are needed for certain small ideals, and technically more intricate theoretical underpinnings for the linear and partial braid categories.

The weight multiplicities of finite dimensional simple Lie algebras can be computed individually using various methods. Still, it is hard to derive explicit closed formulas. Similarly, explicit closed formulas for the multiplicities of maximal weights of affine Kac–Moody algebras are not known in most cases. In this paper, we study weight multiplicities for both finite and affine cases of classical types for certain infinite families of highest weights modules. We introduce new classes of Young tableaux, called the

BACKGROUND: Loneliness is often treated as an individual or psychological experience, but it also reflects broader demographic and structural shifts. Population aging, declining fertility, migration, and shifting household arrangements have reshaped the ways individuals connect, care, and live. Despite its growing social relevance, loneliness remains under-theorized and under-measured in demographic research. OBJECTIVE: This essay outlines why loneliness should be understood as both a consequence and a driver of demographic change. It examines how relational disconnection interacts with fertility, mortality, and migration across the life course, and offers a demographic framework for studying loneliness as a population process. RESULTS: Loneliness emerges from demographic transitions (declining fertility, rising childlessness, increased longevity, and new migration regimes) that shape how people connect and care. In turn, loneliness influences demographic behaviors and outcomes, from reproductive intentions and union formation to survival, health, mobility, and return migration. These dynamics operate within cohorts and can accumulate across generations. CONCLUSIONS: Integrating loneliness into demographic models and data systems enhances the field’s ability to characterize population vulnerability and inform policy. Counting connection and disconnection alongside births, deaths, and moves improves our understanding on how people live and whether they thrive. CONTRIBUTION: This essay contributes to demographic theory by articulating a framework that situates loneliness as both a cause and a consequence of demographic change. It bridges literatures on demography, well-being, and population dynamics to propose a demographic lens for understanding social disconnection across different contexts and life stages.

Do political connections buffer firms from or bind firms to the government? To examine this theoretical puzzle, we distinguish two types of managerial political connections, ascribed and achieved , and theorize that these different types of ties either buffer firms from or bind firms to government demands. Furthermore, we propose that these effects are contingent on both industrial and regional institutional conditions. We test our framework with a unique panel data set of privately controlled listed firms’ charitable donations in China from 2001 to 2012. We find that firms whose executives have ascribed bureaucratic connections are more likely to use their connections as a buffer from governmental donation pressure, particularly in competitive industries and less market-oriented regions, whereas in state-monopolized industries this buffering effect is reduced. In contrast, achieved political connections are more likely to serve a binding function that facilitates donation, particularly in state-monopolized industries and more market-oriented regions, but in less market-oriented regions, they buffer firms from the pressure to donate. Our research contributes to the literatures on the effects of political connections, the institutional contingencies of political connections, and the relationship between corporate social responsibility (CSR) and corporate political activities (CPA).

The simplicial volume is a homotopy invariant of manifolds introduced by Gromov in his pioneering paper The first aim of this paper is to lay the foundation of the theory of multicomplexes. After setting the main definitions, we construct the singular multicomplex In the second part of this work we apply the theory of multicomplexes to the study of the bounded cohomology of topological spaces. Our constructions and arguments culminate in the complete proofs of Gromov’s Mapping Theorem (which implies in particular that the bounded cohomology of a space only depends on its fundamental group) and of Gromov’s Vanishing Theorem, which ensures the vanishing of the simplicial volume of closed manifolds admitting an amenable cover of small multiplicity. The third and last part of the paper is devoted to the study of locally finite chains on non-compact spaces, hence to the simplicial volume of open manifolds. We expand some ideas of Gromov to provide detailed proofs of a criterion for the vanishing and a criterion for the finiteness of the simplicial volume of open manifolds. As a by-product of these results, we prove a criterion for the

In this paper, we classify affine Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections and perturbed canonical connections and perturbed Kobayashi-Nomizu connections on three-dimensional Lorentzian Lie groups with some product structure.

A dualistic structure on a smooth Riemaniann manifold M is a triple (M,g,∇) with g a Riemaniann metric and ∇ an affine connection generally assumed to be torsionless. From g and ∇, dual connection ∇* can be defined. In this work, we give conditions on the basis of this notion for a manifold to admit an almost contact structure and some related structures: almost contact metric, contact, contact metric, cosymplectic, and co-Kähler in the three-dimensional case.

We introduce and study the bounded word problem and the precise word problem for groups given by means of generators and defining relations. For example, for every finitely presented group, the bounded word problem is in

Purpose Brands are investing heavily in content marketing within digital communication channels, yet there is limited understanding of the effectiveness of this content on consumer engagement. This paper aims to examine how consumer engagement with branded content is created through consumer-initiated online brand communities (OBCs) and brand-initiated digital content marketing (DCM) communications. Self-brand connections are examined as an important antecedent to the cognitive, affective, behavioural and social dimensions of consumer engagement and the subsequent impact of engagement on loyalty is explored across these two channels. Design/methodology/approach A survey approach was used with two consumer samples for one focal retail brand, namely, a consumer-initiated OBC (Facebook) and email subscribers of the retail brand’s DCM communications. A multi-group analysis of structural invariance procedure was used to comparatively examine the formation of engagement for consumers within the OBC and DCM channels. Findings This study demonstrates the different ways in which engagement forms across different digital communication channels. Self-brand connection (SBC) was found to strongly drive behavioural, cognitive, affective and social engagement. The cognitive, affective and behavioural engagement was found to mediate the self-brand connection and consumer loyalty relationship. Overall, this relationship was most strongly and significantly mediated by affective and cognitive engagement within the OBC channel when compared to the DCM channel. Research limitations/implications The findings of this study should be interpreted with several limitations in mind. First, the research was conducted within the confines of one OBC, within one social networking site platform characterised by self-selected membership based on a passion and immersion with the brand. This means that consumers within the OBC were highly connected to one another and the retail brand and highly socialised in-group norms and mores. This type and intensity of connection may not be the case for all forms of OBCs. Second, this study was limited to one retail brand, from one brand category. Future research should examine OBCs across a range of utilitarian and hedonic brands to comprehensively contextualise the dimensions of engagement. Third, the data for this study was cross-sectional. The use of netnographic analysis and qualitative interviews across a range of OBCs would support the triangulation of the findings of this research, especially with regard to the narrative that consumers’ express when discussing how their SBC manifests through the dimensions of engagement. Fourth, this study explored a single antecedent of engagement, namely, self-brand connections. Future research may consider how SBC operates in conjunction with other complementary factors to enhance consumers’ affective, cognitive, social and behavioural engagement such as brand awareness, satisfaction and participation/interactivity. In addition, future research could examine an expanded array of engagement outcomes such as purchase intention, the share of wallet and reputation. Finally, future research should examine the operationalisation and validation of the dimensions of engagement using multiple competing scales to assess the suitability of these engagement scales across multiple brand categories and contexts. Practical implications Given the increasing investment in branding within social media and the fragmentation of brand communications across multiple communications platforms, the management of effective brand communications remains a significant challenge. This study found that the relationship between self-brand connections, affective, social, behavioural and cognitive engagement and loyalty was context-specific and moderated by a digital communication channel (OBC vs DCM email marketing), thus providing insights as to the effectiveness of OBCs and DCMs as two tools for enhancing consumer loyalty. Originality/value This study makes a novel contribution to the engagement literature by examining the antecedent role of self-brand connections in predicting consumers’ engagement; the moderating role of digital communication platforms (OBC vs DCM) on the formation of cognitive, affective, behavioural and social engagement; and the mediating effect of these dimensions on loyalty.