Structures where we have both a contravariant (pullback) and a covariant (pushforward) functoriality that satisfy base change can be encoded by functors out of ($\infty$-)categories of spans (or correspondences). In this paper, we study the more complicated setup where we have two pushforwards (an ‘additive’ and a ‘multiplicative’ one), satisfying a distributivity relation. Such structures can be described in terms of bispans (or polynomial diagrams). We show that there exist $(\infty,2)$-categories of bispans, characterized by a universal property: they corepresent functors out of $\infty$-categories of spans where the pullbacks have left adjoints and certain canonical 2-morphisms (encoding base change and distributivity) are invertible. This gives a universal way to obtain functors from bispans, which amounts to upgrading ‘monoid-like’ structures to ‘ring-like’ ones. For example, symmetric monoidal $\infty$-categories can be described as product-preserving functors from spans of finite sets, and if the tensor product is compatible with finite coproducts our universal property gives the canonical semiring structure using the coproduct and tensor product. More interestingly, we encode the additive and multiplicative transfers on equivariant spectra as a functor from bispans in finite $G$-sets, extend the norms for finite étale maps in motivic spectra to a functor from certain bispans in schemes, and make $\mathrm {Perf}(X)$ for $X$ a spectral Deligne–Mumford stack a functor of bispans using a multiplicative pushforward for finite étale maps in addition to the usual pullback and pushforward maps. Combining this with the polynomial functoriality of $K$-theory constructed by Barwick, Glasman, Mathew, and Nikolaus, we obtain norms on algebraic $K$-theory spectra.

Let $S=\{s_{1}, s_{2}, \ldots \}$ be an unbounded sequence of positive integers with $s_{n+1}/s_{n}$ approaching $\alpha $ as $n\rightarrow \infty $ and let $\beta>\max (\alpha , 2)$. We show that for all sufficiently large positive integers l, if $A\subset [0, l]$ with $l\in A$, $\gcd A=1$ and $|A|\geq (2-{k}/{\lambda \beta })l/(\lambda +1)$, where $\lambda =\lceil {k}/{\beta }\rceil $, then $kA\cap S\neq \emptyset $ for $2<\beta \leq 3$ and $k\geq {2\beta }/{(\beta -2)}$ or for $\beta>3$ and $k\geq 3$.

This study examined how linguistic complexity features contribute to second language (L2) processing effort by analyzing the Dutch English-L2 learners’ eye movements from GECO and MECO, two eye-tracking corpora. Processing effort was operationalized as reading rate, mean fixation duration, regression rate, skipping rate, and mean saccade amplitude. In Study 1, the lexical, syntactic, and discoursal indices in 272 snippets of a novel in GECO were regressed against these eye-movement measures. The results showed that the one-component partial least square regression (PLS-R) models could explain 11%–37% of the variance in these eye-movement measures and outperformed eight readability formulas (six traditional and two recent cognitively inspired formulas based on the readers’ perception on text difficulty) in predicting L2 processing effort. In Study 2, the eye-tracking data from MECO were used to evaluate whether the findings from Study 1 could be applied more broadly. The results revealed that although the predictability of these PLS-R components decreased, they still performed better than the readability formulas. These findings suggest that the linguistic indices identified can be used to predict L2 text processing effort and provide useful implications for developing systems to assess text difficulty for L2 learners.

We show large flows of workers into the real estate agent (REA) occupation during the early 2000s from virtually all parts of the skill, wage, and education spectrums. We find those entering REA in Metropolitan Statistical Areas (MSAs) with house price bubbles end up in occupations paying significantly less in the long-run as compared to similar REA entrants in non-bubble areas. Even in 2017, when house prices and employment return to their pre-crisis levels, REA entrants in Bubble MSAs are in occupations earning about 6% less. These results point to lasting effects of labor allocation decisions in response to distorted price signals.

In this article I argue for the relevance of examining tolerance in Victorian literature in its specificity and particularly as distinct from sympathy. Following a recent reconceptualization of tolerance in philosophy and political theory, I argue for recovering the relevance of its roots in the Latin term tolerare, which means “to suffer,” “to endure,” and suggest that Victorian novels explore the cognitive, emotional, and physical pains involved in tolerance. Victorian studies thus add an important perspective to current discussions of tolerance, while, conversely, a focus on tolerance in Victorian literature reveals an important category as yet overshadowed by the strong focus on sympathy in the field.

Anglo-American socialism has reached a curious impasse. Levels of economic inequality not seen since the nineteenth century would lead one to anticipate an upsurge in socialist affiliation and activism. Nevertheless, socialism has stagnated as a political force. This essay argues that, as a collectivist ideology, socialism demands a degree of communal identification that modern individuals find uncomfortable, even threatening, to the integrity of the self. Investigating this discomfort leads me to argue that, while socialism is not a religion (pace the claim of Gareth Stedman Jones), it nonetheless flourishes in eras of spiritual and societal foment. In such periods, communal effervescence softens the boundaries of the self, rendering collectivist ideologies like socialism more intuitive and appealing. Given desacralizing trends in Anglo-America, a contemporary socialist revival would seem unlikely. Consequently, I argue that it is time for progressives to embrace, rather than critique, the strong sense of individual autonomy and deep subjectivity that characterizes bourgeois individualism. The Victorian socialist canon provides many resources for a reconciliation of individualism and collectivism, which I illustrate via reference to H. G. Wells, John Stuart Mill, and Oscar Wilde.

Nineteenth-century piracy expressed a winking attitude toward many widespread forms of unauthorized reuse and thus conditioned the emergence of vast, innovative, and dynamic pan-media and transnational networks of aesthetic communications.

This “Keywords Redux” article examines the significance of drugs (psychoactive substances) in Victorian culture and literature, theorizing their potential for both performance enhancement and for pleasure.

The Hamiltonian shape invariant of a domain $X \subset \mathbb {R}^4$, as a subset of $\mathbb {R}^2$, describes the product Lagrangian tori which may be embedded in $X$. We provide necessary and sufficient conditions to determine whether or not a path in the shape invariant can lift, that is, be realized as a smooth family of embedded Lagrangian tori, when $X$ is a basic $4$-dimensional toric domain such as a ball $B^4(R)$, an ellipsoid $E(a,b)$ with ${b}/{a} \in \mathbb {N}_{\geq ~2}$, or a polydisk $P(c,d)$. As applications, via the path lifting, we can detect knotted embeddings of product Lagrangian tori in many toric $X$. We also obtain novel obstructions to symplectic embeddings between domains that are more general than toric concave or toric convex.

We study the étale cohomology of Hilbert modular varieties, building on the methods introduced by Caraiani and Scholze for unitary Shimura varieties. We obtain the analogous vanishing theorem: in the ‘generic’ case, the cohomology with torsion coefficients is concentrated in the middle degree. We also probe the structure of the cohomology beyond the generic case, obtaining bounds on the range of degrees where cohomology with torsion coefficients can be non-zero. The proof is based on the geometric Jacquet–Langlands functoriality established by Tian and Xiao and avoids trace formula computations for the cohomology of Igusa varieties. As an application, we show that, when $p$ splits completely in the totally real field and under certain technical assumptions, the $p$-adic local Langlands correspondence for $\mathrm {GL}_2(\mathbb {Q}_p)$ occurs in the completed homology of Hilbert modular varieties.

We study configurations of disjoint Lagrangian submanifolds in certain low-dimensional symplectic manifolds from the perspective of the geometry of Hamiltonian maps. We detect infinite-dimensional flats in the Hamiltonian group of the two-sphere equipped with Hofer's metric, prove constraints on Lagrangian packing, find instances of Lagrangian Poincaré recurrence, and present a new hierarchy of normal subgroups of area-preserving homeomorphisms of the two-sphere. The technology involves Lagrangian spectral invariants with Hamiltonian term in symmetric product orbifolds.

We study two models of discrete height functions, that is, models of random integer-valued functions on the vertices of a tree. First, we consider the random homomorphism model, in which neighbours must have a height difference of exactly one. The local law is uniform by definition. We prove that the height variance of this model is bounded, uniformly over all boundary conditions (both in terms of location and boundary heights). This implies a strong notion of localisation, uniformly over all extremal Gibbs measures of the system. For the second model, we consider directed trees, in which each vertex has exactly one parent and at least two children. We consider the locally uniform law on height functions which are monotone, that is, such that the height of the parent vertex is always at least the height of the child vertex. We provide a complete classification of all extremal gradient Gibbs measures, and describe exactly the localisation-delocalisation transition for this model. Typical extremal gradient Gibbs measures are localised also in this case. Localisation in both models is consistent with the observation that the Gaussian free field is localised on trees, which is an immediate consequence of transience of the random walk.

Although grammar is often associated with schematic approaches to education, it was a hotly contested subject in the nineteenth century. Considering nineteenth-century grammar texts, as well as the recent turn to “grammar” as a theoretical lens in nineteenth-century studies, this keywords entry proposes that grammar, far from reflecting fixed and incontrovertible precepts, serves as a powerful tool for querying and renegotiating disciplinary structures.

The article surveys the significance of the dandy as a central figure in nineteenth-century constructions of masculinity and social class.

In order to solve the problems of low loading capacity and low driving efficiency for the powered exoskeleton, this paper presents a bionic multi-chamber pneumatic actuator based on muscle scale mechanism. Firstly, the bionic muscle scale mechanism and multi-chamber structure design for the novel pneumatic actuator are introduced. Afterward, the driving characteristics of the multi-chamber actuator are analyzed theoretically, including analysis of output force and analysis of energy efficiency. Then, the load matching control strategy for the novel actuator is optimized, and the load matching performance, displacement tracking accuracy, and energy efficiency are studied by simulation. Finally, the prototype of the multi-chamber actuator is developed, and the exoskeleton testing platform is built, experiment and discussion are conducted for the driving characteristics, which realized the high energy efficiency and the feasibility of load matching.