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Trigonal curves provide an example of Brill–Noether special curves. Theorem 1.3 of Larson (Invent Math 224(3):767–790, 2021) characterizes the Brill–Noether theory of general trigonal curves and the refined stratification by Brill–Noether splitting loci, which parametrize line bundles whose push-forward to $$\mathbb {P}^1$$ has a specified splitting type. This note describes the refined stratification for all trigonal curves. Given the Maroni invariant of a trigonal curve, we determine the dimensions of all Brill–Noether splitting loci and describe their irreducible components. When the dimension is positive, these loci are connected, and if furthermore the Maroni invariant is 0 or 1, they are irreducible.

Rationale & objectiveObservational studies have reported a U-shaped association between blood pressure (BP) before a hemodialysis session and death. In contrast, because a linear association between out-of-dialysis-unit BP and death has been reported, home BP may be a better target for treatment. To test the feasibility of this approach, we conducted a pilot trial of treating home versus predialysis BP in hemodialysis patients.Study designA 4-month, parallel, randomized, controlled trial.Settings & participants50 prevalent hemodialysis patients in San Francisco and Seattle. Participants were randomly assigned using 1:1 block randomization, stratified by site.InterventionsTo target home systolic BP (SBP)of 100-200mm Hg; 0.2% vs 0%) or low (defined as

Let $$f:C \rightarrow \mathbb {P}^1$$ be a degree k genus g cover. The stratification of line bundles $$L \in {{\,\mathrm{Pic}\,}}^d(C)$$ by the splitting type of $$f_*L$$ is a refinement of the stratification by Brill–Noether loci $$W^r_d(C)$$ . We prove that for general degree k covers, these strata are smooth of the expected dimension. In particular, this determines the dimensions of all irreducible components of $$W^r_d(C)$$ for a general k-gonal curve (there are often components of different dimensions), extending results of Pflueger (Adv Math 312:46–63, 2017) and Jensen and Ranganathan (Brill–Noether theory for curves of a fixed gonality, arXiv:1701.06579 , 2017). The results here apply over any algebraically closed field.

The development of a functional vertebrate musculoskeletal system requires the combination of contractile muscle and extracellular matrix (ECM)-rich tendons that transmit muscle-generated force to bone. Despite the different embryologic origins, muscle and tendon integrate at the myotendinous junction (MTJ) to seamlessly connect cells and ECM across this interface. While the cell-cell signaling factors that direct development have received considerable attention, how and when the ECM linking these tissues is deposited remains unknown. To address this gap, we analyzed the 3D distribution of different ECM and the influence of skeletal muscle in forelimbs from wildype (WT) and muscle-less Pax3Cre/Cre mice. At E11.5, prior to MTJ integration, an aligned ECM was present at the presumptive insertion of the long triceps into the WT ulna. Mechanically robust tendon-like and muscle compartmentalization structures, positive for type I collagen, type V collagen, and fibrillin-2, still formed when muscle was knocked out. However, MTJ-specific ECM was not observed when muscle was absent. Our results show that an ECM-based template forms independent of muscle, but muscle is needed for the proper assembly of ECM at the MTJ.Summary statementAn aligned ECM template connects tendon and muscle during limb development, independent of muscle progenitor migration into the limb; however, the assembly of MTJ-specific ECM requires the presence of muscle.

Poyang Lake is China's largest freshwater lake, and it has been an internationally important wintering ground for migratory waterbirds. Based on waterbird censuses from 2001 to 2016, community structure and abundance trends of waterbirds were analyzed, as well as the potential correlations with the water level of Poyang Lake. The results showed that the annual average number of waterbirds in Poyang Lake was 426,707 ± 150,170, with 111 species from 17 families. Waterfowl was the most abundant family accounting for 74.4 ± 8.8% of all individuals, followed by shorebirds (14.8 ± 8.5%), wading birds (6.5 ± 1.8%) and open-water/waterbirds (4.4 ± 1.9%). Although waterbird abundance fluctuated dramatically, there were no significant trends in the abundance of most guilds or in total waterbird abundance; only geese significantly increased among the eight groups. Analysis of trends of 37 relatively abundant or regularly occurring species indicated that population trends appeared to be species-specific. As for the correlations between water level and waterbird abundance, only shorebirds showed significant correlations with average July water level, average water level, and high water level duration in wet season among four guilds, i.e. waterfowl, wading bird, shorebird and, open-water and waterbird. At the group level, abundances of swans, geese, and ducks were significantly correlated with monthly average water level during the wet season, and wader abundance was significantly correlated with average water levels and high water level duration during the wet season. Correlations between population abundance and monthly water level also exhibited species-specific patterns.

Recent work of Kass--Wickelgren gives an enriched count of the $27$ lines on a smooth cubic surface over arbitrary fields. Their approach using $\mathbb{A}^1$-enumerative geometry suggests that other classical enumerative problems should have similar enrichments, when the answer is computed as the degree of the Euler class of a relatively orientable vector bundle. Here, we consider the closely related problem of the $28$ bitangents to a smooth plane quartic. However, it turns out the relevant vector bundle is not relatively orientable and new ideas are needed to produce enriched counts. We introduce a fixed "line at infinity," which leads to enriched counts of bitangents that depend on their geometry relative to the quartic and this distinguished line., 17 pages, comments welcome!

Biological systems are typically dependent on transportation networks for the efficient distribution of resources and information. Revealing the decentralized mechanisms underlying the generative process of these networks is key in our global understanding of their functions and is of interest to design, manage and improve human transport systems. Ants are a particularly interesting taxon to address these issues because some species build multi-sink multi-source transport networks analogous to human ones. Here, by combining empirical field data and modelling at several scales of description, we show that pre-existing mechanisms of recruitment with positive feedback involved in foraging can account for the structure of complex ant transport networks. Specifically, we find that emergent group-level properties of these empirical networks, such as robustness, efficiency and cost, can arise from models built on simple individual-level behaviour addressing a quality-distance trade-off by the means of pheromone trails. Our work represents a first step in developing a theory for the generation of effective multi-source multi-sink transport networks based on combining exploration and positive reinforcement of best sources.

The extracellular matrix (ECM) is a complex network of biomolecules that mechanically and biochemically directs cell behavior and is crucial for maintaining tissue function and health. The heterogeneous organization and composition of the ECM varies within and between tissue types, directing mechanics, aiding in cell-cell communication, and facilitating tissue assembly and reassembly during development, injury and disease. As technologies like 3D printing rapidly advance, researchers are better able to recapitulate in vivo tissue properties in vitro; however, tissue-specific variations in ECM composition and organization are not given enough consideration. This is in part due to a lack of information regarding how the ECM of many tissues varies in both homeostatic and diseased states. To address this gap, we describe the components and organization of the ECM, and provide examples for different tissues at various states of disease. While many aspects of ECM biology remain unknown, our goal is to highlight the complexity of various tissues and inspire engineers to incorporate unique components of the native ECM into in vitro platform design and fabrication. Ultimately, we anticipate that the use of biomaterials that incorporate key tissue-specific ECM will lead to in vitro models that better emulate human pathologies. STATEMENT OF SIGNIFICANCE: Biomaterial development primarily emphasizes the engineering of new materials and therapies at the expense of identifying key parameters of the tissue that is being emulated. This can be partially attributed to the difficulty in defining the 3D composition, organization, and mechanics of the ECM within different tissues and how these material properties vary as a function of homeostasis and disease. In this review, we highlight a range of tissues throughout the body and describe how ECM content, cell diversity, and mechanical properties change in diseased tissues and influence cellular behavior. Accurately mimicking the tissue of interest in vitro by using ECM specific to the appropriate state of homeostasis or pathology in vivo will yield results more translatable to humans.

The rational Chow ring of the moduli space $\mathcal{M}_g$ of curves of genus $g$ is known for $g \leq 6$. Here, we determine the rational Chow rings of $\mathcal{M}_7, \mathcal{M}_8,$ and $\mathcal{M}_9$ by showing they are tautological. One key ingredient is intersection theory on Hurwitz spaces of degree $4$ and $5$ covers of $\mathbb{P}^1$, as developed by the authors in [1]. The main focus of this paper is a detailed geometric analysis of special tetragonal and pentagonal covers whose associated vector bundles on $\mathbb{P}^1$ are so unbalanced that they fail to lie in the large open subset considered in [1]. In genus $9$, we use work of Mukai [23] to present the locus of hexagonal curves as a global quotient stack, and, using equivariant intersection theory, we show its Chow ring is generated by restrictions of tautological classes., 47 pages, comments welcome!

Let $\mathcal{H}_{k,g}$ be the Hurwitz stack parametrizing degree $k$, genus $g$ covers of $\mathbb{P}^1$. We define the tautological ring of $\mathcal{H}_{k,g}$ and we show that all Chow classes, except possibly those supported on the locus of "factoring covers," are tautological up to codimension roughly $g/k$ when $k \leq 5$. The set-up developed here is also used in our subsequent work, wherein we prove new results about the structure of the Chow ring for $k \leq 5$., 30 pages. This paper is part of our work on low-degree Hurwitz spaces, which has been split off from a previous version because of length

Let $X \subset \mathbb{P}^n$ be a smooth hypersurface. Given a sequence of integers $\vec{a} = (a_1, \ldots, a_{n-2})$ with $a_1 \leq \cdots \leq a_{n-2}$, let $F_{\vec{a}}(X)$ be the parameter space of lines $L$ on $X$ such that $N_{L/X} \cong \mathcal{O}(a_1) \oplus \cdots \oplus \mathcal{O}(a_{n-2})$. The loci $F_{\vec{a}}(X)$ form a stratification of the Fano scheme of lines on $X$. We show that for general hypersurfaces, the $F_{\vec{a}}(X)$ have the expected dimension and, in this case, compute the class of $\overline{F_{\vec{a}}(X)}$ in the Chow ring of the Grassmannian of lines in $\mathbb{P}^n$. For certain splitting types $\vec{a}$, we also provide non-trivial upper bounds on the dimension of $F_{\vec{a}}(X)$ that hold for all smooth $X$.

While there is much work and many conjectures surrounding the intersection theory of the moduli space of curves, relatively little is known about the intersection theory of the Hurwitz space $\mathcal{H}_{k, g}$ parametrizing smooth degree $k$, genus $g$ covers of $\mathbb{P}^1$. Let $k = 3, 4, 5$. We prove that the rational Chow rings of $\mathcal{H}_{k,g}$ stabilize in a suitable sense as $g$ tends to infinity. In the case $k = 3$, we completely determine the Chow rings for all $g$. We also prove that the rational Chow groups of the simply branched Hurwitz space $\mathcal{H}^s_{k,g} \subset \mathcal{H}_{k,g}$ are zero in codimension up to roughly $g/k$. In subsequent work, results developed in this paper are used to prove that the Chow rings of $\mathcal{M}_7, \mathcal{M}_8,$ and $\mathcal{M}_9$ are tautological., Comment: 45 pages, split off from arXiv:2103.09902v1 because of length

To operate in traffic, autonomous vehicles must plan long trajectories (e.g., unprotected left turns across traffic) and validate that they are not-at-fault in a collision. Reachability-based Trajectory Design for Dynamic environments (RTD-D) is a method that plans validated trajectories, which are then tracked a feedback controller—in this work linear MPC. RTD-D computes a Forward Reachable Set (FRS) of the ego vehicle’s motion offline, then uses the FRS online to plan by discretizing time and buffering obstacles (i.e., artificially increasing their size) to compensate for the discretization; this requires choosing a conservative buffer so that the discretization is coarse enough for real-time planning. This paper eliminates the discretization-dependent buffer with a new method of computing the FRS over a prespecified choice of time intervals, allowing for a much coarser time discretization that reduces both computational cost and conservatism at runtime. The new method, RTD-Interval (RTD-I), is shown in simulation on a vehicle described by a nonlinear bicycle model in comparison to RTD-D and a State Lattice planner in unstructured dynamic environments. RTD-I is also compared to RTD-D in unprotected left turns across busy intersections, and we demonstrate that the MPC controller tracking the road centerline is unsafe on its own. RTD-I plans faster and less conservatively than RTD-D, and causes no at-fault collisions.

There is a minor typographical error on page 3 which says that the hyperbolicity of the polynomial P(X) is equivalent.

Given an arithmetic function $$a: \mathbb {N}\rightarrow \mathbb {R}$$ , one can associate a naturally defined, doubly infinite family of Jensen polynomials. Recent work of Griffin et al. shows that for certain families of functions $$a: \mathbb {N}\rightarrow \mathbb {R}$$ , the associated Jensen polynomials are eventually hyperbolic (i.e., eventually all of their roots are real). This work proves Chen et al. conjecture that the partition Jensen polynomials are eventually hyperbolic as a special case. Here, we make this result explicit. Let N(d) be the minimal number such that for all $$n \ge N(d)$$ , the partition Jensen polynomial of degree d and shift n is hyperbolic. We prove that $$N(3)=94$$ , $$N(4)=206$$ , and $$N(5)=381$$ , and in general, that $$N(d) \le (3d)^{24d} (50d)^{3d^{2}}$$ .

As autonomous robots increasingly become part of daily life, they will often encounter dynamic environments while only having limited information about their surroundings. Unfortunately, due to the possible presence of malicious dynamic actors, it is infeasible to develop an algorithm that can guarantee collision-free operation. Instead, one can attempt to design a control technique that guarantees the robot is not-at-fault in any collision. In the literature, making such guarantees in real time has been restricted to static environments or specific dynamic models. To ensure not-at-fault behavior, a robot must first correctly sense and predict the world around it within some sufficiently large sensor horizon (the prediction problem), then correctly control relative to the predictions (the control problem). This paper addresses the control problem by proposing Reachability-based Trajectory Design for Dynamic environments (RTD-D), which guarantees that a robot with an arbitrary nonlinear dynamic model correctly responds to predictions in arbitrary dynamic environments. RTD-D first computes a Forward Reachable Set (FRS) offline of the robot tracking parameterized desired trajectories that include fail-safe maneuvers. Then, for online receding-horizon planning, the method provides a way to discretize predictions of an arbitrary dynamic environment to enable real-time collision checking. The FRS is used to map these discretized predictions to trajectories that the robot can track while provably not-at-fault. One such trajectory is chosen at each iteration, or the robot executes the fail-safe maneuver from its previous trajectory which is guaranteed to be not at fault. RTD-D is shown to produce not-at-fault behavior over thousands of simulations and several real-world hardware demonstrations on two robots: a Segway, and a small electric vehicle., Comment: 10 pages, 3 figures

In a recent paper, Duncan, Griffin and Ono provide exact formulas for the coefficients of McKay-Thompson series and use them to find asymptotic expressions for the distribution of irreducible representations in the moonshine module V ♮ = ⨁ n V n ♮ V^\natural = \bigoplus _n V_n^\natural . Their results show that as n n tends to infinity, V n ♮ V_n^\natural is dominated by direct sums of copies of the regular representation. That is, if we view V n ♮ V_n^\natural as a module over the group ring Z [ M ] \mathbb {Z}[\mathbb {M}] , the free part dominates. A natural problem, posed at the end of the aforementioned paper, is to characterize the distribution of irreducible representations in the non-free part. Here, we study asymptotic formulas for the coefficients of McKay-Thompson series to answer this question. We arrive at an ordering of the series by the magnitude of their coefficients, which corresponds to various contributions to the distribution. In particular, we show how the asymptotic distribution of the non-free part is dictated by the column for conjugacy class 2A in the monster’s character table. We find analogous results for the other monster modules V ( − m ) V^{(-m)} and W ♮ W^\natural studied by Duncan, Griffin, and Ono.

In recent work, Folsom presents a family of modular units as higher-level analogues of the Rogers-Ramanujan q q -continued fraction. These units are constructed from analytic solutions to the higher-order q q -recurrence equations of Selberg. Here, we consider another family of modular units, which are quotients of Hall-Littlewood q q -series that appear in the generalized Rogers-Ramanujan type identities in the work of Griffin, Ono, and Warnaar. In analogy with the results of Folsom, we provide a formula for the rank of the subgroup these units generate and show that their specializations at the cusp 0 0 generate a subgroup of the cyclotomic unit group of the same rank. In addition, we prove that their singular values generate the same class fields as those of Folsom’s units.

In an important paper, Zagier proved that certain half-integral weight modular forms are generating functions for traces of polynomials in the j-function. It turns out that Zagier's work makes it possible to algorithmically compute Hilbert class polynomials using a canonical family of modular forms of weight [Formula: see text]. We generalize these results and consider Hauptmoduln for levels 1, 2, 3, 5, 7, and 13. We show that traces of singular values of polynomials in Hauptmoduln are again described by coefficients of half-integral weight modular forms. This realization makes it possible to algorithmically compute class polynomials.

A fusion category of rank 4 has either four self-dual simple objects or exactly two self-dual simple objects. We study fusion categories of rank 4 with exactly two self-dual simple objects, giving nearly a complete classification of those based rings that admit pseudo-unitary categorification. More precisely, we show that if C is such a fusion category, then its Grothendieck ring K ( C ) must be one of seven based rings, six of which have known categorifications. In doing so, we classify all based rings associated with near-group categories of the group Z / 3 Z .

In a recent paper, Griffin, Ono and Warnaar present a framework for Rogers-Ramanujan type identities using extended Hall-Littlewood polynomials Pλ(x1,x2,…;q) to arrive at expressions of the form $$\sum\limits_{\lambda : \lambda_{1} \leq m} q^{a|\lambda|}P_{2\lambda}(1,q,q^{2},\ldots ; q^{n}) = \text{{\textquotedblleft}Infinite product modular function{\textquotedblright}} $$

The partition function p(n), which counts the number of partitions of a positive integer n, is widely studied. Here, we study partition functions pS(n) that count partitions of n into distinct parts satisfying certain congruence conditions. A shifted partition identity is an identity of the form $${{p_{S_1}} (n - H) = {p_{S_2}} (n)}$$ for all n in some arithmetic progression. Several identities of this type have been discovered, including two infinite families found by Alladi. In this paper, we use the theory of modular functions to determine the necessary and sufficient conditions for such an identity to exist. In addition, for two specific cases, we extend Alladi’s theorem to other arithmetic progressions.

In their work, Serre and Swinnerton-Dyer study the congruence properties of the Fourier coefficients of modular forms. We examine similar congruence properties, but for the coefficients of a modified Taylor expansion about a CM point $\tau$. These coefficients can be shown to be the product of a power of a constant transcendental factor and an algebraic integer. In our work, we give conditions on $\tau$ and a prime number $p$ that, if satisfied, imply that $p^m$ divides the algebraic part of all the Taylor coefficients of $f$ of sufficiently high degree. We also give effective bounds on the largest $n$ such that $p^m$ does not divide the algebraic part of the $n^{\text{th}}$ Taylor coefficient of $f$ at $\tau$ that are sharp under certain additional hypotheses., Comment: 16 pages, to appear in Int. J. Number Theory

In a recent paper, Rose proves that certain generalized sum-of-divisor functions are quasi-modular forms for some congruence subgroup and conjectures that these forms are quasi-modular for Γ 1(n). Here, we prove this conjecture.