Papers

We formulate and solve a $G$-equivariant analogue of the Seiberg-Witten equations on any irreducible Hermitian symmetric space $G/K$. We use a canonical $K$-invariant form to define a twisted Hodge operator generalising self-duality to higher dimensions. The resulting equivariant Seiberg-Witten equations arise, for a compact $G/K$, as conditions for the absolute minima of an energy functional. Their linear part has a representation theoretic interpretation: the same Weyl group data control the highest-weight realisation of the positive Dirac kernel in the compact case and its discrete-series realisation on the non-compact dual. This space of harmonic spinors collapses almost entirely by the quadratic conditions of the monopole equation. In the compact case, the equivariant moduli set, when non-empty, is a single gauge class, identifying with the $G$-invariant part of Bradlow's trichotomy for vortices, while in the non-compact case, every irreducible solution sits outside the discrete series.

For five complex points $z_1,\ldots,z_5$ satisfying $\sum_i|z_i|^2=5$, Mordell's discriminant inequality states that $\prod_{i<j}|z_i-z_j|^2\leq5^5$, with equality precisely for a unit regular pentagon, up to rotation and permutation. We organize its proof around a direct comparison of discriminant values. A global maximizer gives a centered stationary configuration, whose coefficient equations reduce the generic case to two quartic branches. On each branch, the absolute value of a cubic polynomial gives the original discriminant objective after accounting for the normalization scale. Exact rational interval bounds exclude both branches from the maximum, and the exceptional coefficient cases leave the regular pentagon. The argument determines the maximizing configuration without reconstructing all stationary shapes. We have completed a full Lean formalization of the five-point inequality and both directions of its equality condition, including all reductions and finite checks used in this proof.

Instructional practices that require student sensemaking of mathematics are considered critical to learning outcomes, yet strategies for measurement of this instruction at scale have yet to be established. This study aims at validating and exploring the predictive validity of three student survey measures: students' opportunity to engage with, comfort engaging with, and utility of engaging with instructional practices targeting sensemaking in mathematics. Using an initial sample of 1,955 US middle school student responses, we conducted exploratory factor analysis to establish sensemaking construct structure, identifying two distinct factors reflecting students' active and passive participation in mathematical sensemaking. We then analyzed data from an additional 3,170 surveys collected over two years in the same context to confirm measurement properties and evaluate the measures' predictive validity. The resulting measures capture students' reported opportunities to engage in sensemaking practices (Sensemaking Opportunity), their comfort with engaging in those practices (Sensemaking Comfort), and their perceived utility of those practices when tackling new or difficult math problems (Sensemaking Utility). While Sensemaking Comfort and Sensemaking Utility both showed strong predictive validity over two years, Sensemaking Utility provided the most reliable estimates of student sensemaking. These findings offer researchers and practitioners a psychometrically sound, scalable toolkit for monitoring sensemaking in mathematics classrooms with implications for both large-scale research on instructional quality and teachers' ongoing efforts to monitor and improve their practice.

We study the set of integers $n \le x$ for which $\sum_{i=1}^{n}\frac{1}{\gcd(n,i)}$ takes an integer value. We show that the number of such integers is $\ll x^{2/3+\epsilon}$ for all fixed $\epsilon > 0$.

The lattice-coset method of Alon, Bucić, and Davies gives real solutions with controlled coordinates. We combine it with integer rounding to prescribe outdegrees and flows from spanning-tree packing. For every integer $q\ge3$, $q$ edge-disjoint spanning trees suffice for every outdegree prescription modulo $q$ whose sum is compatible with the number of edges. The tree count is optimal even for simple graphs. A common interval theorem extends the argument to totally unimodular matrices; for graphs, its partition inequalities admit a sharp allowance of two. Fractional packing characterizes when every prescription has a strongly connected realization and gives circular-flow bounds. In particular, for every integer $k\ge1$, every $(4k+2)$-edge-connected graph has circular flow number strictly below $2+1/k$, with optimal connectivity for this strict inequality. As a structural application, every four-edge-connected graph with no bond of size six, seven or eight has a nowhere-zero three-flow. Further consequences concern star decompositions, antisymmetric flows and contraction.

For a fixed integer $k\ge 2$, let $kK_3$ denote the vertex-disjoint union of $k$ triangles. A recent fixed-size spectral theorem of Das and Yamini asserts that, for all sufficiently large $m$, every $kK_3$-free graph $G$ of size $m$ satisfies $\lambda(G)\le (k-1)+\sqrt{m-k(k-1)},$ and equality holds if and only if $(2k-1)\mid m$ and $ G\cong \bigl(K_{2k-1}\vee qK_1\bigr)\cup tK_1,\, q=\frac{m}{2k-1}-(k-1) $ for some $t\ge 0$. They explicitly posed the open problem: Let $k\ge 2$ be fixed and $\ell$ be a residue in $\{1,\dots,2k-2\}$. For all sufficiently large integers $m\equiv \ell\pmod{2k-1}$, determine the exact value of $\max\bigl\{\lambda(G): e(G)=m,\ G\text{ is }kK_3\text{-free}\bigr\}, $ and characterize all graphs attaining this maximum. In this paper, using the positive-defect version of the bounded-core method for divisible sizes together with several new ideas developed in this paper, we give a complete solution to the aforementioned open problem.

We prove that the categories of strictly unital (resp.\ non unital) A$_{\infty}$-algebras and A$_{\infty}$-categories, with strictly unital (resp.\ non unital) A$_{\infty}$-morphisms and A$_{\infty}$-functors have no coequalizers.

Modeling the channel in a single-antenna fluid antenna system (FAS) using extreme value distributions (EVDs) provides an accurate and tractable framework for FAS performance evaluation. When the objective of FAS channel fitting is outage probability (OP) evaluation, accurate characterization of the low-probability left-tail region becomes crucial, while existing fitting strategies that emphasize global fitting accuracy may fail to capture the critical tail behavior required for precise OP evaluation. In this paper, we propose an OP-oriented channel fitting strategy with a left-tail-sensitive target distribution and fitting criterion. Specifically, a combined EVD (CEVD) is introduced as the target distribution, where a generalized Pareto distribution (GPD) is employed to characterize the left tail and a generalized extreme value (GEV) distribution is used to model the global behavior. Furthermore, a modified mean-square-error (MMSE) criterion is developed, which employs logarithmic-domain errors to enhance sensitivity to left-tail discrepancies. Meanwhile, the evaluation points are constructed via uniform discretization on the logarithm of the cumulative distribution function, ensuring uniform sampling across all probability scales. This mitigates the under-representation of tail errors in the overall MMSE, which cannot be effectively addressed by error amplification alone due to the sparsity of tail samples. Simulation results demonstrate that the proposed fitting strategy significantly improves the OP evaluation accuracy in the ultra-low-OP regime.

Let $N\ge1$, $p\in[1,\infty)$, and $\Omega\subset\mathbb R^N$ be a bounded open interval when $N=1$ or a bounded Lipschitz domain when $N\ge2$. For any given $\gamma\in\mathbb R\setminus\{0\}$ and $\lambda\in(0,\infty)$, consider the weak-type nonlocal functional \begin{align*} F_{\lambda,p,\gamma}(u;\Omega):=\lambda^p \underset{|u(x)-u(y)|>\lambda|x-y|^{1+\frac{\gamma}{p}}} {\int_\Omega\int_\Omega}|x-y|^{\gamma-N}\,dy\,dx. \end{align*} In this article, we prove that, for $\gamma\in(0,\infty)$ as $\lambda\to\infty$ or for $\gamma\in(-\infty,-1]$ as $\lambda\to0^+$, the family $F_{\lambda,p,\gamma}(\cdot;\Omega)$ converges, in the sense of $\Gamma$-convergence in $L^p(\Omega)$, to the functional \begin{align*} \Psi_{p,\gamma}^{\Omega}(u) :=\begin{cases} C_{N,p,\gamma}^cell\displaystyle\int_\Omega|\nabla u|^p\,dx, &p\in(1,\infty)\ \hbox{and}\ u\in W^{1,p}(\Omega),\\[2mm] C_{N,1,\gamma}^cell|Du|(\Omega), &p=1\ \hbox{and}\ u\in BV(\Omega),\\[1mm] \infty,&\hbox{otherwise}. \end{cases} \end{align*} where the positive constants $C_{N,p,\gamma}^{\mathrm{cell}}$ are independent of $\Omega$. As applications, we further show that, for $\gamma\in(0,\infty)$ as $\lambda\to\infty$ or for $\gamma\in(-\infty,-1]$ as $\lambda\to0^+$, the weak-type nonlocal functional $F_{\lambda,p,\gamma}(\cdot;\mathbb R^N)$ converges, in the sense of $\Gamma$-convergence in $L^1_{\rm loc}(\mathbb R^N)$, to a similar functional; this gives an complete answer to the problem posed by Brezis et al. in [Section~7C, Anal. PDE 2024].

We study coded caching with joint file and demand privacy: each user recovers its requested file while learning nothing about the remaining files and the other users' requests jointly. Let $N$ and $K$ be the numbers of files and users, and let $M$ and $R$ denote the cache memory and delivery rate, both normalized by the file size. For every $N,K\ge2$ and every feasible cache size, we give a scheme whose worst-case delivery rate is at most $9/2$ times the optimum. To reduce the memory used for file shares, the scheme secret-shares $N-1$ differences relative to a reference file. Cached masks supply the correction needed to recover the requested file from its difference. For each integer $t\in\{0,\ldots,K-1\}$, the scheme achieves $M=1+(N-1)t/(K-t)$ and $R=K/(t+1)$. To lower-bound the delivery rate, we compare the joint cache entropy of user groups under alternative demand vectors, using one fixed user's privacy constraint. Two choices of groups determine the optimal delivery rate up to constants at the scale $\min\{K,1+(N-1)/(M-1)\}$ for $M>1$. At $M=1$, the exact optimum is $K$. A complementary bound on joint broadcast entropy gives the minimum memory for unit delivery rate, $1+(N-1)(K-1)$, and the exact tradeoff on an interval ending at that memory. All schemes and bounds apply to repeated as well as distinct requests.

This paper studies the penalization method for a general class of time-inconsistent stopping problems. Inspired by some existing results on penalization for optimal stopping problems, we introduce the penalized PDE problem and the penalized control problem for time-inconsistent stopping problems, uncovering some interesting discrepancy stemming from the time-inconsistency. As the penalization tends to infinity, we develop some novel arguments to rigorously establish the local $W^{2,1}_{p}$ convergence results for the solution of the penalized PDE problem to a solution of the variational inequality associated to the time-inconsistent stopping problem. Under an additional concavity assumption, we further derive the convergence of the penalized time-inconsistent control problem to the time-inconsistent stopping problem. We revisit two examples of time-inconsistent stopping to numerically illustrate the convergence theory.

We present a structure-aware construction of quantum BPX preconditioners for Q1 finite element discretizations of constant-coefficient reaction–diffusion equations on the unit cube, including the Poisson equation. Building on existing quantum finite element and finite difference methods, we express the preconditioned operator through explicit one-dimensional maps, tensor products, and norm-preserving interlevel transfers. Symmetric preconditioning gives multilevel coefficients, and its favorable conditioning alone does not ensure efficient recovery of a state in the original finite element nodal basis. We instead use the local $L^2$-orthonormal basis underlying the construction as the output representation of the same finite element function. Its coefficient norm equals the continuous $L^2$ norm. Assuming direct access to the preconditioned right-hand side, we prepare this orthonormal coefficient state with a represented function error at most $\varepsilon$. The expected query complexity is \[ \mathcal{O}\!\Big(dL^3\frac{\|f\|_{L^2}}{\|u\|_{L^2}} \log\!\Big(2+\frac{L\|f\|_{L^2}}{\varepsilon}\Big)\Big), \] where $u$ is the exact solution, $f$ is the right-hand side, $d$ is the spatial dimension, and $L=\log_2(1/h)$ for mesh width $h$. The bound holds for $0<\varepsilon\le\|u\|_{L^2}/2$ and depends only polylogarithmically on mesh refinement for fixed continuous data. We also analyze linear-functional estimation and illustrate the construction and continuous error bounds with a one-dimensional UnitaryLab experiment.

For every $N\ge36$, we realise any prescribed monic irreducible binary polynomial of degree $N$ as the characteristic polynomial of a linear hybrid cellular automaton on a near-square region of exactly $N$ cells. The transition matrix is Moore-local with a null boundary and has at most $3N-1$ directed nonself dependencies. The dependency graph retains a bidirectional Hamilton path; its indegree, outdegree and underlying undirected degree are at most six. The underlying graph contains an explicit square grid minor of side proportional to $\sqrt N$. The deterministic synthesis takes $O(N^3)$ bit operations. The construction combines a local similarity transformation with a transport potential and joint routing across consecutive row gaps. Exact verification of finite certificates, followed by induction, proves the required routing inequalities for every admissible width. We also give an entirely analytic construction with fewer than $7N/2$ dependencies and prove a lower bound of $5N/2-O(\sqrt N)$ for the retained path and full rectangular grid minor. This lower bound is sharp when the characteristic polynomial is unrestricted.

In this paper, we study the multiplicity of smooth limit interfaces for the anisotropic Allen-Cahn equation on closed Riemannian manifolds. We prove that an isolated smooth limit interface has multiplicity one whenever it is a closed, embedded, two-sided, strictly \(F\)-stable \(F\)-minimal hypersurface. The proof uses anisotropic barriers and a sliding argument to confine the nodal sets to a neighborhood of width \(O(\varepsilon)\). We then classify the nonconstant rescaled limits as one-dimensional solutions and obtain an energy upper bound near the limiting hypersurface. Exponential decay controls the remaining energy, yielding the multiplicity-one conclusion.

We prove that the absolute Galois group of every discretely valued field of mixed characteristic is ab-faithful. We also prove ab-faithfulness for higher local fields of positive dimension and give further examples using finite split embedding problems. Finally, we give a formal Hilbert~90 criterion for center-freeness and slimness of finite-step solvable quotients. We apply this criterion to ab-faithful absolute Galois groups and obtain slimness for several classes of fields, including higher local fields of positive dimension.

We study overdetermined problems for normalized sum Hessian equations and their quotients in the space forms. The equations and boundary conditions imply admissibility and ellipticity. For quotient equations, we prove that the domain is a geodesic ball if it is strictly star-shaped, or if the Dirichlet value satisfies an upper bound. We also prove the corresponding result for sum Hessian equations without star-shapedness. The proofs use two auxiliary functions and a Rellich–Pohožaev identity.

The Guided Local Hamiltonian Problem (GLHP) is the problem of ground state energy estimation and preparation given a guiding state having good overlap with the true ground state. The problem is known to be BQP-complete for small error but is classically tractable for larger error. In this work, we identify an orthogonal regime in which the problem is classically tractable, namely when the true ground state $\psi$ has polynomially bounded $\ell_1$-norm. In particular, we show that the Truncated Power Method (TPM), a classical algorithm introduced by Kirby et al. in the quantum context, has a runtime that is polynomially comparable to quantum algorithms for GLHP in all relevant parameters, except for an extra factor of $\|\psi\|_1^2$. In other words, if $\|\psi\|_1^2$ is polynomially bounded, then TPM dequantizes quantum algorithms for GLHP.

Lyapunov functions have been a cornerstone of dynamical systems theory ever since the early 1900s. Moreover and akin to Morse theory, Lyapunov functions have been the key in linking topology to dynamical systems theory. In this note we continue along these lines. Suppose that some vector field on Euclidean space has a unique equilibrium point that is globally asymptotically stable (GAS). Under these conditions, it is known that there is always a smooth Lyapunov function to certify stability. One may wonder if this function can always be chosen to be Morse. We show that if the equilibrium point is hyperbolic, this is indeed true. Moreover, we show that if this would be true in general, then the generalized 4-dimensional smooth Poincaré conjecture (SPC4) must be true. Even so, we provide an example that is GAS, but does not admit a Morse Lyapunov function, closing this route of proving SPC4.

A Hilbert space is usually described using an orthonormal basis, but a more general description can be defined using a tight frame. In the case of continuous-variable Gaussian states, the coherence is usually described using the Hermite-Gauss basis. We present an alternative description using the tight frame of all canonical coherent states. The description of Gaussian states in terms of coherent states is simpler than in terms of Hermite-Gauss basis, and this allows us to obtain an explicit formula describing the frame-dependent coherence of a mixed Gaussian state as a function of the determinant and the trace of covariance matrix. For each mixed Gaussian state, we construct a sequence of discrete-variable quantum states rapidly convergent to it (as concern the purity and coherence) when the dimension of Hilbert space becomes large enough (15 or more). In the discrete case, the coherence is measured with respect to the tight frame of all discrete coherent states.

We study the finite-sample properties of estimators of a broad class of normalized scale-invariant inequality indices (NSIIs) under finite Gamma mixture models. The class includes the Gini coefficient, the $m$th Gini index, the Extended $m$th Gini index, linear order-statistic inequality indices, and several entropy- and dispersion-based measures. Building on the Gamma–Dirichlet representation, we derive exact expressions for the expectation and finite-sample bias of the natural estimator of an NSII based on a $U$-statistic when the population follows a finite Gamma mixture with a common rate parameter. The resulting bias formula explicitly characterizes the effects of the mixture weights and component shape parameters. We also establish that the bias converges to zero as the sample size increases, yielding asymptotic unbiasedness. Furthermore, we derive exact finite-sample expressions for the variance and mean squared error of the estimator by accounting for the overlap structure among the subsets entering the $U$-statistic. In addition, we establish strong consistency and asymptotic normality. When the mixture degenerates to a single Gamma distribution, the finite-sample bias vanishes, recovering the exact unbiasedness result previously obtained for NSIIs under the Gamma model. Numerical illustrations and simulation studies are provided to investigate the finite-sample behavior of the estimator and to illustrate the theoretical findings. Overall, the proposed framework provides a unified characterization of finite-sample estimation under heterogeneous Gamma populations and quantifies the effects of mixture heterogeneity on bias, variance, and mean squared error.

Every balanced instance of the stable marriage problem with strict complete preferences has a unique finest partition into prime blocks, and that single partition simultaneously factors three different structures: the reachable execution digraph as a Cartesian product, the proposal-prefix antimatroid as a direct sum, and the stable-matching lattice as a direct product. The converse fails, and fails at every size from two on: two explicit families share the identical Boolean-cube execution while one is maximally decomposable with a single stable matching and the other is prime with n. Uniqueness yields an exact census, a recursion counting the prime instances at every size, under which exactly 88,478,208 of the 110,075,314,176 profiles with four agents on each side are decomposable and the decomposable fraction is asymptotically n! / n^(2n). The blocks are characterised as the square components of the mutual-rank filtration, so the partition is computable in polynomial time and the factorisation is a tool rather than only a fact.

This paper analyzes the quantized asymmetric transport along the boundary of a two-dimensional topological insulator occupying a half plane, or along the interface separating two insulators. In contrast with problems posed on the Euclidean plane, terminating a sample requires a choice of self-adjointrealization, $H_L$, and we show that a quantized transport depends on that choice. The physical observable modeling the asymmetric transport is the edge, or interface, current $\sigma_I[H_L]=\Tr\,i[H_L,P]\varphi'(H_L)$. It is shown to be well defined along two routes. For Shapiro–Lopatinski elliptic realizations, we use estimates from Boutet de Monvel's calculus to show that the current observable is a well-defined trace, that $\mathcal I=2\pi\sigma_I[H_L]$ is a Fredholm index, and that it is stable under deformations and compactly supported changes of the operator and of its termination. For translation invariant operators along the edge, a spectral analysis defines $\mathcal I$ when the edge spectrum escapes a bulk gap at large momenta. $\mathcal I$ is then also a spectral flow and a Maslov index. Our main theorem is a bulk–edge correspondence \[ \mathcal I=\cR-\cR_L+\eta\bigl(L_\infty,\Lm^\Phi(+)\bigr)-\eta\bigl(L_\infty,\Lm^\Phi(-)\bigr), \] We determine the range of $\mathcal I$. We extend the formula to interfaces with arbitrary transmission conditions. At the transparent condition, the invariant equals the bulk-difference invariant of the Euclidean plane. Eight examples illustrate the theoretical findings.

Alongside three established presentations of elementary complex analysis, we propose an approach centered on one organizing question: when is contour convolution well defined on homotopy classes? From this starting point, we develop the foundational results of the subject. A supporting conceptual distinction identifies those elements of real and complex analysis that share the same operational and symbolic rules. We refer to this collection as the "Toolbox." Examples and structural connections illustrate the explanatory value and reach of the approach across different aspects of the theory. We also examine the relationship of our approach to the established presentations.

We show that for each countable ordinal $\alpha$, there is a $\mathbf{\Delta}^0_\alpha$ operator that, for all $A$ of Scott rank at most $\alpha$, takes copies of $T_A$ to copies of $A$, where $T_A$ is the tree of tuples corresponding to $A$. For finite $\alpha$, we show that this is best possible. The proof that there is a $\mathbf{\Delta}^0_\alpha$ operator uses Harrington's method of workers. The sharpness result, for finite $n$, uses recent work of Harrison-Trainor on “unfriendly jump inversion.”

For a prescribed classical profile and a fixed positive regularization parameter, we construct an Itô diffusion whose interior generator reproduces the regularized $p(x)$-Laplacian when applied to that profile. We determine the exact tangent spectra of the unregularized and regularized fluxes, characterize their behavior at zero gradient, and establish quantitative ellipticity bounds on bounded gradient ranges. The full divergence expansion identifies a covariance equal to twice the gradient derivative of the regularized flux, evaluated at the profile gradient, and a logarithmic drift generated by spatial variation of the variable exponent $p(x)$. Under explicit classical regularity assumptions, we establish boundedness and local Lipschitz continuity of the coefficients and prove strong existence and pathwise uniqueness up to domain exit. We identify the interior generator of the boundary-stopped process and verify the profile-reproduction identity. Further consequences include initial-point stability for a fixed global extension, localized martingale identities, and Dynkin's formula at bounded stopping times. An explicit nonconstant exponent example illustrates the construction at a critical point, and the coefficient selection is shown to be canonical under the flux-based prescriptions rather than determined uniquely by profile reproduction alone.

We establish dimension bounds and an exact dimension formula for equilibrium measures of degree $d$ holomorphic endomorphisms of $\mathbb P^2=\mathbb P^2(\mathbb C)$ satisfying suitable expansion and domination conditions. Write $\lambda_1>\lambda_2>0$ for their Lyapunov exponents. If two inverse histories ending at the same point determine different fast directions, we prove that ${\mathrm dim}_H \mu_F>\log d/\lambda_1+\log d/\lambda_2$ when $\lambda_2>\log d$, and that ${\mathrm dim}_H \mu_F=2\log d/\lambda_2$ when $\lambda_2\ge2\log d$. Applying these results to an explicit quadratic family, we disprove the Binder–DeMarco conjecture on a non-empty open set of holomorphic endomorphisms. The proof adapts the projection growth strategy of Li–Pan–Tong–Xu to the holomorphic setting, combining strong leaf geometry and the Ledrappier–Young theory for endomorphisms with a multidimensional extension of Wu's restricted sum estimate.

We show that unlike $\textsf{PFA}$, $\textsf{MA}_{\omega_{1}}$ is consistent with the existence of $2^{\omega_{1}}$-many Countryman lines which pairwise contain no uncountable isomorphic or reverse-isomorphic suborders. This clarifies an issue regarding the claim that $\textsf{MA}_{\omega_{1}}$ implies that there is a finite basis for the Countryman lines. The family of Countryman lines comes from a family of $2^{\omega_{1}}$-many coherent trees which pairwise contain no club-isomorphic subtrees. To prove this, we first construct such a family of trees using $\diamondsuit$, and then we iterate ccc forcings while preserving this property.

Subset currents on free groups, on surface groups and, more generally, on word-hyperbolic groups, were introduced by Kapovich and Nagnibeda, and further studied by Sasaki. Subset currents extend ordinary geodesic currents and provide measure-theoretic generalizations of conjugacy classes of infinite quasiconvex subgroups. Their supports can be viewed as “branching laminations". For free groups and surface groups, we construct non-rational subset currents with genuinely branching supports that carry no other subset currents except scalar multiples. For every nonabelian finite rank free group and every closed hyperbolic surface group, we show that such a current can be chosen so that its support is uncountable, is disjoint from the locus of two-point boundary subsets, and consists entirely of Cantor subsets of the boundary. More generally, we prove that every nonempty metrizable Choquet simplex occurs as a compact convex base for the cone of currents carried by a minimal genuinely branching subset lamination. The same universality phenomenon extends, apart from the usual rational-current terminology, to every non-elementary torsion-free hyperbolic group. We also study finite-pattern complexity. We prove that if $R_L(n)$ counts the allowed round patterns of radius $n$ in a minimal subset lamination $L$, then $R_L(n)=O(n)$ forces the carried-current cone to be finite-dimensional, while infinite-dimensionality forces $R_L(n)/n\to\infty$; this universal threshold is sharp. We show that the coarse growth type of $R_L$ is independent of the chosen free basis and, more generally, of the chosen finite marked graph. We also show that a refined switch system recovers the exact dimension of the carried-current cone.

We study which structural results about Aronszajn trees require the Proper Forcing Axiom and which require only MA(aleph1). In particular, we show that MA(aleph1) is consistent with the existence of 2^aleph1 pairwise non-club-isomorphic coherent A-trees.

For an arbitrary prime $p$ and arbitrary positive integers $r$ and $d_{1}, \ldots, d_{r}$ with $d_{r} > \cdots > d_{1}$, we show that $x^{d_{r}} + x^{d_{r-1}} + \cdots + x^{d_{2}} + x^{d_{1}} + p$ removed of all of its monic irreducible reciprocal factors is irreducible. As a consequence, we establish a number of results related to a conjecture of Charles Nicol that, over the rationals, the sum of two cyclotomic polynomials $\Phi_{n}(x) + \Phi_{m}(x)$ is a product of cyclotomic polynomials times either $2$ or an irreducible polynomial, where $n$ and $m$ are integers exceeding $1$. We also establish similar results for $\Phi_{n}(x)\Phi_{m}(x)+1$.