Papers

We give two new lower bounds on the rank of categorified link invariants: one on the link Floer homology of fibered links in terms of the fractional Dehn twist coefficients of their monodromies, and another, as a corollary, on the annular Khovanov homology of braid closures in terms of the fractional Dehn twist coefficient of the braid. The most important technical component of the proof is that we determine the behaviour of the link Floer homology of fibered links under adding boundary Dehn twists to their monodromies in the next to top Alexander grading.

Let $G$ be a group. A bijection $f\colon G\to G$ is called inverse ambiguous if $f^{-1}(x)=f(x)^{-1}$ for every $x\in G$. We prove that every infinite group admits an inverse ambiguous function. An inverse ambiguous automorphism can exist only on an abelian group. We classify the finitely generated and divisible abelian groups admitting such automorphisms. If $A\cong\Z^r\oplus T$, where $T$ is finite, then $A$ admits an inverse ambiguous automorphism if and only if $r$ is even and $T$ admits one. Together with Toborg's finite classification, this gives an explicit classification of the finitely generated abelian groups admitting such automorphisms. For a divisible abelian group $A\cong\mathbb Q^{(\kappa_0)}\oplus \bigoplus_p C_{p^\infty}^{(\kappa_p)}$, such an automorphism exists if and only if none of $\kappa_0$, $\kappa_2$, or $\kappa_p$ for $p\equiv3\pmod4$ is a finite odd cardinal. We also prove that if every proper subgroup of an infinite locally finite group $G$ admits an inverse ambiguous automorphism, then $G$ admits one that leaves every subgroup invariant. Finally, we prove that a residually finite group whose finite quotients admit inverse ambiguous automorphisms is abelian. However, we prove the existence of infinite residually finite abelian groups whose finite quotients all admit such automorphisms, but the group itself does not.

A framework is developed for the inference of dynamics described by a generalized system of ordinary differential equations. A stochastic gradient method is coined that infers dynamics from observed marginal probability density functions using the joint probability density function of the observable and latent variables. Diffusion and other irreversible processes observed in a low-dimensional state can be recast as deterministic, reversible flows in a sufficiently augmented state space, where the joint density satisfies the hyperbolic Liouville equation. The marginal distribution observed is the projection of these hyperbolic dynamics onto the observed coordinates, with the latent components carrying the randomness and memory. This reframing allows inference for irreversible or stochastic dynamics into the recovery of a deterministic Ordinary Differential Equation (ODE) from marginal observations. Instead of solving the high-dimensional Liouville equation for the joint density, the algorithm exploits its characteristic representation. Particles sampled from the initial distribution are transported along characteristic lines. The Eulerian sensitivity with respect to parameters is obtained by sensitivity propagation along the characteristic lines, with a crossed U-statistic producing an unbiased gradient estimator, which enables stochastic gradient descent. Four experiments validate the method: recovery of a three-mode linear system observed through the marginal of a single mode; a nonlinear Gompertz growth model with a hidden mode; a bistable system whose hidden mode turns a unimodal marginal bimodal; and Stokes–Oseen drag law recovery for particles in a cellular flow. Convergence behavior is analyzed across these settings.

We develop a stable framework for exterior homotopy theory and apply it to proper homotopy theory. Starting from a known simplicial model structure on the category of exterior spaces we then pass to the proper, cellular model category of retractive exterior spaces over the ray, and establish the associated stable homotopy theory of exterior spectra. The resulting framework provides exterior analogues of several classical constructions in stable homotopy theory and sheds new light on proper homotopy theory, where the development of a satisfactory stable theory has been hindered by the lack of suitable categorical properties. As an application, we establish a Brown representability theorem for reduced exterior cohomology theories. This yields representability results for proper cohomology theories arising from the exterior setting, including cohomology theories with compact supports associated with classical generalized cohomology theories.

We study stochastic extragradient (SEG) methods for solving monotone variational inequality problems (VIPs) over a feasible set. Although extragradient is a foundational algorithm for VIPs and its deterministic convergence theory is well developed, its stochastic counterpart remains less understood. Most existing analyses focus on independent-sample SEG (I-SEG) and assume either that the domain is compact or that the variance of the stochastic operator is uniformly bounded. The behavior of same-sample SEG (S-SEG), a natural variant with materially different properties, has received far less attention. In this work, we address these gaps in the literature. We first show that S-SEG is sensitive to samplewise Lipschitz parameters: mean Lipschitzness and bounded variance alone do not ensure convergence, even on a compact set. Then, for possibly unbounded domains, we establish a high-probability restricted-gap convergence for each SEG variant under a relaxed set of assumptions, and show that certain fundamental improvements to these results are impossible in general. Finally, we show that a known asymmetric double step-size selection that guarantees almost sure last-iterate convergence for I-SEG can fail for S-SEG: there exists a stochastic monotone VIP for which S-SEG diverges almost surely even under the modified step-sizes.

The Intersection Euler Characteristic Profile (Intersection ECP) of $k$ colored point clouds $X_1, \ldots, X_k \subset \mathbb{R}^d$ is the Euler characteristic $\chi(\bigcap_{i=1}^k \mathcal{U}(X_i; t_i))$ of the overlap of their ball unions—an integer-valued, multiparameter invariant of their topological interaction across scales. Its organizing framework is the Euler calculus on constructible functions: the profile is equally the Euler integral $\int \prod_{i=1}^k \mathbf{1}_{\mathcal{U}(X_i; t_i)} \, d\chi$ of the product of the $k$ data-dependent offsets, and this identity—our Intersection Theorem—is a commuting square interchanging geometric intersection and algebraic product. The invariant is rigid-motion invariant, scale-equivariant, and $L^1$-stable, and it is canonical: among pointwise-Euler interaction profiles it is the one forced by separation and normalization, the top floor of a spectrum of descriptors graded by how many clouds meet. For $n$ points a single sorted Alpha-complex sweep computes it in $O(n^{\lceil d/2 \rceil} \log n)$ time with no persistence reduction, worst-case optimal in even dimensions. Where the Euler characteristic cancels, a relative-homology refinement resolves the finer interaction and is stable in the two-parameter interleaving distance. Finally, for increasingly dense samples the profile and its refinement are consistent, recovering the (relative) homology and Euler characteristic of the underlying shapes—in the inverse limit for compact sets, and, under positive reach, persistently and with explicit sample complexity.

Consider the oscillatory integral operators \begin{equation} T_{\lambda}f(y)=\int_{\mathbb{R}^{2}}e^{i\lambda S\left( x_{1},x_{2}% ,y\right) }\Phi(x_{1},x_{2},y)f(x_{1},x_{2})dx_{1}dx_{2},\nonumber \end{equation} where $\Phi(x_{1},x_{2},y)\in C_{0}^{\infty}\left( \mathbb{R}^{3}\right) $, $S\left( x_{1},x_{2},y\right) \in C_{0}^{\infty}\left( \mathbb{R} ^{3}\right) $ is real valued, and $\lambda$ is a large real number. We prove that, if $S\left( x_{1},x_{2},y\right) =y^{n_{1}}h_{n-n_{1}}\left(x_{1},x_{2}\right) +\cdots+y^{n_{s}}h_{n-n_{s}}\left( x_{1},x_{2}\right) $ is a homogeneous polynomial of degree $n,$ where $0<n_{1}<n_{2}<\cdots <n_{s}<n$, and $h_{n-n_{1}}\left( x_{1},x_{2}\right) $ and $h_{n-n_{s}}\left( x_{1},x_{2}\right) $ are non-degenerate in the sense that there are no multiple factors when they are factored into linear terms over complex numbers, then for $\delta=\max \left( \frac{n}{3},\frac{n-n_{s}}{2},n_{1}\right) >1,$ $\left \Vert T_{\lambda}\right \Vert _{L^{2}\rightarrow L^{2}}=O\left( \lambda^{-1/\left( 2\delta \right) }\right) $, while in the endpoint case $\delta=1$ the bound becomes $\left \Vert T_{\lambda}\right \Vert_{L^{2}\rightarrow L^{2}}=O\left( \lambda^{-1/2}\log \lambda \right) $. The decay rate is sharp, up to a power of $\log \lambda$ when $\delta=1$. We further show that $\delta$ is exactly the modified Newton distance for the phase function, thus verifies the conjecture of \citet{Greenleaf07} in this case.

We investigate the extent to which Whitehead's theorem remains valid for minimal finite models. We show that it fails in this setting, answering negatively a question posed by Barmak. More precisely, for every $n\geq 2$, we construct a weak homotopy equivalence between two $(2n+4)$-point minimal finite models of $S^n\vee S^{n-1}\vee S^{n-1}$ which are not homotopy equivalent. The minimality of these examples follows from a near-extremal classification theorem: if a connected finite space has at most $2n+3$ points and nonzero $n$th homology over a field, then its order complex is homotopy equivalent either to $S^n$ or to $S^n\vee S^k$ for some $1\leq k\leq n$. Finally, we prove a positive Whitehead-type result: under a natural cohomological rigidity hypothesis, every weak homotopy equivalence between minimal finite models is a homotopy equivalence.

The physics of spreading in static networks is well understood through mappings to percolation. We show that spreading dynamics on temporal networks can analogously be mapped to reachability in temporal event graphs. This provides a theoretical and computational framework for a class of processes, such as variants of the susceptible-infected-susceptible model. Without explicit simulations, through the component analysis of event graphs, we obtain epidemic prevalence and derive epidemic thresholds for temporal networks with arbitrary degree and inter-event time distributions, with significant computational advantages as compared to explicit simulations.

We investigate the vanishing viscosity limit for the one-dimensional compressible Navier-Stokes equations in the regime of two interacting shock waves from the same characteristic family of the underlying Euler equations. Unlike the interaction of two shocks from distinct families, which produces two outgoing shocks in their respective original families, the collision of two same-family shocks generates an outgoing shock in the same family together with a rarefaction wave in the other family. This configuration is more singular because the collision occurs on a larger time scale that depends inversely on the wave strength, making it challenging to justify the vanishing viscosity limit not only in the processes before and after the shock collision but, more crucially, at the collision point. Furthermore, the emergence of both shock and rarefaction waves after the collision introduces an additional difficulty. To overcome these obstacles, we first employ the anti-derivative method before the collision, which allows us to fix the locations of both viscous shocks precisely up to the collision point. However, uniform estimates with respect to the viscosity cannot be closed up to the collision time; we therefore introduce a carefully constructed approximate collision time to obtain the required higher-order energy bounds. After the collision, we apply a weighted relative entropy method, combined with time-dependent shifts, to handle the composite wave structure arising from the coexistence of shock and rarefaction waves. Our main result establishes that, for suitably small wave strengths and viscosity coefficients, there exists a family of global smooth solutions to the Navier-Stokes equations that converge to the entropy solution of the Euler equations with an explicit convergence rate. The techniques are expected to be applicable to other related problems in vanishing viscosity theory.

Collective decisions in agent networks emerge from repeated local interactions, as agents update their opinions (colors) based on their neighbors, potentially leading to consensus. In sparse communication graphs, repeated interactions can amplify both the initial majority preference and the random fluctuations caused by individual agents. Understanding when the initial majority determines the final preference and how quickly an agreement emerges is, therefore, a basic question in collective decision-making. We study binary majority dynamics as an idealized model of this process. At each round, the interaction graph is independently resampled from the Erdős–Rényi model $\mathbb{G}(N,p)$, with $p=b\log N/N$ and fixed $b>1$. Each agent adopts the majority opinion among its neighbors, retaining its current opinion if there is a tie. For a fixed initial configuration, let $\Delta_0$ denote the initial difference between the number of blue agents and the number of red agents. We identify three regimes for the consensus time, determined by $|\Delta_0|$. First, when $\Delta_0\sqrt p=O(1)$, the probability of blue unanimity is $$ \Phi\left(\Delta_0\sqrt{2p/\pi}\right) + o(1), $$ where $\Phi$ is the cumulative distribution function of a standard Gaussian. Moreover, consensus is reached after $(1+o(1))\log N/\log\log N$ rounds with high probability. Second, in the intermediate regime $1\ll\Delta_0 \sqrt{p}\lesssim \sqrt{N}$, we establish explicit high probability upper and lower bounds on the time to blue-unanimity. Lastly, when the initial blue advantage is above an explicit constant multiple of $N/\sqrt{\log N}$, blue unanimity is guaranteed within two rounds with high probability. Our findings provide a complete description of how the initial advantage influences winner selection and the speed of consensus in this type of majority dynamics with sparse interactions.

Conditional expectation operators (CEOs) and their associated conditional mean embeddings (CMEs) play a central role across applied mathematics and machine learning, appearing in nonparametric regression, Bayesian inverse problems, and Koopman operator theory. A fundamental question is when a CEO maps a function space on $\mathcal{Y}$ into a prescribed function space on $\mathcal{X}$, particularly a reproducing kernel Hilbert space (RKHS). We show that such mapping properties are characterized by the regularity of the Radon–Nikodym density of the conditional law, and establish a simple, verifiable sufficient condition under which the CEO is bounded and Hilbert–Schmidt. For RKHSs norm-equivalent to Sobolev spaces, this condition reduces to Sobolev regularity of the conditional density. The result yields a direct route to validate CME representations and error bounds for Galerkin-type and CME-based estimators. We verify the regularity condition in three settings: nonparametric regression, Bayesian inverse problems, and Koopman operator theory for stochastic dynamical systems. We show in each case that classical regularity results on the underlying probabilistic model imply the required mapping properties. The resulting framework offers a unified perspective on conditional expectation operators across probability, operator theory, kernel methods, and stochastic dynamics.

We classify equisingular families of pencils of curves on smooth simply connected projective surfaces whose dual varieties have degree at most 6, and which contain sufficiently large number of members which irreducible components are either hyperplane sections or irreducible components of hyperplane sections. As a consequence, we describe the fundamental groups of complements to curves on such surfaces whose irreducible components satisfy the above condition and whose fundamental groups admit an essential surjection onto a free group of rank greater than 6.

In the $3$-neighbor bootstrap percolation process, a vertex becomes (and remains) infected if at least three of its neighbors are infected. We say that an initial configuration of infected vertices percolates if eventually all vertices are infected. We exactly determine the size of the minimum percolating set for the $3$-neighbor bootstrap percolation process on all remaining open cases for rectangular grid graphs $P_m\square P_n$. This extends earlier work of Dukes, Noel, and Romer. Additionally, we consider the same question for the toroidal grids $C_m\square C_n$, proving upper and lower bounds which are at most one apart and determining the answer precisely in many divisibility cases.

Mechanistic chromatography models are valuable for process development, but gradient-based parameter estimation and optimization can be hindered by computational cost and the effort of deriving objective-function gradients. To address this concern, a fully differentiable Julia chromatography solver, ChromOps.jl, is presented that combines high-order spatial discretization with discrete forward sensitivity analysis (DFSA). Two high-order spatial discretizations, finite difference summation-by-parts (FD-SBP) and the discontinuous Galerkin spectral element method (DG-SEM), are compared on a 6-component ion-exchange chromatography problem with steric mass action kinetics. Both attain their theoretical convergence rates on manufactured problems, while FD-SBP shows favourable work-precision performance for both forward (primal) and dual-valued simulations and is conceptually simpler to implement. DFSA scales as tDFSA approx 1.4(1+Np)tfwd for up to Np = 24 parameters propagated in a single ForwardDiff.jl chunk, i.e. about 1.4 forward solves per additional parameter. Because the primal and dual-valued solves share the same code path, DFSA provides accurate gradients of user-defined objectives without any hand-derived chain-rule code, enabling accessible gradient-based parameter estimation and optimization.

Motivated by the need to reason about case statements quantitatively, we extend quantalic linear lambda-calculus with additive disjunction. We show that the resulting equational system is sound. We also show that when certain continuity properties (of the underlying quantale) are adopted, it is additionally (approximately) complete. We present several models of the extended calculus, involving for example meta-theoretical properties in categorical logic (gluing), probabilistic, and quantum computation. As a concrete application, we illustrate how a probabilistic model, based on Banach spaces, can be synergistically used with the calculus' equational system to reason about Cauchy sequences of random walks. This highlights the emergent shift from "program semantics as the science of program equivalence" to flexible, quantitative perspectives, involving functional analysis and beyond.

The extended-onion and C-vine constructions of Lewandowski, Kurowicka and Joe (2009) are standard methods for sampling from the LKJ_n(eta) distribution on correlation matrices. We show that both arise from the simpler row-normalized Bartlett construction associated with the restricted-Wishart representation of Wang, Wu and Chu (2018), which exposes redundancies hidden in standard gamma-based implementations of the classical constructions. Two exact row-wise couplings establish this: the squared norm of the Gaussian vector supplying the onion's direction has exactly the Gamma law required for one component of the Beta radius, and the same vector, with one chi-squared variate, generates the entire row of mutually independent C-vine partial correlations with their required symmetric-Beta laws. We also show that, relative to flat off-diagonal measure, the LKJ family maximizes entropy at fixed expected log-determinant; the dual natural parameter is eta-1. Under gamma-ratio accounting, normalized Bartlett, the onion, and the conventional symmetric-Beta C-vine require n-1, 2(n-1), and n(n-1) Gamma-equivalent calls. Controlled benchmarks confirm a low-dimensional advantage over the onion implementation and a persistent advantage over the C-vine implementations examined; direct Bartlett normalization also avoids subtractive complements, moving the small-eta zero-diagonal threshold from machine-epsilon scale toward the subnormal range. The sampler is valid for every real eta > 0 and requires only standard normal and chi-squared variates.

Machine learning offers a promising avenue to accelerate physical simulations by replacing computationally expensive traditional Partial Differential Equation (PDE) solvers with fast, differentiable surrogate models. However, standard auto-regressive ML emulators often suffer from error accumulation over long horizons and struggle to capture the stochasticity of complex physical systems. In this paper, we propose Kastor, a comprehensive methodology to adapt a deterministic physics foundation model into a highly efficient and accurate generative surrogate. First, we introduce a two-stage inference scheme that combines a large-stride causal auto-regressive model with a non-causal temporal super-resolution network, significantly reducing error accumulation while minimizing computational cost. Second, we present Mean prediction regularization (MPR), a novel training objective that constrains the generative model to predict the deterministic distribution mean under null noise conditioning. This regularization dramatically improves the performance and stability of both Functional Generative Networks (FGN) and diffusion-based emulators. Finally, we demonstrate that incorporating spatial gradient matching improves the accuracy and physical fidelity of the simulations as measured by power spectrum density. Extensive evaluations on diverse simulation datasets of the benchmark The Well show that with these components, our model outperforms competing methods in forecasting accuracy, spectral consistency, and computational efficiency. Our model achieves a 42.9% average reduction in forecasting compared to our reference based on the Walrus finetuning methodology, and outperforms Walrus for 8 out of 10 datasets on variance-normalized RMSE (VRMSE).

For each prime $p$ and integer $n \geq 1$, we construct a $p$-local spectrum $X_n$ with $X_n^{\otimes n} \not\simeq 0$ but $X_n^{\otimes (n+1)} \simeq 0$. This refutes the retract conjecture of Hovey-Palmieri. Our construction also allows us to determine the cardinality of the Bousfield lattice to be $2^{2^{\aleph_0}}$, answering a question of Dwyer-Palmieri.

A compact metrizable system is null if its topological sequence entropy vanishes along every sequence of times. We prove that nullness is equivalent to polynomial maximal pattern complexity for every finite open cover, while equicontinuity is equivalent to sublinear maximal pattern complexity. The first characterization is obtained from finite fat-shattering at every positive scale and polynomial empirical covering of orbit-distance classes. We also construct transitive nonminimal null systems with properties excluded in the minimal setting: one is uniformly rigid and has two fixed points, and another is two-scattering. These results settle several long-standing open problems from the literature on polynomial maximal pattern growth and on the structure of transitive nonminimal null systems.

This work develops a thermodynamically consistent phase-field model for tumor growth based on the energetic variational framework. The model couples the Cahn-Hilliard equations for tumor evolution and nutrient transport with the incompressible Navier-Stokes equations. A first-order time discretization scheme based on the Multiple Scalar Auxiliary Variables (MSAV) approach together with a pressure-correction strategy is proposed to efficiently handle the nonlinear and coupled structure of the system. The proposed scheme is rigorously proved to be unconditionally energy stable and mass conservative. Furthermore, optimal first-order temporal error estimates are established for the tumor phase-field variable, the nutrient concentration, and the fluid velocity. Finally, numerical experiments demonstrate the effectiveness and robustness of the proposed method and verify the theoretical convergence rates.

We study the existence and multiplicity of principal eigenvalues and eigenfunctions for a periodically heterogeneous nonlocal dispersal model with advection. The operator we consider is resolvent-positive but not resolvent-compact; therefore, the classical Krein-Rutman theory cannot be applied directly. When the advection coefficient has a constant sign, we prove the existence and uniqueness of the principal eigenvalue and the corresponding normalized eigenfunction. In sharp contrast, when the advection does not have a constant sign, the problem is more involved and leads to surprising results. Depending on the coefficients of the equation, the principal eigenproblem can either have a unique normalized solution or a continuum of solutions, at the boundary of which there exists a principal eigenvector with a singular measure component. In the latter situation, all the constructed eigenvalues are embedded in the continuous spectrum of our operator. We completely characterize the eigenvalues associated with positive eigenvectors, even when the eigenvector is a Radon measure. Finally, we discuss an application to a nonlinear KPP-type equation with nonlocal dispersal, which possesses a continuum of nontrivial stationary solutions, a different behavior from the classical KPP equation with local diffusion.

We introduce an empirical version of dual volumes within dual Brunn–Minkowski theory. These provide a simple model for computation and analysis under symmetrization, including moment inequalities. As an application, we derive a new proof of the Busemann intersection inequality that relies on counting points in slabs rather than working directly with volumes of slices.

We establish a criterion relating smooth and contact characterizing slopes under a uniqueness assumption. Let $L$ be a Legendrian representative of a knot $K\subset S^3$ with standard contact structure, and assume that the isotopy class of $L$ is uniquely determined by its classical invariants: the Thurston–Bennequin invariant $tb(L)$ and the rotation number. Then, for any non-zero rational number $r$, if $r+tb(L)$ is a smooth characterizing slope for $K$, it becomes a contact characterizing slope for $L$. As applications, we study the characterizing slopes for Legendrian representatives of the unknot, trefoil, figure-eight knot, cinquefoil, $5_2$, and $\overline{5_2}$.

A finite nonempty set $F\subset\mathbb{N}$ is Schreier if $\min F\ge |F|$. First, we prove a linear recurrence relation and compute initial counts for Schreier sets consisting of intervals. Two intervals of integers are separated if their union is not an interval. If $\mathcal J_{k,n}$ is the collection of Schreier sets that are the union of exactly $k$ separated intervals, then the sequence $(|\mathcal{J}_{k,n}|)_{n=1}^\infty$ satisfies the characteristic polynomial $p_k(x) = (x-1)^{2k+1}(x+1)^k$. Furthermore, we introduce the new concept of $k$-super-Schreier sets and let $\mathcal{S}_{k,n}$ denote the collection of $k$-super-Schreier sets whose maximum is $n$. We show that the sequence $(|\mathcal{S}_{k,n}|)_{n=1}^\infty$ satisfies a Fibonacci-type recurrence with a remainder term expressible as a polynomial of $n$.

Let $u(n)$ be the largest integer $d$ such that every $n$-vertex claw with at most $d$ branches occurs in every tournament on $n$ vertices, and let $c_{\mathrm{claw}}=\limsup_{n\to\infty}u(n)/n$. In 1998, Lu, Wang and Wong proved that $19/50\le c_{\mathrm{claw}}\le11/23$, and these have remained the best bounds known. We improve them to $2/5\le c_{\mathrm{claw}}\le10/21$. We also isolate two parameters $\theta$ and $\sigma$ which place the lower- and upper-bound arguments in a common framework: we show $\sigma\le\theta$ and $1/21\le\sigma\le\theta\le1/5$, our two bounds being the images of the endpoints under $\alpha\mapsto\frac12-\frac{\alpha}{2}$, and $\sigma=\theta$ would force $\lim u(n)/n$ to exist.

Network coding allows intermediate nodes to encode received messages before transmission. The multiple-unicast conjecture asserts that coding has no throughput advantage over fractional routing for independent unicast sessions in any undirected network. Despite more than two decades of sustained study, this central open problem remains unresolved. The conjecture is deeply connected to computational complexity: a proof would yield long-sought lower bounds for fundamental problems. To study the conjecture, this paper develops a unified metric framework from the perspective that the basic objects behind the comparison between coding and routing are not cuts alone, but graph metrics. Using this framework, we prove the conjecture for three new classes of undirected networks: (a) networks with at most five terminal locations; (b) planar networks whose terminal locations lie on the boundaries of at most three designated faces, with each session's endpoints on one such face; and (c) networks with arbitrarily many nodes and terminal locations under a structural restriction on session endpoints. We give a new proof that the conjecture holds for networks with at most six coding nodes, without computer-aided search, and show that if the conjecture holds on $\Gamma_{3,3}$, then it holds whenever no three sessions have six distinct terminal locations.

A fusion category $\mathcal{C}$ is said to be $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$-quadratic if the group $G(\mathcal{C})$ of invertible objects is isomorphic to $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$, and the remaining simple objects form an orbit under the action of $G(\mathcal{C})$. In this paper, we give a partial classification of $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$-quadratic fusion categories of rank six. More precisely, we show that its Grothendieck ring $\mathcal{K}_0(\mathcal{C})$ must be one of nine fusion rings if the fusion rule multiplicities are less than $20$, and the categorifications of five of them are previously known. We prove that one of the last four fusion rings can be realized as de-equivariantization of a near-group fusion category of type $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/4\mathbb{Z}+8$.

Beisegel et al. [SWAT 2024] introduced the concept of simultaneous $\mathcal{C}$-numbers which associate a graph class $\mathcal{C}$ with a graph parameter. Given a graph $G$, the simultaneous $\mathcal{C}$-number is the smallest number $d$ for which there is a graph $H \in \mathcal{C}$ and a function $L : V(G) \to \mathcal{P}(\{1,\dots,d\})$ such that two vertices $u$ and $v$ are adjacent in $G$ if and only if they are adjacent in $H$ and their sets $L(u)$ and $L(v)$ are not disjoint. We study the relation of these simultaneous $\mathcal{C}$-numbers to other graph parameters. In particular, we investigate which parameters fulfill the following property: Parameter $p$ is bounded on class $\mathcal{C}$ if and only if $p$ is bounded on the class of graphs of simultaneous $\mathcal{C}$-number $d$ for any fixed $d$. We show that many well-known graph parameters have this property. Examples are cliquewidth, twin-width, mim-width, tree independence number, thinness as well as boxicity. We furthermore present some parameters, including modular-width and tree-length, that do no have this property. We also study when a parameter forms an upper bound on a simultaneous $\mathcal{C}$-number. We characterize those graph classes $\mathcal{C}$ for which the parameters treewidth, pathwidth, bandwidth, and treedepth upper bound the simultaneous $\mathcal{C}$-number. Furthermore, we present sufficient conditions on a class $\mathcal{C}$, such that $\mathcal{P}$-modular cardinality upper bounds the simultaneous $\mathcal{C}$-number, where $\mathcal{P}$ is replaced by the complete graphs, the edgeless graphs, cographs, or the class $\mathcal{C}$ itself. On the contrary, we show that modular width never forms an upper bound on a non-trivial simultaneous $\mathcal{C}$-number. Finally, we present some general algorithmic results on the clique problem and computation of simultaneous $\mathcal{C}$-numbers.

Let \(\pi_E:X_0(M)\to E\) be the strong Weil parametrization with Manin constant \(c_E\). We prove \(\deg \pi_E\mid c_E^{\Omega(N/M)}\deg g\) for every multiple \(N\) of \(M\) and every nonconstant morphism \(g:X_0(N)\to E'\) over \(\mathbb{Q}\), where \(E'\) is \(\mathbb{Q}\)-isogenous to \(E\) and \(\Omega\) counts prime factors with multiplicity. When \(c_E=1\), as is known for squarefree \(M\), the modular degree at level \(M\) therefore divides every such degree at every higher level. As an application of the divisibility theorem, we prove that no \(X_0(N)/\mathbb{Q}\) admits a morphism over \(\mathbb{Q}\) of positive odd degree at most \(1645\) to an elliptic curve of positive \(\mathbb{Q}\)-rank. For a fixed target \(E'\) and a generator \(u:E\to E'\), we also prove that if the Manin constant \(c_{u\circ\pi_E}=1\), the old homomorphisms induced by degeneracy maps form an integral basis of \(\operatorname{Hom}_{\mathbb{Q}}(J_0(N),E')\), and the old degree matrix determines the exact morphism degrees. The proofs bound denominators in the rational old basis. The divisibility and lattice results extend to compatible towers of intermediate modular curves, including the \(X_1\)-tower.