Fang, Lan, and Xiao established the geometric Green's formula as a categorical isomorphism for arbitrary semisimple complexes. In this short note, we generalize their work to multi-step compositions. Specifically, we establish the iterated geometric Green's formulas for the composition of an $(n-1)$-fold restriction and an induction, as well as its dual.

Let $T(n)=n+τ(n)$, where $τ$ is the divisor function. We study the Erdos-Graham coalescence problem by encoding finite-level obstructions in the divisor-successor graph and in square-annular transfer maps. Coalescence is equivalent both to connectedness of this graph and to synchronization along an infinite non-autonomous sequence of finite annular systems. The basic identities are \[ \operatorname{im}(\mathcal A_k)=E_{k+1}, \qquad \mathcal F_{k^2}=k^2+E_k, \] where $E_k$ is the set of square-crossing overshoots from below $k^2$. We prove a transfer parity law, dynamic frontier bounds for the widths $W_{k,s}$, and the criterion that $\liminf_k|\mathcal A_k(E_k)|=1$ would imply connectedness. Unconditionally, \[ R(X)\le \log X+2γ+O(X^{-1/4}), \] and the exit sets are residue-universal, satisfy $|E_k|\le k^{o(1)}$, and obey \[ \frac94K+O(1)\le \sum_{k\le K}|E_k|\ll K(\log K)^3. \] Using the shifted-square estimate HST, obtained from the corrected Henriot--Nair--Tenenbaum theorem in the specialized form of Proposition 8.4 and from separate square-shift estimates, we obtain fixed-moment bounds \[ \sum_{k\le K}|E_k|^m\ll_m K(\log K)^{C_m}\quad(m\ge2). \] A further first-moment refinement to $K(\log K)^2$ is conditional on the additional, currently unproved, uniform quadratic Euler-product mean-value hypothesis HQE. We also prove quantitative large-jump and lower-runner race theorems, isolate interval filling, and formulate a square-gated two-branch criterion. No proof of the full Erdos-Graham problem is claimed.

In this short note, we prove that if $A \in \mathbb C^{N \times M}$ with $N=4M-5$ has i.i.d.\ standard complex Gaussian entries, then the probability that the phase retrieval map generated by $A$ is not injective is positive. This proves Part (1) of a conjecture of Cynthia Vinzant, which was later restated by Afonso S. Bandeira in \cite{BDL+26}. The main result of this paper was obtained using generative AI, in particular the Rethlas system.

We develop and analyze a conforming Virtual Element Method (VEM) for the fourth-order Poisson-Nernst-Planck-Navier-Stokes (PNP-NS) system. The proposed scheme is based on compatible discretizations of each component: an \(H^2\)-conforming VEM for a fourth-order electrostatic potential equation, an \(H^1\)-conforming VEM for the Nernst--Planck equations, and a \textit{divergence-free} and \textit{pressure-robust} VEM for the Navier--Stokes equations. Time integration is performed using a backward Euler scheme to ensure stability. We establish the well-posedness of the continuous problem up to three-dimensions and also establish existence and uniqueness of the fully discrete solution via a fixed-point argument. Further, we derive a priori error estimates showing that the electrostatic potential, concentration and velocity converge optimally in Bochner norms \(L^\infty\bigl(0,T; H^2(Ω)\bigr)\), \(L^2\bigl(0,T; H^1(Ω)\bigr)\), and \(L^2\bigl(0,T; \bm{H}^1(Ω)\bigr)\), respectively. The analysis requires a sophisticated argument to avoid any restrictive assumptions on the coercivity and continuity constants and to handle the trilinear form involving three different variables. The pressure-robust design permits the use of lowest-order pressure approximations without compromising convergence rates of the other variables, reducing computational cost. Numerical experiments confirm the theoretical convergence rates for various polynomial orders and demonstrate the scheme's robustness, including in low-viscosity regimes.

We study the asymptotic enumeration of labelled connected multitype graphs in the sparse regime, where both the number of vertices and edges grow linearly and the excess is proportional to the size of the graph. Extending the classical theory of connected graph enumeration to the multitype setting, we consider graphs with prescribed numbers of vertices of each type and prescribed edge counts between each pair of types. Our approach is probabilistic and relies on the theory of inhomogeneous random graphs. In particular, we exploit large-deviation principles and asymptotic estimates for connectedness probabilities to relate the counting problem to the emergence of giant components in suitably tuned supercritical random graphs. From large deviation asymptotics of connected components of inhomogeneous random graphs, we recognize that a connected graph with a given edge statistics corresponds to the (unique) giant component of larger inhomogeneous random graph with a suitably chosen connection kernel. This correspondence allows us to derive the leading exponential asymptotics for the number of connected multitype graphs with fixed type profile and edge matrix. The resulting formula generalizes the asymptotic enumeration results of Bender, Canfield, and McKay for connected sparse graphs to the multitype framework. More broadly, the paper illustrates how probabilistic techniques can provide transparent and effective tools for addressing new combinatorial enumeration problems.

For a graph $G$ put $h_r(G)=\max{χ(H):H\subseteq G,\operatorname{girth}(H)\ge r}.$ Erdős and Hajnal asked whether $h_r(G)\to\infty$ as $χ(G)\to\infty$, for every fixed $r\ge4$. We prove this in every fixed polynomial edge-density regime: for all $r\ge4$, $k\ge2$, $P,C>0$, there is $M=M_{r,k}(P,C)$ such that $χ(G)\ge M,\ e(G)\le Cχ(G)^P\Longrightarrow h_r(G)\ge k.$ Quantitatively, after replacing $P$ by $P\vee2$ and $C$ by $C\vee2$, $M_{r,k}(P,C)\le \exp!\left(O_{r,k}\bigl((P+2+\log(C\vee2))^2\bigr)\right),$ and consequently the same conclusion holds throughout the quasi-polynomial range $e(G)\le \exp\bigl(C_0(\logχ(G))^a\bigr),\ 1 < a < 3/2,$ for all sufficiently large $χ(G)$. In each fixed polynomial-density regime we also obtain $f_{P,C}(k,r)\le k^{O_{r,P,C}(1)}.$ The proof combines a chromatic-defect random extraction lemma, compact and near-quadratic sparse-core bases, and a peeling/thinning bootstrap increasing the admissible edge exponent by $1/(r-1)$. We also prove structural saturation results for possible counterexamples, including Moore-strength exact-cycle packings and quadratic saturation in projected colour-pair space. Finally, writing $h_r^{\mathrm f}(G)=\max{χ_{\mathrm f}(H):H\subseteq G,\operatorname{girth}(H)\ge r},$ we develop a fractional random-extraction framework based on Mohar-Wu preservation. We prove sufficient cheap-cycle-killing criteria and verify them for several structured families, including clique-organised families, line graphs of incidence graphs of equal-order generalized quadrangles and generalized hexagons, and the Bohman-Keevash tracking-time triangle-free-process graph. We also isolate a density-free obstruction that any proof using this fractional surgery route must overcome.

Let us consider a random infinite lower triangular matrix, where the entries on and below the diagonal are i.i.d. uniform random elements of a fixed finite field. We investigate the evolution of the span of the first $n$ rows of this matrix as $n$ grows. Many properties of this evolving subspace can be captured with the help of the verbose persistence diagram, which is a standard tool in stochastic topology and topological data analysis. We give an explicit formula for the distribution of the persistence diagram. We prove a law of large numbers for the distribution of lifetimes. We also describe the fluctuations of the persistent Betti numbers.

A cyclic square matrix (and its conjugacy class) C over a field K is called (m,k)-cyclic if it has a decomposition $C = A \oplus B$, where $\dim A = m, \dim B = k$ and $m, k \ne 0$. It is shown that the product of two nonsingular (m,k)-cyclic conjugacy classes $Ω$ and $Ψ$ of GL(m+k,K) contains all nonscalar matrices P in GL(m+k,K) with determinant $\det P = \det ΩΨ$.

We define naive atomic decompositions of smooth projective varieties. We show that they satisfy a naive version of Iritani's blowup formula in several examples that are complicated enough to show most interesting features of the general theory while being simple enough to be computable by elementary methods.

We study the pair correlation statistics of orbits generated by maps on the interval. We show that under suitable mixing and multifractal assumptions, the pair correlation statistics of an orbit will almost surely exhibit the same asymptotic behaviour as a suitable sequence of i.i.d. random variables. We will also show that, under suitable hypotheses, the pair correlation statistics defined by two orbits will almost surely exhibit the same behaviour as two suitable sequences of i.i.d. random variables. Specific dynamical systems to which our results apply to include Gibbs-Markov maps and the Gauss map. We also give an example of a slowly mixing system for which the pair correlation statistics of an orbit almost surely behave distinctly to an i.i.d. sequence.

We propose a continuous time dynamical system model for tracing the evolution of confidence in a small decision making group by consensus with the possibility that a forcing factor is exerted onto the confidence of the participants. We experimentally check whether a forcing factor appears, where it begins and ends and how it affects the evolution. We find that the equilibrium value of the confidence is lower in the model with the forcing factor than without it, and that the forcing factor can be identified and induces additional oscillations of the confidence level. This is probably one of the first times when a mathematical model is able to speak about visible effects on the confidence process tracing under alterations of its levels. Pragmatically we find a model that captures influence of participants' confidence level by observing oscillations and equilibrium and we experimentally test it with measures of individual "confidence that the decision is correct" throughout the group decision-making.

For $0<q<1$ we compare two natural $q$-analogs of the Laguerre--Pólya class. The first one is a coefficient-side class, defined as the inverse image of the classical Laguerre--Pólya class under the normalized $q$-Borel transform \[ \Bq\left(\sum_{k\ge 0}a_k\frac{z^k}{k!}\right) =\sum_{k\ge 0}a_k\frac{q^{k(k-1)/2}(1-q)^k}{(q;q)_k}z^k . \] The second one is a zero-side class, defined as the locally uniform closure of real polynomials whose nonzero zeros are logarithmically $q$-separated on each side of the origin. We prove that the normalized $q$-Borel transform maps the classical Laguerre--Pólya class, and its type-I subclass, into themselves. This yields a $q$-Jensen-polynomial criterion and shows that the coefficient-side class strictly contains the classical Laguerre--Pólya class. On the zero side, we prove a genus-zero product representation. The logarithmic separation condition prevents zeros escaping to infinity from producing a residual exponential factor; consequently no nonconstant exponential factor can occur. For every $q\in(0,1)$ we obtain the strict chains \[ \qLPs\subsetneq \LP\subsetneq \qLPw, \qquad \qLPIs\subsetneq \LPI\subsetneq \qLPIw . \]

In this paper, we study the global dynamics of the 3D ionic Euler-Poisson equations with the parameter of Mach number $\varepsilon$. We first establish the global well-posedness and scattering for the high Mach number regime $0<\varepsilon\leq1$ and pressureless case $\varepsilon=0$. Moreover, we prove the high Mach number limit, showing that the profile of the solution for ionic Euler-Poisson equations converged to that of the pressureless equation as $\varepsilon\rightarrow0$. Our approach combines energy estimates, dispersive estimates and the normal form method. The major difficulty lies in establishing the uniform estimates with respect to the parameter, as the dispersive or resonance structure degenerates when $\varepsilon$ tends to 0. A crucial observation is that despite the disappearance of the pressure ($\varepsilon\rightarrow0$), dispersive phase function always remains a wave-type structure in zero frequencies, which enables us to derive linear and bilinear multiplier estimates adapted to the uniformity of Mach number parameter.

Consider the diffusion process \begin{equation*} dX_ε(t) = \mss b(X_ε(t)) \, dt + \sqrt{2\, ε\, \mss a(X_ε(t))} \, dW_{t}, \end{equation*} on the one-dimensional torus $\bb T = [0,1)$. Here $ε$ is the temperature, $W_{t}$ a Brownian motion on $\bb T$ and $\mss a$, $\mss b$ functions of class $C^{2}(\bb T)$ satisfying further conditions. Denote by $\mss P(\bb T)$ the set of probability measures on $\bb T$ equipped with the weak topology, and by $\ms I_ε\colon \mss P(\bb T)\to [0,+\infty)$ the level two large deviation rate functional of the diffusion $X_ε(\cdot)$. We derive a full $Γ-$expansion of $\ms I_ε$, as $ε\to 0$, expressing it as \begin{equation*} \ms I_ε = \frac{1}ε \;\ms J^{(-1)} \; +\; \ms J^{(0)} \;+\; \sum_{p=1}^{\widehat{\mf q}}\frac{1}{θ^{(p)}_ε}\;\ms J^{(p)}\,, \end{equation*} where $\ms J^{(-1)}$, $\ms J^{(0)}$, $\ms J^{(p)} \colon \mss P(\bb T)\to [0,+\infty]$ represent rate functionals, independent of $ε$, and $θ^{(p)}_ε$ are the time-scales at which the Markov process $X_ε(\cdot)$ exhibits a metastable behaviour.

This article surveys the mathematical contributions of Manfred Denker, with a focus on themes that connect ergodic theory, probability theory, dynamical systems, fractal geometry, and statistics. Denker's highly influential work includes a systematic study of the statistical properties of dynamical systems, the development of limit theorems for dependent processes, and the use of thermodynamic formalism to relate geometric and measure-theoretic properties. Particular emphasis is placed on the emergence of probabilistic behavior in deterministic systems, including central limit theorems, invariance principles or local limit theorems, under weak dependence assumptions or in infinite measure. Further topics include equilibrium states and transfer operator methods, the role of conformal measures in fractal geometry, and the asymptotic theory of statistical procedures for dependent data, such as rank statistics and U-statistics. In addition to these theoretical developments, the survey highlights contributions connecting rigorous analysis with computational and statistical methods. Taken together, these works illustrate a unifying perspective in which ergodic, probabilistic, geometric, and statistical methods interact in the study of dynamical systems.

This paper initiates a mathematical investigation of a PDE model for the transport of high voltage direct current via overhead lines. We prove the existence of infinitely many solutions, give necessary conditions for existence, explicitly compute the continuum of all radial solutions, and develop a new numerical algorithm for this problem.

We investigate the uniqueness of radial bound state solutions to the sublinear elliptic equation \[ \begin{cases} -Δu - u + |u|^{q-2}u = 0 & \text{in } \mathbb{R}^n, \\ u(x) \to 0 & \text{as } |x| \to \infty, \end{cases} \] where $q\in(1,2)$ and $n\geq 2$. We prove that for any prescribed integer $k\geq 1$, the equation admits exactly one radial bound state solution with $k$ simple zeros. Furthermore, we consider the superlinear equation \[ \begin{cases} -Δu + u -|u|^{p-1}u = 0 & \text{in } \mathbb{R}^n, \\ u(x) \to 0 & \text{as } |x| \to \infty. \end{cases} \] While the uniqueness of radial bound state solutions for this equation was established by Tang (2026) for $n\geq 3$ and $1<p<\frac{n+2}{n-2}$, we provide the necessary arguments to show that this uniqueness result remains valid for the case $n=2$ with $p>1$.

Two well-known decoding algorithms for BCH codes are conventional decoding, based on the Berlekamp-Massey algorithm in combination with Chien search, and direct decoding, which uses direct solutions to find the error locator polynomial and its roots. We introduce hardware architectures for conventional and direct decoding of extended BCH codes. Both architectures support implementation for any blocklength. Our conventional decoder supports any error-correction capability, whereas direct decoding is supported up to error correcting capability t = 4. To the best of our knowledge, our work is the first to implement a direct BCH decoder with an error-correction capability 4. We synthesize for the Xilinx Ultrascale+ XCZU48DR field-programmable gate-array and 16 nm FinFET for blocklengths up to 1024 bits and t = 4. We show that the direct decoder outperforms the conventional decoder in area efficiency for t = 2, t = 3, and for t = 4 for blocklengths longer than 256. Post-synthesis results for 16 nm FinFET show codeword per clock-cycle throughput at 1 GHz, achieving 239 Gb/s for the (256, 239) eBCH code and 223 Gb/s for (256, 223) eBCH code at 2 ns and 8 ns latency, respectively.

We introduce and study a synthetic notion of timelike total curvature for curves in Lorentzian length spaces with upper curvature bounds. In particular, we prove that our notion agrees with its smooth counterpart, and we show that timelike curves of finite total curvature are rectifiable. As the main application, we provide a sharp lower bound for the length of timelike curves solely in terms of the time separation between their endpoints and their total curvature.

Let $G$ be a graph with vertex set $\{v_1,\dots,v_n\}$. The Seidel matrix of $G$ is an $ n\times n$ matrix whose diagonal entries are zero, $ij$-th entry is $-1$ if $v_i$ and $v_j$ are adjacent, and otherwise is $1$. The $p$-Seidel energy of the graph $G$ is defined as the sum of the absolute values of the $p$-th powers of all eigenvalues of the Seidel matrix of $G$ and introduced in [European Journal of Combinatorics, (86) (2020), 103078]. In this article, we characterize the graph that minimizes the $p$-Seidel energy among all graphs with fixed order $n$, for $p>2$. We also characterize the graph that maximizes the $p$-Seidel energy among all graphs with fixed order $n$, for $0<p<2$. In addition, for every $p>2$, we characterize the graph that minimizes the $p$-Seidel energy among all $r$-regular graphs with fixed order $n$, where $n$ is a prime power with $n\equiv 1\pmod 4$, $r=\frac{n-1}{2}$. For every $p>2$, we also characterize the graph that maximizes the $p$-Seidel energy among all $r$-regular graphs with fixed order $n=2r$. Finally, we pose several open problems concerning the $p$-Seidel energy for different values of $p$.

The shifted Jack Littlewood-Richardson coefficients $g^λ_{μν}(α)$, first studied by Alexandersson-Féray, are Laurent polynomials in the Jack parameter $α$ attached to triples of partitions, which generalize the classical Jack Littlewood-Richardson coefficients investigated by Stanley, et al. In a previous work of the author's, it was conjectured that the Littlewood-Richardson coefficients for two triples, in which one of the partitions differ by a single box move, are congruent modulo the $α$-hook length of the pivot box for that move. In this note we prove that conjecture. We also investigate the extension of that conjecture to shifted Macdonald functions, which remains open pending two properies of Lassalle's shift map in that case.

The Seidel energy is defined as the sum of the absolute values of the eigenvalues of the Seidel matrix of a hypergraph. We first characterize the k-uniform hypergraphs of fixed order n with minimum and maximum Frobenius norms of Seidel matrices and then derive bounds for the Seidel energy. Building on these results, we obtain a negative answer to the hypergraph analogue of Haemers Conjecture by showing that the complete k-uniform hypergraph does not, in general, minimize Seidel energy. Motivated by the theory of hypoenergetic and non-hypoenergetic graphs, we define Seidel hypoenergetic and Seidel non-hypoenergetic hypergraphs and prove that almost all k-uniform hypergraphs are Seidel non-hypoenergetic.

We study observability and controllability for constant-coefficient first-order hyperbolic systems on the real line when only part of the state is observed or controlled. Even when the Kalman rank condition holds, the usual \(L^2(\mathbb R)^N\)-observability estimate may fail because some components are detected only through the dynamics. We show that the Kalman structure determines the appropriate observability estimate. A component that becomes visible after \(k\) algebraic steps is measured at low Fourier frequencies with a weight of order \(|ξ|^{2k}\) in the Fourier variable. This yields a natural Kalman-adapted observation space for the system. We also prove localized observability for diagonalizable systems on observation sets with uniformly bounded gaps and, separately, extend the whole-line construction to systems with real spectrum and Jordan blocks.

Jointly inferring latent node states and unknown network topology in nonlinear graphical dynamical systems is a fundamental yet largely unsolved problem, where the mutual entanglement of continuous states and discrete structure renders accurate recovery of either quantity critically dependent on the other. We propose \textbf{Koopman-GKFA} (Koopman Group-sparse Kalman Filter--ADMM), a unified framework that lifts nonlinear network dynamics into an approximately linear system via Koopman operator embedding with a separable node-wise dictionary, enabling optimal linear filtering for state estimation and provably convergent convex optimization for topology inference. Three theoretical contributions underpin the framework: (i)~a \emph{structural homomorphism lemma} proving that, under a separable-dictionary condition, block sparsity of the lifted coupling operator is isomorphic to the graph topology, providing the rigorous foundation for group-sparse regularization; (ii)~a block-structured group-sparse ADMM topology subproblem with certified linear convergence, extended by an exponential forgetting factor to track time-varying topologies; and (iii)~a \emph{three-term certified mean-squared error bound} that decomposes total estimation error into Koopman truncation, observation noise, and topology residual components, with monotone consistency established as the dictionary dimension grows. Extensive experiments on synthetic benchmarks (Kuramoto oscillators, Hill-kinetics gene-regulatory networks) and real-world datasets (NGSIM US-101, DREAM4) demonstrate that Koopman-GKFA consistently outperforms EKF-, UKF-, and particle-filter-based joint estimators in both state estimation and topology recovery, while exhibiting polynomial computational scaling and strong robustness in high-dimensional nonlinear settings.

In this article we examine the cross-connections of the normal category of finite dimensional subspaces of a Hilbert space and it's dual space. Further, we describe the cross-connection semigroup of cones, which is a normed algebra isomrphic to the normed algebra of finite rank bounded operators on a Hilbert space. We also characterize compact operators and their spectrum by the normal cones in the normal category of proper subspaces of a Hilbert space.

We prove that if a holomorphic diffeomorphism of a compact complex manifold is bi-Lipschitz conjugate to an ergodic affine automorphism $A$ on $Γ\backslash G$, then the conjugacy is $C^\infty$. Moreover, if $A$ is weakly mixing, then the induced complex structure on $Γ\backslash G$ is left-invariant. As applications, we establish a regularity bootstrap result for holomorphic Anosov diffeomorphisms bi-Lipschitz conjugate to affine models, as well as a holomorphic analogue of the rigidity theorem for higher-rank abelian Anosov actions by Hertz--Wang. The key observation is that the condition for a diffeomorphism to preserve a complex structure has the same form as the cocycle compatibility relation appearing in the study of centralizers. This places invariant complex structures and centralizers within a common $\mathbb Z^2$-cocycle framework. From this viewpoint, our main result may be regarded as a holomorphic counterpart of the Lipschitz centralizer rigidity theorem of Damjanović--Wilkinson--Wu--Xu for affine automorphisms.

What is the effect of randomly removing material from an infinite stretched membrane? Under what conditions can the membrane still sustain tension? This problem was introduced by Robert Connelly in connection with applications of rigidity theory in the natural sciences, and was later studied in M. V. Menshikov, K. A. Rybnikov, and S. E. Volkov, "The loss of tension in an infinite membrane with holes distributed according to a Poisson law" (2002); a discrete version was also considered in Robert Connelly, Konstantin Rybnikov, and Stanislav Volkov, "Percolation and the Loss of Tension in an Infinite Triangular Lattice" (2001). We study a mathematical framework based on a non-homogeneous Poisson point process whose intensity $λ$ tends to zero at infinity. The hole shapes are i.i.d.\ and independent of their locations. We show that if the intensity does not decay too quickly, then tension is still lost throughout the whole plane, as in the homogeneous model studied in 2002. Conversely, we give sufficient conditions under which complete loss of tension does not occur. Thus, both destruction and non-destruction regimes are possible even when the intensity tends to zero, indicating a phase transition in the model. The processes studied here are closely related to bootstrap percolation.

We study coefficient identification problems for elliptic partial differential equations with total variation regularization and control constraints. Existing related literature relies on continuity and differentiability properties of the control-to-state operator with respect to the $L^\infty$-norm. While this is sufficient for deriving optimality conditions, it is not well-suited for numerical algorithms, as it neglects the spatial extent of perturbations and leads to a qualitative discrepancy compared to $L^q$-norms with $q < \infty$. In this work, we address this gap by exploiting $W^{1,s}$-regularity results to establish differentiability properties of the control-to-state operator with respect to $L^q$-norms for finite $q$. Based on this framework, we derive first- and second-order differentiability results for the reduced objective functional and establish first-order optimality conditions involving a restricted subdifferential characterization of the total variation seminorm and corresponding regularity of the associated multipliers. Building on this, we analyze a nonsmooth trust-region method based on an $L^r$-trust region for $r > 0.5 q$ and prove its convergence to first-order stationary points.

In this article, we describe and analyse a novel mechanism for symmetry-breaking in minimal symmetrically coupled identical slow/fast oscillator networks with strong nonlinear mutually inhibitory coupling. We show that the symmetry-breaking, surprisingly, originates from the canard dynamics of a folded node that lies on the axis of symmetry. By applying geometric singular perturbation theory and the blow-up technique to a normal form, we determine the geometric mechanisms by which the {\em symmetric folded node} induces symmetry-breaking. More specifically, we show that (i) the fold curve of the coupled system is orthogonal to the axis of symmetry at the symmetric folded node; (ii) there is only one primary maximal canard (either strong or weak, depending on parameters), which always lies on the axis of symmetry and is the axis of rotation for the twisting of solutions; and (iii) the number of rotations is the key local diagnostic feature that breaks the symmetry. Our work is closely related to that of Kristiansen and Pedersen [SIAM J. Appl. Dyn. Syst., {\bf 22} (2023)] on symmetrically coupled FitzHugh-Nagumo oscillators with strong linear inhibitory gap junctional coupling, however, we consider nonlinear coupling and we identify and study multiple sub-types of their `cusped singularities'. We demonstrate our theoretical results by applying them to a model of the eukaryotic cell cycle in which the symmetric folded node plays a key role in rhythmogenesis. More specifically, we study periodic and quasi-periodic symmetry-breaking mixed-mode oscillatory attractors of the cell cycle model. We show that the local twisting induced by the symmetric folded node is the local mechanism that both breaks the symmetry and generates the small-amplitude oscillations in the mixed-mode dynamics.

We prove that the Hilbert scheme of conics on $X = G/P$, a projective rational homogeneous space with Picard rank one, is $G$-spherical if and only if $P$ is associated to a long root of the Dynkin diagram.