Papers

Learned communication systems may evaluate stochastic channel surrogates millions of times inside differentiable training loops, making diffusion-style reverse sampling expensive. This paper proposes condition-wise Sinkhorn drifting, a one-shot channel surrogate that preserves the transmitted symbol and transports only the conditional output laws \(p(y\mid x)\). We formulate a conditional Sinkhorn objective over repeated outputs at the same transmitted symbol and train the generator with finite-sample barycentric velocities followed by detached particle regression. Experiments on additive white Gaussian noise (AWGN), Rayleigh fading, solid-state power amplifier (SSPA) nonlinearity, and a compact tapped-delay-line (TDL) channel compare direct drifting, joint Sinkhorn drifting, condition-wise Sinkhorn drifting, conditional denoising diffusion probabilistic modeling (DDPM), denoising diffusion implicit modeling (DDIM), and Wasserstein generative adversarial network (WGAN) references. Within the evaluated one-shot drifting-family variants, condition-wise Sinkhorn is strongest under conditional diagnostics and symbolic-coding checks, while diffusion remains strongest on the hardest downstream symbol-error-rate (SER) curves. The resulting operating point is a condition-preserving one-shot simulator for settings where repeated channel calls make diffusion-style sampling too costly.

Collective migration of heterogeneous cell populations is central to many biological and physiological processes, including development and immune response. Recent experimental and theoretical advances have shown how asymmetric interactions with self-generated chemical gradients shape the spatial distribution of distinct cell types within migrating collectives. However, the principles governing robust spatial organisation of heterogeneous cell populations remain poorly understood. Here, we use asymptotic analysis to systematically derive a nonlinear analytical theory for heterogeneous cell collectives guided by self-generated chemotaxis. Our theory disentangles how heterogeneity in cell diffusivity, chemoattractant consumption, and chemotactic sensitivity shape the density profiles of migrating heterogeneous collectives, revealing four distinct dynamical behaviours that together capture all possible regimes. We calibrate our framework to experimental data on the co-migration of dendritic and T cells. We predict that this system operates in a parameter regime that balances intercellular mixing with T-cell localisation at the leading front of the migrating collective. Our theory reveals that this behaviour is enabled by intermediate long-range chemoattractant signalling generated through strong chemoattractant consumption by dendritic cells. Overall, our framework provides general principles for understanding how non-reciprocal chemical interactions shape robust collective migration in heterogeneous cell populations.

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 $\Omega$ and $\Psi$ of GL(m+k,K) contains all nonscalar matrices P in GL(m+k,K) with determinant $\det P = \det \Omega \Psi$.

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.

Equalization-enhanced phase noise is avoided by applying phase noise compensation (PNC) before chromatic dispersion compensation. Feedforward and iterative PNC algorithms based on expectation propagation are proposed. Both achieve information rates close to channels without phase noise for 100 GBaud 64-QAM and 10,000 km of fiber.

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.

In this article, we develop a multi-dimensional two-layer thin film model extending the thin film model proposed in \cite{barthwal2025hyperbolic}. The model considered in \cite{barthwal2025hyperbolic} considered a very specific Marangoni scale by choosing Marangoni numbers in both layers to be $1$. We relax this condition here and prove that the obtained system possesses a full set of Riemann invariants. Based on these findings, we develop a Riemann Invariant-based Local Characteristic Decomposition WENO (RI-WENO) method for the two-layer thin film model in one and two dimensions. The method is built upon a specially designed variable transformation constructed from the derived Riemann invariants of the system. This transformation partially diagonalizes the governing equations and yields a sparse structure in the transformed eigenvector matrices. As a result, the proposed RI-WENO framework significantly reduces the computational cost of the standard Local Characteristic Decomposition WENO approach while retaining its strong capability to suppress spurious oscillations. Numerical experiments, including new benchmark test cases, demonstrate that the RI-WENO method achieves an effective balance between accuracy and computational efficiency, making it a promising and practical choice for solving the two-layer thin film model.

Consider the diffusion process \begin{equation*} dX_{\epsilon}(t) = \mss b(X_{\epsilon}(t)) \, dt + \sqrt{2\, \epsilon\, \mss a(X_\epsilon(t))} \, dW_{t}, \end{equation*} on the one-dimensional torus $\bb T = [0,1)$. Here $\epsilon$ 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 $\ms I_{\epsilon}\colon \mss P(\bb T)\to [0,+\infty)$ the level two large deviation rate functional of the diffusion $X_{\epsilon}(\cdot)$. We derive a full $\Gamma-$expansion of $\ms I_{\epsilon}$, as $\epsilon \to 0$, expressing it as \begin{equation*} \ms I_{\epsilon} = \frac{1}{\epsilon} \;\ms J^{(-1)} \; +\; \ms J^{(0)} \;+\; \sum_{p=1}^{\widehat{\mf q}}\frac{1}{\theta^{(p)}_{\epsilon}}\;\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 $\epsilon$, and $\theta^{(p)}_{\epsilon}$ are the time-scales at which the Markov process $X_{\epsilon}(\cdot)$ 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.

Wegner conjectured in 1965 that every finite family $\mathcal R$ of axis-parallel rectangles satisfies $\tau(\mathcal R)\le 2\nu(\mathcal R)-1$, where $\tau(\mathcal R)$ is the minimum number of piercing points and $\nu(\mathcal R)$ is the maximum size of a pairwise-disjoint subfamily. We disprove the conjecture by an explicit triangle-free family of $64$ rectangles with $\nu=16$ and $\tau\ge 32$. More generally, for every $\varepsilon>0$, we construct triangle-free rectangle families for which the standard clique-LP relaxation for maximum independent set of rectangles has integrality gap at least $5/2-\varepsilon$. The same families satisfy $\tau(\mathcal R)\ge (5/2-\varepsilon)\nu(\mathcal R)$. We also prove that, on triangle-free rectangle families, this LP has gap at most $3$. Our approach gives an example with axis-parallel segments instead of rectangles with integrality gap tending to $2$. We also give a relatively small $4092$-rectangle triangle-free family with chromatic number $6$ improving the construction of Asplund and Grünbaum (On a coloring problem, Mathematica Scandinavica, 1960) that required more than $10^8$ rectangles.

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 \[ \begin{cases} -\Delta u - u + |u|^{q-2}u = 0 & \text{in } \mathbb{R}^n,\cr u(x) \to 0 & \text{as } |x| \to \infty, \end{cases} \] where $q\in(1,2)$ and $n\geq 2$. A distinctive feature of this problem is the non-Lipschitz singularity of the nonlinearity at the origin, which gives rise to compactly supported ground states and bound states. Using a shooting argument together with a detailed analysis of the linearized variation with respect to the initial value, we prove that for every prescribed integer $k\geq 1$, the equation admits exactly one radial bound state solution with $k$ simple zeros, up to sign reflection and spatial translation. In addition, our analysis yields a classification of radial solutions according to the initial value and describes their behavior near the finite support boundary.

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.

Heat conduction in semiconductor crystals is fundamentally governed by the linearized Peierls-Boltzmann equation (LPBE) for phonon transport, that arises out of a kinetic theory for phonon quasiparticles. Yet, continuum theories such as the Fourier's heat diffusion, weakly quasiballistic and hydrodynamic heat equations are often used to explain the experimental observations of heat flow in these materials. Here, we show that a systematic reduction of the LPBE into such equivalent continuum descriptions are possible only for the limiting values of a set of generalized Knudsen numbers. We further show that all of these continuum heat flow regimes, along with the ballistic heat flow, can be described by a single continuum equation for the temperature field that originates from the eigenmode analysis of the LPBE, thus offering a unified picture of all possible heat flow regimes in semiconducting crystals. Using quantitative examples on twenty three technologically important semiconductors, we show that several previously-unidentified features of the non-Fourier heat flow regimes emerge from this generalized Knudsen number framework such as (1) the mutual exclusivity of the weakly quasiballistic and the hydrodynamic heat flow regimes, (2) length-dependent velocity of the hydrodynamic second sound temperature wave and a characteristic heating length for the strongest hydrodynamic second sound, (3) characteristic frequency-domain temperature response distinguishing the hydrodynamic second sound from the ballistic heat flow regime and, (4) a new non-oscillatory signature of transient hydrodynamic heat flow. Our work formally bridges the continuum and the particulate descriptions of heat flow, and provides insights into the important signatures of temperature dynamics in each of these heat flow regimes, that will aid in their unambiguous experimental observations in the future.

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 generalize the ordinary Jack coefficients and are Laurent polynomials in the Jack parameter $\alpha$. We prove a previously conjectured congruence: coefficients indexed by triples differing by a single box move are congruent modulo the shared $\alpha$-hook at the pivot. We also prove a shifted Macdonald analogue, with a power-of-$t$ twist, and establish that the normalized shifted Macdonald coefficients are Laurent polynomials in $q$ and $t$. The proofs combine coincidences of shifted coordinates with Laurent-preserving shift transforms, and the Macdonald input uses Knop's inversion formula and the integrality of the Bergeron-Garsia-Haiman-Tesler operators. Finally, we realize the Jack and Macdonald congruences as necessary edge conditions on Hilbert schemes of points, in equivariant cohomology and equivariant $K$-theory, respectively. In the Macdonald case a tautological determinant twist accounts for the power-of-$t$ normalization

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 \(|\xi|^{2k}\) 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 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 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 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 $\Gamma\backslash G$, then the conjugacy is $C^\infty$. Moreover, if $A$ is weakly mixing, then the induced complex structure on $\Gamma\backslash G$ 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 $\lambda$ 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 develop a decision-theoretic framework for distributed Bayesian experimental design in which local agents evaluate candidate experiments using expected information gain and transmit their local design decisions to a fusion center. Unlike centralized Bayesian design, where all likelihood components and information-gain values are available to a single planner, the fusion center in the distributed setting chooses a global experiment from compressed local recommendations. We derive the Bayes-optimal fusion rule, which selects the experiment with largest conditional expected centralized information gain given the observed local design decisions. This rule is analogous in spirit to optimal fusion rules in distributed detection, but differs fundamentally because the underlying utility is expected information gain and the resulting loss is information-gain regret rather than classification error. We also establish information-loss bounds and identify conditions under which the decision-only fusion rule is asymptotically equivalent to the centralized design. Numerical experiments show that Bayes-optimal fusion closely approximates the centralized oracle, whereas majority voting can be highly suboptimal when a minority of sites carry disproportionate information.

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 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., 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.