Papers

For more than half a century, iterated Pearson correlation has underpinned methods for network blockmodeling, clustering, seriation, and information visualization. Despite its continued use and the longstanding assumption of convergence, the global convergence problem remained open even in dimension three. We resolve this problem completely for the $3\times3$ correlation map: every admissible orbit is well defined for all forward iterates and converges to one of exactly seven fixed points. In arbitrary dimension, we establish an exact row-wise Gram factorization and the identity $\operatorname{rank}C(A)=\operatorname{rank}(AH_n)$, which gives the precise rank-reduction mechanism and forces every fixed point to be singular. We identify the all-ones matrix as the unique Pearson-degenerate correlation matrix, prove forward invariance of the Pearson-nondegenerate elliptope, and classify the $2^{n-1}-1$ nondegenerate sign-valued fixed points. For $n=3$, we prove a complete analytical fixed-point classification: three rank-one patterned points, three rank-two mixed points, and one rank-two equicorrelation point. We also prove the complete relative Lyapunov stability classification: precisely the patterned points are locally asymptotically stable, while the mixed and equicorrelation points are unstable. The global proof reduces the rank-two dynamics to a one-dimensional projective kernel coordinate; an exact order identity produces monotone projective ratios, excludes nontrivial periodic and recurrent limit sets, and forces convergence to a fixed point. Finally, the initial conditions converging to the four unstable fixed points form a Lebesgue-null set. Hence almost every admissible initial condition converges to a patterned fixed point; the three patterned basins are relatively open and permutation-equivalent, and each has Lebesgue measure exactly one third of that of the elliptope.

We study the geometry of mixed fields, consisting of a connection and a $B$-field on a principal circle bundle over a Riemann surface, from the perspective of gauge theory and T-duality. Motivated by the foundational work of Atiyah–Bott and Segal, we introduce a twisted Yang–Mills functional whose critical locus, in the flat case, is governed by the simultaneous vanishing of the curvature and the $H$-flux. We show that the gauge group is a semi-direct product of abelian groups parametrised by an integer $\lambda$. The moduli spaces are constructed by presymplectic reduction and shown to be Heisenberg contact manifolds for $\lambda \neq 0$, whose topology we characterise completely. We show that T-duality preserves the twisted Yang–Mills functional and acts on the configuration space of flat mixed fields. We identify the subgroups of gauge transformations that are compatible with the T-duality map and describe the induced action on the singular quotient of T-dualizable flat mixed fields. Precisely at $\lambda =1$ does T-duality descend to an involutive contactomorphism of the moduli space.

We establish a simple characterization of amenability of second countable locally compact quantum groups $\mathbb{G}$ in terms of the long-time behavior of convolution semigroups of states on $\mathbb{G}$. We then investigate the deeper problem of combining amenability of $\mathbb{G}$ with other approximation properties of the dual quantum group $\hat{\mathbb{G}}$. Specifically, we characterize the combination of ("strong") amenability of $\mathbb{G}$ with either the failure of property (T) or the presence of the Haagerup property for $\hat{\mathbb{G}}$. These results, which appear to be new even in the classical setting of locally compact groups, produce convolution semigroups on $\mathbb{G}$ whose behavior is controlled both as time goes to zero and as time goes to infinity.

Casolo asked whether a torsion-free group in which every subgroup is subnormal of defect at most $n$ must be nilpotent of class at most $n$. The answer is known to be positive for $n\leq 4$. We construct a $2$-generated torsion-free nilpotent group $G$ such that $$ \cl(G)=6 \qquad\text{and}\qquad [G,{}_5H]\leq H \quad\text{for every }H\leq G. $$ Thus Casolo's question has a negative answer for $n=5$.

We investigate the geometric structure of the set of spectra of bipartite absolute separable states ($\mathrm{ASEP}_{m,n}$) and absolute positive partial transpose states ($\mathrm{APPT}_{m,n}$), i.e., bipartite quantum states that remain separable or PPT, respectively, under all global unitary transformations. First, we establish general geometric properties of absolute convex sets of matrices, their spectra and extreme points. Regarding absolute separability, we present a permutation-symmetric reformulation of the absolute PPT criterion and use it to demonstrate that $\mathrm{APPT}_{m,n}$ is a spectrahedron for all $m\leq n$: in particular, all its faces are exposed. While $\mathrm{ASEP}_{2,n}$ is a spectrahedron, too, for $\mathrm{ASEP}_{m,n}$ we can in general only show that it is a semialgebraic set for all $m,n\geq 3$. Furthermore, we provide a complete characterization of the faces and extreme points of $\mathrm{APPT}_{m,n}$ and demonstrate that the dimension of a face is determined by the rank of a certain matrix, with maximal proper faces having dimension $mn-m-1$. In the quantitative setting, we provide a rigorous lower bound on the maximal attainable purity of $\mathrm{APPT}_{m,n}$ via an inscribed polytope $\mathcal{P}_{m,n}$ and conjecture that the maximal purity of $\mathrm{APPT}_{m,n}$ (along with its spectra) coincides with the polytope for arbitrary dimensions except when $m=n=2$. Additionally, we also provide a rigorous upper bound on the minimal von Neumann entropy of $\mathrm{APPT}_{m,n}$ and demonstrate numerically that the minimum entropy eventually coincides with the polytope $\mathcal{P}_{m,n}$ as the local system dimension $n$ increases. Finally, we show that the relative spectral volume of $\mathrm{APPT}_{m,n}$ decays exponentially in $n$ by a constant multiplicative factor of the relative volume of the inscribed polytope $\mathcal{P}_{m,n}$.

We propose a framework for constructing approximate Nash equilibria in mean-field games (MFG) with common noise based on a two-time-scale structure. In our model, the common noise is carried by a fast variable evolving under ergodic dynamics, while the slow variable is either optimally controlled or stopped. The fast variable enters the dynamics of the slow one through the drift coefficient. The key idea is to construct an approximation to the solution of the full MFG with common noise using an “effective” MFG without common noise, where the coefficients are averaged with respect to the stationary measure of the fast-scale process. We construct an explicit $\varepsilon$-MFG equilibrium for the full MFG from the equilibrium for the effective MFG with randomized control and stopping. To this end, we obtain new results on existence of MFG equilibria with randomized stopping. We rely on strong convergence results for two-scale diffusions under structural assumptions on the MFG, and show that the time-scale separation parameter controls the error in the Nash equilibrium condition.

We consider the Erdős-Moser equation $1^k+2^k+\cdots+(m-1)^k=m^k$ in arithmetic progressions. We prove among other things that when $k=2$, for any solution to exist, the above sum in arithmetic progression must consist of two or four terms. In either case, there are infinitely many solutions that can be completely characterized.

We extend several characterizations of the $L^p$-solvability of the homogeneous Dirichlet problem to (possibly degenerate) elliptic operators $L=-\textrm{div}(wA\nabla)$ defined on a large class of open sets $\Omega$ in $\mathbb{R}^n$. This framework encompasses not only uniformly elliptic operators in Lipschitz domains, but also Caffarelli-Sylvestre-type operators, and boundaries $\partial\Omega$ that are not $(n-1)$-dimensional, for instance. We prove that, for $p\in (1,\infty)$, the $L^p$-solvability of the homogeneous Dirichlet problem is equivalent to the solvability of the Poisson-Dirichlet problem $Lu=wf-\textrm{div}(wF)$, i.e. to the existence of a solution $u$ satisfying a $L^p$ non-tangential estimate whenever $f$ and $F$ belong to suitable weighted $L^p$ tent spaces. Furthermore, it is also equivalent to the solvability of the Poisson-Regularity problem for data in appropriate weighted $L^{p'}$ tent spaces, where estimates are obtained for $\nabla u$ rather than $u$.

We give a method to compute the primitive central idempotents of a semi-simple finite group algebra associated with an irreducible character $\chi$ such that for every $g\in G$, either $\chi(g) = 0$ or all eigenvalues of $\rho(g)$ have the same order. We then show several consequences which highlight the significance and possible applications of our approach.

Environmental DNA (eDNA) has emerged as a novel tool for quantifying the seasonal abundance of aquatic species in water bodies; however, its mathematical modeling is still at a germinating stage because of its mechanistic uncertainties. We propose a first-step mathematical and computational framework for the eDNA dynamics of migratory fish based on a novel stochastic partial differential equation model with a delayed source input. The model governs spatiotemporal eDNA concentration in rivers where the source comes from a stochastic differential equation for the migration dynamics of the fish. The affine nature of the model facilitates its theoretical analysis, including the guarantee of well-posedness and the closed-form derivation of the Laplace functional despite the proportionality coefficient of the multiplicative noise term being non-Lipschitz. We also propose a discretization scheme for the model that theoretically generates nonnegative numerical solutions. We finally apply the proposed model to eDNA concentration data sampled from midstream reaches of a river system and perform sensitivity analysis.

As is well known the Catalan numbers can be interpreted as numbers of Dyck paths. The Narayana polynomials and the q-Narayana polynomials for q=-1 can analogously be interpreted as certain weights of these paths. In the present note we try to get some information about the corresponding weights of bounded Dyck paths. This also leads to some number sequences and analogues of Pascals triangle which previously have occurred in other contexts. We consider some examples and obtain some results and conjectures.

We construct new families of complete minimal submanifolds of codimension two in the non-compact duals to the classical compact Lie groups $\SOC n/\SO n$, $\SLC n/\SU n$ and $\SpC n/\Sp n$.

For a Markov kernel $T$ with an invariant probability measure $\pi$, we give a self-contained proof of the Markov chain convergence theorem via a criterion called asymptotic equivalence with the target. It assumes two parts about the Lebesgue decompositions of $T^n_x$ and $\pi$ for every starting point $x$: 1.) asymptotic absolute continuity: the singular mass $\mathrm{sing}(T^n_x \mid \pi)$ tends to $0$; and, 2.) asymptotic domination of the target: the singular mass $\mathrm{sing}(\pi \mid T^n_x)$ tends to $0$, as $n \to \infty$. Assuming a jointly measurable density for the absolutely continuous part of each iterate $T^n$ w.r.t. $\pi$, this criterion is sufficient and necessary for convergence. A positive minorant density version of it is verified in three cases: i.) $T$ has a positive transition density w.r.t. $\pi$; ii.) $T$ consists of an absolutely continuous part with positive transition density together with an atom at the starting point, which covers the Metropolis-Hastings algorithm; iii.) the transition density is positive only after a finite number of steps that may depend on the starting point $x$. To demonstrate our general criterion, we investigate the Gibbs sampler with random scan and the parallel tempering algorithm. Furthermore, we show that in all mentioned settings Birkhoff's ergodic theorem applies, so as to obtain the strong law of large numbers. Throughout this paper, neither irreducibility, nor aperiodicity, nor recurrence, nor couplings, nor splitting constructions, nor small sets are used. In all results, the state space is a general measurable space with no structure beyond a $\sigma$-algebra. That joint measurability is assumed of the Markov kernel, not of the space; countable generation supplies it. None of the theorems proved here is new; what is offered is a short route to a single, widely applicable Markov chain convergence criterion.

The modular product $G\diamond H$ of graphs $G$ and $H$ is a graph on vertex set $V(G)\times V(H)$. Two vertices $(g,h)$ and $(g',h')$ of $G\diamond H$ are adjacent if $g=g'$ and $hh'\in E(H)$, or $gg'\in E(G)$ and $h=h'$, or $gg'\in E(G)$ and $hh'\in E(H)$, or (for $g\neq g'$ and $h\neq h'$) $gg'\notin E(G)$ and $hh'\notin E(H)$. The independence number $\alpha(G)$ of a graph $G$ is the maximum cardinality of a set of pairwise nonadjacent vertices in $G$. In this paper, we study the independence number of the modular product of graphs. We first structurally characterize all independent set of $G\diamond H$ which lead to the exact result on $\alpha(G \diamond H)$. Special cases of this result lead to several sharp bounds and some exact results for $\alpha(G \diamond H)$. Finally, we introduce a partition graph associated with $G \diamond H$ that provides a framework for constructing independent sets of the modular product from independent sets of its substructures.

This paper proposes an efficient and scalable preconditioning strategy for eddy current problems involving coupled external circuits. The approach, named Electro-Magnetic Decoupling (EMD) preconditioner, decomposes the discrete system into vector and scalar potential components and applies tailored preconditioners to each. In particular, strong preconditioning is applied to the scalar component to address the spectral degradation induced by the discrete Laplacian. The method was evaluated on four eddy current models with varying frequencies, conductor topologies, and excitation types. Compared to the conventional incomplete Cholesky preconditioner, the EMD approach achieved up to 20 times fewer iteration counts and up to 20 times faster iterative solver time. Moreover, the method supports physics-level parallelization, allowing efficient treatment of independently excited conductor domains. The EMD framework is compatible with algebraic multigrid, domain decomposition, and direct solvers, offering flexibility and robustness for large-scale electromagnetic simulations.

In this work, we present a novel dynamic proximal point algorithm for unconstrained optimization. The method generates a sequence of proximal subproblems, where the quadratic regularization term is weighted by a diagonal matrix that is updated adaptively at each iteration. Each subproblem is solved using an inner Newton's method combined with a line search, which provides a global convergence mechanism for the nonlinear solver. At the outer level, the algorithm updates the reference point and adjusts the regularization parameter based on the performance of the inner Newton solver. We derive the reduced linear system used to compute the Newton step, define the corresponding merit function, and discuss practical approaches for constructing the diagonal scaling matrix from derivative information. The paper also provides implementation-oriented pseudocode and stopping criteria that are consistent with the proposed method.

We give an analytic construction of a tau function for the $q$-difference sixth Painlevé equation ($q$PVI) as a Fredholm determinant through the general Riemann-Hilbert problem associated with it. We show that the tau function is an analytic function on its domain of definition that vanishes at a particular point if and only if the corresponding Riemann-Hilbert problem is point-wise not solvable there. We express the corresponding $q$PVI transcendents in terms of the tau function as well as three copies of it with some of the parameters shifted. Then the vanishing of any of these four tau functions corresponds to the transcendents taking value in a specific corresponding exceptional line on the initial value space of $q$PVI. Finally, we derive an asymptotic expansion of the tau function for small times $t$.

The classic embedding problem for finite-dimensional Markov matrices has a natural counterpart for sub-stochastic matrices, which is analysed and discussed here. One necessary and sufficient characterisation of embeddability can be given via the unique extension of a sub-stochastic matrix to a stochastic one with one added state in conjunction with the embedding of this extension. This is then explicitly treated for $d\leqslant 3$.

Let $\mathcal{M}=h\mathcal{K}_u$ be a nearly $S^*$-invariant subspace of $H^2$, where $\mathcal{K}_u=H^2\ominus uH^2$ and $h$ is the extremal multiplier. For $\vp\in L^\infty(\T)$, we study the compression \[ D_\vp^\mathcal{M} = \restr{P_{\mathcal{M}^\perp}M_\vp}{\mathcal{M}^\perp}, \] called a weighted dual truncated Toeplitz operator. When $h\equiv1$, this reduces to the classical dual truncated Toeplitz operator. Using the Hartmann–Ross projection formula, we prove \[ \|D_\vp^\mathcal{M}\|=\|\vp\|_\infty, \] characterize compactness, and show that the natural multiplication map by $h$ identifies the weighted and classical theories precisely when $h$ is inner. We also establish complex symmetry and obtain block matrix, defect, and semi-commutator identities via weighted truncated Hankel operators. As a main algebraic consequence, we prove \[ D_\vp^\mathcal{M} D_\psi^\mathcal{M}=0 \quad\Longleftrightarrow\quad \vp=0\ \text{or}\ \psi=0 \quad\text{a.e. on }\T. \] Finally, we derive a a rank-at-most-two correction formula and a finite-rank displacement identity for the generalized dual shift $D_z^\mathcal{M}$.

We prove stable self-similar blowup for the harmonic map heat flow in dimensions 3 to 6 for target manifolds which are slightly perturbed versions of the round sphere. Starting from the known blowup solution for the sphere, we construct self-similar blowup solutions for this class of target manifolds and prove that these solutions are asymptotically nonlinear stable. Although the blowup solution for the sphere is not explicitly known in this case, we can still use perturbative methods similar to those developed by Donninger, Schörkhuber, and the author in \cite{DonSchWit26} who consider a related problem for the wave maps equation.

Enhanced Gabidulin Matrix Codes (EGMC), introduced by Aragon, Couvreur, Dyseryn, Gaborit, and Vincotte at Asiacrypt 2024, were designed to hide the algebraic structure of Gabidulin matrix codes while enabling very compact McEliece- and Niederreiter-type encryption schemes, with ciphertexts as small as 65 bytes at the claimed 128-bit security level. Their security relies on the assumption that a masked EGMC code is hard to distinguish from a random matrix code. We show that this enhanced construction leaves enough structure for an equivalent code of the secret key to be recovered. Unlike previous cryptanalysis, our attack combines combinatorial and algebraic techniques to recover a Gabidulin-equivalent compressed code. This code can then be extended to a full-length equivalent secret key in polynomial time. As a result, the attack provides both a distinguisher and a key-recovery attack against the EGMC encryption schemes. The attack breaks all 16 proposed EGMC parameter sets by large margins. For example, for the claimed 128-bit parameter set (2,17,37,4,0), it reduces the security level from 186 bits to 35 bits and in our implementation, the equivalent secret key of this parameter set is recovered in less than 10 minutes.

In the present paper, we introduce formulas for the logarithm function of abelian, uniformizable Anderson modules. Combining our formulas with a motivic map introduced recently by the second author, we relate special values of dual Goss $L$-functions of Drinfeld modules to the rigid analytic trivialization of their corresponding $t$-comotives. We apply these formulas to relate certain $L$-values to special values of matrix valued modular forms, generalizing the vector valued Drinfeld modular forms of Pellarin and Pellarin and Perkins in the rank two setting.

Let $\Omega\subset\mathbb R^4$ be a bounded domain of class $C^6$, let $0<\alpha<4$, and let $K\in C^4(\overline\Omega)$ be positive. We study the Navier problem \[ \Delta^2u=\varepsilon^{8-\alpha}K(x)e^{u(x)} \left(\int_\Omega\frac{K(y)e^{u(y)}}{|x-y|^\alpha}\,dy\right), \qquad u=\Delta u=0\quad\text{on }\partial\Omega. \] Let $G$ be the Navier Green function, let $H$ be its regular part, and put $M_\alpha=8\pi^2(8-\alpha)$. The concentration points are governed by \[ \mathcal F_m(\boldsymbol\xi) =\sum_{i=1}^m \left[\log K(\xi_i)+\frac{M_\alpha}{2}H(\xi_i,\xi_i)\right] +M_\alpha\sum_{i<j}G(\xi_i,\xi_j). \] Every $C^1$-stable critical point of $\mathcal F_m$ produces a positive $m$-bubble solution whose scales are of order $\varepsilon^{-1}$ and whose nonlinear source converges to $M_\alpha\sum_i\delta_{\xi_i^*}$. If the critical point is nondegenerate, the corresponding $m$-bubble solution is locally unique, modulo permutations, in a fixed scaled modulation neighborhood. The linearized operator is nondegenerate on $H^2(\Omega)\cap H_0^1(\Omega)$, and \[ \operatorname{ind}(u_\varepsilon) =m+\operatorname{ind}\!\left(-D^2\mathcal F_m(\boldsymbol\xi^*)\right). \] A critical four-dimensional capacity controls the scale directions. The dilation block of the reduced Hessian is positive and equals $8\pi^2b_\alpha^2|\log\varepsilon|^{-1}I_m+o(|\log\varepsilon|^{-1})$, where $b_\alpha=(8-\alpha)/2$.

We study cutting planes for nonconvex quadratically constrained quadratic programs (QCQPs) through a project-then-lift approach inspired by mixed-integer rounding (MIR) inequalities. Given two base valid inequalities for the extended QCQP formulation, we project the associated two-row relaxation into a two-dimensional set and analyze the joint range of quadratic functions in two base inequalities. For the nonconvex joint range, we give a closed-form convex hull description of the projected set; for the convex joint range, we give its semidefinite representation. This yields a new family of joint-range inequalities, which can be lifted back to the extended QCQP formulation. MIR inequalities can handle “mixed” terms: continuous variables or fractional linear combinations of integer variables. Similarly, we propose more flexible secant mixed-joint-range inequalities, which better expose and exploit the nonconvex joint range. The proposed approach preserves sparsity, since the support of each lifted inequality is controlled by that of the base inequalities. In preliminary geometric experiments, the joint-range inequalities yield substantial area reduction of the projected relaxation constructed via reformulation-linearization-technique.

A well-known result of Hesselink gives a formula for the $q$-character of the nilpotent cone of a semisimple Lie algebra in terms of a $q$-analog of the Kostant partition function. For a reductive Lie algebra $\mathfrak{g}$, we define a class of Hesselink-type representations, for which we prove an analog of Hesselink's formula under certain additional freeness assumptions. Moreover, we prove that the formula still holds for certain examples of Hesselink-type representations without the freeness assumptions. Using these methods, we obtain $q$-character formulas for the nilpotent cone of a representation of a cyclic quiver with equal dimensions, a representation of a cyclic quiver with two vertices, and a representation of a product of copies of $\mathfrak{sl}_2$ we call an extended quiver representation of trivial type. In the process, we give a general framework for proving formulas of this type for representations that are not necessarily Hesselink-type. We also provide some counterexamples to natural questions regarding Hesselink-type representations.

We introduce a notion of an admissible pair, and prove a variation to Pollock's conjectures on icosahedral and dodecahedral numbers.

For a rectangular box with edges $a_1,a_2,a_3$, the width in a uniform random direction $u \in S^2$ is $w=\sum_i a_i|u_i|$. Under a bijective change of parameters this is also the projected area of a rectangular parallelepiped, whose distribution was derived by Walters. We give a direct co-area derivation that recasts the law as one global positive-part formula and integrates it to an elementary closed cumulative distribution function. The density is real-analytic except at the edge lengths, face diagonals and space diagonal. We classify every singularity as a corner, square-root fold, superposition or terminal jump and compute its coefficient. A Gaussian representation gives all moments from one generating function. The degree of the $n$th cumulant in $\pi^{-1}$ is exactly $\lfloor n/2 \rfloor$; at every order the mean-normalised cumulant is a polynomial in two scale-invariant combinations of the intrinsic volumes, with explicit formulas given through fifth order. The first three cumulants determine the box, and we characterise the admissible triples. Beyond boxes, intrinsic volumes determine neither width variance nor brightness variance. The sign-pattern representation extends to bodies whose central symmetral is a zonotope, while the cellwise co-area method applies to every full-dimensional polytope.

For fixed $d\geq 2$, let $\tau_d(n)$ be the minimum size of a set $S\subseteq\{0,\ldots,n\}^d$ such that the affine lines determined by pairs of distinct points of $S$ cover the grid. Let $\sigma_d(n)$ be the analogous minimum when every grid point must lie on the closed segment joining two distinct points of $S$. A celebrated result of Alon [GAFA, 1991] proved that $\tau_d(n)$ is of order between $\Omega_d(n^{\alpha_d})$ and $O_d(n^{\alpha_d}\log n)$, where $\alpha_d=\frac{d(d-1)}{2d-1}$, and asked whether the logarithm term is necessary. We prove that $$c_d n^{\alpha_d}\leq\tau_d(n)\leq\sigma_d(n)\leq C_d n^{\alpha_d}$$ for every fixed $d\geq 2$, thereby resolving Alon's problem in a stronger form.

Self-dual corings and self-dual algebra extensions are classified and these (self-)dualities are compared with Ringel (self-)duality, revealing fundamental differences. To each coring $\mathcal C$, two algebras are associated, known as the left and the right dual algebra of $\mathcal C$. It is shown that these two algebras coincide in a natural way if and only if $\mathcal C$ is a Frobenius coring. Various other equivalent characterisations of $\mathcal C$ being self-dual are given, in terms of certain algebra extensions being Frobenius extensions and in terms of certain forgetful or restriction functors being Frobenius functors. Ringel self-duality however is shown to be rather different, for which a homological explanation is given.

We study some geometric consequences of the discrepancy between weak and total variation convergence of probability measures. We consider a sequence of probability measures on $\mathbb R^d$, admitting densities with respect to the Lebesgue measure, that converge weakly to a limiting measure but stay bounded away from it in total variation distance. We show that the sets on which the sequence passes from below to above the limiting density must grow unboundedly in perimeter, as measured by the $(d-1)$-dimensional Hausdorff measure. Moreover, this growth persists within a fixed compact set, so that it must reflect an increase in the geometric complexity of these sets rather than only an unbounded expansion in ambient space. We further provide a sufficient condition under which the number of connected components of the sets diverges, recovering a behavior that is closely reminiscent of the one-dimensional case, in which the number of oscillations of the sequence of densities around the limit grows without bound. We also show that this condition cannot be dispensed with in general, by means of an explicit sequence of measures in the plane whose passing sets remain connected in a single component at every stage while growing in length and complexity. Another sequence, built from cosine oscillations, illustrates the complementary behavior, in which the number of components diverges.