Papers

We study a class of robust forward–backward McKean–Vlasov variational systems under model uncertainty represented by a non-dominated family of probability measures. Mean-field interactions are described through nonlinear collective observables acting on the laws of the forward and backward components. To model asymmetric loss aversion, we introduce a nonsmooth convex functional whose subdifferential defines a law-dependent maximal monotone operator acting on the forward state. We establish existence, uniqueness, and stability of the robust forward dynamics by a fixed-point argument in Wasserstein space. The backward component is formulated as a selected backward variational system rather than a classical backward stochastic variational inequality. Our analysis relies on Yosida regularization, uniform a priori estimates, convergence of the regularized solutions, and a Minty–Brézis identification argument, yielding a canonical solution associated with the minimal norm selection. We further construct a particle approximation and prove propagation of chaos for the forward dynamics with explicit convergence rates uniformly over the non-dominated family. For each fixed regularization parameter, we also establish quantitative propagation of chaos for the regularized backward component and explain why estimates uniform in both the number of particles and the regularization parameter require additional non-contact assumptions near the nonsmooth threshold.

We study a two-dimensional Choquard equation driven by the mixed local and nonlocal operator $L:=-\Delta+(-\Delta)^s$, where the nonlinearity has critical exponential growth of Trudinger–Moser type. Under a coercive assumption on the potential and suitable one-sided assumptions on the nonlinearity, we prove the existence of a least energy positive solution. The proof combines Nehari manifold minimization, compactness below the critical Trudinger–Moser threshold, local regularity, and a strong maximum principle.

We provide necessary and sufficient conditions for convergence towards a unique IPVT on any proper pointed measured metric space. The conditions are that the volume function, when composed with $\log$, is regularly varying and that the limit of the uniform probability measure on a large ball exists in the horocompactification. As an application we prove convergence towards a unique IPVT for higher rank symmetric spaces, which solves an open problem of \cite{MiMe23}. Versions of this theorem are provided for graphs and edge-measured graphs, where a natural parameter $\xi$ appears. We prove independence on $\xi$ in a specific sense under mild assumptions, which answers an open problem of~\cite{IPVT}. As a main example, we show that the latter holds for the IPVT of Diestel-Leader graphs. We also focus on further properties of this example, in particular, that its IPVT cells are distinguishable, providing the first Cayley graph with this property.

We introduce a cohomology theory for cyclic associative algebras, a subclass of shift associative algebras defined by the identity $(xy)z = x(yz) = y(zx)$. This cohomology, denoted $H^\bullet_{\mathrm{cyc}}(A, M)$, is a subtheory of Hochschild cohomology obtained by restricting to cochains that satisfy a cyclic compatibility condition derived from the defining identity. We prove that $H^2_{\mathrm{cyc}}(A, M)$ classifies cyclic associative extensions of $A$ by a cyclic bimodule $M$. The universal derivation and the module of differential forms $\Omega^\bullet_{\mathbb{F}}(A)$ are constructed, and $(\Omega^\bullet_{\mathbb{F}}(A), d)$ is shown to be the universal cyclic differential graded algebra over $A$. For trivial coefficients, we establish natural inclusions $HC^n(A) \hookrightarrow H^n_{\mathrm{cyc}}(A, \mathbb{F}) \hookrightarrow HH^n(A, \mathbb{F})$, placing our theory intermediate between Connes' cyclic cohomology and Hochschild cohomology.

Let $G$ be a finite group with $|G|=p^m$ where $p$ is a prime and $m$ is a positive integer. Let $k<p$. Let $a_1,\ldots,a_k\in G$ be pairwise distinct and let $b_1,\ldots,b_k\in G$. Then there exists a permutation $\sigma$ on $1,\ldots,k$ such that $a_1b_{\sigma(1)},\ldots,a_kb_{\sigma(k)}$ are pairwise distinct. This extends a theorem of Feng, Sun and Xiang, who proved that the conclusion holds in abelian $p$-groups.

Burr, Erd\H os, Graham, and Sós introduced the maximal anti-Ramsey function $\chi_{\mathrm{S}}(n,e,L)$, the minimum number of colors required over all $n$-vertex graphs with at least $e$ edges such that every copy of $L$ is rainbow. In \cite{BEGS1989}, they posed the following two problems: (i) Is it true that there exists $C>0$, such that for all $u\ge 1$, $\chi_{\mathrm{S}}\left(n,\lfloor un \rfloor,P_4 \right)<Cu$ holds for all sufficiently large $n$? (ii) Is it true that for all $\epsilon >0$, there exists $c(\epsilon)>0$ such that for all sufficiently large $n$, \\ $\chi_{\mathrm{S}}\left(n,\binom{n}{2}-\lfloor n^{2-\epsilon} \rfloor,P_4 \right)>c(\epsilon)n^{2}$? In this note, we give an affirmative answer to the first problem and a negative answer to the second problem. For the first problem, our proof uses a local density inequality with strong edge-colorings of odd Kneser graphs. In particular, our proof uses the characterization by Lužar, Máčajová, Škoviera, and Soták of~$k$-regular graphs whose strong chromatic index equals~$2k-1$. For the second result, our main tool is the construction of Alon, Moitra, and Sudakov. We show that for every fixed~$0<\epsilon<1/2$ there exist~$\gamma>0$ and arbitrarily large~$n$ such that~$\chi_{\mathrm{S}}\bigl(n,\tbinom{n}{2}-\lfloor n^{2-\epsilon}\rfloor,P_4\bigr)\;\le\; n^{2-\gamma}=o(n^{2}).$

A Pisot numeration system $U$ for $\mathbb N$ is a sequence of natural numbers generated by an integral homogeneous linear recurrence whose characteristic polynomial is the minimal polynomial of a Pisot number. The purpose of this paper is to introduce the analogue of the group of $p$-adic integers for such numerations when they preserve zeros, which is equivalent to the `Condition F' introduced by Frougny and Solomyak for $\beta$-numerations. We show that these topological groups $\mathbb Z_U$ project homomorphically onto a torus. Equipping $\mathbb Z_U$ with the appropriate topology, we also show that if $U$ is unimodular, then $\mathbb Z_U$ is continuously isomorphic to a torus.

Solving heterogeneous Helmholtz equations at high wavenumbers remains challenging because the discretized operator is indefinite, pollution degrades phase accuracy, and scalar coarse-grid correction can discard the local phase and propagation-direction information carried by oscillatory errors. We propose Multi-channel Multigrid (McMg), a learned phase-space multigrid preconditioner for heterogeneous Helmholtz equations. Rather than predicting the solution directly, McMg maps residuals to corrections within an iterative framework. Its central idea is to coarsen physical space while retaining unresolved local wave information in the channel dimension: each coarse node carries a learned packet of amplitude, phase, direction, and scattering coefficients rather than a single scalar unknown. The architecture combines linear multi-channel transfer operators with locally adaptive stencils, neural PDE operators, and medium-dependent smoothers whose coefficients are generated from the wave speed. For a fixed medium, the V-cycle is linear in the residual; nonlinear physical features are computed once in a setup phase and cached, so each online iteration reduces to convolutions with fixed coefficients. We further study generalization across scales. Models trained on small domains transfer directly to larger domains and higher effective wavenumbers, and a Layer-by-Layer Progressive Finetuning (LLPF) strategy improves large-domain scalability by adding new coarse levels while finetuning only the newly introduced parameters. Numerical experiments on high-frequency, high-contrast, and large-scale three-dimensional problems demonstrate that McMg requires substantially fewer iterations and less wall-clock time than strong classical baselines, while consistently outperforming existing neural preconditioners.

Zigzag persistence tracks topological features in spatio-temporal data through combinatorial invariants called barcodes. For binary videos, existing methods are bottlenecked by the construction of prohibitively large cubical complexes and performing Gaussian elimination on large boundary matrices, rendering high-resolution videos out of reach. We show that the $H_0$ and $H_1$ barcodes can be extracted directly from connected-component dynamics. By encoding these dynamics in a graph, we bypass cubical complexes entirely and are able to leverage the near-linear time barcode decomposition algorithm by Dey and Hou, leading to significant speedups. The total runtime of our pipeline is dominated by the construction of the underlying graph structures, which scales linearly with pixel count and is embarrassingly parallel across frames, ensuring excellent scalability. We demonstrate how this approach enables zigzag persistence on 4k video at real-time rates on consumer hardware.

This paper introduces the Kalman-Bucy-Koopman (KBK) filter, a novel framework for nonlinear state estimation grounded in Koopman operator spectral theory. The nonlinear estimation problem is formulated as a maximum-likelihood (Mortensen) estimator whose solution is characterized by a Hamilton-Jacobi (HJ) partial differential equation. The proposed KBK filter provides a spectral, operator-theoretic realization of this nonlinear filtering problem by parameterizing the HJ value function in terms of principal Koopman eigenfunctions. This transformation converts the nonlinear estimation problem into a Riccati-type evolution in Koopman coordinates, yielding a linear-operator analogue of the classical Kalman-Bucy filter while preserving nonlinear structure in the original state variables. We develop a path-integral formulation for computing principal Koopman eigenfunctions and introduce a dynamics-informed, characteristics-inspired basis construction for their approximation. Theoretical error bounds are derived for value-function and state-estimation approximations. Simulation results demonstrate improved performance over the extended Kalman filter and illustrate the ability of the KBK framework to operate in data-driven settings without explicit model linearization.

Let $P=\{x_1,\ldots,x_n\}$ be a multiset consisting of $n\ge 2$ real numbers such that $\sum_{i=1}^{n}x_i=0$ and $\sum_{i=1}^{n}|x_i|>0$, and let $k <n$ be a positive integer. We sample $k$ elements from $P$ without replacement and set $X_P$ be the sum of the elements in our sample. It is shown that the Shannon entropy of $X_P$ satisfies \[ \mathbf{H}(X_P) \ge \mathbf{H}(\text{Ber}(k/n)) \, , \] where $\text{Ber}(k/n)$ is a Bernoulli random variable of mean $k/n$. The result is sharp, and may be seen as an entropic analogue of the Manickam-Miklós-Singhi (MMS) conjecture.

Motivated by some recent studies on higher order Markov chains and well-known characterizations for irreducibility and primitivity of nonnegative matrices, we propose in this paper an alternative framework for irreducibility and primitivity of nonnegative tensors, giving rise to two new notions: s-irreducibility and s-primitivity. Our framework includes the relevant results on nonnegative matrices as its special cases, yet it expands existing results regarding irreducibility and primitivity of nonnegative tensors. In addition to its tensor theoretic significance, such a framework has important implications for applied problems such as higher order Markov chains.

Let $G=GL(n)$ be the $n\times n$ complex general linear group and let $\B_{n}$ be its flag variety. A Borel subgroup $B$ of $G$ acts on $\B_{n}\times \mathbb{P}^{n-1}$ diagonally with finitely many orbits. In this paper, we give an embedding of the $B$-orbits on $\B_{n}\times \mathbb{P}^{n-1}$ into the $B$-orbits on the flag variety $\B_{n+1}$ of $GL(n+1)$ and show that this correspondence respects closure relations and preserves monoid actions. As a consequence both closure relations and monoid actions on the set of all $B$-orbits on $\B_{n}\times\mathbb{P}^{n-1}$ can be understood via the Bruhat order on the symmetric group on $n+1$ letters by using our results in \cite{Shpairs}. This amplifies work of Magyar \cite{Magyar} by making the closure relation more transparent and allows us to compute the monoid action using Demazure products. If $S_i$ is the stabilizer in $B$ of the line through the ith standard basis vector, we give an embedding of the $S_i$-orbits on $\B_n$ into the $B$-orbits in a single $G$-orbit in $\B_{n+1},$ and this embedding plays an essential role in the above results. We extend results from our papers \cite{CE21I}, \cite{CE21II}, and \cite{Shpairs}, and in particular show that for $S_i$-orbits on $\B_n,$ the closure ordering is given by the Richardson-Springer standard order.

We consider a random recursive DAG $G_n$ on the vertex set $[n]$ where every vertex $i\geq 2$ has out-degree $d$, with the targets chosen uniformly at random among the earlier $i-1$ vertices. For this model, we propose a novel way to investigate the descendants of $n$ (which have recently been studied in a paper by Janson) through what we call ancestry processes. The ancestor process $a_i(n)$ of a vertex $i$ is defined as the number of ancestors of $i$ in $G_n$, and is closely related to the evolutions of multi-draw Pólya urns. Results on the descendants can then be obtained via asymptotic results on functionals of the ancestry processes, generally leading to technical integral expressions. This method yields the answer to two questions posed by Janson, the first on the size of the joint descendants of vertices $n$ and $n+1$, and the other on the location of the earliest non-descendant. We further prove limit theorems for the ancestry processes $a_i(n)$ depending on $i$, determine the location of the earliest source node, and provide an alternative proof of a first-moment result contained in Janson's work.

We study the existence of stable invariant measures for operators and strongly continuous semigroups of operators on Banach spaces admitting either a dense bilateral backward orbit or a sufficiently rich family of eigenvectors. These invariant measures are realized as the distributions of stochastic integrals with respect to stable random measures. We also discuss invariant measures with other classes of distributions for such operators and semigroups.

The standard convergence analysis of mini-batch stochastic gradient descent (SGD) models gradient noise using a single variance term that treats all parameter directions equally, ignoring the fact that noise in high-curvature directions has less impact because learning rates are already constrained there. We introduce Curvature-Weighted Gradient Diversity (CWGD), a geometry-aware measure that weights per-sample gradient diversity by the inverse square root of the Hessian, providing a tighter proxy for the effective optimization noise. For strongly convex quadratic objectives with diagonal Hessians and isotropic noise, we prove that a CWGD-modulated cosine learning-rate schedule can reduce the asymptotic optimization error floor by up to a factor of two compared with standard cosine annealing. We implement this idea as CWGD-Cosine using a Hutchinson-based diagonal Hessian estimator that is exact for quadratic objectives. Across a range of condition numbers, batch sizes, and noise structures, CWGD-Cosine consistently achieves approximately 20% lower final optimization error than standard cosine annealing while incurring negligible overhead in the quadratic setting. We also identify and correct a degenerate curvature estimator, analyze the robustness of the proposed estimator, and explicitly discuss the limitations of the method, including Hessian staleness in non-convex optimization. These results establish CWGD as a principled geometry-aware measure of optimization noise and motivate future extensions to more general learning problems.

In this short note we establish the Schur–Zassenhaus Theorem and Sylow's Third Theorem for finite skew braces. More precisely, we prove that every Hall ideal of a finite skew brace admits a sub-skew brace complement, and more generally that every left ideal whose order is coprime to that of the Hall ideal can be embedded in such a complement. Using similar ideas we show that every left ideal of prime-power order is contained in a Sylow sub-skew brace. Finally, we prove that the number of Sylow $p$-sub-skew braces is congruent to $1$ modulo $p$, and provide examples showing that the corresponding containment property fails for arbitrary sub-skew braces.

Many solution methods for linear discrete ill-posed problems with error-contaminated data (right-hand side) apply Tikhonov regularization to compute a meaningful approximate solution. This solution depends on a regularization parameter. It is well known that iterated Tikhonov regularization often determines an approximate solution of higher quality than (standard) Tikhonov regularization. We consider the situation when an estimate of the norm of the error in the data is known and would like to apply iterative Tikhonov regularization to determine an approximate solution that satisfies the discrepancy principle. This requires a suitable choice of a regularization parameter. The standard approach to determine this parameter is to compute solutions for several values of the regularization parameter and choose a computed approximate solution that satisfies the discrepancy principle. This paper discusses iterated Tikhonov regularization based on partial Golub-Kahan bidiagonalization and describes how the regularization parameter can be determined without computing several approximate solutions by using the connection between Golub-Kahan bidiagonalization and Gauss quadrature. This approach reduces the computational effort required to compute a desired solution.

We present a quantum spectral solver for the steady incompressible Stokes equations on a two-dimensional periodic domain. The method uses the Quantum Fourier Transform as a coherent change of basis and exploits the resulting spectral structure of the Stokes operator: the Laplacian becomes diagonal, while incompressibility is enforced mode by mode through a Helmholtz projection. In two dimensions, this projection is realized by a mode-dependent rotation from Cartesian velocity components to longitudinal–transverse coordinates, followed by component-conditioned inverse-Laplacian scaling. The velocity and pressure fields are encoded as quantum states over Fourier modes and physical components, and the corresponding spectral factors are implemented through polynomially encoded amplitude blocks. The construction extends recent quantum spectral methods in computational mechanics to an incompressible flow operator with explicit pressure–velocity splitting and divergence-free projection. The approach is also compatible with multiscale finite-element architectures in which quantum parallelism can simultaneously update all representative volume element (RVE) states. Numerical verification includes a steady vortex, a regularized periodic force-dipole benchmark, and an RVE-inspired Kolmogorov-like fluctuation benchmark. The latter illustrates how the circuit can recover a homogenized kinetic-energy observable without reconstructing the full velocity field, consistent with the role of averaged quantities in multiscale flow calculations. Subject to standard assumptions about efficient state preparation and observable estimation, the circuit has polylogarithmic dependence on the grid resolution, with the polynomial degree and tile count appearing as explicit approximation and implementation parameters.

We consider local hypersurface measures in ${\mathbb R}^3$ whose density is allowed to have a weight function constructed from real analytic functions in a broad sense. We prove $L^p$ Sobolev smoothing theorems for convolutions with such surface measures and Fourier transform decay rate results for these measures, generalizing and subsuming earlier results for smooth densities. Our theorems are sharp in an appropriate sense and can be described in terms of relatively simple properties of the surfaces and weight functions.

We study a two-dimensional Ornstein-Uhlenbeck system where only the first coordinate is observed, whereas the second coordinate remains hidden. Our goal is the estimation of the coupling parameter in the drift of the observed coordinate. The core novelty lies in accounting for the influence of the observed component on the unobserved one, making the system fully coupled. Using linear filtering, we derive the likelihood under partial observation and establish local asymptotic normality of the statistical model. Within the Ibragimov-Hasminskii framework (1981), we prove consistency, asymptotic normality, convergence of moments and asymptotic efficiency of the MLE under stability and identifiability assumptions as the time horizon tends to infinity.

We introduce a generalized continuous-time compartment model of ethanol metabolism in the human body that extends a recently developed framework. In the proposed model, we replace the Michaelis-Menten mechanism of the liver's ethanol metabolism rate with a general class of nonlinear rate functions. This modification provides greater modeling flexibility and enables the model to capture a wider range of hepatic ethanol metabolism dynamics. The qualitative behavior of the proposed ethanol metabolism model is analyzed rigorously. More specifically, we investigate the positivity and boundedness of solutions, as well as the global asymptotic stability (GAS) of the unique equilibrium point using an appropriate quadratic Lyapunov function. Second, we formulate a discrete-time counterpart of the proposed continuous-time model and investigate its dynamical properties. We show that, under an appropriate condition on the time step size, the discrete-time model faithfully reproduces the qualitative dynamical behavior of the corresponding continuous-time system. Lastly, we conduct a series of numerical experiments employing several ethanol metabolism rate functions to support the theoretical results.

We study the spatially homogeneous, massless Boltzmann equation in Minkowski spacetime for a certain range of hard and soft interactions. For hard interactions, we derive a Povzner-type inequality for massless particles and show that solutions exist for all time into the future. For soft interactions, we employ singular weights to control singularities at $ p = 0 $, which arise from the masslessness of particles, to obtain local existence. These results, which are among rather few proofs of existence for the massless Boltzmann equation, are motivated by our earlier work on the massless Einstein–Boltzmann system in certain cosmological settings.

Let $Q=\{q_n\}_{n\geq1}$ be a sequence of integers with $q_n\geq2$ for every $n\geq1$. Every $x\in[0,1)$ admits a Cantor series expansion of the form $$ x = \frac{\varepsilon_1(x)}{q_1} + \frac{\varepsilon_2(x)}{q_1q_2} +\cdots+ \frac{\varepsilon_n(x)}{q_1q_2\cdots q_n} +\cdots.$$In this paper, we study exact approximation of real numbers by the $n$-th partial sums of their Cantor series expansions. More precisely, for a non-increasing function $\psi:\mathbb{N}\to(0,\infty)$ satisfying $\psi(n)\to0$, let $E_c(\psi)$ be the set of points that are $\psi$-approximable by their $n$-th partial sums but are not $b\psi$-approximable for any $0<b<1$. We determine the Hausdorff dimension of $E_c(\psi)$.

We construct a simple and useful sufficient condition, based on actions on a lattice of idempotents, for monoids admitting homomorphisms to the monogenic free inverse monoid $\mathrm{FIM}(1)$ to not be of type $\mathrm{FP}_2$. This recovers a result of Gray and Steinberg that free inverse monoids are not of type $\mathrm{FP}_2$. The same technique is then used to show that a finitely generated submonoid of $\mathrm{FIM}(1)$ is of type $\mathrm{FP}_2$ if and only if it is finitely presented, answering a question of Cho & Ruškuc.

Lu conjectured that, for each dimension and codimension, there exists a positive gap above the first pinching value for the quantity $S+\lambda_2$ on closed minimal submanifolds of the unit sphere, where $S$ is the squared norm of the second fundamental form and $\lambda_2$ is the second eigenvalue of Lu's fundamental matrix. We disprove this second-gap conjecture for minimal surfaces in every codimension $q\ge3$. More precisely, in every odd codimension $q\ge3$ we construct linearly full closed embedded flat minimal tori with $S\equiv2$ for which the constant values of $S+\lambda_2$ are dense in $(2,3)$. Thus the first pinching value $2$ can be approached from above by closed embedded minimal surfaces, and no uniform second gap exists in codimension at least three. Together with the known positive results in codimensions one and two, our examples complete the codimension picture for Lu's second-gap problem for minimal surfaces: the conjecture holds precisely in codimensions $1$ and $2$,

We use the Banerjee–Mandal–Sahoo dipole-current Ward identity for the one-loop logarithmic soft-photon theorem as input and determine its finite-energy action on Mellin-difference hard currents. The commutator with such hard currents has a scheme-independent hard-hard residue that survives every one-particle redefinition. With the meromorphic continuation stated explicitly below, a two-particle Plancherel transform identifies this residue with an analytic two-particle primary module, and the coefficient map is a hard-current one-cocycle. The cocycle defines a minimal filtered abelian extension. It has a canonical two-particle primitive and integrates to an affine action. For scalar hard legs, the fixed-leg operator agrees coefficient by coefficient with the symmetry-governed long-range logarithmic tower of Choi, Kadhe, and Puhm. Applied to a tree-level scalar-QED photon-exchange block, the finite-energy analysis determines the logarithmic two-particle coefficient functional from the ordinary hard amplitude and the Banerjee–Mandal–Sahoo ordered-pair soft kernel. This gives a finite-energy relation between the Banerjee–Mandal–Sahoo dipole-current Ward identity and the exponentiated long-range celestial OPE.

We consider the problem of estimating numerically integrals of the shape $$ \int_P \frac{dt}{t_1 \dotsb t_k} $$ where $P \in {\mathbb R}_{>0}^k$ is a convex polytope, $t=(t_1,\dotsc, t_k)$ and $d t$ is the Lebesgue measure. This type of integral appears frequently in main terms of sieve theory. We propose a simple method, based on the LattE software for integration of polynomials over polytopes, which computes rigorous bounds on this integral in polynomial time with respect to the precision (in bits). We test the method on several examples from the literature of sieve theory. We apply our results to compute numerical approximations to the natural density $$ h(\alpha, \beta) := \operatorname{density}\{n\in{\mathbb N}, \exists d\mid n, d\in [n^\alpha, n^\beta]\}, \qquad (0<\alpha<\beta<1) $$ of integers having a localized divisor, in the region $\beta - \alpha \geq 0.02$. One ingredient involved is a refined formula for $h(\alpha, \beta)$ which involves a manageable number of terms for these $\alpha, \beta$. As a corollary, we give a numerical approximation of the leading constant in a theorem of Haddad and Koukoulopoulos on the average of the logarithm of middle-divisors of integers.

We consider the problem of transmitting sparse local updates to the server in a distributed learning system. Specifically, the system consists of $n$ clients, each possessing a $k$-sparse $d$-dimensional local model, and a central server responsible for aggregating the clients' models into a global model. The goal is to characterize the tradeoff between the communication cost in the transmission from the clients to the server and the accuracy in aggregating the global model. We propose a compression scheme for sparse local models by concatenating a covering method and a sketching method. We also present a converse based on f-divergence, which strengthens the conventional Fano-type lower bounds. The proposed lower bound is tight for the frequency estimation case, that is, each coordinate takes values in a binary alphabet. For general alphabets, the proposed achievable schemes remain suboptimal relative to the converse bounds, indicating that a complete characterization of the communication-accuracy tradeoff requires further investigation.

We examine how the Rasmussen invariant, satellite operations, and null-homologous twists can be used to establish infinite order of knots in the smooth concordance group. As an application, we show that the Conway knot has infinite concordance order.