Papers

This paper revisits from a mathematical perspective the modeling of locomotion in snake-like bodies in fluid environement as given in \cite{boyer2010poincare} and \cite{boyer2011macrocontinuous}. Using a Cosserat rod theory, we derive several systems of PDEs depending on the type of constraint and actuation.

In this paper, we study several properties concerning cut points and non-cut points. First, we show that the property of being a continuum consisting entirely of shore points is preserved under refinable maps. Next, we show that having a block point, having a non-shore point, and having a strong center point are not Whitney-reversible properties. Consequently, \cite[Question 2.9]{nonweak} has a negative answer. Furthermore, we show that none of the classes of continua witnessing the failure of the sequential strong Whitney-reversible property for these three properties admits a common model. Finally, in connection with the fact that colocal connectedness is not a Whitney-reversible property, we show that the class of nonaposyndetic continua having a cut point and such that every positive Whitney level is colocally connected has no common model.

We establish a Chouinard-type theorem for modules over algebras of cochains on the classifying spaces of a broad class of infinite groups, allowing commutative ring spectra as coefficients. We then use this result to study finiteness properties of the integral cohomology rings of certain infinite groups, as well as stratification of the corresponding module categories. In particular, we complete the (co)stratification of module categories associated to homotopical groups over fields of positive characteristic.

We classify the real characters of the Grossman-Larson Hopf algebra that are nonnegative on the rooted-tree basis. They vanish on trees with branching away from the root, and the normalized characters are parametrized by Kingman paintboxes. Two-sided Pieri states are mixtures of the normalized characters, with unique mixing measures. The corresponding laws form a proper exposed Bauer face of the simplex of central measures on the Hoffman rooted-tree graph. For the path-forest subgraph, the full and minimal Martin boundaries coincide and are homeomorphic to the Kingman simplex. The limiting ranked root-branch frequencies generate the completed central tail.

Let S^{n+1}_{+} be the open hemisphere of the unit sphere S^{n+1} centred at o. We study the non-homogeneous curvature flow X_t=-f(r) sigma_k^{alpha} {nu} of smooth, closed, strictly convex hypersurfaces enclosing o, where r is the geodesic distance to o. We consider both the supercritical regime beta>1+k {alpha} and the critical regime beta=1+k{alpha}, where beta is the growth order of the profile at the origin, f(r) almost equals r^{beta} as r descends to 0. Under the structural condition that f^{1/(1+k{alpha})} is convex, we prove long-time existence and preservation of strict convexity. The normalized radial function converges smoothly and exponentially to a constant: to 1 and to R_{infty}>0, resp. in different two cases. Thus the normalized radial graphs become round, while the original hypersurfaces contract to o.

The study of preordered semirings is motivated by applications in computer science, graph theory, and information theory, and provides tools for understanding the asymptotic preorder, which compares large powers of a pair of elements. This paper studies sequences which behave approximately as sequences of powers, but are not necessarily equivalent to geometric sequences. Our main result is that preordered semirings admit completions where such sequences, that we call approximately geometric, become equivalent to geometric sequences, and that existing characterizations of the asymptotic preorder extend to the completion. We provide several classes of examples of approximately geometric sequences in the semiring of tensors, and in the semiring of graphs. As a concrete application, we determine the strong converse exponent for binary hypothesis testing with composite Markov hypotheses.

We establish an exact additive formula expressing the Lelong number of a plurisubharmonic function in terms of the limiting difference between its spherical and solid means. Building on this identity, we derive several equivalent criteria for the vanishing of the Lelong number, formulated in terms of fixed-scale increments and the asymptotic agreement of radial maxima, spherical and solid means, and radial regularizations. We also give a pointwise $L^1$ mean-oscillation formulation of the VMO characterization for plurisubharmonic functions with zero Lelong number, together with a streamlined proof of the corresponding global result of Biard and Wu. Finally, we establish a nonlinear characterization in terms of the relative extremal depth and the Bedford-Taylor capacity of deep sublevel sets.

The Teukolsky Master Equation (TME) is a powerful technique used to study perturbations of a spacetime with gravitational, electromagnetic, and neutrino components. The well-known Kerr metric is a suitable candidate for this formalism because it is a Petrov type D metric. However, the Kerr metric represents the exterior spacetime of a perfect sphere and only black holes are described using this metric. For another astrophysical object, such as fast-spinning neutron stars, the Kerr metric is not valid due to the rotation induced deformations to the gravitational source. The main goal of the present work is to show a TME for a Kerr-like metric (KL) with a mass quadrupole moment parameter $q$. This new parameter will represent deformations from a perfect sphere; therefore, KL is a better candidate to study perturbations surrounding spinning stars.

A connected graph $G$ with at least two vertices is matching covered if each of its edges lies in a perfect matching. A matching covered graph is minimal if the removal of any edge results in a graph that is no longer matching covered. Lovász and Plummer [J. Combin. Theory, Ser. B 23 (1977) 127–138] proved by ear decompositions that every minimal matching covered bipartite graph $G$ different from $K_2$ has at most $(3|V(G)|-6)/2$ edges, and this bound is sharp for all $|V(G)|\ge4$. In this paper, we prove that every minimal matching covered nonbipartite graph $G$ with at least 6 vertices has at most $5(|V(G)|-2)/2$ edges, and this bound is sharp for all $|V(G)|\ge6$.

This paper investigates the canonical Koszul one-form at the interface of Jean Marie Souriau's Lie group thermodynamics and Paulette Libermann's symplectically complete foliations to describe geometric dissipation. On the Poincaré half-plane $\mathbb{H}$ under $SL(2, \mathbb{R})$, the equivariant momentum map level sets form a regular Libermann foliation framing the conservative boundaries. The Koszul-derived dissipative vector field acts strictly transverse to these leaves, driving a metriplectic flow that breaks Noether conservation laws and generates entropy. Quantum wise, we develop Berezin quantization on para-Kähler symmetric spaces, where the Koszul potential acts as a weight determining state density via the Fisher Souriau metric, and formalize Kostant's geometric quantization using the real dual polarizations of a bi-Lagrangian web.

Recently, Schatten class membership of commutators arising from Connes' quantised calculus has been characterised in a general framework, which covers multiple concrete situations of interest. In contrast to this, related results dealing with the asymptotic behaviour of the singular values of these commutators have been restricted to a relatively short list of specific examples only. In this work, we show that a version of the spectral asymptotic formula, involving comparable lower and upper limits in place of an exact limit, remains valid in great generality of singular integrals over metric measure spaces. A key proof ingredient of independent interest is a new asymptotic version of the Marcinkiewicz interpolation theorem. We also present a new approach to commutator upper bounds at the critical index; while not quite as general as more elaborate methods, it is general enough to reproduce the known upper bounds in Heisenberg and Carnot groups in a much simpler way.

The empirical success of pretraining large language models has inspired a deeper investigation into the underlying loss landscapes and the optimization dynamics. Recent empirical and theoretical study suggest that the training loss landscape often exhibits a "river-valley" structure, which features a low-loss manifold (river) flanked by sharp orthogonal directions with higher loss (mountains). In the long term, the optimization progress is determined primarily by the progress along the river. Within such a landscape, gradient descent with large learning rates can move faster along the river despite high apparent loss due to vertical oscillations, while a subsequent sharp decay in the learning rate suppresses these oscillations, revealing genuine optimization progress. This explains the recent success of warmup-stable-decay (WSD) learning rate scheduler which, unlike cosine scheduling, keeps stable high learning rate and decays before producing intermediate checkpoints. Building on this foundation, in this work we take a step further and study the role of momentum within such a loss landscape. We establish theoretical analysis that characterizes how momentum accelerates optimization by stabilizing large learning rates that can not be tolerated by vanilla GD without deviating significantly from the river. The enabled large learning rate in-turn gives greater speed along the river and makes faster essential progress in the long run. Another intriguing observation from theory is that for a river-valley landscape with very flat and slow-spinning river, the momentum itself does not contribute directly to acceleration in terms of the speed of tracking the river, while the main acceleration comes from the admissible larger learning rate.

We study regularity for the variable exponent quasilinear $\mathfrak{p}\left(x\right)$-Laplace type equation on domains in Heisenberg groups and Euclidean spaces. We establish borderline continuity estimates for the appropriate first order derivatives of solutions. More precisely, we show that for any weak solution $u \in \mathbb{E}W^{1, \mathfrak{p}(\cdot)}\left(\Omega\right)$ of \begin{align*} \operatorname{div}_{\mathbb{E}} \left( \mathfrak{a}(x) \lvert \nabla_{\mathbb{E}} u \rvert^{\mathfrak{p}\left(x\right)-2} \nabla_{\mathbb{E}} u \right) = f \qquad \text{ in } \Omega, \end{align*} where $\Omega \subset \mathbb{E}^{n}$, $\mathfrak{a}$ is a uniformly positive bounded scalar function and the exponent function $\mathfrak{p}$ is uniformly bounded away from $1$ and $\infty,$ $\nabla_{\mathbb{E}}u$ is continuous in $\Omega$ as soon as $f \in L^{\left( Q_{\mathbb{E}}, 1\right)}\left(\Omega\right)$ and $\mathfrak{a}, \mathfrak{p}$ satisfies some conditions regarding the summability of their mean-oscillations. Here $\mathbb{E}^{n}$ is either the Heisenberg group $\mathbb{H}_{n}$ or the Euclidean space $\mathbb{R}^{n}$ and $\operatorname{div}_{\mathbb{E}}$, $\nabla_{\mathbb{E}}$, $Q_{\mathbb{E}}$ stands for the corresponding divergence, gradient and homogeneous dimension, respectively. We treat the elliptic and subelliptic cases in a unified manner using Euclidean techniques and our conditions on $\mathfrak{a}$ and $\mathfrak{p}$ are new and weaker than all the known sufficient conditions even in the Euclidean case. However, all the known sufficient conditions imply our conditions, achieving yet another unification.

Quantum-inspired evolutionary optimization (QIEO) represents design variables as a set of qubits and searches a continuous, multi-dimensional landscape through rotation of the qubit's amplitude pair. Every generation rotates those amplitudes toward a single elite, which corresponds to that generation's best. The update is cheap, almost parameter-free, and well-suited for massive parallel implementation, which has encouraged its adoption in engineering, design, and planning applications. However, there are critical issues with this formulation, principally, the treatment of design variables as independent probability components which make it incapable of exploiting local curvature, anisotropy, or variable coupling. Despite this, QIEO is believed to hold promise, and has been used extensively to solve real-world problems, with significant qualitative and computational advantage over its classical counterpart, Genetic Algorithm (GA). A collection of 256 (actually 508; 256 unshifted + 252 shifted, 4 could not be shifted) continuous function are selected from the prior works, in such a way that they represent eleven landscape characteristics, namely continuity, differentiability, separability, scalability, modality, convexity, conditioning, symmetry, maximum dimensionality, dimension dependency, and the coupling pattern of the design variables. These functions are then solved by three QIEO variants, two GA encodings and Hansen's Covariance Matrix Adaptation Evolution Strategy (CMA-ES). The results are evaluated in terms of computational cost, solution precision, and specialization across landscape characteristics. They identify the conditions under which QIEO provides competitive performance, clarify where its independent-variable representation becomes limiting, and establish whether particular QIEO variants offer advantages for specific landscape characteristics.

This work studies general sequences of discrete orthogonal polynomials on linear lattices, including the so-called discrete Laguerre–Hahn orthogonal polynomials. We deduce the structure relations involving the orthogonal polynomials and their associated, as well as the explicit form of the fourth–order linear difference equation satisfied by these Laguerre–Hahn orthogonal polynomials. Furthermore, the particular cases corresponding to semiclassical and classical families of orthogonal polynomials, which lead to second–order difference equations, are also analyzed. Several examples illustrating the computation of the explicit coefficients of these difference equations for various families of orthogonal polynomials are also presented.

We prove that every water transport problem on an $n$-vertex graph has an optimal strategy of length at most $n^{(2+o(1))n}$. More strongly, the convex hull of all strategy operators stabilizes within the same bound. We also give a five-vertex instance in which every optimal strategy repeats a nontrivial connected averaging set.

We consider an interpolation with the parameter $p\in [0,1]$ between the discrete-time quantum ($p=0$) and its intrinsic random ($p=1$) walks on a finite connected graph. Its time evolution is defined by the convex combination of the Kraus (CPTP) maps of the quantum and random walks. The Kraus map of the random walk is represented by taking a kind of projection of that of the quantum walk. The time evolution of the interpolation is interpreted as the combination of the following two dynamics of $2$ walkers on the same graph correlated with each other: $2$ walkers move independently of each other (no-correlation) with probability $1-p$, while $2$ walkers always move into the same position (the maximal correlation) with probability $p$ at each time step. In this paper, we generalize the random walk to a new walk, namely, the correlated walk, which preserves the intrinsic property and relaxes the maximal correlation of the random walk. This correlation is determined by a partition of the arc sets in the underlying graph. We show that the eigenvalues of the interpolating walk between the correlated and quantum walks live in $\{ z\in \mathbb{C}\;|\;1-p\leq |z|\leq 1 \}$ and that the absorption state coincides with that of the correlated walk, which is characterized by the underlying graph structures.

Let $k\ge3$ and $1\le \ell\le k-1$. We study balanced $k$-partitions of a graph for which the union of any $\ell$ parts induces few edges. We show that every graph $G$ with $n$ vertices and $m$ edges admits a balanced partition $V_1,\ldots,V_k$ such that \begin{equation*} \max_{\substack{A\in\binom{[k]}{\ell}}}e_G\left(\bigcup_{i\in A}V_i\right)\le\frac{\ell^2}{k^2}m+\frac{\ell^2(k-\ell)}{k^2}(n-1)+\frac{\ell(k-\ell)}{k(k-1)}\sqrt{\left(\binom{k}{\ell}-1\right)m}. \end{equation*} In the case $\ell=2$, our result confirms a conjecture of Bollobás and Scott in a stronger form.

For every $2\le m\le n$, we construct Hölder continuous functions on the closed unit ball that are minimal in the Cegrell class $\F_m$, although their Hessian measures are not maximal in the $m$-subharmonic ordering. This answers a question posed by the second author. We also obtain a minimality criterion based on partial pluricomplex energy and an explicit family of Monge–Ampère examples on a product domain. For $m=1$, minimality and maximality are equivalent on a ball.

The Duffin-Schaeffer Conjecture, proven in 2020, gives a zero-full dichotomy for the approximation of irrational numbers with reduced fractions for arbitrary approximation functions. We construct approximation functions such that the inhomogeneous analogue of the Duffin-Schaeffer Conjecture fails. The counterexamples are obtained for all non-zero rationals and certain Liouville numbers.

We establish the global well-posedness for the energy-subcritical defocusing cubic nonlinear Schrödinger equation (NLS) on the three-dimensional Friedlander model domain \( \Omega=\{x\in\mathbb{R}:x\geq0\}\times\mathbb{R}^2_{y,z}, \) subject to homogeneous Dirichlet boundary conditions, for arbitrarily large initial data in the coercive energy space $H_0^1(\Omega)$. The phase-space geometry of this configuration features a strictly convex boundary that admits a non-empty glancing set, trapping high-frequency wave packets within a boundary layer via generalized whispering-gallery caustics. These concentration phenomena induce an intrinsic derivative loss in the sharp linear Strichartz estimates, establishing a major microlocal obstruction to closing the nonlinear Duhamel iteration directly at the energy level via classical perturbative frameworks. To bridge the regularity deficit between the conservation laws and the linear theory, we construct a boundary-adapted family of continuous–discrete Bourgain–Strichartz restriction spaces $X^{s,b}$ built from the spectral decomposition of the Dirichlet realization of the model Friedlander operator \( \Delta_g=\partial_x^2+(1+x)\partial_y^2+\partial_z^2. \) The normal variable is resolved through discrete Airy spectral modes, while the tangential variables are continuously Fourier analyzed. Within this functional framework, we implement a refined high–low frequency decomposition to formulate a perturbed nonlinear equation for the high-frequency remainder in the sub-energy space $H_0^s(\Omega)$ for \( \frac{1}{2}<s<1. \) The initial high-frequency datum satisfies a quantitative sub-energy decay estimate of the form \( \|P_{>\lambda}u_0\|_{H_0^s(\Omega)} \label{eq:abs_decay} \lesssim \lambda^{s-1}\|u_0\|_{H_0^1(\Omega)}, \) providing a small parameter to counteract the derivative loss.

Shallit's law of leap years in Pierce expansions gives, for Lebesgue-almost every $x\in[0,1]$, the upper and lower limits of a normalized difference between $Nx$ and the number of leap years up to the year $N$ determined by the Pierce expansion digits of $x$. In this paper, we show that, for every $x\in[0,1]$, both limits are determined by the lower limit of a normalized logarithm of the product of the first $n$ digits of $x$. In particular, the upper and lower limits always have the same absolute value. As applications, we characterize the sets of points with prescribed upper and lower limits, and show that each of these sets, if non-empty, is dense in $[0,1]$ and its intersection with any non-empty open subset of $[0,1]$ has full Hausdorff dimension. We also compare these sets with the sets defined by the growth rate of the digits, and show that, for each finite parameter, their difference has full Hausdorff dimension.

Quantum-inspired algorithms emulate quantum mechanical principles, such as, superposition, interference, and probabilistic amplitude evolution, on classical hardware by representing candidate solutions as qubit vectors and evolving them through rotation-gate operators. This approach offers higher optimization performance without physical qubits, and has been shown to achieve order-of-magnitude speedups (10–80$\times$) over traditional solvers on combinatorial, high-dimensional NP-hard problems. A critical barrier to adoption, however, is the lack of a unified execution framework that delivers both algorithmic performance and hardware portability. We present Cross-Backend Quantum Inspired Evolutionary Optimizer (QIEO), the runtime core of BQP's BQPhy solver, which addresses this gap through a single-source-of-truth architecture. One C++ implementation of the QIEO algorithm is compiled once per hardware target and exposed to multiple high-level languages via thin binding layers. The framework dispatches to CPU (sequential), OpenMP~5 (multi-core), CUDA (NVIDIA), and HIP (AMD) backends at runtime, adapting kernels to each device's memory hierarchy and warp/wavefront execution model. The framework's real-world utility is validated through binding demonstrations that share the identical C++ runtime. BQPhy's Python library is demonstrated on a neural network hyperparameter optimisation achieving 88.60% test accuracy on MNIST. BQPhy's MATLAB's Toolkit is tested on wind farm layout optimisation attaining $365\,399 \pm 4\,552$~MWh/yr, which is statistically indistinguishable from particle swarm optimisation and $+7.6\%$ above genetic algorithms on a 32-variable constrained engineering problem. The Julia package tackles the Lotka–Volterra parameter estimation where BQPhy replaces native Julia solvers on the same residual, cutting mean SSE by $2.1\times$.

We consider the large time asymptotic behavior of solutions to the initial value problem for the incompressible Navier-Stokes equations in the whole space $\mathbb{R}^d$ ($d\ge 2$). When the initial velocity belongs to $L^1(\mathbb{R}^d)$, it is well-known that its spatial integral vanishes as a consequence of the divergence-free compatibility condition. Taking this property into account, Carpio (1996) and Fujigaki-Miyakawa (2001) derived higher-order asymptotic formulas for the strong solutions. In this paper, we investigate the large time behavior of solutions when the initial velocity belongs to the Fourier-Herz space $\widehat{L}^1(\mathbb{R}^d)$, which provides a broader framework than $L^1(\mathbb{R}^d)$ and imposes an additional condition in the low-frequency region. In particular, in the two-dimensional case, we show that an asymptotic profile different from those obtained by Carpio and Fujigaki-Miyakawa arises.

Odd Cycle Transversal is a classic $\mathsf{NP}$-hard graph optimization problem asking for a minimum-weight set of vertices whose deletion makes the input graph bipartite, or equivalently, a maximum-weight induced bipartite subgraph. We show that Odd Cycle Transversal is quasi-polynomial-time solvable on $kP_4$-free graphs, for every fixed $k \in \mathbb{N}$. In fact, we provide an $n^{O_k(\log n)}$-time algorithm for the more general \textsc{Max-Weight List $2$-Colorable Induced Subgraph}, where the notation $O_{k}(\cdot)$ hides factors depending on $k$. Paired with known results from the literature, this allows us to obtain a complete complexity dichotomy for these two problems on $H$-free graphs into cases solvable in quasi-polynomial time and cases which are $\mathsf{NP}$-hard, in particular resolving an open problem of Agrawal, Lima, Lokshtanov, Saurabh, and Sharma [SODA 2024]. Our algorithms are based on a new structural tool that may be of independent interest. We introduce the notion of $H$-amiable family and show that, for every fixed graph $H$ without isolated vertices and every fixed $k\ge2$, every $kH$-free graph admits an $H$-amiable family of quasi-polynomial size that can be constructed in quasi-polynomial time. Besides yielding the aforementioned algorithms, this result gives, for every fixed connected graph $H$ and every fixed $k\ge2$, a reduction from Max-Weight Independent Set on $kH$-free graphs to the same problem on $H$-free graphs with $n^{O_{H,k}(\log n)}$ overhead. In this setting, it improves the $n^{O_{H,k}(\log^3 n)}$ overhead obtained by specializing the general reduction of Gartland and Lokshtanov [FOCS 2020].

In this paper, we consider bilevel variational inequality problems, where the feasible set is the solution set of another variational inequality. We propose a Bregman inertial regularized extragradient method for solving these problems. By performing the inertial extrapolation in the dual space, the proposed method aligns the inertial step with the structure of the Bregman three-point identity. We also derive explicit non-asymptotic bounds for the generalized gap function. More precisely, using diminishing regularization, we establish convergence rates of $\mathcal{O}(1/k^{1-b})$ for the optimality gap and $\mathcal{O}(1/k^b)$ for the feasibility gap with $ 0 < b< 1$. For constant regularization parameter $\eta>0$, we establish an $\mathcal{O}(1/k)$ rate for the optimality gap and an $\mathcal{O}(1/k)+\mathcal{O}(\eta)$ bound for the feasibility gap. Furthermore, numerical experiments on traffic networks and multi-portfolio Nash equilibrium problems demonstrate the practical effectiveness of the proposed framework.

Let $\mathbb{P}(x\prec y)$ be the probability that $x$ precedes $y$ in a uniformly random linear extension of an $n$-element poset $P$, and define the balancing coefficient to be $\delta(x,y)=\min(\mathbb{P}(x\prec y),\mathbb{P}(y\prec x))$ with $\delta(P)=\max_{x,y}\delta(x,y)$. We prove (Theorem 1) that sufficiently large width forces $\delta(P)$ to be arbitrarily close to $1/2$, answering a long-standing conjecture of Kahn and Saks. In fact, we prove the stronger result (Theorem 2) that large width forces one of two configurations in our poset: either a nearly uniform order on $k$ vertices, or an almost fixed order on $t$ vertices with one further vertex inserted uniformly among the $t+1$ slots. We also show that, for fixed $k$, the first possibility must occur within any antichain $X$ of size $\Omega(n^{2/3})$.

We combine two coloring aspects that work in opposite directions. One can 2-color the vertices of a cubic graph such that each monochromatic component is very small. One can also 2-color the vertices of a cubic graph such that each monochromatic component has degree at least 1. As an intended tool for solving a special case of Wegner's conjecture, Thomassen formulated a conjecture that combined the two previous properties. This led to the concept of a crumby coloring. However it turned out that there are cubic graphs without such coloring. Here we try to see what natural relaxations of the original concept might hold for each cubic graph. We show there exists a constant $c$ such that every cubic graph has a vertex 2-coloring such that every monochromatic component has at least 2 and at most $c$ vertices. We also prove an unbalanced version, which is the natural relaxation of the crumby coloring.

Given a family of genus $g$ curves $f:\Psi\to S$ and a relative degree $d$ line bundle $L$ on $\Psi$ such that $f_*L$ is a vector bundle, the projectivization $\mathbb{P}(f_*L)$ is stratified by relative de Jonquières loci, which parameterize sections of $L$ with prescribed vanishing orders. We give universal formulas for the classes of these loci in terms of the relative $O(1)$ on $\mathbb{P}(f_*L)$ and tautological classes pulled back from the universal Picard stack $\mathrm{Pic}_g^d$. When $f$ is the family of lines in $\mathbb{P}^r$, we use the universal polynomials to compute $\mathrm{GL}_{r+1}$-equivariant classes of loci of hypersurfaces and complete intersections admitting lines with prescribed contact orders.

The balance constant is a poset parameter measuring how well random linear extensions can be split according to their relative order on two elements. We give an $\varepsilon$-improvement for the balance constant over the bound by Brightwell, Felsner and Trotter (1995), for some small $\varepsilon>0$. This is the first general result towards the $\frac{1}{3}$–$\frac{2}{3}$ conjecture in over 30 years.