Papers

In the first half of this paper, we describe the homotopy groups of the equivariant automorphism group of Kirchberg algebras with isometrically shift-absorbing actions of compact groups in terms of equivariant KK-theory. This provides an equivariant version of Dadarlat's result. In the second half, we present a unified treatment of the equivariant Dadarlat-Pennig theory for strongly self-absorbing actions.

For dynamical systems in the plane, there can be many periodic attractors coexisting in a bounded region. They become easier to find in systems with small dissipation, which we call “almost-conservative”. We ask what happens when there are many periodic attractors. That is the vague question we start with. For a test study, we chose the Hénon map with a tiny dissipation. We tuned the other parameter to yield a case with 50 attracting periodic orbits. They have a total of 4259 periodic points. We describe how these orbits can be organized into families. In addition to two low-period orbits, the remaining 48 orbits can be classified into three families, which we describe in detail.

Let BMO and BLO denote the spaces of all locally integrable real-valued functions on $\mathbb{R}^n$ with bounded mean oscillation and bounded lower oscillation, respectively. It is well known that $$L^\infty(\mathbb{R}^n)\subsetneqq {\rm BLO}\subsetneqq {\rm BMO}.$$ In 1978, Garnett and Jones gave distance formulas of $f\in {\rm BMO}$ to $L^\infty(\mathbb{R}^n)$ and recently, Angrisani studied the distance of $f\in {\rm BLO}$ to $L^\infty(\mathbb{R}^n)$. In this paper, we characterize the distance from any given function $f \in {\rm BMO}$ to BLO via the Muckenhoupt weight class $A_p$ as follows \begin{center} dist\,($f$,\ BLO)\,$\sim\inf\left\{\xi\in(0,\infty):\ e^{-\frac f\xi}\in A_p\ \mathrm{for\ some\ }p\in(1,\infty)\right\}$. \end{center} Two equivalent representations of this distance are also established in terms of exponential form and the infimum of the constant in a variant of John–Nirenberg inequality, respectively.

In this paper, we develop an equivariant version of the classical theory of deformations of germs of complex spaces in the presence of complex Lie groups. As a tradition, the main result is the existence of equivariant semi-universal deformations of germs of complex spaces in the reductive case and the non-existence in general in the non-reductive case. The latter is justified by an explicit counterexample. In particular, it generalizes Grauert-Donin's existence theorem to the equivariant settings and Ferrer-Puerta-Slodowy's one to the case of reductive complex Lie groups. Several applications to deformations of pairs, rigidity and actions on Milnor fibers are given.

Classical Euclidean instances for the Vehicle Routing Problem with Time Windows (VRPTW) have floating-point arc costs derived from node coordinates. This raises reproducibility and numerical consistency issues. Hence, a common practice is to scale and round data to integer values, yet the impact of these choices is poorly understood. This paper studies the effect of integer scaling by a factor P and compares three rounding schemes with distinct feasibility and optimality guarantees. We analyze how scaling and rounding influence empirical performance of two well-known VRPTW solvers: Hybrid Genetic Search (HGS) and OR-Tools. We evaluate the quality of solutions mapped back to the original floating-point instances. Our results show that scaling and rounding can significantly affect both solver performance and solution quality. Based on these findings, we provide practical recommendations for scaling and rounding in VRPTW benchmarks to improve efficiency, robustness and reproducibility.

Despite growing concerns over energy security, infrastructure planning and modelling for emerging green fuel supply chains often neglect risks from supply disruptions. Using a stochastic optimisation model of EU hydrogen imports, we show that 'naive' infrastructure planning results in welfare losses of 12 % (24 billion EUR) compared to risk-aware planning that anticipates supply disruptions. Despite requiring higher upfront investments, anticipatory planning achieves welfare levels close to those of an idealised system without disruptions, but entails a markedly different infrastructure configuration. Two complementary resilience strategies emerge: diversification across import corridors and strategic over-investment. This leads to increased intra-European transport capacity, a broader set of import pipelines, and investments in costly shipping terminals for hydrogen carriers. Our results show that incorporating supply risk considerations into infrastructure planning helps prevent the structural vulnerabilities seen in fossil fuel systems when designing future hydrogen supply chains.

We construct here global in time mild solutions to the forced fractional Navier-Stokes-Boussinesq system in some critical Fourier-Herz spaces which will be based on Lorentz norms in the time and the frequency variables. For the initial data and for the external force we will consider Fourier-Besov spaces that will also be based on Lorentz spaces. Moreover, we will show, by performing a separate study of the velocity field and the temperature, that it is possible to consider one of the largest functional space -in the Fourier-Besov-Lorentz framework-for the initial velocity field.

Michael Farber introduced a generalization of the Lusternik–Schnirelmann category for closed 1-forms. In this paper, we introduce and study a corresponding version of topological complexity for closed 1-forms. We establish analogues of the basic properties of ordinary topological complexity, including inequalities relating this invariant to the Lusternik–Schnirelmann category for closed 1-forms. We also prove a closed 1-form analogue of the navigation-function method: upper bounds for our invariant can be obtained from the dynamical properties of gradient-like flows of closed 1-forms.

Social Learning is a decentralized decision-making paradigm in which spatially dispersed agents collect streaming observations regulated by one of a finite number of models (the hypotheses). The agents are interested in assigning probability scores (the beliefs) to the possible hypotheses. To this end, the agents exchange their beliefs according to a certain communication graph. It has been shown that, under reasonable conditions on the identifiability of the decision model and the network connectivity, each agent ultimately places all the belief mass on the true hypothesis governing the data. However, several questions remain unanswered regarding the evaluation of the social learning performance. One recently adopted performance metric is the rejection rate, i.e., the rate at which the beliefs about the erroneous hypotheses vanish. One contribution of this work is to establish that the rejection rate leads to several paradoxes, which make it unsuitable as a valid performance measure. We then focus on studying the error probability measure. For a binary Gaussian problem, we derive an analytical formula characterizing the ratio between the individual agents' probabilities and the optimal Bayesian probability. The formula shows that this ratio is expressed by the product of two terms quantifying the effect of the network connectivity and the role of the prior information. As a result, an irreducible gap emerges between the decentralized and the centralized error probabilities, which is agent-dependent and does not disappear asymptotically.

We establish sharp, uniform-in-time strong error estimates for a nonlinearity-explicit tamed finite element method (FEM) applied to a class of superlinear stochastic partial differential equations (SPDEs) driven by multiplicative noise, including the stochastic Allen–Cahn equation with a moderately thick interface. This tamed-FEM was first introduced in Z. Liu and J. Shen, J. Sci. Comput., 109:Paper No. 61, 26, 2026 to ensure long-time unconditional stability and to preserve the Lyapunov structure of this class of SPDEs. We further prove that the scheme is exponentially ergodic and derive the convergence rate of the numerical invariant measure to the exact one in the Wasserstein-2 distance. Finally, we present numerical experiments that verify the ergodicity, sharpness, and time-independence of the strong convergence rates for this tamed-FEM.

In this paper, we continue to study the barcode entropy of relative symplectic cohomology $SH_M(K)$ of a Liouville domain $K$ embedded in a symplectic manifold $M$. This barcode entropy measures the exponential growth rate of the number of not-too-short bars in the persistence module $SH_M(K)$. We prove that this Floer-theoretic invariant admits a nontrivial lower bound in terms of the topological entropy of the Reeb flow on $\partial K$ when the Reeb flow possesses a hyperbolic invariant set. More precisely, we show that the barcode entropy of $SH_M(K)$ is bounded below by the topological entropy of the Reeb flow restricted to a hyperbolic invariant set.

The structure of locally differentially private (LDP) mechanisms can be understood through the geometry of the corresponding privacy polytope. While the extreme points of the \( (\epsilon,0)\)-LDP polytope are well characterized (Kairouz {et al.}, 2014; Holohan {et al.}, 2017), comparatively little is known for the \((\epsilon,\delta)\)-LDP polytope with \(\delta>0\). Recent work (Elangovan and Jog, 2024) has shown that even in the special case \(\epsilon=0\), the \( (0,\delta) \)-LDP privacy polytope exhibits fundamentally different behaviour. In this work, we provide complete characterizations of the extreme points for the low-input-alphabet regime \(k=2\) and \(k=3\) and with arbitrary output alphabet size \(m \). We also identify new extreme mechanisms for larger input alphabet sizes \(k\), of the star configuration type, as introduced by Elangovan and Jog (2024).

The interest in studying the size of the intersection of multiple $q$-ary Hamming balls has grown due to the recent advances in DNA-based data storage systems. We present an exact formula for the cardinality of the intersection of $s$ Hamming balls of varying radii over a $q$-ary alphabet. It is known that the distances between the center points of the Hamming balls are not enough, in general, to determine the size of the intersection. Based on our formula, we are able to find more refined structural properties of the center points for determining the exact size of the intersection. Moreover, we also analyze the size of the intersection of Hamming balls with equal radii $t$ for sufficiently large $n$ when $t$ and $q$ are constants. When $s=3$, we give the necessary and sufficient conditions (for all $q\ge 2$, $q\neq 6$ and sufficiently large $n$) to obtain the maximum size of the intersection when the centers of the Hamming balls have a given minimum distance and demonstrate how to compute it using our general formula.

This paper studies the asymptotic behavior of sample canonical directions in a finite-rank spiked high-dimensional canonical correlation analysis model under a Gaussian population assumption. Under the asymptotic regime in which the dimensions of the two data blocks grow proportionally with the sample size, sample canonical directions are generally not consistent estimators of their population counterparts, even when the corresponding sample canonical correlations separate from the bulk spectrum. To quantify directional recovery, we investigate the squared alignment between a sample canonical direction and its associated population direction. For a supercritical simple population spike, we establish an explicit nonzero deterministic limit for the squared alignment. In contrast, for population non-spikes, we show that the alignment with the corresponding population signal directions vanishes asymptotically. Thus, the same phase transition that governs eigenvalue separation also marks a transition in directional recovery. For a supercritical simple spike, we further prove a central limit theorem for its fluctuations around the deterministic limit, with an explicit asymptotic variance expressed through deterministic limits of resolvent trace functionals. To make the theoretical quantities computable from data, we further construct plug-in estimators for both the deterministic alignment limit and the asymptotic variance by inverting the deterministic outlier eigenvalue map, and prove their consistency. Numerical simulations and a real-data illustration support the theoretical results and demonstrate how the proposed estimators assess the recovery quality of sample canonical directions.

For a real variety with smooth point, we construct a complex of sheaves on its real points which behaves like intersection cohomology. This real geometric extension is (shifted) Verdier self-dual, and occurs as a direct summand within any resolution of singularities. The stalks of this object provide lower bounds for the Betti numbers of the real fibres in any resolution of singularities. On the real flag variety, the category of real geometric extensions along the Schubert stratification is equivalent to the corresponding category of even parity sheaves on the complex flag variety.

We study Fourier coefficients of the Siegel-Eisenstein series $E_{m+1}^{(2m)}(Z)$ for $m\equiv1\pmod4$ and $m\geq5$. Using the Fourier expansion formula due to Mizumoto, we determine the constant term and the exceptional non-zero Fourier coefficients. The result gives a higher-degree analogue of the degree-two formulas of Kohnen and Nagaoka, which were later rederived by Haruki.

We study finite expected-signature information for mixed-fBm paths with Hurst indices above $1/4$. Up to level three, the only parameter-dependent expected features are the variance transform $q_\theta$ and the time-ordered transform $R_\theta$. We prove the scale tradeoff $2K$ level-two scales versus $K$ selected level-two/three scales, together with separation and local inverse bounds.

In this paper, we introduce the braided cochain complex and the braided cohomology of braided coalgebras in linear monoidal categories, and compare those of braided coalgebras living in different linear monoidal categories using relative morphisms. The symmetric cohomology was introduced for groups by Staic, and was generalized to cocommutative Hopf algebras by Shiba, Sanada, and the second author. This notion involves degreewise actions of the symmetric groups on a cochain complex, which come from the symmetry on the monoidal category of group representations. We generalize this framework by dealing with arbitrary linear monoidal categories, and by replacing symmetries with braidings defined merely on objects. We first describe relative morphisms in this general setting, and apply this result to prove that the braided cochain complex of quasi-triangular bialgebras is a braided Morita invariant under some hypotheses.

In \cite{PO1} and \cite{PO2}, Pfante defined a noncommutative Chern-Simons action for 3-summable spectral triples and computed the Chern-Simons action for $\mathrm{SU}_q(2)$ and the noncommutative 3-torus. In this paper,we compute the Chern-Simons action for well-know perturbed Dirac triples.

We establish a unified operator-theoretic framework for understanding the results on the Weil quadratic form obtained by Yoshida (1992), Bombieri (2001, 2003), Connes–Consani (2023), and Connes–Consani–Moscovici (2025+) from the perspective of the screw function introduced in Suzuki (2023). An advantage of the approach via the screw function is that it provides a method to study the Weil quadratic form, which is originally defined in terms of distributions, by means of continuous functions and brings the theory of de Branges spaces into the analysis of the Weil quadratic form via the Fourier transform. Based on this framework, we formulate a conjecture stating that a self-adjoint operator whose eigenvalues are the imaginary parts of the nontrivial zeros of the Riemann zeta function can be obtained as the limit, as $a \to \infty$, of self-adjoint operators arising from nonlocal realizations of the first-order differential operator on the finite interval $[-a,a]$. All these results are obtained without assuming the Riemann Hypothesis. This conjecture may be compared with the limit formula for the Riemann zeta function expressed in terms of zeta-regularized products proposed by Connes, Consani, and Moscovici, and it sheds new light on the spectral-theoretic interpretation of the nontrivial zeros of the Riemann zeta function.

We establish the generalized canonical bundle formula in the numerical setting.

We prove local centralizer rigidity results for elements of Weyl chamber flows and twisted Weyl chamber flows on compact homogeneous spaces. For generic elements in the twisted setting, sufficiently large dimension of the centralizer forces smooth conjugacy to an algebraic model. For many non-generic elements, we prove analogous rigidity under a virtual centralizer-isomorphism hypothesis.

This paper proposes a novel Adaptive Decentralized Quasi-Newton (AdaDQN) method for solving smooth nonconvex optimization problems over undirected networks. While standard decentralized algorithms with multiple fixed local updates typically admit a convergence stepsize inversely proportional to the number of local updates, we show that this scaling is worst-case tight for the typical unscaled fixed-local-update scheme. We revisit a class of gradient-tracking methods with this scheme from a surrogate-based perspective and establish a Robust Inexact Algorithm (RIA) framework. Inspired by this framework, AdaDQN integrates a safeguarded consensus-aware termination criterion, a standard event-triggered communication protocol, and a scalable memoryless BFGS update. We establish an $\mathcal{O}(1/T)$ best-iterate rate for first-order stationarity. For a squared stationarity tolerance $\delta$, the guaranteed gradient complexity of the fixed scheme is $\mathcal{O}(nK_g/\delta)$, whereas AdaDQN attains $\mathcal{O}(n/\delta+n\alpha\tilde\varepsilon^{-2})$, independent of the maximum local-update budget ($K_g$). Numerical experiments demonstrate that AdaDQN achieves a superior computation-communication tradeoff, outperforming state-of-the-art decentralized methods across various performance metrics.

Let $H$ be a finite-dimensional, non-semisimple Hopf algebra over an algebraically closed field $\mathbf{k}$. This paper investigates the asymptotic behavior of the core of left $H$-modules through the lens of the gamma invariant $\gamma_{\mathfrak{X}}$ relative to a representation ideal $I_{\mathfrak{X}}$. We establish an equivalent characterization for the quotient of the Green ring $R_{\mathfrak{X}}$ to be a transitive fusion ring, demonstrating that transitivity is synonymous with the non-degeneracy of a naturally induced bilinear form and the collapse of the ideals $P_{+}$, $P_{-}$ and $I_{\operatorname{max}}$ into a single ideal. Furthermore, we prove that the Green ring exhibits the structure of a representation ring in the sense of Benson, provided that the square of the antipode is an inner automorphism and the equality $I_{\operatorname{max}}=I_{\operatorname{proj}}$ holds. As an explicit application of these frameworks, we analyze the Drinfeld double $D(H_4)$ of the Sweedler algebra, identifying an infinite family of distinct representation ideals and proving that the maximal gamma invariant $\gamma_{\operatorname{max}}$ induces a genuine ring homomorphism. Finally, for Hopf algebras of finite representation type under the assumption $P_{+} = P_{-} = I_{\operatorname{max}}$, we show that $\gamma_{\operatorname{max}}$ coincides precisely with the Frobenius–Perron dimension, and we explicitly compute the gamma invariants for the standard basis elements of the Green ring of the Taft algebra $H_n(q)$.

We study the behaviour of the smallest possible constant $d(a,b, p,\epsilon)$ in Hardy inequality $$ \int_a^b\left(\frac{1}{x}\int_a^xf(t)dt\right)^px^{\epsilon}\,dx\leq d(a,b,p,\epsilon)\,\int_a^b [f(x)]^px^{\epsilon}\, dx, \quad 2\le p<\infty. $$ The exact rate of convergence of $d(a,b,p,\epsilon)$ is established and the “almost extremal” function is found.

A bicoloration of a graph $G=(V,E)$ is a map $\beta:V\to\{-1,1\}$. A local inversion at a vertex $v$ complements the subgraph induced by the neighbors of $v$ and simultaneously reverses the colors of all neighbors of $v$. Sabidussi (Discrete Mathematics, 1987) showed that every bicolored graph on $n$ vertices without isolated vertices admits a color reversal using at most $6n+3$ local inversions, and that any two bicolorings of such a graph can be transformed into each other using at most $9n$ local inversions. Recently, Porte, Sandeep, and Santra (CALDAM 2026) improved these bounds to $4n-3$ and $\lfloor(11n-3)/2\rfloor$, respectively. We prove the tight bound $3n$ by showing that, for every graph on $n$ vertices without isolated vertices, any bicoloring can be transformed into any other bicoloring using at most $3n$ local inversions. We also show that this bound is best possible: for complete graphs and stars on $n$ vertices, at least $3n$ local inversions are required to reverse the colors of all vertices. Moreover, the proof of the upper bound is constructive: given two bicolorings, it produces, in polynomial time, a sequence of at most $3n$ local inversions transforming one into the other.

This paper addresses the problem of quantitative rapid stabilization via the linear quadratic (LQ) theory. Specifically, for non-self-adjoint parabolic equations, by virtue of the LQ theory, we derive the quantitative rapid stabilization by selecting a special cost functional and combining it with a quantitative observability inequality. Furthermore, this approach reveals the equivalence between the quantitative rapid stabilization and the quantitative observability inequality for linear systems. In addition, we apply this framework to the Navier–Stokes equations and establish the quantitative rapid stabilization around nontrivial steady states. Finally, we extend this methodology to finite-dimensional feedback laws in self-adjoint cases.

We study the behaviour of the smallest possible constant $d_n$ in weighted Hardy inequality $$ \sum_{k=1}^{n}\Big(\frac{1}{k}\sum_{j=1}^{k}a_j\Big)^2 k^\epsilon\le d(n,\epsilon)\,\sum_{k=1}^{n}{a_k^2}\,k^\epsilon $$ The exact rate of convergence of $d_n$ is established and the “almost extremal” sequence is found.

Numerical computation of stochastic control problems under partial observation is challenging because the dynamic programming formulation is naturally posed on the conditional distribution of the hidden state. We propose particle-based methods that replace this infinite-dimensional filtering state by a finite-dimensional weighted particle system, building on recent limit theory for mean-field control with common-noise-adapted controls. We prove, under suitable assumptions, convergence of the fully discretized particle approximation to the original continuous-time partially observed control problem. The particle reformulation is high-dimensional but permutation-invariant, a structure that can be exploited by symmetric neural network architectures. We develop two deep learning algorithms: a direct optimization method for feedback controls and a Deep BSDE method for particle problems admitting a backward stochastic differential equation representation. We also extend the computational framework to partially observed mean-field control problems, which have been studied theoretically but remain less developed numerically. Numerical experiments on a linear–quadratic benchmark, a nonlinear partially observed mean-field control problem, and two financial applications, portfolio liquidation and asset allocation, demonstrate the accuracy and practical utility of the approach.

The correspondence between Morse flow trees and $J$-holomorphic discs was established by Fukaya–Oh and Ekholm. We revisit this correspondence and present an alternative approach, designed to generalize naturally to the equivariant setting and to certain Morse graph configurations. The central ingredient is a gluing construction that produces $J$-holomorphic discs from Morse flow trees. A well-known difficulty is that this gluing is of Morse–Bott type, equivalently, in an appropriate Fredholm framework, pieces to be glued together are obstructed. We resolve this via the obstruction bundle gluing technique of Hutchings–Taubes. Given a rigid, transversely cut-out Y-shaped Morse flow tree, we show that for every sufficiently small $\epsilon > 0$ there exists at least one corresponding $J$-holomorphic discs in the cotangent bundle, with boundaries inside corresponding Lagrangian submanifolds of height $\epsilon$. This is the first paper in a series; subsequent work will extend the result to all ribbon trees and to moduli spaces of all dimensions and establish the injectivity and surjectivity of the correspondence.