Papers

Let $s^+(G)$ and $s^-(G)$ denote the sums of the squares of the positive and negative adjacency eigenvalues of a graph $G$, respectively. We prove the conjecture of Elphick, Farber, Goldberg, and Wocjan that every connected graph $G$ on $n$ vertices satisfies $$ \min\{s^+(G),s^-(G)\}\ge n-1. $$ The proof introduces a new framework for square-energy estimates, in which the Hadamard squares of positive semidefinite matrices that encode these spectral quantities are relaxed to the full doubly nonnegative cone.

Differential privacy (DP) imposes fundamental trade-offs between privacy and statistical fidelity in synthetic data generation. While access to public data has been shown to improve these trade-offs empirically, existing approaches use public data only indirectly, through pre-processing (e.g., using pre-trained generative models) or post-processing steps (e.g., matching target statistics estimated from public datasets), while relying on domain-agnostic DP mechanisms. In this work, we lay the theoretical framework to study the principled incorporation of public data into DP mechanisms themselves. We consider normalized histograms as distribution estimators and characterize the asymptotically optimal domain-aware privacy mechanism within a specific class of DP mechanisms. We introduce PubMix, a public-data-aware DP mechanism that can be used in histogram-based data synthesis pipelines. Our experiments demonstrate that PubMix significantly improves synthetic data generation quality compared to domain-agnostic privacy mechanisms.

We propose two new approaches for constructing families of rational functions with shared real poles that nearly uniformly approximate the functions $\exp(-tz)$ for $z\geq 0$ and $t$ in a positive time interval. The first result concerns the case where all real poles coalesce into a single point. With an appropriate choice of a weight function we are able to derive a closed formula for the asymptotically optimal location of such a pole. We then discuss the more general case where all real poles are distinct. Using Zolotarev's construction of certain optimal rational functions, we present a simple algorithm to derive nearly optimal poles efficiently. We analyze the stability of the numerical evaluation of the resulting rational matrix functions in floating-point arithmetic. By controlling the growth of potential ill-conditioning arising from partial fractions, reliable and highly parallelizable exponential propagators are obtained.

We study $2$-step nilpotent Lie groups with naturally reductive left-invariant Lorentzian metrics with respect to the presentation group $N\rtimes H^{\operatorname{aut}}$. Replacing the standard non-degenerate center assumption with the weaker condition that the commutator ideal be non-degenerate, we develop a framework that extends the representation-theoretic construction to the Lorentzian setting and covers both the non-degenerate and degenerate center cases. In the degenerate case, we show that the associated Lie algebra is a central extension of a semidirect product whose Riemannian factor is naturally reductive. Furthermore, we obtain invariant decompositions of the defining representation in the non-degenerate case. In the degenerate case, we introduce a Riemannian quotient representation and show that the kernel of this representation is a central abelian ideal whose elements act by null rotations. Finally, we use these results to characterize the isotropy algebra of the group of isometric automorphisms.

We prove the analogue of Schramm's locality conjecture for long-range percolation on transitive graphs of polynomial growth with $\alpha \in (0,2)$. In this setting, we also prove the joint continuity of the percolation probability $\theta$ with respect to three parameters: the underlying graph with respect to the local topology, the connectivity kernel, and the percolation parameter $\beta$ for all values of $\beta \in \mathbf{R}_+$, including the critical parameter $\beta_c$. We also prove a number of results related to the supercritical sharpness of long-range percolation: the long-range order decay of the distribution of finite clusters, the truncation problem, the anchored isoperimetric dimension and the transience of the infinite percolation cluster, and the smoothness of the percolation characters. We obtain these results from proving the local existence-and-uniqueness of the linear-sized (giant) cluster. As an immediate corollary of the local existence-and-uniqueness of the giant we obtain the law of large numbers, which answers a special case of a question of Nekrashevych and Pete \cite[Question 1.3]{nekrashevych_scale-invariant_2011}. The main technical contribution is the construction of a renormalisation scheme combining iteratively merged Voronoi tiles with scale-invariant nets, related to the scale-invariant groups of Benjamini.

We give an elementary proof of power Fourier decay for Patterson-Sullivan measures associated to convex co-compact Schottky groups of dimension $\delta>\frac12$, obtaining the explicit decay exponent $\frac{\delta(2\delta - 1)}{(2\delta + 1)(3 - \delta)}$. The proof replaces the technical machinery used in previous works, such as sum-product estimates, renewal theory, Dolgopyat methods, and $L^2$ flattening, by elementary oscillatory integral estimates, hyperbolic geometry, and a duality argument based on the transpose Schottky group.

We investigate the asymptotic behavior and turnpike properties of graphon mean field control (GMFC) problems in the linear-quadratic setting. We consider both a finite-horizon GMFC problem and its associated ergodic counterpart, in which the controlled dynamics are governed by a graphon mean field stochastic differential equation with heterogeneous interactions. The optimal controls and state trajectories for both problems are characterized by systems of Riccati equations together with systems of generalized differential and algebraic equations on suitable Hilbert spaces. Under a stabilizability condition and appropriate positivity assumptions on the graphon-induced operators, we establish the unique solvability of the ergodic control problem and derive exponential convergence estimates for the finite-horizon system to its stationary limit. As a consequence, we establish an exponential turnpike property for the optimal pair and prove the convergence of the time-averaged value function for the finite-horizon GMFC problem.

We introduce a family of polynomials satisfying the natural duality relations for Hermite interpolation, analogous to the classical Lagrange interpolation polynomials. They yield an explicit closed formula for the Hermite interpolant with arbitrary multiplicities, without recourse to divided differences, recursive corrections, or auxiliary Bézout identities. We also give a detailed account of Hermite's original approach to the problem, based on an integral formula, which he used both to derive the interpolating polynomial and to estimate the interpolation error for holomorphic data.(To appear in L'Enseignement Mathématique.)

Prediction markets are attracting growing attention as trading volumes rise and their practical relevance increases. To ensure efficient price discovery, liquidity provision becomes ever more important. Due to the binary settlement structure in prediction markets, optimal market making leads to an optimization problem that is fundamentally different from the ones studied in classical settings. In this paper, we develop a stochastic control framework for prediction markets in which the market price is modeled as a conditional probability of the outcome that is generated by a transformed latent belief diffusion. A market maker selects bid and ask quotes to maximize expected terminal wealth while controlling both mark-to-market inventory risk and the settlement risk of remaining positions at resolution. We derive the associated Hamilton–Jacobi–Bellman equation and characterize the unique optimal bid and ask quotes. By transforming the equation to the latent belief space and using a fixed-point argument, we prove existence and uniqueness of a classical solution and verify the resulting optimal quoting strategy. In addition, we provide a numerical analysis, which reveals how optimal liquidity provision in prediction markets depends on inventory, market beliefs, time to resolution, and risk aversion. Further, we demonstrate that the optimal quoting strategy substantially improves downside protection while preserving most of its expected profit relative to a myopic benchmark that maximizes the instantaneous expected mark-to-market profit.

Highly nonlinear chaotic dynamical systems remain difficult to model due to fundamental trade-offs between complexity, expressivity, and data efficiency. Modern machine learning methods achieve strong predictive performance but often rely on a-priori system knowledge or curated data with limited interpretability. Koopman operator theory offers a promising direction via linear representation in an infinite-dimensional observable space. However, many data-driven Koopman methods seek globally valid operators for which useful finite-dimensional spectral embeddings remain difficult to identify under these constraints. To overcome associated limitations, we introduce Fuzzy Spectral Region Decomposition (fSRD), a fully automated learning framework for estimating finite Koopman representation via multiple operators. The proposed method realizes a data-adaptive framework for assembling locally invariant embeddings, termed Invariant Decomposition. fSRD achieves highly accurate linear reconstructions of nonlinear systems while learning finite-dimensional representations of their induced evolution operators, bridging interpretable operator-theoretic models with expressive data-driven sequence learning. These embeddings are adaptively constructed via a global fuzzy tree model, drawing inspiration from fuzzy neural architectures to learn the induced dynamics while prioritizing parsimonious solutions. Empirical results across canonical chaotic systems (e.g., Lorenz and Duffing) and high-dimensional real-world data demonstrate strong predictive accuracy, interpretability, and robust expressivity across data-rich and data-limited regimes, highlighting the method's generality.

A feedback vertex set of a graph is a set of vertices whose deletion leaves a forest. In 2016, Dross, Montassier, and Pinlou conjectured that every planar graph $G$ of girth at least $g$ admits a feedback vertex set of size at most $e(G)/g$. In this note, we confirm this conjecture by connecting this problem with signed graphs. The frustration index of a signed graph $(G,\Sigma)$ is defined as the minimum number of negative edges among all signatures on $G$ that are switching-equivalent to $\Sigma$. Equivalently, it is the minimum number of edges whose deletion results in a balanced subgraph of $(G,\Sigma)$. We show that the minimum size of a feedback vertex set of a planar graph is bounded above by the maximum frustration index over all signatures of the graph, and thereby provide a tight upper bound on the size of the minimum feedback vertex set, which resolves the conjecture of Dross, Montassier, and Pinlou (2016).

In this paper, on a bounded domain $\Omega\subset {\bf R}^n$, we consider a nonlocal problem of the type $$\cases{-\Delta u=Q(u,\lambda)f(u) & in $\Omega$ \cr & \cr u=0 & on $\partial\Omega$\cr}$$ where $Q:H^1_0(\Omega)\times {\bf R}\to {\bf R}$, proving, under suitable assumptions, the existence of at least three weak solutions for each $\lambda$ running in a suitable interval.

We study the behavior of projective dual varieties and discriminants under flat degenerations. For Gröbner degenerations, we show that the conormal variety admits a Gröbner degeneration with opposite weights on the dual coordinates. Using Whitney stratifications and Sabbah's formula, we describe the irreducible components and multiplicities of the special fiber of this degeneration, and hence of the limiting dual hypersurface. We then extend these results to higher associated hypersurfaces in Grassmannians. As applications, we recover classical formulas for hypersurfaces with isolated singularities, analyze degenerations of generic complete intersections and reciprocal linear spaces, and relate the theory to mixed discriminants of parametrized polynomial systems.

High-speed LDPC decoding is a major bottleneck in CV-QKD and motivates hardware acceleration with fixed-point arithmetic. This work compares SPA, MSA, and NMS under a unified low-SNR fixed-point framework using common graph, matrix, and quantization settings. Multiple formats are evaluated through FER, and average iterations. The results show that performance depends strongly on the interaction between decoder rule and numerical precision. SPA achieved the best overall performance. For reduced-complexity decoders, Q16.8 was the lowest consistent precision, with NMS outperforming MSA. Practically, SPA with Q8.4 offered the best balance between reliability and hardware efficiency for large-scale implementations.

Radio maps, which estimate spatial radio-frequency characteristics from spectrum measurements, are essential for applications such as spectrum management and network planning. With the continuous arrival of spectrum measurements, conventional batch processing methods for radio map reconstruction become computationally prohibitive, as they require reprocessing all accumulated measurements for each radio map update. To address this, we propose a memory-based online sparse variational Gaussian process (M-OSVGP) method that efficiently updates radio maps from streaming spectrum measurements. Our method employs sparse variational inference and updates the posterior online by minimizing a hybrid objective that integrates newly received measurements and a memory subset of previous ones to mitigate catastrophic forgetting. To further improve posterior approximation as measurements accumulate over spatially diverse regions, we extend M-OSVGP with a grid-assisted online inducing point selection (GOIPS) algorithm. GOIPS dynamically adapts the number and locations of inducing points based on measurement density and spatial correlation, providing a more informative inducing set while maintaining computational efficiency. Extensive simulations demonstrate the effectiveness of our proposed methods in reconstruction accuracy, computational efficiency, and uncertainty quantification, compared to existing batch and online baselines across various scenarios.

Due to the importance of invariant, in the present paper, we provide a unified approach to study various kinds of operators on Lie triple systems such as crossed homomorphisms, relative Rota-Baxter operators, Reynolds operators etc. For this purpose, we examine two kinds of deformation maps on quasi-twilled Lie triple systems. A quasi-twilled Lie triple system is a Lie triple system which can be decomposed into two subspaces, one of which is a subalgebra. Moreover, we establish cohomologies of two kinds of deformation maps, which recover those of several kinds of operators on Lie triple systems.

We study the emergence of (cross-scale) fluxes associated with energy and enstrophy in a stochastic version of Zeitlin's $SU(N)$ approximation of 2-d Euler dynamics. Motivated by the Euler-Arnold formulation, we interpret the nonlinear transport as motion along coadjoint orbits, reflecting an underlying symmetry that preserves all Casimir invariants. We introduce a fast-slow stochastic framework in which rapid mixing is modeled by a structured fast stochastic forcing acting along selected directions. We show that when the fast dynamics preserves the full coadjoint orbit symmetry given by the natural symplectic structure, the averaged system exhibits no nontrivial flux. In contrast, when this symmetry is broken by tangential non-Hamiltonian vector fields on the coadjoint orbit, we identify conditions that yield nonzero fluxes carried by the Euler-Arnold nonlinearity. Thus, coadjoint symmetry suppresses averaged nonlinear flux, whereas its breaking can select a preferred direction of cross-scale transfer.

We present a column adsorption model that couples a Pseudo-First-Order (PFO) kinetic formulation with the Sips isotherm framework. Using a traveling wave approximation, we derive analytical solutions for specific operating conditions. Qualitatively, these solutions deviate significantly from their pure Sips counterparts: instead of a smooth, continuous increase in concentration at the column outlet, the PFO-Sips model predicts an abrupt, sudden breakthrough. We validate these analytical solutions against diverse experimental datasets from the literature. The results reveal that the PFO-based model consistently underperforms compared to the original Sips formulation. Furthermore, this validation exposes fundamental inconsistencies within the PFO framework. We demonstrate that despite its widespread use in the literature for almost a century, the PFO model is inherently flawed and structurally unfit for describing column adsorption dynamics.

We present CARDAL (Curvature-Aware Rank-Adaptive Distributed Augmented Lagrangian), a distributed multi-GPU solver for large-scale semidefinite programs (SDPs) based on a rank-adaptive Burer-Monteiro factorization and an augmented Lagrangian method. At fixed ranks, a matrix-free L-BFGS method with negative-curvature corrections targets an approximate Euclidean second-order stationary point of the factored augmented Lagrangian. A reverse multiplier shift turns a negative dual-slack direction into exact negative curvature after rank expansion, and a small joint rank-lift problem selects a batched low-rank correction. A verified slack lower bound provides an a posteriori approximate KKT certificate. Our analysis establishes generic global-optimality guarantees for heterogeneous products of PSD cones at per-block ranks near the Barvinok-Pataki scale, together with a finite-accuracy counterpart under blockwise cost smoothing. For scalable execution, CARDAL distributes constraint rows, factor columns, and PSD blocks over a Constraint x Rank x Cone device mesh. The primal residual, gradient, Hessian-vector products, and slack matrix-vector products are evaluated using device-local operations and axis-wise collectives. On the Mittelmann benchmark, CARDAL exhibits stronger robustness than existing low-rank GPU approaches under a uniform accuracy standard. Experiments on large-scale SDP relaxations from robotics, electronic structure, and Max-Cut demonstrate the complementary scaling regimes of the three distribution axes, with observed wall-clock speedups of up to 4x on four H100 GPUs.

We consider products and semiproducts of propositional modal logics L with S5 and present new examples of product and semiproduct logics axiomatized in the minimal way and enjoying the product (or semiproduct) FMP. An essential part of the proof is local tabularity of these (semi)products for L of finite depth; it is obtained by using bisimulation games. These results readily imply decidability for 1-variable fragments of predicate modal logics QL and QL+Barcan formula. We also present new counterexamples, i.e. (semi)products not axiomatizable in the simplest way.

This paper is devoted to the study of dimonoids, i.e., algebraic structures equipped with two associative binary operations satisfying a given system of axioms. We introduce several new classes of dimonoids and determine their automorphism groups and halos. In particular, we construct examples of iso-dual nonabelian dimonoids, as well as nonabelian dimonoids with nonempty halo. Using these constructions, we obtain a complete classification, up to isomorphism, of all nonabelian noncommutative nontrivial dimonoids of order 3, thereby resolving the problem concerning this classification. Consequently, all three-element dimonoids are classified up to isomorphism, yielding exactly 52 pairwise nonisomorphic dimonoids of order 3. Finally, we present the results of computer computations determining the numbers of all pairwise nonisomorphic dimonoids of orders up to 5, as well as all pairwise nonisomorphic commutative, abelian, and rectangular dimonoids of orders up to 6, obtained using GAP, Python, and C++.

The impact of an externally imposed magnetic field on numerical simulations of two-dimensional Rayleigh-Bénard convection (RBC) is investigated. Initially, the RBC model is examined in the absence of a magnetic field to establish a baseline. Then, a background magnetic field is introduced, and its influence on the transition to chaos is explored. For the purely hydrodynamic case and a range of the reduced Rayleigh number, the system exhibits traveling rolls which, after an attractor-merging crisis, give way to chaotic traveling rolls. Upon imposing a background magnetic field, there is a notable increase in the occurrence of traveling roll dynamics. Furthermore, the presence of the magnetic field favors the splitting/breaking of convective rolls, indicating a possible mechanism for transition to two-dimensional turbulence, with the structure of the convection cell being disrupted. A detailed analysis of the velocity field reveals that the collision between a saddle point and the center of a convective roll restores the system's original topology, with two symmetric kinetic vortices. During this collision, a magnetic vortex splits in two as a result of a magnetic reconnection. This behavior occurs intermittently in time.

We carry on a study of the point-wise regularity theory for the nonlocal Trudinger equation, with measurable and bounded singular kernel. We establish refined quantitative upper bounds, weak and strong parabolic Harnack inequalities, both under optimal tail conditions. Our analysis relies on the adaptation of a refined De Giorgi-Moser machinery based on the parabolic approach á la Di Benedetto, specifically designed to overcome the technical intricacies of the nonlocal, doubly nonlinear framework.

High-stakes decisions in oncology clinical trials must often be made while survival data remains immature: progression-free survival (PFS) and overall survival (OS) are heavily censored, few events have accumulated, and the primary endpoint may be months or years from reading out. What is available at interim data cut-offs is information-rich longitudinal tumour measurements and baseline covariates. We present PIONEER, a Bayesian joint modelling framework that couples a mechanistic two-component state-space submodel of longitudinal tumour size dynamics to a multistate proportional-hazard submodel for competing clinical events, fitted simultaneously under a single posterior. The mechanistic submodel infers latent per-patient tumour trajectories - decomposed into treatment-responsive and refractory compartments with Gompertz-attenuated growth - from sparse, noisy sum-of-longest-diameter (SLD) observations. These latent trajectories feed the multistate hazard as time-varying covariates, while the event data simultaneously refines the tumour dynamics through the joint likelihood. All clinical endpoints (PFS, OS, objective response rate) are derived from the joint posterior in a single forward simulation pass, propagating full parameter uncertainty without any two-stage plug-in. Applied to a case study in extensive-stage small-cell lung cancer (two trials, N = 497), leave-future-out cross-validation demonstrates that at month 4 of enrolment (9 patients) the model produces calibrated PFS forecasts covering the mature month-19 Kaplan-Meier curve, and at month 11 (39 patients) the OS forecast converges - representing at least 8 months of advance forecasting with properly quantified uncertainty. We hope this work paves the way for broader adoption of Bayesian mechanistic state-space frameworks in clinical development, enabling earlier and more informed decision-making from immature trial data.

The maximization of statistical complexity has long been associated with the emergence of probability distributions lying between perfect order and complete disorder. While previous studies have shown that complexity-maximizing distributions exhibit a two-level structure in finite discrete systems, an analogous unified treatment for both discrete and continuous probability spaces has remained unavailable. In this work, we develop a general variational framework for a generalized statistical complexity constructed from Shannon and Renyi entropies. We derive a common stationary equation governing both discrete probability masses and continuous probability densities and prove that every stationary solution necessarily possesses exactly two probability levels, establishing a universal core-halo structure. We further demonstrate that the optimization problem reduces to a single multiplicity parameter and prove that the global complexity maximum is attained by the smallest admissible core, corresponding to a single dominant state in the discrete case and an infinitesimal core in the continuous limit. These results provide a complete analytical characterization of the complexity-maximizing distributions and reveal a common mathematical structure underlying complexity optimization in both discrete and continuous settings. The framework establishes a unified foundation for generalized statistical complexity with potential applications in statistical mechanics, information theory, and the analysis of complex systems.

We formulate and solve the reflected sub-Riemannian Schrödinger bridge (SB) problem: minimum-energy transport of probability distributions on a bounded domain for underactuated dynamics with degenerate diffusion and hard state constraints. The main difficulties are geometric and degeneracy: Euclidean normal reflection is generally incompatible with the horizontal subbundle, since it may push the process in directions not generated by the admissible control and noise fields. Also, the reference measure may not have full support. We address these by introducing an intrinsic oblique reflection mechanism that is compatible to the sub-Riemannian structure. Under Hörmander and non-characteristic boundary assumptions, we prove that the reflected degenerate reference process admits a smooth, strictly positive transition density. The resulting optimal control is characterized by a forward–backward PDE system with asymmetric boundary conditions: an oblique Neumann condition for the backward factor and a normal no-flux condition for the forward factor. Since explicit transition densities are generally not available in this setting, we develop a PDE-based Sinkhorn iteration that enforces these boundary conditions directly. Our examples reveal that SB depends on the topology of the domain and the geometry of the diffusion.

In an incomplete round robin tournament, teams play against different sets of opponents. Given varying strengths of the teams, this raises a fairness issue. We establish the complexity of determining whether a fair draw exists under various scenarios involving the presence of pots (i.e., strength based groups of teams), teams coming from the same association, connectivity requirements, and others. We also show experimentally how fair incomplete round robin tournaments can be generated.

This work establishes an improved lower bound for the spectral radius of a graph given its average degree. The new bound follows from an exact solution of the fractional relaxation of the problem. Our findings lead to an affirmative answer to a conjecture by Hong (1993) for graphs with specific average degrees – as the extremal graphs that meet our bound are proven to have a minimal and maximal degree that differ by at most one. Furthermore, we provide an exact characterization of the conditions that permit such discrete realizations. We prove that for a fixed number of vertices $n$, the number of valid edge configurations grows at least linearly with $n$, achieving an average asymptotic order of $\Theta(n\log n)$.

This paper addresses the problem of energy-efficient thermal management in Battery Electric Vehicles (BEVs), with the goal of extending driving range while ensuring passenger comfort and battery thermal safety. A hierarchical Nonlinear Model Predictive Control (NMPC) framework is proposed, in which the supervisory layer explicitly balances energy consumption and cabin temperature reference tracking over a long horizon and translates this trade-off into optimal actuator command sequences for the integrated heating, ventilation, and air conditioning (HVAC) system and the battery thermal management (BTM) system. These command sequences are then enforced by a lower-layer NMPC, which introduces local corrections to compensate disturbances and modeling uncertainty while tracking an externally specified cabin temperature reference trajectory. To enable predictive control of the highly nonlinear and strongly coupled thermal dynamics, data-driven neural network models are employed as prediction models within both control layers. The proposed approach is developed and validated using a high-fidelity BEV thermal simulator developed at Politecnico di Torino in collaboration with Stellantis N.V., and benchmarked against an industrial rule-based (RB) control strategy over the WLTC driving cycle. Simulation results demonstrate a significant reduction in energy consumption compared to the baseline strategy, while satisfying comfort and battery temperature constraints.

This paper studies the Białynicki–Birula decomposition associated with the natural $\mathbb C^*$-action on Nakajima quiver varieties given by scaling the maps along the reversed arrows of the doubled quiver. Fixed points of this action are described in terms of representations with relations of auxiliary quivers, called quiver chains. Several aspects of the geometry of the resulting Białynicki–Birula decomposition are also investigated, including the dimensions of their attracting fibers and the corresponding motivic decomposition of the Nakajima quiver variety in terms of quiver-chain moduli spaces. Finally, the general framework is specialized to star-shaped quivers, where, for particular dimension vectors, the fixed-locus components are classified and the motivic class of the full Nakajima quiver variety is computed in a suitable localization of the Grothendieck ring of varieties.