Papers

The evaluation of the Riemann zeta function at negative integers is a classical result typically obtained through analytic continuation or contour integration. In this paper, we present a novel and concise derivation of these special values by employing the theory of Cesàro limit of distributions, a generalized limit concept developed by Estrada, Kanwal, and Fulling. We use this tool to give a quick proof of the result that \[ \zeta(-n)=-\frac{B_{n+1}}{n+1}, \] for $n\in\mathbb{N}^+.$ We also give a short discussion on $\zeta^{\prime }(\alpha)$ and compute the value of $\zeta^{\prime}(0)$.

We introduce a new method for computing plus-pure thresholds, a mixed-characteristic analogue of both log canonical thresholds and $F$-pure thresholds. We obtain some necessary conditions and some sufficient conditions for BCM-regularity of Fermat-type hypersurfaces. We also establish lower bounds for plus-pure thresholds of diagonal hypersurfaces in mixed characteristic. Furthermore, we give bounds for plus-pure thresholds of hypersurfaces in mixed characteristic $(0,2)$ using splitting-order sequences, introduced by Yoshikawa. As an application, we classify BCM-regular diagonal hypersurfaces in mixed characteristic $(0,2)$.

We introduce a functor from cochain complexes to bicomplexes, called inflation functor, which sends quasi-isomorphisms to the class of pluripotential weak equivalences. We show this functor is part of a Quillen adjunction. Its right adjoint is a well-known construction in complex geometry, which gives a sheaf-theoretic presentation of Bott-Chern and Aeppli cohomologies. The inflation functor plays a key role in pluripotential Koszul duality theory for operads and allows us to construct the infinity-category of bicomplexes in the pluripotential sense.

We study the deterministic reiterated homogenization of the non-stationary Navier-Stokes type equations in fixed domains with periodically rapidly varying coefficients. One convergence theorem and a corrector result are proved, and we derive the macroscopic homogenized model.

Let K be a complete, non-archimedean valued field with a residue field of characteristic different from 2. A Whittaker group G is a discontinuous subgroup of PGL(2,K), freely generated by elements s_0,...,s_g of order two, each defined by a pair of fixed points {a_0,b_0},...,{a_g,b_g}. These fixed points are called “in good position”. A subgroup W in G of index 2 is a Schottky group and produces a hyperelliptic Mumford curve Omega/W –> Omega/G = P^1, called `Whittaker curve', of genus g and with branch locus B in P^1(K). An explicit parametrization of Whittaker curves in terms of theta functions for W and G and the data of the fixed points, is developed. In particular, this allows one to express the branched points (and other data such as p-adic periods and p-adic heights) in terms of values of theta functions. A central theme of this paper is the relation between the fixed points and the branch locus. For a given configuration (P,m) of $g+1$ pairs of points in P^1, one defines a rigid space Fix_{P,m} of fixed points in good position with that configuration and a rigid space of branched points $ Branch_{P,m} in that configuration. A main result is that the natural morphism FB: Fix_{P,m} –> Branch_{P,m} is a rigid etale covering with Galois group {\pm 1}^{d-1} for some d>0. For all cases of genus g=2,3 (and for some more), an approximation of FB is computed which confirms the main result. Classification of Whittaker groups and analytic reductions of Whittaker curves is another important issue of this paper. The background material in this paper complements the work of L.~Gerritzen, G.~Van Steen, F.~Herrlich and others. It involves re-examination of some proofs, the derivation of properties of semi-stable analytic reductions and studying good position of fixed points.

We construct a scalar invariant of flat principal 2-bundles over 3-manifolds, with structure 2-group $\mathcal{G}$, from an involutory Hopf algebra graded by $\mathcal{G}$. Expressing $\mathcal{G}$ in terms of a crossed module $\chi$ and using the classification of such 2-bundles via the classifying space $B\chi$, this amounts to constructing a homotopy invariant of maps from 3-manifolds to $B\chi$. The construction of the invariant relies on a combinatorial description of such maps by $\chi$-colored Heegaard diagrams. When the corresponding map to $B\chi$ is nullhomotopic or, equivalently, when the associated flat principal $\mathcal{G}$-bundle is trivializable, the invariant reduces to the Kuperberg invariant of the underlying 3-manifold.

In \cite{BCO25}, Bruzzo, Capasso and Otero extended the notion of ampleness of vector bundles to the more general context of Higgs bundles. But the ampleness of Higgs bundles did not coincide with the ampleness of vector bundles when the Higgs field is zero. We modify the definition of ample Higgs bundles that results in removal of this discrepancy. Invoking this definition, we study various properties of Higgs ample vector bundles. In particular, we prove a Barton-Kleimann type criterion to characterize the Higgs ample vector bundles.

Let $\mathcal{F}_5$ denote the $3$-uniform hypergraph on the vertex set $\{f_1,f_2,\dots,f_5\}$ with hyperedges $\{f_1f_2f_3,f_1f_2f_4,f_3f_4f_5\}$. Recently, Balogh, Clemen and Luo determined the Turán number of a one-vertex blow-up of $\mathcal{F}_5$, more specifically, they blow up the vertex $f_5$ to $t$ vertices, the resulting hypergraph is denoted by $\mathcal{F}_5(f_5;t)$. They show that for infinitely many $t$, $\mathcal{F}_5(f_5;t)$ has exponentially many extremal constructions and positive Turán density. In this paper, we determine the exact Turán number of the hypergraph obtained by blowing up $f_3$ of $\mathcal{F}_5$ to $t$ vertices and show that it also has exponentially many extremal constructions. We also give a general upper bound and lower bound of the Turán number of every blow-up of $\mathcal{F}_5$. For some special blow-ups of $\mathcal{F}_5$, for example, $t$-disjoint copies of $\mathcal{F}_5$, we determine the exact Turán number. We construct a hypergraph $\mathcal{F}_{sim}(t)$ which is a subgraph of a blow-up of $\mathcal{F}_5$, and is contained in the hypergraph obtained by adding any new hyperedge to the Turán hypergraph (the balanced complete $3$-partite hypergraph), but its extremal construction is not the Turán hypergraph. We also determine the exact Turán number of $\mathcal{F}_{sim}(t)$.

Biconservative submanifolds arise as a natural relaxation of the biharmonic condition and play an important role in the submanifold theory. In this paper, we study non-CMC biconservative surfaces with parallel normalized mean curvature vector field (PNMC surfaces) in the four-dimensional hyperbolic space $\mathbb{H}^4$, for which we consider the hyperboloid model. We provide a local extrinsic description of such surfaces, showing that they are generated by a directrix curve lying in a totally geodesic hypersurface $\mathbb{H}^3$ of $\mathbb{H}^4$, through a certain normal flow. This extrinsic classification of non-CMC, PNMC biconservative surfaces in $\mathbb{H}^4$ splits naturally into three cases according to the type of a certain vector field, which can be non-zero null, spacelike or timelike. We also prove that these surfaces are invariant under the action of a parabolic, elliptic, and hyperbolic one-parameter group of isometries of $\mathbb{H}^4$, respectively. Moreover, their full groups of ambient isometries preserving the surfaces are determined. Together with the previous results, the classification of non-CMC, PNMC surfaces in four-dimensional space forms is now complete, from both intrinsic and extrinsic points of view.

A well-known result of Burr, Erdős and Spencer [Transactions of the American Mathematical Society, 1975] determines the $2$-colour Ramsey number for any sufficiently large collection of vertex-disjoint copies of a fixed graph $H$ without isolated vertices. A focus of this paper is to give analogous results for the corresponding $r$-colour Ramsey problem. More precisely we determine, up to an additive constant, this Ramsey number in the case when $r=3$ and also in the case when the chromatic number of $H$ is at least $r$. In these cases our results depend on a parameter which, roughly speaking, describes the corresponding class of potential extremal colourings of the complete graph. Our proofs rely on our notion of $(H,r)$-gadgets, which is crucial to obtain results that are best-possible up to additive constant terms. We exploit this notion using linear programming and the Poincaré-Miranda theorem. We also determine, up to a linear error term, the corresponding $r$-colour Ramsey number in the case when $H$ is a complete bipartite graph. Here, the corresponding class of potential extremal examples exhibits a connection with the well-known clique-edge-covering problem. We also prove random versions of some of our results. In particular, we prove a random version of the Burr-Erdős-Spencer theorem, thereby generalising the random Ramsey theorem of Rödl and Ruciński [Journal of the American Mathematical Society, 1995]. Our proofs make use of coloured versions of the (sparse) regularity lemma and the KLR conjecture for random graphs.

We study aggregation-fragmentation processes in which pairs of clusters can aggregate, and each cluster can break into two fragments. If the rates of aggregation and fragmentation do not depend on the masses, detailed balance does not hold, but nonequilibrium steady states can still be deduced from an exact solution for the Laplace transform. For models in which aggregation rates remain constant but fragmentation rates scale as $(\text{mass})^\beta$, detailed balance holds only when $\beta=1$. Away from this solvable case, we employ asymptotic techniques and show that when $\beta\geq 0$, the steady states share similarities with those from the mass-independent ($\beta=0$) model. An instantaneous shattering transition with continuous mass loss occurs when $\beta<0$.

We describe an efficient method for computing the Ehrhart polynomial of Gelfand–Tsetlin polytopes arising from Kostka coefficients. The key idea is to exploit Ehrhart–Macdonald reciprocity: evaluating the Ehrhart polynomial at negative integers reduces to counting strict Gelfand–Tsetlin patterns, which are often zero or very small for low dilations. Combined with an adaptive strategy that chooses the cheapest evaluation point (positive or negative) at each step, this yields substantial practical speedups compared to general-purpose polytope software. We benchmark against $\mathtt{OSCAR}$/$\mathtt{polymake}$, and illustrate the broader applicability of the method through order polytopes and permutation posets. The implementation is available in the Rust kostka package, with related optimizations also incorporated in the new lrcalc-rs replacement for lrcalc.

Let $k$ be a positive integer and let $X_k$ be the cubic hypersurface defined by the equation $x^3-(y_1^2+\cdots+y_{4k}^2)z=0$. In this paper, we give an asymptotic formula for the counting function of semi-integral points on $X_k$. We also prove that this asymptotic formula agrees with Manin's conjecture for $\mathcal{M}$-points [Moe26a, Conjecture 1.4] on the $a$-invariant and the $b$-invariant.

Vehicle-to-vehicle (V2V) energy trading enables decentralized peer-to-peer energy exchange among electric vehicles (EVs), reducing grid dependency while monetizing surplus capacity. However, coordinating self-interested EV agents with diverse charging needs and uncertain arrival-departure schedules remains challenging. Existing approaches either require centralized optimization with computational limitations or lack fairness guarantees. This paper integrates Nash Bargaining Solution into Multi-Agent Deep Deterministic Policy Gradient, namely Nash-MADDPG, for incentive-aligned V2V energy trading. Nash bargaining determines efficient bilateral pricing, while Nash-guided price proximity rewards align agent learning toward bargaining-optimal strategies. Evaluation over 30-day continuous operation demonstrates an improvement of 61.6% in social welfare and 62.9% improvement in trading volume over Double Auction, while achieving superior fairness, such as 40.1% improvement in Jain's index. Testing across 6-100 agents over a 30-day horizon with continuous vehicle turnover confirms scalability across population size and empirically stable pricing near the Nash Bargaining benchmark.

Highly pathogenic avian influenza A(H5N1) has expanded geographically and ecologically, affecting wild birds, mammalian wildlife, domestic animals, and humans. Wildlife surveillance provides critical early warning for One Health preparedness, yet national-scale analyses integrating host ecology, spatial patterns, seasonality, viral lineage, and risk factors remain limited. This study analysed Canadian wildlife HPAI A(H5N1) surveillance records from 2022 to 2026 to characterise spatiotemporal dynamics and identify factors associated with detection counts. A retrospective analysis of 2,657 detections across 13 provinces and territories was conducted using descriptive epidemiology, spatial clustering methods, and Negative Binomial mixed models. Detections were predominantly avian, with waterfowl and raptors as the major host groups, while mammals accounted for a smaller but epidemiologically important proportion. Detection burden was highest in 2022, with increased activity in autumn and spring. Ontario, Alberta, and British Columbia were identified as major hotspots, with evidence of local clustering in parts of the Prairie region. Reassortant Eurasian-North American lineages dominated detections and were strongly associated with higher detection counts. Modelling results identified year, season, and lineage as key predictors. These findings support risk-based One Health surveillance prioritising high-burden regions, migration-associated periods, key avian host groups, reassortant viral lineages, and continued monitoring of mammalian wildlife.

We prove uniform diameter estimates, volume non-collapsing estimates and Gromov-Hausdorff convergence for the normalized Chern-Ricci flow on smooth complex minimal surfaces of general type, starting from an arbitrary Hermitian metric. This removes the local Kahler assumption near the null locus used in our previous work and confirms the Tosatti-Weinkove conjecture in complex dimension two. The main analytic ingredients are a surface torsion estimate, a uniform total variation bound for Delta |G|, a Green-weighted L^2 estimate for the torsion, and a linear iteration of real Poisson equations, which together give the required Green function estimates.

We study the existence of large (boundary blow-up) solutions to semilinear equations involving the Finsler p-Laplacian on bounded domains with sufficiently smooth boundaries. We establish a Keller-Osserman-type condition that ensures the existence of such solutions, and show that this condition retains the same integrability as that of the p-Laplacian. We examine the influence of the anisotropic norm underlying the Finsler p-Laplacian on the boundary behaviour of the solution, then derive asymptotic estimates for large solutions near the boundary of the domain. Using these boundary asymptotics, we prove uniqueness results for power-type nonlinearities.

Some time ago, we published the full count of degrees of freedom in the linearised weak gravity limit of arbitrary New GR models. We did it by considering the linear weak gravity equations and presented a thorough analysis with no ambiguity left. A bit later, we generalised it to linear cosmological perturbations and discussed the strong coupling issues that appear already at this level. Recently, there were claims that some dynamical modes had been missed in our work. However, the authors of the new claims did not look at the equations of motion and analysed the quadratic Lagrangian densities instead. In this paper, I take one of the most elementary cases, namely the vector modes in New GR of Type 3, and show what was their mistake that had led them to claiming that those were dynamical. The main message: Do not substitute constraint equations into a Lagrangian.

We develop a general theory of Cartesian and non-Cartesian polynomials on products of complex spaces $\mathbb{C}^{n_1} \times \cdots \times \mathbb{C}^{n_k}$. We prove that, for any fixed degree $d \ge 2$, a (Zariski) generic polynomial is non-Cartesian in a broad range of dimensions, establishing that Cartesian structure is highly exceptional. We further introduce effective sufficient criteria for a polynomial to be non-Cartesian. Moreover, we show that being (non)-Catersian can be decided algorithmically via Gröbner basis methods and quantitative forms of Hilbert's Nullstellensatz. As an application, we connect the non-Cartesian condition to incidence geometry, obtaining sharp intersection bounds and constructing extremal configurations that demonstrate the optimality of these estimates.

Pricing American options is more complicated than pricing European options, because they can be exercised at any time, and one thus needs to solve a linear complementarity problem instead of simply doing time stepping for computing European options. We introduce a new Schwarz modulus-based splitting method for solving such linear complementarity problems, and further accelerate them using Modified Polynomial Extrapolation, a non-linear vector sequence acceleration technique, which is very much related to Krylov methods in the linear case. Numerical experiments on a model problem show that our new solver can have close to an order of magnitude lower iteration counts than the classically used modulus-based matrix splitting technique.

Answering a question of Goode, we show that $k$-triviality collapses to (1-)triviality among simple theories. In particular, every stable theory with quantifier elimination in a relational language of bounded arity is trivial. We use our collapse result, along with other facts about $k$-triviality and $k$-total triviality, to generate examples of (strongly) $k$-distal theories. The collapse result immediately implies that no stable theory can be strictly $k$-distal for some $k\geq 3$, partially answering a question of Walker. Moreover, all known examples of non-distal (strongly) $k$-distal theories are $k$-ary, rendering (strong) $k$-distality moot as a $(k+1)$-ary dividing line; we give four classes of examples that are not $k$-ary. We also show that just as distality is not preserved under taking reducts, neither is (strong) $k$-distality.

This paper focuses on the application of time domain decomposition to solve partial differential equations constrained optimization problems and controllability problems. After clarifying the link between these two types of problems, we show that applying time domain decomposition to both problems leads to the same convergence behavior. Our numerical experiments also confirm these theoretical findings.

In this paper we prove that every coefficient of twisted Alexander polynomials of torus knots associated with irreducible $\mathrm{SL}_n(\Bbb C)$-representations is an $\Bbb A$-valued locally constant function on the $\mathrm{SL}_n(\Bbb C)$-character variety, where $\Bbb A$ is the ring of all algebraic integers over $\Bbb C$. Moreover, as a generalization of a recent result of Kitano and Nozaki, we show that $\mathrm{SL}_n(\Bbb C)$-Reidemeister torsions are algebraic integers for many Seifert fibered spaces. Also, we discuss the power sums of Reidemeister torsions of torus knots for low-dimensional irreducible representations that provide a mysterious relation to TQFT.

The weak $k$-metric dimension of a graph is roughly understood as the cardinality of a smallest set of vertices $S$ of the graph with the property of uniquely recognizing all the vertices of the graph throughout summations of differences of distances to the vertices of $S$. The weak $k$-metric dimension of the direct product of two isomorphic complete graphs is considered in this work. Specifically, the value of such parameter is computed for almost all possibilities of these products and a bound is provided in the remaining case.

We prove a structure theorem for finite perfect two-sided skew braces. The main tool is a central product theory for skew braces, developed here in both external and internal form; we show that these two constructions are equivalent. Our main result states that every finite perfect two-sided skew brace \(B\) admits the canonical decomposition $B=B^2\circ B^{2,\operatorname{op}},$ where \(B^2\) is almost trivial with perfect additive group, while \(B^{2,\operatorname{op}}\) is trivial with perfect additive group. Thus finite perfect two-sided skew braces are classified, up to central amalgamation, by trivial and almost trivial skew braces arising from perfect groups. This decomposition has strong consequences for the underlying groups: for finite two-sided skew braces, perfectness of the skew brace is equivalent to perfectness of either the additive or the multiplicative group. In the trivial-center case the central product becomes a direct product, recovering Trappeniers' classification of finite simple two-sided skew braces. We also show that quasi-simple two-sided skew braces are necessarily either trivial or almost trivial. Finally, we prove that this rigidity is genuinely two-sided by constructing a quasi-simple skew brace which is not two-sided and is neither trivial nor almost trivial.

We show how the recent extension of spectral submanifold (SSM) theory to delay differential equations (DDEs) enables data-driven model reduction of nonlinear delay systems. First, using a scalar DDE with a single discrete delay, we compare equation-based and data-driven SSM reductions, to illustrate the need for the latter. We then use the same algorithm to obtain purely data-driven, SSM-reduced, delay-free ODE models for several nonlinear delayed systems. Our approach requires no information about the form of the underlying DDE, or about the number and magnitude of the delays it contains. Our SSM-reduced, low-dimensional models remain predictive even for chaotic dynamics. We also illustrate the use of parametric SSM-reduction to capture bifurcations in systems with both distributed and discrete delays. Finally we extend the theoretical underpinning of delayed SSM-reductions to non-autonomous systems with periodic delays, and apply these results to experimental data from a control system with feedback delay and quantization.

We study the $L^{\infty}$ discrepancy of point sets generated by determinantal point processes on all compact, connected two-point homogeneous spaces, namely spheres and projective spaces. Using concentration inequalities and variance estimates for the number of points in metric balls, we derive general upper bounds for the discrepancy of homogeneous determinantal point processes. In the particular case of the harmonic ensemble, we show that the discrepancy of $N$ points is $O((N^{1-1/D})^{1/2}\log N)$ with high probability, where $D$ denotes the real dimension of the manifold. For the projective ensemble on $\mathbb{CP}^d$, we obtain the sharper bound $O((N^{1-1/D}\log N)^{1/2})$. These results extend previously known discrepancy estimates for determinantal point processes on the sphere to all compact, connected two-point homogeneous spaces.

An $(r, s)$-${\textit set}$ in ${\rm PG}(n, q)$ is a set of points, say $\mathcal X$, such that each $s$-dimensional projective subspace contains at most $r$ points of $\mathcal X$. We investigate $(n, n-2)$-sets and $(n-2, n-3)$-sets in ${\rm PG}(n, q)$, $n \le 6$. We show that the trivial upper bounds on $(n, n-2)$-sets in ${\rm PG}(n, q)$, $4 \le n \le 6$, $(4, 3)$-sets in ${\rm PG}(6, q)$ and $(3, 2)$-sets in ${\rm PG}(5, q)$ are essentially sharp. A $(3, 2)$-set in ${\rm PG}(13, q)$ of size $\frac{q^6-1}{q-1}$ is also constructed.

In this paper, we investigate a multi-cell six-dimensional movable antenna (6DMA) network for enhancing downlink communication performance under inter-cell interference (ICI). Each base station (BS) is equipped with multiple 6DMA surfaces, and the 6DMA rotations affect both the desired-signal enhancement for in-cell users and the interference leakage toward neighboring cells, which makes the antenna-rotation design and transmit precoding intrinsically coupled across BSs. To address this issue, we formulate an average weighted sum-rate maximization problem for the multi-cell system by jointly optimizing the short-term downlink precoders and long-term 6DMA rotations under practical antenna geometric constraints. To tackle the resulting nonconvex problem, we propose a distributed two-timescale design based on inter-cell interference power constraint (IPC) coordination among neighboring BSs, under which each BS performs local short-term precoder optimization based on instantaneous channel state information (CSI) and long-term 6DMA rotation update according to statistical CSI with limited inter-BS information exchange. In particular, an edge-wise IPC coordination mechanism based on two-stage one-dimensional grid search and random maximal matching is developed to enable scalable distributed implementation. A centralized offline benchmark is also provided for performance comparison. Numerical results show that the proposed distributed design achieves performance close to the centralized benchmark under different interference conditions, while maintaining favorable scalability as the network size increases.

LSQR and LSMR are iterative methods, based on the Golub-Kahan bidiagonalization algorithm, widely used for large-scale linear least squares problems. FLSQR and FLSMR are flexible variants of LSQR and LSMR, respectively, based on a flexible Golub-Kahan (Arnoldi-like) factorization algorithm, which naturally allow modifications of the solution approximation subspace and/or handling inexact matrix-vector multiplications with the (transpose of the) coefficient matrix, thereby enabling to enforce prior information into the computed solution. The goal of this paper is to introduce sFLSQR and sFLSMR, i.e., sketched variants of FLSQR and FLSMR, respectively, where randomization becomes particularly effective, as it allows to recover short recurrences for the solution approximation. In particular, this paper explores applications to large-scale inverse problems, showing the ability of the new randomized solvers to alleviate computational bottlenecks while preserving reconstruction quality. A theoretical analysis of sFLSQR and sFLSMR is provided, and their performance is validated through numerical experiments.