Papers

Let $E$ be an elliptic curve over $\mathbb{Q}$ having split multiplicative reduction at a prime number $p$. We describe the tame part of the $\mathcal{L}$-invariant of $E$ at $p$ in terms of automorphic $p$-adic periods introduced in the work of Darmon. More precisely, we prove an equality of refined $\mathcal{L}$-invariants using twisted versions of refined exceptional zero conjectures. When the conductor of the elliptic curve is exactly $p$ and the automorphic period is attached to an optimal embedding of conductor $1$ then we prove this equality unconditionally by using the work of de-Shalit.

This paper investigates a single species reaction-diffusion system incorporating a spatiotemporal delay memory kernel, which models the cognitive map of animals, under Neumann boundary conditions. The model can be used to describe the process in which individuals are influenced by historical information during spatial diffusion. An equivalent system construction method with auxiliary variables is introduced to transform the original system into a delay-free coupled reaction-diffusion equation. By employing Fourier modal decomposition and eigenvalue analysis, we conduct stability and bifurcation analyses for both the exponentially decaying weak kernel and the peak type strong kernel, obtaining explicit expressions for the steady state and Hopf bifurcation points. Compared with the model in which the memory term of the continuous-time integral kernel using its own population density, our model exhibits Hopf bifurcations and steady state bifurcations even under a weak kernel because of the introduce of a dynamic cognitive map. This implies that a dynamic cognitive map introduces sufficient flexibility to generate both steady state bifurcations and Hopf bifurcations across a broader range of temporal kernels. Numerical simulations are presented to demonstrate the influence of stable, steady state and Hopf bifurcation regions on the spatiotemporal distribution of solutions.

We give generators and relations for the monoidal categories of crystals generated by the fundamental crystals of a simple complex Lie algebra. We also spell out several small-rank examples.

Let $n$ be an integer not equal to $-2$, $0$ or $2$. We consider the unit equation $\varepsilon + \delta = n$ in units $\varepsilon, \delta$ of cubic fields. We show that this equation has no solutions for 100% of cubic fields, when ordered by discriminant. This is consistent with a recent conjecture of the authors.

This paper investigates a class of Schrödinger-Poisson systems in $\mathbb R^3$ featuring the (p,q)-Laplacian operator and a combination of critical and subcritical nonlinearities in the Schrödinger equation while the m-Laplacian and a power type nonlinearity in the Poisson's one. We consider both the attractive and repulsive cases, which correspond to different signs in front of the nonlocal term. While most existing literature relies on auxiliary functionals or specialized techniques to overcome the lack of compactness and ensure the boundedness of Palais-Smale sequences, we employ a direct variational approach. By applying the Mountain Pass Theorem and concentration compactness principles, we establish the existence of positive solutions. A careful analysis is conducted to identify the parameter ranges for which the Mountain Pass level falls within the compactness threshold, despite the technical challenges posed by the unbalanced growth of the operator and the nonlocal interaction.

We study prime endomorphisms of the free group $F_2$. The main results provide lower and upper bounds for the logarithmic density $\delta_{\mathrm{np}}$ of non-prime endomorphisms: $$ \frac34 \leq \delta_{\mathrm{np}}\leq \frac{27}{28}. $$ Equivalently, we prove corresponding lower and upper bounds for the exponential decay of the probability that a random endomorphism is non-prime, aligning with a heuristic argument of Ian Agol.

Estimating the generative mechanism of large-scale networks is a fundamental challenge in statistical machine learning. It requires the identification of the latent connectivity structure, which is in general an NP-hard combinatorial problem due to the absence of canonical node labels. We address this challenge by allowing for probabilistic couplings, thereby relaxing the assignment problem. Our estimation framework can be formulated as a semi-relaxed Gromov-Wasserstein objective and provides a low-dimensional representation of the generative structure. We solve this via a block-coordinate conditional gradient algorithm. Despite the relaxation, the resulting solution is typically deterministic: in fact, we show that the optimality gap between the relaxed solution and the deterministic assignment vanishes at rate $O(1/n)$, where $n$ is the number of nodes. This allows for tractable recovery of the underlying model and enables rigorous statistical analysis: we establish consistency and minimax-optimal convergence rates for both stochastic block models and Holder-smooth graphons. Our implementation scales efficiently with $n$, as demonstrated on both synthetic and real-world datasets.

Balogh, Clemen, and Lidický proved that the $\ell_{2}$-norm Turán problem for $K_{5}^{3}$ is asymptotically solved by the balanced bipartite construction, and they further conjectured that this construction is uniquely extremal for all sufficiently large $n$. We confirm this conjecture. We also determine exactly the maximum number of cliques in an $n$-vertex $K_{5}^{3}$-free $3$-uniform hypergraph for all sufficiently large $n$, thereby verifying the corresponding case of a conjecture of Frankl, Gryaznov, and Talebanfard. The main ingredients are Turán-type theorems for vertex-colored graphs forbidding balanced cliques, including an edge bound, an $\ell_{2}$-norm bound, and a sharp crossing-triangle theorem in the two-colored balanced $K_{4}$-free case. We also use a local modification procedure within the stability method. This reduces the exact hypergraph problems to proving that the relevant objective function increases under suitable local changes near the bipartite construction.

A graph pair $(\Gamma, \Sigma)$ is called stable if every automorphism of the direct product $\Gamma\times\Sigma$ is induced componentwise by automorphisms of $\Gamma$ and $\Sigma$. A graph is twin-free if no two distinct vertices share the same neighbourhood in the graph. Two graphs $\Gamma$ and $\Sigma$ are coprime with respect to the direct product if there is no graph $\Delta$ of order greater than $1$ such that $\Gamma\cong\Gamma'\times\Delta$ and $\Sigma\cong\Sigma'\times\Delta$ for some graphs $\Gamma'$ and $\Sigma'$. A graph pair $(\Gamma,\Sigma)$ is nontrivial if $\Gamma$ and $\Sigma$ are coprime connected twin-free graphs and exactly one of them is bipartite. In this paper, we prove that if $\Gamma$ is non-bipartite, stable, and factor-loopless, then each nontrivial graph pair $(\Gamma,\Sigma)$ is stable. This gives a partial answer to [Question~19, Qin, Xia and Zhou, Discrete Math., 113856, (2024)] and proves the factor-loopless case of [Conjecture~1.3, Wang, Qin and Xia, arXiv:2509.26170]. We also give affirmative answers to [Questions~3.5, 3.6, Gan, Liu and Xia, J. Combin. Theory Ser. B, 140–164, (2025)] and a negative answer to [Question~3.7, Gan, Liu and Xia, J. Combin. Theory Ser. B, 140–164, (2025)].

Let Gamma be an additive abelian group with a commutative factor omega. We describe the simple and untwisted affine Lie colour algebras which admit a Cartan - Weyl description. For both classes, we construct the quantised universal enveloping colour algebras, and establish their quasi-triangular Hopf colour algebraic structure. These colour quantum groups generalise the Drinfeld - Jimbo quantum groups, which are recovered when Gamma is trivial. They provide an algebraic foundation for investigations in knot theory and soluble models in statistical mechanics.

Decarbonizing urban energy systems requires optimization approaches capable of handling the operational complexity of large-scale district heating networks. However, existing studies typically focus on a single real-world network, limiting comparability and the transferability of insights. To address this, realistic synthetic instances provide controlled, reproducible environments for testing optimization algorithms independent of specific case studies while preserving key structural and temporal characteristics of real systems. Such instances enable systematic benchmarking, methodological development, and comparative studies across algorithms and modeling choices. In this work, we generate a suite of large-scale synthetic instances for multi-objective optimization of district heating systems. The instances are openly available as network topologies in JSON format and as mixed-integer programs (MPS files) for benchmarking. They are constructed via a transparent procedure that allows reproduction, extension, and transfer to other network-based problems. We apply the method to Berlins district heating network, the most complex in Western Europe, formulating a tri-objective mixed-integer model for unit commitment over up to 25 years with 4-hour temporal resolution. A computational study provides a detailed comparison between the synthetic instances and the real-world Berlin data, showing under which conditions the generated instances reproduce realistic optimization characteristics. Furthermore, we investigate which features make the resulting models computationally challenging. The findings highlight how well-designed synthetic instances can support robust benchmarking practices and enable meaningful assessment of (multi-objective) optimization methods for large-scale district heating systems.

Low Earth orbit (LEO) satellite links experience rapid angular variation due to high orbital velocities, which causes severe beam misalignment and array gain degradation under conventional fixed-antenna architectures. In this letter, we propose a rotatable antenna (RA)-enabled LEO communication framework, where RA arrays are deployed at both the satellite and the ground node (GN) to exploit antenna boresight reconfiguration as an additional spatial degree-of-freedom (DoF) for maintaining directional alignment under high mobility. By leveraging the rank-one line-of-sight (LoS) channel structure inherent to satellite links, we derive closed-form solutions for the joint design of the transmit/receive beamforming and antenna boresight directions, revealing that optimal performance can be achieved via decoupled alignment across antennas with low computational complexity. To enable practical operation under dynamic conditions, we further develop a channel estimation and beam tracking protocol that exploits the predictable satellite orbit to continuously update boresight directions with low training overhead. Simulation results demonstrate that the proposed RA-enabled design significantly outperforms fixed and random boresight baselines in terms of achievable rate and robustness to angular variations, highlighting the effectiveness of rotational spatial reconfiguration in high-mobility satellite communications.

A symmetric-tensor distributional mixed method for a fourth-order elliptic singular perturbation problem is developed in this paper. The moment variable is approximated by normal-normal continuous symmetric tensor elements, while the scalar variable is represented by an H^1-nonconforming virtual element space coupled with a polynomial multiplier on interior codimension-two subsimplices. Optimal parameter-uniform error estimates are derived, independent of the presence of boundary layers. A hybridized form is further shown to be equivalent to stabilization-free weak Galerkin and H^2-nonconforming virtual element formulations. In two dimensions, we establish a close connection between the distributional mixed method and the classical Hellan-Herrmann-Johnson (HHJ) method by identifying the scalar virtual element-multiplier pair with the Lagrange finite element space. Consequently, the proposed method extends the two-dimensional HHJ framework to any spatial dimension d >= 2. Three-dimensional numerical experiments support the theoretical convergence and robustness estimates. A two-dimensional adaptive constant-load benchmark on an L-shaped polygonal domain tests the method on a non-manufactured nonsmooth problem and shows mesh concentration near the reentrant corner and, for small epsilon, boundary refinement at the expected O(epsilon) scale.

We study $O(2)$-equivariant spectral flow for Dirac operators on a finite warped cylinder equipped with fixed admissible regularized APS boundary conditions. The twisting bundle is a real higher-rank orthogonal bundle, and reflection symmetry is implemented by a fiber involution. After complexifying the twisting bundle and diagonalizing the orthogonal twist, the Dirac equation decomposes into a scalar Fourier-mode radial equation, with moving rotating blocks and stationary neutral blocks. After regrouping conjugate and reflection-paired blocks, the crossing contributions define real $RO(O(2))$-classes. Consequently, we obtain an explicit blockwise formula for the $RO(O(2))$-valued spectral flow of the resulting regularized APS family. Under the standard self-adjoint Fredholm, endpoint-invertibility, and regular-crossing hypotheses, together with a fixed neutral-sector convention, this formula is obtained by assembling the local crossing contributions of the separated blocks. It refines ordinary integer-valued spectral flow and shows explicitly how the dimension map $RO(O(2))\to\mathbb Z$ loses representation-theoretic information. We also discuss the rank-three case, including the role of the fixed neutral sector, and the corresponding endpoint $\eta$/APS index interpretation.

In this paper, we prove universality of the distribution of the cokernels of random $p$-adic matrices with inhomogeneously balanced columns. More precisely, let $u \ge 0$ be an integer, and for each positive integer $n$, let $A(n)$ be a random $n \times (n+u)$ matrix over $\mathbb{Z}_p$ whose $i$-th column is $\alpha_n(i)$-balanced. We prove that if $\sum_{i=1}^{n+u} \exp(-\epsilon \alpha_n(i)n) \to 0$ as $n \to \infty$ for every $\epsilon>0$, then the cokernels of $A(n)$ converge in distribution, as $n \to \infty$, to the same limiting law as the cokernels of Haar-random $n \times (n+u)$ matrices over $\mathbb{Z}_p$. This extends a universality theorem of Nguyen and Wood to random $p$-adic matrices with inhomogeneously balanced columns.

This work introduces L0PathBrex, a novel method for estimating the solution path of L0-regularized problems through the use of L0 Bregman relaxations (B-rex). Recently introduced and analyzed in the literature, these relaxations provide continuous reformulations of the original objective, are applicable to possibly non-quadratic data fidelity terms, and depend on a family of functions designed to preserve the global minimizers while eliminating part of the undesirable local minima. Given any numerical solver for the relaxation, the proposed approach dynamically constructs a collection of local minimizers that are candidates for the L0-solution path. It exploits warm-start strategies and identifies ranges of the regularization parameter for which each minimizer remains valid under the corresponding relaxation. Experiments on sparse least-squares and logistic regression problems demonstrate that L0PathBrex systematically outperforms state-of-the-art baselines across both synthetic and real-world datasets in terms of various evaluation metrics; additionally, the study investigates how the choice of the B-rex affects the quality of the estimated path in the sparse Poisson regression setting.

In wholesale electricity markets, electricity producers and the independent system operator (ISO) play a central role. The ISO is responsible for minimizing production costs while satisfying supply-demand balance and capacity constraints. In this paper, we study a continuous-time problem in which the ISO seeks to minimize the joint cost of operation and investment in an electricity network. The problem is formulated in terms of operational and investment control variables. We analyze the hierarchy between these controls and use the so-called Day-Ahead Problem to find an explicit form of the optimal operation. This allows us to reformulate the investment problem as a stochastic control problem with state constraints. We extend the results of state-constrained stochastic control to fit our setting. In particular, we use a version of the Pointing Inward Condition to fully characterize the value of the problem as the unique viscosity solution of a constrained HJB equation. We then assign a specific dynamic to the capacity-demand process and discuss how the assumptions for the HJB characterization result in a budget constraint for the planning. Finally, we run simulations for a three-node setting that resembles the Chilean market. We analyze short-, medium-, and long-term planning scenarios and discuss how to transition toward a system with high penetration of renewable energy.

We consider the skein algebra of an unpunctured marked surface. Thurston previously constructed its band basis topologically. We show that this band basis coincides with the common triangular basis, which is a Kazhdan-Lusztig type basis for quantum cluster algebras analogous to the dual canonical basis of quantum groups. Our result confirms a conjecture of Thurston. It also provides new cases for the existence of the common triangular basis. In addition, we discover a phenomenon where certain unknots are arranged in configurations resembling beads on a necklace.

We determine the sharp minimum-degree threshold for Hamiltonicity in graphs perturbed by a uniformly random $K_r$-factor, resolving a conjecture of Espuny Díaz and Girão [Random Structures Algorithms, 2023]. In fact, we prove the stronger pancyclic statement. Let $\alpha^*(K_r)$ and $\alpha_{\text{pan}}^*(K_r)$ denote the Hamiltonicity and pancyclicity thresholds, respectively. We show that $\alpha^*(K_r)=\alpha_{\text{pan}}^*(K_r)=\rho_r$, where $\rho_r$ is the unique positive solution of $x^r+rx-1=0$. The proof is obtained from a general framework for perturbations by a uniformly random $F$-factor, where $F$ is an arbitrary fixed connected graph.

Lambert-Cole and Meier showed that the elliptic surface $E(n)$ admits a $(12n-2,0)$-trisection, considering the property that $E(n)$ is a certain double branched cover of $S^2 \times S^2$, which is a minimal genus trisection. In this paper, we clarify a way to construct an explicit $(12n-2,0)$-trisection diagram of $E(n)$ from its handle diagram arising from its Lefschetz fibration.

We consider the reciprocal cost function $ J(x)=\frac12(x+x^{-1})-1 $ and its $n$-dimensional extension $J(x_1,\ldots,x_n) = \frac12(R+R^{-1})-1, R=\prod\limits_{i=1}^n x_i^{\alpha_i}, \alpha=(\alpha_1,\ldots,\alpha_n)\in\mathbb{R}^n\setminus\{0\}.$ In logarithmic coordinates $t_i=\log x_i$, the Hessian of $J$ has a rank of one at every point. The associated Hessian geometry is degenerate and does not define a Riemannian metric. To obtain a nondegenerate geometric structure, we introduce a family of Hessian metrics $h_\lambda$. Combining the rank-one tensor with the Hessian metric $h_\lambda$, we construct a $(1,1)$-tensor field $A_\lambda$. Its trace normalization defines a projector $P_\lambda$, which induces an almost product structure and the corresponding golden and metallic structures. We study several properties of the projector $P_\lambda$ and the induced structures, including eigendistributions, parallelism, integrability, and curvature. The construction is given in an arbitrary dimension, and explicit formulas are obtained in the two-dimensional case. In particular, we show that the projector $P_\lambda$ is generally not parallel with respect to either the canonical flat affine connection or the Levi-Civita connection $\nabla^\lambda$ of the Hessian metric $h_\lambda$.

We investigate support thresholds for fully smeary and directionally smeary absolutely continuous probability measures on the sphere \(\mathbb{S}^m\). The motivation is inferential: smeariness is caused by degeneracy of the Hessian of the Fréchet function, and such degeneracy can invalidate the classical central limit theorem (CLT) for Fréchet means and the corresponding Wald-type \(\chi^2\) inference. For rotationally symmetric densities, we show that full and directional smeariness are equivalent. The Hessian and fourth-order terms are governed by two explicit geometry-dependent radii \(R_m<S_m\). In dimensions \(m=2,3\), rotationally symmetric smeariness cannot occur. For \(m\ge4\), support contained in the geodesic ball of radius \(S_m\), centered at the Fréchet mean, rules out smeariness; conversely, for every \(\varepsilon>0\) with \(S_m+\varepsilon<\pi\), we construct examples of rotationally symmetric \(2\)-smeary densities supported in the ball of radius \(S_m+\varepsilon\). For general densities, closed hemispherical support rules out both full and directional smeariness. Support contained in the closed ball of radius \(S_m\) rules out full smeariness, while we construct explicit, directionally \(2\)-smeary examples supported in balls of radius \(\pi/2+\varepsilon\). As a byproduct, the explicit Hessian formulas in this paper also provide a practical diagnostic for detecting proximity to the Hessian-degenerate, non-classical regime.

In this paper, we study the existence of normalized solutions to the mixed nonlinear Choquard equation with exponential growth \begin{equation} \left\{ \mathcal{L}u+\lambda u = \Lambda (I_{\alpha}*F(u))f(u) \text{ in }\mathbb{R}^{2}; \;\; \int_{\mathbb{R}^{2}}|u|^{2}\,dx = a^{2}, \right. \end{equation} where $\mathcal{L}=-\Delta+(-\Delta)^s$, $0<s<1$, $a>0$, $I_{\alpha}$ is the Riesz potential of order $\alpha\in(0,2)$, $\Lambda>0$, and $\lambda\in\mathbb{R}$ arises as a Lagrange multiplier. The nonlinearity has critical exponential growth in the sense of the Trudinger-Moser inequality. Using variational methods, the associated Pohožaev manifold, and suitable Moser-type estimates, we establish the existence of a normalized solution of mountain-pass type with associated non zero Lagrange multiplier. Under an additional monotonicity assumption on the nonlinearity, we further show that the obtained solution is a normalized ground state. We also establish the regularity properties required for the proof of the corresponding Pohožaev identity.

In this work, we determine the generator polynomials for the Hermitian hulls and Hermitian sums of cyclic codes defined over the composite ring $\mathbb{F}_2 \times (\mathbb{F}_2 + v\mathbb{F}_2)$, where $v^2 = v$. Based on these structures, we develop quantum error-correcting (QEC) codes by applying the Hermitian dual version of Quantum Construction~X to the obtained Hermitian hulls and sums. Moreover, by employing matrix product code methods on linear complementary dual (LCD) codes defined over the same ring, we derive families of entanglement-assisted quantum error-correcting (EAQEC) codes.

We exhibit, in a systematic way, connections between hypergraph Turán problems and extremal set theory. More specifically, we construct natural families of uniform hypergraphs for which the upper bounds on their Turán densities reduce to classical problems in extremal set theory, including the Erdős–Ko–Rado theorem, $L$-intersecting families, and the Erdős matching problem.

Trace-compatible polynomial extensions are a recurring local ingredient in high-order finite element analysis on conforming hexahedral meshes. They are needed whenever prescribed edge and face traces must be preserved while a polynomial is extended into a neighboring cell or boundary patch. The main contribution of this paper is the construction of p-robust polynomial liftings on nonsingular conforming hexahedral boundary patches, with stable control of both the H^1 norm and the H^1-seminorm estimates needed for energy arguments. These liftings imply H^1-seminorm stable discrete harmonic extensions of polynomial Dirichlet traces. They also serve as boundary corrections for the conforming hp Clement interpolant, yielding trace-preserving interpolation operators for functions with only H^1 regularity. Under the uniform boundary-degree condition the constants are p-uniform; in the non-uniform case the stated logarithmic loss appears. We also treat meshes that may contain conforming singular boundary patches, where the loss remains polylogarithmic in the maximal local degree. Trace-preserving interpolation on reference cells and vertex-supported decompositions are developed as local tools for these patch and mesh-level constructions.

Maesaka, Seki and Watanabe proved a formula for multiple harmonic sums. Yamamoto generalized it to Schur-type multiple harmonic sums, and the second author proved a $q$-analogue of this generalization. In this paper, we give two applications of the $q$-analogue formula. The first is an alternative proof of the duality of a $q$-analogue of multiple zeta values. The second is a proof of an identity for a $q$-analogue of the Kawashima function.

Geodesic regression generalizes classical regression models to manifold-valued data by replacing affine models in Euclidean spaces with geodesic models on Riemannian manifolds. In this paper, we set up a framework for regression based on retractions instead of the Riemannian exponential map and its corresponding retraction-based distance. The associated optimization problem is posed on a subset of the tangent bundle which is why we additionally construct retractions on the tangent bundle induced by retractions on the underlying manifold. Our approach yields a more flexible formulation which is applicable beyond settings where the exponential map can be computed efficiently. As a proof of concept, we apply the developed framework to the (n-1)-dimensional p-norm sphere using the retraction by normalization to define the regression problem. The resulting optimization problem is solved using the Riemannian steepest descent method.

This paper develops a switched event-triggered adaptive boundary control for a class of reaction-diffusion PDE-ODE cascade systems, where the system and input matrices in the ODE as well as the spatially-varying reaction coefficient in the PDE are uncertain. A two-step backstepping transformation is constructed to derive the continuous-time control law. Then a novel dynamic event-triggered control strategy for the PDE-ODE cascade is proposed based on a switched event-triggering mechanism, ensuring global exponential stability of the closed-loop system in place of the exponential convergence commonly achieved with backstepping-based classical dynamic ETC, while inherently excluding Zeno behavior. To address the uncertainties in the PDE-ODE cascade, adaptive update laws are developed, leading to time-varying gain kernels that are adaptively scheduled through the event-triggered control mechanism. Furthermore,to facilitate efficient real-time implementation, deep neural operators (DeepONet) are employed to approximate the backstepping kernels as mappings from the estimated parameters to kernel functions, thereby eliminating the need to repeatedly solve kernel PDEs online. Through a Lyapunov analysis that incorporates the effects of the event-triggering mechanism, parameter adaptation, and kernel approximation errors, we prove the $L^2$ global asymptotic regulation of the resulting closed-loop system. In summary, the key contributions of the paper are threefold: (i) developing an adaptive DeepONet-based framework for reaction-diffusion PDE-ODE cascade systems; (ii) extending the existing adaptive event-triggered control design for reaction-diffusion PDEs to the case with more complex uncertainties; and (iii) generalizing switched dynamic ETC with global exponential stability to PDE-ODE cascades. The effectiveness of the proposed approach is demonstrated through numerical simulations.

Let $\mathcal{H}$ be an $r$-uniform hypergraph. For $S\in \binom{V(\mathcal{H})}{j}$, let $\mathrm{deg}(S)$ be the number of edges of $\mathcal{H}$ containing $S$, and define the $(j,p)$-norm of $\mathcal{H}$ by $\|\mathcal{H}\|_{j,p}=\left(\sum_{S\in \binom{V(\mathcal{H})}{j}}\mathrm{deg}(S)^p\right)^{1/p}$. Motivated by a problem of Chao, Dong, Shen and Yang, we determine the maximum number of $t$-cliques in an $n$-vertex $r$-graph with bounded $(j,p)$-norm in the range $p>(t-j)/(r-j)$. The proof uses an entropy argument adapted to hypergraphs, together with a continuous interpolation step. The bound is sharp whenever the corresponding Steiner systems exist.