The most general Ruijsenaars-van Diejen-Takemura Hamiltonians are characterized as Heun operators defined as second order $q$-difference operators with a raising action on elementary rational functions with poles on the Askey-Wilson grid.

Marine biogeochemical models are widely used to study nutrient dynamics, water quality, and climate-related processes in coastal and estuarine systems. However, developing models that reliably represent specific environments remains computationally demanding, which makes their application to complex systems such as river plumes and estuarine environments challenging. In addition, these models contain several parameters that must be calibrated for the region of interest, a process that is often performed empirically using limited observational data. This thesis advances the development and calibration of marine biogeochemical models in the Brazilian context through three main contributions. First, we develop a conceptual model describing nutrient-phytoplankton dynamics in the Paranagua Estuarine Complex (PEC) in southern Brazil. The model is intentionally simple and computationally inexpensive, allowing simulations to be performed on standard personal computers. Second, we propose a systematic calibration framework based on tracer datasets and derivative-free optimization techniques. Finally, we demonstrate the practical application of this approach by calibrating the PEC model using in situ observations. Results show that, despite its simplicity, the model can reproduce observed nutrient dynamics when properly calibrated. The proposed framework is general and can be extended to multi-parameter calibration, seasonal parameter variation, and the coupling of biogeochemical models with higher-fidelity hydrodynamic models.

Given a hypersurface defined by $f$ in a smooth complex algebraic variety $X$, and a point $P$ on this hypersurface, we consider the invariant $β_P(f)$ given by the log canonical threshold at $P$ of ${\mathfrak m}_P\cdot J_f$, where ${\mathfrak m}_P$ is the ideal defining $P$ and $J_f$ is the Jacobian ideal of $f$. We show that this invariant satisfies most of the formal properties of the log canonical threshold of $f$ and give some examples. Dano Kim asked whether this invariant always gives an upper bound for the minimal exponent of $f$ at $P$. Motivated by this, we raise another question about minimal exponents, give a positive answer to a weaker version, and discuss some examples.

This paper develops new variance-reduction techniques for the forward-reflected-backward splitting (FRBS) method to solve a class of possibly nonmonotone stochastic composite inclusions. Unlike unbiased estimators such as mini-batching, developing stochastic biased variants faces a fundamental technical challenge and has not been utilized before for inclusions and fixed-point problems. We fill this gap by designing a new framework that can handle both unbiased and biased estimators. Our main idea is to construct stochastic variance-reduced estimators for the forward-reflected direction and use them to perform iterate updates. First, we propose a class of unbiased variance-reduced estimators and show that increasing mini-batch SGD, loopless-SVRG, and SAGA estimators fall within this class. For these unbiased estimators, we establish a $\mathcal{O}(1/k)$ best-iterate convergence rate for the expected squared residual norm, together with almost-sure convergence of the iterate sequence to a solution. Consequently, we prove that the best oracle complexities for the $n$-finite-sum and expectation settings are $\mathcal{O}(n^{2/3}ε^{-2})$ and $\mathcal{O}(ε^{-10/3})$, respectively, when employing loopless-SVRG or SAGA, where $ε$ is a desired accuracy. Second, we introduce a new class of biased variance-reduced estimators for the forward-reflected direction, which includes SARAH, Hybrid SGD, and Hybrid SVRG as special instances. While the convergence rates remain valid for these biased estimators, the resulting oracle complexities are $\mathcal{O}(n^{3/4}ε^{-2})$ and $\mathcal{O}(ε^{-5})$ for the $n$-finite-sum and expectation settings, respectively. Finally, we conduct two numerical experiments on AUC optimization for imbalanced classification and policy evaluation in reinforcement learning.

We apply the technique of jet differentials to establish a Gauss curvature estimate for an open Riemann surface $M$, equipped with a conformal metric induced from a nonconstant holomorphic map that is highly ramified over a generic hypersurface of sufficiently high degree.

We establish global stability for a chemotaxis-growth model with logarithmic sensitivity under dynamic Dirichlet boundary conditions on a 1D domain. We analyze both parabolic-parabolic and parabolic-hyperbolic systems. The key challenge is handling time-dependent boundary data for the unknown functions. We overcome this by introducing dynamic reference profiles which suitably interpolate boundary values. Using an expanded entropy functional measuring deviation from these profiles, we prove energy estimates the uniform boundedness of solutions and global asymptotic stability of perturbations.

We present a substitution rule for a rhomb tiling with 10-fold rotational symmetry. The tiling is closely related to the Penrose rhomb tilings and can be obtained from the pentagrid construction. We introduce a finite set of marked prototiles and describe an explicit substitution rule with inflation factor phi^3. Our main result is that the substitution is recognizable, so that the hierarchical structure of the tiling can be uniquely recovered from local configurations. Finally, we describe the relation between the tiling and the pentagrid construction.

We prove that two-step nilpotent $p$-extensions of rational global function fields of characteristic $p$ satisfy a quantitative local-global principle when they are counted according to their largest upper ramification break ("last jump"). We had previously shown this only for $p\neq2$. Compared to our previous proof, this proof is also more self-contained, and may apply to heights other than the last jump. As an application, we describe the distribution of last jumps of $D_4$-extensions of rational global function fields of characteristic $2$. We also exhibit a counterexample to the analogous local-global principle when counting by discriminants.

We investigate the Drazin invertibility of adjacency matrices associated with a class of oriented graphs known as oriented Dutch windmill graphs. By analyzing walks of prescribed lengths and exploiting the structure of the minimal polynomial, we obtain explicit expressions for the Drazin inverse and determine its index. The approach combines combinatorial enumeration with algebraic matrix analysis, offering a constructive characterization that generalizes known results for paths, cycles, and bipartite graphs. Beyond its intrinsic theoretical value, the framework provides insight into discrete models governed by cyclic feedback and may serve as a basis for symbolic computation of generalized inverses in structured networks.

We study spectral constants for convex domains $Ω$ containing the spectrum of an operator. We extend the Crouzeix--Palencia framework by obtaining bounds depending on a parameter $γ$ and relating these bounds to geometric properties of $Ω$ and the numerical range $W(A)$. We generalise the proof that the numerical range is a $1+\sqrt{2}$-spectral set to scaled $q$-numerical ranges. We also propose a generalisation of Crouzeix's Conjecture in the context of $q$-numerical ranges.

The purpose of these notes is to provide background on understanding the primal-dual algorithm of Chambolle and Pock [1] for imaging scientists. The presentation focuses on providing intuition and an algorithmic system that is amenable to pre-conditioning. The document aims to be self-contained, providing background on the essential facts of non-smooth convex analysis.[2]

We introduce the annex of an element $x$ in a Coxeter group as the set of elements $y$ such that $x \nleq y$ with respect to Bruhat order. This notion provides a complementary perspective to the study of Bruhat intervals and their interpretation via folded galleries. We establish general properties of annexes and show that in affine Coxeter groups the annex of any fixed element is finite. In rank-two affine Coxeter complexes, we further describe the geometric structure of annex boundaries using descent sets and configurations of parallel reflections. These results offer a new geometric viewpoint on the structure of the Bruhat order.

Given a reference risk measure, the risk budgeting is the portfolio where each asset contributes a predetermined amount to the total risk. We propose a novel approach, alternative to the ones proposed in the literature, for the calculation of the risk budgeting portfolio. This different perspective on the problem has several interesting consequences. For the calculation of the portfolio, we define a Cauchy sequence within the simplex of R^n, whose limit corresponds to the risk budgeting portfolio. This construction allows for the straightforward implementation of an efficient algorithm, avoiding the need to solve auxiliary, equivalent optimization problems, which may be computationally challenging and hard to interpret in the decision theory context. We compare our algorithm with the standard optimization-based methods proposed in the literature. From a theoretical point of view, starting from the Cauchy sequence, we define a function for which the risk budgeting portfolio is a fixed point. Therefore, sufficient conditions for the existence and uniqueness of the fixed point can be used. The methodology is developed for general risk measures and implemented in detail in the case of standard deviation.

This technical note documents the implementation and use of the Primal-Dual Conic Programming Solver (PDCS), a first-order solver for large-scale conic optimization problems introduced by Lin et al. (arXiv:2505.00311). It describes the algorithmic and implementation details underlying PDCS, including the restarted primal-dual hybrid gradient method framework, adaptive step-size selection, adaptive reflected Halpern iterations, adaptive restarts, and diagonal preconditioning. It also provides practical instructions for using PDCS, including its interfaces with JuMP and CVXPY, solver options, and illustrative code examples. PDCS is available at https://github.com/ZikaiXiong/PDCS under the Apache License 2.0.

Let $X$ be a smooth projective complex curve, $P\subset X$ a reduced effective divisor, and $X^{0}=X\setminus P$. We study logarithmic $V$-twisted Higgs bundles arising from a logarithmic Hecke compactification of a rank-two bundle on $X^{0}$. We show that a pair of induced logarithmic line-twisted fields lifts uniquely exactly under explicit local Hecke conditions, and that the lift is integrable precisely when the fields commute. Fixing the compactified spectral curve $Y$, we classify such Higgs bundles by pairs $(F,\,\vartheta)$, where $F$ is a rank-one torsion-free sheaf on $Y$ and $\vartheta$ satisfies a marked spectral condition on a finite subscheme $Z\subset Y$. This gives a logarithmic extension of the compact rank-two spectral correspondence of~\cite{ABK} to the punctured case. On the line-bundle locus, the moduli stack is canonically equivalent to $\mathrm{Pic}^{d}(Y)\times A_Z$.

Without randomization, escaping the saddle points of $f \colon \mathbb{R}^d \to \mathbb{R}$ requires at least $Ω(d)$ pieces of information about $f$ (values, gradients, Hessian-vector products). With randomization, this can be reduced to a polylogarithmic dependence in $d$. The prototypical algorithm to that effect is perturbed gradient descent (PGD): through sustained jitter, it reliably escapes strict saddle points. However, it also never settles: there is no convergence. What is more, PGD requires precise tuning based on Lipschitz constants and a preset target accuracy. To improve on this, we modify the time-tested trust-region method with truncated conjugate gradients (TR-tCG). Specifically, we randomize the initialization of tCG (the subproblem solver), and we prove that tCG automatically amplifies the randomization near saddles (to escape) and absorbs it near local minimizers (to converge). Saddle escape happens over several iterations. Accordingly, our analysis is multi-step, with several novelties. The proposed algorithm is practical: it essentially tracks the good behavior of TR-tCG, with three minute modifications and a single new hyperparameter (the noise scale $σ$). We provide an implementation and numerical experiments.

We study linear backward stochastic Volterra integral equations (BSVIEs) on the infinite time horizon. By introducing weighted function spaces with exponential decay, we establish existence and uniqueness of adapted M-solutions. We construct an infinite-horizon resolvent kernel and derive explicit formulas for the solution components (Y,Z,K) using a Girsanov transformation and Hida-Malliavin calculus. The results extend the finite-horizon theory of Hu and Oksendal to the infinite horizon framework.

In this paper, we investigate a class of Mean Field Games (MFGs) in which the state dynamics are governed by multidimensional reflected stochastic differential equations (SDEs). We establish the existence of an equilibrium and show that it can be approximated by the equilibrium of MFGs with non-reflected SDE. This approximation is constructed via a penalization method combined with the relaxed control approach introduced in [21]. Under a uniform ellipticity condition, and by applying the penalization method together with the mimicking theorem, we prove the existence of a Markovian MFG. Furthermore, under an additional convexity assumption, we demonstrate the existence of a strict-Markovian MFG. In the general case, we prove that relaxed MFG solutions with reflected dynamics can be approximated by strict controls whose dynamics are governed by penalized SDEs.

Accurate channel state information (CSI) prediction is crucial for next-generation multiple-input multiple-output (MIMO) communication systems. Classical prediction methods often become inefficient for high-dimensional and rapidly time-varying channels. To improve prediction efficiency, it is essential to exploit the inherent low-rank tensor structure of the MIMO channel. Motivated by this observation, we propose a dynamic mode decomposition (DMD)-based prediction framework operating on the low-dimensional core tensors obtained via a Tucker decomposition. The proposed method predicts reduced-order channel cores, significantly lowering computational complexity. Simulation results demonstrate that the proposed approach preserves the dominant channel dynamics and achieves high prediction accuracy.

We study the notion of tangent-like maps, which is a transcendental analogue of polynomial-like maps. We introduce a model family analogous to quadratic polynomials, with only one free asymptotic value, and define the "Tandelbrot set" as the analogue of the Mandelbrot set. We prove a Straightening Theorem for tangent-like maps, with uniqueness of the model map in the case where the filled-in Julia set is connected, and a parameter version of the Straightening Theorem for suitable holomorphic families of tangent-like maps. As a consequence, we prove the existence of topological copies of the Tandelbrot set in bifurcation loci of numerous families of meromorphic maps.

This paper proposes a light-emitting reconfigurable intelligent surface (LeRIS) architecture that integrates vertical cavity surface-emitting lasers (VCSELs) to jointly support user localization and mmWave communication. By leveraging the directional Gaussian beams and dual-mode diversity of VCSELs, we derive a closed-form method for estimating user position and orientation using only three VCSEL sources. These estimates are then used to configure LeRIS panels for directional mmWave beamforming, enabling optimized wave propagation in programmable wireless environments. Simulation results demonstrate that the proposed system achieves millimeter-level localization accuracy and maintains high spectral efficiency. These findings establish VCSEL-integrated LeRIS as a scalable and multifunctional solution for future 6G programmable wireless environments.

This article introduces a quantized chip-firing model with close connections to the theory of rational lattice paths and rational parking functions. Given a graph with a sink and positive integers a,b,c with gcd(a,b)=1, a set S of vertices fires by the following rule. Each vertex in S provisionally sends c chips to the sink and a/b chips to each non-sink neighbor outside of S. The novel feature is that the total number of chips leaving from or arriving at any vertex gets rounded down to the nearest integer before being finalized. We define the notions of chip configurations being superstable, k-stable, or k-skeletal in this model. When c=1 and the graph is complete, superstable configurations correspond to rational parking functions. There is a bijection between superstable configurations and k-skeletal configurations for each k. We establish these results by building a combinatorial theory of k-skeletal rational lattice paths (both unlabeled and labeled) and translating that theory to chip configurations. There is a group structure on the set of chip configurations modulo firing and borrowing moves. We show that this group is isomorphic to the product of b-1 copies of the integers modulo a; and, for each k, each coset of chip configurations in this group contains a unique k-skeletal representative.

We construct $p$-adic $L$-functions interpolating critical $L$-values of algebraic Hecke characters for arbitrary unramified primes $p$ and any totally imaginary field. For non-ordinary primes, the only previously known case was that of imaginary quadratic extensions of $\mathbb{Q}$. One of the main ingredients is a new $p$-adic Fourier theory relating generic fibers of $p$-divisible groups to a general class of character varieties. Combining this with equivariant cohomology classes constructed in a previous paper allows us to construct the $p$-adic $L$-function.

We design a deep reinforcement learning algorithm to explore a high-dimensional integer lattice with sparse rewards, training a feedforward neural network as a dynamic search heuristic to steer exploration toward reward dense regions. We apply this to the discovery of Fano 4-fold hypersurfaces with terminal singularities, objects of central importance in algebraic geometry. Fano varieties with terminal singularities are fundamental building blocks of algebraic varieties, and explicit examples serve as a vital testing ground for the development and generalisation of theory. Despite decades of effort, the combinatorial intractability of the underlying search space has left this classification severely incomplete. Our reinforcement learning approach yields thousands of previously unknown examples, hundreds of which we show are inaccessible to known search methods.

Foundation models for partial differential equations (PDEs) have emerged as powerful surrogates pre-trained on diverse physical systems, but adapting them to new downstream tasks remains challenging due to limited task-specific data and distribution shifts. While fine-tuning has proven transformative in natural language processing, best practices for adapting PDE foundation models remain underexplored. Although physics-informed training has successfully trained accurate solvers across a wide range of PDE problems, its potential for fine-tuning data-based foundation models has not been systematically studied. In this work, we introduce a physics-informed fine-tuning framework that adapts pre-trained PDE foundation models by incorporating physical constraints (PDE residuals and boundary conditions) directly into the fine-tuning objective. This enables effective adaptation in data-scarce regimes while promoting physical consistency. We evaluate our method on a downstream task composed of an unseen PDE class and compare it with data-driven finetuning counterparts. Our results demonstrate that physics-informed fine-tuning achieves competitive accuracy without requiring PDE solutions for training. Furthermore, a hybrid fine-tuning strategy yields superior generalization to out-of-distribution scenarios when only minimal training data is available. These findings establish physics-informed fine-tuning as a scalable and data-efficient paradigm, providing a physically interpretable pathway for adapting foundation models in scientific machine learning.

We construct a class of infinite-order multisoliton solutions of the Benjamin-Ono equation on the line, for which the initial data exhibits slow spatial decay. We prove that in the long-time asymptotics, such a solution decouples as an infinite superposition of independent soliton solutions with different velocities and no radiation term.

In this paper, we analyze a multiserver Markovian queue with customer abandonment i.e. the Erlang-A queue uder a novel framework, i.e. the random impulsive differential equations (RIDEs). This framework captures systems that evolve continuously while experiencing sudden, discrete interventions. The combination of such framework with Erlang-A queue give rise to multiple real-life applications, such as improving efficiency at call centers and designing optimal timing to apply quantum error correction in trying to shun away from decoherence in quantum computing. We derive closed-form expressions for steady-state amplitude bounds and average queue lengths under impulse, and we identify the impulse timings that optimize system performance.

We develop a Bohmian analysis of a two-dimensional ghost Hamiltonian and its mapping to the degenerate Pais-Uhlenbeck model. Using Gaussian wavepackets, we derive the corresponding guidance equations, the centre and width evolution, and the quantum potential. We use these quantities to characterise bounded, quasi-semiclassical, spiral, and runaway regimes. The Bohmian trajectories provide a direct dynamical diagnostic of coherence, packet deformation, and quantum-classical separation. We then compare a bi-Hamiltonian pair consisting of the ghost Hamiltonian and a classically equivalent alternative formulation. While the two descriptions produce identical classical trajectories, they lead to different Bohmian trajectories and different quantum potentials evaluated along those trajectories. This demonstrates that classical equivalence need not extend to Bohmian quantum dynamics and identifies a concrete quantum ambiguity in the degenerate higher-derivative system.

This paper characterizes self-oscillations in discrete-time linear time-invariant (LTI) relay feedback systems with nonnegative dead zone. Specifically, we aim to establish existence criteria for unimodal self-oscillations, defined as periodic solutions where the output exhibits a single-peaked period. Assuming that the linear part of system is stable, with a strictly monotonically decreasing impulse response on its infinite support, we propose a novel analytical framework based on the theory of total positivity to address this problem. We demonstrate that unimodal self-oscillations subject to mild variation-based constraints exist only if the number of positive and negative values of the system's loop gain coincides within a given strictly positive period, i.e., the self-oscillation is sign-symmetric. Building upon these findings, we derive conditions for the existence of such self-oscillations, establish tight bounds on their periods, and address the question of their uniqueness.

In this paper, we introduce the point-exact condition for a Koszul algebra $A$, which is useful for characterizing the (G1) condition of $A$ in the sense of Mori. Let $B = A/(f)$, where $f \in A_2$ is a regular normal element. We show that if $A$ satisfies the (G1) condition and is point-exact up to degree $\ell \geq 2$, then $B$ also satisfies the (G1) condition and is point-exact up to degree $\ell$. Moreover, we show that skew polynomial algebras satisfy the point-exact condition.