Papers

RaNNDy is a randomized neural network architecture for the data-driven approximation of transfer operators associated with complex dynamical systems. The weights and biases of the hidden layers of the network are randomly initialized and kept fixed, only the output layer is trained. This has several advantages over fully optimized neural networks, notably a closed-form solution for the output layer and significantly lower training costs. Despite these advantages, RaNNDy is restricted to the initial selection of weights and biases that parametrize the basis functions required for the operator approximation. Since the basis functions are determined by the activation function, choosing an appropriate activation function for the hidden layers is crucial. In this work, we propose an algorithm that optimizes the activation function itself, while keeping the weights and biases in the randomized neural network fixed, providing a more suitable dictionary. We illustrate the efficacy of the approach using various benchmark problems, including stochastic differential equations and random walks on graphons.

We prove a tropical analogue of Cartan's second main theorem for holomorphic curves intersecting hyperplanes in general position–a setting that was not fully resolved by previous tropical Nevanlinna theory. Two versions are obtained. The first (Theorem 1.7) requires subnormal growth and involves the tropical Casorati determinant. The second and main version (Theorem 1.9) is completely free of growth conditions and exceptional sets; it replaces the Casorati term by the sum of the counting functions of the curve's components, yielding an inequality valid for every r. The proof uses a tropical Cramer theorem, bypassing the logarithmic derivative lemma. This improves upon previous results by Korhonen-Tohge and Cao-Zheng, where the coefficient could be suboptimal even under the general position hypothesis. We also clarify the relation between different notions of linear independence, and present the first counterexample to the truncated second main theorem in the tropical setting (Example 5.4).

In this paper, we investigate an intelligent reflecting surface (IRS)-assisted multi-user system, where the base station (BS) employs rotatable antennas (RAs) and the IRS can adjust the panel orientation.To alleviate the severe multiplicative path loss of the cascaded channel, the IRS is deployed near the BS, while the user-BS and user-IRS links remain in the far field. We formulate a sum-rate maximization problem by jointly optimizing the receive beamforming, IRS phase shifts, BS antenna boresights, and IRS panel orientation. To tackle the resulting highly coupled and non-convex problem, we first study a single-user case to reveal the structure of the dual-rotation gain, which is shown to be multiplicatively separable in the far field but coupled in the near field. For the general multi-user case, we develop an alternating optimization algorithm, where the receive beamforming is updated in closed form, the IRS phase shifts are optimized by an FP-assisted Riemannian conjugate gradient method, and the BS antenna boresights and IRS panel orientation are updated via projected gradient methods. Simulation results demonstrate the significant sum-rate gains achieved by the proposed coordinated rotation design over fixed-orientation and single-rotation benchmark schemes, and provide useful insights into near-field dual-rotation design.

In this paper, we propose a tri-domain reconfigurable multiuser multiple-input multiple-output (MIMO) communication system that integrates the electromagnetic (EM) reconfigurable antenna (EMRA) with the spatially movable antenna (SMA), termed the spatial-EM reconfigurable antenna (SEMRA). The proposed system offers EM, spatial, and digital domain degrees of freedom (DoFs) for joint channel reconfiguration, yet introduces new challenges in channel estimation (CE) and precoding optimization. Specifically, for multiuser orthogonal frequency division multiplexing (OFDM) downlink, the precoding design is formulated as a tri-domain optimization problem over antenna positions, EM-domain radiation-pattern weights, and digital precoders. We first develop a zero-forcing (ZF)-based baseline algorithm to decouple the design of spatial reconfiguration, and then propose a weighted minimum mean square error (WMMSE)-based tri-domain joint optimization algorithm for further improving the spectral efficiency (SE). Furthermore, we propose a low-overhead movement-aided channel estimation scheme in which coordinated antenna repositioning across pilot slots synthesizes a denser virtual array, enabling more accurate angle-of-departure (AoD) estimation and EM-domain channel state information (eCSI) reconstruction under the same per-user pilot overhead as the EMRA baseline. The resulting parametric representation enables eCSI assembly at desired antenna positions without additional pilots. Simulation results show that the proposed CE scheme improves eCSI estimation accuracy and the proposed SEMRA achieves higher SE than the EMRA baseline under the same pilot overhead.

In this paper we completely classify semifields of order $2^8=256$ containing a nucleus of order $2^4=16$. We introduce new invariants for semifields, and apply new computational techniques for calculating old invariants. Together these make the computational classification significantly quicker.

The geometrisation theorem of 3-manifolds was conjectured by Thurston the 1980s and proved by Perelman in the 2000s. This is an overview on the subject. We explain the content of the theorem and describe its effects in various situations.

The moment-sum-of-squares hierarchy provides a powerful framework for solving non-convex optimal control problems by constructing a sequence of convex semidefinite relaxations. However, when extending these methods to nonlinear partial differential equations (PDEs), a fundamental challenge is the potential existence of a relaxation gap, where the solution to the linear measure formulation using occupation measures fails to correspond to a classical physical solution of the original PDE. In this paper, we prove the absence of a relaxation gap for scalar semilinear parabolic PDEs of the reaction-diffusion type. We do so by showing that each solution to the linear measure equation gives rise to an energy measure-valued (emv) solution in the space of Young measures satisfying suitable energy identities. We then prove that any such emv solution concentrates on the solution to the nonlinear PDE, provided the latter exists and is unique. To the best of our knowledge, this is the first concentration result of this kind for measure-valued solutions of reaction-diffusion PDEs.

In scheduling problems, deterministic task durations are often assumed. This usually does not capture reality and may lead to schedules that are not robust to (small) changes to these task lengths. The use of stochastic task durations therefore seems preferable. Including these in local search, which is the way to find good solutions for difficult scheduling problems, is not straightforward, though. The objective value becomes stochastic then too, and computing the expected value is often not possible. One way out of this it to approximate this value by using simulation. This is quite easy to implement in a local search algorithm, but it may require many simulations each iteration to get a reliable estimate. Hence such an approach comes with a performance penalty. In this paper, we study techniques to limit the number of simulations. Besides comparing known techniques, we propose our own method for this, which is based on $t$-tests. We evaluate these techniques on the Stochastic Parallel Machine Scheduling Problem and the Stochastic Electric Vehicle Scheduling Problem. In these case studies, we show the effectiveness of using such methods to reduce runtime while retaining solution quality. Our method using $t$-tests turns out to be most effective in both problems.

Real-time hierarchical energy-sharing markets are promising to coordinate large numbers of prosumers. Still, most existing clearing methods rely on linearized or DC power-flow models and do not explicitly handle reactive power or voltage-security constraints. With AC network constraints, the problem becomes a large-scale bilevel Mathematical Program with Equilibrium Constraints (MPEC) that is difficult to solve in real time. This paper develops a non-iterative clearing algorithm for two-layer energy-sharing markets with voltage constraints. We first derive an efficient best-response function for each lower-layer energy-sharing market and reduce the equilibrium search to one dimension by exploiting the pricing-coupling structure. We then embed this function into the upper-layer network-constrained problem and reformulate the bilevel MPEC as a single-level mixed-integer second-order cone program (MISOCP), which is computationally tractable. Case studies on the IEEE 123-bus system with 12,300 prosumers show that the proposed method preserves nodal voltages within prescribed limits and delivers solutions with maximum errors below 0.01% in 0.829 s.

Various decays of the B mesons are here used to establish the performances of an ultra-granular electromagnetic calorimeter for heavy flavour physics at an electron positron accelerator running at the Z peak. The silicon-tungsten electromagnetic calorimeter of the ILD concept is used for this purpose, enhanced by a timing capability. When possible, a $\pi$ 0 mass fit of $\gamma$$\gamma$ system is performed to improve the $\pi$ 0 energy resolution. It is also shown that in the presence in the final state of a photon without a $\pi$ 0 , the identification of genuine photon(s) versus fake photon(s) coming from K L 's, neutron's, debris of hadronic shower or other high energy $\pi$ 0 , is essential. It allows for a good precision and a good signal over background ratio for this kind of physics. The possible impact for the tau physics is discussed.

In this paper, we establish optimal a priori $C^{1,\alpha}$ regularity estimates for the ratio $w = v/u$ of two solutions to the same elliptic equation $-\operatorname{div}(A \nabla u )=0$ with Lipschitz coefficients $A$, under the assumption that their nodal sets satisfy $Z(u) \subseteq Z(v)$. We specifically address the case where the zero set $Z(u)$ exhibits a product structure of $2$-dimensional nodal sets, namely $Z(u)=Z(u_1)\times \cdots \times Z(u_{m})$, where the $u_i$ are $2$-dimensional functions. This result extends the regularity estimates previously proved in dimension $2$ by [Logunov and Malinnikova, 2016] and by [Terracini, Tortone, and Vita, 2026].

We study the small-time approximate controllability of bilinear Schrödinger equations, where the drift is a magnetic Schrödinger operator and the control is an electric potential. We prove this property in two circumstances: (i) in $\mathbb{R}^d$, with a quadratic and an additional generic bounded electric potential in the control, and with a uniform magnetic field in the drift; (ii) in $\mathbb{R}^d$ or $\mathbb{T}^d$, with control electric potentials supported on a finite number of Hermite or Fourier eigenfunctions, and with any differentiable magnetic potential in the drift.

An output feedback model predictive control (MPC) framework with adaptive tubes is proposed for linear time-invariant systems subject to parametric and additive uncertainties. An adaptive observer provides point estimates of the system state, model parameters, and initial condition, while jointly updating the corresponding sets containing the true parameters and initial state. These estimates parameterize the constrained optimal control problem, enabling constraint tightening, terminal ingredients, and tube geometry to be updated as the estimates evolve. In contrast to standard robust tube-based MPC formulations, the proposed approach does not require a common quadratically stabilizing linear feedback gain across the parametric uncertainty set. As the available uncertainty information improves, the tube geometry evolves accordingly, resulting in an adaptive tube MPC framework with improved performance over time. Recursive feasibility and robust exponential stability are established, and a numerical example is presented.

We study an excursion-dependent generalization of inverse-subordinator time changes for recurrent regenerative processes. Subordinators from an independent family are assigned to measurable classes of excursions and to the distinguished state at which the process is regenerative, and the random clock is constructed from the corresponding occupation times. Although the time-changed process is not necessarily regenerative or Markovian, we show that its transformed excursion point process is again Poisson and is obtained from the original excursion point process by a marking-and-mapping procedure. We also identify the transformed stagnancy component, on the original local-time scale, as a subordinator independent of this point process. We use this representation to study occupation- and waiting-time asymptotics. Under regular variation assumptions on the tails of the Lévy measures of the cumulative excursion lifetimes and stagnancy components, we prove a multiscale joint occupation-time limit theorem, including generalized arcsine laws and Darling–Kac-type limits. We further obtain joint scaling limits for the last and next visit times to the distinguished state.

Let $(X,d)$ be a nonempty metric space and let $n\in \mathbb N^+$. We shall say that $T\colon X\to X$ is a graphic contraction of order $n$ if there exists $\alpha\in (0,1)$ such that the inequality $$ d(T^n x,T^{2n}x) \leqslant \alpha d(x,T^nx) $$ holds for all $x\in X$. In the case $n=1$ these mapping are known as graphic contractions and are well studied. In the present paper, we establish a theorem on the existence of periodic points for a graphic contraction of order $n$. Examples of such mappings having different properties are constructed.

Most existing integrated sensing and communication (ISAC) studies focus on enabling a base station (BS) to support sensing and communication over shared resources through advanced waveform design and power allocation. In contrast, the object-side perspective remains underexplored. For example, an object may wish to remain difficult to detect for security reasons, while another object in close proximity may generate dominant reflections that confuse the BS and impair sensing reliability for the intended target. These challenges motivate the fluid antenna system (FAS) paradigm which introduces a reconfigurable spatial degree of freedom (DoF) that can reshape sensing signatures via port selection, beyond what waveform and power control alone can provide. In this paper, we devise diffusion FAS, a generative artificial intelligence (AI)-driven framework that exploits spatial agility to steer ISAC performance over the electromagnetic fading manifold. Instead of optimizing ISAC solely in the power domain, diffusion FAS casts ISAC as a dynamic spatial selection problem in which antenna states (i.e., ports) are chosen to shape sensing signatures while maintaining communication objectives. To work under sparse measurements, we employ a conditional denoising diffusion probabilistic model (DDPM) to reconstruct the latent spatial correlation structure from a small set of observed ports, enabling efficient exploration of the reconfigurable aperture. We demonstrate two FAS-enabled ISAC modes: (1) generative spatial stealth, which identifies localized deep fades to suppress a target's sensing visibility by up to two orders of magnitude, and (2) target isolation, which synthesizes spatial nulls that reject interference from adjacent objects.

Graph coloring with preferences offers a powerful framework for constraint satisfaction problems in which fulfilling every request is impossible but satisfying a guaranteed positive fraction is highly desirable. A request on a graph $G$ equipped with a list assignment $L$ assigns to each vertex of some subset $dom(r)\subseteq V(G)$ a preferred color from its list. Following Dvořák, Norin, and Postle (2019), $G$ is \emph{$\varepsilon$-flexibly $k$-choosable} if, for every $k$-list assignment $L$ and every request $r$, there is an $L$-coloring of $G$ that agrees with $r$ on at least $\varepsilon|dom(r)|$ vertices. The corresponding notion for DP-coloring (correspondence coloring) was formalized by Bradshaw, Choi, and Kostochka (2025). Choi, Clemen, Ferrara, Horn, Ma, and Masařík (2022) proved that every planar graph without $4$-cycles and with $3$-cycle distance at least $2$ is $\varepsilon$-flexibly $4$-choosable. We improve the result in two respects: weakening the hypothesis from $3$-cycle distance $\geq 2$ to vertex-disjoint triangles, and strengthening the conclusion from list flexibility to weighted DP-flexibility: \emph{Every simple planar graph without $4$-cycles and without intersecting triangles is weighted $\varepsilon$-flexibly DP-$4$-colorable.} The list size $4$ is sharp: Montassier, Raspaud, and Wang constructed a planar graph without $4$-cycles, $5$-cycles, and intersecting triangles that is not $3$-choosable.

We study intersection size distributions arising in finite geometries and vector spaces. For a projective plane $\Pi_q$ of order $q$ with line set $\mathcal L$, we show that \[ \min_{S\subseteq\Pi_q}\max_k |\{\ell\in\mathcal L:|S\cap\ell|=k\}|=\Theta(q^{3/2}). \] The arguments behind these bounds admit a more general formulation. We prove a similar lower bound for balanced incomplete block designs, together with probabilistic upper bounds for uniform block systems: one for fixed block size under a weak dependency condition and another for growing block size under bounded pairwise intersections. This yields corresponding results for subspaces of vector spaces over finite fields, equivalently, for affine subspaces of an affine space. We also study explicit constructions in affine planes of prime order and relate their intersection size distributions to character sum estimates; an elliptic curve construction gives a bound within a polylogarithmic factor of the optimal order. Finally, we relate balanced intersection size distributions to legitimate colorings and prove that every n-uniform linear hypergraph with $n$ edges admits a legitimate 2-coloring.

The problem of computing the cardinality of the intersection of multiple balls in the Hamming space has attracted a lot of attention recently due to their applications in the list reconstruction problem and information retrieval in Associative Memories. In previous work, most of the results are for the cases where the radii of each ball, $r$ and the distance between the centers of these balls, $k$ are fixed when the length $n$ of each codeword tend to infinity. In this work, we focus on the case where $r = \alpha n$ and $k=\beta n$ for some constants $\alpha$ and $\beta$ and compute the maximum asymptotic rate of the cardinality of the intersection of three balls. We provide the maximum asymptotic rate as a function of two parameters $\alpha$ and $\beta$. We also provide numerical results and compare these results with the intersection of two balls.

In this article we provide an integration formula enabling us to integrate random variables defined on the moduli space of hyperbolic surfaces which involve the lengths of closed geodesics of any prescribed topological type, at fixed genus and number of boundaries. We then compute the general form of length functions of closed loops in Fenchel-Nielsen coordinates. Using this expression together with our integration formula, we express the expectations of geometric random variables as integrals over R with almost polynomial density. As a consequence, given an arbitrary closed loop $\gamma$, we obtain an improvement of Mirzakhani's asymptotic equivalent of the Weil-Petersson expectation E[N$\gamma$(a)], when a $\rightarrow$ $\infty$, of the number of geodesics with same topological type as $\gamma$ of length at most a. We finally provide, for a loop $\gamma$ filling a one-holed torus or a four-holed sphere, an interpretation of the density function associated with $\gamma$ as the period of the Hamiltonian flow of the length function of $\gamma$.

We identify the dominant part of the Frenkel-Reshetikhin $q$-character with a natural invariant arising from the Langlands/Zelevinsky parameterization for affine Hecke algebras. We introduce the reciprocal character of a module over a $GL_n$-type affine Hecke algebra, defined in terms of multiplicities within parabolic restriction. The main theorem claims that the reciprocal character matches, under quantum affine Schur–Weyl duality, with the dominant $q$-character for finite-dimensional modules over quantum affine algebras. This result gives a type $A$ realization of the Nakajima expectation that the dominant monomials in the $q$-character should play the role of monomial-basis coordinates in Lusztig's framework for finite quantum groups. Indeed, under the affine Hecke categorification of $U_q(\mathfrak{sl}_\infty)^+$, we prove that the reciprocal character is the specialization at $q=1$ of the coordinate map attached to a monomial basis. As a consequence, dominant $q$-character multiplicities for simple (or standard) modules are described by transition coefficients between monomial and canonical (or PBW) bases. Our methods rely on the development of explicit tableau-counting formulas for such dominant multiplicities, or equivalently for the reciprocal characters of standard modules over affine Hecke algebras.

This paper investigates the optimal transport problem within the framework of Linear Quadratic optimal control systems. We establish the well-posedness of the Monge problem and analyze the regularity of the resulting optimal transport map, extending the results obtained in [Hindawi, Pomet, Rifford, 2011] for non-negative costs. Furthermore, we study the displacement interpolation of measures and derive general interpolation inequalities for entropy functionals. Our analysis is motivated by the role of these systems as natural model spaces for comparison theory in sub-Riemannian geometry.

We prove several abstract results giving general conditions under which subspaces of linear or multilinear operators on Banach spaces or Banach lattices are closed. Each of these abstract results is followed by concrete applications, concerning classes of linear/multilinear operators already studied as well as new classes.

In 1948, Erdös and Straus formulated a conjecture : for any positive integer $n>2$, there exist positive integers $n_1,n_2$ and $n_3$ such that \begin{equation}\frac{4}{n}=\frac{1}{n_1}+\frac{1}{n_2}+\frac{1}{n_3},\nonumber\end{equation} which is still open. It is known that one only needs to prove the conjecture for any prime number $n$ such that $n\equiv 1\;(\mbox{mod}\;24)$. If $n=24m+1$ and $n_1\leq n_2,n_3$, then $n_1=6m+k$ with $1\leq k\leq 12m$. A solution $(n_1,n_2,n_3)$ of the above equation is called a tame solution if $n_2$ and $n_3$ are factors of $(6m+k)(24m+1)$. We call $n=24m+1$ wild if it does not have any tame solution. In this paper, we derive twenty-eight families of tame solutions of the above equation. Numeric test shows that they cover all the tame primes.

This paper studies the validity of Stokes' theorem for differential subcomplexes naturally adapted to the noncommutative geometry of positively graded Lie groups, with particular emphasis on Carnot groups. We introduce geometric conditions under which Stokes-type formulae hold for the Rumin complex and for a new family of spectral complexes associated with the homogeneous weight filtration of the de Rham complex. In particular, the spectral complexes allow us to recover the validity of Stokes' theorem on locally smooth intrinsic graphs. This is achieved by showing that the corresponding Stokes' formulae are governed entirely by the degree of the underlying submanifolds. Our approach also reveals that both the Rumin complex and the spectral complexes can be interpreted directly in terms of the classical de Rham complex through the Leibniz rule and integration over suitable classes of submanifolds, namely R-manifolds and spectral manifolds, respectively. Finally, motivated by this interaction between homogeneous weights and degrees of submanifolds, we propose a notion of current naturally adapted to these subcomplexes.

We introduce a model for neural scaling laws under sparse activations. In the model, test loss is often dominated by rare coordinates that are never observed in the training input. This mechanism induces a novel bottleneck absent from dense models. We derive the asymptotic population loss in both the underparameterized and overparameterized regimes, and show that the loss exhibits a double-descent peak near the interpolation threshold – where the number of parameters is just sufficient to fit the training data – resulting in a loss curve governed by two distinct scaling exponents – one for the overparameterized regime and one for the underparameterized regime – with a gap determined by the degree of sparsity. Additionally, we derive a compute-optimal frontier that favors increasing dataset size over model capacity under fixed compute budgets. We also analyze gradient-descent dynamics and identify a scaling law for the probability that fixed-step gradient descent becomes unstable. We further show that the sparsity-induced effect persists under nonlinear activations.

This paper addresses the generation expansion planning (GEP) problem, formulated as a mixed-integer linear programming model with intertemporal storage constraints. Being generally NP-hard, the problem's computational complexity grows sharply with the planning horizon and the number of binary variables. While previous research has tackled this challenge using heuristic time series aggregation (TSA) methods, we propose a theoretically grounded marginal-cost-based TSA, designed to construct an aggregated model that preserves the active constraints of its full-scale counterpart, thereby explicitly targeting exact temporal aggregation. This TSA method is embedded within solution algorithms that iteratively refine theoretically validated bounds on the maximum error introduced by the temporal aggregation relative to conventional full-scale optimization, thus offering a formal performance guarantee to the decision-maker. Numerical results highlight the computational advantages of the proposed algorithms, which notably recover tractability whereas full-scale optimization proves intractable.

We propose a framework for suboptimal model predictive control (MPC) based on the interconnection of monotone dynamical systems, such as port-Hamiltonian systems. In contrast to classical MPC formulations, where the optimizer is treated as an instantaneous mapping, we model both the plant and the optimizer as dynamical systems and couple them through a structured interconnection. This leads to a continuous-time closed-loop formulation governed by (quasi-)monotone operators. Within this setting, we establish well-posedness of the coupled optimizer-plant dynamics and provide a unified interpretation of suboptimal MPC schemes. In particular, we reveal a direct connection between iterative optimization algorithms and dynamical control systems theory by showing that standard suboptimal MPC algorithms can be understood as time discretizations of the underlying continuous-time dynamics via operator splitting methods.

Bogoliubov's 1947 approximation, originally developed in the microscopic theory of superfluidity, laid the foundation for solving previously intractable quantum models and later became part of "quantum mathematics". Regarding mathematically rigorous results, one of its most advanced forms - the only one that handles quantum equilibrium states - was published in the Memoirs of the AMS in 2013. Building on key results from convex analysis, the present work significantly extends it to obtain a general mathematical theory that enables nonlinear variational problems on convex compact spaces to be fully studied via a linearization process, referred to here as the "Bogoliubov linearization". This problem is particularly timely, given the current development of quantum algorithms and computers, which are inherently linear machines. A deep connection with the optimal transport is also proven. As a paradigmatic example of application, the approach proposed here is applied to the nonlinear thermodynamic formalism - an emerging field that can have important impacts on various fields of mathematics, such as ergodic transport, the fractals and multifractal formalism, discrete-time linear dynamics, C*-algebras, etc. Notably, even in the case of finite alphabets the obtained results go beyond the scope of the existing literature in nonlinear thermodynamic formalism.

We present an open problem about non-colliding freely moving hard disks in the Euclidean plane, together with related positive and negative partial results. The open problem is stated in a non-degenerate form: velocities are required to be pairwise distinct and their speeds are required to be uniformly bounded away from infinity. The positive deterministic result gives a bounded, injective, non-colliding velocity assignment for the integer lattice; after a common velocity shift, the speeds are also bounded away from zero. The negative result shows that no bounded continuous vector field on the whole plane can serve as a universal assignment satisfying the same separation inequality for all pairs of points at distance greater than one. We also record a space-time interpretation of the problem, relate it to packings by nonparallel cylinders in three dimensions, and formulate a corresponding topological-dynamical question for cylinder packings.