The type-II see-saw mechanism based on the annexation of the Standard Model by weak gauge triplet scalar field proffers a natural explanation for the very minuteness of neutrino masses. Noting that the phenomenology for the non-degenerate triplet Higgs spectrum is substantially contrasting than that for the degenerate one, we perform a comprehensive study for an extensive model parameter space parametrised by the triplet scalar vacuum expectation value (VEV), the mass-splitting between the triplet-like doubly and singly charged scalars and the mass of the doubly charged scalar. Considering all Drell-Yan production mechanisms for the triplet-like scalars and taking into account the all-encompassing complexity of their decays, we derive the most stringent 95% CL lower limits on the mass of the doubly charged scalar for a vast model parameter space by implementing already existing direct collider searches by CMS and ATLAS. These estimated limits are beyond those from the existing LHC searches by approximately 50-230 GeV. However, we also find that a specific region of the parameter space is not constrained by the LHC searches. Then, we forecast future limits by extending an ATLAS search at high-luminosity, and we propose a search strategy that yields improved limits for a part of the parameter space.

In this paper, we consider representations of integers as sums of at most four distinct $m$-gonal numbers (allowing a fixed number of repeats of each polygonal number occurring in the sum). We show that the number of such representations with non-negative parameters (hence counting the number of points in a regular $m$-gon) is asymptotically the same as $\frac{1}{16}$ times the number of such representations with arbitrary integer parameters (often called generalized polygonal numbers).

We study a model which generates Majorana neutrino masses at tree-level via low-energy effective operator with mass-dimension-9. Introduction of such a higher dimensional operator brings down the lepton number violating mass scale to TeV making such model potentially testable at present or near future colliders. This model possesses several new $SU(2)_L$ fermionic multiplets, in particular, three generations of triplets, quadruplets and quintuplets, and thus a rich phenomenology at the LHC. As the lepton flavour violation arises very naturally in such setup, we put constraints on the Yukawa couplings and heavy fermion masses from the current experimental bounds on lepton flavour violating processes. We also obtain 95% CL lower bounds on the masses of the triplets, quadruplets and quintuplets using a recent CMS search for multilepton final states with 137 inverse femtobarn integrated luminosity data at 13 TeV center of mass energy. The possibility that the heavy fermions could be long-lived leaving disappearing charge track signatures or displaced vertex at the future colliders like LHeC, FCC-he, MATHUSLA, etc. is also discussed.

We provide a written proof of a result due to H. Minakawa, which states that all suspension Anosov flows generated by hyperbolic matrices with positive trace are pairwise almost equivalent. The proof relies on constructing, for any given suspension flow, a genus-one Birkhoff section whose first-return map has fewer fixed points than the original map. We improve Minakawa's result by explicitly calculating the first return map onto this section, which leads to explicit bounds on the distances between suspension Anosov flows within the graph of Anosov flows.

The main result of this paper is, that if we suppose that a function is absolutely continuous and uniformly Hölder continuous and that its finite difference function does not oscillate infinitely often on a bounded interval, then the decay rate of its Fourier coefficients can be estimated exactly. This rate of decay predicts the same uniform Hölder continuity but the two other conditions are not necessary. Several examples from literature and by the author show that none of the assumptions can be relaxed without weakening the decay for some functions. The uniform Hölder continuity of chirps and the decay of their Fourier coefficients are studied. The main result is then applied in the estimation of the error of numerical Weyl fractional derivatives calculated using the discrete Fourier transform. The main result is also extended to Fourier transforms.

The Dirac-Moshinsky oscillator is an elegant example of an exactly solvable quantum relativistic model that under certain circumstances can be mapped onto the Jaynes-Cummings model in quantum optics. In this work we show, how to do this in detail. Then we extend it by considering its coupling with an external (isospin) field and find the conditions that maintain solvability. We use this extended system to explore entanglement in relativistic systems and then identify its quantum optical analog: two different atoms interacting with an electromagnetic mode. We show different aspects of entanglement which gain relevance in this last system, which can be used to emulate the former.

In this work we accurately consider quantum dynamical interaction (or entanglement) between single photon and usual photographic plate as photon coordinate detector in well-known experiments of single photon interference at beam splitter (and analogous experiments). Also, we here consider final measurement or detection procedure (collapse) as a quantum-classical continuous phase transition with spontaneous (non-dynamical) unitary symmetry (superposition) breaking (effective hiding). Practically, collapse as an effective (non-absolute) phenomena can be considered as an especial case of the general formalism of spontaneous symmetry breaking (with applications in many different domains of the physics, e.g. in elasticity of rigid bodies, quantum theory of ferromagnetism, quantum theory of electro-weak interactions as well as in chaotic inflation cosmology).

We investigate the representation of lattices as sublattices of the lattice of all convex subsets (intervals) of a linearly ordered set $(X,\le)$. We introduce the purely lattice-theoretic notion of a \textit{loc-lattice} and prove that every loc-lattice is representable as a lattice of intervals. Furthermore, we provide the complete, unabridged construction for the general representation theorem, establishing that a well-separated lattice is faithfully representable as a lattice of intervals if and only if it is a loc-lattice. Finally, we apply these results to general topology, obtaining novel algebraic characterizations for the bases of weakly orderable and completely orderable topological spaces.

Future integrated sensing and communication (ISAC) systems require simultaneous multibeam operation with low-latency hardware and robust isolation under synchronization error and fading. Conventional code-division multiplexing using Walsh-Hadamard codes is extremely time-sensitive. This paper demonstrates that conventional temporal-only coded multibeam arrays suffer from inter-beam sidelobe level (SLL) collapse to within a few dB of the main lobe, with variations exceeding 10-20 dB over delay. By embedding moderate-length Gold sequences into a spherical spatial codebook, the proposed Spherical-Gold scheme leverages both temporal and spatial correlation bounds, achieving effective inter-beam isolation without increasing RF complexity. Measurement results and verifications are performed using an Analog Devices ADAR3002 Ka-band 256-element receiver with four simultaneous beams. The proposed scheme demonstrates at least 15 dB rejection with less than 2.5 dB variation in SLL under time error and fading, whereas temporal-only CDMA degrades to approximately -5 to -7 dB SLL with nearly 8 dB variation under time delay.

Many biological processes exhibit oscillatory behavior. Among these, glycolytic oscillations have been extensively studied due to their well-characterized biochemical reaction networks. However, the complexity of these networks necessitates low-dimensional ordinary differential equation (ODE) models to identify core mechanisms and perform stability analysis. While previous studies proposed reduced ODE models, these were typically introduced from deterministic descriptions rather than the underlying stochastic dynamics, which more accurately represent discrete reaction events occurring at random times. In this paper, we develop a rigorous probabilistic framework for deriving a reduced Othmer-Aldridge model of the glycolytic pathway from its stochastic formulation. The full system is modeled as a multiscale continuous-time Markov chain with different time and abundance scales. Under an appropriate scaling regime and specific structural conditions, we prove that the dynamics of the slow components are approximated by a two-dimensional ODE. The proof is technically involved due to the network's complexity and strong coupling between its components. We further consider the problem of parameter estimation when observations are limited to the slow species: fructose-6-phosphate and ADP. The reduced system yields a tractable loss function depending solely on these variables. We prove that the resulting estimators are statistically consistent when the data originate from the full stochastic reaction network. Together, these results provide a mathematically rigorous framework linking stochastic biochemical reaction networks, reduced deterministic dynamics, and statistically reliable parameter estimation.

ATLAS J1138-5139 is a newly detected ultra-compact double white dwarf (DWD) system which is composed of a $1.02\,M_{\odot}$ carbon-oxygen white dwarf (CO WD) and a $0.24\,M_{\odot}$ helium (He) WD with an orbital period of about 27.68 min, making it one of the shortest-period DWD systems known. The future evolution and final fate of this system remain unexplored. In this work, we investigate the evolution of ATLAS J1138-5139 with the one-dimensional stellar evolution code Modules for Experiments in Stellar Astrophysics (MESA). We find that ATLAS J1138-5139 will evolve into an AM Canum Venaticorum (AM CVn) system in about \sim 6.3 Myr. Afterwards, the transferred material from the He WD companion start to build up to form a He shell near the surface of the CO WD. This accumulated He-shell masses can be up to approximately $0.12\,M_{\odot}$, which is likely to trigger a double-detonation (DDet) explosion of the CO WD. We therefore expect that ATLAS J1138-5139 will likely explode as a type Ia supernova eventually through the DDet explosion mechanism. Moreover, our calculations show that ATLAS J1138-5139 will be a promising target for gravitational-wave (GW) detection by future detectors like LISA, Tianqin and Taiji.

Against rising global loneliness, AI companions promise connection, yet accumulating evidence suggests that, for some users and contexts, intensive companion-style use can correlate with increased loneliness and reduced offline socialisation. This position paper challenges the dominant "AI as companion" paradigm by proposing a shift: from AI that simulates relationships with humans to AI that supports relationships between humans. We introduce Relational AI Translation, positioning AI as cultural-relational infrastructure that scaffolds human connection across cultural, generational, and geographical divides. Using first-generation East Asian migrants as a theoretically productive critical case, we outline a multi-agent architecture instantiating three translation operations: emotion-intent decoding, contextual reframing, and relational scaffolding. We articulate design provocations around measurement, safety architecture, and the tension between technological intervention and structural justice, and explicitly frame success as graduation toward renewed human-to-human support rather than sustained engagement with the system.

Live programming systems aim to quickly show programmers the dynamic impacts of program edits. To do so, they re-execute the program whenever it is edited, which poses a computational challenge when programs become large or complex. This has led to the need for incrementality in the implementation of live program interpreters. This paper introduces Chordata, an incremental program interpreter based on shortcut memoization, which learns repeated patterns of computation, called shortcuts, by observing executions of previous versions of a program. It can then apply these shortcuts when the same or a structurally similar program fragment is re-executed. This paper contributes a formal semantics of shortcut memoization for any language with a rewrite-based semantics, with mechanized proofs of key correctness properties. We then express a variant of the Hazel live programming system, expressed as a CEK machine, in Chordata, and develop a number of practical heuristics to learn high-value shortcuts. We evaluate the resulting system on editing traces of students solving simple programming problems. Chordata achieves a speedup of 13.03\times compared to baseline with a 19.97\times memory overhead. For smaller changes and for more complex programs, Chordata achieves even greater speedups. Furthermore, we show that Chordata is capable of providing a speedup even within a single execution, with a faster speedup on a larger input.

We report new results obtained in our lattice studies of the $Sp(4)$ gauge theory coupled to two fundamental Dirac fermions. This theory provides a candidate for the dynamical origin of dark matter models within the strongly interacting massive particle paradigm. We employ Lüscher's formalism to analyse finite-volume energy levels and study the scattering amplitude of two pseudoscalar states in the spin-1 channel. We present our preliminary findings for a set of ensembles generated within a broad range of (Wilson) fermion masses.

Work zone navigation remains one of the most challenging manoeuvres for autonomous vehicles (AVs), where constrained geometries and unpredictable traffic patterns create a high-risk environment. Despite extensive research on AV trajectory planning, few studies address the decision-making required to navigate work zones safely. This paper proposes a novel game-theoretic framework for trajectory generation and control to enhance the safety of lane changes in a work zone environment. By modelling the lane change manoeuvre as a non-cooperative game between vehicles, we use a game-theoretic planner to generate trajectories that balance safety, progress, and traffic stability. The simulation results show that the proposed game-theoretic model reduces the frequency of conflicts by 35 percent and decreases the probability of high risk safety events compared to traditional vehicle behaviour planning models in safety-critical highway work-zone scenarios.

Astronomical spectra, which encode rich astrophysical and chemical information, are fundamental to understanding celestial objects and universal laws. The advent of large-scale spectroscopic surveys, generating tens of millions of spectra, presents significant challenges for efficient data processing and analysis. To address these challenges, we develop an AI-powered platform (named ``SpecZoo'') for spectral visualization and analysis. This platform integrates modern information technology and machine learning to lower the barrier to spectral data utilization and enhance research efficiency. Its core functionalities include interactive visualization, automated spectral classification, physical parameter measurement, spectral annotation, and multi-band/multi-modal data fusion, all supported by flexible user and data management systems. It has become an essential tool for the National Astronomical Data Center, directly supporting spectral data processing and research for major projects including LAMOST, SDSS, DESI, and so on. Furthermore, the platform demonstrates strong potential for science-education integration, providing a novel resource for cultivating talent in astronomy and data science.

We introduce the \emph{Parry order} $\mathrm{Ord}_P(β)$, defined as the largest integer $n$ for which $β^n$ is a Parry number. This leads to a natural partition of the set of Perron numbers as follows: \[ \mathcal{P} = \left( \bigcup_{n \geq 0} H_n \right) \cup H_\infty, \] where $H_n$ is the class of Perron numbers with Parry order $n$, and $H_\infty = S \cup T$ consists exactly of all Pisot and Salem numbers. We show that a Perron number has infinitely many Parry powers if and only if it is Pisot or Salem. For every other Perron number, only finitely many powers can be Parry. We give an explicit upper bound on $\mathrm{Ord}_P(β)$ in terms of algebraic properties of~$β$. We provide explicit examples of non-Parry Perron numbers whose powers become Parry, demonstrating that several $H_n$ are non-empty and structurally rich. We give an infinite family of cubic non-Pisot numbers, all of which have finite Parry order, but where the family has unbounded Parry order. These results establish a new dynamical perspective on Perron numbers, connecting $β$-expansion theory with classical questions surrounding Salem numbers and Lehmer-type conjectures.

We provide a general notion of induced structures of operated algebras in the context of unary-binary operads. This notion fully captures the binary quadratic relations encoded by a unary-binary operad, thereby unifying and formalizing the various constructions that have appeared in the literature under the informal term of ``induced structures''. As an application, we show that the Novikov algebra and the recently introduced multi-Novikov algebra are the induced structures of the differential commutative algebra and the multi-differential commutative algebra respectively. We also explicitly determine the induced structure of the noncommuting multi-differential commutative algebra.

We investigate the lineshape of the $e^{+}e^{-}\to J/ψ\,π^{+}π^{-}$ cross section in the vicinity of the $D^{*}\bar{D}+c.c.$ threshold, where the ``so-called" $G(3900)$ is observed in the $e^+e^-\to D\bar{D}$ channel. To take into account the possible $D^{*}\bar{D}+c.c.$ open channel effects or possible contributions from $G(3900)$, we include the intermediate meson loop transitions in $e^{+}e^{-}\to J/ψ\,π^{+}π^{-}$. As a consequence, a triangle singularity (TS) is fulfilled which can produce nontrivial structures in the $j/ψπ$ invariant mass spectrum. Moreover, the TS transition also allows access to exotic quantum number of $(I,J^{P(C)})=(1,1^{-(-)})$ in the $J/ψπ$ invariant spectrum. We present predictions for the $J/ψ\,π$ invariant-mass spectrum and our results clarify the different manifestations of the kinematic effects and genuine resonances. In particular, we show that resonance structures arising from the $P$-wave $D\bar{D}$ scatterings or hidden charm tetraquark state with $(I,J^{P(C)})=(1,1^{-(-)})$ can be identified by the $J/ψπ$ invariant mass spectrum. It can provide a theoretical guidance for future experimental search for these exotic candidates.

Near-IR Paschen lines are potentially an excellent tracer of Type 1 AGNs that is hardly affected by dust extinction. JWST allows us, for the first time, to explore Paschen-line objects at redshift z>1. Here we present a study of 62 AGNs with broad Pa$α$ and Pa$β$ lines at 1<z<3 using data from the JWST COSMOS-3D program. These AGNs are efficiently selected and identified using NIRCam imaging and grism slitless spectroscopic data. We separate the AGN-host emission with image decomposition and quantify dust attenuation with multi-band data. We construct a calibration sample with optical spectroscopy and use single-epoch, Mg II-based black hole masses ($M_{\mathrm{BH}}$) as an anchor to derive new, Paschen-based $M_{\mathrm{BH}}$ estimators. We obtain three sets of $M_{\mathrm{BH}}$ estimators based on Paschen line luminosities and AGN continuum luminosities at 1 and 2 $μ$m, respectively. After dust corrections, they are well consistent with each other, and also broadly agree with previous results. With this AGN sample, we further construct the first Pa$α$ and Pa$β$ luminosity functions (LFs) of Type 1 AGNs. The derived LFs are 3-5 times higher than those of UV/optical-selected AGNs, indicating that Paschen-selected Type 1 AGNs are more complete. In addition, the intrinsic properties of our AGNs show no dependence on dust reddening, suggesting that the observed reddening is unrelated to the central engine and is thus likely caused by line-of-sight obscuration.

Turbulent heat and freshwater transport at ice-ocean interfaces controls glacier and iceberg melt rates, yet the underlying physics remains poorly constrained. Parameterizations that assume shear boundary layer scaling are commonly used, which neglects meltwater buoyancy-driven convective processes. Using Direct Numerical Simulations with realistic salt diffusivity, which is critical for representing the thin solutal boundary layer (deltaS ~ 0.4 mm) and resulting convective instabilities, we investigate ice-ocean boundary layer turbulence across varying temperature, salinity, stratification, external velocity, and interfacial slope angles. Our simulations agree with laboratory measurements of melt rate and interfacial temperature. In the absence of external flows, we find no transition from buoyancy-controlled to shear-controlled regimes and convection is important even at near-horizontal slopes. External shear becomes significant only when it is strong enough to thin the thermal and solutal boundary layers, which starts influence melting substantially above background flow speeds of 5 cm/s. Understanding how shear and convection compete to determine the ice-ocean diffusive boundary layer enables accurate melt rate predictions across the parameter space relevant to ice shelves and marine-terminating glaciers.

Vera C. Rubin Observatory's Prompt Processing system will automatically process 10 TB of raw images to produce up to 10 million transient alerts per night. We summarize how Prompt Processing meets its throughput, latency, and reliability requirements and present results from Rubin Observatory Commissioning.

We prove the Davies-Gaffney (i.e., integrated Nash-Aronson) type diffusive upper bounds on the propagators of parabolic equations in $L^p$-sense for all $1\le p\le\infty$. Our approach is based on a simple exponential deformation argument that does not require hypoellipticity. It provides a unified approach to diffusive upper bounds that covers a wide class of problems including degenerate, non-autonomous, and non-linear equations.

Governing equations are fundamental for describing and predicting dynamic urban geographic systems. Unlike physical systems guided by first principles, urban spatiotemporal phenomena emerge from coupled geographic processes that lack deterministic theoretical foundations, making the discovery of governing equations elusive and largely heuristic. Spatiotemporal dynamics in urban systems are often observed as sequential snapshot data of spatial distribution, while the cause of such dynamics is often implied or unknown. In this study, we propose a unified differential equation formalism that decomposes urban dynamics into a time-invariant spatial interaction process and a self-dynamic component. Building on this formalism, we introduce the Urban Discovery Framework (U-Discovery), which integrates hypothesis generation, neural fitting, and governing equation identification for the discovery of governing spatial interaction laws. U-Discovery leverages Large Language Models and literature-based reasoning to propose differential equation candidates. Each candidate was calibrated from the observed spatiotemporal dynamics using a neural fitting method. The candidates are evaluated and ranked based on the fitting error and mathematical complexity. Our synthetic experiments prove that U-Discovery can find the sole governing equation from the simulated dynamics. Empirical experiments in Hennepin County, Minnesota, further demonstrate the potential of U-Discovery in identifying optimal governing laws from real-world human activity dynamics.

Wildfires require governments to communicate under conditions of urgency, uncertainty, and intense public scrutiny, yet such communication now unfolds within a digitally mediated environment shaped by polarization and engagement-based amplification. We analyze over 1.3 million wildfire-related social media posts from California (2016-2025) to examine how institutional actors are evaluated within this landscape. Users' stance toward government is actor-specific: individual political officials are discussed more negatively than operational agencies across federal, state, and local levels, and this gap widens during extreme wildfire events. Moreover, interaction networks become increasingly modular over time, consolidating into polarized communities in which negativity concentrates within cohesive clusters. Engagement-weighted measures show that highly interactive negative content disproportionately shapes visible discourse, while crisis periods redirect attention from emergency agencies to high-profile political figures, reinforcing reputational divergence. These findings indicate that wildfire communication operates within a polarized, engagement-ranked ecosystem in which evaluative tone, network structure, and visibility dynamics jointly shape institutional perception. Effective disaster communication should therefore account for the structural conditions of contemporary digital public communities.

We study carrier transport, distribution, and recombination in dual-color c-plane InGaN/GaN LEDs using spectral analysis and small-signal electroluminescence (SSEL). The emissions from green and blue quantum wells (QWs) were experimentally separated and analyzed. Spectral analysis and SSEL independently demonstrate that emission from the green QW is dominant at low current densities due to its narrower bandgap, while emission from the blue QW is more significant at higher current densities due to its reduced quantum confined stark effect (QCSE) and larger wavefunction overlap. In addition, we demonstrate that the carrier recombination in the QWs is non-uniform, with carrier transport dramatically affecting the carrier distribution between QWs and the recombination in a specific QW. Finally, we also show that the effective active region in these V-pit-engineered InGaN/GaN LEDs is roughly 2 to 3 QWs on the p-GaN side, with limited interwell carrier transport and recombination in additional QWs.

We propose a market design for real-time electricity markets that utilizes a two-layered dispatch mechanism to systematically incorporate carbon accounting into grid operations. In this mechanism, ``dispatch'', the centralized allocation of generation resources to meet system load, is executed via a hierarchical structure where the first layer minimizes financial costs to maintain economic efficiency, while the second layer minimizes system emissions strictly within the set of cost-optimal solutions. We define locational marginal emissions (LMEs) as the marginal rate of system emissions derived from the dual variables of the two-layered formulation. Unlike standard marginal prices which correspond to right-hand-side constraint relaxations, LMEs must account for the requirement of economic optimality which introduces demand parameters into the problem's constraint structure. Under the framework, we establish that LMEs satisfy properties analogous to the first and second fundamental theorems of welfare economics. We prove that (1) decentralized ``carbon profit'' maximization by individual grid entities guarantees a system-wide emission profile consistent with the economic dispatch, and (2) any optimal low-carbon economic dispatch is supported by a corresponding set of LME signals acting as a decentralized equilibrium. Furthermore, we establish a general carbon accounting theorem, called the Carbon Footprint Theorem, showing that these market-consistent LMEs ensure the sum of carbon accounts across all grid components (loads, generators, transmission, and storage) equals the total physical carbon emissions. This completes the theoretical foundation of the LME. Finally, we investigate and validate the empirical properties of LMEs and LME-based carbon accounting through case studies on a realistic Texas grid model.

Shape-morphing solutions (SMS) refer to a class of approximate solutions of partial differential equations (PDEs) with the distinguishing feature that they depend nonlinearly on a set of time-dependent parameters. They generalize Galerkin truncations by allowing the basis (or trial functions) to evolve in time in order to adapt to the solution of the PDE. As such, SMS are particularly suitable for reduced-order modeling as well as high fidelity simulation of multiscale systems which exhibit localized time-dependent features, such as vortices, dispersive wave packets, and shocks. Furthermore, being mesh-free, SMS is scalable for solving PDEs in higher spatial dimensions. As a special case, SMS allows the approximation of the PDE's solution by a neural network whose weights and biases depend on time. Such neural networks are known as evolutional neural networks or neural Galerkin schemes. The evolution of SMS parameters is dictated by the SMS equation, a set of ordinary differential equations derived from the Dirac-Frenkel variational principle. Over the past five years, contributions to the theory and computation of SMS have been growing rapidly. Here, we survey these developments, showcase some applications of SMS, and highlight important open problems for future research. At the same time, this review is structured to serve as a tutorial for applied mathematicians, physicist, and engineers who wish to enter this field.

We introduce homotopy lattice gauge fields (HLGFs), a version of gauge fields over a discretized base, based on a notion of higher parallel transport that enriches the usual parallel transport along paths on a lattice to also consider higher dimensional paths. Higher dimensional data keeps information about the parallel transport along homotopies of curves. With this data, a HLGF on a base space of dimension two or three determines a principal bundle over the base manifold. This data is also responsible for our formulas for the topological charge on two-dimensional bases. Our framework is an application of a nonabelian algebraic topology framework developed to solve the local to global problem in higher dimensional homotopy. No previous knowledge of higher category theory is assumed. The second part will be devoted to the space of fields as an arena for doing Quantum Field Theory, and to give the first examples of how our framework refines standard lattice gauge theory.

This note establishes a theoretical framework for finding (potentially overparameterized) approximations of a function on a compact set with a-priori bounds for the generalization error. The approximation method considered is to choose, among all functions that (approximately) interpolate a given data set, one with a minimal Lipschitz constant. The paper establishes rigorous generalization bounds over practically relevant classes of approximators, including deep neural networks. It also presents a neural network implementation based on Lipschitz-bounded network layers and an augmented Lagrangian method. The results are illustrated for a problem of learning the dynamics of an input-to-state stable system with certified bounds on simulation error.