The semiring of discrete dynamical systems is a simple algebraic model for modularity in deterministic systems. The objects of the semiring are finite transformations (viewed as directed graphs and regarded up to isomorphism), the sum of two transformations corresponds to applying them independently on distinct sets, and the product corresponds to applying both transformations in parallel. In this paper, we extend this semiring to include partial transformations; the sum and product are natural generalisations. Each (partial) transformation can be viewed as a sum (over $\mathbb{N}$) of connected (partial) transformations. We generalise this idea by working in semirings of formal sums over any semiring $\mathbb{S}$. Here we consider the case where $\mathbb{S} = \mathbb{F}_2$, the binary field, and we focus on injective partial transformations, i.e. sums of chains and cycles. While no efficient algorithm for the division problem for sums of cycles in the original semiring of discrete dynamical systems is known, we give a concise characterisation of all the solutions of the division problem for sums of cycles over $\mathbb{F}_2$. We then extend this characterisation to dividing any injective partial transformations, i.e. sums of chains and cycles over $\mathbb{F}_2$.

We study the integral means spectrum associated with the analytic function whose derivative is the so-called randomized Riemann zeta-function, introduced some time ago by Bagchi. The randomized $ζ$-function, $ζ_{\mathrm{rand}}(σ+ih)$, is known to represent the asymptotic statistical behaviour of the random vertical shifts of the actual $ζ$-function in the critical strip, $1/2 <σ\leq 1, h\in \mathbb R$, and appears in a number of recent works on the asymptotic behavior of the moments and maxima of the $ζ$-function on short intervals along the critical axis $σ=1/2$. Using probability and basic analytic number theory, we show that the complex integral means spectrum of the primitive of $ζ_{\mathrm{rand}}$ is almost surely of the form conjectured 30 years ago by Kraetzer, for the so-called universal integral means spectrum of univalent functions in the disc. The Riemann $ζ$-function and its random version have recently been rigorously related to the so-called Gaussian multiplicative chaos (GMC), initiated by Kahane 40 years ago. In the case of the holomorphic multiplicative chaos on the unit disc -- an important stochastic object closely related to Liouville quantum gravity on the unit circle -- we prove that the integral means spectrum of the primitive is almost surely also of the same Kraetzer form. However, we establish that neither the primitive of the random function $ζ_{\mathrm{rand}}$, nor that of the holomorphic GMC are injective. Building on earlier work by one of the authors and Webb on the convergence of Riemann $ζ$-function on the critical line to a holomorphic GMC distribution, we finally provide an alternative derivation of the integral means spectrum for the random Riemann $ζ$-function.

We propose an efficient numerical method to compute the $k$-space second Chern number in four-dimensional (4D) topological systems. Our approach employs an adaptive mesh refinement scheme to evaluate the Brillouin-zone integral, which automatically increases the grid density in regions where the Berry curvature is sharply peaked. We compare our method with the 4D lattice-gauge extension of the Fukui-Hatsugai-Suzuki method and a direct uniform grid integration scheme. Compared with these approaches, our method (i) achieves the same accuracy with substantially fewer diagonalizations, and thus runs faster; (ii) requires minimal memory to execute, enabling calculations for larger systems; and (iii) remains accurate even near topological phase transitions where conventional methods often face challenges. These results demonstrate that the adaptive subdivision strategy is a practical and powerful tool for calculating the $k$-space second Chern number.

We study online temporal voting, where a group of voters submit 0/1 approvals on sets of alternatives that arrive online over multiple rounds and a single alternative is chosen in each round. We introduce online variants of two well-known game theoretic properties, strategyproofness (SP) and independence of irrelevant alternatives. We show that online independence of irrelevant alternatives (OIIA) is a sufficient condition for online strategyproofness (OSP), and that several known online voting rules satisfy OIIA and thus OSP, but that they are not SP. In particular, we show that Perpetual Phragmen, the only known online voting rule to satisfy PJR, satisfies OSP. The Method of Equal Shares (MES), a semi-online voting rule knwon to satisfy wEJR, also satisfies OSP. We then introduce the price of manipulability, which quantifies the effect of strategic behaviour on proportional representation guarantees. Finally, we introduce asymptotic satisfaction of proportional representation and show that an online voting rule, Serial Dictator, is fully strategyproof and satisfies proportional justified representation (PJR) up to an additive constant.

In recent years, there has been growing interest in causal machine learning estimators for quantifying subject-specific effects of a binary treatment on time-to-event outcomes. Estimation approaches have been proposed which attenuate the inherent regularisation bias in machine learning predictions, with each of these estimators addressing measured confounding, right censoring, and in some cases, left truncation. However, the existing approaches are found to exhibit suboptimal finite-sample performance, with none of the existing estimators fully leveraging the temporal structure of the data, yielding non-smooth treatment effects over time. We address these limitations by introducing surv-iTMLE, a targeted learning procedure for estimating the difference in the conditional survival probabilities under two treatments. Unlike existing estimators, surv-iTMLE accommodates both left truncation and right censoring while enforcing smoothness and boundedness of the estimated treatment effect curve over time. Through extensive simulation studies under both right censoring and left truncation scenarios, we demonstrate that surv-iTMLE outperforms existing methods in terms of bias and smoothness of time-varying effect estimates in finite samples. We then illustrate surv-iTMLE's practical utility by exploring heterogeneity in the effects of immunotherapy on survival among non-small cell lung cancer (NSCLC) patients, revealing clinically meaningful temporal patterns that existing estimators may obscure.

We discuss local-global principles for the existence of Levi factors (i.e., complements to the unipotent radical) for linear algebraic groups over one-variable function fields. We give examples of disconnected groups that fail the local-global principle, and prove a strong local-global principle in the presence of Levi descent.

In this text, we outline a theory of schemes associated with a site, which generalizes a variety of geometries, such as manifolds, schemes, analytic spaces, simplicial complexes, and more. We present an abstract process of gluing model spaces via sheaf theory and recover a posteriori the underlying topological spaces that are often present in the construction of such geometric objects. We apply this formalism to semiring schemes and reason why the usual definition of semiring schemes has to be considered as the good approach to the geometry of semirings.

Reentrancy remains one of the most critical classes of vulnerabilities in Ethereum smart contracts, yet widely used detection tools and datasets continue to reflect outdated patterns and obsolete Solidity versions. This paper adopts a dependability-oriented perspective on reentrancy detection in Solidity 0.8+, assessing how reliably state-of-the-art static analyzers and AI-based techniques operate on modern code by putting them to the test on two fronts. We construct two manually verified benchmarks: an Aggregated Benchmark of 432 real-world contracts, consolidated and relabeled from prior datasets, and a Reentrancy Scenarios Dataset (RSD) of \chadded{143} handcrafted minimal working examples designed to isolate and stress-test individual reentrancy patterns. We then evaluate 12 formal-methods-based tools, 10 machine-learning models, and 9 large language models. On the Aggregated Benchmark, traditional tools and ML models achieve up to 0.87 F1, while the best LLMs reach 0.96 in a zero-shot setting. On the RSD, most tools fail on multiple scenarios, the top performer achieving an F1 of 0.76, whereas the strongest model attains 0.82. Overall, our results indicate that leading LLMs outperform the majority of existing detectors, highlighting concerning gaps in the robustness and maintainability of current reentrancy-analysis tools.

As networking systems become increasingly complex, achieving disruptive innovation grows more challenging. At the same time, recent progress in Large Language Models (LLMs) has shown strong potential for scientific hypothesis formation and idea generation. Nevertheless, applying LLMs effectively to networking research remains difficult for two main reasons: standalone LLMs tend to generate ideas by recombining existing solutions, and current open-source networking resources do not provide the structured, idea-level knowledge necessary for data-driven scientific discovery. To bridge this gap, we present SciNet, a research idea generation system specifically designed for networking. SciNet is built upon three key components: (1) constructing a networking-oriented scientific discovery dataset from top-tier networking conferences, (2) simulating the human idea discovery workflow through problem setting, inspiration retrieval, and idea generation, and (3) developing an idea evaluation method that jointly measures novelty and practicality. Experimental results show that \system consistently produces practical and novel networking research ideas across multiple LLM backbones, and outperforms standalone LLM-based generation in overall idea quality.

We study electromagnetic radiation from classical sources near a planar interface separating a topological and a trivial insulator, modeled within axion electrodynamics. The system features a piecewise constant $θ$-term that encodes the magnetoelectric response of topological surface states. Treating this coupling perturbatively, we derive analytical corrections to the standard Liénard-Wiechert potentials and obtain modified radiation fields in the far zone. As applications, we analyze the emission from linear antennas and the bremsstrahlung radiation of accelerated charges near the interface. For antennas, the surface Hall response breaks axial symmetry and produces azimuthal modulations that grow with the electrical length, leading to distinct scaling behaviors in the total and angular radiated power. {For accelerated charges, the emitted intensity is uniformly reduced by a factor $1 - (σ_{\mathrm{Hall}} / 2εv)^2$, which we interpret as a process-specific attenuation of the radiative strength due to interference with its image magnetic monopole inside the topological medium.} These results reveal how topological surface states mediate measurable modifications to classical radiation, establishing a link between axion electrodynamics, topological phases, and field theories with spatially varying couplings.

We study a nonlinear Schrödinger equation with mixed dispersion in the mass competition regime, namely mass-supercritical for the Laplacian and mass-subcritical for the Bilaplacian. In this setting, the existence of a critical value of the mass $c_\varepsilon$, which divides existence and nonexistence of energy ground state solutions, was established in [Bonheure, Castéras, dos Santos, Nascimento, SIAM J. Math. Anal. 50 (2018)]. In this work, we strengthen these results by investigating the relationship between the energy ground states with critical mass, and the optimizers of mixed Gagliardo-Nirenberg-type inequalities. Moreover, we discuss the equivalence between energy and action ground states solutions.

We study the tensor-triangular geometry of the category of Voevodsky motives generated by real quadrics. At the prime 2, we determine its Balmer spectrum, and find that it is a countably infinite, non-Noetherian space of Krull dimension 2. We detail the relationship between this space, the real Artin-Tate spectrum computed by Balmer-Gallauer, and Vishik's isotropic points. We conclude by combining our computation with Balmer-Gallauer's results on Artin-Tate motives to obtain a full description of the spectrum of integral motives of quadrics over real algebraic numbers.

This article proposes a new class of risk-sharing rules by exploring the relationship between capital allocation and risk sharing. While the former is concerned with ex-ante allocating capitals to different lines of business within a corporation based on the relationship among the individual risks, often also through the aggregate risk, the latter is an arrangement which collects risks from and allocates them to, also ex-ante, a group of participants. Drawing on this analogy, we introduce a novel idea of inducing risk-sharing rules by randomizing existing capital allocation principles. Such an approach derives new risk-sharing rules complementing known results in the literature, which were largely based on economic principles and Pareto optimality.

Topological properties of solid-state materials arise when crossings occur in their band-structure eigenvalues, which give rise to discontinuities in the associated Bloch-function eigenvectors once these are mapped over the whole Brillouin zone. These nonanalytic properties have direct consequences on the spatial decay of the corresponding Wannier functions, leading to what is nowadays referred to as the "obstruction to finding symmetric Wannier functions" for a given set of bands, as well as on the need for shifting the Wannier functions to interstitial positions, related to what is nowadays known as the "bulk-boundary correspondence." The importance of nonanalytic points of Bloch eigenfunctions and their consequences for the spatial decay of Wannier functions were historically anticipated back in 1978 [G. Strinati, Phys. Rev. B 18, 4104-4119 (1978)], somewhat before the work of Berry on what came to be referred to as the "Berry phase" [M. V. Berry, Proc. R. Soc. London, Ser. A 392, 45 (1984)]. In particular, the former paper identified key precursors and physical insights that are now understood, in hindsight, to be closely related to the later developments mentioned above. Here, we recap the essential features of these key issues in a rather pedagogical way, by considering in full details two instructing examples for which the origin of the discontinuities in the eigenvectors can be readily traced and mapped out, and the rate of the spatial falloff of the associated Wannier functions can be fully determined. For this analysis to be as complete as possible, two cases, one for noninteracting and one for interacting fermions, are considered on equal footing.

Generative AI (GenAI) is playing an increasingly important role in open source software (OSS). Beyond completing code and documentation, GenAI is increasingly involved in issues, pull requests, code reviews, and security reports. Yet, cheaper generation does not mean cheaper review - and the resulting maintenance burden has pushed OSS projects to experiment with GenAI-specific rules in contribution guidelines, security policies, and repository instructions, even including a total ban on AI-assisted contributions. However, governing GenAI in OSS is far more than a ban-or-not question. The responses remain scattered, with neither a shared governance framework in practice nor a systematic understanding in research. Therefore, in this paper, we conduct a multi-stage analysis on various qualitative materials related to GenAI governance retrieved from 67 highly visible OSS projects. Our analysis identifies recurring concerns across contribution workflows, derives three governance orientations, and maps out 12 governance strategies and their implementation patterns. We show that governing GenAI in OSS extends well beyond banning - it requires coordinated responses across accountability, verification, review capacity, code provenance, and platform infrastructure. Overall, our work distills dispersed community practices into a structured overview, providing a conceptual baseline for researchers and a practical reference for maintainers and platform designers.

Spin-orbit interactions of light couple polarization and spatial degrees of freedom, underpinning phenomena such as the spin Hall effect of light. Although widely explored at interfaces and in tightly focused beams, their impact in nominally paraxial confocal systems remains largely unexamined. Here we show that a single quarter-wave plate embedded in a simple confocal geometry between polarizers can strongly reshape the transverse structure of a Gaussian beam. We observe an enhancement of the polarization extinction ratio by more than two orders of magnitude, accompanied by a transformation of the Gaussian intensity profile into a first-order Hermite-Gaussian-like two-lobe mode. The orientation of this pattern is continuously tunable via rotation of the wave plate, evidencing polarization-controlled reorientation of the transverse field. To explain these observations, we introduce a minimal extension of Jones matrix formalism incorporating complex parameters that quantitatively reproduces the measurements. Our results uncover a previously overlooked form of spin-orbit-mediated mode control in standard confocal optics and establish a simple route to on-demand spatial mode engineering for applications in resonant spectroscopy, optical imaging and quantum optics.

An approximation of a real is a sequence of rational numbers that converges to the real. An approximation is left-c.e. if it is computable and nondecreasing and is d.c.e. if it is computable and has bounded variation. A real is computably approximable if it has some computable approximation, and left-c.e. and d.c.e. reals are defined accordingly. An approximation $\{a_s\}_{s \in ω}$ is speedable if there exists a nondecreasing computable function $f$ such that the approximation $\{a_{f(s)}\}_{s \in ω}$ converges in a certain formal sense faster than $\{a_s\}_{s \in ω}$. This leads to various notions of speedability for reals, e.g., one may require for a computably approximable real that either all or some of its approximations of a specific type are speedable. Merkle and Titov established the equivalence of several speedability notions for left-c.e. reals that are defined in terms of left-c.e. approximations. We extend these results to d.c.e. reals and d.c.e. approximations, and we prove that in this setting, being speedable is equivalent to not being Martin-Löf random. Finally, we demonstrate that every computably approximable real has a computable approximation that is speedable.

The presence of chaos in classical Hamiltonian systems is witnessed by its maximal Lyapunov exponent, that quantifies the instability of motion through the exponential growth of indicators such as the trace of the stability matrix or the out-of-time-ordered correlator. On the other hand, integrable dynamics near unstable fixed points, which are in turn characterized by a stability exponent, can also induce such exponential growth. Following the paradigm of integrability-breaking as driven by nonlinear resonances that hallmarks the genesis of chaos, the integrability-chaos transition is universally described by a periodic perturbation applied to a generic pendulum. Remarkably, this means that within the corresponding separatrix dynamics, which is an unavoidable a consequence of the resonance scenario, both instability exponents must play a role as both dynamical regimes coexist. We report here the universality of the transition from instability to Lyapunov exponents, thus completing the resonance scenario at the level of indicators based on exponential growth. To achieve this goal we obtain an analytical expression for the time evolution near separatrices, which enables us to derive an analytical expression for the exponent that characterises chaos and its transition from local instability to global chaos. We support our claim for the universality of this mechanism by studying two paradigmatic examples of the integrability-to-chaos transition, namely the kicked rotor and the driven pendulum.

In this work the heavy flavor conserving (HFC) hadronic weak decays of bottom baryons are studied in the framework of the nonrelativistic constituent quark model (NRCQM). We show that the pole terms play an indispensable role in the description of the branching ratio of $Ξ_b^-\to Λ_b^0 π^-$. With the pole terms included we can make reliable predictions for $Ξ_b^0\to Λ_b^0 π^0$. A combined study of the HFC hadronic weak decays allows us to make a reasonable prediction for $Ω_b^-\toΞ_b^{-(0)}π^{0(-)}$, which can be searched for at LHCb and Belle-II experiments.

We investigate the Stirling-cycle performance of a Heisenberg--Kitaev magnonic medium with Dzyaloshinskii--Moriya (DM) interactions. Using linear spin-wave theory, we show the DM interaction preserves spectral symmetry, yielding even caloric responses and symmetric Stirling engine efficiency. In contrast, bond-dependent Kitaev exchange asymmetrically distorts the magnonic density of states, enabling distinct direct and inverse caloric effects. Consequently, Kitaev-driven cycles achieve significantly higher efficiencies than DM-driven protocols, approaching a high-performance saturation regime for negative couplings. This establishes exchange-anisotropic magnets as highly tunable platforms for nanoscale solid-state energy conversion.

We report on the control and characterization of the isomer population of ions inside superfluid helium nanodroplets, using two-color operation of a dual-oscillator infrared free-electron laser. The timing of both lasers is highly synchronized, and their frequencies (or "colors") can be tuned independently over a wide range. Interaction of the singly deuterated proton-bound dimer of dihydrogen phosphate and formate inside helium nanodroplets with both colors enables the control over its isomer population and the recording of - one-color hidden - infrared spectra of individual isomers.

We present a general, discovery-grade framework for searching for weakly coupled new particles emitted in nuclear de-excitation following neutron capture. Rather than relying on isolated spectral features, the method exploits correlated ``satellite-line combs'': multiple weak $γ$-ray lines appearing at a common energy offset $Δ$ below known capture transitions. By combining likelihood information across many parent lines and multiple target nuclei, the approach strongly suppresses nuclear-structure ambiguities and instrumental artifacts. We also discuss optimal target selection and practical experimental implementation with high-resolution HPGe detectors.

Machine learning interatomic potentials (MLIPs) enable large-scale atomistic simulations but remain challenged in describing mixed-valence materials where charge ordering strongly influences thermodynamic stability. Here we investigate the role of electronic entropy in MLIP structural optimization of the battery cathode material \ce{NaFePO4}. We show that conventional MLIPs fail to reproduce the correct stability of intermediate \ce{Na} concentrations because structural optimization leads to incorrect \ce{Fe^{2+}}/\ce{Fe^{3+}} charge assignments, resulting in erroneous energy ordering and convex-hull predictions. Analysis of magnetic moments during structural optimization reveals that MLIPs are unable to capture electronic entropy associated with charge ordering. To address this limitation, we introduce an approach that embeds charge-state information directly into the MLIP representation by distinguishing between \ce{Fe^{2+}} and \ce{Fe^{3+}} environments during training. Retraining CHGNet, cPaiNN, and MACE with this representation enables accurate structural optimization, correct identification of charge ordering, and improved agreement with density functional theory convex hulls. Our results demonstrate that incorporating electronic entropy into MLIP representations is essential for modeling charge-disordered materials and provide a practical framework for extending MLIP simulations to mixed-valence transition-metal systems.

Entanglement is a key property of quantum systems. In this Letter the first measurements of quantum entanglement between spins in pairs of $Z$ bosons are reported, using proton-proton collision data from the Large Hadron Collider (LHC) at center-of-mass energies of 13 TeV and 13.6 TeV, recorded with the ATLAS detector. Measurements of angular observables sensitive to $ZZ^*$ spin-density-matrix elements in the $H\rightarrow ZZ^* \rightarrow \ell^+\ell^-\ell^+\ell^-$ process yield coefficients $C_{2,1,2,-1} = -0.71 \pm 0.45$ and $C_{2,2,2,-2}=0.08 \pm 0.44$, consistent with their Standard Model predictions. A complementary hypothesis test using the full angular distribution, and relying on several Standard Model assumptions in the decays, provides substantially higher sensitivity to quantum correlations and disfavors the separable-state hypothesis at a significance of 4.7 standard deviations (expected $4.9σ$) relative to the entangled Standard Model hypothesis. These results provide strong evidence of quantum entanglement between massive bosons (spin qutrits) at the electroweak scale.

We investigate the asymptotic properties of Bayesian bivariate causal discovery for Gaussian Linear Structural Equation Models (SEMs) with heteroscedastic noise. We demonstrate that with purely observational data, the posterior distribution over the models fails to consistently identify the true causal structure - a consequence of the fundamental non-identifiability within the Markov Equivalence Class. Specifically, if the true generating mechanism corresponds to a connected graph (A -> B or B -> A), the asymptotic behavior of the posterior is given by the ratio between the prior on the true model and the push-forward prior of the alternative. In contrast, for the independence model, we establish that the posterior concentrates at a stochastic polynomial rate of O_p(n^{-1/2}). To resolve this non-identifiability, we incorporate m interventional samples and characterize the concentration rates as a function of the observational-to-total sample ratio, η. We identify a sharp concentration dichotomy: while the independence graph maintains a polynomial O_p(N^{-1/2}) rate (where N = n+m), connected graphs undergo a phase transition to exponentially fast convergence. This highlights an exponential relative importance between the two data types, as altering the amount of one data type directly changes the exponent governing the concentration speed. We derive explicit formulae for the exponential decay rates and provide precise conditions under which mixing observational and interventional data optimizes concentration speed. Finally, our theoretical findings are validated through empirical simulations in Bayesian Gaussian equivalent (BGe)-style prior specifications offering a principled foundation for experimental design in Bayesian causal discovery.

Recently a one-parameter family of deformed $ \mathcal{L} w_{1+\infty}$ soft symmetry algebras, denoted $ \mathcal{L}_Λw_{1+\infty}$, acting on tree-level gravitational theories in AdS$_4$ has been discovered. Here we show that all CFT$_3$s, including those dual to quantum gravity on AdS$_4$, admit an $\mathcal{L}_Λw_{1+\infty}$ action generated by the ANEC operator, its conformal descendants and their commutators. This extends the previous tree-level results on these soft symmetries to the strongly-coupled quantum regime.

Cd(Zn)O stands out as probably the best plasmonic material in the mid-IR, but it is usually grown on sapphire or other passive substrates. In this work we introduce SiC as a novel, highly polar, dopable substrate for Cd(Zn)O. The Cd(Zn)O/SiC system is analyzed as a function of the Zn concentration and thin film thickness, and the results are compared to those obtained in the Cd(Zn)O/sapphire system. XRD and reflectance measurements show that the alloy with 10 % Zn nominal concentration has the best crystalline and plasmonic quality, with optical losses as good as 13 % of the plasma frequency. The thin films show two surface polariton modes: a purely plasmonic symmetric mode at higher energies with negligible frequency dispersion and pinning at the plasma frequency for the thinnest films, characteristic of an ideal epsilon-near-zero mode; and a plasmonic-phononic hybridized antisymmetric mode at lower energies, which thanks to the large value of the higher frequency dielectric constant of SiC compared to sapphire, shows much lower frequency dispersion, indicative of the stronger epsilon-near-zero character. Hence, Cd(Zn)O/SiC offers a promising platform for the development of ENZ devices on an active substrate.

We combine our previous results on magnetic pseudo-differential operators for Hörmander symbols dominated by tempered weights [arXiv:2511.07184] with the magnetic Weyl super calculus of Lee and Lein [arXiv:2201.11487, arXiv:2405.19964]. This allows us to extend some previous results on the semi-super and super Moyal algebra, as well as to prove boundedness, compactness, and Schatten-class properties of super operators. Moreover, we prove a Beals-type commutator criterion for super operators and we also formulate sufficient conditions on super symbols in order to give rise to completely positive and trace preserving maps. For most of the proofs we use decompositions of operators and super operators based on Parseval frames of smoothing operators.

This letter investigates the channel estimation problem in THz wireless communications where a RIS is employed to assist wireless transmission between different devices. Unlike existing studies, we consider a novel scenario where specific devices are all located in the radiative NF region of the RIS. Meanwhile, we also account for the impact on channel estimation of the random electromagnetic interference occurring at the RIS location. A linear minimum mean-square error estimator is employed, where the estimation error is fully determined by the RIS configuration. Optimizing the RIS involves solving a non-convex problem, which is addressed using an alternating optimization approach based on the diagonally scaled gradient descent algorithm. Numerical results in the THz band highlight the importance of leveraging NF channel statistics over far-field approximations and demonstrate that the proposed estimator achieves substantial improvements in normalized mean-square error compared to existing methods.

This paper studies an analog of the classical Schwartz space $ \mathscr{S}(\mathbb{R}^N) $ in the framework of $ (k, a) $-deformed harmonic analysis associated with the $ (k, a) $-generalized Fourier transform $ \mathscr{F}_{k, a} $. Motivated by the observation that $ \mathscr{S}(\mathbb{R}^N) $ coincides with the space of smooth vectors for the Segal--Shale--Weil representation, we define the $ (k, a) $-generalized Schwartz space $ \mathscr{S}_{k, a}(\mathbb{R}^N) $ as the space of smooth vectors for the unitary representation associated with $ \mathscr{F}_{k, a} $. Since this definition is intrinsic to the representation, it follows immediately that $ \mathscr{S}_{k, a}(\mathbb{R}^N) $ is preserved by $ \mathscr{F}_{k, a} $. As main results, we explicitly determine $ \mathscr{S}_{k, a}(\mathbb{R}^N) $ for $ N = 1 $, as well as for general $ N $ when $ k = 0 $ and $ a $ is rational. We also explicitly determine the space of smooth vectors for the $ L^2 $-model of the minimal representation of the conformal group $ \widetilde{\mathit{SO}}_0(N + 1, 2) $ studied by Kobayashi--Mano.