The classical Goodstein process, defined via hereditary base-$k$ exponential normal form, is a well-known example of a principle unprovable in Peano Arithmetic. In this paper, we generalize this framework by constructing a new Goodstein process based on the Hardy hierarchy. We develop an ordinal notation system utilizing a two-step collapsing procedure, which yields a proof-theoretic ordinal of $ψ_0ψ_1(\varepsilon_{Ω_2+1})$. By defining $k$-normal forms for natural numbers within this system, we introduce a Goodstein-type process and demonstrate that the theory of non-iterated positive inductive definitions for two operators ($ID_2$) cannot prove its termination. This result establishes a new independence result at the second proof-theoretic threshold, further extending the reach of Goodstein-type principles beyond the Bachmann-Howard level.

We discuss a Wakimoto-like free field realisation of ${\frak d}(2,1;α)_1$, whose $\frak{sl}(2,\mathbb{R})$ subalgebra has level $k=1$, that requires no gauging, i.e., realises the current algebra exactly. We then compute the partition function of the theory and show that, by combining this with the ghost contribution, the full, on-shell projected string partition function reproduces precisely the single-particle partition function of the ${\rm Sym}^N({\cal S'}_0^2)$ theory, i.e. the symmetric orbifold theory of 8 free fermions, 1 compact free boson, and 1 non-compact free boson. We also discuss other aspects such as DDF operators and BRST and physical state conditions.

The approximate coherent state rank is the minimal number of (classical) coherent states required to approximate a continuous-variable bosonic quantum state and directly relates to the classical complexity of simulating bosonic computations. Despite its importance, little is known about lower bounds on this quantity, even for basic families of states. In this work, we initiate a systematic study of lower bounds on the approximate coherent state rank. Our contributions are as follows. (i) We introduce a technique based on low-rank approximation theory yielding generic lower bounds on the approximate coherent state rank of arbitrary single-mode states. (ii) Using this technique, we find a complete characterization of all single-mode states of finite approximate coherent state rank, and we obtain in particular analytical expressions for the approximate coherent state rank of squeezed states and of finite superpositions of Fock states. (iii) We further show that our single-mode lower bounds can be lifted to multimode lower bounds for finite superpositions of multimode Fock states. (iv) Finally, we prove a super-polynomial lower bound on the approximate coherent state rank of the $n$-mode Fock state $|1\rangle^{\otimes n}$, by exploiting a connection to the permanent. To do so, we show that the algebraic complexity of approximate multi-linear formulas for the permanent is super-polynomial, building upon the proof of a lower bound for exact formulas due to [Raz, JACM 2009]. Our results establish an unconditional barrier to efficient classical simulation of Boson Sampling via coherent state decompositions and connect non-classicality of bosonic quantum systems to central questions in algebraic complexity.

We show the existence of a coherent frame in the orbit of a connected, simply connected unimodular solvable Lie group of exponential growth for which the lower Beurling density of its index set is zero.

The oxygenation process at constant temperature of YBa$_2$Cu$_3$O$_{7-δ}$ (YBCO) was systematically investigated in the temperature range from 300 $^o$C to 800 $^o$C. With this purpose, fully deoxygenated powder samples was exposed to an oxygen saturated atmosphere, and the evolution of their mass was recorded as a function of time using a thermogravimetric balance. Results reveal a strong dependence of both: the oxygenation kinetics and the final oxygen saturation level, on the temperature used for oxygenation. Moreover, results show that higher oxygen temperatures promote faster oxygen absorption but lead to lower saturation levels (higher final $δ$ values), whereas lower oxygen temperatures result in slower kinetics but enable the system to approach better oxygenation conditions in order to improve the final superconducting properties of the material. In addition to our measurements, a comparative analysis between oxygenation levels at the oxygen temperatures under study was performed in the range around oxygen saturation ($δ$ $<$ 0.3). Consequently, an oxygenation protocol based on a combination of several oxygenation temperatures is proposed. As a first approach, results from a protocol with just two different oxygenation temperatures is compared with results coming from using just one oxygenation temperature. Outstandingly, a protocol with a first oxygenation step at high temperatures and a second one at low temperatures demonstrates to improve oxygenation times in near a 30 \% for reaching $δ$ values below 0.1 and in near a 60 \% for reaching $δ$ values around 0.12. Finally, we trust that our results are of direct application on industry since size of grain used herein are in scales of typical thickness of superconductor tapes.

This paper presents an error analysis of classical and learned Tikhonov regularization schemes for inverse problems. We first demonstrate, both theoretically and numerically, that using a fixed regularization parameter across varying noise levels-which is a common miss-specification in practice-has only a mild impact on the reconstruction error. As a special case, we then investigate scenarios where the true data resides in an unknown finite-dimensional subspace. Here, our results lead to an empirically supported strategy for estimating the unknown dimension based on numerical experiments. Finally, we examine the approach that motivated this study: a method where a sparsity-promoting term is learned from denoising tasks and subsequently applied to general inverse problems via a simple heuristic parameter selection. The corresponding error analysis is initially developed using classical concepts and subsequently refined through a more detailed investigation of the discretized setting.

Rapid development of Large Language Models (LLMs) and similar automated approaches for translation tasks is increasingly affecting the landscape of translation technologies. As concerns about the outsourcing of translator work to these automated translation tools grow, it becomes increasingly crucial to gather insights from the translation community directly. To this end, we conduct an interview study with 19 professional translators working across 11 languages and 11 domains to understand their perspectives, experiences, and concerns with using translation technologies in their work. We find that translators are cautious when incorporating new tools into their workflow, with several expressing concerns machine translation (MT) and LLMs are infringing on the necessary human aspects and verification steps of translation, worried that these tools have potential for harmful downstream effects due to compromising the human aspect of translation work. These findings demonstrate the need to develop translation technologies that directly serve translators' needs rather than replacing human translation. This can be done by focusing more on the assistive, rather than the automating aspects of these tools.

Stochastic reaction networks are mathematical models with a wide range of applications in biochemistry, ecology, and epidemiology, and are often complex to analyze. Except for some special cases, it is generally difficult to predict how the abundances of all considered species evolve over time. A possible approach to address this issue is to develop tools to compare the model under study with a similar one whose behavior is better understood. The main contribution of our work is to provide direct and computable conditions that can be used to ensure the existence of an ordered coupling between two stochastic reaction networks and to identify which parameter changes in a given model lead to an increase or decrease in the count of certain species. We also make available an algorithm that implements our theory, and we illustrate it with several applications.

Ge/SiGe quantum well heterostructures confining a high-mobility two-dimensional hole gas (2DHG) have emerged as a compelling platform for hybrid superconductor(S)-semiconductor(Sm) quantum devices. Here, we investigate the low-temperature transport properties of split-gate quantum point contacts (QPC) defined in one such heterostructure and positioned at different distances from an aluminum superconducting contact. We observe ballistic one-dimensional transport evidenced by conductance quantization with at least four clearly visible plateaus. Andreev reflection at the S/Sm interface induces a 40% enhancement of the conductance steps relative to the normal-state conductance staircase measured under a 100-mT out-of-plane magnetic field. This result is in excellent agreement with the theoretical expectation for an interface transparency of 0.88. By operating the QPCs in the tunneling regime, we probe the local density of states of the proximitized 2DHG. We report direct experimental evidence of an induced superconducting gap, demonstrating that its magnitude can be tuned by a gate voltage acting on the carrier density in the 2DHG.

We construct black hole geometries coexisting with anisotropic matter in (anti)-de Sitter spacetime. We specifically focus on the black hole phase transitions that occur in anti-de Sitter spacetime and analyze the effects of the incorporated matter fields. Its local stability is examined by evaluating the heat capacity, while global stability is investigated in greater detail through phase transition analysis. The black hole system coexisting with the matter field allows for a phase transition from a small black hole to a large black hole. This demonstrates that the constructed geometry with the matter field would resemble that of the Reissner-Nordström black hole. We examine null geodesics, particularly unstable homoclinic orbits, which allow us to obtain Lyapunov exponents that characterize sensitivity to initial conditions. Finally, we analyze the relationship between the different black hole phases and the behavior of these Lyapunov exponents.

We introduce tropical matroid Schubert varieties, a tropical analogue of arrangement Schubert varieties associated with realisable matroids. We prove that the tropical cohomology ring of the tropical matroid Schubert variety associated to any matroid $M$ is isomorphic to the graded Möbius algebra $\operatorname{B}^\bullet(M)$. This yields a geometric model for $\operatorname{B}^\bullet(M)$, extending the geometric setting of arrangement Schubert varieties to arbitrary matroids, including non-realisable ones.

A coarse-grained molecular simulation approach originally developed for entangled polymeric liquids is extended to model the mechanical behavior of long-fiber networks. The model, based on the slip-link picture of chain entanglements, resolves the force balance at contact points and accounts for fiber slippage under these topological constraints. Two key governing equations describe the time evolution of contact-point positions and the local fiber fraction between adjacent contact points. A yield-force criterion determines whether contact points are displaced or remain pinned, as well as whether fiber slippage occurs at contact points. Uniaxial compression simulations corresponding to press molding of fiber-reinforced thermoplastics were performed for networks with varying fiber lengths and compression rates. The results were qualitatively consistent with experimental observations of long-fiber thermoplastics. The model captures physics inaccessible to the classical van Wyk theory of fiber network compression, which is quasi-static and insensitive to fiber length. This work demonstrates that the slip-link framework, already validated for polymer melts, provides a promising mesoscale simulation tool for understanding and predicting the processing behavior of non-thermal fiber networks.

This paper studies efficient data management and timely information dissemination for real-time monitoring of an $N$-state Markov process, enabling accurate state estimation and reliable actuation decisions. First, we analyze the Age of Incorrect Information (AoII) and derive closed-form expressions for its time average under several scheduling policies, including randomized stationary, change-aware randomized stationary, semantics-aware randomized stationary, and threshold-aware randomized stationary policies. We then formulate and solve constrained optimization problems to minimize the average AoII under a time-averaged sampling action constraint, and compare the resulting optimal sampling and transmission policies to identify the conditions under which each policy is most effective. We further show that directly using reconstructed states for actuation can degrade system performance, especially when the receiver is uncertain about the state estimate or when actuation is costly. To address this issue, we introduce a cost function, termed the Cost of Actions under Uncertainty (CoAU), which determines when the actuator should take correct actions and avoid incorrect ones when the receiver is uncertain about the reconstructed source state. We propose a randomized actuation policy and derive a closed-form expression for the probability of taking no incorrect action. Finally, we formulate an optimization problem to find the optimal randomized actuation policy that maximizes this probability. The results show that the resulting policy substantially reduces incorrect actuator actions.

We discuss how canonical and universal constructions, properties and characterizations interact with equality in the framework of Homotopy Type Theory, comparing it with Grothendieck's use of equality and shedding further light on (efficient) formalisation of mathematics. This is achieved by investigating examples that range from monoids, groups, rings and modules to cohomology theories in the category of modules over commutative rings and culminate in a cohomological criterion of flatness.

Engineering synthetic dimensions, where the physics of additional spatial dimensions is simulated within the internal states of a quantum system, allows the realisation of phenomena not otherwise accessible in experiments. Ultracold ground-state polar molecules are an ideal platform to encode synthetic dimensions, offering access to large Hilbert spaces of long-lived internal states associated with the rotational and hyperfine degrees of freedom, that can be coupled together with microwave fields to simulate tunnelling. Here, to benchmark the advantages of ultracold molecules, we encode a 1D synthetic lattice in the rotational states of ultracold RbCs molecules and use it to investigate the well-known Su-Schrieffer-Heeger (SSH) model, a minimal model displaying topological properties. To probe the system, we perform spectroscopy using an auxiliary rotational state and study the time dynamics after deterministic state preparation. We demonstrate long coherence times, typically ~500 times the lattice tunnelling period, even for a synthetic lattice using 8 rotational states. Observations of dynamics at long times with full site-resolved readout of the synthetic dimension allow us to test the effects of chiral and non-chiral perturbations on the topologically protected edge states. Our work lays the foundation for further quantum simulations using the rich internal structure of molecules, including dipolar string phases in interacting samples of molecules, and adiabatic state preparation of many-body Hamiltonians.

The nature of low-lying scalar and axial-vector charmed mesons has long been debated, specifically whether they are best explained as hadronic molecules or compact tetraquark systems. These two scenarios exhibit quite different features for the accessible $SU(3)$ multiplets in the scalar and axial-vector sectors. To resolve this debate, we performed $N_f=3+1$ lattice simulations and calculated the energy levels of the $SU(3)$ $[6]$ and $[\overline{15}]$ multiplets for both the scalar and axial-vector mesons in an $SU(3)$ flavor-symmetric setting. In both sectors we find attractive states for the [6] and repulsive interactions for the $[\overline{15}]$. This is consistent with the hadronic molecule picture, but not the compact tetraquark picture which predicts a low-lying $[\overline{15}]$ states in the axial-vector sector but not in the scalar sector.

We introduce and analyze a long-memory continuous-time Markov chain on $\mathbb{R}_{+}$ whose jump mechanism depends explicitly on a state in the past. From the present state $x_0$, the process jumps to $x_0\left(1+\frac{1}{N}\right)$ or $x_0\left(1-\frac{x_{-\lfloor τN \rfloor}}{N^2}\right)$, each at rate $\tfrac{1}{2}$, where $x_{-\lfloor τN \rfloor}$ denotes the state located $\lfloor τN \rfloor$ jumps backward in time. Here the delay $τ> 0$ is fixed and $N$ is the scaling parameter. The initial condition is prescribed by a vector of length $\lfloor τN \rfloor + 1$, all of whose entries are equal to $μN$. Using a genuine space-time replacement lemma, we prove that, as $N \to \infty$, the rescaled process converges to a deterministic limit governed by the Delayed Logistic Equation (also known as the Hutchinson equation) with delay $τ$ and initial condition $ρ(t) \equiv μ$ for $t \in [-τ, 0]$.

Random Number Generators (RNGs) are crucial for applications ranging from cryptography to simulations. Depending on the source of randomness, RNGs are classified into Pseudo-Random Number Generators (PRNGs), True Random Number Generators (TRNGs), and Quantum Random Number Generators (QRNGs). This work presents the end-to-end development of a high-speed, high-efficiency, phase-noise-based QRNG system that taps into the quantum phase noise of a single-frequency laser, with randomness originating from spontaneous emission. Using a self-heterodyne measurement with a semiconductor laser (linewidth $\approx$ 5.23 $GHz$) operated near threshold and a $\sim$48 $cm$ fiber delay line, a raw data generation rate of 2.0 $Gbps$ is achieved. To ensure uniform randomness in the QRNG output, robust extraction techniques developed in-house, such as the Toeplitz Strong Extractor (TSE), are used. Randomness validation using the NIST and Diehard test suites confirms that all statistical tests pass at standard confidence levels. The developed system achieves a post-processed generation rate of 1.0 $Gbps$ in operation and attains a Technology Readiness Level (TRL) of 7, approaching TRL 8, making it suitable for real-time secure applications such as cryptographic key generation and stochastic modeling.

We study Serrin's overdetermined boundary value problems in bounded domains on weighted Riemannian manifolds. When the closure of the domain is compact, we establish a rigidity result that characterizes both the solution and the geometry of the ambient manifold. We further address the case of domains with non-compact closure for manifolds conformally equivalent to the Euclidean space, possibly degenerating or becoming singular at a point, where both the weight and the conformal factor are radial functions.

Network slicing across multiple administrative domains raises two coupled challenges: enforcing slice-specific trust constraints while enabling fast online admission and placement decisions. This paper considers a multi-domain infrastructure where each slice request specifies a VNF chain, resource demands, and a set of (un)trusted operators, and formulates the problem as a Node-Link (NL) integer program to obtain an optimal benchmark, before proposing a Path-Link (PL) formulation that pre-generates trust and order-compliant candidate paths to enable real-time operation. To mitigate congestion, resource prices are made dynamic using a Kleinrock congestion function, which inflates marginal costs as utilization approaches capacity, steering traffic away from hotspots. Extensive simulations across different congestion levels and slice types show that: (i) PL closely tracks NL with negligible gaps at low load and moderate gaps otherwise, (ii) dynamic pricing significantly reduces blocking under scarce resources, and (iii) PL reduces computation time by about 3x-6x compared to NL, remaining within a few seconds even at high load. These results demonstrate that the proposed PL and dynamic pricing framework achieves near-optimal performance with practical runtime for online multi-domain slicing under trust constraints.

Semileptonic $b\to cτ\barν_τ$ decays are powerful probes of non-Standard-Model effects within an effective-field-theory (EFT) framework, but fully exploiting them in current and future data demands combinations that maximise sensitivity while controlling biases from Standard-Model-based modelling and from theory inputs that are shared, and therefore correlated, across analyses in different experiments. We present a first sensitivity study of a combined extraction of Wilson coefficients in $\bar{B}\to D^{(*)}τ\barν_τ$ decays using LHCb- and Belle~II-like analysis configurations. Detector simulations for signal and backgrounds are typically generated under Standard Model assumptions; if non-SM contributions are present, this can bias the fitted Wilson coefficients. In addition, hadronic inputs such as form-factor parameters of signal and background components are common across analyses, requiring a consistent treatment of fully correlated effects in combinations. To avoid repeating large-scale detector simulation for each EFT hypothesis, we use event-by-event reweighting to map simulated samples to arbitrary combinations of Wilson coefficients. We then compare a simultaneous fit across multiple $\bar{B}\to D^{(*)}τ\barν_τ$ channels and datasets with a combination based on post-fit averages. Sharing Wilson coefficients and common form-factor parameters in the simultaneous fit reduces model-induced biases and improves sensitivity relative to independent fits, providing a robust and scalable strategy for precision EFT constraints in $b\to cτ\barν_τ$ transitions using forthcoming LHCb and Belle~II datasets.

We prove that the forgetful functor from algebras over an $N_{\infty}$-operad to equivariant spaces preserves sifted homotopy colimits

Test collections are essential for evaluating retrieval and re-ranking models. However, constructing such collections is challenging due to the high cost of manual annotation, particularly in specialized domains like Algerian legal texts, where high-quality corpora and relevance judgments are scarce. To address this limitation, we propose STCALIR, a framework for generating semi-synthetic test collections directly from raw legal documents. The pipeline follows the Cranfield paradigm, maintaining its core components of topics, corpus, and relevance judgments, while significantly reducing manual effort through automated multi-stage retrieval and filtering, achieving a 99% reduction in annotation workload. We validate STCALIR using the Mr. TyDi benchmark, demonstrating that the resulting semi-synthetic relevance judgments yield retrieval effectiveness comparable to human-annotated evaluations (Hit@10 \approx 0.785). Furthermore, system-level rankings derived from these labels exhibit strong concordance with human-based evaluations, as measured by Kendall's τ (0.89) and Spearman's \r{ho} (0.92). Overall, STCALIR offers a reproducible and cost-efficient solution for constructing reliable test collections in low-resource legal domains.

This paper investigates a system of nonlocal continuity equations modelling the interaction of two species coupled through Riesz-type potentials. The model incorporates self- and cross-interaction kernels of possibly different fractional orders. By exploiting optimal transportation theory and the theory of gradient flows in Wasserstein spaces, we establish the existence of weak solutions under singularity assumptions on all interaction potentials, provided the cross-interaction ones satisfy a symmetry condition. Our analysis extends previous results available for either single-species equations or multi-species systems with smoother cross-interaction kernels.

Learning the graph Laplacian from observed data is one of the most investigated and fundamental tasks in Graph Signal Processing (GSP). Different variants of the Laplacian, such as the combinatorial, signless or signed Laplacians have been considered depending on the type of features to be extracted from the data. The main contribution of this paper is the introduction of a parametric Laplacian, called the deformed Laplacian, defined as a quadratic matrix polynomial that provides a parametric dictionary for graph signal processing. The deformed Laplacian can be interpreted as the generator of a parametric linear reaction-diffusion dynamics on graphs, capturing the interplay between diffusive coupling and nodal reaction effects. It is a parametric polynomial matrix that enables the design of novel topological operators tailored to both the underlying graph structure and the observed signals. Interestingly, we show that several Laplacian variants proposed in the literature arise as special cases of the deformed Laplacian. We then develop a method to jointly learn the deformed Laplacian and the graph signals from data, showing how its use improves signal representation across a broad class of graphs compared to standard Laplacian forms. Through extensive numerical experiments on both synthetic and real-world datasets, including financial and communication networks, we assess the benefits of the proposed method in terms of graph signal reconstruction error and sparsity of the representation.

This paper presents a phase-difference-based scheme for three-dimensional (3D) line-of-sight (LoS) user localization using a uniform planar array (UPA), applicable to both near-field and far-field regimes under the exact spherical-wave model. Unlike the previously studied two-dimensional (2D) uniform linear array (ULA) case, the 3D UPA case requires jointly exploiting the two array axes in order to recover the user's range, azimuth, and zenith angle. Adjacent-antenna phase-differences are first estimated from uplink pilots and then summed along the array axes to obtain unwrapped phase-differences between widely separated antenna elements. These summed phase-differences enable the construction of multiple three-equation systems whose solutions yield the user's range, azimuth, and zenith angle. We quantify the number of such equation systems, provide a representative closed-form estimator that uses only three phase-difference sums, and propose an all-data nonlinear least-squares estimator that exploits all available sums. Numerical results show that the least-squares estimator, when initialized by the closed-form estimate, achieves Cramér--Rao bound accuracy. Moreover, unlike state-of-the-art baseline schemes, whose performance depends on well-tuned hyperparameters, the proposed estimators are hyperparameter-free.

Traditional econometric analyzes represent observations as vectors despite the inherent complexity of empirical data structures. When data are organized along dual classification dimensions, a matrix representation provides a more natural and interpretable framework. Building on recent advances in matrix autoregressive (MAR) modeling, this study introduces a novel error correction representation tailored for matrix-structured data. Through comparative analysis with existing methodologies, we demonstrate two critical advancements. First, the proposed model preserves the interpretative foundations of conventional cointegration analysis, with coefficients that explicitly capture dynamics rooted in adjustment toward steady-state positions. Second, in contrast to previous formulations, our error correction framework allows for an equivalent matrix autoregressive representation, preserving the fundamental structure of the data in both specifications. This ensures that the matrix representation reflects an intrinsic characteristic of the data.

In this study, we tackle the problem of finding a maximum \emph{co-3-plex}, which is a subset of vertices of an input graph, inducing a subgraph of maximum degree 2. We focus on the class of chordal graphs. By observing that the graph induced by a co-3-plex in a chordal graph is a set of isolated triangles and induced paths, we reduce the problem of finding a maximum weight co-3-plex in a graph $G$ to that of finding a maximum stable set in an auxiliary graph $\mathcal{A}(G)$ of exponential size. This reduction allows us to derive an exponential variable-sized linear programming formulation for the maximum weighted co-3-plex problem. We show that the pricing subproblem of this formulation reduces to solving a maximum vertex and edge weight induced path. Such a problem is solvable in polynomial time; therefore, this exhibits a polynomial time column generation algorithm solving the maximum co-3-plex problem on chordal graphs. Moreover, this machinery exhibits a new application for the maximum vertex and edge weighted induced path problems.

We provide a mathematically rigorous definition of local approximation and demonstrate its applicability to some interesting classes of structures. In particular, we prove that any compact simple Lie group is locally approximated by finite groups. The definition and main examples are motivated by physics but the techniques are of model theory. Namely, we introduce the ultraproduct of emerging metric structures, which generalises the ultraproduct in metric model theory.

We examine the origin of the short-lived radionuclides (SLRs, defined as having half-lives between 0.1 and 100 Ma) present in the early Solar System (ESS) by investigating how predictions of their abundances in the interstellar medium (ISM) from steady-state equilibrium relate to their ESS values. For this, we take into account the non-negligible time $t_{\mathrm{iso}}$ elapsed between the isolation of the pre-solar molecular cloud and the formation of the ESS, during which the SLRs decayed freely. We also consider the alternative scenario in which the pre-solar molecular cloud remained partially mixed with the ISM, with a mixing timescale $t_{\mathrm{mix}}$. We find that the ESS abundances of $^{107}$Pd and $^{182}$Hf produced by \textit{slow} neutron captures (\textit{s}-process), and of $^{53}$Mn and $^{60}$Fe produced by explosive nucleosynthesis, can be consistently explained within these scenarios. Their required $t_{\mathrm{iso}}$ is 9-12 Ma, and their required $t_{\mathrm{mix}}$ is 11-14 Ma (with one potential exception of $t_{\mathrm{mix}}$ = 38 Ma), depending on galactic uncertainties, such as the galactic star formation history and efficiency and the star-to-gas mass ratio. Another \textit{s}-process SLR, $^{205}$Pb has a more uncertain ESS value, and falls within only some of these time values. The same applies to the SLRs produced by the $p$-process ($^{92}$Nb and $^{146}$Sm), depending on the latter's half-life. In agreement with previous studies, we find that the ESS abundances of the \textit{rapid} neutron-capture isotopes ($^{129}$I, $^{244}$Pu, and $^{247}$Cm) and of the most short-lived radionuclides ($^{26}$Al, $^{36}$Cl and $^{41}$Ca) cannot be explained by assuming steady-state equilibrium in the ISM.