This paper presents an investigation of a novel class of stealthy and selective reactive jamming attacks targeting the Physical Uplink Shared Channel (PUSCH) in 5G New Radio (NR) networks. We design and implement STORM-RJ (Stealthy Timing Obstruction and Radio Manipulation -- Reactive Jamming), a Software-Defined Radio (SDR)-based adversarial framework that enables highly precise, time-frequency aligned interference by dynamically adapting the bandwidth and center frequency of injected noise bursts in real time. STORM-RJ leverages decoded Downlink Control Information (DCI) to identify Uplink-Grants (UL-Grants) and synchronizes interference exactly with the resource blocks allocated to a target User Equipment (UE). We further characterize and mitigate the dominant latency sources -- both at the software processing and hardware Radio Frequency (RF) frontend levels -- to achieve a rapid jamming response upon grant detection. We conduct a comparative analysis of high-level versus low-level radio control strategies, demonstrating that only low-level tuning provides the microsecond-scale responsiveness necessary to meet 5G-NR timing constraints for effective reactive jamming. We analyze the practical feasibility of such selective jamming under realistic hardware and timing constraints, highlighting key trade-offs between SDR flexibility, processing latency, and synchronization accuracy. Finally, we discuss potential mitigation strategies, including Hybrid Automatic Repeat reQuest (HARQ) anomaly detection.

In this note, we show that, for all $n\ge 2$, the number of distinct rooted binary phylogenetic $X$-trees displayed by a binary tree-child network $\mathcal{N}$ on $X$ with $n$ leaves is at most $2^{n-1}-1$ and that this upper bound is sharp. Furthermore, if $\mathcal{N}$ displays exactly $2^{n-1}-1$ such trees, then exactly one rooted binary phylogenetic $X$-tree is displayed twice, and this tree can be canonically found by iteratively replacing a reticulated cherry with a cherry.

This study investigates demographic differences in cybersecurity culture in a large global organisation supporting safety critical and critical infrastructure sectors to target CSC improvement. A global survey was administered to all internal and external employees of a total of 21148 employees, with 6502 responses. The questionnaire evaluates nine CSC dimensions such as Password Management, Governance, Email Use. Anonymous survey responses were analysed using Kruskal-Wallis tests and Dunns post hoc comparisons to identify differences across demographic variables including employment, recruitment paths, managerial role, gender, age, tenure, and work base. CSC was broadly consistent across the organisation, with statistically significant but small to moderate demographic effects. CSC variations were observed across employment, age, recruitment paths, and line managerial role. In general, fulltime, internal, permanent, older employees, Merge and Acquisition recruits, and line managers consistently scored higher across multiple CSC dimensions. Parttime, younger, external employees, and those with 6 to 20 years of tenure in general scored lower. These patterns highlight higher-scoring groups that may act as CSC carriers and lower-scoring groups that may benefit from tailored improvement measures, enabling organisational learning. Our study offers a practical, scalable way to assess CSC, generating meaningful insights despite industrial constraints. It enables organisations to benchmark maturity, identify gaps, and prioritise targeted improvements using workforce diversity as a guide.

The main result of this paper is a generalization of a theorem of Nakajima-Landweber-Stong to the modular invariant rings of transvection groups over Dedekind domains. More precisely, let $A$ be a Dedekind domain and $K$ be its field of fractions. Assume that $A$ contains a finite field $\mathbb{F}_q$ with $q=p^r$ elements for a prime $p$. Let $n\geq 2$ and consider a finite subgroup $G$ of $\mathrm{GL}(A^n)$ such that every non-identity element of $G$ inside $\mathrm{GL}(K^n)$ is a transvection. Consider the ring $A[X_1,X_2,\dots, X_n]$ and let $G$ act linearly on the ring (fixing $A$). Then $(A[X_1,X_2,\dots, X_n])^{G}$ is regular.

We study an inverse problem for the time-dependent nonlinear fourth-order Schrödinger equation on both compact Euclidean domains and compact Riemannian manifolds, (say) $M$. This model arises in nonlinear fiber optics and the theory of optical solitons in gyrotropic media. Our main objective is the identification of unknown coefficients from the associated source-to-solution map, which assigns to each source term $f$, supported in $(0, T)\times \Gamma$, the corresponding solution $u$ restricted to the same set, where $\Gamma \subset M$ is a neighborhood of $\partial M$. We prove that the zeroth-order term, the second-order coefficient, and the nonlinear coefficient are uniquely determined by this map. Moreover, the recovery of the symmetric second-order tensor reduces to the inversion of a divergent beam transform.

In this paper, we establish Riesz--Fejér type inequalities for the $\alpha$-harmonic functions $f=P_{\alpha}[f^*]$ in $\mathbb B^n$, where $f^*\in L^{p}(\mathbb{S}^{n-1})$ and $1<p<\infty$. More precisely, for $n\geq2$ and $\alpha>-1$, we prove the existence of a constant $\mathcal{C}_{n,p,\alpha}$ such that $\int_{-1}^{1} |f(r\eta)|^p(1-r^2)^{n-2}\,dr \leq \mathcal{C}_{n,p,\alpha} \int_{\mathbb S^{n-1}}|f^*(\xi)|^p\,d\sigma(\xi)$. Moreover, in the range $\alpha>\max\left\{ -\frac{n-1}{p},\,n-2-\frac{2(n-1)}{p} \right\}$, we determine the sharp constant explicitly. The result generalize and extend the corresponding results of Ahmed et al. (J. Math. Anal. Appl., 563:13, 2026), Hu et al. (Anal. Math. 51:15, 2025) and Long (arXiv: 2410.12137 ).

In a world of various competing quantum computing architectures, hardware-agnostic, full-stack platforms are necessary to bring the full power of quantum computing hardware to domain experts via the cloud. QCI Connect and its Software Development Kit provide a reference architecture for a full-stack platform with a modular design and open-source interface definitions, built to facilitate a community-driven application ecosystem. Here, we present its overall design and features, central interfaces, and lessons learned, both for users of the platform and as a reference guide for future developments.

Accurate and efficient decoding is a crucial component for achieving fault-tolerant quantum computing. Realistic circuit-level noise introduces temporal correlations and degeneracy, making optimal (maximum-likelihood) decoding computationally intractable in general. As a result, practical decoders rely on heuristic approximations, and it is generally difficult to quantify how suboptimal they are, as this strongly depends on the code and noise model considered. In this work, we study the accuracy of practical decoding algorithms under circuit-level noise by comparing them against a maximum likelihood decoding benchmark. Our approach propagates the density matrix through the full memory experiment and computes the optimal decoding decision for each syndrome history. We introduce pruning techniques with rigorous bounds, allowing us to access larger numbers of syndrome-extraction rounds. We apply this framework to small instances of the repetition code and a cellular automaton code, and benchmark minimum-weight perfect matching (MWPM), belief propagation with ordered statistics decoding (BP+OSD), Tesseract, and Planar decoders against optimal decoding. While standard decoders remain close to optimal for the repetition code, we find significant deviations for the cellular automaton code, with BP+OSD deteriorating already in experimentally relevant noise regimes. Moreover, the pruning method developed here highlights that, at low physical error rates, only a narrow fraction of syndrome histories contributes significantly to the logical error rate.

We study entanglement measures for oscillating neutrinos in the wave-packet formalism and confirm that an oscillating neutrino exhibits genuine tripartite entanglement and can be characterized as a member of the W class of states. We also show that this feature survives in the decoherence limit, and discuss the effect of CP phases on the entanglement. These findings provide new insights into the quantum nature of neutrino oscillations and their potential role in quantum information science, especially for neutrinos propagating large distances such as the astrophysical neutrinos observed at neutrino telescopes.

Independent Redistricting Commissions (IRCs) are a promising tool for bottom-up redistricting, but their public testimony processes are vulnerable to adversarial manipulation. We propose using differential privacy to draw redistricting plans that incorporate community of interest (COI) testimonies while remaining robust to adversarial input. Treating individual testimonies as data points, we use the marked edge walk to sample from differentially private distributions of redistricting plans via the exponential mechanism. We introduce two score functions and demonstrate that both can be targeted by MEW across a range of privacy budgets. Applying this method to Missouri's mid-cycle redistricting using 808 COI testimonies, we show that COI-informed sampling outperforms an uninformed baseline and the enacted plan. An adversarial experiment demonstrates that the method can be robust to attacks under certain privacy budgets and may perform better in practice than formal group privacy guarantees imply. We also find that stronger COI preservation tends to spread minority and Democratic representation more evenly across districts.

Multistability is a well-known feature of the inertial Kuramoto system (KMI). Here, we demonstrate that an interplay of phase lag and triadic interactions in KMI leads to distinct hysteresis widths corresponding to different stable states. This phenomenon becomes more pronounced with increasing inertia. Theoretical calculations for the backward branch based on self-consistent analysis show that these multiple widths arise from saddle-node bifurcation occurring at different coupling strengths. Moreover, the forward branch corresponds to oscillatory state and does not admit steady-state solution. The study of multiple hysteresis widths may be useful in modeling power grid systems, information storage, and memory selection in real-word systems.

Hybrid baryons, in which gluonic degrees of freedom play an explicit dynamical role, provide a key testing ground for nonperturbative quantum chromodynamics. In this work, we investigate the mass spectrum of light hybrid baryons composed of identical quarks within a phenomenological constituent framework, applied to a quark core-gluon approximation. In this approach, the hybrid baryon is described as a bound state of a color-octet three-quark core and a constituent gluon, allowing the original four-body problem to be reduced to a three-body calculation followed by an effective two-body treatment. The spectrum of the color-octet quark core is obtained by solving a semirelativistic three-quark Hamiltonian with linear confinement, Coulomb, and regularized hyperfine interactions using an oscillator basis expansion. Finite-size effects of the core are incorporated through the convolution of the effective core-gluon interaction with the spatial quark density. The resulting two-body problem, whose associated Hamiltonian has the same shape as the one of the core, is solved applying the helicity formalism and using the Lagrange mesh method. Our results predict the lightest hybrid baryons to occur at energies above $3~\mathrm{GeV}$, with negative-parity states generally lying below their positive-parity counterparts. The predicted spectra are compared with lattice QCD and QCD sum-rule calculations, showing qualitative agreement although the lowest-lying lattice QCD results are significantly lighter than the present ones. Possible extensions of the model and implications for future experimental searches are discussed.

Let \((w_{ij})_{i,j\ge1}\) be a single infinite array of independent identically distributed real- or complex-valued entries of mean zero, variance \(\sigma^2\), and finite fourth moment. Set \(W_n=(w_{ij})_{1\le i,j\le n}\) and \(X_n=n^{-1/2}W_n\). For every fixed \(k\ge1\), we identify the almost sure limiting operator norm of several fixed products built from this family. Define the \(k\)-th freeness coefficient by \[ \gamma_k:=\sqrt{\frac{(k+1)^{k+1}}{k^k}}. \] Then we prove \[ \|X_n^k\|\to\sigma^k\gamma_k \qquad \text{almost surely}. \] The same limit holds for products sampled with replacement from any fixed finite pool of independent copies of \(X_n\); in particular, it holds for the product of \(k\) independent copies. Thus, the freeness coefficient captures the non-commuting characteristic between large random matrices %powers and independent or fixed-pool sampled products under the finite fourth moment assumption. The improvement of the classical Bai--Yin-type power estimate from the scale \(\sigma^k(k{+}1)\) to \(\sigma^k \sqrt{k{+}1}\) is a direct corollary of our result. The main technical challenge is to prove the upper bound using a high-moment expansion of %the upper bound is proved by a high-moment expansion of \(\E\Tr((X_n^kX_n^{*k})^m)\). The leading zero-defect trace words are tree-like and are counted by the Fuss--Catalan number \[ F_{k,m}= \frac1{km+1}\binom{(k+1)m}{m}. \] The combinatorial tool helps to devise a defect-sensitive global enumeration: if \(L=km\) and \[ r=(L+1-v)+(L-q), \] then the number of admissible word classes with defect \(r\) is at most \(F_{k,m}(Cm)^{Dr}\). This polynomial-in-\(m\) loss, with degree proportional to the defect, is summable in the logarithmic moment range.

Cryogenic electron microscopy images sample individual biomolecules from their conformational landscapes, offering a route to infer the distributions underlying molecular mechanisms. However, because images are indirect measurements, they limit which features of an underlying landscape are statistically identifiable. In ensemble reweighting, this problem appears as a choice of resolution: conformational space is discretized into representative structures whose population weights are inferred from images. Adding structures increases nominal resolution, but nearby conformations may generate overlapping image distributions and indistinguishable weights. Here, we develop an information-theoretic framework that selects representative conformations by maximizing mutual information between ensemble weights and images under a probabilistic forward model. Analytically, we show in a one-dimensional Gaussian model that measurement noise sets the optimal spacing. Applied to molecular conformations sampled from simulation, the framework constructs near-optimal ensembles that span heterogeneity while avoiding redundancy. Thus, the measurement process induces a maximally learnable coarse graining of conformation space.

Cooperative massive multiple-input multiple-output (MIMO) promises large gains over cellular deployments, but existing comparisons of different architectures often mix antenna distribution, inter-site coordination, and processing assumptions. This paper introduces a graph-based framework for fair comparison of cellular, coordinated, and cell-free massive-MIMO systems. We differentiate between two key properties, namely antenna distribution and inter-site cooperation, which yields seven representative system types. We derive compatible uplink and downlink spectral efficiency (SE) expressions, including an uplink bound for detectors with mixed instantaneous and statistical effective channel state information (CSI), and adapt scalable user association and processing rules to all considered architectures. We evaluate these systems using extensive numerical simulations and show that for a fair comparison much larger simulation areas (at least 2.5 $\times$ 2.5 km2) than commonly used are required. We introduce the relative capacity, which measures how closely each architecture approaches centralized cell-free processing. The results show that coordinated, phase-aligned beamforming across spatially distributed antennas is the main source of cooperation gains. In dense deployments with few antennas per access point (AP), coordinated Distributed Antenna System (DAS) and hybrid cell-free architectures achieve much of the centralized cell-free performance while requiring substantially weaker midhaul assumptions.

Shallow-depth QAOA on sparse, high-diameter MaxCut instances faces a locality bottleneck: at depth \(p\), local observables can depend only on a bounded neighborhood of the circuit interaction graph. We propose a transport-augmented QAOA that keeps the MaxCut cost Hamiltonian unchanged but enriches the mixer with optimized, unweighted shortcut couplings (scheduled \(XX+YY\)) to collapse the effective interaction-graph diameter. Using exact finite-depth support recursions, we relate optimal shortcut placement to bounded-diameter graph augmentation, and show in benchmarks that (unlike ma-QAOA) performance becomes effectively size-invariant once the diameter is reduced. For bipartite families (base diameter 4), reducing the interaction path to \(d=1\) raises the ensemble-averaged approximation ratio from 0.7378 (ma-QAOA) to 0.9767 at \(p=1\) (\(\sigma=0.0251\), nine system sizes); on random trees (base diameter 10), at \(p=2\) it improves from 0.9226 to 0.9997 (\(\sigma=0.0001\)).

We consider a hydrodynamic model of flocking-type with pressure on the torus, with integrable interaction kernel and density bounded away from zero. We prove that, if an entropy weak solution exists, then its $L^2$ norm decays exponentially fast in time towards the mean values on the period. The proof relies on the study of a suitable energy functional that combines a strictly convex entropy for the system and a potential term, and this allows us to treat the nonlocal source term for a class of strictly positive convolution kernels in $L^1$.

We study the logarithmic Schrödinger equation \[ -\Delta u+V(x)u=u\log u^2,\qquad x\in\mathbb R^N,\ N\ge3. \] Since the logarithmic energy is not \(C^1\) on the natural space \(H_V^1(\mathbb R^N)\), direct invariant-set minimax arguments for sign-changing solutions are not available. We introduce an \(L^p\)-regularization perturbation, which restores a \(C^1\) variational structure while preserving the logarithmic nonlinearity, and prove via a limiting argument that the original equation admits infinitely many sign-changing weak solutions.

We study the semiclassical limit of the Polyakov-Liouville measure $\boldsymbol{\nu}_\gamma$, which is a non-Gaussian measure on $H^{-\eps}(M)$ that has recently been extended from Riemann surfaces to general Riemannian manifolds $(M,g)$ of even dimension. We show that under an appropriate rescaling in the semiclassical limit as $\gamma\to0$, the normalized Polyakov-Liouville measure $\Q_\gamma$ concentrates on the unique smooth weight $u$ for which the conformal metric $e^{2u}g$ on $M$ has constant $Q$-curvature.

Inclusion minimal convex bodies $K$ with the property that the integer translates of $K$ cover the space are studied. Such bodies are referred to as minimal covering bodies and it is shown that, while they are not necessarily tiles, they are polytopes with at least $2d$ facets, if $d$ is the dimension of $K$. Moreover, minimal covering bodies are related to covering properties of Minkowski combinations of convex bodies. Two sharp Brunn Minkowski type inequalities are established for the covering radius of the Minkowski sum of planar convex bodies.

We compute the decay width of neutrinos in a thermal medium using finite-temperature quantum field theory, focusing on non-standard decays into lighter neutrinos and a scalar or light vector boson. We derive general expressions for the thermal decay rate and show that finite-temperature effects can dramatically enhance neutrino decays when the parent and daughter states are nearly degenerate in mass. In this regime, the emitted boson is kinematically soft and undergoes strong Bose enhancement, leading to decay widths that can exceed their vacuum values by a couple of orders of magnitude. We demonstrate that this effect is largely insensitive to the Lorentz structure of the underlying interaction and arises generically from the interplay of thermal occupation factors and quasi-degenerate kinematics. Our results highlight a previously underappreciated feature of neutrino decay in thermal environments and provide a general framework applicable to a broad class of fermionic decay processes.

Solar energetic particle (SEP) events, defined by proton flux exceeding 10 pfu in the $>$ 10 MeV channel, pose major risks to spacecraft operations, astronaut safety, and high-latitude aviation. Due to the complexity and rarity of SEP events, reliable operational SEP forecasting remains an important challenge in space weather. Here we present a novel 24-hour-ahead operational forecasting framework based on a multi-task learning structure with a thoroughly constructed list of features from multiple sources that span over multiple solar cycles. It extends an earlier-introduced \texttt{SEPNET}-based models by integrating a broader range of solar observations, including active region magnetic parameters from SHARP and SMARP, solar flares information, coronal mass ejections, soft X-ray flux, and historical $>$ 10 MeV proton flux. The inclusion of SMARP data expands the temporal coverage of magnetic-field predictors to earlier solar cycles, while flux-based inputs provide additional precursor information. Building on the multi-task architecture, we develop \texttt{SEPNET-Ov2}, which jointly predicts SEP event occurrence together with future proton flux and soft X-ray flux. Evaluation on the CLEAR SEP benchmark dataset shows improved classification performance over the earlier \texttt{SEPNET-O} (operational version of \texttt{SEPNET}) on the newly aligned dataset. The best operational model is obtained when magnetic, radiative, and proton-flux predictors are combined, highlighting the value of expanded historical coverage and complementary precursor information for improving operational SEP forecasting.

Determining accurate wave dispersion relations is a central problem in plasma physics. Recent advances have enabled the numerical computation of linear dispersion relation in plasmas with arbitrary particle velocity distribution functions (VDFs), using two distinct solvers, BO and ALPS. Their reliability and mutual consistency, however, have not been systematically tested for a broad range of VDFs. Here we compare the dispersion relations obtained from BO and ALPS for several representative distributions. We find that the two solvers give consistent unstable modes for kappa distributions with large values of $\kappa$, as well as for ring-beam, shell, and proton core-beam distributions. BO, however, becomes unreliable for kappa distributions with $\kappa < 4$. For an observationally derived VDF, the two solvers give similar real frequencies for the unstable waves but substantially different growth rates. This difference is mainly caused by the imperfect fitting of the input distribution required by BO. Despite this limitation, BO has a clear computational advantage because it can obtain all roots in a single run. Considering the complementary strengths of the two solvers, their combined use can provide a more reliable and effective framework for investigating instabilities in non-Maxwellian plasma environments.

Quantum projection noise is the fundamental noise source for the population measurement of spin ensembles. While projection-noise-limited measurements have been extensively studied in atomic systems, corresponding experiments on solid-state spin ensembles remain challenging due to dominant classical readout noise. Here, we report direct measurement of the quantum projection noise of mesoscopic ensembles of nitrogen-vacancy (NV) spin defects at room temperature. Our experiment is enabled by a high optically-detected magnetic resonance (ODMR) contrast of over 20% for a single crystallographic orientation of the defect spins, obtained by combining polarization-selective optical excitation with spin-to-charge conversion. We use our protocol to demonstrate projection noise measurements and spin counting from nanoscale NV ensembles of up to 43 spins. We further demonstrate that the protocol allows for significant gains in sensitivity for magnetometry applications without need for cryogenic operation or high bias magnetic fields.

We introduce a new convex optimization framework for logistic scalar-on-matrix regression which incorporates nuclear and $\ell_1$ norm penalties to enforce simultaneously low-rank and sparse structures in the estimated coefficient matrix. The proposed method enables interpretable modeling of high-dimensional matrix-valued predictors in the presence of binary responses. We derive a custom algorithm based on the Alternating Direction Method of Multipliers (ADMM) to efficiently solve the resulting convex optimization problem and establish the theoretical properties of the obtained solution. Numerical experiments clearly demonstrate the effectiveness of our method in recovering meaningful predictive patterns. Finally, we apply our method to the brain imaging data to identify structures in functional brain connectivity matrices that are characteristic of subjects with a family history of alcohol use disorders (AUDs).

Good and Meddaugh proved that every compact metric dynamical system with shadowing is a factor of the inverse limit of an inverse sequence of shifts of finite type. We show first that, for this factor representation alone, both assumptions are unnecessary: every compact Hausdorff dynamical system is a factor of the inverse limit of an inverse system of shifts of finite type. In particular, the mere existence of such a symbolic inverse-limit representation is not specific to shadowing. The main contribution of the paper is to identify the additional stability which shadowing provides in the metric case. We prove that every compact metric system with shadowing is a factor of the inverse limit of an inverse sequence of shifts of finite type whose bonding maps are surjective. Hence the inverse sequence satisfies the Mittag-Leffler condition, and the corresponding zero-dimensional extension still has shadowing. This strengthens the metric representation theorem of Good and Meddaugh and completes their characterization in terms of ALP factors of Mittag-Leffler inverse sequences of shifts of finite type. Finally, for arbitrary compact Hausdorff spaces, we show that every compact shadowing system is conjugate to the inverse limit of metrizable shadowing systems with factor bonding maps. In this sense, compact shadowing systems are generated from shifts of finite type by applying, at most three times, the two shadowing-preserving operations of taking Mittag-Leffler inverse limits and passing to ALP factors.

A nonlinear orbital station-keeping solution for the circular and elliptic versions of the Earth-Moon Restricted Three-Body Problem (R3BP) is developed via a backstepping technique. Formal guarantees for global asymptotic stability (GAS) are attained, as shown through Lyapunov's stability theory. The adequacy of the proposed control law is evaluated through the means of numerical trials over closed periodic solutions of the circular and elliptic R3BPs. The ramifications of the control gain choice are carefully studied and simulated.

Based on a variational characterization of the local slope of a functional, we extend the celebrated Brezis-Ekeland-Nayroles null-minimization principle to curves of maximal slope in metric spaces. In the time continuous setting, we check that the ensuing global-in-time functionals admit minimizers.

In dual-gated bilayer graphene, a perpendicular displacement field opens a band gap that modifies both the channel conductance and the electronic compressibility, motivating measurements that resolve resistive and capacitive responses on the same device. We integrate an hBN-encapsulated bilayer graphene heterostructure with an on-chip superconducting Nb lumped-element LC resonator and carry out RF reflectometry near 4.25 GHz. DC transport and finite-bias spectroscopy on the same device provide a transport reference. Top and bottom gates independently set the carrier density and displacement field. The DC and RF gate maps share the same gate-dependent features, with finite-bias measurements revealing a region of suppressed conductance whose bias extent grows with the displacement field, consistent with a field-induced gap. The gate-dependent resonance-frequency shift is converted to the effective capacitance seen by the resonator using an equivalent-circuit model. The capacitance shows a minimum near the conductance-suppressed region, consistent with reduced electronic compressibility in the gapped bilayer graphene, and exhibits an electron-hole asymmetry. The on-chip configuration probes the gate-dependent admittance of a dual-gated van der Waals heterostructure, providing capacitance-sensitive information that complements DC transport within a single device.

Partial differential equations (PDEs) regulate the behaviour of countless spatiotemporal systems in the physical and life sciences. In many cases, they encode the coupling between the system's degrees of freedom, leading to nonlinear equations whose solution space is challenging to explore exhaustively. Systematic approaches to PDE model exploration are a holy grail of computational science. In this article, we formulate a criterion for increasing the diversity of a search campaign, based on the PDE residual behaviour under solution deformation. We develop a practical formalism to compute this property and illustrate its role in a few cases of interest.