The aim of this article is to find appropriate definitions for shifts of finite type and sofic shifts in a general context of symbolic dynamics. We start showing that the classical definitions of shifts of finite type and sofic shifts, as they are given in the context of finite-alphabet shift spaces on the one-dimensional monoid $\mathbb{N}$ or $\mathbb{Z}$ with the usual sum, do not fit for shift spaces over infinite alphabet or on other monoids. Therefore, by examining the core features in the classical definitions of shifts of finite type and sofic shifts, we propose general definitions that can be used in any context. The alternative definition given for shifts of finite type inspires the definition of a new class of shift spaces which intersects with the class of sofic shifts and includes shifts of finite type. This new class is named finitely defined shifts, and the non-finite-type shifts in it are named shifts of variable length. For the specific case of infinite-alphabet shifts on the lattice $\mathbb{N}$ or $\mathbb{Z}$ with the usual sum, shifts of variable length can be interpreted as the topological version of variable length Markov chains.
In an atom, the interaction of a bound electron with the vacuum fluctuations of the electromagnetic field leads to complex shifts in the energy levels of the electron, with the real part of the shift corresponding to a shift in the energy level and the imaginary part to the width of the energy level. The most celebrated radiative shift is the Lamb shift between the $2S_{1/2}$ and the $2P_{1/2}$ levels of the hydrogen atom.~The measurement of this shift in 1947 by Willis Lamb Jr. proved that the prediction by Dirac theory that the energy levels were degenerate was incorrect. Hans~Bethe's calculation of the shift demonstrated the renormalization process required to deal with the divergences plaguing the existing theories and led to the understanding that it was essential for theory to include interactions with the zero-point quantum vacuum field. This was the birth of modern quantum electrodynamics (QED). Other calculations of the Lamb shift followed by Welton and Power in an effort to clarify the physical mechanisms leading to the shift. We have done a calculation of the shift using a group theoretical approach which gives the shift as an integral over frequency of a function, which
Light shift and vapor-cell temperature shift are the two most significant factors dominating the long-term instability of compact atomic clocks. Due to the different physical mechanisms, there is not yet a solution that can effectively suppress the frequency shifts induced by these two effects. Here, we propose a 'resonance-offset' locking approach that compensates for the two physical frequency shifts. In this approach, the additional offset locking shift can effectively counteract the atomic resonance shifts arising from changes in vapor-cell temperature and light power, reducing the net impact on the clock's frequency to nearly zero. We have demonstrated this strategy on the 778 nm Rubidium two-photon optical frequency standard, successfully compensating for light shift and cell-temperature shift, respectively. This general method is particularly appealing for compact vapor-cell microwave and optical atomic clocks designed for the excellent stability rather than accuracy.
Safety classifiers deployed in production operate under a stationarity assumption that fails silently: when input distributions drift, accuracy degrades with no error signal until ground-truth labels arrive. We present an online monitor that detects distributional shift in classifier scores via a sliding-window KS statistic with empirically calibrated alarm thresholds. In a pre-registered factorial evaluation (4 classifiers $\times$ 5 shift conditions $\times$ 20 seeds $\times$ 2 window sizes; 800 cells), the monitor achieves 86.6% valid detection (mean latency 39.5 steps) across synthetic-onset, real-jailbreak, and adversarial regimes; a classifier $\times$ shift interaction ($η^2 = 0.185$) shows that monitoring must be tuned per classifier. Attempting to recover post-detection coverage via weighted conformal prediction exposes a failure mode: density-ratio estimation collapses for generative classifiers because logistic regression separates source from target perfectly in 3584-4096-dimensional embedding space, clipping all importance weights to zero; projecting to $\leq 32$ dimensions restores coverage. We then extend the framework to gradient-based evasion and give the first thr
This paper is a corrigendum to the article 'Some notes on the classification of shift spaces: Shifts of Finite Type; Sofic Shifts; and Finitely Defined Shifts'. In this article we correct Lemma 5.3. Therefore, we follow correcting statements and proofs of subsequent results that depend on Lemma 5.3.
Natural distribution shift causes a deterioration in the perception performance of convolutional neural networks (CNNs). This comprehensive analysis for real-world traffic data addresses: 1) investigating the effect of natural distribution shift and weather augmentations on both detection quality and confidence estimation, 2) evaluating model performance for both classification and object localization, and 3) benchmarking two common uncertainty quantification methods - Ensembles and different variants of Monte-Carlo (MC) Dropout - under natural and close-to-natural distribution shift. For this purpose, a novel dataset has been curated from publicly available autonomous driving datasets. The in-distribution (ID) data is based on cutouts of a single object, for which both class and bounding box annotations are available. The six distribution-shift datasets cover adverse weather scenarios, simulated rain and fog, corner cases, and out-of-distribution data. A granular analysis of CNNs under distribution shift allows to quantize the impact of different types of shifts on both, task performance and confidence estimation: ConvNeXt-Tiny is more robust than EfficientNet-B0; heavy rain degra
This paper introduces Gate-Shift-Pose, an enhanced version of Gate-Shift-Fuse networks, designed for athlete fall classification in figure skating by integrating skeleton pose data alongside RGB frames. We evaluate two fusion strategies: early-fusion, which combines RGB frames with Gaussian heatmaps of pose keypoints at the input stage, and late-fusion, which employs a multi-stream architecture with attention mechanisms to combine RGB and pose features. Experiments on the FR-FS dataset demonstrate that Gate-Shift-Pose significantly outperforms the RGB-only baseline, improving accuracy by up to 40% with ResNet18 and 20% with ResNet50. Early-fusion achieves the highest accuracy (98.08%) with ResNet50, leveraging the model's capacity for effective multimodal integration, while late-fusion is better suited for lighter backbones like ResNet18. These results highlight the potential of multimodal architectures for sports action recognition and the critical role of skeleton pose information in capturing complex motion patterns. Visit the project page at https://edowhite.github.io/Gate-Shift-Pose
We investigate the computability (in the sense of computable analysis) of the topological pressure $P_{\rm top}(φ)$ on compact shift spaces $X$ for continuous potentials $φ:X\to {\mathbb R}$. This question has recently been studied for subshifts of finite type (SFTs) and their factors (Sofic shifts). We develop a framework to address the computability of the topological pressure on general shift spaces and apply this framework to coded shifts. In particular, we prove the computability of the topological pressure for all continuous potentials on S-gap shifts, generalized gap shifts, and particular Beta-shifts. We also construct shift spaces which, depending on the potential, exhibit computability and non-computability of the topological pressure. We further prove that the generalized pressure function $(X,φ)\mapsto P_{\rm top}(X,φ\vert_{X})$ is not computable for a large set of shift spaces $X$ and potentials $φ$. In particular, the entropy map $X\mapsto h_{\rm top}(X)$ is computable at a shift space $X$ if and only if $X$ has zero topological entropy. Along the way of developing these computability results, we derive several ergodic-theoretical properties of coded shifts which are
Search-Based Software Testing (SBST) automates test input generation but is frequently hindered by challenging fitness landscapes characterized by numerous deceptive local optima that impede search progress, as well as extended plateaus where informative fitness signals are scarce. To address this bottleneck, we propose SHIFT (Sigmoid-Based Heuristic Invertible Fitness-Landscape Transformation for Accelerating SBST), a method designed to compress local landscapes and facilitate escape from stagnant regions without altering global semantics. By systematically contracting dense regions where search points cluster, the approach preserves mapping invertibility while enabling optimization algorithms to traverse more effectively toward global coverage with the same step size. When evaluated against established baselines, including pure hill climbing and genetic algorithms, under a normalized experimental protocol, the proposed technique yields consistent improvements in convergence speed and search efficiency. These results demonstrate that sigmoid compression constitutes a lightweight yet effective mechanism for achieving more reliable coverage discovery in complex testing environments.
Africa's green industrialization imperative demands reliable infrastructure for monitoring air quality. We present a satellite-reanalysis PM2.5 fusion system trained on 2,068,901 records from 404 monitoring locations in 29 African countries (OpenAQ, 2017-2022), combining LightGBM with leakage-resistant spatial cross-validation and conformal prediction to quantify predictions and their geographic applicability limits. Under 5-fold location-grouped spatial cross-validation, LightGBM achieves RMSE = 30.83 +/- 5.07 ug/m3, MAE = 14.54 +/- 1.66 ug/m3, R2 = 0.134 +/- 0.023, and macro F1 = 0.336 +/- 0.018. This R2 is substantially below random-split benchmarks (>0.90) but reflects true geographic generalisation difficulty rather than model failure. Split conformal prediction targeting 90% marginal coverage reveals severe East Africa degradation (actual PICP = 65.3% vs. nominal 90%), consistent with medium-strength covariate shift (humidity KS = 0.2237, sat_pblh KS = 0.2558). We operationalise these findings through regional reliability flags (High/Medium/Low/Unreliable) and a monitor prioritisation score directing infrastructure expansion toward highest-burden unmonitored populations, d
We study transmission stabilization of optical solitons against emission of radiation in nonlinear optical waveguides in the presence of weak linear gain-loss, cubic loss, and the collisional Raman frequency shift. We first show how the collisional Raman frequency shift perturbation arises in three different physical setups. We then show by numerical simulations with a perturbed nonlinear Schrödinger (NLS) model that transmission in waveguides with weak frequency-independent linear gain is unstable. The radiative instability is stronger than the radiative instabilities that were observed in earlier studies for soliton transmission in the presence of weak linear gain, cubic loss, and various frequency-shifting physical mechanisms. In particular, the Fourier spectrum of the radiation is significantly more spiky and broadband than the radiation's Fourier spectra in earlier studies. Moreover, we demonstrate by numerical simulations with another perturbed NLS model that transmission in waveguides with weak frequency-dependent linear gain-loss, cubic loss, and the collisional Raman frequency shift is stable. Despite the stronger radiative instability in the corresponding waveguide setup
We investigate the feasibility of extracting infinite volume scattering phase shift on quantum computers in a simple one-dimensional quantum mechanical model, using the formalism established in Ref.~\cite{Guo:2023ecc} that relates the integrated correlation functions (ICF) for a trapped system to the infinite volume scattering phase shifts through a weighted integral. The system is first discretized in a finite box with periodic boundary conditions, and the formalism in real time is verified by employing a contact interaction potential with exact solutions. Quantum circuits are then designed and constructed to implement the formalism on current quantum computing architectures. To overcome the fast oscillatory behavior of the integrated correlation functions in real-time simulation, different methods of post-data analysis are proposed and discussed. Test results on IBM hardware show that good agreement can be achieved with two qubits, but complete failure ensues with three qubits due to two-qubit gate operation errors and thermal relaxation errors.
Positive topological entropy and distributional chaos are characterized for hereditary shifts. A hereditary shift has positive topological entropy if and only if it is DC2-chaotic (or equivalently, DC3-chaotic) if and only if it is not uniquely ergodic. A hereditary shift is DC1-chaotic if and only if it is not proximal (has more than one minimal set). As every spacing shift and every beta shift is hereditary the results apply to those classes of shifts. Two open problems on topological entropy and distributional chaos of spacing shifts from an article of Banks et al. are solved thanks to this characterization. Moreover, it is shown that a spacing shift $Ω_P$ has positive topological entropy if and only if $\mathbb{N}\setminus P$ is a set of Poincaré recurrence. Using a result of Kříž an example of a proximal spacing shift with positive entropy is constructed. Connections between spacing shifts and difference sets are revealed and the methods of this paper are used to obtain new proofs of some results on difference sets.
We consider estimating the proportion of random variables for two types of composite null hypotheses: (i) the means or medians of the random variables belonging to a non-empty, bounded interval; (ii) the means or medians of the random variables belonging to an unbounded interval that is not the whole real line. For each type of composite null hypotheses, uniformly consistent estimators of the proportion of false null hypotheses are constructed for random variables whose distributions are members of a Type I location-shift family. Further, uniformly consistent estimators of certain functions of a bounded null on the means or medians are provided for the random variables mentioned earlier; these functions are continuous and of bounded variation. The estimators are constructed via solutions to Lebesgue-Stieltjes integral equations and harmonic analysis, do not rely on a concept of p-value, and have various applications.
Goos-Hanchen (GH) and Imbert-Fedorov (IF) shifts are lateral and transverse displacements of a wavepacket reflecting off a surface. A dramatic real-space manifestation of wavepacket phases, they have traditionally been analyzed in a model dependent fashion. Here we argue that GH and IF shifts admit a general geometrical description and arise from a gauge invariant geometric phase. In particular, we show GH/IF shifts can be naturally captured by a shift vector, analogous to the shift vector from shift currents in the bulk photovoltaic effect. Employing Wilson loops to visualize the scattering processes contributing to the shift vector, we separate the shift into an intrinsic (depends solely on the system bulk) and an extrinsic part. This enables to establish a clear model-independent link between symmetry and the presence/absence of intrinsic and extrinsic GH/IF shifts.
This paper presents a novel theoretical derivation of the Lamb shift in the hydrogen atom, based solely on fundamental constants and the stochastic (Brownian) motion of the proton. Unlike conventional quantum electrodynamics (QED), the proposed approach introduces no experimentally fitted parameters, offering a fully self-consistent explanation grounded entirely in known physical quantities.
Blurring mean shift (BMS) algorithm, a variant of the mean shift algorithm, is a kernel-based iterative method for data clustering, where data points are clustered according to their convergent points via iterative blurring. In this paper, we analyze convergence properties of the BMS algorithm by leveraging its interpretation as an optimization procedure, which is known but has been underutilized in existing convergence studies. Whereas existing results on convergence properties applicable to multi-dimensional data only cover the case where all the blurred data point sequences converge to a single point, this study provides a convergence guarantee even when those sequences can converge to multiple points, yielding multiple clusters. This study also shows that the convergence of the BMS algorithm is fast by further leveraging geometrical characterization of the convergent points.
Building on our previous work, we give a thorough presentation of the techniques developed for synchronizing dynamical systems in the special case of synchronizing shift spaces. Following work of Thomsen, we give a construction of the homoclinic, the heteroclinic, and synchronizing heteroclinic $C^\ast$-algebras along with the synchronizing ideal of a shift space in terms of Bratteli diagrams. The algebras introduced in our previous work (the synchronizing ideal, and synchronizing heteroclinic algebra) are discussed in detail. In the sofic shift case, these algebras are shown to be related to the $C^\ast$-algebras of its minimal left and minimal right presentations. Several specific examples are discussed to demonstrate these techniques. For the even shift we give a complete computation of all the associated invariants. We discuss these algebras for a sofic shift that is not of almost finite type and for a number of strictly non-sofic synchronizing shifts. In particular we discuss the rank of the $K$-theory of the homoclinic algebra of a shift space and its synchronizing ideal and its implications. We also give a construction for producing from any minimal shift a synchronizing shi
Tuning the density of resident electrons or holes in semiconductors provides crucial insight into the composition of excitonic complexes that are observed as absorption or photoluminescence resonances in optical studies. Moreover, we can change the way these resonances shift and broaden in energy by controlling the quantum numbers of the resident carriers with magnetic fields and doping levels, and by selecting the quantum numbers of the photoexcited or recombining electron-hole (e-h) pair through optical polarization. We discuss the roles of distinguishability and optimality of excitonic complexes, showing them to be key ingredients that determine the energy shifts and broadening of optical resonances in charge-tunable semiconductors. A distinguishable e-h pair means that the electron and hole undergoing photoexcitation or recombination have quantum numbers that are not shared by any of the resident carriers. An optimal excitonic complex refers to a complex whose particles come with all available quantum numbers of the resident carriers. All optical resonances may be classified as either distinct or indistinct depending on the distinguishability of the e-h pair, and the underlying
Analysis of the first-order corrections to higher-spin equations is extended to homotopy operators involving shift parameters with respect to the spinor $Y$ variables, the argument of the higher-spin connection $ω(Y)$ and the argument of the higher-spin zero-form $C(Y)$. It is shown that a relaxed uniform $(y+p)$-shift and a shift by the argument of $ω(Y)$ respect the proper form of the free higher-spin equations and constitute a one-parametric class of vertices that contains those resulting from the conventional (no shift) homotopy. A pure shift by the argument of $ω(Y)$ is shown not to affect the one-form higher-spin field $W$ in the first order and, hence, the form of the respective vertices.