Time Evolution Research Articles

Combinatorial optimization with quantum imaginary time evolution

Combinatorial optimization with quantum imaginary time evolution

Estimation of battery temperature during drive cycle operation by the time evolution of voltage and current

During the operation of electric vehicles, accurate temperature monitoring of lithium-ion batteries by battery management system is crucial to their safety. However, temperature sensors possess limitations as they necessitate regular calibration to avoid inaccuracies, which can be challenging to accomplish in certain scenarios. As an alternative, a general sensorless temperature estimation framework is developed in this study to either serve as a complementary approach alongside sensors or function independently to ensure reliable temperature measurement. The proposed framework relies solely on data regarding battery voltage and current measurements, and this information is employed to determine the parameters of the equivalent circuit model via the vector-type recursive least squares algorithm in the first step. In the second step, these parameters and the voltage/current data serve as the input to the deep neural network (DNN) to estimate the instantaneous battery surface temperature during dynamic driving cycles under various ambient temperatures and degradation scenarios. To reflect the effect of past battery operation on the current temperature, a newly defined current integral function is proposed, by which the input time length of the DNN model is suggested. Four combinations of the model input are studied, and the best input option yields an average estimation error of the battery temperature that falls below 0.5 °C.

Two telomere-to-telomere gapless genomes reveal insights into Capsicum evolution and capsaicinoid biosynthesis

Chili pepper (Capsicum) is known for its unique fruit pungency due to the presence of capsaicinoids. The evolutionary history of capsaicinoid biosynthesis and the mechanism of their tissue specificity remain obscure due to the lack of high-quality Capsicum genomes. Here, we report two telomere-to-telomere (T2T) gap-free genomes of C. annuum and its wild nonpungent relative C. rhomboideum to investigate the evolution of fruit pungency in chili peppers. We precisely delineate Capsicum centromeres, which lack high-copy tandem repeats but are extensively invaded by CRM retrotransposons. Through phylogenomic analyses, we estimate the evolutionary timing of capsaicinoid biosynthesis. We reveal disrupted coding and regulatory regions of key biosynthesis genes in nonpungent species. We also find conserved placenta-specific accessible chromatin regions, which likely allow for tissue-specific biosynthetic gene coregulation and capsaicinoid accumulation. These T2T genomic resources will accelerate chili pepper genetic improvement and help to understand Capsicum genome evolution.

Wiring between close nodes in molecular networks evolves more quickly than between distant nodes.

As species diverge, a wide range of evolutionary processes lead to changes in protein-protein interaction networks and metabolic networks. The rate at which molecular networks evolve is an important question in evolutionary biology. Previous empirical work has focused on interactomes from model organisms to calculate rewiring rates, but this is limited by the relatively small number of species and sparse nature of network data across species. We present a proxy for variation in network topology: variation in drug-drug interactions (DDIs), obtained by studying drug combinations (DCs) across taxa. Here, we propose the rate at which DDIs change across species as an estimate of the rate at which the underlying molecular network changes as species diverge. We computed the evolutionary rates of DDIs using previously published data from a high throughput study in gram-negative bacteria. Using phylogenetic comparative methods, we found that DDIs diverge rapidly over short evolutionary time periods, but that divergence saturates over longer time periods. In parallel, we mapped drugs with known targets in protein-protein interaction and co-functional networks. We found that the targets of synergistic DDIs are closer in these networks than other types of DCs and that synergistic interactions have a higher evolutionary rate, meaning that nodes that are closer evolve at a faster rate. Future studies of network evolution may use DC data to gain larger-scale perspectives on the details of network evolution within and between species.

About the Possible Solar Nature of the ~200 yr (de Vries/Suess) and ~2000–2500 yr (Hallstadt) Cycles and Their Influences on the Earth’s Climate: The Role of Solar-Triggered Tectonic Processes in General “Sun–Climate” Relationship

(1) Introduction: The subject of the present study concerns the analysis of the existence and long time evolution of the solar ~200 yr (de Vries/Suess) and ~2400 yr (Hallstadt) cycles during the recent part of the Wurm ice epoch and the Holocene, as well as their forcing on the regional East European climate during the last two calendar millennia. The results obtained here are compared with those from our previous studies, as well as with the results obtained by other authors and with other types of data. A possible scenario of solar activity changes during the 21st century, as well as different possible mechanisms of solar–climatic relationships, is discussed. (2) Data and methods: Two types of indirect (historical) data series for solar activity were used: (a) the international radiocarbon tree ring series (INTCAL13) for the last 13,900 years; (b) the Schove series of the calendar years of minima and maxima and the magnitudes of 156 quasi 11 yr sunspot Schwabe–Wolf cycles since 296 AD and up to the sunspot cycle with number 24 (SC24) in the Zurich series; (c) manuscript messages about extreme meteorological and climatic events (Danube and Black Sea near-coast water freezing), extreme summer droughts, etc., in Bulgaria and adjacent territories since 296 and up to 1899 AD, when the Bulgarian meteorological dataset was started. A time series analysis and χ2-test were used. (3) Results and analysis: The amplitude modulation of the 200 yr solar cycle by the 2400 yr (Hallstadt) cycle was confirmed. Two groups of extremely cold winters (ECWs) during the last ~1700 years were established. Both groups without exclusion are concentrated near 11 yr sunspot cycle extremes. The number of ECWs near sunspot cycle minima is about 2 times greater than that of ECWs near sunspot cycle maxima. This result is in agreement with our earlier studies for the instrumental epoch since 1899 AD. The driest “spring-summer-early autumn” seasons in Bulgaria and adjacent territories occur near the initial and middle phases of the grand solar minima of the Oort–Dalton type, which relate to the downward phases and minima of the 200 yr Suess cycle. (4) Discussion: The above results confirm the effect of the Sun’s forcing on climate. However, it cannot be explained by the standard hypothesis for total solar irradiation (TSI) variations. That is why another hypothesis is suggested by the author. The mechanism considered by Svensmark for galactic cosmic ray (GCR) forcing on aerosol nuclei was taken into account. However, in the hypothesis suggested here, the forcing of solar X-ray flux changes (including solar flares) on the low ionosphere (the D-layer) and following interactions with the Earth’s lithosphere due to the terrestrial electric current systems play a key role for aerosol nuclei and cloud generation and dynamics during sunspot maxima epochs. The GCR flux maximum absorption layer at heights of 35–40 km replaces the ionosphere D-layer role during the sunspot minima epochs.

Monitoring Density Redistribution at the Excited State in a Dual Emitting Molecule: An Analysis Based on Real-Time Density Functional Theory and Density Descriptors.

In this work, we computed and analyzed, by means of density-based descriptors, the real-time evolution of both the locally excited (LE) and charge-transfer (CT) excited states for the planar and twisted conformations of the DMABN (4-(N,N-dimethylamino)benzonitrile) molecule using real-time time-dependent density functional theory (DFT) and three different exchange-correlation energy functionals (EXC) belonging to the same family (the PBE one). Our results based on the analysis of density-based descriptors show that the underlying EXC modifies the evolution in time of the density. In particular, comparing the frequency of density reorganization computed with the three functionals (PBE, PBE0, and LC-PBE), we found that the frequency of electronic interconversion of the individual determinants involved during the dynamics increases from PBE to PBE0 and to LC-PBE. This allows us to show that there is a correlation between the delocalization of the electronic density and the frequency of reorganization. In particular, the greater the mean hole-electron distance during the dynamics, the lower is the frequency of density reorganization.

Looking back, moving forward: protein corona of lipid nanoparticles.

Lipid nanoparticles (LNPs) are commonly employed for drug delivery owing to their considerable drug-loading capacity, low toxicity, and excellent biocompatibility. Nevertheless, the formation of protein corona (PC) on their surfaces significantly influences the drug's in vivo fate (such as absorption, distribution, metabolism, and elimination) upon administration. PC denotes the phenomenon wherein one or multiple strata of proteins adhere to the external interface of nanoparticles (NPs) or microparticles within the biological milieu, encompassing ex vivo fluids (e.g., serum-containing culture media) and in vivo fluids (such as blood and tissue fluids). Hence, it is essential to claim the PC formation behaviors and mechanisms on the surface of LNPs. This overview provided a comprehensive examination of crucial aspects related to such issues, encompassing time evolution, controllability, and their subsequent impacts on LNPs. Classical studies of PC generation on the surface of LNPs were additionally integrated, and its decisive role in shaping the in vivo fate of LNPs was explored. The mechanisms underlying PC formation, including the adsorption theory and alteration theory, were introduced to delve into the formation process. Subsequently, the existing experimental outcomes were synthesized to offer insights into the research and application facets of PC, and it was concluded that the manipulation of PC held substantial promise in the realm of targeted delivery.

Prognostic factors in childhood vitiligo.

Vitiligo is a multifactorial disease characterized by the progressive loss of melanocytes. The worldwide prevalence ranges from 0.5% to 2%, and in children from 0% to 2.16%. The objective of this study was to determine the variables associated with progression of vitiligo. A retrospective cohort was carried out where a random sample of records of pediatric patients with vitiligo from January 2016 to December 2020 was analyzed. The variables were studied: age at onset, sex, hereditary family history, personal history of thyroid diseases, time of evolution, classification, Köebner phenomena, mucosal vitiligo, halo nevus, premature graying and the presence of other dermatoses. The final state was classified as progression, stability, partial remission and complete remission. 574 children with vitiligo; 290 (50.5%) women, 284 (49.5%) men. Non-segmental vitiligo in 324 (56.4%), segmental vitiligo in 250 (43.6%). Mean age of onset 8.7 years (SD: 4.54). Median evolution time 6 months (25th percentile of 3 months and 75th percentile of 24 months). Family history 27 (4.70%). Thyroid disease 7 (1.21%). Evolution remained stable in 44 (7.7%), 68 (11.8%) had progression, 32 (5.6%) complete remission, 222 (38.7%) partial remission and 208 (36.2%) one consultation. Non-segmental vitiligo was obtained p < 0.028, younger age of onset p < 0.000, and none skin comorbidities p < 0.009. The variables that were associated with a more progression were non-segmental vitiligo, early ages at the onset of the disease, and not presenting with other skin diseases.

Entangled states dynamics of moving two-level atoms in a thermal field bath

Abstract We consider a two-level atom that follows a wordline of constant velocity, while interacting with a massless scalar field in a thermal state through: (i) an Unruh-DeWitt coupling, and (ii) a coupling that involves the time derivative of the field. We treat the atom as an open quantum system, with the field playing the role of the environment, and employ a master equation to describe its time evolution. We study the dynamics of entanglement between the moving atom and a (auxiliary) qubit at rest and isolated from the thermal field. We find that in the case of the standard Unruh-DeWitt coupling and for high temperatures of the environment the decay of entanglement is delayed due to the atom's motion. Instead, in the derivative coupling case, the atom's motion always causes the rapid death of entanglement.

Deceleration of kicked objects due to the Galactic potential

Various stellar objects experience a velocity kick at some point in their evolution. These include neutron stars and black holes at their birth, or binary systems when one of the two components goes supernova. For most of these objects, the magnitude of the kick and its impact on the object dynamics remains a topic of debate. We investigate how kicks alter the velocity distribution of objects born in the Milky Way disc, both immediately after the kick and at later times, and whether these kicks are encoded in the observed population of Galactic neutron stars. We simulated the Galactic trajectories of point masses on circular orbits in the disc after being perturbed by an isotropic kick, with a Maxwellian distribution where $ km/s. Then, we simulated the motion of these point masses for $200$ Myr. These trajectories were then evaluated, either for the Milky Way population as a whole or for those passing within two kiloparsecs of the Sun, to get the time evolution of the velocities. During the first $20$ Myr, the bulk velocity of kicked objects becomes temporarily aligned with the cylindrical radius, implying an anisotropy in the velocity orientations. Beyond this age, the velocity distribution shifts towards lower values and settles to a median of $ km/s. Around the Sun, the distribution also loses its upper tail, primarily due to unbound objects escaping the Galaxy. We compared this to the velocities of Galactic pulsars and find that pulsars show a similar evolution with characteristic age. The shift in the velocity distribution is due to bound objects spending most of their orbits at larger radii after the kick. They are, therefore, decelerated by the Galactic potential. We find the same deceleration for nearby objects and the total population, and conclude that it is also observed in Galactic pulsars. Because of this effect, the (scalar) speeds of old neutron stars provide little information about their kicks at birth.

Hydric environment and chemical composition shape non-avian reptile eggshell absorption.

The amniotic egg fulfils a critical role in reproduction by serving as an interface between the external environment and the embryo. Because non-avian reptiles are rarely incubated, they must be heated by, and absorb water from, the oviposition site for the developing embryo. The mechanisms by which they absorb sufficient, but not excess, water and how these mechanisms vary with local habitat is largely unknown, despite its significance to their evolution. Here, we first performed histology, Fourier transform infra-red spectroscopy and dynamic vapor sorption experiments to elucidate the mechanisms of eggshell absorption for 56 reptile species. Then, we used phylogenetic comparative analyse to test the hypothesis that absorptive capacity of reptile eggshells increases with aridity of the environment. We found that water absorption increases in the presence of a superficial mucopolysaccharide layer and decreases with increased calcium content. We found that eggs from arid environments have highly absorbent eggshells, but only in species with weakly calcified shells. Our results suggest that reptile eggshells have over evolutionary time tuned absorptive capacity to environmental moisture level. Since these eggs often must sustain conflicting constraints, they may serve as inspirations for new biomimetic materials, such as water filtering membranes or humidity sensors.

A unifying quantum speed limit for time-independent Hamiltonian evolution

Abstract Quantum speed limit (QSL) is the study of fundamental limits on the evolution time of quantum systems. For instance, under the action of a time-independent Hamiltonian, the evolution time between an initial and a final quantum state obeys various mutually complementary lower bounds. They include the Mandelstam-Tamm bound, the Margolus-Levitin bound, the Luo-Zhang&amp;#xD; bound, the Lee-Chau bound together with the dualML bound introduced by Ness and coworkers. Here we show that the Mandelstam-Tamm bound can be obtained by optimizing the Lee-Chau bound over a certain parameter. More importantly, we report a QSL that includes all the above bounds as special cases before optimizing over the physically meaningless reference energy level of a quantum system. This unifying bound depends on a certain parameter $p$. For any fixed $p$, we find all pairs of time-independent Hamiltonian and initial pure quantum state that saturate this unifying bound. More importantly, these pairs allow us to compute this bound accurately and efficiently using an oracle that returns certain $p$th moments related to the absolute value of&amp;#xD; energy of the quantum state. Moreover, this oracle can be simulated by a computationally efficient and accurate algorithm for finite-dimensional quantum systems as well as for certain infinite-dimensional quantum states with bounded and continuous energy spectra. This makes our computational method feasible in a lot of practical situations. We find that if the dimension of&amp;#xD; the underlying Hilbert space is $\lesssim 2000$, our unifying bound optimized over $p$ can be computed accurately in a few minutes using Mathematica code with just-in-time compilation in a typical desktop. Besides, this optimized unifying QSL is at least as good as all the existing ones combined and can occasionally be a few percent to a few times better.

Optimal compression of constrained quantum time evolution

Optimal compression of constrained quantum time evolution

Cosmological implications of the Weyl geometric gravity theory

We consider the cosmological implications of the Weyl geometric gravity theory. The basic action of the model is obtained from the simplest conformally invariant gravitational action, constructed, in Weyl geometry, from the square of the Weyl scalar, the strength of the Weyl vector, and a matter term, respectively. The total action is linearized in the Weyl scalar by introducing an auxiliary scalar field. To maintain the conformal invariance of the action the trace condition is imposed on the matter energy–momentum tensor, thus making the matter sector of the action conformally invariant. The field equations are derived by varying the action with respect to the metric tensor, the Weyl vector field, and the scalar field, respectively. We investigate the cosmological implications of the theory, and we obtain first the cosmological evolution equations for a flat, homogeneous and isotropic geometry, described by Friedmann–Lemaitre–Robertson–Walker metric, which generalize the Friedmann equations of standard general relativity. In this context we consider two cosmological models, corresponding to the vacuum state, and to the presence of matter described by a linear barotropic equation of state. In both cases we perform a detailed comparison of the predictions of the theory with the cosmological observational data, and with the standard Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document} CDM model. By assuming that the presence of the Weyl geometric effects induce small perturbations in the homogeneous and isotropic cosmological background, and that the anisotropy parameter is small, the equations of the cosmological perturbations due to the presence of the Weyl geometric effects are derived. The time evolution of the metric and matter perturbations are explicitly obtained. Therefore, if Weyl geometric effects are present, the Universe would acquire some anisotropic characteristics, and its geometry will deviate from the standard FLRW one.

Shadow Hamiltonians of structure-preserving integrators for Nambu mechanics

Abstract Symplectic integrators are widely implemented numerical integrators for Hamiltonian mechanics, which preserve the Hamiltonian structure (symplecticity) of the system. Although the symplectic integrator does not conserve the energy of the system, it is well known that there exists a conserving modified Hamiltonian, called the shadow Hamiltonian. For the Nambu mechanics, which is one of the generalized Hamiltonian mechanics, we can also construct structure-preserving integrators by the same procedure used to construct the symplectic integrators. In the structure-preserving integrator, however, the existence of shadow Hamiltonians is non-trivial. This is because the Nambu mechanics is driven by multiple Hamiltonians and it is non-trivial whether the time evolution by the integrator can be cast into the Nambu mechanical time evolution driven by multiple shadow Hamiltonians. In this paper we present a general procedure to calculate the shadow Hamiltonians of structure-preserving integrators for Nambu mechanics, and give an example where the shadow Hamiltonians exist. This is the first attempt to determine the concrete forms of the shadow Hamiltonians for a Nambu mechanical system. We show that the fundamental identity, which corresponds to the Jacobi identity in Hamiltonian mechanics, plays an important role to calculate the shadow Hamiltonians using the Baker-Campbell-Hausdorff formula. It turns out that the resulting shadow Hamiltonians have indefinite forms depending on how the fundamental identities are used. This is not a technical artifact, because the exact shadow Hamiltonians obtained independently have the same indefiniteness.

Visualizing Electrochemical CO2 Conversion via the Emerging Scanning Electrochemical Microscope: Fundamentals, Applications and Perspectives.

With the rapid development and maturity of electrochemical CO2 conversion involving cathodic CO2 reduction reaction (CO2RR) and anodic oxygen evolution reaction (OER), conventional ex situ characterizations gradually fall behind in detecting real-time products distribution, tracking intermediates, and monitoring structural evolution, etc. Nevertheless, advanced in situ techniques, with intriguing merits like good reproducibility, facile operability, high sensitivity, and short response time, can realize in situ detection and recording of dynamic data, and observe materials structural evolution in real time. As an emerging visual technique, scanning electrochemical microscope (SECM) presents local electrochemical signals on various materials surface through capturing micro-current caused by reactants oxidation and reduction. Importantly, SECM holds particular potentials in visualizing reactive intermediates at active sites and obtaining instantaneous morphology evolution images to reveal the intrinsic reactivity of active sites. Therefore, this review focuses on SECM fundamentals and its specific applications toward CO2RR and OER, mainly including electrochemical behavior observation on local regions of various materials, target products and onset potentials identification in real-time, reaction pathways clarification, reaction kinetics exploration under steady-state conditions, electroactive materials screening and multi-techniques coupling for a joint utilization. This review undoubtedly provides a leading guidance to extend various SECM applications to other energy-related fields.

Notes on time entanglement and pseudo-entropy

Following arXiv:2210.12963 [hep-th], we investigate aspects of the time evolution operator regarded as a density operator and associated entanglement-like structures in various quantum systems. These involve timelike separations and generically lead to complex-valued entropy, although there are interesting real subfamilies. There are many parallels and close relations with reduced transition matrices and pseudo-entropy, which we discuss and clarify. For instance, a related quantity involves the time evolution operator along with a projection onto some initial state, which amounts to analysing pseudo-entropy for the initial state and its time-evolved final state.

Simultaneous Water Sorption and Crystallization in ASDs 2: Modeling Long-Term Stabilities.

The physical stability of amorphous solid dispersions (ASDs) is a major topic in the formulation research of oral dosage forms. To minimize the effort of investigating the long-term stability using cost- and time-consuming experiments, we developed a thermodynamic and kinetic modeling framework to predict and understand the crystallization kinetics of ASDs during long-term storage below the glass transition. Since crystallization of the active phrarmaceutical ingredients (APIs) in ASDs largely depends on the amount of water absorbed by the ASDs, water-sorption kinetics and API-crystallization kinetics were considered simultaneously. The developed modeling approach allows prediction of the time evolution of viscosity, supersaturation, and crystallinity as a function of drug load, relative humidity, and temperature. It was applied and evaluated against two-year-lasting crystallization experiments of ASDs containing nifedipine and copovidone or HPMCAS measured in part I of this work. We could show that the proposed modeling approach is able to describe the interplay between water sorption and API crystallization and to predict long-term stabilities of ASDs just based on short-term measurements. Most importantly, it enables explaining and understanding the reasons for different and sometimes even unexpected crystallization behaviors of ASDs.

Work statistics for quantum spin chains: Characterizing quantum phase transitions, benchmarking time evolution, and examining passivity of quantum states

We study three aspects of work statistics in the context of the fluctuation theorem for quantum spin chains up to 1024 sites by numerical methods based on matrix-product states (MPSs). First, we use our numerical method to evaluate the moments/cumulants of work done by the sudden quench process on the Ising or Haldane spin chains, and we study their behaviors across the quantum phase transitions. Our results show that, up to the fourth cumulant, the work statistics can indicate the quantum phase transition characterized by the local order parameters but barely for purely topological phase transitions. Second, we propose to use the fluctuation theorem, such as Jarzynski's equality, which relates the real-time correlator to the ratio of the thermal partition functions, as a benchmark indicator for the numerical real-time evolving methods. Third, we study the passivity of ground and thermal states of quantum spin chains under some cyclic impulse processes. We show that the passivity of thermal states and ground states under the Hermitian actions is ensured by the second laws and variational principles, respectively, and we also verify this by numerical calculations. In addition, we also consider the passivity of ground states under non-Hermitian actions, for which the variational principle cannot be applied. Despite that, we find no violation of passivity from our numerical results for all the cases considered in the Ising and Haldane chains. Overall, we demonstrate that the work statistics for the sudden quench and impulse processes can be evaluated precisely by the numerical MPS method to characterize quantum phase transitions and examine the passivity of quantum states. We also propose to exploit the universality of the fluctuation theorem to benchmark the numerical real-time evolutions in an algorithmic and model-independent way. Published by the American Physical Society 2024

Signature of f(R) gravity via Lemaître–Tolman–Bondi inhomogeneous perturbations

We analyze inhomogeneous cosmological models in the local Universe, described by the Lemaître–Tolman–Bondi (LTB) metric and developed using linear perturbation theory on a homogeneous and isotropic Universe background. Focusing on the different evolution of spherical symmetric inhomogeneities, we compare the Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}LTB model, in which the cosmological constant Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document} is included in the LTB formalism, with inhomogeneous cosmological models based on fR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f\\left( R\\right) $$\\end{document} modified gravity theories viewed in the Jordan frame. We solve the system of field equations for both inhomogeneous cosmological models adopting the method of separation of variables: we integrate analytically the radial profiles of local perturbations, while their time evolution requires a numerical approach. The main result of the analysis concerns the different radial profiles of local inhomogeneities due to the presence of a non-minimally coupled scalar field in the Jordan frame of fR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f\\left( R\\right) $$\\end{document} gravity. While radial perturbations follow a power-law in the Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}LTB model, Yukawa-like contributions appear in the fR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f\\left( R\\right) $$\\end{document} theory. Interestingly, this latter peculiar behavior of radial profile is not affected by the choice of the fR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f\\left( R\\right) $$\\end{document} functional form. The numerical solution of time-dependent perturbations exhibits a non-diverging profile. This work suggests that investigations about local inhomogeneities in the late Universe may allow us to discriminate if the present cosmic acceleration is caused by a cosmological constant term or a modified gravity effect.