Qualitative networks, an extension of Boolean networks. With this framework, we use formal verification methods to check whether a model is consistent with the laboratory experimental observations on which it is based. If the model does not conform to the data, we suggest a revised model and the new hypotheses are tested in-silico.A central goal of Systems Biology is to model and analyze biological signaling pathways that interact with one another to form complex networks. Here we introduce 86 states. We illustrate the usefulness of this approach through a model of the interaction between the Notch and the Wnt signaling pathways in mammalian skin, and its extensive analysis.We consider networks in which elements range over a small finite domain allowing more flexibility than Boolean values, and add target functions that allow to model a rich set of behaviors. We propose a symbolic algorithm for analyzing the steady state of these networks, allowing us to scale up to a system consisting of 144 elements and state spaces of approximately 10We introduce an approach for constructing computational models of biological systems that extends the framework of Boolean networks and uses formal verification methods for the analysis of the model. This approach can scale to multicellular models of complex pathways, and is therefore a useful tool for the analysis of complex biological systems. The hypotheses formulated during in-silico testing suggest new avenues to explore experimentally. Hence, this approach has the potential to efficiently complement experimental studies in biology. Considering biology from a systems point of view also allows the use of methods designed for the engineering of artificial systems. A wide array of formalisms originating from computer science have already been used to model biological systems. These include process calculi  as the restriction of s to the variables in V. Projection is generalized in the natural way to sets of states. The function post (σM) maps the region σM to the union of the successors of all states in σM. This function is constructed according to the definition of the module M.We want to find the set of all infinitely visited states of a module M, we compute the set of states that is mapped to itself by the post function. This set is called the fix point of the function. We use the following algorithm to obtain this fix point:In order to obtain the set of infinitely visited states of i := 0repeati := i + 1until σi contains all states that can be visited at time step i or later. Therefore the region σin f contains all states that are infinitely visited. In order to show that this algorithm terminates, it is sufficient to show that post function is monotonous: The region p holds on all infinitely visited states. We define ⟦p⟧ ⊆ ∑M as the set of all states for which the predicate is true. Therefore we can say that the predicate holds on all infinitely visited sets iff p⟧. If this is not the case, we are interested in knowing if there is at least one attractor for which the property holds for all states. We modify the algorithm in order to consider only infinitely visited states for which the property holds:We are interested in knowing if a predicate i := 0repeati := i + 1until p⟧. The kind of analysis we are interested in does not need explicit computation of the individual attractors. Obtaining the explicit trace of each attractor would need to be done in an enumerative way, and thus requires O . The set of possible behaviors of an external variable is therefore a superset of the behavior of any building block. Using the algorithm above, we compute the infinitely visited region M1. We consider the projection M onto the variables of M1. Since the set of behaviors of the non-deterministic building block are a superset of the behavior of any other building block we have that M2, M. By repeating this iteratively we can handle larger compositions.The general algorithm described above can be applied to every kind of module. We use a more specific algorithm tailored for Qualitative networks. Each module consists of an atom computing the target function and a controller atom which enforces the transition rules specified in Equation (2). Rather than directly computing the infinitely visited region of a composed module partitioned transition relations [post function for the whole model, we define one function per cell, and apply them successively.When applying this method to a multicellular model, we use elations . Rather The performance tests are performed on a 3 GHz Intel Xeon computer running Linux Fedora Core 3 (Kernel version 2.6.11). The CrocoPat tool can use up to 3 GB of memory.MS carried out the modeling work, conceived the idea of the Qualitative network framework, and performed the analysis of the model. JF conceived the study and participated in its design. TAH participated in the design of this study. All authors collaborated for the elaboration of the manuscript, read, and approved the final manuscript.


P = .012). In this study, we developed a real-time endoscopic image orientation correction system, which significantly improved physician performance during a video endoscopic exam.The discrepancy between spatial orientations of an endoscopic image and a physician’s working environment can make it difficult to interpret endoscopic images. In this study, we developed and evaluated a device that corrects the endoscopic image orientation using an accelerometer and gyrosensor. The acceleration of gravity and angular velocity were retrieved from the accelerometer and gyrosensor attached to the handle of the endoscope. The rotational angle of the endoscope handle was calculated using a Kalman filter with transmission delay compensation. Technical evaluation of the orientation correction system was performed using a camera by comparing the optical rotational angle from the captured image with the rotational angle calculated from the sensor outputs. For the clinical utility test, fifteen anesthesiology residents performed a video endoscopic examination of an airway model with and without using the orientation correction system. The participants reported numbers written on papers placed at the left main, right main, and right upper bronchi of the airway model. The correctness and the total time it took participants to report the numbers were recorded. During the technical evaluation, errors in the calculated rotational angle were less than 5 degrees. In the clinical utility test, there was a significant time reduction when using the orientation correction system compared with not using the system (median, 52 vs. 76 seconds;  Video endoscopes, such as laparoscopes, gastroenteroscopes, and bronchoscopes, are used in many fields of medicine to help clinicians diagnose and treat patients less invasively with shorter hospital stays. However, video endoscopes are difficult to operate, and it takes time for clinicians to learn how to interpret endoscopic images. One of the difficulties is the discrepancy between the spatial orientation of the endoscopic images and the endoscopist’s working environment , 2. For The gyrosensor has a fast response time that is not influenced by translational acceleration. Because the accelerometer fails to differentiate acceleration from gravity, the gyrosensor measures the rotational angle more accurately than the accelerometer when the endoscope rotates. Moreover, the integration error of the rotational angle from the gyrosensor that accumulates over time can be easily corrected using an accelerometer signal. Therefore, these two sensors are often used to complement each other.In this study, we developed and evaluated a real-time system for correcting endoscopic image orientation using an accelerometer and gyrosensor.To obtain the direction of gravity and angular velocity, an accelerometer and gyrosensor were attached to the handle of the endoscope. We assumed that the endoscope handle was a rigid body and that there would be no twisting from the handle to the tip of the endoscope. Quaternion, direction cosine matrix, Euler angles, and axis-angle representation can be used to signify the posture of a rigid body. Due to its simplicity and clear physical meaning, we used the axis-angle to represent the posture of the endoscope handle. The axis was defined as a unit vector indicating the direction of the advancing endoscope. The rotational angle θ was defined as 0 when the sensor was in the opposite direction of gravity and increased when the endoscope rotated in the clockwise direction. Because a singularity occurs when the operator holds the endoscope handle perpendicular to the ground, this action was prohibited during the study.x axis pointed to the right as seen by the operator, the y axis pointed toward the head of the endoscope, and the z axis point in the direction of the advancing endoscope  is a function that returns the arc tangent of y/x within the range of −π and π depending upon the sign of x and y.The rotational angle with respect to the direction of gravity was calculated as:θKalman is a state variable–the rotational angle after the Kalman filter , rk is the angular velocity from the gyrosensor, T is the time from the last measurement, wk is the processing error, and θaccel is the rotational angle from the accelerometer.The rotational angles from the accelerometer and gyrosensor were calibrated using a Kalman filter , 13. Theθaccel is a measurement variable–the rotational angle from the accelerometer, θKalman,k is a state variable, and zk is the measurement error.Eq  represenk and zk were assumed to be zero mean Gaussian white noise with a covariance of Qk and Rk. The smaller the Qk value, the greater the importance of the gyrosensor output; the smaller the Rk value, the greater the importance of the accelerometer output. We set the Rk value to give less weight to the accelerometer output if an additional acceleration was observed.ak is the acceleration from the accelerometer and g is the acceleration of gravity.In the Kalman filter algorithm, wKalman,k is the estimated rotational angle from the Kalman filter, rk is the angular velocity from the gyrosensor, and Pk is the estimated covariance (P0 = 1).The predicted rotational angle was calculated using k is the Kalman gain.Finally, the states and the estimated covariance were updated for the next iteration as shown in θpostproc,k is the final rotational angle after postprocessing, rk is the angular velocity from the gyrosensor, and Td is the time difference between the last available sensor output and image frame. Td was determined by a least square error method using the pre-examination data from the system with r is the vector of the measured angular velocity, θKalman is the rotational angle vector from the Kalman filter, and θoptical is the optical rotational angle vector .Subsequently, the rotational angle from the Kalman filter was postprocessed to correct for the delay between the image acquisition and sensor measurement . Because2 and 0.070°/s, respectively. The sensor device also has a magnetometer, although it was not used in this study. The video was captured by a charge-coupled device (CCD) camera attached to the eyepiece of the endoscope and transferred to a PC at 30 frames per second using a USB-ECPT video capture device .A 3-Space Sensor™ Bluetooth  was used for the system. According to the manufacturer’s specifications, the accelerometer and gyrosensor precisions are 0.0024 m/sThe image correction software was developed in C++. The sensor data were acquired by a polling method.θoptical) was automatically calculated from the image by recognizing the black and white boundaries.To evaluate the performance of the image orientation correction system, a board with a black upper half and a white lower half was placed 30 cm from the PC camera , and the sensors were attached to the camera. The rotational angle (θaccel), gyrosensor (θgyro), and Kalman filter , the postprocessed rotational angle (θpostproc), and the optical rotational angle  were recorded. Every measurement was repeated using the sensor’s wired and wireless connection.While the camera was rotated randomly for 30 seconds, the rotational angles from the accelerometer . P < .05 was considered statistically significant. Data are expressed as median (interquartile).For a statistical power of 0.8 and a significance level of 0.05, the estimated sample size was 12, with the assumption that the mean difference and standard deviation between two measurements was 20 seconds and 25 seconds, respectively. Fifteen participants were needed to compensate for possible dropouts. The times required to finish the exam with and without using the image orientation correction system were compared using the Mann-Whitney The optical rotational angle measured from the captured image during the random rotation of the camera ranged from −83.1° to 88.0° with the wired sensor connection and from −108.0° to 90.9° with the wireless connection. The optical angular velocity of the camera ranged from −184°/s to 182°/s with the wired sensor connection and from −251°/s to 201°/s with the wireless connection. The mean sampling rate of the sensor was 333.4 Hz and 27.7 Hz for the wired and wireless connection, respectively.θoptical and θaccel was 12.79° with the wired connection and 15.08° with the wireless connection .All participants successfully completed the experiments. The time required to finish the bronchoscopic exam decreased significantly using the image orientation correction system  compared with not using the system , when there is no information about the image orientation , or when communicating with another specialist for consultation or education.This study does have some limitations. A camera was used instead of an endoscope during the technical evaluation phase. However, measuring the optical rotational angle from with an actual endoscope may have induced an error due to the lower resolution. Second, an error may have occurred during the technical evaluation if the center of the camera did not align exactly with the center of the target board. Third, an airway model was used instead of an animal or human airway for the clinical utility test. During a bronchoscopic exam of a human patient, the shape of tracheal rings and muscle stripes can help orient the operator; however, the airway model used in this study did not have such detailed structures.In conclusion, the endoscopic image orientation correction system using both an accelerometer and a gyroscope resulted in greater accuracy than using an accelerometer alone. This system also significantly decreased the time needed to perform a bronchoscopic exam, which would be quite valuable in the clinical setting. Although these results were obtained in a limited situation, this orientation correction system will likely help clinicians interpret and analyze endoscopic images in clinical practice.S1 Video FileAs the endoscope rotates, the image also rotates, making it difficult to interpret the direction.(AVI)Click here for additional data file.S2 Video FileThe orientation of the image is fixed to the direction of gravity while the endoscope is rotating. A wired connection was used for capturing the video.(AVI)Click here for additional data file.


As the two dimensional version of the functional wavefront manipulation metamaterial, metasurface has become a research hot spot for engineering the wavefront at will with a subwavelength thickness. The wave scattered by the gradient metasurface, which is composed by the periodic supercells, is governed by the generalized Snell’s law. However, the critical angle that derived from the generalized Snell’s law circles the domain of the incident angles that allow the occurrence of the anomalous reflection and refraction, and no free space scattering waves could exist when the incident angle is beyond the critical angle. Here we theoretically demonstrate that apparent negative reflection can be realized by a gradient acoustic metasurface when the incident angle is beyond the critical angle. The underlying mechanism of the apparent negative reflection is understood as the higher order diffraction arising from the interaction between the local phase modulation and the non-local effects introduced by the supercell periodicity. The apparent negative reflection phenomena has been perfectly verified by the calculated scattered acoustic waves of the reflected gradient acoustic metasurface. This work may provide new freedom in designing functional acoustic signal modulation devices, such as acoustic isolator and acoustic illusion device. Yu, N et al.24681011215212426Tailoring the wavefront into arbitrary desired shape with a metasurface, the two dimensional metamaterial with subwavelength inclusions, has attracted tremendous attention in recent years. Since metasurface itself can be regarded as the application of the Huygens’ principle, arbitrary wavefront manipulation can be realized by designing these artificial secondary sources according to the detailed information of the desired wavefront. The milestone work done by Xie Y. et al. can be understood by revisiting the influence of the supercell periodicity as well as the local phase modulation on the acoustic gradient metasurface. Thereafter, a more comprehensive discussion about the negative refraction that occurs in optical region when the incident angle is beyond the critical angle has been demonstratedXie Y. et al.Gradient metasurface is a periodic array of the supercell with a linearly varying phase modulation ranging from 0 to 2π, and the wave scattered by the gradient metasurface is governed by the generalized laws of reflection and refraction. According to the generalized laws of reflection and refraction, there exists a critical angle which defines the domain that the anomalous reflection or refraction can occur, and beyond the region defined by the critical angle the incident beam would be transformed into the surface bounded wave, and thus the metasurface works as the anti-scattering coating28In this paper, we theoretically study the influences of the supercell periodicity on the reflection behaviors of the gradient acoustic metasurface. The apparent negative reflection occurs when the incident angle is beyond the critical angle, which has been perfectly verified by the calculated reflected acoustic field of the reflected gradient acoustic metasurface. The theoretical study shows that the generalized law of reflection still works within the region defined by the critical angle, however, when the incident angle is beyond the critical angle, the generalized law of reflection would be modified with the reciprocal lattice vector corresponding to the higher order diffraction. As for the possible higher order diffraction, it depends on the amplitude of the metasurface’s surface phase gradient and the incident angle, which can be regarded as a more general case discussed in ref. φs is the position dependent reflected phase modulation along the gradient metasurface, the generalized law of reflection provides a complete description about the reflection behaviors of the metasurface under the plane wave illumination:Since the gradient metasurface introduces the phase discontinuity across the scattered interface, the local graded phase modulation of the periodic supercell resets the classical Snell’s law by adding a surface phase gradient term which now is known as the generalized law of reflection and refraction. For a reflected gradient metasurface with a surface phase gradient θi is the incident angle, θre is the anomalous reflected angle and k0 is the amplitude of the free space wave vector. Since the gradient metasurface offers a linear phase modulation from 0 to 2π over a supercell period, the surface phase gradient term can be further written as ps is the supercell period length and σ = 1 or −1, which indicates the direction of the surface phase gradient. According to formula (1), the critical angle where nG indicates the corresponding diffraction order. According to the classical diffraction theory, nG is equal to 0 as long as the incident angle is inside the region defined by the formula (1). When the incident angle is beyond the critical angle, nG would take some other values to maintain the validity of formula (2) , dν = 0.044 mm, dth = 0.05 mm and the minimal geometrical size is 1 mm  solutions of the allowed ks, and the generalized law of reflection that is amended by the reciprocal lattice vector of the supercell period can be visually demonstrated as shifting the Snell’s law of reflection within the region encompassed by the pink dashed square as well. The blue solid line shown in θc, the surface bounded waves supported by the linear local phase modulation of the subunits would interact with the Floquet-Bloch modes offered by the supercell lattices and reradiate into the free space as the high order diffractions. Under this case, the relation between the incident angle and reflected angle can be described by formula (3), and there could be several possible nG values as long as the momentum matching condition over the metasurface is satisfied. It is apparent that the reduced surface phase gradient ks of smaller absolute value will have more possible nG values, and thus, to simplify our discussion, we firstly study the case that nG values can be solved by maintaining the validity of formula(3) and consequently nG could take the value of 0, −1, −2 and −3. Since the lateral wave vector component of the impinging wave nG value would be incident angle dependent as well. When the incident angle gradually varies from 90° to −90°, the value of nG can be determined by the ‘jump-up’ rule in our paper. Here we firstly treat the reflection behaviors of the gradient acoustic metasurface as several reflection states θi varies beyond the critical angle θc, reflection state nG value would be superior to that of a smaller one. For example, where ks = −0.762. The critical angle θc is solved to be −13.8° and the shift critical angle θi < 90°, the reflected angle can be predicted by θi < −13.8°, the Floquet-Bloch mode with 2G propagation constant amplitude would strongly scatter the surface bounded waves and forms the free space propagating waves, the reflected angle can be solved by θi < −31.6°, the Floquet-Bloch mode with 3G propagation constant amplitude scatters the surface bounded waves intensively and the reflected angle nG = −2 and nG = −3, nG is allowed. For example, when ks =−0.5, there are five possible reflection states: ks = −0.508, which slightly diverts from −0.5, for the full-wave simulation. However, such minor difference would shift θc from −30° to −29.5°, and thus the apparent negative reflection state Considering the situation that ngle see , becausek0. When When ks = −1.307. The shift critical angle is When ate  see . Such reThe ‘jump’ between the different reflection states is the redistribution of the reflected acoustic energy of different diffraction orders, which originates from the interaction between the surface bounded waves and the effect of the non-local supercell period. It can be seen from the calculated reflected acoustic field corresponding to the apparent negative reflection, the total scattered field is the superposition of a surface bounded waves and the free space propagation wave see . When weIt should be noted that such apparent negative reflection and the anomalous reflection phenomenon of the gradient acoustic metasurface could distort the scattered field of the object encompassed by it and thus the gradient acoustic metasurface has the potential to be applied as the illusion devices. Furthermore, the full-angle negative reflection phenomena illustrated in In summary, we demonstrate the apparent negative reflection phenomenon that takes place when the incident angle is beyond the critical angle, which should be forbidden according to the generalized law of reflection, can be realized with the reflected gradient acoustic metasurface. The underlying mechanism of such anomalous phenomenon can be understood by treating the apparent negative reflection as the high order diffraction which is produced via the interaction between the surface bounded waves and the supercell periodicity. Integrating the reciprocal lattice vector term, which represents the influences of the supercell periodicity, to the generalized law of reflection offers perfect prediction about the reflected angle values of the apparent negative reflection. Full-wave simulation of the scattered acoustic fields of the gradient acoustic metasurface illustrates versatile apparent negative reflection phenomenon perfectly, and the nontrivial full-angle negative reflection, which combines the anomalous reflection and the apparent negative reflection behaviors of the gradient acoustic metasurface, offers a simple way for isolating the acoustic waves incident from opposite directions. This study may provide more freedom for designing the acoustic signal modulation devices and the potential applications in acoustic beam steering.In this paper, we take advantage of the Finite Element Method (FEM) based on the commercial software COMSOL Multiphysics to conduct the full-wave simulations. The acoustic pressure model is used, the background medium is set to be air and the corresponding acoustic speed is chosen as 343 m/s. The gradient metasurface is studied as the infinite period by applying the periodic boundary condition on the right and left sides of the simulation region. The reflected angle is obtained according to the far-field acoustic pressure distribution. As for the influences of the viscosity and thermal acoustic effects on the reflection behaviors of the metasurface, we utilize two different methods to testify the robustness of the apparent negative reflection by introduce an imaginary part to the acoustic refractive index of the air How to cite this article: Liu, B. et al. Apparent Negative Reflection with the Gradient Acoustic Metasurface by Integrating Supercell Periodicity into the Generalized Law of Reflection. Sci. Rep.6, 38314; doi: 10.1038/srep38314 (2016).Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


Geant4 hadronic physics models, of the calorimeter response to isolated charged hadrons in the central calorimeter region. The description of the response to anti-protons at low momenta is found to be improved with respect to previous analyses. The electromagnetic and hadronic calorimeters are also examined separately, and the detector simulation is found to describe the response in the hadronic calorimeter well. The jet energy scale uncertainty and correlations in scale between jets of different momenta and pseudorapidity are derived based on these studies. The uncertainty is 2–5% for jets with transverse momenta above 2 A measurement of the calorimeter response to isolated charged hadrons in the ATLAS detector at the LHC is presented. This measurement is performed with 3.2 nb The proton–proton collisions measured by the ATLAS detector at the Large Hadron Collider (LHC) produce quarks and gluons that are observed as collimated sprays of hadrons, called jets. The hadrons in jets are measured as charged-particle tracks and showers of particles in the calorimeters. Uncertainties in the measurement of jet energies and the modelling of the calorimeter response to hadrons often dominate systematic uncertainties in measurements at the LHC.The measurement of the calorimeter response to single charged hadrons provides an important validation of the modelling of hadronic showers in the calorimeters and of the detector geometry implemented in the ATLAS simulation . It is oentclass1pt{minima applied  and has entclass1pt{minima applied . The res-leptons .Geant4 [This paper describes the updated measurement of the response to isolated charged hadrons using data from both 2010 and 2012 with the most recent detector simulation. Between 2010 and 2012, the centre-of-mass energy was increased from 7 to 8  [Geant4 , the decThe calibration of jets based on the energy deposited by individual particles involves a number of steps that can be separately tested. Many particles are not sufficiently energetic to reach the calorimeter, and some particles interact before reaching the calorimeter and do not deposit a significant amount of energy. The fraction of particles not depositing energy in the calorimeter is the first important test of the geometry  and simulation of hadrons, discussed in Sect. A jet is a complex object, however, and good modelling of the average properties of jets does not always indicate that jets would be well described in more extreme configurations. Some jets may be composed of more positively or more negatively charged particles, resulting in differences in response. The separate modelling of positively and negatively charged particles is discussed in Sect. The studies of the calorimeter response to isolated charged hadrons are then used to construct a jet energy scale uncertainty in Sect. The paper is organised as follows. The ATLAS detector is briefly introduced in Sect. The ATLAS detector  is a genmentclass2pt{minimA three-level trigger system is used to select events for offline analysis. The first level is hardware-based, while the second two levels are implemented in software. The minimum-bias trigger scintillators, used for selecting events in this analysis, are two sets of 16 thin scintillators covering The primary data sample consists of eight million proton–proton collision events corresponding to an integrated luminosity of 0.1 nbPythia8.160 [The primary 2012 data sample is compared to 20 million simulated single-, double-, and non-diffractive proton–proton collision events, generated using hia8.160  using thhia8.160  and the hia8.160 . ThroughGeant4 9.4 [The simulated events are passed through the ATLAS detector simulation  based onant4 9.4 .2 Two saant4 9.4  are usedant4 9.4 –19 with  GHEISHA  from 9.5 cascade –23 and nof model –27 with Pythia8 are overlaid on top of one another to mimic the simultaneous collisions in the detector, including the bipolar pulse shape of the calorimeter electronics. The MC simulation samples with pile-up are only included using the QGSP_BERT set of hadronic physics models.In all cases, the simulated detector conditions match those of the data-taking period, and the simulated events and data are passed through the same trigger and reconstruction software. Where the data include pile-up, minimum-bias events generated with In order to be selected by the trigger system, events in the low-To collect data during data-taking periods with pile-up, three triggers are used. Only a fraction of events passing the selection criteria of any of the triggers are written out, as the rates are above the maximum bandwidth for the trigger. The first trigger is random and requires only crossing proton bunches in the detector. The events accepted by this trigger correspond to an integrated luminosity of 24 nbEach event is required to have a well-reconstructed vertex with at least four associated tracks with y vertex . This rele level .Although it does not provide the same level of precision tracking information as the pixel and SCT layers, the TRT provides additional information to efficiently reject the tracks originating from hadronic interactions in the ID material. Tracks interacting in the ID produce a range of secondary particles, often including ions and neutrons, which can be difficult to model correctly. Moreover, the kinematics and species of hadrons resulting from these interactions is generally poorly known and may not be well modeled. For most of the results in this paper, in the region In each event, the calorimeter cells’ energies are topologically clustered using a 4–2–0 algorithm . This alest beam . After tThe noise thresholds used in the clustering procedure for the low-E, to the momentum of its track, p, as measured by the ID. The ratio is denoted The calorimeter response to charged hadrons is studied using the ratio of the energy deposited by the isolated charged particle in the calorimeter, The energy of a topological cluster in a certain layer of the calorimeter is matched to the track if the energy-weighted position of the cells associated with the topological cluster in that layer is situated within ow level .The construction of the mentclass2pt{minimE and the non-compensating response of the calorimeter. The large fraction of tracks with E but not to p. In events with significant amounts of pile-up, such as those shown in Fig. E tail from noise; however, this tail is only a very small fraction of the distribution. In the same Figure  cluster . The negentclass1pt{minimaentclass1pt{minimaThe fraction of the distribution with Detector noise, which is largely symmetric, drives the portion of the response distribution with E. While the isolation requirement on the charged-hadron track is efficient at reducing the effect from other charged particles to negligible levels, there is no equivalent method for eliminating the contribution from neutral particles, such as neutral hadrons or photons from Energy deposits from close-by particles bias the calorimeter measurement of Since neutral particles, which are mostly photons with some low-energy hadrons, deposit their energy mostly in the EM calorimeter, it is possible to measure the background in situ by considering late showering charged hadrons, which behave like minimally-ionising particles (MIP) in the EM layers of the calorimeter. Such late showering hadrons are selected by requiring that they leave less than 1.1 The While this method accounts for the average energy deposited by the neutral-particle background, it cannot account for per-event background fluctuations. This is particularly important when considering threshold effects, since a small background energy deposit might be sufficient to raise the signal in a cell above the threshold required to seed a topological cluster. In events with large background, this can lead to a positive bias in Pythia8, as well as the difficulty of correctly modelling the calorimeter response to low-momentum neutral particles. They do not strongly indicate a deficiency in the detector description, which would typically be isolated in a narrow region of Pythia8, as the background is consistently and significantly higher in the MC simulation than in the data. Provided the neutral-particle background is correctly accounted for separately in data and MC simulation, however, any imperfection in the modelling of neutral particles can be removed from the comparison of calorimeter response. The neutral-particle background is calculated separately for each dataset and calorimeter configuration considered in this paper.Figures Part of the difference between the rate of tracks with no associated energy in the simulation and the data  technique  applies The mentclasspt{minimaThis difference is primarily due to the difference in the modelling of the response to anti-protons, as is suggested by Fig. Historically, the calorimeter response to isolated single particles has been measured using events with only a single proton–proton collision in the event. Pile-up contributes additional neutral-particle background to the event that is normally only removed on average from the topological clusters. The charged-particle background from pile-up can still be removed using the track isolation requirement. Moreover, fluctuations of the neutral-particle background significantly worsen the energy resolution for low-momentum particles. Nonetheless, as the background subtraction technique in this paper depends only on the average background distributions, the (pile-up and background-corrected) There are two response issues to be addressed in these events. The first is the dependence of the response on the number of reconstructed vertices, which is an excellent proxy for the in-time pile-up. The raw response, background, and background-corrected response to isolated charged hadrons as a function of the number of reconstructed vertices is shown in Fig. The ATLAS calorimeter is additionally sensitive to out-of-time pile-up, collisions in bunch crossings close in time to the one that was selected by the trigger, although this dependence is mitigated somewhat by the bipolar pulse shape of the calorimeter electronics. A bunch-dependent correction is applied to the calorimeter energy measured in each calorimeter cell to correct for the residual average energy shift per bunch due to the bunch train structure and the fluctuations in the luminosity per bunch crossing. Energy deposits up to 100 ns after, and up to 800 ns before the collision of interest may affect the energy measured in a calorimeter cell. Thus, an equally important test is the stability of the response to isolated charged hadrons against the average number of proton–proton collisions per bunch crossing, To measure the response of the hadronic calorimeter, only tracks that deposit an amount of energy in the EM calorimeter consistent with a MIP are selected. The criteria for selecting a MIP are identical to those described in Sect. Figure In order to examine the response of the EM calorimeter alone, particles are selected that deposit most of their energy in the EM calorimeter. In this case, tracks are required to have no associated energy in the hadronic calorimeter. Such a selection is inherently more sensitive to neutral-particle backgrounds, which deposit most of their energy in the EM calorimeter. A comparison of During the low-Figure The change in threshold settings affects the EM calorimeter in particular, because particles from pile-up tend to be low-energy and deposit most of their energy in the EM calorimeter, leading to more similar threshold settings in the hadronic calorimeter when calculated with and without pile-up. For tracks leaving significant energy in the tile calorimeter and only minimal energy in the EM calorimeter, therefore, the two topological cluster settings are expected to provide comparable results. This comparison is shown in Fig. In addition to the calorimeter response to an inclusive set of isolated charged hadrons, uncertainties in jet energy scale calibration rely on an understanding of the modelling of the detector response to specific particle species. A jet includes a variety of hadrons that need to be modelled. A measurement of the response to individual hadron species can be used to validate that the MC simulation correctly models each component of the jet shower. This study uses decays of identified particles that have long enough lifetimes to be identified via a secondary vertex. A sample of individual particle species is extracted to provide separate measurements of the calorimeter response to each. Only the 2012 data are used for these comparisons, as the 2010 and 2012 data show consistent features.Decays of In addition to the event selection required for the inclusive tracks listed in Sect. To match the energy available to be deposited in the calorimeter, the ratio The selection of particle decays in the ID is based on previous ATLAS results . The decExample mass distributions for reconstructed mentclass2pt{minim in 2010 .Table 1TThere are three primary sources of charged backgrounds in the identified particle Another charged background for identified candidates comes from combinatoric sources. The purity of The final charged background for the proton (anti-proton) There is also a contribution to the Thus, all of the systematic uncertainties arising from backgrounds are found to be negligible compared to the statistical errors of the available data sample.Examples of the uncorrected distributions of not from the hadronic-interaction model for one particle species, but is caused by an effect present for all particle species.A significant feature of the In order to reduce the effect of the neutral background, the average responses, ith Ref. , where tFigure or pions , 34. ThiFigure s models , the ress models .Fig. 28Tp. Although the accessible range of momenta of identified tracks is limited, these distributions suggest that the data and MC simulation differ in the Insofar as the neutral background is independent of the species of the particle of interest, the neutral background estimate described in Sect. Pythia8 is dominated by charged pions (60–70%), with lesser components from charged kaons (15–20%) and protons and anti-protons (5–15%). The largest component of this composition that has not been measured in ATLAS is the charged-kaon component. Discriminants based on ionisation energy loss in thin detectors are only applicable at low momenta  are also visible in this figure: these jets appear more like gluons at low As shown in Fig. For the jet response and uncertainty, the important quantity is the fraction of jet energy carried by particles in several momentum ranges. This determines what the most important measurements are, through the convolution of particle-level response and uncertainties into jet-level response and uncertainties. The fractions for several jet Within the MC simulation, it is possible to isolate the contribution to the energy deposited in the calorimeter from each individual particle in the jet. Figure cf. Fig. , but thip and E / p”, from Sect. The statistical uncertainties in the main inclusive The uncertainty in p and E / p Zero Fraction”, from Sect. The difference in the zero-fraction between data and MC simulation, with a calorimeter setting of E / p Threshold”, a parameterisation based on the studies in Sect. The uncertainty in the EM calorimeter response from the potential mis-modelling of threshold effects in topological clustering, derived using the comparison of response calculated with cells to that calculated using topological clusters , LAr prGeant4 .p measurement from misalignment of the ID .The uncertainty in the p and The main rom Ref. ).The EM energy scale in the combined test beam of the LAr calorimeter (0.7%) and tile calorimeter (0.5%), for rom Ref. ).The response uniformity in the combined test beam of the LAr calorimeter (0.4%) and tile calorimeter (1.5%), for rom Ref. ).A flat 10% uncertainty added to all particles above the energy range probed with the combined test beam (i.e. for The same 10% uncertainty is applied in the combined test beam momentum range when examining regions higher in No explicit uncertainty term is derived to account for possible dependence of the jet energy scale and uncertainty in the composition of the jets in the MC simulation, either in terms of partonic origin or particle content. The uncertainty derived in this manner is applicable to data taken with a calorimeter threshold setting of In order to derive a jet energy scale response and uncertainty, the calorimeter response to single particles shown in this paper is combined with the response measured in the combined test beam , and addBased on these effects, the jet energy scale uncertainty is extracted. Each term is treated as an independent Gaussian-distributed uncertainty, and pseudo-data are used to determine both the full uncertainty and the size of the uncertainty correlations between jets with different entclass1pt{minima methods .4 HoweveThe pseudo-data are also used to explore the correlations in uncertainty of jets at different Geant4-based MC simulation is consistent with the data.A measurement of the calorimeter response to isolated single charged hadrons in the ATLAS detector with data at Displaced decays are used to construct samples of pions, protons, and anti-protons. These samples suggest that the description of response to anti-protons by the hadronic physics models in the FTFP_BERT set of hadronic physics models is consistent with the data below 5 The jet energy scale uncertainty is derived using these calorimeter response observables, along with results from the ATLAS test beam and additional MC simulation. The uncertainty derived in this manner is 2–5% for jets with


We investigate modified theories of gravity in the context of teleparallel geometries with possible Gauss–Bonnet contributions. The possible coupling of gravity with the trace of the energy-momentum tensor is also taken into account. This is motivated by the various different theories formulated in the teleparallel approach and the metric approach without discussing the exact relationship between them. Our formulation clarifies the connections between different well-known theories. For instance, we are able to formulate the correct teleparallel equivalent of Gauss–Bonnet modified general relativity, amongst other results. Finally, we are able to identify modified gravity models which have not been studied in the past. These appear naturally within our setup and would make a interesting starting point for further studies. One possible approach to motivating a geometrical theory of gravity is to compare the geodesic equation of differential geometry with Newton’s force law. This suggests the identification of the gravitational forces with the components of the Christoffel symbols which in turn yields the identification of the gravitational potentials with the metric. Assuming a geometrical framework with a metric compatible covariant derivative without torsion gives the building block of Einstein’s theory of general relativity, one can speak of the metric approach.Soon after the original formulation of this geometrical theory of gravity, it was noted that there exists an alternative geometrical formulation which is based on a globally flat geometry with torsion. The key mathematical result to this approach goes back to Weitzenböck who noted that it is indeed possible to choose a connection such that the curvature vanishes everywhere. This formulation gives equivalent field equations to those of general relativity and we refer to this as the teleparallel formulation. This naming convention stems from the fact that the notion of parallelism is global instead of local on flat manifolds; see for instance , 2 and tR is the Ricci scalar, T is the torsion scalar and B is a total derivative term which only depends on torsion. Clearly, the Einstein–Hilbert action can now be represented in two distinct ways, either using the Ricci scalar or the torsion scalar, and consequently giving identical equations of motion. A popular modification of general relativity is based on the Lagrangian f(R) and can be viewed as a natural non-linear extension that results in fourth order field equations [f(T) which gives second order field equations [T itself is not invariant [B but the particular combination f(R) gravity and f(T) gravity and not equivalent and correspond to physically different theories. Now, considering the more general family of theories based on f one can establish the precise relationship between these theories and it turns out that f(R) gravity is the unique locally Lorentz invariant theory while f(T) gravity is the unique second order theory.One of the basic equations of general relativity and its teleparallel equivalent is quations , 4. On tquations . Howevernvariant . Neithersee also . Hence, The principal aim of this paper is to extend these results to take into account the Gauss–Bonnet term and its teleparallel equivalent. The Gauss–Bonnet scalar is one of the so-called Lovelock scalars  which onThe teleparallel equivalent of the Gauss–Bonnet term was first considered in , 14 who nt years .There exists a large body of literature dealing with the various modified theories of gravity that have been investigated; see for instance the reviews , 16–18.Our conventions  Greek indices denote spacetime coordinates, Latin indices are frame or tangent space indices. entclass1pt{minimaThe teleparallel formulation of general relativity is well known and provides some interesting insight into this theory. The fundamental objects of this formulation are the tetrad fields K is the contortion tensor. The contortion tensor can be expressed using the torsion tensor as follows:R(e) stands for the metric or Levi-Civita Ricci scalar; see also [The starting point of the teleparallel formulation of general relativity is the objectentclass1pt{minimaT so that Eq.  gravity [In contrast to the standard teleparallel approach, complete Lorentz invariance is preserved when considering metric-affine theories  in which gravity  which isTeleparallel geometries have been well understood for many decades. Perhaps more surprising is the fact that the Gauss–Bonnet term was not studied in this context until quite recently . The GauFollowing the procedure outlined in the above, we can again compute the (complete) Gauss–Bonnet term using the connection  and decoG is a Lorentz scalar, these second derivative terms must be cancelled by terms coming from On the other hand, the Gauss–Bonnet boundary term ced from  who do nfinition  containsIn the following we will show some simple examples of the Gauss–Bonnet term and its teleparallel equivalent.G is independent of the Cartesian coordinates. The unique linear combination Let us begin with the FLRW metric and the diagonal tetrad given bytline in , 24 we wt, x, y, z) to avoid coordinate issues with the tetrads. Including time dependence is straightforward, however, the resulting equations are too involved. We chooseR, T and B are given byIn this example we consider static and spherically symmetric spacetimes and work with isotropic coordinates  gravity, for instance, is rather involved, see [The tetrads used in  and 29)29) serveved, see . In geneved, see .a(t) is the scale factor of the universe and Let us next consider FLRW metric in spherical coordinates given byf(T) gravity it implies an off-diagonal field equation which is highly restrictive, namely the condition t and r. Doing this means we will work with the rotated tetradT and the boundary term B becomesr and t, respectively. In order to have T and B position independent we must choose our function T and B time dependent only. Therefore, the rotated tetrad  one can formulate the teleparallel equivalent of f(R) gravity and identify those parts of the field equations which are part of f(T) gravity, the second order part of the equations which is not locally Lorentz invariant. In analogy to this one can also make the relationships between various modified theories of gravity clear which are based on the Gauss–Bonnet term.Let us begin these conclusions with a discussion of the relationship between the various modified theories of gravity which are governed by the function shown in , by consThe top left corner of Fig. If we now include the trace of the energy-momentum tensor to our approach, things get slightly more complicated as the number of possible theories increases quite dramatically. We tried to visualise the entire set of possible theories in Fig. Now, the left half of the figure corresponds to the metric approach while the right half corresponds to the teleparallel framework. The four main theories of the previous discussions are highlighted by boxes. Many of these theories were considered in isolation in the past and their relationship with other similarly looking theories was only made implicitly. We should also point out that our representation of these theories is only one of the many possibilities and moreover, Fig. f should be viewed as a special case of the teleparallel gravity theory f. To see this, simply recall the principal identity f theory and also that the teleparallel framework should be viewed as the slightly more natural choice for this theory.In constructing the diagram we also made the interesting observation that the theory based on the function We provide a short list of theories in Table Of the many possible theories one could potentially construct from It is also useful to make explicit the limitations of the current approach put forward by us. In essence, we are dealing with modified theories of gravity which are based on scalars derived from tensorial quantities of interest, for instance the Ricci scalar or the trace of the energy-momentum tensor. However, theories containing the square of the Ricci tensor or theories containing the term mentclasspt{minimacall Eq. . TheorieT which is motivated by its close relation to the Ricci tensor. However, in principle one could follow the work of [The current approach is entirely based on the torsion scalar  work of  and decof(R) gravity for instance can be formulated as a theory based on a non-minimally coupled scalar field. Hence, many of the theories in Fig. For many years now, an ever increasing number of modifications of general relativity has been considered. In this work we focussed only on theories where the gravitational field can either be modelled using the metric of the tetrad. Hence, we excluded all types of metric-affine theories where the metric and the torsion tensor are treated as two independent dynamical variables. It would be almost impossible to present a visualisation that encompasses all those theories as well. Even this would represent only a fraction of what is referred to as modified gravity. It would still exclude higher dimensional models, Einstein–Aether models, Hořava–Lifshitz theory and many others. It is also interesting to note that This discussion motivates the process of classifying the different families of modifications of general relativity and their possible interrelations, followed by a broad investigation of which theories should not be studied further due to incompatibilities with well-established observational bounds. It appears that we are reaching the point where we possibly do not need more theories but rather an improved sense of direction for future developments.


The concept of feedback is key in assessing whether a perturbation to a system is amplified or damped by mechanisms internal to the system. In polar regions, climate dynamics are controlled by both radiative and non-radiative interactions between the atmosphere, ocean, sea ice, ice sheets and land surfaces. Precisely quantifying polar feedbacks is required for a process-oriented evaluation of climate models, a clear understanding of the processes responsible for polar climate changes, and a reduction in uncertainty associated with model projections. This quantification can be performed using a simple and consistent approach that is valid for a wide range of feedbacks, offering the opportunity for more systematic feedback analyses and a better understanding of polar climate changes. Estimating the magnitude of radiative and non-radiative feedbacks is key for understanding the climate dynamics of polar regions. Here the authors propose an inclusive methodology to quantify the influence of all those feedbacks, stimulating more systematic analyses in observational and model ensembles. In contrast, the southern polar regions have warmed less rapidly with some regions experiencing cooling and sea ice advance and others experiencing warming and sea ice loss6.The climate of polar regions is highly sensitive to changes in climate forcing, but also displays large internal variability. Over recent decades, northern polar regions have warmed more than twice the global average with sea-ice decreasing trends for all months of the year, especially in late summer6, large internal climate variability9 and the large biases of climate models in these regions10.Observed changes in polar regions result from numerous interactions involving the atmosphere, land surfaces, ocean and sea ice. Due to the complexity of the underlying processes, we do not fully understand them. Advancing scientific understanding in polar regions is particularly challenging due to a short and incomplete observational record12 offers a standard method to analyze such complex dynamics , is termed the ‘radiative forcing’ and is due, for instance, to a change in the atmospheric concentration of carbon dioxide (CO2) or in solar irradiance. Consider as an example the response to a positive radiative forcing resulting from an increase in greenhouse gas concentrations. This will initially lead to a decrease in outgoing longwave radiation, resulting in a TOA radiative imbalance and accumulation of energy within the climate system117. This in turn will trigger changes in the climate, in particular a temperature increase that leads to larger emissions of infrared radiation by the Earth. Ultimately, those larger emissions will compensate for the additional energy input due to the forcing. After some time, the climate system will come into a new equilibrium characterized by higher temperatures than before the perturbation was applied.The radiative feedback framework is based on the analysis of changes to the energy balance at the top of the atmosphere (TOA) caused by a perturbation. An initial perturbation to TOA radiation, Ts (in K). The imbalance of the energy budget at the TOA averaged over the whole Earth at any time  is then expressed asλ is the net climate feedback parameter (W m−2 K−1), which is a key characteristic of the climate system response, and ΔTs is the global surface air temperature change following the perturbation. λ is negative for a stable climate and a larger absolute value corresponds to a less sensitive climate characterized by a smaller temperature change for a specific forcing. At equilibrium, when by definition the heat budget is balanced at the TOA , the surface temperature change in response to the perturbation is simply ΔTs = −F/λ. The equilibrium climate sensitivity, estimated as the global mean temperature change in response to a doubling of the CO2 concentration in the atmosphere, which corresponds to a radiative forcing F of roughly 3.7 W m−2, is thus equal to −3.7/λ. The transient imbalance at the TOA corresponds to heat storage, which is mainly accounted for by the ocean, so the term ΔR is often approximated by the ocean heat uptake66.When studying the energy budget of the whole Earth, it is convenient to assume that the modification of the radiative fluxes emitted by the Earth is proportional to changes in global mean surface temperature λ can be separated into contributions from changes in surface albedo, clouds, water vapor and temperature, referred to as the feedback variables. The feedback related to temperature is itself the sum of a contribution from vertically homogenous warming or cooling  and one from changes in vertical temperature gradient (the lapse rate feedback). For each process, a specific feedback parameter λi can be computed. Their sum approximatively gives back the net climate feedback parameter ε is a small residual accounting for non-linearities. A positive value of the feedback parameter λi corresponds to a positive feedback, a negative one to a negative feedbackThe net climate feedback parameter 12). With no feedback, the reference response of the system to a radiative forcing F is considered to be ΔT0 = −F/λ0. A feedback, with a feedback parameter λi, will induce a change in the radiative balance λiΔTs that will reinforce or dampen the effect of the radiative forcing, leading to a response of the system of ΔTs = −(F + λiΔTs)/λ0.Schematic illustration of the radiative feedback framework  radiative fluxes due to changes in surface and tropospheric temperatures Table . It can 22. This water vapor feedback is largest in the tropics where the climatological temperature is higher and the increase in water vapor is at its maximum. In polar regions, the positive water vapor feedback is weaker than in the tropics but it still plays a relevant role in the polar response to the forcing24.As the surface warms, additional water vapor amplifies the greenhouse effect and induces further warming28. As the climate warms, snow and ice cover melt, exposing underlying surfaces that typically have much lower albedos. This leads to an increased absorption of shortwave radiation by the surface, and as a result amplifies the initial warming. When melting, the snow covering Arctic sea ice contributes to forming melt ponds. increasing the absorption of solar radiation and amplifying the surface albedo feedback29. Melt ponds do not form in the Southern Ocean as surface melting is very limited there, providing an illustration of different ways snow and ice interactions affect the surface albedo feedback30.The surface albedo feedback is a first-order visible (shortwave) positive radiative climate feedback mechanism in polar regions32. The sign of any cloud feedback depends on the balance of shortwave cooling and longwave heating by the clouds. Cloud feedbacks are the most uncertain of all the radiative feedbacks as the cloud radiative effect depends on several factors that can be modified by the initial response to the perturbation35. Among all mechanisms involved, two polar-specific cloud feedback examples are listed in Table 39 and the cloud optical depth feedback40. When sea ice melts and new open water is exposed, surface turbulent heat fluxes can increase humidity in the lower atmosphere and increase low-level clouds. During polar night, increasing low cloud increases downwelling longwave radiation, leading to further sea ice loss and thus to a positive feedback. Observational evidence shows that this cloud-sea ice feedback operates in non-summer months in both the Arctic39 and the Antarctic41. The cloud optical depth feedback operates both at mid- and high- latitudes. Cloud liquid particles are smaller than cloud ice particles, and are therefore more efficient at reflecting solar radiation back to space. As the climate warms, the total amount of cloud water in mixed phase clouds increases, which increases the amount of reflected solar radiation , acting as a negative feedback32. At the same time, the fraction of cloud water that is liquid also increases, enhancing the effect of this cloud optical depth feedback . In winter, when brine is released by sea ice formation, surface mixed layer density increases, inducing a mixed layer deepening and the entrainment of relatively warm water into the surface oceanic layer. This warm water reduces ice formation and can even melt ice, which partly compensates for the initial ice formation, leading to a negative ice production–entrainment feedback53. If ice production is very large during a particular year, the mixed layer will be deeper and the salt released by ice formation will be distributed over a larger depth range. In summer, the mixing is weaker and the freshwater input due to ice melting will be spread over a shallower layer, leading to a net downward vertical salt transport. This can restrain the vertical exchanges in the water column the subsequent winter, leading to less heat transfer to the surface and more heat storage at depth. Finally, the weaker heat flux at the ice–ocean interface would favor additional ice formation, leading to a positive feedback amplifying the initial perturbation. The heat storage at depth can also be reinforced by a net freshwater input at the surface (due for instance to a net transport of sea ice to the region) that further stabilizes the water column.The ice production–entrainment feedback is acting at the seasonal scale but the magnitude of the ice–ocean flux can also be modulated at inter-annual to decadal timescales, leading to the positive ice production heat-storage feedback54. This positive soil moisture–temperature feedback is less active at high latitudes but, despite the low temperatures, evapotranspiration strongly contributes to moderate the summer warming over land55.At low and mid-latitudes, a drying of the soils in response to an initial temperature rise can amplify the warming as the evapotranspiration that normally cools the surface is reduced28. They also give rise to a number of specific feedback mechanisms. Three important ones are discussed here. Compared to the feedbacks mainly involving the atmosphere and sea ice, ice sheets generally, but not exclusively, play a role on longer time scales. In the positive surface mass balance–elevation feedback, increased air temperature leads to ice melting, which lowers the surface elevation of the ice sheet, exposing the ice to warmer air temperatures and thus further melting57. This positive feedback is mostly relevant for the Greenland ice sheet where surface melting is substantial, while currently the Antarctic ice sheet hardly experiences it due to very low surface temperature.Ice sheets, glaciers and snow cover over land provide key components to the surface albedo feedback60. The stability of a marine ice sheet such as the West Antarctic Ice Sheet is determined by the position of the grounding line, i.e., the boundary between the grounded ice sheet and the floating ice shelf. If it is located on a bedrock sloping downward toward the interior of the ice sheet, an initial retreat of the grounding line, for instance due to basal ice melting, leads to an increase in ice discharge, which results in a further retreat of the grounding line inland until a new stable position is reached. Rapid changes in ice sheets may also be linked to the acceleration of the ice transport due to basal lubrication caused by meltwater penetration at the bed or to breakup of the ice shelves because of a weakening of the ice due to surface melting62.The marine ice sheet instability has the potential to destabilize large ice sheet regions64.Another feedback mainly acting in the Southern Ocean is related to the interactions between floating ice shelves, sea ice and the surrounding ocean. A subsurface Southern Ocean warming leads to increased basal ice shelf melting, and the upper ocean layers get fresher due to the resulting cold freshwater input. This results in lower heat flux from the ocean interior to the surface, sea ice expansion and reduced ocean surface warming, providing a negative ice shelf melting sea ice feedback68. One example is the bio-optical feedback, which occurs when climate warming and sea ice retreat in the Arctic Ocean lead to intense phytoplankton blooms69. These blooms trap the penetrating solar heat flux at the ocean surface, which increases sea surface temperature. As a result, sea ice concentration decreases, which leads to enhanced absorption of solar energy into the ocean and further warming of the Arctic70.While this Perspective focuses on feedbacks that act through physical processes in polar regions Table , Fig. 1,75. This indicates a coupling between radiative feedbacks and atmospheric heat transport77. Moreover, radiative feedbacks at low latitudes may influence polar warming through their effect on poleward energy transport, while changes in polar regions may affect dynamics in the lower latitudes78. This is an area of ongoing research, and it is not yet clear if the response of the system has a direct impact on the original perturbation itself such that a closed feedback loop can be identified.The response to a perturbation also implies a redistribution of the energy between different latitudes. First, the warming of the tropics under greenhouse gas forcing leads to enhanced poleward energy transport by the atmospheric circulation to higher latitudes, contributing to warming there81 and decreased poleward heat transport into the Southern Ocean79 under global warming. Here too, it is unclear whether these changes can be represented in terms of a closed feedback loop  or should be classified as important drivers of polar climate change that cannot be expressed within a feedback framework.Second, the ocean heat transport has been found to strongly shape polar climate change, with increased poleward heat transport into the Arctic−2 K−1. The net climate feedback parameter λ, which is equal to the sum of all the parameters for the individual feedbacks, can be estimated by measuring all the terms of the equation describing the global mean radiative balance  the Planck feedback λ0: γi = λi/−λ0. An advantage of this approach is that the feedback factor γi is dimensionless because it is expressed relative to the reference system. It can then be used to compare the impact of very different processes, bearing in mind that its specific value depends on the reference system chosen12(for more details see the methods).In parallel to feedback parameters, other expressions can sometimes be easier to interpret or be more convenient. One option is to diagnose the temperature change that can be attributed to each feedback explicitly, known as a warming contribution (see the methods). It is also instructive to compare the temperature changes due to a particular feedback to changes of a reference system in which the feedbacks of interest are inactive. In the radiative feedback framework, the reference system is traditionally chosen as the Planck response. The feedback factor 89 (CFRAM). It has also been proposed to decompose the feedbacks in ways that differ90 from the traditional one described in Box In addition to the approaches focused on the top of the atmosphere or the surface, it is possible to analyze the origin of three-dimensional temperature changes such as in the climate feedback response analysis method92. Moreover, assuming linearity in feedbacks fails in many cases, as can be expected for a system as complex as the Earth’s climate. For example, the magnitude of the climate feedback parameter λ generally decreases with time in climate models after a rise in atmospheric CO2 concentration, corresponding to increasing climate sensitivity as equilibrium is approached92. λ may also depend on the magnitude of the perturbation95.The standard radiative feedback framework assumes that the response of the system can be expressed as a linear function of the surface temperature. It is a very useful approximation but some processes cannot be expressed in terms of functions of single variables and the radiative feedback framework has to be adjusted to capture changes in the system not directly related to surface temperatureγ is not constant. Non-linearities can be caused by several processes. The strength of the feedback can be a function of the state of the system. This state dependence can often be expressed as a time dependence when the state changes with time. Furthermore, the different processes controlling the response to a perturbation may have different time scales. Their relative contribution to local and global scale feedbacks may thus evolve leading to spatially or temporally non-constant feedback factors96.The non-linearity of the feedbacks can be described in different ways. A simple definition will be used here: the feedback is non-linear if the feedback factor 40. Furthermore, feedbacks related to the cryosphere generally depend on the surface area covered by snow or ice. As temperatures rise, this area decreases and the feedback strength is reduced. This is illustrated87 in Fig. 2 in the Community Climate System Model version 3 (CCSM3). At many latitudes, the value of the feedback factor is smaller for the third doubling (8 × CO2–4 × CO2) than it is for the first (2 × CO2–CNTL). Between 50°S and 60°S the feedback approaches zero for the third doubling, since the Southern Ocean is already ice-free at these latitudes in the 4xCO2 climate, and no further melting can occur. On the other hand, the value of the feedback factor increases at northern high latitudes (75°N–90°N), as the sea ice edge retreats within the central Arctic at high warming.In polar regions, the presence of different phases of water implies that many feedback parameters display a particularly strong dependence on the state of the system near the freezing point and are thus highly non-linear. For instance, phase changes play an important role in polar clouds leading to non-linearities in the cloud feedback89. This approach has also been successful for some biogeochemical and biogeophysical feedbacks66.The traditional radiative feedback framework has been extended to additional processes that influence the energy balance of the Earth, offering an effective way to evaluate and compare the strength of the different feedbacks49, but this definition has not been widely used so far. The effectiveness of the ice production-entrainment feedback can be measured51 as the ratio between the melting immediately caused by the entrainment of warmer water in the surface layer to the initial ice growth. The ice production-ocean heat storage feedback has been estimated by the ratio between the heat losses associated with sea-ice volume changes to the heat storage below the surface level53. Both quantities can be evaluated directly from observations or model results.  a relatively large and positive lapse rate feedback, due to a different vertical distribution of the temperature change compared to the tropics; (2) a relatively weak negative Planck response, due to smaller blackbody emissions per unit warming at lower temperatures (Stefan–Boltzmann law); and (3) a large positive surface albedo feedback, due to the loss of high albedo snow and ice-covered surfaces, as well as a contribution from atmospheric heat transport  in polar regions than in the tropics. This explains the larger temperature changes experienced in polar regions in response to a perturbation Fig. , a phenoort Fig. . In the ort Fig. .Fig. 3Co98.Note that if feedbacks are defined using global mean rather than local surface temperature change, the Planck feedback appears strongly negative in the Arctic because the local temperature change exceeds the local mean. Additionally, the polar amplification has a large seasonal cycle, displaying over the Arctic a minimum in summer and a maximum in fall/winter. In summer, the influence of the large positive surface albedo feedback is compensated by a strong oceanic heat uptake and negative cloud feedbacks while in fall/winter the heat released by the ocean, the contribution of lapse rate feedback and cloud feedbacks induce a large warming100. Yet, perhaps the most important challenge is identifying the climate system processes that must be represented in order to consider a model realistic enough for mechanistic studies and projections. More generally, simple comparisons between model results and observations do not allow estimating the origin of model biases or their impact. Process-based model evaluation offers the possibility of exploring the causes of discrepancies more deeply, identifying the links between various variables and ultimately suggesting model improvements103.In polar regions, many studies have identified model strengths and weaknesses in reproducing observationsTo illustrate this point, we compare the ice production–entrainment feedback in three existing simulations for all the sectors of the Southern Ocean with estimates derived from observations and a reanalysis Fig. . A clearOne justification of the development of the radiative feedback framework is to determine the processes that can explain the range of model projections for a specific scenario of future changes in radiative forcing. As expected, the models displaying the largest surface temperature changes are the ones for which the radiative feedbacks have the largest (most positive) values. The same approach can be applied to the non-radiative feedbacks investigated here as they are related to the magnitude of the response to any type of perturbation.104. In other words, there is no a priori reason to believe that a model which reproduces the present global mean temperature well will provide more reasonable projections of future climate than a model that has larger biases. Indeed, a model may have a global mean temperature close to observations due to compensations between many factors that may not necessarily balance in a projected climate105.Although it is better if a model is able to reproduce the observations with a bias that is as small as possible, it is not clear for many variables, such as the global mean temperature, that the response to a perturbation is a function of this bias107. Furthermore, it has been argued that a model displaying a more realistic mean state in polar regions will also have a better representation of key processes and thus will provide a more likely estimate of future climate changes than a model with larger biases. This has then been used to justify the selection of models based on their mean state109. This idea appears useful in principle but is hard to generalize and is subject to criticism. In particular, it is not always clear to determine how to evaluate models, which variable should be used to select models, and if the currently available model sample is adequate to apply a meaningful selection. This has led to strong debate in the community about the justification of this approach, which may artificially reduce the uncertainty range by discarding model results that are as likely as the others110. We propose to use feedback quantification more extensively to evaluate model behavior, and foresee that this can contribute to more robust estimates of the likelihood of projections.The situation is distinctive for polar regions, where the feedbacks are strongly non-linear and thus state dependent. This provides an instructive way to interpret the range of model responses as a function of the value of some variables for present-day conditionsγ.The review above illustrates that many definitions and evaluation methods have been proposed for the various radiative and non-radiative feedbacks. Nevertheless, all the feedbacks can be described and quantified using a simple and consistent framework, based on the definition of a feedback factor γi is the ratio of a particular feedback parameter to minus the Planck feedback parameter. An analogous expression can be written for any other feedback. When only one feedback is operating (see the methods for the case of multiple feedbacks), the feedback factor γ can be quantified as the ratio between the additional changes specifically due to the feedback and the response of the full system including all the feedbacks . This additional change  is itself computed as the difference between the response of the full system and the one of a reference system in which the feedback under consideration does not operate (Reference response):For radiative feedbacks, the feedback factor 12. Let’s examine radiative feedbacks as one example (see the methods): (1) the perturbation is the radiative forcing F, (2) the response variable is Ts, (3) the full system includes one or more radiative feedbacks plus the Planck response referenced to Ts, and (4) the reference system is the Planck response only.The methodology requires explicitly identifying (1) a perturbation or a class of perturbations, (2) a response variable involved in the feedback loop, (3) the full system with all processes operating and its response to the perturbation, and (4) the reference system with the process of interest not operating and the reference system response to the perturbation. While the framework is general, a clear definition of this system is required, as the value of the feedback factor depends on the way the reference system is chosenAs it is based on the same principles, analysis of non-radiative processes using this feedback factor retains the main advantages of the radiative feedback framework. First, each feedback can be associated with a well-defined conceptual model describing the mechanisms and interactions involved. This is essential in order to allow each feedback to be firmly rooted in a process-based analysis that is straightforward to apply and understand. Secondly, it is possible to evaluate the magnitude of the feedbacks using a dimensionless factor, ideally both in models and observations. This is required to assess the contributions of the different feedbacks to the total response and to compare the role of each feedback in various Earth System models to determine which is responsible for their distinctive sensitivities.89.As in the example below, the feedback factor can in some cases be evaluated using observations or model outputs only, but it may also require specific additional calculations. This is illustrated using a simple model in Supplementary Note 50. For this negative feedback, (1) the perturbation is a given amount of ice production, (2) the reference variable is ice thickness, (3) the full system is the sea ice plus ocean column with the entrainment process, and (4) the reference system is the sea ice plus ocean column but without entrainment. The intensity of this feedback can then be evaluated using the ratioWe illustrate the methodology with the ice production–entrainment feedback50 and used for Fig. Despite a different form, this expression is strictly equivalent to the original formulation proposed in ref. γθ increases with ice formation  until the end of winter  and the heat input grows. Consequently, the absolute value of the feedback factor ter Fig. . This no2 concentrations  it is compatible with the radiative feedback framework which has proven useful over the past three decades, (2) it is based on a clear physical interpretation of key physical processes, (3) the framework is simple to articulate, and (4) an unambiguous quantification can be obtained allowing an objective evaluation of the processes that control the response to a perturbation.Using a feedback factor provides the clear theoretical advantages of being consistent with the feedback theory, general and easy to interpret. Nevertheless, based on data availability or on the specific goal of a study, other parameters will continue to be used to diagnose the magnitude of some feedbacks. This may of course be perfectly justified. However, the limitations and merit of such an approach compared to a more general view need to be discussed and taken into account when interpreting the results.Even though the feedbacks discussed here are well-known, some are often only used to provide a qualitative interpretation or a narrative framework to explain the changes occurring in polar regions. In particular, the quantitative evaluation of non-radiative feedbacks in models is rare, with only few and, in some cases, no publications on this topic for some feedbacks. This Perspective motivates the use of more systematic approaches to analyze past and upcoming model results. In particular, the framework presented here allows tracing the origin of model diversity back to physical considerations. Identifying the feedbacks that are critical for correctly simulating the mean state and variability of polar climate will ultimately promote the development of targeted observational campaigns, by means of which models will be evaluated. Such campaigns already exist or are underway: the Year of Polar Prediction (YOPP) or the Multidisciplinary drifting Observatory for the Study of Arctic Climate (MOSAiC) are two examples. In summary, advancing evaluation of feedbacks will require dedicated model experiments and careful analyses, complemented with the collection of dedicated observations that can constrain model feedbacks. Some of those elements are clearly challenging, but this will be strongly beneficial for our understanding of polar climate dynamics and of the future changes expected at high latitudes.F (in W m−2) at equilibrium can be written asTS (in K) is the surface temperature change, λ0 (~−3.2 W m−2 K−1) the Planck response and the λi (in W m−2 K−1) correspond to the radiative feedback parameters related to the response of surface albedo, clouds, water vapor and vertical temperature gradient (lapse rate feedback).The global mean radiative balance at the TOA in response to a radiative forcing 2 would be governed by only the Planck response, given by12 1.2 K. This temperature change is amplified or damped by individual feedbacks, depending on whether they contribute to a positive or negative additional radiative perturbation to the TOA radiative balance in response to warming. The overall equilibrium warming resulting from a CO2 doubling is thus greater than ΔT0, likely14 between about 2 and 4.5 K.In the absence of these feedbacks, the equilibrium surface temperature change in response to a doubling of COγi = λi/−λ0. In turn, the surface temperature change in response to forcing can be cast in terms of the feedback factors asThe primary advantage of the feedback framework is that it allows a process-based analysis of the adjustment of the system to a radiative perturbation and a quantification of the importance of each process. Traditionally, the magnitude of each feedback is compared to that of the Planck response, giving dimensionless feedback factors 72, we define the warming contribution as the energetic contribution (in W m−2) associated with a particular feedback (λiΔTs) or atmospheric forcing  divided by the magnitude of the global-mean Planck response Ts ; λ diagnosed using radiative kernels85 averaged over years 85–115 of abrupt CO2 quadrupling simulations of 13 models72; ΔOHU is diagnosed as the anomalous net surface heat flux and ΔAHT as the difference between ΔOHU and net TOA radiation flux anomalies. Figure The formalism can be extended to assess the relative contributions of individual feedbacks to local surface warming by use of the local energy budget equation:λ0 toWhen several feedbacks are operating, their contribution to the changes in the radiative balance is additive. Using the same notations as above, the radiative balance at equilibrium isGThis is not the case for feedback gain γg can be simply evaluated byWhen only one feedback is acting or if only the sum of all feedbacks is considered, 85 to estimate the individual γi. They generally require to perform specific analyses to extract the contribution of a particular feedback. One example is to perform an experiment when only the investigated feedback is operating and comparing the changes ΔTsi in this experiment to the one of the reference system ΔT0.Several techniques are availableSupplementary Information