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[email protected] Corresponding author Zernike Institute for Advanced Materials,University of Groningen, Nijenborgh 4, NL-9747AG Groningen, The Netherlands Institute for Advanced Simulation, Jülich Supercomputing Centre,Forschungszentrum Jülich, D-52425 Jülich, Germany RWTH Aachen University, D-52056 Aachen, Germany Center for Advanced Study, University of Illinois, Urbana, Illinois, USA Recent Einstein-Podolsky-Rosen-Bohm experiments [M. Giustina et al. Phys. Rev. Lett. 115, 250401 (2015); L. K. Shalm et al. Phys. Rev. Lett. 115, 250402 (2015)] that claim to be loophole free are scrutinized and are shown to suffer a photon identification loophole. The combination of a digital computer and discrete-event simulation is used to construct a minimal but faithful model of the most perfected realization of these laboratory experiments. In contrast to prior simulations, all photon selections are strictly made, as they are in the actual experiments, at the local station and no other “post-selection” is involved. The simulation results demonstrate that a manifestly non-quantum model that identifies photons in the same local manner as in these experiments can produce correlations that are in excellent agreement with those of the quantum theoretical description of the corresponding thought experiment, in conflict with Bell's theorem. The failure of Bell's theorem is possible because of our recognition of the photon identification loophole. Such identification measurement-procedures are necessarily included in all actual experiments but are not included in the theory of Bell and his followers.The photon identification loophole in EPRB experiments:computer models with single-wing selection Karl Hess December 30, 2023 ===================================================================================================The debate between Einstein and Bohr, about the foundations of quantum mechanics, resulted in a Gedanken- experiment suggested by Einstein-Podolsky-Rosen (EPR) <cit.> that was later modified by Bohm (EPRB) <cit.>. The schematics of this experiment is shown in Fig. <ref> and involves two wings and two measurement stations. EPR used the quantum mechanical predictions for possible outcomes of this experiment to show that quantum mechanics was incomplete.Many years later, John S. Bell derived an inequality for the possible outcomes of EPRB experiments that he perceived to be based only on the physics of Einstein's relativity <cit.>. Bell's inequality, as it was henceforth called, seemed to contradict the quantum results for EPRB experimental outcomes altogether <cit.>.Experimental investigations, following Bell's theoretical suggestions, provided a large number of data that were violating Bell-type inequalities and climaxed in the suspicion of a failure of Einstein's physics and his basic understanding of space and time <cit.>.Central to these discussions and questions are the correlations of space-like separated detection events, some of which are interpreted as the observation of a pair of entities such as photons. The problem of classifying events as the observation of a “photon” or of something else is not as simple as in the case of say, billiard balls. The particle identification problem is, in fact, key for the understanding of the epistemology of correlations between events.What do we know about such correlations of space-like separated events? Popular presentations of Bell's work typically involve two isolated persons (Alice, Bob) at separated measurement stations (Tenerife, La Palma), who just collect data of local measurements. But how does Alice know that she is dealing with a particle of a pair of which Bob investigates the other particle? She is supposed to be totally isolated from Bob's wing of the experiment in order to fulfill Einstein's separation and locality principle! The answer is that neither Alice nor Bob know they deal with correlated pairs if their stations are completely separated from each other and have no space-time knowledge of the other wing ever.In his theoretical work on the EPRB experiment, Bell did not address this fundamental question but considered correlated pairs as given, without any trace of the tools of measurement and of space-time concepts that are both necessary to accomplish the identification of events. He then claimed to have discovered a conflict between his theoretical description and the quantum theoretical description of the EPRB thought experiment <cit.>. As a consequence of this discovery, much research was devoted to * the actual derivation of Bell-type inequalities from Einstein's framework of physics (particularly his separation principle that derives from the speed of light in vacuum (c being the limit of all speeds) and Kolmogorov's probability theory <cit.>,* designing and performing laboratory experiments that provide data that are in conflict with the Bell-type inequalities,* constructing mathematical-physical models, at times supported by computer simulations that entirely comply with Einstein's relativity framework and separation principle, that do not rely on concepts of quantum theory, and are nevertheless in conflict with the Bell's theorem. Bell's theory, and the theories of all his followers, including Wigner, do not deal with the identification of the correlated particles and assume that the measurement pairs corresponding to the correlated particles are known automatically, so to speak per fiat. But the knowledge of pairing requires additional data or additional channels of information. These additional data may be measurement times, certain thresholds for detection and many other elements of the physical reality of the experiments in both wings. It is important to note that these data must involve measurements in both stations and are necessarily influencing the possible knowledge of the correlations of the single measurements in these stations.In the case of atomic or subatomic measurements the measurement equipment does not only influence the single outcomes as Bohr has taught us, but correlated measurement equipment (such as synchronized clocks or instruments that determine thresholds) also influence the knowledge of correlations of these single measurements. It is this extension of the Copenhagen view that leads to a loophole in Bell's Theorem, the photon identification loophole. The violations of Bell-type inequalities described in this paper are based on this loophole.Bell and followers envisage that the correlations may be measured in the laboratory in complete separation and, therefore, physical models of the Bell opponents must only use the measurements of two completely separated wings operated by Alice and Bob who know nothing of each other. As just explained, correlations of spatially separated events can only be conceived by involving the human-invented space-time system in order to demonstrate the pairing, the knowledge that measurements of particles belong together. Therefore, correlations can only be determined in a self-consistent way under the umbrella of a given space-time system that encompasses the two or more measurement stations. This space-time system that all experimenters subscribe to enables us to ban spooky influences out of science, particularly for the EPR experiments that Einstein constructed for the purpose to show this fact.We maintain that it is highly non-trivial to identify (correlated) photons by experimental methods and that this identification involves, at least in some way, a space-time system, a system developed by the human mind and agreed upon by all experimenters and evaluators of the Bell-type experiments. In fact, the identification of particle pairs requires certain knowledge of the space-time properties of all the experimental equipment involved. This knowledge must necessarily extend to measurement stations far from each other and is, therefore, “non-local”. Of course, this non-local knowledge does not imply that there are non-local physical influences on the data measured in the two wings. In EPRB experiments <cit.>, great care is taken to rule out the possibility that the observed correlations are due to physical influences that travel with velocities not exceeding the speed of light in vacuum. But without that non-local knowledge, only spooky influences are left as a possibility for connecting events in the two experimental wings. A space-time knowledge of all involved equipment is nevertheless required to apply the scientific method. § AIM AND STRUCTURE OF THE PAPER As explained above, it is essential that the identification of photons is included into any meaningful theoretical model of an EPRB experiment because otherwise, the model is too simple to describe this experiment. Failing to do so, only spooky influences can explain the observed pair correlations. Specifically, omitting the inclusion of data which select the photons and/or pairs opens an Einstein-local loophole, which we call henceforth the photon identification loophole. By design, Bell-type models for the recent experiments that claim to be loophole free <cit.> suffer from this loophole.The main aim of this paper is to show that by exploiting this loophole, or formulated more positively, by constructing a model that captures the essence of these recent laboratory experiments <cit.>, a manifestly non-quantum computer simulation of an Einstein-local model that employs the same local photon identification method in each wing of the EPRB experiments as in recent EPRB experiments <cit.>, yields the photon pair correlation of a pair of photons in the singlet state (of their polarizations), in blatant contradiction with Bell's theorem.The paper is structured as follows. Section <ref> argues that a simulation on a digital computer is a perfect laboratory experiment with a physically existing device and can therefore be used as a metaphor for other laboratory experiments. The material in this section forms the conceptual basis for developing computer simulation models of the recent EPRB experiments <cit.>.Section <ref> discusses the relevance of counterfactual definiteness (CFD) in relation to the derivation of Bell-type inequalities and hence also for computer simulation models that yield data that are in conflict with these inequalities. In Section <ref>, we introduce a CFD-compliant computer simulation model of the recent EPRB experiments <cit.>. We discuss the correspondence of the essential elements of the latter with those of the simulation model but refrain from plunging into the details of the algorithm itself.Section <ref> gives a simple, rigorous proof that operationally but not conceptually, photon identification in each wing by a local voltage threshold <cit.>, photon identification in each wing by a local time window, and photon identification by time-coincidence counting are all mathematically equivalent. In Section <ref>, we discuss the consequences of excluding from a model, at least one feature that is essential for an experiment to yield useful data. We argue that Bell-type models, which are believed to be relevant for the description of recent EPRB experiments <cit.>, suffer from the photon identification loophole in a dramatic manner.Section <ref> is devoted to a simple proof that the simulation model introduced in Section <ref> is CFD-compliant. In Section <ref>, we give a simple derivation of the Bell <cit.>, Clauser-Horn-Shimony-Holt (CHSH) <cit.>, Eberhard <cit.>, and Clauser-Horn (CH) <cit.> inequalities for real data, not for the imagined data produced by probabilistic models (which are discussed in the Appendix) and in Section <ref> we present a more general inequality that accounts for the local photon identification procedure, employed in the laboratory experiments.In section <ref>, we specify the computer simulation algorithm and the simulation procedure in full detail. A representative collection of simulation results is presented in section <ref>. The main conclusion from these simulations is that a non-quantum model that employs the same photon identification method in each wing of the EPRB experiments as the one used in recent EPRB experiments <cit.>, reproduces the results of quantum theory of the EPRB thought experiment.In section <ref>, we argue that all EPRB laboratory experiments with photons can be viewed as a tool to characterize the response of the observation stations, leading to the conclusion that this response, in particular the local photon identification rate, depends on the settings of these stations, consistent with the assumptions made in constructing the simulation model and debunking the hypothesis that the observed pair correlations can be explained by non-local influences only. The paper ends with section <ref> which contains our conclusions.§ METAPHOR FOR A PERFECT LABORATORY EXPERIMENT An important, characteristic feature of digital computers is that their logical operation does not depend on the technology that is used to construct the machine. These days, the first thing that comes to mind when talking about digital computers are the electronic machines based on semiconductor technology but it is a fact that, although not cost-effective nor particularly useful in practice, digital computers can also be built from mechanical parts, e.g. LegoTM elements.In former times, one could read off the state of the computer's internal registers from a LED display. Although not practical at all, in principle one could use a huge LED display to show the internal state of the whole computer. This is only to say that there is a one-to-one mapping from the state of the computer to sense impressions (e.g. light on/off). Therefore, the metaphor also offers unique possibilities to confront man-made concepts and theories with actual facts, i.e. real perfect experiments, because it guarantees that we have a well-defined, precise representation of the concepts and algorithms (both in terms of bits) involved that directly translate into sense impressions.In the analysis of laboratory EPRB experiments, it is essential that all the important degrees of freedom that affect the data analysis are identified and included, otherwise the conclusions drawn from an incomplete analysis may be wrong <cit.>. Computer simulation puts us in the position to perform experiments under the same mathematical conditions for which e.g. Bell-type inequalities can be derived, simply because we can carry out real, perfect experiments that are void of any unknown elements that may affect the results and analysis.A digital computer is a physical (electronic or mechanical) device that changes its physical state (by flipping bits) according to well-defined rules (the algorithm). Therefore, assuming that the machine is operating flawless for the time period of interest (a very reasonable assumption these days), executing an algorithm on a digital computer is a physics experiment in which there are no unknown elements of physical reality that might affect the outcome. In this sense, the “digital computer + algorithm” can be viewed as a metaphor for a perfected laboratory experiment, a discrete-event simulation that represents the so called “loophole-free” EPRB experiments <cit.>.Starting with Bell's work <cit.>, most theoretical work on the subject matter is based on probability theory. This mathematical framework contains conceptual elements (probability measures and infinitesimals) that are outside the domain of our sensory experiences and have no counterpart in our physical world. Therefore, to avoid pitfalls, we first devise an algorithm that simulates the perfect laboratory experiment and then construct a probabilistic model of this algorithm. A graphical representation of the modeling philosophy that we adopt in this paper is shown in Fig. <ref>. Note that our approach starts from the experiments and results in a theory, which we believe is the only direction one should go. In contrast, Bell's approach was the design of an experiment starting from his theoretical point of view.§ COUNTERFACTUAL DEFINITENESS Counterfactual reasoning <cit.> plays a significant role in the literature related to Bell's work and is seen by many a conditia sine qua non to derive Bell-type inequalities. However, as explained below, the actual EPRB experiments do not permit any proof of CFD compliance. This fact demonstrates an unexpected conceptual advantage of computer experiments. We can turn on and off CFD compliancy at will in our algorithm and simulate the consequences and thus distinguish the precise conditions that may or may not lead to violations of Bell-type inequalities. This is our reason to dedicate significant sections of this paper to counterfactual definiteness (CFD) as defined below, and include CFD-compliant models at the side of more faithful models of the actual EPRB experiments.So called counterfactual “measurements” yield values that have been derived by means other than direct observation or actual measurement, such as by calculation on the basis of a well-substantiated theory. If one knows an equation that permits deriving reliably, output values from a list of inputs to the system under investigation, then one has “counterfactual definiteness” (CFD) in the knowledge of that system <cit.>.The word “counterfactual” is a misnomer <cit.> but is well established. It is therefore helpful to have a clear-cut operational definition of what is meant with CFD. In essence, CFD means that the output state of a system, represented by a vector of values y, can be calculated using an explicit formula, e.g. y = f(x) where f(.) is a known vector-valued function of its argument x. If x denotes a vector of values, the elements of this vector must be independent variables for the mathematical model to be CFD-compliant <cit.>.In laboratory EPRB experiments, every trial takes place under different conditions, different settings etc.which may or may not affect the outcome of a single trial. Therefore, data produced by laboratory EPRB experiments (or any other laboratory experiments) can, as a matter of principle, not be cast in the form of data generated by a CFD-compliant model. On the other hand, performing computer experiments in a CFD-compliant manner is not difficult nor is it much work to change a CFD-compliant algorithm into one that does not meet all the requirements of CFD simulation. In other words, computer experiments can be carried out in both CFD and non-CFD mode, providing quantitative information about the differences of these two modes of modeling.In the realm of finite sets of two-valued data, a strict derivation of Bell-type inequalities <cit.>, such as the Bell-CHSH <cit.> and Eberhard's inequalities <cit.> require that these data are generated in a CFD-compliant manner <cit.>. In other words, CFD is a prerequisite for deriving Bell-type inequalities. Therefore, to test whether or not a simulation model produces data that do not satisfy such inequalities, it is necessary to perform a CFD-compliant simulation. Otherwise, there is no mathematical justification for the hypothesis that these data should satisfy Bell-type inequalities in the first place. Of course, we can always revert to the non-CFD algorithm and check if e.g. averages exhibit the same features as the averages obtained from the CFD-compliant algorithm (see section <ref>).In an earlier paper <cit.>, we have adopted this strategy to demonstrate that in the case of EPRB experiments, * CFD-compliant simulations can reproduce the averages and correlation of two particles in the singlet state,* CFD does not distinguish classical from quantum physics because our computer models do not contain any quantum concepts, yet yield results that lead to conclusions (e.g. entanglement) that are commonly regarded as signatures of quantum physics. In this paper, we adopt the same strategy. We construct a CFD-compliant simulation model of the laboratory experiments <cit.>, meaning that we simulate a perfected, idealized realization of these laboratory experiments. Of course, this does not mean that we omit essential features of the laboratory experiments. These features have to be included, otherwise the simulation model is not applicable to these laboratory experiments. see section <ref> for a general discussion of this aspect. § COMPUTATIONAL MODEL OF THE LABORATORY EXPERIMENTS <CIT.> In this section, we introduce the essential elements of the simulation model of the laboratory experiments reported in Ref. <cit.>. The details of the simulation algorithm are given in section <ref>. For concreteness, we adopt the terminology that is used in Ref. <cit.> when we connect the elements of the simulation model to those of the laboratory experiments.As shown in Fig. <ref>, in a typical EPRB experiment there are three different components. There is a source and there are two observation stations. The algorithm that simulates the source is described in full detail in section <ref>. In this section, we focus on the observation stations which, because we are performing “perfected” experiments, are assumed to be identical. §.§ Observation stationIn Fig. <ref>, we show a graphical representation of the function of an observation station. Input to an observation station is the setting a (representing the orientation of the polarizer), two numbers 0≤ r,r<1 taken from a list of uniform random numbers (see section <ref> for further details) and an angle 0≤ϕ < 2π (representing the polarization of the photon). Output of the observation station is a two-valued variable x=±1 and a detector-related variable v_min≤ v≤ v_max.The correspondence between the data produced by the experimental realization of an observation station and those generated by the computational model is as follows. The variable x encodes the detector outcomes (either D_+,i or D_-,i in station i=1,2) that fired. In the laboratory experiments <cit.> there is only one detector in each station but in the computer experiment we can easily simulate the complete EPRB experiment (see Fig. <ref>), hence we consider both the “+” and “-” events. The variable v represents the voltage signal produced by the electronics that amplifies the transition-edge detector current (see the description in section IV of the supplementary material to Ref. <cit.>).If necessary, we label different events by attaching the subscript i=1,2 of the observation station and/or the subscript k where k=1,…,N and N denotes the total number of input events to a station. In full detail, for the kth input at station i, the observation station i generates the output values x_i=x_i(a_i,ϕ_i,k,r_i,k) and v_i=v_i(a_i,ϕ_i,k,r_i,k) according to the rules which will be specified in full detail in section <ref>.Occasionally, we use the notation x_i(a_i)=x_i(a_i,ϕ_i,k,r_i,k) and v_i(a_i)=v_i(a_i,ϕ_i,k,r_i,k) to simplify the writing while still emphasizing that the x's and v's only depend on variables that are local to the respective station. §.§ Photon identification In the following, we use the term detection event whenever the negative voltage signal produced by the electronics that amplifies the transition-edge detector current is smaller (we are dealing with negative voltages) than the “trigger threshold” (terminology from Ref. <cit.> (supplementary material)), and speak of the observation of a photon whenever the same negative voltage signal is smaller than the “photon identification threshold” (about 4/3 times the “trigger threshold”) (terminology from Ref. <cit.> (supplementary material)).From the description of the laboratory EPRB experiments under scrutiny, it follows immediately that not every detection event is regarded as the observation of a photon <cit.>. Indeed, after all the voltage traces of an experimental run have been recorded, a part of the collected trace is analyzed by software, the photon identification thresholds are “calibrated” and assuming that the relevant properties of the whole set of traces is stationary in real time, the remaining set of traces is analyzed <cit.>. In Ref. <cit.> there is no specification of the cost function that is being minimized by the calibration procedure whereas Ref. <cit.>(supplementary material) explicitly states that “Because the experiment was calibrated to maximize violation of the CH inequality...”. This seems to suggest that the software is designed to adjust the photon identification thresholds such that the desired result, namely a violation of a Bell-type inequality, is obtained.In our simulation approach, we may assume that all units are identical. Therefore, unlike in Ref. <cit.>, one and the same value of photon identification threshold, denoted by V, can be used to identify photons. The effect of the photon identification threshold is captured by the functionw(a_i) = w_i(a_i,ϕ_i,k,r_i,k)=Θ( V-v_i), i=1,2 ,where Θ(x) is equal to one if x>0 and is zero otherwise. Recall that, as in Ref. <cit.>, V is negative. In the simulation, we do not “calibrate” V but simply generate the data and analyze the results as a function V.The correspondence with the data collected in the laboratory experiment is as follows: a detection event is represented by x_i(a_i)=+1 and w_i(a_i)=0 and the observation of a photon in station i=1,2 is represented by x_i(a_i)=+1 (because there is only one, not two, transition-edge detectors at each station) and w_i(a_i)=+1 (implemented in software), both exactly as in the simulation model. Recall, and this is new and important, that also in the simulations the photon identification is performed locally, i.e. without communication between the observation stations. § EQUIVALENCE OF LOCAL TIME-WINDOW AND TIME-COINCIDENCE PROCESSING In this section we show that in spite of the conceptually very different setup, from an operational point of view, employing local photon identification thresholds is equivalent to local time-window selection and also to time-coincidence counting that is used in most EPRB experiments with photons <cit.>.As explained above, in the laboratory experiments a detection event is classified as being a photon if the (negative) voltage signal, denoted by v, produced by the electronics that amplifies the transition-edge detector current (see the description in section IV of the supplementary material to Ref. <cit.>) is smaller than the photon identification threshold V. This rejection criterion is implemented through Eq. (<ref>) from which it follows directly that the criterion to observe a photon in these laboratory experiments is v≤ V. Recall that we adopted the convention of the laboratory experiments <cit.> in which V takes negative values.In practice, we have v_min≤ v_i ≤ v_max and v_min≤ V≤ v_max with finite v_min and v_max, hence the condition for counting a detection event as photon may be written as 0≤v_i-v_min/v_max-v_min≤ V-v_min/v_max-v_min, i=1,2 . Defining a dimensionless “time” t_i≡ (v_i-v_min)/(v_max-v_min) and a dimensionless “time window” W=( V-v_min)/(v_max-v_min), Eq. (<ref>) reads 0≤ t_i ≤ W, i=1,2 ,which expresses the condition to observe a photon at station i=1,2 in terms of locally defined time slots of size W. From Eq. (<ref>) we have -t_2 ≤ t_1-t_2 ≤ W - t_2 and using -W ≤ -t_2we find|t_1-t_2| ≤ W ,which is exactly the same criterion as the one used in most EPRB experiments with photons <cit.> and in computer simulation models thereof <cit.>.In summary: although physically very different, local voltage thresholds, local time windows or time-coincidence counting are mathematically equivalent and all serve the same purpose, namely to give an operational meaning to the statement “a single photon (pair)” has been identified.§ LOOPHOLES IN EXPERIMENTAL TESTS OF BELL'S THEOREM A useful physical theory of an experiment needs to encompass all relevant parameters that affect the experimental outcomes and, of course, the most important elements of physical reality, namely the data itself. Specifically, a physical theory that describes pair-correlations of space-like separated systems, must account for and include all procedures that determine the detection of the particles and the knowledge which pair of particles and data belongs together. Therefore, any model which purports to describe the laboratory experiments <cit.> that we consider in this paper must necessarily account for the photon identification threshold mechanism that is instrumental in the data-processing step of these experiments, see section <ref>. Likewise, the earlier generation of EPRB experiments that employ time coincidence to identify pairs <cit.> can only be faithfully be described by models that incorporate the time-coincidence window selection process that is an essential component of this class of experiments <cit.>.Drawing a conclusion about a world view from models (such as those of Bell and his followers, see the Appendix) that do not properly account for the photon identification threshold mechanism which, in the laboratory experiments <cit.>, is essential for identifying the photons, requires a drastic departure from rational reasoning. If we allow for such a departure, we might equally well wonder what it means for our wold view when we construct and analyze a model of an airplane that excludes the engines and then observe that a real airplane can take off by itself. Any reasonable person would rightfully question our ability to represent the airplane (or laboratory experiments) by such a model and regard the idea that we may have to change our world because of the contradictions to such a model as unfounded. In other words, the only logically correct conclusion that one can draw from the failure of Bell-type models to describe the qualitative features of the experimental data is that these models are too simple, which in this case is obvious as they miss at least one important ingredient: the photon identification mechanism.The photon-identification loophole that we introduce in this paper accounts for * the fact that laboratory experiments <cit.> employ a threshold to decide whether or not a detection event is considered to be a photon,* the assumption that voltage signals produced by the detection equipment may depend on the analyzer setting (see Eq. (<ref>)). Regarding this latter assumption, it is of interest to recall that since the early days of the Bell-test experiments, it is well-known that application of Bell-type models requires at least one extra assumption. We reproduce here the relevant passage from Ref. <cit.> (p.1890): “The approach used by CHSH is to introduce an auxiliary assumption, that if a particle passes through a spin analyser, its probability of detection is independent of the analysers orientation. Unfortunately, this assumption is not contained in the hypotheses of locality, realism or determinism.”* the requirement that a relevant model of an experiment needs to encompass all elements that affect the experimental outcomes. There is a large body of theoretical work that considers all kinds of loopholes in experiments that must be closed before a definite conclusion about the consequences of Bell inequality tests for certain wold views can be drawn. A detailed, comprehensive discussion of a large collection of loopholes is given in Ref. LARS14. Also in this respect, the digital computer – laboratory experiment metaphor offers unique possibilities because we can open and close loopholes at will. As this and our earlier paper <cit.> demonstrate, computer simulation models of EPRB experiments can easily be engineered to be free of e.g. detection, coincidence, and memory loopholes <cit.> and, in addition, include features such as CFD compliance that close the contextuality loophole <cit.>.Wrapping up: in this paper we construct a minimal model of the perfected version of the laboratory experiment <cit.>. With the exception of the photon-identification loophole, this minimal model is free of the known loopholes and reproduces the quantum results of the EPRB thought experiment, from which violations follow automatically. This approach offers the unique possibility to confront all kinds of reasonings and assumptions, such as the (ii) above, with actual facts.§ CFD COMPLIANCE In section <ref>, the operation of the simulation model of an observation station has been defined such that for every input event (a,ϕ,r,r), we know the values of all outputs variables x=x(a,ϕ,r) and v=v(a,ϕ,r). Therefore, the input-output relation of this unit, represented by the diagram of Fig. <ref>, satisfies the requirement of a CFD-compliant model.The computational equivalent of the EPRB experiments <cit.> is shown in Fig. <ref>. Each time the source 𝐒 is activated, it sends one entity carrying the data ϕ_1 to station 1 and another entity carrying the data ϕ_2 to station 2. The procedure for generating the ϕ's, r's and r's is specified in section <ref>.Upon arrival of the entities, observation stations i=1,2 execute their internal algorithm (completely specified by Eqs. (<ref>) and (<ref>)) and produces output in the form of the pair (x_i,v_i). The scheme represented by Fig. <ref> computes the vector-valued function([ x_1; v_1; x_2; v_2 ])= ([ x_1=x_1(a_1,ϕ_1,r_1); v_1=v_1(a_1,ϕ_1,r_1); x_2=x_2(a_2,ϕ_2,r_2); v_2=v_2(a_2,ϕ_2,r_2);])=𝐅(a_1,a_2,ϕ_1,ϕ_2,r_1,r_2,r_1,r_2) ,which clearly defines a CFD-compliant model. Nevertheless, with this CFD-compliant model we cannot construct the quadruple (x_1,x_2,x_1^',x_2^') in a CFD-compliant manner. Indeed, by construction, there is no guarantee that the (ϕ_i,k,r_i,k)'s that determine say the x_1's for the pair of settings (a_1,a_2) will be the same as the (ϕ_i,k,r_i,k)'s that determine that values of the x_1^''s for the pair of settings (a_1^',a_2). Of course, with a simulation on a digital computer being an ideal, fully controllable experiment, we could enforce CFD-compliance by re-using the same (ϕ_i,k,r_i,k,r_i,k)'s for every pair of settings. This would make the simulation CFD-compliant. However, in this paper we do not sobut instead generate new values of the (ϕ_i,k,r_i,k,r_i,k)'s for every new instance of input.The layout of a CFD-compliant computer model of the EPRB experiment is depicted in Fig. <ref>. It uses the same units as the model shown in Fig. <ref>, the only difference being that the input ϕ_i is now fed into an observation station with setting a_i and into another one with setting a_i^', something which, for obvious reasons, is impossible to realize in laboratory experiments with photons. As each of the four units operates according to the rules given by Eq. (<ref>) and (<ref>), we have (x_1,x_1^',x_2,x_2^')=𝐗(a_1,a_1^',a_2,a_2^',ϕ_1,ϕ_2,r_1,r_1^',r_2,r_2^') and (v_1,v_1^',v_2,v_2^')=𝐓(a_1,a_1^',a_2,a_2^',ϕ_1,ϕ_2,r_1,r_1^',r_2,r_2^'). As the arguments of the functions 𝐗 and 𝐓 are independent and may take any value out of their respective domain, the whole system represented by Fig. <ref> satisfies, by construction, the criterion of a CFD theory.Note that the actual EPRB experiments produce only pairs of data. The three pairs of data considered by Bell involve, therefore, six local measurements and the four pairs of CHSH involve eight local measurements. Our CFD compliant model considers only quadruple (= four local) measurements to simulate the actual eight possible measurement outcomes of a CHSH type experiment.§ BELL-TYPE INEQUALITIES It is evident from the formulation of his model that Bell and all his followers, including Wigner, do not deal with the issue of identifying particles and take for granted that the measured pairs correspond to the correlated particles. The common prejudice that additional variables cannot possibly defeat Bell-type inequalities is based on the assumption that all sent out correlated pairs, or a representative sample of them, are measured. This reasoning does not account for the photon identification or pair-modeling loophole: the necessary particle or pair identification may necessarily select in a way that is not representative for all possible measurements of all possible pairs emanating from the source. In this section, we adopt Bell's viewpoint by ignoring the v-variables and demonstrate that CFD-compliance and the existence of Bell-type inequalities are mathematically equivalent.Figure. <ref> shows the CFD compliant arrangement of the computer experiment. The two stations on the left of the source S receive the same data ϕ_1 from the source. The settings a_1 and a_1^' are fixed for the duration of the N repetitions of the experiment. The same holds for the two stations on the right of the source, with subscript 1 replaced by 2. Clearly, the algorithm represented by Fig. <ref> generates quadruples of output data (x_1^',x_1^',x_2^',x_2^') in a CFD-compliant manner.For any such quadruple (x_1^',x_1^',x_2^',x_2^') in which the x's only take values +1 and -1, it is straightforward to verify that the following equalities hold: b_1 =x_1 x_1^' + x_1 x_2 + x_1^' x_2 ={[ -1; +3 ]. b_2 =x_1 x_1^' + x_1 x_2^' + x_1^' x_2^' ={[ -1; +3 ]. b_3 =x_1 x_2 + x_1 x_2^' + x_2 x_2^' ={[ -1; +3 ]. b_4 =x_1^' x_2 + x_1^' x_2^' + x_2 x_2^' ={[ -1; +3 ]. s =x_1 x_2 - x_1 x_2^' + x_1^' x_2 + x_1^' x_2^' ={[ -2; +2 ]. .Other equivalent sets of equalities can be obtained by replacing e.g. x_1 by -x_1 etc. Note that e.g. Eq. (<ref>) follows from Eq. (<ref>) if we set x_2^'=x_1.In a non-CFD setting, the data is collected as four pairs which we may denote as (x_1^',x_2^'), (x_1^',x_2^'), (x_1^',x_2^'), and (x_1^',x_2^') where the tilde is used to indicate that the value of e.g. x_1^' obtained with setting (a_1,a_2^') may be different from the one obtained with setting (a_1,a_2). Instead of Eq. (<ref>), we now consider the expression s= x_1 x_2 - x_1 x_2^' + x_1^'x_2 + x_1^'x_2^'=-4,-2,0,+2,+4 and similar ones for b_1,…,b_4, each of them taking values -3,-1,+1,+3. If we now impose that s=-2,+2 and b_1…,b_4=-1,3, simple enumeration of all possible values of the x's and the x's shows that in order for all equalities to be satisfied simultaneously we must have x_1^'=x_1^', x_1^'=x_1^', x_2^'=x_2^', x_2^'=x_2^'. In other words, imposing the constraints s=-2,+2 and b_1=-1,3,…,b_4=-1,3 on data obtained in a non-CFD setting forces this data to form quadruples, i.e. to be CFD compliant. It then follows immediately that CFD is necessary and sufficient for the equalities Eqs. (<ref>) – (<ref>) to hold.Attaching the subscript k (k=1,…,N) to label the events, the algorithm generates the set of quadruples {(x_1,k^',x_1,k^',x_2,k^',x_2,k^') | k=1,…,N}. Introducing the Bell-CHSH function S = 1/N∑_k=1^N s_k ,it follows immediately from |s_k|=2 (see Eq. (<ref>)) that |S|≤2 for all N≥1, that is we obtain the Bell-CHSH inequality constraining four correlations of pairs of actual data. Put differently, if the output consists of quadruples of two-valued data generated by the setup shown in Fig. <ref> and we ignore the v-variables then the Bell-CHSH inequality |S|≤2 is always satisfied, independent of the number of events N≥1 considered.Similarly, from the fact that for example b_1,k=-1,3 we obtain the Leggett-Garg inequality <cit.> for three correlations of pairs of actual data and by combining b_1,k=-1,3 with the equalities obtained by substituting x_1→ -x_1 we obtain the Bell inequality involving three correlations of pairs of actual data <cit.>. In other words, Bell-type inequalities follow directly from the fact that quadruples of data satisfy rather trivial arithmetic identities such as Eq. (<ref>).It then also follows immediately that CFD is a necessary and sufficient condition for the data (x_1,k^',x_2,k^'), (x_1,k^',x_2,k^'), (x_1,k^',x_2,k^'), and (x_1,k^',x_2,k^') with k=1,…,N to satisfy simultaneously for all N≥1, all Bell-type inequalities involving three and four different correlations of pairs. We emphasize that this conclusion follows from elementary arithmetic only. Concepts such as “locality” or any other physical argument are irrelevant for establishing this result.Similar reasoning yields Eberhard's inequality which differs from the Bell-CHSHinequality in the sense that it can account for reduced detector efficiencies <cit.>. For convenience of comparison with the original work, we temporarily adopt Eberhard's parlance and notation. Central to Eberhard's derivation is the so-called fate of a photon. This fate can be either detected in the ordinary beam (labeled o), or detected in the extraordinary beam (labeled e), or undetected (labeled u). For counting purposes, we represent the fate of a photon by the symbol f, taking the values +1, 0, and -1 corresponding to o, u and e, respectively. Introducing the variables n_o=f(f+1)/2, n_e=f(f-1)/2, and n_u=(1-f^2), it is clear that one of them takes the value 1 with the other two taking the value 0. For a given pair of settings, say (α_1,β_2), the number of pairs with both photons suffering fate (o) is then given by n_oo(α_1,β_1)=n_o(α_1)n_o(β_1)=f_1,1(f_1,1+1)f_2,1(f_2,1+1)/4 where f_1,1=f_1,i(α_i) and f_2,i=f_2,i(β_i) for i=1,2. There are similar expressions for n_eo(α_1,β_2), n_uo(α_1,β_2), etc. Following Eberhard, we consider the expression <cit.> j =n_oe(α_1,β_2)+n_ou(α_1,β_2)+n_eo(α_2,β_1)+n_uo(α_2,β_1) +n_oo(α_2,β_2)-n_oo(α_1,β_1) .It is straightforward to enumerate all possible 81 values of the 4 different f-variables that appear in Eq. (<ref>). This enumeration proves that j≥0, independent of the values of the settings. Attaching the subscript k (k=1,…,N) to label the events as we did to derive the Bell-CHSH inequality and introducing the Eberhard function J_Eberhard = ∑_k=1^N j_k ,it follows immediately that J_Eberhard≥0 for all N≥1.In the laboratory experiments <cit.> there is only one detector per observation station. Hence it makes sense to regard also say, the e photons, as undetected. In terms of the “fate” variables f introduced above this amounts to letting f taking the values +1 and 0 corresponding to o and u, respectively. Instead of Eq. (<ref>), we now consider the expression j_CH = n_ou(α_1,β_2)+n_uo(α_2,β_1) +n_oo(α_2,β_2)-n_oo(α_1,β_1) .Enumerating all possible 16 values of the 4 different f-variables that appear in Eq. (<ref>) proves that j_CH≥0, independent of the values of the settings. Attaching the subscript k (k=1,…,N) to label the events as before and introducing the CH function J_CH = ∑_k=1^N j_CH,k ,it follows immediately that the CH inequality <cit.> J_CH≥0 holds for all N≥1.In short: if for all N, the x's (f's) are generated according to a CFD-compliant procedure the Bell-CHSH (the Eberhard and CH) inequality is (are) satisfied. In essence, this result is embodied in the work of George Boole <cit.>, see also Ref. <cit.>. Moreover, as CFD implies that all Bell-type inequalities hold for all N≥1, there is no room for speculating without violating at least one of the rules of Aristotelian logic that something “spooky” is going on if we encounter data that violate a Bell-type inequality. The logically correct conclusion that one can draw from such a violation is that these data have not been generated in a CFD-compliant manner. § AN INEQUALITY ACCOUNTING FOR PHOTON IDENTIFICATION In this section, we address the modifications to the inequality |S|≤2 that ensue when we take into account the fact that laboratory experiments employ the photon identification threshold to decide whether or not a detection event corresponds to the observation of a photon.The average detection event counts and detection event pair correlation are given byE_i(a_i) = 1/N∑ x_i(a_i), i=1,2E(a_1,a_2) = 1/N∑ x_1(a_1)x_2(a_2) , respectively, and we have similar expressions for the other choices of settings. In Eq. (<ref>) and in the equations that follow, it is understood that ∑ means ∑_k=1^N, i.e. the sum over all input events, represented by values of the ϕ's. As shown in section <ref>, if the x's that enter Eq. (<ref>) have been obtained by a CFD-compliant procedure, the correlations E(a_1,a_2),… satisfy Bell-type inequalities.In contrast to Eq. (<ref>), the average photon counts and photon pair correlation for the settings (a_1,a_2) are given by E_i(a_i) = ∑ w(a_i)x_i(a_i)/∑ w(a_i), i=1,2 E(a_1,a_2) = ∑ w(a_1)w(a_2)x_1(a_1)x_2(a_2)/∑ w(a_1)w(a_2) , where, as explained in section <ref>, the w's in Eq. (<ref>) account for the effect of the photon identification thresholds and take values 0 or 1. Clearly, Eq. (<ref>) is very different from Eq. (<ref>) unless all the w's that appear in Eq. (<ref>) are equal to 1, in which case the photon identification threshold mechanism is superfluous and unlike as in the laboratory experiment <cit.>, the number of photon and detection events is the same.In the analysis of the experimental data, the photon identification threshold is chosen such that many of the w's are zero <cit.>. Hence from the discussion in section <ref>, it follows immediately that with some w's zero, it is impossible to prove that the Bell-CHSH function S= S(a_1,a_2,a_1^',a_2^') ≡E(a_1,a_2)- E(a_1,a_2^')+E(a_1^',a_2)+ E(a_1^',a_2^') satisfies the inequality |S|≤2.However, it directly follows from the proof given in our earlier paper <cit.> that if the x's and w's have been generated by a CFD-compliant procedure, the Bell-CHSH function S can never violate the inequality |S|=| E(a_1,a_2)-E(a_1,a_2^')+E(a_1^',a_2)+E(a_1^',a_2^') |≤ 4-2δ.The term 2δ in Eq. (<ref>) is a measure for the number of paired events that have been rejected relative to the number of emitted pairs. In detail, 0≤δ≡ N'/N_max≤ 1 where N^' denotes the number of input events for which the negative voltage signal of all the photons is smaller than the photon identification threshold V and N_max is the maximum number of contributing pairs per setting. If all paired events would be regarded as photon pairs then δ=1 and then, and only then we recover the Bell-CHSH inequality |S|≤2. If the x's and w's have not been generated by a CFD-compliant procedure, there is only the trivial bound |S|≤4.The inequality Eq. (<ref>) is a rigorous mathematical fact that holds if, for all N≥1, the x's and v's are generated in a CFD-compliant manner and none of the denominators in Eq. (<ref>) is identically zero (in which case no photon pairs have been detected). Conversely, if we find a set of x's and v's that yields a value of |S| that exceeds 4-2δ, we can only conclude that these data have not been obtained from a CFD-compliant procedure. Any other conclusion would not be logically justified.In analogy with the derivation of Eq. (<ref>), one may derive an Eberhard-type or CH-type inequality that accounts for the w's but as such inequalities do not add anything to the discussion that follows, we do not discuss them any further.§ DISCRETE-EVENT SIMULATION ALGORITHM In this section, we specify the algorithm and the simulation procedure in full detail. The algorithm that mimics the operation of the particle source is very simple. For each event k=1,…,N, a uniform random generator is used to generate a floating-point number 0≤ϕ_1,k≤ 2π. This number is input to the stations with setting (a_1,a_1^') and another number ϕ_2,k=ϕ_1,k+π/2 is input to the stations with setting (a_2,a_2^'). Because of ϕ_2,k=ϕ_1,k+π/2, the kth event simulates the emission of a photon pair with maximally correlated, orthogonal polarizations. In this respect, we deviate from what is done in the laboratory experiments <cit.> in the following sense. Unlike in the computer simulation, the detectors used in these laboratory experiments are not perfect. As already mentioned, Eberhard's inequality can account for reduced detector efficiencies and this feature can be put to good use through minimizing the value of J_Eberhard with respect to the correlation <cit.>. This is what is done in the laboratory experiments <cit.>. However, in our simulation model, the detectors are perfect. Hence the minimum value of J_Eberhard will be obtained by choosing maximally correlated, orthogonal polarizations <cit.>. Recall that our aim here is to simulate the most ideal, perfect experiment that accounts for all the essential features of the laboratory experiments, not to simulate a real laboratory experiment including trams passing by <cit.>etc.Upon receiving the input ϕ an observation station (see Fig. <ref>) executes the following two steps. First it retrieves two uniform random numbers r and r from a list of such numbers (or, more conveniently, generates these numbers on the fly) and then 1.computesc=cos[2(a-ϕ)] ,s=sin[2(a-ϕ)] 2.setsx=(1+c-2*r),v=r |s|^d (V_max-V_min)-V_max ,where d, is an adjustable parameter to be discussed later and 0≤ V_max, and 0≤ V_min≤ V_max set the range of the voltage signal. From Eq. (<ref>) it follows that-V_max≤ v, V≤ -V_min, as in the laboratory experiment <cit.>.Our choice for the specific functional forms of x=x(a,ϕ,r) and v=v(a,ϕ,r) is inspired by previous work in which it was shown that a similar model, which employs time-coincidence to identify pairs, exactly reproduces the single particle averages and two-particle correlations of the singlet state if the number of events becomes very large <cit.>.Equations (<ref>) and (<ref>) form the core of the simulation algorithm which has the following key features: * For any fixed value of ϕ and uniformly distributed random numbers r, the unit generates a sequence of randomly distributed x's such that the average of the x's agrees (within statistical fluctuations) with Malus' law, i.e. the normalized frequencies to observe x=+1 and x=-1 are given by cos^2(a-ϕ) and sin^2(a-ϕ), respectively.* The presence of an output variable v which serves to mimic the detector traces recorded in the laboratory experiments. Note that the explicit expression of v=v(a,ϕ,r) shows a dependence on the local setting of the station. Such a dependence cannot be ruled out a posterioribut finds a post-factum justification in the fact that the simulations reproduce the results for two particles in a singlet state, see section <ref>.* The use of the random numbers 0 ≤ r,r≤ 1 mimics the uncertainties about the outcomes, as observed in experiments. Thereby, it is implicitly understood that for every instance of new input, new values of the uniform random numbers r and r have been generated.* By construction, the algorithm is a metaphor for Einstein-local experiments: changing a_1 (a_2) does not affect the present, past or future values of x_2 (x_1) or v_2 (v_1). In plain words, the output of one particular unit depends on the input to that particular unit only. For the settings of the observation stations, we take a_1=θ+π/8, a_1^'=a_1+π/4, a_2=π/8, a_2^'=3π/8 and let θ vary from 0 to π. For this choice of settings, quantum theory for a system in the singlet state predicts E(a_1,a_2)=E(a_1^',a_2^')=cos 2θ, E(a_1,a_2^')=E(a_1^',a_2)=sin 2θ and S(a_1,a_2,a_1^',a_2^')=-2√(2)cos(2θ+π/4), the latter reaching its maximum 2√(2) at θ=3π/8. When we operate the computer model in non-CFD mode, random numbers are used to make a choice between the settings a_i and a_i^', for i=1,2, exactly as in the experiments <cit.>.The simulation procedure is quite simple. We choose a fixed photon identification threshold V, generate input pairs k=1,…,N, collect corresponding outputs in terms of x's and w's, and compute the single- and two-particle averages according to Eq. (<ref>), the Bell-CHSH function S(a_1,a_2,a_1^',a_2^')=E(a_1,a_2)-E(a_1,a_2^')+E(a_1^',a_2)+E(a_1^',a_2^'), and the Eberhard function J given by Eq. (<ref>) .Because the computer experiment is “perfect”, it differs from the laboratory experiment in the sense that all pairs are created “on demand” and all emitted pairs create one detection event in each station (there are no “false” detection events) but exactly as in the laboratory experiment, the local photon identification threshold at each observation station serves to decide whether a photon has arrived or not. § COMPUTER SIMULATION RESULTS This section reports the results of simulations with N=10^5 events for the CFD-compliant and N=10^5 events per setting for the non-CFD model, with V_min=1/2 and V_max=1. Note that the “time-tag threshold” and a “trigger threshold” (terminology from Ref. <cit.> (supplementary material)) are important to the laboratory implementation but are superfluous, meaning that they do not affect the results of our computer experiments in any way. Indeed, in our perfect experiments, there is no ambiguity in determining when a particle arrives at the observation station. Nevertheless, to counter possible (pointless) critique that we have not incorporated into our simulation model the two thresholds that are essential to the laboratory implementation, we have chosen V_min=1/2 in order to leave room for introducing these thresholds.We limit the discussion to the case d=4 because we know from earlier work <cit.>, which uses time-coincidence selection, that for d=4 the computer model reproduces the quantum theoretical result of the correlation of two particles in the singlet state, Malus' law for the single-particle averages etc. if N→∞ followed by V→ - V_max.It is not difficult to see that single-particle averages E_1(a_1,a_2), E_2(a_1,a_2) etc. are expected to be zero, up to fluctuations. The reason is that ϕ→ϕ+π/2 changes the sign of the x's but has no effect on the values of the v's (see Eq. (<ref>)). Therefore, if the ϕ's uniformly cover [0,2π[, the number of times that x=+1 and x=-1 appear is about the same. All our simulation results are in concert with this prediction.In Fig. <ref>, we present the simulation data of the correlation E(a_1,a_2) (◯), the single-particle averages E_1(a_1,a_2) (△) and E_2(a_1,a_2) (▽) as a function of θ=a_1-a_2, as obtained from a CFD-compliant (Fig. <ref>(a)) and non-CFD (Fig. <ref>(b)) simulation. All the simulation data are in excellent agreement with the quantum theoretical description of a two-particle system in the singlet state which predicts E_1(a_1,a_2)=E_2(a_1,a_2) =0 and E(a_1,a_2) =-cos2θ. Within statistical fluctuations, it is difficult to distinguish between CFD-compliant and non-CFD simulation data, in concert with our earlier work <cit.>.In Fig. <ref> we show the data of the Bell-CHSH function S(θ+π/8,θ+3π/8,π/8,3π/8) as a function of θ. Clearly the simulation results are in excellent agreement with the quantum theoretical prediction S(θ+π/8,θ+3π/8,π/8,3π/8)=-2√(2)cos(2θ+π/4). In both Figs. <ref> and Fig. <ref>, there are deviations from the quantum theoretical prediction which are not due to statistical fluctuations. These deviations can be reduced systematically and eventually vanish by lettingV→ -V_max(V→ >-V_max), a fact that can be proven rigorously for the probabilistic version of the simulation model <cit.>.We note in passing that the observation that the frequency distribution of many events agrees with the probability distribution of a singlet state is a post-factum characterization of the repeated preparation and measurement process only, not a demonstration that at the end of the preparation stage, each pair of particles actually is in an entangled state. The latter describes the statistics, not a property of a particular pair of particles <cit.>.Results of the Eberhard function Eq. (<ref>) as a function of θ are given in Fig. <ref>(a). The correspondence between the symbols used in Eberhard's and this paper are as follows: o⇔+1, e⇔-1, α_1⇔ a_1^', α_2⇔ a_1, β_1⇔ a_2, and β_2⇔ a_2^'. As expected from the requirements to derive Eq. (<ref>) (see section <ref>), the CFD-compliant simulation without the photon identification threshold satisfies J_Eberhard≥0 for all θ whereas processing the data as in the laboratory experiment, i.e. by employing a photon identification threshold, yields J_Eberhard<0 for a non-zero interval of θ's. As is clear from Fig. <ref>(a), the results of Eberhard function Eq. (<ref>) do not change significantly if we use replace the CFD-compliant simulation model by its non-CFD version. The reason for this apparent violation is that the data obtained through the application of the photon identification mechanism do not satisfy the mathematical requirements for deriving Eq. (<ref>).For completeness, in Fig. <ref>(b) we present results for the function δ which determines the upperbound to the Bell-CHSH function in the case that the photon identification threshold is being used to discard detection events (see Eq. (<ref>)). From Fig. <ref>(b), it follows that δ<0.8. Hence Eq. (<ref>) predicts an upperbound that is not smaller than 3.4, large enough to include the maximum value of 2√(2)≈2.83 predicted by the quantum theory of the polarizations of two photons (or, equivalently, two spin-1/2 particles).Finally, it is instructive to compare the number of detection events that the photon identification threshold rejects as being a photon. In the laboratory experiment <cit.>, the number of trials is about 3.5×10^9 and the total number of so-called “relevant counts”, i.e. the number of times that at least one photon was identified by means of the photon identification thresholds (by software), is about 1.8×10^5. Thus, in this experiment the overall number of events considered to be relevant for the physics, relative to the number of detection events is about 0.005%. For comparison, in the simulations, a photon identification threshold V=-0.995 identifies about 23% of the detection events as photons, several orders of magnitude larger than in the laboratory experiment. Clearly, the quality of the data collected in the laboratory experiments are not on par with the quality of the data produced by the computer experiments but obviously, the latter is much easier to realize and use than the former. § POST-FACTUM JUSTIFICATION OF THE SIMULATION MODEL We have already drawn attention to the fact that the explicit expression of the voltage v=v(a,ϕ,r) shows a dependence on the local setting of the station through the factor |sin[2(a-ϕ)]|^d (see Eqs. (<ref>) and (<ref>)) and mentioned that such a dependence cannot be ruled out a posteriori. In this section, we examine the consequences of removing this dependence. In Fig. <ref>, we show results for d=0, in which case the random variations of the voltage signals v_1, v_1^', v_2, and v_2^' do not depend on a_1, a_1^', a_2, and a_2^', respectively. Instead of E(a_1,a_2)≈ -cos2θ for d=4, the simulation for d=0 yields E(a_1,a_2)≈ -(1/2)cos2θ and, as Fig. <ref>(b) shows, |S(θ+π/8,θ+3π/8,π/8,3π/8)| ≤ 2. Therefore, the only way to have simulation models of the laboratory experiment <cit.> reproduce the quantum theoretical prediction of the polarizations of two photons (or, equivalently, two spin-1/2 particles) is to assume that v_1, v_1^', v_2, and v_2^' depend on a_1, a_1^', a_2, and a_2^'. Of course, there is no good argument why, in a particular experiment, this dependence should be of the form Eq. (<ref>). We repeat that we have chosen the form Eq. (<ref>) because our main goal is to reproduce by a CFD-compliant, manifestly non-quantum model, the quantum theoretical predictions of the polarizations of two photons (or, equivalently, two spin-1/2 particles), which as a by-product, yields |S(θ+π/8,θ+3π/8,π/8,3π/8)|>2. Disregarding the original motivation to perform the Bell-test experiments, the experimental setup shown in Fig. <ref> can be regarded as a tool to characterize the response of the observation stations to the incoming signals. In the case at hand, what is under scrutiny is the response of the observation station, i.e. of its optical components, the transition-edge detector and the electronics that amplifies its current, under the condition that the incident light is extremely feeble. Viewed from this perspective, our simulations support the hypothesis that the laboratory experiments <cit.> convincingly demonstrate that the statistics of the observed photons, as defined by the photon detection threshold, depends on the settings (and hence on the polarizations assigned to the photons) of the observation stations.It is of interest to mention here that since the early days of the Bell-test experiments, it is well-known that application of a Bell-type model requires at least one extra assumption. We reproduce here a passage from Ref. <cit.> (p.1890): “The approach used by CHSH is to introduce an auxiliary assumption, that if a particle passes through a spin analyser, its probability of detection is independent of the analysers orientation. Unfortunately, this assumption is not contained in the hypotheses of locality, realism or determinism.” It is stunning that although there is at least one auxiliary assumption involved in testing e.g. the CHSH inequality with Bell-test data, the possibility that this assumption is not valid is, to the best of our knowledge, ignored in the experimental studies. As a matter of fact, as we have argued above,all Bell-test experiments with photons performed up to this day can be regarded as direct experimental proof that this auxiliary assumption is invalid. In view of the intricate atomic-scale processes that are involved when light passes through a material, this conclusion seems very reasonable but is, of course, way less spectacular than the conclusion that Bell-type experiments can be used to rule out certain world views. § CONCLUSION The general message of this paper is that a model that purports to describe the data produced by an experiment should account for all the data that are relevant for the analysis of the experimental results. In the case at hand, the situation is as follows:* experimental data <cit.> are interpreted in terms of a Bell-type model that uses only half of the variables (the x's),* in the actual experiments <cit.> the other half of the variables (the v's) is essential for the identification of the photons but are ignored in Bell-type models,* the failure of Bell-type models to describe the experimental data is taken as a proof that “local realism” (local in Einstein's sense) is incompatible with quantum theory and is therefore is declared dead.We believe that it requires an exotic form of logic to reconciliate the last statement (iii) with the second one (ii).To head off possible misunderstandings, the authors of this paper do not necessarily subscribe to all or any forms of what is called local realism, CFD theories, or ... We are of the opinion that the arguments based on Bell's theorem in conjunction with Bell-type experiments suffer from what we earlier called the photon identification loophole. One simply cannot blame a model that only accounts for part of the data for not describing all of them. Regarding the previous sentence, Albert Einstein's quote “make it as simple as possible, but not simpler” is more pertinent than ever.The challenge for the Bell-experiments community is, therefore, to construct an EPRB-type experiment with a photon (pair) identification that cannot, from the perspective of the simple Bell models, be turned into a “loophole”. Our general proofs of the derivation of Bell-type inequalities for actual data (see section <ref>), indicate that this challenge cannot be met.§ ACKNOWLEDGEMENTSWe like to thank D. Willsch, F. Jin, and M. Nocon for useful comments and discussions. * § PROBABILISTIC MODELS Traditionally, mathematical models of the EPRB experiments are formulated in terms of probabilistic models <cit.>, often without explicitly mentioning Kolmogorov's axiomatic framework of probability theory <cit.>. However, there is a considerable,conceptual gap between a laboratory EPRB experiment and a probabilistic model thereof. The (over)simplifications required to come up with a tractable, proper probabilistic model of a laboratory EPRB experiment are key to the understanding of the phenomena involved.When we use the “digital computer – laboratory experiment” metaphor, both the simplification and replacement are made during the formulation of the computer model. A computer simulation algorithm entails a complete specification of how the data are generated. In this respect, all “physically relevant” processes are well-defined (by construction) and known explicitly in full detail. There are no uncertainties or unknown influences. Note that the reverse operation, i.e. to construct an algorithm for a digital computer out of a probabilistic model is, as a matter of principle, impossible. At most, a probabilistic model can serve as a guide to construct an algorithm.In the particular case where the observed phenomena take the form of data generated by simulations of EPRB experiments on a digital computer, the transition from the observed phenomena to suitable mathematical models does not suffer from the many “uncertain” factors that may or may not play an essential role in the laboratory experiments and is, therefore, a fairly simple transition. In this section, we start from a computer simulation model and construct the probabilistic model thereof. We start by showing that the simplest of these models are identical to those proposed and analyzed by Bell <cit.> and then move on to the construction of a probabilistic model for the computer model of the recent Bell-type experiments <cit.> that we use in our simulation work presented in section <ref>. §.§ Bell-type models Consider the CFD simulation model in which we explicitly ignore the v-variables. For a given input to the observation stations, the outcome is one of the so-called elementary events <cit.>, in this case one of the 16 different quadruples (x_1,x_1^',x_2,x_2^'). In the language of probability theory, the set of these 16 different quadruples is called the sample space Ω <cit.>, the set of elementary events, from which we construct the so-called σ-field F of subsets of Ω, containing the impossible (null) event and all the (compound) events in whose occurences we may be interested <cit.>. In modeling the computer experiments, we only need to consider finite sets, hence we do not have to worry about the mathematical subtleties that arise when dealing with infinite sets <cit.>.The next step is to assign a real number, a probability measure, between zero and one that expresses the likelihood that an element of the set Ω occurs <cit.>. We denote this (conditional) probability measure by P(x_1,x_1^',x_2,x_2^'|a_1,a_1^',a_2,a_2^', Z), the part |a_1,a_1^',a_2,a_2^', Z) indicating that the settings and all other conditions, denoted collectively by Z, do not change during the imaginary probabilistic experiment. By definition, the probability measure satisfies ∑_(x_1,x_1^',x_2,x_2^')∈Ω P(x_1,x_1^',x_2,x_2^'|a_1,a_1^',a_2,a_2^', Z) =1 <cit.>.Note that a probability measure is a purely mental construct <cit.>. If it were not, we could interchange the experiment/computer simulation, the results of which directly connect to our senses, with the imaginary world of mathematical models and prove theorems, not only about the mathematical description, but also about our sensory experiences, a tantalizing possibility. One such example that exploits intricate features of set theory is given in Ref. <cit.>, in which it is explicitly stated that there does not exist an algorithm to actually calculate the relevant functions. In other words, this example cannot be realized on a physical device such as a digital computer, not even approximately. Moreover, unlike the simulation algorithm executing on a digital computer, the probabilistic description does not contain a specification of the process that actually produces an event: we have to call up Tyche to do this for us. In other words, a probabilistic model is incomplete in that it only describes the outcomes of the simulation procedure, not the procedure itself. However, this incompleteness is partially compensated for by the fact that the calculation of averages and correlations no longer involves the number of events N. We have for instance E_12(a_1,a_1^',a_2,a_2^') = ∑_Ω x_1x_2P(x_1,x_1^',x_2,x_2^'|a_1,a_1^',a_2,a_2^', Z),and we have similar expressions for the other two-particle averages and also for the single-particle averages. Here and in the following, we use the shorthand notation ∑_Ω=∑_(x_1,x_1^',x_2,x_2^')∈Ω. We have written E instead of E to emphasize that the former have been calculated within a probabilistic model whereas the latter involve calculations with actual data. From Eq. (<ref>), we have S(a_1,a_2,a_1^',a_2^') = ∑_Ω( x_1x_2 - x_1^' x_2 + x_1x_2^' + x_1^' x_2^') to 1cm× P(x_1,x_1^',x_2,x_2^'|a_1,a_1^',a_2,a_2^', Z) ,and because the elementary events are quadruples, it follows directly from Eq. (<ref>) that |S(a_1,a_2,a_1^',a_2^')|≤ 2. Thus, in the probabilistic realm, not in the world of the observed two-valued data, the existence of the Bell-CHSH inequality follows from the existence of a probability measure for the elementary events of quadruples (x_1,x_1^',x_2,x_2^'). Moreover, it can be shown that with some additional requirements on its marginals, the existence of such a probability measure is necessary and sufficient for Bell-type inequalities to hold <cit.>.It is clear from Fig. <ref> that x_i and x_i^' for i=1,2 only depend on the corresponding a_i and a_i^', respectively. However, the probability measure for quadruples, P(x_1,x_1^',x_2,x_2^'|a_1,a_1^',a_2,a_2^', Z), does not express this basic property of the computer model, nor does it explicitly express the dependence on the ϕ's.A simple way to incorporate all these features of the simulation model in a probabilistic description is to define a new joint probability measure for quadruples by P^'(x_1,x_1^',x_2,x_2^'|a_1,a_1^',a_2,a_2^', Z) = ∫ P(x_1|a_1,_1, Z) P(x_1^'|a_1^',_1, Z) P(x_2|a_2,_2, Z) to 2cm× P(x_2^'|a_2^',_2, Z) μ(_1,_2) d_1 d_2 ,where P(x_1|a_1,_1, Z) etc. are the “local” probabilities to observe x_1 etc.,the integration is over the whole domain of _1 and _2 and μ(_1,_2) is a non-negative, normalized density. With the new probability measure Eq. (<ref>), Eq. (<ref>) simplifies considerably. For instance, for the detection events we have E_12(a_1,a_1^',a_2,a_2^')=E(a_1,a_2) where E(a,b) =∑_x,y=±1∫ xy P(x|a,_1, Z)P(y|b,_2, Z) μ(_1,_2) d_1 d_2 .Instead of Eq. (<ref>), we now have |S(a_1,a_2,a_1^',a_2^')|= |E(a_1,a_2)-E(a_1,a_2^')+E(a_1^',a_2)+E(a_1^',a_2^')|≤ 2 ,which is the Bell-CHSH inequality in probabilistic form <cit.>.From Eq. (<ref>), it follows directly that P^'(x_1,x_2|a_1,a_2, Z)= ∑_x_1^'=±1∑_x_2^'=±1 P^'(x_1,x_1^',x_2,x_2^'|a_1,a_1^',a_2,a_2^', Z) = ∫ P(x_1|a_1,_1, Z) P(x_2|a_2,_2, Z) μ(_1,_2) d_1 d_2 ,which expresses the probability measure P^'(x_1,x_2|a_1,a_2, Z) in terms of the single-variable probability measures P(x_1|a_1,_1, Z) and P(x_2|a_2,_2, Z) and the measure μ(_1,_2) of the variables _1 and _2.The factorized form Eq. (<ref>) is the landmark of the so-called “local hidden-variable models” introduced by Bell <cit.>. Although “local” is often used to express the notion that physical influences do not travel faster than the speed of light it is, in the context of a probabilistic model (computer model), an expression of statistical (arithmetic) independence only. Bell's theorem uses the factorized form Eq. (<ref>) to state that quantum mechanics is incompatible with local realism, the world view in which physical properties of objects exist independently of measurement and where physical influences cannot travel faster than the speed of light <cit.>.In one respect, Eq. (<ref>) is grossly deceiving, namely it does not reflect the elementary fact that the parent probability measure Eq. (<ref>) from which Eq. (<ref>) follows, concerns quadruples, not pairs. Without the knowledge that Eq. (<ref>) is in fact a marginal distribution of the probability measure Eq. (<ref>) for quadruples, one is inclined to think, as Bell did and his followers still seem to do, that there are “physical” assumptions involved in justifying the factorized form Eq. (<ref>). However, this is not the case because the Bell-type inequalities hold if and only if there exists a joint probability measure for the quadruples <cit.>. This mathematical statement is void of any physical meaning.In summary: in this subsection we have shown that a probabilistic model of the CFD computer simulation model that does not account for the photon identification mechanism of the EPRB laboratory experiment, automatically leads to the models introduced by Bell <cit.>. Within this framework, the existence of Bell-type inequalities and the corresponding joint probability measures are mathematically equivalent <cit.>. The latter statement, which relates to imaginary data only, corresponds to the statement that in the realm of actual two-valued data, the existence of Bell-type inequalities and CFD-compliant generation of all the quadruples are mathematically equivalent, see section <ref>. §.§ Incorporating the photon identification threshold Referring to Eq. (<ref>), the extension of the construction outlined in section <ref> to incorporate the local photon identification mechanism is of purely technical nature. Instead of Eq. (<ref>), we now introduce P”(x_1,v_1,x_1^',v_1^',x_2,v_2,x_2^',v_2^'|a_1,a_1^',a_2,a_2^', Z) = ∫ P(v_1|a_1,_1, Z) P(x_1|a_1,_1, Z)P(v_1^'|a_1^',_1, Z) P(x_1^'|a_1^',_1, Z) to 1.5cm× P(v_2|a_2,_2, Z) P(x_2|a_2,_2, Z) P(v_2^'|a_2^',_2, Z) P(x_2^'|a_2^',_2, Z) to 2cm×μ(_1,_2)d_1 d_2 dv_1 dv_2 , where P(v_1|a_1,_1, Z) etc. are the “local” probability densities to pick v_1 etc., and all other symbols have the same meaning as in Eq. (<ref>).It is now straightforward to write down the probabilistic expressions that incorporate in exactly the same manner as in the analysis of the laboratory experiment data <cit.>, the effect of the photon identification threshold. For instance, we have for the photon counts E(a_1,a_2) = A(a_1,a_2)/B(a_1,a_2) , where A(a_1,a_2) = ∑_x_1,x_2=±1∫ x_1x_2 Θ( V-v_1) Θ( V-v_2) P(v_1|a_1,_1, Z) to 1.5cm× P(x_1|a_1,_1, Z) P(v_2|a_2,_2, Z) P(x_2|a_2,_2, Z) to 2cm×μ(_1,_2)d_1 d_2 dv_1 dv_2 , and B(a_1,a_2) = ∫Θ( V-v_1) Θ( V-v_2) P(v_1|a_1,_1, Z) P(v_2|a_2,_2, Z) to 2cm×μ(_1,_2)d_1 d_2 dv_1 dv_2 ,and, as before, we have similar expressions for the other expectations in Eq. (<ref>) and for the single-particle averages.The expressions for the single- and two-particles averages that derive from the probabilistic model Eq. (<ref>) all have the form that is characteristic of a genuine “local hidden-variable model”, as exemplified by Eq. (<ref>). Only “local” detection and photon identification are involved. The values of the variables local to one observation station do not depend on variables that are local to another observation station. The only form of “communication” between the stations is through the “hidden” variables _1 and _2.It directly follows from the general discussion of section <ref> that A(a_1,a_2) and B(a_1,a_2) can be expressed in terms of both the local time-windowand time-coincidence selection. In detail, for the local time-window selection we haveA(a_1,a_2) = ∑_x_1,x_2=±1∫ x_1x_2 Θ(W-t_1) Θ(W-t_2) Θ(t_1) Θ(t_2) P(t_1|a_1,_1, Z) to 1.5cm× P(x_1|a_1,_1, Z) P(t_2|a_2,_2, Z) P(x_2|a_2,_2, Z) to 2cm×μ(_1,_2)d_1 d_2 dt_1 dt_2 , and B(a_1,a_2) = ∫Θ(W-t_1) Θ(W-t_2) Θ(t_1) Θ(t_2) P(t_1|a_1,_1, Z) P(t_2|a_2,_2, Z) to 2cm×μ(_1,_2)d_1 d_2 dt_1 dt_2 , for the time-coincidence selection we haveA(a_1,a_2) = ∑_x_1,x_2=±1∫ x_1x_2 Θ(W-|t_1-t_2|) P(t_1|a_1,_1, Z) to 1.5cm× P(x_1|a_1,_1, Z) P(t_2|a_2,_2, Z) P(x_2|a_2,_2, Z) to 2cm×μ(_1,_2)d_1 d_2 dt_1 dt_2 , and B(a_1,a_2) = ∫Θ(W-|t_1-t_2|) P(t_1|a_1,_1, Z) P(t_2|a_2,_2, Z) to 2cm×μ(_1,_2)d_1 d_2 dt_1 dt_2 . From earlier work based on representation Eq. (<ref>) <cit.> and from the simulation results presented in section <ref>, it follows directly that the probabilistic model defined by Eq. (<ref>) is capable of reproducing the predictions of quantum theory for the single- and two-particles averages of two photon polarizations in the singlet state. This then should stop spreading the misconception that Bell has proven that quantum theory is incompatible with all “local hidden-variable models”. Of course, “local hidden-variable models” that do not include the, for the laboratory experiment essential, mechanism to identify single or pairs of photons are incapable to describe the salient features of the experimental data but as explained in section <ref>, that is hardly more than a platitude.In summary, in this appendix we have shown how to construct probabilistic descriptions of computer simulation models of EPRB experiments that rely on photon identification thresholds to decide whether or not a photon has been detected <cit.>. 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"authors": [
"H. De Raedt",
"K. Michielsen",
"K. Hess"
],
"categories": [
"quant-ph",
"physics.comp-ph"
],
"primary_category": "quant-ph",
"published": "20170726072710",
"title": "The photon identification loophole in EPRB experiments: computer models with single-wing selection"
} |
Department of Physics, Zhejiang University, Hangzhou 310027, P. R. China Department of Physics, Zhejiang University of Science and Technology, Hangzhou 310023, P. R. China Department of Physics, Zhejiang University, Hangzhou 310027, P. R. China Institute for Natural Sciences, Westlake Institute for Advanced Study, Hangzhou 310013, P. R. ChinaDepartment of Physics, Zhejiang University, Hangzhou 310027, P. R. China [][email protected] Department of Physics, Zhejiang University, Hangzhou 310027, P. R. China State Key Lab of Silicon Materials, Zhejiang University, Hangzhou 310027, P. R. China Collaborative Innovation Centre of Advanced Microstructures, Nanjing 210093, P. R. ChinaSpontaneous vortex phase (SVP) is an exotic quantum matter in which quantized superconducting vortices form in the absence of external magnetic field. Although being predicted theoretically nearly 40 years ago, its rigorous experimental verification still appears to be lacking. Here we present low-field magnetic measurements on single crystals of the iron-based ferromagnetic superconductor Eu(Fe_0.91Rh_0.09)_2As_2 which undergoes a superconducting transition at T_sc = 19.6 K followed by a magnetic transition at T_m = 16.8 K. We observe a characteristic first-order transition from a Meissner state within T_m<T<T_sc to an SVP below T_m, under a magnetic field approaching zero. Additional isothermal magnetization and ac magnetization measurements at T≪ T_sc confirm that the system is intrinsically in a spontaneous-vortex ground state. The unambiguous demonstration of SVP in the title material lays a solid foundation for future imaging and spectroscopic studies on this intriguing quantum matter.74.70.Xa, 74.25.Ha, 75.30.-mEvidence of Spontaneous Vortex Ground State in An Iron-Based Ferromagnetic Superconductor Guang-Han Cao December 30, 2023 =========================================================================================Spontaneous vortex (SV) phase, originally predicted by theoretical investigations <cit.>, is an exotic quantum matter in which superconducting vortices form in the absence of external magnetic field, which can be qualitatively different from those induced by an external field <cit.>. While SV state was also predicted to be present in the pseudogap phase of cuprates due to local spins of the paramagnetic phase <cit.>, self-induced vortices are mostly generated by an internal magnetic field, H_int = 4π M, due to the spontaneous magnetization M. This means that the prerequisite of realization of an SV state is that superconductivity (SC) coexists with magnetic order, the latter of which at least gives rise to a ferromagnetic component. Such a coexistence is rare because of the antagonism between SC and ferromagnetism (FM). Additional requirements for observation of the SV phenomenon include yet, are not limited to, (1) the SC alone (a hypothetical nonmagnetic analog without any internal field) belongs to the second type with intrinsic lower and upper critical fields (H_c1^* and H_c2^*); and (2) the internal magnetic field strength lies in the range of H_c1^* < H_int <H_c2^*.Materials that bear both SC and FM generally have distinct superconducting critical temperature T_sc and magnetic transition temperature T_m. If T_sc > T_m, they are traditionally called “ferromagnetic superconductors" (FSCs) <cit.>. Otherwise, the terminology “superconducting ferromagnets" (SFMs) are often employed <cit.>. According to this classification and, with the consideration of the relative strength between H_c1^* and H_int, possible existence of an SV phase is schematically depicted in Fig. <ref> for an extremely type-II superconductor in which H_c2^*(0) ≫ H_c1^*(0) holds. In the cases of H_int(0) > H_c1^*(0), as shown in the panels (a) and (b), the SV phase can be realized as a ground state. If H_int(0) < H_c1^*(0), however, possible SV phase appears only at finite temperatures above zero, as is seen in the panels (c) and (d).There have been a few systems that may host an SV state. In the SFMs, Ru-containing cuprates <cit.> and U-based UCoGe <cit.>, which can be categorized into scenarios (c) and (a) in Fig. <ref>, respectively, were argued to have an SV state on the basis of magnetic measurements. For FSCs, however, evidence of SV phase from bulk magnetic measurements is still lacking, although theoretical <cit.> and experimental <cit.> investigations suggested an SV state in the weakly ferromagnetic superconductor ErNi_2B_2C. In fact, rigorous demonstration of a bulk SV phase by magnetic measurements is challenging primarily because an external field, which by itself induces vortices and, possibly changes the magnetic state, has to be applied. In general, one needs to demonstrate the existence of SV state as the external field approaches zero, which requires a sufficiently high measurement precision. This issue becomes more stringent in the cases above where the internal field generated by the small ferromagnetic component is very weak (e.g., the spontaneous magnetization of UCoGe is ∼0.04 μ_B/U, corresponding to ∼30 Oe field). Furthermore, the magnetic measurements always encounter the interferences of ferromagnetic domains <cit.>.As was first pointed out by Ng and Varma <cit.>, nevertheless, the SV phase can be manifested by the unique first-order phase transition from a Meissner state to an SV phase, which can be possibly seen in cases of Fig. <ref>(b-d). The first-order transition is expected to accompany with a thermal hysteresis that may be easily captured experimentally. Indeed, a thermal hysteresis in magnetic susceptibility was observed in the SFM RuSr_2GdCu_2O_8, which is interpreted as a characteristic of SV state <cit.>. However, the observed phenomenon was sample dependent and, the polycrystalline samples employed expose the flaw: the magnetic-flux pinning by grain boundaries might also account for the phenomenon <cit.>. Therefore, detection of a Meissner-to-SV transition should be done at least using single crystalline samples.In this context, the recently discovered FSCs in doped EuFe_2As_2 systems <cit.>, which show a remarkable coexistence of SC and strong FM in a broad temperature range (note that the temperature window for probing an SV phase is mostly below 2.5 K in previous systems <cit.>), provide a desirable platform to look into the SV state. Through either P doping at As site <cit.> or transition-metal (such as Ru, Co, Rh, and Ir) doping at Fe site <cit.>, SC can be induced with a T_ sc between 20 and 30 K, and the Eu^2+ spins (with S = 7/2) become ferromagnetically ordered at T_ m a few kelvins lower. Although there were debates on details of the magnetic order <cit.>, recent x-ray resonant magnetic scattering and neutron diffraction studies <cit.> show that the Eu^2+ spins always align ferromagnetically along the c axis with an ordered moment of about 7 μ_ B. The ferromagnetic ordering gives rise to a large spontaneous magnetization that generates an internal field of H_int≈ 9,000 Oe along the c axis, well above the expected H_c1^*(0) of ∼150 Oe <cit.>. Additional important advantage of the iron-based FSC is that the high-quality single crystals are easily accessible <cit.>. Note that the internal-field direction induces superconducting vortices within the FeAs layers. As such, the magnetic measurements can be limited to those under external fields parallel to the c axis, which greatly simplifies the interpretation of the measurement result.ResultsWe employed an optimally Rh-doped single crystal of Eu(Fe_0.91Rh_0.09)_2As_2 with T_sc = 19.6 K, T_m = 16.8 K, and a saturation magnetization M_sat = 6.5 μ_B/Eu <cit.>. The saturation magnetization is close to gS = 7.0, which tells that the Eu^2+ spins align ferromagnetically, similar to other Eu-containing FSC <cit.>, as shown in Fig. <ref>(a). The superconducting transition of in-plane resistivity is plotted in Fig. <ref>(b). The relatively broadened resistive transition seems to be related to the Eu-spin exchange field which suppresses the T_sc value (note that T_sc is 21.9 K for the optimally Rh-doped SrFe_2As_2 <cit.>). Below T_m, the ferromagnetic ordering leads to a re-appearance of resistivity [Fig. <ref>(b)]. The maximum of the reentrant resistivity is only 1/40 the normal-state value, indicating that it is by no means a recovery of the normal state, instead, it is associated with the SV formation. Specifically speaking, the revival of resistivity comes from the vortex flow in an SV liquid state. With decreasing temperature, H_irr^* surpasses H_int, as shown in Fig. <ref>(c), making the vortices frozen, hence zero-resistance state is achieved below ∼8 K. Note that the SV scenario naturally explains various resistivity states below T_m <cit.>, some of which show absence of the resistivity reentrance <cit.>, depending on the doping levels and physical pressures. As is seen, the absence of reentrant behaviour is more easily to be observed in P-doped EuFe_2As_2 <cit.> where T_sc is significantly higher than T_m such that H_irr^*>H_int is satisfied.The dc magnetic susceptibility shows a kink for the field-cooling (FC) protocol and a peak for the zero-field-cooling (ZFC) protocol at T_ m, as shown in Fig. <ref>(d). This can be interpreted as the formation of antiparallel ferromagnetic domains <cit.>. Because of the proximity between SC and FM, the superconducting transition is not distinctly seen in the dc magnetic measurements (although it was directly observable at very low fields <cit.>). Nevertheless, χ_c^FC and χ_c^ZFC bifurcate just at T_ sc, owing to the magnetic-flux pinning effect. The superconducting magnetic shielding effect below T_ sc is confirmed by the following ac susceptibility measurement.Since the internal field generated by the Eu^2+-spin FM is much stronger than the expected H_c1^*(0), as described above, the SV state is stabilized once the FM develops. On the other hand, the internal field vanishes for T > T_ m, hence it is in a Meissner state at zero external field in the temperature range T_ m < T < T_ sc, as shown in Fig. <ref>(c). Therefore, a transition from the Meissner state to the SV phase definitely occurs as temperature decreases. During the transition, the spontaneous vortices (SVs) suddenly penetrate the crystal's interior, which gives rise to a unique first-order transition with a magnetization discontinuity at around T_ m for the ideal case with single magnetic domain. In the case of a large sample with multi-domains, nevertheless, a “continuous" change with an obvious thermal hysteresis is expected because of the latent heat in the first-order transition. The possible thermal hysteresis from the domain-wall depinning can be avoided by employing magnetic fields that are much lower than the coercive field (∼200 Oe <cit.>).As shown in Fig. <ref>(a), the FC magnetization data indeed show a thermal hysteresis in the vicinity of T_ m, demonstrating the nature of first-order transition. In the cooling process, Meissner state is first stabilized, which expectedly gives a lower value of magnetization because of Meissner effect <cit.>. On the other hand, In the FCH process from T_ m to T_ sc, some “superheated" spontaneous vortices survive accompanying with the “polarization" of Eu spins until T_ sc, which gives rise to a higher magnetization value. The hysteresis regime extends up to T_ sc, suggesting that the SV state could be stabilized by the Eu-spin ferromagnetic fluctuations. The magnetization differences of the cooling and warming data, Δ M_c = M_FCC - M_FCH, are plotted in Fig. <ref>(b). One sees that Δ M_c drops at T_ sc, and it increases rapidly till T_ m. The maximum of |Δ M_c| increases with the applied field. Fig. <ref>(c) plots the Δ M_c value at T_ m (Δ M_c^T_ m) as a function of external field. Remarkably, Δ M_c^T_ m is exactly proportional to the field (note that the field accuracy is self-checked by the field-dependent magnetization at 30 K shown on the right axis). In fact, Δ M_c can be fully scaled with the applied field, as shown in Fig. <ref>(d). Here we emphasize that the thermal hysteresis is always observable, even at very low magnetic fields, for different pieces of the sample. By contrast, no thermal hysteresis is seen in overdoped samples where only a ferromagnetic transition takes place. This further rules out the possibility that the domain-wall depinning could be responsible for the large thermal hysteresis.The magnetization difference at T_ m, Δ M_c^T_ m, can be understood as follows. For T→ T_ m^- (FCH data), the SV state dominates. The magnetic contribution of SVs is always accompanied with the ferromagnetic domains. Owing to the existence of multi-domain, the magnetic fluxes from SVs cancels out at zero field <cit.>, and with applying fields, the moment appears to be proportional to the external field. When temperature exceeds T_ m, SVs still survive (superheating effect) although the FM vanishes. Namely, the external magnetic field penetrates the sample where superconducting layers contain SVs and, the Eu^2+ spins are basically in the Curie-Weiss paramagnetic state [see the two right-side cartoons in Fig. <ref>(a)]. Thus, the FCH magnetic susceptibility at T_ m is approximately equal to the Curie-Weiss paramagnetic susceptibility, i.e., χ_FCH^T_ m≈χ_CW^T_ m. For T→ T_ m^+ (FCC data), on the other hand, the Meissner state dominates, which gives an additional diamagnetic susceptibility of χ_sc^T_ m, yielding χ_FCC^T_ m≈χ_CW^T_ m + χ_sc^T_ m. Thus we have, Δχ_c^T_ m = χ_FCC^T_ m - χ_FCH^T_ m≈χ_sc^T_ m, which simply reflects the superconducting magnetic expulsion (see the cartoon pictures). The Meissner volume fraction can be estimated to be, 4πΔ M_c^T_ m/H ≈ 15%, which is not surprising because of the unavoidable flux pinning effect.Above we demonstrate the first-order transition from a Meissner state to an SV phase with decreasing temperature. This suggests that the SV phase represents the ground state in Eu(Fe_0.91Rh_0.09)_2As_2. If this is the case, one expects that the lower critical field at zero temperature, H_c1(0), would be zero <cit.>. Fig. <ref> shows the low-temperature isothermal magnetization, M_c(H), for Eu(Fe_0.91Rh_0.09)_2As_2, in comparison with that of the nonmagnetic superconducting analog, Ba(Fe_0.9Co_0.1)_2As_2. The latter shows an essentially linear M_c(H) since the applied fields are much lower than the H_c1(0). In contrast, Eu(Fe_0.91Rh_0.09)_2As_2 displays a non-linear virgin M_c(H) curve and an obvious magnetic hysteresis loop. This means that, in addition to the superconducting magnetic shielding effect, the external field always penetrates the sample, even if the field is around zero. In other words, Eu(Fe_0.91Rh_0.09)_2As_2 is intrinsically in a mixed state below T_ m.Another piece of evidence for the mixed state at zero field comes from the ac magnetic susceptibility measurements. As shown in the main panel of Fig. <ref>, one can clearly distinguish T_ sc and T_ m from the real part of the ac susceptibility, χ'. The magnetic shielding effect below T_ sc is much more obvious than that of the dc magnetic measurement above. The imaginary part of the susceptibility, χ”, which is sensitive to dissipations, shows two sharp peaks below T_ sc and T_ m, respectively. An additional large broad peak appears below the re-entrant spin-glass temperature T_ sg≈ 13.5 K <cit.>. Note that this χ” peak may also be contributed from the SV liquid-to-solid transition.Remarkably, the χ” value at the lowest temperature of 1.90 K in our measurements remains considerably high at the driving field H_ac = 2.5 Oe, verifying that it is in a mixed state. To examine if there is a lower limit of the ac field, we performed a field-dependent ac magnetization measurement, the imaginary part (m_c”) of which is shown in the inset of Fig. <ref>. One sees that m_c” is exactly proportional to H_ac. According to the critical-state model <cit.>, m_c” will be zero for H_ac < H_ c1; while m_c” = β(H_ac - H_ c1)^2/H_ac (β is the sample's geometrical factor) for H_ac > H_ c1. Both the non-zero m_c” and the linearity of m_c”(H_ac) through the origin indicate that H_ c1 must be zero. Similar observation is seen in the SFM UCoGe <cit.>. DiscussionThe above results allow us to arrive at the following picture for the Eu(Fe_0.91Rh_0.09)_2As_2 FSC in the absence of external magnetic field. At T>T_ sc, the [(Fe,Rh)_2As_2]^2- and Eu^2+ layers are Pauli paramagnetic and Curie-Weiss paramagnetic, respectively. When cooled below T_ sc, the [(Fe,Rh)_2As_2]^2- layers become superconducting, showing a Meissner state coexisting with the Curie-Weiss paramagnetism of Eu^2+ spins. With further decreasing temperature to below T_ m, the Eu^2+ spins are ferromagnetic ordered along the c axis, which generates an internal field far above the expected H_c1^*(0). Superconducting vortices then form spontaneously in the [(Fe,Rh)_2As_2]^2- layers. In the temperature range of 8 K <T<T_ m, the SVs are mobile, which leads to the revival of resistance. The subsequent solidification of the SVs below 8 K gives rise to the zero-resistance state. Therefore, the ground state is an SV solid, which reconciles SC and FM in Eu(Fe_0.91Rh_0.09)_2As_2.Finally we note that, apart from the formation of the SV phase, an alternative that allows to reconcile the SC and FM is the so-called Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state characterized by a spatial modulation of the superconducting order parameter <cit.>. Nevertheless, in general, realization of an FFLO state at zero external field needs more rigorous conditions. Among them are Pauli-limited H_c2^⊥ (for H⊥ ab) with a large Maki parameter and clean limit for the SC, which cannot be satisfied in the present system. The H_c2^⊥(T) curve in Eu(Fe_0.91Rh_0.09)_2As_2 keeps linear down to 0.2T_ sc <cit.>, indicating that the orbital-limiting effect dominates. Besides, the large residual resistivity (∼60 μΩ cm) as well as the small residual resistivity ratio (RRR = 2.6) <cit.> suggests a dirty limit. Both properties actually favor the SV scenario. Nevertheless, here we note that the recently discovered 1144-type FSC <cit.> could be the candidate for an FFLO state, because their nonmagnetic analog, CaKFe_4As_4, indeed shows a large Maki parameter together with a clean limit for the SC <cit.>.In summary, we have studied the low-field magnetic properties for the iron-based ferromagnetic superconductor Eu(Fe_0.91Rh_0.09)_2As_2. We observed a remarkable thermal hysteresis around the ferromagnetic transition in the superconducting state, even under a vanishingly small field, demonstrating the unique first-order transition from a Meissner state to an SV phase. The SV ground state is further corroborated by the non-linear virgin dc magnetization as well as the non-zero imaginary part of ac magnetic susceptibility under extremely low external fields at T ≪ T_sc. The unambiguous demonstration of the SV ground state in the iron-based FSC lays a solid foundation for future studies. For example, it is of great interest to see whether the SV solid behaves like a glassy or a lattice state. The imaging observations such as magnetic-force microscopy as well as the small-angle neutron scattering technique may help to clarify this interesting issue.Methods Crystal growth. High-quality crystals of Eu(Fe_0.91Rh_0.09)_2As_2 were grown by a self-flux method <cit.>. First, mixtures of Eu (99.9%), Fe (99.998%), Rh (99.9%), and As (99.999%) powders in a molar ratio of Eu:Fe:Rh:As = 1:4.4:0.6:5 reacted at 973 K for 24 h in a sealed evacuated quartz ampoule. The precursor was ground, and then was loaded into an alumina crucible. The crucible was sealed in a stainless steel tube by arc welding under an atmosphere of argon. The assembly was subsequently heated up to 1573 K and, holding for 5 h, in a muffle furnace with the flow of argon gas. The crystal growth took place during the slow cooling down to 1223 K at the rate of 4 K/h. Large crystals with typical size of 3×3×0.5 mm^3 were harvested.Structural and compositional characterizations. We checked the as-grown crystal flakes by x-ray diffraction using a PANAlytical x-ray diffractometer (using Cu K_α1 monochromatic radiation) at room temperature. All the crystals show only (00l) reflections with even l values, similar to the previous report <cit.>. The c axis is then determined to be 12.016(1) Å. The crystal structure is analogous to EuFe_2As_2 (c = 12.136 Å) <cit.>, yet it consists of superconducting [(Fe,Rh)_2As_2]^2- layers separated by magnetic Eu^2+ ions. The full width at half maximum (FWHM) of the reflection peaks is typically 2θ = 0.06^∘, verifying the high quality of the crystals. The real composition of the crystal was determined by energy dispersive x-ray spectroscopy, which gives the chemical formula of Eu(Fe_0.91Rh_0.09)_2As_2.Physical properties. The electrical and magnetic properties of the Eu(Fe_0.91Rh_0.09)_2As_2 crystals were reported previously <cit.>, which demonstrate a superconducting transition at T_sc = 19.6 K, followed by a ferromagnetic transition at T_m = 16.8 K. The isothermal magnetization loops below T_m show characteristic features for both FM and SC. The saturation magnetization achieves M_sat = 6.5 μ_B/Eu, confirming that the Eu spins align ferromagnetically.Low-field magnetic measurements. We selected a free-standing crystal for all the measurements in this paper. Magnetic measurements were carried out on a Quantum Design Magnetic Property Measurement System. The residual field in the superconducting magnet, after being removed by a degaussing procedure prior to the measurements, is less than ±0.05 Oe. The crystal was carefully mounted into the sample holder with the applied field perpendicular to the crystal plate, such that the external field is either parallel or antiparallel to the internal field. The FC data were collected in both heating and cooling procedures. In the ac susceptibility measurement, the frequency was set to 1.0 Hz. The demagnetization effect is taken into account on the basis of the sample's geometry in respect to the field direction. Acknowledgments This work is supported by the National Science Foundation of China (Nos.11474252 and 11504329), the National Key Research and Development Program of China (No.2016YFA0300202), and the Fundamental Research Funds for the Central Universities of China.Author contributions W.H.J. and Y.L. grew and characterized the crystals. W.H.J. and Q.T. performed the magnetic measurements. W.H.J., Z.R. and G.H.C. analyzed and interpreted the data, and wrote the paper. This work was coordinated and designed by G.H.C.Additional Information The authors declare no competing financial interests. | http://arxiv.org/abs/1707.08420v1 | {
"authors": [
"Wen-He Jiao",
"Qian Tao",
"Zhi Ren",
"Yi Liu",
"Guang-Han Cao"
],
"categories": [
"cond-mat.supr-con",
"cond-mat.other"
],
"primary_category": "cond-mat.supr-con",
"published": "20170726130928",
"title": "Evidence of Spontaneous Vortex Ground State in An Iron-Based Ferromagnetic Superconductor"
} |
STN-OCR: A single Neural Network for Text Detection and Text Recognition C. Conti^4,5 December 30, 2023 ======================================================================== OCROptical Character Recognition CNNConvolutional Neural Network RNNRecurrent Neural Network DNNDeep Neural Network BLSTMBidirectional Long-Short Term Memory CTCConnectionist Temporal Classification FSNSFrench Street Name Signs SGDStochastic Gradient Descent Detecting and recognizing text in natural scene images is a challenging, yet not completely solved task. In recent years several new systems that try to solve at least one of the two sub-tasks (text detection and text recognition) have been proposed. In this paper we present STN-OCR, a step towards semi-supervised neural networks for scene text recognition that can be optimized end-to-end. In contrast to most existing works that consist of multiple deep neural networks and several pre-processing steps we propose to use a single deep neural network that learns to detect and recognize text from natural images in a semi-supervised way. STN-OCR is a network that integrates and jointly learns a spatial transformer network <cit.>, that can learn to detect text regions in an image, and a text recognition network that takes the identified text regions and recognizes their textual content. We investigate how our model behaves on a range of different tasks (detection and recognition of characters, and lines of text). Experimental results on public benchmark datasets show the ability of our model to handle a variety of different tasks, without substantial changes in its overall network structure. § INTRODUCTION Text is ubiquitous in our daily lifes. Text can be found on documents, road signs, billboards, and other objects like cars or telephones. Automatically detecting and reading text from natural scene images is an important part of systems that can be used for several challenging tasks such as image-based machine translation, autonomous cars or image/video indexing. In recent years the task of detecting text and recognizing text in natural scenes has seen much interest from the computer vision and document analysis community. Furthermore recent breakthroughs <cit.> in other areas of computer vision enabled the creation of even better scene text detection and recognition systems than before <cit.>. Although the problem of OCR can be seen as solved for printed document texts it is still challenging to detect and recognize text in natural scene images. Images containing natural scenes exhibit large variations of illumination, perspective distortions, image qualities, text fonts, diverse backgrounds, .The majority of existing research works developed end-to-end scene text recognition systems that consist of complex two-step pipelines, where the first step is to detect regions of text in an image and the second step is to recognize the textual content of that identified region. Most of the existing works only concentrate on one of these two steps.In this paper, we present a solution that consists of a single DNN that can learn to detect and recognize text in a semi-supervised way. This is contrary to existing works, where text detection and text recognition systems are trained separately in a fully-supervised way. Recent work <cit.> showed that CNN are capable of learning how to solve complex multi-task problems, while being trained in an end-to-end manner. Our motivation is to use these capabilities of CNN and create an end-to-end scene text recognition system that behaves more like a human by dividing the task at hand into smaller subtasks and solving these subtask independently from each other. In order to achieve this behavior we learn a single DNN that is able to divide the input image into subtasks (single characters, words or even lines of text) and solve these subtasks independently of each other. This is achieved by jointly learning a localization network that uses a recurrent spatial transformer <cit.> as attention mechanism and a text recognition network (see <ref> for a schematic overview of the system). In this setting the network only receives the image and the labels for the text contained in that image as input. The localization of the text is learned by the network itself, making this approach semi-supervised. Our contributions are as follows: [label=(*)] * We present a system that is a step towards solving end-to-end scene text recognition by integrating spatial transformer networks. * We train our proposed system end-to-end in a semi-supervised way. * We demonstrate that our approach is able to reach state-of-the-art/competitive performance on a range of standard scene text detection and recognition benchmarks. * We provide our code[<https://github.com/Bartzi/stn-ocr>] and trained models[<https://bartzi.de/research/stn-ocr>] to the research community. This paper is structured in the following way: In <ref> we outline work of other researchers related to ours. Section <ref> describes our proposed system in detail and provides best practices on how to train such a system. We show and discuss our results on standard benchmark datasets in <ref> and conclude our findings in <ref>. § RELATED WORKOver the course of years a rich environment of different approaches to scene text detection and recognition have been developed and published. Nearly all systems use a two-step process for performing end-to-end recognition of scene text. The first step is to detect regions of text and extract these regions from the input image. The second step is to recognize the textual content and return the text strings of these extracted text regions. It is further possible to divide these approaches into three broad categories: [label=(*)] * Systems relying on hand crafted features and human knowledge for text detection and text recognition. * Systems using deep learning approaches together with hand crafted features, or two different deep networks for each of the two steps. * Systems that do not consist of a two step approach but rather perform text detection and recognition using a single deep neural network.We will discuss some of these systems for each category below. Hand Crafted Features In the beginning methods based on hand crafted features and human knowledge have been used to perform text detection. These systems used features like MSERs <cit.>, Stroke Width Transforms <cit.> or HOG-Features <cit.> to identify regions of text and provide them to the text recognition stage of the system. In the text recognition stage sliding window classifiers <cit.> and ensembles of SVMs <cit.> or k-Nearest Neighbor classifiers using HOG features <cit.> were used. All of these approaches use hand crafted features that have a large variety of hyper parameters that need expert knowledge to correctly tune them for achieving the best results. Deep Learning Approaches More recent systems exchange approaches based on hand crafted features in one or both steps of end-to-end recognition systems by approaches using DNN. Gómez and Karatzas <cit.> propose a text-specific selective search algorithm that, together with a DNN, can be used to detect (distorted) text regions in natural scene images. Gupta <cit.> propose a text detection model based on the YOLO-Architecture <cit.> that uses a fully convolutional deep neural network to identify text regions. The text regions identified by these approaches can then be used as input for further systems based on DNN that perform text recognition. Bissacco <cit.> propose a complete end-to-end architecture that performs text detection using hand crafted features. The identified text regions are binarized and then used as input to a deep fully connected neural network that classifies each found character independently. Jaderberg <cit.> propose several systems that use deep neural networks for text detection and text recognition. In <cit.> Jaderberg propose a sliding window text detection approach that slides a convolutional text detection model across the image in multiple resolutions. The text recognition stage uses a single character CNN, which is slided across the identified text region. This CNN shares its weights with the CNN used for text detection. In <cit.> Jaderberg propose to use a region proposal network with an extra bounding box regression CNN for text detection and a CNN that takes the whole text region as input and performs classification across a pre-defined dictionary of words, making this approach only applicable to one given language. Goodfellow <cit.> propose a text recognition system for house numbers, that has been refined by Jaderberg <cit.> for unconstrained text recognition. This system uses a single CNN, which takes the complete extracted text region as input, and provides the text contained in that text region. This is achieved by having one independent classifier for each possible character in the given word. Based on this idea He <cit.> and Shi <cit.> propose text recognition systems that treat the recognition of characters from the extracted text region as a sequence recognition problem. He <cit.> use a naive sliding window approach that creates slices of the text region, which are used as input to their text recognition CNN. The features produced by the text recognition CNN are used as input to a RNN that predicts the sequence of characters. In our experiments on pure scene text recognition (see section <ref> for more information) we use a similar approach, but our system uses a more sophisticated sliding window approach, where the choice of the sliding windows is automatically learned by the network and not engineered by hand. Shi <cit.> utilize a CNN that uses the complete text region as input and produces a sequence of feature vectors, which are fed to a RNN that predicts the sequence of characters in the extracted text region. This approach generates a fixed number of feature vectors based on the width of the text region. That means for a text region that only contains a few characters, but has the same width as a text region with sufficently more characters, this approach will produce the same amount of feature vectors used as input to the RNN. In our pure text recognition experiments we utilized the strength of our approach to learn to attend to the most important information in the extracted text region, hence producing only as many feature vectors as necessary. Shi <cit.> improve their approach by firstly adding an extra step that utilizes the rectification capabilities of Spatial Transformer Networks <cit.> for rectifying the extracted text line. Secondly they added a soft-attention mechanism to their network that helps to produce the sequence of characters in the input image. In their work Shi make use of Spatial Transformers as an extra pre-processing step to make it easier for the recognition network to recognize the text in the image. In our system we use the Spatial Transformer as a core building block for detecting text in a semi-supervised way. End-to-End trainable Approaches The presented systems always use a two-step approach for detecting and recognizing text from scene text images. Although recent approaches make use of deep neural networks they are still using a huge amount of hand crafted knowledge in either of the steps or at the point where the results of both steps are fused together. Smith <cit.> propose an end-to-end trainable system that is able to detect and recognize text on french street name signs, using a single DNN. In contrast to our system it is not possible for the system to provide the location of the text in the image, only the textual content can be extracted. Furthermore the attention mechanism used in our approach shows a more human-like behaviour because is sequentially localizes and recognizes text from the given image. § PROPOSED SYSTEMA human trying to find and read text will do so in a sequential manner. The first action is to put attention on a line of text, read each character sequentially and then attend to the next line of text. Most current end-to-end systems for scene text recognition do not behave in that way. These systems rather try to solve the problem by extracting all information from the image at once. Our system first tries to attend sequentially to different text regions in the image and then recognize the textual content of each text region. In order to this we created a simple DNN consisting of two stages: [label=(*)] * text detection * text recognition . In this section we will introduce the attention concept used by the text detection stage and the overall structure of the proposed system. We also report best practices for successfully training such a system. §.§ Detecting Text with Spatial TransformersA spatial transformer proposed by Jaderberg <cit.> is a differentiable module for DNN that takes an input feature map I and applies a spatial transformation to this feature map, producing an output feature map O. Such a spatial transformer module is a combination of three parts. The first part is a localisation network computing a function f_loc, that predicts the parameters θ of the spatial transformation to be applied. These predicted parameters are used in the second part to create a sampling grid that defines which features of the input feature map should be mapped to the output feature map. The third part is a differentiable interpolation method that takes the generated sampling grid and produces the spatially transformed output feature map O. We will shortly describe each component in the following paragraphs. Localization Network The localization network takes the input feature map I ∈ℝ^C × H × W, with C channels, height H and width W and outputs the parameters θ of the transformation that shall be applied. In our system we use the localization network (f_loc) to predict N two-dimensional affine transformation matrices A^n_θ, where n ∈{0, …, N - 1}: f_loc(I) = A^n_θ = [ θ^n_1 θ^n_2 θ^n_3; θ^n_4 θ^n_5 θ^n_6; ] N is thereby the number of characters, words or textlines the localization network shall localize. The affine transformation matrices predicted in that way allow the network to apply translation, rotation, zoom and skew to the input image, hence the network learns to produce transformation parameters that can zoom on characters, words or text lines that are to be extracted from the image. In our system the N transformation matrices A^n_θ are produced by using a feed-forward CNN together with a RNN. Each of the N transformation matrices is computed using the hidden state h_n for each time-step of the RNN: c= f^conv_loc(I) h_n= f^rnn_loc(c, h_n-1) A^n_θ = g_loc(h_n) where g_loc is another feed-forward network, and each transformation matrix A^n_θ is conditioned on the globally extracted convolutional features (f^conv_loc) together with the hidden state of the previously performed time-step. The CNN in the localization network used by us is a variant of the well known ResNet by He <cit.>. We use a variant of ResNet because we found that with this network structure our system learns faster and more successful, as compared to experiments with other network structures like the VGGNet <cit.>. We argue that this is due to the fact that the residual connections of the ResNet help with retaining a strong gradient down to the very first convolutional layers. In addition to the structure we also used Batch Normalization <cit.> for all our experiments. The RNN used in the localization network is a BLSTM <cit.> unit. This BLSTM is used to generate the hidden states h_n, which in turn are used to predict the affine transformation matrices. We used the same structure of the network for all our experiments we report in <ref>. <ref> provides a structural overview of this network. Grid Generator The grid generator uses a regularly spaced grid G_o with coordinates y_h_o,x_w_o, of height H_o and width W_o, together with the affine transformation matrices A^n_θ to produce N regular grids G^n_i of coordinates u^n_i,v^n_j of the input feature map I, where i ∈ H_o and j ∈ W_o: [ u^n_i; v^n_j ] = A^n_θ[ x_w_o; y_h_o; 1 ] = [ θ^n_1 θ^n_2 θ^n_3; θ^n_4 θ^n_5 θ^n_6; ][ x_w_o; y_h_o; 1 ] During inference we can extract the N resulting grids G^n_i which contain the bounding boxes of the text regions found by the localization network. Height H_o and width W_o can be chosen freely and if they are lower than height H or width W of the input feature map I the grid generator is producing a grid that performs a downsampling operation in the next step. Image Sampling The N sampling grids G^n_i produced by the grid generator are now used to sample values of the feature map I at their corresponding coordinates u^n_i,v^n_j for each n ∈ N. Naturally these points will not always perfectly align with the discrete grid of values in the input feature map. Because of that we use bilinear sampling that extracts the value at a given coordinate by bilinear interpolating the values of the nearest neighbors. With that we define the values of the N output feature maps O^n at a given location i,j where i ∈ H_o and j ∈ W_o: O^n_ij = ∑^H_o_h ∑^W_o_w I_hw max(0, 1 - | u^n_i - h |) max(0, 1 - | v^n_j - w |) This bilinear sampling is (sub-)differentiable, hence it is possible to propagate error gradients to the localization network by using standard backpropagation.The combination of localization network, grid generator and image sampler forms a spatial transformer and can in general be used in every part of a DNN. In our system we use the spatial transformer as the first step of our network. The localization network receives the input image as input feature map and produces a set of affine transformation matrices that are used by the grid generator to calculate the position of the pixels that shall be sampled by the bilinear sampling operation. §.§ Text Recognition Stage The image sampler of the text detection stage produces a set of N regions that are extracted from the original input image. The text recognition stage (a structural overview of this stage can be found in <ref>) uses each of these N different regions and processes them independently of each other. The processing of the N different regions is handled by a CNN. This CNN is also based on the ResNet architecture as we found that we could only achieve good results if we use a variant of the ResNet architecture for our recognition network. We argue that using a ResNet in the recognition stage is even more important than in the detection stage, because the detection stage needs to receive strong gradients from the recognition stage in order to successfully update the weights of the localization network. The CNN of the recognition stage predicts a probability distribution ŷ over the label space L_ϵ, where L_ϵ = L ∪{ϵ}, with L = {0-9a-z} and ϵ representing the blank label. Depending on the task this probability distribution is either generated by a fixed number of T softmax classifiers, where each softmax classifier is used to predict one character of the given word:x^n= O^nŷ^n_t= softmax(f_rec(x^n))ŷ^n= ∑_t=1^ T ŷ^n_twhere f_rec(x) is the result of applying the convolutional feature extractor on the sampled input x.Another possibility is to train the network using CTC <cit.> and retrieve the most probable labeling by setting ŷ to be the most probable labeling path π, that is given by:p(π|x^n)= ∏^T_t=1ŷ^n_π_t, ∀π∈ L^T_ϵ ŷ^n_t= argmax p(π|x^n)ŷ^n= B(∑^T_t=1ŷ^n_t)with L^T_ϵ being the set of all labels that have the length T and p(π|x^n) being the probability that path π∈ L^T_ϵ is predicted by the DNN. B is a function that removes all predicted blank labels and all repeated labels (e.g. B(-IC-CC-V) = B(II–CCC-C–V-) = ICCV). §.§ Model Training The training set X used for training the model consists of a set of input images I and a set of text labels L_I for each input image. We do not use any labels for training the text detection stage. This stage is learning to detect regions of text only by using the error gradients obtained by either calculating the cross-entropy loss or the CTC loss of the predictions and the textual labels. During our experiments we found that, when trained from scratch, a network that shall detect and recognize more than two text lines does not converge. The solution to this problem is to perform a series of pre-training steps where the difficulty is gradually increasing. Furthermore we find that the optimization algorithm chosen to train the network has a great influence on the convergence of the network. We found that it is beneficial to use SGD for pre-training the network on a simpler task and Adam <cit.> for finetuning the already pre-trained network on images with more text lines. We argue that SGD performs better during pre-training because the learning rate η is kept constant during a longer period of time, which enables the text detection stage to explore the input images and better find text regions. With decreasing learning rate the updates in the detection stage become smaller and the text detection stage (ideally) settles on already found text regions. At the same time the text recognition network can start to use the extracted text regions and learn to recognize the text in that regions. While training the network with SGD it is important to note that choosing a too high learning rate will result in divergence of the model early on.We found that using initial learning rates between 1^-5 and 5^-7 tend to work in nearly all cases, except in cases where the network should only be fine-tuned. Here we found that using Adam is the more reliable choice, as Adam chooses the learning rate for each parameter in an adaptive way and hence does not allow the detection network to explore as radically as it does when using SGD. § EXPERIMENTSIn this section we evaluate our presented network architecture on several standard scene text detection/recognition datasets. We present the results of experiments for three different datasets, where the difficulty of the task at hand increases for each dataset. We first begin with experiments on the SVHN dataset <cit.>, that we used to prove that our concept as such is feasible. The second type of dataset we performed experiments on were datasets for focused scene text recognition, where we explored the performance of our model, when it comes to find and recognize single characters. The third dataset we exerimented with was the FSNS dataset <cit.>, which is the most challenging dataset we used, as this dataset contains a vast amount of irregular, low resolution text lines that are more difficult to locate and recognize than text lines from the SVHN dataset. We begin this section by introducing our experimental setup. We will then present the results and characteristics of the experiments for each of the aforementioned datasets. §.§ Experimental Setup Localization Network The localization network used in every experiment is based on the ResNet architecture <cit.>. The input to the network is the image where text shall be localized and later recognized. Before the first residual block the network performs a 3 × 3 convolution followed by a 2 × 2 average pooling layer with stride 2. After these layers three residual blocks with two 3 × 3 convolutions, each followed by batch normalization <cit.>, are used. The number of convolutional filters is 32, 48 and 48 respectively and ReLU <cit.> is used as activation function for each convolutional layer. A 2 × 2 max-pooling with stride 2 follows after the second residual block. The last residual block is followed by a 5 × 5 average pooling layer and this layer is followed by a BLSTM with 256 hidden units. For each time step of the BLSTM a fully connected layer with 6 hidden units follows. This layer predicts the affine transformation matrix, that is used to generate the sampling grid for the bilinear interpolation. As rectification of scene text is beyond the scope of this work we disabled skew and rotation in the affine transformation matrices by setting the according parameters to 0. We will discuss the rectification capabilities of Spatial Transformers for scene text detection in our future work. Recognition Network The inputs to the recognition network are N crops from the original input image that represent the text regions found by the localization network. The recognition network has the same structure as the localization network, but the number of convolutional filters is higher. The number of convolutional filters is 32, 64 and 128 respectively. Depending on the experiment we either used an ensemble of T independent softmax classifiers as used in <cit.> and <cit.>, where T is the maximum length that a word may have, or we used CTC with best path decoding as used in <cit.> and <cit.>. Implementation We implemented all our experiments using MXNet <cit.>. We conduted all our experiments on a work station which has an Intel(R) Core(TM) i7-6900K CPU, 64 GB RAM and 4 TITAN X (Pascal) GPUs. §.§ Experiments on the SVHN datasetWith our first experiments on the SVHN dataset <cit.> we wanted to prove that our concept works and can be used with real world data. We therefore first conducted experiments similar to the experiments in <cit.> on SVHN image crops with a single house number in each image crop, that is centered around the number and also contains background noise. <ref> shows that we are able to reach competitive recognition accuracies.Based on this experiment we wanted to determine whether our model is able to detect different lines of text that are arranged in a regular grid or placed at random locations in the image. In <ref> we show samples from two purpose build datasets[datasets are available here: <https://bartzi.de/research/stn-ocr>] that we used for our other experiments based on SVHN data. We found that our network performs well on the task of finding and recognizing house numbers that are arranged in a regular grid. An interesting observation we made during training on this data was that we were able to achieve our best results when we did two training steps. The first step was to train the complete model from scratch (all weights initialized randomly) and then train the model with the same data again, but this time with the localization network pre-initialized with the weights obtained from the last training and the recognition net initialized with random weights. This strategy leads to better localization results of the localization network and hence improved recognition results.During our experiments on the second dataset, created by us, we found that it is not possible to train a model from scratch, that can find and recognize more than two textlines that are scattered across the whole image. It is possible to train such a network by first training the model on easier tasks first (few textlines, textlines closer to the center of the image) and then increase the difficulty of the task gradually. In the supplementary material we provide short video clips that show how the network is exploring the image while learning to detect text for a range of different experiments.§.§ Experiments on Robust Reading DatasetsIn our next experiments we used datasets where text regions are aleady cropped from the input images. We wanted to see whether our text localization network can be used as an intelligent sliding window generator that adopts to irregularities of the text in the cropped text region. Therefore we trained our recognition model using CTC on a dataset of synthetic cropped word images, that we generated using our own data generator, that works similar to the data generator introduced by Jaderberg <cit.>. In <ref> we report the recognition results of our model on the ICDAR 2013 robust reading <cit.>, the Street View Text (SVT) <cit.> and the IIIT5K <cit.> benchmark datasets. For evaluation on the ICDAR 2013 and SVT datasets, we filtered all images that contain non-alphanumeric characters and discarded all images that have less than 3 characters as done in <cit.>. We obtained our final results by post-processing the predictions using the standard hunspell english (en-US) dictionary. Overall we find that our model achieves state-of-the-art performance for unconstrained recognition models on the ICDAR 2013 and IIIT5K dataset and competitive performance on the SVT dataset. In <ref> we show that our model learns to follow the slope of the individual text regions, proving that our model produces sliding windows in an intelligent way. §.§ Preliminary Experiments on the FSNS datasetFollowing our scheme of increasing the difficulty of the task that should be solved by the network, we chose the FSNS dataset by Smith <cit.> to be our third dataset to perform experiments on. The results we report here are preliminary and are only meant to show that our network architecture is also applicable to this kind of data, although it does not yet reach state-of-the-art results. The FSNS dataset contains images of french street name signs that have been extracted from Google Streetview. This dataset is the most challenging dataset for our approach as it[label=(*)] * contains multiple lines of text with varying length embedded in natural scenes with distracting backgrounds and * contains a lot of images that do not include the full name of the streets. During our first experiments with that dataset we found that our model is not able to converge, when trained on the supplied groundtruth. We argue that this is because the labels of the original dataset do not include any hint on which words can be found in which text line. We therefore changed our approach and started with experiments where we tried to find individual words instead of textlines with more than one word. We adapted the groundtruth accordingly and used all images that contain a maximum of three words for our experiments, which leaves us with approximately 80 of the original data from the dataset. <ref> shows some examples from the FSNS dataset where our model correctly localized the individual words and also correctly recognized the words. Using this approach we were able to achieve a reasonably good character recognition accuracy of 97 on the test set, but only a word accuracy of 71.8. The discrepancy in character recognition rate and word recognition rate is caused by the fact that the model we trained for this task uses independent softmax classifiers for each character in a word. Having a character recognition accuracy of 97 means that there is a high probability that at least one classifier makes a mistake and thus increases the sequence error. § CONCLUSIONIn this paper we presented a system that can be seen as a step towards solving end-to-end scene text recognition, using only a single multi-task deep neural network. We trained the text detection component of our model in a semi-supervised way and are able to extract the localization results of the text detection component. The network architecture of our system is simple, but it is not easy to train this system, as a successful training requires extensive pre-training on easier sub-tasks before the model can converge on the real task. We also showed that the same network architecture can be used to reach competitive or state-of-the-art results on a range of different public benchmark datasets for scene text detection/recognition.At the current state we note that our models are not fully capable of detecting text in arbitrary locations in the image, as we saw during our experiments with the FSNS dataset. Right now our model is also constrained to a fixed number of maximum textlines/characters that can be detected at once, in our future work we want to redesign the network in a way that makes it possible for the network to determine the number of textlines in an image by itself. ieee | http://arxiv.org/abs/1707.08831v1 | {
"authors": [
"Christian Bartz",
"Haojin Yang",
"Christoph Meinel"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170727122234",
"title": "STN-OCR: A single Neural Network for Text Detection and Text Recognition"
} |
Anytime Exact Belief Propagation Gabriel Azevedo Ferreira, Quentin Bertrand, Charles Maussion, Rodrigo de Salvo BrazArtificial Intelligence CenterSRI InternationalMenlo Park CA, USADecember 30, 2023 =================================================================================================================================================================Statistical Relational Models and, more recently, Probabilistic Programming, have been making strides towards an integration of logic and probabilistic reasoning. A natural expectation for this project is that a probabilistic logic reasoning algorithm reduces to a logic reasoning algorithm when provided a model that only involves 0-1 probabilities, exhibiting all the advantages of logic reasoning such as short-circuiting, intelligibility, and the ability to provide proof trees for a query answer. In fact, we can take this further and require that these characteristics be present even for probabilistic models with probabilities near 0 and 1, with graceful degradation as the model becomes more uncertain. We also seek inference that has amortized constant time complexity on a model's size (even if still exponential in the induced width of a more directly relevant portion of it) so that it can be applied to huge knowledge bases of which only a relatively small portion is relevant to typical queries. We believe that, among the probabilistic reasoning algorithms, Belief Propagation is the most similar to logic reasoning: messages are propagated among neighboring variables, and the paths of message-passingare similar to proof trees. However, Belief Propagation is either only applicable to tree models, or approximate (and without guarantees) for precision and convergence. In this paper we present work in progress on an Anytime Exact Belief Propagation algorithm that is very similar to Belief Propagation but is exact even for graphical models with cycles, while exhibiting soft short-circuiting, amortized constant time complexity in the model size, and which can provide probabilistic proof trees.§ INTRODUCTION Statistical Relational Models <cit.> and, more recently, Probabilistic Programming, have been making strides towards probabilistic logic inference algorithms that integrate logic and probabilistic reasoning. These algorithms perform inference on probabilistic logical models, which generalize regular logical models by containing formulas that have a probability of being true, rather than always being true.While Statistical Relational Models and Probabilistic Programming focus on higher-level representations such as relations and data structures, even regular graphical models such as Bayesian networks can be thought of as probabilistic logic models, since conditional probability distributions (or factors, for undirected models) can be described by formulas and therefore thought of as probabilistic formulas. This paper focuses on regular graphical models that will later serve as a basis for higher-level probabilistic logic models (as further discussed in the conclusion).Naturally, probabilistic inference algorithms must be able to perform inference on purely logic models, since these can be seen asprobabilistic logic models whose formulas have probability 1 of being true. In this case, it is desirable that the probabilistic inference algorithmsreduce to logic reasoning in a way that exploits the logic structure of the model nearly as efficiently as pure logic reasoning algorithms would. In fact, we should expect even more: if a model (or part of it) is near-certain (with probabilities close to 0 and 1), then the model is very close to a purely logical model and it is reasonable to expect a probabilistic inference algorithm to exploit the logical structure to some degree, with graceful degradation as the model becomes more uncertain.Short-circuiting is a type of structure that is an important source of efficiency for logic reasoning. A formula is short-circuited if its value can be determined from the value of only some of its sub-formulas. For example, if a model contains the formula A ⇐ B ∨ C ∨ D and B happens to be true, then a logic reasoning algorithm can conclude that A is true without having to decide whether C and D are true. However, if a probabilistic reasoning algorithm knows P(A | B ∨ C ∨ D) = 1 and that P(B) = 0.9, it will typically still need to compute P(C) and P(D) in order to compute P(A), even though it is already possible to affirm that P(A) ≥ 0.9 without any reasoning about C and D. Providing such a bound can be considered a soft short-circuiting that approximates logical short-circuiting as probabilities get closer to 0 and 1, but such ability is absent from most probabilistic inference methods.Soft short-circuiting serves as a basis for an anytime, incremental algorithm that trades bound accuracy for time. Given more time, the algorithm may determine that P(C) ≥ 0.8 independently of B, perhaps by recursively processing another rule P(C | E ∨ F), which allows it to increase its lower bound P(A) ≥ 0.98, a tight bound obtained without ever reasoning about some potentially large parts of the model (in this case, rules involving D or adding information about C).There are several probabilistic inference algorithms, discussed in Section <ref>, that produce bounds on query probabilities. However, we find that most of these algorithms do not exhibit another important property of logic reasoning algorithms: a time complexity for inference that is amortized constant in the size of the entire model (although still exponential on the induced width of the portion of the model that is relevant for computing the current bound). This is achieved by storing formulas in a model in hash tables indexed by the variables they contain, and looking up formulas only as needed during inference as their variables come into play. An important application forprobabilistic logic reasoning in the future is reasoning about thousands or even millions of probabilistic rules (for example in knowledge bases learned from the Web such as NELL <cit.>), for which this property will be essential.Finally, we are also interested in a third important property of logic reasoning algorithms: the ability to produce an intelligible trace of its inference (such as a proof or refutation tree) that serves as a basis for explanations and debugging. In probabilistic reasoning, Belief Propagation is perhaps the closest we get to this, since local message-passing is easily understood and the tree of messages can be used asa proof tree. However, BP only returns correct marginal probabilities for graphical models without cycles,and non-guaranteed approximation and convergence for general graphical models. In this paper, we present work in progress on Anytime Exact Belief Propagation, an algorithm that exhibits the three properties described above: it incrementally computes bounds on marginal probabilities that can be provided at any time, whose accuracy can be traded off for time, and that eventually converge to the exact marginal; it has time complexity amortized constant in the size of the entire model; and it produces a tree of local messages that can be used toexplain its conclusion.§ RELATED WORKThe most obvious candidates for probabilistic logic reasoning approaches that exhibit logic properties with graceful degradation are the ones based on logic programming:Bayesian Logic Programs <cit.>, PRISM <cit.>,Stochastic Logic Programs <cit.>, and ProbLog <cit.>. While these approaches can be used with sampling, they typically derive (by regular logic programming methods) a proof tree for the query and evidence in order to determine which portion of the model is qualitatively relevant to them. Only after that does the probabilistic reasoning starts. This prevents selecting portions of the model based on quantitative relevance.More recently, the ProbLog group has proposed two methods for anytime inference: <cit.> successively finds multiple explanations for a query, each of them narrowing bounds on the latter's probability. However, finding an explanation requires inference on the entire model, or at least on a qualitatively relevant portion that may include quantitatively irrelevant knowledge. <cit.> proposes a method based on forward reasoning with iterative deepening. Because the reasoning goes forward, there is no clear way to limit the inference to the portion most relevant to a query (which is a form of backward inference), and no selection for more likely proofs.Our calculation of bounds is equivalent to the one presented in<cit.>, but that work does not attempt to focus on relevant portions of a model and does not exploit the graphical model's factorization as much as our method and Variable Elimination do. Box propagation<cit.>propagates boxes, which are looser bounds than ours and Leisink & Kappen's bounds. In fact, their method can easily use these tighter bounds, but in any case it does not deal with cycles, stopping the unrolling of the model (the Bethe tree) once a cycle is found. Ihler <cit.> presents a similar method that does not stop when a cycle is found, but is not guaranteed to converge to the exact marginal.Liu et al <cit.> present a method very similar to ours based on growing an AND-OR search tree from the query and bounding the not-yet-processed remaining of the model. As more of the tree is expanded, the better the bounds become. The main difference from our method is that the AND-OR search tree has a child per value of each random variable, making it arguably less intelligible and harder to use as a proof tree and to generalize to richer representations such as Statistical Relational Models.§ BACKGROUND§.§ Graphical Models Graphical models are a standard framework for reasoning with uncertainty. The most common types are Bayesian networks and Markov networks. In both cases, a joint probability distribution for each assignment tuple to N random variables is defined as a normalized product of non-negative real functions {ϕ_i}_i∈ 1..K, where 1..K is short for {1,…,K}, each of them applied to a subtuple _i of :[For simplicity, we use the same symbols for both random variables and their values, but the meaning should be clear.]P() = 1/Z∏_i=1 ^K ϕ_i(_i),where Z is a normalization constant equal to ∑_∏_i ϕ_i(_i). Functions ϕ_i are called factors and map each assignment on their arguments to a potential, a non-negative real number that represents how likely the assignment _i is. This representation is called factorized due to its breaking the joint probability into this product. In Bayesian networks, K = N and factors are conditional probabilities P(X_i | Pa_i), for each random variable X_i in , where Pa_i are its parents in a directed acyclic graph.For succinctness, we often do not explicitly write the arguments to factors:P() = 1/Z∏_i ϕ_i(_i) = 1/Z∏_i ϕ_i. The marginal probability (MAR) problem consists of computingP() = ∑_∖ P(),whereis a subtuple of containing queried variables, and ∑_∖ is the summation over all variables in but not in . It can be shown that P()=1/Z_∑_∖∏_i ϕ_i for Z_ a normalization constant over . Therefore, because Z_ is easily computable if || is small, the problem can be simply reduced to computing a summation over products of factors, which the rest of the paper focuses on.We denote the variables (or neighbors) of a factor ϕ or set of factors M as Var(ϕ) and Var(M). The neighbors neighbors_M(V) of a variable V given a set of factors M is defined as the set of factors {ϕ∈ M : V ∈ Var(ϕ) }. We call a set of factors a model. The factor graph of a model M is the graph with variables and factors of M as nodes and with an edge between each factor and each of its variables. §.§ Belief Propagation Belief Propagation <cit.>is an algorithm that computes the marginal probability of a random variable given a graphical model whose factor graph has no cycles. Let M be a set of factors and P_M bethe probability distribution it defines. Then, for a set of variables Q ⊆ Var(M), we define: P_M(Q) ∝ m^M_. ← Q (note that m^M_ϕ← Q does not depend on ϕ).m^M_V ←ϕ = ∑_Sϕ∏_S^j ∈ S m^M^j_ϕ← S^j where {S^1,…,S^n} Var(ϕ) ∖ V, M^jis the set of factors in M ∖{ϕ} connected to S^j,m^M_ϕ← V = ∏_ϕ^j ∈ neighbors_M(V) m^M^j_V ←ϕ^j,where {ϕ^1,…,ϕ^n} neighbors_M(V), M^jis the set of factors in M connected to ϕ^j. Note that each message depends on a number of sub-messages. Since the factor graph is a tree (it has no cycles), each sub-message involves a disjoint set of factors M^j. This is crucial for the correctness of BP because it allows the computation to be separately performed for each branch.If a graphical model has cycles, an iterative version of BP, loopy BP, can still be applied to it <cit.>. In this case, since a message will eventually depend on itself, we use its value from a previous iteration, with random or uniform messages in the initial iteration. By iterating until a convergence criterion is reached, loopy BP provides distributions, called beliefs, that in practice often approximate the marginal of the query well. However, loopy BP is not guaranteed to provide a good approximation, or even to converge.§.§ Anytime Belief Propagation Even though BP is based on local computations between neighboring nodes, it only provides any information on the query's answer once it has analyzed the entire model, even if some parts of the model have a relatively small influence on the answer. This goes against our initial goalproviding information on the query's answer even after analyzing only a (hopefully more relevant) portion of the model. Anytime BP <cit.> is an algorithm based on (loopy) BP that computes iteratively improved bounds on a message. A bound (following definitions in <cit.>) on a message m is any set of messages to which m is known to belong. Anytime BP (and, later, Anytime Exact BP) only use bounds that are convex sets of messages, and that can therefore be represented by a finite number of messages (the bounds extrema), the convex hull of which is the entire bound. Initially, the bound on a message on a variable V is the simplex (V), the set of all possible probability distributions on V, and whose extremes are the distributions that place the entire probability on a single value. For example, if V is a boolean random variable, its simplex is the set {V = true10, V = false10}.It turns out that the computation of a message m given its sub-messages is a convex function. Therefore, given the bounds on sub-messages represented by their extremes, we can compute a bound b(m) on m by computing the extremes of this bound, each extreme being equal to the message computed from a combination of extremes to the sub-messages. This provides a finite set of extremum messages that define b(m) and can be used to compute further bounds.Figure <ref> shows an example of Anytime Belief propagation on a factor network. The algorithm provides increasingly improving bounds on the belief m(A) on query A, by first returning the simplex (A) as a bound, then returning the bound computed from simplex sub-messages, and then successively refining this bound by selecting one or more of the sub-messages, obtaining tighter bounds on these sub-messages, and recomputing a tighter bound for m(A). At every step from (b) to (d), factors are included in the set so as to complete some random variable's blanket (we do not show the expansions from (d) to (e), however, only their consequences). We include the table for factorϕ_1 but omit the others. For simplicity, the figure uses binary variables only and shows bounds as the interval of possible probabilities for value 1, but it applies to multi-valued variables as well.This incrementally processes the model from the query, eventually processing it all and producing an exact bound on the final result. Like BP, Anytime BP is exact only for tree graphical models, and approximate for graphical models with cycles (in this case, the bounds are exact for the belief, that is, they bound the approximation to the marginal). The main contribution of this paper is Anytime Exact BP, which is a bounded versions of BP that is exact for any graphical models, presented in Section <ref>. §.§ Cycle Cutset Conditioning If a graphical model has cycles, an iterative version of BP, loopy BP, can still be applied to it <cit.>. However, loopy BP is not guaranteed to provide a good approximation, or even to converge.Cycle cutset conditioning <cit.> is a way of using BP to solve a graphical model M with cycles. The method uses the concept of absorption: if a factor ϕ_i(_i) has some of its variables ⊆_i set to an assignment , it can be replaced by a new factor ϕ', defined on ∖ and ϕ'(_i ∖) = ϕ_i(_i ∖, ). Then, cutset conditioning consists of selecting C, a cycle cutset random variables in M such that, when fixed to a value c, gives rise through absorption in all factors involving variables in C to a new graphical model M_c without cycles and defined on the other variables ∖ C such that P_M_c(∖ C) = P_M(∖ C, c). The marginal P_M(Q) can then be computed by going over all assignments to C and solving the corresponding M_c with BP:P_M(Q) = ∑_∖ Q P_M()=∑_c ∑_∖ (Q ∪ C) P_M(∖ C, c)=∑_c ∑_∖ (Q ∪ C) P_M_c(∖ C) =∑_c P_M_c(Q)(using BP on M_c). Figure <ref> (a) shows a model with a cycle. Panel (b) shows how cutset conditioning for cutset {A} can be used to compute P(Q): we successively fix A to each value a in its domain, and use absorption to create two new factors ϕ_1' = ϕ_1(C,a) and ϕ'_2(E) = ϕ_2(E,a). This new model does not contain any cycles and BP computes P(Q,a). The overall P(Q) is then computed as ∑_a P(Q,a). Now, consider that the messages computed across the reduced model that depend on a can be thought of as functions of a. From that angle, the multiple applications of BP for each a can be thought of as a single application of BP in which A is a fixed, free variable that is not eliminated and becomes a parameter in the propagated messages (panel (c)). This has the advantage of computing all messages that do not depend on a only once.While cutset conditioning solves graphical models with cycles exactly, it has some disadvantages. Like standard BP, cutset conditioning requires the entire model to be processed before providing useful information. In fact, simply finding a cutset already requires going over the entire model, before inference proper starts. Besides, its cost grows exponentially in the size of the cutset, which may be larger than the induced tree width. Our main proposal in this paper, Anytime Exact Belief Propagation, counters those disadvantages by processing the model in an incremental way, providing a hard bound around the exact solution, determining the cutset during inference, and summing out cutset variables as soon as possible as opposed to summing them out only at the end.§ ANYTIME EXACT BELIEF PROPAGATIONWe are now ready to present the main contribution of this paper, Anytime Exact Belief Propagation (AEBP). Like cutset conditioning, the algorithm applies to any graphical models, including those with cycles. Unlike cutset conditioning, it does not require a cutset to be determined in advance, and instead determines it on the fly, through local message-passing. It also performs a gradual discovery of the model, providing bounds on the final result as it goes. This is similar to Anytime BP, but Anytime Exact BP, as the name implies, provides bounds on the exact query marginal probability and eventually converges to it.We first provide the intuition for Anytime Exact BP through an example. Consider the graphical model in Figure <ref> (the full model is shown in (e)). If we simply apply Anytime BP to compute P(Q), messages will be computed in an infinite loop. This occurs because Anytime BP has no way of identifying loops. AEBP, on the other hand, takes an extra measure in this regard: when it requests a new bound from one of the branches leading to a node, it also provides the set of factors known so far to belong to the other branches. Any variable that is in the branch and is connected to these external factors must necessarily be a cutset variable. Upon finding a cutset variable C, AEBP considers it fixed and does not sum it out, and resulting messages are functions of C (as well we the regular variable for which we have a message). Branches sharing a cutset variable C will therefore return bounds that are functions of C. Cutset variables are summed out only after messages from all branches containing C are collected.While the above procedure is correct, delaying the sum over cutset variables until the very end is exponentially expensive in the number of them. Figure <ref> shows an example in which a cutset variable (G) that occurs only in an inner cycle is summed out when that cycle is processed (at node E), while the more global cutset variables (C and F) are summed out at the end, when the more global cycle is processed (at node A). Algorithm 1 presents the general formulation. It works by keeping track of each component, that is, a branch of the factor graph rooted in either a variable or factor, its Node, and computing a message from Node to some other requesting node immediately outside the component. Each component is initially set with the external factors that have already been selected by other components. The message is on a variable V (this is Node itself if Node is a variable, and some argument of Node if it is a factor). In the first update, the component sets the bound to the simplex on V and creates its children components: if Node is a variable, the children components are based on the factors on it that are not already external; if it is a factor, they are based on its argument variables.From the second update on, the component selects a child, updates the child's external factors by including that child's siblings external factors, updates the child's bound, updates its own set of factors by including the child's newly discovered factors, and computes a new bound. If Node is a factor, this is just the product of the bounds of its children. If Node is a variable, this is obtaining by multiplying ϕ and children bounds, and summing out the set of variablesS. S is the set of variables that occur only inside , (which excludes V and cutset variables connected to external factors, but does include cutset variables that occur only inside this component. This allows cutset variables to be eliminated as soon as possible in the process. To compute the marginal probability for a query Q, all that is needed is to create a component for Q without external factors and update it successively until it converges to an exact probability distribution.During the entire process, even before convergence, this component tree can be used as a trace of the inference process, indicating how each message has been computed from sub-messages so far, similarly to a probabilistic proof or refutation tree in logic reasoning. Update(𝒞) is a component, defined as a tuple (V,Node, Bound, M, ExteriorFactors, Children) where: V: the variable for which a message is being computed Node: a variable or factor from which the message on variable V is being computed Bound: a bound on the computed message Factors: the set of factors selected for this message already ExternalFactors: set of factors already observed outside the component, and used to identify new cutset variables. Children: components for the sub-messages of this message. first update Bound (V) Node is variable Factors factors with Node as argument and not in ExternalFactors Children components based on each factor in Factors and ExternalFactors set to .ExternalFactors // Node is factor ϕ Bound ∑_C ∪ Sϕ∏_Ch ∈ Children Children components based on each variableargument (that is, neighbor) of ϕand ExternalFactors set to .ExternalFactors ∖{ϕ} Child ← chooseNonConvergedChild(Children) Child.ExternalFactors ExternalFactors ∪ ⋃_Ch ∈ Children ∖{Child} Ch.Factors Update(Child) FactorsFactors ∪ Child.Factors ChildrenBoundProduct ∏_Ch ∈ Children Ch.Bound Node is variable BoundChildrenBoundsProduct // Node is factor ϕ S variables in ChildrenBoundProduct not in any factor in ExternalFactors Bound ∑_Sϕ∏_Ch ∈ Children Ch.Bound Algorithm 1: Anytime Exact Belief Propagation.neuron/.style=shape=circle, minimum size=.9cm, inner sep=0, draw, font=, io/.style=neuron, fill=gray!20 § CONCLUSION We presented our preliminary work on Anytime Exact Belief Propagation, an anytime, exact inference method for graphical models that provides hard bounds based on a neighborhood of a query. The algorithm aims at generalizing the advantages of logic reasoning to probabilistic models, even for dependencies that are not certain, but near certain.Future work includes finishing the implementation, evaluating it on benchmarks, and generalizing it higher-level logic representations such as relational models and probabilistic programs. To achieve that, we will employ techniques from the lifted inference literature <cit.> as well as probabilistic inference modulo theories <cit.>.aaai§ BELIEF PROPAGATION WITH SEPARATOR CONDITIONING: AN EXACT MESSAGE-PASSING INFERENCE ALGORITHM FOR ANY GRAPHICAL MODEL Belief Propagation is equivalent to the inference calculationwhen the graphical model does not present any cycles. The same can not be stated for general graphical models, where loopy belief propagation algorithm may not converge. If it does,not necessarily it converges to an optimal solution.We based our anytime algorithm, therefore, upon an adaptation of the belief propagation, where we “break” the cycles by summingout some variables in a later step of the process. This provides us with an exact elimination algorithm. §.§ DefinitionsWe provide some definitionsthatare useful in defining the algorithm. §.§.§ PartitionLet M be a set of factors. A partition Pt_M of M id a tuple of nonempty sets of factorsPt_M = (Pt^1,…,Pt^k)such that each element of M is exactly in one of the subsets Pt^i. That is to say that :* for i = 1,…,kPt^i ≠∅* for i,j = 1,…,ki≠ j, Pt^i∩ Pt^j = ∅* ⋃_i=1,…,kPt^i=M§.§.§ Separator A separator of set of a partition Pt_M is defined as following:Sep(Pt_M) ={V:∃ i, j ,i≠ j,s.t.V∈ Var(Pt^i)∩ Var(Pt^j) } §.§ BP with conditioning Let M be a set of factors. Then, for a set of variables Q ⊆ Var(M), we can employ Belief Propagation with Separator Conditioning (S-BP) to compute P(Q): P_M(Q) ∝ m^M_. ← Q (note that m^M_ϕ← Q does not depend on ϕ). m^M_V ←ϕ = ∑_C ∑_S∖ Cϕ∏_S^j ∈ S m^Pt^j_ϕ← S^j ∪ C where {S_1,…,S_n} = Var(ϕ) ∖ V,(Pt^1,…,Pt^n) is any n-partition of M ∖{ϕ},C is the Sep(Pt∖{ϕ}), m^M_ϕ← V = ∑_C ∏_ϕ^j ∈ neighbors_M(V) m^Pt^j_V ∪ C ←ϕ^j,whereneighbors_M(V)are the factors on V in M, (Pt^1,…,Pt^n) is any n-partition of M such that ϕ^j ∈ Pt^j,C is Sep(Pt)∖ V. We prove that m^M_ϕ← V∝ P_M(V) and m^M_V ←ϕ∝ P_M(V) by induction on the size of M. If |M|=0, m^M_ϕ← V = 1, which when normalized is equal to Uniform(V) = P_M(V). m^M_V ←ϕ is undefined for |M|=0, but for |M|=1 (that is, M={ϕ}), C=∅ and m^M_V ←ϕ = ∑_S ϕ(V,S) ∝ P_M(V). If |M|>0, m^M_ϕ← V= ∑_C ∏_ϕ^j ∈ neighbors_M(V) m^Pt^j_V ∪ C ←ϕ^j∝∑_C ∏_ϕ^j ∈ neighbors_M(V) P_Pt^j(V ∪ C) = ∑_C ∏_ϕ^j ∈ neighbors_M(V)∑_Var(Pt^j)∖(V ∪ C)Pt^j = ∑_C ∑_Var(M)∖(V ∪ C)∏_ϕ^j ∈ neighbors_M(V) Pt^j = ∑_Var(M)∖ V∏_ϕ^j ∈ neighbors_M(V) Pt^j = ∑_Var(M)∖ V M∝ P_M(V) m^M_V←ϕ= ∑_C ∑_S ∖ Cϕ∏_S^j ∈ S m^Pt^j_ϕ← S^j ∪ C∝∑_C ∑_S ∖ Cϕ∏_S^j ∈ S P_Pt^j(S^j ∪ C) = ∑_C ∑_S ∖ Cϕ P_M ∖ϕ(S ∪ C)∝∑_C ∑_S ∖ Cϕ∏_ϕ' ∈ M ∖ϕϕ'∝∑_C ∑_S ∖ C P_M(S ∪ C) = ∑_S ∪ C P_M(S ∪ C) = ∑_Var(M) ∖ V P_M(Var(M) ∖ V) = P_M(V). S-BP reduces to regular BP if the model does not contain loops. Otherwise, it uses separators to ensure incoming messages are computed on components (sets of factors) that are disjoint given a separator set. The separator set for a message is marginalized out at the end of its computation. Unlike BP, S-BP is not a deterministic algorithm, since there may be several choices for Pt. It can be made deterministic by representing the set of factors M as a tree of all partitions to be used in all message computations. Later, we will use partial specifications of M in this manner to represent bounds on M. § BOUNDING FUNCTIONS The following definition introduces the concept of iteratively bounding the value f(a) of a function f for a unknown domain element a. A bound on some mathematical object α is simply some set containing α. A bounding function does two things given a bound on a: it computes a bound on f(a), and computes a strictly tighter bound (a smaller set) on a in the process. This yields a bounding sequence generated by an iterative process in which partial or complete ignorance on a and therefore f(a) leads to tighter and tighter bounds on a and f(a) until we converge to the exact values. When we apply this framework toour particular application of exact BP, a will be the model M and the cutset Q, and f will be the messages on them. The bounds on (M,Q) will be based keep partial descriptions of M and Q and convergence is obtained when they are fully analyzed. We will have a bound on the messages during the entire process, making the algorithm anytime. However, we define the notions of bounding functions and sequences in an abstract way that applies to any functions and sets, and those in probabilistic inference are just one case. The intuition behind a bounding function is that, instead of receiving an exact argument a, it receives a set of possible arguments A' ⊆ A containing a (a bound on a), and returns the possible results B = f(A') (and thus guaranteed to contain f(a)), and also a strictly smaller A”⊂ A' determined during the processing. A” can then be fed again to f̅ to obtain an even better bound on f(a). Let f : A → B be any function, and a ∈ A. A function f̅ : 2^A → (2^A, 2^B) is a bounding function of f with respect to a if, for any A' ⊆ A for which a ∈ A', f̅(A') = (A”, B) satisfies a ∈ A”, A”⊂ A', and B = { f(a”) : a”∈ A”}. Let A be a finite, discrete set, f : A → B be any function with domain A, and f̅ a bounding function of f. Then there is n ∈ℕ and a sequence defined by A_0 = A; B_0 = B(A_i , B_i) = f̅(A_i - 1), i = 1,…,n such that, for all i ≥ 1, B_i ⊂ B_i-1, f(a) ∈ B_i, and B_n = { f(a) }. The existence of the sequence and its properties, except for B_n = {f(a)}, comes directly from the definition of a bounding function. We prove B_n = {f(a)} by induction on |A_i|. If |A_i|=1, A_i={a} and B_i = {f(a)} by the definition of bounding function. If |A_i| > 1, then |A_i+1| < |A_i|, and by induction it defines a sequence from (A_i+1, B_i+1) to (A_n, B_n) such that B_n = { f(a) }. Therefore, A_i defines a sequence from (A_i, B_i), (A_i+1, B_i+1) to (A_n, B_n)such that B_n = { f(a) }. § BOUNDED ANYTIME EXACT BP We now define bounding functions for the message functions in BP. For a carrier set 𝕄, we inductively define partition tree on 𝕄 as either of the following objects: * a set 𝕄∖Φ^ext, where Φ^ext⊆𝕄 (that is to say, a set defined by exclusion). * a tuple (Pt_1,…,Pt_n) of partition trees on 𝕄. A complete partition tree is a partition tree in which all the leaves are equal to 𝕄∖𝕄 = ∅. When 𝕄 is a set of factors, we can annotate a partition tree with a set of external factors Φ^ext that do not occur in it, a set of external separator variables D^ext present in more than one of its partition sub-trees and also in the external factors Φ^ext, and a set of internal separator variables D present in more than one of its partition subtrees but not in the external factors Φ^ext. A partition tree Pt induces a set completion(Pt) all the full partition trees consistent with Pt. A set of full partition trees can be used as a bound on a single, unknown full partition tree. We will use such bounds, represented by a partial partition tree, as the bounds A_i on an unknown full partition tree passed to bounding functions for the message functions m_ϕ← V and m_V ←ϕ . The bounding function obtains a partial partition tree Pt_i representing an unknown full partition tree, chooses an j-th incoming message that has not converged yet, obtains a tighter bound Pt^j and a tighter bound B^j_i+1 on the respective incoming message, and generates a tighter Pt_i+1 and bound B_i+1 on message as a result. For fixed V and ϕ, the message calculations m^Pt,D,D^ext_V ←ϕ and m^Pt,D,D^ext_ϕ← V are functions on a complete Pt. The bounding functions can be defined as functions on partial Pt as follows: Given a set of factors M and a cutset Q, m̅_V and m̅_V ←ϕ are bounding functions with respect to P_M(Q). (Note that A_0 = 2^𝕄× compl(Q) and B_0 = ℙ(V).) m̅_V ←ϕ^Pt_i, D_i, D^ext_i = ((Pt_i+1,D_i+1,D^ext_i+1),B_i+1)where if Pt_i is of the form 𝕄∖Φ^ext_i,{S_V ←ϕ^1,…,S_V ←ϕ^n} is Var(ϕ) ∖ (V ∪ D^ext_i)Pt^j_i+1 is 𝕄∖ (Φ_i^0 ∪{ϕ}), j = 1,…,nPt_i+1 is (Pt^1_i+1,…,Pt^n_i+1)B_i+1 is 𝒫(V)else j =index of incoming message not yet exhausted ((Pt_i+1^j, D_i+1^j, D_i+1^ext,j), B_i+1^j) = m̅_ϕ← S_i^j ∪ D_i^Pt_i^j, D_i^j, D_i^ext ((Pt_i+1^k, D_i+1^k, D_i+1^ext,k), B_i+1^k) = ((Pt_i^k, D_i^k, D_i^ext,k), B_i^k), k ≠ jD^ext_i+1 = ( ⋃_j D^j_i+1) ∩ Var(Φ^ext_i) D_i+1 = ( ⋃_j D^j_i+1) ∖ D^ext_i+1 B_i+i = ∑_D_i+1∑_S_V ←ϕ∖ D_i+1ϕ∏_j=1^n B_i+1^jm̅_ϕ← V^Pt_i, D_i, D^ext_i = ((Pt_i+1,D_i+1,D^ext_i+1),B_i+1) where if Pt_i is of the form 𝕄∖Φ^ext_i,{ϕ^1_ϕ← V,…,ϕ_ϕ← V^n} is neighbors_Pt_i(V) ∖{ϕ}Pt^j_i+1 is 𝕄∖ (Φ_i^ext∪⋃_k≠ jϕ_ϕ← V^k), j = 1,…,nPt_i+1 is (Pt^1_i+1,…,Pt^n_i+1)B_i+1 is 𝒫(V)else j =index of incoming message not yet exhausted ((Pt_i+1^j, D_i+1^j, D_i+1^ext,j), B_i+1^j) = m̅_V ←ϕ_ϕ← V^j^Pt_i^j, D_i^j, D_i^ext ((Pt_i+1^k, D_i+1^k, D_i+1^ext,k), B_i+1^k) = ((Pt_i^k, D_i^k, D_i^ext,k), B_i^k), k ≠ jD^ext_i+1 = ( ⋃_j D^j_i+1) ∩ Var(Φ^ext_i) D_i+1 = ( ⋃_j D^j_i+1) ∖ D^ext_i+1Pt_i+1 is (Pt_1,…,Pt_n) B_i+i = ∑_D_i+1∏_j=1^n B_i+1^j § COMPUTING WITH BOUNDS In the algorithm described on section <ref>, we make two operations that can be specially expensive: B_i+1 = {∑_N ϕ(V, N) ∏_N_j ∈ N b^j : b^j ∈ B_i^j }andB_i+i = {∏_j = 1..N b^j : b^j ∈ B_i^j }One can easily see that the computation of such expressions is not computationally possible in the general case: if we consider bounds with an infinite number of elements there would be an infinite number of computations to do. However, this can be a relatively inexpensive computation in certain conditions, such as in the case where the B's are convex sets with a finite number of extreme points. In this section we show that these conditions are fulfilled in our case and provide a way to preform this operations. §.§ The simplex bound We define a bound on a query Q as a set of non-normalized probabilities for that query. The simplest and most general bound we can assign to a query is the simplex relative to Q, noted as 𝒫_Q. The simplex is defined as: 𝒫_Q ≡{ϕ(Q) : Q →ℝ,∑_q∈ Qϕ(q) = 1} One can easily see that a simplex is a convex set. Its extreme points are: ext(_Q) = {ϕ : ∃ q∈ Qs.t. ϕ(q) = 1and ∀ p ≠ qϕ(p) = 0 }Which is a finite set. §.§ Operations with sets (definitions) This means that the probability distribution of Q can be any function whose sum over the values of Q is equal to one. That makes 𝒫_Q a trivial bound for _M(Q).We define the product of bounds as the set of products of each term from both sets. The product between a bound and a term is defined the same way. B_1 × B_2 ≡{ϕ_1×ϕ_2 : ϕ_1 ∈ B_1, ϕ_2 ∈ B_2}B ×ϕ≡{ϕ×ϕ' : ϕ' ∈ B}We define the normalization operatorin the following manner. Given a convex set functions B: (B) = {ϕ/∑_domain(ϕ)ϕ : ϕ∈ B }§.§ Proving convexity Consider a function ϕ : D_ϕ→ℝ, with D_ϕ finite, and B a convex set of functions f:D_B→ℝ with finite extreme points. Let A ⊂ D_ϕ.. We prove that: * (∑_A ϕ B) is convex * ext((∑_A ϕ B)) ⊂∑_A ϕ ext(B) We split the proof in two parts:§.§.§ 1we want to prove that: * ∑_A ϕ B is convex :Consider ψ_1,...,ψ_k ∈∑_A ϕ B. Then ψ_i = ∑_A ϕψ'_i, i = 1,...,k, ψ'_i ∈ B. consider α_j>0, j= 1,..,k such that ∑_jα_j=1. Then: ∑_i= 1,...,kα_iψ_i=∑_i= 1,...,kα_i∑_Aϕψ_i=∑_Aϕ∑_i= 1,...,kα_iψ_i==∑_Aϕ(ψ”) , with ψ”∈ B This implies that ∑_i= 1,...,kα_iψ_i ∈∑_A ϕ B, i.e, that ∑_A ϕ B is convex. * ext(∑_A ϕ B) ⊂∑_A ϕ ext(B)The proof follows by contradiction.Consider ψ∈ ext(∑_Aϕ B) and suppose that ψ∉∑_Aϕ ext(B). Then ψ = ∑_Aϕψ', with ψ' = ∑_iα_iψ'_i, ψ'_i ∈ B and α_i>0 summing to 1. Then: ψ = ∑_Aϕ∑_iα_iψ'_i = ∑_iα_i(∑_Aϕψ'_i) = ∑_iα_iψ_i with ψ_i ∈∑_Aϕ B. Then ψ is a convex combination of elements in ∑_Aϕ B, which s a contradiction on the assumption that it is an extreme point.§.§.§ 2we want to prove that: * (B) is convex We note ∑_domain(ϕ)ϕ = |ϕ| for any functionϕ.Let ψ = ∑_i α_iψ_i be a convex combination of ψ_i∈(B). One can easily see that ψ∈(B)|ψ| = 1and ψ∝ψ'for some ψ' ∈ B We prove that |ψ| = 1 through the following equation: |ψ| = |∑_iα_iψ_i| = ∑_iα_i|ψ_i| = ∑_iα_i× 1 = 1 Now we prove the second part of the equivalence stament:We have ∀ i ,ψ_i = ψ_i'/|ψ_i'| for some ψ'_i ∈ B. Then: ψ = ∑_i α_iψ_i = ∑_iα_i/|ψ_i'|ψ'_i = ∑_iβ_iψ'_i Where β_i = α_i/|ψ_i'|. Calling N = ∑_iβ_i, we have that ∑_i β_i/Nψ_i ∈ B because it is a convex combination of elements in S. Therefore: ψ = N × (∑_i β_i/Nψ_i)and ∑_i β_i/Nψ_i ∈ B Then: ψ∝ψ' for some ψ'∈ B Which proves that (B) is convex. * ext((B)) ⊂ ext(B)Consider ψ∈(B). Then, ∃ψ' ∈ Bs.t.ψ = ψ'/|ψ'|. According to Krein-Milman theorem, ψ' = ∑_i=1,...,kα_iψ'_i , α_i>0, ∑_iα_i = 1 and ψ_i'∈ext(B). Then: ψ = ∑_i (|ψ'_i|α_i)(ψ'_i/|ψ'_i|)/|∑_iα_iψ'_i| = ∑_ic_iψ'_i/|ψ'_i|,withc_i =|ψ'_i|α_i /|∑_iα_iψ'_i| We can easily see that c_i > 0 and ∑_ic_i = 1 and that ψ'_i/|ψ'_i|∈(ext(B)). Also, (ext(B)) ⊂(B), since ∀ψ' ∈ ext(B),ψ'/|ψ'|∈(B). Then we have:(B) = C.Hull((ext(B)))Then, because the extreme points of a convex se S are those that are not convex combinations of any others (except 1 × themselves), we have:ext((B)) ⊂ ext(B)§.§ Storing and computing the following bounds The operations with bounds can be resumed by applying the expression (∑_Aϕ B) for a certain set of variables A, a factor ϕ and a previous bound B. The last section showed that it suffices to store the extreme points of previous bounds and compute and store the new bound as a result of (∑_Aϕψ) for each ψ∈ ext(B). | http://arxiv.org/abs/1707.08704v1 | {
"authors": [
"Gabriel Azevedo Ferreira",
"Quentin Bertrand",
"Charles Maussion",
"Rodrigo de Salvo Braz"
],
"categories": [
"cs.AI"
],
"primary_category": "cs.AI",
"published": "20170727043134",
"title": "Anytime Exact Belief Propagation"
} |
(Technical Report)^∗Department of Computing, The Hong Kong Polytechnic University ^† School of Software, Tsinghua University young, lin, hui, [email protected] Deployment ofbillions of Commercial off-the-shelf (COTS) RFID tags has drawn much of the attention of the research community because of the performance gaps of current systems. In particular, hash-enabled protocol (HEP)is one of the most thoroughly studied topics in the past decade.HEPs are designed for a wide spectrum of notable applications (missing detection) without need to collect all tags. HEPs assume that each tag contains a hash function, such that a tag can select a random but predicable time slot to reply with a one-bit presence signal that shows its existence. However, the hash function has never been implemented in COTS tags in reality, which makes HEPs a 10-year untouchable mirage.This work designs and implements a group ofanalog on-tag hash primitives (called Tash) for COTS Gen2-compatible RFID systems,which moves prior HEPs forward from theory to practice.In particular, we design three types of hash primitives, namely, tash function, tash table function and tash operator. All of these hash primitives are implemented through selective reading, which is a fundamental and mandatory functionality specified in Gen2 protocol, without any hardware modification and fabrication. We further apply our hash primitives in two typical HEP applications (cardinality estimation and missing detection) to show the feasibility and effectiveness of Tash. Results from our prototype, which iscomposed of one ImpinJ reader and 3,000 Alien tags, demonstrate that the new design lowers 60% of the communication overhead in the air. The tash operator can additionally introduce an overhead drop of 29.7%.<ccs2012> <concept> <concept_id>10003033.10003106.10003112</concept_id> <concept_desc>Networks Cyber-physical networks</concept_desc> <concept_significance>500</concept_significance> </concept> <concept> <concept_id>10010520.10010553</concept_id> <concept_desc>Computer systems organization Embedded and cyber-physical systems</concept_desc> <concept_significance>500</concept_significance> </concept> </ccs2012> [500]Networks Cyber-physical networks [500]Computer systems organization Embedded and cyber-physical systemsAnalog On-Tag Hashing: Towards Selective Reading as Hash Primitives in Gen2 RFID Systems Lei Yang^∗,Qiongzheng Lin^∗, Chunhui Duan^∗^†,Zhenlin An^∗December 30, 2023 ========================================================================================§ INTRODUCTIONRFID systems are increasingly used in everyday scenarios, which range from object tracking, indoor localization <cit.>, vibration sensing <cit.>, to medical-patient management, because of the extremely low cost of commercial RFID tags (as low as 5 cents per tag). Recent reports show that many industries like healthcare and retailing are moving towards deploying RFID systems for object tracking, asset monitoring, and emerging Internet of Things <cit.>. §.§ The State-of-the-ArtThe Electronic Product Code global is an organization establishedtoaccomplish the worldwide adoption and standardization of EPC technology. It published the Gen2 air protocol <cit.> for RFID system in 2004. A Gen2 RFID system consists of a reader and many passive tags. The passive tags without batteries arepowered up purely by harvesting radio signals from readers.This protocol has become the mainstream specification globally, and has been adopted as a major part of the ISO/IEC 18000-6 standard. Embedding Gen2 tags into everyday objects to construct ubiquitous networks has been a long-standing vision. However, a major problem that challenges this vision is that the Gen2 RFID system is not efficient <cit.>. First, the RFID system utilizessimple modulations (e.g., ON-OFF keying or BPSK) due to the lack of traditional transceiver <cit.>, which prevents tags from leveraging a suitable channel to transmit more bits per symbol and increase the bandwidth efficiency.Second, tags cannot hear the transmissions of other tags. They merely reply on the reader to schedule their medium access with the Framed Slotted ALOHA protocol, which results in manyempty and collided slots. This condition also retards the inventory process. These two limitations force a reader to go through along inventory phase when it collects all the tags in the scene. §.§ Ten-Year Mirage of HEP Motivated by the aforementioned performance gaps, the research community opened a new focus on HEP design approximately 10 years ago.The key idea that underlies HEPs is that each tag selects a time slot according to the hash value of itsand a random seed. It then replies a one-bit presence signal rather than the entirenumber in the selected slot. HEPs treat all tags as if they were a virtual sender, which outputs a gimped hash table(a presence bitmap) when responding to a challenge (a random seed). Most importantly, HEPs assume the backend server and every tag share a hash function, and the resulting bitmap is random but predicable when the s and seeds are known.Fig. <ref> shows a toy example with n=8 tags, each of which contains a uniquenumber presented in binary format (101010_2), to illustrate the HEP concept. The reader divides the time into d time slots (d=8.) and challenges these tags with the random seed r. Each tag selects the (h_d(,r))^th time slot to reply the one-bit signal, where h(·) is a common hash function (MD5, SHA-1) and h_d(·) = h(·) d. The reader can recognize two possible results for each time slot, namely, empty and non-empty[Some work assume the reader can recognize the signal collision, obtaining three results: empty, single and collision. ]. The reader abstracts the reply results into a bitmap (B=[0,1,1,0,1,1,1,0]), where each elementcontains two possible values, that is, 0 and 1, that corresponds to empty and non-empty slots, respectively.The upper layer then utilizes this returned bitmap to explore many notableapplications. We show the following two typical applications as examples to drive the key point:∙ Cardinality estimation. Estimating the size of a given tag population is required in many applications, such as privacy sensitive systems and warehouse monitoring. Kodialam <cit.> presented a pioneer estimator. Given that tags select the time slots uniformly because of hashing, the expected number of `0's equals n_0=d (1-1/d)^n ≈ de^-n/d. Counting n_0 in an instance yields a “zero estimator”, n≈ -dln(n_0/d). For example, n=-8×ln(3/8)=7.8 in our toy example.∙ Missing detection. Consider a major warehouse that stores thousands of apparel, shoes, pallets, and cases. How can a staff immediately determine if anything is missing?Sheng and Li <cit.> conducted the early study on the fast detection of missing-tag events by using the presence bitmap. They assumed all s were known in a closed system. Given that hash results are predicable, the system can generate an intact bitmap at the backend. We can identify the missing tags in a probabilistic approach by comparing the intact and instanced bitmaps.For example, if the second entry equals 0 (which is supposed to be 1), the the tag 101010_2 must be missing in our toy example. HEPs are advantageous in terms of speed and privacy. HEPs are faster than all prior per-tag reading schemes for two reasons. First, collecting all the EPCs of the tags is time consumingbecause of the aforementioned low-rate modulation, whereas one-bit presence signals of HEPs save approximately96× of the time (thelength equals 96 bits in theory [Actual case in practice would be less than this estimate due to other extra jobs, such as setup time, query time, ]). Second, collisionsare considered as one of the major reasons that drag down the reading. On the contrary, HEPs tolerate and consider collisions as informative.When privacy issues are considered, the tag's identification may be unacceptable in certain instances.HEPs allow tags to send out non-identifiable information (one-bit signals).HEPs are very promising. However, after 10 years of enthusiastic discussion about the opportunities that HEPs provide, the reality is beginning to settle: the functionality of hashing (hash function and hash table function) has never been implementedin any Gen2 RFID tags and considered by any RFID standard. No hint shows that this function will be widely accepted in the near future. §.§ Why Not Support Hashing? A large number of recent work have attempted to supplement hash functionality to RFID tags, which can be categorized into three groups. First group, like <cit.>, modifies the common hash functions to accommodate resource-constrained RFID tags. The second group <cit.> designs new lightweight and efficient hash functions dedicatedly for RFID tags. The third group seeks new design of RFID tags like WISP<cit.> and Moo <cit.>, which gives tags more powerful computing capabilities (hashing <cit.>). Unfortunately, as far as we know, none of these work has been really applied in COTS RFID systems yet. Why is the hash function unfavored? A term called as Gate Equivalent (GE) is widely used to evaluate a hardware design with respect to its efficiency and availability. One GE is esquivalient to the area which is required by the two-input NAND gate with the lowest deriving strength of the corresponding technology.A glance at Table. <ref> shows the available designs of hash functions for RFID tags require a significant number of GEs, which are completely unaffordable by current COTS tags. For example, the most compact hash functions requires thousands of GEs(1,075 GEs for PRESENT-80), whichincur extremely high energy consumption and manufacture cost. Thus, relatively few RFID-oriented protocols that appeal to a hash function can be utilized. RFID was expected to be one of the most competitive automatic identification technologiesdue to its many attractive advantages (simultaneous reading, NLOS, ) compared with others (barcode). However, this progress has been hindered for many years by the final obstacle that the industry is attempting to overcome (the price). The industry is extremely sensitive to the cost being doubled or tripledby the hash, although HEPs actually introduce significantoutperformance. §.§ OurContributions This work designs a group ofhash primitives, Tash,which takes advantage of existing fundamental function of selective reading specified in Gen2 protocol, without any hardware modification and fabrication. Our design and implementation both strictly follow the Gen2 specification, so it can work in any Gen2-Compatible RFID system. These mimic (or analog) hash primitives act as we embedded real hashcircuits on tags[This work does not target at designing any analog circuit on readers or tags, but offers a mimic hash function acting as weembed a hash circuit on each tag.], while we actually implement them in application layer. Specifically, we design the following three kinds of hash primitives to revive prior HEPs:∙ We design a hash function (aka tash function)over existing COTS Gen2 tags. The hash function outputs a hash value associated with theof the tag and a random seed, as HEPs require.∙ We design a hash table function (aka tash table function) over all tags in the scene. It can produce ahash table (aka tash table), which is more informative than a bitmap, over the all tags in the scene. In particular, each entry indicates the exact number of tags hashed into this entry.∙ Major prior HEPs require multiple acquisitions of bitmaps to meet an acceptable confidence. We design three tash operators (tash AND, OR and XOR) to perform entry-wise set operations over multiple tash tables on tag in the physical layer, which offers a one-stop acquisition solution.Summary. Ithas been considered that HEPs are hardly applied in practice because of the `impossible mission'of implementing hash function on COTS Gen2 tags <cit.>. In this work, our main contribution lies in the practicality and usability, that is, enabling billions of deployed tags to benefit performance boost from prior well-studied HEPs, with our hash primitives.To the best of our knowledge,this is the first work to implement the hash functionality over COTS Gen2 tags.Second, we provide an implementation of and show its feasibilityand efficiency in two typical usage scenarios.Third, we investigate several leading RFID products inmarketincluding 18 types of tags and 10 types of readers, in terms of their compatibility with Gen2, and conduct an extensive evaluation on our prototype with COTS devices. § RELATED WORKWe review the related work from two aspects: the designs of hash functions and hash enabled protocols.Design of hash function. Feldhofer and Rechberger <cit.> firstly point that current common hash functions (, , ), are not hardware friendly and unsuitable at all for RFID tags, which have very constrained computing ability. Such difficulty has spurred considerable research <cit.>. We sketch the primary designs and their features in Table. <ref>. For example, Bogdanov <cit.> propose a hardware-optimized block cipher, PRESENT, designed with area and power constraints.The follow-up work <cit.> presents three different architectures of PRESENT and highlights their availability for both active and passive smart devices. Their implementations reduce the number of GEs to 1,000 around. Another follow-up work <cit.> extends the design of PRESENT and gives 8 variants to fulfill different requirements, DM-PRESENT-80, DM-PRESENT-128, H-PRESENT-128, The work <cit.> suggests to choose DES as hash function for RFID tags due to relatively low complexity, and presents a variant of DES, called asDESXL. Lim and Korkishko <cit.> present a 64-bit hash function with three key size options (64 bits, 96 bits and 128 bits),which requiresabout 3,500 and 4,100 GEs.In summary, despite these optimized designs, majority are still presented in theory andnone of them are available for the COTS RFID tags. On contrary, our work explores hash function from another different aspect, that is, leveraging selective reading to mimic equivalent hash primitives. `-' means the algorithm is presented in theory and does not havespecific power consumption.Design of hash enabled protocol. To drive our key point, we conduct a brief survey of previous related works. We list several key usage scenarios that we would like to support. Our objective is not to complete the list, but to motivate our design. (1) Cardinality estimation. Dozens of estimators <cit.> have been proposed in the past decade. For example, Qian <cit.> proposed an estimation scheme called lottery frame. Shahzad and Liu <cit.> estimated the number based on the average run-length of ones in a bit string received using the FSA. In particular, they claimed that their protocol is compatible with Gen2 systems. However, their scheme still requires modifying the communication protocol, and thus, it fails to work with COTS Gen2 systems. By contrast, our prototype can operate in COTS Gen2 systems as demonstrated in this study. (2) Missing detection. The missing detection problemwas firstly mentioned in <cit.>. Thereafter,many follow-up works <cit.> have started to study the issue of false positives resulting from the collided slotsby using multiple bitmaps. Additionaldetails regardingthis application are introduced in <ref>. (3) Continuous reading. The traditional inventory approach starts from the beginning each time it interrogates all the tags, thereby making it highly time-inefficient. These works <cit.> have proposed continuous reading protocols that can incrementally collect tags in each step using the bitmap. For example, Sheng <cit.> aimed to preserve the tags collected in the previous round and collect only unknown tags. Xie <cit.> conducted an experimental study on mobile reader scanning. Liu <cit.> initially estimated the number of overlapping tags in two adjacent inventories and then performed an effective incremental inventory. (4) Data mining. These works <cit.> discuss how to discover potential information online through bitmaps. For example,Sheng <cit.> proposed to identify the popular RFID categories using the group testing technique.Xie found histograms over tags through a small number of bitmaps<cit.>. Luo <cit.> determined whether the number of objects in each group was above or below a threshold. Liu <cit.> proposed a new online classification protocol for a large number of groups. (5) Tag searching. These works<cit.> have studied the tag searching problem that aims to find wanted tags from a large number of tags using bitmaps in amultiple-reader environment. Zheng <cit.> utilized bitmaps to aggregate a large volume of RFID tag information and to search the tags quickly.Liu <cit.>first used the testing slot technique to obtain the local search result by iteratively eliminating wanted tags that were absent from the interrogation region. (6) Tag polling. <cit.> consider how to quickly obtain the sensing information from sensor-augmented tags. The system requires to assign a time slot to each tag using the presence bitmap. In summary, all the aforementioned HEP designs have allowed RFID research to develop considerably in the past decade.All the work can be boosted by our hash primitives. § OVERVIEWTash is a software framework that provides a group of fundamental hash primitives for HEPs.This section presents its usage scope and formally defines our problem domain. §.§ Scope Despite clear and certain specifications, the implementation of the Gen2 protocol still varies with readers and manufacturers because of firmwarebugs or compromises,especially in early released reader devices, according to our compatibility report presented in<ref>.Here, we firmly claim that our design and implementation strictly follow the specifications of the Gen2 and LLRP protocols(refer to <ref>). The frameworkworks with any Gen2-compatible readers and tags. Theperformance losses caused bydefects in devices are outside the scope of our discussion. §.§ Definitions of Hash PrimitivesBefore delving intodetails, we formally define the hash primitives that the HEPs require, from a high-level.An l-bit tash function is actually a hash function f_l(t,r): 𝒯×ℛ→ 2^l, where 𝒯 and ℛ are the domains ofs of the tags and random seeds. Tash function and tash value. As the above definition specifies, an l-bit tash function takes ant and a random seed r as input and outputs an l-bit integer i, denotedby:i=f_l(t,r)We call l the dimension of tash function (l=0,1,2,…). The tash value i is an integer ∈ [0,2^l-1]. Similar to other common hash function, the tash function has three basic characteristics. First, the output changes significantly when the two parameters are altered.Second, its output is uniformly distributed within the given range, and predicable if all inputs are known. Third, the hash values are accessible. An l-bit tash table function can assign each tag t from a given set into the i^th entry of a hash table (aka tash table) with a random seed r, where i=f_l(t,r). Each entry of the tash table is the number of tags tashed into it. Tash table function and tash table. Let B and ℱ_l denote atash table and a tash table function respectively. The tash table function takes a set of tags (T={t_1,t_2, …, t_n}) and a random number r asinput andoutputs a tash table B, denote by:B=ℱ_l(T,r)where B[i]=|{t|f_l(t,r)=i}| (the number of tags tashed into the i^th entry) for ∀ t∈ T.Let L=2^l, which is defined as the size of the tash table. The tash table function is the core function that HEPs expect.HEPs consider the reader as well as all tags as a black box equipping with tash table function. When inputing a random seed, the box would output a tash table. HEPs then utilize such table to provide various services (missing detection or cardinality estimation.). It worths noting that superior to the bitmap employed in prior HEPs, our tash table is a perfect table that contains the exact number of tags tashed into each entry.Clearly, the table is completely backward compatible with prior HEPs because it can be forcedly converted into a presence bitmap.Tash operators. Most prior HEPs adopt probabilistic ways and their results are guaranteed with a given confidence level. To meet the level, they usually combine multiple bitmaps, which are acquired through multiple rounds and challenged by different seeds. We abstract such combination into three basic tash operators, namely, tash AND, OR and XOR. These operators can comprise other complex operations. Let B_1= ℱ_l(T, r_1) and B_2=ℱ_l(T, r_2) denote two tash tables acquired twice with two different seeds, r_1 and r_2.The tash AND(denoted by ⊕) of two tash tables is to obtain the intersection oftwo corresponding entry sets. Formally,B =B_1⊕ B_2, where B[i]=|{t|f_l (t,r_1)=i& f_l(t,r_2)=i}|.The tash AND is aimed at obtaining the common intersection of corresponding entries from two tash tables. For example,as shown in Fig. <ref>,B_1[1] and B_2[1] count{t_1,t_2} and {t_2} respectively. However, (B_1⊕ B_2)[1]=|{t_2}|=1, which counts t_2 only.The tash OR (denoted by ||) of two tash tables is to merge two corresponding entry sets. Formally, B = B_1|| B_2, where B[i]=|{t | f_l(t,r_1) =i||f_l(t,r_2)=i }|.The tash OR is aimed at obtaining the total number of tags mapped into the corresponding entries in two tash tables.Note tash OR is not the same as the entry-wise sum, B_1 || B_2 ≠ B_1+B_2 because the tags twice mapped into a same entry are counted only once.As shown in Fig. <ref>, (B_1||B_2)[5]=|{t_5,t_6,t_7}|=3 although B_1[5]+B_2[5]=5 becauset_6 and t_7 appear twice in the two tash tables.The tash XOR (denoted by ⊗) is to remove the intersection of two corresponding entry sets from the first entry set. Formally, B = B_1 ⊗ B_2 such that B[i]=|{t| f_l(t,r_1)=i & f_l(t,r_2)≠ i}|.The tash XOR is aimed at obtaining the total number of the set difference. As Fig. <ref> shows, B_1[5]=|{t_5,t_6,t_7}| and B_2[5]=|{t_6, t_7}|. Then (B_1 ⊗ B_2)[5]=|{t_5}|=1. The above operators can be applied in a series of tash tables with the same dimension for a hybrid operation, B_1⊕ B_2 || B_3. Tash AND and ORsatisfy operational laws such as associative law and commutative law, B_1⊕ B_2 = B_2 ⊕ B_1.The design of tash operators is one of the attractive features of the tash framework, and it has never been proposed before. More importantly, we design and implement these operators in the physical layer to provide one-stop acquisition solution.§.§ Solution Sketch Tash is designed to reduce the overhead for air communications. It runs in the middle of the reader and upper application. The upper application passes a pair of arguments (r and l), or pairs of arguments (as well as operators) to .On the basis of the arguments, generates one or more configuration files to manipulate the reader's reading. Finally, abstracts the reading results to a tash table, which is returned to the upper application.The rest of the paper is structured as follows.We firstly present the tash design in <ref>. We next demonstrate the usage of our hash primitives in two classic applications in <ref>. We then introduce the tash implementation using LLRP interfaces in <ref>. In <ref> and <ref>, we present the microbenchmark and the usage evaluation. Finally, we conclude in <ref> and present future directions.§ TASH DESIGNIn this section, we introduce the background of selective reading in Gen2, and then present the technical details of our designs.§.§ Background of Gen2 Protocol The Gen2 standard defines air communication between readers and tags.On the basis of <cit.>, we introduce its four central functions we will employ:F1: Memory Model. Gen2 specifies a simple tag memory model (pages 44 ∼ 46 of <cit.>). Each tag contains four types of non-volatile memory blocks (called memory banks): (1)is reserved for password associated with the tag. (2)stores thenumber. (3)stores thethat specifies the unchangeable tag and vendor specific information. (4)is a customized storage that contains user-defined data.F2: Selective Reading. Gen2 specifies that each inventory must be started withcommand (pages 72∼73 of <cit.>). The reader can use this command to choose a subset of tags that will participate in the upcoming inventory round. In particular, each tag maintains a flag variable . The reader can use thecommand to turn theflags of tags into(true) or(false).Thecommand comprises six mandatory fields and one optional field apart from the constant cmd code (1010_2), as shown in Fig. <ref>. The following fields are presented for this study.∙ . This field allows a reader to change flags or the inventoried flags of the tags. The inventoried flags are used when multiple readers are present. Such scenario is irrelevant to our requirements. Thus, we aim to changeflags only by setting .∙ . This field specifies an action that will be will performed by the tags. Table. <ref> lists eight action codes to which the tag makes different responses. For example,the matching or not-matching tags assert or deassert theirflagswhen . Weleverage this useful feature to design tash operators.∙ , ,and .These four fields are combined to compose a bitmask. The bitmask indicates which tags are matched or not-matched for an . Thecontains a variable length binary string that should match the content of a specific position in the memory of a tag. Thefield defines the length of thefield in bits. Thefield can be compared with one of the four types of memory banks in a tag. Thefield specifies which memory bank thewill be compared with. Thefield specifies the starting position in the memory bank where thewill be compared with. For example, if we use a tuple (b,p,l, m) to denote the four fields, then only the tags with data starting at the p^th bit with a length of l bits in the b^th memory bank that is equal to m are matched. To visually understand the selective reading, we show an example in Fig. <ref> in which 4 out of 7 tags are selected to participate in the incoming inventory. Complex and multiple subsets of tags can be facilitated by issuing a group ofcommands to choose a subset of tags before an inventory round starts. For example, we can issue twocommands: onefor division and another for one-bit reply. Note theenabledcommandmust be the last one if multiple selection commands are issued <cit.>. F3: Truncated Reply. Gen2 allows tags to reply a truncated reply (replying a part of) through a specialcommand with an enabledfield, making a one-bit presence signal possible. Whenis enabled (set to 1), then the corresponding bitmask is not used for the division of tags, but lets tags reply with a portion of their s following the pattern specified by the bitmask. Note that whenis enabled, themust be set to thebank (=1) and suchcommand must be the last one. F4: Query Model. Followed by a group ofcommands, command (see page 76∼80 of <cit.>.) starts a new inventory round over a subset of tags, chosen by the previouscommands. There are 7 fields in thecommand. We only focus onfield, which is most tightly relevant to the selective reading. As mentioned above, thecommand has divided the tags into two opposite subsets with asserted and deassertedrespectively. Thefield specifies which subset will reply in the current inventory round. If , the tags with assertedreply. If , the tags with deassertedreply. We choose the tags with assertedby default. §.§ Design of the Tash Function An l-bit tash function is essentially a hash function that is indispensable to HEPs. We design the tash function while following the three principles outlined as follows. The first principle requires that the tash result must be dependent on the inputand the seed. Moreover, it must be predictable as long as all the input parameters are known. The second principle requires the output values to be random, uniformly distributed in [0, 2^l-1]. Even a one bit difference in the input will result in a completely different outcome. The third principle requires a method that can access the tash result of a tag directly or indirectly. We have constructed the tash function as follows by applying the aforementioned principles:given a tag with anof t, we firstly calculate the hash value of theoffline, using a common perfect hash function like 128-bitor . Let h(t) denote the calculated hash value. We then writeh(t) into the tag's user-defined memory bank of the tag, , for later use. The l-bit tash value oftag t challenged byseed r is defined as the value of the sub-bitstring starting from the r^th bit and ending at the (r+l-1)^th bit in theof the tag. Evidently, f_l(t,r) is actually a portion of h(t), and thus,the parameter r∈ [0,ℒ-1] and l∈ [1, ℒ-r], where ℒ is the length of the hash value (128 bits).Fig. <ref> shows an example wherein theandof the tag store itst and the hash value h(t), respectively. When r=5 and l=4 are inputted,the tash value that this tag outputs is 1010_2, which is the sub-bitstring of h(t) starting from the 5^th bit and ending at the 8^th bit in , f_4(t,5)=1010_2. Our design does not require a tag to equip a real hash function or the engagement of its chip. It clearly applies the preceding principles. First,f_l(t,r) is evidently repeatable, predicable and dependent on the inputs. Second, the randomness of f_l(t,r) is derived from h(t) and r, which are supposed to have a good randomness quality. Third,we have two ways to access the tash value. We can use the memorycommand to access of a tag directly, or use the selective reading function to access the tash value indirectly (discussed later). Discussion:A few points are worth-noting about the design:∙ As the tash value is a portion of the hash value, if two random numbers may cover a common sub-string. For example, if r_1 and r_2 differ by 1, there exist l-1 same bits with 50% of probability that two hash values are same,although such case occurs with a small probability, ≈ 127/(128× 128)× 0.5=0.0039. If some upper applications require extremely strong independence, we should generate the second random number r_2 meeting the condition of r_2<r_1-l and r_2≥ r+l, so as to avoid the common coverage and potential relevance.∙ The design of tash function involves the , the user-defined storage. We can usecommand to store any data into this memory bank. Our compatibility report (shown in <ref>) suggests that almost all types of tags support bothandcommand except one read-only type (ImpinJ Monza R6). Our approach is generally practicable. ∙ Our design targets at enabling COTS tags, billions of which have been deployed in recent years,to obtain performance advantages from well-studied hash based protocols, instead of enhancing their security or privacy preservation. Our design still follows the current COTS tag's security mechanism, password protected memory access.∙ Tash function also offers a good feature that the computation is one way and irreversible, the output reveals nothing about the input. This feature is inherited from the hash function. It may be useful for privacy protection in practice. However, this topic is beyond the scope of this work.§.§ Design of the Tash Table FunctionThe tash table function treats a reader and multiple tags as if they were a single virtual node, outputting a tash table.For simplicity, we useS ( a_, b^, p_, l^, m_, u^)to denote a selection command () with an(a), a(b), a(p), a(l), a(m) and a(u). The command aims to select a subset oftags with a sub-bitstring that starts from the p^th bit and ends at the (p+l-1)^th bit in the b^th memory bank that is equal to m. These selected tags are requested to take an action a. The action codes are shown in Table. <ref>. In particular, if u=1, then each tag will reply with a truncatednumber.The tash table functionis designed as follows.An l-bit table B consists of a total of2^l entries, each of which contains the amount of tags mapped into it. In particular, the index number of each entry, which ranges from 0 to 2^l-1, is actually the tash values of the tags mapped into this entry, B[i]=|{t|f_l(t,r)=i}|. When constructing the i^th entry, the reader performs selective reading with two selection commands as follows:S_1(0, 3, r, l, i, 0)andS_(1, 1, 1, 1, 1, 1)Command S_1 selects a subset oftags with a sub-bitstring that starts from the r^th bit and ends at the (r+l-1)^th bit in thethat is equal to i.Notably, the involved sub-bitstring is the tash value of a tag, f_l(t,r), which refers to Definition. <ref>. Consequently, onlytags with tash values equal to i are selected to participate in the incoming inventory, counted by the i^th entry.The second command S_ enables the selected tags to reply with the first bit of theirnumbers for the one-bit signals. We call such inventory round as an entry-inventory. In this manner, we can obtain the whole tash table by launching 2^l entry-inventories. To visually understand the procedure, we illustrate an example in Fig. <ref>, where r=5 and l=2. The tash table contains 2^2 entries; hence,four entry-inventories are launched. Their selection commands are defined as follows:182 S_1(0, 3, 5, 2, 0, 0)andS_(1, 1, 1, 1, 1, 1) 183S_1(0, 3, 5, 2, 1, 0)andS_(1, 1, 1, 1, 1, 1) 184S_1(0, 3, 5, 2, 2, 0)andS_(1, 1, 1, 1, 1, 1) 185S_1(0, 3, 5, 2, 3, 0)andS_(1, 1, 1, 1, 1, 1)For the third entry-inventory, thefield is set to 2 because the index of the third entry is 2.Four tags (t_5, t_6, t_7 and t_8) are selected to join in this entry-inventory. Thus,ℱ_2(T,5)[2]=4.For a tash table, note that (1) the sum of all its entries is equal to the total number of tags, and (2) it allows an application to selectively construct the entries of a tash table becaues each entry-inventory are independent of each other and completely controllable. For example, we can skip the inventories of these entries that are predicted to be empty. §.§ Design ofTash Operators A tash operator is connected to two tash tables, which have the same dimensions but are constructed using two different seeds. When two seeds, r_1 and r_2, are given,we can obtain two l-bit tash tables: B_1=ℱ_l(T,r_1) and B_2=ℱ_l(T,r_2). Our objective is to obtain a final tash table B by performing one of the subsequent tash operators on B_1 and B_2.Tash AND. If B=B_1⊕ B_2, then each entry of B denotesthe number of tags that are concurrently mapped into the corresponding entries ofB_1 and B_2. The selection commands for the i^th entry-inventory are defined as follows:S_1(0,3,r_1,l,i,0), S_2(2,3,r_2,l,i,0), S_From the action codes shown in Table. <ref>, the purpose of S_1 with action code of 0is to select tags ∈ B_1[i] and deselect tags ∉ B_1[i]. S_2 with action code of 2deselects tags∉ B_2[i] and results in tags ∈ B_2[i]doing nothing.After S_1 is received, each tag exhibits one of two states, selected or deselected. Then, S_2 will make the selected tags remain in their selected states if they match its condition (doing nothing); otherwise, it changes their states to the deselected states (selected → deselected), which is equivalent to removing tags ∉ B_2[i] from tags ∈ B_1[i]. Meanwhile, the tags deselected by S_1 remain in their states regardless of whether they match (do nothing) or not match (deselected → deselected) the condition of S_2. Finally, S_ is reserved for the one-bit presence signal.Tash OR. If B = B_1 || B_2, then each entry of B is the number of tags that mapped into the corresponding entry of either B_1 or B_2. The selection commands for the i^th entry-inventory are defined as follows:S_1(0,3,r_1,l,i,0), S_1(1,3,r_2,l,i,0),S_Similarly,S_1 selects tags ∈ B_1[i] and deselect tags ∉ B_1[i].S_2 with action code of 1 (see Table. <ref>) allows tags ∈ B_2[i] to be selected as well, but tags ∉ B_2[i] remain in their states (do nothing), some ofthese tags may have been selected by S_1. The process is equivalent to holding the tags selected by S_1 and incrementallyadding the new tags selected by S_2.Tash XOR. If B = B_1 ⊗ B_2, then each entry of B is the number of tags that are mapped into the corresponding entry of B_1 but not into the entry of B_2. The selection commands for the i^th entry-inventory are defined as follows:S_1(0,3,r_1,l,i,0), S_2(5,3,r_2,l,i,0),S_Similarly,S_2 allows tags ∈ B_2[i] to be deselected (removed from tags ∈ B_1[i]) and tags ∉ B_2[i] to do nothing. This process is equivalent to removing tags ∈ B_2[i] from tags ∈ B_1[i].Tash hybrid. The aforementionedthree operators can be further applied to a hybrid operation. When k seeds (r_1,⋯, r_k) are given, we can obtain k tash tables. The selection commands for the i^th entry-inventory can be designed as follows:S_1(0,3,r_1,l, i,0), S_2(,3,r_2,l, i,0), ⋯, S_k(,3,r_k,l, i,0),S_whererepresents thecode, which is set to 2, 1 and 5 for tash AND, OR and XOR, respectively. The action code of the first command is always set to 0.For example, the selection commands in the i^th entry-inventory for ℱ_l(T, r_1) ⊕ℱ_l(T, r_2)|| ℱ_l(T,r_3) ⊗ℱ_l(T,r_4) are given by:S_1(0,3,r_1,l, i,0), S_2(2,3,r_2, l, i,0),S_3(1,3,r_3,l, i,0), S_4(5,3,r_4, l, i,0),S_We leverage the action of a selection command to perform an operation in the physical layer before an entry-inventory starts, therefore, we introduce minimal additional communication overhead, broadcasting multiplecommands. Compared with the multiple acquisitions of bitmaps used by prior HEPs, our solution provides a one-stop solution that can significantly reduce the total overhead in such situation.§.§ Discussion Comparison with bitmap. A tash table evidently takes a considerably longer time to obtain than a bitmap because a bitmap requires only one round of inventory, whereas a tash table requires multiple rounds. The additional time consumption is the trade-off for practicality because the reply of a COTS tag at the slot level is out of control. Nevertheless, this additional cost brings an additional benefit, a tash table has the exact number of tags mapped onto its each entry, which cannot be suggested by a bitmap. Moreover, a one-stop operator service can save more time.Embedded pseudo-random function. Qian <cit.> and Shahzad <cit.> proposed a similar concept of utilizing a pre-stored random bit-string to construct a lightweight pseudo-random function. These studies have inspired our work.However, their main objective of these previous researchers is to accelerate the calculation of a random number, which still requires the engagementwith the chip of a tag, and thus,has never been implemented in practice. In the present work, we do not require additional efforts on changing the logics ofa tag chip and we associate this concept with the function of selective reading, moving the main task from a tag to a reader. Our design not only preserves the good features of the hash function but also gives a practical solution. This process has never been performed before.Channel error. Channel error is one of the most notorious problems of HEPs because pure one-bit signal transmission is vulnerable to ambient interference. Thus, an additional error control mechanism is expected to be applied to HEPs. In the Gen2 protocol, the CRC8 code is automatically appended to the data transmitted between a reader and a tag for error detection, even when one bit of EPC is transmitted. The corrupt data will be retransmitted. Therefore, we should not be concerned with channel error.§ TASH USAGEThis section revisits two classic problems of HEPs for usage study. We propose two practical solutions that use tash primitives for these problems. Note that in spite of two demonstration presented in this section, our tash primitives especially the tash table can serve any kind of HEPs.§.§ Usage I: Cardinality Estimation Cardinality estimation aims to estimate the total number of tags by using one-bit presence signals that are received without collecting each individual tag. The problemis formally defined as follows:When a tag population of an unknown size n,a tolerance of β∈ (0,1), and a required confidence level of α∈(0,1) is given, how can the number of tags n be estimated such that (|n-n|≤β n)≥α?A naive method would be to add all the entries of a tash table together or let all tags reply at the first entry.Since each tag participates in one and only one entry-inventory,the final number is exactly equal to n.Keep in mind that our each entry corresponds to a complete round inventory. The naive method is equivalent to collecting them all, which is extremely time-consuming. We subsequently provide a reliable solution in a probabilistic way.Proposed Estimator. We leverage the number of tags mapped into the first entry of a tash table to estimate n. Let X be the random variable to indicate the value of the first entry of a tash table. Since n tags are randomly and uniformly assigned into 2^l entires, we have(X=m) = nmp^m (1-p)^n-mwhere p=1/2^l. Evidently, variable X follows a standard Binomial distribution with the parameters n and p, X∼ B(n,p). Therefore the expected value μ=np and variance δ=np(1-p). By equating the expected value and an instanced value m, our estimator n is given by:n= m/p = m 2^lThe estimator only requires the first entry of the hash table, so it skips inventories of other entries. We must choose an appropriate l to ensure the estimation error within the given tolerance level β with a confidence of greater than α.Analysis. For the sake of simplicity, we use a Gaussian model to approximate the above distribution due to the central limit theory. Let the random variable Y=(X-μ)/δ∼𝒩(0,1).We can always find a constant c, which satisfiesα = (-c≤ Y ≤ c) = erf(c/√(2))where erf is the Gaussian error function. Since we require @size9@̌mathfonts(|n-n|≤β n) = ((1-β)n≤n≤(1+β)n) =((1-β)n≤m/p≤ (1+β)n)) =((1-β)np-μ/δ≤m-μ/δ≤(1+β)np-μ/δ) ≥α = (-c ≤ Y ≤c), we can find a constant c by subjecting to the below inequality: @size9@̌mathfontsc≤ max{(1+β)np-μ/δ, -(1-β)np-μ/δ}= (1+β)np-μ/δ = β/1-pBy substituting p=1/2^l into the above inequality, we obtainl≤log_2 c/c-βwhere c=√(2)·erf^-1(α). Therefore, we can obtain the following theorem. The optimal dimension of the tash table is equalto ⌈log_2 √(2)·erf^-1(α)/√(2)·erf^-1(α)-β⌉, which results in an estimation error ≤β with a probability of at least α. §.§ Usage II: Missing Tag Detection The purpose of missing tag detection is to quickly find out the missing tags without collecting all the tags in the scene. Such detection is very useful, especially whenthousands of tags are present. We formally define the problem of detecting missing tags in Problem <ref>. We assume that the s of all the tags in a closed system arestored in a database and known in advance. This assumption is reasonable and necessary, because it is impossible for us to tell that a tag is missing without any prior knowledge of its existence.How to quickly identify m missing out of n tags with a false positive rate of γ at most? Proposed detector. The underlying idea is to compare two tash tables B and B.B is an intact tash tablecreated by tashing all the known s which are stored in the database, while B is an instance tash table obtained from the tags in the scene. We can detect the missing tags through comparing the difference between B and B. If the residual table B-B (entry-wise subtraction) equals 0, no missing tag event happens. Otherwise, the tags mapped into the non-zero entries of the residual table are missing.Fig. <ref> illustrates an example in which three tags, t_1, t_2 and t_3, are mapped into the intact tash table B. B is an instance table where tag t_2 is missing, and thus B[4]=1. Consequently, (B-B)[4]=1, we can definitely infer that one tag is missing. However,it is impossible for us to tell which tag is missing because t_2 and t_3 are simultaneously mapped into the fourth entry.Inspired by the Bloom filter<cit.>, we perform k tashings to identify the missing tags as follows:B = ℱ_l(T, r_1) || … || ℱ_l(T,r_k)The final B after tash ORs is considered to use k independent hash functions (induced by k random seeds) to map each tag into B for k times, as shown in Fig. <ref>.The residual table of B-B is therefore viewed as a Bloom filter which represents the missing tags. Thereafter,to answer a query of whether a tag t is missing,we check whether all entries set by f_l(t,r_1), ⋯ and f_l(t,r_k) in the residual table have a value ofnon-zero. If the answer is yes, then tag t is the missing one. Otherwise, it is not the missing tag. Fig. <ref> illustrates an example in which each tag is tashed twice. The missing tag t_2 can be identified because both the 2^rd and the 4^th entry in the residual tablehave value ofnon-zero. Despite multiple tashings,the query may yield a false positive, where it suggests a tag is missing even though it is not.Analysis. To lower the rate of false positive rate, it is necessary to answer two questions.(1) How many tash functions do we need? Given the table dimension l, we expect to optimize the number of tash functions. There are two competing forces: using more tash functions gives us more chance to find a zero bit for a missing tag, but using fewer tash functions increases the fraction of zero bits in the table.After m missing tags are tashed into the table, the probability that a specific bit is still 0 is (1-1/L)^km≈ e^-km/L where L=2^l. Correspondingly, the probability of a false positive p is given byp = (1-e^-km/L)^kNamely,a missing tag falls into k non-zero entries.Lemma. <ref> suggests that the optimal number of tash functions is achieved when k=ln 2 · (L/m). The false positive rate is minimized when p=(1/2)^k or equivalently k=ln 2 · (L/m). Please refer to <cit.> for the proof. (2) How large tash table is necessary to represent all m missing tags?Recall that the false positive rate achieves minimum when p=(1/2)^k. Let p≤γ. After some algebraic manipulation, we findL≥mlog_2 (1/γ)/ln2 = mlog_2 e ·log_2(1/γ) = 1.44m log_2(1/γ)Finally, putting the above conclusions together, we have the subsequent theorem. Setting the table dimension to ⌈log_2(1.44m log_2 (1/γ)) ⌉ and using ⌈ln 2 · (2^l/m) ⌉ random seeds allow the false positive rate of identifying m missing tags lower than a given tolerance γ.§ TASH IMPLEMENTATIONOur implementation involves two kinds of protocols: UHF Gen2 air interface protocol (Gen2) and Low Level Reader Protocol (LLRP). As shown in Fig. <ref>,Gen2 protocol defines the physical and logical interaction between readers and passive tags, while LLRP allows a client computer to control a reader.Each client computer connects one ore more RFID readers via Ethernet cables.LLRP is the driver program (or driver protocol) forGen2 readers. We leverage LLRP to manipulate a reader to broadcast Gen2 commands that we need.Notice that we do not need particularly implement Gen2 protocol, which has been implemented in the COTS RFID devices that we are using.Specifically, LLRP specifies two types of operations: reader operation (RO) and access operation (AO). Both operations are represented inXML document form and transported to a reader through TCP/IP.Reader operation. RO defines the inventory parameters specified in the Gen2 protocol, such as bitmask, antenna power, and frequency. Fig. <ref> shows a simplified instance of an .Anis composed of at least one . Eachis used for an antenna setting. Anconsists of more than one s. The filter functions as a bitmask. We can set multiple selection commands by adding multiple s.Access operation. AO defines the access parameters for writing or reading datato and from a tag.We leverage the inside anto write the hash value of theinto a user-defined memory bank. As the s are highly related to the products the tags attached, the writing of hash values should be accomplished by the product manufactures or administrators.There is almost no overhead to write data intosince it is allowed to write a batch of tags simultaneously usingcommands specified in one , without physically changing tags' positions. Tash framework. Our framework is developed by using Java language and the LLRP Toolkit<cit.>, which is an open-source library for handlingand .Fig. <ref> shows the primary interfaces provided by the tash framework. The classmakes the first selection through its construction method and allows the calls of three operators to be chained together in a single statement. The methodconverts aobject ora chain of objects into an . The entry-inventories are physically executed in the connected reader when the methodis invoked.This method allows users to make selective entry-inventories by passing an index array. For example, the operation ℱ_l(T, r_1) ⊕ℱ_l(T, r_2)|| ℱ_l(T,r_3) ⊗ℱ_l(T,r_4) can be coded in a manner similar to that shown at the bottom of Fig. <ref>.§ MICROBENCHMARKWe start with a few experiments that provide insight to our hash primitives. §.§ Experimental SetupWe evaluate the framework using COTS UHF readers and tags. We use a total of 3 models of ImpinJ readers (R220, R420 and R680), each of which is connected to a 900MHz and 8dB gain directional antenna.In order to better understand the feasibility and effectiveness of in practice, we test a total of 3,000 COTS tags with different models. We divide these tags into 10 groups of 300 tags each. The tags of each group are densely attached to a plastic board which is placed in front of a reader antenna. As shown in Fig. <ref>, three hundreds is the maximum number of tags that can be covered by one directional antenna in our laboratory. We store the 3,000numbers in our database as the ground truth.The 128-bit MD5 is employed as the common hash function to generate the hash values of s. The experiments with the same settings are repeated across the 10 groups, and the average result is reported.§.§ Compatibility InvestigationFirst, we investigate the compatibility of Gen2 across 10 differenttypes of readers and 18 different types oftags in terms ofthe functions or commands that requires. The readers and tags may come from different manufacturers but work together in practice. These investigated productsare all publicly claimed to be completely Gen2-compatible.Reader compatibility. We investigate the R220, R420, and R680 models from ImpinJ<cit.>, the Mercury6, Sargasand M6e models from ThingMagic<cit.>, as well as the ALR-F800, 9900+, 9680 and 9650 models from Alien<cit.>. We perform the investigation through real tests for the first three models of readers (the ImpinJ series), and investigate the other readers through their data sheets or manuals (because we are limited by the lack of hardware). The Gen2-compatibility of readers is briefly summarized in Table. <ref>. Consequently, we have the subsequent findings. (1) All the readers do supportcommand, which Tash uses for writing or reading hash values ofnumbers. (2) All the readers do support thecommand, which Tash uses for the selective reading. (3) However, our practical tests suggest that none model of the ImpinJ series supports the command, which Tash uses to hear the one-bit presence signal. The serviceabilityof other readers is not clearly indicated in the manuals of those readers.(4) The Gen2 protocol does not specify how many s and sthat a reader should support. Our practical tests suggest that the ImpinJ series supports 4 s and 16 s, which means that we can only use a maximum of four tash operators each time.Tag compatibility.We investigate 9 chip models from ImpinJ Monza series and 9 additional models from Alien ALN series. The majority of tags on the market contain these 18 models of chips and customized antennas. Table. <ref> summarizes the result of our investigation, from which we have the subsequent findings. (1) Tags reserve 96∼ 480 bits of memory for storingnumbers, among which the size of 96 bits has become the de facto standard.(2) requiresto store the hash values. The results of the investigation show that almost all tags allow to write to and read from the third memory bank, with an exception of ImpinJ Monza R6, which does not have the user-defined memory. The size of the third memory bank fluctuates around 32∼ 512 bits. The de facto standard has become 128 bits. (4) All tags are claimed to support thecommand according to their public data sheets. However, we have no idea about their real serviceability due to the lack of -supportable reader available for practical tests. In our future work, we plan to utilize USRP for further tests.Summary. Despite positive and public claims, our investigation shows that current COTS RFID devices, regardless of readers or tags and models, have some defects in their compatibility with Gen2, especially with regard to.The reason, we may infer, is that these commands are seldom used in practice and therefore never receive attention from manufacturers. The partial compatibility of such devices cannot fully achieve the performance brings. Even so, we are obliged to make the claim, again, that our design strictly follows the Gen2 protocol. We hope this work can encourage manufacturers to upgrade their products (reader firmware) to achieve full compatibility. §.§ Tash FunctionSecond, we evaluate the tash function with respect to the randomness and the accessibility. Randomness. Randomness is the most important metric for a hash function. It requires that the outputs of a hash function must be uniformly distributed.To validate the randomness of the tash function, we collect 99,886 realnumbers from our partner (an international logistics company), which introduced RFID technology for sorting tasks five years ago. Eachnumber has a length of 96 bits and encodes the basic information about the package, such as sources, destinations, serial numbers, and so on. We employ the 128-bit MD5 to create the hash values of these s. As the minimum size of theis 32 bits (see Table. <ref>), we choose to use only the first 32 bits for our tests. We traverse r and l from 0∼31 and 1∼ 32-r respectively. For each pair of r and l, we obtain 99,886 tash values over all the s. Across these tash values, we further conduct the following two analysis: (1) We merge 100 tash values, which are randomly selected from the above results, into a long bit string. We then calculate the percents of `0' and `1' emerged in that bit string. This operation is repeated for 100 times. Finally, totally 100 pairs of percents are obtained. Their CDFs are plotted inFig. <ref>. Ideally, each bit has a equal probability of 0.5 to be zero or one if a hash function makes a good randomicity. From the figure, we can figure out that the percents distributed between 0.4 and 0.6. In particular,percents of `0' and `1' have means of 0.49 and 0.50 with standard deviations of 0.043 and 0.044 respectively.(2)We shuffle these values into 100 groups, and employ the χ^2-test with a significance level of 0.05 to test each group's goodness-of-fits of the uniform distribution (passed or failed). Then, we finally calculate the pass rate for a pair of setting. In this manner, we totally obtain 496 pass rates. More than 60% of the pass rates are over than 0.95. In particular, three sets of the results with r=16, 20 and 26 and a variable l, are selected to show in Fig. <ref>. We find that 90% of the pass rates exceed 0.95 for the three cases, and their median pass rates are around 0.97. Thus, the two above statistical results suggest that our tash function has a very good quality of randomness.Accessibility. Accessibilityrefers to the ability to get access to a tash value from a tag. As aforementioned, we have two ways to acquire the tash values. The first way is to use thecommand. The second way is to indirectly access a tash value through a selective reading. We choose the second method since it is the basis of our design. Specifically, we perform a selective reading to determine whether the tags are collected as expected, when given random inputs and a possible tash value. We intensively and continuously perform such readings across the 10× 300 tags using three 4-port ImpinJ readers for three rounds of 24 hours in a relatively isolated environment (an empty room without disturbance). Surprisingly, we find all the reading results faithfully conform to our benchmarks without any exceptions. This shows that the selective reading is well supported by the manufactures and is both stable and reliable. §.§ Tash Table Function Third, we evaluate the performance of the tash table function in terms of its balance and gathering speed.Balance. A good hash table function will equally assign each key to a bucket. We expect the output tash table to be as balanced as possible.To show this feature, we generate 100 different 4-bit tash tables (each includes 16 entries) across 300 tags using 100 different random seeds. If the tash table is well balanced, the expected number of each entry should be very close to 300/16=18.7. Fig. <ref> shows the mean number of tags in each entry as well as their standard deviations. The average number across 16 entries equals 18.75, which is very close to the expectedtheoretical value. The average standard deviation equals 0.44. Thus, the good randomness quality of tash functions results in output tash tables being well balanced.Gathering speed. We then consider the time consumption of gathering a tash table. Fixing the random seed, we vary the table dimension l from 0 to 6. We then measure thetime taken on gathering a tash table with the deployed 300 tags.Fig. <ref> shows the resulting time as a function of the table dimension. From the immediately above-mentionedfigure,we can observe the subsequent findings.(1)When l=0 without truncating reply, the result is equivalent to collecting 300 complete s of all the tags. Such time consumption (4,524ms) is viewed as our baseline.(2) By contrast, when l>0 without truncating a reply, the collection amounts to dividing all the tags into 2^l groups “equally” and then collecting each group independently.In this manner, when l≤ 4, such “divide and conquer” approach is better than “one time deal”, a drop in overhead of about 10%. The Gen2 reader uses a Q-adaptive algorithm for the anti-collision. This algorithm is able to adaptively learn the best frame length from the collision history. Due to the division,a smaller number of tags can make reader's learning relatively quicker and improve the overall performance.(3) However, when l>4, the performance of “divide and conquer” approach starts to deteriorate. The ImpinJ readersupports 16 s at most (see Table. <ref>). We have to re-send anotherfor the remaining selective readings when the number of entry-inventory is above 16 (l>4), which introduces additional time consumption. (4) We then consider the case where the reply is truncated to a one-bit presence signal as assumed by HEPs. Due to the defects of ImpinJ readers in the implementation of thecommand, we cannot measure the actual time spent on collecting truncated s. We can only utilize the least-square algorithm to estimate the transmissiontime for a one-bit presence signal. Our fitting results show that truncating reply would introduce about 60% drop of the overhead at least. §.§ Tash Operators Finally, we investigate the performance of tash operators.Superior to existing HEPs, these operators allow us to perform set operations on-tag and conduct a one-stop inventory. In particular, we show the performance of OR as a representative across 300 tags. The tests for other operators are similar and omitted due to the space limitation. In the experiments, we fix the two random seeds but change the dimension of tash table. Fig. <ref> shows the results of three cases. In Case 1, we independently produce 2 tash tables without truncating a reply and conduct the OR in the application layer.In Case 2 and Case 3, we conduct on-tag OR function as provides without and with truncating a reply respectively. Consequently, when the dimension equals 2, Case 1 takes 6,511ms on collecting two tables. On the contrary,the amount of time taken is reduced to 4,578ms (29.7% drop) if we perform an on-tag OR function even without truncation (Case 2).Ideally, the amount of time taken could be further reduced to 50.97ms by using a truncating reply (Case 3), which offers a staggering drop in time usage by 99.22%. Our experiments relate only to the amount of time spent on ORing two tables. It may be predicted thatmuch more outperformance will be gained if multiple tables are involved. The tash operators that we design in this work have never been proposed before. § USAGE EVALUATION We then use our prototype to demonstrate the benefits and potentials of in two typical applications. §.§ Usage I: Cardinality Estimation We evaluate our estimation scheme through the testbed as well as large-scale simulations.Testbed based. Our scheme only uses the first entry of the tash table for the estimation, thereby we only need one entry-inventory. Fig. <ref> shows the CDF of estimation results across 300 tags. We define the error rate as |n-n|/n where n is the estimated number.As a result, 90% of the estimations have an error rate less than 0.1 and a median of 0.04 when setting the dimension l=1. In this case, almost half tags follow into the first entry so the rate could be pretty high, at the price of longer inventory time. As l increases, the error rate also increases because less samples are acquired for the estimation. These experiments show the feasibility of using tash table for cardinality estimation.Simulation based. We then perform the evaluation through large-scale simulations for two reasons: (1) ensuring its scalability when meeting a huge number of tags. (2) making comparisons with prior work, which are all simulation-based. We numerically simulate in Matlab using tash scheme as well as other five prior RFID estimation schemes: UPE<cit.>, EZB<cit.>, FNEB<cit.>, MLE<cit.>, ART<cit.>. We implement these schemes by referring to the RFID estimation tool developed by Shahzad<cit.>. Fig. <ref> shows the time cost with a varying n given α=0.9 and β=0.08. We observe that our scheme is 5× faster than the others on average when n<1000. Thus our scheme is suitable for the estimation with a small number of tags. When n>1000, the performance of our scheme starts to vibrate between ART and MLE, due to two reasons. First, our scheme is not collision-free so that more efforts are required to deal with the collisions incurred by more tags. Second,the size of a tash table can only increase in the power of two, making the size always vibrate around the optimal one.Even so, the advantage of our scheme is still clear: it is the first RFID estimation scheme that can work in real life. Notice that ART claimed to work with RFID systems because they are theoretically compatible with ALOHA protocols. Actually, the current COTS RFID systems do not allow user to control the low-level access, like fined-grained adjustment offrame length and obtaining slot-level feedback, which are necessary to implement ART. Thus, there is no way for ART to implement their algorithms over COTS RFID systems without any hardwaremodification and fabrication.§.§ Usage II: Missing Detection Finally, we evaluate the effectiveness of missing detection in real case. We randomly remove m tags from the testbed. Since we only have 300 tags in total, we fix the number of random seeds to 2, k=2. The performance is evaluated in term of the false positive rate (FPR), which is the ratio of number of mistakenly detected as missing tags to the total number of really missing tags. Our scheme is able to successfully find out all the missing tags because the residual table always contains the entries that missing tags are tashed into. Fig. <ref> shows the results of the first case in which we use an 8-bit hash table (l=8) to detect the missing tags. Consequently, the FPR is maintained around 0.01 when m<14 (5% of the tags are missing).Fig. <ref> shows the second case in which we remove 10 tags and detect the missing tags by changing the dimension of tash table.As Theorem. <ref> suggests, we should set l=5,6,7 to guarantee the FPR γ< 0.2,0.1,0.01. From the figure, we can find that the results of our experiments completely conform to this theorem. The real FPRs equal 0.21, 0.07 and 0.008 in the three cases. Tash enabled missing detection works well in practice. § CONCLUSIONThis work discusses a fundamental issue that how to supplement hash functionality to existing COTS RFID systems, which is dispensable for prior HEPs. A key innovation of this work is our design of hash primitives, which is implemented using selective reading. Tash not only makes a big step forward in boosting prior HEPs, but also opens up a wide range of exciting opportunities. § ACKNOWLEDGMENTSThe research issupported by GRF/ECS (NO. 25222917), NSFCGeneral Program (NO. 61572282) and Hong Kong Polytechnic University (NO. 1-ZVJ3). We thank all the reviewers for their valuable comments and helpful suggestions, and particularly thank Eric Rozner for the shepherd.ACM-Reference-Format | http://arxiv.org/abs/1707.08883v1 | {
"authors": [
"Lei Yang",
"Qiongzheng Lin",
"Chunhui Duan",
"Zhenlin An"
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"primary_category": "cs.NI",
"published": "20170727143144",
"title": "Analog On-Tag Hashing: Towards Selective Reading as Hash Primitives in Gen2 RFID Systems"
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[pages=1-last,fitpaper=true]paper.pdf | http://arxiv.org/abs/1707.08718v1 | {
"authors": [
"Saman Naderiparizi",
"Mehrdad Hessar",
"Vamsi Talla",
"Shyamnath Gollakota",
"Joshua R. Smith"
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"categories": [
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"cs.CV"
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"primary_category": "cs.ET",
"published": "20170727064318",
"title": "Ultra-low-power Wireless Streaming Cameras"
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Universidad de los Andes ColombiaUniversità della Svizzera italiana SwitzerlandCollege of William and MaryUnited StatesCollege of William and MaryUnited StatesUniversity of Sannio ItalyCollege of William and MaryUnited StatesCollege of William and MaryUnited StatesCollege of William and MaryUnited States Mutation testing has been widely used to assess the fault-detection effectiveness of a test suite, as well as to guide test case generation or prioritization. Empirical studies have shown that, while mutants are generally representative of real faults, an effective application of mutation testing requires “traditional" operators designed for programming languages to be augmented with operators specific to an application domain and/or technology. This paper proposes , a framework for effective mutation testing of Android apps. First, we systematically devise a taxonomy of types of Android faults grouped in categories by manually analyzing software artifacts from different sources (bug reports, commits). Then, we identified a set of mutation operators, and implemented an infrastructure to automatically seed mutations in Android apps with 35 of the identified operators. The taxonomy and the proposed operators have been evaluated in terms of stillborn/trivial mutants generated and their capacity to represent real faults in Android apps, as compared to other well know mutation tools. <ccs2012> <concept> <concept_id>10011007.10011074.10011099</concept_id> <concept_desc>Software and its engineering Software verification and validation</concept_desc> <concept_significance>500</concept_significance> </concept> </ccs2012> [500]Software and its engineering Software verification and validationEnabling Mutation Testing for Android Apps Denys Poshyvanyk December 30, 2023 ==========================================§ INTRODUCTION In the last few years mobile apps have become indispensable in our daily lives. With millions of mobile apps available for download on Google Play <cit.> and the Apple App Store <cit.>, mobile users have access to an unprecedentedly large set of apps that are not only intended to provide entertainment but also to support critical activities such as banking and health monitoring. Therefore, given the increasing relevance and demand for high quality apps, industrial practitioners and academic researchers have been devoting significant effort to improving methods for measuring and assuring the quality of mobile apps. Manifestations of interest in this topic include the broad portfolio of mobile testing methods ranging from tools for assisting record and replay testing <cit.>, to automated approaches that generate and execute test cases <cit.>, and cloud-based services for large-scale multi-device testing <cit.>.Despite the availability of these tools/approaches, the fieldof mobile app testing is still very much under development; as evidenced by limitations related to test data generation <cit.>, and concerns regarding effective assessment of the quality of mobile apps' test suites. One way to evaluate test suites is to seed small faults, called mutants, into source code and asses the ability of a suite to detect these faults <cit.>. Such mutants have been defined in the literature to reflect the typical errors developers make when writing source code <cit.>. However, existing literature lacks a thorough characterization of bugs exhibited by mobile apps. Therefore, it is unclear whether such apps exhibit a distribution of faults similar to other systems, or if there are types of faults that require special attention. As a consequence, it is unclear whether the use of traditional mutant taxonomies <cit.> is enough to asses test quality and drive test case generation/selection of mobile apps.In this paper, we explore this topic focusing on apps developed for Android, the most popular mobile operating system. Android apps are characterized by GUI-centric design/interaction, event-driven programming, Inter Processes Communication (IPC), and interaction with backend and local services. In addition, there are specific characteristics of Android apps—such as permission mechanisms, Software Development Kit (SDK) version compatibility, or features of target devices—that can lead to a failure. While this set of characteristics would demand a specialized set of mutation operators that can support mutation analysis and testing, there is no available tool to date that supports mutation analysis/testing of Android apps, and relatively few (eight) mutation operators have been proposed by the research community <cit.>. At the same time, mutation tools for Java apps, such as Pit <cit.> and Major <cit.> lack any Android-specific mutation operators, and present challenges for their use in this context, resulting in common problems such as trivial mutants that always crash at runtime or difficulties automating mutant compilation into Android PacKages (APKs). Paper contributions. This paper aims to deal with the lack of (i) an extensive empirical evidence of the distribution of Android faults, (ii) a thorough catalog of Android-specific mutants, and (iii) an analysis of the applicability of state-of-the-art mutation tools on Android apps. We then propose a framework, , that relies on a catalog of mutation operators inspired by a taxonomy of bugs/crashes specific for Android apps, and a profile of potential failure points automatically extracted from APKs.As a first step, we produced a taxonomy of Android faults by analyzing a statistically significant sample ofcandidate faults documented in (i) bug reports from open source apps, (ii)bug-fixing commits of open source apps;(iii) Stack Overflow discussions, (iv) the Android exception hierarchy and APIs potentially triggering such exceptions; and (v) crashes/bugs described in previous studies on Android <cit.>.As a result, we produced a taxonomy of types of faults grouped in categories, four of which relate to Android-specific faults, five to Java-related faults, and five mixed categories (fig:taxonomy). Then, leveraging this fault taxonomy and focusing on Android-specific faults, we devised a set of Android mutation operators and implemented aplatform to automatically seed of them. Finally, we conducted a study comparing with other Java and Android-specific mutation tools. The study results indicate that , as compared to existing competitive tools, (i) is able to cover a larger number of bug types/instances present in Android app, (ii) is highly complementary to the existing tools in terms of covered bug types, and (iii) generates fewer trivial and stillborn mutants.§ RELATED WORK This section describes related literature and publicly available, state-of-the-art tools on mutation testing. We do not discuss the literature on testing Android apps <cit.>, since proposing a novel approach for testing Android apps is not the main goal of this work. For further details about the concepts, recent research, and future work in the field of mutation testing, one can refer to the survey by Jia and Harman <cit.>. Mutation Testing. It is a technique in which faults are automatically injected into program source code or bytecode <cit.>; a program with an injected fault is considered as a mutant of the original version, therefore, each mutant is a version of the Software Under Test (SUT), if 1K faults are injected, then there would be 1K different versions (mutants) of the original system. Note that the mutations are changes to the code that are normally performed at a statement level (small changes), by following pre-defined rules known as mutation operators.The primary goals of mutation testing are (i) to guide the design of test cases that will kill (detect) as many mutants as possible, and (ii)to evaluate the quality/effectiveness of a predefined test suite by measuring how many faulty versions(mutants) of the SUT can be detected.The quality/effectiveness of the test suite is often measured with the mutation score that computes the ratio between the killed mutants and the size of the mutant set; other versions of the mutation score have been proposed to take into account equivalent and redundant mutants <cit.>, and approaches to discard equivalent mutants have been proposed too <cit.>. Also, different researchers have proposed approachesto measure the effectiveness and efficiency of mutation testing <cit.>,to devise strategies for reducing the effort required to generate effective mutant sets (mutant set minimization) <cit.>, and to define theoretical frameworks <cit.>. Mutation Operators. Since the introduction of mutation testing in the 70s <cit.>, researchers have tried not only to define new mutation operators for different programming languages and paradigms (mutation operators have been defined for Java <cit.> and Python <cit.>) but also for specific types of software like Web applications <cit.> and data-intensive applications <cit.> either to exercise their GUIs <cit.> or to alter complex, model-defined input data <cit.>. The aim of our research, which we share with prior work, is to define customized mutation operators suitable for Android applications, by relying on a solid empirical foundation. To the best of our knowledge, the closest work to ours is that of Deng , <cit.>, which defined eight mutant operators aimed at introducing faults in the essential programming elements of Android apps, intents, event handlers, activity lifecycle, and XML files (GUI or permission files). While we share with Deng the need for defining specific operators for the key Android programming elements, our work builds upon it by (i) empirically analyzing the distribution of faults in Android apps by manually tagging documents, (ii) based on this distribution, defining a mutant taxonomy—complementing Java mutants—which includes a total of operators tailored for the Android platform.Mutation Testing Effectiveness and Efficiency. Several researchers have proposed approachesto measure the effectiveness and efficiency of mutation testing <cit.>to devise strategies for reducing the effort required to generate effective mutant sets <cit.>, and to define theoretical frameworks <cit.>. Such strategies can complement our work, since in this paper we aim at defining new mutant operators for Android, on which effectiveness/efficiency measures or minimization strategies can be applied.Mutation Testing Tools. Most of the available mutation testing tools are in the form of research prototypes. Concerning Java, representative tools are μJava <cit.>,Jester <cit.>,Major <cit.>, Jumble <cit.>, PIT <cit.>, and javaLanche <cit.>. Some of these tools operate on the Java source code, while others inject mutants in the bytecode. For instance, μJava,Jester, and Major generate the mutants by modifying the source code, while Jumble, PIT, and javaLanche perform the mutations in the bytecode. When it comes to Android apps, there is only one available tool, namely muDroid <cit.>, which performs the mutations at byte code level by generating one APK (one version of the mobile app) for each mutant. The tools for mutation testing can be also categorized according to the tool's capabilities (the availability of automatic tests selection). A thorough comparison of these tools is out of the scope of this paper. The interested reader can find more details on PIT's website <cit.> and in the paper by Madeysky and Radyk <cit.>.Empirical Studies on Mutation Testing.Daran and Thévenod-Fosse <cit.> were the first to empirically compare mutants and real faults, finding that the set of errors and failures they produced with a given test suite were similar. Andrews <cit.> studied whether mutant-generated faults and faults seeded by humans can be representative of real faults. The study showed that carefully-selected mutants are not easier to detect than real faults, and can provide a good indication of test suite adequacy, whereas human-seeded faults can likely produce underestimates. Just <cit.> correlated mutant detection and real fault detection using automatically and manually generated test suites. They found that these two variables exhibit a statistically significant correlation. At the same time, their study pointed out that traditional Java mutants need to be complemented by further operators, as they found that around 17% of faults were not related to mutants. § A TAXONOMY OF CRASHES/BUGS IN ANDROID APPS To the best of our knowledge there is currently no (i) large-scale study describing a taxonomy of bugs in Android apps, or (ii) comprehensive mutation framework including operators derived from such a taxonomy and targeting mobile-specific faults (the only framework available is the one with eight mutation operators proposed by Deng <cit.>).In this section, we describe a taxonomy of bugs in Android apps derived from a large manual analysis of (un)structured sources. Our work is the first large-scale data driven effort to design such a taxonomy. Our purpose is to extend/complement previous studies analyzing bugs/crashes in Android apps and to provide a large taxonomy of bugs that can be used to design mutation operators. In all the cases reported below the manually analyzed sets of sources—randomly extracted—represent a 95% statistically significant sample with a 5% confidence interval. §.§ Design To derive such a taxonomy we manually analyzed six different sources of information described below: * Bug reports of Android open source apps. Bug reports are the most obvious source to mine in order to identify typical bugs affecting Android apps. We mined the issue trackers of 16,331 open source Android apps hosted on GitHub. Such apps have been identified by locally cloning all Java projects (381,161) identified through GitHub's API and searching for projects with an AndroidManifest.xml file (a requirement for Android apps) in the top-level directory. We then removed forked projects to avoid duplicated apps and filtered projects that did not have a single star or watcher to avoid abandoned apps. We utilized a web crawler to mine the GitHub issue trackers. To be able to analyze the bug cause, we only selected closed issues (those having a fix that can be inspected) having “Bug” as type. Overall, we collected 2,234 issues from which we randomly sampled 328 for manual inspection.* Bug-fixing commits of Android open source apps. Android apps are often developed by very small teams <cit.>. Thus, it is possible that some bugs are not documented in issue trackers but quickly discussed by the developers and then directly fixed. This might be particularly true for bugs having a straightforward solution. Thus, we also mined the versioning system of the same 16,331 Android apps considered for the bug reports by looking for bug-fixing commits not related to any of the bugs considered in the previous point (the ones documented in the issue tracker). With the cloned repositories, we utilized the git command line utility to extract the commit notes and matched the ones containing lexical patterns indicating bug fixing activities, “fix issue”, “fixed bug”, similarly to the approach proposed by Fischer <cit.>. By exploiting this procedure we collected 26,826 commits, from which we randomly selected a statistically significant sample of 376 commits for manual inspection. * Android-related Stack Overflow (SO) discussions. It is not unusual for developers to ask help on SO for bugs they are experiencing and having difficulty fixing <cit.>. Thus, mining SO discussions could provide additional hints on the types of bugs experienced by Android developers. To this aim, we collected all 51,829 discussions tagged “Android” from SO. Then, we randomly extracted a statistically significant sample of377 of them for the manual analysis. * The exception hierarchy of the Android APIs. Uncaught exceptions and statements throwing exceptions are a major source of faults in Android apps <cit.>. We automatically crawled the official Android developer JavaDoc guide to extract the exception hierarchy and API methods throwing exceptions. We collected 5,414 items from which we sampled 360 of them for manual analysis. * Crashes/bugs described in previous studies on Android apps. 43 papers related to Android testing[The complete list of papers is provided with our online appendix <cit.>.] were analyzed by looking for crashes/bugs reported in the papers. For each identified bug, we kept track of the following information: app, version, bug id, bug description, bug URL. When we were not able to identify some of this information, we contacted the paper's authors. In the 43 papers, a total of 365 bugs were mentioned/reported; however, we were able (in some cases with the authors' help) to identify the app and the bug descriptions for only 182 bugs/issues (from nine papers <cit.>). Given the limited number, in this case we considered all of them in our manual analysis.* Reviews posted by users of Android apps on the Google Play store. App store reviews have been identified as a prominent source of bugs and crashes in mobile apps <cit.>.However, only a reduced set of reviews are in fact informative and useful for developers <cit.>. Therefore, to automatically detect informative reviews reporting bugs and crashes, we leverage CLAP, the tool developed by Villarroel <cit.>, to automatically identify the bug-reporting reviews. Such a tool has been shown to have a precision of 88% in identifying this specific type of review. We ran CLAP on the Android user reviews dataset made available by Chen <cit.>. This dataset reports user reviews for multiple releases of ∼21K apps, in which CLAP identified 718,132 reviews as bug-reporting. Our statistically significant sample included 384 reviews that we analyzed. The data collected from the six sources listed above was manually analyzed by the eight authors following a procedure inspired by open coding <cit.>. In particular, the 2,007 documents (bug reports, user reviews, ) to manually validate were equally and randomly distributed among the authors making sure that each document was classified by two authors. The goal of the process was to identify the exact reason behind the bug and to define a tag (null GPS position) describing such a reason. Thus, when inspecting a bug report, we did not limit our analysis to the reading of the bug description, but we analyzed (i) the whole discussion performed by the developers, (ii) the commit message related to the bug fixing, and (iii) the patch used to fix the bug (source code diff). The tagging process was supported by a Web application that we developed to classify the documents (to describe the reason behind the bug) and to solve conflicts between the authors. Each author independently tagged the documents assigned to him by defining a tag describing the cause behind a bug. Every time the authors had to tag a document, the Web application also shows the list of tags created so far, allowing the tagger to select one of the already defined tags. Although, in principle, this is against the notion of open coding, in a context like the one encountered in this work, where the number of possible tags (cause behind the bug) is extremely high, such a choice helps using consistent naming and does not introduce a substantial bias. In cases for which there was no agreement between the two evaluators (∼43% of the classified documents), the document was automatically assigned to an additional evaluator. The process was iterated until all the documents were classified by the absolute majority of the evaluators with the same tag. When there was no agreement after all eight authors tagged the same document (four of them used the tag t_1 and the other four the tag t_2), two of the authors manually analyzed these cases in order to solve the conflict and define the most appropriate tag to assign (this happened for ∼22% of the classified documents). It is important to note that the Web application did not consider documents tagged as false positive (a bug report that does not report an actual bug in an Android app) in the count of the documents manually analyzed. This means that, for example, to reach the 328 bug reports to manually analyze and tag, we had to analyze 400 bug reports (since 72 were tagged as false positives).It is important to point out that, during the tagging, we discovered that for user reviews, except for very few cases, it was impossible (without internal knowledge of an app's source code) to infer the likely cause of the failure (fault) by only relying on what was described in the user review. For this reason, we decided to discard user reviews from our analysis, and this left us with 2,007-384=1,623 documents to manually analyze.After having manually tagged all the documents (overall, = 1,623 + 400 additional documents, since 400 false positives were encountered in the tagging process), all the authors met online to refine the identified tags by merging similar ones and splitting generic ones when needed. Also, in order to build the fault taxonomy, the identified tags were clustered in cohesive groups at two different levels of abstraction, categories and subcategories. Again, the grouping was performed over multiple iterations, in which tags were moved across categories, and categories merged/split.Finally, the output of this step was (i) a taxonomy of representative bugs for Android apps, and (ii) the assignment of the analyzed documents to a specific tag describing the reason behind the bug reported in the document.§.§ The Defined Taxonomyfig:taxonomy depicts the taxonomy that we obtained through the manual coding. The black rectangle in the bottom-right part of fig:taxonomy reports the number of documents tagged as false positive or as unclear. The other rectangles—marked with the Android and/or with the Java logo—represent the 14 high-level categories that we identified. Categories marked with the Android logo (Activities and Intents) group together Android-specific bugs while those marked with the Java logo (Collections and Strings) group bugs that could affect any Java application. Both symbols together indicate categories featuring both Android-specific and Java-related bugs (see I/O). The number reported in square brackets indicates the bug instances (from the manually classified sample) belonging to each category. Inner rectangles, when present, represent sub-categories,Responsiveness/Battery Drain in Non-functional Requirements.Finally, the most fine-grained levels, represented as lighter text, describe the specific type of faults as labeled using our manually-defined tags, the Invalid resource ID tag under the sub-category Resources, in turn part of the Android programming category. The analysis of fig:taxonomy allows to note that:* We were able to classify the faults reported in 1,230 documents (bug reports, commits, ). This number is obtained by subtracting from the tagged documents the 400 tagged as false positives and the 393 tagged as unclear.* Of these 1,230, 26% (324) are grouped in categories only reporting Android-related bugs. This means that more than one fourth of the bugs present in Android apps are specific of this architecture, and not shared with other types of Java systems. Also, this percentage clearly represents an underestimation. Indeed, Android-specific bugs are also present in the previously mentioned “mixed” categories (in Non-functional requirements 25 out of the 26 instances present in the Responsiveness/Battery Drain subcategory are Android-specific—all but Performance (unnecessary computation)). From a more detailed count, after including also the Android-specific bugs in the “mixed" categories,we estimated that 35% (430) of the identified bugs are Android-specific. * As expected, several bugs are related to simple Java programming. This holds for 800 of the identified bugs (65%).Take-away. Over one third (35%) of the bugs we identified with manual inspection are Android-specific. This highlights the importance of having testing instruments, such as mutation operators, tailored for such a specific type of software. At the same time, 65% of the bugs that are typical of any Java application confirm the importance of also considering standard testing tools developed for Java, including mutation operators, when performing verification and validation activities of Android apps. § MUTATION OPERATORS FOR ANDROID Given the taxonomy of faults in Android apps and the set of available operators widely used for Java applications, a catalog of Android-specific mutation operators should (i) complement the classic Java operators, (ii) be representative of the faults exhibited by Android apps, (iii) reduce the rate of still-born and trivial mutants, and (iv) consider faults that can be simulated by modifying statements/elements in the app source code and resources (the strings.xml file). The last condition is based on the fact that some faults cannot be simulated by changing the source code, like in the case of device specific bugs, or bugs related to the API and third-party libraries.Following the aforementioned conditions, we defined a set of operators, trying to cover as many fault categories as possible (10 out of the 14 categories in fig:taxonomy), and complementing the available Java mutation operators. The reasons for not including operators from the other four categories are:* API/Libraries: bugs in this category are related to API/Library issues and API misuses.The former will require applying operators to the APIs; the latter requires a deeper analysis of the specific API usage patterns inducing the bugs;* Collections/Strings: most of the bugs in this category can be induced with classic Java mutation operators;* Device/Emulator: because this type of bug is Device/Emulator specific, their implementation is out of the scope of source code mutations;* Multi-threading: the detection of the places for applying the corresponding mutations is not trivial; therefore, this category will be considered in future work. The list of defined mutation operators is provided in tab:operators and these operators were implemented in a tool named . In the context of this paper, we define a Potential Failure Profile (PFP) that sipulates locations of the analyzed apps—which can be source code statements, XML tags or locations in other resource files—that can be the source of a potential fault, given the faults catalog from sec:taxonomy. Consequently, the PFP lists the locations where a mutation operator can be applied. In order the extract the PFP, statically analyzes the targeted mobile app, looking for locations where the operators from tab:operators can be implemented. The locations are detected automatically by parsing XML files or through AST-based analysis for detecting the location of API calls.Given an automatically derived PFP for an app, and the catalog of Android-specific operators, generates a mutant for each location in the PFP. Mutants are initially generated as clones (at source code-level) of the original app, and then the clones are automatically compiled/built into individual Android Packages (APKs). Note that each location in the PFP is related to a mutation operator. Therefore, given a location entry in the PFP, automatically detects the corresponding mutation operator and applies the mutation in the source code. Details of the detection rules and code transformations applied with each operator are provided in our replication package <cit.>.It is worth noting that from our catalog of Android-specific operators only two operators(DifferentActivityIntentDefinition and MissingPermissionManifest) overlap with the eight operators proposed by Deng , <cit.>. Future work will be devoted to cover a larger number of fault categories and define/implement a larger number of operators.§ APPLYING MUTATION TESTING OPERATORS TO ANDROID APPS The goal of this study is to: (i) understand and compare the applicability of and other currently available mutation testing tools to Android apps; (ii) to understand the underlying reasons for mutants—generated by these tools—that cannot be considered useful for the mutant analysis purposes, mutants that do not compile or cannot be launched. This study is conducted from the perspective of researchers interested in improving current tools and approaches for mutation testing in the context of mobile apps. The study addresses the following research questions: * RQ_1:Are the mutation operators (available for Java and Android apps) representative of real bugs in Android apps?* RQ_2:What is the rate of stillborn mutants (those leading to failed compilations)and trivial mutants (those leading to crashes on app launch) produced by the studied tools when used with Android apps?* RQ_3:What are the major causes for stillborn and trivial mutants produced by the mutation testing tools when applied to Android apps?To answer RQ_1, we measured the applicability of operators from seven mutation testing tools (Major <cit.>, PIT <cit.>, μJava <cit.>, Javalanche <cit.>, muDroid <cit.>, Deng <cit.>, and ) in terms of their ability of representing real Android apps' faults documented in a sample of software artifacts not used to build the taxonomy presented in sec:taxonomy. To answer RQ_2, we used a representative subset of the aforementioned tools to generate mutants for 55 open source Android apps, quantitatively and qualitatively examining the stillborn and trivial mutants generated by each tool. Finally, to answer RQ_3, we manually analyzed the mutants and their crash outputs to qualitatively determine the reasons for trivial and stillborn mutants generated by each tool.§.§ Study Context and Data Collection To answer RQ_1, we analyzed the complete list of mutation operators from the seven considered tools to investigate their ability to “cover” bugs described in 726 artifacts[With “cover” we mean the ability to generate a mutant simulating the presence of a give type of bug.] (103 exceptions hierarchy and API methods throwing exceptions, 245 bug-fixing commits from GitHub, 176 closed issues from GitHub, and 202 questions from SO). Such 726 documents were randomly selected from the dataset built for the taxonomy definition (see sub:taxdesing) by excluding the ones already tagged and used in the taxonomy. The documents were manually analyzed by the eight authors using the same exact procedure previously described for the taxonomy building (two evaluators per document having the goal of tagging the type of bug described in the document; conflicts solved by using a majority-rule schema; tagging process supported by a Web app—details in sub:taxdesing). We targeted the tagging of ∼150 documents per evaluator (600 overall documents considering eight evaluators and two evaluations per document). However, some of the authors tagged more documents, leading to the considered 726 documents. Note that we did not constrain the tagging of the bug type to the ones already present in our taxonomy (fig:taxonomy): The evaluations were free to include new types of previously unseen bugs. We answer RQ_1 by reporting (i) the new bug types we identified in the tagging of the additional 726 documents (the ones not present in our original taxonomy), (ii) the coverage level ensured by each of the seven mutation tools, measured as the percentage of bug types and bug instances identified in the 726 documents covered by its operators. We also analyze the complementarity of with respect to the existing tools.Concerning RQ_2and RQ_3, we compare with two popular open source mutation testing tools (Major and PIT), which are available and can be tailored for Android apps, and with one context-specific mutation testing tool for Android called muDroid <cit.>.We chose these tools because of their diversity (in terms of functionality and mutation operators), their compatibility with Java, and their representativeness of tools working at different representation levels: source code, Java bytecode, and smali bytecode (Android-specific bytecode representation).To compare the applicability of each mutation tool, we need a set of Android apps that meet certain constraints: (i) the source code of the apps must be available, (ii), the apps should be representative of different categories, and (iii) the apps should be compilable (including proper versions of the external libraries they depend upon).For these reasons, we use the Androtest suite of apps <cit.>, which includes 68 Android apps from 18 Google Play categories. These apps have been previously used to study the design and implementation of automated testing tools for Android and met the three above listed constraints. The mutation testing tools exhibited issues in 13 of the considered 68 apps, the 13 apps did not compile after injecting the faults. Thus, in the end, we considered 55 subject apps in our study. The list of considered apps as well as their source code is available in our replication package <cit.>.Note that while Major and PIT are compatible with Java applications, they cannot be directly applied to Android apps. Thus, we wrote specific wrapper programs to perform the mutation, the assembly of files, and the compilation of the mutated apps into runnable Android application packages (APKs).While the procedure used to generate and compile mutants varies for each tool, the following general workflow was used in our study: (i) generate mutants by operating on the original source/byte/smali code using all possible mutation operators; (ii) compile or assemble the APKs either using the , , ortools; (iii) run all of the apps in a parallel-testing architecture that utilizes Android Virtual Devices (AVDs); (iv) collect data about the number of apps that crash on launch and the corresponding exceptions of these crashes which will be utilized for a manual qualitative analysis.We refer readers to our replication package for the complete technical methodology used for each mutation tool <cit.>.To quantitatively assess the applicability and effectiveness of the considered mutation tools to Android apps, we used three metrics: Total Number of Generated Mutants (TNGM), Stillborn Mutants (SM), andTrivial Mutants (TM).In this paper, we consider stillborn mutants as those that are syntactically incorrect to the point that the APK file cannot be compiled/assembled, and trivial mutants as those that are killed arbitrarily by nearly any test case. If a mutant crashes upon launch, we consider it as a trivial mutant. Another metric one might consider to evaluate the effectiveness of a mutation testing tool is the number of equivalent and redundant mutants the tool produces. However, in past work, the identification of equivalent mutants has been proven to be an undecidable problem <cit.>, and both equivalent and redundant mutants require the existence of test suites (not available for the Androtest apps). Therefore, this aspect is not studied in our work. After generating the mutants'using each tool, we needed a viable methodology for launching all these mutants in a reasonable amount of time to determine the number of trivial mutants. To accomplish this, we relied on a parallel Android execution architecture that we call the Execution Engine (EE).EE utilizes concurrently running instances of Android Virtual Devices based on theproject <cit.>. Specifically, we configured 20 AVDs with thev4.4.2 image, a screen resolution of 1900x1200, and 1GB of RAM to resemble the hardware configuration of a Google Nexus 7 device.We then concurrently instantiated these AVDs and launched each mutant, identifying app crashes.§.§ ResultsRQ_1: fig:tag-second-phase reports (i) the percentage of bug types identified during our manual tagging that are covered by the taxonomy of bugs we previously presented in fig:taxonomy (top part of fig:tag-second-phase), and (ii) the coverage in terms of bug types as well as of instances of tagged bugs ensured by each of the considered mutation tools (bottom part). The data shown in fig:tag-second-phase refers to the 413 bug instances for which we were able to define the exact reason behind the bug (this excludes the 114 entities tagged as unclear and the 199 identified as false positives).87% of the bug types are covered in our taxonomy. In particular, we identified 16 new categories of bugs that we did not encounter before in the definition of our taxonomy (sec:taxonomy). Examples of these categories (full list in our replication package) are: Issues with audio codecs, Improper implementation of sensors as Activities, and Improper usage of the static modifier. Note that these categories just represent a minority of the bugs we analyzed, accounting all together for a total of 21 bugs (5% of the 413 bugs considered). Thus, our bug taxonomy covers 95% of the bug instances we found, indicating a very good coverage. Moving to the bottom part of fig:tag-second-phase, our first important finding highlights the limitations of the experimented mutation tools (including ) in potentially unveiling the bugs subject of our study. Indeed, for 60 out of the 119 bug types (50%), none of the considered tools is able to generate mutants simulating the bug. This stresses the need for new and more powerful mutation tools tailored for mobile platforms. For instance, no tool is currently able to generate mutants covering the Bug in webViewClient listener and the Components with wrong dimensions bug types. When comparing the seven mutation tools considered in our study, clearly stands out as the tool ensuring the highest coverage both in terms of bug types and bug instances. In particular, mutators generated by have the potential to unveil 38% of the bug types and 62% of the bug instances. In comparison, the best competitive tool (the catalog of mutants proposed by Deng <cit.>) covers 15% of the bug types (61% less as compared to ) and 41% of the bug instances (34% less as compared to ). Also, we observe that covers bug categories (and, as a consequence, bug instances) missed by all competitive tools. Indeed, while the union of the six competitive tools covers 24% of the bug types (54% of the bug instances), adding the mutation operators included in increases the percentage of covered bug types to 50% (73% of the bug instances). Examples of categories covered by and not by the competitive tools are: Android app permissions, thanks to the MissingPermissionManifest operator, and the FindViewById returns null, thanks to the FindViewByIdReturnsNull operator.Finally, we statistically compared the proportion of bug types and the number of bug instances covered by , by all other techniques, and by their combination, using Fisher's exact test and Odds Ratio (OR) <cit.>. The results indicate that: * The odds of covering bug types using are 1.56 times greater than other techniques, although the difference is not statistically significant (p-value=0.11). Similarly, the odds of discovering faults with are 1.15 times greater than other techniques, but the difference is not significant (p-value=0.25);* The odds of covering bug types using combined with other techniques are 2.0 times greater than the other techniques alone, with a statistically significant difference (p-value=0.008). Similarly, the odds of discovering bugs using the combination of and other techniques are 1.35 times greater than other techniques alone, with asignificant difference (p-value=0.008). RQ_2: Figure <ref> depicts the achieved results as percentage of (a) Stillborn Mutants (SM), and (b) Trivial Mutants (TM) generated by each tool on each app. On average, 167, 904, 2.6k+, and 1.5k+ mutants were generated by , Major, PIT, and muDroid, respectively for each app. The larger number of mutants generated by PIT is due in part to the larger number of mutation operators available for the tool. The average percentage of stillborn mutants (SM) generated by , Major and muDroid over all the apps is 0.56%, 1.8%, and 53.9%, respectively, while no SM are generated by PIT (Figure <ref>). produces significantly less SM than Major (Wilcoxon paired signed rank testp-value<0.001 – adjusted with Holm's correction <cit.>, Cliff's d=0.59 - large) and than muDroid (adjusted p-value<0.001, Cliff's d=0.35 - medium).These differences across the tools are mainly due to the compilation/assembly process they adopt during the mutation process. PIT works at Java bytecode level and thus can avoid the SM problem, at the risk of creating a larger number of TM. However, PIT is the tool that required the highest effort to build a wrapper to make it compatible with Android apps. Major works at the source code level and compiles the app in a “traditional" manner. Thus, it is prone to SM and requires an overhead in terms of memory and CPU resources needed for generating the mutants. Finally, muDroid operates onandcode, reducing the computational cost of mutant generation, but significantly increasing the chances of SM. All four tools generated trivial mutants (TM) (mutants that crashed simply upon launching the app). These instances place an unnecessary burden on the developer, particularly in the context of mobile apps, as they must be discarded from analysis. The mean of the distribution of the percentage of TM over all apps for , Major, PIT and muDroid is 2.42%, 5.4%, 7.2%, and 11.8%, respectively (Figure <ref>). generates significantly less TM than muDroid (Wilcoxon paired signed rank test adjusted p-value=0.04, Cliff's d=0.61 - large) and than PIT (adjusted p-value=0.004, Cliff's d=0.49 - large), while there is no statistically significant difference with Major (adjusted p-value=0.11).While these percentages may appear small, the raw values show that the TM can comprise a large set of instances for tools that can generate thousands of mutants per app. For example, for the Translate app, 518 out of the 1,877 mutants generated by PIT were TM. For the same app, muDroid creates 348 TM out of the 1,038 it generates.For the Blokish app, 340 out of the 3,479 mutants generated by Major were TM. Conversely, while may generate a smaller number of mutants per app, this also leads to a smaller number of TM, only 213 in total across all apps. This is due to the fact that generates a much smaller set ofmutants that are specifically targeted towards emulating real faults identified in our empirically derived taxonomy, and are applied on specific locations detected by the PFP. RQ_3:In terms of mutation operators causing the highest number of stillborn and TM we found that for Major, the Literal Value Replacement (LVR) operator had the highest number of TM, whereas the Relational Operator Replacement (ROR) had the highest number of SM. It may seem surprising that ROR generated many SM, however, we discovered that the reason was due to improper modifications of loop conditions.For instance, in the A2dp.Vol app one mutant changed this loop:and replaced the condition “" with “", causing the compiler to throw an unreachable code error.For PIT, the Member Variable Mutator (MVM) is the one causing most of the TM; for muDroid, the Unary Operator Insertion (UOI) operator has the highest number of SM (although all the operators have relatively high failure rates), and the Relational Value Replacement (RVR) has the highest number of TM. For , the WrongStringResource operator had that highest number of SM, whereas the FindViewByIdReturnsNull operator had the highest number of TM. To qualitatively investigate the causes behind the crashes, three authors manually analyzed a randomly selected sample of 15 crashed mutants per tool. In this analysis, the authors relied on information about the mutation (applied mutation operator and location), and the generated stack trace. Major. The reasons behind the crashing mutants generated by Major mainly fall in two categories. First, mutants generated with the LVR operator that changes the value of a literal causing an app to crash. This was the case for the wikipedia app when changing the “1” in the invocation setCacheMode(params.getString(1)) to “0”. This passed a wrong asset URL to the method setCacheMode, thus crashing the app. Second, the Statement Deletion (STD) operator was responsible for app crashes especially when it deleted needed methods' invocations. A representative example is the deletion of invocations to methods of the superclass when overriding methods, when removing the super.onDestroy() invocation from the onDestroy() method of an Activity. This results in throwing of an android.util.SuperNotCalledException. Other STD mutations causing crashes involved deleting a statement initializing the main Activity leading to a NullPointerException.muDroid. This tool is the one exhibiting the highest percentage of stillborn and TM. The most interesting finding of our qualitative analysis is that 75% of the crashing mutants lead to the throwing of a java.lang.VerifyError. A VerifyError occurs when Android tries to load a class that, while being syntactically correct, refers to resources that are not available (wrong class paths). In the remaining 25% of the cases, several of the crashes were due to the Inline Constant Replacement (ICR) operator. An example is the crash observed in the photostream app where the “100” value has been replaced with “101” in bitmap.compress(Bitmap.CompressFormat.PNG, 100, out). Since “100” represents the quality of the compression, its value must be bounded between 0 and 100. PIT. In this tool, several of the manually analyzed crashes were due to (i) the RVR operator changing the return value of a method to null, causing a NullPointerException, and (ii) removed method invocations causing issues similar to the ones described for Major.MDroid+. tab:mplus-stats lists the mutants generated by across all the systems (information for the other tools is provided with our replication package).The overall rate of SM is very low in , and most failed compilations pertain to edge cases that would require a more robust static analysis approach to resolve.For example, the ClosingNullCursor operator has the highest total number of SM (across all the apps) with 13, and some edge cases that trigger compilation errors involve cursors that have been declared , thus causing the reassignment to trigger the compilation error.The small number of other SM are generally other edge cases, and current limitations of can be found in our replication package with detailed documentation. The three operators generating the highest number of TM are NullIntent(41), FindViewByIdReturnsNull(40), and InvalidIDFindView(30).The main reason for the NullIntent TM are intents invoked by the Main Activity of an app (the activity loaded when the app starts). Intents are one of the fundamental components of Android apps and function as asynchronous messengers that activate Activities, Broadcast Receivers and services.One example of a trivial mutant is for the A2dp.Vol app, in which a bluetooth service, inteneded to start up when the app is launched, causes a NullPointerException when opened due to NullIntent operator.To avoid cases like this, more sophisticated static analysis could be performed to prevent mutations from affecting Intents in an app's MainActivity.The story is similar for the FindViewViewByIdReturnsNull and InvalidIDFindView operators:TM will occur when views in the MainActivity of the app are set to null or reference invalid Ids, causing a crash on startup. Future improvements to the tool could avoid mutants to be seeded in components related to the MainActivity. Also, it would be desirable to allow developers to choose the activities in which mutations should be injected.Summary of the RQs. outperformed the other six mutation tools by achieving the highest coverage both in terms of bug types and bug instances. However, the results show that Android-specific mutation operators should be combined with classic operators to generate mutants that are representative of real faults in mobile apps (RQ_1). generated the smallest rate of both stillborn and trivial mutants illustrating its immediate applicability to Android apps.Major and muDroid generate stillborn mutants, with the latter having a critical average rate of 58.7% stillborn mutants per app (RQ_2). All four tools generated a relatively low rate of trivial mutants, with muDroid again being the worst with an 11.8% average rate of trivial mutants (RQ_3). Our analysis shows that the PIT tool is most applicable to Android apps when evaluated in terms of the ratio between failed andgenerated mutants. However, is both practical and based on Android-specific operations implemented according to an empirically derived fault-taxonomy of Android apps.§ THREATS TO VALIDITY This section discusses the threats to validity of the work related to devising the fault taxonomy, and carrying out the study reported in sec:tools.Threats to construct validity concern the relationship between theory and observation. The main threat is related to how we assess and compare the performance of mutation tools, by covering the types, and by their capability to limit stillborn and trivial mutants. A further, even more relevant evaluation would explore the extent to which different mutant taxonomies are able to support test case prioritization. However, this requires a more complex setting which we leave for our future work.Threats to internal validity concern factors internal to our settings that could have influenced our results. This is, in particular,related to possible subjectiveness of mistakes in the tagging of sec:taxonomy and for RQ_1. As explained, we employed multiple taggers to mitigate such a threat.Threats to external validity concern the generalizability of our findings. To maximize the generalizability of the fault taxonomy, we have considered six different data sources. However, it is still possible that we could have missed some fault types available in sources we did not consider, or due to our sampling methodology. Also, we are aware that in our study results of RQ_1 are based on the new sample of data sources, and results of RQ_2 on the set of 68 apps considered <cit.>.§ CONCLUSIONS Although Android apps rely on theJava language as a programming platform, they have specific elements that make the testing process different than other Java applications. In particular, the type and distribution of faults exhibited by Android apps may be very peculiar, requiring, in the context of mutation analysis, specific operators.In this paper, we presented the first taxonomy of faults in Android apps, based on a manual analysis of software artifacts from six different sources. The taxonomy is composed of categories containing types. Then, based on the taxonomy, we have defineda set of Android-specific mutation operators, implemented in an infrastructure called , to automatically seed mutations in Android apps. To validate the taxonomy and , we conducted a comparative study with Java mutation tools. The study results show that operators are more representative of Android faults than other catalogs of mutation operators, including both Java and Android-specific operators previously proposed.Also is able to outperform state-of-the-art tools in terms of stillborn and trivial mutants.The obtained results make our taxonomy and ready to be used and possibly extended by other researchers/practitioners. To this aim, and the wrappers for using Major and Pit with Android apps are available as open source projects <cit.>. Future work will extend by implementing more operators, and creating a framework for mutation analysiss. Also, we plan to experiment with in the context of test case prioritization.§ ACKNOWLEDGMENTSBavota was supported in part by the SNF project JITRA, No. 172479.ACM-Reference-Format | http://arxiv.org/abs/1707.09038v3 | {
"authors": [
"Mario Linares-Vásquez",
"Gabriele Bavota",
"Michele Tufano",
"Kevin Moran",
"Massimiliano Di Penta",
"Christopher Vendome",
"Carlos Bernal-Cárdenas",
"Denys Poshyvanyk"
],
"categories": [
"cs.SE"
],
"primary_category": "cs.SE",
"published": "20170727204904",
"title": "Enabling Mutation Testing for Android Apps"
} |
1,2]Henri [email protected] 2]Jean-Luc [email protected][author] Corresponding author. E-mail address: [email protected] [1]LIDYL, CEA, CNRS, Université Paris-Saclay, CEA Saclay, 91 191 Gif-sur-Yvette, France [2]Lawrence Berkeley National Laboratory, Berkeley, CA, USA The advent of massively parallel supercomputers, with their distributed-memory technology using many processing units, has favored the development of highly-scalable local low-order solvers at the expense of harder-to-scale global very high-order spectral methods. Indeed, FFT-based methods, which were very popular on shared memory computers, have been largely replaced by finite-difference (FD) methods for the solution of many problems, including plasmas simulations with electromagnetic Particle-In-Cell methods. For some problems, such as the modeling of so-called “plasma mirrors” for the generation of high-energy particles and ultra-short radiations, we have shown that the inaccuracies of standard FD-based PIC methods prevent the modeling on present supercomputers at sufficient accuracy. We demonstrate here that a new method, based on the use of local FFTs, enables ultrahigh-order accuracy with unprecedented scalability, and thus for the first time the accurate modeling of plasma mirrors in 3D. Electromagnetic Particle-In-Cell method; Massively parallel pseudo-spectral solvers; Relativistic plasma mirrors; Pseudo-Spectral Analytical Time Domain solver; Finite-Difference Time-Domain solver§ INTRODUCTION §.§ Challenges in the modeling of Ultra-High Intensity (UHI) physicsThe advent of high power petawatt (PW) femtosecond lasers has paved the way to a new, promising but still largely unexplored branch of physics called Ultra-High Intensity (UHI) physics <cit.>. Once such a laser is focused on a solid target, the laser intensity can reach values as large as 10^22W.cm^-2, for which matter is fully ionized and turns into a “plasma mirror” that reflects the incident light <cit.> (See Fig. <ref>). The corresponding laser electric field at focus is so high, that “plasma mirror” particles (electrons and ions) get accelerated to relativistic velocities upon reflection of the laser on its surface. A whole range of compact “tabletop” sources of high-energy particles (electrons, protons, highly charged ions) and radiations ranging from X-rays to γ-rays may thus be produced from the interaction between this plasma mirror and the ultra-intense laser field at focus <cit.>.The success of PW laser facilities presently under construction worldwide, which aim at understanding and controlling these promising particle and light sources for future application experiments <cit.>, will rely on the strong coupling between experiments and large-scale simulations with Particle-In-Cell (PIC) codes.Nevertheless, standard PIC codes currently in use partly fail to accurately describe most of UHI laser-plasma interaction regimes because the finite-difference time domain (FDTD) Maxwell solver produces strong instabilities and noise when the accelerated particles move at relativistic velocities <cit.> or when the produced short-wavelength radiations span broad emission angles and frequencies <cit.>. With standard PIC codes, the mitigation of these instabilities often requires spatial and temporal resolutions that are so high that they are not practical for realistic 3D modeling on current petascale supercomputers and, it is projected, even on upcoming exascale machines.§.§ Goal and outline of the paper To address this challenge, the solution that we propose here is to use highly precise pseudo-spectral methods to solve Maxwell's equations. Despite their high accuracy,legacy pseudo-spectral methods employing global Fast Fourier Transforms (FFT) on the whole simulation domain have hardly been used so far in large-scale 2D/3D simulations due to their difficulty to efficiently scale beyond 10,000s of cores <cit.>, which is not enough to take advantage of the largest supercomputers required for 3D modeling. To break this barrier a pioneering grid decomposition technique was recently proposed for pseudo-spectral FFT-based electromagnetic solvers <cit.>. The new technique was first validated by an extensive analytical work <cit.> and then implemented in our PIC code Warp+PXR. In this paper, we will first demonstrate that the new technique enables, for the first time, the scaling of pseudo-spectral solvers on up to a million cores. We will then compare the speedup brought by our pseudo-spectral solvers against FDTD solvers in terms of time-to-solution, on a 3D simulation of relativistic plasma mirrors.The paper is divided in 4 sections: * In section 2: we briefly present the standard PIC method and detail its limitations in the modeling of UHI laser-plasma interactions, * In section 3: we describe the new parallelization technique of pseudo-spectral solvers that we implemented in Warp+PXR and that enabled their scaling on up to a million cores, * In section 4: we present scaling tests of our new implementation on the MIRA cluster at Argonne National Laboratory and the Cori cluster at the National Energy Research Scientific Computing Center in Berkeley. We also present performance benefits in terms of time-to-solution of the new solvers against FDTD solvers in the 3D modeling of plasma mirrors.* In section 5: we present future implications of this work on UHI physics and beyond. § LIMITS OF THE STANDARD PARTICLE-IN-CELL METHOD The electromagnetic Particle-In-Cell (PIC) method follows the evolution of a collection of charged macro-particles that evolve self-consistently with their electromagnetic fields. The core algorithm involves four operations at each time step: 1) evolve the velocity and position of the particles using the Newton-Lorentz equations, 2) deposit the charge and/or current densities through interpolation from the particles distributions onto the grid, 3) evolve Maxwell's electromagnetic wave equations on the grid, 4) interpolate the fields from the grid onto the particles for the next particle push. The most popular algorithm for solving Maxwell's wave equations is the Finite-Difference Time-Domain (or FDTD) solver:D_t𝐁= -∇×𝐄D_t𝐄=∇×𝐁-𝐉 where the spatial differential operator is defined as ∇=D_x𝐱̂+D_yŷ+D_zẑ and the finite-difference operators in time and space are defined respectively as D_tG|_i,j,k^n=(G|_i,j,k^n+1/2-G|_i,j,k^n-1/2)/Δ t and D_xG|_i,j,k^n=(G|_i+1/2,j,k^n-G|_i-1/2,j,k^n)/Δ x, where Δ t and Δ x are respectively the time step and the grid cell size along x, n is the time index and i, j and k are the spatial indices along x, y and z respectively. The difference operators along y and z are obtained by circular permutation.Even at relatively high resolution in space and time, the finite difference (FDTD) solver (used in the standard PIC formulation to integrate Maxwell's equations <cit.>) can generate strong non-physical instabilities, which would affect the physics at play. One of the most popular FDTD solver uses the Yee scheme <cit.>, which places fields on a staggered grid, giving second-order accuracy in space and time. Variations include non-standard FDTD schemes that average in the direction orthogonal to the stencils' derivative, for added benefits <cit.> (labeled FDTD-CK in the section 4). Those FDTD solvers use only spatially local information and must hence only exchange a few cells at the margin between each processor's assigned domain neighbors (guard cells, cf. Fig <ref> (a)), thus achieving efficient parallelization up to millions of cores, as required for the simulation of large-scale problems. Nevertheless, it can produce significant unphysical degradation from discretization errors that are highly detrimental in the simulation of relativistic laser-plasma interactions. Using finite-difference solvers, the practically achievable level-of-accuracy is indeed strongly limited by numerical dispersion <cit.> and numerical heating or noise, which are particularly critical for laser-plasma accelerator experiments where small unphysical errors can spoil the required high beam quality, or for simulations where accurate description of a large band of frequencies is required (e.g Doppler harmonics generated on relativistic plasma mirrors). The staggering of the electromagnetic field components on the Yee mesh also leads to errors due to inaccurate cancellation of self electric and magnetic fields components with charged particles moving at relativistic velocities <cit.>. For a large majority of application experiments (e.g plasma harmonic generation spanning hundreds of harmonic orders) where a very accurate description of electromagnetic waves is required on a very large band of frequencies and angles, the resolution needed with finite difference solvers to accurately describe the physics would be so high that it would not be practical to perform a realistic 3D modeling on existing petascale supercomputers and, it is projected, not even on upcoming exascale machines <cit.>.§ NEW TECHNIQUE TO BUILD MASSIVELY PARALLEL PSEUDO-SPECTRAL PIC CODES To address this challenge, our solution is to use ultrahigh-order (p) solvers (up-to the infinite order limit p→∞) pseudo-spectral solvers to solve Maxwell's equations, which advance electromagnetic fields in Fourier space (rather than configuration space) and offer a number of advantages over standard FDTD solvers in terms of accuracy and stability. §.§ Pseudo-spectral solvers for better accuracy In particular, Haber et al. <cit.> showed that under weak assumptions, Fourier transforming Maxwell's equation in space yields an analytical solution for electromagnetic fields in time, called the Pseudo-Spectral Analytical Time Domain (PSATD) solver, which is accurate to machine precision for the electromagnetic modes resolved by the calculation grid. As a consequence, this solver enables infinite order (p→∞), imposes no Courant time step limit in vacuum and has no numerical dispersion. In addition, it represents naturally all field values at the nodes of a grid, thus eliminating errors associated with staggered field quantities. The PSATD algorithm advances the equations in Fourier () space as follows (see <cit.> for the original formulation and <cit.> for a more detailed derivation):^n+1= C^n+iS×^n-S/k^n+1/2+ (1-C)(·^n) +(·^n+1/2)(S/k-Δ t), ^n+1= C^n-iS×^n +i1-C/k×^n+1/2.where ã is the Fourier Transform of the quantity a,=/k, C=cos(kΔ t) and S=sin(kΔ t). As it turns out, the utilization of the wavenumbersin the PSATD algorithm corresponds to taking an infinite order approximation to the spatial derivative operators. In Fourier space, finite order p approximations are obtained simply by substituting the formula k_u→∑_j=1^p/2w_jsin(jk_uΔ u)/Δ u/2 for the wavenumber component along the direction u={ x,y orz} for centered finite differences on a nodal grid, where the w_j are the coefficients of the finite-differenciation at order p at position j in real space. This substitution enables efficient Maxwell's solve at ultrahigh order (e.g. 100 or higher) much more efficiently than a finite-difference-based solver at the same order <cit.>, while significantly reducing the footprint of the stencil, as compared to infinite order <cit.>. §.§ New technique to scale pseudo-spectral solvers to a million cores and beyond Nevertheless, despite significant advantages in terms of accuracy, ultrahigh-order pseudo-spectral solvers have not been widely adopted so far for large-scale simulations because of their poor scalability with increasing number of processors, which is due to the requirement of global inter-processor communications in the computation of global Fourier transforms (cf. Fig <ref> (b)). Hence, while efficient strong scaling to millions of cores has been demonstrated with FDTD solvers, strong scaling with standard pseudo-spectral methods employing global FFTs have previously been reported only up to around 10,000s cores <cit.> and would thus not be adequate for realistic 3D modeling of UHI laser-plasma interactions using modern computers.Recently, our team initiated a change of paradigm for simulation codes solving time-dependent problems where physical information propagates at a finite speed (e.g. Maxwell's equations). This new paradigm is based on using domain decomposition (standard for finite-difference solvers but not for spectral solvers) with spectral (FFT-based) solvers <cit.>. This technique implies a small numerical approximation that falls off very rapidly with the number of guard cells surrounding each subdomain (as explained in more details below), allowing strong scaling of pseudo-spectral solvers to hundreds of thousands of cores and beyond. As in the case of low-order schemes, this technique divides the simulation domain into several subdomains (see Fig <ref> (a)) with guard regions at their borders and Maxwell's equations are solved locally on each subdomain using local FFTs.For large order p solvers and a finite number of guard cells n_g such that p/2>n_g or even p/2≫ n_g, Fig <ref> (a) illustrates that this technique however implies some stencil truncations at subdomain boundaries that could generate spurious errors and that need to be characterized and controlled. The fundamental argument still legitimating this method is that physical information cannot travel faster than the speed of light (since we are solving Maxwell's equations). Choosing large enough guard regions should therefore ensure that spurious signal coming from these stencil truncations at subdomain boundaries would remain in guard regions and would not enter the simulation domain. This is illustrated in Fig. <ref> where a unit pulse is initialized at the center of a domain that has been decomposed into two subdomains (of unequal sizes). While some spurious signal is created in the guard cells' regions because of the domain truncation, it remains confined to the guard regions and does not enter the computational domain, thanks to the finite-speed of light.In 2016, we performed a comprehensive analytical and numerical study <cit.> that derived the exact expression of stencil truncation errors as a function of various numerical parameters (stencil order, number of guard cells, mesh resolution) and demonstrated that truncation errors are not growing even at very high orders p and with a moderately low number of guard cells n_g. In particular, thanks to this study, we are now able to compute the number of guard cells required at a given order to have truncation error amplitudes lower than a given precision, including the zero machine precision. For instance, our model (validated by early numerical benchmarks) shows that 8 guard cells only are required at order 100 to get a negligible error that does not affect the UHI physics. This new paradigm opened the way to the use of these ultrahigh-order pseudo-spectral solvers at large scale for the accurate modeling of 3D laser-plasma interaction regimes that were previously out of reach of previous codes. Collaborators in Europe have also implemented the technique in a code co-developed with our team <cit.> (but not used in this study). § SCALING TESTS AND PERFORMANCE BENEFITS OF THE NEW TECHNIQUE §.§ How Performance Was MeasuredPerformances were measured on (a) full physic simulations of plasma mirrors and (b) Maxwell solver only. The performance simulations (b) were typically performed without I/O, but full scale physics simulations (a) with I/O were performed on up-to 270,000 cores. The full physics simulations (a) were performed on Argonne's National Laboratory Mira supercomputer as part of our 'PICSSAR' 2017 INCITE allocation, while simulations (b) were performed on Mira and U.S. DOE NERSC supercomputer Cori. This section presents the physical and numerical parameters used in simulations (a) as well as applications and kernels that were used for the simulations, the timing procedure, and the platforms Mira and Cori.§.§.§ Full physic simulations of plasma mirrors Recent experiments performed with the 100TW laser UHI 100 at CEA Saclay revealed a crucial feature of the emission from plasma mirrors <cit.>. Plasma mirrors act as injectors of attosecond electron bunches in the specularly reflected laser field that are further accelerated over distances of the Rayleigh length by Vacuum Laser Acceleration (VLA). The spatial pattern observed on the electron beam (see central panel on Fig <ref>) shows a hole in the electron beam spatial profile in the direction of the reflected laser beam in the far field. This was shown as a clear signature of the laser-electron beam interaction in vacuum and provided some of the most direct evidence of VLA. This first experiment opened the way for the first time to the investigation of dynamics of free relativistic electrons in ultra-intense laser fields. 3D Simulations of this process are extremely challenging because the reflected electric field carries a large high harmonic content (see Fig. <ref>) at broad angles and any spurious numerical dispersion induced by FDTD Maxwell solvers will inevitably affect the spatio-temporal phase of harmonic components by deforming the reflected field and therefore significantly impact the properties of the accelerated VLA electrons. Effects of numerical dispersion on high harmonics have already been extensively discussed and we demonstrated that our PSATD local implementation can bring up to two orders of magnitude speed-up over FDTD solvers ro reach convergence <cit.>. Here we will focus on VLA electron properties. Reproducing accurately the features that were observed in the experiment has remained elusive with standard FDTD PIC codes. Hence, the reproduction of the experimental features and the numerical convergence was used as a metric of success for our new pseudo-spectral PIC code.Thanks to the high performance implementation of the PSATD-local solver in our PIC code, we could for the first time use the PSATD solver at very large scale on over 260k cores (16384 nodes) on MIRA to benchmark and quantify its huge benefit over standard solvers in terms of time-to-solution and memory for achieving a given precision, as presented in the remainder of this section. Physical parameters/configuration used in simulations were comparable to the experimental ones in <cit.>: the femtosecond high-intensity laser (intensity I≈10^19W.cm^-2) reflects at 45^o on the plasma mirror and ejects VLA electrons recorded on a detector normal to the specular reflection direction (see central panel on Fig. <ref>). 3D simulation box dimensions are 50λ× 30λ× 70λ along x,y,z directions where λ is the laser wavelength and (x,z) the plane of incidence of the laser on the target. 16 plasma pseudo-particles per cell (electrons and ions) were used in 3D. Spatial resolutions was varied from 66 cells per λ to 330 cells per λ. §.§.§ Applications and timersWarp :Warp <cit.> is an extensively developed open-source 3D Particle-In-Cell (PIC) code designed to simulate a rich variety of physical processes including laser-plasma interactions at high laser intensities. Warp is written in a combination of 1) Fortran for efficient implementation of computationally intensive tasks 2) Python for high level specification and control of simulations and 3) C for interfaces between Fortran and Python.Warp has now been routinely used for many years on NERSC supercomputers[ MCurie/Seaborg/Bassi/Franklin/Hopper/Edison/Cori], on Mira at Argonne National Laboratory and other platforms by many scientists worldwide. The last developments of Warp added the advanced ultrahigh-order scalable Maxwell solver described in the preceding section, that is based on domain decomposition with local FFTs. PICSAR: Under the auspices of the NERSC Exascale Science Application Program (NESAP), and now DOE's Exascale Project, a full Fortran 90 high-performance PIC library PICSAR (“Particle-In-Cell Scalable Application Ressource”) was recently developed by our team <cit.>. This library contains optimized versions of the Warp electromagnetic PIC kernel subroutines. PXR includes numerous optimization strategies to fully benefit from the three levels of parallelisms (Internode, Intranode, Vectorization) offered by current and upcoming architectures (exascale). In particular, thanks to the developments made in PXR (some developments are detailed in <cit.>), Warp+PXR is now a highly optimized code and includes MPI dynamic load balancing at the internode level, optimized MPI stencil communications, hybrid MPI/OpenMP parallelization of the PIC loop, particle tiling and sorting for optimal cache reuse/memory locality and good shared memory OpenMP scaling/intra-node load-balancing, threaded FFTW <cit.> for the advanced Maxwell solvers, as well as cutting edge SIMD algorithms for efficient vectorization of hotspots routines <cit.>. Other optimizations notably include use of MPI-IO for efficient parallel dumping of particles and fields. PXR has been coupled back to Warp through a python layer, by defining a python class that re-defines most of the time consuming Warp methods of the PIC loop. PXR has also now been entirely ported to the new Intel KNL architectures and shows very good performances in the early benchmarks done on NERSC's Cori phase 1 and 2. The Warp+PXR simulation tool is now routinely used on NERSC supercomputers in support of laser-plasma experiments performed at LBNL on the BELLA PW laser and also at CEA Saclay in France on the 100 TW laser UHI100. The PIC loop in Warp+PXR can be run as a set of python routines calling Fortran HPC routines using Forthon or as a standalone full Fortran code.We used the second option for scaling tests on simplified physics problems on Mira and Cori. Timers: Timings that are reported in this section were performed with calls to MPI_WTIME().§.§.§ System and environment PXR simulations have been run on two large-scale systems (i) The Mira supercomputer at the Argonne Leadership Computer Facility (ALCF) and (ii) The Cori supercomputer at the National Energy Research Scientific Computing Center (NERSC). Mira: Mira <cit.>, an IBM Blue Gene/Q supercomputer at the Argonne Leadership Computing Facility, is equipped with 786,432 cores, 768 terabytes of memory and has a peak performance of 10 petaflops. Mira's 49,152 compute nodes have a PowerPC A2 1600 MHz processor containing 16 cores, each with 4 hardware threads, running at 1.6 GHz, and 16 gigabytes of DDR3 memory. A 17th core is available for the communication library.IBM's 5D torus interconnect configuration, with 2GB/s chip-to-chip links, connects the nodes, enabling highly efficient computation by reducing the average number of hops and latency between compute nodes. Environment use to compile/link the code: MPICH3, OpenMP 3.0, powerpc-gnu-linux-gcc-cnk v4.4.7 (bgqtoolchain-gcc447), FFTW v3.3.5, Compiler options: '-O3 -fopenmp'. Cori: U.S. DOE NERSC's newest supercomputer, named Cori <cit.> and ranked five of the 500 most powerful supercomputers in the world, includes the Haswell partition (Phase I) and the KNL partition (Phase II). We used the KNL partition, which has a (theoretical) peak performance of 27.9 petaflops/sec, 9,688 compute nodes (658,784 cores in total), and 1 PB of memory. Each node is a single-socket Intel Xeon Phi Processor 7250 ("Knights Landing") processor with 68 cores per node at 1.4 GHz. Each core has two 512-bit-wide vector processing units. Each core has 4 hardware threads (272 threads total). Two cores form a tile. The peak flops counts are 44 GFlops/core, 3 TFlops/node and 29.1 PFlops total.Concerning memory, each node has 96 GB DDR4 2400 MHz memory, six 16 GB DIMMs (102 GB/s peak bandwidth), for a total aggregate memory (combined with MCDRAM) of 1 PB. Each node also has 16 GB MCDRAM (multi-channel DRAM), > 460 GB/s peak bandwidth. Each core has its own L1 caches, with 64 KB (32 KB instruction cache, 32 KB data). Each tile (2 cores) shares a 1MB L2 cache. The interconnect is a Cray Aries with Dragonfly topology with 45.0 TB/s global peak bisection bandwidth.The operating system is a lightweigh Linux based on the SuSE Linux Enterprise Server distribution.The batch scheduler is SLURM. On Cori, PXR has been compiled using the Intel compiler version 17.0.1.132.The MPI implementation is developped by CRAY based on MPICH. We use a specific option to have memory page size of 2 Mb (huge page) instead of the default 4 Kb page size enabling fastest communications with less fluctuations.The code is compiled with the following arguments on KNL: -O3 -xMIC-AVX512 -align array64byte. Libraries/API: FFTW v3.3.5, OpenMP 4.0. §.§ Performance Results§.§.§ Maxwell solver only Fig. <ref> presents performance results (weak/strong scaling) of global and local implementations of the pseudo-spectral PSATD Maxwell solver on Cori-KNL and Mira machines.The global implementation (PSATD-global) employs the global 3D distributed FFTs/IFFTs of the FFTW-MPI library (parallelized using MPI. Threading was activated). As detailed in section 3, our new local implementation (PSATD-local) instead uses a cartesian MPI domain decomposition with local 3D FFTs/IFFTS performed on each subdomain using FFTW threaded version. We used 8 guard cells at the margin of each MPI-subdomain that are exchanged between neighboring subdomains each time step. For order p=100 PSATD (used here), we can demonstrate <cit.> that 8 guard cells are enough toreduce truncation errors amplitude and their growth to a negligible level in the simulations that were conducted. The results detailed in Fig. <ref> demonstrate excellent strong and weak scaling of our PSATD-local implementation (red squares on panels (a) and (b)) on up to 800k cores on the full MIRA machine (96% efficiency for the weak scaling on the full machine). As a comparison, the common PSATD-global implementation (blue dots on panels (a) and (b)) performs very poorly at very large scale. On panel (a), the strong scaling efficiency drops at around 20k cores. On panel (b) the weak scaling efficiency drops dramatically with the number of cores at large scale due to the increase in the volume of exchanged data between MPI processes as required by the transposition in global FFTs. For the strong scaling, we could not scale the PSATD-global implementation beyond 120k cores because the FFTW implementation only allows CPU split along one dimension in space (last dimension in Fortran). For the weak scaling, we could not obtain scaling data beyond 120k cores because FFTW-MPI was requiring too much memory. On Cori-KNL, the PSATD-local solver demonstrated excellent strong scaling on up to 100k cores (red/magenta curves on panel (c)) and very good weak scaling (red squares on panel (d)) with 98% efficiency on 260k cores (half machine).On the contrary, the PSATD-global implementation exhibits again very poor performance. The strong scaling efficiency drops at 10k nodes and execution time even increases with the number of MPI processes due to MPI exchanges in the global FFT. At 20k nodes, we already have almost 2 orders of magnitude speed-up between the local and global PSATD in terms of time-to-solution.Notice that for panels (a) and (c), the PSATD-global (red curve) is slower than the PSATD-local (blue curve) for the same problem size even for a low number of cores. This is due to the fact that the FFT complexity varies as N log N and the global-FFT is performed on an array size N larger than the one for each individual local FFTs, in addition to performing a global transpose that is not needed with PSATD-local. §.§.§ Full physic simulations of plasma mirrors Fig. <ref> shows 3D PIC simulation results of relativistic electrons accelerated by the Vacuum Laser Acceleration mechanism. Here we assess convergence rate of FDTD and PSATD solvers on VLA electron properties.Panels (a) and (b) show angular distribution of VLA electrons obtained with Yee and PSATD solvers at similar resolutions. One can see that the PSATD solver reproduces all characteristic features of the experimental measurements on panel (c):the VLA electron beam is located between the target normal and specular direction (θ_>0) and there is a clear hole in the electron distribution projected along θ_⊥ and θ_. On the contrary the simulation performed with the Yee solver leads to wrong results (electrons located in the hole, much more electrons below the specular direction i.e for θ_<0). These spurious artefacts are induced by numerical dispersion of the Yee scheme. Convergence tests were performed at different resolutions (cf. Fig. <ref>) in 2D and 3D.Fig. <ref> (b-c) anddemonstrate convergence of the PSATD at order 100 already for 66 cells per laser wavelength λ (max resolution carried out in 3D), while Yee and CK solvers would at least require ≈300 cells/λ (cf. Fig. <ref> (a)) .In addition, we also observe that PSATD at order 64 converges more slowly than at order 100, justifying the use of ultrahigh order. These simulation results thus enable real measurement (PSATD solver) and estimates (Yee,CK) of resources-to-solution needed for convergence, which are given in table <ref>, with speedups ranging between 240× to 430× for PSATD (order 100) over standard FDTD solvers. Note that we provide estimates for Yee and CK solvers based on projections from 2D simulations of Fig. <ref> (a), as we could not carry 3D simulations on up to ≈300 cells/λ on even the largest available machines at time of writing. § FUTURE IMPLICATIONSThese results may have a huge fundamental impact on UHI physics, as 3D accurate PIC simulations are essential to the detailed understanding of these new laser-plasma interaction regimes, where particle motion is highly relativistic and very short-wavelength radiations can be emitted at broad angles. In particular, by accurately capturingthe spatio-temporal properties of ultra-compact attosecond electron and X-UV light sources from plasma mirrors, the new method can be used to identify optimal regimes of productions of these sources for performing promising application experiments such as attosecond pump-probe experiments or time-resolved diffraction imagery. Beyond plasma mirrors, our general approach may also impact other applications relying on the self-consistent modeling of plasmas or charged particle beam. For instance, by eliminating dispersion and minimizing heating errors, the algorithm proposed here is also especially important to the conception of next generation PW laser wakefield acceleration experiments where the orders of magnitude higher electron beam quality (in emittance and/or energy spread) that are required will necessitate a similar increase in numerical accuracy. In fact, the new algorithm is one of the key innovations that are at the heart of DOE's Exascale application project “Exascale modeling of advanced particle accelerators” <cit.>. Another innovation, that is key to the efficient modeling of laser-plasma accelerators, is the reduction of the number of time steps that are required to model the propagation of a laser beam through an under-dense plasma by orders of magnitude, by choosing an optimal relativistic Lorentz boosted frame of reference for the calculation <cit.>. As it turns out, the PSATD solver is paramount to a new method <cit.> eliminating the so-called “numerical Cherenkov instability” <cit.>, and thus enabling the orders of magnitude speedup of the Lorentz boosted frame method to its full potential.In addition, by reducing the number of space and time steps required for a given accuracy to the solution of a plasma physics problem, the new method can considerably reduce the time to solution in both the design of devices and the study of fundamental science in the area of plasma and electro-energetic physics, including -but not limited to- laser plasma acceleration and relativistic optics (e.g. filamentation, high harmonic generation, ion acceleration). As we demonstrated in this paper, this renders for the first time converged 3D simulations of plasma mirrors accessible on existing supercomputers and 2D simulations on a local workstation. Finally, the method presented here has the potential for having a broader impact on a large class of computational physics problems, as the underlying concept is applicable in principle to various initial value problems that can be treated by FFT-based pseudo-spectral methods. The integration of Maxwell's equations is just one example of the so-called `initial value' problems that are some of the most important in science, where differential equations are integrated numerically in time based on the initial state of the system and time-dependent source terms. Examples of initial value problems include the diffusion equation, Vlasov equation, general relativity, Schrdinger equation, etc (indeed our team successfully tested the method on the modeling of the heat equation). When these problems can be treated by FFT-based pseudo-spectral methods, the new parallelization method demonstrated here may in some cases apply and enable better scalability, especially on future exascale supercomputers and beyond, where scaling to ultrahigh concurrency will be paramount.§ ACKNOWLEDGEMENT We thank Guillaume Blaclard, Dr. Remi Lehe, Dr. Brendan Godfrey, Dr. Irving Haber and Dr. Fabien Quere for fruitful discussions. We are also very grateful to Dr. A. Leblanc who provided us with the experimental results performed on the UHI100 laser at CEA Saclay. This work was supported by the European Commission through the Marie Skłowdoska-Curie actions (Marie Curie IOF fellowship PICSSAR grant number 624543) as well as by the Director, Office of Science, Office of High Energy Physics, U.S. Dept. of Energy under Contract No. DE-AC02-05CH11231, the US-DOE SciDAC program ComPASS, and the US-DOE program CAMPA. An award of computer time (PICSSAR_INCITE) was provided by the Innovative and Novel Computational Impact on Theory and Experiment (INCITE) program. This research used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC02-06CH11357 as well as resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.This document was prepared as an account of work sponsored in part by the United States Government. 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"authors": [
"Henri Vincenti",
"Jean-Luc Vay"
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"published": "20170726153615",
"title": "Ultrahigh-order Maxwell solver with extreme scalability for electromagnetic PIC simulations of plasmas"
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A minimaj-preserving crystal on ordered multiset partitions [ December 30, 2023 ===========================================================fancyplaingobble Accepted final version. To appear in Proc. of the 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems.2017 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.emptyMulti-robot transfer learning allows a robot to use data generated by a second, similar robot to improve its own behavior. The potential advantages are reducing the time of training and the unavoidable risks that exist during the training phase. Transfer learning algorithms aim to find an optimal transfer map between different robots. In this paper, we investigate, through a theoretical study of single-input single-output (SISO) systems, the properties of such optimal transfer maps. We first show that the optimal transfer learning map is, in general, a dynamic system. The main contribution of the paper is to provide an algorithm for determining the properties of this optimal dynamic map including its order and regressors (i.e., the variables it depends on). The proposed algorithm does not require detailed knowledge of the robots' dynamics, but relies on basic system properties easily obtainable through simple experimental tests. We validate the proposed algorithm experimentally through an example of transfer learning between two different quadrotor platforms. Experimental results show that an optimal dynamic map, with correct properties obtained from our proposed algorithm, achieves 60-70% reduction of transfer learning error compared to the cases when the data is directly transferred or transferred using an optimal static map.§ INTRODUCTION Machine learning approaches have been successfully applied to a wide range of robotic applications. This includes the use of regression models, e.g., Gaussian processes and deep neural networks, to approximate kinematic/dynamic models <cit.>, inverse dynamic models <cit.>, and unknown disturbance models <cit.> of robots.It also includes the use of reinforcement learning (RL) methods to automate a variety of human-like tasks such as screwing bottle caps onto bottles and arranging lego blocks <cit.>. Nevertheless, machine learning methods typically require collectinga considerable amount of data from real-world operation or simulations of the robots, or a combination of both <cit.>.Transfer learning (TL) reduces the burden of a robot to collect real-world data by enabling it to use the data generated by a second, similar robot <cit.>. This is typically carried out in two phases <cit.>. In the first phase, both robots generate data, and an optimal mapping between the generated data sets is learned. In the second phase, the learned map is used to transfer subsequent learning data collected by the second robotic system, called the source system, to the first robotic system, called the target system (see Figure <ref>). The goal of transfer learning is toreduce the time needed for teaching robots new skills and to reduce the unavoidable risks that usually exist in the training phase, particularly for the cases where the target robotic platform is more expensive or more hazardous to operate than the source robotic platform.The use of transfer learning in robotics can be classified into (i) multi-task transfer learning, in which the data gathered by a robot when learning a particular task is utilized to speed up the learning of the same robot in other similar tasks <cit.>, and (ii) multi-robot transfer learning, where the data gathered by a robot is used by other similar robots <cit.>. The latter is the main focus of this paper. Multi-robot transfer learning has received less attention in the literature, cf. <cit.>. In <cit.>, task-dependent disturbance estimates are shared among similar robots to speed up learning in an iterative learning control (ILC) framework, while in <cit.>, polices and rules are transferred between simple, finite-state systems in an RL framework to accelerate robot learning.For a similar configuration, skills learned by two different agents in <cit.> are used to train invariant feature spaces instead of transferring policies. One typical approach for transfer learning, used in many applications including robotics, is manifold alignment, which aims to find an optimal, static transformation that aligns datasets <cit.>. In <cit.>, manifold alignment is used to transfer input-output data of a robotic arm to another arm to improve the learning of a model of the second arm. As partially stated in <cit.>, although multi-robot transfer learning has been successfully applied in some robotic examples, there is still an urgent need for a general, theoretical study of when multi-robot transfer learning is beneficial, how the dynamics of the considered robots affect the quality of transfer learning, what form the optimal transfer map takes, and how to efficiently identify the transfer map from a few experiments. To fill this gap, the authors of <cit.> recently initiated a study along these lines for two first-order, linear time-invariant (LTI), single-input single-output (SISO) systems. In particular, in <cit.>, a simple, constant scalar is applied to align the output of the source system with the output of the target system, and then an upper bound on the Euclidean norm of the transformation error is derived and minimized with respect to (w.r.t.) the transformation parameter. The paper <cit.> also utilizes the derived, minimized upper bound to analyze the effect of the dynamics of the source and target systems on the quality of transfer learning. In this paper, we study how the dynamical properties of the two robotic systems affect the choice of the optimal transfer map. This paper generalizes <cit.>, as we consider higher-order, possibly nonlinear dynamical systems and remove the restriction that the transformation map is a static gain. The contributions of this paper may be summarized as follows. First, while many transfer learning methods in the literature depend on finding an optimal, static map between multi-robot data sets <cit.>, we show through our theoretical study that the optimal transfer map is, in general, a dynamic system. Recall that in the time domain, static maps are represented by algebraic equations, while dynamic maps are represented by differential or difference equations. Second, we utilize our theoretical study to provide insights into the correct features or properties of this optimal, dynamic map, including its order and regressors (i.e., the variables it depends on). Third, based on these insights, we provide an algorithm for selecting the correct features of this transformation map from basic properties of the source and target systems that can be obtained from few, easy-to-execute experimental tests. Knowing these features greatly facilitates learning the map efficiently and from little data. Fourth, we verify the soundness of the proposed algorithm experimentally for transfer learning between two different quadrotor platforms. Experimental results show that an optimal, dynamic map, with correct features obtained from our proposed algorithm, achieves 60-70% reduction of transfer learning error, compared to the cases when the data is directly transferred or transferred through a static map.This paper is organized as follows. Section <ref> provides preliminary, dynamic-systems definitions. In Section <ref>, we define the transfer learning problem studied in this paper. In Section <ref>, we provide theoretical results on transformation maps that achieve perfect transfer learning, and then utilize these results to provide insights into the correct features of optimal transfer maps. In Section <ref>, we present our proposed, practical algorithm. Section <ref> includes a robotic application, and Section <ref> concludes the paper.§ BACKGROUND In this section, we review basic definitions from control systems theory needed in later sections, see <cit.>. We first introduce these definitions for linear systems, and then generalize them to an important class of nonlinear systems, namely control affine systems. To that end, consider first the LTI, SISO, n-dimensional state space model,ẋ(t) =Ax(t) + Bu(t)y(t) =Cx(t), where x(t)∈^n is the system state vector, u(t)∈ is its input, and y(t)∈ is its output. It is well known that the input-output representation of (<ref>) is the transfer functionG(s)=Y(s)/U(s)=C(sI-A)^-1B=:N(s)/D(s), where N(s) and D(s) are polynomials in s, and we assume without loss of generality (w.l.o.g.) that they do not have common factors.Evidently, the system (<ref>) is bounded-input-bounded-output (BIBO) stable if and only if all the roots of D(s) are in the open left half plane (OLHP).The relative degree of the system (<ref>) is deg(D(s))-deg(N(s)), that is the order of the denominator polynomial D(s) minus the order of the numerator polynomial N(s).The definition of relative degree remains the same for discrete-time linear systems Y(z)/U(z)=N(z)/D(z), where z is the forward shift operator. For (<ref>), it can be shown using the series expansion of the transfer function (<ref>) that the relative degree is the smallest integer r for which CA^r-1B≠ 0, and consequently, the relative degree is also the lowest-order derivative of the output that explicitly depends on the input recalling y^(r)(t)=CA^rx(t)+CA^r-1Bu(t), where y^(r)(t) represents the r-th derivative of y(t) w.r.t. t. The relative degree r can be calculated from the step response of the system. For continuous-time systems, it is the lowest-order derivative of the step response y that changes suddenly when the input u is suddenly changed. For discrete-time systems, it is the number of sample delays between changing the input and seeing the change in the output. We now extend the relative degree definition to nonlinear systems. Let C^∞ denote the class of smooth functions whose partial derivatives of any order exist and are continuous.The Lie derivative of a smooth function λ(x) w.r.t. a smooth vector field f(x), denoted L_fλ, is the derivative of λ in the direction of f; that is, L_fλ:=∂λ/∂ xf(x). The notation L_f^2λ is used for the repeated Lie derivative; that is, L_f^2λ=L_f(L_fλ(x))=∂ L_fλ(x)/∂ xf(x). Similarly, one can derive an expression for L_f^kλ, where k>1.Now consider the SISO control affine system,ẋ(t) = f(x(t)) + g(x(t))u(t)y(t) =h(x(t)), where x(t)∈ D ⊂^n, u(t)∈, y(t)∈, and f, g, h are C^∞, nonlinear functions. Analogous to linear systems, the relative degree of the system (<ref>) is the smallest integer r for which L_gL_f^r-1h(x)≠ 0, for all x in the neighborhood of the operating point x_0.By successive derivatives of the output y, it can be shown that y^(r)=L_f^r h(x)+L_gL_f^r-1h(x)u.Hence, the relative degree again represents the lowest-order derivative of the output that explicitly depends on the input. For example, the nonlinear dynamics θ̈=-cos(θ)+u, θ∈ (-π,π), with output θ and input u, have relative degree 2 for all θ in the operating range. We next review the left inverse of the dynamics (<ref>), which is used in the literature to reconstruct input u from the output y, see <cit.>, and which we utilize in our discussion in Section <ref>. Note that the inverse dynamics of linear systems can be easily derived from the transfer function, and its stability is determined by the zeros of the original transfer function (the roots of the polynomial N(s)). Suppose that (<ref>) has a well-defined relative degree r in the operating range. Recall that y^(r)=L_f^r h(x)+L_gL_f^r-1h(x)u, where by definition L_gL_f^r-1h(x) ≠ 0. By reordering this equation and from (<ref>), we obtain the inverse dynamics ẋ = (f(x)-g(x)L_f^rh(x)/L_gL_f^r-1h(x)) + g(x)/L_gL_f^r-1h(x)y^(r)u =-L_f^rh(x)/L_gL_f^r-1h(x)+ 1/L_gL_f^r-1h(x)y^(r),with input y^(r) and output u. A necessary condition for the stability of (<ref>) is that the dynamics of (<ref>) when y(t)=0 uniformly (consequently, y^(r)(t)=0) are stable in the Lyapunov sense; this is called the zero dynamics of the system (<ref>). While it appears from (<ref>) that the inverse dynamics have n states, this is not the minimum realization of the inverse dynamics. Instead, for dynamic systems (<ref>) with well-defined relative degree, one can always find a nonlinear coordinate transformation to convert (<ref>) into a special form, called the Byrnes-Isidori normal form. Using this form, a minimum realization of (<ref>) can be derived with (n-r) states, inputs y,ẏ,⋯,y^(r), and output u, refer to <cit.>. § PROBLEM STATEMENT In this paper, we study transfer learning between two robotic systems from a dynamical system perspective. We use our theoretical results to provide insights into the properties of optimal transfer maps. These insights facilitate the identification of this optimal map from data using, for instance, system identification algorithms.In particular, as shown in Figure <ref>, we consider a source SISO dynamical system _S with input reference signal d and output y_s, representing the source robot, and a target SISO dynamical system _T with the same input d and output y_t, representing the target robot. Assuming that d is an arbitrary bounded signal, the transfer learning problem is to find a transfer map _TL with input y_s and output y_TL such that the error e between y_t and y_TL is minimized.To make the transfer learning problem tractable, we assume that both the source system _S and the target system _T are input-output stable (this is typically characterized by the BIBO stability notion for LTI systems and by the Input to Output Stability (IOS) notion for nonlinear systems <cit.>). This is a reasonable assumption, given that input-output stability is necessary for the safe operation of the robot and transfer learning is only efficient for stable systems <cit.>. § MAIN RESULTS In this section, we assume that the source system dynamics _S and the target system dynamics _T are known, and then provide a theoretical study on when it is possible to identify a dynamic map _TL that achieves perfect transfer learning from _S to _T, i.e., it perfectly aligns y_t and y_TL resulting in zero transfer learning error (e(t)=0). From this theoretical study, we provide insights into the correct properties of the dynamic map _TL, including its order, relative degree, and input-output variables. We then show that these properties can be determined from basic properties of the source and target systems, which can be identified through short, simple experiments. There is no need to know the source/target system dynamics a priori. Knowing the properties of the optimal transfer map greatly facilitates the identification of this map from data using standard system identification tools, as we will show in Section <ref>.For simplicity, we first present our theoretical study and insights for linear systems, and then show that these insights remain valid for nonlinear systems. To that end, in this paper, we say that an LTI system is minimum-phase if the dynamics of the system and its inverse dynamics are BIBO stable. Consider two continuous-time, BIBO stable, SISO, LTI systems, with rational transfer functions G_S(s) and G_T(s), and suppose that G_S(s) is minimum-phase. Then, there exists a causal, BIBO stable map from the source system G_S(s) to the target system G_T(s) that achieves perfect transfer learning if and only if the relative degree of G_S(s) ≤ the relative degree of G_T(s).(⇒) Let G_S(s):=N_S(s)/D_S(s) and G_T(s):=N_T(s)/D_T(s). By assumption, there exists a causal function G_α(s) such that for any bounded input u, (G_α(s)G_S(s)-G_T(s))U(s)=0. Since u is arbitrary, then clearly G_α(s)G_S(s)-G_T(s)=0. Equivalently, G_α(s)G_S(s) = G_T(s). Let G_α(s):=N_α(s)/D_α(s). Then, we haveN_α(s)/D_α(s)N_S(s)/D_S(s) = N_T(s)/D_T(s). Even in the presence of pole-zero cancellations, it can be shown that the above equation implies(deg(D_α(s))+deg(D_S(s)))-(deg(N_α(s))+deg(N_S(s)))=(deg(D_T(s))-deg(N_T(s))).By reordering the terms on the left hand side (LHS), the summation of the relative degrees of G_α(s) and G_S(s) is equal to the relative degree of G_T(s). Since G_α(s) is causal, the relative degree of G_α(s)≥0, and the result follows.(⇐) Suppose that the relative degree of G_S(s) ≤ the relative degree of G_T(s). We construct a causal, stable map G_α(s) that achieves perfect transfer learning. LetG_α(s):=D_S(s)/N_S(s)N_T(s)/D_T(s).Since the relative degree of G_S(s) ≤ the relative degree of G_T(s), we havedeg(D_S(s))-deg(N_S(s))≤ deg(D_T(s))-deg(N_T(s)). Equivalently,deg(D_S(s))+deg(N_T(s))≤ deg(D_T(s))+deg(N_S(s)).This implies G_α(s) is a causal function. Notice that the poles of G_α(s) are a subset of the roots of D_T(s) and N_S(s). Then, since G_T(s) is BIBO stable and G_S(s) is minimum-phase by assumption,the roots of D_T(s) and N_S(s) are all in the OLHP, and G_α(s) is a BIBO stable transfer function. Next, one can verify that for the selected G_α(s), we have G_α(s)G_S(s)-G_T(s)=0, and consequently G_α(s) achieves perfect transfer learning.Equation (<ref>) and its associated discussion are similar to standard methods in linear control synthesis. Similar results can be derived for discrete-time LTI systems.Consider two discrete-time, BIBO stable, SISO, LTI systems, with rational transfer functions G_S(z), G_T(z), and suppose that G_S(z) is minimum-phase. Then, there exists a causal, BIBO stable map from the source system G_S(z) to the target system G_T(z) that achieves perfect transfer learning if and only if the relative degree of G_S(z) ≤ the relative degree of G_T(z).Theorems <ref> and <ref> provide the following insights into the properties of the optimal transfer maps between systems.Insight 1: From (<ref>), one can see that the optimal transfer map is, in general, a dynamic system. Therefore, limiting the transfer map to be static <cit.> may be restrictive; see also Section <ref>.Insight 2: To be able to identify the optimal transfer learning map from data using system identification algorithms, it is important to decide on the right order of the dynamic map. From (<ref>), the order of the optimal map that achieves zero transfer learning error is in general deg(N_S(s))+deg(D_T(s)). Equivalently, the correct order of the map is n_s-r_s+n_t, where n_s is the order of the source system, n_t is the order of the target system, and r_s is the relative degree of the source system, which can be identified experimentally from the system step response as stated in Section <ref>.Insight 3: From (<ref>), the relative degree of the optimal transfer learning map is r_t-r_s, where r_s, r_t are the relative degrees of the source and target systems, respectively. The relative degree of the transfer map is also needed for standard system identification algorithms. By knowing the order and the relative degree of the transfer learning map, the regressors of the map are determined. For instance, for a discrete-time transfer learning map of order 3 and relative degree 1, the map relates the output y_TL(k) to the inputs y_TL(k-1), y_TL(k-2), y_TL(k-3),y_s(k-1),y_s(k-2),y_s(k-3).Insight 4: From Theorems <ref> and <ref>, if the relative degree of the source system r_s is greater than the relative degree of the target system r_t, then we cannot find a causal map satisfying perfect transfer learning (zero transfer learning error). Nevertheless, since we have the complete input-output data of the source robot available before carrying out the transfer learning, the causality requirement can be relaxed. For instance, although a discrete-time transfer learning map from {y_s(k-1),y_s(k),y_s(k+1),y_TL(k-1)} to y_TL(k) is non-causal, it can be implemented since all the future values of y_s are saved before using the transfer learning map for transferring the source data to the target system.However, system identification computer tools such as MATLAB's identification toolbox are typically used for identifying causal models such as causal transfer functions (MATLAB: ), nonlinear autoregressive exogenous (NARX) models (MATLAB: ), and recurrent neural networks, among others. One possible trick to solve this problem is to tailor the input of the dynamic transfer learning map as follows.First, for continuous-time systems, we know from (<ref>) that the optimal transfer map is Y_TL(s)/Y_s(s)=N_α(s)/D_α(s), where deg(D_α)=n_s+n_t-r_s, deg(N_α)=n_s+n_t-r_t, and for this case deg(N_α)>deg(D_α) (non-causal map). Instead of identifying this non-causal map, we use standard system identification computer tools to identify the causal map Y_TL(s)/s^(r_s-r_t)Y_s(s)=N_α(s)/s^(r_s-r_t)D_α(s), which represents the Laplace transform of the map from y_s^(r_s-r_t)(t), the (r_s-r_t)-th derivative of the source robot's output y_s(t), to y_TL(t). Hence, from the y_s(t) response, we calculate y_s^(r_s-r_t)(t) (and possibly low-pass filter y_s(t) to avoid noise amplification). We then use y_s^(r_s-r_t)(t) as the input to the dynamic transfer map to be identified. Notice that this is not the only choice. One can, for example, use the system identification tools to identify the causal map Y_TL(s)/P(s)Y_s(s)=N_α(s)/P(s)D_α(s), where P(s) is a known (r_s-r_t)-th order polynomial in s, with all its roots in the OLHP. Since both y_s and the polynomial P(s) are known, one can define the data column for the tailored input of the dynamic map to be identified. For instance, if for r_s-r_t=1, one selects the polynomial P(s)=s+1, then the tailored input to the dynamic map, to be identified, is y_s(t)+ẏ_s(t), and so on. Similarly, for discrete-time systems, we use system identification tools to identify the causal map Y_TL(z)/z^(r_s-r_t)Y_s(z)=N_α(z)/z^(r_s-r_t)D_α(z). For this tailored, causal map, the input is y_s,mod, which is obtained by shifting each element in the data column for y_s forward in time by (r_s-r_t) samples, and the output is y_TL. We now show that these insights remain valid for nonlinear systems. Theorems<ref>, <ref> and the related insights mainly depend on the definition of relative degree, which is also defined for control affine nonlinear systems as discussed in Section <ref>. Hence, suppose that we have two smooth, control affine nonlinear systems of the form (<ref>): a source system with order n_s and a well-defined relative degree r_s in the operating range, and a target system with order n_t and a well-defined relative degree r_t in the operating range. Also, suppose that the source system dynamics and its inverse dynamics are both input-output stable, and that the target dynamics are input-output stable <cit.>.From (<ref>), one can see that the optimal transfer map, that achieves zero transfer learning error, is composed of two cascaded systems: the inverse of the source system dynamics and the target system dynamics. Intuitively, the inverse of the source dynamics is utilized to successfully reconstruct the input d from the source output response y_s, and then the target system dynamics are applied to exactly obtain the target output response y_t from d. A similar approach can be utilized for nonlinear systems to get zero transfer learning error. From the last paragraph in Section <ref>, we know that the minimum realization of the inverse dynamics of the source system has order n_s-r_s, while the order of the target system dynamics is by definition n_t. Therefore, the correct order of the optimal dynamic map, composed of these two cascaded systems, is n_s+n_t-r_s, which is the same conclusion we reached in Insight 2 for linear systems. Then, for this optimal map, we have from the relative degree definition for the source system with internal state x (subscript S is dropped from f, g, h for notational simplicity)y_s^(r_s)=L_f^r_s h(x)+L_gL_f^r_s-1h(x)d, where L_gL_f^r_s-1h(x)≠ 0, and for the target dynamics with internal state v (subscript T is dropped from f, g, h)y_TL^(r_t)=L_f^r_t h(v)+L_gL_f^r_t-1h(v)d, where L_gL_f^r_t-1h(v)≠ 0.By getting d from the first equation and substituting it in the second one, we havey_TL^(r_t)=L_f^r_t h(v)-L_gL_f^r_t-1h(v) L_f^r_sh(x)/L_gL_f^r_s-1h(x)+ L_gL_f^r_t-1h(v)/L_gL_f^r_s-1h(x)y_s^(r_s),i.e., y_TL^(r_t) explicitly depends on y_s^(r_s), and consequently, it is reasonable to select the relative degree of the optimal map from y_s to y_TL to be r_t-r_s as in Insight 3.§ ALGORITHM Inspired by the insights presented in the previous section, we provide an algorithm for getting the correct properties of the optimal, dynamic transfer learning map between two robotic systems from simple experiments. As discussed before, we assume that both systems are input-output stable and that the source system has stable inverse dynamics.Once the properties of the map are determined, one can utilize any system identification tool, such as MATLAB's identification toolbox, to identify the map from collected data as we will show in our practical examples in Section <ref>. The identified map can then be used to transfer any subsequent learning data from the source system to the target system. The main steps are summarized in Algorithm <ref>. Notice that step 2 of the algorithm directly follows from Insights 2, 3 in Section <ref>, while steps 3, 4 directly follow from Insight 4.To better understand the algorithm, suppose, as a toy numerical example, that we have two minimum-phase, discrete-time systems with zero initial conditions and orders n_s=5 and n_t=3.We assume that this is the only available information about the systems. To identify the relative degrees of the systems, we apply at time step k=0 a step input to both systems. From the step response of the source system, we found that the output only changes at k=4, and consequently, r_s=4. Similarly, we found that r_t=3. We then follow the steps of our algorithm: (1) since r_s > r_t, we jump to step 3; (3) we construct the y_s,mod data column by shifting each element in the step response y_s forward in time by r_s-r_t=1 sample; (4) the input training data is y_s,mod, the output training data is y_t, the map order is 5, and its relative degree is 0. The transfer learning map should relate y_TL(k) to y_TL(k-1),⋯,y_TL(k-5),y_s,mod(k),⋯,y_s,mod(k-5) to best fit the output data y_t. One advantage of the proposed algorithm is that it does not require precise knowledge of the robots' dynamics and/or parameters. Instead, it only requires the knowledge of basic properties of the robotic systems, namely the system order and the relative degree. The order of the robotic system can be determined from approximate physics models, or even from general information about the robot structure.For instance, an N-link manipulator has a dynamical model of order 2N. Similarly, the relative degree of the system may be determined from physics models, or experimentally from the step response of the system as discussed in Section <ref>. Another advantage of the proposed algorithm is that it is generic in the sense that it can be combined with any system identification model/algorithm. For instance, one can utilize the proposed algorithm to determine the correct properties of both linear and nonlinear dynamic transfer maps.§ APPLICATION In this section, we utilize the proposed algorithm to identify a dynamic transfer learning map between two different quadrotor platforms, namely the Parrot AR.Drone 2.0 and the Parrot Bebop 2.0 (see Figure <ref>), and then verify through experimental results the effectiveness of our proposed map.Quadrotor vehicles have six degrees of freedom: the translational position of the vehicle's center of mass (x,y,z), measured in an inertial coordinate frame, and the vehicle's attitude, represented by the Euler angles (ϕ,θ,ψ), namely the roll, pitch, and yaw angles, respectively. The full state of the vehicle also includes the translational velocities (ẋ,ẏ,ż) and the rotational velocities (p,q,r), resulting in a dynamic model of the vehicle with 12 states. Detailed description of the quadrotor's dynamic model can be found in <cit.>. In our experiments, the quadrotor's states are all measured by the overhead motion capture system, which consists of ten 4-mega pixel cameras running at 200 Hz. In our study, the two quadrotor platforms utilize a control strategy that consists of two controllers: (i) an on-board controller that runs at 200 Hz, receives the desired roll ϕ_d, pitch θ_d, yaw velocity r_d and the z-axis velocity ż_d, and outputs the thrusts of the quadrotor's four motors, and (ii) an off-board controller that is implemented using the open-source Robot Operating System (ROS), runs at 70 Hz, receives the desired vehicle's position, and outputs the commands (ϕ_d,θ_d,r_d,ż_d) to the on-board controller.For the off-board controller, we utilize a nonlinear control strategy to stabilize the z-position of the vehicle to a fixed value and the yaw angle to zero, and then manipulate ϕ_d and θ_d to control the vehicle's motion in the x-, y-directions. In particular, we select ϕ_d and θ_d to implement a nonlinear transformation that decouples the dynamics in the x-, y-directions into approximate, linearized, second-order dynamics in each direction, and then utilize a proportional-derivative (PD) controller for each direction. More details can be found in <cit.>.In this application, we identify a transfer learning map from the Parrot AR.Drone 2.0 platform, the source system, to the Parrot Bebop 2.0 platform, the target system, for each of the x-, y-directions. We first stabilize the vehicle's y- and z-positions to constant values, and study the motion in the x-direction. For this case, the input to each system is the desired x-value reference, while the output is the actual x-value of the quadrotor. We start by collecting data for both vehicles in the x-direction. In particular, we apply the same desired reference x_d to both vehicles and detect their outputs (see Figure <ref>). We then utilize this collected data to identify a continuous-time transfer learning map with the aid of our proposed algorithm. Following the previous paragraph, we know that under the applied control strategy the x-direction dynamics for the quadrotors have approximately order 2, and by analyzing the dynamic equations, we have found that the relative degree for the quadrotors in this case is 1. We have verified this value experimentally from the collected data in Figure <ref> as discussed in Section <ref>. To sum up, we have n_s=n_t=2 and r_s=r_t=1. By following the steps of our algorithm, the correct input to the transfer learning map is the x-output of the source system, y_s, its output is the x-output of the target system, y_t, its dynamic order is 3, and its relative degree is 0. Since the applied control strategy turns the closed loop into an approximately linear behavior in the x-direction as discussed in the previous paragraph, we identify a linear transfer learning map with the desired properties using MATLAB'sfor identifying transfer functions. The obtained transfer function fits the training data (y_t) with 95.79%, measured based on the well-known normalized root mean square error (NRMSE) fitness value (fit=100(1-NRMSE)%), see Figure <ref>. For comparison, we have also identified an optimal, static gain from y_s to y_t using MATLAB'swith the function's orders set to (0,0); the gain is 0.6925, and it fits the data with 27.28%. We next test the identified transfer learning maps for transferring six minutes of collected data from the Parrot AR.Drone 2.0 to the Parrot Bebop 2.0. Figure <ref> shows the actual output of the target quadrotor and the transferred output using our proposed map.The proposed map achieves an RMS error of 0.2142 m, compared to 0.7297 m for direct transfer learning (identity map), and 0.6946 m for the identified, optimal, static map. The proposed map achieves 70.65% reduction in error over the direct transfer learning, while the optimal, static map achieves only 4.81% reduction. We similarly identify a transfer learning map from the Parrot AR.Drone 2.0 to the Parrot Bebop 2.0 in the y-direction. We omit the details for brevity. The proposed, identified map is a transfer function with order 3 and zero relative degree, and it fits the training data with 96.59%. The optimal, static TL gain is 0.518, and it fits the data with 21.73%. We then test the proposed, dynamic map for transferring 330-second y-direction data from the Parrot AR.Drone 2.0 to the Parrot Bebop 2.0.For this testing data, our proposed map has an RMS error of 0.192 m, which achieves 68.29% error reduction compared to direct transfer learning and 62% error reduction compared to the optimal, static gain. Figure <ref> shows how optimal transfer functions with different orders and zero relative degree fit the training and testing data. The order 3, proposed by our algorithm, best fits the training data, and it achieves the second highest fit to the 330-second testing data after the order 2. However, using this testing data as training data for identifying new transfer functions again shows that the third-order transfer function outperforms the second-order one in fitting this testing data.While it is expected due to overfitting that higher-order transfer functions have lower fit on the testing data, this is less obvious for the training data. The explanation is likely that the orders in Figure <ref> are not high enough to overfit the 25-second training data (5000 data points). Indeed, for order 50, the obtained transfer function fits the training data with 98.14%, but it completely fails to transfer the testing data. We then test the proposed, identified TL maps in the x-, y-directions for transferring the x-, y-data from the Parrot AR.Drone 2.0 to the Parrot Bebop 2.0, for the case where both vehicles are required to track a unit circle in the (x,y)-plane (with frequency 0.14 Hz, which is different from the frequencies of the references used in the training data). Table <ref> summarizes the transfer learning errors for both the proposed map and the direct transfer learning.Our proposed, dynamic maps achieve significant reduction of the direct transfer learning errors. However, the total improvement is less than in the previous examples. This is likely due to the unmodeled coupling in the x-, y-directions. The optimal TL map for this case should be a (2×2) matrix of dynamic maps to account for the coupling between the two directions.§ CONCLUSIONS We have studied multi-robot transfer learning (TL) from a dynamical system perspective for SISO systems. While many existing methods utilize static TL maps, we have shown that the optimal TL map is a dynamic system and provided an algorithm for determining the properties of the dynamic map, including its order and regressors, from knowing the order and relative degree of the systems. These basic system properties can be obtained from approximate physics models of the robots or from simple experiments.Our results show that for the testing data, dynamic maps, with correct features from our proposed algorithm, achieve on average 66% reduction of TL errors compared to direct TL, while optimal, static gains achieve only 15% reduction. 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Seborg (Eds), Nonlinear Process Control, Prentice Hall, 1997.Isidori A. Isidori. Nonlinear Control Systems, 3rd edition. Springer-Verlag London Limited, 1995.Sontag E. Sontag, Y. Wang. A notion of input to output stability. European Control Conf., 1997, pp. 3862-3867. angela_quad A. P. Schoellig, M. Hehn, S. Lupashin, R. D'Andrea. Feasibility of motion primitives for choreographed quadrocopter flight. American Control Conf., 2011, pp. 3843-3849. HS16 M. K. Helwa, A. P. Schoellig. On the construction of safe controllable regions for affine systems with applications to robotics. IEEE Conf. on Decision and Control, 2016, pp. 3000-3005. Available at ArXiv, arXiv:1610.01243 [cs.SY]. | http://arxiv.org/abs/1707.08689v1 | {
"authors": [
"Mohamed K. Helwa",
"Angela P. Schoellig"
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"categories": [
"cs.RO",
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"primary_category": "cs.RO",
"published": "20170727024819",
"title": "Multi-Robot Transfer Learning: A Dynamical System Perspective"
} |
Irreducible Artin groups of finite type can be parametrized via their associated Coxeter diagrams into six sporadic examples and four infinite families, each of which is further parametrized by the natural numbers. Within each of these four infinite families, we investigate the relationship between elementary equivalence and isomorphism. For three out of the four families, we show that two groups in the same family are equivalent if and only if they are isomorphic; a positive, but weaker, result is also attained for the fourth family. In particular, we show that two braid groups are elementarily equivalent if and only if they are isomorphic. The(∀∃∀)^1 fragment suffices to distinguish the elementary theories of the groups in question.As a consequence of our work, we prove that there are infinitely many elementary equivalence classes of irreducible Artin groups of finite type. We also show that mapping class groups of closed surfaces - a geometric analogue of braid groups - are elementarily equivalent if and only if they are isomorphic.Twonniers: Interaction-induced effects on Bose–Hubbard parameters Thomas Busch December 30, 2023 ===================================================================§ INTRODUCTION Understanding when two non-isomorphic groups have distinct elementary theories has been a long-standing problem of interest in both group theory and model theory. In general, this problem is fairly difficult. Much of the current literature considers families of groups parametrised in a certain fashion, and attempts to determine to what extent these parameters are determined by the elementary theories of the groups in question. A particularly celebrated result, which follows from the work of of Sela <cit.> and independently, Kharlampovich-Myasnikov <cit.>, is that the elementary theory of a non-abelian free group is independent of its rank: this resolved a famous question of Tarski. Some other important classes of groups for which something is known are listed below (this list is not meant to be exhaustive): * Finitely generated free abelian groups, by W. Szmielew <cit.> (1955).* Ordered abelian groups, by A. Robinson and E. Zakon <cit.> (1960), M. Kargapolov <cit.> (1963) and Y. Gurevich <cit.> (1964).* Classical linear groups, by A. Maltsev <cit.> (1961).* Linear groups over the integers, by V. Durlev <cit.> (1995).* Some linear and algebraic groups, by E. Bunina and A. Mikhalëv <cit.> (2000).* Chevalley groups, by E. Bunina <cit.> (2001).* Right-angled Coxeter groups and graph products of finite abelian groups, by M. Casals-Ruiz, I. Kazachkov and V. Remeslennikov <cit.> (2008).For a detailed survey of what is known about the elementary theory of various classes of groups see <cit.> (also see <cit.>). By studying examples of groups with different elementary theories, we can gain insight into the nature of first order statements in group theory.In this paper, we are primarily concerned with irreducible Artin groups of finite type: archetypical examples of such groups are braid groups. To every Coxeter matrix C we can associate two groups: the Artin group G_C and the Coxeter group G̅_C. The Artin group G_C is said to be of finite type if G̅_C is finite. The Coxeter diagrams corresponding to irreducible Artin groups of finite type have been completely classified, and can be organized into four infinite families (indexed by the natural numbers, so A_n, B_n, D_n and I_2(n)) and six sporadic examples (the reader should note that for very small n, isomorphism classes of groups in the four infinite families may overlap). Details will be provided in Section <ref>.In this paper, we study elementary equivalence classes within (not between!)[The proofs of Lemma <ref> and Theorem <ref>, however, are strong enough to distinguish some groups between these classes.]these four families. Our main result, Theorem <ref>, is that within three out of these four families (A_n, B_n and D_n), elementary equivalence class determines isomorphism class, and thus the parameter n. For the family I_2(n) a weaker statement is attained. An immediate consequence of our work here is that there are infinitely many classes of elementary theories amongst Artin groups of finite type. Moreover, we show that all the above results hold true within the (∀∃∀)^1 fragment of the elementary theory. Braid groups are examples of irreducible Artin groups of finite type of particularly significant interest. The following result follows as a corollary of Theorem <ref>: Any two braid groups are elementarily equivalent if and only if they are isomorphic. We prove the above results by explicitly constructing a class of first-order sentences {Φ_n}_n∈ℕ to help us distinguish elementary theories; Φ_n expresses the notion that every central element has an n^th root. Irreducible Artin groups of finite type can be treated as algebraic generalizations of braid groups. The natural generalization in terms of geometric group theory would be the mapping class groups, as braid groups occur as mapping class groups of punctured discs. Mapping class groups of surfaces with non-empty boundary and puncturesalong with mapping class groups of closed surfaces are two of the most significant classes ofgroups in geometric group theory. Using a result <cit.> about cyclic subgroups of such groups, it becomes straightforward to prove the following result: Let Mod(S_g) denote the mapping class group of a closed surface S_g of genus g. Mod(S_g) is elementarily equivalent to Mod(S_h) if and only if g=h. This paper is organized along the following lines. Section <ref> contains a preliminary introduction to the first-order theory of groups; we also provide a short proof of the above proposition on elementary equivalence in the context of mapping class groups of closed surfaces. Section <ref> is an overview of the theory of Artin groups of finite type: we list the basic definitions and results that will be used over the course of this document. A reader familiar with the theory of Artin groups may skip reading this section in detail, but we still recommend they take a quick glance in order to familiarise themselves with the notation used. Section <ref> and <ref> contain some key lemmas about Artin monoids and roots of central elements in irreducible Artin groups of finite type. Section <ref> contains our main results concerning elementary equivalence in irreducible Artin groups of finite type. Acknowledgements: All three authors would like to thank the Institute of Mathematical Sciences, Chennnai and the Chennai Mathematical Institute for their support and hospitality. The first author is supported by the Department of Science & Technology (DST): INSPIRE Faculty. The second author would like to thank Siddhartha Gadgil and Igor Rivin for (independently) suggesting the question considered here in the context of braid groups. § LOGICAL PRELIMINARIES AND MAPPING CLASS GROUPS§.§ First Order Logic This section contains a very brief introduction to first order logic in the context of group theory. It contains only those definitions which are pertinent to our work and context. For a broader and more comprehensive introduction to first-order logic, the reader is referred to <cit.>.The first-order language of groups ℒ_G is the tuple (·, ^-1, 1), where · refers to the multiplication, ^-1is the multiplicative inverse and 1 is the multiplicative identity.An atomic formula with variables x_1,…,x_n in ℒ_Gis astatement of the form: x_1^ε_1·x_2^ε_2·…·x_n^ε_n =1,where eachε_i ∈{±1 }.A quantifier free formula in ℒ_G is recursively defined as either an atomic formula, the negation of a quantifier free formula, the conjunction of finitely many quantifier free formulas, or a disjunction of finitely many quantifier free formulas. A sentence in ℒ_G is astatement of the following form:Q_1 x_1. Q_2 x_1 … Q_n x_n. ( Φ(x_1, x_2, x_3,… ,x_n) ),where each Q_i ∈{∃, ∀}, and Φ(x_1, x_2,… , x_n) is a quantifier free formula with variables x_1, x_2,… ,x_n. The set of all sentences which are hold in the group G is called the elementary theory of G. It is denoted by Th(G).The elementary theory of a finite group determines the group up to isomorphism. This is no longer true for infinite groups: for instance, all finitely generated free groups have the same elementary theory (see<cit.>, <cit.>).The class of sentences in the first-order language of groups is strong enough to describe admission of roots of central elements in a group. Consider the following statement:Φ_n =∀ x. ∃ y. ∀ z. ( (xz = zx) ∨ (x = y^n)).Φ_n is true in a group G precisely when every central element admits an n^th root. One can also describe the existence of a finite cyclic subgroup of order n in a group by the following sentence: Ψ_n =∃ x. ((x^n = 1) ∧_k=1^n-1 (x^k ≠ 1)). A sentence is of the class (∀∃∀)^1 if it is of the form ∀ x ∃ y∀ z(Ψ (x,y,z)). This is a well studied class of sentences called the Kahr class (see Chapter 3.1 of <cit.>).§.§ Mapping Class GroupsIn this subsection, we discuss elementary equivalence in the context of mapping class groups of closed surfaces: using a result from the literature, we are able to provide a short proof of the fact that elementary equivalence determines isomorphism class. The reader may consider the material here as motivation for our results on Artin groups of finite type - it is an easy example of how explicit first order sentences can be used to distinguish elementary equivalence classes.Let S_g be a closed orientable surface of genus g≥ 2. The mapping class group Mod(S_g) of the surface S_g is the group of all homotopy classes of orientation preserving homeomorphisms of S_g. In this section we show that if g≠ h, then the elementary theories of Mod(S_g) and Mod(S_h) are not equivalent; indeed, the (∃)^1 fragment of the elementary theory suffices to distinguish the elementary theories in question. To prove this result we need the following theorem about finite cyclic subgroups of mapping class groups. The second part of the theorem follows from the statement after Theorem 7.5 in <cit.>.<cit.> The order of a finite cyclic subgroup of the mapping class groupMod(S_g) is at most 4g+2. Moreover for every g≥ 2, there exists an element of order 4g+2 in the mapping class group Mod(S_g). The elementary theories of Mod(S_g) and Mod(S_h) are distinct for g≠ h. Consider the following first order statement statement: Ψ_4g+2 = ∃ x. ((x^4g+2=1) ∧(x^k≠ 1 for k∈{1,2,… , 4g+1})). By Theorem <ref>, the above statement is true in Mod(S_g) but false in Mod(S_h) for g≠ h. § ARTIN GROUPSWe recall the definition and basic algebraic properties of Artin groups of finite type and Coxeter groups. For detailed exposition and proofs of the results mentioned here see <cit.> and <cit.>.Suppose C=(m_i,j) denotes a n× nsymmetric matrix with (i,j)^th entry m_i,j,where m_i,i=1 and m_i,j∈{2,3,… ,∞} for i≠ j. Such a matrix is called a Coxeter matrix. To every Coxeter matrix C we can associate a labelled graph. If C is an n× n-matrix its associated graph has n labelled ordered vertices, say x_1,…,x_n. There is an edge between x_i and x_j with label m_i,j if and only if m_i,j≥ 3; it is a convention to drop the label if m_i,j = 3. Such a labelled graph is called a Coxeter graph or Coxeter diagram. Coxeter matrices are in a canonical one-to-one correspondence with Coxeter diagrams. We will treat these two notions interchangeably in this paper. Let ⟨ x,y⟩^m denote the alternating product of x and y of length m starting with x (e.g. ⟨ x,y⟩^3=xyx). By convention, ⟨ x,y⟩^∞ is the empty product. Let C be a Coxeter matrix with (i,j)^th entry m_i,j. The Artin group corresponding to C, G_C, is the group presented by the following presentation: ⟨ x_1,x_2,…,x_n |⟨ x_i, x_j ⟩^m_i,j=⟨ x_j, x_i ⟩^m_j,i,i,j∈{1,…, n}⟩.The presentation used in the definition above is called the standard presentation of an Artin group G_C. We shall from here on assume that unless stated otherwise, the terms presentation, generators and relations used in the context of an Artin group refer to the standard presentation and the associated generators and relations respectively. Let C be a Coxeter matrix with (i,j)^th entry m_i,j. The Coxeter group corresponding to C, G̅_C, is the group presented by the following presentation: ⟨ x_1,x_2,…,x_n | (x_i x_j)^m_i,j=1, i,j∈{1,…, n}andm_i,j≠∞⟩.As in the Artin group case, we call the above presentation the standard presentation of a Coxeter group. The terms presentation, generators, and relations used in the context of a Coxeter group will refer to the standard ones. It is straightforward to check that a Coxeter group G̅_C with generators x_1,x_2,… ,x_n is the quotient of the Artin group with the same set of generators by the relation x_i^2=1 for all i∈{1,2,… ,n}. The Coxeter diagram C can be recovered from the Coxeter group G̅_Cand its standard generators. The vertices of the graph correspond to the generators, and m_i,j can be recovered from the order of x_ix_j in G̅_C. An Artin group G_C is said to be of finite type if the Coxeter group G̅_C associated to C is finite. An Artin group G_C and the corresponding Coxeter group G̅_C are called irreducible if the associated Coxeter diagram C is connected. Throughout this paper we assume all Artin groups to be irreducible and of finite type unless otherwise mentioned, though we may mention this hypothesis explicitly on occasion for the sake of clarity. Coxeter classified all irreducible Artin groups of finite type. In this case the Coxeter diagram is always a tree. There are four infinite families A_n,B_n=C_n,D_n, and I_2(n), and six distinct groups E_6, E_7, E_8, F_4, H_3, H_4 (Figure <ref>). In this paper, we will restrict our attention to groups within the four infinite families. We will denote the Artin groups associated to these families by the same notation i.e. by A_n, not G_A_n. When the diagram C is either clear from the context or irrelevant, we may suppress it and denote the Artin group by G and the corresponding Coxeter group by G̅. However, we are obliged to remind the reader once more: in this paper, an Artin group means a group presented by a presentation associated to one of the diagrams in Figure <ref> along with the data of its presentation. Let G be an irreducible Artin group of finite type with generating set I={x_1,x_2,… , x_n}. Up to ordering, there is a unique partition of I into two maximal disjoint subsets J_1 and J_2 such that the elements in J_1 (respectively J_2) commute pairwise in G̅. Let 𝒥_1=∏_x_i∈ J_1x_i,𝒥_2=∏_x_j∈ J_2x_jand𝒥=𝒥_1𝒥_2. For every irreducible Artin group of finite type, there exists a corresponding natural number called the Coxeter number. In Table <ref> below, we list theCoxeter numbers associated to some Artin groups of finite type; for more information about this number see <cit.>. Let h be the Coxeter number of G, and define Δ:=𝒥^h/2ifhis even,𝒥^h-1/2𝒥_1=𝒥_2𝒥^h-1/2ifhis odd. The following theorem follows from <cit.> and the proposition following it in loc. cit. For any irreducible Artin group of finite type G with Coxeter number h we have Δ^2=𝒥^h. Furthermore, if Δ is in the center of G then Δ=𝒥^h/2.If Δ is in the center of an irreducible Artin group of finite type then h is necessarily even: see Table <ref>.Indeed, we can say even more. The center of an irreducible Artin group of finite type G, Z(G), is cyclic and is generated by either Δ or Δ^2. Moreover, we know exactly which of these two elements generates the centre in each of the cases that we care about. We will always refer to this choice of generator of Z(G) by c_G.The followingtheorem follows from the Corollary at the end of Section 7 of <cit.>. The center of an irreducible Artin group G of finite type is infinite cyclic. (1) For B_n, D_2n and I_2(2n) the center is generated by Δ. (2) For A_n, D_2n+1 and I_2(2n+1) the center is generated by Δ^2 In the following table we collect the numerics associated to the irreducible Artin groups of finite type required for our calculations: Group Rank Coxeter number (h) Generator of the center (c_G) Word length of c_G A_k k k+1 Δ^2 k^2+k B_k k 2k Δ k^2 D_2k+1 2k+1 4k Δ^2 8k^2+4k D_2k 2k 4k-2 Δ 4k^2-2k I_2(2k+1) 2 2k+1 Δ^2 4k+2 I_2(2k) 2 2k Δ 2k Table <ref>. Observe that irrespective of whether n is odd or even, the Coxeter numbers corresponding to D_n and I_2(n) are 2n-2 and n respectively.The following lemma is a straightforward consequence of Theorem <ref>. Let G be an irreducible Artin group of finite type. Let c_G denotes the generator of the center of G. There is a word containing all the generators of G which is equal to c_G in G. Furthermore, (1) c_A_n admits an (n+1)^th root in A_n. (2) c_B_n admits an n^th root in B_n. (3) If n is odd, c_D_n admits an (2n-2)^th root in D_n. (4) If n is even, c_D_n admits an (n-1)^th root in D_n. (5) If n is odd, c_I_2(n) admits an n^th root in I_2(n). (6) If n is even, c_I_2(n) admits an (n/2)^th root in I_2(n). § KEY LEMMASThis section containsresults which we will later use in the proof of Theorem <ref>. Section <ref> includes material about the shape of words in Artin monoids. Section <ref> contains the technical heart of this paper: Lemma <ref>. §.§ Artin Monoids Let G be an irreducible Artin group of finite type. Denote the monoid of positive words (with respect to the standard presentation) in G by G^+. We call this monoid the Artin monoid.For the benefit of the reader (and the authors!), we briefly recall the notion of a monoid presentation. Consider the presentation P = ⟨ g_1,…, g_n | r_1=r'_1,…, r_m=r'_m⟩ where r_i and r'_i are positive words in the g_j. Consider the free monoid on the letters g_j; denote this by F. The monoid associated to the presentation P is the quotient of F by the smallest equivalence relation containing the relation {(xr_iy, xr'_iy)}_x,y∈ F, i∈{1,m}; the set of equivalence classes clearly carries a natural monoid structure.Let G be an irreducible Artin group of finite type. Let w and w' be positive words in the generators of G. Suppose w=w' in G^+. If a generator x_i appears in w, then it must appear in w' as well.By <cit.>, the Artin monoid G^+ is isomorphic to the monoid presented by the Artin presentation associated to G via the obvious isomorphism. If x_i appears as a letter on one side of any of the relations defining G^+, it appears on the other side as well. From this, and the definition of a monoid presentation, the result follows.The following lemma is a direct consequence of the above proposition and the definition of Δ.Let G be an irreducible Artin group of finite type. Every generator of G appears in every positive word representing Δ or Δ^2. We also have the following lemma about the appearance of generators in powers of words.Let G be an irreducible Artin group of finite type. Let x ∈G^+. If for some n ∈ℕ a generator x_i appears in x^n, then it also appears in x. Assume that a generator x_i does not appear in a word representing x. Then, it does not appear in a word representing x^k. This contradicts Proposition <ref>.§.§ Roots in Artin groups of finite type Artin groups of finite type are, in particular, examples of Garside groups. As there are many good references for theory of Garside groups we will not define this notion here, instead directing the reader towards <cit.> (where these objects were first introduced) for a comprehensive overview. Briefly, however: a Garside group is a group that can be realized as the group of fractions of a Garside monoid. Garside monoids, in turn, are a class of cancellative monoids with good divisibility properties (<cit.>). The fact that Artin groups of finite type are examples of Garside groups was essentially proved by Brieskorn and Saito in <cit.> (also see <cit.>). More precisely, Brieskorn and Saito prove the following: suppose C is a Coxeter matrix such that G̅_C is finite. Then, the Artin monoid G_C^+ is a Garside monoid with group of fractions G_C. In this paper, we are concerned with root extraction in Artin groups of finite type. An algorithm for extracting roots in Garside groups was given by Siebert in <cit.>. It lies at the core of our proof of the following proposition.Let G be an irreducible Artin group of finite type. Suppose a∈Z(G)∩ G^+, and n∈ℕ. Then, the equation x^n=a has a solution in G if and only if it has a solution in G^+.This follows from the root extraction algorithm for Garside groups - see <cit.>. Understanding how this algorithm works in our particular context is particularly easy because of our hypotheses on a: the fact that a∈ G^+ and a∈Z(G) leads to a substantial simplification. However, there is one point that we wish to elaborate. The hypothesis in <cit.> requires the Garside group G to be the group of fractions of a Garside monoid M with finite positive conjugacy classes. This hypothesis is satisfied in our case, by <cit.> and <cit.>. Let G be an irreducible Artin group of finite type. Let c_G denote the generator of the center of G. The following statements hold: (1) For k> n+1, c_A_n does not have a k^th root in A_n. (2) For k> n, c_B_n does not have a k^th root in B_n. (3) For D_n, the following holds: (a) If n is even and k> n-1, c_D_n does not have a k^th root in D_n. (b) If n is odd and k> 2n-2, c_D_n does not have a k^th root in D_n. (4) For I_2(n), the following holds: (a) If n is even and k > n/2, c_I_2(n) does not have a k^th root in I_2(n). (b) If n is odd and k>n, c_I_2(n) does not have k^th root in I_2(n). Every Artin group G admits a group homomorphism λ: G→ℤ defined by sending each generator in the standard presentation of G to 1; a moments thought will convince the reader that this map is well-defined. λ restricts to a monoid homomorphism λ: G^+→ℕ. We will call the restriction of λ to G^+ the word length function on G^+. We will use Δ_n for the element from <ref> in the groups A_n, B_n, D_n, and I_2(n): the group we are working in should be clear from the context. (1): See Table <ref>: the center of A_n is generated by Δ_n^2. Suppose Δ_n^2 has a k^th root for k> n+1.By Proposition <ref>, there therefore exists x∈ A_n^+ such that x^k=Δ_n^2. Evaluating both sides by λ, we havekλ(x)=λ(Δ_n^2)=n(n+1).As k>n+1, this implies that λ(x)<n. We know from Lemma <ref> that each x_i appears in Δ_n^2. Since x^k = Δ_n^2,as a consequence of Lemma <ref> and Lemma <ref> each x_i must also appear in any word representing x. Since x is a positive word in which each and every generator appears, λ(x)≥ n, a contradiction. (2): See Table <ref>: the center of B_n is generated by Δ_n. Suppose Δ_n has a k^th root for k>n. Again by Proposition <ref>, there exists x∈ B_n^+ such that x^k=Δ_n. Evaluating both sides by λ, we see thatkλ(x)=λ(Δ_n)=n^2.As k>n, we have λ(x)<n. But x is a root of Δ_n. Arguing as in the previous case, Lemma <ref>, Lemma <ref> and Lemma <ref> tell us that λ(x)≥ n. This is a contradiction. (3): In this case we need to be slightly more careful. From Table <ref>, we see that Z(D_n) is generated by Δ_n when n is even and by Δ_n^2 when n is odd. We also have λ(Δ_n)=n(n-1) ifnis even,λ(Δ_n^2)=n(2n-2) ifnis odd. We consider the cases when n is odd and even separately. (a) 𝑛 is even: From Lemma <ref>, if there exists an k^th root for c_D_n in D_n there is one in D_n^+. Assume that x^k = Δ_n, for some k> n-1 and x∈D_n^+. Analogous to earlier cases, we have the following equality: kλ(x)=λ(Δ_n)=n(n-1). It thus follows that λ(x) < n. By arguments similar to what we have done above, we derive a contradiction. (b) 𝑛 is odd: Let k> 2n-2.As before, we assume the existence of a k^th root for the generator of the center, and thus the existence of a root in the positive monoid D_n^+, which we denote by x. As earlier we have the following equality kλ (x)=λ (Δ_n^2)=n(2n-2). Thus it follows that λ(x) < n. This is a contradiction. (4):Gaze once more upon Table <ref>: Z(I_2(n)) is generated by Δ_n when n is even and by Δ_n^2 when n is odd. We also have λ(Δ_n)=n ifnis even,λ(Δ_n^2)=2n ifnis odd. As in the previous case, we divide it into cases where n is even and odd: (a) 𝑛 is even: Let k> n/2. Suppose there is a k^th root, and thus a positive k^th root, for the generator of the center: let this positive root be x. k λ(x) = λ(Δ_n) = n. Since k> n/2, this implies that λ(x) < 2. As before, we derive a contradiction. (b) 𝑛 is odd: Given k>n, if we assume an k^th root for the generator of the center in I_2(n), it follows that there exists a k^th root in I_2(n)^+ from Proposition <ref>: call this positive root x. By similar arguments as before, we obtain the following equality: kλ(x) = λ(Δ_n^2). It thus follows that λ(x) < 2.This is a contradiction. § ELEMENTARY EQUIVALENCEThe elementary theories of irreducible Artin groups of finite type can be characterized as follows: (1)Th(A_n) = Th(A_m) if and only if n = m. (2) Th(B_n) = Th(B_m) if and only if n = m. (3) Th(D_n) = Th(D_m) if and only if n = m. (4) For the family I_2(n), the following holds: (a) If m,n are odd, Th(I_2(n)) = Th(I_2(m)) if and only if m=n. (b)If m,n are even, Th(I_2(n)) = Th(I_2(m)) if and only if m=n. (c) If n is even, then for any m>n, Th(I_2(m)) ≠Th(I_2(n)). Each of the above results continues to hold in the (∀∃∀)^1 fragment of the elementary theory.Consider the following family of first order sentence, first introduced in Section 1: Φ_k = ∀ x. ∃ y. ∀ z.((xz = zy) ∨ (x = y^k)). Φ_k holds in a group precisely when every element in the center of the group has an k^th root; these are the sentences we will use to distinguish the elementary theories of the groups under consideration. Whenever m,n appear in this proof assume that m>n. (1): The family A_k: Lemma <ref> implies that Δ_m^2 has an (m+1)^th root in A_m. As Z(A_m) is generated by Δ_m^2, every element in Z(A_m) has an (m+1)^th root. This implies thatΦ_m+1 holds in A_m. As m+1>n+1, Lemma <ref>(1) tells us that Δ_n^2 does not admit an (m+1)^th root. This shows that Φ_m+1 does not hold in A_n. Thus Th(A_n) ≠Th(A_m). (2): The family B_k:Z(B_m) is generated by Δ_m. Δ_m has an m^th root in B_m by Lemma <ref>, thus every element in Z(B_m) has an m^th root. This implies that Φ_m holds in B_m. As m>n,we see from Lemma <ref>(2) that Δ_n does not admit an m^th root. Thus Φ_m does not hold true in B_n. (3): The family D_k:We subdivide the problem into following cases: (i)Both 𝑚 and 𝑛 are odd: As Z(D_m) is generated by Δ_m^2, Lemma <ref> shows thatΦ_2m-2 holds in D_m. As m>n, Lemma <ref>(3(b)) shows that Φ_2m-2 does not hold in D_n. (ii)Both 𝑚 and 𝑛 are even:Here, Z(D_m) is generated by Δ_m, by Lemma <ref>. Thus Φ_m-1 holds in D_m. As m>n it follows from Lemma <ref>(3(a)) that Φ_m-1 does not hold in D_n. (iii) One out of 𝑚 and 𝑛 is odd and the other is even:Suppose first that m is odd and n is even. By examining parts 3(i) and 3(ii) of this proof, we note the following fact: the largest integer l such that Φ_l holds in D_m is even while the largest integer j such that Φ_j holds in D_n is odd. Thus Th(D_m)≠Th(D_n). The same argument easily adapts to the case where m is even and n is odd. (4): The family I_2(n): As in the case of D_n, here we have sub-cases: (a)Both 𝑚 and 𝑛 are odd: By Lemma <ref>, we know that Φ_n and Φ_m are hold in I_2(n) and I_2(m) respectively. From Lemma <ref>(4(a)), it follows that Φ_m does not hold in I_2(n). (b) Both 𝑚 and 𝑛 are even: By Lemma <ref>, we see that Φ_n/2 and Φ_m/2 are true in I_2(n) and I_2(m) respectively. From Lemma <ref>(4(b)), it follows that Φ_m/2 does not hold true in I_2(n).(c) If n is even and m > n, then by parts 4(a) and 4(b) of this proof, we see that there is an integer l > n/2 such that Φ_l holds in I_2(m). By an argument nearly identical to those used above, Φ_l does not hold in I_2(n).This completes the proof.Two braid groups have the same elementary theory if and only if they are isomorphic. The result continues to hold even if we only consider the (∀∃∀)^1 fragment of the elementary theory. The class {A_n}_n∈ℕ represents braid groups: the group A_n is the braid group on n+1 strands.99 [Bor01]BGG E. Börger, E. Grädel, and Y. Gurevich, The classical decision problem, Springer Science & Business Media, 2001. [Bou02]Bourbaki N. Bourbaki, Elements of Mathematics, Lie Groups and Lie Algebras, Chapter 4-6, Springer-Verlag, 2002. [BS72]Artin-Cox E. Brieskorn and K. Saito, Artin-Gruppen und Coxeter-Gruppen, Inventiones Math., 17, (1972), 245-271. [BM00]BunM E.I. Bunina, A.V. Mikhalev, Elementary equivalence of linear and algebraic groups, Fundam. Appl. Math., 6 (3), (2000), 707-722. [Bu01]Bun E.I. Bunina, Elementary equivalence of Chevalley groups, Usp. Mat. Nauk, 155 (1), (2001), 157-158. [BM04]BM E.I. Bunina and A.V. Mikhalev, Elementary properties of linear groups and related problems, Journal of Mathematical Sciences, 123 (2), (2004), 3921-85. [MKR08]MKR M. Casals-Ruiz, I. Kazachkov, and V. Remeslennikov, Elementary equivalence of right-angled Coxeter groups and graph products of finite abelian groups, Bull. London Math. Soc., 42 (1), (2010), 130–136. [CCCEHJR]CCCEHJR C. Coleman, R. Corran, J. Crisp, D. Easdown, R. Howlett, D. Jackson and A. Ram, A translation, with notes, of: E. Brieskorn and K. Saito, Artin-Gruppen und Coxeter-Gruppen, Inventiones math. 17, (1972), 245-271. 1996. <http://www.ms.unimelb.edu.au/ ram/Resources/finbs5.12.97.pdf> [DP99]Deh-P P. Dehornoy and L. Paris, Gaussian Groups and Garside Groups, Two Generalisations of Artin groups, Proc. London Math. Soc., 79 (3),(1999), 569-604. [Du95]Dur V.G. Durlev, On elementary theories of integer-valued linear groups, Izv. Ross. Akad. Nauk. Ser. Mat., 59 (5), (1995), 41-58. [FM12]PMCGB. Farb and D. Margalit, A primer on mapping class groups, Princeton Mathematical Series 49. Princeton University Press, 2012. [Gu63]Gur Y.S. Gurevich, Elementary properties of ordered Abelian groups, Algebra Logika, 3 (1), (1964), 5-40. [Hod06]HW W. Hodges, Model theory, Encyclopedia of Mathematics and its Applications 42. Cambridge University Press, 1993. [Ma61]Mal A.I. Maltsev, On elementary properties of linear groups, Problems of Mathematics and Mechanics, Novosibirsk, (1961), 110-132. [Ka63]Kag M.I. Kargapolov, On elementary theory of abelian groups, Algebra Logika, 1 (6), (1963), 26-36. [KM06]KM O. Kharlampovich and A. Myasnikov, Elementary theory of free non-abelian groups, J. Algebra, 302 (2), (2006), 451-552. [Ro60]Rob A. Robinson, E. Zakon, Elementary properties of ordered abelian groups, Trans. Amer. Math. Soc., 96, (1960), 222-236. [Sel06]Sela Z. Sela, Diophantine geometry over groups. VI. The elementary theory of a free group, Geom. Funct. Anal., 16 (3), (2006), 707-730. [Si04]sibtame H. Sibert, Tame Garside monoids, J. Algebra, 281, (2004), 487-501. [Si02]sibgar H. Sibert, Extraction of roots in Garside groups., Comm. Algebra, 30(6), (2002), 2915-2927. [Sz55]Szw W. Szmielew, Elementary properties of Abelian groups, Fund. Math. 41, (1955), 203-271. | http://arxiv.org/abs/1707.08353v2 | {
"authors": [
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"Rishi Vyas"
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"published": "20170726101621",
"title": "Elementary equivalence in Artin groups of finite type"
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The Representation Theorem of Persistence Revisited and Generalized René CorbetGraz University of Technology, [email protected] Michael KerberGraz University of Technology, [email protected] 30, 2023 ==================================================================================================================================== The Representation Theorem by Zomorodian and Carlsson has been the starting point of the study of persistent homology under the lens of representation theory. In this work, we give a more accurate statement of the original theorem and provide a complete and self-contained proof. Furthermore, we generalize the statement from the case of linear sequences of R-modules to R-modules indexed over more general monoids. This generalization subsumes the Representation Theorem of multidimensional persistence as a special case. § INTRODUCTIONPersistent homology, introduced by Edelsbrunner et al. <cit.>, is a multi-scale extension of classical homology theory. The idea is to track how homological features appear and disappear in a shape when the scale parameter is increasing. This data can be summarized by a barcode where each bar corresponds to a homology class that appears in the process and represents the range of scales where the class is present. The usefulness of this paradigm in the context of real-world data sets has led to the term topological data analysis; see the surveys <cit.> and textbooks <cit.> for various use cases.A strong point of persistent homology is that it can be defined and motivated both in geometric and in algebraic terms. For the latter, the main object are persistence modules. In the simplest case, such a persistence module consists of a sequence of R-modules indexed over ℕ and module homomorphisms connecting consecutive modules, as in the following diagram: M_0[r]^φ_0M_1[r]^φ_1 …[r]^φ_i-1M_i[r]^φ_iM_i+1[r]^φ_i+1 … A persistence module as above is of finitely generated type if each M_i is finitely generated and there is an m∈ℕ such that φ_i is an isomorphism for all i≥ m. Under this condition, Zomorodian and Carlsson <cit.> observed that a persistence module can be expressed as single module over the polynomial ring R[t]:[Theorem 3.1 in <cit.>] Let R be a commutative ring with unity. The category of persistence modules of finitely generated type[In <cit.>, the term “finite type” is used instead, but we renamed it here as we will define another finiteness condition later.] over R is equivalent to the category of finitely generated graded modules over R[t]. The importance of this equivalence stems from the case most important for applications, namely if R is a field. In this case, graded R[t]-modules, and hence also persistence modules of finitely generated type, permit a decomposition (⊕_i=1^nΣ^α_iR[t])⊕(⊕_j=1^mΣ^β_jR[t]/(t^n_j))) where Σ^· denotes a shift in the grading. The integers α_i,β_j,n_j give rise to the aforementioned barcode of the persistence module; see <cit.> for details. Subsequent work studied the property of more general persistence modules, for instance, for modules indexed over any subset of ℝ (and not necessarily of finite type) <cit.> and for the case that the M_i and φ_i are replaced with any objects and morphisms in a target category <cit.>.Given the importance of the ZC-Representation Theorem, it is remarkable that a comprehensive proof seems not to be present in the literature.In <cit.>, the authors assign an R[t]-module to a persistence module of finite type and simply state: The proof is the Artin-Rees theory in commutative algebra (Eisenbud, 1995). In Zomorodian's textbook <cit.>, the same statement is accompanied with this proof (where α is the assignment mentioned above): It is clear that α is functorial. We only need to construct a functor β that carries finitely generated non-negatively graded k[t]-modules [sic] to persistence modules of finite[ly generated] type. But this is readily done by sending the graded module M=⊕_i=0^∞M_i to the persistence module {M_i,φ_i}_i∈ℕ where φ_i:M^i→ M^i+1 is multiplication by t. It is clear that αβ and βα are canonically isomorphic to the corresponding identity functors on both sides. This proof is the Artin-Rees theory in commutative algebra (Eisenbud, 1995).While that proof strategy works for the most important case of fields,it fails for “sufficiently” bad choices of R, as the following example shows: Let R=ℤ[x_1,x_2,…] and consider the graded R[t] module M:=⊕_i∈ℕM_i with M_i=R/<x_1,…,x_i> where multiplication by t corresponds to the map M_i→ M_i+1 that assigns p mod x_i to a polynomial p. M is generated by { 1}. However, the persistence module β(M) as in Zomorodian's proof is not of finitely generated type, because no inclusion M_i→ M_i+1 is an isomorphism.This counterexample raises the question: what are the requirements on the ring R to make the claimed correspondence valid? In the light of the cited Artin-Rees theory, it appears natural to require R to be a Noetherian ring (that is, every ascending chain of ideals becomes stationary), because the theory is formulated for such rings only; see <cit.>. Indeed, as carefully exposed in the master's thesis of the first author <cit.>, the above proof strategy works under the additional assumption of R being Noetherian.We sketch the proof in Appendix <ref>. Our contributions.As our first result, we prove a generalized version of the ZC-Representation Theorem. In short, we show that the original statement becomes valid without additional assumptions on R if “finitely generated type” is replaced with “finitely presented type” (that is, in particular, every M_i must be finitely presented).Furthermore, we remove the requirement of R being commutative and arrive at the following result.Let R be a ring with unity. The category of persistence modules of finitely presented type over R is isomorphic to the category of finitely presented graded modules over R[t]. The example from above does not violate the statement of this theorem because the module M is not finitely presented. Also, the statement implies the ZC-Representation Theorem for commutative Noetherian rings, because if R is commutative with unity and Noetherian, finitely generated modules are finitely presented.Our proof follows the same path as sketched by Zomorodian, using the functors α and β to define a (straight-forward) correspondence between persistence modules and graded R[t]-modules. The technical difficulty lies in showing that these functors are well-defined if restricted to subclasses of finitely presented type. It is worth to remark that our proof is elementary and self-contained and does not require Artin-Rees theory at any point. We think that the ZC-Representation Theorem is of such outstanding importance in the theory of persistent homology that it deserves a complete proof in the literature. As our second result, we give a Representation Theorem for a more general class of persistence modules. We work over an arbitrary ring R with unity and generalize the indexing set of persistence modules to a monoid (G,⋆).[Recall that a monoid is almost a group, except that elements might not have inverses.]We consider a subclass which we call “good” monoids in this work (see Section <ref> for the definition and a discussion ofrelated concepts). Among them is the case ℕ^kcorresponding to multidimensional persistence modules, but also other monoids such as (_≥ 0,+), (∩(0,1],·) andthe non-commutative word monoid as illustrated in Figure <ref>.It is not difficult to show that such generalized persistence modules can beisomorphically described as a single module over the monoid ring R[G].Our second main resultis that finitely presented graded modules over R[G] correspond again to generalized persistence modules with a finiteness condition. Specifically, finiteness means that there exists a finite set S of indices (i.e., elements in the monoid) such that for each monoid element g with associated R-module R_g, there exists an s∈ S such that each map R_s→ R_g̃ is anisomorphism, whenever g̃ lies between s and g. For G=^k, we prove thatthis condition is equivalent to the property that all sequences in our persistence module are of finite type(as a persistence module over ℕ), see Figure <ref>,but this equivalence fails for general (good) monoids.Particularly, our second main result implies the first one, because for G=, the monoid ring R[] is precisely the polynomial ring R[t]. | http://arxiv.org/abs/1707.08864v3 | {
"authors": [
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Institut für Physik, Universität Rostock, Albert-Einstein-Str. 23,18059 Rostock, [email protected] Quantum Systems Unit, Okinawa Institute of Science and Technology Graduate University, Okinawa 904-0495, JapanQuantum Systems Unit, Okinawa Institute of Science and Technology Graduate University, Okinawa 904-0495, JapanQuantum Systems Unit, Okinawa Institute of Science and Technology Graduate University, Okinawa 904-0495, Japan We study the effects of the repulsive on-site interactions on the broadening of the localized Wannier functions used for calculating the parameters to describe ultracold atoms in optical lattices. For this, we replace the common single-particle Wannier functions, which do not contain any information about the interactions, by two-particle Wannier functions (“Twonniers") obtained from an exact solution which takes the interactions into account. We then use these interaction-dependent basis functions to calculate the Bose–Hubbard model parameters, showing that they are substantially different both at low and high lattice depths, from the ones calculated using single-particle Wannier functions. Our results suggest that density effects are not negligible for many parameter ranges and need to be taken into account in metrology experiments. 67.85.Hj, 67.85.-dTwonniers: Interaction-induced effects on Bose–Hubbard parameters Thomas Busch December 30, 2023 =================================================================== § INTRODUCTION Ultracold atoms in optical lattices have been a recent topic of significant interest, as they can be used to perform quantum simulations of fundamental models of many-body physics, which are often difficult to access using traditional condensed matter systems <cit.>. The perfect periodicity ofoptical lattices allows tomimic the crystalline environments electrons experience in solids and unprecedented control over the kinetic properties of the atoms is possible by tuning the lattice depths. Furthermore, the interaction properties between the ultracold atoms can be changed using techniques like Feshbach resonances. This has opened up many new avenues of research, particularly in the field of condensed matter and atomic physics, and made it possible to study quantum phases and quantum phase transitions over a wide range of parameters <cit.>. Theoretically, ultracold atoms in optical lattices can be described by a Bose–Hubbard model <cit.>, which stems from a mapping of the continuous system to the lattice by using site localized single-particle Wannier functions.The static and dynamics properties of the gas are then described by two main parameters: the hopping term, which accounts for bosons tunneling between neighboring sites, and the on-site interaction term, which accounts for the repulsive energy when two particles sit at the same lattice site. The competition between these parameters (commonly determined by calculating overlap integrals using single-particle Wannier functions) characterizes the Mott-insulator/superfluid transition <cit.>.However, while mathematically convenient, single particle Wannier functions neglect certain physical effects, such as the broadening of the localized wave functions due to repulsive on-site interactions when two or more bosons occupy the same lattice site. This can have significant effects when trying to make precision measurements <cit.> or when using optical lattices for metrology <cit.>, as the energy scales that govern the behavior of the atoms are typically small.Recently, a number of theoretical efforts have been made to incorporate the effects of interaction on the Wannier functions using mean-field and numerical approaches <cit.>. In addition, there has been strong experimental evidence of the broadening of Wannier function at high fillings, when high-resolution spectroscopy showed non-uniform frequency shifts for different occupation numbers per site <cit.>.It is therefore important to include the effects of modified densities due to the repulsive interactions when calculating the Bose–Hubbard parameters. In this work we suggest to do this by using the exact two-particle wave functions (“Twonniers"), obtained after solving the two-particle Schrödinger equation with contact interaction. For comparison, we also perform calculations using the single particle Wannier functions. To the best of our knowledge, this is the first study where the expansion is directly performed in terms of the two-particle wave functions, which has an implicit dependence on repulsive atom-atom interactions.Our presentation is organized as follows. In Sec. <ref> we provide a brief review of the conventional way of calculating the Hubbard parameters using the single-particle Wannier function approach. Then, in Sec. <ref> we introduce the two-particle wave functions that include the interaction effects by solving the two-particle Schrödinger equation with contact interaction. These wave functions are used in Sec. <ref> to calculate the parameters of the modified Bose–Hubbard Hamiltonian, which are interpreted in Sec. <ref> in comparison to those obtained from single-particle Wannier functions. Finally, we discuss possible applications and conclude in Sec. <ref>. § THE BOSE–HUBBARD MODEL The starting point for our analysis is the Hamiltonian for a Bose gas, given by Ĥ = Ĥ_SP + Ĥ_I , where the single-particle term includes the kinetic energy and the optical lattice potential,Ĥ_SP = ∫d Ψ̂^† () [-ħ^2/2m∇^2+V_L()]Ψ̂ ().Here m is the atomic mass. The term including the point-like interactions is given byĤ_I = g/2∫d Ψ̂^† ()Ψ̂^† ()Ψ̂ ()Ψ̂ () ,where g=4πħ^2 a_s/m is the interaction strength related to the s-wave scattering length, a_s. The bosonic field operators, Ψ̂ and Ψ̂^†, can be expanded into a series of orthonormal functions, f_i(), and bosonicannihilation and creation operators, â_i and â_i^†, for each lattice site asΨ̂ ()=∑_i f_i()â_i with∫d f_i^*()f_j()=δ_ij.A convenient and common choice for the orthonormal functions in a lattice potential are the well-known Wannier functions <cit.>, which are localized at the individual lattice sites.The single-particle Wannier function at lattice site i in the Bloch band α is defined as w^α_i() = w_i,x^α(x) w_i,y^α(y) w_i,z^α(z) ,and the components in each direction can be written in terms of the Bloch functions ϕ^α_k(x) as w_i,x^α(x) = 1/√(N_x)∑_ke^-i k x_iϕ^α_k(x),where N_x is the number of lattice sites along the x-direction (equivalent expressions exist for the other spatial directions), and x_i is the center of the i-th trap. It is important to note that the Wannier functions are not eigenfunctions of the system and that, as single-particle functions, they do not contain any information about possible scattering effects due to multi-particle occupancy of a site. Also, for small interaction energies the particles can be considered to be confined in the lowest Wannier orbitals because the energy separation between the lowest and first excited band is quite large compared to interaction energy. We work in this regime and from now onwards will drop the band index α. The hopping amplitude in the Bose–Hubbard model can then be calculated asJ=∫dw_i^*() [-ħ^2/2m∇^2+V()]w_i(),where only the nearest-neighbor overlaps are taken into account, and the interaction part of the Hamiltonian leads to the onsite interaction amplitudeU=g∫d|w_i()|^4 .§ TWO-PARTICLE WAVE FUNCTIONSThe effect of the repulsive scattering interaction depends on both the interaction strength g and the density distribution of the wave function (see Eq. (<ref>)). Therefore, it is important to choose the correct form for the orthonormal functions with which one performs the expansion: since the interactions are local and the functions are localized the density distribution should take theinteraction into account if two (or more) particles are at the same lattice site. We will therefore in the following replace terms of the form f_i()f_i() by two-particle Wannier functions, but leave terms of the form f_i()f_j() (i≠ j) to be described by single-particle Wannier functions. To find the two-particle Wannier functions we solve the Schrödinger equation for two particles in a sinusoidal potential, V_L(), interacting via a point-like potential. The Hamiltonian is given byĤ = ∑_k=1^2[-ħ^2/2m∇^2_k+V_L(_k)]+g/2δ(_1-_2),and its corresponding delocalized eigenfunctions Φ_j(_1,_2)can be used as a basis to construct the localized(two-particle) functionsW_i(_1,_2)=∑_j c_jΦ_j(_1,_2) with ∑_j|c_j|^2=1.Since the interactions raise the energies, we use the eigenfunctions of the two lowest bands. To determine the coefficients c_j, we assume that the particles are well localized at each lattice site, using as the criteria for localization the minimization of the second moment <cit.>M_i=∫d_1d_2 W_i^*(_1,_2)(_1^2+_2^2)W_i(_1,_2).This allows us to define the single-particle single-site densities from the two-particle wave functions as |W_i(,)|. In order to fulfill the orthogonality condition in Eq. (<ref>) this density needs to be normalized as∫d |W_i(,)|!=1,which also assures the fulfilment of the particle statistics,[a_i,a_j^†] = δ_i,jand[a_i,a_j]=[a_i^†,a_j^†]=0.To compare the single particle and two-particle Wannier functions, we show in Fig. <ref> their respective densities computed in a one-dimensionalpotential V_L(x)=V_0sin^2(π x/a).One can clearly see that, as expected, the repulsive interaction leads to a broadening of two-particle Wannier function, which eventually results in significant change in the Bose–Hubbard parameters. However, one can also see that the wings of the two particle Wannier function at the position of the neighbouring lattice sites are suppressed, which is due to the orthogonality requirement between two of the modified Wannier functions.In the next section, we use this two-particle wave function and density to construct the different terms in the Hamiltonian and compare them to the ones using only single-particle Wannier function solutions.§ MODIFIED BOSE–HUBBARD HAMILTONIANThe effects of the interactions between the particles are fully contained in the interaction term Ĥ_I, which, after inserting the expansion of Eq. (<ref>), takes the formĤ_I =g/2∑_ijkl∫df_i^*()f_j^*()f_k()f_l()â_i^†â_j^†â_k â_l =1/2∑_ijklU_ijklâ_i^†â_j^†â_k â_l.As we are only interested in the ground state, the Wannier functions and the two-particle wave functions based on Eq. (<ref>) can be chosen to be real and we will therefore neglect the complex conjugates below.The parameters U_ijkl can then be calculated using the substitutionf_i()f_j()|W_i(,)|if i=j, w_i()w_j()if i≠ j,which should be compared to the standard way of calculating using single-particle Wannier functionsf_i()f_j()w_i()w_j() ∀ i,j.Here we have introduced the labels W and w which will be used below to distinguish, respectively, terms calculated from the two-particle Wannier function density or from single-particle Wannier functions. The hopping term in the Bose–Hubbard model depends only on the single-particle Wannier functionsas it comes from the non-interacting part of the Hamiltonian (<ref>), and it is therefore is not affected by these substitutions. To explicitly identify the different physical processes that are summarized in the interaction term, we will in the following group the different terms into four categories. The first one is the one where two particles are at the same site and interact with each other. The associated terms include â_i^†â_i^†â_i â_i and their corresponding amplitude is given byU_iiii = g∫df^4_i() ,which under the substitutions of Eqs. (<ref>) and (<ref>) becomesU_iiii^W =g∫d |W_i(,)|^2 , U_iiii^w =g∫d |w_i()|^4 . The second group corresponds to terms with operators â_i^†â_i^†â_j â_j, (i≠j), which describe the joint tunneling of two particles between two neighbouring lattice sites, i.e. the particles hop together from one lattice site to another. The coupling amplitudes associated with this process are given byU_iijj = g∫df^2_i()f^2_j() ,and become after substitutionU_iijj^W =g∫d |W_i(,)||W_j(,)| , U_iijj^w =g∫d |w_i()|^2 |w_j()|^2 . The next effect is associated with terms including â_i^†â_j^†â_i â_j, and it can be interpreted as two indistinguishable processes: the interaction between particles at neighbouring sites or cross tunneling of particles. As these processes only involve a single particle at each site, one gets U^W_ijij = U^w_ijij = U^w_iijj.Finally, the last effect is associated with terms including â_i^†â_i^†â_i â_j, which describes single-particle tunneling between an empty and an already occupied neighbouring trap. The coupling amplitudes for this process are given byU_iiij = g∫df^3_i()f_j() ,which, after the substitutions, becomeU_iiij^W =g∫d |W_i(,)| |w_i()| |w_j()| , U_iiij^w =g∫d |w_i()|^3 |w_j()| . § RESULTS AND DISCUSSIONSIn the following we will numerically compute and compare the interaction parameters for the single-particle and the two-particle Wannier function approach. To avoid complications from the regularized delta function in three dimensions, all calculations are done in one dimension, assuming a tight harmonic confinement of the atoms in the transverse direction (of frequency ω_⊥). However, all calculations are conceptually straightforward to extend to higher dimensions. Adjusting the coupling constant g to one dimension can be done via g_1D=-2ħ^2/ma_1D, with a_1D=-d_⊥^2/2a_s(1-Ca_s/d_⊥), where C ≃ 1.4603 and d_⊥=√(2ħ/mω_⊥) <cit.>. In the following we choose ω_⊥=2π× 10^4 Hz.The results fortwo different values of the scattering length (a_s=100 a_0 and a_s=400 a_0) and as a function of the lattice depth are shown in Fig. <ref>. It can be seen that the overlap integrals U_iiii, which describe the on-site interaction, are generally in good agreement with each other for both approaches. The biggest deviations appear for shallow lattices (see Fig. <ref>(c)), whereU_iiii^W is smaller than U_iiii^w. The difference stems from the fact that the repulsive interaction leads to a broadening of two particle density and consequently a reduction in its maximal amplitude, which directly translates into a smaller magnitude of the interaction coefficient for the two-particle Wannier approach. For deeper lattices, i.e. larger potential energies, the broadening is reduced and the two quantities have similar values. The crossing between U_iiii and J, which is visible in the inset of Fig. <ref>(a), corresponds to the parameter range where tunneling starts to dominate over the interaction effects. Since at the crossing point the two relevant values of U_iiii differ by about 10%, an effect on the Mott-transition point can be expected. Similar differences between the two methods can also be noted for the overlap integrals for the correlated pair tunneling, U_iijj, where for shallow lattices the integral based on the two-particle Wannier functions is larger than the one based on the single-particle functions. Here the extended size of the localised functions due to the repulsive interactions leads directly to a larger overlap between neighboring sites. On the other hand, for deeper lattices, the pair-tunneling coupling calculated from the two-particle functions becomes an order of magnitude smaller than that from the single-particle functions. This is due to the fact that even at higher lattice depths the single particle Wannier function density and the two particle density have different behaviour in their tails, although their bulk density becomes almost identical. In this regime, the magnitude of the tail of the single particle Wannier density is higher than the one of the two particle density, leading to a larger overlap between neighboring densities, and thus to higher values of U_iijj (see alsoFig. <ref>(d)). Finally, the density dependent couplings U_iiij show a difference for shallow lattices, which can be explained in the same way as for the interaction terms above (see Fig. <ref>(e)).These results are consistent with the situation where the interaction strength is changed while keeping the lattice depth constant (see Fig. <ref>). The on-site interaction and interaction-mediated tunneling terms, U_iiii and U_iiij, do not show much difference between the two methods, but the two-particle tunneling coupling, U_iijj is much more severely affected.For a comparatively deep lattice (V_0=20E_r, Fig. <ref>(b)) the two-particle tunneling amplitude calculated using the two-particle Wannier approach increases faster than the one based on the single-particle Wannier functions,and the two methods do not coincide anywhere in the plotted parameter regime. However, for a shallower lattice (V_0=10E_r, Fig. <ref>(a)) a crossing can be seen, as the two curves associated to U_iijj are closer together. This leads to the conclusion that the effects of the interactions can have significant influence on the parameters of the Bose–Hubbard model, and should be taken into account in particular in metrology experiments. It also provides justification for the use of extended Bose–Hubbard models <cit.>, which take the two-particle tunnelling and the cross tunnelling terms into account <cit.>.§ POSSIBLE APPLICATIONS AND CONCLUSIONSTo summarize, we have calculated the parameters for the Bose–Hubbard model by consistently including on-site density effects. This was done by replacing the commonly used single-particle Wannier functions by two particle Wannier functions, which result in a broadening of the density due to repulsive interactions. Given the experimental control parameter of the optical lattice depth and the scattering lengths, we have shown that in certain regimes the Bose–Hubbard parameters show substantial deviation from the results using single-particle Wannier functions and that terms such as the correlated pair tunnelling can be become important, even though they are usually neglected.These results are hence of principle interest for current and future experiments in the field of ultracold atoms in optical lattices, especially to account for non-uniform shifts in atomic clock frequencies due to the collision of atoms.In a recent experiment by Campbell et al. <cit.>, the atomic clock shift of ^87Rb was measured, and found to decrease with increasing number of atoms per site. Other works have also shown that the clock frequency shift is directly proportional to the onsite interaction strength <cit.>. When calculated using single particle Wannier functions, the onsite interaction term is independent of the occupancy of lattice sites, and hence cannot explain the decrease of the clock shift with increasing occupancy. However, the presented technique takes into account the effect of repulsive interactions implicitly, and the resulting broadening of the two-particle single-site density and the decrease of the magnitude of onsite interaction term U_iiii, can explain the decrease of clock shift. Greiner:02 M. Greiner, O. Mandel, T. Esslinger, T.W. Hänsch, and I. Bloch, Nature 415, 39 (2002). Yukalov:09 V. Yukalov, Laser Phys. 19, 1 (2009). lewenstein_book M. Lewenstein, A. Sanpera, and V. Ahufinger, Ultracold Atoms in Optical Lattices, (Oxford Univ. Press, 2012). Jordens:08 R. Jördens, N. Strohmaier, K. Günter, H. Moritz, and T. Esslinger, Nature 455, 204 (2008). Fisher:89 M.P.A. Fisher, P.B. Weichman, G. Grinsten, and D.S. Fisher, Phys. Rev. B 40, 546 (1989). Jaksch:98 D. Jaksch, C. Bruder, J.I. Cirac, C.W. Gardiner, and P. Zoller, Phys. Rev. Lett. 81, 3108 (1998). vanOosten:01 D. van Oosten, P. van der Straten, and H.T.C. Stoof, Phys. Rev. A 63, 053601 (2001). Zwerger:03 W. Zwerger, J. Opt. B: Quantum Semiclass. Opt. 5, S9 (2003). Campbell:06 G.K. Campbell, J. Mun, M. Boyd, P. Medley, A.E. Leanhardt, L.G. Marcassa, D.E. Pritchard, and W. Ketterle, Science 313, 649 (2006). latticeclock:14 B.J. Bloom, T.L. Nicholson, J.R. Williams, S.L. Campbell, M. Bishof, X. Zhang, W. Zhang, S.L. Bromley, and J. Ye, Nature 506, 71 (2014). Li:06 J.B. Li, Y. Yu, A. Dudarev, and Q. Niu, New. J. Phys. 8, 154 (2006). Liang:09 Z.X. Liang, B. Hu, and B. Wu, arXiv:0903.4058 [cond-mat.str-el] (2009). Schneider:09 P.-I. Schneider, S. Grishkevich, and A. Saenz, Phys. Rev. A 80, 013404 (2009). Luhmann:12 D.S. Lühmann, O. Jürgensen, and K. Sengstock, New. J. Phys. 14, 033021 (2012). Zhu:15 S. Zhu and B. Wu, Phys. Rev. A 92, 063637 (2015).wan37 G.H. Wannier, Phys. Rev. 52, 191 (1937). kohn73 W. Kohn, Phys. Rev. B 7, 4388 (1973). Stollenwerk:11 T. Stollenwerk, D.N. Chigrin, and J. Kroha, J. Opt. Soc. Am. B 28, 1951 (2011). olshanii98 M. Olshanii, Phys. Rev. Lett. 81, 938 (1998).rashi1 R. Sachdeva, S. Johri, and S. Ghosh, Phys. Rev. A 82, 063617 (2010).rashi2 R. Sachdeva and S. Ghosh, Phys. Rev. A 85, 013642 (2012).benseny A. Benseny, J. Gillet, and Th. Busch, Phys. Rev. A 93, 033629 (2016). reshodko I. Reshodko, A. Benseny, and Th. Busch, Phys. Rev. A (2017); arXiv:1703.02189 (2017).ebhm:lewenstein M. Maik, P. Hauke, O. Dutta, M. Lewenstein, and J. Zakrzewski, New J. Phys. 15, 113041 (2013). Cornell:02 D. M. Harber, H. J. Lewandowski, J. M. McGuirk, and E. A. Cornell, Phys. Rev. A 66, 053616 (2002). Baym:96 G. Baym and C. J. Pethick, Phys. Rev. Lett. 76, 6 (1996). | http://arxiv.org/abs/1707.08355v1 | {
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Prospects of dynamical determination of General Relativity parameter β and solar quadrupole moment J_2⊙ with asteroid radar astronomy [ December 30, 2023 =====================================================================================================================================In strongly correlated electron systems such as heavy fermions, cuprates, and organic materials, superconductivity often emerges when the antiferromagnetic (AFM) order is suppressed through control parameters such as pressure and chemical composition<cit.>. A striking feature in these materials is that, in several cases, physical properties that deviate from the conventional Fermi-liquid theory (i.e., non-Fermi liquid properties) also appear when the AFM transition is tuned to zero temperature (T), suggesting the existence of an AFM quantum critical point (QCP). Although it is widely believed that quantum-critical fluctuations originating from the QCP are closely related to the superconductivity through unconventional pairing mechanisms<cit.>, it remains unclear whether the QCP actually exists inside the superconducting dome. The recently discovered iron pnictides<cit.> also exhibit superconductivity in the vicinity of AFM order accompanying tetragonal-to-orthorhombic structural transitions<cit.>. These magneto-structural transitions can be suppressed by pressure or chemical substitution<cit.>, but the quantum criticality is often avoided by a first-order transition in several systems<cit.>. Among the iron pnictides, Phosphorus(P)-substituted BaFe_2As_2 is a particularly clean system, and moreover, is unique in the fact that there is growing evidence for the existence of a QCP inside the superconducting dome near the optimal composition<cit.>. Although a QCP located at the maximum T_c naturally leads to the consideration that the quantum-critical fluctuations help to enhance superconductivity, there has been no direct evidence against a scenario that it is just a coincidence. A direct test to address this issue is to investigate how the superconducting dome traces when the AFM phase is shifted. However, it has been quite challenging to perform such experiments without changing the carrier numbers or bandwidth, whose effects on the QCP and superconductivity are nontrivial. In fact, in heavy-fermion superconductors, it was highlighted that chemical substitution of dopant atoms may prevent the appearance of quantum criticality altogether<cit.>. Recent advances in the study of the effects of atomic-scale point defects in superconductors using high-energy electron beams allows us to investigate the evolution of the electronic states with increasing impurity scattering in a controlled manner. Through the use of successive electron irradiation, we can perform systematic measurements on a given sample with a gradual introduction of impurity scattering induced by point defects, and without changing the carrier concentration or band width<cit.>. In general, impurity scattering reduces the T_c in unconventional superconductors, where the suppression rate depends on the gap structures<cit.>. Indeed, the superconducting dome shrinks with the introduction of scattering via chemical substitution in cuprates and heavy-fermion superconductors<cit.>. Here, we report on the changes of the magneto-structural transition temperature (T_N) and superconducting transition temperature (T_c) in the T-dependence of resistivity ρ(T) with increasing defect density across the entire superconducting dome of BaFe_2(As_1-xP_x)_2, revealing a monotonic decrease of T_N and highly composition-dependent changes of T_c. In particular, the superconductivity initially exhibits an unusual enhancement at low P concentrations with increasing disorder. After irradiation, the superconducting dome exhibits a shift of the optimal composition toward a lower P concentration. This implies that the superconducting dome tracks the AFM phase, supporting the suggested crucial role of quantum-critical fluctuations on the superconductivity in these high-T_c superconducting materials. Single crystals of BaFe_2(As_1-xP_x)_2 were synthesized by the self-flux method<cit.>. The quality of the single crystals was confirmed by their sharp superconducting transitions<cit.> and quantum oscillation measurements<cit.>. In order to introduce uniform point defects into the BaFe_2(As_1-xP_x)_2 single crystals, we irradiated the sample with an electron beam with an incident energy of 2.5 MeV, which is far above the threshold energy required for the formation of vacancy-interstitial (Frenkel) pairs<cit.>. The sample was kept at 20 K to prevent defect migration and clustering effects during the irradiation. In order to evaluate the change in ρ(T) with irradiation accurately, we repeated the process of ρ(T) measurement and irradiation on the same crystal without removing the electrodes for each composition. During the irradiation process, ρ(T) was monitored to confirm the increase of ρ(T) induced by defects. In Figure <ref>(a)–(h), ρ(T) curves are shown at several irradiation levels for x = 0, 0.05, 0.16, 0.24, 0.28, 0.29, 0.3, and 0.45, respectively. For the x = 0 and 0.05 curves, we observe clear kinks at T_N which correspond to 130 K (Figure <ref>(a)) and 122 K (Figure <ref>(b)) in pristine samples, respectively. Irradiating the sample with electrons causes the observed kinks to split into an upturn and subsequent downturn upon cooling, which is similar to the doping dependence of ρ(T) in BaFe_2As_2<cit.>. The T_N of each composition is monotonically suppressed with increasing irradiation. Above T_N, ρ(T) shows an almost parallel shift with irradiation that is caused by T-independent impurity scattering, indicating the non-magnetic nature of the point defects. On the other hand, below T_N, the change in ρ(T) exhibits some T-dependent term whose magnitude becomes larger upon cooling. Such T-dependent impurity scattering was also observed with increasing irradiation in the previous α-particle irradiation experiment on iron-based superconductor NdFeAs(O,F)<cit.>, and can be understood with the assumption that the magnetic moments of the defects induce Kondo-like scattering, similar to the case of heavy-fermion materials. Although no discernible magnetic moment was observed after irradiation in the paramagnetic state down to 0.1 K in our system<cit.>, this observation indicates that non-magnetic holes created in the AFM networks induce a T-dependent scattering process that deserves further investigation to elucidate its origin. For x = 0.16(Figure <ref>(c)), and 0.24(Figure <ref>(d)), ρ(T) exhibits a reduction at low temperatures due to the onset of superconductivity. Here, T_c is defined as the temperature where ρ(T) starts to drop from the extrapolated linear curve, as indicated in Figure <ref>(c). Upon the introduction of disorder, we observe a remarkable feature at several initial stages of irradiation: T_c gradually increases by ≈ 1-2 K with increasing irradiation dosage up to ≈ 3 C/cm^2. Although ρ(T) does not reach zero for x = 0.16, the initial increase of T_c can be clearly seen for both x = 0.16 and x = 0.24 when ρ(T) is shifted vertically to compare the T_c at different impurity levels, as shown in the inset of Figure <ref>(c) and (d). For x = 0.28(Figure <ref>(e)) and 0.29(Figure <ref>(f)), which are both near the optimal composition level, T_N decreases with increasing irradiation and even disappears above ≈ 1 C/cm^2 for x = 0.29. At the optimal composition x = 0.30 (Figure <ref>(g)) and high-P composition x = 0.45 (Figure <ref>(h)), no T_N is evident, and we observe the monotonic suppression of T_c with increasing irradiation dosage.Figure <ref>(a) shows the T-dependence of the Hall coefficient R_H(T) for x = 0, 0.24, 0.29, and 0.30 at several irradiation levels. In the AFM state, the value of R_H(T) after electron irradiation exhibits a slight change for x = 0.24. The origin of this change may be related to a T-dependent scattering process, as in the case for ρ(T). Here, it should be noted that in the paramagnetic state, the change of R_H(T) with irradiation is almost negligible compared to the reported change induced by chemical substitution<cit.>. This result indicates that electron irradiation does not essentially change the carrier concentration, and mainly introduces impurity scattering. In Figure <ref>(b)–(d), the T-derivative of the resistivity, dρ/dT(T), is shown for the samples near the optimal compositions x = 0.28, 0.29, and 0.30 at different irradiation levels. The presented data was obtained by differentiating the ρ(T) data shown in Figures <ref>(e)–(g). For the x = 0.28 and 0.29 cases, dρ/dT(T) exhibits a sharp dip due to magneto-structural transitions but otherwise remains constant at high temperatures. For x = 0.30, dρ/dT(T) is constant across a wide T range (reflecting the T-linear dependence of ρ(T)), and is not affected by the irradiation level. This result demonstrates that electron irradiation does not induce T-dependent inelastic scattering, which is in sharp contrast to carrier-doped systems in iron-based superconductors where the change of T_c is concomitant with the drastic evolution of ρ(T). Thus, the change of T_N and T_c with irradiation is not due to a change in carrier number or electron correlations, but is mainly due to an increase in impurity scattering. To see the changes in T_N and T_c caused by irradiation in entire compositions, the dependence of T_N and T_c on the irradiation level is shown in Figure <ref>. In Figure <ref>(a) (<ref>(b)), the change of T_N (T_c) from its pristine value T_N0 (T_c0), Δ T_N = T_N - T_N0 (Δ T_c = T_c - T_c0), is normalized by T_N0 (T_c0). Although T_N is reduced by electron irradiation in all compositions, the change of T_c with the irradiation dose displays large composition dependence. For low P concentrations, x = 0.16 and 0.24, as we mentioned earlier, T_c is initially increased and then further levels of irradiation tend to suppress the superconductivity. On the other hand, T_c is monotonically reduced following irradiation for all other compositions, where the magnitude of suppression is larger for higher P concentrations. Figure <ref>(c) shows the change of T_c with irradiation dose for compositions near the optimal composition, x = 0.28, 0.29, and 0.30. In pristine samples, the highest T_c is attained for x = 0.30, and irradiation monotonically suppresses T_c for all three compositions. For increased irradiation dose levels, the T_c of x = 0.29 case surpasses the x = 0.30 case above a dose level of 2.0 C/cm^2, which can be seen by the crossing of the two curves. Moreover, the T_c for the x = 0.28 case also becomes comparable to the x = 0.30 case around 2.5 C/cm^2. These results originate from the fact that the suppression rate of superconductivity becomes gradually larger in cases with high P concentrations, as seen in Figure <ref>(b). In Figure <ref>(a), we illustrate the phase diagram obtained from the dose dependence of T_N and T_c in Figure <ref>. In the phase diagram of the pristine sample, the optimal P concentration coincides with the extrapolated end point of the AFM phase, where the AFM QCP is considered to be located<cit.>. Here we make the phase diagram for the 2.0 C/cm^2 case by interpolating the data points linearly in the dose dependence of T_N and T_c. The monotonic decrease of T_N for each composition leads to a shift of the AFM phase. On the other hand, if we look at the change in T_c, it displays a strong variation in the magnitude of the suppression, as we mentioned when discussing Figure <ref>(b). T_c is increased for low P concentrations, but largely suppressed at high P concentration, as shown by the T_c curve for 2.0 C/cm^2 in Figure <ref>(a). It is worth noting that there is a clear shift of the optimal composition toward a lower P concentration when we consider the 2.5 C/cm^2 phase diagram, as shown in Figure <ref>(b). Recently, the effect of point defects on T_c on the entire superconducting dome has been reported in hole-doped Ba_1-xK_xFe_2As_2<cit.>. Although the suppression of superconductivity in this system is minimal at optimal doping, and increases away from the optimal doping level, T_c is monotonically reduced at all doping levels for an increasing number of defects. This can be understood in terms of the suppression of superconductivity, which is governed by the magnitude of the gap anisotropy. In the BaFe_2(As_1-xP_x)_2 case, however, the change of the phase diagram on irradiation is qualitatively different. Here, the superconductivity is enhanced at low P concentrations. Although an increase of T_c due to the introduction of scattering was experimentally reported in Zn-doped LaFeAs(O_1-xF_x)<cit.>, it is not obvious whether Zn substitution introduces only impurity scattering, or whether it involves additional effects such as carrier doping and changes in the lattice parameters. More recently, electron-irradiated FeSe exhibited a slight increase of T_c ≈ 0.4 K<cit.>. However, the effect of irradiation in FeSe with very small Fermi energies<cit.> is not well understood, and further investigation is needed to confirm the effect of impurity scattering. Therefore, our result is the first clear observation of a significant increase in T_c merely by impurity scattering. Indeed, it was already pointed out theoretically that the superconductivity may be enhanced in the AFM regime with the introduction of disorder based on a spin-fluctuation-mediated pairing, if there is competition between the AFM ordering and superconductivity<cit.>. When the enhancement of T_c due to the suppression of AFM order surpasses the reduction of T_c purely from impurity scattering, T_c may be increased as a result of the competing effects. In this scenario, it is expected that the suppression of T_c is largely enhanced when the AFM order is absent. However, we do not observe any significant difference in the suppression rate of T_c between x = 0.28 and 0.29, as shown in Figure <ref>(c). Here, magnetism is always present in the x = 0.28 with irradiation case, but T_N disappears rapidly for x = 0.29. This fact indicates that the change of the phase diagram with irradiation cannot be explained merely by the simple competition between AFM order and superconductivity. In fact, the effect of impurity scattering on superconductivity remains, and causes the superconducting dome to shrink. In addition to this effect, if we assume that the entire superconducting dome shifts toward a lower P composition, then we can naturally explain the change of T_c for the entire phase diagram. It should be noted here that the monotonic decrease of T_N naturally leads to the fact that the QCP may also shift its location toward a lower P concentration. Indeed, this is implied by the constant dρ/dT, reflecting the fact that the T-linear dependence in ρ(T) is extended toward lower temperatures with irradiation for x = 0.28 and 0.29, as shown in Figures <ref>(b), and <ref>(c). Here, the constant dρ/dT value is universal between x = 0.28, 0.29, and 0.30. This indicates that the T-dependence of ρ(T) in the x = 0.28 and 0.29 cases approaches that of the quantum-critical composition, x=0.30, with increasing irradiation. These observations imply that the introduction of disorder results in the shift of the AFM phase toward low P compositions, and that the superconducting dome traces its movement, suggesting that the quantum-critical fluctuations play an essential role in enhancing superconductivity in these iron-based high-T_c superconductors. Such a change of the phase diagram has not been observed in cuprates, which may be related to the fact that the pseudogap temperature does not change significantly with disorder<cit.>.We thank H. Kontani, V. Mishra, and R. Prozorov for fruitful discussions. We also thank B. Boizot, O. Cavani, J. Losco, and V. Metayer for technical assistance. This work was supported by the Grants-in-Aid for Scientific Research (KAKENHI) program from the Japan Society for the Promotion of Science (JSPS), and by the “Topological Quantum Phenomena” (No. 25103713) Grant-in Aid for Scientific Research on Innovative Areas from the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) of Japan. The irradiation experiments were supported by the EMIR network, proposal no. 11-10-8071 and no. 15-1580.99Park06 T. Park, F. Ronning, H. Q. Yuan, M. B. Salamon, R. Movshovich, J. L. Sarrao, and J. D. Thompson, Nature 440, 65 (2006).Keimer15 B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida, and J. Zaanen, Nature 518, 179 (2015).Gegenwart08 P. Gegenwart, Q. Si, and F. Steglich, Nature Phys. 4, 186 (2008).Broun08 D. M. Broun, Nature Phys. 4, 170 (2008).Hosono15 H. Hosono, and K. Kuroki, Physica C 514, 399 (2015).Dai15 P. Dai, Rev. Mod. Phys. 87, 855 (2015).Paglione10 J. Paglione, and R. L. Greene, Nature Phys. 6, 645 (2010).Shibauchi14 T. 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"authors": [
"Yuta Mizukami",
"Marcin Konczykowski",
"Kohei Matsuura",
"Tatsuya Watashige",
"Shigeru Kasahara",
"Yuji Matsuda",
"Takasada Shibauchi"
],
"categories": [
"cond-mat.supr-con"
],
"primary_category": "cond-mat.supr-con",
"published": "20170727001126",
"title": "Impact of Disorder on the Superconducting Phase Diagram in BaFe$_2$(As$_{1-x}$P$_x$)$_2$"
} |
Electron and Nucleon Localization Functions of Oganesson: Approaching the Thomas-FermiLimit Witold Nazarewicz============================================================================================= We present a new solution framework to solve the generalized trust region subproblem (GTRS)of minimizing a quadratic objectiveover a quadratic constraint. More specifically, we derive a convex quadratic reformulation (CQR)via minimizing a linear objective over two convex quadratic constraints for the GTRS. We show that an optimal solution of the GTRS can be recovered from an optimal solution of the CQR. We further prove that this CQR is equivalent to minimizing the maximum of the two convex quadratic functions derived from the CQR for the case under our investigation. Although the latter minimax problem is nonsmooth, it is well-structured and convex. We thus develop twosteepest descent algorithms corresponding to two different line search rules. We prove for both algorithms their global sublinear convergence rates. We also obtain a local linear convergence rateof the first algorithm by estimating theKurdyka-Łojasiewicz exponent at any optimal solution under mild conditions.We finally demonstrate the efficiency of our algorithms in our numerical experiments.§ INTRODUCTIONWe consider the following generalized trust region subproblem (GTRS),(P)min f_1(x):=1/2x^⊤ Q_1x+b_1^⊤ x s.t. f_2(x):=1/2x^⊤ Q_2x+b_2^⊤ x+c≤ 0,where Q_1 and Q_2 are n× n symmetric matrices (not necessary to be positive semidefinite), b_1,b_2∈ℝ^n and c∈ℝ.Problem (P) is known as the generalized trust region subproblem (GTRS)<cit.>. When Q_2 is an identity matrix I and b_2=0, c=-1/2, problem (P) reduces to the classical trust region subproblem (TRS). The TRS first arose in the trust region method for nonlinear optimization <cit.>, andhas found manyapplications including robust optimization <cit.> and the least square problems <cit.>. As a generalization, the GTRS also admits its own applications such as time of arrival problems <cit.> and subproblems of consensus ADMM in signal processing <cit.>.Over the past two decades, numerous solution methods have been developed for TRS (see <cit.> and references therein).Various methods have been developed forsolving the GTRS under various assumptions (see <cit.> and references therein). Although it appears being nonconvex, the GTRS essentially enjoys its hidden convexity. The GTRS can be solved via a semidefinite programming (SDP) reformulation, due to the celebrated S-lemma<cit.>, which was first established in<cit.>. However, suffering from relatively large computational complexity, the SDP algorithm is not practical for large-scaleapplications. To overcome this difficulty, several recent papers <cit.> demonstrated that the TRS admits a second order cone programming (SOCP) reformulation.Ben-Tal and den Hertog<cit.> further showed an SOCP reformulation for the GTRS under asimultaneously diagonalizing (SD) procedure of the quadratic forms. Jiang et al. <cit.> derived an SOCP reformulation for the GTRS when the problem has a finite optimal value and further derived a closed form solution when the SD condition fails. On the other hand, there is rich literature on iterative algorithms to solve the GTRS directly under mild conditions, for example, <cit.>.Pong and Wolkowicz proposed an efficient algorithm based on minimum generalized eigenvalue of a parameterized matrix pencil for the GTRS, which extendedthe results in <cit.> and <cit.> for the TRS.Salahi and Taati <cit.>also deriveda diagonalization-based algorithm under the SD condition of the quadratic forms. Recently, Adachi and Nakatsukasa <cit.> also developed a novel eigenvalue-based algorithm to solve the GTRS. Our main contribution in this paper is to propose a novel convex quadratic reformulation(CQR) for the GTRS that is simpler than <cit.>and further a minimax problem reformulation and develop an efficient algorithmto solve the minimax problem reformulation. Numerical results demonstrate that our method outperforms all the existing methods in the literature for sparse problem instances. We acknowledge that our CQR was inspired bythe following CQR in Flippo and Janson <cit.> for the TRS,min_x{1/2x^⊤(Q_1-λ_min(Q_1)I)x+b_1^⊤ x+1/2λ_min(Q_1): x^⊤ x≤ 1},where λ_min(Q_1) is the smallest eigenvalue of matrix Q_1. Unfortunately, this CQR was underappreciated in that time. Recently, people rediscovered this result; Wang and Xia <cit.>and Ho-Nguyen and Kilinc-Karzan <cit.> presented a linear time algorithm to solve the TRS by applying Nesterov's accelerated gradient descent algorithm to (<ref>).We, instead, rewritethe epigraph reformulation for (<ref>) as follows,min_x,t{t: 1/2x^⊤(Q_1-λ_min(Q_1)I)x+b_1^⊤ x+1/2λ_min(Q_1)≤ t, x^⊤ x≤ 1}.Motivated by the above reformulation, we demonstrate that the GTRS is equivalent to exact one of the following two convex quadraticreformulations under two different conditions,(P_1) min_x,t {t: h_1(x)≤ t, h_2(x)≤ t}, (P_2) min_x,t {t: h_3(x)≤ t, f_2(x)≤ 0},where h_1(x), h_2(x) and h_3(x), which will be defined later in Theorem <ref>, and f_2(x) defined in problem (P),areconvex but possibly not strongly convex, quadratic functions. To our best knowledge, our proposed CQRs are derived the first time for the GTRS. Thereformulation (P_2) only occurs when the quadratic constraint is convex and thus can be solved by a slight modification of <cit.> in the accelerated gradient projection method by projecting, in each iteration, the current solution to the ellipsoid instead of the unit ball in the TRS case.In this paper we focus on the problem reformulation (P_1). Although our CQR can be solvedas an SOCP problem <cit.>, it is not efficient when the problem size is large. Our main contribution is based on a recognition that problem (P_1) is equivalent to minimizing the maximum of the two convex quadratic functions in(P_1),(M) min H(x):=max{h_1(x),h_2(x)}. We further derive efficient algorithms to solve the above minimax problem. To the best of our knowledge, the current literature lacks studies on such a problem formulation for a large scale setting except using a black box subgradient method with an O(1/ϵ^2) convergence rate <cit.>, which is really slow. Note thatSection 2.3 in Nesterov's book <cit.> presents a gradient based method with linear convergence rate for solving the minimizationproblem (M) under the condition that both h_1(x) and h_2(x) are strongly convex. However, Nesterov's algorithms cannot be applied to solve our problem since in our problem setting at least one function of h_1(x) and h_2(x) is not strongly convex. By using the special structure of problem (M),we derivea steepest descent method in Section 3. More specifically, we choose either the negative gradient when the current point is smooth,oravector in the subgradient set with the smallest norm (the steepest descent direction) when the current point is nonsmoothas the descent direction, and derive two steepest descent algorithms with two different line search rules accordingly. In the first algorithm we choose a special step size, and in the second algorithm we propose a modified Armijo line searchrule. We also prove the global sublinear convergence rate for both algorithms.The first algorithm even admits a global convergence rate of O(1/ϵ), in the same order as the gradient descent algorithm, which is faster than the subgradient method. In addition, we demonstrate that the first algorithm also admitsa local linear convergence rate,by a delicate analysis on the Kurdyka-Łojasiewicz (KL) <cit.> property for problem (M). We illustrate in our numerical experiments the efficiency of the proposed algorithmswhen compared with the state-of-the-art methods for GTRS in the literature.The rest of this paper is organized as follows. In Section 2, we derive an explicit CQR for problem (P) under different conditions and show how to recover an optimal solution of problem (P) from that of the CQR. In Section 3, we reformulate the CQR to a convex nonsmooth unconstrained minimax problem and derive two efficient solution algorithms. We provide convergence analysis for both algorithms. In Section 4, we demonstrate the efficiency of our algorithms from our numerical experiments. We conclude our paper in Section 5.NotationsWe use v(·) to denote the optimal value of problem (·).The matrix transpose of matrix A is denoted by A^⊤ andinverse of matrix A by A^-1, respectively. § CONVEX QUADRATIC REFORMULATIONIn this section, we derive a novel convex quadratic reformulation for problem (P). To avoid some trivial cases, we assume, w.o.l.g., the Slater condition holds for problem (P), i.e., there exists at least one interior feasible point. When both f_1(x) and f_2(x) are convex,problem (P) is already a convex quadratic problem. Hence, w.l.o.g., let us assume that not both f_1(x) and f_2(x) are convex.We need to introduce the following conditions to exclude some unbounded cases.The set I_PSD:={λ:Q_1+λ Q_2≽0}∩ℝ_+ is not empty, where ℝ_+ is the nonnegative orthant.The common null space of Q_1 and Q_2 istrivial, i.e., (Q_1)∩(Q_2)={0}. Before introducing our CQR, let us first recall the celebrated S-lemma by defining f̃_1(x)=f_1(x)+γ with an arbitrary constant γ∈ℝ.The following two statements are equivalent:1. The system of f̃_1(x)<0 and f_2(x)≤0 is not solvable;2. There exists μ≥0 such that f̃_1(x)+μ f_2(x)≥0 for all x∈ℝ^n. Using the S-lemma, the following lemma shows a necessary and sufficient condition under which problem (P) is bounded from below. Problem (P) is bounded from below if and only if the following system has a solution for λ:Q_1+λ Q_2≽0, λ≥0, b_1+λ b_2∈(Q_1+λ Q_2). We make Assumption <ref> without loss of generality, because otherwise we can prove an unboundedness from below of the problem(see, e.g., <cit.> and <cit.>). Under Assumption <ref>, if Assumption <ref> fails, there exists no nonnegative λ such that Q_1+λ Q_2≽0 and problem (P) is unbounded from below due to Lemma <ref>.So both Assumptions <ref> and<ref> are made without loss of generality.It has been shown in <cit.> that {λ:Q_1+λ Q_2≽0} is an interval and thus {λ:Q_1+λ Q_2≽0}∩ℝ_+ is also an interval (if not empty). Under Assumptions <ref> and <ref>, we have the following three cases for I_PSD.The set I_PSD=[λ_1,λ_2] with λ_1<λ_2. The setI_PSD=[λ_3,∞). The set I_PSD={λ_4} is a singleton. Note that Condition <ref> occurs only when Q_2 is positive semidefinite. Under Condition <ref>, Q_1 andQ_2 may not be SD and may have 2×2 block pairs in a canonical form under congruence <cit.>. In this case, when λ is given,the authors in <cit.> showed how to recover an optimal solution if the optimal solution is attainable, and how to obtain an ϵ optimal solution if the optimal solution is unattainable. So in the following, we mainly focus on the cases where either Condition <ref> or <ref> is satisfied.Under Condition <ref> or <ref>, problem (P) is bounded from below. Under Condition <ref> or <ref>, there exists λ_0 such that Q_1+λ_0 Q_2≻0 and λ_0≥0, which further implies b_1+λ_0 b_2∈(Q_1+λ_0 Q_2) as Q_1+λ Q_2 is nonsingular. With Lemma <ref>, we complete the proof. §.§ Convex quadratic reformulation for GTRSIt is obvious that problem (P) is equivalent to its epigraph reformulation as follows,( P_0) min{t: f_1(x)≤ t, f_2(x)≤ 0}.To this end, we are ready to present the main result of this section. Under Assumption <ref>, by defining h_i(x)=f_1(x)+λ_if_2(x), i=1,2,3, we can reformulate problem (P)to a convex quadratic problem under Conditions <ref> and <ref>, respectively: * Under Condition <ref>, problem (P) is equivalent to the following convex quadratic problem,(P_1) min_x,t{t: h_1(x)≤ t, h_2(x)≤ t}; * Under Condition <ref>,problem (P) is equivalent to the following convex quadratic problem,(P_2) min_x,t{t: h_3(x)≤ t, f_2(x)≤ 0}=min_x{h_3(x): f_2(x)≤ 0}. Let us first consider the case where Condition <ref> holds. Due to Lemma <ref>, (P_1) is bounded from below. Together with the assumed Slater conditions,problem (P_1)admits the same optimal value as itsLagrangian dual <cit.>. Due to the S-lemma, problem (P) also has the same optimal value as its Lagrangian dual <cit.>,(D) max_μ≥0min_x f_1(x)+μ f_2(x).Under Condition <ref>, i.e., I_PSD=[λ_1,λ_2] with λ_1<λ_2, it is easy to show that (P_1) is a relaxation of (P_0) since they have the same objective function and the feasible region of (P_1) contains that of (P_0) (note that f_1≤ t and f_2≤0 imply that f_1(x)-t+uf_2(x)≤0 for all u≥0). Thus,v( P_1)≤ v( P_0)=v( P).The Lagrangian dual problem of (P_1) is(D_1) max_s_1,s_2≥0min_x,t t+(s_1+s_2)(f_1(x)-t)+(λ_1s_1+λ_2s_2) f_2(x).For any primal and dual optimal solution pair (x^*,u^*) of (P) and (D), due tou^*∈[λ_1,λ_2] as Q_1+μ^* Q_2≽0 from Lemma <ref>, we can always find a convex combination λ_1s̅_1+λ_2s̅_2=μ^* with s̅_1+s̅_2=1. Hence (x^*,s̅,t), with an arbitrary t, is a feasible solutionto(D_1) andthe objective value of problem (D_1) at(x^*,s̅,t) is the same with the optimal value of (D). This in turn impliesv(D_1)≥ v(D).Since (P_1) is convex and Slater condition is satisfied (because (P_1) is a relaxation of (P) and Slater condition is assumed for (P)), v(P_1)=v(D_1). Finally, by combining (<ref>) and (<ref>), we have v(P_1)=v(D_1)≥ v(D)=v(P)=v(P_0)≥ v(P_1). So all inequalities above become equalities and thus (P_1) is equivalent to (P).Statement 2 can be proved in a similar way and is thus omitted. Reformulation (P_2)generalizes the approaches in <cit.> for the classical TRS with the unit ball constraint to the GTRS with a general convex quadratic constraint.To our best knowledge, there is no method in the literature to compute λ_1 and λ_2in Condition <ref> for general Q_1 and Q_2. However, there existefficient methods in the literature to compute λ_1 and λ_2when a λ_0 is given such that Q_1+λ_0Q_2≻0 is satisfied. More specifically, the method mentioned in Section 2.4.1 in <cit.> gives a way to compute λ_1 and λ_2: first detect a λ_0 such that Q_0 瞿繙:=Q_1+λ_0Q_2≻0, and then compute λ_1 and λ_2 by some generalized eigenvaluesfor a definite matrix pencil that are nearest to 0. Please refer to <cit.> for one of the state-of-the-art methods for detecting λ_0. We can also find another iterative method in Section 5 <cit.> to compute λ_0∈ int(I_PSD) by reducing the length of an interval [λ̅_1,λ̅_2]⊃ I_PSD. We next report our new method to compute λ_1 and λ_2, which is motivated by <cit.>.Our first step is also to find a λ_0 such thatQ_0:=Q_1+λ_0Q_2≻0. Then we compute the maximum generalized eigenvalues for Q_2+μ Q_0 and -Q_2+μ Q_0, denoted by u_1 and u_2, respectively. Note that bothu_1>0 and u_2>0 due toQ_0≻0 and Q_2 has a negative eigenvalue.So we haveQ_1+(1/u_1+λ_0) Q_2≽0 and Q_1+(-1/u_2+λ_0) Q_2≽0.Thus Q_1+η Q_2≽0 for all η∈[λ_0-1/u_2,λ_0+1/u_1], which implies λ_1=λ_0-1/u_2 and λ_2=λ_0+1/u_1. In particular, when one of Q_1 and Q_2 is positive definite, we can skip the step of detecting the definiteness, which would save significant time in implementation. In fact, when λ_0 is given, we only need to compute one extreme eigenvalues, either λ_1 or λ_2, to obtain our convex quadratic reformulation.Define x(λ)=-(Q_1+λ Q_2)^-1(b_1+λ b_2) for all λ∈ int(I_PSD) and define γ(λ)=f_2(x(λ)).After we have computed λ_0 such that λ_0∈ int(I_PSD), under Assumption <ref>,we further have Q_1+λ_0Q_2≻0, which makes (Q_1+λ Q_2)^-1 well defined. In fact, there are Newton type methods in the literature (e.g., <cit.>) for solving the GTRS by finding the optimal λ throughγ(λ)=0.However, each step in <cit.> involves solving a linear system -(Q_1+λ Q_2)^-1(b_1+b_2), which is time consuming for high-dimension settings. Moreover, the Newton's method does not converge in the so calledhard case[The definition here follows <cit.>. In fact, the definitions of hard case and easy case of the GTRS are similar to those of the TRS. More specifically, if the null space of the Hessian matrix, Q_1+λ^* Q_2, withλ^* being the optimal Lagrangian multiplier of problem (P), is orthogonal to b_1+λ^*b_2, we are in the hard case; otherwise we are in the easy case.]. On the other hand, for easy case, an initial λ inI_PSD is also needed as a safeguard to guarantee the positive definiteness of Q_1+λ Q_2 <cit.>. It is shown in <cit.> that γ(λ) is either a strictly decreasing function or a constant in int(I_PSD). Following<cit.>, we have the following three cases: if γ(λ_0)>0, the optimal λ^* locates in [λ_0,λ_2]; if γ(λ_0)=0, x(λ_0) is an optimal solution; and if γ(λ_0)<0, the optimal λ^* locates in [λ_1,λ_0]. Hence we have the following corollary, whose proof is similar to that in Theorem <ref> and thus omitted. Assume that Assumption <ref> holds and define h_i(x)=f_1(x)+λ_if_2(x), i = 0, 1, 2.Under Condition <ref>, the following results hold true. * If γ(λ_0)>0, problem (P) is equivalent to the following convex quadratic problem,(P_1) min_x,t{t: h_0(x)≤ t, h_2(x)≤ t}. * If γ(λ_0)=0, x(λ_0)=-(Q_1+λ_0 Q_2)^-1(b_1+λ_0 b_2)is the optimal solution.* If γ(λ_0)<0, problem (P) is equivalent to the following convex quadratic problem,(P_1) min_x,t{t: h_1(x)≤ t, h_0(x)≤ t}. Since both (P_1)and (P_1) have a similar form to (P_1) and can be solved in a way similar to the solution approach for (P_1), we only discuss how to solve (P_1) in the following.§.§ Recovery of optimal solutionsIn this subsection, we will discuss the recovery of an optimal solution to problem (P) from an optimal solution to reformulation (P_1). Before that, we first introduce the following lemma. Let us assume from now on h_i(x)=1/2x^⊤ A_ix+a_i^⊤ x+r_i, i=1,2.If Condition <ref> holds, A_1 and A_2 are simultaneously diagonalizable. Moreover, we have d^⊤ A_1d>0for all nonzero vector d∈(A_2).Note that Condition <ref> and Assumption <ref> imply that Q_1+λ_1+λ_2/2Q_2≻0, i.e.,A_1+A_2/2≻0. Let A_0=A_1+A_2/2 and A_0=L^⊤ L be its Cholesky decomposition, where L is a nonsingular symmetric matrix. Also let (L^-1)^⊤ A_1L^-1=P^⊤ DP be the spectral decomposition, where P is an orthogonal matrix and D is a diagonal matrix. Then we have(L^-1P^-1)^⊤ A_1L^-1P^-1=D and(L^-1P^-1)^⊤ A_2L^-1P^-1=(L^-1P^-1)^⊤ A_0L^-1P^-1-(L^-1P^-1)^⊤ A_1L^-1P^-1=I-D. Hence A_1 and A_2 are simultaneously diagonalizable by the congruent matrix L^-1P^-1.Now let us assume S=L^-1P^-1 and thus S^⊤ A_1S=(p_1,…,p_n) and S^⊤ A_2S=(q_1,…,q_n) are both diagonal matrices. DefineK={i:q_i=0,i=1,…,n}.Since A_1+A_2≻0, p_i>0 for all i∈ K. Let e_i bethe n-dimensional vector with ith entry being 1 and all others being 0s. We have (Se_i)^⊤ A_1Se_i=p_i>0 for all i∈ K. On the other hand, A_2Se_i=0 for all i∈ K. Henced^⊤ A_1d>0 holds for all nonzero vector d∈(A_2). From Lemma <ref>, Condition <ref> implies the boundedness of problem (P) and thus the optimal solution is always attainable <cit.>.In the following theorem, we show how to recover the optimal solution of problem (P) from an optimal solution of problem (P_1). Assume that Condition <ref> holds and x^* is an optimal solution of problem (P_1). Then an optimal solution of problem (P) can be obtained in the following ways: * If h_1(x^*)= t and h_2(x^*)≤ t, then x^* is an optimal solution to (P);* Otherwise h_1(x^*)<t and h_2(x^*)= t. For any vector v_l∈(A_2), let θ̃ be a solution of the following equation, h_1(x^*+θ v_l)=1/2v_l^⊤ A_1v_lθ^2 +(v_l^⊤ A_1x^*+a_1^⊤ v_l)θ+h_1(x^*)=t.Then {x̃: x̃=x^*+θ̃v_l,v_l∈(A_2), θ̃ is a solution of (<ref>)} forms the set of optimal solutions of(P).Note that at least one of h_1(x^*)≤ t and h_2(x^*)≤ t takes equality. Then we prove the theorem for the following two cases:* If h_1(x^*)= t and h_2(x^*)≤ t,then f_1(x^*)+λ_2f_2(x^*)≤ f_1(x^*)+λ_1f_2(x^*). Hence f_2(x^*)≤0 due to λ_2-λ_1>0. * Otherwise, h_1(x^*)<t and h_2(x^*)=t. In this case, for all d∈(A_2) we have d^⊤ A_1d>0 due to Lemma <ref>. We also claim that a_2^⊤ d=0. Otherwise, setting d such that a_2^⊤ d<0(This can be done since we have a_2^⊤(-d)<0 if a_2^⊤ d>0.) yieldsh_2(x^*+d)=h_2(x^*)+1/2d^⊤ A_2d+(x^*)^⊤ A_2 d+a_2^⊤ d=h_2(x^*)+a_2^⊤ d<t,where the second equality is due to d∈(A_2) and h_1(x^*+d)<t for any sufficiently small d. This implies that (x^*,t) is not optimal, which is a contradiction. Equation (<ref>) has two solutions due to the positive parameter before the quadratic term, i.e., v_l^⊤ A_1v_l>0 and the negative constant, i.e., h_1(x^*)-t<0. With the definition of θ̃, we knowh_1(x̃)=t and h_2(x̃)=t. This further impliesf_1(x̃)=t and f_2(x̃)=0, i.e., x̃ is an optimal solution to (P). In Item2 of the above proof, a_2^⊤ d=0 indicates that problem (P) is in the hard case. We next illustrate our recovery approach for the following simple example,min{3x_1^2-1/2x_2^2-x_2:-x_1^2+1/2x_2^2+x_2+1≤0}.Note that, for this example, Condition <ref> holds, λ_1=1 and λ_2=3. Then we have the following CQR,min{t:2x_1^2+1≤ t, x_2^2+2x_2+3≤ t}.An optimal solution of the CQRis x=(0,-1)^⊤ , t=2. However, this x isnot feasible to (P). Using the approach in Theorem <ref>, we obtain an optimal solution x̃=(√(2)/2,-1)^⊤ to problem (P). In fact, this instance is in the hard case since the optimal Lagrangian multiplier, λ^*=3, is at the end of the interval {λ:Q_1+λ Q_2≽0, λ≥0} and a-λ^*b∈(Q_1+λ^* Q_2). We finally point out that our method can be extended to the following variants of GTRS with equality constraint and interval constraint,[ (EP)minf_1(x):=1/2x^⊤ Q_1x+b_1^⊤ x (IP)minf_1(x):=1/2x^⊤ Q_1x+b_1^⊤ x;s.t. f_2(x):=1/2x^⊤ Q_2x+b_2^⊤ x+c=0, s.t. c_1≤ f_2(x):=1/2x^⊤ Q_2x+b_2^⊤ x≤ c_2. ]It is shown in <cit.> that (IP) can be reduced to(EP) with minor computation. It is obvious that all our previous results for inequality constrained GTRS hold for (EP) if we remove the non-negativity requirement for λ in I_PSD, i.e., I_PSD={λ:Q_1+λ Q_2≽0}. We thus omit detailed discussion for (EP)to save space.In the last part of this section, we compare the CQR in this paper with CQR for general QCQP in <cit.>.The authors in <cit.> considered the following general QCQP,(QP) minb̃_0^⊤ x s.t. 1/2x^⊤Q̃ _ix+b̃_ix+c̃_i≤0, i=1,…,m, x∈ X,where X is a polyhedron. They further showed that the SDP relaxation of (QP) is equivalent tothe following CQR for (QP):(CQP) minb̃_0^⊤ x s.t. x∈ G,where G={x:F_s(x)≤0 for every s∈ T},F_s(x)=∑_i=1^ms_i(1/2x^⊤Q̃_ix+b̃_ix+c̃_i) andT:={s∈ℝ^m: s≥0,τ∈ℝ,[∑_i=1^ms_ic̃_i 1/2(∑_i=1^ms_ib̃_i); 1/2(∑_i=1^ms_ib̃_i^⊤)∑_i=1^ms_i/2Q̃_i ]≽0}.For thequadratic problem (P_1), because the variable t is linear in the objective and the constraints, we can reduce TtoT:={s:s_1+s_2=1,s≥0,∑_i=1^2s_i/2Q_i≽0,∑_i=1^2s_ib_i∈(∑_i=1^2s_iQ_i)},where the restriction s_1+s_2=1 does not affect the feasible region G since F_s(x)≤0 is equivalent tok F_s(x)≤0 with any positive scaling k for s.Note that h_1(x)=F_s^1(x) and h_2(x)=F_s^2(x) with s^1=(1,0)^⊤ and s^2=(0,1)^⊤. For any s∈ T, h_1(x)≤0 and h_2(x)≤0 imply F_s(x)≤0 because F_s(x) is a convex combination of f_1(x)+λ_1f_2(x) and f_1(x)+λ_2f_2(x). Hence, by the strong duality and with analogous proof to that in Theorem <ref>, the twofeasible regionsof problems (P_1) and (CQP) are equivalent and we further have v(P_1)= v(CQP).§ EFFICIENT ALGORITHMS IN SOLVING THE MINIMAX PROBLEM REFORMULATION OF THE CQRIn this section, we propose efficient algorithms to solve the GTRS under Condition <ref>. As shown in Theorem <ref> and Corollary <ref>,the GTRS is equivalent to (P_1) or either (P_1) or (P_1). The three problems have similar forms and can be solved by the following proposed method in this section. Hence, to save space, we only considersolution algorithms for (P_1) in this section.The convex quadratic problem (P_1)can be cast as an SOCP problem and solved by many existing solvers, e.g.,CVX <cit.>, CPLEX <cit.> and MOSEK <cit.>. However, the SOCP reformulation is not very efficient when the dimension is large (e.g., the SOCP solver will take about 1,000 seconds to solve a problem of dimension 10,000). Fortunately, due to its simplestructure, (P_1) is equivalent to the followingminimax problem of two convex quadratic functions (M) min{H(x):=max{h_1(x),h_2(x)}}.Hence we aim to derive an efficient method to solve the above minimax problem, thus solving the original GTRS.Our method is a steepest descent method to find a critical point with 0∈∂ H(x). It is well known that such a critical point is an optimal solution of problem (M). The following theorem tells us how to find the steepest descent direction. Let g_1=∇ h_1(x) and g_2=∇ h_2(x). If g_1 and g_2 have opposite directions, i.e., g_1=-tg_2 for some constant t>0 or if g_i=0 and h_i(x)≥ h_j(x) for i≠ j, i,j∈{1,2}, then x is a global optimal solution. Otherwise we can always find the steepest descent direction d in the following way: * whenh_1(x)≠ h_2(x),d=-g_1 if h_1(x)>h_2(x) and otherwise d=-g_2;* when h_1(x)=h_2(x), d=-(α g_1+(1-α)g_2), where α is defined in the following three cases: * α=0, ifg_1^⊤ g_1≥ g_1^⊤ g_2≥ g_2^⊤ g_2,* α=1, if g_1^⊤ g_1≤ g_1^⊤ g_2≤ g_2^⊤ g_2, * α=g_2^⊤ g_2-g_1^⊤ g_2/g_1^⊤ g_1+g_2^⊤ g_2-2g_1^⊤ g_2, if g_1^⊤ g_2≤ g_2^⊤ g_2 andg_1^⊤ g_2≤ g_1^⊤ g_1. Ifh_1(x)=h_2(x) and g_1=-tg_2, then0∈∂ H(x). Hence, by the definition of subgradient, we have H(y)≥ H(x)+0^⊤ (y-x)=H(x), ∀ y, which further implies that x is the optimal solution.If g_i=0 and h_i(x)≥ h_j(x) for i≠ j, i,j∈{1,2}, then for all y≠ x, we have H(y)≥ h_i(y)≥ h_i(x)=H(x), i.e.,x is a global optimal solution.Otherwise we have the following three cases: * Whenh_1(x)≠ h_2(x), (suppose, w.l.o.g., h_1(x)>h_2(x)),for all y∈ℬ(x,δ) with ℬ(x,δ)⊂ {x:h_2(x)<h_1(x)}), H(x)=h_1(x) and thus H(x) is differentiable at x and smooth in its neighbourhood. Hence, d=-g_1 if h_1(x)>h_2(x). Symmetrically, the case with h_2(x)>h_1(x) can be proved in the same way.* When h_1(x)=h_2(x), the steepest descent direction can be found by solving the following problem:min_y=1max_g∈∂ H(x)g^Ty.The aboveproblem is equivalent to min_g∈∂ H(x)g^2 <cit.>, which is exactly the following problem in minimizing a quadratic function of α,min_0≤α≤1 (α g_1+(1-α)g_2)^⊤(α g_1+(1-α)g_2).The first order derivativeof the above objective function is g_2^⊤ g_2-g_1^⊤ g_2/g_1^⊤ g_1+g_2^⊤ g_2-2g_1^⊤ g_2. Then if the derivative is in the interval [0,1], the optimal α is given by g_2^⊤ g_2-g_1^⊤ g_2/g_1^⊤ g_1+g_2^⊤ g_2-2g_1^⊤ g_2. Otherwise, (<ref>) takes its optimal solution on its boundary. In particular, * when g_2^⊤ g_2-g_1^⊤ g_2/g_1^⊤ g_1+g_2^⊤ g_2-2g_1^⊤ g_2>1, i.e., g_1^⊤ g_1<g_1^⊤ g_2 and g_2^⊤ g_2>g_1^⊤ g_2,we have α=1,* wheng_2^⊤ g_2-g_1^⊤ g_2/g_1^⊤ g_1+g_2^⊤ g_2-2g_1^⊤ g_2<0, i.e., g_1^⊤ g_2>g_2^⊤ g_2, we have α=0. The above theorem shows that the descent direction at each point with h_1(x)=h_2(x) is either the one with the smaller norm between ∇ h_1(x) and ∇ h_2(x)or the negative convex combination d of∇ h_1(x) and ∇ h_2(x)such that ∇ h_1(x)^⊤ d=∇ h_1(x)^⊤ d.We next present an example in Figure <ref> to illustrate the necessity of involving the subgradient (in some cases, both gradients are not descent directions). Consider h_1(x)=x_1^2+x_2^2 and h_2(x)=(x_1-1)^2+x_2^2. The optimal solution of this problem is (0.5,0)^⊤. The gradient method can only converge to some point in the intersection curve of h_1(x)=h_2(x), i.e., x_1=0.5, but not the global optimal solution. For example, when we are at x̅=(0.5,0.1)^⊤, the gradients for h_1(x̅) and h_2(x̅) are g_1=(1,0.2)^⊤ and g_2=(-1,0.2)^⊤, respectively. Neither -g_1 nor -g_2 is a descent direction at H(x̅); H(x̅+ϵ g_i)>H(x̅) for any small ϵ>0, i=1,2, due to g_1^⊤ g_2=-0.96<0 and h_1(x̅)=h_2(x̅). (The direction -g_1 is a descent direction, at x̅, for h_1(x)but ascent for h_2(x) and thus ascent for H(x); the same analysis holds for -g_2.) The way we use to conquer this difficulty is to choose the steepest descent direction in the subgradient set atpoints in the intersection curve. If we use the subgradient direction d=-1/2(g_1+g_2)=-(0,0.2)^⊤, then d is a descent direction since h_1(x̅+ϵ d)=H(x̅)+2ϵ g_1^⊤ d+ϵ^2 d^⊤ d<H(x̅) and h_2(x̅+ϵ d)=H(x̅)+2ϵ g_1^⊤ d+ϵ^2 d^⊤ d<H(x̅)for any ϵ with 0<ϵ<2. Using the descent direction presented in Theorem <ref>, we propose twoalgorithms to solve the minimax problem (M), respectively, in Algorithms <ref> and <ref>:we first compute a descent direction by Theorem <ref>, apply then two different line search rules for choosing the step size, and finally terminate the algorithm if some termination criterion is met. The advantage of our algorithms is that each iteration isvery cheap, thus yielding, with an acceptable iterate number, a low cost in CPU time. The most expensive operationin each iteration is to compute several matrix vector products, which could become cheapwhen the matrices are sparse.§.§ Line search with a special step sizeIn the following, we first derive a local linear convergence rate for Algorithm <ref> and then demonstrate aglobal sublinear convergence rate for Algorithm <ref>. We analyze the local convergence rate by studying the growth in the neighbourhood of any optimal solution to H(x) in (M). In fact, H(x) belongs to a more general class of piecewise quadratic functions. Error bound and KL property, which are two widely used techniques for convergence analysis, have been studied in the literature,for several kinds of piecewise quadratic functions, see <cit.>. However, these results are based on piecewise quadratic functions separated by polyhedral sets, which is not the case of H(x). Li et al. <cit.> demonstratedthat KL property holds for the maximum of finite polynomials, but their KL exponent depends on the problem dimension and is close to one, whichleads to a very weak sublinear convergence rate. Gao et al. <cit.> studied the KL exponent for the TRS with the constraint replaced by an equality constraint x^⊤ x=1. However, their technique depends on the convexity of the function x^⊤ x and cannot be applied to analyze our problem. Up to now,the KL property or error bound for H(x) has not been yet investigated in the literature related to the linear convergence of optimization algorithms. A significant result of this paper is to estimate the KL exponent of 1/2 for function H(x) when min_x H(x)> max_i{min_x h_1(x),min_x h_2(x)}. With this KLexponent, we are able to illustrate the linear convergence of our first algorithm with the proposed special step size.For completeness, we give a definition of KL property in the following. By letting ℬ(x,δ)={y:y-x≤δ}where · denotes the Euclidean norm of a vector, we have the following definition of KL inequality. <cit.> Let f:ℝ^n→ℝ∪{+∞} be a proper lower semicontinuous function satisfying that the restriction of f to its domain is a continuous function.The function f is said to have the KL property if for any∀ x^*∈{x:0∈∂ f(x)}, there exist C,ϵ>0 and θ∈[0,1) such thatCy≥|f(x)-f^*(x)|^θ, ∀ x∈ B(x^*,ϵ), ∀ y∈∂ f(x),where θ is known as the KL exponent. Under Condition <ref>, we know that there exists λ_0≥0 such that Q_1+λ_0Q_2≻0 and thusb_1+λ_0b_2∈( Q_1+λ_0Q_2) due to the non-singularity of Q_1+λ_0Q_2. Hence from Lemma <ref>, problem (P) (and thus problem (P_1)) is bounded from below. It is shown in <cit.> that when the two matrices are SD and problem (P) is bounded from below,the optimal solution of problem (P) is attainable. This further implies that problem (P_1) isbounded from below with its optimal solution attainable. Assuming that x^* is an optimal solution, the following theorem shows that the KL inequality holds with an exponent of 1/2at x^* under some mild conditions.Assume that min h_1(x)<min H(x) andmin h_2(x)<min H(x). Then the KL property in Definition <ref> holds with exponent θ=1/2. Note that min h_1(x)<min H(x) andmin h_2(x)<min H(x) imply that, for any x^*∈{x:∂ H(x)=0}, ∇ h_1(x^*)≠0 and ∇ h_2(x^*)≠0, respectively. Assume L= max{λ_max(A_1),λ_max(A_2)}. We carry out our proof by considering the following two cases. * For any point with h_1(x)≠ h_2(x), w.l.o.g., assuming h_1(x)>h_2(x) gives rise to∂ H(x)={∇ h_1(x)}. Hence|H(x)-H(x^*)| = 1/2(x-x^*)^⊤ A_1(x-x^*)+(x^*)^⊤ A_1(x-x^*)+a_1^⊤(x-x^*)≤ 1/2Lx-x^*^2+∇ h_1(x^*)x-x^*.On the other hand, ∇ h_1(x)=A_1x+a_1 and∇ h_1(x)^2 = ∇ h_1(x)-∇ h_1(x^*)+∇ h_1(x^*)^2= (x-x^*)^⊤ A_1A_1(x-x^*)+∇ h_1(x^*)^2+2(∇ h_1(x^*))^⊤ A_1(x-x^*) ≥ ∇ h_1(x^*)^2-2L∇ h_1(x^*)x-x^*.Defineϵ_0=min{1,∇ h_1(x^*)/4L}. As ∇ h_1(x^*)≠0, for all x∈ℬ(x^*,ϵ_0),we then have|H(x)-H(x^*)|≤1/2Lϵ_0^2+ ∇ h_1(x^*)ϵ_0≤9/32L∇ h_1(x^*)^2and∇ h_1(x)^2≥∇ h_1(x^*)^2-2L∇ h_1(x^*)ϵ_0≥1/2∇ h_1(x^*)^2.Hence |H(x)-H(x^*)|^1/2≤√(9/32L)∇ h_1(x^*)≤3/4√(L)∇ h_1(x). So we have the followinginequality,|H(x)-H(x^*)|^θ≤ C_0y,for all y∈∂ H(x) (here {∇ h_1(x)}=∂ H(x)) with θ=1/2, C_0=3/4√(L). * Consider next a point x with h_1(x)=h_2(x). Define h_α(x)=α h_1(x)+(1-α)h_2(x), for some parameter α∈[0,1]. Let I={i|(∇ h_1(x^*))_i≠0}.The optimality condition 0∈∂ H(x^*) implies that there exists some α_0∈[0,1] such that α_0∇ h_1(x^*)+(1-α_0)∇ h_2(x^*)=0.Note that∇ h_1(x^*)≠0 and ∇ h_2(x^*)≠0 as assumed and thus α_0∈(0,1). Define j= argmax_i{|(∇ h_1(x^*))_i|,i∈ I}, M_1=(∇ h_1(x^*))_j and M_2=(∇ h_2(x^*))_j. Note that ∂ H_α_0(x^*)=0 implies that α_0 M_1=(1-α_0) M_2.W.o.l.g, assumeM_1≥ M_2 and thus α_0≤1/2. Since A_1x (A_2x, respectively) is a continuous function of x, there exists an ϵ_1>0 (ϵ_2>0, respectively) such that for any x∈ℬ(x^*,ϵ_1) (x∈ℬ(x^*,ϵ_2), respectively), 3/2M_1≥|(∇ h_1(x))_j| >1/2M_1 (3/2M_2≥|(∇ h_2(x))_j| >1/2M_2, respectively). Let ϵ_3=min{ϵ_1,ϵ_2}. Then we have the following two subcases. * For all x∈ℬ(x^*,ϵ_3) and α∈[0,1/4α_0], wehave∇ h_α(x) ≥ -α |(∇ h_1(x))_j|+(1-α) |(∇ h_2(x))_j|≥ -3/2α M_1+1/2(1-α)M_2≥ -3/8α_0 M_1+3/8(1-α_0)M_2+(1/8+1/8α_0)M_2= (1/8+1/8α_0)M_2.The third inequality is due to the fact that -3/2α M_1+1/2(1-α)M is a decreasing function of α and the last equality is due to α_0 M_1=(1-α_0) M_2. Symmetrically,for α∈[1-1-α_0/4,1], we have |(∇ h_α(x))|≥(3/8-1/4α_0)M_1. Combining these two cases andα_0≤1/2 yields ∇ h_α(x)≥1/8M_2.On the other hand|H(x)-H(x^*)| = 1/2(x-x^*)^⊤ A_1(x-x^*)+(x^*)^⊤ A_1(x-x^*)+a_1^⊤(x-x^*)≤ 1/2Lx-x^*^2+∇ h_1(x^*)x-x^*≤ (1/2Lϵ_3^2+∇ h_1(x^*))x-x^*.Letting ϵ_4=min{ϵ_3,1} leads to M_2^2/32Lϵ_3^2+64∇ h_1(x^*)|H(x)-H(x^*)|≤∇ h_α(x)^2. So|H(x)-H(x^*)|^θ≤ C_1∇ h_α(x), ∀α∈[0,1/4α_0]∪[1-1-α_0/4,1], ∀ x∈ℬ(x^*,ϵ_4) where θ=1/2 and C_1=√(32Lϵ_3^2+64∇ h_1(x^*))/M_2. * Next let us consider the case with α∈[α_0/4,1-1-α_0/4]. In this case,defining A_α=α A_1+(1-α)A_2 and a_α=α a_1+(1-α)a_2 gives rise to∇ h_α(x)^2 = ∇ h_α(x)-∇ h_α(x^*)+∇ h_α(x^*)^2= (x-x^*)^⊤ A_α A_α(x-x^*)+∇ h_α(x^*)^2+2(∇ h_α(x^*))^⊤ A_α(x-x^*)and since h_1(x)=h_2(x) and h_1(x^*)=h_2(x^*),|H(x)-H(x^*)| = 1/2(x-x^*)^⊤ A_α(x-x^*)+(x^*)^⊤ A_α(x-x^*)+a_α^⊤(x-x^*)= 1/2(x-x^*)^⊤ A_α(x-x^*)+(∇ h_α(x^*))^⊤(x-x^*).Define μ_0=λ_min(A_α). Then∇ h_α(x)^2-2μ_0|H(x)-H(x^*)|= (x-x^*)^⊤ A_α (A_α-μ_0 I)(x-x^*)+∇ h_α(x^*)^2 +2(∇ h_α(x^*))^⊤(A_α-μ_0 I)(x-x^*) = (A_α-μ_0 I)(x-x^*)+∇ h_α(x^*)^2+μ_0(x-x^*)^⊤(A_α-μ_0 I)(x-x^*)≥0. We next show that μ_0 isbounded from below. Define μ_1 (μ_2, respectively) as the smallest nonzero eigenvalue of A_1 (A_2, respectively). Note that α A_1+(1-α)A_2 is positive definite for all α∈[α_0/4,1-1-α_0/4] as assumed in Condition <ref>. Then A_1 and A_2 are simultaneously diagonalizable as shown in Lemma <ref>. Together with the facts that A_1≽0 and A_2≽0, there exists a nonsingular matrix P such that P^⊤ A_1P=D_1≽μ_1(δ ) and P^⊤ A_2P=D_2≽μ_2(δ ), where δ_i=1 if D_ii>0 and δ_i=0 otherwise. Since α∈[α_0/4,1-1-α_0/4], λ_min(A_α)≥min{αμ_1, ㄗ(1-α)μ_2}≥min{α_0/4μ_1,1-α_0/4μ_2}>0. From∇ h_α^2-2μ_0|H(x)-H(x^*)|≥0, we know ∇ h_α^2-μ_0|H(x)-H(x^*)|≥0.Let θ=1/2, C_2=√(1/(2μ_0)). We haveC_2∇ h_α(x)≥|H(x)-H(x^*)|^θ, ∀α∈[α_0/4,1-1-α_0/4], x∈ℬ(x^*,ϵ_4). Combining cases (a) and (b) gives rise to|H(x)-H(x^*)|^θ≤ C_3∇ h_α(x)with θ=1/2, C_3=max{C_1,C_2}, for all x∈ℬ(x^*,ϵ_4).Combining cases 1 and 2 yields that the KL inequality holdswith θ=1/2 and C=max{C_0,C_3} for all x∈ℬ(x^*,ϵ) with ϵ=min{ϵ_0,ϵ_4}. Note that the assumptionmin h_1(x)<min H(x) andmin h_2(x)<min H(x) means that we are in the easy case of GTRS as in this case λ^* is an interior point of I_PSD andQ_1+λ^* Q_2 is nonsingular, where λ^*is the optimal Lagrangian multiplier of the GTRS <cit.>. However, there are two situations for the hard case. Let us consider the KL property for H(x) at the optimal solution x^*. When h_i(x^*)>h_j(x^*), for i=1 or 2 and j={1,2}/{i}, in the neighbourhood x^*, H(x) is just h_i(x), and the KL is also 1/2 <cit.>. In such a case, our algorithm performs asymptotically like the gradient descent method for unconstrained quadratic minimization. However, whenh_i(x^*)=h_j(x^*) (note that min h_j(x)<H(x^*) can still hold in this situation),the KL exponent is not always 1/2 for H(x).Consider the following counterexample with h_1(x)=x_1^2 and h_2(x)=(x_1+1)^2+x_2^2-1. The optimal solution is (0,0) and is attained by both h_1 and h_2. Let x_2=-ϵ, where ϵ is a small positive number. Consider the curve where h_1(x)=h_2(x), which further implies x_1=-ϵ^2/2. Then we have(1-β ) ∇ h_1+β∇ h_2=2[ -(1-β) ϵ^2/2+β(-ϵ^2/2+1); βϵ ]=2[ - ϵ^2/2+β;βϵ ],and thusmin_y∈∂ H(x)y^2 = min_β4(β^2ϵ^2+β^2-ϵ^2β+ϵ^4/4)= min_β4((1+ϵ^2)(β-ϵ^2/2(1+ϵ^2))^2-ϵ^4/4(1+ϵ^2)+ϵ^4/4)= ϵ^6/2(1+ϵ^2)=ϵ^6/2+O(ϵ^8).Thus, min_y∈∂ H(x)y= O(ϵ^3). On the other hand,H(x)-H(x^*)=x_1^2=ϵ^4/4.The KL inequality cannot hold with θ=1/2, but it holds withθ=3/4 since min_y∈∂ H(x)y= O(ϵ^3) and H(x)-H(x^*)=O(ϵ^4).Itis interesting to compare our result with a recent result on KL exponent of the quadratic sphere constrained optimization problem <cit.>. In <cit.>, the authors showed that the KL exponent is 3/4 in general and 1/2 in some special cases, for the following problem,(T) min1/2x^⊤ Ax+b^⊤ x s.t. x^⊤ x=1.The above problem is equivalent to the TRS when the constraint of the TRS is active, which is the case of interest in the literature. For the TRS, the case that the constraint is inactive is trivial: Assuming x^* being the optimal solution, (x^*)^⊤ x^*<1 if and only if the objective function is convex and the optimal solution of the unconstrained quadratic function 1/2x^⊤ Ax+b^⊤ x locates in the interior of the unit ball. The authors in <cit.> proved that the KL exponent is 3/4 in general and particularly the KL exponent is 1/2 if A-λ^*I is nonsingular, where λ^* is the optimal Lagrangian multiplier. The later case is a subcase ofthe easy casefor the TRS and the case that KL exponent equals 3/4 only occurs in some special situations of the hard case.On the other hand, our result shows the KL exponent is1/2for the minimax problem when the associated GTRS is in the easy case. So our result can be seen as an extension of the resents on KL exponent forproblem (T) in <cit.>. One of our future research is to verify if the KL exponent is 3/4 for H(x) when theassociated GTRS is in the hard case. For convergence analysis with error bound or KL property, we still need a sufficient descent property to achieve the convergence rate. We next propose an algorithm with such a property. We further show that our algorithm converges locally linearly with the descent directionchosenin Theorem <ref> and the step size specifiedin the following theorem. Assume that the conditions in Theorem <ref> hold and that the initial point x^0∈ℬ(x^*,ϵ).Assume that h_i is the active function when h_1(x_k)≠ h_2(x_k) and h_j, j={1,2}/{i}, is thus inactive.Let the descent direction be chosen in Theorem <ref> and the associated step size be chosen as follows. * Whenh_1(x_k)= h_2(x_k), * ifthere exists g_α=α∇ h_1(x_k)+(1-α)∇ h_2(x_k) with α∈[0,1] such that ∇ h_1(x_k)^⊤ g_α=∇ h_2(x_k)^⊤ g_α, then set d_k=-g_α and β_k=1/L, where L=max{λ_max(A_1),λ_max(A_2)};* otherwiseset d_k=-∇ h_i(x_k) for i such that∇ h_1(x_k)^⊤∇ h_2(x_k)≥∇ h_i(x_k)^⊤∇ h_i(x_k), i=1,2, and β_k=1/L.* When h_1(x_k)≠ h_2(x_k) and the following quadratic equation for γ,𝐚x^2+𝐛x+𝐜=0,where 𝐚=1/2γ^2∇ h_i(x_k)^⊤ (A_i-A_j)∇ h_i(x_k), 𝐛=(∇ h_i(x_k)^⊤ -∇ h_j(x_k)^⊤ )∇ h_i(x_k) and 𝐜= h_i(x_k)-h_j(x_k),has no positive solution or any positive solution γ≥1/L, set d_k=-∇ h_i(x_k) withand β_k=1/L;* When h_1(x_k)≠ h_2(x_k) and the quadratic equation (<ref>)has a positive solution γ<1/L,set β_k=γand d_k=-∇ h_i(x_k). Thenthe sequence {x_k} generated by Algorithm <ref> satisfies, for any k≥1,H(x_k)-H(x^*)≤(√(%s/%s)2C^2L-12C^2L)^k-1(H(x^0)-H(x^*)),and dist(x_k,X)^2≤2/L(H(x_k)-H(x^*)≤2/L(√(%s/%s)2C^2L-12C^2L)^k-1(H(x^0)-H(x^*)) . For simplicity, let us denote g_i=∇ h_i(x_k)for i=1,2. We claim the following sufficient descent property forsteps 1,2 and 3:H(x_k)-H(x_k+1)≥L/2x_k-x_k+1^2.Hence, if the step size is 1/L (i.e., steps 1 and 2), we have H(x_l)-H(x^*)≤ Cd_l^2= C^2L^2x_l-x_l+1^2≤ 2C^2L(H(x_l)-H(x_l+1)),where the first inequality is due to the KL inequality in Theorem <ref>, the second equality is due to x_l+1=x_l-1/Ld_l and the last inequality is due to the sufficient descent property. Rearranging the above inequality yieldsH(x_l+1)-H(x^*)≤2C^2L-1/2C^2L(H(x_l)-H(x^*)).And since our method is a descent method, we have H(x_l+1)-H(x^*)≤ H(x_l)-H(x^*) for all iterations. Suppose that there are p iterates of step size 1, q iterates of step size 2, and r iterates of step size 3. From the definitions of the steps, every step 3 is followed bya step 1 and thus r≤ p+1 if we terminate our algorithm at step 1 or 2. So for all k≥1, after k=p+q+r steps, we haveH(x_k)-H(x^*)≤(2C^2L-1/2C^2L)^p+q(H(x^0)-H(x^*))≤(2C^2L-1/2C^2L)^k-1/2(H(x^0)-H(x^*)). The sufficient descent property further implies thatL/2∑_k^∞x_k-x_k+1^2≤ H(x_k)-H(x^*).Hence, with ∑_k^∞x_k-x_k+1^2≥ dist(x_k,X)^2, we have L/2 dist(x_k,X)^2≤ H(x_k)-H(x^*). Thus dist(x_k,X)^2≤2/L(H(x_k)-H(x^*)). By noting g_i=A_ix_k+a_i, we haveh_i(x_k+1)-h_i(x_k) =1/2(x_k+d_k)^⊤ A_i(x_k+d_k)+a_i^⊤(x_k+d_k)-[1/2(x_k)^⊤ A_ix_k+a_i^⊤ x_k]= 1/2d_k^⊤ A_id_k+(A_ix_k+a_i)^⊤ d_k= 1/2d_k^⊤ A_id_k+g_i^⊤ d_k.We next prove our claim (<ref>) according to the three cases in our updating rule:* When h_1(x_k)= h_2(x_k), noting that h_iis active at x_k+1 as assumed,we have H(x_k)-H(x_k+1)= h_i(x_k)-h_i(x_k+1).* If there exists an α such that g_α^⊤ g_1=g_α^⊤ g_2, we have g_α^⊤ g_i=g_α^⊤ g_α. And by noting that d_i=-g_α, we furtherhaveh_i(x_k+1)-h_i(x_k) = 1/2L^2d_k^⊤ A_id_k+1/Lg_i^⊤ d_k≤ 1/2Lg_α^⊤ g_α-1/Lg_α^⊤ g_α= -1/2Lg_α^⊤ g_α.Substituting g_α=L(x_k-x_k+1) to the above expression, we have the following sufficient descent property,H(x_k)-H(x_k+1)=h_i(x_k)-h_i(x_k+1)≥L/2x_k-x_k+1^2. * If there does not exist an α such that g_α^⊤ g_1=g_α^⊤ g_2, then we must have g_1^⊤ g_2>0. And thus we must have g_1^⊤ g_1≥ g_1^⊤ g_2≥ g_2^⊤ g_2 or g_2^⊤ g_2≥ g_1^⊤ g_2≥ g_1^⊤ g_1. If g_i^⊤ g_i≥ g_i^⊤ g_j≥ g_j^⊤ g_j, we set d_k=-g_j. ThenH(x_k+1)-H(x_k) ≤ max{h_i(x_k+1)-h_i(x_k), h_j(x_k+1)-h_j(x_k)}≤ max{1/2L^2g_j^⊤ A_ig_j-1/Lg_i^⊤ g_j, 1/2L^2g_j^⊤ A_ig_j-1/Lg_j^⊤ g_j}≤ max{1/2L^2g_j^⊤ A_ig_j-1/Lg_j^⊤ g_j, 1/2L^2g_j^⊤ A_ig_j-1/Lg_j^⊤ g_j}≤ max{1/2Lg_j^⊤ g_j-1/Lg_j^⊤ g_j, 1/2Lg_j^⊤ g_j-1/Lg_j^⊤ g_j}= -1/2Lg_j^⊤ g_j=-L/2x_k-x_k+1^2. Symmetrically, if g_i^⊤ g_j>0 and g_j^⊤ g_j≥ g_i^⊤ g_j≥ g_i^⊤ g_i, settingd_k=-g_i yields the same sufficient descent property. * When h_1(x_k)≠ h_2(x_k) and the quadratic equation (<ref>) for γ has no positive solution or has a positive solution γ≥1/L, we have h_i(x_k+1)>h_j(x_k+1)for x_k+1=x_k+β_kd_k, where d_k=-∇ h_i(x_k)and β_k=1/L. Moreover,H(x_k+1)-H(x_k) =h_i(x_k+1)-h_i(x_k)= 1/2L^2g_i^⊤ A_ig_i-1/Lg_i^⊤ g_i≤ -1/2Lg_i^⊤ g_i.Hence H(x_k)-H(x_k+1)≥1/2Lg_i^⊤ g_i≥L/2x_k-x_k+1^2.* When h_1(x_k)≠ h_2(x_k) and the quadratic equation (<ref>)has a positive solution γ<1/L. With β_k=γand d_k=-∇ h_i(x_k),it is easy to see that the step size γ makes h_1(x_k+1)=h_2(x_k+1). Then we haveH(x_k+1)-H(x_k) = h_i(x_k+1)-h_i(x_k)= 1/2γ^2d_k^⊤ A_id_k+γ g_i^⊤ d_k≤ 1/2Lγ^2g_i^⊤ g_i-γ g_i^⊤ g_i=(L/2-1/γ)x_k-x_k+1^2, which further implies H(x_k)-H(x_k+1)≥L/2x_k-x_k+1^2 due to γ≤1/L.It is worth to note that Step 3 in our algorithm is somehow similar to the retraction step in manifold optimization <cit.>. In manifold optimization, in every iteration, each point is retracted to the manifold. In Step 3, everypointis drawn to the curve that h_1(x)=h_2(x). We will next show that in general a global sublinear convergence rate, in the same order with the gradient descent algorithm,can also be theoretically guaranteed for Algorithm 1. Assume that x^* is an optimal solution. Then we haveH(x_N)-H(x^*)≤L/Nx_0-x^*^2.That is, the required iterate number for H(x_N)-H(x^*)≤ϵ is at most O(1/ϵ). From the proof in Theorem <ref>, for any step size γ≤1/L, we haveH(x_k+1)-H(x_k)≤-γ g^⊤ g+1/2Lγ^2g^⊤ g≤-γ/2g^⊤ g.From the convexity of H(x) and g∈∂ H(x_k), we haveH(x_k+1) ≤H(x_k)-γ/2g^⊤ g≤ H(x^*)+g^⊤(x_k-x^*)-γ/2g⊤ g= H(x^*)+1/2γ(x_k-x^*^2-x_k-x^*-γ g^2)= H(x^*)+1/2γ(x_k-x^*^2-x_k+1-x^*^2).Since H(x_k+1)≥ H(x^*), we have x_k-x^*^2-x_k+1-x^*^2≥0. Let us useindices i_k, k=0,…,K to denote the indices in Steps 1 and 2. By noting that γ=1/L, we haveH(x_i_k+1)≤ H(x^*)+L/2(x_i_k-x^*^2-x_i_k+1-x^*^2).Note that every Step 3 is followed by S tep 1. Hence N≤ 2K+1.Adding the above inequalities from i_0 to i_K, we have∑_k=0^KH(x_i_k)-H(x^*)≤ L/2∑_k=0^K(x_i_k-x^*^2-x_i_k+1-x^*^2)≤ L/2(x_i_0-x^*^2-x_i_K+1-x^*^2+∑_k=1^K(-x_i_k-1+1-x^*^2+x_i_k-x^*^2))≤ L/2(x_i_0-x^*^2-x_i_K+1-x^*^2)≤ L/2x_i_0-x^*^2≤ L/2x_0-x^*^2,where in the second inequality we use the fact,-x_i_k-1+1-x^*^2+x_i_k-x^*^2≤-x_i_k-1+1-x^*^2+x_i_k-1-x^*^2≤⋯≤0 .Since H(x_k) is non-increasing, by noting that N≤ 2K+1, we haveH(x_N)-H(x^*) ≤ 1/K+1∑_k=0^KH(x_i_k)-H(x^*)≤ L/Nx_0-x^*^2.§.§ Line search with the modified Armijo ruleAn alternative way to choose the step size in the classical gradient descent type methodsis the line search with the Armijo rule. A natural thought is then to extend the Armijo rule in our minimax problem (M) asin the proposed Algorithm <ref>. In particular, we set the following modified Armijo rule to choose the smallest nonnegative integer k such that the following inequality holds for the step size β_k=ξ s^k with 0<ξ≤1 and 0<s<1,f(x_k+β_k p_k)≤ f(x_k)+σβ_k p_k^⊤ g,where 0≤σ≤0.5, g=-d andd is the steepest descent direction defined in Theorem <ref>. Particularly, we set the search direction p_k=d at iterate k. Our numerical result in the next section shows that Algorithm <ref> has a comparable performance when compared with(or even better than) Algorithm <ref>. For the sake of completeness, we present the convergence resultfor Algorithm <ref> in the following.Before that, we generalize the definition of a critical point to a (ρ,δ) critical point.A point x is called a (ρ,δ) critical point of H(x)=max{h_1(x),h_2(x)} if∃g<δ, for some g∈∂ H_ρ(x), where ∂ H_ρ(x) is defined as follows:* ∂ H_ρ(x)={α∇ h_1(x)+(1-α)∇ h_2(x):α∈[0,1]}, if |h_1(x)-h_2(x)|≤ρ;* ∂ H_ρ(x)={∇ h_1(x)}, if h_1(x)-h_2(x)> ρ;* ∂ H_ρ(x)={∇ h_2(x)}, if h_2(x)-h_1(x)> ρ.The following proposition shows the relationship of a critical pointanda (ρ,δ) critical point. As this result is pretty obvious, we omit its proof.Assume that{x_k} is a sequence in ℝ^n and that (ρ^t,δ^t)→(0,0), for t→∞and that there exists a positive integerK(t), such thatx_k is a (ρ^t,δ^t) critical point of H(x) for all k≥ K(t) and t≥1. Then, every accumulation point of thesequence{x_k} isa critical point of H(x). Slightly different from Algorithm <ref>, our goal in Algorithm <ref> is to find a (ρ,δ) critical point. With Proposition <ref>, we conclude that Algorithm <ref> outputs a solution that is sufficiently close to a critical point of H(x). Assume thati) d= argmin_y∈∂H_ρ(x_k)y with ρ>0,ii) the termination criterion is d<δ for some δ>0 and iii) x^* is an optimal solution. Then for any givenpositive numbers ρ and δ, Algorithm <ref> generates a (ρ,δ) critical point in at mostH(x_0)-H(x^*)/σ smin{1/L,ξ,ρ/2G^2}δ^2iterations, where G is some positive constant only depending on the initial point and problem setting.Consider the following different cases with d≥δ. * If |h_1(x_k)-h_2(x_k)|<ρ, then as assumedd>δ andfrom Theorem <ref>, we know that d= argmin_α∈[0,1]α∇ h_1(x_k)+(1-α)∇ h_2(x_k) is just the parameter α which we choose in Algorithm <ref>. It suffices to show that the step size β_k is bounded from below such thatH(x_k+1)-H(x_k)≤ -σβ_kd^⊤ d.This further suffices to show thatβ_k is bounded from below such that for i=1 or 2,h_i(x_k+1)-h_i(x_k)= -β_k∇ h_i(x_k)^⊤ d+1/2β_k^2d^⊤ A_id≤ -σβ_kd^⊤ d.By noting that ∇ h_i^⊤ d≥ d^⊤ d from Remark <ref>, thesecond inequality in (<ref>) holds true for all β_k≤2(1-σ)/L.Then the step size chosen by the modified Armijo rule satisfies β_k≥ smin{2(1-σ)/L,ξ}, which further implies thatH(x_k)-H(x_k+1)≥σβ_kg^⊤ g=σβ_k g^2≥σ smin{2(1-σ)/L,ξ}δ^2. * If h_1(x_k)-h_2(x_k)>ρand ∇ h_1(x_k)>δ, we have g=∇ h_1(x_k). Because H(x_k) is decreasing, under Condition <ref>, h_1(x_k)+ h_2(x_k)=1/2x_k^⊤ (A_1+A_2)x_k+(a_1+ a_2)^⊤ x_k≤ 2h_1(x_k)=2H(x_k)≤2H(x_0) and thus x_k is bounded due to A_1+A_2=2(Q_1+λ_1+λ_2/2Q_2)≻0. This further implies that ∇ h_i(x_k)=A_ix_k+b_iis bounded for all k. So there exists some positive constant only depending on the initial point and problem parameters such that ∇ h_i(x_k)≤ G, i=1,.Hence d≤ G because d is a convex combination of ∇ h_1(x_k) and ∇ h_2(x_k). Then we haveh_1(x_k+1)-h_1(x_k)≤ -β_k∇ h_1(x_k)^⊤ d+1/2β_k^2d^⊤ A_1dand for any β_k≤1/L, h_2(x_k+1)-h_2(x_k) ≤-β_k∇ h_2(x_k)^⊤ g+1/2β_k^2g^⊤ A_ig≤ β_kGg+1/2β_k^2Lg^2≤ β_kG^2(1+1/2β_kL)≤ 3/2β_kG^2.On the other hand,when β_k≤1/L,h_1(x_k+1)-h_1(x_k)≤ -β_k∇ h_1(x_k)^⊤ g+1/2β_k^2g^⊤ A_1g≤ -β_kg^⊤ g+1/2β_k^2Lg^⊤ g≤-1/2β_kg^⊤ g.Note that for all β_k≤ρ/2G^2,3/2β_kG^2+1/2β_kg^⊤g≤ρ. Thus for β_k≤min{1/L, ρ/2G^2}, we haveh_1(x_k+1)≤ h_1(x_k)-1/2β_kg^⊤ g, h_2(x_k+1)≤ h_2(x_k)+3/2β_kG^2≤ h_1(x_k)-ρ+3/2β_kG≤ h_1(x_k)-1/2β_kg^⊤ g.Hence we haveH(x_k+1)-H(x_k) = max{h_1(x_k+1),h_2(x_k+1)}-h_1(x_k)= max{h_1(x_k+1)-h_1(x_k), h_2(x_k+1)-h_1(x_k)}≤ max{h_1(x_k+1)-h_1(x_k), h_2(x_k+1)-h_2(x_k)}≤ -1/2β_kg^⊤ g.So the Armujo rule impliesβ_k≥ smin{1/L,ξ,ρ/2G^2}, i.e., β_k is lower bounded. Then according to the modified Armijo rule, we haveH(x_k)-H(x_k+1)≥σβ_kg^⊤ g≥σ smin{1/L,ξ,ρ/2G^2}δ^2. * Symmetrically, the case with h_2(x_k)-h_1(x_k)>ρyields the same result as in (<ref>).The above three cases show that H(x_k)-H(x_k+1)≥σ smin{1/L,ξ, ρ/2G^2}δ^2 (as 1-σ≥1/2, the decrease in case 1 also admits this bound). Since the decrease in each iterate is larger than σ smin{1/L,ξ,ρ/2G^2}δ^2, the total iterate number is bounded byH(x_0)-H(x^*)/σ smin{1/L,ξ,ρ/2G^2}δ^2. At the current stage, we cannot demonstrate a theoretical convergence rate for Algorithm <ref>as good as the sublinear rate O(1/ρ) for Algorithm 1 in Theorem <ref>. But our numerical tests show that Algorithm <ref>converges as fast as Algorithm <ref>. Proposition <ref> and Theorem <ref> offer our main convergence result for Algorithm <ref> as follows.Assume that (ϕ_k,ψ_k)→0 andthat {x^(k)} is a sequence ofsolutions generated by Algorithm <ref> with ρ=ϕ_k and δ=ψ_k. Then any accumulation point of {x^(k)} is an optimal solution of problem (M). § NUMERICAL TESTSIn this section, we illustrate the efficiency of our algorithm with numerical experiments. All the numerical tests were implemented in Matlab 2016a, 64bit and were run on a Linux machine with 48GB RAM, 2600MHz cpu and 64-bit CentOS release 7.1.1503.We compare both Algorithms <ref> and <ref>with the ERW algorithm in <cit.>. Wedisable the parallel setting in the Matlab for fair comparison. If the parallel setting is allowed, our algorithm has a significant improvement, while the ERW algorithm does not.We use the following sametest problem as <cit.> to show the efficiency of our algorithms, (IP) min x^⊤ Ax-2a^⊤ x s.t c_1≤ x^⊤ Bx≤ c_2,where A is an n× npositive definite matrix and B is ann× n (nonsingular) symmetric indefinite matrix. We first reformulate problem (IP) toa formulation of problem (P) in the following procedure, which is motivated from <cit.> (the proof in <cit.> is also based on the monotonicity of γ(λ), which is defined in Section 2.1), in order toapply the CQR for problem (P) andthen invoke Algorithms <ref> and <ref> to solve the CQR.Let x_0=-A^-1a. Then the followings hold.* If x_0^⊤ Bx_0<c_1, problem (IP) is equivalent to(IP_1) min{ x^⊤ Ax-2a^⊤ x: s.t. c_1≤ x^⊤ Bx}; * Else if c_1≤ x_0^⊤ Bx_0≤ c_2, problem (IP)admits an interior solution x_0;* Otherwise c_2<x_0^⊤ Bx_0,problem (IP) is equivalent to (IP_2) min{ x^⊤ Ax-2a^⊤ x: s.t. x^⊤ Bx≤ c_2}.Item 2 is obvious. Item 1 and Item 3 are symmetric. So in the following, we only prove Item 1.In our problem set, matrix A is positive definite and B is indefinite. Hence, in the definition I_PSD={λ:Q_1+λ Q_2≽0}, we have λ_1<0, λ_2>0. Thus from Case 1 in Section 2.2.2 in <cit.> we know, when x_0^⊤ Bx_0<c_1,problem (IP) is equivalent to( EP_1) min{ x^⊤ Ax-2a^⊤ x: s.t. c_1=x^⊤ Bx}.Sincex_0^⊤ Bx_0<c_1, the optimal solution of ( IP_1) must be at its boundary <cit.>. This further yieldsthat problem (IP) is equivalent to ( IP_1). Theorem 4.1 helps ussolve problem (IP) as an inequality constrained GTRS instead of solving two GTRS with equality constraints. Before showing the numerical results, let us illustrate some functions used in our initialization. To obtain the CQR, the generalized eigenvalue problem is solvedby in Matlab, which was developed in <cit.> for computing the maximum generalized eigenvalues for sparse definite matrix pencils. In our numerical settingis usually faster than the Matlab function , thoughwill outperform when the condition number is large or the density is low. We use the Matlab commandandto generate Q_1 and Q_2. We set the density of matricesat 0.01 and use three levels of condition number for matrix Q_1, i.e., 10, 100 and 1000 and, in such settings,always dominates (this may be becauseis developed for computing extreme generalized eigenvalues for arbitrary matrices and does not utilize the definiteness and symmetry properties of the matrix pencils in our problem setting). In general, the main cost in estimating L is to compute the maximum eigenvalues of matrices A_1 and A_2, which may be time consuming for large-scale matrices. To conquer this difficulty, we canestimate a good upper bound with very cheap cost instead. Specially, we can run the functionwith precision 0.1, which is much more efficient thancomputing the true maximum eigenvalue with, and, assuming M is the output,M+0.1 is then a good upper bound for L. In our numerical tests, we just use to estimate L since our main goal is to illustrate the efficiency of Algorithm 2. In Algorithm <ref>, to avoid some numerical accuracy problem, we approximateh_1(x_k)=h_2(x_k) by |h_1(x_k)-h_2(x_k)|/(|h_1(x_k)|+|h_2(x_k)|)≤ϵ_1. Also we use |h_1(x_k)-h_2(x_k)|/(|h_1(x_k)|+|h_2(x_k)|)≤ϵ_1 instead of|h_1(x_k)- h_1(x_k)|≤ρ in Algorithm 2 for stableness consideration. In our numerical tests for both Algorithms <ref> and <ref>, we use the following termination criteria(if any one of the following three conditions is met, we terminate our algorithm), which are slightly different from the presented algorithms for robust consideration: * H(x_k-1)-H(x_k)<ϵ_2,* |h_1(x_k)-h_2(x_k)|/(|h_1(x_k)|+|h_2(x_k)|)≤ϵ_1, α∇ h_1(x_k)+(1-α)∇ h_2(x_k)≤ϵ_3,* ∇ h_i(x_k)≤ϵ_3 and |h_1(x_k)-h_2(x_k)|/(|h_1(x_k)|+|h_2(x_k)|)>ϵ_1, where i≠ j and i,j∈{1,2},ㄛwhere ϵ_1, ϵ_2 and ϵ_3>0 are some small positive numbers for termination of the algorithm. Particularly, we set ϵ_1=10^-8, ϵ_2=10^-11 and ϵ_3=10^-8 in Algorithm 1, and ϵ_1=10^-8, ϵ_2=10^-11, ϵ_3=10^-8,σ=10^-4 and ξ=1 (for the modified Armijo rule) in Algorithm 2. To improve the accuracy of the solution, we apply the Newton refinement process in Section 4.1.2 in <cit.>. More specifically, assuming x^* is the solution returned by our algorithm, we update x^* byδ=(x^*)^⊤ Bx^*/2Bx^*^2Bx^*, x^*=x^*-δ. In general, the ERW algorithm can achieve a higher precision than our method (after the Newton refinement process); the precision in their method is about 10^-14, while ours is slightly less precise than theirs. Letting v_1 denote the optimal value of ERW algorithm and v_2 denote the optimal value of our algorithm, wehave at least |v_2-v_1|/|v_1|≈10^-10 for most cases. The iteration number reduces to 1/5 if we reduce the precision of from ϵ_1=10^-8, ϵ_2=10^-11, ϵ_3=10^-8 to ϵ_1=10^-5, ϵ_2=10^-8, ϵ_3=10^-5. This observation seems reasonable as our method is just a first order method.We report our numerical results in Table 1.We use “Alg1” and “Alg2” to denote Algorithms <ref> and <ref>, respectively. For each n and each condition number, we generate 10 Easy Case and 10 Hard Case 1 examples. Please refer to Table 1 in <cit.> for the detailed definitions of Easy Case and Hard Cases 1 and 2. There is a little difference about the definitions of easy and hard cases between <cit.> and<cit.>. Our analysis in the above sections uses the definitions in<cit.>. In fact, the Easy Case and Hard Case 1 are the easy caseand Hard Case 2 is the hard case mentioned in the above sections and <cit.>. We use the notation “time" to denote the average CPU time (in unit of second) and “iter” to denote the average iteration numbers for all the three algorithms. For “Alg1” and “Alg2”, “time" isjust the time for Algorithms <ref> and <ref>, respectively. The notation “time_eig" denotes the average CPU time for computing the generalized eigenvalue for our algorithm. So the total time for solving problem (P) should be the summation of the time of reformulate (P) into (M) and the time of Algorithm 1 or 2, whose main cost is just “time"+“time_eig". And “fail"denotes the failure times in the 10 examples in each case for the ERW algorithm. One reason of the failures may be that the ERW algorithm terminates in 10 iterations even when it does not find a good approximated solution. We point out that for randomly generated test examples, our method always succeeds in finding an approximated solution to prescribed precision whilethe ERW algorithm fails frequently in Hard Case 1. Anotherdisadvantage of the ERW algorithm is the requirement of an efficient prior estimation of the initialization, which is unknown in general. In our numerical test, we assume that such an initialization is given as the same as <cit.> does.We also need to point out that in the Hard Case 2,our algorithms do not outperform the ERW algorithm which uses theshift and deflation technique.The main time cost of shift and deflate operation is the computation of the extreme generalized eigenvalue of the matrix pencil (A,B) and its corresponding generalized eigenvectors. In the test instances, as the dimension of the eigenspace of the extreme generalized eigenvalue isone, the shift and deflation technique directly finds the optimal solution by calling once. Our algorithm reduces to an unconstrained quadratic minimization in Hard Case 2. However, the condition number of this unconstrainedquadratic minimization is so large that our algorithm performs badly as the classical gradient method. To remedy this disadvantage, we can add a step with almost free-time cost that claims that eitherwe are in Hard Case 2 and output an optimal solution or we are in Easy Case or Hard Case 1. Recall that the hard case (or equivalently, Hard Case 2)states that b_1+λ^*b_2 is orthogonal to the null space of Q_1+λ^*Q_2 which means thatλ^*must be a boundary point of I_PSD. Suppose λ_i=λ^*. Then we must havethatx^*=min H(x) and H(x^*)=h_i(x^*) for some i=1 or 2. In fact, if ∇ h_i(x)=0 and h_i(x)≥ h_j(x), j∈{1,2}/{i} for some x, then x is optimal and we are in the hard case. So ∇ h_i(x)=0 and h_i(x)≥ h_j(x) is sufficient and necessary for x to be optimal to (M)and be in the hard case. Hence we can construct an optimal solution for problem (M) asx̅=(Q_1+λ_i Q_2)^†(b_1+λ _ib_2)+∑_i^k α_j v_j(where A^† denotes the Moore–Penrose pseudoinverse of A) if v_j,j=1,…,kare the generalized eigenvectors of matrix pencil (Q_1,Q_2) with respect to the generalized eigenvalue λ_i such that h_i(x̅)≥ h_j(x̅) and α≥0. This equals to identifying if asmall dimensional convex quadratic programming problem (with variable α) has an optimal value less than h_i((Q_1+λ_i Q_2)^†(b_1+λ _ib_2)). And if such α does not exist, we are in the easy case (or equivalently, Easy Case or Hard Case 1). This technique is very similar to the shift and deflation technique in <cit.>. Hence we cansolve Hard Case 2 within almost the same CPU time as the ERW algorithm. So we do not make further comparison for Hard Case 2. Our numerical tests show thatboth Algorithms 1 and 2 are much more efficient than the ERW algorithm in Easy Case and for most cases in Hard Case 1.The efficiency of our algorithms is mainly due to thatwe only call the generalized eigenvalue solver once and every iteration only involves several matrix vector products (which are very cheap for sparse matrices).We also note that, in Easy Case, Algorithm 1 is faster than Algorithm 2 when the condition number is small and slower than Algorithm 2 when the condition number is large. This may be because thatAlgorithm 2is equipped with the modified Armijo rule, which makes it more aggressive in choosing the step size and thus yields a fast convergence. In Hard Case 1, Algorithm 2 is still much more efficient than the ERW algorithm while Algorithm 1 is slower than the ERW algorithm in about half the cases. This is because Algorithm 2 has a moderate iterate number due to the aggressiveness in choosing the step size and Algorithm 1 has a much large iterate number for these cases.Moreover, our algorithms always succeed, while the ERW algorithm fails frequently in Hard Case 1. A more detailed analysis with condition number for Algorithm 1 will be given in the following.We note that several examples (of the 10 examples) in Easy Cases admit a much larger iteration number than average. This motivates us to analyze the main factor that affects the convergence rate (reflected by the iteration number) of Algorithm 1 (the analysis for Algorithm 2 seems harddue to the non-smoothness of the problem). We then find that the main factoris √(λ_maxα/2λ_min nnzα^2), as evidenced by the fact that examples in Easy Case and Hard Case 1 with more iterates all have a larger √(λ_maxα/2λ_min nnzα^2), whereλ_maxα denotes the maximum eigenvalue of matrix α A_1+(1-α)A_2 and λ_min nnzα denotes the smallest nonzero eigenvalue of matrix α A_1+(1-α)A_2with α being defined in Theorem <ref> in the last iteration. In fact, when x^k→ x^*∈{x:∂ H(x)=0} (in our examples, the optimal solution is unique), let the value of α at iterate k be α^k, then α^k→α^*, where α^*is the solution of α∇ h_1(x^*)+(1-α)∇ h_2(x^*)=0. From the definition of KL exponent,we haveC×min_αα∇ h_1(x^k)+(1-α)∇ h_2(x^k)≥|H(x^k)-H(x^*)|^1/2.Intuitively, the smallest value ofC should be at least|H(x^k)-H(x^*)|^1/2/min_αα∇ h_1(x^k)+(1-α)∇ h_2(x^k)→|α (h_1(x^k)-h_2(x^*))+(1-α) (h_1(x^k)-h_2(x^*))|^1/2/min_αα∇ h_1(x^k)+(1-α)∇ h_2(x^k)which is upper bounded by √(λ_maxα/2λ_min nnzα^2). Thus, the asymptotic value of C can beroughly seen as √(λ_maxα/2λ_min nnzα^2). Hence both Easy Case and Hard Case 1 admit local linear convergence and the convergence rate is(√(1-1/2C^2L))^k= (√(1-λ_min nnzα^2/Lλ_maxα))^kfrom Theorem <ref>. We also observefrom our numerical tests that in most cases the values of λ_maxα are similar and that λ_min nnzα in Easy Case is much larger than λ_min nnzα in Hard Case 1 and λ_maxα in Easy Case is very close to λ_maxαin Hard Case 1. Hence, √(1-λ_min nnzα^2/(Lλ_maxα)) in Easy Caseis usually much smaller than that in Hard Case 1. (As Q_2 is random in our setting, the larger the condition number of Q_1 is, the larger expectation of√(1-λ_min nnzα^2/(Lλ_maxα)) is.) Thisexplains why the condition number of matrix Q_1 measures, to a large degree, the hardness of our algorithms in solving problem (M). Since Easy Case has a smaller √(1-(λ_min nnzα^2/Lλ_maxα)) than Hard Case 1 for the same condition number and problem dimension, Easy Case can be solved faster than Hard Case 1.This coincides with our numerical results, i.e., Easy Case admits a smaller iterate number thanHard Cases 1. We also tried to apply MOSEK <cit.> to solve the CQR. But our numerical results showed that MOSEK is much slower than both our algorithms and the ERW algorithm, which took about 833 seconds for Easy Case and 960 second for Hard Case 1 with n=10000 and cond=10. So we do not run further numerical experiments with MOSEK. We also tested the SOCP reformulation <cit.> under the simultaneous digonalization condition of the quadratic forms of the GTRS and the DB algorithm in <cit.> based on the simultaneous digonalization condition of the quadratic forms. The simultaneous digonalization condition naturally holds for problem (IP) when A is positive definite. Our preliminary result shows that our method is much more efficient than the two methods based onsimultaneous digonalization when n≥10000 and density=0.01 and thus we also do not report the numerical comparison in this paper. We believe this is mainly because thesimultaneously digonalization procedure of the matrices involvesmatrix inverse, matrix matrix product, a full Cholesky decomposition and a spectral decomposition (of a dense matrix),which is more time consuming than the operations ofmatrix vector products in our algorithm.Hence we do not report the numerical results based on the simultaneous digonalization technique. § CONCLUDING REMARKSIn this paper, we have derived a simple convex quadratic reformulation for the GTRS, which only involves a linear objective function and two convex quadratic constraints under mild assumption.We further reformulate the CQR to an unconstrained minimax problem under Condition <ref>, which is the case of interest. The minimax reformulation is a well structured convex, albeit non-smooth, problem.By investigating its inherent structure, we have proposed two efficient matrix-free algorithms to solve this minimax reformulation. Moreover, we have offered a theoretical guarantee of global sublinear convergence rate for both algorithms anddemonstrate a local linear convergence rate for Algorithm 1 by provingthe KL property for the minimax problem with an exponent of 1/2 under some mild conditions. Our numerical results have demonstrated clearly out-performance of our algorithms over the state-of-the-art algorithm for the GTRS.As for our future research, we wouldlike to show whether the CQR and the minimax reformulation and the algorithms for the minimax problem can be extended to solve GTRSwith additional linear constraints. Asthe analysis in numerical section indicates that our algorithms have similar performance with unconstrained quadratic minimization, i.e., both algorithms admit a locally linear convergence rate with the steepest descent method, we would like to generalize existing algorithms that are efficient in solving unconstrained quadratic minimization to solve our minimax reformulation, e.g., the conjugate gradient method or Nesterov's accelerated gradient descent algorithm. Another line of future research is to investigate whether our algorithm can be extended to general minimax problems with more (finite number of) functions. It is also interesting to verify whether the KL property still holds and whether the KL exponent is still 1/2 when more functions are involved.§ ACKNOWLEDGEMENTSThis research was partially supported by Hong Kong Research Grants Council under Grants 14213716 and 14202017. The second author is also grateful to the support from Patrick Huen Wing Ming Chair Professorship of Systems Engineering and Engineering Management. The authors would also like to thank Zirui Zhou and Huikang Liufor their insightful discussions.abbrv | http://arxiv.org/abs/1707.08706v3 | {
"authors": [
"Rujun Jiang",
"Duan Li"
],
"categories": [
"math.OC"
],
"primary_category": "math.OC",
"published": "20170727050152",
"title": "Novel reformulations and efficient algorithms for the generalized trust region subproblem"
} |
Deep Learning Models for Wireless Signal Classification with Distributed Low-Cost Spectrum SensorsSreeraj Rajendran, Student Member, IEEE, Wannes Meert, Member, IEEE Domenico Giustiniano, Senior Member, IEEE, Vincent Lenders, Member, IEEE and Sofie Pollin, Senior Member, IEEE. December 30, 2023 ===============================================================================================================================================================================================================================================This paper looks into the modulation classification problem for a distributed wireless spectrum sensing network. First, a new data-driven model for amc based on long short term memory (LSTM) is proposed. The model learns from the time domain amplitude and phase information of the modulation schemes present in the training data without requiring expert features like higher order cyclic moments. Analyses show that the proposed model yields an average classification accuracy of close to 90% at varying SNR conditions ranging from 0dB to 20dB. Further, we explore the utility of this LSTM model for a variable symbol rate scenario. We show that a LSTM based model can learn good representations of variable length time domain sequences, which is useful in classifying modulation signals with different symbol rates. The achieved accuracy of 75% on an input sample length of 64 for which it was not trained, substantiates the representation power of the model. To reduce the data communication overhead from distributed sensors, the feasibility of classification using averaged magnitude spectrum data and on-line classification on the low-cost spectrum sensors are studied. Furthermore, quantized realizations of the proposed models are analyzed for deployment on sensors with low processing power.Deep learning, Modulation classification, LSTM, CNN, Spectrum sensing.§ INTRODUCTION[HBCI]3gpp[3GPP]3rd Generation Partnership Program cnn[CNN]Convolutional Neural Network fbmc[FBMC]Filter Bank Multicarrier phy[PHY]physical layer pu[PU]Primary User rat[RAT]Radio Access Technology rfnoc[RFNoC]RF Network on Chip sdr[SDR]Software Defined Radio su[SU]Secondary User toa[TOA]Time of Arrival tdoa[TDOA]Time Difference of Arrival usrp[USRP]Universal Software Radio Peripheral amc[AMC]Automatic Modulation Classification lstm[LSTM]Long Short Term Memory soa[SoA]state-of-the-art fft[FFT]Fast Fourier Transform wsn[WSN]Wireless Sensor Networks iq[IQ]in-phase and quadrature phase snr[SNR]signal-to-noise ratio sps[sps]samples/symbol awgn[AWGN]Additive White Gaussian Noise ofdm[OFDM]Orthogonal Frequency Division Multiplexing rnn[RNN]Recurrent Neural Networks svm[SVM]Support Vector Machines psd[PSD]Power Spectral DensityWireless spectrum monitoring over frequency, time and space is important for a wide range of applications such as spectrum enforcement for regulatory bodies, generating coverage maps for wireless operators, and applications including wireless signal detection and positioning. Continuous spectrum monitoring over a large geographical area is extremely challenging mainly due to the multidisciplinary nature of the solution. The monitoring infrastructure requires proper integration of new disruptive technologies than can flexibly address the variability and cost of the used sensors, large spectrum data management, sensor reliability, security and privacy concerns, which can also target a wide variety of the use cases. Electrosense was designed to address these challenges and support a diverse set of applications <cit.>. Electrosense is a crowd-sourced spectrum monitoring solution deployed on a large scale using low cost sensors.One of the main goals of Electrosense is to accomplish automated wireless spectrum anomaly detection, thus enabling efficient spectrum enforcement. Technology classification or specifically Automatic Modulation Classification (amc) is an integral part of spectrum enforcement. Such a classifier can help in identifying suspicious transmissions in a particular wireless band. Furthermore, technology classification modules are fundamental for interference detection and wireless environment analysis. Considering the aforementioned large application space this paper looks into two key aspects: Is efficient wireless technology classification achievable on a large scale with low cost sensor networks and limited uplink communication bandwidth? If possible, which are the key classification models suitable for the same. The number of publications related to amc appearing in literature is large <cit.> mainly due to the broad range of problems associated with amc and huge interest in the problem itself for surveillance applications. amc helps a radio system for environment identification, defining policies and taking actions for throughput or reliability improvements. It is also used for applications like transmitter identification, anomaly detection and localization of interference <cit.>. Various approaches for modulation classification discussed in literature can be brought down to two categories <cit.>, one being the decision theoretic approach and the other the feature based approach. In decision-theoretic approaches the modulation classification problem is presented as a multiple hypothesis . The maximum likelihood criterion is applied to the received signal directly or after some simple transformations such as averaging. Even though decision-theoretic classifiers are optimal in the sense that they minimize the probability of miss-classifications, practical implementations of such systems suffer from computational complexity as they typically require buffering a large number of samples. These methods are also not robust in the presence of unknown channel conditions and other receiver discrepancies like clock frequency offset. Conventional feature-based approaches for amc make use of expert features like cyclic moments <cit.>. Spectral correlation functions of various analog and digital modulation schemes covered in <cit.> and <cit.> respectively are the popularly used features for classification. Detailed analysis of various methods using these cyclostationary features for modulation classification are presented in <cit.>. Various statistical tests for detecting the presence of cycles in the kth-order cyclic cumulants without assuming any specific distribution on the data are presented in <cit.>. In <cit.> authors used a multilayer linear perceptron network over spectral correlation functions for classifying some basic modulation types. Another method makes use of the cyclic prefix <cit.> to distinguish between multi-carrier and single carrier modulation schemes which is used for ofdm signal identification.All these aforementioned model driven approaches exploit knowledge about the structure of different modulation schemes to define the rules for amc. This manual selection of expert features is tedious which makes it difficult to model all channel discrepancies. For instance, it is quite challenging to develop models which are robust to fading, pathloss, time shift and sample rate variations. In addition, a distributed collection of iq data over frequency, space and time is expensive in terms of transmission bandwidth and storage. Furthermore, most of these algorithms are processor intensive and could not be easily deployed on low-end distributed sensors. Recently, deep learning has been shown to be effective in various tasks such as image classification, machine translation, automatic speech recognition <cit.> and network optimization <cit.>, thanks to multiple hidden layers with non-linear logistic functions which enable learning higher-level information hidden in the data. A recently proposed deep learning based model for amc makes use of a cnn based classifier <cit.>. The cnn model operates on the time domain iq data and learns different matched filters for various snr. However, this model may not be efficient on data with unknown sampling rates and pulse shaping filters which the model has never encountered during the training phase. Also being a fixed input length model, the number of modulated symbols the model can process remains limited across various symbol rates. Furthermore, the training and computational complexity of the model increases with increasing input sample length. In <cit.>, the authors extended the analysis on the effect of cnn layer sizes and depths on classification accuracy. They also proposed complex inception modules combining cnn and lstm modules for improving the classification results. In this paper we show that simple lstm models can itself achieve good accuracy, if input data is formatted as amplitude and phase (polar coordinates) instead of iq samples (rectangular coordinates).This paper proposes a lstm <cit.> based deep learning classifier solution, which can learn long term temporal representations, to address the aforementioned issues. The proposed variable input length model can capture sample rate variations without explicit feature extraction. We first train the LSTM model to classify 11 typical modulation types, as also used in <cit.>, and show our approach outperforms the soa. Being a variable input length model we also show that the model enables efficient classification on variable sample rates and sequence lengths. Even though these deep learning models can provide good classification accuracies on lower input sample lengths, their computational power requirements are still high preventing them from low-end sensor deployment as Electrosense.The wireless sensing nodes deployed in the Electrosense network consist of a low-cost and bandwidth limited sdr interfaced with a small sized embedded platform <cit.>. psd and iq pipelines are enabled in the sensor to support various applications producing data in the order of 50-100 Kbps and 50 Mbps respectively. First, the embedded hardware of the sensors is not powerful enough to handle performance intensive amc algorithms. Second, transferring iq samples to the backend for classification by enabling the iq pipeline is not a scalable solution as it is expensive in terms of data transfer and storage. Finally, the sensors are bandwidth limited which prevents them from acquiring wideband signals.To enable instantiation of the newly proposed lstm model for modulation classification in a large distributed network of low cost sensor nodes, we compare various approaches to decrease the implementation cost of the classifier. In the first approach we study the advantages and limitations of classification models for modulation classification on a deployed distributed sensor network with limited bandwidth sensors based on averaged magnitude fft data which decreases the communication cost by a factor 1000. Moreover, quantized versions of the proposed models are studied in detail for sensor deployment. These quantized versions can be run on a low cost sensor and do not require the instantiation of the classifier in the cloud. As a result, the sensor should only communicate the decision variable, which further decreases the communication cost. The code and datasets for all the deep learning models are made public for future research[<https://github.com/zeroXzero/modulation_classif>]. The models are also available for use through Electrosense. The contribution of this paper is thus threefold. First, we develop a new lstm based deep learning solution using time domain amplitude and phase samples which provides soa results for high snrs on a standard dataset. Second, we explore the use of deep learning models for technology classification task in a distributed sensor network only using averaged magnitude fft data. Finally, we explore the model performance by quantizing the deep neural networks for sensor deployment.The rest of the paper is organized as follows. The classification problem is clearly stated in Section <ref>. A brief overview of the modulation dataset and the channel models used are presented in Section <ref>.Section <ref> explains the lstm model used for classification and the parameters used for training along with other implementation details. Section <ref> details the classification results and discusses the advantages of the proposed model. Low-implementation cost models are discussed in Section <ref>. Conclusions and future work are presented in Section <ref>.§ PROBLEM STATEMENT Technology or modulation recognition can be framed as a N-class classification problem in general. A general representation for the received signal is given byr(t) =s(t)*c(t)+n(t),where s(t) is the noise free complex baseband envelope of the received signal, n(t) is awgn with zero mean and variance σ_n^2 and c(t) is the time varying impulse response of the transmitted wireless channel. The basic aim of any modulation classifier is to give out P(s(t)∈ N_i| r(t)) with r(t) as the only signal for reference and N_i represents the ith class. The received signal r(t) is commonly represented in iq format due to its flexibility and simplicity for mathematical operations and hardware design. The in-phase and quadrature components are expressed as I = A cos(ϕ) and Q = A sin(ϕ), where A and ϕ are the instantaneous amplitude and phase of the received signal r(t).The RadioML and modified RadioML datasets used for testing the proposed model, presented in the next section of this paper, follow the signal representation as given in equation <ref>. These datasets make a practical assumption that the sensor's sampling rate is high enough to receive the full-bandwidth signal of interest at the receiver end as r(t). The datasets also take into account complex receiver imperfections which are explained in detail in Section <ref>. The samples per symbol parameter used in the tables <ref> and <ref> specify the number of samples representing each modulated symbol which is a modulation characteristic. Similarly sample length parameter specifies the number of received signal samples used for classification. § MODULATION DATASETS A publicly available dataset used for evaluating the performance of the proposed model is detailed in this section. The standard dataset is also extended to evaluate the sample rate dependence of the proposed model. §.§ RadioML datasetA standard modulation dataset presented in <cit.> is used as the baseline for training and evaluating the performance of the proposed classifier. The used RadioML2016.10a dataset is a synthetically generated dataset using GNU Radio <cit.> with commercially used modulation parameters. This dataset also includes a number of realistic channel imperfections such as channel frequency offset, sample rate offset, additive white gaussian noise along with multipath fading. It contains modulated signals with 4 sps and a sample length of 128 samples. Used modulations along with the complete parameter list can be found in Table <ref>. Detailed specifications and generation details of the dataset can be found in <cit.>. §.§ Modified RadioML datasetThe standard radioML dataset is extended using the generation code[https://github.com/radioML/dataset] by varying the samples per symbol and sample length parameters for evaluating the sample rate dependencies of the lstm model. The extended parameters of the used dataset are listed in the Table <ref>. The extended dataset contains signals with 4 and 8 samples per symbol. This dataset is generated to evaluate the robustness of the model in varying symbol rate scenarios. § MODEL DESCRIPTIONThe proposed lstm model, that works on the time domain amplitude and phase signal, is introduced in the following subsection. In addition, the baseline cnn model used for comparisons is also detailed.§.§ LSTM primer rnn are heavily used for learning persistent features from time series data. lstm <cit.> is a special type of rnn which is efficient in learning long-term dependencies. The block diagram of a basic version of a lstm cell is presented in Figure <ref> along with the corresponding equations (2-7). * Gatesi_t= σ(W_xix_t + W_hih_t-1 + b_i)f_t= σ(W_xfx_t + W_hfh_t-1 + b_f)o_t= σ(W_xox_t + W_hoh_t-1 + b_o) * Input transformc_in_t = tanh(W_xcx_t+W_hch_t-1+b_c_in) * State updatec_t= f_t · c_t-1+i_t · c_in_th_t= o_t · tanh(c_t)lstm cells have an internal state or memory (c_t) along with three gates namely input date (i_t), forget gate (f_t) and output gate (f_t). Based on the previous state and the input data the cells can learn the gate weights for the specified problem. This gating mechanism helps lstm cells to store information for longer duration thereby enabling persistent feature learning.§.§ Model for complex signalsA lstm network with different layers is used for complex data classification as shown in Figure <ref>. The amplitude and phase of the time domain modulated signal are fed to all cells of the lstm model as a two dimensional vector, at each time step for classification. The amplitude vector is L2 normalized and the phase, which is in radians is normalized between -1 and 1. The first two layers are comprised of 128 lstm cells each. The final output from the second lstm layer, a vector of dimension 128, after all time steps, is fed to the last dense layer of the model. The final layer is a dense softmax layer which maps the classified features to one of the 11 output classes representing the modulation schemes. The two layer model is selected after detailed analysis varying the cell size and layer depths which are detailed in Section <ref>. The intuition to use a lstm model for classification is based on the fact that different modulation schemes exhibit different amplitude and phase characteristics and the model can learn these temporal dependencies effectively. Even though fading and other real world effects may slightly hamper the characteristics of the signal, we expect the model to classify signals efficiently by learning good fading resistant representations. Since the proposed model can work on variable length input time domain samples, we expect the model to learn useful symbol rate independent representations for classification. In addition, the importance of the number of lstm cells and layer depth are further investigated by varying these parameters. Model classification accuracies are analyzed with varying layer depth from 1 to 3 and number of cells from 16 to 256. We further analyze these aspects in detail in Section <ref>. §.§ Baseline iq modelThe two layer CNN 8 tap model presented in <cit.> is used as the baseline model for further comparisons. The baseline model uses 256 and 80 filters in the first two convolutional layers respectively. A publicly available training model is used for generating the baseline performance graph <cit.>.§.§ Model training and testingEach of the datasets mentioned in Tables <ref> and <ref> are split into two, one training set and the other testing set. A seed is used to generate random mutually exclusive array indices, which are then used to split the data into two ascertaining the training and testing sets are entirely different. The number of the training and testing vectors are listed in the corresponding tables. A softmax cross entropy with logits[https://www.tensorflow.org/api_docs/python/tf/nn/softmax_cross_ entropy_with_logits], that measures the probability error in discrete classification tasks in which the classes are mutually exclusive, is used as the loss function. Stochastic gradient descent with a minibatch size of 400 vectors is used to avoid local optima. We use the Adam optimizer <cit.>, a first-order gradient based optimizer with a learning rate of 0.001. The complex two layer lstm network is trained for 70 epochs which takes around an hour of training time on a x86 PC with Nvidia GeForce GTX 980 Ti graphics card. We use a basic lstm cell with starting training forget bias set to one. While initializing the network, it is helpful to keep the scale of the input variance constant, so that it does not explode or diminish by reaching the final layer. To achieve this lstm weights are initialized with a default uniform unit scaling initializer which generates weights with a uniform variance. All the models use the same training parameters unless specified explicitly. §.§ Implementation detailsThe neural network is implemented using TensorFlow <cit.>, a data flow graph based numerical computation library from Google. Python and C++ bindings of Tensorflow makes the usage of the final trained model easily portable to host based SDR frameworks like GNU Radio <cit.>. The trained model can be easily imported as a block in GNU Radio which can be readily used in practice with any supported hardware front-end. § RESULTS AND DISCUSSION The classification accuracies of the model for the aforementioned datasets along with the learned representations are discussed in the following subsections. §.§ Classification accuracy on RadioML datasetThe two layer amplitude-phase lstm model, shown in Figure <ref>, is trained on SNR ranges from -10dB to 20dB. Training vectors with SNR ranges below -10dB were not used as the model was converging slowly when those vectors were used. Alternate models with varying lstm layer depths are also trained to understand the performance improvements provided by the different layer depths. The classification accuracy of all the four models are presented in Figure <ref>. The two layer lstm model gave an average accuracy of 90% in snr ranges from 0dB to 20dB. It can be noticed that the single layer lstm also reaches a high accuracy at high snrs, 6% less than the two layer model. It was also noticed that the classification accuracy saturates for layer depths of two. Hence, layer depth of two is selected for the final model and its parameters are fine tuned (dropout = 0.8 and learning rate = 0.001) to achieve the best test performance as shown in Figure <ref>. Rigorous fine tuning was not performed on layer depths other than two accounting for a slightly lower accuracy levels, for instance the accuracy level for layer depth 3 is slightly lower than layer depth two.The performance of the baseline cnn model was shown to be much better on the low snr regions in <cit.>. We were not able to reproduce the reported results on the low snr regions after various attempts, which may be because of the difference in hyper-parameter tuning. Though, the high snr results of the baseline model matches with that of the reported ones in the paper. Detailed discussions on the effect of layer depth and number of lstm cells are presented in Section <ref>.Classification performance of other standard machine learning models such as svm, random forest, k-nearest neighbors and Gaussian Naive Bayes are also summarized in Figure <ref>. All models are fed with the same amplitude-phase training and test data for this comparison. Random forest with 150 decision trees is able to provide close to 70% of accuracy at very high snr conditions while others could reach only around 26%. It could be clearly noticed that the deep learning models perform superior to the other standard techniques when fed with the raw sensed data. The deep learning models can classify signals very efficiently with a very low number of symbols, usually with hundreds of samples (tens of modulated symbols) when compared to the classical cyclostationary based expert feature models which requires samples in thousands range (hundreds of modulated symbols) for averaging. Similarly extracting expert cyclostationary features using tens of symbols is very suboptimal, which substantiate the use of deep learning models.To understand the results better confusion matrices for the two layer lstm model for various snrs are also included. It can be seen in Figure <ref> that at a high snr of 18dB the diagonal is much more sharp even though there are difficulties in separating AM-DSB and WBFM signals. This is mainly due to the silence periods of audio as the modulated signals are generated from real audio streams.Similarly in Figure <ref>, at 0dB snr it is noticed that there is some level of confusion regarding QAM16 and QAM64 as the former is a subset of the the latter. The confusion increases further at low snrs as shown in Figure <ref>. From these basic analysis it is clear that deep complex structures as mentioned in <cit.> are not required to achieved good soa classification accuracy at high snrs. However, use of convolutional layers might turn useful at low snrs as reported in <cit.>. In our experiments we also noticed that simply providing iq samples to the lstm model yielded poor results while normalized amplitude and phase interpretation provided good results. The models even failed to reduce the training loss when fed with time domain iq samples, giving a constant accuracy of 9% on the radioML dataset, as the lstms were not able to extract any meaningful representations. Similarly feeding amplitude-phase information to the cnn model did not provide any accuracy improvements over the iq-cnn model. The classification accuracy improvement is achieved from the combined benefits of using amplitude-phase information along with 2-layer lstm model. §.§ Classification accuracy on modified RadioML datasetThe same two layer lstm model is trained on SNRs ranging from -20dB to 20dB and input sample lengths from 128 to 512 samples. The accuracy of the model is tested on the full range of SNRs and also on input sample length that is smaller than the training set (e.g, 64). It is evident from the results in Figures <ref> and <ref> that the classification accuracy improves as the model sees more modulated symbols. Even though the model is trained on varying data lengths from 128 to 512 samples, it gives an average accuracy of 75% with 64 samples and 4 samples per symbol scenario for which it was not trained, which confirms the model's generalization capabilities. To further analyze the generalization capabilities of the model on unseen sample lengths, four balanced folds of data each containing sequences with sample lengths of 64, 128, 256 and 512 are created. The model is then trained only on three folds, and the left-out fold is used to test generalization to the unseen length. This process is repeated for all four sample lengths and the results are presented in Figure <ref>. The model consistently gives an average accuracy above 70% for high SNR conditions.§.§ Learned representations The inherent non-linearity and deep structures makes understanding the representations learned by lstms difficult. In order to obtain some good insights we use visualization techniques similar to the ones presented in <cit.>. These visualizations can help to understand how lstm cells behave for an input signal, for instance which cells gets activated at each time step and how long each gate remains open. Figures <ref> and <ref> presents the gate activation and saturation of the trained two layer lstm model for a QAM64 input signal with 18dB snr. As explained in Section <ref> the gates of lstm cells have sigmoid activation functions, giving an output value between 0 and 1. A gate is said to be left saturated if its activation is less than 0.1 and right saturated if the activation is greater than 0.9. The fraction of time for which the gate is in left or right saturated mode in the entire 128 samples time is plotted in Figure <ref>. On the first layer, it can be noticed that all the three gates are confined close to the origin showing that they are not highly left or right saturated. The absence of right saturation in the first layer forget gates, confirms that the cells do not store information for long term. There are no cells in the first layer that function in purely feed-forward fashion, since their forget gates would show up as consistently left-saturated. The output gate plots in the first layer also show that there are no cells that are revealed or blocked to the hidden state. This is also visible in the activation plots of the first layer in Figure <ref>. The activations are short when compared to the second layer and it can be noticed in Figure <ref> that many cell activations follow the input amplitude and phase changes in the input waveform. The second layer stores much long term dependencies from the fine grained representations generated from the first layer.§.§ Effect of cell size and layer depth A comprehensive study is also performed to understand the effect of the number cells and layer depth on the model performance. The number of lstm cells and layer depth are varied from 16 to 256 and1 to 3 respectively. The models are trained on RadioML dataset on all snrs. The accuracy levels for various layer depths are presented in Figures <ref>, <ref> and <ref>. An initial analysis clearly shows that the model accuracy increases with increasing layer depth for mostly all cell sizes. It can be also noted that as the depth of the model increases, increasing the number of cells doesn't give much performance improvements. For instance, at depths 2 and 3 increasing the cell numbers from 128 to 256 doesn't provide any performance improvements. § RESOURCE FRIENDLY MODELS As mentioned earlier in the introduction, it is quite difficult to deploy soa amc algorithms on low-end distributed sensors such as the ones in Electrosense. We extend our study in two directions to reduce the resource requirements in terms of data transfer rate to the cloud, data storage and computational power. First, a study is conducted to understand to what extent technology classification based on amc can be done using averaged magnitude fft data, as the psd pipeline being the default enabled one in the Electrosense sensors with medium data transfer costs. As the sensors are sequentially scanning the spectrum, they are capable of generating magnitude spectrum information for wideband signals which is an added advantage. Second, the performance of quantized versions of the proposed deep learning models are analyzed which can reduce the computational cost of the models enabling deployment of these models on the sensors itself. The averaged magnitude fft signal sent by the sensor, the selected dataset for testing the model, averaged magnitude fft classification model, other quantized models and the classification results are detailed in the following subsections. §.§ Received averaged magnitude fft signalElectrosense sensors scan the wireless spectrum sequentially. The sensor samples the wireless spectrum at a fixed sampling rate N = 2.4 MS/s tuned to a particular centre frequency f_x.As the sensor's sampling rate is limited, a wideband signal's magnitude spectrum can be received only by sequential scans to cover the entire bandwidth as given in equation <ref>.R(f) =1M∑_m=0^M|FFT_m(e^-j2π f_1(t_0+mD_t)s(t_0+mD_t)*c(t_0+mD_t)+n(t_0+mD_t))|||1M∑_m=0^M|FFT_m(e^-j2π f_2(t_1+mD_t)s(t_1+mD_t)*c(t_1+mD_t)+n(t_1+mD_t))|||… In equation <ref> || represents the concatenation operation where the full bandwidth of the signal of interest is captured by a sequentially scanning sensor sampling at a lower sampling rate, similar to the Electrosense dataset mentioned in the following subsection. The averaged magnitude fft signal at centre frequencies f_i, where f_i ∈ (50 MHz, 6 GHz) based on the sensor sampling rate and frequency range, are sent to the cloud where they are concatenated together. In equation <ref>, M represents the magnitude-fft averaging factor and t_x the sequential sampling time. For instance, t_n = t_n-1+T, where T=MD_t is the amount of time spent at a particular centre frequency and D_t being the time for collecting fft_size samples for a single FFT input.§.§ Electrosense dataset Six commercially deployed technologies are selected to validate the classification accuracy using averaged magnitude fft data as given in Table <ref>. Over-the-air data from multiple Electrosense sensors are retrieved through the Electrosense API[https://electrosense.org/open-api-spec.html] with a spectral resolution of 10 kHz and time resolution of 60 seconds. The data is collected from sensors with omni-directional antennas which are deployed indoors. The sensors follow sequential scanning of the spectrum with an fft size set to 256 giving a frequency resolution close to 10 kHz. With a fft size of 256 and sensor ADC bit-width of 8, we get an effective bitwidth of 12 resulting in a theoretical dynamic range of 74dB. Practical dynamic range depends on the ADC frontend stages and the noise level, which may vary between 60 to 65dB. Five fft vectors are averaged for reducing the thermal noise of the receiver. Some of the selected technologies such as LTE and DVB have an effective bandwidth which is higher than the sampling bandwidth of the of the low-end sdr. As the sensor is sequentially scanning, full spectrum shapes of these wideband signals are obtained by combining fft outputs of these sequential scans. The entire data is split into two, one half for training and the other half for testing the model.§.§ Averaged Magnitude fft modelSequentially sensed frequency spectrum data from the sensors contain signals of different bandwidth. The model should be able to process this variable length dataand classify them to proper groups. We use the same lstm model used for classifying complex input data as shown in Figure <ref>.The averaged magnitude fft signal is fed to the model as a sequence as presented in Figure <ref>. The same lstm model is chosen as it can handle variable length input signals and is also good at learning long term dependencies. The final output of the lstm model after feeding n frequency binsis given as input to the softmax layer through a fully connected linear layer. The softmax layer outputs the probability P(y=l|a;θ) for l ∈{0,1,..,5} where a denotes the input averaged fft bins, θ the model parameters and l the enumerated label for one of the six technologies as listed in Table <ref>. §.§ Classification resultsAn initial study is conducted to understand the technology classification accuracy of the averaged magnitude fft model when compared to full IQ information. On the Electrosense dataset (Section <ref>) the proposed model achieves a classification accuracy of 80%. The confusion matrix for the same is shown in Figure <ref>.From the confusion matrix it is clear that there is a large confusion between LTE and DVB. This is expected as the power spectra of both DVB and LTE looks very similar as both of them are based on ofdm. As multiple technologies might share the same modulation types, the assumption that a modulation classifier can be used for technology classification is not always valid with the current deployed technologies. We also investigated the effect of number of lstm cells and layer depth on this dataset whose results are summarized in Table <ref>. Increasing the layer depth did not contribute significantly to the classification accuracy as there might be no more low level abstract features to be learned from the magnitude spectrum information. Furthermore, there are a large number of modulation schemes which exhibit the same power spectral density making averaged spectrum a sub-optimal feature for classification. For instance, the power spectral densities of different modulations schemes such as 8PSK, QAM16 and QPSK are identical once passed through a pulse shaping filter with a fixed roll-off factor. This can be theoretically shown and easily verified with manual inspection <cit.>. To further validate the argument, magnitude-fft is calculated on the same RadioML dataset which was used for testing the performance of the amplitude-phase model. As the RadioML dataset consists of modulations with same bandwidths passed through the same pulse shaping filter, their magnitude-ffts looks identical giving very low classification accuracy of only 19% for all 11 modulations even at high snrs. To get a better understanding of the generated magnitude-fft dataset, a visualization of a subset of the data in two dimensions is provided. For reducing the dimensionality of the data to 2 and for the ease of plotting, the t-SNE technique <cit.> is used. A small subset of the radioML dataset of 5000 vectors, containing 128 point magnitude FFT of all generated modulation schemes with varying SNR ranges from +20dB to +20dB are fed to the t-SNE algorithm. t-SNE is useful for a preliminary analysis to check whether classes are separable in some linear or nonlinear representation. The representation generated by t-SNE on the data subset is presented in Figure <ref>. It can be seen that the representation overlap is very high and t-SNE could not generate any meaningful clustering as the phase information is completely lost when computing magnitude-fft, leaving identical magnitude spectrum for many modulation schemes. The obvious solution is to switch to iq pipeline and deploy optimized versions of complex input signal models on the sensors itself, thus reducing the uplink data transfer rate. This is further investigated in the following subsection. §.§ Quantized modelsDeep learning models are processor intensive which makes them quite difficult to be deployed on low-end sensor networks. As mentioned in the introduction, transferring pure iq data to the backend server is not a scalable solution. In addition, our results indicate that some signals require iq information for correct classification. To enable low-end sensor deployment, a feasibility study is conducted by quantizing the weights and activations of the newly proposed as well as baseline neural network models. Binarized networks can exceptionally reduce the required memory size for holding intermediate results and replace most of the arithmetic operations with bitwise operations <cit.>. For instance, when compared to a full precision network with 32 bits weights and activation, a binarized network only needs 32 times smaller memory resulting in a reduced required memory size and memory access cost. In <cit.> the authors had already noticed that binarizing lstms results in very poor accuracy. We confirm the same observation with our lstm models too. However, models with binarized cnns have been reported to provide accuracy close to their full precision variants. To validate this, the performance of the binarized baseline cnn model is also investigated on the radioML modulation dataset. Furthermore, by allowing more quantization levels on the lstm models a higher accuracy can be achieved while still reducing the computational cost. Two quantized lstm model variants are tested, one with ternary weights (-1, 0, +1) and full precision activation (TW_FA) and the other with ternary weights and four bits activation (TW_4BA). The accuracy results of these models are summarized in Figure <ref>. Results show that lstm models with ternary weights and 4bit activation can provide close to 80% accuracy reducing the very high computational power required for full precision models. Binary cnn models also provided an accuracy level 10% below the full precision variants. We believe the classification accuracy can be further improved by proper hyper-parameter tuning and longer training.The theoretical memory requirements for the trained weights along with number of multiplications required for the entire model, excluding activations, are summarized in Table <ref>. A binarized neural network can drastically reduce the processing power requirements of the model. For instance, in a binarized network all weights and activation are either -1 or +1, replacing all multiply operations by XNORs. The multipy-accumulate, which is the core operation in neural networks, can be replaced by 1-bit XNOR-count operation <cit.>. Convolutions also comprises of multiply and accumulate operations which can also be replaced by its binarized variants. Thus the baseline cnn model can provide very good performance improvements on the general purpose ARM based Electrosense sensors. For the CNN models the convolutional layer output numbers are high, as we are not using any pooling layers, which accounts for the larger memory size in the succeeding dense layers. We would like to emphasize the fact that the given memory sizes are for the entire model and the weights that should be hold in the memory might vary based on practical implementations. As binarized lstm models did not provide good accuracy, we are forced to use 4-bit quantized variants of the same. Even though the performance improvements are not that extreme similar to binarized models, quantized lstms can also reduce the resource consumption. First of all, as no large dynamic range is required all the 4-bit multiply-accumulate operations can be implemented in fixed point arithmetic, which is much more faster in ARM CPUs when compared to their floating point versions. Secondly, routines can also be implemented to reduce the space requirements to hold intermediate results and the activations can be implemented as look-up tables. We would also like to emphasize the fact that on a special purpose hardware, such as FPGAs, quantized models can obviously reduce the space usedand power-consumption as the multiply-accumulate units have smaller bit-widths.Most of the machine learning frameworks such as Tensorflow have started supporting quantized models for low-end processor deployment. The quantized kernels are under active development at the time of writing this paper which currently only supports a minimum quantization of 8 bit. Quantized kernels for all operations in all platforms are not available in these libraries resulting in low performance than expected. The full iq information model classification performances on various platforms such as Nvidia GPUs, Intel and ARM processors for full precision and 8bit precision models are summarized in Table <ref>. To avoid implementation mismatches across various models and platforms all the comparisons are done using quantization tools provided by Google's Tensorflow which is currently under active development. These tools allow to freeze and compress a trained model to a single file and then test it on various platforms easily. These values in the table are performance indicators in number of classifications per second for 128 sample length vectors. It can be noticed that the quantized models perform very bad on GPUs and Intel PCs due to lack of support. The quantized models currently provide performance very close to floating point variants on ARM processors. Quantized kernel support is improving for ARM processors due to increased demand for deploying these models on mobile devices. The aforementioned advantages of quantized models is expected to be available in the near future through these standard libraries. § CONCLUSION AND FUTURE WORK Wireless spectrum monitoring and signal classification over frequency, time and space dimensions is still an active research problem. In this paper we proposed a new lstm based model which can classify signals with time domain amplitude and phase as input. soa results on high snrs (0 to 20dB) is achieved without using complex cnn-lstm models as mentioned in <cit.>. Being a recurrent model we showed that the model can handle variable length inputs thus can capture sample rate variations effectively. Though neural networks are good at function approximation, our experiments emphasize the fact that data preprocessing and proper representation are equally important. This claim is substantiated from our experiments with the lstm model where the model gave poor results when fed with time domain iq samples while it gave accuracies close to 90% for high snrs when provided with time domain amplitude and phase information. As shown by various soa models in speech and image domains, performance improvements are seen with increasing layer depth which saturates after a few levels. In addition, we showed that basic technology classification is achievable by only using averaged magnitude fft information over a distributed set of sensors complying with the uplink bandwidth resource constraints. Furthermore, experiments showed that quantized lstm models can achieve good classification results thus reducing the processing power requirements at the cost of 10% accuracy loss. This allows the deployment of these models on low cost sensors networks such as Electrosense enabling a wide area deployment. It is also remarkable that these deep learning models can classify signals with a fewer number of samples when compared to the expert feature variants, such as cyclic frequency estimators, enabling faster classification. Furthermore, deep learning allows for incremental learning, thus it would not be required to retrain the entire network from scratch for the new wireless non-idealities like antenna patterns and sensitivity. In addition, dedicated hardware is gaining popularity to reduce the deep learning model's energy and memory footprints which demands quantized versions of the models.Although the lstm models perform very well at high snr conditions, CNN models seems to provide an additional 5-10% accuracy on the low snr conditions (SNRs below -2dB) as shown in <cit.>. Even though we are not able to replicate the results in <cit.> ( because of hyperparameter tuning), it is reasonable to conclude that the learned filters in CNN for a fixed sample rate might give performance improvements for low SNR values. Furthermore, all the implemented code for the proposed models are made publicly available for reproducing and verifying the results presented in this paper and for supporting future research.Low snr performance of these soa deep learning models could be further improved with the help of efficient blind denoising models. Models which can perform automated channel equalization and compensate receiver imperfections such as frequency offset can further improve the classification performance. The current radio deep learning models make use of layers which basically applies non-linearity after simple multiply-accumulate-add operations while it is well established in the research community that cyclic cumulants, which are generated by time-shifted multiplication and averaging of the input itself, performs well in the expert feature space. Deep learning models which can extract features similar to cyclic cumulants might improve the performance metrics.The analysis would not be complete without emphasizing the limitations of the soa deep learning models. First, the current complex models are tested on a dataset with normalized bandwidth parameters. Real life transmitted signals generally have varying symbol rates and bandwidths. Even though the variable length lstm model is shown to be capable of adapting to these scenarios, further analysis is required to validate the claim. In future, models that can handle all possible spread spectrum modulations should be also tested. Second, the generalization capabilities of these models should be further investigated, in terms of performance of these models in unknown channel conditions and modulation parameters. Finally, most of the successful soa models are supervised models which requires labeled training data. Labeling is a very tedious task which projects the importance of semi-supervised models for classification tasks. Published studies <cit.> on semi-supervised machine learning models for cellular network resource management validates the need for more semi-supervised models, which is also an active direction for future research. We believe deep learning models adapted to radio domain can help in understanding, analyzing and decision making in future complex radio environments. IEEEtran [ < g r a p h i c s > ] Sreeraj Rajendran received his Masters degree in communication and signal processing from the Indian Institute of Technology, Bombay, in 2013. He is currently pursuing the PhD degree in the Department of Electrical Engineering, KU Leuven, Belgium. Before joining KU Leuven, he worked as a senior design engineer in the baseband team of Cadence and as an ASIC verification engineer in Wipro Technologies. His main research interests include machine learning algorithms for wireless and low power wireless sensor networks. [ < g r a p h i c s > ] Wannes Meert received his degrees of Master of Electrotechnical Engineering, Micro-electronics (2005), Master of Artificial Intelligence (2006) and Ph.D. in Computer Science (2011) from KU Leuven. He is currently research manager in the DTAI research group at KU Leuven. His work is focused on applying machine learning, artificial intelligence and anomaly detection technology to industrial application domains. [ < g r a p h i c s > ] Domenico Giustiniano is Research Associate Professor at IMDEA Networks Institute and leader of the Pervasive Wireless Systems Group. He was formerly a Senior Researcher and Lecturer at ETH Zurich and a Post- Doctoral Researcher at Disney Research Zurich and at Telefonica Research Barcelona. He holds a Ph.D. from the University of Rome Tor Vergata (2008). He devotes most of his current research to visible light communication, mobile indoor localization, and collaborative spectrum sensing systems. He is an author of more than 70 international papers, leader of the OpenVLC project and co-founder of the non-profit Electrosense association. [ < g r a p h i c s > ] Vincent Lenders is a research director at armasuisse where he leads the cyber and information sciences research of the Swiss Federal Department of Defense. He received the M.Sc. and Ph.D. degrees in electrical engineering from ETH Zurich. He was postdoctoral research fellow at Princeton University. Dr.Vincent Lenders is the cofounder and in the board of the OpenSky Network and Electrosense associations. His current research interests are in the fields of cyber security, information management, big data, and crowdsourcing. [ < g r a p h i c s > ] Sofie Pollin obtained her PhD degree at KU Leuven with honors in 2006. From 2006-2008 she continued her research on wireless communication, energy-efficient networks, cross-layer design, coexistence and cognitive radio at UC Berkeley.In November 2008 she returned to imec to become a principal scientist in the green radio team. Since 2012, she is tenure track assistant professor at the electrical engineering department at KU Leuven. Her research centers around Networked Systems that require networks that are ever more dense, heterogeneous, battery powered and spectrum constrained. Prof. Pollin is BAEF and Marie Curie fellow, and IEEE senior member. | http://arxiv.org/abs/1707.08908v2 | {
"authors": [
"Sreeraj Rajendran",
"Wannes Meert",
"Domenico Giustiniano",
"Vincent Lenders",
"Sofie Pollin"
],
"categories": [
"cs.NI"
],
"primary_category": "cs.NI",
"published": "20170727154125",
"title": "Distributed Deep Learning Models for Wireless Signal Classification with Low-Cost Spectrum Sensors"
} |
Institut de Physique Théorique, CEA, Université Paris-Saclay, Saclay, FranceInstitut de Physique Théorique, CEA, Université Paris-Saclay, Saclay, FranceDepartment of Physics, Royal Holloway, University of London, Egham, Surrey, United KingdomInstitut de Physique Théorique, CEA, Université Paris-Saclay, Saclay, France Topological states of matter are at the root of some of the most fascinating phenomena in condensed matter physics. Here we argue that skyrmions in the pseudo-spin space related to an emerging SU(2) symmetry enlighten many mysterious properties of the pseudogap phase in under-doped cuprates. We detail the role of the SU(2) symmetry in controlling the phase diagram of the cuprates, in particular how a cascade of phase transitions explains the arising of the pseudogap, superconducting and charge modulation phases seen at low temperature. We specify the structure of the charge modulations inside the vortex core below T_c, as well as in a wide temperature region above T_c, which is a signature of the skyrmion topological structure. We argue that the underlying SU(2) symmetry is the main structure controlling the emergent complexity of excitations at the pseudogap scale T^*. The theory yields a gapping of a large part of the anti-nodal region of the Brillouin zone, along with q=0 phase transitions, of both nematic and loop currents characters. Pseudo-spin Skyrmions in the Phase Diagram of Cuprate Superconductors C. Pépin December 30, 2023 ===================================================================== The pseudo-gap (PG) phase in the under-doped region of cuprate superconductors remains one of the most mysterious known states of matter. First observed as a depression in the Knight shift of nuclear magnetic resonance (NMR) <cit.>, it was soon established that, for a region of intermediate dopings around 0.08<x<0.20, part of the Fermi surface was gapped in a region close to the (0,π) and (π,0) points of the Brillouin zone, called anti-nodal region because of its remoteness from the point were the d-wave superconducting gap changes sign on the (0,0)-(π,π) segment of the Brillouin zone. In this anti-nodal region, the Fermi surface was found to be “wiped out”, and only some lines of massless quasiparticles known as Fermi arcs to be left out <cit.>.This puzzling situation became more complex with the observation of a reconstruction of the Fermi surface by quantum oscillation and other transport measurements in the same doping region <cit.>. This was attributed to the presence of incipient charge modulations with incommensurate wave vectors developing along the crystallographic axes: 𝐐_x,𝐐_y≃0.3×(2π/a), where a is the lattice spacing in a tetragonal structure, detected by X-ray scattering <cit.>. In real space, patches of charge modulation of a size of the order of twenty lattice sites have been observed at low temperatures (T∼4 K) using both scanning tunneling microscopy (STM) <cit.> and nuclear magnetic resonance (NMR) <cit.> measurements. These take the form of oscillations of the charge density on the copper oxide planes of a frequency comparable to twice the lattice spacing. The amplitude of these oscillations decreases away from its centerpoint in real space and disappears around ten lattice lengths away from it.Charge modulations were observed at the core of the superconducting vortices, below the superconducting transition temperature (T_c). When voltage bias is increased, these modulations persist until the applied voltage reaches the energy scale corresponding to the formation of the pseudogap: Δ_PG <cit.>.Below the pseudogap onset temperature T^*, loop currents have been detected <cit.>, and the areas exhibiting charge modulations coexist with zones with long-range nematic order <cit.>, reminiscent of the vicinity of a smectic-nematic transition. The latter are more and more numerous compared to charge-modulated areas when the temperature approaches T^* <cit.>. Simultaneous measurements of the real and reciprocal space spectral functions however established that the opening of the pseudogap is correlated with the presence of charge modulations in real space <cit.>. The whole real space picture has led to the image of an “ineluctable complexity” inherent to cuprate superconductors and driven by strong quantum fluctuations in the vicinity of the Mott transition in two dimensions <cit.>.Here we argue that the presence of an underlying SU(2) symmetry in the under-doped region sheds light on the variety of observed phenomena and clarifies the mysteries of the real space picture. This SU(2) symmetry relates the charge and superconducting orders, similarly to the U(1) symmetry between the two components of a superconducting order parameter. First, we describe the starting short-range antiferromagnetic model and its order-by-disorder treatment, which gives rise to a pseudogap phase governed by O(4) SU(2) fluctuations which stabilise d-wave superconducting, nematic and axial orders. Then we geometrically interpret the proliferation of local defects by introducing an SU(2) order parameter which follows naturally from the previous derivation. Finally, we describe its topological structure and the cascade of phase transitions it generates. § SHORT-RANGE ANTIFERROMAGNETIC MODEL We start by describing how an order-by-disorder treatment of a simple short-range antiferromagnetic model was shown to give rise to d-wave superconducting, nematic and charge orders <cit.>.Our starting point is that short-range antiferromagnetic interactions, strongly coupled to conduction electrons, are the main ingredient of the physics of the cuprates above 5% doping. This leads to the most simple Hamiltonian:H =∑_i,j,σc_i,σ^†t_ijc_j,σ+J∑_⟨ i,j⟩𝐒_i·𝐒_jwhere t_ij is the hopping matrix from one site to another, c_𝐤,σ^† creates an electron of momentum 𝐤 and spin σ, 𝐒_i=∑_α,βc_i,α^†σ_αβc_iβ is the on-site spin operator and ⟨ i,j⟩ denotes the summation over nearest neighbours.One can perform a Hubbard-Stratonovich transformation on this Hamiltonian in order to decouple the interaction term in two channels. That is, transform the interacting term of the Hamiltonian in a sum of two terms, each corresponding to an order parameter. The first one is the usual d-wave superconducting channel described byΔ^†=1/2∑_𝐤,σd_𝐤c_𝐤,σ^†c_-𝐤,-σ^†The second one is the d-wave charge modulations channel at momentum 𝐐_0, described byχ=1/2∑_𝐤,σd_𝐤c_-𝐤+𝐐_0^†c_-𝐤where d_𝐤= 2 cos(2θ_𝐤) is the d-wave factor, and θ_𝐤 the angle spanning the Brillouin zone. It corresponds to a charge density wave order with an ordering wave vector 𝐐_0, and a d-wave modulation, meaning in particular that its gap exhibits a d-wave modulation in momentum space. The charge modulation wave vectors, shown in Fig. <ref>A, are typically incommensurate, and taken either parallel to the crystal axes <cit.> or diagonal <cit.>. Indeed, it can take all the values connecting two hot-spots, which are the points where the Fermi surface crosses the line where the antiferromagnetic fluctuations diverge: 𝐐_0=(0,0);(± Q_x,0);(0,± Q_y);(± Q_x,± Q_y);(±π,±π) (Fig. <ref>A).This decoupling yields various possible order parameters which are all degenerate in magnitude at the Fermi surface hot-spots in the strong coupling limit, i.e. for J much larger than the energy of the bottom of the electronic band in the anti-nodal region, as depicted in Fig. <ref>D-F <cit.>. Note that one could also consider decoupling the interaction term in the spin-sector in the antiferromagnetic channel, in particular for the characterization of the low-doping properties of this model. It is however likely that the antiferromagnetic fluctuations would be gapped by the emergent orders which also compete for the hot-spots.All the previously cited possible orders are d-wave symmetric. This is dictated by the fact that antiferromagnetic interactions correspond to a 𝐐=(π,π) wave vector. Therefore when writing the gap equation corresponding to the ladder diagram in Fig. <ref>C, it couples adjacent anti-nodal regions of the Brillouin zone:Θ_𝐤=-T∑_ω,𝐪J_𝐪Θ_𝐤+𝐪/Θ_𝐤+𝐪^2+ξ_𝐤+𝐪^2+(ϵ+ω)^2,with Θ=χ,Δ, ξ_𝐤 the electronic dispersion and ϵ and ω the fermionic and bosonic Matsubara frequencies and 𝐪=q+𝐐. The solutions of this equation are plotted in Fig. <ref>D-F for χ with three different choices for Q_0. In order to stabilise the gap equation there needs to be a change of sign between two such adjacent regions leading to Θ_𝐤=-Θ_𝐤+𝐪, the simplest case of which is a d-wave order <cit.>.The approach presented here has two specificities: (i) we consider a strong coupling regime where J is larger than the bottom of the band in the anti-nodal region, which yields “hot-regions" instead of hot-spots. (ii) we consider the whole frequency dependence in the gap equation (<ref>). J appears as a weight in the frequency sum of Eq. (<ref>) and therefore corresponds effectively to a bosonic frequency cutoff.A similar decoupling has been used on the spin-fermion model in the vicinity of an antiferromagnetic quantum critical point to obtain a model simpler than the one presented here <cit.>. It is named the eight hot-spots model because the conduction electrons are only considered at the hot-spots, and the electronic dispersions are linearised close to the Fermi surface. In this eight hot-spots model, the fact that there is an instability in the charge modulation channel has been the subject of intense discussion. Indeed, if the charge modulation wave-vector Q_0 is diagonal, the Fermi surface is said to be nested, and k and k+Q_0 are parallel, at least to first approximation. In this case, the gap equation (<ref>) features two poles in different half-planes and performing the sum yields a Cooper logarithm and therefore a charge instability. This divergence disappears when the curvature of the Fermi surface is included, except in the case of superconductivity where the logarithm is independent of the curvature. If Q_0 is parallel, the Fermi surface is said to be anti-nested, and k and k+Q_0 are anti-parallel. In this case, the gap equation features a double pole, and performing the sum gives zero, and therefore no instability <cit.>. However, it was found that this is not true if we consider the frequency dependence of the gap equation, or if we go to second order in the interaction, where there is a back-and-forth scattering between k and k+Q_0 <cit.>. Therefore, in the weak coupling regime, the charge instability only arises if Q_0 is diagonal. One needs to be away from weak coupling to see an instability for a parallel wave-vector. Here we are in the strong coupling regime, and therefore we obtain an instability even if Q_0 is parallel. § ORDER-BY-DISORDER TREATMENT It has been shown that the charge and superconducting order parameters χ and Δ are related by an SU(2) symmetry in a region of the Brillouin zone, in the sense that one can define an SU(2) algebra relating the two <cit.>. This symmetry is exact on a line of the Brillouin zone joining the hot-spots, and is broken away from it <cit.>. This naturally causes the arising of the fluctuations associated to this symmetry, which we call SU(2) fluctuations <cit.>.The degeneracy of the various channels at the hot-spots introduced above has been shown to be lifted by considering the SU(2) fluctuations through the diagram in Fig. <ref>B <cit.>, similarly to what happens in the order-by-disorder formalism, first described by Villain in the classical context <cit.>.Remarkably, the choice of the starting charge modulation wave vector becomes irrelevant at this point, since it was found that the SU(2) fluctuations select three wave vectors characterizing respectively d-wave nematic ordering at 𝐐_0=(0,0) and axial modulations with or without C_4 symmetry breaking at 𝐐_0=(± Q_x,0) and (0,± Q_y) <cit.> (Fig. <ref>A). Both nematic and axial orders are therefore naturally selected by the SU(2) fluctuations.In order to define the set of operators relating Δ and χ, which form the SU(2) algebra, we use the order parameter χ=1/2∑_𝐤,σd_𝐤c_𝐤^†c_-𝐤 with 𝐤=-𝐤+𝐐_0 and d_𝐤 = (d_𝐤 + d_𝐤)/2. The operation 𝐤 is constrained by the closure condition of the SU(2) algebra to satisfy 𝐤=𝐤 and (-𝐤) = -𝐤 <cit.>. This causes 𝐐_0 to be k-dependent, that is, this causes the charge modulations to be multi-k <cit.>. The lack of k-dependence would cause the arising of harmonics at multiples of 𝐐_0 <cit.>. This 𝐐_0(k) can be chosen in several ways, for example by dividing the Brillouin zone in quadrants and assigning a single vector per quadrant, as mentioned implicitely above <cit.>. It can also correspond to 2k_F charge modulations responsible for both the gapping of the anti-nodal region of the Brillouin zone and the formation of excitonic patches, as studied in <cit.>.These excitonic patches have been shown to proliferate in real space in some regions of the phase diagram <cit.>. In the following section, we give a topological interpretation of the proliferation of local objects in real space, by introducing the SU(2) order parameter, which enables us to encompass many aspects of the phase diagram of the cuprates in an integrated manner.§ SU(2) ORDER PARAMETER The order parameter that naturally emerges from the previous discussion to describe the pseudogap is a composite of Δ and χ, which can be cast into the form: Δ̂_SU2=([χΔ; -Δ^*χ^* ]),where Δ_SU2^2=|χ|^2+|Δ|^2, which is the constraint enforcing the SU(2) symmetry. Since χ and Δ are complex fields, this constraint can be written as:Δ_SU2^2=χ_R^2+χ_I^2+Δ_R^2+Δ_I^2.where the indices R and I denote the real and imaginary parts of the operators, respectively. In this picture, Δ_SU2 represents the energy scale below which the fluctuations between the two fields χ and Δ are dominant; this scale is thus doping dependent. Notice that, by construction, this composite SU(2) order parameter is non-abelian.At every doping x, equation (<ref>) describes a three dimensional hypersphere 𝕊^3 in a four-dimensional space. The transverse fluctuations of the order parameter on this hypersphere are naturally described by an O(4) non-linear σ-model <cit.> S =∫ d^2x∑_α=1,41/2[ρ/T(∂_μn_α)^2+∑_αm_αn_α^2]where α=1,4 are the four-component vector subject to the constraint 𝐧^2=1, with n_1,2=χ_I,χ_R, n_3,4=Δ_I,Δ_R, where χ=χ/Δ_SU2, Δ=Δ/Δ_SU2 and the sign of the masses m_α depends on the presence or absence of an applied magnetic field. The amplitude modes, or massive modes, can be safely neglected since the energy difference between the charge and superconducting states is much smaller than both their energies.In the specific context of the 𝕊_3 sphere, no topological defect is generated, since a careful examination of the corresponding homotopy class gives π_2(𝕊^3)=0 <cit.>. In the following, we discuss the case where one degree of freedom is lost, allowing for topological defects to appear.§ TOPOLOGICAL DEFECTS We now argue that, as the temperature is lowered, the phase of the charge modulations is frozen in some real space regions, as measured by phase-resolved STM <cit.>. This reduces this phase to a few integer values ± iπ/n, with n an integer and thus reduces the fluctuation space from 𝕊^3 to a set of halves of 𝕊^2, indexed by ℤ_2n <cit.>. These regions are thus characterised by Δ_SU2^2=χ^2+Δ_R^2+Δ_I^2, and the effective non-linear σ-model becomes O(3). The space of the fluctuations is depicted in Fig. <ref>A where a fluctuating hemi-sphere is shown; ℤ_2n has been reduced to ℤ_2 for the clarity of the representation, with phase +1 and -1 corresponding to the upper and lower hemi-spheres, respectively.The second homotopy class of the 𝕊^2 sphere is π_2(𝕊^2)=ℤ, which yields the spontaneous generation of skyrmions <cit.>. In our case, in the O(3) regions of real space, Δ_SU2^2 fluctuates on a hemisphere and the boundary conditions corresponding to superconducting vortices give us half-skyrmions bearing a half-integer topological charge <cit.>:Q_top = 1/4 π∫n·∂_x n×∂_y ndx dyThey are also called merons and correspond to a variation of the vector 𝐧 over one hemisphere, as illustrated in Fig. <ref>, <ref>A and <ref>C. They take two equivalent typical forms, of an edgehog and a vortex, and the proliferation pattern involves meron/anti-meron pairs such as the one presented in Fig. <ref>. Note that, contrary to the magnetic skyrmions observed in magnetic systems (see e.g. <cit.>), here the topological structure acts on the pseudo-spin sector, with the three quantization axes (S_x,S_y,S_z) corresponding respectively to (Δ_R,Δ_I,χ) (Fig. <ref>B). The choice of the quantization axis z to be parallel to the charge modulation parameter χ is arbitrary but convenient, since the superconducting phase then corresponds to a simple easy plane situation (Fig. <ref>C).Non-zero topological numbers are associated with the arising of edge currents. One can therefore imagine isolating one topological defect in order to examine these currents. In the most simple case of a single meron, such as displayed in Fig. <ref>, the SU(2) order parameter along the edge is purely in the superconducting plane, with a superconducting phase varying by 2π when going around the full edge, exactly like in the case of a superconducting vortex. One can however think of a different situation, such as the particular meron depicted in Fig. <ref>. It corresponds to a rotation of the axes of the simple case considered in Fig. <ref> and <ref> such that Δ_I would now be along z. Note that this would mean that the phase of the charge order parameter changes along a line dividing the meron in two, on which it has zero magnitude. Experimentally, this would give rise to a supercurrent along this line, as well as a peculiar charge pattern, measureable for example via STM. Moreover, one can define a specific order parameter along the edge as ϕ = Δ_R + i χ. The phase of this order parameter then rotates by 2 π when going round the sample, giving rise to a finite phase gradient, and hence to a peculiar type of current.The mechanism that allows one to see these topological defects is the freezing of the phase of the charge order parameter component of the SU(2) order parameter. This is caused by pinning to local defects or superconducting vortices and coupling of the charge order parameter to the lattice. In this formalism there is therefore coexistence, in the pseudogap phase, of two types of regions in real space: regions where the phase of χ is continuous, and which exhibit a nematic response, and regions where the phase of χ is quantised, where merons proliferate.Measurements of superconducting vortices below T_c found that they bear a very specific structure where charge modulations are observed at the core <cit.>. This corresponds to a pseudo-spin meron in whose centre the pseudo-spin vector is oriented along the z-axis, producing charge modulations while the superconductivity order parameter vanishes, as detailed in Fig. <ref>C. The energy associated to the creation of this vortex is intrinsically of the order of the energy splitting between the superconducting and charge modulation orders, which is precisely the typical energy scale of the superconducting coherence Δ_c∼1/2k_BT_c. Hence pseudo-spin merons will proliferate around T_c in the under-doped region of the phase diagram, acting as a Kosterlitz-Thouless (KT) transition towards the pseudogap state <cit.>. Here, this proliferation is the driving mechanism behind the transition from the superconducting state to the pseudogap state, and it will be strongest close to T_c. Note that the system is phase-coherent at low temperature and is driven by this transition to the phase-incoherent pseudogap phase.The size of the merons can be obtained from similar considerations. Indeed, if one only consider one mass for each field (setting, e.g., m_1=m_2 and m_3=m_4), the energy associated with this topological defect is |m_1 - m_3|, which gives us the size of the meron: L=ħ v_F / √(|m_1 - m_3|), written in Fig. <ref>C. Note that if superconductivity dominates, Δ_c = |m_1 - m_3|.Note that such a pseudospin analogy has also been derived in the case where the three components of the pseudospin are the density fluctuations, the ampliton and the phason operators. The dynamics of this model were studied in analogy to the spin transfer torque effect in magnetic systems and it was found that a similar non-equilibrium superconducting effect could be induced by gradients such as electric or heat gradients <cit.>.We have introduced the general framework of the SU(2) theory, described how SU(2) fluctuations stabilise a pseudogap state, and how some regions in real space can freeze the charge modulation phase, causing the proliferation of pseudo-spin merons. We now turn to the consequences of this formalism on experimental observations. We start by discussing the simultaneous arising of nematic and loop current orders at T^*, then we proceed to charge modulations in real space under applied electric field, and finally we examine the phase diagram under applied magnetic field. § MULTIPLICITY OF ORDERS AT T^*: AN INELUCTABLE COMPLEXITY? It is a longstanding issue whether the pseudogap temperature T^*, sketched in Fig. <ref>, corresponds to a phase transition or a cross-over. Solving this issue is made harder by the fact that the cause of the gapping of the Fermi surface also generates “collateral” orders which typically break discrete symmetries. Disintricating this cause and its collateral orders remains an outstanding problem.Recently, the pseudogap line has been associated with the presence of q=0 orders in the form of intra unit cell orbital currents <cit.>, and to the breaking of the C_4 rotational symmetry leading typically to a nematic signal <cit.>. Ultra-sound experimental data following the pseudogap line have been strengthening the case for a phase transition <cit.>. Although q=0 orders cannot open a gap in the Fermi surface, they can induce time-reversal symmetry breaking, as in the case of loop currents, or C_4 symmetry breaking, as in the case of nematicity.The SU(2) model is in very good posture to bring an ample clarification to the situation. Indeed it does predict the simultaneous gapping of the Fermi surface in the anti-nodal region and enhancement of the nematic susceptibility at T^*, both emerging from the SU(2) fluctuations. Nematicity can then be stabilised as a collateral order below T^*, for example by a small amount of C_4 symmetry breaking due to the oxygen chains in YBCO <cit.>. One can remark, interestingly, that the q=0 breaking of the C_4-symmetry is not necessarily in competition with the charge modulations.We now turn to the possible generation of loop current orders by the SU(2) fluctuations. For this, we rely on symmetry considerations, following reference <cit.>. The SU(2) symmetry involves strong coupling between χ and Δ, which leads us to consider the scalar field ϕ_Q_0=χ_Q_0Δ, (ϕ_Q_0^*=χ_Q_0Δ^*). We consider the influence of both time-reversal (𝒯) and parity (𝒫) transformations on this order parameter (see Supplemental Material for details):ϕ_Q_0ϕ_-Q_0^*, ϕ_Q_0ϕ_-Q_0We now form a loop current order parameter out of the field ϕ_Q_0: l=|ϕ_Q_0|^2-|ϕ_-Q_0|^2. Strikingly, this new order parameter transforms as (see Supplemental Material for details):l-l,l-l,llwhich is precisely the signature of the loop current observed experimentally. Following the argument in <cit.>, the field ϕ_Q_0 has the same symmetries as a pair density wave (PDW) order parameter, and by symmetry such a field can sustain loop currents described by l. These are nil outside the pseudogap phase, since |Δ_SU2|=0. But inside the pseudogap phase, |Δ_SU2| ≠ 0 hence both χ and Δ are finite, except when Δ_SU2 is either in the superconducting state or in the charge ordered state. The detailed study of the Ginzburg-Landau formalism for this field theory will be clarified elsewhere, and follow closely the study of the PDW detailed in reference <cit.>. § CHARGE MODULATIONS IN REAL SPACE Charge modulations (CM) were observed up to the pseudogap energy scale by STM experiments under applied electric field <cit.>. Bulk X-ray probes and NMR, however, reported the presence of charge-modulated areas over a doping dome which doesn't follow T^* but decreases with T_c as doping decreases <cit.>. Here lies a remarkable discrepancy : are the CM associated to the pseudogap energy scale Δ_PG, or to the superconducting coherence energy scale Δ_c∼1/2k_BT_c?To answer this question we first notice that the STM experiments are done at low temperature inside the superconducting phase (T=4 K) whereas the X-ray probes can directly reach T^*. The SU(2) scenario provides a simple explanation for this unusual situation. Indeed, in the SU(2) picture, vortices inside the ordered superconducting phase have a different structure than in standard superconductors where the normal state is a Landau metallic state. Fig. <ref>C and <ref>D show that the charge order coordinate of the SU(2) order parameter is finite at the center of the vortex core. This is similar to the case of SU(2) rotations between d-wave superconductivity and a π-flux phase <cit.>. Here the SU(2) symmetry constrains charge modulations to be present inside the vortex core, as it has been actually seen by STM and NMR <cit.>. In terms of topological defects, it is as if a meron was sitting at the center of the vortex, with the quantization axis z locked to the direction of the charge modulations (Fig. <ref>C). The emergence of a finite charge order parameter in the center of superconducting vortices is caused by the existence of this third degree of freedom of the SU(2) order parameter. This gives a natural explanation to the observation by bulk probes of charge modulations in the region of the phase diagram where vortices are present, below T_c and within a dome above T_c <cit.>.At the operation temperature of STM experiments (T=4 K), the quantization axis of the merons is still locked to the charge modulation direction, up to the energy scale Δ_SU2≃Δ_PG, typical of the pseudogap. As the temperature is raised above the superconducting fluctuations dome, the distribution of quantisation axes across merons becomes fully random, as depicted in Fig. <ref>, and the CM are impossible to observe. Since the SU(2) theory features the coexistence of O(3) and O(4) real space regions, the first of which becoming less and less numerous when the temperature is raised, we can see that there are less and less CM as we get closer to T^*, which elucidates experimental data <cit.>. This disappearance of the CM patches when raising the temperature close to T^* was previously explained by dislocations of the phase of the charge order, interpreted in the framework of a nematic/smectic transition where the nematic order competes with the CM patches <cit.>. Note that in our case, there is no competition between the two, and that the disappearance of the topological defects when temperature is raised, as O(3) regions make way for O(4) regions, simply free the phase of the charge order.§ PHASE DIAGRAM UNDER MAGNETIC FIELDThe measured phase diagram as a function of applied magnetic field (H) and temperature (T) is particularly remarkable because it shows an abrupt phase transition at H_0=17 T from a superconducting phase to an incommensurate charge ordered phase <cit.>. Both phases have a transition temperature of the same order of magnitude, and co-exist in a small region of the phase diagram. The abruptness of the transition at H_0 is reminiscent of a spin-flop transition, typical of non-linear σ-models <cit.>, which could be related to a “pseudo-spin-flop” transition of our SU(2) order parameter, sketched in Fig. <ref>.In the presence of an applied magnetic field, the O(3) non-linear σ-model can be written as: S=∫ d^2x[ρ_1/2(∂_μn_1)^2+ρ_⊥/2∑_α=2^3 (∂_μn_α)^2+H^2/8π. . +∑_αm_αn_α^2],where ∂_μ=∂_μ-qA_μ, A_μ is the electromagnetic vector potential, H is the applied magnetic field and we have taken c=1 and q=2e/c. Here again, the vector n_α=1,3 with 𝐧^2=1 describes the pseudo-spin states with n_1=χ, n_2,3=Δ,Δ^*, respectively. ρ_1 and ρ_⊥ are the phase stiffnesses which usually are temperature dependent. The masses m_α can be taken at zero field such as to favour the superconducting state with for example m_1=0, and m_2,3<0.When H is large enough, vortices with charge modulation core are created until H_c2 <cit.>. Meanwhile, the superconducting state Δ becomes less favourable compared to the charge modulations χ. An effective homogeneous approximation of the gradient term <cit.> yields the spin-flop transition (Fig. <ref>) at a the magnetic field H_0 = m_1-m_2,3. Interestingly, for fields in the vicinity of the spin-flop transition, the SU(2) symmetry is almost perfectly realised and the model can be mapped onto the attractive Hubbard model at half-filling, known to possess an exact SU(2) particle-hole symmetry <cit.>.A prediction of our model is that inside the charge order phase for H>H_c2, vortices disappear and dislocations are the only type of topological defects left. The charge order parameter cancels along these dislocations, which means that the superconducting order parameter is then maximal. These “filaments” of superconductivity could also be caused by the application of a current <cit.>, which would be reminiscent of transport measurements where glimpses of superconductivity have been seen at very high fields and low temperatures <cit.>.§ CONCLUSION The complex phenomenology of the cuprates has led to the rise of more and more involved theoretical descriptions, some even untractable analytically. Here, we discussed a simple formalism which naturally gives birth to the wealth of observed phenomena, and enables us to embrace it all at once. We described the SU(2) parameter, which is a non-abelian composite of a charge and a superconducting order parameters, and its derivation from a short-range antiferromagnetic model. This order parameter is constrained by an SU(2) symmetry, which means that it sits on a three-dimensional hypersphere. As the temperature is lowered within the pseudogap phase, some real-space regions lose one charge degree of freedom, and the order parameter thus sits on a two-dimensional sphere. The SU(2) order parameter can then be seen as a pseudo-spin order parameter in these regions. This naturally leads to the formation of pseudo-spin skyrmions, which account for many puzzling features of STM, X-ray, and NMR data. Nematic and time-reversal symmetry-breaking features arise in the other regions where all the degrees of freedom still remain. Finally, we discussed the phase diagram under magnetic field, which could correspond to a pseudo-spin-flop transition.In summary, general considerations on the SU(2) order parameter allow us to grasp the phase diagram of the cuprate superconductors in its complexity. We think this approach is an important step towards finally unravelling the mysteries of these wonderful materials.§ ACKNOWLEDGMENTS We thank Y. Sidis for stimulating discussions. This work has received financial support from the ANR project UNESCOS ANR-14-CE05-0007, and the ERC, under grant agreement AdG-694651-CHAMPAGNE. The authors also like to thank the IIP (Natal, Brazil), where parts of this work were done, for hospitality. apsrev4-1Supplemental material for: Pseudo-spin Skyrmions in the Phase Diagram of Cuprate Superconductors§ LOOP CURRENTS Here, for convenience, we give the details of the calculation of the loop current order parameter defined in the main text under the time-reversal (𝒯) and parity (𝒫) transformations. It closely follows the argument given in <cit.>. We start by considering the charge order parameter, which was derived from the t-J model in a previous work (<cit.> Eq. 7b):χ_Q_0 = 1/2∑_k,σ[ cos (2 θ_k) + cos (2 θ_k̅ ) ] ψ^†_k̅,σψ_-k,σwhere θ_k is the angle spanning the Brillouin zone and k̅ is the involution:k̅ = -k + Q_0where Q_0 is the ordering wave vector of the charge order parameter defined in the main text.The time-reversal operation acts on annihilation operators following <cit.>:𝒯( ψ_k,σ) = ∑_σ '( -i σ^y )_σσ 'ψ^†_-k,σ'where σ and σ' are spin indices, and σ^y is a Pauli matrix. This can be written more simply as:𝒯( ψ_k,σ) = -sgn(σ) ψ^†_-k,-σwhere sgn(↑) = 1 and sgn(↓) = -1. Applying time-reversal to the charge order parameter gives:𝒯( χ_Q_0) =-1/2∑_k,σ[ cos (2 θ_k) + cos (2 θ_-k + Q_0 ) ] sgn(σ)^2 ψ^†_k,-σψ_k - Q_0,-σ=-1/2∑_k',σ'[ cos (2 θ_-k') + cos (2 θ_k' + Q_0 ) ] ψ^†_-k',σ'ψ_-k' - Q_0,σ'=1/2∑_k',σ'[ cos (2 θ_k') + cos (2 θ_-k' - Q_0 ) ] ( ψ^†_-k' - Q_0,σ'ψ_-k',σ')^†=χ^†_-Q_0Let us now consider the d-wave superconducting order parameter:Δ_Q_0^† = 1/√(2)∑_k, σ[ cos (2 θ_k) + cos (2 θ_-k + Q_0 ) ] ψ^†_k,σψ^†_-k,-σHere we obtain:𝒯( Δ_Q_0^†) =1/√(2)∑_k, σ[ cos (2 θ_k) + cos (2 θ_-k + Q_0 ) ] ψ_k,σ( -ψ_-k,-σ) =-1/√(2)∑_k', σ'[ cos (2 θ_-k') + cos (2 θ_k' + Q_0 ) ] ψ_-k',-σ'ψ_k',σ'=1/√(2)∑_k', σ'[ cos (2 θ_k') + cos (2 θ_-k' - Q_0 ) ] ( ψ_k',σ'ψ_-k',-σ')^†=Δ_-Q_0 Note that here the operator Δ is not Hermitian. Gathering these two results gives:𝒯(ϕ_Q_0) = 𝒯 (χ_Q_0 Δ_Q_0) = Δ^†_-Q_0χ^†_-Q_0 = ϕ^†_-Q_0The case of parity is much simpler and yields:𝒫( ϕ_Q_0) = ϕ_-Q_0We now define the loop current order parameter as:l = |ϕ_Q_0|^2 - |ϕ_-Q_0|^2which therefore transforms as:l-l,l-l,ll | http://arxiv.org/abs/1707.08497v2 | {
"authors": [
"Corentin Morice",
"Debmalya Chakraborty",
"Xavier Montiel",
"Catherine Pépin"
],
"categories": [
"cond-mat.supr-con"
],
"primary_category": "cond-mat.supr-con",
"published": "20170726153223",
"title": "Pseudo-spin Skyrmions in the Phase Diagram of Cuprate Superconductors"
} |
Hybrid Precoding in Millimeter Wave Systems: How Many Phase Shifters Are Needed? Xianghao Yu^*, Jun Zhang^*, and Khaled B. Letaief^*†, Fellow, IEEE ^*Dept. of ECE, The Hong Kong University of Science and Technology, Hong Kong ^†Hamad Bin Khalifa University, Doha, Qatar Email: ^*{xyuam, eejzhang, eekhaled}@ust.hk, ^†[email protected] This work was supported by the Hong Kong Research Grants Council under Grant No. 16210216. December 30, 2023 ================================================================================================================================================================================================================================================================================================================================================================================== Hybrid precoding has been recently proposed as a cost-effective transceiver solution for millimeter wave (mm-wave) systems. The analog component in such precoders, which is composed of a phase shifter network, is the key differentiating element in contrast to conventional fully digital precoders. While a large number of phase shifters with unquantized phases are commonly assumed in existing works, in practice the phase shifters should be discretized with a coarse quantization, and their number should be reduced to a minimum due to cost and power consideration. In this paper, we propose a new hybrid precoder implementation using a small number of phase shifters with quantized and fixed phases, i.e., a fixed phase shifter (FPS) implementation, which significantly reduces the cost and hardware complexity. In addition, a dynamic switch network is proposed to enhance the spectral efficiency. Based on the proposed FPS implementation, an effective alternating minimization (AltMin) algorithm is developed with closed-form solutions in each iteration. Simulation results show that the proposed algorithm with the FPS implementation outperforms existing ones. More importantly, it needs much fewer phase shifters than existing hybrid precoder proposals, e.g., ∼10 fixed phase shifters are sufficient for practically relevant system settings. § INTRODUCTION Uplifting the carrier frequency to millimeter wave (mm-wave) bands has been proposed to meet the capacity requirement of the upcoming 5G networks, and it thus has drawn extensive attention from both academia and industry <cit.>. Thanks to the small wavelength of mm-wave signals, large-scale antenna arrays can be leveraged at transceivers to support directional transmissions. As equipping each antenna element with a single radio frequency (RF) chain is costly, hybrid precoding has been put forward as a cost-effective solution, which utilizes a limited number of RF chains to incorporate a digital baseband precoder and an analog RF precoder <cit.>. In contrast to the conventional fully digital precoder, the additional component in the hybrid one is the analog precoder, which is usually implemented by a bunch of phase shifters in the RF domain. Several hybrid precoder structures and implementations have been proposed in existing works, e.g., the fully- and partially-connected structures <cit.>, as well as the single phase shifter (SPS) <cit.> and double phase shifter (DPS) <cit.> implementations, to provide trade-off between spectral efficiency, energy efficiency, and algorithmic complexity. The main differences among them are the connecting strategies from RF chains to antennas and the number of phase shifters in use to compose the beamforming gain for each of the connected paths. While existing hybrid precoder structures and implementations enjoy a small number of RF chains, the number of phase shifters scales linearly with the antenna size, which is a huge number and thus causes prohibitively high cost and power consumption. On the other hand, various hybrid precoding algorithms have been proposed assuming phase shifters with arbitrary precision, e.g., orthogonal matching pursuit (OMP) <cit.>, manifold optimization <cit.>, and successive interference canceling <cit.>. Although considering phase shifters with programmable high resolution eases the hybrid precoder design, it will weaken the practicality of the results since adaptively carrying out arbitrary phase shifts at mm-wave frequencies is highly impractical <cit.>. Therefore, it is of critical importance to develop effective design methodologies for hybrid precoders with a small number of quantized phase shifters. There are a few works that attempted to consider quantized phase shifters <cit.>. The main approach is either to determine all the phases at once <cit.> or update one phase at a time <cit.> by ignoring the quantization effect at first. Then the phases are heuristically quantized into the finite feasible set according to a certain criterion. However, a simple quantization step is far from satisfactory, and the optimality and convergence of the proposed algorithms cannot be guaranteed <cit.>. On the other hand, the number of phase shifters in use was to some extent reduced in <cit.>, which was determined for achieving a certain precision of the unquantized ones. Unfortunately, a large number of phase shifters are still needed for practical settings, i.e., 40 phase shifters for each RF chain, and the number will vary according to the precision requirement. More importantly, in existing works, the phases need to be adapted to the channel states, which brings high complexity for hardware implementation and also increases power consumption. To overcome the above limitations, in this paper we propose a novel hybrid precoder implementation for general multiuser orthogonal frequency-division multiplexing (OFDM) mm-wave systems, where only a small number of phase shifters with fixed phases are available <cit.>, namely the fixed phase shifter (FPS) implementation. To compensate the performance loss induced by the fixed phases, a switch network is proposed to provide dynamic mappings from phase shifters to antennas, which is easily implementable with adaptive switches <cit.>. With the proposed FPS implementation, we develop an alternating minimization (AltMin) algorithm to design the hybrid precoder <cit.>, where an upper bound of the objective function is derived as an effective surrogate. In particular, the large-scale binary constraints introduced by the switch network are delicately tackled with the help of the upper bound, which leads to closed-from solutions for both the dynamic switch network and the digital baseband precoder, and therefore enables a low-complexity hybrid precoding algorithm. Simulation results shall demonstrate that the proposed FPS-AltMin algorithm outperforms existing ones and approaches the performance of the fully digital precoder. What deserves a special mention is the sharp reduction of the number of phase shifters compared to existing hybrid precoder implementations, which indicates that the proposed FPS implementation is a promising candidate for hybrid precoding in 5G mm-wave communication systems. § SYSTEM MODEL §.§ Signal Model Consider the downlink transmission for a multiuser mm-wave MIMO-OFDM system as shown in Fig. <ref>. A base station (BS) leverages an -size antenna array to serve K users over F subcarriers using OFDM. Each user is equipped withantennas and receives N_s data streams from the BS on each subcarrier. The numbers of available RF chains areandfor the BS and each user, respectively, which are restricted as KN_s≤< and N_s≤<. The received signal of the k-th user on the f-th subcarrier is given by 𝐲_k,f=𝐖^H_BBk,f𝐖^H_RFk(𝐇_k,f∑_k=1^K_k,f𝐬_k,f+𝐧_k,f), where 𝐬_k,f is the transmitted signal to the k-th user on the f-th subcarrier such that 𝔼[𝐬_k,f𝐬_k,f^H]=P/KN_sF𝐈_N_s, and 𝐧_k,f is the circularly symmetric complex Gaussian noise with power as σ_n^2 at the users. The digital baseband precoders and combiners are denoted as _k,f and _k,f, respectively, with dimensions × N_s and × N_s. Since the transmitted signals for all the users are mixed together by the digital precoders, and analog RF precoding is a post-IFFT operation, the RF analog precoderwith dimension × is a common component for all the users and subcarriers. Correspondingly, the × RF analog combiner _k is subcarrier-independent for each user. §.§ FPS Implementation In earlier works on hybrid precoding <cit.>, a single phase shifter is adopted to adjust the phase of each of the paths from RF chains to antennas. Therefore,phase shifters are required, commonly assumed with arbitrary precision. Recently, it was shown in <cit.> that the performance of the hybrid precoder can be greatly improved by passing each signal through two unquantized phase shifters and then combining the outputs, which, however, induces high hardware complexity by employing 2 adaptive phase shifters. In this paper, we propose a hybrid precoder implementation using N_c phase shifters with fixed phases <cit.>, where N_c≪, as shown in Fig. <ref>. Nevertheless, the limited number of fixed phase shifters, which cannot be adaptively adjusted according to the channel states, inevitably entail performance loss. To overcome this drawback brought by the simplified hardware implementation, we propose to cascade a dynamic switch network after the fixed phase shifters, which is adapted to the channel states. In particular, N_c multichannel (-channel) fixed phase shifters <cit.> are deployed in the phase shifter network, each of which simultaneously processes the output signals fromRF chains, i.e., in a parallel fashion. To clearly illustrate the proposed FPS implementation, we focus on one signal flow from an RF chain to an antenna, as shown in Fig. <ref>. The N_c fixed phase shifters generate N_c signals with different phases for the output signal of the given RF chain. Inspired by the idea of doubling phase shifters to achieve high spectral efficiency, as demonstrated in <cit.>, we propose to adaptively combine a subset of the N_c signals to compose the analog precoding gain from the RF chain to the antenna, which is implemented with N_c adaptive switches. As N_c switches are needed for each RF chain-antenna pair, in total N_c switches are required in the proposed FPS implementation. Note that adaptive switches with binary states are easier to implement in mm-wave bands than adaptive phase shifters with arbitrary precision <cit.>. Accordingly, the analog RF precoding matrixcan be expressed as =𝐒𝐂, where 𝐒∈{0,1}^× N_c is the switch matrix, and the boolean constraints are induced by the switches with binary states. The matrix 𝐂∈ℂ^N_c× stands for the phase shift operation carried out by the available fixed phase shifters, given by a block diagonal matrix as 𝐂=diag(𝐜,𝐜,⋯,𝐜_), where 𝐜=1/√(N_c)[e^θ_1,e^θ_2,⋯,e^θ_N_c]^T is the normalized phase shifter vector containing all N_c fixed phases {θ_i}_i=1^N_c. §.§ Problem Formulation It has been shown in <cit.> that minimizing the Euclidean distance between the fully digital precoder and the hybrid precoder is an effective and tractable alternative objective for maximizing the spectral efficiency of mm-wave systems. In this paper, we resort to this approach and the hybrid precoder design is correspondingly formulated as 𝒫_1: 𝐒,𝐅_BBminimize‖-𝐒𝐂‖_F^2subject to𝐒∈ℬ ‖𝐒𝐂‖_F^2≤ KN_sF, where =[_1,1,⋯,_k,f,⋯,_K,F] is the combined fully digital precoder with dimension × KN_sF, and =[_1,1,⋯,_k,f,⋯,_K,F] is the concatenated digital baseband precoder with dimension × KN_sF. The set of binary matrices is denoted as ℬ, and the second constraint is the transmit power constraint. Note that the combiners at the user side can be designed in the same way without the power constraint <cit.> and thus are omitted due to space limitation. Remark 1: Since the switch matrix 𝐒 is with finite possibilities, the cardinality of the constraint set foris finite, which means the OMP algorithm <cit.> is applicable to this problem 𝒫_1. However, the dimension of the dictionary in the OMP algorithm is oversize, i.e., [∑_i=1^N_cN_ci]^, which prevents its practical implementation. Remark 2: Alternating minimization can be directly applied to 𝒫_1 where the binary constraints can be tackled with semidefinite relaxation <cit.>. However, an N_c+1-dimension semidefinite programming (SDP) problem should be solved in each iteration, which causes prohibitive computational complexity. Moreover, the optimality of the relaxation in each iteration cannot be ensured and hence the overall convergence of the AltMin algorithm cannot be guaranteed. As illustrated above, the main difficulty to solve 𝒫_1 is the binary constraints of 𝐒, and it is the main obstacle for designing an efficient algorithm with performance guarantee. In this paper, by deriving an effective surrogate and adopting alternating minimization, we shall come up with a low-complexity hybrid precoding algorithm that well tackles the binary constraints. § HYBRID PRECODER DESIGN IN SINGLE-CARRIER SYSTEMS WITH THE FPS IMPLEMENTATION In this section, we first present the hybrid precoder design in single-carrier systems[In this paper, single-carrier systems refer to single-carrier transmissions assuming flat-fading channels. The choice of such systems is for the ease of presentation, and the algorithm will be later extended to the more realistic multicarrier case with frequency-selective fading channels.], i.e., when F=1. In particular, an upper bound of the objective function is firstly derived, based on which an alternating minimization algorithm is then developed. §.§ Objective Upper Bound In <cit.>, imposing a semi-orthogonal structure foris shown to achieve near-optimal performance. Inspired by these results, we take a similar approach. In single-carrier systems, the digital precoder matrixis with dimension × KN_s. Recall that the number of RF chains is limited as KN_s≤<, which forcesas to be tall matrix, and thus the semi-orthogonal constraint is specified as ^H=α^2^H=α^2𝐈_KN_s, where =α andis a semi-unitary matrix. Then, the objective function in 𝒫_1 can be rewritten as ‖‖_F^2-2α(^H𝐒𝐂)+α^2‖𝐒𝐂‖_F^2. Note that, according to (<ref>), the phase shifter matrix 𝐂 is also a semi-unitary matrix, i.e., 𝐂^H𝐂=𝐈_. Therefore, we can derive an upper bound for the last term in (<ref>), given by ‖𝐒𝐂‖_F^2 = (^H𝐂^H𝐒^H𝐒𝐂)(a)=( [ 𝐈_KN_s; 0;]𝐊^H𝐒^H𝐒𝐊)< ( 𝐊^H𝐒^H𝐒𝐊)=‖𝐒‖_F^2, where (a) follows the singular value decomposition (SVD) of 𝐂^H𝐂^H=𝐊diag(𝐈_KN_s,0)𝐊^H by utilizing the semi-unitary property of 𝐂. Thus, we obtain an upper bound for the original objective function, expressed as ‖‖_F^2-2α(^H𝐒𝐂)+α^2‖𝐒‖_F^2. §.§ Alternating Minimization By adopting the upper bound (<ref>) as the surrogate objective function and dropping the constant term ‖‖_F^2, the hybrid precoder design problem is reformulated as 𝒫_2: α,𝐒,𝐅_DDminimizeα^2‖𝐒‖_F^2-2α(^H𝐒𝐂)subject to𝐒∈ℬ ^H=𝐈_KN_s. Alternating minimization, as an effective tool for optimization problems involving different subsets of variables, has been widely applied and shown empirically successful in hybrid precoder design <cit.>. In this section, we apply it this effective rule of thumb to the hybrid precoder design with the FPS implementation. In each step of the AltMin algorithm, one subset of the optimization variables is optimized while keeping the other parts fixed. When the switch matrix 𝐒 and α are fixed, the optimization problem can be written as 𝐅_DDmaximizeα(^H𝐒𝐂)subject to^H=𝐈_KN_s. According to the definition of the dual norm <cit.>, we have α(^H𝐒𝐂) ≤|(α^H𝐒𝐂)|(b)≤‖^H‖_∞‖α^H𝐒𝐂‖_1=‖α^H𝐒𝐂‖_1=∑_i=1^KN_sσ_i, where ‖·‖_∞ and ‖·‖_1 stand for the infinite and one Schatten norms <cit.>, and (b) follows the Hölder's inequality. The equality is established only when =𝐕_1𝐔^H, where α^H𝐒𝐂=𝐔Σ 𝐕_1^H follows the SVD and Σ is a diagonal matrix with non-zero singular values σ_1,⋯,σ_KN_s. While we can divide the optimization of the two variables α and 𝐒 into two separate subproblems, we propose to update them in parallel to save the number of subproblems involved in the AltMin algorithm and therefore reduce the computational complexity. By adding a constant term ‖(^H𝐂^H)‖_F^2 to the objective function of 𝒫_2, the subproblem of updating α and 𝐒 can be recast as α,𝐒minimize‖(^H𝐂^H)-α𝐒‖_F^2subject to𝐒∈ℬ. The optimal solution to (<ref>) is given by α^⋆={x̃_i,x̅_i}_i=1^nmin{f(x̃_i),f(x̅_i)}, 𝐒^⋆=1{(^H𝐂^H)>α/21_× N_c} α>0 1{(^H𝐂^H)<α/21_× N_c} α<0, where 𝐱=vec{(^H𝐂^H)}, 1(·) is the indicator function, and 1_m× n denotes an m× n matrix with all entries equal to one. The objective function in (<ref>) can be rewritten as f(α) as (<ref>) in the proof. In addition, x̃_i is the i-th smallest entry in 𝐱, and[f(α) is a coercive function, i.e., f(+∞)→+∞.] x̅_i≜∑_j=1^ix̃_j/i α<0 and ∑_j=1^ix̃_j/i∈[2x̃_i,2x̃_i+1] ∑_j=i+1^nx̃_j/n-i α>0 and ∑_j=i+1^nx̃_j/n-i∈[2x̃_i,2x̃_i+1] +∞ otherwiese. See Appendix A. Basically, the optimal α^⋆ is obtained via a closed-form solution by comparing the optimal solutions of α in all the intervals {ℛ_i}_i=1^n, where ℛ_i≜[2x̃_i,2x̃_i+1]. Nevertheless, since the number of intervals that need to compare is n= N_c, it will incur high computational complexity whenis large in mm-wave systems. In the following lemma, we show that there is no need to compute the optimal α in all the intervals ℛ_i, which will further reduce the complexity of the proposed algorithm. The optimal α^⋆ is obtained at one of the points {x̅_i}_i=1^n. See Appendix B. Lemma <ref> indicates that any endpoint of the intervals cannot be the optimal solution for α. Therefore, we only need to pick the x̅_i's that have finite values of f(x̅_i), i.e., the ones that satisfy the first two conditions in (<ref>), denoted as a set 𝒳, and the optimal solution for α is given by α^⋆=x̅_i∈𝒳min f(x̅_i). By Lemma <ref>, the number of intervals we need to compare to obtain the optimal α^⋆ is shrunk from n to |𝒳|, which is empirically shown to be less than 5 via simulations in Section V and hence further reduces the computational complexity of the proposed AltMin algorithm. Remark 3: It is shown that, with the help of the upper bound derived in (<ref>), the large-scale binary switch matrix 𝐒 can be efficiently optimized by a closed-form solution, which verifies the benefits and superiority of the surrogate objective function adopted in 𝒫_2. With the closed-form solutions derived in (<ref>), (<ref>), and (<ref>) at hands, the AltMin algorithm for the FPS implementation is summarized as FPS-AltMin Algorithm. The FPS-AltMin algorithm is essentially a block coordinate descent (BCD) algorithm with two blocks that have globally optimal solutions in Steps 3 and 4, and the algorithm is guaranteed to converge to a stationary point of 𝒫_2 <cit.>. The algorithm may be sensitive to the initial point ^(0). Note that the fully digital precoding matrixcan be decomposed as follows according to its SVD =𝐔Σ 𝐕^H, i.e, =[ 𝐔Σ𝐅 ][ 𝐕^H; 0 ], where 𝐔Σ is an × KN_s full rank matrix, 𝐕^H is a KN_s dimension square matrix, and 𝐅 is an arbitrary ×(-KN_s) matrix. In (<ref>), the fully digital precoding matrixis decomposed into two matrices that satisfy the dimensions ofand , respectively. In this way, we propose to construct the initial point ^(0) as ^(0)= [𝐕 0_KN_s×(-KN_s) ]^H. To cancel the inter-user interference, similar to <cit.>, we cascade an additional block diagonal precoder at the baseband in the Step 7 based on the effective channel including the hybrid precoder and physical channel. In the final step, we normalize the digital precoder to maximize the signal to noise ratio (SNR) while satisfying the transmit power constraint. algorithmFPS-AltMin Algorithm: § HYBRID PRECODER DESIGN IN MULTICARRIER SYSTEMS WITH THE FPS IMPLEMENTATION Multicarrier techniques such as OFDM are often utilized to overcome the multipath fading caused by the large available bandwidth in mm-wave systems. Compared with the narrowband hybrid precoder design in Section III, the main difference in OFDM systems is that the analog precoder is shared by all the subcarriers. In particular, the digital precoding matrix ∈ℂ^× KN_sF in 𝒫_1 is no longer a tall matrix since KN_sF≥ for practical OFDM system settings. In this section, we modify the FPS-AltMin algorithm for OFDM systems. Similar to (<ref>), we enforce a semi-orthogonal constraint on the digital precoding matrix, i.e., ^H=α ^2^H=α^2𝐈_. In this way, the upper bound of the objective function derived in (<ref>) still holds since ‖𝐒𝐂‖_F^2 = (𝐂^H𝐒^H𝐒𝐂)(c)=( [ 𝐈_; 0 ]𝐊^H𝐒^H𝐒𝐊)< ( 𝐊^H𝐒^H𝐒𝐊)=‖𝐒‖_F^2, where (c) comes from the SVD of 𝐂𝐂^H since 𝐂 is a semi-unitary matrix. In the AltMin algorithm, the update of α and 𝐒 is the same as that in Section III-B. Since the dimension ofis different in OFDM systems, the optimization ofis modified as =𝐕𝐔_1^H, where ^H𝐒𝐂=𝐔_1Σ 𝐕^H and Σ is a diagonal matrix with non-zero singular values σ_1,⋯,σ_, which is the SVD of ^H𝐒𝐂. Correspondingly, the construction of the initial ^(0) is given by ^(0)=𝐕^H_[1:], where =𝐔Σ 𝐕^H is the SVD ofand the subscript [1:n] denotes the first to the n-th columns of a matrix. By substituting (<ref>) and (<ref>) into the Steps 1 and 4, we obtain the FPS-AltMin algorithm for OFDM mm-wave systems. § SIMULATION RESULTS In this section, we will evaluate the performance of the proposed FPS-AltMin algorithm via simulations. The BS and each user are equipped with 144 and 16 antennas, respectively, while all the transceivers are equipped with uniform planar arrays (UPAs). Four users and 128 subcarriers are assumed when considering multiuser OFDM systems. To reduce the cost and power consumption, the minimum number of RF chains is adopted according to the assumptions in Section II-A, i.e., =KN_s and =N_s. The phases of the available fixed phase shifters are uniformly separated within [0,2π] by N_c equal length intervals. Furthermore, the Saleh-Valenzuela model is adopted in simulations to characterize mm-wave channels <cit.>. The nominal SNR is defined as P/KN_sFσ_n^2, and all the simulation results are averaged over 1000 channel realizations. §.§ Single-User Single-Carrier (SU-SC) Systems As a great number of previous efforts have been spent on point-to-point systems, it is intriguing to test the performance of the proposed algorithm by comparing with existing works as benchmarks. The OMP algorithm proposed in <cit.> has been widely used as a low-complexity algorithm with the analog precoder selected from a predefined set. The MO-AltMin algorithm was then proposed in <cit.> to improve the performance of the OMP algorithm, yet with high computational complexity of performing the manifold optimization. Both of these algorithms are applied with the SPS implementation. Fig. <ref> shows that the proposed FPS-AltMin algorithm achieves the highest spectral efficiency with the simulation time comparable to the OMP algorithm. The performance gain is mainly attributed to the proposed FPS implementation, where the unit modulus constraints in the SPS implementation are relaxed. Furthermore, the proposed algorithm leads to an effective design of the dynamic switch network, and provides a better approximation of the fully digital precoder than existing algorithms. §.§ Multiuser Multicarrier (MU-MC) Systems In <cit.>, the DPS implementation was proposed for MU-MC systems to approach the performance of the fully digital precoder by sacrificing the hardware complexity of employing a large number of phase shifters, i.e., 2 phase shifters. As shown in Fig. <ref>, the proposed FPS-AltMin algorithm only entails little performance loss compared to the DPS implementation when only 30 fixed phase shifters are adopted. On the other hand, it enjoys significant improvement in terms of spectral efficiency compared to the OMP algorithm. This result demonstrates the effectiveness of both the newly proposed implementation and algorithm. In addition, it indicates that the number of phase shifters can be sharply reduced even if the analog precoder is shared by all the subcarriers and users in MU-MC systems. §.§ How Many Phase Shifters Are Needed? Fig. <ref> plots the spectral efficiency achieved with different numbers of fixed phase shifters, i.e., N_c. The simulation parameters are the same as those in Figs. <ref> and <ref> for SU-SC and MU-MC systems, respectively. Fig. <ref> shows that in SU-SC systems 15 phase shifters are enough for achieving a satisfactory performance as the spectral efficiency almost saturates when we further increase the number of fixed phase shifters. By contrast, 576 phase shifters are needed in the SPS implementation. Moreover, the OMP algorithm achieves a lower spectral efficiency and the MO-AltMin algorithm suffers from the high computational complexity. A similar phenomenon is found in MU-MC systems, i.e., around 10 fixed phase shifters are sufficient, which has not been revealed in existing works. Although the performance of the DPS implementation slightly outperforms the proposed FPS-AltMin algorithm, it employs 200 times more phase shifters. This illustrates that the proposed FPS implementation is much more cost-effective than existing hybrid precoder implementations, and with satisfactory performance. § CONCLUSIONS In this paper, we proposed a cost-effective hybrid precoder implementation with a small number of fixed phase shifters. To enhance the performance, a dynamic switch network was adopted, for which a low-complexity AltMin algorithm was developed. The proposed implementation is able to approach the performance of the fully digital precoder, remarkably, with small numbers of RF chains and phase shifters. Thus, this proposal stands out as a promising candidate for hybrid precoders for 5G mm-wave systems. § PROOF OF PROPOSITION 1 TempEqCnt Note that each entry in the switch matrix 𝐒 is either 0 or 1, and we discover that they can be optimally determined individually once α is given. In particular, to minimize the objective function, s_m,n should take value 1 if the corresponding (m,n)-th entry in the matrix (^H𝐂^H) is closer to α than 0 in the Euclidean space, and take value 0 otherwise, as given in (<ref>). The remaining problem is to choose an optimal α that minimizes the objective function. Since 𝐒∈ℬ is an element wise constraint, to simplify the notations, it is equivalent to consider the vectorization version of (<ref>), given by α,𝐬minimize‖𝐱-α𝐬‖_2^2subject to𝐬∈{0,1}^n, where n= N_c, 𝐱≜vec{(^H𝐂^H)}, and 𝐬=[s_1,s_2,⋯,s_n]≜vec{α𝐒}. First, we sort the entries of 𝐱 in the ascending order as 𝐱̃=[x̃_1,x̃_2,⋯,x̃_n], where x̃_1≤x̃_2≤⋯≤x̃_n. Then all the entries split the real line into n+1 intervals {ℐ_i}_i=0^n, where ℐ_i≜[x̃_i,x̃_i+1]. Furthermore, we can obtain some insights from (<ref>) to optimize α. Specifically, if α/2 falls into a certain interval ℐ_i, the corresponding optimal 𝐬 can be determined as {s_k}_k=1^i-1= 0 α>0 1 α<0, {s_k}_k=i^n= 1 α>0 0 α<0. equation1 Therefore, the objective function in (<ref>) can be rewritten as (<ref>) at the top of this page. Note that within each interval ℛ_i=[2x̃_i,2x̃_i+1], the objective function is a quadratic function in terms of α, and hence it is easy to give the optimal solution for α in Proposition 1. § PROOF OF LEMMA <REF> We prove Lemma 1 by contradictory. Since in each interval ℛ_i the objective function is a quadratic function of α. The optimal α^⋆ can only be obtained at the two endpoints of ℛ_i or at the axis of symmetry if the objective is not monotonic in ℛ_i. When α<0, the axis of symmetry of the quadratic function is given by x̅_i=∑_j=1^ix̃_j/i, which is the mean value of the first i entries in 𝐱̃. A hypothesis is firstly made that a certain endpoint x̃_i is the optimal solution to α. It means that the axis of symmetry of the objective function in ℛ_i-1 is on the right hand side of x̃_i, and the axis of symmetry of the objective function in ℛ_i is on the left hand side of x̃_i, i.e., x̅_i<x̃_i<x̅_i-1. Note that the entries in 𝐱̃ are ordered in the ascending order. Hence, x̅_i, as the mean value of the first i entries in 𝐱̃, is an increasing function with respect to i, i.e., x̅_i≥x̅_i-1, which is contradictory with (<ref>) and completes the proof for α<0. The scenario of α>0 can be similarly proved. IEEEtran | http://arxiv.org/abs/1707.08302v1 | {
"authors": [
"Xianghao Yu",
"Jun Zhang",
"Khaled B. Letaief"
],
"categories": [
"cs.IT",
"math.IT"
],
"primary_category": "cs.IT",
"published": "20170726064420",
"title": "Hybrid Precoding in Millimeter Wave Systems: How Many Phase Shifters Are Needed?"
} |
Online Wideband Spectrum Sensing Using Sparsity Lampros Flokas and Petros Maragos, Fellow, IEEE Lampros Flokas is with the Columbia University in the City of New York, Department of Computer Science, New York, USA, .Petros Maragos is with the National Tech. University of Athens, School of ECE, Greece, Email:December 30, 2023 ======================================================================================================================================================================================================================================================================================== Wideband spectrum sensing is an essential part of cognitive radio systems. Exact spectrum estimation is usually inefficient as it requires sampling rates at or above the Nyquist rate. Using prior information on the structure of the signal could allow near exact reconstruction at much lower sampling rates. Sparsity of the sampled signal in the frequency domain is one of the popular priors studied for cognitive radio applications. Reconstruction of signals under sparsity assumptions has been studied rigorously by researchers in the field of Compressed Sensing (CS). CS algorithms that operate on batches of samples are known to be robust but can be computationally costly, making them unsuitable for cheap low power cognitive radio devices that require spectrum sensing in real time. On the other hand, online algorithms that are based on variations of the Least Mean Squares (LMS) algorithm have very simple updates so they are computationally efficient and can easily adapt in real time to changes of the underlying spectrum. In this paper we will present two variations of the LMS algorithm that enforce sparsity in the estimated spectrum given an upper bound on the number of non-zero coefficients. Assuming that the number of non-zero elements in the spectrum is known we show that under conditions the hard threshold operation can only reduce the error of our estimation. We will also show that we can estimate the number of non-zero elements of the spectrum at each iteration based on our online estimations. Finally, we numerically compare our algorithm with other online sparsity-inducing algorithms in the literature. signal processing, sparse representations, LMS, cognitive radio.§ INTRODUCTION Wireless telecommunications spectrum is a limited resource and with the rapid increase of telecommunication applications, static allocation of spectrum for each case is not a viable solution. Additionally, static allocation of spectrum is also not effective as the primary users of the spectrum may use it from time to time and only in some locations. To overcome this limitation cognitive radio devices try to dynamically manage the spectrum by detecting which part of the spectrum is unused by its primary users and temporarily using it for their own needs.In order to be effective, these devices would need to check a wide band of frequencies to increase the possibility of finding unused spectrum. If cognitive devices used sampling rates that are equal or above the Nyquist rate, their cost would be prohibitive for most applications. In order to reduce the sampling rate needed as well as the computational effort, we will need to use some prior information on the structure of the received signal. This prior is that the same one that enables the usage of cognitive radio devices in the first place: Primary users do not use their share of the spectrum all the time so the received signal should be sparse in the frequency domain. The area of compressed sensing (CS) has provided several celebrated algorithms for the reconstruction of undersampled signals with sparse representations <cit.>. Classic algorithms of CS assume a batch setting where the device is assumed to collect a number of observations and operate on themin an iterative manner. Therefore it is of great importance to provide algorithms that reduce the number of iterations needed in order to reduce the computational burden on the cognitive radio devices and provide real time spectrum estimations. CS based approaches have been adapted by many researchers in the area of spectrum sensing for cognitive radio applications <cit.>. On the other hand, online algorithms, based on variations of Least Mean Squares introduced by Widrow and Hoff <cit.>, have also been adapted for the CS setting. Algorithms like the ones presented in <cit.> have been shown to estimate sparse signals with faster convergence and smaller steady state errors than methods that do not exploit sparsity. Additionally, they have much simpler updates based on a single sample at a time. This allows them not only to be more computationally efficient but also to be adaptive to the changes of the estimated signal.Here we will propose two new variations of the classical LMS algorithm. The first is a variation of the Zero Attracting LMS <cit.> that does not penalize the s algebraically largest coeffients of the estimation where s is an upper bound on the number of non-zero elements in the estimated vector. The second one alternates the standard LMS update with shrinkage using a hard threshold operator. The hard threshold operator will keep the s algebraicly largest components of the estimated vector where s is again an upper bound on the number of non-zero elements in the estimated vector. This algorithm is the online version of the iterative hard thresholding studied in <cit.> and <cit.> and <cit.>. The sparsity of the estimated vector or even an upper bound on it may not be known in advance so we will also propose a way to estimate it in an adaptive manner. Even though we are going to apply the proposed algorithms for the problem of spectrum estimation, they can also be applied in other telecommunications and general machine learning applications where the incoming signals have a known sparse representation. The structure of the paper is as follows. In Section 2 we will define the problem of sparse spectrum reconstruction using below Nyquist rate sampling frequencies. In Section 3 we will present the properties of online sparsity aware estimation techniques in the literature and in Section 4 we will introduce our hard thresholding based algorithms. In Section 5 numerical simulations comparing our algorithm with other sparsity aware algorithms are provided. Finally, Section 6 contains concluding remarks and discusses possible directions for future research. § PROBLEM STATEMENTLet 𝐳∈ℝ^N be the full signal that the cognitive radio device would receive if it was sampling it at the Nyquist rate. We would like to undersample 𝐳, taking just M observations from 𝐳 where M<N. Let us call 𝐔 the undersampling matrix whose rows are a subset of the rows of the identity matrix including only the rows where the respective observation of 𝐳 is sampled. Let us call 𝐲=[y_0,y_1,…,y_M-1]^T∈ℝ^M the resulting vector. If each observation y_i is corrupted by an additive error term v_i and 𝐯=[v_0,v_1,…,v_M-1]^T ∈ℝ^M Then we obviously have that 𝐲=𝐔𝐳 + 𝐯Of course without any further assumptions the lost information cannot be recovered and important information about the spectrum of 𝐳 cannot be estimated. However, in our case we can assume that the underlying spectrum of the signal is sparse as a large portion of the spectrumwill be left unused by its primary users. Let𝐰∈ℂ^N be the complex vector representing the Discrete Fourier Transform (DFT) of 𝐳 and Φ be the Inverse Discrete Fourier Transform (IDFT) matrix so that 𝐳=Φ𝐰. Given our assumption on the sparsity of the spectrum of 𝐳, we have that 𝐰 is a sparse vector and therefore we are interested in solving the following problem:min𝐰_0: 𝐲-(𝐔Φ)𝐰_2 ≤δwhere the ℓ_0 norm is the count of non-zero elements of the vector i.e 𝐰_0=|support(𝐰)|, where support(𝐰)={ i ∈{0,1,..,N-1}: w_i ≠ 0},|S| denotes the cardinality of set S and δ is an upper bound on 𝐯_2. In general this problem is NP-hard and therefore computationally intractable. However, researchers in the area of CS have developed several algorithms that recover the solution of problem described by <ref> when the matrix 𝐔Φ satisfies the Restricted Isometry Property and vector 𝐰 is sparse enough. Out of all the algorithms probably the most popular one is Lasso regression. One of its equivalent formulations ismin𝐰_1: 𝐲-(𝐔Φ)𝐰_2 ≤δ The resulting optimization problem can be solved with standard convex optimization methods. The limiting factor for Lasso and other classical approaches to CS is that they may require several iterations to converge to the optimal solution. This makes them unsuitable for the low power cognitive radio devices that need real time spectrum estimations in potentially highly volatile settings. In contrast, online estimation algorithms have much simpler update rules that involve one sample at a time and are robust to changes in the estimated signal. In the online setting there is a stream of measurements of 𝐲 and the corresponding rows of Φ that are fed one by one to the online algorithm. There are at least two options when it comes to forming this stream. * The first option is to use an online algorithm as a drop in replacement of a batch algorithm. We can collect M out of N samples of the signal and feed them one by one to the online algorithm. Of course the online algorithm may not converge in a single pass over the data so we can augment the stream by feeding the same measurementsto the algorithm multiple times in order to achieve convergence. The advantage of the online algorithms over batch algorithms in this setting is that they have simpler update rules than their batch counterparts and so they could be more easily implementable in low power cognitive radio devices.* The second option is to form a stream by continuously incorporating new measurements. One way to do this is to split the incoming signal in non overlapping windows of length N, randomly sample M measurements in each window and feed the resulting measurements to the online algorithm. The advantage of the online algorithms over batch algorithms in this setting is that they can track the spectrum changes in the signal in real time.In Section 5 we shall provide experimental results for both settings.§ RELATED WORK§.§ The LMS algorithmThe algorithms proposed in this paper are based on the updates of the LMS algorithm. To better understand the procedure we review the steps of the classical LMS algorithm. Let y(n) be a sequence of observations of the output of a system following the modely(n)=𝐰^H𝐱(n)+v(n)where 𝐰=[w_0,w_1,…,w_N-1]^T ∈ℂ^Nis the parameter vector to be estimated, 𝐱(n)∈ℂ^N is taken from the rows of the Φ^* that correspond to the observed samples and v(n) is the additive observation noise. Let also 𝐰(n) be the estimation we have up to time n for the unknown vector 𝐰 and e(n) be the sample error. Thene(n)=y(n)-𝐰^H(n)𝐱(n)The LMS update rule is recursive and produces a new estimation given the previous one, following the rule𝐰(n+1)=𝐰(n)+μ e^*(n)𝐱(n)where μ is a an appropriately chosen constant. If 𝐑_x=𝔼[𝐱(n)𝐱^H(n)] is the uncentered covariance matrix of 𝐱(n), here assumed constant over time, and λ_max is its maximum eigenvalue then <cit.> shows that LMS will converge in the mean sense if:0<μ<2/λ_maxOf course the simple LMS algorithm has the same guarantees for all estimated signals, sparse and dense alike. Using the sparsity assumption can increase the speed of convergence and yield much lower steady state estimation errors than the classical algorithms.§.§ Zero Attracting LMSThe Zero Attracting LMS algorithm (ZA-LMS) <cit.> is a modification of the standard LMS algorithm that specializes in sparse system identification.This algorithm follows the spirit of the equivalence of the ℓ_1 and ℓ_0 regularization problems in the batch case. Therefore the objective minimized at every iteration becomesJ_ZA(n) = 1/2e(n)^2 +γ𝐰(n)_1for some parameter γ. Taking the gradient descent update one can adapt the LMS update scheme to the following𝐰(n+1)=𝐰(n)+μ e^*(n)𝐱(n)-ρsgn(𝐰(n))where ρ=μγ and sgn(x) is the component wise sign function defined as sgn(x)= x/x , x ≠ 0 0,otherwiseIt is clear that smaller coefficients of the estimated vector are quickly drawn to zero making the vector sparse while larger coefficients remain mostly unaffected for small values of ρ. Thus the update rule converges to sparse vectors.§.§ ℓ_0-LMSℓ_0-LMS <cit.> takes a different approach to sparse system identification by trying to minimize the objectiveJ_ℓ_0(n) = 1/2e(n)^2 +γ𝐰(n)_0Of course simply doing a gradient descent on the objective directly is not possible and in general the problem is known to be NP-hard. Instead the ℓ_0 norm is approximated by 𝐰(n)_0 ≈∑_i=0^N-1(1-e^-βw_i(n))The parameter β here controls the quality of the approximation of the ℓ_0 norm and as β tends to infinity the formula becomes exact. Taking the gradient on the modified objective leads to the following update rule𝐰(n+1)=𝐰(n)+μ e^*(n)𝐱(n)-ρsgn(𝐰(n))e^-β𝐰(n)where the exponentiation and the sign is applied element-wise. The same observations as in the previous algorithms apply here also. The difference is that the attraction to zero is even weaker for the coefficients that have large magnitudes so we expect that the convergence should be faster in general. § NEW ONLINE ALGORITHMS §.§ Selective Zero Attracting LMS In the previous two sub sections we saw two regularized objectives of the standard LMS objective. In this paper we will try to solve a constrained version of the LMS objective. We will try to minimize J(n) = 1/2e(n)^2 but under the restriction that 𝐰(n)_0 ≤ swhere s, a positive integer less than N, is an upper bound on the sparsity of the vector under estimation that we know in advance. Let us define the operator H_s that outputs a vector having zeros in all coefficients except for the ones with the s largest absolute values that remain the same as in the input vector. For example if 𝐱_0=[2,-2,1,0]^T then H_2(𝐱_0)=[2,-2,0,0]^T. In case of ties we can take a conservative approach and allow all tying coefficients to be nonzero in the resulting vector so that H_1(𝐱_0)=[2,-2,0,0]^T. Thus |support(H_s(𝐱))| ≥ s and therefore it is not guaranteed that the output will always be s-sparse. The operator will give as output vectors that are not s-sparse when there are multiple coefficients in the vector that their absolute value is equal to the s largest absolute value in the vector. However, in most cases such ties will be nonexistent and the result will be an s-sparse vector.Given the definition of H_s one could easily see a connection with ℓ_0-LMS. Specifically we could relax the objective just like in the previous subsection. Here we will use however different β_i for each coefficient. Let us approximate the ℓ_0 norm as𝐰(n)_0 ≈∑_i=0^N-1(1-e^-β_iw_i(n))Then if we want to make the estimate to converge to an s-sparse vector we can do the following: For the s algebraically largest coefficients we will use β_i=∞ whereas for all the others we will use β_i=0. This can be interpreted as the following penaltyP_s(𝐱)_i=0,i ∈support(H_s(x)) sgn(x_i), otherwisewhich then leads to the following update rule𝐰(n+1)=𝐰(n)+μ e^*(n)𝐱(n)-ρ P_s(𝐰(n))This is the same concept of the ℓ_1 penalization presented in <cit.> but applied only to the possibly superfluous coefficients given the a priori estimation of sparsity. We shall call this algorithm Selective Zero Attracting LMS.Based on this fact we can prove a similar convergence result Let us have a zero mean observation noise v(n) independent of 𝐱(n) and given that 𝐱(n) and 𝐰(n) are independent. Let us also assume that 𝔼[𝐱(n)𝐱^H(n)] is constant over time, invertible and equal to 𝐑_x. Then the algorithm described by (<ref>) converges in the mean sense provided that the condition of (<ref>) holds. The limiting vector satisfies the equation𝔼[𝐰(∞)]=𝐰-ρ/μ𝐑_x^-1𝔼[P_s(𝐰(∞))] The proof of the theorem can be found in Appendix <ref> and is similar to the proof for the ZA-LMS. The interested reader can find an in depth analysis of a similar approximation scheme in <cit.>. The difference is in the choice of coefficients that get penalized and those who do not. In the update scheme presented we choose not to penalize the s largest coefficients. In <cit.> the coefficients that do not get penalized are those who are greater than a predefined threshold. As we can see in Equation (<ref>), the expected value of the the estimation does not converge necessarily to 𝐰. In fact there might be a O(ρ) deviation per coefficient just like in the simple Zero Attracting LMS. However, if 𝐰 is an s sparse vector and the algorithm identifies the support correctly then the bias for the leading s coefficients should be eliminated as the penalty term will be zero for those coefficients, a property that the Zero Attracting LMS does not have. For the rest of the coefficients, unless the estimate for those does not converge exactly to 0 we will still incur the O(ρ) deviation, which should be negligible for small values of ρ.§.§ Hard Threshold LMS The contribution of this paper is the study of the properties of the following update scheme𝐰(n+1)=H_s(𝐰(n)+μ e^*(n)𝐱(n))It is easy to see the similarity of our proposed algorithm with the iterative hard thresholding algorithm studied in <cit.>, <cit.> and <cit.>. There, since the algorithm is developed in a batch setting where all the data are known in advance, the relation between the observations 𝐲 and the estimated vector 𝐰 is 𝐲=𝐀𝐰 where 𝐀 is M× N matrix with M<N; thus the problem is undetermined. The update of the iterative hard thresholding under similar assumptions for the sparsity of 𝐰 is𝐰(n+1)=H_s(𝐰(n)+μ𝐀^H𝐞(n))where 𝐞(n)=𝐲-𝐀𝐰(n). It must be noted that the complexity of implementing such an operator is still linear in N as finding the s largest value in a vector does not require sorting it first. As a result it is clear that the proposed algorithm is closely related to the special case of iterative hard thresholding having M=1. It is also clear that we cannot use the rigorous proofs found in <cit.>, <cit.> and <cit.> to show that the proposed algorithm also converges since for M=1 it is impossible to fulfill the strict properties needed. However, it is still possible to prove some interesting properties of the hard threshold operator. The main contribution of the operator is to let us focus our attention on the support of the estimated vector. If the algorithm does not provide a correct estimation of the support of the estimated vector then this could have a negative effect on the convergence of the algorithm. So one of the key properties that need to be studied is under which conditions is the estimation of the support using the hard threshold operator correct. Let 𝐰=[w_0,w_1,…,w_N-1]^T ∈ℂ^N with 𝐰_0=s and 𝐰̂ be an approximation. Let q=min_w_i ≠ 0w_i. Then if 𝐰-𝐰̂_2^2 < q^2/2 the following will be true support(H_s(𝐰̂))= support(𝐰) The proof of the theorem is quite involved and can be found in Appendix <ref>. The essence of the proof however is rather simple. In order to have the minimal error and still incorrectly specify the support of the vector, the error must be concentrated in two coefficients, one that belongs in support(𝐰) and one that does not. The one coefficient that belongs to the correct support must end up having a smallermagnitude than the one that should not. Since the first coefficient has at least magnitude q in 𝐰 and the other coefficient must have magnitude 0, the minimal error is achieved when both have magnitude q/2 in𝐰̂ which leads to the bound of the error that we have in the proof. In order to understand the significance of the theorem we need to see some equivalent bounds having to do with the signal to error ratio that is needed so that the result in relation (<ref>) still holds. The true vector 𝐰 has s nonzero valueseach with an absolute value of at least q. Thus 𝐰_2^2≥ sq^2 and hence we needSER=𝐰_2^2/𝐰-𝐰̂_2^2> sq^2/q^2/2=2sInequality (<ref>) is a necessary condition so that the required conditions of the theorem are true. Even if it is not sufficient it gives us the intuition that for small values of s it will be easier to come up with an estimate 𝐰̂ for which relation (<ref>) is true. On the other hand the conditions of Theorem <ref> are just sufficient for the relation (<ref>) so in practice relation (<ref>) could be true even with much lower signal to error ratios. To further relax the conditions of our theorem we could allow the estimate to be less sparse. In order to do this we could use H_d instead of H_s with N>d>s>0 where N is the size of the estimated vector. What happens here is a trade off. On the one hand, the result now is less attractive since we have more nonzero coefficients than what is actually needed and that may lead to excessive estimation error that could possibly be avoided. On the other hand, the estimation error of the input to the threshold operator can be greater without risking of loosing an element of support(𝐰) after the application of the operator. The next theorem quantifies the gain in allowable estimation error. Let 𝐰 be a vector in ℂ^N with 𝐰_0=s and 𝐰̂ be an approximation. Let q=min_w_i ≠ 0w_i and d=s+τ with d<N and τ>0 where s, τ, d are integers. Then if 𝐰-𝐰̂_2^2≤ q^2(1-1/τ+2) and 𝐰̂_0 ≥ d, the following will be truesupport(H_d(𝐰̂))⊇support(𝐰)The proof of this theorem, found in the Appendix <ref>, is similar to the previous one. The difference in the result comes from the fact that τ+1 coefficients that are not part of support(𝐰) must have significant magnitudes in 𝐰̂ in order to miss a coefficient of support(𝐰).The analogous inequality of relation (<ref>) for this theorem isSER≥s/(1-1/τ+2)which is less strict as we have expected. Given the last theorem one can overcome the need to have an initialization that is too close to the vector to be estimated. If we have an estimate that has an error 𝐰-𝐰̂_2^2 at most q^2, we can use the hard threshold operator to reduce its sparsity up to a degree that depends on the error without loosing an important coefficient and thus reducing the error in the process. Of course this is a worst case analysis and the conditions are sufficient but not necessary. Therefore in practice we should be able to to use the update rule of (<ref>) without waiting to converge so close to the solution.§.§ Estimating SparsityIn some applications knowing an upper bound on sparsity, the parameter s in our algorithms, may be an acceptable assumption. For example, in echo cancellation one can assume that there will be a small batch of tens of coefficients that are non-zero. In spectrum estimation we can calibrate s based on prior knowledge about how many primary and secondary users of the spectrum are usually utilizing the spectrum. In general however, we would like our algorithm to adapt in different settings and therefore we need to be able to estimate the parameter s in an online fashion. To achieve that we will assume that we have a knowledge of lower bound on q, the minimum magnitude of the non-zero coefficient in the estimated vector. One such lower bound could be the minimum magnitude required to consider the corresponding frequency occupied in the cognitive radio application. Let us call this value q^*. One naive way to estimate the sparsity could be to count the number of coefficients in the current estimate 𝐰(n) that have magnitude greater than q^* and use this as an estimate for the sparsity.Unfortunately, the current estimation may not be suitable to use for sparsity estimation when the underlying spectrum is changing. For example, let us assume that the number of non zero coefficients increases. To increase our estimation of s based on 𝐰(n) at least one coefficient's magnitude would need to go from 0 to above q^* in a single iteration. Waiting for multiple iterations does not help if hard thresholding is used to remove negligible coefficients. But for such a significant jump to happen in a single iteration one would need either a small q^* or a large μ both of which are undesirable as the first one reduces the accuracy of our sparsity estimate and the second one may make the estimation unstable.Instead we will try to approximate the error of our current estimate in order to construct a more suitable vector for the aforementioned procedure. The intuition behind this is that if we track an the error of our current estimate we can then use it to trigger increases in the parameter s when the error increases significantly. Let 𝐰 be once gain the true vector and 𝐰(n) our current estimate. We want to approximate 𝐰̅(n)=𝐰(n)-𝐰. From equation (<ref>) we can get by taking the expectation and assuming that the noise has zero mean that𝔼[e^*(n)𝐱(n)]=-𝔼[𝐱(n)𝐱^H(n)]𝔼[𝐰̅(n)]𝐱(n) correspond to rows of Φ^*. Since they are chosen uniformly at random we know that 𝔼[𝐱(n)𝐱^H(n)]=Φ^*Φ^T=𝐈_N where 𝐈_N is the N× N identity matrix. This equality is based on the properties of the IDFT matrix. Therefore the equation becomes𝔼[e^*(n)𝐱(n)]=-𝔼[𝐰̅(n)]Let 𝐞𝐫𝐫(n) be our approximation of 𝐰̅(n). Ideally, we could take a set of new or even past measurements and calculate e for them in every iteration to approximate the right hand side. This however would be wasteful and it would invalidate the online nature of the proposed algorithms. To avoid that we can reuse the estimations of the errors of the previous iterations. However, as our algorithm progresses, errors that were calculated many iterations ago in the past are not representative for our current estimate so they should be down-weighted compared to errors in recent iterations.To overcome this we can take an exponentially weighted window average of the errors. Let λ∈ (0,1] be the forgetting factor of the window and 𝐛(n)=e^*(n)𝐱(n). Then we can write the following equationsκ_n+1 = λκ_n +1𝐞𝐫𝐫(n+1)= ( 1 -1/κ_n+1) 𝐞𝐫𝐫(n) -1/κ_n+1𝐛(n)where 𝐞𝐫𝐫(0) is all zeros andκ_0 is zero as well.In the end we will get a 𝐰'(n)=𝐰(n)-𝐞𝐫𝐫(n) and we will compare each coefficient magnitude and compare it to the threshold q^*. The number of coefficients that pass this test is the estimate of s. Optionally we can use the coefficients that pass the test as the support of 𝐰(n+1) in the next iteration in order to reduce the computational effort.The advantage of using this process instead of operating directly on𝐰(n) is that we can increase the importance of errors only for sparsity estimation and thus we avoid making our estimate unstable. In general we can even scale the importance of the error correction 𝐰'(n)=𝐰(n)-ξ𝐞𝐫𝐫(n)where ξ is a positive real number. Holding q^* fixed we can increase ξ to make our sparsity estimation more sensitive to the error estimate.§ EXPERIMENTATIONIn this section we will compare the performance of the various algorithms discussed previously. Let us first define the signals on which we will compare these algorithms. The signals of interest are going to be sums of sines affected by additive white Gaussian noise. Specifically the signals of interest here will have the formg(n)= ∑_i=1^k A_isin(2π f_i t(n)) + v(n)where k will be the number of the signals added, f_i is the randomly chosen frequency of each sine, A_i is the amplitude of each sine wave and v(n) is the white zero mean noise. Therefore the spectrum of these signals will be sparse with s=2k non-zero coefficients. The sampling positions t(n) are spread uniformly in a time interval T and the corresponding sampling frequency is equal to the Nyquist frequency. This results in a vector of N samples per time interval T out of which we will sample M of those. Here we will assume for simplicity that A_i=1.The first thing we would like to show is that using sparsity aware techniques for spectrum estimation is a necessity when we are undersampling. We will therefore compare the spectrum estimations of the Hard Threshold LMS and the classical LMS. We will use the sum of k=10 sine waves of length N=1000 samples out of which we collect only M=300 samples corrupted by noise so that the SNR is equal to 20db. In order for the algorithms to converge we will make 10 iterations over the data. For the Hard Threshold LMS (HARD-LMS) algorithm we will use s=20 and we will refrain from thresholding during the first pass over the data. For both HARD-LMS and LMS we will use μ=1. The results can be seen in Figure <ref>. As we can clearly see the LMS algorithm does not converge to the true sparse vector that generated the measurements but simply to one of the many non-sparse solutions. In contrast HARD-LMS identified the support of the spectrum perfectly and the error is minimal compared to the one of LMS.Moreover, we would like to show experimentally how the parameter s in our Hard Threshold LMS algorithm influences the speed of convergence as well as the steady state error. We set N to be equal to 1000 and M=200 leading to a one to 5 undersampling ratio. We set k to be equal to 10 and set the noise power so that the SNR of the observed samples is equal to 20db. We collect the M samples and repeat them K=100 times to test the convergence of the algorithms. We repeat the whole process of choosing the different frequencies f_i and the random M samples for R=200 times. The algorithms that are compared are the following: The Hard Threshold LMS algorithm for values s equal to 20 (HARD-20), 40 (HARD-40) and 80 (HARD-80) as well as the version of the Hard Threshold LMS with sparsity estimation (HARD-EST). For the sparsity estimation we use λ=0.99, q^* equal to one tenth of the magnitude of any of the non zero coefficients of the spectrum (all have equal magnitude in this case) and ξ=1. For all those algorithms we refrain from hard thresholding for the first 2M samples so that we get a good enough approximation. Additionally, for all algorithms μ is set equal to 1. We also include the standard LMS (LMS) as a baseline comparison. The results we get from Figure <ref> are quite interesting. What is evaluated is the relative Mean Square Error (r-MSE). For every run of the algorithm and for every iteration we calculater-MSE=𝐰-𝐰(n)^2/𝐰^2and then we take the average performance in db. As we can see selecting s being exactly equal to the true sparsity is not always optimal. The algorithm for s=20 quickly converges to a suboptimal solution with high steady state error. This is because the algorithm has made a wrong estimation of the spectrum's support. In contrast allowing more non-zero coefficients allows the algorithm to include the true support of the spectrum as well as some superfluous coefficients. This allows both s=40 and s=80 to achieve much lower steady state errors. However, increasing the parameter s will tend to significantly decrease the speed of convergence of the algorithm. On the other hand the hard thresholding algorithm with sparsity estimation by making better estimates of the true spectrum and using a conservative magnitude threshold gradually decreases the sparsity estimate in order to smoothly and quickly converge. This aligns with what we proved in the previous section especially with Theorem <ref>. Of course the classical LMS algorithm had no hope of finding the true spectrum as the problem is undetermined and LMS gets stuck in a non sparse spectrum that could give the observed measurements. Since HARD-EST achieved the best performance compared to all other methods we will compare it with other online sparsity aware algorithms from the literature.Specifically we will also compare with Zero Attracting LMS (ZA-LMS) and Reweighted Zero Attracting LMS (RZA-LMS) from <cit.> as well as with ℓ_0-LMS from <cit.>. We will set the parameter ρ of ZA-LMS and RZA-LMS equal to 0.005 and ϵ=2.25. For the ℓ_0-LMS we will setβ=0.5 and κβ=0.005. We will also include the Selective Zero Attracting LMS (SZA-LMS) that we proposed in this paper using the true sparsity s=20 and ρ=0.005. Additionally, the proposed hard thesholding scheme with sparsity estimation can be combined with other more complicated update rules to further improve performance. So for this experiment we will combine it with the update rule of ℓ_0-LMS using the same parameters to show that we can get improved performance over each method alone. We shall call this algorithm HARD-ℓ_0 and its update rule will be𝐰(n+1)=H_s(𝐰(n)+μ e^*(n)𝐱(n)-ρsgn(𝐰(n))e^-β𝐰(n)) where s is estimated the same way as for HARD-EST. For HARD-EST we will refrain from hard thesholding for the first M samples and for HARD-ℓ_0 for the first 2M samples. The experimental settings are the same as in the previous experiment.The results can be seen in Figure <ref> where we show again the r-MSE. Clearly, we can see that all algorithms manage to reach nearly the same level of r-MSE after some iterations with SZA-LMS and ℓ_0-LMS achieving a slightly smaller r-MSE than the other methods. So it makes sense to compare them in terms of speed of convergence. The fastest convergence belongs to SZA-LMS. SZA-LMS has ground truth knowledge of the sparsity of the vector just like the hard thresholding algorithms in the previous experiments but uses it slowly but steadily in order not to reduce coefficients of the true support to 0.Then we have HARD-ℓ_0 which combines the quickly convergent update rule of ℓ_0-LMS with hard thresholding improving the convergence speed of an already fast algorithm like ℓ_0-LMS.Then ℓ_0-LMS with an update rule that uses two parameters to tune the amount of zero attraction to use for each coefficient manages to converge faster than the simpler HARD-EST algorithm. HARD-EST then manages to converge faster than RZA-LMS. Finally the simple ZA-LMS algorithm fails to achieve a low steady state error.The third experiment that we will present has to with the robustness of the proposed procedures with varying degrees of undersampling. We evaluate the sparsity aware algorithms for M=100, 200, … 1000 samples where 1000 samples corresponds to the full measurement of the signal. In each setting we take 50 instantiations of the random sampling procedure. Then we calculate the steady state r-MSE after K=50 iterations over all the available measurements. The results are shown in Figure <ref>. We compare once again the same algorithms with the same parameters as in the previous experiment. We can clearly see that reducing the number of samples to 100 which corresponds to a 1 over 10 undersampling ratio is prohibitive for all algorithms except maybe SZA-LMS which has ground truth knowledge. However, once we get to 200 samples, which was the case in the previous experiment all algorithms improve their predictions considerably. Adding even more samples leads to better performance although with diminishing returns. One pattern that may seem counter-intuitive is that the hard thresholding algorithms, HARD-EST and HARD-ℓ_0, manage to outperform all other methods by a small margin after 200 samples which is in contrast to what we saw in the previous experiment. The reasoning behind this is that HARD-EST and HARD-ℓ_0 have no misalignment with the ground truth for the coefficients that are exactly zero. In contrast for the other methods these weights oscillate around zerodue to the zero attraction term for the same reasons we analyzed for the case of SZA based on Equation (<ref>). This difference in performance is quite small so it is only observable when HARD-EST and HARD-ℓ_0 have reached their optimal performance. In the fourth experiment we are going to validate that HARD-EST is capable of tracking changing sparsity patterns in real time. In this experiment, the incoming signal will change over time. At first, the incoming signal consists of 10 sine waves just like before. The pattern of N=1000 samples is repeated 150 times. Then the incoming signal is augmented with another 10 sine waves of different frequency. Then the new pattern is repeated for 150 times. The incoming signal is split in non overlapping windows of N samples and we randomly sample M=200 measurements corrupted by additive noise in each window. The SNR is 20db. To help HARD-EST perform well in scenarios where the incoming signal is changing abruptly we are going to change the algorithms configuration. We are going to set λ=0.98, q^* equal to one hundredth of the magnitude of any of the non zero coefficients of the spectrum (again all the coefficients have equal magnitude) and ξ=20.The smaller λ allows us to forget previous error estimates more quickly whereas the combination of the smaller q^* and the higher ξ allows us to adapt more quickly to changes in the sparsity of the spectrum. We can see the results in Figure <ref>. The algorithm converges very close to the true spectrum using the new samples it gets in every window. When the change in the spectrum happens the estimation is initially far away from the new spectrum. Then, the increased error estimates trigger the increase of the estimated sparsity from 20 towards 40 non zero coefficients. This allows the estimate to adapt to the less sparse spectrum and eventually converge to it. The r-MSE in the end is higher than before the spectrum change but this is to expected since the spectrum now is less sparse. To understand the effect of the parameter ξ in Equation <ref> and the significance of our sparsity estimation procedurewe add an additional algorithm HARD-EST-SIMPLE which is the same as HARD-EST with the only difference being that for HARD-EST-SIMPLE we set ξ=0. In this setting the sparsity estimation is successful in the first half of the simulation yielding the same approximation error with HARD-EST. However, in the second half while HARD-EST manages to increase its estimate s to 40, HARD-EST-SIMPLE does not manage to adapt resulting in s being equal to 20 in the second half as well and in an r-MSE of -3 db. Therefore, it is clear that, when the underlying patterns of sparsity are changing, setting a positive value for ξ is crucial for the adaptation of the sparsity estimation. § CONCLUSIONIn this paper we studied the problem of online sparse spectrum estimation for cognitive radio applications using sub-Nyquist sampling rates. To solve the problem, we analyzed the properties of two algorithms that try to minimize the squared error at each iteration while maintaining the ℓ_0 norm of the estimated vector under a predefined threshold s. Moreover, we analyzed the convergence properties of the Selective Zero Attracting LMS as well as the properties of the Hard Thresholding operator. Specifically, we proved that if our current estimation is close enough to the solution we can use the Hard Thresholding operator to reduce the error without risking to loose an important coefficient of the spectrum especially when we allow the operator to use more non-zero coefficients. Additionally, we proposed a way to estimate in an adaptive way the parameter s so that the estimation can gradually become sparser without misspecifying the support of the estimated spectrum. Further, in the experimentation section we analyzed the importance of the parameter s for the steady state error as well as the speed of convergence. Then we compared our results with other online sparsity aware algorithms in the literature. We also showed that the two proposed algorithms have robust performance even when the sampling rate is low and that they can produce even better estimates when the number of samples increases. Finally, we showed experimentally that the proposed sparsity estimation technique is robust to signal changes. Of course spectrum estimation for cognitive radio applications is only one of the many possible applications of the proposed algorithms. Obviously an a priori estimation of the sparsity of the estimated vector may not be available in all use cases, even though we showed that this estimate must not be exact in order to actually take benefit. However, there are other use cases where the algorithms proposed here could make a difference. The standard LMS algorithm has been used in many adaptive machine learning tasks like neural network training and others as discussed in <cit.> so taking advantage of sparsity could be advantageous. For example, in the case of training a perceptron with an abundance of available features one could begin training with all the features but then proceed to use one of the proposed algorithms to impose feature selection through sparsity. By increasing the imposed sparsity one can then train several classifiers and then compare them using criteria like the Bayesian information criterion.§ PROOF OF THEOREM <REF>Let us define 𝐰̅(n) as the difference between the estimation 𝐰(n) and the true vector 𝐰. Subtracting 𝐰 from both sides of the equation (<ref>) gives𝐰̅(n+1) =𝐰(n)+μ e^*(n)𝐱(n)-𝐰 -ρ P_s(𝐰(n))+v(n)=𝐰̅(n)+μ e^*(n) 𝐱(n) -ρ P_s(𝐰(n))+v(n)After some calculations, which are the same as in the case of the classical LMS, we have thate^*(n)𝐱(n)=-𝐱(n)𝐱^H(n)𝐰̅(n)+v(n)𝐱(n)Taking the mean under the independence assumptions made and given that the observation noise mean is zero will yield𝔼[e^*(n)𝐱(n)]=-𝐑_x𝔼[𝐰̅(n)]Then from equation (<ref>) we obtain 𝔼[𝐰̅(n+1)]=(𝐈_N-μ𝐑_x)𝔼[𝐰̅(n)]-ρ𝔼[P_s(𝐰(n))]where 𝐈_N is the N× N identity matrix. Given the bound in (<ref>) the algebraically largest eigenvalue of 𝐈_N-μ𝐑_x is less than one. Further the term induced by the penalty is bounded by the vectors -ρ1 and ρ1 where 1 is the vector of ℝ^N whose every element is one. Thus we can conclude that the𝔼[𝐰̅(n)] converges and as a result so does𝔼[𝐰(n)]. Therefore the algorithm provided by equation (<ref>) converges. The limiting vector cannot be found in a closed form but is guaranteed to be the solution of equation (<ref>).§ PROOF OF THEOREM <REF>The proof will be completed in three distinct cases. (i) First, we assume that H_s(𝐰̂)_0<s which can be true only if 𝐰̂_0<s. We can easily see that, since𝐰_0=s, there is at least one coefficient index i such that ŵ_i=0 and w_i≠ 0, which from the hypothesis also means that w_i≥ q. As a result we have that𝐰-𝐰̂_2^2 ≥w_i-ŵ_i^2= w_i^2 ≥ q^2which contradicts the hypothesis; so this case is impossible. (ii) Now we have that H_s(𝐰̂)_0 = s. Let us assume that relation (<ref>) does not hold. Then since the two sets have the same number of nonzero elements, it is clear that there is a coefficient index ℓ∈support(𝐰) but ℓ∉support(H_s(𝐰̂)) and a coefficient index k so that k ∈support(H_s(𝐰̂)) but k ∉support(𝐰). We directly know that w_k=0 and that w_ℓ≥ q. We can also deduce that ŵ_k > ŵ_ℓ since k belongs in support(H_s(𝐰̂)) but ℓ does not. Then, for the error norm we have 𝐰-𝐰̂_2^2 ≥w_k-ŵ_k^2+w_ℓ-ŵ_ℓ^2Since w_k-ŵ_k^2 = ŵ_k^2 > ŵ_ℓ^2, it follows that 𝐰-𝐰̂_2^2 > 2ŵ_ℓ^2-w^*_ℓŵ_ℓ -w_ℓŵ^*_ℓ + w_ℓ^2 Therefore we can also write that𝐰-𝐰̂_2^2 >min_ŵ_ℓ∈ℂ2ŵ_ℓ^2-w^*_ℓŵ_ℓ -w_ℓŵ^*_ℓ + w_ℓ^2 The minimum value of the RHS is attained for ŵ_ℓ=w^*_ℓ/2 and equals w_ℓ^2/2; hence 𝐰-𝐰̂_2^2 >w_ℓ^2/2≥q^2/2 This once again contradicts the hypothesis and so relation (<ref>) is true in this case. (iii) Finally, we assume that H_s(𝐰̂)_0 > s. This can happen only if there are ties for the s largest absolute values in 𝐰̂. Let us denote as B the set of tying coefficients, A= support(H_s(𝐰̂)) ∖ B and finally C=(support(H_s(𝐰̂))^c. It is evident that A≤ s-1. We shall prove that this case is impossible. There are two subcases: (a) B ∩support(𝐰)= ∅. Since A≤ s-1 and w_0=s, support(𝐰) must have an element in common with C. Let us call that element ℓ. Let us also take an element k from B.Then just like in the second case ŵ_k > ŵ_ℓ since k belongs in support(H_s(𝐰̂)) but ℓ does not. Following the rest of the steps in case (ii) we reach a contradiction. (b) B ∩support(𝐰)≠∅. Let ℓ a common element of the two sets. Since H_s(𝐰̂)_0 >𝐰_0 there is an element k so that k ∈support(H_s(𝐰̂)) but k ∉support(𝐰). Since ℓ is one of the indexes tying for the last spot, we have ŵ_k≥ŵ_ℓ. Following the steps of case (ii) yields 𝐰-𝐰̂_2^2 ≥w_ℓ^2/2≥q^2/2 and therefore we get a contradiction.§ PROOF OF THE THEOREM <REF>Let us assume that relation (<ref>) does not hold. Just like in the proof of Theorem <ref> it is clear that there is a coefficient index so that ℓ∈support(𝐰) but ℓ∉support(H_d(𝐰̂)). This time however the set support(H_d(𝐰̂)) has at least d=s+τ elements but support(𝐰) has at most s-1 elements that could exist insupport(H_d(𝐰̂)). As a result we are sure that there are at least τ+1 indexes k_i so that k_i ∈support(H_s(𝐰̂)) but k_i ∉support(𝐰). Once again we know that w_k_i=0 and that w_ℓ≥ q and we can deduce that ŵ_k_i > ŵ_ℓ since k_i exists in support(H_d(𝐰̂)) but ℓ does not. Like in the the proof of Theorem <ref> we can deduce about the error norm that𝐰-𝐰̂_2^2 ≥∑_i=1^τ+1w_k_i-ŵ_k_i^2+w_ℓ-ŵ_ℓ^2We bound the first term just like in the previous proof so that it becomes∑_i=1^τ+1w_k_i-ŵ_k_i^2 = ∑_i=1^τ+1ŵ_k_i^2≥ (τ+1) ŵ_ℓ^2Thus, we end up𝐰-𝐰̂_2^2 > (τ+2)ŵ_ℓ^2-2w_ℓŵ_ℓ + w_ℓ^2Taking the minimum on the right side with respect to ŵ_ℓ will lead once again to finding the minimum value of a quadratic function. The minimum is found for ŵ_ℓ= w_ℓ/τ+2 and equals to w_ℓ^2(1-1/τ+2); hence𝐰-𝐰̂_2^2 >w_ℓ^2(1-1/τ+2) ≥ q^2(1-1/τ+2)which once again contradicts the hypothesis so the proof is completed. § ACKNOWLEDGMENTWe wish to thank the anonymous reviewers whose constructive comments helped us improve this paper.IEEEtran | http://arxiv.org/abs/1707.08291v3 | {
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Arithmetic Circuits for Multilevel QuditsBased on Quantum Fourier Transform A. Pavlidis and E. Floratosempty 1ARITHMETIC CIRCUITS FOR MULTILEVEL QUDITS BASED ON QUANTUM FOURIER TRANSFORM ARCHIMEDES PAVLIDIS Department of Informatics, University of Piraeus =10pt 80, Karaoli & Dimitriou str., GR 185 34, Piraeus, Greece Department of Informatics and Telecommunications, National and Kapodistrian University of Athens =10pt Panepistimiopolis, Ilissia, GR 157 84, Athens, Greece =10pt e-mail: [email protected] EMMANUEL FLORATOS Department of Physics, National and Kapodistrian University of Athens =10pt Panepistimiopolis, Ilissia, GR 157 84, Athens, Greece =10pt Institute of Nuclear and ParticlePhysics, N.C.S.R. Demokritos =10pt 27, Neapoleos str., Agia Paraskevi, GR 153 41, Athens, Greece =10pt e-mail: [email protected] We present some basic integer arithmetic quantum circuits, such as adders and multipliers-accumulators of various forms, as well asdiagonal operators,which operate on multilevel qudits. The integers to be processed are represented in an alternative basis after they have been Fourier transformed. Several arithmetic circuits operating on Fourier transformed integers have appeared in the literature for two level qubits. Here we extend these techniques on multilevel qudits, as they may offer some advantages relative to qubits implementations. The arithmetic circuits presented can be used as basic building blocks for higher level algorithms such as quantum phase estimation, quantum simulation, quantum optimization etc., but they can also be used in the implementation of a quantum fractional Fourier transform as it is shown in a companion work presented separately.§ INTRODUCTIONThe common representation of the elementary quantum information is the qubit, where its state is a superposition a|0⟩+b|1⟩ which belongs to a two-dimensional Hilbert space with two basis states |0⟩ and |1⟩ known as the computational basis. A quantum computer is a finite dimensional quantum system composed of aqubits collection, performing various unitary operations on the qubits (quantum gates) and quantum measurements. Accordingly, there is a correspondence between a qubit and a classical bit, in the sensethat the basis states of a qubit follow the binary logic. We can extend this correspondence to multivalued logic instead of two values only by enlarging the dimension of the elementary Hilbert space used. The qudit is a generalization of the qubit to a larger Hilbert space of dimension d>2. The state of a qudit is a superposition a_0|0⟩+a_1|1⟩+⋯ + a_d-1|d-1⟩, where |0⟩, |1⟩, …|d-1⟩ are the computational basis states. Qutrit is a special name for the cased=3, while ququart corresponds to d=4. In many cases, the employment of a multivalued quantum logic is more natural. E.g. in ion traps we could exploit more than two energy levels. Multiple laser beams could be used to manipulate the transitions between these levels <cit.>. Working with qudits instead of qubits may offer some advantages. The required number of qudits is smaller by a factor log _2d than the corresponding number of qubits for the same dimension a quantum computer has to explore. E.g. the dimensions of a composite system of n qubitsis 2^n, while the same dimensioncan be reached with only log_d2^n=log_22^n / log_2d=n/log_2d qudits. Such as reduction of the required number of physical carriers of quantum information is advantageous, considering the difficulty of reliably controlling a large number ofcarriers. Also, when fewer quantum information carriers are used, a decrease in the overall decoherence is expected and this fact favors the scalability issues <cit.>. Another advantage, which is also related to the adverse effect of decoherence, is that fewer multilevel qudit gates are required to construct a quantum circuit implementing a given unitary operation compared to the case of using two-level gates <cit.>. Fewer gates reduce the number of steps needed to complete the circuit operation (circuit depth), and consequently less errors are accumulated during the overall operation of the circuit. Even so, protection of quantum information against environmental interaction is inevitable. Quantum error correcting codes and fault tolerant gate constructions to combat decoherence on multilevel qudits have been proposed and they are similar to the ones used for the qubit case <cit.>. At a higher level, generalizations of known quantum algorithms and circuits using d-level qudits may offer improvements with respect to their qubits implementation counterparts. E.g., quantum phase estimation, which is the core part of Shor's algorithm <cit.> and also it is used in quantum simulation <cit.>, is improved in terms of success probability when multilevel qudits are incorporated <cit.>. Multiple-valued version of Deutsch-Josza algorithmhas been reported in <cit.> while an implementation proposal for five level superconducting qudit appeared in <cit.>. Qudits version for Grover's algorithm <cit.> has been reported in <cit.>. The high dimensional Deutsch-Josza algorithm may find applications in image processing, while the high dimensional Grover's algorithm offers a trade-off between space and time. An assortment of quantum gates operating on qudits have been proposed or experimentally realized on various technologies. Single and two qudit d-level gates proposed in <cit.> for the ion trap technology. Single qudit gates for d=5 implemented in superconducting technology and used to emulate spins of 1/2, 1 and 3/2 in <cit.>. Proposals for single and two qudit gates based on superconducting technology appeared in <cit.>. Single qudit gates based on optical technology reported in <cit.>. Three dimensional entanglement between photons observed in <cit.>.In this work we present some quantum arithmetic circuits operating on d-level qudits by extending results given in prior works <cit.>. These circuits exploit the quantum Fourier transform and various single qudit and two qudit rotation gates to perform the desired calculations. Processing in the Fourier domain may offer some advantages related to speed <cit.> and robustness to decoherence <cit.>. Among the proposed circuits are various versions of adders (adder with constant, generic adder, adder with constant controlled by single qudit) and multipliers(multiplier with constant and accumulator, multiplier with constant). Such circuits are useful in many quantum algorithms, e.g. quantum phase estimation, quantum simulation. The increased interest in quantum information processing exploiting d-level qudits, both in theoretical and experimental aspects, was one of the stimulation for this work.However, the main motivation was the particular application targeted by the quantum circuits presented in this manuscript, which is a new definition of the fractional Fourier transform. Unitary operations on high dimensional d-level qudits fit more naturally for this specific application because the proposed fractional Fourier transform operates on a Hilbert space of dimension d^n, where d ≠ 2 is a prime. The development of the quantum fractional Fourier transform and its implementation on qudits is presented in a separate work <cit.>.The rest of the paper is organized as follows: A short background about design and synthesis of qudits quantum circuits is given in section <ref>. The elementary and basic qudit gates used in the proposed designs are given in section <ref>. The quantum Fourier transform definition and its circuit for q qudits of d levels is presented in section <ref>. Section <ref> introduces integer arithmetic circuits like adder with constant, adder of two integers, controlled adder with constant, multiplier with constant and accumulator, and multiplier with constant. All of the arithmetic units accept one of their operands after it has been Fourier transformed.In section <ref> a method to implement a diagonal operator on q qubits is analyzed where the diagonal elements are some powers of roots of unity. A quantum multiplier of two integers is introducedand then a quantum squarer is built upon this multiplier. It is demonstrated how to introduce relative phases between the basis states of superposition which depend quadratically on the index of the basis state. It can be generalized for a function that is polynomial in the states index. This operation is a necessary part ofthe quantum fractional Fourier transform presented in the companion article and also it may find other applications, such as quantum simulation algorithms and Grover's search algorithm. Appendix A gives the decomposition of a three qudits rotation gate introduced in section <ref>. Complexity analysis in terms of quantum cost, depth and width is reported in section <ref>. In Appendix B we discuss how it is possible to use a discrete library of components to approximate the proposed designs and the impact to cost and depth. A discrete library of gates is necessary if fault tolerance is to be incorporated. Finally, we conclude in section <ref>.§ BACKGROUND AND RELATED WORK The construction of a complex quantum circuit operating on multilevel qudits is based on the selection of a set of elementary qudit gates and their interconnection so as to achieve the target operation. A multilevel gate operates on a single qudit, on two qudits or more qudits. A single d-level qudit gate is represented by a unitary matrix U of dimensions d× d. It transforms an initial qudit state |ψ⟩ _in to |ψ⟩ _out=U|ψ⟩ _in. As an example consider thegeneral superposition qutrit state |ψ⟩ _in=a|0⟩+b|1⟩+c|2⟩. The application of the gate U= [ 0 0 1; 1 0 0; 0 1 0 ] =|1⟩⟨ 0| + |2⟩⟨ 1|+ |0⟩⟨ 2| on this state results to the state |ψ⟩ _out= c|0⟩+a|1⟩+b|2⟩.Two qudit gates operate on states of two qudits which are d^2 dimensional, so their representing unitary matrices have dimensions of d^2 × d^2. Two single qudit gates V_1 and V_2 operating on two different qudits can be seen as a two qudit gate which is their tensor product U=V_1⊗ V_2. However, not every two qudit gate can be decomposed as a tensor product of two single qudit gates, in which case we have an entangling gate.Consecutive application of single, two or more qudit gates to a collection of q qudits results in a quantum circuit which is represented by a unitary matrix of dimensions d^q × d^q. The design of a quantum circuit is the procedure of interconnecting various elementary gates so as to fulfill the given specifications. These specifications are given in the form of a unitary matrix or the relationship between the desired input-output state relation in the computational basis.It is proven that single qudit gatesand a two qudit gate alone are adequate to form a universal set of gates, provided that the two qudit gate is an entangling gate <cit.>. A universal set of gates can be used to approximate any targetquantum circuit with arbitrary precision. Various sets of qudit gates (gate libraries) and methods to exploit them to build more complex unitaries have been introduced in the literature. The library used in<cit.> consists of single and two qudit gates with continuous parameters and the synthesis method is based on spectral decomposition of the target unitary matrix. Cosine-Sine decomposition is another method used in<cit.>. A discrete set of single qudit gates and a single two qudit gate is used in<cit.>to synthesize the large unitary matrix using QR decomposition. In <cit.> a different two qudit gate and a set of single qudit gates is used along with quantum Shannon decomposition to synthesize the target unitary. The previous methods and results are similar to the two-level qubits synthesis cases.It is proven that the cost of the resulting circuit in terms of two qudit gates isupper bounded by O(d^2n) where n is the qudits number <cit.>. Thus, these automated methods are suitable only for small quantum circuits due to the exponential cost increase. When the target circuit is an arithmetic or logic block where its unitary matrix is a permutation matrix consisting of 0 and 1 elements, then multiple-valued reversible synthesis methods could be applied. These methods are extension of the binary reversible logic case and may be applied to a specific value of d, e.g. <cit.> (d=3), or applied to any value of d <cit.>. Similarly to the quantum synthesis case, these algorithms are not suitable for large circuits.As many algorithms widely use quantum arithmetic blocks like adders or multipliers recurrently, it is crucial to have available efficient arithmetic and logic blocks. Ad hoc design of such blocks usually offers better results compared to the automated synthesis methods. One can exploit the iterative and regular structure of these arithmetic blocks or extend known classical designs to the quantum case. A diversity of ad hoc designed quantum arithmetic and logic circuits for two-level qubits can be found in the literature <cit.>, but few (usually adders) are known for multilevel qudits and also they are mostly designed for a specific value of d. In contrast, the proposed designs are parametrized for any value of d.One of the first ternary (d=3) quantum adder for 3-inputs only appeared in <cit.> as an application example of the proposed synthesis method. Ternary quantum adders/subtractors ad hoc designed for any number of inputs is given in <cit.>. In <cit.> a ternary extension of the well known VBE ripple-carry adder <cit.> is reported. Quaternary (d=4) comparators proposed in <cit.>. Improved designs of ternary ripple carry and carry look-ahead adders along with modifications that lead to subtractors and comparators are given in<cit.>. The previous ternary ripple carry adder is a modification of the CDKM binary quantum adder appeared in<cit.>and it has also a depth of O(n) using one ancilla qutrit. Similarly, the previous ternary carry look-ahead quantum adder is an extension of the DKRS binary quantum adder appeared in <cit.> and thus it offers a depth O(log(n)) using O(n) ancilla qutrits. Several of the previous multilevel qudits (usually qutrits) designs are modifications of binary quantum adders. The gates libraries used are differentiated among each design. This is justifiable as the implementation technology for multilevel qudits is far apart to be considered matured. However, gates of one library can be expressed as gates of another one, provided that the libraries are universal. Diagonal operator circuits on qubits or qudits don't change the absolute value of the amplitudes of a superposition, but rearrange their relative phases. Such circuits are useful in quantum algorithms <cit.> like quantum optimization, quantum simulation, Grover's search etc.Synthesis methods for diagonal unitary matrices of two-level qubitshave been developedfor diagonals of special structure <cit.> or for any diagonal <cit.>. Recently, diagonal synthesis for multilevel qudits reported in <cit.>. In general, the synthesis cost is related to the dimensionality of the Hilbert space covered (that is exponential in the number of the qubits or qudits) and the number of the distinct phases of the diagonal. In this work, based on ideas of <cit.> and the arithmetic circuits designed,we develop a diagonal operator circuit which has a special structure, that is the phases are quadratic functions of the coordinates, with polynomial cost and depth. Using same techniques, other powers, instead of quadratic, can be achieved.The gates used in our proposed designs are the ones introduced in <cit.> where also physical implementation directions are given. The proposed designs use extensively various rotation gates. As the rotation angles of the gates vary with the size of the circuit and also small angles are required, implementation and fault tolerance issues are addressed in Appendix B, using results of the binary quantum case. Many of the arithmetic multilevel quantum designs of this manuscript are direct extension or modifications of binary quantum designs appeared in<cit.> which use QFT before applying the rotation gates to one of the two integer operands and then applying the inverse QFT to bring back the result in the computational basis. The following arithmetic units are presented:* Adder of two integers of q qudits with depth of O(q) and width of 2q qudits(Subsection <ref>). * Adder of an integer of q qudits with a constant integer. Its depth is O(q) including the direct and inverse QFT blocks or O(1) without the QFT blocks. Itswidth is q qudits(Subsection <ref>). * Single state controlled adder of an integer of q with a constant. The adder is enabled if the control qudit is in a particular basis state of the different d possible states, otherwise it acts as an identity. Its depth is O(q) and its width is q+1 qudits(Subsection <ref>). * Generalized controlled adder of an integer of q qudits with a constant. It adds a multiple of the constant to the integer. The multiple depends on the state of control qudit, being between 0 and d-1. Its depth is O(q) and its width is q+1 qudits (Subsection <ref>). * Multiplier with constant and accumulator.It multiplies an integer of q qudits with a constant and adds the product to a second integer of q qudits. Its depth is O(q) and its width is 2q qudits (Subsection <ref>). * Multiplier with constant. It multiplies an integer of q qudits with a constant provided that the constant is relative prime with d^q, which is always the case when p is prime.Its depth is O(q) and its width is 2q qudits of which q qudits are ancilla initialized to the zero state and then are reset back to the zero state (Subsection <ref>). * Multiplier of two integers and accumulator. It multiplies two integer of q qudits and adds the product to a third integer of q qudits. Its depth is O(q^2)and its width is 3q qudits (Subsection <ref>). * Squarer/Multiplier with constant/Accumulator. It performs the transform |x⟩ |z⟩→ |x⟩|z + γ x^2⟩, where γ is the integer constant. Its depth is O(q^2) and its width is 4q qudits (Subsection <ref>). * General diagonal operator. It operates diagonally on a general superposition state ofq qudits and changes the phases of the superposition amplitudes by applying the matrix ∑_k=0^d^q-1 e^i2π/d^qf(k) |k⟩⟨ k|, where f(k) is a function of k. The specific diagonal operator presented here is based on the previous squarer and some other blocks as it applies the function f(k)=γ k^2. It can be generalized for other powers of k or even for polynomial functions on k by exploiting similar techniques. It has a depth of O(q^2) and its width is 4q qudits of which 3q qudits are ancilla (Section <ref>).Detailed complexity analysis in terms of quantum cost and depth is given in section <ref>, where the parameter d enters the previous rough approximations. This is because many gates like the basic rotation gates used and introduced in subsection <ref> are synthesized using more elementary gates with a cost (and consequently depth) which depends on the dimension d of the qudits.§ ELEMENTARY AND BASIC GATES ON QUDITSWe followed a hierarchical bottom-up approach to design the arithmetic circuits. At the lowest level, elementary gates operating in a two dimensional subspace of the d-dimensional space of a qudit are used. Upon them, more complex gates (which are basic for the designs) operating in the whole d-dimensional space are built. Some of the basic gates are reported in <cit.>, while others like the generalized controlled and doubly controlled rotation gates are introduced here (subsection <ref>, subsection <ref>and Appendix A ).§.§ Generalized X gatesThe X^(jk) gates <cit.> operate on a two-dimensional subspace of a d-level qudit by exchanging the basis states |j⟩,|k⟩, and leaving intact the other basis states, thus they are a generalization of the wellknown X gate for qubits which exchanges the basis states |0⟩ and |1⟩. They are defined by the d× d matrix[ X^(jk)=|j ⟩⟨ k| + |k ⟩⟨ j| + ∑_l n=0 n ≠ j n ≠ k ^d-1 |n ⟩⟨ n|j,k=0 … d-1 ]It holds that X^(jk)= X^(kj), so there are d(d-1)/2 different such gates in this family. §.§ Rotation gates of two levelsThese gates perform a rotation on a two dimensional subspace <cit.> of a d-level qudit and are defined as[ R_a^jk(θ)=exp(-iθσ _a^(jk)/2 ),0 ≤ j,k ≤ d-1,a ∈{x,y,z} ] where σ_x^(jk)=|j⟩⟨ k| +|k⟩⟨ j|, σ_y^(jk)=-i|j⟩⟨ k| +i|k⟩⟨ j| and σ_z^(jk)=|j⟩⟨ j| - |k⟩⟨ k| for j,k=0… d-1 arematrices of dimensions d × d. Parameter θ is the rotation angle, while i=√(-1).§.§ Generalized Controlled X gatesThe GCX_(m)^(jk) gates are generalization in the qudits of the CNOT gates acting on qubits <cit.>. Thus, they are gates which operate on a control and a target qudit. A GCX gate has three parameters, m, j and k, which define its operation. A GCX_(m)^(jk) acts like a X^(jk) on the target qudit iff the control qudit is on the basis state |m ⟩. Consequently, the definition matrix of such a gate is block diagonal with dimension d^2× d^2 consisting of dblocks of d× d dimensions each and it is given by [ GCX_(m)^(jk)= |m ⟩⟨ m| ⊗( |j ⟩⟨ k| + |k ⟩⟨ j| +∑_ln=0n ≠ jn ≠ k^d-1 |n ⟩⟨ n| ) +∑_ln=0n ≠ m ^d-1 |n ⟩⟨ n| ⊗ I_d j,k,m=0 … d-1 ] where I_d is the identity matrix of dimensions d× d. Equation (<ref>) can be equivalently written as GCX_m^(jk)= diag(I_d,I_d,…,m-th blockX^(jk) ,…,I_d)§.§ Hadamard gate The Hadamard gate H^(d)on d-level qudits is defined by the matrix H^(d)=1/√(d)[1111 ;1 e^i2π1/d e^i2π2/d⋯ e^i2πd-1/d;1e^i2π 2 1/de^i2π 2 2/d⋯e^i2π 2 d-1/d;⋮⋮⋮⋱⋮;1e^i2π(d-1)1/d e^i2π (d-1)2/d⋯ e^i2π (d-1)d-1/d ] =1/√(d)[1111 ;1 e^i2π(0.1) e^i2π(0.2)⋯ e^i2π(0.d-1);1 e^i2π 2(0.1)e^i2π 2 (0.2)⋯e^i2π 2 (0.d-1);⋮⋮⋮⋱⋮;1e^i2π(d-1)(0.1) e^i2π (d-1)(0.2)⋯ e^i2π (d-1)(0.d-1) ]In the above equation the notation (0.n) is the fractional representation of n/d in the base-d arithmetic system. The application of the H^(d) gateto a basis state |j ⟩is shown below H^(d)|j ⟩ =1/√(d)[ 1e^i2π(0.j)e^i2π 2(0.j) … e^i2π(d-1)(0.j) ] ^T = 1/√(d) ( |0⟩+e^i2π(0.j)|1⟩+⋯ +e^i2π(d-1)(0.j) |d-1⟩ )The Hadamard gate for qudits essentially performs the order-d Fourier transform, likewise the Hadamard gate for qubits performs the order-2 Fourier transform. Methods for implementation of the H^(d)gate are proposed in <cit.>.The symbols that will be used throughout the text for the three families of elementary gates defined and the H^(d)gate are shown in Figure <ref>.§.§ Diagonal Gates of one and two qudits The qudit elementary gates of the previous section affect a 2-dimensional subspace of the whole d-dimensional Hilbert space of a single qudit. In this section single and two qudit diagonal basic gates affecting the whole d-dimensional space of one of the qudits are described and synthesized using elementary gates of the previous section.§.§.§ Diagonal D^'(a_1,a_2,…,a_d-1) and D(φ_1,φ_2,…,φ_d-1) gates The diagonal D^'(a_1,a_2,…,a_d-1)gate <cit.> is defined by the equationD^'(a_1,a_2,…,a_d-1)=e^iφ diag( e^-i(a_1+a_2+… +a_d-1) , e^ia_1, e^ia_2, … , e^ia_d-1 )It can be easily proved that such a gate can be constructed by sequentially applying d-1R_z^(jk)(θ) gates as shown in the following equationD^'(a_1,a_2,…,a_d-1)=e^iφ R_z^(01)(a_1) R_z^(02)(a_2) ⋯R_z^(0(d-1))(a_d-1)A related gate is the D(φ_1,φ_2,…,φ_d-1) defined asD(φ_1,φ_2,…,φ_d-1) = diag(1,e^iφ_1,e^iφ_2,…, e^iφ_d-1 )The D(φ_1,φ_2,…,φ_d-1)gate is identical with theD^'(a_1,a_2,…,a_d-1) gate if we set[ a_j= φ_j-1/d∑_k=1^d-1φ_kj=1 … d-1 ] and add a global phase of angle φ=1/d∑_k=1^d-1φ_k to every diagonal element ofD^'(a_1,a_2,…,a_d-1).§.§.§ Controlled Diagonal CD^'(a_1,a_2,…,a_d-1) and CD(φ_1,φ_2,…,φ_d-1) gatesThe diagonal gates of the previous subsection can be extended to operate on two qudits, where the first is the control qudit and the second is the target qudit, in the following manner: A diagonal gateD^'(a_1,a_2,…,a_d-1) or D(φ_1,φ_2,…,φ_d-1)is applied on the target qudit iff the control qudit is in state |m⟩, otherwise no operation is effective on the target. Thus, the d^2× d^2 matrices representing such gates have the following block diagonal formCD_m^'(a_1,a_2,…,a_d-1)= diag(I_d, … ,I_d,m-thblockD^'(a_1,a_2 ,…,a_d-1) ,I_d, … ,I_d) and CD_m(φ_1,φ_2,…,φ_d-1)= diag(I_d, … ,I_d,m-thblockD(φ_1,φ_2 ,…,φ_d-1) ,I_d, … ,I_d)A construction of a CD_m^'(a_1,a_2,…,a_d-1) gate using 4(d-1) elementary GCX_(m)^(jk) andR_z^(jk)(θ) gates is shown inFigure <ref>. Single qudit gateS_m=diag(1, … ,1, m-th pose^iφ,1,… , 1) is a phase gate which is identical to a D' gate up to a global phase. §.§ Generalized Controlled Rotation gate R_k^(d) The controlled diagonal gates CD_m^' and CD_m of the previous subsection are activated whenever the control state is equal to one of the d possible basis states, e.g. |m⟩. We define a basic controlled diagonal gate, R_k^(d), such that each one of the d possible control states have a different effect on the target qudit. Such gates will be useful in the QFT and arithmetic circuits presented in the following sections. The R_k^(d)gate is parametrized by the integer k. The matrix defining this gate is block diagonal of the formR_k^(d)=diag (( Φ_k^(d)) ^0,( Φ_k^(d)) ^1,… , ( Φ_k^(d)) ^d-1) where the matrix Φ_k^(d) is diagonal too, and defined with Φ_k^(d)=diag (1,e^iφ_1,e^iφ_2, … , e^iφ_(d-1))The angles φ_1,φ_2,…,φ_(d-1) depend on the parameter k as follows[ φ_m=2π/d^km,m=1,…,d-1 ]The R_k^(d)gates can be equivalently written in a more detailed form consisting of a sum of tensor products of the basis states of the two qudits asR_k^(d)= ∑_j=0^d-1∑_m=0^d-1e^i2π/d^kjm |j⟩⟨ j| ⊗ |m⟩⟨ m| = ∑_j=0^d-1∑_m=0^d-1e^i2π (0.00… 0_k-1j)m |j⟩⟨ j| ⊗ |m⟩⟨ m|We can see by inspecting Eq. (<ref>)that an R_k^(d)gate is a generalization on qudits of the controlled rotation gates R_k=R_z(2π/2^k)=diag(1,1,1,e^i2π/2^k) for the qubit case (where d=2) and this generalization will be exploited when constructing the QFT and various arithmetic circuits based on the QFT. To understand this, it is useful to see what is the effect of an R_k^(d) gate when the control qudit is on a basis state |j_1⟩ (j_1=0,1,…,d-1) and the target qudit is in a superposition of equal amplitudes, but with different phases, such as |b⟩=1/√(d)∑_l=0^d-1e^iφ_l|l⟩. The joint state of the two qudits after the application of the R_k^(d) gate isR_k^(d)( |j_1⟩ |b ⟩)=1/√(d)∑_j=0^d-1∑_m=0^d-1e^i2π (0.00… 0_k-1j)m |j⟩⟨ j|j_1⟩_=δ_jj_1⊗|m⟩⟨ m| ∑_l=0^d-1e^iφ_l|l⟩ _=e^iφ_m|m⟩=1/√(d)∑_m=0^d-1e^i2π (0.00… 0_k-1j_1)m |j_1⟩ e^iφ_m |m⟩=1/√(d) |j_1⟩∑_m=0^d-1e^i2π[ (0.00… 0_k-1j_1)m +φ_m]|m⟩ Thus, an angle 2π(0.00… 0_k-1j_1)m=2π/d^k j_1m is added to every component |m⟩ of the target qudit superposition and this angle is proportional to the value |j_1⟩ of the control qudit and also proportional to the |m⟩ component of target qudit superposition.The implementation of an R_k^(d) can be achieved by sequentially combining d-1 controlled diagonal gates CD_m(φ_1,φ_2,…,φ_d-1) for m=1 … d-1 and different angles for each case of m as shown below (see also Eqs. (<ref>),(<ref>) and (<ref>) )R_k^(d) = CD_(1)(2π/d^k,2π/d^k2, … , 2π/d^k(d-1) ) · CD_(2)(2π/d^k2,2π/d^k4, … , 2π/d^k2(d-1) ) ⋯ CD_(d-1)(2π/d^k(d-1),2π/d^k(d-1)2, … , 2π/d^k(d-1)(d-1) ) Taking into account that aCD_(m)(φ_1,φ_2,…,φ_d-1) gate is composed by 4(d-1) elementary GCX_(m)^(jk) and R_z^(jk)(θ) gates, then we conclude that an R_k^(d) gate requires 4(d-1)^2 elementary gates.The symbol used for the R_k^(d)gate in this text is shown in Figure <ref>.§ QUANTUM FOURIER TRANSFORM The Quantum Fourier Transform on the N-dimensional computational basis{ |0⟩ ,|0⟩, … ,|N-1 ⟩}is defined by |j⟩1/√(N)∑_k=0^N-1e^i2π/Njk|k⟩ Using q qudits of d levels <cit.>,<cit.>, and setting N=d^q, the q qudits basis consists of |j⟩ = |j_1… j_q⟩ = |j_1⟩… |j_q⟩ where for l-th qudit it holds |j_l⟩∈{ |0⟩ , … , |d-1 ⟩}. Then, the QFT action on a basis state |j⟩ (j=0 … d^q-1) is |j⟩ = |j_1… j_q⟩ 1/√(N)∑_k_1=0^d-1⋯∑_k_q=0^d-1e^i2π/d^qj ∑_l=1^qk_ld^q-l |k_1… k_q⟩= ∑_k_1=0^d-1⋯∑_k_q=0^d-1⊗_l=1^qe^i2π jk_ld^-l |k_l⟩ = ⊗_l=1^q∑_k_l=0^d-1 e^i2π jk_ld^-l |k_l⟩ = (∑_m=0^d-1 e^i2π (0.j_q) m |m⟩) (∑_m=0^d-1 e^i2π (0.j_q-1j_q) m |m⟩) ⋯(∑_m=0^d-1 e^i2π (0.j_1j_2… j_q-1j_q) m |m⟩)The d-ary representation (j_1j_2… j_q) of the integer j=j_1d^q +j_2d^q-1+ ⋯ +j_q as well as the fractional d-ary representation (0.j_1j_2…j_q)= j_1/d +j_2/d^2+ … +j_q/d^q are used in the above definition. This tensor product form is similar to the form of the QFT of order 2^n implemented using n qubits of two levels. Thus, the structure of a QFT circuit implemented with qudits is similar to the binary QFT case as depicted in Figure <ref>. Indeed, comparing the state ∑_m=0^d-1 e^i2π (0.j_lj_l+1… j_q-1j_q) m |m⟩of the l-th qudit after the transformation of Eq.(<ref>) with Eq.(<ref>) and (<ref>) we can conclude that this state can be generated by applying at the basis state |j_l⟩of the l-th qudit a Hadamard gate H^(d)and a sequence of q-l generalized rotation gates R_k^(d) , with k=2… q-l+1,controlled by the qudits l+1 … q, respectively. At the end, the order of the qudits must be reversed with swap gates as in the case of the QFT operated on qubits. This swapping of the qudits is not shown in Figure <ref>. The inverse QFT circuit is derived by reversing horizontallythe direct QFT circuit of Figure <ref> (including the SWAP gates not shown) withopposite signs in the angles of the rotation gates. § ARITHMETIC CIRCUITS The integer arithmetic circuits presented in this section are developed in a bottom up succession, starting from the simpler ones and proceeding gradually to more complex ones. The arithmetic operation are assumed to be modulo d^q where d are the qudit levels and q is the number of qudits used to represent the integers. All the adders can be easily converted to subtractorsby using opposite sign in the angles of the rotation gates while retaining the same circuit structure. §.§ Adder of two integers (ADD) A basic arithmetic operation block is an adder of two integers of q d-ary digits each, e.g a=(a_1a_2… a_q) and b=(b_1b_2… b_q) or two superpositions of integers. Followingthe previous sections, the most significant d-ary digit of an integer is indexed with 1 while the least significant digit is indexed with q. The circuit in Figure <ref>operates on 2q qudits, the state |b_1… b_q⟩of the q upper qudits (upper register) represents integer b while the state of the lower q qudits (lower register) represents the Fourier transformed state of the other integer a, that is |φ_1(a)⟩ |φ_2(a)⟩⋯ |φ_q(a)⟩, where |φ_l(a)⟩=∑_m=0^d-1e^i2π ( 0.a_la_l+1… a_q)m (see Eq. (<ref>)). It is a generalization on qudits of the circuit proposed in <cit.>.The first qudit of the lower register is initially in the state |φ_1(a)⟩. The effect of the first rotation gate R_1^(d) controlled by state |b_1⟩ to this qudit (step[1,1]), taking into account Eq. (<ref>), is to evolve itin the state|φ_1(a)⟩ |φ_1(a)⟩ _1,1 = 1/√(d)∑_m=0^d-1 e^i2π[ ( 0.a_1a_2… a_q) +( 0.b_1) ]m |m⟩The effect of the second gate R_2^(d)controlled by |b_2⟩ is to further evolve it (step[1,2]) in the state |φ_1(a)⟩ _1,1 |φ_1(a)⟩ _1,2 = 1/√(d)∑_m=0^d-1 e^i2π[ ( 0.a_1a_2… a_q) +( 0.b_1) + ( 0.0b_2) ]m |m⟩Proceeding in a similar way up to gate R_q^(d)controlled by |b_q⟩, we find the final state (step[1,q]) of the first qudit which becomes |φ_1(a)⟩ _1,q-1 |φ_1(a)⟩ _1,q = 1/√(d)∑_m=0^d-1 e^i2π[ ( 0.a_1a_2… a_q) +( 0.b_1b_2… b_q) ]m |m⟩In general, the final state of the l-th qudit of the lower register is found to be |φ_l(a)⟩ _l,q-l+1 = 1/√(d)∑_m=0^d-1 e^i2π[ ( 0.a_la_l+1… a_q) +( 0.b_lb_l+1… b_q) ]m |m⟩Applying Eq. (<ref>) to each lower register qudits we can find that thelower register has the final joint state|φ(a)⟩ _1 |φ(a)⟩ _2⋯ |φ(a)⟩ _q⊗_l=1^q1/√(d)∑_d=0^d-1 e^i2π[ ( 0.a_la_l+1… a_q) +( 0.b_lb_l+1… b_q) +]m |m⟩ = |φ (a+b) ⟩This is the quantum Fourier transform of the sum state |a+bd^q⟩. By applying the inverse QFT at the lower register we can get the desired sum in the computational basis, while the upper register remains in the initial state |b ⟩. The required direct and inverse QFT blocks are not shown in Figure <ref>.§.§ Adder of an integer with constant (ADDC_b) Whenever one of the integers is constant, e.g. b=(b_1b_2… b_q), then the upper register in Figure <ref> is not necessary and all the controlled rotation gates become single qudit rotation gateswith their angles defined by the constant integer b. Thus (see Eqs. (<ref>) and (<ref>)), we must apply on the l-th qudit of the lower registera sequence of q-l+1 rotation gates ( Φ_k^(d)) ^b_k+l-1 = ∑_m=0^d-1 e^i2π/d^kmb_k+l-1 |m⟩⟨ m|, for k=1… q-l+1. This product of gates can be merged into one gate of the formB_l(b) =∏_k=1^q-l+1( Φ_k^(d)) ^b_k+l-1=∏_k=1^q-l+1( ∑_m=0^d-1 e^i2π m/d^k |m⟩⟨ m| ) ^b_k+l-1 =∑_m=0^d-1( ∏_k=1^q-l+1 e^i2π m/d^kb_k+l-1) |m⟩⟨ m|These are diagonal gates of the form of Eq. (<ref>), and their angles depend on the constant b by the relationφ_l,m(b)= ∑_k=1^q-l+12π/d^kmb_k+l-1 so they can be constructed with elementary R_z^(jk)(θ)gatesusing the procedure described insubsection <ref>. Figure <ref> shows the constant b adder (direct and inverse QFT blocks not included in the diagram). Likewise the general adder ADD, this adder performs the addition modulo d^q.§.§ Single State Controlled Adder of an integer with constant (C_cADDC_b)The constant adder ADDC_b can be easily converted to a constant adder controlled by the stateof an additional control qudit so as to perform the transformation C_cADDC_b( |e⟩ |a⟩)= |e⟩ |a+bδ_ce⟩ where δ_ce is the Kronecker delta function. Consequently, the addition is performed iff the control stateequals |c⟩, otherwise the target state |a⟩ remains unaltered. The one state controlled constant adder C_cADDC_b can be constructed as shown in Figure <ref> if the one qudit rotation gates B_l(b) of Figure <ref> are converted to the respective two qudits diagonal gates controlled by state |c⟩. These gates are exactly the CD_(c) gates ofsubsection <ref>.§.§ Generalized Controlled Adder of an integer with constant (GCADDC_b)A useful generalization of the previous C_cADDC_b circuit can be achieved if we permit all the basis states of the control qudit to have an influence on the result of the addition. Such a circuit will be named Generalized Controlled Adder with constant b and is defined bythe relation GCADDC_b( |e⟩ |a⟩)= |e⟩ |a+be⟩The above equation can be rewritten asGCADDC_b( |e⟩ |a⟩)= |e⟩ |a+bδ _1e+2bδ _2e+ ⋯+(d-1)bδ _(d-1)e⟩ Equation (<ref>) directly leads to the implementation of Figure <ref> where d-1 consecutive applications of C_cADDC_bc (c=1… d-1) adders are employed. §.§ Multiplier with constant and Accumulator (MAC_b)A Multiplier with constant and Accumulator MAC_b multiplies a q qudits integer x with a constant b of q d-ary digits, and accumulates the product bx to a q qudits integer a (modulo d^q). Namely, the MAC_b circuit consists of two q qudits registers holding initially the states |x⟩and |a⟩and transforms them asMAC_b( |x⟩ |a⟩)= |x⟩ |a+bx⟩Taking into account that x can be written as (x_1x_2… x_q)=∑_l=1^qx_ld^q-l then Eq. (<ref>) can be written asMAC_b( |x⟩ |a⟩)= |x⟩ |a+b∑_l=1^qx_ld^q-l⟩ =|x⟩ |a + x_qb + x_q-1db + ⋯ + x_1(d^q-1b) ⟩This means that the above transformation can be implemented by applying q GCADDC circuits, wherethe control is done consecutively by the qudits x_q,x_q-1,…,x_1 and the constant parameter for each one GCADDC block is b,db,…,d^q-1b (modulo d^q), respectively, as shown in Figure <ref>.§.§ Multiplier with constant(MULC_b)A multiplier (modulo d^q) with constant b implements the function f:0… d^q-1→0… d^q-1 with y=f(x)=bx q^q. When cosntant b is relative prime to d^q then there exists the inverse b^-1d^q and consequently there exists the inverse function f^-1(y)=b^-1y d^q = b^-1bx d^q=x. This always happens when d is a prime number.Figure <ref> shows how to construct a Multiplier with constant b using two MAC_b blocks and the necessary direct and inverse QFT blocks. It requires a q qudits register initially holding the integer x and another q qubits ancilla register initially in zero state. At the end, one register is set to the state |bx d^q⟩ while the other register is set to state zero, so effectively the ancilla register is reset back and can be reused.In the diagram of Figure <ref>, the boxes with the black strip at their right side are the "direct" blocks while these with the black strip at their left side are the respective inverses. The operation of the inverse MAC with parameter b^-1 is to perform substraction instead of accumulation, that is referring to Figure <ref>,we have the operation MAC_b^-1^-1 | bx⟩ |φ(x) ⟩ = | bx⟩ |φ (x-b^-1(bx)) ⟩ = | bx⟩ |0⟩.The inverse MAC^-1 has the same internal topology as the direct MAC of Figure <ref> (of course with parameter b^-1 instead of b) with the only difference that the angles of its rotation gates have a minus sign. By inspecting the labels at the qudit buses of Figure <ref> describing the respective states we can conclude that the circuit implements the multiplicationMULC_b( |x⟩ |0⟩)= |bx⟩ |0⟩Excluding the ancilla register, which is in the zero state before and after the operation and thus it remains unentangled, we conclude that this circuit performs the desired multiplication operation.§ DIAGONAL OPERATORS ON Q QUDITS The diagonal operator on q qudits of d levels, as its name implies, is a circuit whose unitary matrix ofdimensions d^q× d^q has a diagonal form. The circuit developed in this section is such that the diagonal elements of the matrix are integer powers of the principal root of unity e^i2π/d^q and the integer powers are a function f(k) of the coordinate k=1… d^q-1 of the elements. In what follows, the diagonal operator circuit developed is for the function f(k)=γ k^2, where γ is an integer constant. Thus, the definition of our diagonal operator on q qudits is Δ_γ^(q)=∑_k=0^Q-1 e^i2π/Qf(k) |k⟩⟨ k| where Q=d^q and f(k)=γ k^2d^q. All the diagonal entries of the above matrix are integer powers of the basic phase ω =e^i2π/Q. The effect of this matrix upon a general superposition state of q qubits will be∑_k=0^Q-1 c_k |k ⟩∑_k=0^Q-1 c_k e^i2π/Qf(k)|k ⟩The circuit that implements the operator of Eq. (<ref>) will be derived by exploiting results of <cit.> which are given for the case of binary quantum circuits. A prerequisite for this construction is a Squarer/Multiplier with constant/Accumulator circuit (SMAC) that computes the function finvolving two q qudits registers as inSMAC_γ(|k⟩ |z⟩ ) = |k⟩ |z + f(k) ⟩= |k⟩ |z + γ k^2Q⟩Such an SMAC circuit will be described insubsection <ref>. Figure <ref> shows the diagonal operator circuit with entries dependable on the function f(k)=γ k^2Q. Twoquantum registers are used, each q qudits wide, namely Reg1 and Reg2. The upper register Reg1 is assumed to be in a general superposition state prior the operator Δ_γ^(q) is applied as described in Eq. (<ref>), while the lower register Reg2 is an ancilla register with zero initial and final state.The first step is to form in the ancilla register Reg2 the uniform superposition state |R⟩ = 1/√(Q)∑_h=0^Q-1 |h⟩. This is accomplished with the application of q Hadamard gates H^(d) on each qudit of the register.Then, we apply on each qudit the diagonal gates D_Q^d^m† for m=0… q-1 . The matrix representing these gates is D_Q^d^m†= diag(1,ω _Q ^-1· d^m,ω _Q ^-2· d^m,… ,ω _Q ^(d-1)· d^m ),m=0… q-1 and it has exactly the same form of the diagonal gates of Eq. (<ref>). The joint affect of these gates at Reg2 is given by their tensor product which is a diagonal matrix too, of dimensions d^q× d^qD_QQ^†= diag(1,ω _Q^-1,ω _Q ^-2, … ,ω _Q ^Q-2, ω _Q ^Q-1 )Then, the state of Reg2becomes |R⟩ _1= 1/√(Q) = ∑_h=0^Q-1ω ^(Q-h) |h ⟩The initial state of Reg1 is assumed to be a general superposition of basis states and can be expressed as |S ⟩ _1= ∑_k=0^Q-1 c_k |k ⟩ =∑_n=0^Q-1∑_k ∈ K_n c_k |k ⟩ , K_n={k: f(k)=n }In the right hand side of Eq. (<ref>) we have grouped all the basis states with value k such that f(k)=n in a set K_n and then sum over all the states belonging to sets K_n. The expediency of this grouping will be clear later. Combining Eq. (<ref>) and (<ref>) we find the joint state of Reg1 and Reg2 just before the application of the SMAC block, which is given by the tensor product|S ⟩ _1⊗ |R ⟩ _1 =1/√(Q)∑_n=0^Q-1∑_k ∈ K_n∑_h=0^Q-1 c_kω ^Q-h |k ⟩ |h ⟩Taking into account the effect of the SMAC block given by Eq. (<ref>) we get the state of the two registers after the application of the SMAC|SR ⟩ _2= SMAC(|S ⟩ _1⊗ |R ⟩ _1) = 1/√(Q)∑_n=0^Q-1∑_k ∈ K_n∑_h=0^Q-1 c_kω ^Q-h |k ⟩ |h + f(k) Q ⟩1/√(Q)∑_n=0^Q-1∑_k ∈ K_n∑_h=0^Q-1 c_kω ^Q-h |k ⟩ |h + n Q ⟩We are going to use m=h+n Q as the index of the inner summation in place of h. We observe that for a particular n, as h takes the values from 0 to Q-1, then m=h+n Q takes one value a time ("1-1" mapping), that is the new index m will be in the same range from 0 to Q-1. Thus the lower and upper limits of the new index m remain the same and we have h=m-n Q and Q-h=n+(Q-m) Q. Also, it holds ω ^Q=1. Then Eq. (<ref>) becomes|SR ⟩ _2=1/√(Q)∑_m=0^Q-1∑_n=0^Q-1∑_k ∈ K_nc_kω ^nω ^Q-m |k ⟩ |m⟩1/√(Q)∑_n=0^Q-1∑_k ∈ K_n c_kω ^n |k ⟩⊗∑_m=0^Q-1ω ^Q-m |m ⟩∑_k=0^Q-1 c_kω ^f(k) |k ⟩⊗1/√(Q)∑_m=0^Q-1ω ^Q-m |m ⟩( Δ_γ^(q) |S ⟩) ⊗ |R ⟩This shows that Reg1 has the desired state of Eq. (<ref>) and it is disentangled with respect to Reg2 which remains in state of Eq. (<ref>).Thus, the ancilla Reg2 can be reset without any effect on the Reg1. The resetting can be accomplished as shown in Figure <ref> by applying in the reverse sequence (a) the inverse of the gates H^(d) and (b) the inverse of D_Q^d^m†,which are the H^(d)†=H^(d)* (conjugate Hadamard) and D_Q^d^m, respectively. An alternative method would be to measure Reg2 and depending on the measurement result to apply GCX gates controlled by the measurement classical result. This measurement would not affect Reg1 as it is disentangled with respect to Reg2. §.§ Multiplier of two integers / Accumulator (MMAC) The construction of the SMAC block requires amultiplier of two integers and accumulator block (MMAC) whose operation is to multiplyinteger x with integer y and accumulate the product xy tointeger z (modulo d^q). This means that the MMAC block is applied on three q qudits registers and performs the transformation MMAC ( |x⟩ |y⟩ |z⟩) =|x⟩ |y⟩ |z+xy⟩ If x=(x_1x_2… x_q)=∑_t=1^q x_td^q-t and y=(y_1y_2… y_q)=∑_s=1^q y_sd^q-s are the d-base representations of the two integers , then their product (modulo d^q) is given byxy=∑_s=0^q-1 d^s∑_t=0^q-1x_q-ty_q-s+tIn Eq. (<ref>) the full product terms corresponding to powers d^s with s ≥ q have not been included, because the product is to be calculated modulo d^q. Also, digits with negative index (e.g. x_-1), as well as with index greater than q (e.g. x_q+1), are assumed zero. The calculation of the product and the accumulation can be performed in an similar way as in the MAC circuit given in subsection <ref>. We assume that the state corresponding to the accumulation register integer |z⟩ is already Fourier transformed and taking into account Eq. (<ref>) which expresses the QFT we expect that the l-th qudit of the accumulation result |z+xy⟩ prior the inverse QFT is|φ_l(z+xy)⟩ =|0⟩ + e^i2π/d^l (z+xy) |1⟩ +e^i2π/d^l (z+xy)2 |2⟩ + ⋯ + e^i2π/d^l (z+xy)(d-1) |d-1⟩Thus, to bring and initial state |φ_l(z)⟩of the l-th qudit to the state of Eq. (<ref>) we mustadd various integer multiples of the basic angle (2π/d^l). Namely, taking into account Eq. (<ref>), the angles that must be added to the amplitude phases of a basis state |r⟩ (r=0… d-1) in the superposition|φ _l(z)⟩ of Eq. (<ref>) areΦ_l,r = 2π/d^lxyr = 2π r∑_s=0^q-1 d^s-l∑_t=0^sx_q-ty_q-s+t= 2π r ∑_s=0^l-1 d^s-l∑_t=0^s x_q-ty_q-s+tThe restriction s<l at the upper limit of the first sum of Eq. (<ref>) comes due to the periodicity exp(φ +2π d^n )= exp(φ ) that holds for any integer d and any non negative integer n. The restriction t≤ s at the upper limit of the second sum results because y_q-s+t=0 for t>s. Replacing with k=l-s, Eq. (<ref>) becomesΦ_l,r =2π r ∑_k=1^l d^-k∑_t=0^l-k x_q-ty_q+k-l+tConsequently, the angles that must be added to the phase amplitude of the |r⟩ component of the superposition are (2π/d^k)x_my_nr and depend on indices m=q-t and n=q+k-l+t. This can be attained if we introduce the notion of a double controlled generalized rotation gate applied to three qudits, two controls and one target. Similarly to Eq. (<ref>) which is the definition of the simply controlled generalized rotation gate, we define the double controlled generalized rotation gate R _k^(d) with the d^3× d^3 matrix R _k^(d) = ∑_m=0^d-1 ∑_n=0^d-1 ∑_r=0^d-1e^i2π/d^kmnr( |m⟩⟨ m| ) ⊗( |n⟩⟨ n| ) ⊗( |r⟩⟨ r| ) Figure <ref> depicts the symbol for this double controlled rotation gate. In Appendix A a construction of R _k^(d) will be presented using some of the elementary and basic gates introduced in Section 3.The topology of the MMAC circuit can be directly concluded from Eq.(<ref>), as this equation describeswhichgates have to applied and which are their control connections to the qudits carrying |x⟩ and |y⟩. Figure <ref>shows an example MMAC for the case of q=4. In this figure the R _k^(d)gates are represented with the value k inside the circle. Generalization for any value of q is obvious.We observe in Figure <ref> and in Eq. (<ref>) that l-k+1 R _k^(d) gates are sequentially applied on the l-th target qudit for a specific k (l=1 … q,k=1 … l). In total, C_l=∑_k=1^l (l-k+1) =l(l+1)/2 R _k^(d)gates are applied on the l-th target qudit. Summing over all target qudits we find the total number of gates usedC_MMAC(q)=∑_l=1^q C_l = ∑_l=1^ql(l+1)/2 =1/6q^3+1/2q^2 + 1/3q. The same value gives the depth of the circuit as arranged in Figure <ref>. Indeed, for the example q=4 we find C_MMAC(4)=20. We can exploit the fact that gates R _k^(d)mutually commute as they are diagonals and rearrange them so as to achieve a parallelization in their execution. The gates that can be executed simultaneously are those that operate on different qudits. An example of the proposed parallelization for the case q=4 is shown Figure <ref>, where below each gate is shown the soonest timestep in which it can be executed. E.g. at the first timestep three gates can be executed in parallel as none of these gates operate on the same qudit as the other two. We can generalize this parallelization scheme and conclude that we can achieve a depth of about q(q+1)/2 which is quadratic instead of cubic without the proposed rearrangement.§.§ Squarer/Multiplier/Accumulator(SMAC) The MMAC circuit allows theconstruction of the SMAC_γ circuit described by Eq. (<ref>) and required for the q qudits diagonal operator Δ_γ^(q). The Squarer/Multiplier with constant γ /Accumulator modulo d^q is presented in block diagram in Figure <ref>. It uses 4q qudits, 2q of which are ancilla qudits with zero initial and final state, where q is the number of qudits used to represent the argument x. The 4q qudits are grouped into four registers of q qudits each. The second register from top holds the argument |x⟩ while the bottom register holds the accumulation value |z+γ x^2 d^q⟩. The first step is to set the state of the top register into the same state as the second one, which is |x⟩. This is accomplished with the adder block sandwiched between two QFT blocks, direct and inverse (This operation could be also achieved using a sequence of GCX gates to "copy" the second's register state to the first). In a second step, the two states |x⟩ and |x⟩of the two top registers are multiplied together by the MMAC andthe product is accumulated to the third register from top, which was initially in the zero state. At this stage, the joint state of the three top registers is |x⟩ |x⟩ |x^2d^q⟩. Next, the MAC_γ block follows to multiply the constant γ with the |x^2d^q⟩ state of the third register. The result is accumulated to the bottom register, which was initially in state |z⟩. At this point the joint state of the four registers can be described by |x⟩ |x⟩ |x^2d^q⟩ |z+γ x^2d^q⟩. What remains is to reset the first and the third ancilla registers. The inverse MMAC resets the third register by performing substraction instead of accumulation of the product |x^2⟩. The inverse MMAC is constructed like the direct MMAC with opposite angles in its rotation gates. Last, the inverse adder resets the top ancilla register. Consequently, the circuit of Figure <ref> implements the transformationSMAC ( |0⟩ |x⟩ |0⟩ |z⟩) = |0⟩ |x⟩ |0⟩ |z + γ x^2d^q⟩which is exactly the transformation of Eq. (<ref>) if the ancilla registers are ignored. § COMPLEXITY ANALYSISThe arithmetic quantum circuits proposed in the previous sections are broken down to the level of elementary gates H^(d), R_z^(jk)(θ), R_x^(jk)(θ) and GCX_m^(jk) introduced in Section <ref>. This decomposition is depicted in Figure <ref> in a tree structure, where the root of of each tree is some of the complete circuits proposed and the leaves of the tree (trapezoids) represent the elementary gates. The edges of each tree are labeled with the number of components needed by each level from one level below (no label stands for 1). The SMAC and the Diagonal operator are not included in this Figure, but their costs and depths can be easily calculated after the calculations of the blocks shown in this figure. A rough complexity analysis in terms of quantum cost (number of elementary gates used) and depth (execution time) can be done with the help of Figure <ref>. The analysis assumes that single and two qudits gates are equivalent in terms of costs and execution time. Exact costs and depths depend on the particular implementations. The total gates count for each block can be found by traversing the tree emerging from the inspected block down to each leaf of the subtree. The labels of the edges for each path are multiplied and then the products of each path used are summed together. E.g. the QFT circuit needs q Hadamard gates, (q^2-q/2)(d-1)(2d-1) GCX_m^(jk)gates and (q^2-q/2)(d-1)2d R_z^(jk)(θ)gates. Similar calculations provide us with the quantum costs shown in Table <ref>, which shows only the highest order terms. The depth calculation will be done in more detail by finding first the depths of QFT, ADD, MAC and MMAC blocks and then the depths of MULC and SMAC. QFT At first glance Figure <ref> exhibits a quadratic depth O(q^2), but it can be easily shown that we can parallelize the execution with an appropriate reordering of the gates and thus achieve a linear depth, namely depth(QFT)=8d^2q. ADD Similarly as in the QFT case, a reordering of gates in Figure <ref> offers a linear depth too, that is depth(ADD)=4d^2q. MAC Concurrent execution of gates is possible in this case, too. It can be easily seen that by flattening the hierarchy MAC-GCADDC-CADDC, q different controlled gates B_l(b) (Eq. (<ref>)) belonging in different GCADDC blocks can be executed concurrently. Thus, the depth of the MAC is of the order O(4d^2q) instead of O(4d^2q^2) as directly calculated by the number of elementary gates. MULC Observing Figure <ref> we find depth(MULC)=3depth(QFT)+2depth(MAC), as the two middle QFT blocks (direct and inverse) can be executed simultaneously. Thus, we derive depth(MULC)= 32d^2q. MMAC The reordering of gates achieves q(q+1)/2 execution steps of double controlled rotation gates. Taking into account the decomposition of these three qudit gates into single and two qudits gates (see Appendix A) we end up in depth(MMAC)=21d^3q^2.SMAC From the previous calculations and Figure <ref> we find that the dominant depth of SMAC in leading order is twice the depth of the MMAC block. Δ _γ^q The depth of the diagonal circuit is essentially the depth of the SMAC. § CONCLUSIONS AND FUTURE WORKIn this paper we presented an assortment of quantum circuits for multilevel qudits. These are basic integer arithmetic operations circuits (like addition, multiplication/accumulation and multiplication) as well as more complex circuits such as squarers. Additional extensions can be applied. E.g., the ADD, ADDC, MAC and MULC circuits can be converted to single qudit controlled versions. Such controlled versions could be useful to the multilevel qudits quantum phase estimation algorithms and quantum simulations. The general diagonal operator has been developed for the special case of a quadratic function f(x)=γ k^2, where k is the coordinate of the diagonal element, however using the same techniques we can easily construct diagonal operators for any power of k and even for a polynomial function on k. E.g. the Squarer/Multiplier/Accumulator can be converted to a circuit that accumulatesthe third power by inserting additional MMAC units in Figure <ref>.The designsare based on the alternative representation of an integer after QFT transformed instead of the usual computational basis representation, a method which has been already exploited in the binary qubits case. QFT based arithmetic circuits design is a versatile method to develop many arithmetic circuits. E.g. there is no need to handle carries which leads to space reduction. Moreover, if it is suitably used, it can offer advantages in terms of speed. This is possible when similar blocks are iterated to act on a datapath whose state follows the QFT representation. The extensive usage of rotation gates (which mutually commute) on such a datapath permits their rearrangement so as they execute concurrently. This capability is observed in the MAC block, where the application of a suitable reordering of gates led to depth reduction from O(q^2) to O(q). Similarly, the depth of the MMAC block reduced from O(q^3) to O(q^2).Another advantage that has been observed in designs adopting the QFT method is their robustness to various kinds of deviations from the ideal operation. E.g. approximate QFT <cit.> or QFT banding is the design procedure of eliminating small angle rotation gates. Studies of the Shor's algorithm which uses the QFT showed that the algorithm still works sufficiently even when a large proportion of the QFT rotation gates are eliminated <cit.>. Recent studies extended to circuits beyond QFT. In <cit.> the simultaneous gate pruning of rotation gates of the QFT circuit and the QFT based modular exponentiator of Beauregard's circuit <cit.> were simulated. The simulation results showed similar robustness of Shor's algorithm to these gates eliminations. This robustness is sustained even if the parameters of the remaining rotation gates are randomly selected <cit.>. The above results suggest that a similar robustness is expected in the multidimensional qudits case and further investigation to be carried.On the other side, there is a drawback related to the requirement of reliable implementing high accuracy small angles rotation gates. Moreover, these gates must belong to a set of fault tolerant gates if large scale quantum computation is considered. Fortunately, as shown in Appendix B, approximation of these gates is possible, albeit with a cost. 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Equation (<ref>) states that the effect of the R _k^(d)to the target qudit state ismultiplication by a diagonal matrix d × d of the form diag(1,e^i2π/d^kmn,e^i2π/d^k2mn, … ,e^i2π/d^k(d-1)mn) iff the two control states are|m⟩ and |n⟩. Consequently, to implement this gate we need (as was the case of thegate CD_m(φ _1, … , φ _d-1 )), the construction of double controlled diagonal gates of the formCCD_(m,n)( φ_1,φ_2,…,φ_d-1)= diag(I_d, … ,I_d,mn-thblockD(φ_1,φ_2 ,…,φ_d-1) ,I_d, … ,I_d) where the diagonal D(φ_1,φ_2 ,…,φ_d-1) is applied to the target qudit iff the two control qudits states are |m⟩ and |n⟩. The angles are φ _l =(2π / d^kmnl) for l=1 … d-1. Thus, the gate R _k^(d) is constructed by successively using (d-1)× (d-1) three qudit gates CCD_(m,n)(φ_1,φ_2,…,φ_d-1) as followsR _k^(d) = ∏_m=0^d-1∏_n=0^d-1 CCD_(m,n) (1,i2π/d^kmn,i2π/d^k2mn, … ,i2π/d^k(d-1)mn )The above decomposition is depicted in Figure <ref>. The parameter k inside the rectangular symbol of the CCD gates corresponds to the parameter k of R _k^(d) gate, while the values m and n inside the small circles of the same gate signify that the (mn)-th block of the diagonal matrix CCD is of the form diag(1,e^i2π/d^kmn,e^i2π/d^k2mn, … ,e^i2π/d^k(d-1)mn) while the rest of the blocks are identity matrices (see Eq. (<ref>)). That is the gate transforms the target qudit with the matrix diag(1,e^i2π/d^kmn,e^i2π/d^k2mn, … ,e^i2π/d^k(d-1)mn) iff the control states are |x⟩ = |m⟩ and |y⟩ = |n⟩.The way to construct a double controlled rotation gate CCD_(m,n)(φ_1,φ_2,…,φ_d-1) is analogous to the one for thesimply controlled gateCD_(m)(φ_1,φ_2,…,φ_d-1) which is equivalent to the CD^'_m(a_1,a_2,…,a_d-1). The difference in this case is thatwe needdouble controlled generalized NOT gates, which will be called GCCX_(m,n)^(jk). They can be thought as an extension of Toffoli gates to the qudit case and their operation is analogous to that ofthe GCX_(m)^(jk), but in three qudits. They are defined by the equation[ GCCX_(m,n)^(jk)= |m ⟩⟨ m| ⊗|n ⟩⟨ n| ⊗( |j ⟩⟨ k| + |k ⟩⟨ j| +∑_lk=0k ≠ j ^d-1 |k ⟩⟨ k| ) + ;∑_ll=0l ≠ ml ≠ n^d-1 |l ⟩⟨ l| ⊗ I_d⊗ I_dj,k,m,n=0 … d-1 ]This description means that they interchange the two target qudit basis states |j⟩ and |k⟩ iff the two control states are |m⟩ and |n⟩. Having available the GCCX_(m,n)^(jk) gates we can construct a CCD^'_(m,n)(a_1,a_2,…,a_d-1) as shown in Figure <ref> (the example depicts the qutrit case of d=3, generalization for other values of d is obvious). The controlled gate S_n is the analogous of the single qudit gate S_m of Figure <ref>. Namely, it is a CD_m(1,…,1, n-th pose^iφ,1,…,1) gate, with angle φ=1/d∑_k=1^d-1a_k. The CCD_(m,n)^' gate is identical with the desired CCD_(m,n) gate if angles redefinition similar to the ones of Eq. (<ref>) are applied.What remains is the GCCX_(m,n)^(jk) gateconstruction. This gate operates in a two dimensional subspace of the target qudit. Thus, extension to the qudit case of a Toffoli gate decomposition into single and two qubits <cit.> can be exploited and this is shown in Figure <ref>. The single and two qudit gates used will be the generalization of the S,T and H qubit gates to the d dimension of the qudits but operating only on a 2-dimensional subspace. Concretely, we define the gates[ S^(jk)=|j ⟩⟨ j| + i|k ⟩⟨ k| + ∑_l n=0 n ≠ j n ≠ k ^d-1 |n ⟩⟨ n| j,k=0 … d-1 ] [ T^(jk)=|j ⟩⟨ j| + e^iπ /4|k ⟩⟨ k| + ∑_l n=0 n ≠ j n ≠ k ^d-1 |n ⟩⟨ n| j,k=0 … d-1 ] [ H^(jk)=1/√(2)( |j ⟩⟨ j| + |j ⟩⟨ k| +|k ⟩⟨ j| - |k ⟩⟨ k| ) + ∑_l n=0 n ≠ j n ≠ k ^d-1 |n ⟩⟨ n| j,k=0 … d-1 ] Gates S^(jk) and T^(jk) are effectively R_z^(jk)(θ) with θ equal to π /2 and π /4, respectively, ignoring some global phase. Also, we can build the H^(jk) gate using the easily proved identity H^(jk)=e^iπ /2 R_z^(jk)(π /2) R_x^(jk)(π /2) R_z^(jk)(π /2), which is similar to the one for the qubit Hadamard case. We have finally achieved to synthesize a three qudits rotation gate R _k^(d) with elementary single qudit rotation R_z^(jk)(θ), R_x ^(jk)(θ) gates and two qudits GCX_(m)^(jk) gates.The design of arithmetic quantum circuits based on the QFT involves a library of elementary gates{ H^(d),R_z^(jk)(θ), R_x^(jk)(θ), GCX_m^(jk)}. The size of this library is not constant as the parameter θ of the rotation gates depends on the size q of the circuit (number of qudits used). However, it is possible to approximate these rotation gates with arbitrary precision using a constant set of gates.Much research has been done recently in this area focused on gates operating on qubits <cit.>. An extension of some of these results can be easily applied to the case of the qudits for the specific elementary gates library and thus we can use a constant library for the synthesis. This is important both for physical implementation reasons and for the fault tolerance aspect of the circuit, as fault tolerance techniques have developed for a restricted set of gates (e.g Clifford + T gates for the case of binary qubits).Before proceeding to the extension of the well established qubit gates approximation methods to the qudit case, some definitions are necessary. A unitary matrix U of dimensions d × d is called a two level matrix if it has the form<cit.> U=[1 ; ⋱;1 ;u_jju_jk;1 ;u_kju_kk;⋱ ; 1;]This kind of matrix leaves invariant a subspace of d-2 dimensions and operates only on the two dimensions corresponding to coordinates j and k. A more compact notation for the above matrix is[U_[ jk ](V), V= [ v_11 v_12; v_21 v_22 ] ]In this notation, we only need to define a unitary 2 × 2 dimensional matrix V and also define which coordinates j and k this matrix operates on. A multiplication homomorphism is valid since U_[ jk ](V_1)·U_[ jk ](V_2) = U_[ jk ](V_1· V_2)The R_z^(jk)(θ) and R_x^(jk)(θ) elementary gates have exactly the form of Eq. (<ref>) with V_z(θ)=[ e^- iθ/20;0e^ iθ/2 ] andV_x(θ)=[ cosθ /2 -isinθ /2; -isinθ /2 cosθ /2 ] , respectively.The V_z(θ) and V_x(θ) gates are in fact qubits rotation gatesand thus we canexploit the known approximation results of the literature for the qubit gates. These results state that an arbitrary rotation gate like V_z(θ) can be approximated by a finite sequence of gates belonging to a discrete set, e.g.V̂_z(θ)=(HT⋯ T)(HT⋯ T)⋯ (HT⋯ T), where H is the Hadamard gate and T is the π/8 gate, such as the approximation error ϵ = V̂_z(θ) - V_z(θ) can be arbitrary small (Solovay-Kitaev Theorem and improvements <cit.>). Using this fact and Eq. (<ref>) we find that every rotation gate R_z^(jk)(θ) can be approximated by another one R̂_z^(jk)(θ) with arbitrary precision asR̂_z^(jk)(θ)= U_[jk ](V̂_z(θ) )= U_[jk ](H) U_[jk ](T) ⋯U_[jk ](T)⋯ U_[jk ](H) U_[jk ](T) ⋯U_[jk ](T)In essence, the U_[jk ](H) and U_[jk ](T) gates are the H^(jk) and T^(jk) gates of Eq. (<ref>) and (<ref>), respectively. On the other side, the R_x^(jk)(θ) gates can be decomposed using the identity R_x^(jk)(θ)=H^(jk) R_z^(jk)(θ) H^(jk) and thus the proposed circuits can be synthesized using the discrete library of constant number of components shown in Table <ref>. The second column shows the number of different gates of the same family, which depends on the family parameters (none, j, k and m). The constant library consists of a total of (3+d)d(d-1)/2+1 gates.The first Solovay-Kitaev algorithms <cit.> generate a sequence of such gates of length O(log^3.97(1/ϵ)) and synthesis time in order of O(log^2.71(1/ϵ)). In the last few years extensive research resulted in great improvements both in terms of the sequence length and synthesis time. They used a diverse set of techniques (usage of ancilla or not, different libraries, approximate or exact synthesis etc). Some of the best results in terms of the generated sequence length can be found in <cit.>. These works offer a length of less than 10log(1/ϵ)T gates (T gates are considered more costly if they are built fault-tolerantly). In the presented circuits, the worst case angle of a R_z^jk(θ)gate is θ=2π / d^q, so the desired approximation error should be of the same order ϵ≈ 2π/d^q. Consequently, each R_z^(jk)(θ) gate can be adequately approximated by a sequence of H^(jk) and T^(jk) gates of length of the order 10log(d^q/2π)≈ 10qlogd. Thus if we have to use a constant library of components due to implementation and/or fault tolerance reasons we have a linearin q multiplicative overhead in the quantum costs and depths calculated in Section <ref>. | http://arxiv.org/abs/1707.08834v2 | {
"authors": [
"Archimedes Pavlidis",
"Emmanuel Floratos"
],
"categories": [
"quant-ph",
"cs.ET"
],
"primary_category": "quant-ph",
"published": "20170727123339",
"title": "Arithmetic Circuits for Multilevel Qudits Based on Quantum Fourier Transform"
} |
The second-order Matsubara Green's function method (GF2) is a robust temperature dependent quantum chemistry approach, extending beyond the random-phase approximation. However, till now the scope of GF2 applications was quite limited as they require computer resources which rise steeply with system size. In each step of the self-consistent GF2 calculation there are two parts: the estimation of the self-energy from the previous step's Green's function, and updating the Green's function from the self-energy. The first part formally scales as the fifth power of the system size while the second has a much gentler cubic scaling. Here, we develop a stochastic approach to GF2 (sGF2) which reduces the fifth power scaling of the first step to merely quadratic, leaving the overall sGF2 scaling as cubic. We apply the method to linear hydrogen chains containing up to 1000 electrons, showing that the approach is numerically stable, efficient and accurate. The stochastic errors are very small, of the order of 0.1% or less of the correlation energy for large systems, with only a moderate computational effort. The first iteration of GF2 is an MP2 calculation that is done in linear scaling, hence we obtain an extremely fast stochastic MP2 (sMP2) method as a by-product. While here we consider finite systems with large band gaps where at low temperatures effects are negligible, the sGF2 formalism is temperature dependent and general and can be applied to finite or periodic systems with small gaps at finite temperatures. § INTRODUCTION Second-order Green's function (GF2) is a temperature-dependent self-consistent perturbation approach where the Green's function is iteratively renormalized. At self-consistency the self-energy which accounts for the many-body correlation effects is a functional of the Green's function, Σ(G). The GF2 approximation as implemented here is described by the diagrams in Fig. <ref> and employs Matsubara Green’s functions that are temperature dependent and expressed on the imaginary axis.<cit.> The implementation we discuss, for total energies, relies on thermal Matsubara Green's functions instead of real time Green's functions.<cit.> This offers advantages in terms of stability and smoothness of the self-energy. Upon convergence the GF2 method includes all second order skeleton diagrams dressed with the renormalized second order Green's function propagators, as illustrated in Fig. <ref>. Specifically, as shown in Ref. , GF2, which at convergence is reference independent, preserves the desirable features of M�ller-Plesset perturbation theory (MP2) while avoiding the divergences that appear when static correlation is important. Additionally, GF2 possesses only a very small fractional charge and spin error,<cit.> less than either typical hybrid density functionals or RPA with exchange, therefore having a minimal many-body self-interaction error. In solids GF2 describes the insulating and Mott regimes and recovers the internal and free energy for multiple solid phases.<cit.> Moreover, GF2 is useful for efficient Green's function embedding techniques such as in the self-energy embedding method (SEET).<cit.>The formal advantages of GF2 come, however, with a price tag. The calculation of the self-energy matrix scales as O(n_τN^5), where n_τ is the size of the imaginary time grid and N the number of atomic orbitals (AOs). This leads to steep numerical costs which prevent application of GF2 to systems larger than a few dozen electrons. The application to larger systems requires therefore a different paradigm and here we therefore develop a statistical formulation of GF2 that calculates the self-energy matrix in linear-scaling.The key to the present development, distinguishing it from previous work <cit.>, is the conversion of nested summations into stochastic averages. Our method draws from previous work on stochastic electronic structure methods, including stochastic- density functional theory (sDFT),<cit.>, sDFT with long-range exact exchange,<cit.> multi-exciton generation,<cit.> Moller-Plesset perturbation theory (sMP2),<cit.> random-phase approximation (sRPA),<cit.> GW approximation (sGW),<cit.> time-dependent DFT (sTDDFT),<cit.> optimally-tuned range separated hybrid DFT <cit.> and Bethe-Salpeter equation (sBSE).<cit.> Among these, the closest to this work are the stochastic version of sMP2 in real-time plane-waves,<cit.> and MO-based MP2 with Gaussian basis sets.<cit.> The stochastic method presented here benefits from the fact that the GF2 self-energy is a smooth function of imaginary time and is therefore naturally amenable to random sampling. § METHOD§.§ Brief review of GF2 Our starting point is a basis of N real single-electron non-orthogonal atomic-orbital (AO) states ϕ_i(𝐫), with an N× N overlap matrix S_ij=⟨ϕ_i|ϕ_j.⟩. Such states could be of any form, Gaussian, numerical, etc., but for efficiency should be localized. We then use second quantization creation a_i^† and annihilation a_i operators with respect to the non-orthogonal basis ϕ_i(𝐫). The non-orthogonality is manifested only in a modified commutation relation,{a_i,a_j^+}=(S^-1)_ij.The Hamiltonian for the interacting electrons has the usual formĤ=∑_ijh_ija_i^†a_j+1/2∑_ijklv_ijkla_i^†a_k^†a_la_j,where h_ij=∫ drϕ_j(r)(-1/2∇^2+v_ext(r))ϕ_i(r) and v_ext(r) is the bare external potential (due to the nuclei), while V̂ is the two electron-electron (e-e) Coulomb interaction described by the 2-electron integrals v_ijkl=∬ϕ_i(r)ϕ_j(r)v(|r-r^'|)ϕ_k(r^')ϕ_l(r^')drdr^',where v(r)=1/r is the Coulomb interaction potential. At a finite temperature β^-1 and chemical potential μ we employ the grand canonical density operator e^-β(Ĥ-μN̂)/Z, where N̂=∑_ijS_ija_i^†a_i is the electron-number operator and Z(β)=[e^-β(Ĥ-μN̂)] is the partition function. The thermal expectation value of any operator  can be calculated as ⟨Â⟩ =[e^-β((Ĥ-μN̂)/Z(β)Â]. For one-body observables Â=∑_ijA_ija_i^†a_j we write ⟨Â⟩ =∑_ijA_ijP_ij where P_ij=⟨ a_i^†a_j⟩ is the reduced density matrix.The 1-particle Green's function G_jk(τ) at an imaginary time τ is a generalization of the concept of the density matrix and obeys an equation of motion that can be solved by perturbation methods. Formally:G_jk(τ)=-⟨ Ta_j(τ)a_k^†⟩ ,where a_j(τ)≡ e^(Ĥ-μN̂)τa_je^-(Ĥ-μN̂)τ with -β<τ<β, and T is the time-ordering symbol: Ta_j(τ)a_k^†≡θ(τ)a_j(τ)a_k^†-θ(-τ)a_k^†a_j(τ).Note that G(τ) is a real and symmetric matrix.Each element G_jk(τ) (and therefore the entire matrix G(τ)) is discontinuous when going from negative to positive times, but this discontinuity is not a problem since we only need to treat explicitly positive times τ>0 while negative τ's are accessible by the anti-periodic relation for G(τ)G(τ)=-G(τ+β), -β<τ<0,as directly verified by substitution in Eq. (<ref>). Hence G(τ) can be expanded as a Fourier series involving the Matsubara frequencies ω_n=(2n+1)π/β :G(τ) =1/β∑_n=-∞^∞G(iω_n)e^-iω_nτwhere: G(iω_n)=∫_0^βG(τ)e^iω_nτdτ. The Green's function of Eq. (<ref>) gives access to the reduced density matrix by taking the imaginary time τ as a negative infinitesimal (denoted as 0^-):[email protected]_kj=2G_kj(0^-)=-2G_kj(β^-) =2/β∑_n=0^∞e^-iω_n0^-G_kj(iω_n). Hence, all thermal averages of one-electron operators are accessible through the sum of the Matsubara coefficients. Perturbation theory can be used to build approximations for G(τ) based on a non-interacting Green's function G_0(τ) corresponding to a reference one-body Hamiltonian Ĥ_̂0̂=∑_ijF_ija_i^†a_j. Here, F is any real symmetric “Fock” matrix such that Ĥ_̂0̂ well approximates the interacting electron Hamiltonian. The derivation of G_0(τ) requires orthogonal combination of the basis set, i.e., finding a matrix X that fulfills XX^T=S^-1. Then it is straightforward to show thatG_0(τ) = Xe^-τ(F̅-μ)[θ(-τ)/1+e^β(F̅-μ)-θ(τ)/1+e^-β(F̅-μ)]X^Twhere F̅=X^TFX is the Fock matrix in the orthogonal basis set. Note that for positive (or negative) imaginary times G_0(τ) is a real, smooth and non-oscillatory Green's function. This is important for us since it much easier to stochastically sample a smooth function. Integration of Eqs. (<ref>) yields: G_0(iω_n)=((μ+iω_n)S-F)^-1.Since we now know how to write down Green's functions for non-interacting systems, we rewrite the unknown part of the exact Green's function by introducing the frequency-dependent self-energy, formally defined by:G(iω_n)=((μ+iω_n)S-F-Σ(iω_n))^-1,and by construction the self-energy fulfills the Dyson equation:G(iω_n)=G_0(iω_n)+G_0(iω_n)Σ(iω_n)G(iω_n).Instead of viewing these equations as a definition of the self-energy Σ(iω_n), we can calculate this self-energy to a given order of perturbation theory in ΔĤ=Ĥ-Ĥ_0. Specifically, the GF2 approximation<cit.> uses a Hartree-Fock ansatz for F, F_ij=h_ij+1/2P_kl(2v_ijkl-v_ilkj),where an Einstein summation convention is used, summing indices that appear in pairs (here, both k and l). The self-energy in imaginary time Σ(τ) is then obtained by second order perturbation theory (see Fig. (<ref>)): Σ_ij(τ) =G_kl(τ)G_mn(τ)G_pq(β-τ)v_impk(2v_jnlq-v_jlnq).Note that Σ(τ) and Σ(iω_n) are connected by exactly the same Matsubara relations connecting G(τ) and G(iω_n), Eqs. <ref>-<ref>.The self-consistent one-body Green's function governs all one-body expectation values. Moreover, even the total two-body potential energy is available, by differentiation of the matrix trace (denoted by []) of the Green's function with respect to τ: ⟨V̂⟩ =-1/2lim_τ→0^-[((∂/∂τ-μ)S+h)G(τ)]. Hence, the total energy is:⟨Ĥ⟩ = Tr[hP-1/2lim_τ→0^-((∂/∂τ-μ)S+h)G(τ)].It is easy to show by plugging the definition of G(τ) to Eq. (<ref>) that this total energy has convenient frequency and time forms: ⟨Ĥ⟩=1/2[(h+F)P]+2/β∑_n[G(iω_n)Σ^T(iω_n)] =1/2[(h+F)P]+2∫_0^βTr[G(β-τ)Σ(τ)]dτ. To conclude, the combination of Eqs. (<ref>), (<ref>), (<ref>) and (<ref>) along with the requirement that the density matrix describes N_e electrons results in the following self-consistent GF2 procedure: * Perform a standard HF calculation and obtain a starting guess for the Fock matrix F=F_HF and the density matrix P=P_HF. Set Σ(iω_n)=0 for the set of N_ω positive Matsubara frequencies ω_n, n=0,1,2,…,N_ω-1., .* Given Σ(iω_n) and F, find μ such that Tr[PS]=N_e, where P is given in Eq. (<ref>) from G(τ=β^-) which depends on μ through the basic definition Eq. (<ref>).* Calculate G(τ) (Eq. (<ref>)) and P (Eq. (<ref>)).* Calculate the Fock matrix F from P (Eq. (<ref>)).* Calculate the self-energy Σ(τ) from Eq. (<ref>) and transform is to the Matsubara frequency domain to yield Σ(iω_n).* Calculate the total energy⟨Ĥ⟩ from Eq. (<ref>).* Repeat steps 2-6 until convergence of the density and the total energy.Once converged, the GF2 correlation energy is defined as the difference E_ corr=⟨Ĥ⟩ -E_ HF between the converged total energy (Eq. (<ref>)) and the initial Hartree-Fock energy, E_HF=1/2[(h+F_HF)P_HF]. Note that in the first iteration GF2 yields automatically the temperature-dependent MP2 energy:E_ MP2^corr=∫_0^βTr[G_0(β-τ;F_HF)Σ_0(τ)]dτ,where Σ_0(τ) is that of Eq. (<ref>) with G_0 replacing G. This expression reduces to the familiar MP2 energy expression at the limit β→∞ (zero temperature limit), when evaluated in the molecular orbital basis set that diagonalizes the matrix F_HF. Finally, a technical point. The representation of the Green's functions in τ-space can be complicated when the energy range of the eigenvalues of F is large since a function of the type e^-τ(f-μ)/(1+e^-β(f-μ)) can be spiky when f>μ and τ→β or when f<μ and τ→0. This requires special techniques for both imaginary time and frequency grids as discussed in Refs. . §.§ sGF2: Stochastic approach to GF2 Most of the computational steps in the above algorithm scale with system size N (number of AO basis functions) as O(N_SC× N_τ× N^3) where N_SC is the number of GF2 self-consistent iterations and N_τ is the number of time-steps. However, the main numerical challenge in GF2 is step <ref> (Eq. (<ref>)) which scales formally as O(N_SC× N_τ× N^5) making GF2 highly expensive for any reasonably sized system. This steep scaling is due to the contraction of two 4-index tensors with three Green's function matrices. To reduce this high complexity, we turn to the stochastic paradigm which represents the matrices G(τ) by an equivalent random average over stochastically chosen vectors. Fundamentally, this is based on resolving the identity operator. Specifically, for each τ we generate a vector η^0 of N components randomly set to +1 or -1. Vectors at different times τ are statistically independent, but we omit for simplicity their τ labeling. Then, the key, and trivial, observation is that average of the product of different components of η^0 is the unit matrix, which we write symbolically asη_k^0η_l^0=δ_kl.We emphasize that the equality in this equation should be interpreted to hold in the limit of averaging over infinitely many random vectors η^0. Given this separable presentation of the unit matrix, it is easy to rewrite any matrix as an average over separable vectors. Specifically, from η^0 we define the two vectors:η=√(|G(τ)|)η^0, η̅=sgn(G(τ))√(|G(τ)|)η^0,and then G_kl(τ)=η̅_kη_l.Here, the square-root matrix is √(|G(τ)|)=A√(|g|)A^T, where A(τ) is the unitary matrix of eigenvectors and g(τ) is the diagonal matrix of eigenvalues of G(τ). As a side note, we have a freedom to choose other vectors; specifically, any two vectors η̅=D̅η^0, η=Dη^0, will work if D̅D^T=G(τ). In principle, we can even use the simplest choice D̅=1, D=G(τ), corresponding to η̅=η^0 and η=G(τ)η^0. But while this latter choice has the advantage that G(τ) does not need to be diagonalized, we find that it is numerically better to use Eq. (<ref>) as it is more balanced and therefore converges faster with the number of stochastic samples. Also note that at the first iteration, where G(τ)=G_0(τ), there is no need to diagonalize G_0(τ) at different times, since it is obtained directly from the eigenstates of F̅ in Eq. (<ref>).Going back to Eq. (<ref>), we similarly separate the other two Green's function matrices appearing in Eq. (<ref>), writing them as G_mn(τ)=ξ̅_mξ_n and G_pq(β-τ)=ζ̅_pζ_q. The self-energy in Eq. (<ref>) is thenΣ_ij(τ) =η̅_kξ̅_mζ̅_pv_impk(2η_lξ_nζ_qv_jnlq-η_lξ_nζ_qv_jlnq),so that it is separable to a product of two termsΣ_ij(τ)=u̅_i[2u_j-w_j],where we defined three auxiliary vectors u̅_i=η̅_kξ̅_mζ̅_pv_impku_j=η_lξ_nζ_qv_jnlqw_i=η_lξ_nζ_qv_jlnq.The self-energy in Eq. (<ref>) should be viewed as the average, over the stochastic vectors ξ^0, η^0 and ζ^0, of the product term (u̅_i times 2u_j-w_j).The direct calculation of the vectors u̅, u, w by Eq. (<ref>) is numerically expensive once M>30 . We reduce the scaling by recalling the definition of v_jnlq in Eq. (<ref>):u_j=η_lξ_nζ_q∬ϕ_j(r)ϕ_n(r)v(|r-r^'|)ϕ_l(r^')ϕ_q(r^')drdr^' =∬ϕ_j(r)ξ(r)v(|r-r^'|)η(r^')ζ(r^')drdr^',where:η(r)=η_lϕ_l(r),and ξ(r) and ζ(r) are analogously defined. We can therefore write u_j=∫ϕ_j(r)ξ(r)v_ηζ(r)dr,wherev_ηζ(r)≡∫ v(|r-r^'|)η(r^')ζ(r^')dr^'is the Coulomb potential corresponding to the random charge distribution η(r)ζ(r). Similar expressions apply for u̅_i and w_j. Equations (<ref>)-(<ref>) are performed numerically using FFT methods on a 3D Cartesian grid with N_g grid points, so Eq. (<ref>) is calculated with O(N_glog N_g) operations. Since the AO basis functions ϕ_i(r) are local in 3D space, the calculations of η(r), ξ(r) and ζ(r) in Eq. (<ref>) scale linearly with system size. Eq. (<ref>) gives an exact expression for Σ_ij(τ), as an expected value over formally an infinite number of stochastic orbitals η^0, ξ^0 and ζ^0. Actual calculations use a finite number I of “stochastic iterations”, where in each such iteration a set of stochastic vectors η^0, ξ^0 and ζ^0 (different at each τ) is generated and Σ_ij(τ) is averaged over them. The overall scaling of this step is therefore I× N_τ×(N_glog N_g+N^2). We note that the typical values of N_τ and I are in the hundreds, see the discussion of the stochastic error below. Finally, we note that while the stochastic vectors (η^0,ζ^0,ξ^0) are statistically independent for each time point τ, the same τ-dependent vectors are used at each GF2 iteration, making it possible to converge these iterations.§ RESULTS§.§ Systems and specifics The algorithm was tested on linear hydrogen chains, (H_M) a nearest neighbor distance of 1�, for several sizes: M=10,100,300 and 1000. The linearity was for convenience and we emphasize that it does not play any role in the algorithm. The smallest chain was used to demonstrate the convergence of the approach to the basis-set deterministic values, and the other three calculations were used to study the dependence of the algorithm on system size. In all calculations, an STO-3G basis was used, so that in this case N=M and obviously the number of electrons is also N_e=M. A periodic spatial grid of 0.5a_0 spacing was used to represent the wave functions, and the grids contained 10×10 points in the direction orthogonal to chain and between 60 and 4000 points along the chain, depending on system size. For the smallest system (H_10) a finer, bigger grid was also used, as detailed below. Other, technical details:* Periodic images were screened using the method of Ref. .* The inverse temperature was β=50E_h^-1. * A Chebyshev-type imaginary-time grid with 128 time points was employed using a spline-fit method <cit.> for the frequency-to-time conversions of G(iω_n) and Σ(iω_n) and for the evaluation of the two-body energy.§.§ Small system In our GF2 and MP2 algorithm, we make two types of numerical discretizations. First, we use a finite number (labeled I) of stochastic iterations to sample the self-energy, so we must show convergence as I grows. Second, we use grids for bypassing the need to sum over O(N^4) two-electron integrals, hence we need to demonstrate convergence with respect to grid quality. We therefore examine in this section a small system, linear H_10, and make four types of GF2/MP2 correlation energy calculations:* DET: fully deterministic calculations based on the analytical 2-electron integrals; * STOC(I)-NG: stochastic calculations based on I stochastic iterations and on the analytical two electron integrals; * STOC(I)-G1 and STOC(I)-G2: stochastic calculations based on I stochastic iterations and on a 3D grid. Here, G1 is the same type of grid we use for the larger calculations, and includes 10×10×60 points with a spacing h=0.5a_0. G2 is somewhat denser and covers more space, with 16×16×100 points and h=0.4a_0. Our strategy is to first show that STOC-NG(I) converges to the deterministic set (DET) as I grows. Then we show that for a given number of stochastic orbitals, I=800, both grid results are quite close to the non-grid result, and that the somewhat better second grid (STOC(I=800)-G2 leads to extremely close results to the non-grid values (STOC(I=800)-NG), so that the convergence with grid is very rapid. We repeat the STOC-NG calculation 10 times determining the average correlation energy E̅ and its standard deviation σ as a function of I. The results are shown in the left panels of Fig. <ref> as error-bars at E̅±σ, which shrink approximately as 1/√(I) and which include the DET result, represented as dashed horizontal lines, showing very small or no bias. For MP2, a bias in the stochastic calculations is not expected since the correlation energy is calculated linearly from the first iteration of the self-energy Σ_0 (Eq. (<ref>)). But for GF2 such a bias may form since the the “noisy” self-energy is used non-linearly to update the Green's function in Eq. (<ref>). However, for this small N=10 system the stochastic MP2 and GF2 energies do not exhibit a noticeable bias. We discuss the bias in larger systems below. Next, we asses the errors associated with using grid calculations replacing the analytical 2-electron integration. In both right panels of Fig. <ref> we show 10 blue dots, each corresponding to a pair of stochastic energies (E_STOC-G1(800),E_STOC-NG(800))_i , i=1,…,10, both calculated with the same random seed s_i (of course s_i and s_j are statistically independent). We also show 10 orange dots, each corresponding to a pair of stochastic energies (E_STOC-G2(800),E_STOC-NG(800))_i, also calculated with the same seed s_i as before. The use of the same seeds for each pair of blue and orange dots allows for comparison of the grid error (which is the horizontal distance of a point from the diagonal) without worrying about the larger statistical error, seen as the spread of the results along the diagonal. We see that the grid error decreases significantly when moving from G1 to G2, but even the error for G1 is already very small (about 0.5meV per electron).§.§ Larger systems In the small system considered above the bias was not noticeable and here we examine the bias in larger systems. In Fig. <ref> we show the STOC-G1(I) correlation energies in three specific systems composed of N=100,300 and 1000 hydrogen atoms placed on a straight line with a nearest neighbor spacing of 1 �. We first study the MP2 correlation energy of each system, appearing in the lower energy range in the figure. The starting point of the GF2 calculation is the Hartree-Fock F_HF and P_HF matrices, so the MP2 energy is half the correlation energy of the first self-consistent iteration (see Eqs. (<ref>) and (<ref>)). The statistical errors in MP2 are pure fluctuations, a random number distributed normally with zero average and with standard deviation given by σ_0/√(I) where σ_0 is independent of I but shrinks with chain length: σ_0∝1/√(L), exhibiting “self averaging”. <cit.> The stochastic MP2 errors are very small and decrease with system size, so for N=1000 the standard deviation of the I=800 iteration calculation is 0.07% of the total correlation energy. For perspective, note that (deterministic) errors of larger or similar magnitude are present in linear scaling local or divide and conquer MP2 methods with density fitting. <cit.>Next, we discuss the stochastic estimates of the self-consistent GF2 correlation energies. These exhibit statistical errors with two visible components. The first is a fluctuation, similar in nature to that of the MP2 calculation, and the second component is a bias which decreases as I grows. In fact, we expect the bias to asymptotically decrease inversely with I, [A bias arises whenever we plug a random variable x, having an expected value μ and variance σ^2, into a nonlinear function f(x). One cannot hope that f(x) will have the expected value of f(μ) unless f is a linear function. A simple example is f(x)=x^2, where from the definition of variance ⟨ f(x) ⟩ = f(μ)+σ^2 . Using the Taylor expansion of f around μ, it is straightforward to show that f(x̅), where x̅=1/I∑_i=1^Ix_i is an average over I samples and when I is sufficiently large, ⟨ f(x̅)⟩≈ f(μ)+f”(x̅)σ^2/2I and so the bias is proportional to the variance of x, the curvature of f at μ and inversely proportional to the number of iterations I. ] so we fit the numerical GF2 results to a straight line in I^-1. Table <ref> shows the estimate of the correlation energies, the fluctuation and the bias as a function of the number of stochastic orbitals. The results are highly accurate, for example when I=800 is used for the largest system (N=1000), the errors in MP2 and in GF2 are smaller than 0.1%.Timings. The measured overall CPU time for the stochastic self-energy calculation (performed on a XEON system) can be expressed asT^Σ≈2.5× N× N_τ× I×10^-7hr,where, as mentioned, N is the number of electrons and I the number of stochastic orbitals in the system. The MP2 wall time calculation is essentially equal to the self-energy time divided by the number of cores n_CORES, since the parallelization has negligible overhead:T_wall^MP2≈T^Σ/n_CORES. GF2 involves an additional step, where the Green's function is constructed from the self-energy and this step scales cubically with system size. Furthermore, there are N_SC self-consistent iterations. The total time is therefore found to be:T_wall^GF2 ≈ N_SC(1.7× N^3× N_τ1hr+T^Σ)/n_CORES.For the H_1000 system, with I=800 stochastic orbitals, the MP2 calculation takes T_wall^MP2=24hr/n_CORES, i.e. about 30min when using 48 cores. The GF2 calculation for this same system involves N_SC=12 iterations and a cubic part which takes about 2 core-hours per iteration, i.e., the cubic part is still an order of magnitude smaller than the self-energy sampling time for this system size. The wall time is therefore T_wall^GF2=6.5hr with 48 cores. For the H_300 system we find T_wall^MP2=12min and T_wall^GF2=2hr while for H_100 we have T_wall^MP2=3.5min and T_wall^GF2=40 min. Note that these timings are for a single calculation. The error estimation uses, as mentioned, ten completely independent runs, and therefore took 10 times longer. For comparison, we note that the CPU time for the deterministic calculation in the H_100 system takes 45 min. on a single core, which is 4 times faster than the stochastic calculation. Since the deterministic algorithm scales steeply as O(N^5), the crossover occurs already at H_150 and at H_1000 the deterministic calculation would take 10^4-10^5/n_CORES wall time hours per SCF iteration, compared to 24/n_CORES hours for the stochastic calculations.§.§ Born Oppenheimer potential curves Potential energy curves can be calculated by correlated sampling, where at each new nuclear configuration one employs the same set of stochastic orbitals η_0,ξ_0 and ζ_0 for the self-energy estimation. For demonstration, the HF, MP2 and GF2 Born Oppenheimer potentials of the H_100 system are shown in Fig. <ref> as a function of the displacement of atom no. 25 (counting from the left). In all three methods the most stable position of the atom is at ∼-0.1a_0, slightly displaced towards the nearest chain end. HF theory produces an energy potential with large variations of up to 1.5eV and large vibrational frequencies of order of 3.4eV. The MP2 curve is much smoother and the vibrational frequencies reduces to ∼2.4eV while the GF2 energy curve is considerably flatter, predicting a vibrational frequency of ∼1.0 eV.§ SUMMARY AND CONCLUSIONS The problem we addressed here is the reduction of the the steep O( N^5) scaling associated with the implementation of self-consistent GF2 calculations. We developed an effective way to reduce complexity to O( N^3) by using stochastic techniques for calculating the self-energy. A detailed derivation was given along with a specific algorithm. The sampling error in the overall algorithm was studied for linear H_N systems, and the simulation showed that the stochastic errors in the correlation energies can be controlled to less than 0.1% for very large systems. While the studied systems were linear, the algorithm makes no use of the linearity and applies equally well to any geometry.As a byproduct, since the first step in GF2 is equivalent to MP2, we obtain a stochastic MP2 method (sMP2) performed on top of an existing HF calculation. This approach too has a formal complexity of O(N^5) which is reduced here to linear O( N), except for a single overall Fock-matrix diagonalization which is often available from the underlying HF or DFT ground-state calculation. The errors in this well-scaling stochastic MP2 method are comparable to those of local MP2 approaches used in quantum chemistry. For GF2, the method has two main stages. The first stage, as in the MP2 case, is a linear scaling calculation of the self-energy. This self-energy is then used in the second stage to construct the Green's function, at an O(N^3) cost. A complication arises in GF2 due to this second stage (but not in MP2!), where the self-energy enters non-linearly into the expression for the Green's function. This non-linearity gives rise to a noticeable bias which is proportional to the system size N. To overcome this bias the number of stochastic orbitals I used in the first step must be increased in proportion to the system size N, and hence the self-energy calculation in GF2 attains an O(N^2) scaling. The overall scaling of the GF2 calculation is unaffected by this bias problem and remains O(N^3).The present calculations give a fully self-consistent Green's function method for a large system with a thousand electrons described by a full quantum chemistry Hamiltonian. Moreover, we demonstrated that the splitting of matrices by a random average over stochastically chosen vectors leads to small variance and that relatively few Monte Carlo samples already yield quite accurate correlation energies. The reason for this excellent sampling dependence is two-fold: the stochastic sampling inherently acts only in the space of atomic orbitals while the actual spatial integrals (Eq. (<ref>)) are evaluated using a deterministic, numerically exact calculation; in addition, since the Green's function matrices are smooth in imaginary time, different random vectors can be used at each imaginary-time point thereby enhancing the stochastic sampling efficiency.We have shown that both sMP2 and sGF2 are suitable for calculating potential energy curves or surfaces. Interestingly, for the H_N systems the potential curve is much smoother and flatter than in HF or MP2. As for future applications, we note that sGF2 and sMP2 methods are automatically suitable for periodic systems, as all the deterministic steps and the time-frequency transforms are very efficient when done in the reciprocal (k) space. The only additional detail is that in periodic systems one needs to choose the random vectors to be in k-space and then convert them to real-space, as detailed in an upcoming article.Finally, we also note that, beyond the results presented here, it should also be possible to achieve further reduction of the stochastic error with an embedded fragment approach, analogous to self-energy embedding approaches, where a deterministic self-energy is calculated for embedded saturated fragments as introduced for stochastic DFT applications. <cit.>Discussions with Eran Rabani are gratefully acknowledged. 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Efficient temperature-dependent Green's functions methods for realistic systems: Compact grids for orthogonal polynomial transforms. J. Chem. Theory Comput. 2016, 12, 564–571 [Kananenka et al.(2016)Kananenka, Welden, Lan, Gull, and Zgid]Kananenka16 Kananenka, A. A.; Welden, A. R.; Lan, T. N.; Gull, E.; Zgid, D. Efficient Temperature-Dependent Green's Function Methods for Realistic Systems: Using Cubic Spline Interpolation to Approximate Matsubara Green's Functions. J. Chem. Theory Comput. 2016, 12, 2250–2259 [Martyna and Tuckerman(1999)Martyna, and Tuckerman]Martyna1999 Martyna, G. J.; Tuckerman, M. E. A reciprocal space based method for treating long range interactions in ab initio and force-field-based calculations in clusters. J. Chem. Phys. 1999, 110, 2810–2821 [Werner et al.(2003)Werner, Manby, and Knowles]Werner2003 Werner, H. J.; Manby, F. R.; Knowles, P. J. Fast linear scaling second-order Moller-Plesset perturbation theory (MP2) using local and density fitting approximations. J. Chem. Phys. 2003, 118, 8149–8160 [Baudin et al.(2016)Baudin, Ettenhuber, Reine, Kristensen, and Kjærgaard]Baudin2016 Baudin, P.; Ettenhuber, P.; Reine, S.; Kristensen, K.; Kjærgaard, T. Efficient linear-scaling second-order Møller-Plesset perturbation theory: The divide–expand–consolidate RI-MP2 model. The Journal of chemical physics 2016, 144, 054102 | http://arxiv.org/abs/1707.08296v2 | {
"authors": [
"Daniel Neuhauser",
"Roi Baer",
"Dominika Zgid"
],
"categories": [
"physics.chem-ph",
"cond-mat.mes-hall",
"cond-mat.mtrl-sci",
"cond-mat.str-el",
"physics.comp-ph"
],
"primary_category": "physics.chem-ph",
"published": "20170726062209",
"title": "Stochastic self-consistent second-order Green's function method for correlation energies of large electronic systems"
} |
Budker,NSU,NSTU]A. Yu. Barnyakov Budker,NSU]M. Yu. Barnyakov MiB]L. Brianza INFNRm]F. Cavallari INFNRm,Roma]M. Cipriani MiB]V. Ciriolo INFNRm,Roma]D. del Re INFNRm,Roma]S. Gelli MiB]A. Ghezzi MiB]C. Gotti MiB]P. Govoni Budker,NSU]A. A. Katcin MiB]M. Malberti MiB]A. Martellinowatcern INFNRm,Roma]B. Marzocchi INFNRm]P. Meridiani INFNRm,Roma]G. Organtini INFNRm,Roma]R. Paramatti MiB]S. Pigazzini INFNRm,Roma]F. Preiato Budker,NSU]V. G. Prisekin INFNRm,Roma]S. Rahatlou INFNRm]C. Rovellimycorrespondingauthor INFNRm,Roma]F. Santanastasio MiB]T. Tabarelli de Fatis[nowatcern]Now at CERN [mycorrespondingauthor]Corresponding author: [email protected][Budker]Budker Institute of Nuclear Physics, Lavrentieva 11, Novosibirsk 630090, Russia [NSU]Novosibirsk State University, Pirogova 2, Novosibirsk 630090, Russia [NSTU]Novosibirsk State Technical University, Karla Marksa 20, Novosibirsk 630073, Russia [MiB]Università di Milano-Bicocca and INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3, 20126, Milano, Italy [INFNRm]INFN, Sezione di Roma, Piazzale A. Moro 2, 00185, Roma, Italy[Roma]Sapienza, Università di Roma, Piazzale A. Moro 2, 00185, Roma, ItalyHundreds of concurrent collisions per bunch crossing are expected at future hadron colliders. Precision timing calorimetry has been advocated as a way to mitigate the pileup effects and, thanks to their excellent time resolution, microchannel plates (MCPs) are good candidate detectors for this goal.We report on the response of MCPs, used as secondary emission detectors,to single relativistic particles and to electromagnetic showers. Several prototypes, with different geometries and characteristics, were exposed to particle beams at theINFN-LNF Beam Test Facility and at CERN. Their time resolution and efficiencyare measured for single particles and as a function of the multiplicity of particles. Efficiencies between 50% and 90% to single relativistic particles are reached,and up to 100% in presence of a large number of particles. Time resolutionsbetween 20 ps and 30 ps are obtained. 29.40.Vj, 85.60.Ha, 79.20.Hx § INTRODUCTIONThe projected beam intensity of future hadron colliders <cit.><cit.> will result in several hundreds of concurrent collisions per bunch crossing, spread over a luminous region of a few centimeters along the beam axis and of about a few 100 ps in time. The scientific program at these colliders, which includes precision characterization of the Higgs boson, measurements of vector boson scattering, and searches for new heavy or exotic particles, will benefit greatly from the enormous dataset. However, particle reconstruction and correct assignment to primary interaction vertices at high vertex densities presents a formidable challenge to the detectors that must be overcome in order to harvest that benefit. Time tagging of minimum ionizing particles (MIPs) and of neutral particles in the calorimeters with a resolution of a few 10 ps provides further discrimination of the interaction vertices in the same bunch crossing, beyond spatial tracking algorithms <cit.>.Among other sensors that are being investigated for precision timing of charged tracks, microchannel plates (MCPs) <cit.>, renowned for their fast response, have been advocated as a candidate detector to time tag electromagnetic showers and MIPs and tested with moderate success <cit.><cit.><cit.><cit.>. In a previous work <cit.>, we reported time resolutions of about 50 ps with cosmic muons and efficiencies of about 50% in thedetection of single relativistic charged particles. The detectors consisted in a stack of two MCP layers with the dual function of seeding the cascade process, via the secondary electrons extracted from the MCP surface by the incoming ionizing particle, and of providing signal amplification. Similar studies <cit.><cit.><cit.> reported results comparable to ours for MIPs and virtuallyfull efficiency in the detection of electromagnetic showers, at shower depths where the track multiplicity is high,with time resolutions at the level of a few tens of ps. In this paper, we further characterize the response of MCPs in the direct detection of ionizing particles,hereafter referred to as `ionization-MCPs' or shortly `i-MCPs'.The potential advantage of i-MCPs consists in the elimination of the photocathode, resulting in a easier andmore robust construction and in a potentially larger radiation tolerance. We report on the dependence of the i-MCP performance on the stack geometry (number of layers or aspect ratio) and the use of MCP with high emissivity layers is also investigated. Several different prototypes of i-MCP detectors were exposed to charged particles beams.After the description of the detectors and of the measurement setup (Sec. <ref> and Sec. <ref>)we present results in terms of efficiency and time resolution in response to single particles (Sec. <ref>).We report also about the behaviour of i-MCP prototypes in response to electromagnetic showers at different depths (Sec. <ref>).Due to their excellent time response, indeed, a layer of MCPs embedded in a calorimeter or in a preshower could be exploited to provide a precise time response for photons.§ DETECTORS DESCRIPTION AND OPERATIONThe usage of MCPs as secondary electron emitters is investigated using either sealed MCP devicesdeveloped at BINP (Novosibirsk) in collaboration with the Ekran FEP manufacturer <cit.> or layers by the Photonis and Incom manufacturers mounted inside a vacuum chamber (with pressure kept below 10^-5 mbar using a turbo molecular pump). The detectors are operated in `i-MCP mode', therefore when a photocathode is present a retarding biasis applied to the gap between the photocathode itself and the MCP, as opposed to the standard `PMT-MCP' mode. This prevents photoelectrons emitted from the photocathode from reaching the MCP surface and triggering an avalanche. In this configuration, the response of the detector is uniquely determined by the secondary emission of electrons from the MCP layers which are crossed by ionizing particles.Devices are built with two, three or four layers of lead glass MCPs, except for one prototype made of borosilicate glass coated with emissive and resistive layers. The channel diameter varies in the range from 3.5 μm to 25 μm, and its ratio to the layer thickness (aspect ratio) ranges from 1:40 to 1:90 depending on the prototype. The channels have a bias angle to the photodetector axis of a few degrees. Two different electrical configurations are tested, which are sketched in Fig. <ref>. In the first one, same as in <cit.>, a voltage divider is used to provide about 90% of the voltage drop through the entire MCP stack (2 or 3 layers). The layers carry out the dual function of seeding the cascade process, via secondary electrons extracted from the MCP, and providing signal amplification. In the second configuration the MCP layers are separated by the presence of electrodes, dividing a stage where the emission happens with large probability from the amplification stage.More details on this approach will be given in the following.The specifications of the detectors are detailed in Tab. <ref> and Tab. <ref> for the two configurations. The prototypes which are mounted inside a vacuum chamber (VC) have slightly less than one radiation length of material in front due to the steel flange closing the chamber. § EXPERIMENTAL SETUPThe i-MCP detectors were exposed to particle beams, in order to characterize their response to single particles and electromagnetic showers in terms of detection efficiency and time resolution.The devices without separation of the emission and amplification stages were tested at the H4 <cit.> andH2 <cit.> beam lines at the CERN North Area. The electron beam at the H4 Area is extracted from the CERN SPS and can be tuned in the momentum range from 10 GeV to 200 GeV. Our efficiency and time resolution measurements were performed with 20 GeV and 50 GeV electrons. The MCP devices were mounted in a box with the optical window orthogonal to the beam direction. Signals were read from a 32-channels digitizer (CAEN V1742) with 5 GHz sampling frequency. A hodoscope was placed along the beam line upstream of the MCPs and readout into a gated-ADC, for beam centering and offline selection purposes. It consisted of four planes with 64 parallel scintillating fibers each, 1 mm in diameter and staggered, covering an acceptance of 20×20 mm^2 in the coordinates transverse to the beam. One of the MCPs along the beam line was operated in PMT-MCP mode, to provide a trigger and a reference for the efficiency and time measurements.The H2 beam is a secondary particle beam extracted from the CERN SPS providing hadrons, electrons or muons of energies between 10 GeV and 360 GeV. In this case, our efficiency measurements were performedwith 150 GeV muons with a similar setup as in H4.The devices with separated emission and amplification stages were tested at the T9 area at CERN <cit.>and at the Beam Test Facility (BTF) of the INFN Laboratori Nazionali di Frascati (Italy) <cit.>. In the T9 Area a secondary beam with momentum range from 1 GeV to 5 GeV is originated from the CERN PS beam impacting on a target.Our measurements were performed on a beam of 2 GeV electrons (20%) and pions (80%).The experimental setup was similar to the one used at the North Area and aCherenkov detector was used to separate the electron and pion components of the beam. The BTF in Frascati provides 10 ns long electron pulses with tunable energy (up to about 500 MeV), repetition rate (up to 49 Hz) and intensity (from 1 to 10^10 particles per pulse). Our measurements were performed with 491 MeV electrons and an intensity tuned to provide an average of about one electron per pulse. A similar beam line equipment as in H4 was used, with only minimal differences. One MCP operated in PMT-MCP mode was placed upstream of the i-MCPs along the beam line and another one downstream, to select events going through the i-MCPs without showering. Also, a 5 mm thick plastic scintillator counter with an area of 24×24 mm^2 was installed in front of the MCPs, together with a hodoscope, to allow selecting events with single electrons impinging on the detectors. A schematic view of the BTF setup is shown in Fig. <ref>. The MCP devices produce very fast signals, with a rise-time of the order of 1 ns. A typical MCP waveform is shown in Fig. <ref>, where the secondary peaks are due to an imperfect matching between the anode and the transmission line to the digitizer.For each event, a coincidence window of 60 ns is opened around the time corresponding to the maximum amplitude of the MCP operated in PMT-MCP mode acting as a trigger. The maximum of the waveform of the i-MCP under test issearched inside this window.The signal time information is extracted from the interpolated waveforms via a constant fraction discriminator (CFD), corresponding to the time when the amplitude is half of its maximum.§ RESPONSE TO SINGLE CHARGED PARTICLESTo characterize the i-MCP detectors behaviour their efficiency and time resolution were first measured in response to single electrons and muons. Events consistent with a single particle are selected in the offline analysis requiring a pulse larger than 200 ADC counts in the reference PMT-MCP device. The hodoscope information is also used when available. At the BTF, the bunched beam structure requires a further selection for single electron events, identified from the pulse-height of the signal in the scintillator counter which is asked to be between 200 and 700 ADC counts. Furthermore, a signal between 200 and 1200 ADC counts is required in the PMT-MCP downstream of the i-MCP (the upper limit is set to discard events in which the electron makes a shower along the beam line).Typical spectra of the scintillator counter and of the PMT-MCP upstream of thetest i-MCP at the BTF are shown in Fig. <ref>. §.§ Efficiency of single-stage MCP stacksThe efficiency of each MCP is defined as the fraction of events with a signal above threshold with respect to the total number of single electron events, selected as discussedin the previous section. The threshold is defined as 5 times the noise of the detector, which is measured in pedestal runs or in events without electrons impacting on the scintillator. The efficiency of some prototypes exposed to the H4 beam and operated in i-MCP mode is shown in Fig. <ref> on the left as a function of the MCP-stack bias. For a comparison, the efficiency of one of them operated in PMT-MCP mode is also shown.In PMT-MCP mode the photoelectrons are extracted from the photocathode and amplified above the detection threshold by the MCP. As it was found in <cit.>, a 100% efficiency is obtained at plateau, where the MCP gain is high enough to supply single photoelectron detection. In i-MCP mode, the MCP layers have the dual function of initiating the cascade process and providing signal amplification. Different channel lengths are involved in the amplification process, depending on where the secondary electron is emitted, and inefficiencies may arise either because of lack of secondary emission or because of insufficient amplification in the cascade following the secondary emission. A model was developed in our earlier work <cit.> in which the detection efficiency is parameterized as ε = s (1-1/b ·ln(V/V_th)+1), (V ≥ V_th)under the hypothesis that the gain has a power-law dependence on the bias voltage. The model parameters V_th, s and b are extracted from data. The parameter s represents the probability of secondary emission over the entire MCP length and it is found to be compatible with one for the detectors under study. The parameter b describes the gain and depends on the device. V_th is the threshold voltage at which secondary electrons generated at the surface of the MCP receive the exact gain needed for detection. The behaviour of the efficiency versus the bias voltage can be explained with a change in the volume of the sensitive region, that is the region where an electron receives enough amplification to be detected as a signal. At the threshold voltage, the whole MCP length is needed in order to amplify the signal over a detectable threshold and the signal is detected only if the secondary emission occurs at the surface of the MCP. The efficiency increases with the bias voltage, since signals overcome the detection threshold also if the secondary emission occurs inside the MCP layers. The curves in Fig. <ref> are extrapolated to voltage values which can not be currently reached due to the MCP electric strength.The prototype with two layers (40×2) reaches a maximum efficiency of about 50% for single electrons, confirming the results in <cit.>. Adding a third layer (40×3) the efficiency raises up to about 70%, since thenumber of channel interfaces crossed by the particle is increased. As suggested in <cit.>, and demonstrated in Fig. <ref>, the detection efficiency increases with the number of layers. However, even with three layers the curve is not at plateau, suggesting that the efficiency gain is too slow.The efficiency is larger, for the same channel length, in i-MCPs with enhanced secondary emission yield. The SEE and MGO prototypes, also shown in Fig. <ref>, are two-layers i-MCPs. The SEE prototype underwent a surface treatment in order to increase the channel walls secondaryemission efficiency. The MGO prototype is made of layers produced from a borosilicate glass substrate with an atomic layer deposition technique (ALD by INCOM Ltd <cit.><cit.>) and a final layer of Magnesium oxide. The SEE device reaches an efficiency of about 50% at a stack bias of about 1750 V, while for regular i-MCPs with 2 layers the same efficiency is reached at about 3000 V. Also, the efficiency raises up to about 70%. The same efficiency is reached for the MGO prototype. Owing to an accident while improving the test setup, we could not study the response of the MGO prototype at biases larger than 2400 V. The analysis however shows that with MgO coatings efficiencies as high as 90% could be attained still at moderate high voltages.For all the i-MCP prototypes, data are well described by the model of Eq. <ref>, suggesting that theefficiency trend with the voltage does not depend on the specific characteristics of the detector.The universality of the model for single stage i-MCPs is further demonstrated in the right panel of Fig. <ref>, where the efficiency is shownas a function of the stack bias normalized to the threshold voltage, extracted from a fit of the functional form of Eq. <ref> to the data. §.§ Efficiency of multi-stage MCP stacksAn alternative way to improve the i-MCP behaviour consists in separating the region where the signal is created from the amplification stage, providing a separate bias for the two layers. This configuration is motivated by the fact that a few photoelectrons entering two or three MCP layers give a clear signal above noise, as demonstrated operating the detectors in PMT-MCP mode.In Fig. <ref>we show the efficiency of the multi-stage prototypes as a function of the MCP-stack bias, which is varied to change the response of the devices. The amplification stage voltage is fixed. For the detectors in the vacuum chamber (80×1+40×1, 80×2+40×1) the probability for a single electron to originate a shower before reaching the MCP is not negligible. A correction factor is therefore computed using the data collected at the T9 beam area, where one of the i-MCPs was exposed to a mixed beam of electrons and pions. The electron and pion components were separated with a Cherenkov detector and the efficiency in response to each component was computed. The i-MCP efficiency in response to electrons islarger than to pions, which means that also electrons produced in the primary electron cascade may generate a signal. Describing the i-MCP response with a binomial per-event probabilityε_n = 1 - (1 - ε_1)^n,where ε_1 is the efficiency to single particles and n is the multiplicity of electrons per electron crossing the MCP, we find an average n of about 1.4. The efficiencies measured at the BTF on the i-MCPs inside vacuum chambers are corrected for this factor, neglecting the difference in moving from 2 GeV to 491 MeV.Three of the multi-stage MCPs have the same amplification region, consisting of two layers with aspect ratio 40:1 (40×2) operated at a voltage around 2100 V. The efficiency of these MCPs in response to charged particles does not reach zero even at low stack bias, the residual efficiency being given by the amplification stage itself. Experimental results suggest that the efficiency is larger using layers with large aspect ratio in the emission stage (90×2+40×2, 90×1+40×2).An efficiency larger than 90% is reached in the configuration 90×2+40×2, which is almost 20% larger than with small aspect ratio, e.g. 40×2+40×2. In the other two chambers the amplification region consists of one layer with aspect ratio 40:1 (40×1). Also in this case, the efficiency increases by 20% adding an extra layer in the emission stage (80×2+40×1 with respect to 80×1+40×1). In summary, the efficiency can be increased either building chambers with thicker layers or adding extra layers. Since the prototypes 40×3, discussed in Section <ref>, and 80×1+40×1 reach similar efficiencies, the relevant factor appears to be only the total thickness, independently of the number of layers which are employed. Finally, the prototypes 90×2+40×2 and 80×2+40×1 reach the same plateau efficiency, while the efficiency at low bias voltage is larger for the stack with an additional amplification layer. This suggests that the addition of extra layers in the amplification stage gives only a small contribution to the total efficiency if the efficiency of the emission stage is large enough.The efficiency as a function of the emission stage bias (HV) for multi-stage i-MCPs is parameterized in Fig. <ref> asε = P_amp(1-P_em(HV))+P_em(HV)P_amp is the contribution of the amplification layer only, which may have an efficiency larger than zero even when no electron is generated in the emission layers and which does not depend on the voltage of the emission stage. P_amp is compatible with zero for the devices where the amplification stage is made of a single layer MCP, and it is slightly above 20% for the devices with double layer amplification stage. P_em may be parameterized as in Eq. <ref>. This model well describes the data as a function of the voltage of the emission stage, which suggests that the parameterization in Eq. <ref> works well for every single stage i-MCP detector. §.§ Time resolution The time resolution of i-MCP detectors exposed to a single particle beam is extracted comparing the time of the detector under study with the time of a reference MCP operated in PMT-MCP mode and put upstream of the i-MCPs along the beam line. Pulses consistent with a single electron entering the test setup are selected in the same way as for the efficiency measurement. The signal time information is estimated with the CFD methodand only the events where the time of the i-MCP detector is compatible with the time of the reference PMT-MCP within 1 ns are retained for the analysis.When a detector is operated in i-MCP mode, differently from the PMT-MCP mode case, time non-linearities as a function of the signal amplitude arise. The time difference between signals with low and large amplitude is compatible with a fraction of the transit time, which is around 200-300 ps, therefore it is reasonable to assume that the signals with low amplitude are in average created closer to the anode than larger signals. This effect, which is shown in Fig. <ref> on the left, is corrected for with an empirical function.The distribution of the time difference between the 90×1+40×2 detector operated in i-MCP mode and the reference PMT-MCP, after the non-linearity correction, is shown in Fig. <ref> on the right in response to 491 MeV electrons and for an operating voltage of 1400 V. To compute the i-MCP time resolution a Gaussian fit to this distribution is performed, thenthe resolution of the reference PMT-MCP is subtracted. The latter is extracted comparing the relative resolution of the time difference with respect to another PMT-MCP detector and with respect to an i-MCP detector.It is found to be 17±2 ps.Fig. <ref> shows the time resolution of the 90×1+40×2 i-MCP detector as a function of the signal over noise ratio, after the resolution of the reference PMT-MCP is subtracted.The trend can be parametrized asσ = a/S/N⊕ b,from which the noise component a and the constant term b of the resolution are extracted. The parameter b determines the best resolution which can be achieved and it mainly depends on the transit time spread, i.e. the time difference due to the different paths followed by the electrons in reaching the anode. Other sources can also contribute, like imperfections in the time reconstruction or in the non-linearity corrections. In Fig. <ref> data are compared to a toy simulation which is based on a pulse shape template built from data. Uncorrelated noise is added to each sample as extracted from a pedestal run. The noise contribution a measured in data is compatible with the expectations from simulation. The current simulation does not contain sources which can contribute to the constant term, soa Gaussian smearing is added to match the data. The time resolution averaged over all the signal amplitudes is 26 ± 2 ps for this device. For a comparison, the time resolution for single photoelectron events of the same device operated in PMT-MCPmode and measured with a laser is 24 ps <cit.>.The average time resolution and the constant term of Eq. <ref> were computed for all the devices under test. The results are compatible among the different prototypes and constant terms between 20 ps and 30 ps are obtained in all cases. Only a small dependence of the time resolution on the number of layers and their aspect ratio is observed. The data collected do not allow a systematic study of the time resolution dependence on the channel diameter and layer thickness.§ RESPONSE TO ELECTROMAGNETIC SHOWERSTo characterize the i-MCP response to electromagnetic showers, data were acquired with a set of absorbers of variable thickness and up to almost 5 radiation lengths X_0 in front of the detector. The tests were performed at the CERN H4 Area.The evolution of the efficiency to detect 20 GeV electrons is shown in Fig. <ref> on the left for a 3 layers i-MCPas a function of the absorber thickness in units of X_0. The efficiency definition is the same as in Sec. <ref>. The device was operated at the maximum voltage used in the bias voltage scan performed in the single particle setup, corresponding to the highest detection efficiency in that configuration. The efficiency increases with the thickness of the absorber, reaching 95% after 1X_0 and close to 100% after 2X_0. These results, which confirm the indications in <cit.>, are promising in view of the usage of the i-MCP detectors in calorimeters at hadron colliders.The time resolution in response to electromagnetic showers initiated by 20 GeV electrons is shown as a function of the shower depth in Fig. <ref> on the right, after the removal of the contribution of the reference MCP resolution. Due to the increased multiplicity of charged particles the digitizer input can saturate and in this case the time to cross a fixed threshold of 500 ADC counts is used as time estimate,instead of the CFD algorithm. The same event selection as in Sec. <ref> is used. Fig. <ref> refers to a different detector with respect to the one used as an example in the previous section, with a larger noise but comparable constant term. This explains why the time resolution at 0X_0 is worse than in Fig. <ref> when averaging over all the signal amplitudes. The time resolution improves as expected with the absorber thickness, because the MCP layers are crossed by a larger multiplicity of particles. An inclusive time resolution of about 20 ps is reached after 5X_0.§ SUMMARY AND OUTLOOKWe report on the response of microchannel plates to single relativistic particles and to electromagnetic showers.Several prototypes of MCPs used as secondary emission detectorswere exposed to particle beams at the INFN-LNF Beam-Test Facility and at CERN. Configurations with multiple MCP stacks, different geometries or layer coatings were compared, to investigate howthese aspects affect the total amount of secondary emission and the channel gain. In the tested configurations, detection efficiencies to singlerelativistic particles between 50% and 90% and constant terms for the time resolution between 20 ps and 30 ps are reached.Measurements with electromagnetic showers sampled at different depths show that, in presence of a large enough number of particles, detection efficiencies up to 100% can be reached with average time resolutions as good as 20 ps, independently of the geometry. Present results suggest that this detection technique is suitable for a precise determination of the time of high energy photons andcharged particles, and could help in the event reconstruction at high luminosity colliders.§ ACKNOWLEDGEMENTS We warmly thank R. Bertoni, R. Mazza, M. Nuccetelli and F. Pellegrino for the preparation of the experimental setup. We are indebted to B. Buonomo, C. Di Giulio, L. Foggetta and P. Valente for their help with the setup of the beam facility at Frascati.We are grateful to the CERN PS and SPS accelerator teams for providing excellent beam quality. This research program is carried out in the iMCP R&D project, funded by the Istituto Nazionale di Fisica Nucleare (INFN) in theCommissione Scientifica Nazionale 5 (CSN5).It has also received funding from the European Union Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 707080. The production of tested BINP design exemplars and caring out their measurements in Novosibirsk was supported by the Russian Science Foundation (Project no.16-12-10221). § REFERENCES200HLLHC G. Apollinari et al., High-Luminosity Large Hadron Collider (HL-LHC):Preliminary Design Report, CERN Yellow Reports: Monographs, 2015 FCCM. Benedikt, F. Zimmermann, Future Circular Colliders, CERN-ACC-2015-164, 2015 Alice B. 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Brianza et al., Response of microchannel plates to single particles and to electromagnetic showers, Nucl.Instrum.Meth. A797:216-221, 2015, doi:10.1016/j.nima.2015.06.057 Apresyan1 R. Ronzhin et al., Development of a new fast shower maximum detector based on microchannel plates photomultipliers (MCP-PMT) as an active element, Nucl.Instrum.Meth A759:65-73, 2014,doi:10.1016/j.nima.2014.05.039 Apresyan2 R. Ronzhin et al., Direct tests of micro channel plates as the active element of a new shower maximum detector, Nucl.Instrum.Meth. A795:52-57, 2015,doi:10.1016/j.nima.2015.05.029 Apresyan3 A. Apresyan et al., Direct tests of a pixelated microchannel plate as the active element of a shower maximum detector, Nucl. Instrum. Meth. A828:1-7, 2016, doi:10.1016/j.nima.2016.05.015 Barnyakov A. Yu. Barnyakov et al., Investigation and development of microchannel plate phototubes, Nucl.Instrum.Meth. A572:404-407, 2007, doi:10.1016/j.nima. 2006.10.276 h4 http://sba.web.cern.ch/sba/BeamsAndAreas/resultbeam.asp?beamline=H4 h2 http://sba.web.cern.ch/sba/BeamsAndAreas/h2/H2manual.html t9 L.Durieu, M.Martini and A.-S.Muller, Optics studies for the T9 beam line in the CERN PS East Area secondary beam facility, Proceedings of the 2001 Particle Accelerator Conference, Chicago btf A. Ghigo et al., Commissioning of the DAFNE beam test facility. Nucl.Instrum.Meth. A515:524-542, 2003 ald S. J. Jokela et al., Secondary Electron Yield of Emissive Materials for Large-Area Micro-Channel Plate Detectors: Surface Composition and Film Thickness Dependencies, Physics Procedia, 37:740-747, 2012, doi:10.1016/j.phpro.2012.03.718 ald2 A. U. Mane et al., An Atomic Layer Deposition Method to Fabricate Economical and Robust Large Area Microchannel Plates for Photodetectors, Physics Procedia, 37:722-732, 2012, doi:10.1016/j.phpro.2012.03.720 barn A. Yu. Barnyakov et al., Micro-channel plates in ionization mode as a fast timing device for future hadron colliders, to be published in JINST, Int. Conf. “Instrumentation for Colliding Beam Physics”, 28 Feb. – 3 Mar., 2017, Novosibirsk, Russia. | http://arxiv.org/abs/1707.08503v1 | {
"authors": [
"A. Yu. Barnyakov",
"M. Yu. Barnyakov",
"L. Brianza",
"F. Cavallari",
"M. Cipriani",
"V. Ciriolo",
"D. del Re",
"S. Gelli",
"A. Ghezzi",
"C. Gotti",
"P. Govoni",
"A. A. Katcin",
"M. Malberti",
"A. Martelli",
"B. Marzocchi",
"P. Meridiani",
"G. Organtini",
"R. Paramatti",
"S. Pigazzini",
"F. Preiato",
"V. G. Prisekin",
"S. Rahatlou",
"C. Rovelli",
"F. Santanastasio",
"T. Tabarelli de Fatis"
],
"categories": [
"physics.ins-det"
],
"primary_category": "physics.ins-det",
"published": "20170726153946",
"title": "Response of microchannel plates in ionization mode to single particles and electromagnetic showers"
} |
M[1]>m#1 OMLcmmbit⟨⟩ł↑̆↓̣ We present a detail theoretical study of the Drude weight andoptical conductivity of 8-Pmmn borophene having tilted anisotropicDirac cones.We provide exact analytical expressions of xx and yy components ofthe Drude weight as well as maximum optical conductivity. We also obtainexact analytical expressions ofthe minimum energy (ϵ_1) required to trigger the optical transitionsand energy (ϵ_2) needed to attain maximum optical conductivity.We find that the Drude weight and optical conductivity are highly anisotropic as aconsequence of the tilted Dirac cone. The tilted parameter can be extracted byknowing ϵ_1 andϵ_2 from optical measurements. The maximumvalues of the components of the optical conductivity do not depend on the carrierdensity and the tilted parameter. The product of the maximum values of the anisotropicconductivities has the universal value (e^2/4ħ)^2. The tilted anisotropic Dirac cones in 8-Pmmn borophene can be realized bythe optical conductivity measurement.Effect of electron-hole asymmetry on optical conductivity in 8-Pmmn borophene Sonu Verma, Alestin Mawrie and Tarun Kanti GhoshDepartment of Physics, Indian Institute of Technology-Kanpur, Kanpur-208 016, India=========================================================================================================================================§ INTRODUCTIONGraphene is the first atomically thin two-dimensional (2D) materialhaving isotropic Dirac cones realized in a laboratory<cit.>.Since then, there have been numerous attempts to synthesizemore and more new 2D materials having Dirac cones. Several quasi-2D materials possessing Dirac cones such assilicene<cit.>, germanene<cit.>, and MoS_2<cit.> have been synthesized experimentally and are being studied theoretically. Recently, there has been intense research interest in synthesis of 2D crystalline boron structures, referred to as borophene. Several attempts have been made to synthesize a stable structure of borophene but only three different quasi-2D structures of borophene havebeen synthesized <cit.>. Various numerical experiments havepredicted a large number of borophene structures with various geometriesand symmetries <cit.>. The orthorhombic 8-Pmmn borophene isone of the energetically stable structures, having ground state energy lowerthan that of the α-sheet structures and its analogues. The Pmmn boron structures have two non-equivalent sub-lattices. The coupling and buckling between two sub-lattices and vacancy give riseto the energetic stability as well as tilted anisotropic Dirac cones <cit.>. The coupling between different sub-lattices enhances the strength of the boron-boron bonds and hence gives rise to structural stability. The finite thickness is required for energetic stability of 2D boron allotropes.The orthorhombic 8-Pmmn borophene possesses tilted anisotropic Dirac cones and is a zero-gap semiconductor. It can be thought of as topologicallyequivalent to the distorted graphene.In the last couple of years, there have been several theoreticalstudies on 8-Pmmn borophene. Very recently, electronic properties of 8-Pmmn borophene have been studied using thefirst-principle calculations and have shown the Dirac cones arising from the p_zorbitals of one of the two inequivalent sub-lattices <cit.>. Zabolotskiy and Lozovik proposed a tight-binding Hamiltonianfor 8-Pmmn borophene and obtained a low-energy effective Hamiltonian in the continuum limit <cit.>. The effective Hamiltonian successfully described all the main features of the quasi-particle spectrum as predicted inab initiocalculation. A similar Hamiltonian has been considered in Ref. <cit.> for studyingquinoid-type graphene due to mechanical deformation andorganic compound α-(BEDT-TTF)_2I_3. The anisotropic plasmon dispersion of borophene is predicted in Ref. <cit.>.The magnetotransport coefficients have been investigated very recently <cit.>.The frequency-dependent optical conductivity is associated with the transitionsfrom a filled band to an empty band, whereas the zero-frequency Drude weight isdue to the intra-band transitions. The real part of the complex optical conductivity is connected with the absorption of the incident photon energy. Its measurement is an important tool forextracting the shape and nature of the energy bands.There are several theoretical and experimental studies on the optical conductivity in various monolayer quantum materials such as graphene<cit.>,silicene <cit.>,MoS_2<cit.>, andsurface states of topological insulators <cit.>.In this paper, we study zero-frequency Drude weight and frequency-dependent opticalconductivity of 8-Pmmn borophene.We find that the Drude weight and optical conductivity are highly anisotropic dueto tilted Dirac cones. We obtain an analytical expression of the minimum photon energyrequired for triggering the optical transitions and of the photon energy at which theconductivities attain maximum value. The maximum value of the optical conductivity along the tilted and perpendicular directions are obtained, which are independent of the carrier density and tilting parameter. The spectroscopic measurement of the absorptive part of the optical conductivity can shed some light on the anisotropic but tilted Dirac cone. This paper is organized as follows. In Sec. II, we provide basicinformation of 8-Pmmn borophene in detail.We discuss the Drude weight and absorptive part of the optical conductivityin Sec. III. An alternate derivation of the optical conductivity usingGreen's function method is provided in the Appendix. We provide a summary and conclusions in Sec. IV. § BASIC INFORMATIONThe massless Dirac Hamiltonian associated with the 8-Pmmn borophenein the vicinity of one of the two independent Dirac points is givenby <cit.> H = v_xσ_x p_x+ v_yσ_yp_y+ v_tσ_0p_y,where p_μ with μ=x,y are the momentum operators, σ_μ arethe 2× 2 Pauli matrices, and σ_0 is the 2 × 2 identity matrix.The velocities are given <cit.> as v_x = 0.86 v_F, v_y= 0.69 v_F,and v_t= 0.32 v_F with v_F = 10^6 ms^-1. The Hamiltonian associated with the second Dirac cone has the opposite sign of v_t. The energy dispersion and the corresponding wave functions are given byE_λ( k) = ħ k[v_tsinθ_ k + λΔ(θ_ k)]andψ_ k^λ( r) = e^ik·r/√(2)[ 1; λ e^i ϕ ],where λ = ± denotes the conduction and valence bands, respectively, θ_ k = tan^-1(k_y/k_x),Δ(θ_ k) = √(v_x^2cos^2θ_ k +v_y^2sin^2θ_ k) describes anisotropy of the spectrum and ϕ = tan^-1(v_y k_y/v_x k_x). The energy difference between the conduction and valence bands at a given k is E_g( k) = E_+( k) - E_-( k)=2 ħ k Δ(θ_ k). Note that the first term in the energy spectrum tilts the Dirac cone and breaks the electron-hole symmetry E_λ( k) = - E_-λ( k),even for the isotropic case v_x = v_y. The tilted Dirac cones are depicted in Fig. 1. The Berry connection of 8-Pmmn borophene is given byA_λ( k) = - v_x v_y/2 Δ^2(θ_ k) θ̂_ k/k,where θ̂_ k = - sinθ_ k x̂ + cosθ_ k ŷis the unit polar angle.The corresponding Berry phase is γ_λ =∮ A_λ( k) · d k= π, exactly the same as in themonolayer graphene case.The chirality operator can be defined asΛ̂ = v_x cosθ_ k σ_x + v_ysinθ_ k σ_y/Δ(θ_ k).It can be easily checked that the chirality operator Λ̂ commutes withthe Hamiltonian H even in the presence of the tilted Dirac cone andthe eigenfunctions ψ_ k^λ( r) are also eigenfunctions of the chiralityoperator with the eigenvalues λ = ± 1, respectively. The role of the Berry phase γ_λ = π and of the chiral symmetry preservationmust be reflected in the scattering process. This can be easily understood by analyzing the angular scattering probability for the borophene.This is given by the squared moduli of the overlap matrix element between the initial spinor(χ_λ(θ_ k)) and the final spinor (χ_λ(θ_ k^')) with |k| = | k^' |. The angular scattering probability is then |f(θ_ k,θ_ k^')|^2 = |⟨χ_λ(θ_ k)|χ_λ(θ_ k^')⟩ |^2=1/2[1 + v_x^2cosθ_ kcosθ_ k^' + v_y^2sinθ_ ksinθ_ k^'/Δ(θ_ k)Δ(θ_ k^')] .It is to be noted that the wave functions do not depend on the tilt parameter v_t. Hence the Berry connection and |f(θ_ k,θ_ k^')|^2 are also independent of the tilt parameter. It also shows that |f(θ_ k,θ_ k^')|^2 is independent of the bands and vanishes exactly whenθ_ k^' - θ_ k = π. It implies that thebackscattering is completely absent, similar to the graphene case. The absence of backscattering survives even for tilted anisotropic energy spectrum. This is due to the conservation of the chirality and/or the ∓π Berry phase.The density of states is given byD(E) = g_s g_v ∫d^2k/(2 π)^2δ(E - E_λ( k))= N_0|E|/π^2ħ^2v_F^2,where the spin degeneracy g_s =2 and the “valley degeneracy" g_v =2<cit.>.Also, the constant N_0 is given byN_0 =∫_0^2π v_F^2d θ_ k/[v_t sinθ_ k +λΔ(θ_ k)]^2 = 15.2263. For a given carrier density n_c, the Fermi energy isE_F =ħṽ_F√(π n_c) with ṽ_F = √(2π/N_0) v_F and the associated anisotropic Fermi wave vectors are obtained as k_F^λ (θ_ k) =E_F /ħ |v_t sinθ_ k + λΔ(θ_ k)|. The components of the velocity operator along the x- and y-directions arev̂_x = v_x σ_x and v̂_y = v_t σ_0 + v_y σ_y. The expectation values of these operators are given by ⟨v̂_x⟩_λ = [ v_x^2/Δ(θ_ k)] cosθ_ kand ⟨v̂_y⟩_λ = v_t + [v_y^2/Δ(θ_ k)] sinθ_ k,respectively. § OPTICAL CONDUCTIVITYWe consider n-doped 8-Pmmn borophene subjected to zero-momentumelectric field E∼μ̂ E_0 e^i ω t with oscillation frequency ω (μ̂ = x̂, ŷ). The complex charge optical conductivity tensor is given byΣ_μν(ω) = δ_μνσ_D(ω) +σ_μν(ω), where μ,ν=x,y, σ_D(ω) = σ_d/(1- i ωτ) isthe dynamic Drude conductivity due to the intra-band transitions, withσ_d being the static Drude conductivity andσ_μν(ω) being the complex optical conductivity due to transitions between valence and conduction bands. It should be mentioned here that Re σ_D and Re σ_μν correspond to the absorption of the photon energy. Drude weight:The Drude weight at vanishingly low-temperature is given by <cit.> D_μν^ = g_s g_v e^2/4π∫ d^2 k⟨v̂_μ⟩_⟨v̂_ν⟩_δ(E( k)-E_F^ ), On further simplification, we obtainD_μν^ = e^2/ħE_F/πħδ_μν(δ_μ x N_1^ + δ_ν y N_2^),where N_1 = 4.686 and N_2 = 2.673. In this case, the Drude weight is anisotropic, unlike the monolayer graphene casewhere the Drude weight D_w^G = (v_F e^2/ħ) √(π n_c) is isotropic. Optical Conductivity: Within the linear response theory,the Kubo formula for the optical conductivity tensor σ_μν(ω)is given byσ_μν(ω) =1/ħ(ω+i η)∫_0^∞dte^i(ω+iη)t⟨ [ĵ_μ(t), ĵ_ν(0) ] ⟩,where⟨ [ĵ_μ(t), ĵ_ν(0) ]⟩ =∑_m,n[f(E_n)-f(E_m)] e^i(E_n-E_m)t/ħj_μ^nmj_ν^mn,ĵ_μ = e v̂_μ is the charge current density with μ = x,y,f(E) is the Fermi-Diracdistribution function and η→ 0^+.Here E_n and E_m are the discrete energy levels of the system.So changing the sum into integration over momentum space, the real part ofthe charge optical conductivity is given by Re σ_μν(ω)=e^2/4πω∫ d^2k [f(E_-(k)) - f(E_+(k))] v_μ^-+( k)v_ν^+-( k)δ(E_g( k) - ħω).The final expression of the real part of the optical conductivity tensor is given byReσ_μν(ω)=e^2/4 πħ∫_0^2πdθ_ kv_x^2 v_y^2/Δ^4(θ_ k )[ f(E_-) - f(E_+) ] [(δ_μ xsin^2θ_ k + δ_ν ycos^2θ_ k) δ_μν-(1-δ_μν)sinθ_ kcosθ_ k ],whereE_±(k_ω(θ_ k),θ_ k) ≡ E_± with k_ω(θ_ k) = ω/[2 Δ(θ_ k)].First of all, we find that the real part of the off-diagonal optical conductivityRe σ_xy(ω) vanishes exactly. For monolayer graphene (v_t=0 and v_x=v_y=v_F), Eq. (<ref>) gives featurelessisotropic optical conductivity which has a step-like shape with a step heightσ_0 = e^2/4ħ at ħω = 2 E_F^0 = 2 ħ v_F √(π n_c). Whereas tilted Dirac cones in borophene provide a distinct anisotropic optical conductivity which can be seen in the subsequent discussion.We analyze the real part of the optical conductivity by solving Eq. (<ref>) numerically for electron density n_c = 1.0 × 10^16 m^-2 at T=0.The plots of Re σ_xx(ω) and Re σ_yy(ω)as a function of photon energy ħω are shown in the lower panel of Fig. 2. It exhibits anisotropic nature of the optical conductivity.We plot ϵ_-(θ_ k) = 2 ħ k_F^-(θ_ k) Δ (θ_ k) in the top panel of Fig. 2. The shaded region in the top panel contributes to theoptical conductivity. The optical transition from the valence band to the conduction band takes place when the photon energy satisfies the inequality ħω≥ϵ_-(θ_ k). The optical transition begins at ħω = 0.113 eV, which corresponds toϵ_1 = ϵ_-(π/2) =2E_F v_y/v_y + v_t < 2 E_F.Moreover, the optical conductivities attain a maximum value whenħω = 0.359 eV, which corresponds to ϵ_2 = ϵ_-(3π/2) = 2E_Fv_y/v_y - v_t > 2E_F. Note that the two energy scales ϵ_1 and ϵ_2 depend on the carrier density, tilted parameter v_t and the velocity v_y along the tilted direction.We have checked numerically thatRe σ_xx(ω) = Re σ_yy(ω) whenħω≃ 2 E_F. By knowing the energies ϵ_1 and ϵ_2 from an experimental measurement, one can extract the tilted parameter v_t using the relation v_t= v_y E_F [ 1/ϵ_1 - 1/ϵ_2].Analyzing the lower panel of Fig. 2, the maximum attainable absorptive part of theconductivity along the tilted direction is σ_yy^ max =σ_0 (v_y/v_x) < σ_0and its orthogonal axis is σ_xx^ max =σ_0 (v_x/v_y) > σ_0. It is interesting to note that σ_xx^ max and σ_yy^ maxdo not depend on the carrier density as well as the tilted parameter v_t. Moreover, σ_xx^ max > σ_yy^ max and the product of these two conductivitiesσ_xx^ max·σ_yy^ max = (e^2/4ħ)^2 is universal. To confirm the results of the optical conductivity, weanalyze the joint density of states which is given byD(ω) = ∫_0^2πdC [f(E_-(k_ω,θ_ k)) - f(E_+(k_ω,θ_ k)) ]/4 π^2 |∂_k E_g( k)|_E_g=ħω,where C is the line element along the contour. In the middle panel of Fig. 2, we show the joint density of states versus the photon energy ħω. One can easily see that that the van Hove singular pointsare at θ_ k = θ_s = π/2,3π/2. The region of zero optical conductivity is nicely captured by the joint density of states. The absorptive part of the optical conductivity arises due to the transitionsfrom the valence band to the conduction band for a given momentum as demonstratedin Fig. 3.The green (dashed) and red (solid) arrows indicate the allowed and forbidden transitions, respectively.One can easily see from this sketch that there are no allowed transitionsfor the photon energy ħω < ϵ_1 as a result of the Pauli blocking.One can also notice that the transitions are allowed even for ħω > ϵ_2.§ SUMMARY AND CONCLUSIONSWe have presented detailed theoretical studies of the Drude weight and optical conductivityof the 8-Pmmn borophene.The exact analytical expressions of the Drude weight, components of the optical conductivity and the onset energy needed for initiating the optical transitions are provided. We also obtain an analytical expression for the photon energy required to attain maximum optical conductivity. We find that the Drude weight and the absorptive parts of the optical conductivityare strongly anisotropic as a result of the tilted Dirac cones. The tilted parameter v_t and the velocity components (v_x,v_y) can be extracted from experimental measurements. We have shown that the product of the maximum values of the anisotropic conductivitiesis always universal.§ ACKNOWLEDGEMENTSWe would like to thank Arijit Kundu and SK Firoz Islam for useful discussion.§ ALTERNATIVE DERIVATION OF THE OPTICAL CONDUCTIVITYWe provide an alternative derivation of the opticalconductivity using Green's function method.. The Kubo formula for the optical conductivityis given by σ_μν(ω)=ie^2/ω1/(2π)^2∫ d^2k × T∑_n Tr⟨v̂_μĜ( k,ω_n)v̂_νĜ( k,ω_n+ω_l) ⟩_iω_n→ω + iδ.Here, T is the temperature and n and l are integers, where ω_l=(2l+1) π Tand ω_n = 2n π T are the fermionic and bosonic Matsubara frequencies, respectively.The Green's function of the Hamiltonian in Eq. (<ref>) is given byĜ( k,ω) =∑_λ[ σ_0 - λ/Δ(θ_ k)(v_xcosθ_ k σ_x + v_y sinθ_ k σ_y) ] ×G_λ( k,ω),whereG_λ( k,ω) = [iħω + μ - E_λ( k)]^-1.Using this Green's function, the following trace is obtained asTr⟨v̂_xĜ( k,ω_n)v̂_xĜ( k, ω_n + ω_l)⟩ = ∑_λ,λ^'[v_x^2/2(1-λλ^')+λλ^'v_x^4cos^2θ_ k/Δ^2(θ_ k)]G_λ( k,ω_l) G_λ^'( k,ω_l+ω_n).Using the well-known identityT ∑_s [1/iħω_n+μ-E_λ·1/iħ(ω_l+ω_n)+μ-E_λ^']=f(E_λ)-f(E_λ^')/iħω_n-E_λ^'+E_λ, if λ≠λ^'0, otherwise.One can further simply the above equation asT ∑_n Tr⟨v̂_x Ĝ( k,ω_n) v̂_xĜ( k,ω_l+ω_n) ⟩ =v_x^2v_y^2sin^2θ_ k/Δ^2(θ_ k)[f[E_-( k)] - f[E_+( k)]/iħω_n-E_+( k) + E_-( k)+f[E_+( k)]-f[E_-( k)]/iħω_n-E_-( k)+E_+( k)]. It can be seen that the second term turns out to be zero asa result of the conservation of energy. Using the result of Eq. (<ref>) into Eq. (<ref>), we haveσ_xx(ω)=i e^2/(2π)^2ω∫_0^∞∫_0^2πdk dθ_ kv_x^2v_y^2 k sin^2θ_ k/Δ^2(θ_ k)f[E_-( k)]-f[E_+( k)]/iħω_n-E_+( k)+E_-( k)|_iω_n→ω+iδ.The real part of the optical conductivity is given byRe σ_xx(ω) =e^2/4π∫ dk dθ_ kv_x^2v_y^2k sin^2θ_ k/Δ^2(θ_ k)[ f(E_-( k)) - f(E_+( k))] δ(ħω - 2ħ kΔ(θ_ k)).The above Eq. (<ref>) can be further simplified to Re σ_xx(ω) =e^2/16π∫_0^2π dθ_ kv_x^2v_y^2sin^2θ_ k/Δ^4(θ_ k)[ f(E_-(k_ω(θ_ k))) - f(E_+ (k_ω(θ_ k)))]where k_ω(θ_ k) = ω/2Δ(θ_ k). Similarly, the yy and yx components of the optical conductivity can be obtained asRe σ_yy(ω) =- e^2/16π∫_0^2πdθ_ kv_x^2v_y^2cos^2θ_ k/Δ^4(θ_ k)[ f(E_+ (k_ω(θ_ k))) - f(E_- (k_ω(θ_ k)))], Re σ_yx(ω) =e^2/16π∫_0^2πdθ_ kv_x^2v_y^2sinθ_ kcosθ_ k/Δ^4(θ_ k)[f(E_+ (k_ω(θ_ k))) - f(E_- (k_ω(θ_ k))) ] . Equations (<ref>), (<ref>) and (<ref>) can be written in acompact form as given in Eq. (<ref>). 55graphene1 A. K. Geim and K. S. Novoselov, Nature Materials6, 183-191 (2007).graphene2 D. Pesin and A. H. 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Piechon, Phys. Rev. B78, 045415 (2008).amit K. Sadhukhan and A. Agarwal, Phys. Rev. B96, 035410 (2017). firoz SK F. Islam and A. Jayannavar, arXiv: 1707.05578 graphene-op-con V. P. Gusynin, S. G. Sarapov, and J. P. Carbotte, Phys. Rev. B75, 165407 (2007).graphene-op-con1 T. Stauber, N. M. R. Peres, and A. K. Geim, Phys. Rev. B,78, 085432 (2008).op-con-exp K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, Phys. Rev. Lett.101, 196405 (2008).silicene-op-con L. Stille, C. J. Tabert, and E. J. Nicol, Phys. Rev. B86, 195405(2012).silicene-op-con1 L. Mathes, P. Gori, O. Pulci, and F. Bechstedt, Phys. Rev. B87, 035438(2013).silicene-op-con2 C. J. Tabert and E. J. Nicol, Phys. Rev. B87, 235426(2013).mos2-op-con Z. Li and J. P. Carbotte, Phys. Rev. B86, 205425 (2012).mos2-op-con1 I. Milosevic, B. Nikolic, E. Dobardzic, M. Damnjanovic, I. Popov, and G. Seifert, Phys. Rev. B76, 234414 (2007). ti-op-con P. D. Pietro, F. M. Vitucci, D. Nicoletti, L. Baldassarre,P. Calvani, R. Cava, Y. S. Hor, U. Schade, and S. Lupi, Phys. Rev. B86, 045439 (2012).ti-op-con1 Z. Li and J. P. Carbotte, Phys. Rev. B87, 155416 (2013).ti-op-con2 X. Xiao and W. Wen, Phys. Rev. B88, 045442 (2013).lozovik1 Y. E. Lozovik (private communication, September 2016). mermin N. W. Ashcroft and N. D. Mermin, Solid State Physics, (Harcourt College Publishes-2001). | http://arxiv.org/abs/1707.08469v3 | {
"authors": [
"Sonu Verma",
"Alestin Mawrie",
"Tarun Kanti Ghosh"
],
"categories": [
"cond-mat.mes-hall"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170726143618",
"title": "Effect of electron-hole asymmetry on optical conductivity in 8-Pmmn borophene"
} |
Aapo Hyvarinen Tensor Regression Networks Jean Kossaifi [email protected] & Imperial College LondonZachary C. Lipton [email protected] Mellon UniversityArinbjörn Kolbeinsson [email protected] Imperial College LondonAran Khanna [email protected] AITommaso Furlanello [email protected] University of Southern CaliforniaAnima Anandkumar [email protected] NVIDIA &California Institute of TechnologyDecember 30, 2023 ============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================ Convolutional neural networkstypically consist of many convolutional layers followed by one or more fully connected layers.While convolutional layers map betweenhigh-order activation tensors,the fully connected layersoperate on flattened activation vectors. Despite empirical success, this approach has notable drawbacks. Flattening followed by fully connected layersdiscards multilinear structure in the activationsand requires many parameters.We address these problems by incorporating tensor algebraic operationsthat preserve multilinear structure at every layer. First, we introduce Tensor Contraction Layers (TCLs) that reduce the dimensionality of their inputwhile preserving their multilinear structure using tensor contraction.Next, we introduce Tensor Regression Layers (TRLs), which express outputs through a low-rank multilinear mappingfrom a high-order activation tensorto an output tensor of arbitrary order.We learn the contraction and regression factors end-to-end,and produce accurate nets with fewer parameters. Additionally, our layers regularize networks by imposing low-rank constraintson the activations (TCL) and regression weights (TRL). Experiments on ImageNet show that,applied to VGG and ResNet architectures, TCLs and TRLs reduce the number of parameters compared to fully connected layers by more than 65%while maintaining or increasing accuracy.In addition to the space savings,our approach's abilityto leverage topological structurecan be crucial for structured data such as MRI.In particular, we demonstrate significant performance improvementsover comparable architectureson three tasks associated with the UK Biobank dataset. Machine Learning, Tensor Methods, Tensor Regression Networks, Low-Rank Regression, Tensor Regression Layers, Deep Learning, Tensor Contraction § INTRODUCTIONMany natural datasets exhibit multi-modal structure.We represent audio spectrograms as 2-order tensors (matrices) with modes correspondingto frequency and time.We represent images as 3-order tensorswith modes corresponding to width, height and the color channels. Videos are expressedas 4-order tensors, and the signal processedby an array of video sensorscan be described as a 5-order tensor. Multilinear structure arises naturally in many medical applications: MRI images are 3-order tensors and functional MRI images are 4-order tensors. Generally, a broad array of multimodal data can be naturally encoded as tensors.Tensor methods extend linear algebrato higher order tensors and are promising tools for manipulating and analyzing such data. The mathematical properties of tensors have long been the subject of theoretical study. Previously, in machine learning, data points were typically assumed to be vectorsand datasets to be matrices.Hence, spectral methods,such as matrix decompositions,have been popular in machine learning.Recently, tensor methods, which generalize these techniques to higher-order tensors, have gained prominence. One class of broadly useful techniqueswithin tensor methods are tensor decompositions,which have been studied for a variety of applications in signal processing and machine learning <cit.>,data mining and fusion <cit.>,blind source separation <cit.>, computer vision <cit.> and learning latent variable models <cit.>.Deep Neural Networks (DNNs) frequentlymanipulate high-order tensors: in a standard deep Convolutional Neural Network (CNN) for image recognition, the inputs and the activationsof convolutional layers are 3-order tensors.And yet, to wit, most architecturesoutput predictions by first flattening the activation tensors and then connecting to the output neuronsvia one or more fully connected layers.This approach presents several issues:(i) we lose multimodal information during the flattening process;and (ii) the fully connected layers require a large number of parameters. In this paper, we propose Tensor Contraction Layers (TCLs)and Tensor Regression Layers (TRLs) as end-to-end trainablecomponents of neural networks.In doing so, we exploit multilinear structurewithout giving up the power and flexibilityoffered by modern deep learning methods.By replacing fully connected layerswith tensor contractions,we aggregate long-range spatial informationwhile preserving multi-modal structure. Moreover, by enforcing low rank,we reduce the number of parameters needed significantly with minimal impact on accuracy.Our proposed TRL expresses the regression weights through the factors of a low-rank tensor decomposition.The TRL obviates the need for flattening, instead leveraging the structure when generating output. By combining tensor regressionwith tensor contraction, we further increase efficiency.Augmenting the VGG and ResNet architectures,we demonstrate improved performance on the ImageNet datasetdespite significantly reducing the number of parameters (almost by 65%).The ability to preserve the topological structure in the datais particularly crucial for prediction from MRI data.In this context, we conduct experiments for 3 different tasks(gender classification, body mass index prediction and age regression) on the UK biobank dataset,the largest available MRI dataset.There, we demonstrate superior performance with our approach and show large performance improvements for all 3 tasks. This is the first paper to presentan end-to-end trainable architecturethat retains the multi-modal tensor structure throughout the network.Related work:Several recent papers apply tensor decomposition to deep learning.One notable line of application is to re-parametrize existing layers using tensor decomposition either to speed these up or reduce the number of parameters. <cit.> propose using CP decompositionto speed up convolutional layers.Similarly, <cit.> propose to use tensor decompositionto remove redundancy in convolutional layers and express theseas the composition of two convolutional layers with less parameters. <cit.> take a pre-trained networkand apply tensor (Tucker) decomposition on the convolutional kernel tensors and then fine-tune the resulting network.<cit.> propose weight sharingin multitask learning and <cit.>propose sharing residual units. <cit.> usethe Tensor-Train (TT) format to impose low-rank tensor structure on weightsof the fully connected layers in order to compress them.However, they still retainthe fully connected layers for the output. In addition, the reshaping to arbitrary higher orders and dimensions does not guarantee that the multilinear structure is preserved.By contrast, we present an end-to-end tensorized network architecturethat focuses on leveraging that structure. Many of these contributions are orthogonal to ours and can be applied together. Despite the success of DNNs,many open questions remain as to why they work so well and whether they really need so many parameters. Tensor methods have emerged as promising tools of analysisto address these questions and to better understand the success of deep neural networks. <cit.>, for example, use tensor methods as tools of analysisto study the expressive power of CNNs,while the follow up work <cit.>focuses on the expressive power of overlapping architectures of deep learning.<cit.> derive sufficient conditions for global optimality and optimization of non-convex factorization problems, including tensor factorization and deep neural network training. <cit.> explorecontraction as a composition operator for NLP. Other papers investigate tensor methodsas tools for devising neural network learning algorithms with theoretical guarantees of convergence <cit.>.Several prior papers address thepower of tensor regression to preserve natural multi-modal structureand learn compact predictive models <cit.>. However, these works typically rely on analytical solutionsand require manipulating large tensors containing the data. They are usually used for small datasets or require to downsample the dataor extract compact features prior to fitting the model, and do not scale to large datasets such as ImageNet. To our knowledge, no prior work combines tensor contraction or tensor regressionwith deep learning in an end-to-end trainable fashion.§ MATHEMATICAL BACKGROUND Notation Throughout the paper, we define tensors as multidimensional arrays,with indexing starting at 0.First order tensors are vectors, denoted v.Second order tensors are matrices, denoted Mandis the identity matrix.By X, we denote tensors of order 3 or greater.For a third order tensor X, we denote its element (i, j, k) as X_i_1, i_2, i_3.A colon is used to denote all elements of a mode, e.g., the mode-1 fibers of Xare denoted as X_:, i_2, i_3.The transpose of M is denoted M.Finally, for any i, j ∈,i < j, ij denotes the set of integers { i, i+1, ⋯ , j-1, j}.Tensor unfolding Given a tensor, X∈^I_0 × I_1 ×⋯× I_N, its mode-n unfolding is a matrix X_[n]∈^I_n × I_M,with I_M = ∏_k=0, k ≠ n^N I_k and is defined by the mapping from element (i_0, i_1, ⋯, i_N) to (i_n, j), withj = ∑_k=0, k ≠ n^N i_k ×∏_l=k+1,l ≠ n^N I_l.We use the definition introduced in <cit.>, which corresponds to an underlying row-wise ordering of the elements.This differs from the definition used by <cit.>, which correponds to an underlying column-wise ordering of the elements. Throughout the paper, we assume the elements are arranged in a row-wise manner, which is reflected in the definitions of unfolding,vectorization, and the resulting formulas.This row-ordering elements matches the actual ordering on GPUs and allows for more efficient implementation. Tensor vectorization Given a tensor, X∈^I_0 × I_1 ×⋯× I_N,we can flatten it into a vector vec(X)of size (I_0 ×⋯× I_N)defined by the mapping from element (i_0, i_1, ⋯, i_N) of X to element j of vec(X), withj = ∑_k=0^N i_k ×∏_m=k+1^N I_m. As for unfolding, we assume a row-ordering of the elements, following <cit.>. n-mode product For a tensor X∈^I_0 × I_1 ×⋯× I_N and a matrix M∈^R × I_n, the n-mode product of a tensoris a tensor of size(I_0 ×⋯× I_n-1× R × I_n+1×·× I_N)and can be expressed using unfolding of Xand the classical dot product asX×_n M = MX_[n]∈^I_0 ×⋯× I_n-1× R × I_n+1×⋯× I_N.Generalized inner-product For two tensors X, Y∈^I_0 × I_1 ×⋯× I_N of same size, their inner product is defined asXY= ∑_i_0=0^I_0-1∑_i_1=0^I_1 - 1⋯∑_i_n=0^I_N - 1X_i_0, i_1, ⋯, i_nY_i_0, i_1, ⋯, i_nFor two tensors X∈^I_x × I_1 × I_2 ×⋯× I_N and Y∈^I_1 × I_2 ×⋯× I_N × I_y sharing N modes of same size, we similarly define a “generalized inner product”along the N last (respectively first) modesof X (respectively Y) as XY_N = ∑_i_1=0^I_1-1∑_i_2=0^I_1 - 1⋯∑_i_n=0^I_N - 1X_:, i_1, i_2, ⋯, i_nY_i_1, i_2, ⋯, i_n, :,with XY_N ∈^I_x × I_y. Tucker decomposition Given a tensor X∈^I_0 × I_1 ×⋯× I_N,we can decompose it into a low rank coreG∈^R_0 × R_1 ×⋯× R_Nby projecting along each of its modes with projection factors( U^(0), ⋯,U^(N)), with U^(k)∈^R_k × I_k, k ∈0N. In other words, we can writeX =G×_0 U^(0)×_1U^(2)×⋯×_N U^(N)= GU^(0), ⋯, U^(N).Typically, the factors and core of the decompositionare obtained by solving a least squares problem.In particular, closed form solutionscan be obtained for the factor by considering the n-mode unfolding of Xthat can be expressed asX_[n] = U^(n)G_[n](U^(-k)),where U^(-k) is defined as follows:U^(-k)=U^(0)⊗⋯U^(n-1)⊗U^(n+1)⊗⋯⊗U^(N).Notice the natural ordering of the factors, from 0 to N,that follows from our definition of unfolding. Similarly, we can optimize the core in a straightforward mannerby isolating it using the equivalent rewriting of the above equality:(X) =( U^(0)⊗⋯⊗U^(N)) (G).We refer the interested reader to the thorough review of the literatureon tensor decompositions by <cit.>. § TENSOR CONTRACTION LAYEROne natural way to incorporatetensor operations into a neural networkis to apply tensor contractionto an activation tensor in order to obtaina low-dimensional representation <cit.>. In this section, we explain how to incorporate tensor contractions into neural networks as a differentiable layer. We call this technique the Tensor Contraction layer (TCL). Compared to performing a similar rank reduction with a fully connected layer, TCLs require fewer parameters and less computation, while preserving the multilinear structure of the activation tensor. §.§ Tensor contraction layers Given an activation tensor X of size ( S_0, I_0, I_1, ⋯, I_N ),the TCL will produce a compact core tensor G of smaller size ( S_0, R_0, R_1, ⋯, R_N ) defined asX' =X×_1 V^(0)×_2V^(1)×⋯×_N+1V^(N),with V^(k)∈^R_k × I_k, k ∈0N.Note that the projections start at the second mode because the first mode S_0 corresponds to the batch.This layer allows us to contract the input activation tensor without discarding its multi-linear structure.By contrast, a flattening layer followedby a fully connected layer would discard that information. One way to see this is to recognise that a fully connected layeris simply a tensor contraction over the second mode of a batchof flattened activations, as exposed in Subsection <ref> below.The projection factors (V^(k))_k ∈ [1, ⋯ N]are learned end-to-end with the rest of the network by gradient backpropagation. In the rest of this paper, we denote size–(R_0, ⋯, R_N) TCL, or TCL–(R_0, ⋯, R_N) a TCL that produces a compact core of dimension (R_0, ⋯, R_N). §.§ Gradient back-propagationIn the case of the TCL,we simply need to take the gradients with respect to the factors V^(k) for each k ∈0, ⋯, N of the tensor contraction. Specifically, we computeX'V^(k) =X×_1 V^(0)×_2V^(1)×⋯×_N+1V^(N)V^(k).By rewriting the previous equality in terms of unfolded tensors, we get an equivalent rewriting where we have isolated the considered factor:X'_[k]V^(k) = V^(k)X_[k](⊗V^(-k))V^(k),with V^(-k) = V^(0)⊗⋯V^(k-1)⊗V^(k+1)⊗⋯⊗V^(N).§.§ Model analysisLink with fully connected layersLet's considering an activation tensor X of size ( S_0, I_0, I_1, ⋯, I_N ). A size–(R_0, R_1, ⋯, R_N ) TCLparameterized by weight factors V^(0), ⋯, V^(N)is equivalent to a fully connected layerparametrized by the weight matrix W = (V^(0)⊗⋯⊗V^(N)) and would computeX_[0]W =X_[0](V^(0)⊗⋯⊗V^(N)),where X_[0] corresponds to the unfolding of Xalong the first mode, e.g., a vectorization of each of the samples in the batch. Number of parameters Considering an activation tensor X of size ( S_0, I_0, I_1, ⋯, I_N ),a size–(R_0, R_1, ⋯, R_N ) TCL parameterized by weight factors V^(0), ⋯, V^(N) and taking X as input will havea total of ∑_k=0^N I_k × R_k parameters.This is to contrast with an equivalent fully connected layer (as presented above), parametrized a weight matrix W = (V^(0)⊗⋯⊗V^(N)), which would have a total of ∏k=0^N I_k × R_k parameters.Notice how the product in number of parameters of the fully connected layer becomes a sum when using a TCL. In other words, in addition to preserving the topological structure in the activation tensor, the TCL has significantly less parameters than a corresponsing fully connected layer.§ TENSOR REGRESSION LAYER In this section, we introduce the Tensor Regression Layer,a new differentiable neural network layer. In order to generate outputs,CNNs typically either flatten the activationsor apply a spatial pooling operation.In either case, they discard all multimodal structure and subsequently apply a fully-connected output layer. Instead, we propose leveraging that multilinear structurein the activation tensorand formulate the output as lying in a low-rank subspacethat jointly models the input and the output.We do this by means of a low-rank tensor regression,where we enforce a low multilinear rankof the regression weight tensor.§.§ Tensor regression as a layerLet us denote by X∈^S × I_0 × I_1 ×⋯× I_N the input activation tensor corresponding to a batch of S samples (X_1, ⋯, X_S) andY∈^ S × O the O corresponding labels for each sample.We are interested in the problemof estimating the regression weight tensorW∈^I_0 × I_1 ×⋯× I_N × O under some fixed low rank (R_0, ⋯, R_N, R_N+1) and a bias b∈^O, such that Y = XW_N + b, i.e.,Y= XW_N + b subject toW = GU^(0), ⋯, U^(N), U^(N + 1),with XW_N = X_[0]×W_[N+1] the contraction of X by W along their N last (respectively first) modes, G∈^R_0 ×⋯× R_N × R_N+1,U^(k)∈^I_k × R_k for each k in 0N and U^(N+1)∈^O × R_N+1. Previously, this setting has been studied as a standalone problem.In that setting, the input data is directly mapped to the output, and the problem solved analytically <cit.>. However, this either limits the model to raw data(e.g., pixel intensities) or requires pre-processing the datato extract (hand-crafted) features to feed the model. For instance, fiducial points that encode the geometry (e.g. facial landmarks) are extracted <cit.>. In addition, analytical solutions are prohibitivein terms of computation and memory usage for large datasets.In this work, we incorporate tensor regressions as trainable layers in neural networks, which allow to learn jointly the features (e.g. via convolutional layers) and the tensor regression. We do so by replacing the traditionalflattening + fully connected layerswith a tensor regression applied directlyto the high-order input and enforcinglow rank constraints on the weights of the regression.We call our layer the Tensor Regression Layer (TRL).Intuitively, the advantage of the TRL comes from leveraging the multi-modal structure in the data and expressing the solution as lying on a low rank manifold encompassing both the data and the associated outputs.§.§ Gradient backpropagation The gradients of the regression weights and the corewith respect to each factorcan be obtained by writing:WU^(k)= G×_0 U^(0)×_1U^(1)×⋯×_N+1U^(N+1)U^(k)Using the unfolded expression of the regression weights, we obtain the equivalent formulation:W_[k]U^(k)= U^(k)G_[k]𝐑U^(k),with𝐑 = U^(0)⊗⋯U^(k-1)⊗U^(k+1)⊗⋯⊗U^(N+1).Similarly, we can obtain the gradient with respect to the coreby considering the vectorized expressions:vec(W)vec(G)= ( U^(0)⊗⋯⊗U^(N+1)) vec(G)vec(G).§.§ Model analysisWe consider as input an activation tensor X∈^S × I_0 × I_1 ×⋯× I_N,and a rank-(R_0, R_1, ⋯, R_N, R_N+1) tensor regression layer,where, typically, R_k ≤ I_k.Let's assume the output is n-dimensional. A fully connected layer taking X (after a flattening layer) as input will have n_FC parameters, withn_FC = n ×∏_k=0^N I_kBy comparison, a rank-(R_0, R_1, ⋯, R_N, R_N+1) TRL taking X as input has a number of parameters n_TRL, with:n_TRL =∏_k=0^N+1 R_k + ∑_k=0^N R_k × I_k + R_N+1× n.§ EFFICIENT IMPLEMENTATION OF TENSOR REGRESSION LAYERS Based on the previous layers, we propose an equivalent, more efficient practical implementation of the tensor regression layer. Equation <ref> can be written:Y = XG×_0 U^(0)×_1U^(1)×⋯×_N+1U^(N+1)_N + bWe can rewrite this equivalently asY = X×_0 (U^(0))×_1(U^(1))×⋯×_N(U^(N))G×_N+1U^(N+1)_N + b.This way, most of the computation is done in the low-rank subspacerather than directly on the dimensions of X.In practice, we use this formulation for implementation,and directly learn the pseudo-inverse of each factorV^(k)∈^I_k × R_k for each k in 0N:Y = X×_0 V^(0)×_1V^(1)×⋯×_NV^(N)G×_N+1U^(N+1)_N + bNote that this is equivalent to first applying a TCL on X, and then applying a TRL on the result to produce Y, with the first factors set to the identity(i.e., the low-rank constraint is applied onlyto the modes corresponding to the output).In addition to this efficient formulation,it is also possible to achieve computational speedup via hardware accelerationby leveraging recent work, e.g., <cit.>, on extending BLAS primitives.This would allow us to reduce the computational overhead for transpositions, which are necessary when computing tensor contractions. § EXPERIMENTSWe empirically demonstrate the effectivenessof preserving the tensor structurethrough tensor contraction and tensor regressionby integrating it into state-of-the-art architecturesand demonstrating similar performanceon the popular ImageNet dataset. We show that a TRL gives equal or greater performanceas compared to flattening followed by fully connected layers, while allowing for large space savings. In particular, we show that our proposed layers allow usto best leverage multi-linear structure in data.This is particularly important for MRI data.On the largest database available, the U.K. biobank,we show that this approach outperforms its traditional counterparts by large margins on three separate prediction tasks.We empirically verify the effectiveness of the TCL on VGG-19 <cit.>.We also conduct thorough experiments and ablation studieswith the proposed layers on VGG-19, AlexNet, ResNet-50 and ResNet-101 <cit.> in various scenarios. §.§ Implementation detailsWe implemented all models using the MXNet library <cit.>as well as the PyTorch library <cit.>. For all tensor methods, we used the TensorLy library <cit.>s. The models were trained with data parallelismacross multiple GPUs on Amazon Web Services,with 4 NVIDIA k80 GPUs.For training on ImageNet, we adopt the same data augmentation procedure as in the original Residual Networks (ResNets) paper <cit.>.When training the layers from scratch,we found it useful to add a batch normalization layer <cit.> before and after the TCL/TRLto avoid vanishing or exploding gradients, and to make the layers more robust to changesin the initialization of the factors.In addition, we constrain the weightsof the tensor regressionby applying ℓ_2 normalization <cit.>to the factors of the Tucker decomposition. When experimenting with the tensor regression layer,instead of retraining the whole network each timeit is possible to start from a pre-trained ResNet.We experimented with two settings:(i) We replaced the last average pooling,flattening and fully connected layersby either a TRL or a combination of TCL + TRLand trained these from scratchwhile keeping the rest of the network fixed; and (ii) We investigate replacing the pooling and fully connected layerswith a TRL that jointly learns the spatial pooling as part of the tensor regression.In that setting, we also explore initializing the TRLby performing a Tucker decompositionon the weights of the fully connected layer.§.§ Large scale image classificationFirst, we report results in the typical settingfor large scale image classification on the the widely-used ImageNet-1K dataset,by learning the spatial pooling as part of the tensor regression. The ILSVRC dataset <cit.> (ImageNet)is composed of 1.2 million images for trainingand 50,000 for validation,all labeled for 1,000 classes.Following <cit.>,we report results on the validation setin terms of Top-1 accuracy and Top-5 accuracy across all 1,000 classes. Specifically, we evaluate the classification erroron single 224 × 224 single center crop from the raw input images.In this setting, we remove the average pooling layerand feed the tensor of size(batch size, number of channels, height, width) to the TRL,while imposing a rank of 1 on the spatial dimensionsof the core tensor of the regression. Effectively, this setting simultaneously learns weightsfor the multilinear spatial pooling as well as the regression. Our experimental results show that our method enables an effective trade-off between performance and space savings (Table <ref>). In particular, small space savings (e.g. about 25%)translate inmarginal increases in performance. It is possible to obtain more than 65% space savings without impacting accuracy,and to reach larger performance space savingswith litte impact on performance (e.g. almost 80% space savingswith less than 1% decrease in Top-1 and Top-5 accuracy). We can express the space savings of a model Mwith n_M total parameters in its fully connected layerswith respect to a reference model Rwith n_R total parametersin its fully connected layersas 1 - n_M/n_R (bias excluded). To study the TRL in isolation, we consider the weights of a pre-trained model and replace the flattening and fully connected layer with a TRLwhile keeping the rest of the network fixed. In practice, to initialize the weights of the TRL in this setting,it is possible to consider the weights of the fully connected layeras a tensor of size(batch size, number of channels, 1, 1, number of classes)and apply a partial Tucker decomposition to itby keeping the first dimension (batch-size) untouched.The core and factors of the decompositionthen give us the initialization of the TRL.The projection vectors over the spatial dimensionare then initialized to1/height and1/width, respectively.The Tucker decomposition was performed using TensorLy <cit.>.In this setting, we show that we can drastically decreasethe number of parameters with little impact on performance. §.§ Large scale phenotypical trait prediction from MRI data A major challenge in neuroscience and healthcare is analysing structure-rich data,such as 3D brain scans <cit.>. The human brain is a highly complex and structured organ. It is composed of interconnected heterogeneous components,each of which is structurally distinct <cit.>.The organizational properties are critical to the brain’s, and the individual’s, overall health. Studies of brain topology have revealed that structural changesare associated with cognitive function <cit.>and diseases such as diabetes <cit.>. Retaining this topological information is thus criticalin order to maximize modelling accuracy.However, flattening layers followed by fully connected layersdiscards that information and are therefore sub-optimal for the task.By contrast, our proposed tensor regression layers naturally leveragesand preserves the topological information, allowing for better performance. We empirically demonstrate this on the UK biobank MRI dataset,the largest imaging study of its kind <cit.>, the results can be seen in Table <ref>.We split the data into a training set containing 11,500 scans,a validation set of 3,800 scans and 3,800 scans for a held-out test set. Each scan is a T1-weighted three-dimensional structural MRI with dimensions 182×218×182.We select three tasks: classifying gender,predicting body mass index (BMI) and predicting age from raw brain MRI scans.Predicting age is an important task as the differencebetween true and predicted age is a biomarkerthat has a number of clinical applications <cit.>. Similarly, learning to predict BMI from MRI is a valuable objective,as influence of obesity on brain structure has already been established <cit.>, but its mechanism is not fully understood.For the third task of gender classification,differences between genders (more precisely biological sex) in brain functional mappingshave already been described <cit.>.The reasons and implications (if any) of these differences are not known, but understanding the biological processes underpinning themcould provide insights into how the brain works. For gender prediction the classes are fairly balanced, the male:female ratio in the UK Biobank is 0.46:0.54. We compare a standard ResNet with 3D convolutions (3D-ResNet)to a 3D-ResNet where the final fully connected layer was replaced with a TRL. The baseline ResNet contains a global-average pooling layerthat compresses all three spatial dimensions,giving an output size of 512 × n_outputs.The models model were implemented using PyTorch <cit.>and TensorLy <cit.> and trained was done on a Tesla P100 GPU. Both networks were trained from random initialization.The same number of iterations was used for all comparisons.After validating the number of iterations,we found the baseline and the TRL model required same number of iterationsin order to reach convergence and the same number of iterationswas subsequently used for all models and all experiments.The results (table. <ref>) demonstrate that our tensor-based architectureis able to learn a mapping between structural MRI and phenotypes.It performs significantly better on all tasks compared to a 3D-ResNet with traditional average pooling operation and a fully connected layer. In particular, the 3D-ResNet achieved a mean absolute error (MAE) of 2.96 yearson the age-regression task compared with 2.70 years for our TRL. Similarly, the TRL out-performed the baseline on the gender classifying task,achieving an error of only 0.53 % (accuracy of 99.47 %)compared to an error of 0.79 % for the baseline.For BMI regression, the TRL also performed with a notable improvement over the baseline.A secondary baseline, 3D-ResNet with no average pooling,failed to train at all due to unstable gradients.These three important tasks demonstrate that the TRL is leveragingadditional information in the structural MRI data beyond that of the baseline model.Our method improves on previously reported brain age accuracy in UK Biobank <cit.>, and published studies <cit.> on other brain MRI datasets.The method also improves on BMI prediction beyond thosepreviously published for UK Biobank data <cit.>. Improved performance gives further support for use of machine learningas a support tool in neuroradiological research and clinical decision making.One question is whether this improvements are due to the ability of the TRLto preserve and leverage topological structure or whetherit is simply a result of the regularizing effect of the TRL.To answer this question, we experimented with a TRL, where the entire 6 × 7 × 6 full-rank activation tensorwas used for the age prediction task.Using the full-rank tensor the network achieved a MAE of 2.71 years, which is similar to the results obtained with lower-rank set ups,and significantly better than the baseline.This empirically confirms the importance of leveraging topological structure. §.§ Ablation studiesSynthetic settingTo illustrate the effectiveness of the low-rank tensor regression,in terms of learning and data efficiency, we first investigate it in isolation. We apply it to synthetic datay = vec(X) ×Wwhere each sample X∈^(6464)follows a Gaussian distribution 𝒩(0, 3).W is a fixed matrix and the labels are generated asy = vec(X) ×W.We then train the data on X + E, where E is added Gaussian noisesampled from 𝒩(0, 3).We compare i) a TRL with squared loss andii) a fully connected layer with a squared loss. In Figure <ref>,we show the trained weight of both a linear regression based on a fully connected layerand a TRL with various ranks, both obtained in the same setting.The TRL is able to better recover the ground-truth weight due to the low-rank structure imposed on these,which allows to leverage the multi-linear structure and acts as an implicit regularizer.This is in line with findings from existing worksfocusing on analytical solution to tensor regression problems <cit.>.Additional regularizers such as Lasso (l1) <cit.> can provide further advantages, especially in the face of noisy inputs.Going forward, we plan to investigate such additional regularization terms.We also compare the data efficiency of a tensor regression layercompared to a linear regression, in the same setting, by varying the number of training samples. As can be observed in Figure <ref>,the TRL is easier to train on small datasets and less prone to overfitting,due to the low rank structure of its regression weights, as opposed to typical(fully connected) linear regression. Impact of the tensor contraction layer We first investigate the effectiveness of the TCLusing a VGG-19 network architecture <cit.>.This network is especially well-suited for our methods because of its 138,357,544 parameters,119,545,856 of which (more than 80% of the total number of parameters)are contained in the fully-connected layers.By adding a TCL to contract the activation tensorprior to the fully connected layers,we can achieve large space saving. Table <ref> presents the accuracy obtainedby the different combinations of TCLin terms of top-1 and top-5 accuracy as well as space saving.By adding a TCL that preserves the size of its inputwe are able to obtain slightly higher performancewith little impact on the space saving (0.21% of space loss)while by decreasing the size of the TCL we got more than 65% space saving with almost no performance deterioration.Overcomplete TRL We first tested the TRL with a ResNet-50 and a ResNet-101 architectures on ImageNet, removing the average pooling layerto preserve the spatial information in the tensor. The full activation tensor is directly passed onto a TRL which produces the outputson which we apply softmax to get the final predictions.This results in more parameters as the spatial dimensions are preserved. To reduce the computational burdenbut preserve the multi-dimensional information,we alternatively insert a TCL before the TRL. In Table <ref>, we present results obtained in this settingon ImageNet for various configurations of the network architecture.In each case, we report the size of the TCL(i.e. the dimension of the contracted tensor)and the rank of the TRL (i.e. the dimension of the core of the regression weights). Choice of the rank of the TRL While the rank of the TRLis an additional parameter to validate,it turns out to be easy to tune in practice.In Figure <ref>,we show the effect on Top-1 and Top-5 accuracyof decreasing the size of the core tensorof the TRL.We also show the corresponding space savings. The results suggest that choosing the rank is easy because there is a large rangeof values of the rankfor which the performance does not decrease.In particular, we can obtain up to 80%space savings with negligible impact on performance.§ CONCLUSIONSDeep neural networks already operate on multilinear activation tensors,the structure of which is typically discardedby flattening operations and fully-connected layers.This paper proposed preserving and leveragingthe tensor structure of the activationsby introducing two new, end-to-end trainable, layersthat enable substantial space savingswhile preserving and leveraging the multi-dimensional topological structure.The TCL that we propose reduces the dimensionof the input without discarding its multi-linear structure,while TRLs directly map their input tensors to the output with low-rank regression weights.These techniques are easy to plug in to existing architectures and are trainable end-to-end.Our experiments demonstratethat by imposing a low-rank constrainton the weights of the regression,we can learn a low-rank manifoldon which both the data and the labels lie.Furthermore these new layers act as an additional type of regularizationon the activations (TCL) and the regression weight tensors (TRL). 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"authors": [
"Jean Kossaifi",
"Zachary C. Lipton",
"Arinbjorn Kolbeinsson",
"Aran Khanna",
"Tommaso Furlanello",
"Anima Anandkumar"
],
"categories": [
"cs.LG"
],
"primary_category": "cs.LG",
"published": "20170726073757",
"title": "Tensor Regression Networks"
} |
National Research Council of Canada, Ottawa, Canada, K1A 0R6 The Hugoniot for Deuterium is calculated using the classical-map hyper-netted-chain (CHNC) approach using several models of the effective temperature that may be assigned to the electron-ion interaction. This effective temperature embodies the exchange-correlation and kinetic energy functional that is assigned to the electron-ion interaction. Deuterium pair distribution functions (calculated using theneutral-pseudo atom method) showing the formation of molecular pre-peaks are displayed to clarify the soft-turning over of the hugoniot in the pressure range of 0.2-0.6 Megabars. This contribution updates a previous CHNC calculation of the deuterium hugoniot given in Phys. Rev. B, 66, 014110 (2002).A calculation of the Deuterium Hugoniot using theclassical-map hypernetted-chain (CHNC) approach. M. W. C. Dharma-wardana[Email address: ][email protected] December 30, 2023 ===================================================================================================§ CALCULATIONS OF THE DEUTERIUM HUGONIOT USING A CLASSICALREPRESENTATION OF THE ELECTRONS. The major problem in simulations of warm dense matter is the quantum nature of electrons and the representation of physical variables by operators defined in function space. The higher the temperature, the larger is the number of excited states needed to properly represent this function space. The large basis sets needed make the usual density-functional theory (DFT) and molecular dynamics (MD) method prohibitive for temperatures T/E_F exceeding about 0.5, where E_F isthe Fermi energy. At low temperatures, the mean free pathsλ_ mfp of electrons become very large, small-k effects become relevant, and large MD simulation cells with the number of ions N exceeding thousands <cit.> become necessary, making the methodimpractical or inaccurate for many physical properties. However, DFT offers the possibility of completely describing at least the static properties, e.g., the equation of state (EOS) of quantum systems using only the one-body electron density n(r), without appeal to wavefunctions. Hence, a classical representation of electrons which interactvia an equivalent effective potential can be legitimately anticipated <cit.> if the effective potentials accurately predict the DFT n(r). Thus the Coulomb interaction V^ee_cou(r) inclusive of quantum diffraction corrections, and an additional Pauli-exclusion interaction P^ee(r,ζ)chosen to exactly recoverthe Fermi hole in the electron-electron pair distribution function (PDF) has to be constructed. That is, the e-ePDF, g_ee(r,ζ) which depends on the spin polarization ζ needs tobe mapped to an effective classical potential V^ee_cls(r)=V^ee_cou(r)+P^ee(r) even at T=0. However, such an equivalent Coulomb system at the physical temperature T will also have an effective classical temperature T^ee_cls which takes account of the non-zero kinetic and correlation effects which manifest even at T=0. In effect, T^ee_cls is an approximate classical solution to the long standing problem of the representation for the kinetic energy functional of DFT.Thus the self-consistent construction of V_cls(r,ζ,n̅), and T^ee_cls(ζ, n̅) for a given mean electron density n̅ and temperature T to accurately recover the one-body density n(r) and the associated PDFs using the classical hyper-netted-chain equationconstitutes the classical map of the quantum electrons of spin polarizationζ, denoted by CHNC. If molecular dynamics is used instead of the HNC, for inverting or calculating the PDFs,we refer to it asCMMD.Pair-distribution functions g_ee(r,ζ) using such a map V^e_cls, T^ee_cls can be generated via MD or even more cheaply via the hyper-netted chain (HNC) equation or its modified form known as MHNC containing an appropriate bridge-diagram correction.Dharma-wardana andPerrot have constructed such maps for 2-D and 3-D electrons at zero and finite T <cit.>, and shown accurate agreement of the PDFs with quantum Monte Carlo (QMC) PDFs available at T=0, even for very high coupling, while Dufty et al. <cit.>have also constructed similar classical maps. Dufty et al. have used additional constraints in constructing potentials rather than using the minimum necessary conditions imposed by DFT and HNC or MD invertible maps. Their methods also link with the older work on effective potentials derived from Slater sums. In the classical map proposed by Dharma-wardana and Perrotwe use an effective temperature T^ee_cls which is chosen so that the equivalentclassical Coulomb fluid has the correct exchange-correlation (XC) energy at T=0. This is found to be sufficiently accurately given by the formT^ee={T_q^2+T^2}^1/2. The “quantum temperature” T_q is of the order of E_F;its form proposed in Ref. <cit.> is a simple function of the electron Wigner-Seitzradius r_s={3/(4πn̅}^1/3. The use of the actual form and the Pauli potential are crucial to the accuracy of the map. This classical map givenby Dharma-wardana and Perrot in 2000 was found to be very accurate when results for finite-T QMC exchange-correlation energies <cit.> and finite-T g_ee(r,ζ)becameavailable a decade later. § CLASSICAL MAP FOR ELECTRON-ION MIXTURES.Once the electrons are mapped to an effective classical system, they can now be used with ions (e.g., protons, carbon nuclei etc.) to study the propertiesof electron-ion systems without the heavy computational burden of purely quantum approaches. However, in including the ions, the electron-ion interaction itself needs to be mapped to a classical form correctly so as to include (i) quantum effects in the effective electron-ion interaction; (ii) quantum effects in the kinetic energy functional applicable to the effective temperature T^ei_cls of the electron-ion interaction. The ions themselves can usually be taken to be classical andpose no problems, with the ion temperature T_i=T.In dealing with V^ei_cls we need to consider(a) that the quantum electron-ion interactions produce a bound-electron spectrum requiring the use of a pseudo-potential with a core radius r_c for V_cls^ei, (b) the fact that transientas well as persistent molecular forms, e.g., H-H, C-H and C-C are formed <cit.>.Examples of H-H or equivalently D-D PDFs showingpre-peaks ing_dd(r) due the formation of transient D-D bonds, calculated using the neutral-pseudo-atom (NPA) approach <cit.>, are shown in Fig. <ref>. Unlike many average-atom models, the NPA approach is capable of providing an average description of all transiently bonded forms that may be found in the fluid, as described in Ref. <cit.>. The CHNC currently does a less satisfactory job of picking up such effects, as this depends on the specification of T^ei.The CHNC is developed within the DFT conceptual framework where one-body densities and XC-functionals rather than wavefunctions determine the physics of the system. We recall <cit.> that an ion-electron system with one-body densities n(r) for electrons and ρ(r) for ions hasa free energy of the form F([n],[ρ]) and hence the XC-functionals and Kohn-Sham potentials need to be calculated via functional derivatives with respect to n(r) and alsoρ(r), and not just with respect to n(r). ObtainingXC-potentials needs takingfurther functional derivatives. Unlike inconventional DFT, the ions are not treated as merely providing a static “external potential” but enter directly into the theory. This leads to twoDFT equations (one for the ions, and one for the electrons), and equations connecting these subsystems. They have distinct XC-functional for electrons, ions and for electron-ion interactions. The ion-ion XC functional does not have an exchange part as the ions are treated classically, andthe correlation part is expressed as a sum of hyper-netted-chain diagrams. The CHNC also has threeequations, where the equations for the densitydistributions are integral equations of the HNC type. The effective classicaltemperatures of these equations should be selected to recover the actual XC-energiesof each subsystem.Thus the he classical map for electron-ion systems requires, in addition to the map for e-e interactions, a map defining β_eiV^ei_cl where β_ei=1/T^ei_cls. Bradow it et al. <cit.> used a diffraction corrected Coulomb potential V^ei_cls(r)= -|e|Z{1-exp(k_eir)}/r,|e|=1 and an effective temperatureT^ei_cls=(T^ee_clsT_i)^1/2. This point-ion interaction uses the effective charge Z of the ion; while the model works successfully in some regimes, it was inaccurate, for example for aluminum and other ions where a finite-core radius becomes necessary.§ CLASSICAL MAP APPLIED TO DEUTERIUM.However, the classical map applied to hydrogen or deuterium should work successfully in regimes where the system is fully ionized, without having to worry about pseudopotential forms. It should also work successfully even in the regime where there are transient H-H bonds since the attractive potential arises mostly via the exchange interaction which splits the singlet (bonding) and triplet (anti-bonding) interactions. We associate the “softening turnover” (STO) of the deuteriumHugoniot between a density ρ of 0.6 g/cm^3 to 0.8 g/cm^3 and pressures of 0.2 Mbar to 0.6 Mbar, to the changing influence of transient D-D bonding over this interval of density and pressure (see Fig. <ref>). In this regime there are no stable (i.e., persistent) H-H or D-D molecules. Thisregime is extremely sensitive to the choice of XC-functionals, finite-T effects etc.,used in DFT calculations. The same uncertainty reappears in the determination of the effective classical temperature T^ei_cls; this is abbreviatedto T^ei etc. in the following. In fact, trying out different simple models for T^ei gives insight into the nature of the STO. Here we consider three models of T^ei and the corresponding CHNC-Deuterium Hugoniots, one of which was published by Dharma-wardana and Perrot in 2002 <cit.>. There we studied the choiceT^ei=(T^ee_cls+T_i)/2 (fordetails see Ref. <cit.>). A more extended plot of the Hugoniot is given in Fig. <ref>. The free energy F(r_s ,T) obtained by coupling-constant integrations over the PDFs obtained from the CHNC methodis used to calculate the deute- rium Hugoniot. The initial state of internal energy, volume and pressure(E_0 ,V_0 , P_0 ), with T= 619.6 K, initial density of 0.171 g/cm^3, E_0=15.886 eV per atom, and P_0 = 0.0 were used. The results are shown in Fig. <ref>.An alternative CHNC model, with T^ei= (T^eeT_i)^1/2 is given in Bredow et al. <cit.> but no Hugoniot was calculated. That model, based on ensuring the appropriate behaviour of the structure factors S(k) as k→ 0 <cit.>may need rectification in the limitT_i→ 0. Since the momentum exchange in an electron-ion collision is determined by the reduced mass m^ei of the colliding pair, one may consider determining T^ei via m^ei/T^ei = m^ee/T^ee+m^ii/T_i.This implies that T^ei=T^ee since the effective mass is determined predominantly by the light particle, with m^ei≃ m_e. The corresponding Hugoniot is also presented in Fig. <ref>. This is clearly incorrect and diverges to high compressibilityat the onset of the STO region. In fact,a proper model of T^ei_cls has to be based on considerations of the electron-ion XC and kineticcomponents and not on collision dynamics which dominate high-energy collisions. A model for which we do not yet have a Hugoniot calculation is where the classical electron-ion interaction β_eiV^ei_cls is obtained directly by an HNC inversion of the NPA charge-density pileup at the proton.Such a more complete map will be considered in future work for the provision of fast, accurate calculations of hugoniots.§ CONCLUSIONAn accurate classical representation of electrons <cit.> is not sufficient to deal with electron-ion systems, since the quantum features contained in the electron-ion interaction delicately depend on including additional quantum effects through an electron-ion exchange-correlation functional which contributes to the e-i interaction and the classical temperature associated with it. The soft feature in the hugoniot is shown to be closely related totemperature-dependent exchange correlation effects and the onset oftransient deuterium-deuterium molecular bonding that manifests as thetemperatures and densities are lowered. The transient bonding is signaledby the appearance of pre-peaks in the deuterium-deuterium pair-distribution functions. Future work will present a more accurate classicalmapping of the electron-ion interaction in the context of the models already presented <cit.>. (Some of the contents of this paper were presented atthe DOE/NNSA Equation of State workshop, 31-May, 2017, Rochester, USA).99Pozzo11 Monica Pozzo etal.,Phys. Rev. B, 84, 054203 (2011).chnc1 M. W. C. Dharma-wardana and F. Perrot, Phys. Rev. Lett. 84, 959 (2000).chnc2 François Perrot and M. W. C. Dharma-wardana,Phys. Rev. Lett. 87,206404 (2001).SandipDufty13 J. Dufty and S. Dutta, Phys. Rev. 87, 032101 (2013). BrownXCT13 W. E Brown, J. L. DuBuois, M. Holzmann and D. M. Ceperley, Phys. Reb. B 88, 081102 (2013).Hungary16 M. W. C. Dharma-wardana, Proceedings of the Conference in Density Functional Theory, Debrecen, Hungary, 2016. Ed. Karlheinz Schwarz and Agnes Nagy. Computation4 (2), 16; 2016;http://arxiv.org/abs/1602.04734TrickeyXC14 Valentin V. Karasiev, Travis Sjostrom, James Dufty, S. B. Trickey, Phys Rev Lett, 112 (7), 076403 (2014). cdw-carbon16 M. W. C. Dharma-wardana, ArXive [cond-mat] 1607.07511 (2017).cdw-cpp15 M. W. C. Dharma-wardana, A review of studies on strongly-coupled Coulomb systems since the rise of DFT and SCCS-1977. Contrib. Plasma Phys. 55, No.2-3, 79-81 (2015).ilciacco93 M. W. C. Dharma-wardana, p 625-650,inE. K. U. Gross, and R. M. Dreizler, Eds. Density Functional Theory,NATO ASI series, 337, 625Plenum Press, New York (1993). Bredow15 R. Bredow, Th. Bornath, W.-D. Kraeft, M.W.C. Dharma-wardana and R. Redmer, Contrib. Plasma Phys., 55, 222-229 (2015). Kerley83 G. I. Kerley, Molecular Based Study of Fluids ͑(ACS, Washington DC, 1883). hug02 M. W. C. Dharma-wardana andF. Perrot, Phys. Rev. B66, 014110 (2002).BonarthPriv15 Th. Bonarth, Private communication (2015) | http://arxiv.org/abs/1707.08880v1 | {
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"title": "A calculation of the Deuterium Hugoniot using the classical-map hypernetted-chain (CHNC) approach"
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=2em An enriched view on the monad–theory correspondence]An enriched view on the extended finitary monad–Lawvere theory correspondence R. Garner]Richard Garner Department of Mathematics, Macquarie University, NSW 2109, Australia [email protected]. Power]John Power Department of Computer Science, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom [email protected][2010]Primary: 18C10, 18C35, 18D20The authors gratefully acknowledge the support of Australian Research Council grants DP160101519 and FT160100393 and of the Royal Society International Exchanges scheme, grant number IE151369. No data was generated in the research for this paper. We give a new account of the correspondence, first established by Nishizawa–Power, between finitary monads and Lawvere theories over an arbitrary locally finitely presentable base. Our account explains this correspondence in terms of enriched category theory: the passage from a finitary monad to the corresponding Lawvere theory is exhibited as an instance of free completion of an enriched category under a class of absolute colimits. This extends work of the first author, who established the result in the special case of finitary monads and Lawvere theories over the category of sets; a novel aspect of the generalisation is its use of enrichment over a bicategory, rather than a monoidal category, in order to capture the monad–theory correspondence over all locally finitely presentable bases simultaneously. Dedicated to Jiří Adámek on the occasion of his retirement [ [ December 30, 2023 =====================§ INTRODUCTIONA key theme of Jiří Adámek's superlative research career has been the study of the subtle interaction between monads and theories, especially within computer science. We hope he might see this paper as a development of the abstract mathematics underlying this aspect of his body of work. Jiří has been an inspiration to both of us, on both a scientific and a personal level, as he has been to many in category theory and beyond, and we are therefore pleased to dedicate this paper to him.The starting point of our development is the well-known fact that categorical universal algebra provides two distinct ways to approach the notion of (single-sorted, finitary) equational algebraic theory. On the one hand, any such theory 𝕋 gives rise to a Lawvere theory whose models coincide (to within coherent isomorphism) with the 𝕋-models. Recall that a Lawvere theory is a small category Ł equipped with an identity-on-objects, strict finite-power-preserving functor 𝔽^op→Ł and that a model of a Lawvere theory is a finite-power-preserving functor Ł→Set, where here 𝔽 denotes the category of finite cardinals and mappings.On the other hand, an algebraic theory 𝕋 gives rise to a finitary (i.e., filtered-colimit-preserving) monad on the category of sets, whose Eilenberg–Moore algebras also coincide with the 𝕋-models. We can pass back and forth between the presentations using finitary monads and Lawvere theories in a manner compatible with semantics; this is encapsulated by an equivalence of categories fitting into a triangle[@[email protected]]Mnd_f(Set)^op[dr]_-Alg()[rr]^(0.5)≃ d Law^op[dl]^-Mod()CATwhich commutes up to pseudonatural equivalence. Both these categorical formulations of equational algebraic structure are invariant with respect to the models, meaning that whenever two algebraic theories have isomorphic categories of models, the Lawvere theories and the monads they induce are also isomorphic. However, the two approaches emphasise different aspects of an equational theory 𝕋. On the one hand, the Lawvere theory Ł encapsulates the operations of 𝕋: the hom-set Ł(m,n) comprises all the operations X^m → X^n which are definable in any 𝕋-model X. On the other hand, the action of the monad T encapsulates the construction of the free 𝕋-model on any set; though since there are infinitary equational theories which also admit free models, the restriction to finitary monads is necessary to recover the equivalence with Lawvere theories.While the equivalence in (<ref>) is not hard to construct, there remains the question of how it should be understood. Clarifying this point was the main objective of <cit.>: it describes a setting within which both finitary monads on Set and Lawvere theories can be considered on an equal footing, and in which the passage from a finitary monad to the associated Lawvere theory can be understood as an instance of the same process by which one associates: * To a locally small category , its Karoubian envelope;* To a ring R, its category of finite-dimensional matrices;* To a metric space, its Cauchy-completion. The setting is that of -enriched category theory <cit.>; while the process is that of free completion under a class of absolute colimits <cit.>—colimits that are preserved by any -functor. The above examples are instances of such a completion, since: * Each locally small category is a Set-category, and splittings of idempotents are Set-absolute colimits;* Each ring can be seen as a one-object Ab-category and the corresponding category of finite-dimensional matrices can be obtained by freely adjoining finite biproducts—which are Ab-absolute colimits;* Each metric space can be seen as an ℝ_+^∞-category—where ℝ_+^∞ is the monoidal poset of non-negative reals extended by infinity, as defined in <cit.>—and its Cauchy-completion can be obtained by adding limits for Cauchy sequences which, again as in <cit.>, are ℝ_+^∞-absolute colimits. In order to fit (<ref>) into this same setting, one takes as enrichment base the categoryof finitary endofunctors of Set, endowed with its composition monoidal structure. On the one hand, finitary monads on Set are the same as monoids in , which are the same as one-object -categories. On the other hand, Lawvere theories may also be identified with certain -categories; the argument here is slightly more involved, and may be summarised as follows.A key result of <cit.>, recalled as Proposition <ref> below, identifies -categories admitting all absolute tensors (a kind of enriched colimit) with ordinary categories admitting all finite powers. In one direction, we obtain a category with finite powers from an absolute-tensored -category by taking the underlying ordinary category; in the other, we use a construction which generalises the endomorphism monad (or in logical terms the complete theory) of an object in a category with finite powers.Using this key result, we may identify Lawvere theories with identity-on-objects strict absolute-tensor-preserving -functors 𝔽^op→Ł, where, overloading notation, we use 𝔽^op and Ł to denote not just the relevant categories with finite powers, but also the corresponding -categories. In <cit.>, the -categories equipped with an -functor of the above form were termed Lawvere -categories.By way of the above identifications, the equivalence (<ref>) between finitary monads and Lawvere theories can now be re-expressed as an equivalence between one-object -categories and Lawvere -categories: which can be obtained via the standard enriched-categorical process of free completion under all absolute tensors. The universal property of this completion also explains the compatibility with the semantics in (<ref>); we recall the details of this in Section <ref> below.We thus have three categorical perspectives on equational algebraic theories: as Lawvere theories, as finitary monads on Set, and (encompassing the other two) as -categories. It is natural to ask if these perspectives extend so as to account for algebraic structure borne not by sets but by objects of an arbitrary locally finitely presentable category . This is of practical interest, since such structure arises throughout mathematics and computer science, as in, for example, sheaves of rings, or monoidal categories, or the second-order algebraic structure of <cit.>.The approach using monads extends easily: we simply replace finitary monads on Set by finitary monads on . The approach using Lawvere theories also extends, albeit more delicately, by way of the Lawvere -theories of <cit.>. If we write _f for a skeleton of the full subcategory of finitely presentable objects in , then a Lawvere -theory is a small category Ł together with an identity-on-objects finite-limit-preserving[Note the absence of the qualifier “strict”; for a discussion of this, see Remark <ref> below.] functor J →Ł; while a model of this theory is a functor Ł→Set whose restriction along J preserves finite limits. These definitions are precisely what is needed to extend the equivalence (<ref>) to one of the form:[@!C@C-3em]Mnd_f()^op[dr]_-Alg()[rr]^(0.5)≃ d Law()^op .[dl]^-Mod()CAT What does not yet exist in this situation is an extension of the third, enriched-categorical perspective; the objective of this paper is to provide one. Like in <cit.>, this will provide a common setting in which the approaches using monads and using Lawvere theories can coexist; and, like before, it will provide an explanation as to why an equivalence (<ref>) should exist, by exhibiting it as another example of a completion under a class of absolute colimits.There is a subtlety worth remarking on in how we go about this. One might expect that, for each locally finitely presentable , one simply replaces the monoidal categoryof finitary endofunctors of Set by the monoidal category _ of finitary endofunctors of , and then proceeds as before. This turns out not to work in general: there is a paucity of _-enriched absolute colimits, such that freely adjoining them to a one-object _-category does not necessary yield something resembling the notion of Lawvere -theory.In overcoming this apparent obstacle, we are led to a global analysis which is arguably more elegant: it involves a single enriched-categorical setting in which finitary monads and Lawvere theories over all locally finitely presentable bases coexist simultaneously, and in which the monad–theory correspondences for eacharise as instances of the same free completion process.This setting involves enrichment not in the monoidal category of finitary endofunctors of a particular , but in the bicategoryof finitary functors between locally presentable categories[In fact, when it comes down to it, we will work not withitself, but with a biequivalent bicategoryof lex profunctors; see Section <ref>.]. The theory of categories enriched in a bicategory was developed in <cit.> and will be recalled in Section <ref> below; for now, note that a one-object -enriched category is a monad in , thus, a finitary monad on a locally finitely presentable category. This explains one side of the correspondence (<ref>); for the other, we extend the key technical result of <cit.> by showing that absolute-tensored -categories can be identified with what we call partially finitely complete categories. These will be defined in Section <ref> below; they are categories , not necessarily finitely complete, that come equipped with a sieve of finite-limit-preserving functors expressing which finite limits do in fact exist in .The relevance of this result is as follows. If J →Ł is a Lawvere -theory, then we may view bothand Ł as partially finitely complete, and so as absolute-tensored -categories, on equipping the former with the sieve of all finite-limit-preserving functors into it, and the latter with the sieve of all finite-limit-preserving functors which factor through J. In this way, we can view a Lawvere -theory as a particular kind of -functor →Ł, where, as before, we overload notation by using the same names for the -enriched categories as for the ordinary categories from which they are derived.If we term the -functors arising in this way Lawvere -categories, then our reconstruction of the equivalence (<ref>) will follow, exactly as in <cit.>, upon showing the equivalence of one-object -categories, and Lawvere -categories; and, exactly as before, we will obtain this equivalence via the enriched-categorical process of free completion under absolute tensors. Moreover, the universal property of this free completion once again explains the compatibility of this equivalence with the taking of semantics; thereby giving an enriched-categorical explanation of the entire triangle (<ref>). § THE ONE-OBJECT CASEIn this section, we summarise and discuss the manner in which <cit.> reconstructs the equivalence of finitary monads on Set and Lawvere theories from an enriched-categorical viewpoint. Most of what we will say is simply revision, but note that the points clarified by Proposition <ref> and Example <ref> are new.We start from the observation that finitary monads on Set are equally monoids in the monoidal categoryof finitary endofunctors of Set, so equally one-object -enriched categories in the sense of <cit.>. More precisely: The category of finitary monads on Set is equivalent to the category of one-object -enriched categories. To understand Lawvere theories from the -enriched perspective is a little more involved. As a first step, note that if J 𝔽^op→Ł is a Lawvere theory, then both 𝔽^op and Ł are categories with finite powers, and J is a finite-power-preserving functor between them. Thus the desired understanding flows from one of the key results of <cit.>, which shows that categories admittings finite powers are equivalent to -enriched categories admitting all absolute tensors in the following sense. Ifis a monoidal category andis a -category, then a tensor of X ∈ by V ∈ comprises an object V · X ∈ and morphism uV →(X, V · X) in , such that, for any U ∈ and Y ∈, the assignationU (V · X, Y) ↦ U ⊗ V (V · X, Y) ⊗(X, V · X) (X, Y)gives a bijection between maps U →(V · X, Y) and U ⊗ V →(X,Y) in . Such a tensor is said to be preserved by a -functor F → if the composite morphism F_X,V· X∘ uV →(X,V· X) →(FX, F(V · X)) inexhibits F(V · X) as V · FX. Tensors by V ∈ are said to be absolute if they are preserved by any -functor. There is a delicate point here. The theory of <cit.> considers enrichment only over a symmetric monoidal closed base; by contrast, ouris non-symmetric and right-closed—meaning that there exists a right adjoint [V, ] to the functor () ⊗ V tensoring on the right by an object V. The value [V,W] of this right adjoint can be computed by first forming the right Kan extension Ran_V W ∈ [Set, Set], and then the finitary coreflection of that. On the other hand,is not left-closed—meaning that there is not always a right adjoint to the functor V ⊗ () tensoring on the left by V—because V ⊗ () will not be cocontinuous if V itself is not cocontinuous.While the non-symmetric, right-closed setting is too weak to allow constructions such as opposite -categories, functor -categories, or tensor product of -categories, it is strong enough to allow for a good theory of -enriched colimits—of which absolute tensors are an example. In particular, we may give the following tractable characterisation of the absolute tensors over a right-closed base. In the statement, recall that a left dual for V ∈ is a left adjoint for V seen as a 1-cell in the one-object bicategory corresponding to . Letbe a right-closed monoidal category. Tensors by V ∈ are absolute if and only if V admits a left dual in . This result originates in <cit.>, though some adaptations to the proof are required in the non-left-closed setting; we defer giving these to Section <ref> below, where we will give a proof which works in the more general bicategory-enriched context.For now, applying this result when =, we see that tensors by an object F ∈—a finitary endofunctor of Set—are absolute just when F has a (necessarily finitary) left adjoint G. In this case, by adjointness we must have F ≅ ()^G1; moreover, in order for F to be a finitary endofunctor, G1 must be finitely presentable in Set, thus, a finite set. So an -category is absolute-tensored just when it admits tensors by ()^n ∈ for all n ∈ℕ. In <cit.>, such -categories were calledrepresentable.The following further characterisation of the absolute-tensored -categories is Proposition 3.8 of ibid.; in the statement, we write -CAT_abs for the 2-category of absolute-tensored -categories and all (necessarily absolute-tensor-preserving) -functors and -transformations, and write FPOW for the 2-category of categories with finite powers and finite-power-preserving functors.The underlying ordinary category of any absolute-tensoredadmits finite powers, while the underlying ordinary functor of anybetween representable -categories preserves finite powers. The induced 2-functor -CAT_abs→FPOW is an equivalence of 2-categories. This result is the technical heart of <cit.>; as there, the task of giving its proof will be eased if we replace the category of finitary endofunctors of Set by the equivalent functor category [𝔽, Set]. The equivalence in question arises via left Kan extension and restriction along the inclusion 𝔽→Set; and transporting the composition monoidal structure on finitary endofunctors across it yields the so-called substitution monoidal structure on [𝔽, Set], with tensor and unit given by (A ⊗ B)(n) = ∫^k Ak × (Bn)^k and I(n) = n. Henceforth, we re-defineto be this monoidal category. Having done so, we see that a general -categoryinvolves objects X,Y,…, hom-objects (X,Y) ∈ [𝔽, Set], and composition and identities notated as follows:∫^k (Y,Z)(k) ×(X,Y)(n)^k→(X,Z)(n) I(n)→(X,X)(n) [g, f_1, …, f_k]↦ g ∘ (f_1, …, f_k) i↦π_i .Note moreover that, since the unit I ∈ [𝔽, Set] is represented by 1, the arrows X → Y in the underlying ordinary category _0 ofare the elements of (X,Y)(1). Supposeis an absolute-tensored -category. Then for each X ∈ and n ∈ℕ, we have y_n · X ∈ and a unit map y_n →(X, y_n · X), or equally, an element u ∈(X, y_n · X)(n), rendering each (<ref>) invertible. When U = y_1, the function (<ref>) is given, to within isomorphism, by(y_n · X,Y)(1)→(X,Y)(n) f↦ f ∘ (u) ;thus, when Y = X we obtain elements p_1, …, p_n ∈(y_n · X, X)(1) with p_i ∘ (u) = π_i in (X,X)(n). It follows by the -category axioms that (u ∘ (p_1, …, p_n)) ∘ (u) = u, and so, by (<ref>) with Y = y_n · X, that u ∘ (p_1, …, p_n) = 𝕀_y_n · X in (y_n · X, y_n · X)(1).We claim that the maps p_iy_n · X → X in _0 exhibit y_n · X as the n-fold power X^n. Indeed, given g_1, …, g_nZ → X in _0, we define gZ → y_n · X by g = u ∘ (g_1, …, g_n), and now p_i ∘ (u) = π_i implies p_i ∘ (g) = g_i. Moreover, if hZ → y_n · X satisfies p_i ∘ (h) = g_i then g = u ∘ (p_1 ∘ (h), …, p_n ∘ (h)) = (u ∘ (p_1, …, p_n)) ∘ h = h.This proves the first claim. The second follows easily from the fact that any -functor preserves absolute tensors, and so we have a 2-functor -CAT_abs→FPOW. Finally, to show this is a 2-equivalence we construct an explicit pseudoinverse. To each categorywith finite powers, we associate the -categorywith objects those of , with hom-objects (X, Y) = (X^(), Y), with composition operations (<ref>) obtained using the universal property of power, and with identity elements π_i given by power projections. Thisis absolute-tensored on taking the tensor of X ∈ by y_n to be X^n, with unit element 1_X^n∈(X, X^n)(n) = (X^n, X^n). It is now straightforward to extend the assignation ↦ to a 2-functor FPOW→-CAT_abs, and to show that this is pseudoinverse to the underlying ordinary category 2-functor. In particular, if J 𝔽^op→Ł is a Lawvere theory, then we can view both 𝔽^op and Ł as -categories, and J as an -functor between them. Defining a Lawvere -category to be a representable -category Ł equipped with an identity-on-objects, strict absolute-tensor-preserving -functor 𝔽^op→Ł, it follows easily that:The category of Lawvere -categories is equivalent to the category of Lawvere theories (where maps in each case are commuting triangles under 𝔽^op). Using Propositions <ref> and <ref>, we may now re-express the the monad–theory correspondence (<ref>) in -categorical terms as an equivalence between one-object -categories and Lawvere -categories. We obtain this using the process of free completion under absolute tensors—a description of which can be derived from <cit.>.Letbe a monoidal category and let _d ⊂ be a subcategory equivalent to the full subcategory of objects with left duals in . The free completion under absolute tensors of a -categoryis given by the -categorywith: * Objects V · X, where X ∈ and V ∈_d;* Hom-objects (V · X, W · Y) = W ⊗(X,Y) ⊗ V^∗ (for V^∗ a left dual for V);* Composition built from composition inand the counit maps ε W^∗⊗ W → I;* Identities built from the unit maps η I → V ⊗ V^∗ and identities in .Taking = and _d to be the full subcategory on the y_n's, we thus arrive at:The category of one-object -categories is equivalent to the category of Lawvere -categories.The free completion under absolute tensors of the unit -categoryis 𝔽^op; whence each one-object -categoryyields a Lawvere -category on applying completion under absolute tensors to the unique -functor →. In the other direction, we send a Lawvere -category J 𝔽^op→Ł to the one-object sub--category of Ł on J1. The conjunction of Propositions <ref>, <ref> and <ref> now yields the equivalence on the top row of (<ref>). More pedantically, it yields an equivalence, which we should check is in fact the usual one:Let T be a finitary monad on Set, and letbe the corresponding one-object -category; thus,has a single object X with (X, X)(n) = Tn, and composition and identities coming from the monad structure of T. The free completionofunder absolute tensors has objects y_n · X for n ∈ℕ—or equally, just natural numbers—and hom-objects given by(n, m) = y_m ⊗(X,X) ⊗ (y_n)^∗≅ (T(n ×))^m.The underlying ordinary category ofis thus the category Ł with natural numbers as objects, and Ł(n,m) = (Tn)^m. Similarly, the underlying strict finite-power-preserving functor of 𝔽^op→ is given by postcomposition with the unit of T, and so is precisely the Lawvere theory corresponding to the finitary monad T. To reconstruct the whole pseudocommutative triangle in (<ref>), we need the following result, which combines Propositions 2.5 and 4.4 of <cit.>; we omit the proof for now, though note that the corresponding generalisations over a general locally finitely presentable base will be proven as Propositions <ref> and <ref> below.Letdenote the -category corresponding to the category-with-finite-powers Set. The embeddings of finitary monads and Lawvere theories into -categories fit into pseudocommutative triangles:[@!C@C-4em]Mnd_f(Set)^op[dr]_-Alg()[rr]^(0.5) d -CAT^op[dl]^--CAT(, )CAT and[@!C@C-3em]Law^op[dr]_-Mod()[rr]^(0.5) d -CAT^op[dl]^--CAT(, ) .CATGiven this, to obtain the compability with semantics in (<ref>), it suffices to show that, for any one-object -categorywith completion under absolute tensors , there is an equivalence between the category of -functors → and the category of -functors →. But since by constructionis absolute-tensored, this follows directly from the universal property of free completion under absolute tensors. § INGREDIENTS FOR GENERALISATIONIn the rest of the paper, we extend the analysis of the previous section to deal with the finitary monad–Lawvere correspondence over an arbitrary locally finitely presentable (lfp) base. In this section, we set up the necessary background for this: first recalling from <cit.> the details of the generalised finitary monad–Lawvere theory correspondence, and then recalling from <cit.> some necessary aspects of bicategory-enriched category theory. We will assume familiarity with the basic theory of lfp categories as found, for example, in <cit.>.§.§ The monad–theory correspondence over a general lfp baseIn extending the monad–theory correspondence (<ref>) from Set to a given lfp category , one side of the generalisation is apparent: we simply replace finitary monads on Set by finitary monads on . On the other side, the appropriate generalisation of Lawvere theories is given by the Lawvere -theories of <cit.>:A Lawvere -theory is a small category Ł together with an identity-on-objects, finite-limit-preserving functor J →Ł. A morphism of Lawvere -theories is a functor Ł→Ł' commuting with the maps from . A model for a Lawvere -theory is a functor F Ł→Set for which FJ preserves finite limits. Here, and in what follows, we write _f for a small subcategory equivalent to the full subcategory of finitely presentable objects in . Note that, whilehas all finite limits, we do not assume the same of Ł; though, of course, it will admit all finite limits of diagrams in the image of J, and so in particular all finite products.Our definition of Lawvere -theory alters that of <cit.> by dropping the requirement of strict finite limit preservation. However, this apparent relaxation does not in fact change the notion of theory. To see why, we must consider carefully what this strictness amounts to, which is delicate, since Ł need not be finitely complete. The correct interpretion is as follows: we fix some choice of finite limits in , and also assume that, for each finite diagram D →, the category Ł is endowed with a choice of limit for JD (in particular, because J is the identity on objects, this equips Ł with a choice of finite products). We now require that J →Ł send the chosen limits into the chosen limits in Ł. However, in this situation, the choice of limits in Ł is uniquely determined by that inso long as J preserves finite limits in the non-algebraic sense of sending limit cones to limit cones. Thus, if we interpret the preservation of finite limits in Definition <ref> in this non-algebraic sense, then our notion of Lawvere -theory agrees with <cit.>.Note also that our definition of model for a Lawvere -theory is that of <cit.>, rather than that of <cit.>: the latter paper defines a model to comprise A ∈ together with a functor F Ł→Set such that FJ = (, A) →Set. The equivalence of these definitions follows sinceis equivalent to the category FL(_f^op, Set) of finite-limit-preserving functors →Set via the assignation A ↦(, A). Even bearing the above remark in mind, it is not immediate that Lawvere Set-theories and their models coincide with Lawvere theories and their models in the previous sense; however, this was shown to be so in <cit.>. The correctness of these notions over a general base is confirmed by the main result of <cit.>, which we restate here as: <cit.> The category of finitary monads onis equivalent to the category of Lawvere -theories; moreover this equivalence is compatible with the semantics in the sense displayed in (<ref>).For a finitary monad T on , let Ł_T be the category with objects those of _f, hom-sets Ł_T(A,B) = (B,TA), and the usual Kleisli composition; now the identity-on-objects J_T _f^op→Ł_T sending f ∈_f(B,A) to η_A ∘ f ∈Ł_T(A,B) is a Lawvere -theory. Conversely, if J _f^op→Ł is a Lawvere -theory, then the composite of the evident forgetful functor Mod(Ł) →FL(_f^op, Set) with the equivalence FL(_f^op, Set) → is finitarily monadic, and so gives a finitary monad T_Ł on . With some care one may now show that these processes are pseudoinverse in a manner which is compatible with the semantics. §.§ Bicategory-enriched category theoryWe now recall some basic definitions from the theory of categories enriched over a bicategory as developed in <cit.>.Letbe a bicategory whose 1-cell composition and identities we write asand _x respectively. A -categorycomprises: * A setof objects;* For each X ∈, an extent ϵ X ∈;* For each X, Y ∈, a hom-object[There are two conventions in the literature: we either take (X,Y) ∈(ϵ X, ϵ Y), as in <cit.> for example, or we take (X,Y) ∈(ϵ Y, ϵ X) as in <cit.>. We have chosen the former convention here, and haveadjusted results from the literature where necessary to conform with this.] (X,Y) ∈(ϵ X, ϵ Y);* For each X, Y, Z ∈, composition maps in (ϵ X, ϵ Z) of the formμ_xyz(Y,Z) (X,Y) →(X,Z); * For each X ∈ an identities map in (ϵ X, ϵ X) of the formι_X _ϵ X→(X,X); subject to associativity and unitality axioms. A -functor F → between -categories comprises an extent-preserving assignation on objects, together with maps F_XY(X,Y) →(FX,FY) in (ϵ X, ϵ Y) for each X,Y ∈, subject to the two usual functoriality axioms. Finally, a -transformation α F ⇒ G → between -functors comprises maps α_x _ϵ X→(FX,GX) in (ϵ X, ϵ X) for each X ∈ obeying a naturality axiom. We write -CAT for the 2-category of -categories, -functors and -transformations. Note that, ifis the one-object bicategory corresponding to a monoidal category , then we re-find the usual definitions of -category, -functor and -transformation. A key difference in the general bicategorical situation is that a -category does not have a single “underlying ordinary category”, but a whole family of them:For any x ∈, we write _x for the -category with a single object ∗ of extent x and with _A(∗, ∗) = _x, and write ()_x for the representable 2-functor -CAT(_x, ) -CAT→CAT. On objects, this 2-functor sends a -categoryto the ordinary category _x whose objects are the objects ofwith extent x, and whose morphisms X → Y are morphisms _x →(X,Y) in (x,x). §.§ Enrichment through variationIt was shown in <cit.> that there is a close link between -categories and -representations. A -representation is simply a homomorphism of bicategories F →CAT, but thought of as a “left action”; thus, we notate the functors F_xy(x,y) →CAT(Fx,Fy) as W ↦ W ∗_F (), and write the components of the coherence isomorphisms for F as maps λ I_x ∗_F X → X and α (V ⊗ W) ∗_F X → V ∗_F (W ∗_F X).Theorem 3.7 of <cit.> establishes an equivalence between closed -representations and tensored -categories. Here, a -representation is called closed if each functor () ∗_F W (x,y) → Fy has a right adjoint W, _FFy →(x,y); while a -category is tensored if it admits all tensors in the following sense: Ifis a bicategory andis a -category, then a tensor of X ∈_x by W ∈(x,y) is an object W · X ∈_y together with a map uW →(X, W · X) in (x,y) such that, for any U ∈(y,z) and Z ∈_z, the assignationU (W · X, Z) ↦ U ⊗ W (V · X, Z) ⊗(X, W · X) (X, Z)establishes a bijection between morphisms U →(W · X, Z) in (y,z) and morphisms U ⊗ W →(X,Z) in (x,z). We note here for future use that a tensor W · X is said to be preserved by a -functor F → if the composite W →(X, W · X) →(FX, F(W · X)) exhibits F(W · X) as W · FX; and that tensors by a 1-cell W are called absolute if they are preserved by any -functor.<cit.>There is an equivalence between the 2-category-CAT_tens of tensored -categories, tensor-preserving -functors and -natural transformations, and the2-category Hom(, CAT)_cl of closed-representations, pseudonatural transformations and modifications. In one direction, the closed -representation C associated to atensored -categoryis defined on objects by C(x) = _x,and with action by 1-cells given by tensors:W ∗_C X = W · X. We do not need the further details here,and so omit them. In the other direction, the tensored -category associated to a closed representationF →CAT has objects of extent a beingobjects of Fa; hom-objects given by (X,Y) = X,Y_F; andcomposition and identities given by transposing the maps (Y,Z_F ⊗X,Y_F) ∗_F X Y,Z_F ∗_F (X,Y_F ∗_F X) Y,Z_F ∗_F Y Z and λ I_a ∗_F X → X under the closureadjunctions. The -categoryso obtained admits all tensors ontaking W · X = W ∗_F X with unitW →X, W ∗_F X_F obtained from the closureadjunctions.We will make use of this equivalence in Section <ref> below, and will require the following easy consequence of the definitions:Let F →CAT be a closed representation, corresponding to the tensored -category , and let Ta → a be a monad in , corresponding to the one-object -category . There is an isomorphism of categories -CAT(, ) ≅ F(T)-Alg, natural in maps of monads on a in .The action on objects of a -functor → picks out an object of extent a in , thus, an object X ∈ Fa. The action on homs is given by a map xT →X,X_F in (a,a), while functoriality requires the commutativity of:[@[email protected]@-0.5em]I_a [dl]_η[dr]^ιT [rr]^x X,X_Fand[@[email protected]] T ⊗ T [r]^-x ⊗ x[d]_μ X,X_F ⊗X,X_F [d]^μT [r]^-xX,X_F .Transposing under adjunction, this is equally to give X ∈ Fa and a map T ∗_F X → X satisfying the two axioms to be an algebra for F(T) = T ∗_F (). Further, to give a -transformation F ⇒ G → is equally to give ϕ I_x →X,Y_F such that[@C+1em] T [r]^y[d]_x Y, Y_F [r]^-φ⊗ 1 X,Y_F ⊗Y,Y_F [d]^μ X, X_F [r]^-1 ⊗φ X,X_F⊗X,Y_F [r]^-μ X, Y_Fcommutes; which, transposing under adjunction and using the coherence constraint I_a ∗ X ≅ X, is equally to give a map X → Y commuting with the F(T)-actions. The naturality of the correspondence just described in T is easily checked.§ FINITARY MONADS AND THEIR ALGEBRASWe now begin our enriched-categorical analysis of the monad–theory correspondence over an lfp base. We first describe a bicategoryof lex profunctors which is biequivalent to the bicategoryof finitary functors between lfp categories, but more convenient to work with; we then exhibit each finitary monad on an lfp category as a -category, and the associated category of algebras as a category of -enriched functors.§.§ Finitary monads as enriched categoriesThe basic theory of lfp categories tells us that for any small finitely-complete 𝔸, the category (𝔸, Set) of finite-limit-preserving functors 𝔸→Set is lfp, and moreover that every lfp category is equivalent to one of this form. Sois biequivalent to the bicategory whose objects are small finitely-complete categories, whose hom-category from 𝔸 to 𝔹 is ((𝔸, Set), (𝔹, Set)), and whose composition is inherited from .Now, since for any small finitely-complete 𝔸, the inclusion 𝔸^op→(𝔸, Set) exhibits its codomain as the free filtered-cocomplete category on its domain, there are equivalences ((𝔸, Set), (𝔹, Set)) ≃ [𝔸^op, (𝔹, Set)]; thus transporting the compositional structure ofacross these equivalences, we obtain:The right-closed bicategoryof lex profunctors has:* As objects, small categories with finite limits. * (𝔸, 𝔹) = [𝔸^op, (𝔹, Set)]; we typically identify objects therein with functors 𝔸^op×𝔹→Set that preserve finite limits in their second variable. * The identity 1-cell I_𝔸∈(𝔸, 𝔸) is given by I_𝔸(a',a) = 𝔸(a',a), while the composition of M ∈(𝔸, 𝔹) and N ∈(𝔹, ℂ) is given by:(N ⊗ M)(a, c) = ∫^b ∈𝔹 N(b,c) × M(a,b) .* For M ∈(𝔸, 𝔹) and P ∈(𝔸, ℂ), the right closure [M,P] ∈(𝔹, ℂ) is defined by[M,P](b,c) = ∫_a[M(a,b), P(a,c)] . By the above discussion,is biequivalent to , and this induces an equivalence between the category of monads onin —thus, the category of finitary monads on —and the category of monads onin . Such monads correspond with one-object -categories of extentand so:For any locally finitely presentable category , the category Mnd_f() of finitary monads onis equivalent to the category of -categories with a single object of extent . §.§ Algebras for finitary monads as enriched functorsWe now explain algebras for finitary monads in the -enriched context. Composing the biequivalence → with the inclusion 2-functor →CAT yields a homomorphism S →CAT which on objects sends 𝔸 to (𝔸, Set), and for which the action of a 1-cell M ∈(𝔸, 𝔹) on an object X ∈(𝔸, Set) is given as on the left in(M ∗_S X)(b) = ∫^a ∈𝔸 M(a,b) × Xa X,Y_S(a,b) = Set(Xa,Yb) .This S is a closed representation, where for X ∈(𝔸, Set) and Y ∈(𝔹, Set) we define X,Y_S as to the right above; and so applying Proposition <ref> gives a tensored -categorywith objects of extent 𝔸 being finite-limit-preserving functors 𝔸→Set, and with hom-objects (X,Y)(a,b) = Set(Xa, Yb).For any locally finitely presentable category , the embedding of finitary monads onas one-object -categories obtained in Proposition <ref> fits into a triangle, commuting up to pseudonatural equivalence:[@!C@C-4em]Mnd_f()^op[dr]_-()-Alg[rr]^(0.5) d(-CAT)^op .[dl]^- -CAT(, )CAT Given T ∈Mnd_f(), which is equally a monad onin , we can successively apply the biequivalences → and → to obtain in turn a monad T' on _f^op∈ and a monad T” on FL(_f^op, Set). It follows easily from the fact of a biequivalence that T-Alg≃ T”-Alg.Now, starting from T ∈Mnd_f(), the functor across the top of (<ref>) sends it to the one-object -category ' corresponding to T'; whereupon by Proposition <ref>, we have pseudonatural equivalences-CAT(', ) ≅ S(T')-Alg = T”-Alg≃ T-Alg .* §.§ General -categoriesBefore turning to the relationship of -categories and Lawvere -theories, we take a moment to unpack the data for a general -category . We have objects X, Y, … with associated extents 𝔸, 𝔹, … in ; while for objects X ∈_𝔸 and Y ∈_𝔹, we have the hom-object (X,Y) 𝔸^op×𝔹→Set, which is a functor preserving finite limits in its second variable. By the coend formula (<ref>) for 1-cell composition in , composition inis equally given by functions(Y,Z)(j,k) ×(X,Y)(i,j)→(X,Z)(i, k) (g,f)↦ g ∘ fwhich are natural in i ∈𝔸 and k ∈ℂ and dinatural in j ∈𝔹. On the other hand, identities inare given by functions ι_X 𝔸(i,j) →(X,X)(i,j), natural in i,j ∈𝔸; if we define 1_X,iι_X(1_i), then the -category axioms forsay that f ∘ 1_X,i = f = 1_Y,j∘ f for all f ∈(X,Y)(i,j) and that the operation (<ref>) is associative. Note that the naturality of each ι_X together with the unit axioms imply that the action on morphisms of the hom-object (X,Y) is given by(X,Y)(φ, ψ) (X,Y)(i,j)→(X,Y)(i',j')f↦ι_Y(ψ) ∘ f ∘ι_X(φ) .Applying naturality of ι_X again to this formula yields the following functoriality equation for any pair of composable maps in 𝔸:ι_X(φ' ∘φ) = ι_X(φ') ∘ι_X(φ) .§ PARTIAL FINITE COMPLETENESSIn the following two sections, we will identify the absolute-tensored -categories with what we call partially finitely complete ordinary categories; this identification will take the form of a biequivalence between suitably-defined 2-categories. We will exploit this biequivalence in Section <ref> in order to identify Lawvere -theories with certain functors between absolute-tensored -categories. §.§ Partially finitely complete categories We begin by introducing the 2-category of partially finitely complete categories and partially finite-limit-preserving functors. By a left-exact sieve on a category , we mean a collectionof finite-limit-preserving functors 𝔸→, each with small, finitely-complete domain, and satisfying the following conditions, wherein we write [𝔸] for those elements ofwith domain 𝔸: * If X ∈[𝔹] and G ∈FL(𝔸, 𝔹), then XG ∈[𝔸];* If X ∈[𝔸] and X ≅ Y 𝔸→, then Y ∈[𝔸];* Each object ofis in the image of some functor in .A partially finitely complete category (, _) is a categorytogether with a left-exact sieve _ on it. Where confusion is unlikely, we may write (, _) simply as . A partially finite-limit-preserving functor (, _) → (, _) is a functor F → such that FX ∈_ for all X ∈_; we call such an F sieve-reflecting if, for all Y ∈_, there exists X ∈_ such that FX ≅ Y. We write PARFL for the 2-category of partially finitely complete categories, partially finite-limit-preserving functors, and arbitrary natural transformations. The following examples should serve to clarify the relevance of these notions to Lawvere theories over a general lfp base.Any finitely completecan be seen as partially finitely complete when endowed with the sieve _ of all finite-limit-preserving functors intowith small domain. Ifis also finitely complete, then any finite-limit-preserving F → is clearly also partially finite-limit-preserving; conversely, if F → is partially finite-limit-preserving, then for any finite diagram D 𝕀→, closing its image inunder finite limits yields a small subcategory 𝔸 for which the full inclusion J 𝔸→ preserves finite limits. As F is partially finite-limit-preserving, the composite FJ 𝔸→ also preserves finite limits; in particular, the chosen limit cone over D in —which lies in the subcategory 𝔸—is sent to a limit cone in . It follows there is a full and locally full inclusion of 2-categories →. If J _f^op→Ł is a Lawvere -theory, then Ł becomes a partially finitely complete category when endowed with the sieve generated by J:_Ł = { F 𝔸→Ł : F ≅ JGfor some finite-limit-preservingG 𝔸→_f^op} .Clearly _Ł satisfies conditions (i) and (ii) above, and satisfies (iii) by virtue of J being bijective on objects. Moreover, a partially finite-limit-preserving Ł→ is precisely a functor F Ł→ such that FJ ∈_; so in particular, a partially finite-limit-preserving Ł→Set is precisely a model for the Lawvere -theory Ł. §.§ Partial finite completeness and -enrichment Towards our identification of absolute-tensored -categories with partially finitely complete categories, we now construct a 2-adjunction@<-4.5pt>[r]_-Γ@[r]|- -CAT@<-4.5pt>[l]_-∫ . Letbe a partially finitely complete category. The -category Γ() has objects of extent 𝔸 given by elements X ∈_[𝔸], and remaining data defined as follows: * For X ∈_[𝔸] and Y ∈_[𝔹], the hom-object Γ()(X,Y) ∈(𝔸, 𝔹) is given by Γ()(X,Y)(i,j) = (Xi,Yj). Note that this preserves finite limits in its second variable since Y and each (Xi, ) do so.* Composition in Γ() may be specified, as in (<ref>), by natural families of functions Γ()(Y,Z)(j,k) ×Γ()(X,Y)(i,j) →Γ()(X,Z)(i,k), which we obtain from composition in .* Identities ι_X 𝔸(i,j) →Γ()(X,X)(i,j) = (Xi,Xj) are given by the action of X on morphisms.The -category axioms for Γ() follow from the category axioms ofand functoriality of each X.If F → is a partially finite-limit-preserving functor, then we define the -functor Γ(F) Γ() →Γ() to have action on objects X ↦ FX (using the fact that FX ∈_ whenever X ∈_). The components of the action of Γ(F) on hom-objects Γ()(X,Y) →Γ()(FX,FY) are functions (Xi,Yj) →(FXi,FYj), which are given simply by the action of F on morphisms. The -functor axioms are immediate from functoriality of F.Finally, for a 2-cell α F ⇒ G → in , we define a -transformation Γ(α) Γ(F) ⇒Γ(G) whose component I_𝔸→Γ()(FX,GX) is given by the dinatural family of elements α_Xi∈(FXi,GXi). The -naturality of Γ(α) amounts to the condition that Gf ∘α_Xi = α_Yj∘ FfFXi → GYj for all fXi → Yj in ; which is so by naturality of α. The data of Definition <ref> comprise the action on 0-, 1-, and 2-cellsof a 2-functor Γ→-CAT, Moreover the 2-functor Γ admits a left 2-adjoint ∫-CAT→.The 2-functoriality of Γ is easy to check, and so it remains to construct its left 2-adjoint ∫. Given a -category , we writefor the category with: * Objects of the form (X,i) where X ∈_𝔸 and i ∈𝔸;* Morphisms f(X,i) → (Y,j) being elements f ∈(X,Y)(i,j);* Identities given by the elements 1_X,i∈(X,X)(i,i);* Composition mediated by the functions (<ref>).Given X ∈_𝔸, we write ι_X 𝔸→ for the functor given by i ↦ (X,i) on objects and by the identities map ι_X 𝔸(i,i') →(X,X)(i,i') ofon morphisms; note this is functorial by (<ref>). By the definition ofand (<ref>), we have that(X,Y) = ()(ι_X(), ι_Y()) 𝔸^op×𝔹→Set ;in particular, as each (X,Y) preserves finite limits in its second variable, each functor ( (X,i), ι_Y()) 𝔹→Set preserve finite limits, whence each ι_Y 𝔹→ preserves finite limits. It follows thatis partially finitely complete when endowed with the left-exact sieve_∫ = { G 𝔸→Γ() : G ≅ι_Y Ffor some Y ∈_𝔹 and F ∈FL(𝔸, 𝔹) } . We now show thatprovides the value atof a left 2-adjoint to Γ; thus, we must exhibit isomorphisms of categories, 2-natural in ∈, of the form:(, ) ≅-CAT(, Γ()) .Now, to give a partially finite-limit-preserving functor F → is to give: * For all X ∈_𝔸 and i ∈𝔸 an object F(X,i) ∈; and* For all f ∈(X,Y)(i,j), a map FfF(X,i) → F(Y,j) in ,functorially with respect to the composition (<ref>) and composition in , and subject to the requirement that Fι_X ∈_[𝔸] for all X ∈_𝔸. On the other hand, to give a -functor G →Γ() is to give: * For all X ∈_𝔸, a functor GX ∈_[𝔸]; and* For all f ∈(X,Y)(i,j), an element of Γ()(GX, GY) = ((GX)i, (GY)j), i.e., a map Gf(GX)i → (GY)j in ,subject to the same functoriality condition. Thus, given F →, we may define F̅→Γ() by taking F̅X = Fι_X (which is in _[𝔸] by assumption) and F̅f = Ff; the functoriality is clear. On the other hand, given G →Γ(), we may define G̅→ by taking G̅(X,i) = (GX)i and G̅f = Gf. Functoriality is again clear, but we need to check that G̅ι_X ∈_[𝔸] for all X ∈_𝔸. In fact we show that G̅ι_X = GX, which is in _[𝔸] by assumption. On objects, G̅ι_X(i) = G̅(X,i) = (GX)i as required. On morphisms, the compatibility of G with identities inand Γ() gives a commuting triangle of sets and functions:[@!C@C-6em]𝔸(i,j)[dl]_-ι_X[dr]^-ι_GX (X,X)(i,j)[rr]^-GΓ()(GX, GX)(i,j) = (GXi, GXj) .The left-hand path maps φ∈𝔸(i,j) to Gι_X(φ) = G̅ι_X(φ); while by definition of Γ() the right-hand path maps φ to GX(φ); whence G̅ι_X = GX as required. It is clear from the above calculations that the assignations F ↦F̅ and G ↦G̅ are mutually inverse, which establishes the bijection (<ref>) on objects.To establish (<ref>) on maps, let F_1, F_2 ⇉. The components of a -transformation α̅F̅_1 ⇒F̅_2 →Γ() comprise natural families of functions α̅_Xij𝔸(i,j) →Γ()(F̅_1X, F̅_2X)(i,j) = (F_1ι_X(i), F_2ι_X(j)) satisfying -naturality. By Yoneda, each α̅_Xij is uniquely determined by elements α̅_X,i = α̅_Xii(𝕀_i) ∈(F_1(X,i), F_2(X,i)) satisfying F_2ι_X(φ) ∘α̅_X,i = α̅_X,j∘ F_1ι_X(φ) for all φ∈𝔸(i,j); their -naturality is now the requirement that the square[@C+2em](X,Y)(i,j)[r]^-F_1[d]_F_2 (F_1(X,i),F_1(Y,j))[d]^α̅_Y,j∘ () (F_2(X,i),F_2(Y,j))[r]^-() ∘α̅_X,i (F_1(X,i),F_2(Y,j))commute for each X,Y,i,j. Note that this implies the earlier condition that F_2ι_X(φ) ∘α̅_X,i = α̅_X,j∘ F_1ι_X(φ) on taking X=Y and evaluating at ι_X(φ); now evaluating at a general element, we get the condition that F_2f ∘α̅_X,i = α̅_Y,j∘ F_1f for all f(X,i) → (Y,j) in —which says precisely that we have a natural transformation α̅ F_1 ⇒ F_2 →. This establishes the bijection (<ref>) on morphisms; the 2-naturality inis left as an easy exercise for the reader.§ ABSOLUTE-TENSORED -CATEGORIESIn this section, we prove the key technical result of this paper, Theorem <ref>, which shows that the 2-adjunction (<ref>) exhibitsas biequivalent to the full sub-2-category of -CAT on the absolute-tensored -categories.§.§ Absolute tensors in -categoriesWe begin by characterising absolute tensors in -categories for an arbitrary right-closed bicategory . Here, right-closedness is the condition that, for every 1-cell W ∈(x,y) and every z ∈, the functor () ⊗ W (y,z) →(x,z) admits a right adjoint [W, ] (x,z) →(y,z). In this setting, we will show that tensors by a 1-cell W are absolute if and only if W is a right adjoint in . Ifwere both left- and right- closed, this would follow from the characterisation of enriched absolute colimits given in <cit.>, but in the absence of left-closedness, we need a different proof. The first step is the following, which is a special case of <cit.>:Letbe a bicategory, letbe a -category, let X ∈_x and let W ∈(x,y). If W admits the left adjoint W^∗∈(y,x), then there is a bijective correspondence between data of the following forms: * A map uW →(X, Y) in (x,y) exhibiting Y as W · X;* Maps uW →(X, Y) in (x,y) and u^∗ W^∗→(Y, X) in (y,x) rendering commutative the squares:I_y [r]^η[d]_ιW ⊗ W^∗[d]^u ⊗ u^∗W^∗⊗ W [r]^ε[d]_u^∗⊗ uI_x [d]^ι (Y,Y) @<-[r]^- μ (X,Y) ⊗(Y,X)(Y,X) ⊗(X,Y) [r]^-μ (X,X) .Given (a), applying surjectivity in (<ref>) to ι_X ∘ε W^∗⊗ W → I_x →(X,X) yields a unique map u^∗ W^∗→(Y,X) making the square right above commute. To see that the left square also commutes, it suffices by injectivity in (<ref>) to check that the sides become equal after tensoring on the right with u and postcomposing with μ(Y,Y) ⊗(X,Y) →(X,Y). This follows by a short calculation using commutativity in the right square and the triangle identities.To complete the proof, it remains to show that if u and u^∗ are given as in (b), then u exhibits Y as W · X. Thus, given gU ⊗ W →(X,Z), we must show that g = μ∘ (f ⊗ u) as in (<ref>) for a unique fU →(Y,Z). We may take f to beUU ⊗ W ⊗ W^∗(X,Z) ⊗(Y,X) (Y,Z) ;now that g = μ∘ (f ⊗ u) follows on rewriting with the right-hand square of (<ref>), the triangle identities and the -category axioms for . Moreover, if f'U →(Y,Z) also satisfies g = μ∘ (f' ⊗ u), then substituting into (<ref>) givesf = UU ⊗ W ⊗ W^∗(Y,Z) ⊗(X,Y) ⊗(Y,X) (Y,Z)which is equal to f' via the category axioms forand the left square of (<ref>). Using this result, we may now prove:Letbe a right-closed bicategory. Tensors by W ∈(x,y) are absolute if and only if the 1-cell W admits a left adjoint in .If W admits a left adjoint, then the data for a tensor by W can be expressed as in Proposition <ref>(b); since these data are clearly preserved by any -functor, tensors by W are absolute. Conversely, suppose that tensors by W are absolute; we will show that W admits the left dual [W,I_x] ∈(y,x). The counit ε is the evaluation map ev [W, I_x] ⊗ W → I_x, and it remains only to define the unit.For each a ∈, we have the -representation (a, ) →CAT which is closed sinceis right-closed. Thus, by the construction of Proposition <ref>, there is a tensored -category (a, ) = a/ whose objects of extent b are 1-cells a → b, whose hom-objects are (a/)(X,Y) = [X,Y], and whose tensors are given by Y · X = Y ⊗ X.Now, for any 1-cell Z ∈(a,b), there is a -functor [Z, ]a/→ b/ given on objects by X ↦ [Z,X] and with action [X,Y] → [ [Z,X], [Z,Y] ] on hom-objects obtained by transposing the composition map in a /. Since tensors by W are absolute, they are preserved by [Z, ]a/→ b/; it follows that the mapθ_ZX W ⊗ [Z,X] → [Z, W ⊗ X]in (b,z) given by transposing W ⊗ev W ⊗ [Z,X] ⊗ Z → W ⊗ X is invertible for all Z ∈(a,b) and X ∈(a, x). In particular, we have θ_W,I_x W ⊗ [W,I_x] ≅ [W,W ⊗ I_x] and so a unique η I_y → W ⊗ [W,I_x] such that θ_VI∘η is the transpose of the morphism ρ_Wλ_WI_y ⊗ W → W → W ⊗ I_x. This condition immediately implies the triangle identity (W ⊗ε) ∘ (η⊗ W) = 1, and implies the other triangle identity (ε⊗ [W,I_x]) ∘ ([W,I_x] ⊗η) = 1 after transposing under adjunction and using bifunctoriality of ⊗. §.§ Absolute -tensorsUsing the above result, we may now characterise the absolute-tensored -categories via the construction ∫ of Proposition <ref>.A -categoryis absolute-tensored if and only if, for all X ∈_𝔹 and all F 𝔸→𝔹 in , there exists Y ∈_𝔸 and a natural isomorphism[@-0.5em@!C]𝔸[rr]^-F[dr]_-ι_Y dυ 𝔹[dl]^-ι_X We write ()_∗^op→ for the identity-on-objects homomorphism sending F 𝔸→𝔹 to the lex profunctor F_∗𝔹𝔸 with F_∗(b,a) = 𝔹(b,Fa). Each F_∗ has a left adjoint F^∗ inwith F^∗(a,b) = 𝔹(Fa,b), and—as all idempotents split in a finitely complete category—the usual analysis of adjunctions of profunctors adapts to show that, within isomorphism, every right adjoint 1-cell inarises thus. So by Proposition <ref>, a -categoryis absolute-tensored just when it admits all tensors by 1-cells F_∗.By Proposition <ref>, this is equally to say that, for all X ∈_𝔹 and F ∈(𝔸, 𝔹), we can find Y ∈_𝔸 and maps uF_∗→(X, Y) and u^∗ F^∗→(Y,X) rendering commutative both squares in (<ref>). To complete the proof, it suffices to show that the data of u and u^∗ are equivalent to those of an invertible transformation υ as in (<ref>). Now, u comprises a natural family of maps 𝔹(j,Fi) →(X,Y)(j, i); equally, by Yoneda, elements υ_i ∈(X, Y)(Fi, i) = (ι_X(Fi), ι_Y(i)) dinatural in i ∈𝔸; or equally, the components of a natural transformation υ as in (<ref>). Similar arguments show that giving u^∗ F^∗→(Y,X) is equivalent to giving a natural transformation υ^∗ι_Y ⇒ι_X F, and that commutativity in the two squares of (<ref>) is equivalent to the condition that υ and υ^∗ are mutually inverse. We now have all the necessary ingredients to prove:The 2-functor Γ→-CAT of (<ref>) is an equivalence on hom-categories, and its biessential image comprises the absolute-tensored -categories. Thus Γ exhibitsas biequivalent to the full and locally full sub-2-category of -CAT on the absolute-tensored -categories.By a standard argument, to say that Γ is an equivalence on homs is equally to say that each counit component ε_∫Γ→ of (<ref>) is an equivalence in . Now, from the definitions, the category ∫Γ has: * Objects being pairs (X ∈[𝔸], i ∈𝔸);* Morphisms (X,i) → (Y,j) being maps Xi → Yj in ;* Composition and identities inherited from ,while ε_∫Γ→ sends (X,i) to Xi and is the identity on homsets. So clearly ε_ is fully faithful; while condition (iii) for a left-exact sieve ensures that it is essentially surjective, and so an equivalence of categories. However, for ε_ to be an equivalence in , its pseudoinverse must also be partially finite-limit-preserving. This is easily seen to be equivalent to ε_ being sieve-reflecting; but for each X ∈[𝔸], the functor ι_X 𝔸→ sending i to (X,i) and φ to Xφ is by definition in the sieve _∫Γ, and clearly ε_∘ι_X = X.This shows that Γ is locally an equivalence; as for its biessential image, this comprises just those ∈-CAT at which the unit η_→Γ() is an equivalence of -categories. Now, from the definitions, Γ() has: * Objects of extent 𝔸 being functors 𝔸→ in the left-exact sieve _∫;* Hom-objects given by Γ()(X,Y) = (X, Y);* Composition and identities inherited from ,while the -functor η_→Γ() is given on objects by X ↦ι_X, and on homs by the equality (X,Y) = ()(ι_X, ι_Y) of (<ref>); in particular, it is always fully faithful. To characterise when it is essentially surjective, note first that isomorphisms in Γ()_𝔸 are equally natural isomorphisms in [𝔸, ]. Now as objects of Γ()_𝔸 are elements of _∫[𝔸], and since by (<ref>) every such is isomorphic to ι_X F for some X ∈_𝔹 and F ∈(𝔸, 𝔹), we see that η_ is essentially surjective precisely when for all X ∈_𝔹 and F ∈(𝔸, 𝔹) there exists Y ∈_𝔸 and a natural isomorphism υι_X F ≅ι_Y: which by Proposition <ref>, happens precisely whenadmits all absolute tensors. We will use this result in the sequel to freely identify absolute-tensored -categories with partially finitely complete categories; note that, on doing so, the left 2-adjoint ∫-CAT→ of (<ref>) provides us with a description of the free completion of an -category under absolute tensors. § LAWVERE -THEORIES AND THEIR MODELSWe are now ready to give our -categorical account of Lawvere -theories and their models. We will identify each Lawvere -theory with what we term a Lawvere -category on , and will identify the category of models with a suitable category of -enriched functors.§.§ Lawvere -theories as enriched categoriesLawvere -categories will involve certain absolute-tensored -categories, which in light of Theorem <ref>, we may work with in the equivalent guise of partially finitely complete categories. In giving the following definition, and throughout the rest of this section, we view the finitely completeas being partially finitely complete as in Example <ref>.Letbe an lfp category. A Lawvere -category overcomprises a partially finitely complete category Ł together with a map J →Ł inwhich is identity-on-objects and sieve-reflecting. A morphism of Lawvere -categories is a commuting triangle in . For any locally finitely presentable , the category of Lawvere -theories is isomorphic to the category of Lawvere -categories over .Any Lawvere -theory J →Ł can be viewed as a Lawvere -category overas in Example <ref>; it is moreover clear that under this assignation, maps of Lawvere -theories correspond bijectively with maps of Lawvere -categories. It remains to show that each Lawvere -category J →Ł overarises from a Lawvere -theory. Because in this context,is equipped with the maximal left-exact sieve, the fact that J is a morphism inis equivalent to the condition that J ∈_Ł—so that, in particular, J is finite-limit-preserving. Moreover, the fact of J being sieve-reflecting implies that _Ł must be exactly the left-exact sieve (<ref>) generated by J. §.§ Models for Lawvere -theories as enriched functors We now describe how models for a Lawvere -theory can be understood in -categorical terms. Recall from Section <ref> that we definedto be the -category whose objects of extent 𝔸 are finite-limit-preserving functors 𝔸→Set, and whose hom-objects are given by (X,Y)(i,j) = Set(Xi,Yj). By inspection of Definition <ref>, this is equally the -category Γ(Set) when Set is seen as partially finitely complete as in Example <ref>.For any locally finitely presentable category , the identification of Lawvere -theories with Lawvere -categories onfits into a triangle, commuting up to pseudonatural equivalence:[@!C@C-5em]Law()^op[dr]_-()-Mod[rr]^(0.5) d r1.2em(-CAT)^op[dl]^- -CAT(, )CATwherein the horizontal functor is that sending a Lawvere -theory J →Ł to the -category Γ(Ł).We observed in Example <ref> above that, if J →Ł is a Lawvere -theory, then the category Ł-Mod of Ł-models is isomorphic to the hom-category (Ł, Set). Since Γ→-CAT is an equivalence on homs, and since Γ(Set) =, we thus obtain the components of the required pseudonatural equivalence as the compositesŁ-Mod(Ł, Set) -CAT(Γ(Ł), ) .* § RECONSTRUCTING THE MONAD–THEORY CORRESPONDENCEWe have now done all the hard work necessary to prove our main result. The process of freely completing a one-object -category under absolute tensors induces, by way of the identifications of Propositions <ref> and <ref>, an equivalence betwen the categories of finitary monads onand of Lawvere -theories. This equivalence fits into a pseudocommuting triangle:[@!C@C-2em]Mnd_f()^op[dr]_(0.4)()-Alg[rr]@[d]|≃ Law()^op .[dl]^(0.4) ()-ModCAT To obtain the desired equivalence, it suffices by Propositions <ref> and <ref> to construct an equivalence between the category of -categories with a single object of extent _f^op, and the category of Lawvere -categories on .On the one hand, given the one-object -category , applying the free completion under absolute tensors ∫-CAT→ to the unique -functor ! _→ yields a Lawvere -category:J_ = ∫ ! →∫ ,where direct inspection of the definition of ∫ tells us that ∫_ = and that ∫ ! is identity-on-objects and sieve-reflecting.On the other hand, if J →Ł is a Lawvere -category on , then we may form the composite around the top and right of the following square, wherein η is a unit component of the 2-adjunction (<ref>):_[r]^-η[d]_F Γ()[d]^Γ(J) _J[r]^-G Γ(Ł) .We now factorise this composite as (identity-on-objects, fully faithful), as around the left and bottom, to obtain the required one-object -category _J.The functoriality of the above assignations is direct; it remains to check that they are inverse to within isomorphism. First, ifis a one-object -category with associated Lawvere -category (<ref>), then in the naturality square__f^op[r]^-η[d]_! Γ()[d]^Γ(∫ !) [r]^-η Γ(∫)for η, the left-hand arrow is identity-on-objects, and the bottom fully faithful (by Theorem <ref>). Comparing with (<ref>), we conclude by the essential uniqueness of (identity-on-objects, fully faithful) factorisations that ≅_J_ as required.Conversely, if J →Ł is a Lawvere -category onwith associated one-object -category _J as in (<ref>), then we may form the following diagram:[dl]_-J__J = ∫F[dr]^-J ∫_J[r]_-∫G ∫Γ(Ł) [r]_-ε Łwhere ε is a counit component of (<ref>). The composite around the left and bottom is the adjoint transpose of GF under (<ref>); but by (<ref>), GF = Γ(J) ∘η which is in turn the adjoint transpose of J. It thus follows that the above triangle commutes. Since both ∫F and J are identity-on-objects, so is the horizontal composite; moreover, ε is an equivalence by Theorem <ref> while ∫G is fully faithful since G is, by inspection of the definition of ∫. So the lower composite is fully faithful and identity-on-objects, whence invertible, so that J ≅ J__J as required.We thus have an equivalence as across the top of (<ref>), and it remains to show that this renders the triangle belowcommutative to within pseudonatural equivalence. To this end, consider the diagram[@!C@C-4em]Mnd_f()^op[d] [rr] drrLaw()^op[d]. -CAT^op[rr]^Γ∫[dr]_(0.4)-CAT(,) @[d]|≃ -CAT^op[dl]^(0.4)-CAT(,)CAT^The top square commutes to within isomorphism by our construction of the equivalence Mnd_f(Set) ≃Law; whilst the lower triangle commutes to within pseudonatural equivalence because Γ∫ is a bireflector of -categories into absolute-tensored -categories, andis by definition absolute-tensored. Finally, by Propositions <ref> and <ref>, the composites down the left and the right are pseudonaturally equivalent to ()-Alg and ()-Mod respectively. The only thing that remains to check is: The equivalence constructed in Theorem <ref> agrees with the equivalence constructed by Nishizawa–Power in <cit.>. We prove this by tracing through the steps by which we constructed the equivalence of Theorem <ref>, starting from a finitary monad S → in LFP.* We first transport S across the biequivalence LFP≃ to get a monad of the form T in . From the description of this biequivalence in Section <ref>, the underlying lex profunctor T _f ×→Set is given by T(i,j) = (j,Si), while the unit and multiplication of T are induced by postcomposition with those of S.* We next form the one-object -categorywhich corresponds to T.* We next construct the Lawvere -category J →∫ corresponding toby applying ∫ to the unique -functor _→. From the explicit description of ∫ in Proposition <ref>, we see that ∫ has the same objects as , hom-sets ∫(i,j) = (j,Si), and composition as in the Kleisli category of S. Moreover, the functor J is the identity on objects, and given on hom-sets by postcomposition with the unit of S.* Finally, the Lawvere -theory associated to this Lawvere -category is obtained by applying the forgetful functor →CAT; which, comparing the preceding description with the proof of Theorem <ref>, is exactly the Lawvere -theory associated to the finitary monad S →.Gar14b[AR94]Adamek1994Locally Jiří Adámek and Jiří Rosický. Locally presentable and accessible categories, volume 189 of London Mathematical Society Lecture Note Series. Cambridge University Press, 1994.[BC82]Betti1982Cauchy-completion R. Betti and A. Carboni. Cauchy-completion and the associated sheaf. Cahiers de Topologie et Géométrie Différentielle, 23(3):243–256, 1982.[FPT99]Fiore1999Abstract Marcelo Fiore, Gordon Plotkin, and Daniele Turi. 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Republished as: Reprints in Theory and Applications of Categories 10 (2005).[Law73]Lawvere1973Metric F. William Lawvere. Metric spaces, generalized logic, and closed categories. Rendiconti del Seminario Matematico e Fisico di Milano, (43):135–166, 1973. Republished as: Reprints in Theory and Applications of Categories 1 (2002).[LP09]Lack2009Gabriel-Ulmer Stephen Lack and John Power. Gabriel-Ulmer duality and Lawvere theories enriched over a general base. Journal of Functional Programming, 19(3-4):265–286, 2009.[NP09]Nishizawa2009Lawvere Koki Nishizawa and John Power. Lawvere theories enriched over a general base. Journal of Pure and Applied Algebra, 213(3):377–386, 2009.[Str83a]Street1983Absolute Ross Street. Absolute colimits in enriched categories. Cahiers de Topologie et Geométrie Différentielle Catégoriques, 24(4):377–379, 1983.[Str83b]Street1983Enriched Ross Street. Enriched categories and cohomology. Quaestiones Mathematicae, 6:265–283, 1983. Republished as: Reprints in Theory and Applications of Categories 14 (2005).[Wal81]Walters1981Sheaves R. F. C. Walters. Sheaves and Cauchy-complete categories. Cahiers de Topologie et Geométrie Différentielle Catégoriques, 22(3):283–286, 1981. | http://arxiv.org/abs/1707.08694v3 | {
"authors": [
"Richard Garner",
"John Power"
],
"categories": [
"math.CT",
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],
"primary_category": "math.CT",
"published": "20170727032231",
"title": "An enriched view on the extended finitary monad--Lawvere theory correspondence"
} |
http://arxiv.org/abs/1707.08865v1 | {
"authors": [
"Parveen Kumar",
"Apoorva Patel"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170727135441",
"title": "Quantum error correction using weak measurements"
} |
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Ignatios Athanasiadis, Simon J. Wheeler and Peter Grassl^*School of Engineering, University of Glasgow, Glasgow, UK^*Corresponding author: Email: [email protected], Phone: +44 141 330 5208Keywords: Microcracking, particulate composites, aggregate restrained shrinkage, mass transport, lattice, hydro-mechanical, network model, periodic boundary conditions§ ABSTRACTDifferential shrinkage in particulate quasi-brittle materials causes microcracking which reduces durability in these materials by increasing their mass transport properties. A hydro-mechanical three-dimensional periodic network approach was used to investigate the influence of particle and specimen size on the specimen permeability. The particulate quasi-brittle materials studied here consist of stiff elastic particles, and a softer matrixand interfacial transition zones between matrix and particles exhibiting nonlinear material responses. An incrementally applied uniform eigenstrain, along with a damage-plasticity constitutive model, are used to describe the shrinkage and cracking processes of the matrix and interfacial transition zones. The results showed that increasing particle diameter at constant volume fraction increases the crack widths and, therefore, permeability, which confirms previously obtained 2D modelling results. Furthermore, it was demonstrated that specimen thickness has, in comparison to the influence of particle size, a small influence on permeability increase due to microcracking.§ INTRODUCTIONMicrocracking due to particle restrained shrinkage significantly increases the permeability of porous quasi-brittle materials, which often reduces the durability of these materials.For instance for cementitious composites, microcracking due to particle (aggregate) restrained shrinkage has been experimentally observed in <cit.> and has shown to increase mass transport properties such as permeability and sorptivity <cit.>. Numerically, the initiation of microcracking due to particle restrained shrinkage was studied in <cit.>. In some of these studies, it was shown that the width of cracks produced by particle restrained shrinkage depends strongly on the size of particles <cit.>. In the two-dimensional numerical modelling, particles were often idealised as cylindrical particles and all cracks were assumed to penetrate completely the specimen thickness. Therefore, in these two-dimensional analyses, increase of crack width resulted in a significant increase in permeability.In experiments of irregular particulate composites such as concrete, permeability was measured by applying a unidirectional pressure gradient of either water or gas across the specimen thickness. In these thicker specimens, it can be assumed that complicated 3D fracture networks are generated due to differential shrinkage. It appears to be reasonable to expect that part of these networks are not connecting the opposite sides of the specimen and, therefore, do not equally contribute to the increase of permeability. Even for crack paths connecting opposite sides of the specimen, the crack widths along the path might vary. Therefore, it can be expected that the thicker the specimen is, the smaller the increase of permeability due to cracking will be. For instance, recent experimental studies in <cit.> for nonuniform drying shrinkage showed that permeability depends on specimen thickness. This dependence on specimen thickness could be due to the nonuniformity of the shrinkage strain, resulting in thickness dependent patterns of microcracking in the specimen or due to the variation of crack openings along random crack planes.The aim of the present study was to investigate numerically the separate influences of specimen thickness and particle size on the increase of permeability due to particle restrained shrinkage induced microcracking. For this purpose, a new coupled hydro-mechanical periodic network approach consisting of coupled structural and transport networks was developed.Periodic cells are known in the area of homogenisation <cit.>, where it has been shown that periodic boundaries result in faster convergence of properties with increasing cell size than boundaries subjected to displacement or traction conditions. One of the new features of the present periodic approach is that not only the periodic displacement/pressure-gradient conditions were applied, but also the three-dimensional network structure has been chosen to be periodic for both the structural and the transport networks. Peviously, combinations of periodic network structure and periodic displacement conditions were developed only for two-dimensional structural networks <cit.>. The new three-dimensional coupled periodic approach allows for describing fracture patterns and resulting increases in conductivity independent of boundaries. For the transport network, the constitutive models were based on Darcy's law combined with a cubic law to model the influence of cracking on permeability <cit.>.§ METHODThe present numerical method for investigating the influence of particle restrained shrinkage induced microcracking on transport properties was based on three-dimensional periodic hydro-mechanical networks of structural and transport elements. For the constitutive model of structural elements, a damage-plasticity model was used <cit.>. The transport constitutive model used was Darcy's law combined with a cubic law <cit.> to model the increase of permeability due to fracture. The new feature of the present method is the extension of the coupled network approach proposed in <cit.> to a periodic cell of the shape of a rectangular cuboid, which uses periodic network structures and periodicity requirements for the displacements, rotations and pressures. In addition to nodal degrees of freedom in the form of displacements, rotations and fluid pressures of nodes inside the cell, average strain and pressure gradient components are used to solve for unknown degrees of freedom of nodes outside the cell. The use of a periodic cell with periodic network structure and periodicity requirements for the degrees of freedom has the advantage that for the mechanical network, crack patterns are independent of the cell boundaries, which is not the case for boundary value problems with either displacements or tractions applied to the boundaries. The formulation of the hydro-mechanical periodic cell approach is conceptually based on work reported in <cit.>. However, the method in <cit.> was limited to a two-dimensional structural network. Here, a three-dimensional coupled hydro-mechanical periodic network approach was proposed.§.§ DiscretisationThe periodic dual network approach was based on Delaunay and Voronoi tessellations of a set of points placed randomly within a rectangular cuboid shown in Figure <ref> with thick lines.The points were placed sequentially while enforcing a minimum distance d_min between all placed points. Trial points that fail the minimum distance criterion were rejected. The placement was terminated once the number of trials for placing one point exceeds the limit N_iter.Once the placement of points was completed, 26 periodic image points were generated for each successfully placed point within the cell using the translation rule𝐱' = 𝐌𝐱where 𝐱 and 𝐱' are the coordinate vector of the original point and one of the image points, respectively. Furthermore, 𝐌 is the translation matrix defined as 𝐌 = diag [1 + k_ x a, 1 + k_ y b, 1 + k_ z c] where k_ x, k_ y, k_ z∈{-1,0,1}. The k_ x, k_ y and k_ z coefficients define the direction of the shift from the original point to the image point. The coordinates of the 26 image points are the result of the coordinate translations in (<ref>) for all k_ x, k_ y, k_ z combinations except for the case where k_ x = k_ y = k_ z = 0. In Figure <ref>, the cell with two of its 26 neighbours is shown. The points I and J are examples of two randomly placed points satisfying the minimum distance requirements. Points I' and J' are one of 26 sets of periodic image points of I and J, respectively, whereby I' was generated by a translation with k_ x = -1 and k_ y = k_ z = 0, and J' with k_ x = 1 and k_ y = k_ z = 0.All points within the cell and all periodic image points are used for the Delaunay and Voronoi tessellations.The Delaunay tessellation decomposes the domain into tetrahedra whose vertices coincide with the randomly placed points. The Voronoi tessellation divides the domain into polyhedra associated with the random points <cit.>. Each polyhedron is the subset of the domain in which points are closer to the placed point that is associated with the polyhedron than all the other placed points. Facets of Voronoi polyhedra form subsets of the 3D space, in which every location is equidistant from a pair of placed points and nearer to these two points than to any other point. The edges of Delaunay tetrahedra connect pairs of placed points of Voronoi polyhedra with common facets.Delaunay and Voronoi tessellations were used to define the structural and transport elements <cit.>. In Figure <ref>a, a Delaunay tetrahedron and the Voronoi facet associated with Delaunay edge i-j are shown. The structural elements were placed on the Delaunay edges with their mid-cross-sections defined by the facets of the Voronoi polyhedra (Figure <ref>b). Analogous to the structural network, the transport elements were placed on the edges of the Voronoi polyhedra, with their cross-sections formed by the facets of the Delaunay tetrahedra (Figure <ref>(c)).Because of the periodic image points, the tessellated space is larger than the main cell. Therefore, edges of Delaunay tetrahedra and Voronoi polyhedra cross the cell boundaries. In Figure <ref>, an example of two elements crossing the cell boundaries is shown. Here, the intersections of the elements I-J' and I'-J with the boundaries of the periodic cell are presented with a circle and a cross in the plane of the boundary. The parts of the elements that lie outside the cell boundaries are shown with dashed lines and those inside by solid lines. In the present periodic cell approach, all degrees of freedom of nodes inside the cell boundaries were involved in the system of equations that was used for determining the unknown degrees of freedom. For elements crossing the boundary, the degrees of freedom of the nodes outside the cell were determined from those of the periodic image of the node inside the cell and additional information in the form of average strain and pressure gradients for the structural and transport problem, respectively. Examples of structural and transport networks generated from the same set of random points are presented in Figures <ref> and <ref>, respectively.For the structural network, the structural elements are shown in Figure <ref>a and the edges of the mid-cross-sections of the structural elements in Figure <ref>b.The transport elements of the transport network are presented in Figure <ref>a and the edges of the transport mid-cross-sections are shown in Figure <ref>b. From these examples of structural and transport networks, several interesting features of this coupled periodic cell are visible. Firstly, both structural and transport elements (Figures <ref>a and <ref>a) are either located inside the cell or cross its boundaries. No elements are entirely placed outside the periodic cell.The edges of the cross-sections of structural and transport elements are either inside the cell, cross the boundary or lie entirely outside the periodic cell. Therefore, the network of structural elements in Figure <ref>a is smaller than the network of edges of the cross-sections of transport elements shown in Figure <ref>b. Analogue to the structural network, the network of transport elements in Figure <ref>a is smaller than the network of the edges of cross-sections of the structural network in Figure <ref>b. For the coupling of the two networks, cross-section edges which are entirely outside the cell require special consideration. In the present network approach, a one way coupling approach was used in which crack openings obtained from the structural network were used to compute the conductivities of transport elements. Therefore, for the transport network, crack openings associated with cross-sectional edges outside the cell were assumed to be equal to those of the corresponding cross-sectional edges located inside the cell <cit.>. Details regarding how the conductivity was calculated are presented in section <ref>. The only input parameters required for the discretisation of the periodic cell are the minimum distance d_min and the maximum number of trials to place one point N_ iter. These parameters control the average lengths of structural and transport elements. The greater N_ iter is, the smaller is the standard deviation of the element lengths up to the stage at which the domain is almost saturated with points and and increase of N_ iter will result in small changes of the number of placed points. §.§ Structural networkThe three-dimensional structural network was designed to approximate the quasi-static equilibrium equation without body force <cit.>, which is∇σ^ c = 0 where ∇ is the divergence operator and σ^ c is the continuum stress.§.§.§ Structural element The structural element formulation for elements which lie entirely in the periodic cell is identical to the one presented in <cit.>. However, for elements crossing the boundary of the periodic cell, a new element formulation was introduced. To be able to explain this new feature, the standard formulation is presented first. The discrete version of (<ref>) for the structural element shown in Figure <ref>(b) is 𝐊𝐮_ e = 𝐟_ s where 𝐊 is the stiffness matrix, 𝐮_ e are the vector of degrees of freedom and 𝐟_ s are the acting forces. The formulation of the structural element is presented in the local coordinate system, i.e. the coordinate system (x, y and z) of the nodal degrees of freedom coincides with the coordinate system (n, p and q) of the quantities used for evaluating the constitutive response. Each node has 3 translational (u_ x, u_ y and u_ z) and 3 rotational (ϕ_ x, ϕ_ y and ϕ_ z) degrees of freedom. The degrees of freedom of a structural element with nodes i and j are grouped in translational and rotational parts as 𝐮_ e = {𝐮_ t^T, 𝐮_ r^T }^T, where 𝐮_ t = {𝐮_ ti^T, 𝐮_ tj^T}^T = { u_ xi, u_ yi, u_ zi, u_ xj, u_ yj, u_ zj}^T and 𝐮_ r = {𝐮_ ri^T, 𝐮_ rj^T}^T = {ϕ_ xi, ϕ_ yi, ϕ_ zi, ϕ_ xj, ϕ_ yj, ϕ_ zj}^T.These degrees of freedom 𝐮_ t and 𝐮_ r are used to determine displacement discontinuities 𝐮_ C = {u_ Cn, u_ Cp, u_ Cq}^T at point C by rigid body kinematics <cit.> as𝐮_ C = 𝐁𝐮_ e = 𝐁_ 1𝐮_ t + 𝐁_ 2𝐮_ r where 𝐁 = {𝐁_1, 𝐁_2}^T, 𝐁_1 and 𝐁_2 are two matrices containing the rigid body information for the nodal translations and rotations, respectively, which are 𝐁_ 1 = [ -𝐈𝐈 ] and 𝐁_ 2 = [ 0 -e_ qe_ p 0e_ q -e_ p;e_ q 0-h/2 -e_ q 0-h/2; -e_ p h/2 0e_ p h/2 0 ] Here, 𝐈 is a 3×3 unity matrix.In (<ref>), e_ p and e_ q are the eccentricities between the midpoint of the network element and the centroid C in the directions p and q of the local coordinate system, respectively (Figure <ref>b). The local coordinate system is defined by the direction n, which is parallel to the axis of the element, and p and q, which are chosen as the two principal axes of the mid-cross-section. The displacement jump 𝐮_ C in (<ref>) is transformed into strains ε = {ε_ n, ε_ p, ε_ q}^T = 𝐮_ C/h, where h is the length of the structural element. The strains are related to stresses σ = {σ_ n, σ_ p, σ_ q}^T by means of a material stiffness 𝐃 = (1-ω) 𝐃_ e, where 𝐃_ e = diag{E, E, E }. Here, E is the Young's modulus and ω is the damage variable, which is further discussed in Section <ref>. For the present elastic stiffness matrix D_ e, Poisson's ratio equal to zero is obtained and the structural network is elastically homogeneous under uniform modes of straining.For the case that the global coordinate system coincides with the local one, the element stiffness matrix is 𝐊 = Ah[ 𝐁_ 1^ T𝐃𝐁_ 1 𝐁_ 1^ T𝐃𝐁_ 2; 𝐁_ 2^ T𝐃𝐁_ 1 𝐁_ 2^ T𝐃𝐁_ 2 ] + [ 0 0; 0 𝐁_ 1^ T𝐊_ 𝐫𝐁_ 1 ] Here, 𝐊_ r is a matrix containing the rotational stiffness at point C defined as 𝐊_ r = (1-ω)Eh[ I_ p00;0I_10;00I_2 ] Here, I_ p is the polar moment of area, and I_1 and I_2 are the two principal second moments of area of the cross-section. The factor (1-ω) in (<ref>) ensures that the rotational stiffness reduces to zero for a fully damaged cross-section (ω=1). The stiffness matrix is then expressed in the global coordinate system by means of rotation matrices as described for instance in <cit.>.The above element formulation for structural elements entirely located in the periodic cell is identical to the one described in <cit.>. For elements crossing the cell boundaries, a special formulation is required. For these elements, the degrees of freedom of the nodes outside the cell are determined from the degrees of freedom of the periodic image inside the cell and the average strain 𝐄 = {E_ x, E_ y, E_ z, E_ yz, E_ zx, E_ yx}^T.Here, E_ x, E_ y and E_ z are the average normal strains in the x, y and z direction, respectively and E_yz, E_zx, E_yx are the average engineering shear strain components. For an illustration of the coordinate system x, y and z, see Figure <ref>. The translations of a node outside the cell is u'_ x = u_ x + ak_ xE_ x + ck_ zE_zx + bk_ yE_yxu'_ y = u_ y + bk_ yE_ y + ck_ zE_yzu'_ z = u_ z + ck_ zE_ zwhere the translation presented without and with the prime symbol are those of the nodes located within and outside the cell, respectively. Note that the contributions of the average shear strains E_zx and E_yx have been included only in the displacements in the x direction and the contribution of E_yz has been included only in the displacement in the y-direction. This is justified because rigid body rotations of the entire cell are arbitrary. One node of the network is fully fixed in order to prevent rigid body rotation and translation.Consider the element IJ' in Figure <ref>. Node J' is outside the cell and its periodic image J is inside the cell. Making use of (<ref>), (<ref>) and (<ref>) and assuming that ϕ_xJ = ϕ_xJ', ϕ_yJ = ϕ_yJ' and ϕ_zJ = ϕ_zJ', the transformation rule giving the translations and rotations of the two ends I and J' of a structural element IJ' crossing a cell boundary is [𝐮_I; 𝐮_J';𝐫_I; 𝐫_J' ] = 𝐓_ m[ 𝐮_I; 𝐮_J; 𝐫_I; 𝐫_J; 𝐄 ] where 𝐮_I, 𝐫_I, 𝐮_J, 𝐫_J, 𝐮_J' and 𝐫_J' are the vectors containing translational and rotational degrees of freedom of nodes I, J and J', respectively. The node J is the periodic image of point J' inside the cell. The transformation matrix 𝐓_ m is of size 12 × 18 and has the form𝐓_ m = [ 𝐈 0 0 0 0 0; 0 𝐈 0 0 𝐤_ 21 𝐤_ 22; 0 0 𝐈 0 0 0; 0 0 0 𝐈 0 0; ]The sub-matrices 𝐤_ 21 and 𝐤_ 22 are 3 × 3 matrices which contain information about the transformation of the nodal translations due to the average strains. They are defined as𝐤_21= [ ak_ x 0 0; 0 bk_ y 0; 0 0 ck_ z; ]and𝐤_22= [ 0 ck_ z bk_ y; ck_ z 0 0; 0 0 0; ] If (<ref>) is combined with (<ref>) for calculating the displacement jump, the transformation matrix 𝐓_ m multiplies matrix 𝐁 from the right. It follows from duality that the internal forces must be multiplied by 𝐓_m^ T from the left, before the evaluation of the equilibrium conditions. Hence, the original 12 × 12 stiffness matrix 𝐊 of the non-periodic element is now transformed into the 18 × 18 matrix 𝐓_ m^ T𝐊𝐓_ m.With the present approach, average stresses and strains are prescribed by means of the additional six average strain components introduced in the formulation of the periodic cell. Finally, the total number of degrees of freedom are six times the number of nodes positioned within the cell boundaries plus six additional degrees of freedom, which correspond to the six average strain components.§.§.§ Structural materialThe constitutive model for the structural material is based on a damage-plasticity framework <cit.>, which is capable of reproducing the important features of the response of quasibrittle materials in tension and compression. The strains are related to the nominal stress σ = {σ_ n, σ_ p, σ_ q}^T as σ = (1-ω) 𝐃_ e(ε-ε_ p - ε_ s) = (1-ω) σ̅where ω is the damage variable, 𝐃_ e is the elastic stiffness, ε_ p = {ε^ p_ n, ε^ p_ p, ε^ p_ q}^T is the plastic strain and σ̅ is the effective stress. Furthermore, ε_ s = {ε_ s, 0, 0}^T is the shrinkage strain which was used in this study to initiate microcracking.The plasticity model used to determine the effective stress is independent of damage. The model is described by the yield function (<ref>), flow rule (<ref>), evolution law for the hardening variable (<ref>) and loading unloading conditions (<ref>): f = F(σ̅,κ)ε̇_ p = λ̇∂ g∂σ̅κ̇ = λ̇ h_κf ≤ 0 λ̇≥ 0 λ̇ f = 0 Here, f is the yield function, κ is the hardening variable, g is the plastic potential, h_κ is the evolution law for the hardening parameter and λ̇ is the rate of theplastic multiplier. The yield function of the two stress variables σ̅_n and σ̅_q = √(σ̅_ s^2 + σ̅_ t^2) is f = {[α^2σ̅_ n^2 + 2 α^2(f_ c - αβ f_ t)1+αβ q σ̅_ n + σ̅_ q^2 - 2 α^2 f_ cf_ t + α^2 (1-αβ) f_ t^21+αβ q^2 ; σ̅_ n^2β^2 + 2 f_ c - αβ f_ t1+αβ q σ̅_ n + σ̅_ q^2 + (1-α^2 β^2) f_ c^2 -2 αβ(1+αβ) f_ c f_ tβ^2 (1+αβ) q^2].where f_ t and f_ c are the tensile and compressive strengths, respectively, and α and β are the friction angles shown in Fig. <ref> for f = 0 and q=1, which controls the hardening.It is defined asq = exp(κA_ h)where A_ h is an input parameter. For the onset of plastic flow κ = 0 and q = 1.The stress dependent parts of the plastic potential g in the non-associated flow rule in (<ref>) are the same as those of the yield surface f except that α is replaced by ψ: g = {[ ψ^2 σ̅_ n^2 + 2ψ^2 (f_ c - ψβ f_ t)1+ψβ q σ̅_ n + σ̅_ q^2; σ̅_ n^2β^2 + 2 f_ c - ψβ f_ t1+ψβ q σ̅_ n + σ̅_ q^2 ].The smaller ψ is, the smaller is the ratio of normal and shear components of plastic strains for σ̅_ n≥f_ c - ψβ f_ t1+ψβ q. The function h_κ in the evolution law in (<ref>) is chosen as h_κ =|∂ g∂σ_ n |which is the absolute value of the normal component of the direction of the plastic flow.The damage variable in (<ref>) is determined by means of the damage history variableκ_ d = ⟨ε_ pn⟩where ⟨ . ⟩ denotes the McAuley brackets (positive part of operator). The function of the damage variable is derived from the stress-crack opening curve in pure tension (σ_ n > 0, σ_ q = 0). For the damage-plasticity constitutive model, the vector of crack opening components is defined as𝐰_ c = h ( ε_ p + ω(ε_ n - ε_ pn))For pure tension, the crack opening simplifies tow_ c = h ( ε_ pn + ω(ε_ n - ε_ pn)) where h is the length of the network element (Figure <ref>). The stress-crack opening curve isσ_ n = f_ texp(-w_ cw_ f)where w_ f controls the initial slope of the exponential softening curve. It is related to the area under the stress-crack opening curve G_ F as w_ f = G_ F/f_ t. Setting (<ref>) equal to the first component of (<ref>), a nonlinear equation of the damage ω is obtained, which is solved using the Newton-Raphson method. For modelling the dependence of transport properties on cracking, permeability, which is part of the transport model described in section <ref>, is made dependent on the absolute value of the crack openingw̃_ c = |w_ c| The structural constitutive model requires eight input parameters. The Young's modulus of the lattice material E controls the macroscopic Young's modulus. The parameters of the plasticity part are f_ t, f_ c, α, β, ψ and A_ h. Finally, G_ F controls the amount of energy dissipated during cracking. §.§ Transport networkFor the transport part of the model, a 3D network of 1D transport elements is used to discretise the stationary transport equation <cit.> div(k grad P_ f) = 0 Here, P_ f is the fluid pressure, k is the conductivity. In (<ref>), gravitational effects are neglected.§.§.§ Transport elementsAnalogue to the structural network, different element formulations are used for elements which are located entirely in the cell and those that cross one of the faces of the cell. For those inside the cell, the discrete form of (<ref>) for a 1D transport element shown in Figure <ref>c is 𝐤_ eP_f = 𝐟_ e where 𝐤_ e is the one-dimensional element conductivity and 𝐟_ e are the nodal flow rate vector <cit.>. The degrees of freedom of the transport elements are the fluid pressures P_ f = {P_ f1, P_ f2}^T. Within the context of a one-dimensional finite element formulation <cit.>, Galerkin's method is used to construct the elemental conductivity matrix as 𝐤_ e = k A_ th_ t[1 -1; -11 ]Here, A_ t is themid-cross-sectional area and h_ t the length of the transport element shown in Figure <ref>c.For elements, which cross the faces of the cell, the element formulation is explained based on Figure <ref> used earlier for the structural element. For instance, for the element I'J in Figure <ref>, the node I' is outside the cell and J is inside the cell. The periodic image of I' is I which is located inside the cell, which is used together with average fluid pressure gradients to determine the fluid pressure of the node outside the cell. The nodal fluid pressures of an element crossing the cell boundary are[ P_f I'; P_fJ ] = 𝐓_ t[ P_fI; P_fJ; Δ P_fx/a; Δ P_fy/b; Δ P_fz/c ]where Δ P_fx/a, Δ P_fy/band Δ P_fz/c are the average fluid pressure gradients along the x, y and z directions respectively. Here, a, b and c are the dimensions of the periodic cell shown in Figure <ref>. Furthermore, 𝐓_ t is a transformation matrix of size 2 × 5 and has the form𝐓_ t = [ 1 0 ak_ x bk_ y ck_ z; 0 1 0 0 0; ]For combining (<ref>) with (<ref>), the transformation matrix 𝐓_ t multiplies matrix 𝐤_e from the right. It follows from duality that the internal flux must be multiplied by 𝐓_ t^ T from the left, before the evaluation of the balance condition. The conductivity matrix is evaluated as 𝐓_ t^ T𝐤_ e𝐓_ t, where the original conductivity matrix 𝐤_ e is transformed from a 2 × 2 matrix to a 5 × 5 one. The global conductivity matrix is assembled normally except for six rows and columns that relate the global degrees of freedom to its conjugate reaction flow rates. As a result, the periodic cell can be subjected to arbitrary combinations of average flux or gradients. The total number of unknown degrees of freedom is the total number of the nodes located in the interior of the periodic cell plus three global degrees of freedom controlling the average flux or pressure gradient in the three directions.§.§.§ Transport materialsThe conductivity matrix for the material of the transport elements isk = k_0 + k_ c where k_0 is the initial conductivity of the undamaged material and k_ c is the change of conductivity due to fracture. The conducticity of the undamaged material isk_0 = ρκ_0μwhere ρ is the density and μ is the dynamic viscosity of the fluid, and κ_0 is the intrinsic permeability. In this work, the density and dynamic viscosity was set to ρ=1000 kg/m^3 and μ=0.001 Pa s, respectively, which corresponds to the values commonly used for water.The second term k_ c in (<ref>) models the increase of conductivity due to cracking using a cubic law based on the concept of flow through parallel plates <cit.> with a reduction factor ξ for the presence of roughness of the wall surface <cit.>. A detailed description of the definition of k_ c and its dependence on the crack openings of the structural network, which was used in this study, has already been presented in <cit.>. However, since this is an important part of the model of the present study, it is shown here once more. The term k_ c is k_ c = ξρ12 μ A_ t∑_i=1^3w̃_ci^3 l_ci where w̃_ci and l_ci are the equivalent crack openings and crack lengths (see Figure <ref>) of neighbouring structural elements, which are located on the edges of the cross-section, and ξ is a reduction factor which considers the reduction of flow for cracks with rough surfaces compared to that between smooth parallel plates.Here, w̃_ c is the magnitude of the crack opening 𝐰_ c defined in (<ref>).The relation in (<ref>) expresses the well known cubic law, which has shown to produce good results for transport in fractured geomaterials <cit.>.The way how crack openings in the structural elements influence the conductivity of a transport element is schematically shown in Figure <ref>. For instance, for the transport element o-p, three structural elements (i-k, k-j and i-j) bound the cross-section of the transport element. Thus, the conductivity will be influenced by these three elements according to (<ref>) in proportion to their equivalent crack widths and the crack lengths. This crack length (shown by blue double lines in Figure <ref>) is defined as the length from the midpoint of the structural element to the centroid C_ t of the transport element cross-section.§ ANALYSES §.§ IntroductionThe coupled structural transport network model described in section <ref> was applied to the analysis of particle restrained shrinkage of prisms made of a particulate quasi-brittle material consisting of particles, matrix and interfacial transition zones. The matrix and interfacial transition zones are made of permeable quasi-brittle cohesive-frictional material, where the interfacial transition zone is weaker and more permeable than the matrix. The particles are elastic, and stiffer and less permeable than the matrix. The input parameters for the three material phases are presented in Table <ref>. The properties of the different phases are mapped onto a coupled periodic network. Network elements with both nodes within the same particle are given the property of particles. Those elements with nodes in different particles or one node in a particle and the other in the matrix are assigned the properties of the interfacial transition zone. Here, it is assumed that the thickness of the interfacial transition zone is much smaller than the length of the element, so that the Young's modulus of the elements crossing the interfacial transition zone is determined as the harmonic mean of those of matrix and particle. The strength of the element is determined by the strength of the interfacial transition zone.Finally, for elements with both nodes outside particles, the matrix material properties are used. For all elements, the same structural and transport constitutive models were used with input parameters shown in Table <ref> according to <cit.>.Three groups of analyses were carried out. Firstly, the structural response of the matrix material subjected to mechanical loading was analysed by means of direct tension and a hydrostatic compression test using the structural periodic cell. With these analyses key factors of the performance of the structural constitutive model were demonstrated. This study was required since the model used here differs from the one in the previous two-dimensional study in <cit.> and it is important that the numerical method provides network-independent results in tension and compression. In the second group, periodic cells with a centrally located single particle were analysed. Particle restrained shrinkage was modelled by subjecting matrix and interfacial transition zones to uniform incrementally increasing eigenstrain, while keeping the force resultants of the entire specimen at zero. The particle, which was not subjected to eigenstrain, restrained the matrix and interfacial transition zones. Therefore, this process is called particle restrained shrinkage. After every increment of applied eigenstrain, the permeability of the specimens was evaluated by a stationary transport analysis with a fluid pressure gradient across the specimen applied in one direction. In these single particle analyses, the particle diameter was varied at constant particle volume fraction to investigate the influence of particle diameter on changes of permeability due to microcracking induced by restrained shrinkage. The last group consists of coupled analyses of specimens with multiple randomly placed particles of constant diameter and volume fraction for varying specimen thickness in the direction in which the pressure gradient is applied. With these analyses, the influence of specimen thickness on changes of permeability due to microcracking induced by particle restrained shrinkage were investigated. In the following sections, the three groups of analyses are discussed in detail. §.§ Uniaxial tension and hydrostatic compression Before investigating microcracking induced by particle restrained shrinkage, the performance of the structural network approach was investigated by means of direct tension and hydrostatic compression analyses. For this, a cubic cell with edge length of a=b=c=5 cm was discretised with the periodic structural network approach using three network sizes of d_min = 8, 4 and 2 mm. The number of iterations for the network generation was N_ iter = 10000. The material parameters for all network elements in this cube were the one of the matrix phase shown in Table <ref>.For direct tension, the periodic cell was subjected to monotonically increasing average axial strain E_ x introduced in section <ref>. The average stress components corresponding to the other average strain components were set to be zero. The resulting normalised stress-displacement curve and crack pattern for the fine network at the stage marked in the stress-displacement curve are shown in Figure <ref>a and b, respectively. The structural network approach exhibits the typical response of cohesive-frictional quasibrittle materials subjected to tension in the form of softening, i.e. decreasing stress with increasing displacement (Figure <ref>a), and localised deformation (Figure <ref>b). The peak stress is larger than the tensile strength input f_ t, because for an irregular network arrangement the individual elements are subjected to a combination of normal and shear stresses. For the present input, the shear strength is greater than the tensile strength. For shear at zero normal stress at the onset of hardening (called here f_ q), the elastic limit is f_ q = 2f_ t for the values of the parameters α and β in Table <ref>.For hydrostatic compression, the normalised average stress σ_ v/f_ c versus strain response and the crack patterns at the stage marked in the stress-strain curve are shown in Figure <ref>a and b, respectively. Here, σ_ v = (σ_ x + σ_ y + σ_ z)/3 and ε_ v = (ε_ x + ε_ y + ε_ z)/3.The overall response in hydrostatic compression is initially elastic followed by elasto-plastic hardening. The deviation from the elastic response occurs at σ_ v = -f_ c, which corresponds to the onset of the yielding in the constitutive model for negative normal stress only (σ_ n<0 and σ_ q = 0). The cracks patterns, using the definition of the crack opening in (<ref>), are distributed within the periodic cell without showing any patterns of localised deformations, which is typical for quasi-brittle materials subjected to hydrostatic compression. The crack opening consists only of plastic strains, since no damage occurs in these hydrostatic compression analyses. The two examples of direct tension and hydrostatic compression demonstrate that the structural constitutive model is capable of describing the response in tension and compression realistically. This is important for the particle restrained shrinkage analyses in sections <ref> and <ref>, in which complex tensile and compressive stress states play an important role. The responses for both tension and compression are insensitive to the element size. For direct tension, the irregularity of the network affects the results, because the inelastic displacements are localised in an element size dependent region. However, global result in the form of the stress-displacement curve is insensitive to the element size. For hydrostatic compression, the results converge with network refinement. For large element sizes, the stiffness is overestimated because elements crossing the boundary contribute stronger to the stiffness of the cell. Since in the analyses in sections <ref> the network size is varied, this insensitivity to the network size is important.§.§ Size of particles In the second part of the study, the influence of size of inclusions on increase of permeability due to cracking was analysed. For these analyses, a coupled periodic cubic cell (a=b=c in Figure <ref>) with a single centrally arranged particle was used. The volume fraction was kept constant at ρ_ p = 0.137, while varying the particle size as d = 4, 8, 16 mm. For the volume of a spherical particle V_ p = π d^3/6 and the cell volume V_ cell = a^3, the volume fraction for n_ p particles is ρ_ p = (n_ p V_ p)/V_ cell. Combining these expressions and solving for the specimen length gives a = (n_ pπ6ρ_ p)^1/3 dTherefore, for constant volume fraction ρ_ p, the size of the periodic cell decreases with decreasing particle size. For instance, for n_ p = 1 and d=16 mm, the specimen length results in a = 25 mm. For the discretisation of the network, the ratio of the size of the cell and minimum distance was chosen as a/d_min = 12.5 for all particle sizes investigated. Thus, the average network element length decreases with decreasing particle size. This change of element size should not affect the results strongly, as it was shown in section <ref>. For all analyses N_ iter = 10000 was used.For the coupled analyses, a uniform shrinkage strain of ε_ s=-0.5% was applied in 100 increments to matrix and interfacial transition zone. The particles were not subjected to the shrinkage strain.For each particle size, ten analyses with random network generations were performed.After every increment of shrinkage strain, the permeability was determined by applying a unit fluid pressure gradient in the y-direction(Δ P_ y/L_ y = 1). Here, Δ P_ y and L_ y = a = b= c are the fluid pressure difference and the length in the y-direction, respectively. The total flow Q_ y in the y-direction resulting from this fluid pressure gradient was used to determine the macroscopic permeability component of the cell in the y-direction asκ_ yy = Q_ yμA_ yρΔ P_ y/L_ ywhere A_ y = a× b is the cross-sectional area in y-direction (Figure <ref>). The permeability component in (<ref>), which is one of nine components of the matrix of permeability <cit.>, was used here to assess the influence of particle size on permeability for microcracking induced by particle restrained shrinkage.In Figure <ref>, the mean of the permeability κ_ yy normalised by the intrinsic permeability of the matrix κ_0^m versus shrinkage strain ε_ s of ten random analyses is shown.The areas next to the curves show plus/minus one standard deviation. The scatter originates from the irregularity of the structural and transport networks.At early stages of the analyses (ε_ s>-0.1 %), microcracking due to particle restraint does not occur, so that there is no visible influence of particles size on permeability on the log-scale used in Figure <ref>. Once cracking has been initiated (ε_ s<-0.1 %), permeability is strongly influenced by the particle size at constant particle volume fraction. The greater the particle is, the greater is the increase of permeability.This strong dependence of permeability on particle size at constant volume fraction is explained by the crack patterns which are generated by the shrinkage of matrix and interfacial transition zone and the restraint that the particle provides. Crack patterns for the largest particle size (d=16 mm) are shown in Figure <ref>a at the final increment of shrinkage strain. In this figure, cracks are visualised by yellow polygons representing mid-cross-sections of elements in which the equivalent crack width w̃_ c defined in (<ref>) is greater than 10 μm. Colours refer to the online version. For all analyses of periodic cells with a single particle size, the shrinkage strain applied to matrix and interfacial transition zone results in overall regular localised crack patterns with three distinct crack planes aligned with the directions of the Cartesian coordinate system used for the periodicity of the cell. In an earlier two-dimensional study in <cit.>, these type of regular crack patterns were obtained by placing multiple layers of inclusions in a regular pattern in a bigger specimen. Here, because a cell with periodic boundary conditions is used, these characteristic crack patterns for a regular inclusion arrangement are obtained for a cell with a single inclusion.The crack openings depend strongly on particle size. The greater the particle size is, the greater is the volume of material associated with this particle, which will be subjected to the eigenstrain and which will crack if the eigenstrain is sufficiently large. Therefore, the greater the particle size, the greater is the crack opening and the smaller is the crack length. This reasoning is in agreement with the two-dimensional numerical results reported in <cit.>. This particle size dependent crack opening results in a strong dependence of conductivity on particle size, since the cubic law in (<ref>) was used to related crack opening to permeability due to cracking. Therefore, the increase in crack openings dominates the decrease in crack length.In addition to the crack patterns, the main flow paths through the crack network is visualised in Figure <ref>b by showing transport elements in blue in which the flow is greater than the threshold Q_0 = 2.5× 10^-16 kg/s. This value is equal to the entire flow through a cell of the same cross-section made of undamaged matrix material and subjected to a unit pressure gradient. In Figure <ref>b, it can be seen that the flow paths are orientated preferentially in the vertical direction. Two of the crack planes in Figure <ref>a are orientated so that they result in an increase of the mass transport in the y-direction. Most of the transport elements exceeding the threshold are located on these two crack planes. The strong dependence of permeability on particle size in Figure <ref> is in agreement with the two-dimensional results in <cit.>. §.§ Specimen thicknessIn the third part, the influence of specimen thickness on random particle arrangements in the direction of the pressure gradient was investigated for microcracking induced by particle restrained shrinkage. The aim of this part of the study was to investigate if potentially disconnected crack networks result in reduction of permeability with increasing specimen thickness. The cross-section of the rectangular periodic cell was chosen as a=b=50 cm and the specimen thickness was varied as c=25, 50 and 75 mm (Figure <ref>). Furthermore, particle diameter and density were chosen as d=16 mm and ρ_ p = 0.137, respectively. The minimum distance for the background network was chosen as d_min = 2 mm. For each specimen thickness, ten analyses with random periodic particle and network arrangements were carried out using the coupled network approach presented in section <ref>. As in the single particle analyses in section <ref>, a shrinkage strain of ε_ s = -0.5 % was applied incrementally to matrix and interfacial transition zone elements. After every increment, the permeability component κ_ yy was determined as explained in (<ref>). The mean of the permeability of ten random analyses versus the shrinkage strain is presented in Figure <ref> in the form of lines with symbols for the three specimen thicknesses.Note that the mean results for c=50 and 75 mm are almost indistinguishable. The coloured areas around the mean curves represent plus/minus one standard deviation. The permeability increases strongly with increasing shrinkage strain, as it was already observed for the single particle analyses. The strong increase of permeability is the result of the creation of random crack networks. These are more complex than the regular patterns obtained in section <ref>. However, the permeability at the final stage is similar for the random and regular particle arrangements. In Figure <ref>, the crack patterns of one of the thick specimens (c=75 mm) is shown for three stages of applied shrinkage strain (ε_ s = -0.25, -0.375 and -0.5 %). The blue spheres indicate the position of the particles. The yellow polygons show mid-cross-sections of elements in which the crack opening increases at this stage of analysis and is greater than 10 μm. Colours refer to the online version. The spacing between the crack planes is determined by the size of the particles and their spatial arrangement. For all three stages of applied shrinkage strains, the crack network connects randomly placed particles. Very similar observations were made in the two-dimensional study in <cit.>. The greater the shrinkage strain, the denser is the crack network and the greater are the crack openings. In addition, the flow network is shown in Figure <ref>. Blue lines indicate transport elements in which the flow is greater than Q_0, which is the threshold used earlier for the single particle analyses. At the first stage in Figure <ref>a, none of the transport elements exhibits flow greater than Q_0. The crack openings of the crack networks in Figure <ref>a are, while already localised, not large enough to increase the flow through the transport element sufficiently to exceed the threshold Q_0. For the second stage in Figure <ref>b, a network of transport elements exceeding the threshold is visible. These transport elements are located on the crack planes shown in Figure <ref>b. Not all transport elements on crack planes exhibit high flow, because of the variation of the crack openings of individual elements. In the final stage, the majority of transport elements located on crack planes shown in Figure <ref>b conduct flow greater than Q_0.At this stage, the crack network is fully developed and the cracks have opened up so much that transport elements along the crack planes provide the majority of the flow through the specimen. The specimen thickness has overall only a small influence on the increase of permeability compared to the influence of the particle size and shrinkage strain. The mean permeability for c=50 and 75 mm are almost identical. Only for the specimen with c=25 mm, a greater permeability than for c=50 and 75 mm was obtained. This difference is smaller than the standard deviation of the analyses with c=25 mm. The greater the thickness is, the smaller is the standard deviation of the permeability, because some of the irregularities of the crack planes are averaged out along the specimen thickness. Crack patterns for specimens of three different thicknesses are shown in Figure <ref> for the final stage (ε_ s = -0.5). For all three thicknesses, crack patterns connecting the particles are fully formed.Qualitatively, there is little difference between these crack patterns.In Figure <ref>, the network of transport elements in which the flow is greater than the threshold Q_0 is shown for a shrinkage strain of ε_ s = -0.5.For all three specimens, the network of transport elements exceeding this threshold have a very similar structure. The high flow elements are positioned on the crack planes as shown in Figure <ref>. For the network with the smallest specimen thickness (c=25 mm) in Figure <ref>a, the number of elements with high flow is smaller than for the other two thicknesses in Figure <ref>b and c. This appears to be in contradiction to the results in Figure <ref>, in which the specimen with the smallest thickness exhibits the greatest increase in conductivity due to cracking. Nevertheless, the crack and flow patterns in Figures <ref> and <ref> are only one of ten random particle arrangements. These illustrations are useful for assessing the type of crack and flow patterns, but cannot be used for a quantitative comparison analyses with different thicknesses. For such a comparison, the mean values shown in Figure <ref> should be used. In addition, for the smallest specimen thickness, the highest standard deviation was reported. § CONCLUSIONSA three-dimensional coupled structural transport network model was used for the analysis of microcracking in heterogeneous materials due to particle restrained shrinkage. A new coupled periodic network approach has been proposed, which combines periodic displacement/pressure-gradient conditions with periodic structural and transport networks. This new approach was applied to investigate the influence of particle size and specimen thickness on permeability for particle restrained shrinkage. The results of the three-dimensional analyses show that a change of particle size at constant volume fraction has a very strong influence on the increase of permeability due to microcracking for the case of particle restrained shrinkage. The influence of a change specimen thickness on increase of permeability for a pressure gradient in the direction of the changed specimen thickness was shown to be small in comparison.This study was limited to shrinkage of matrix and ITZ applied uniformly across the periodic cells. In many applications, the shrinkage strain will be nonuniform through the specimen because of for instance a humidty gradient away from the specimen surface <cit.>. This influence of nonuniform shrinkage strain on microcracking and conductivity changes will be investigated in future studies. § ACKNOWLEDGEMENTSThe numerical analyses were performed with the nonlinear analyses program OOFEM <cit.> extended by the present authors. The authors acknowledge funding received from the UK Engineering and Physical Sciences Research Council (EPSRC) under grant EP/I036427/1 and funding from Radioactive Waste Management Limited (RWM) (http://www.nda.gov.uk/rwm), a wholly-owned subsidiary of the Nuclear Decommissioning Authority. RWM is committed to the open publication of such work in peer reviewed literature, and welcomes e-feedback to [email protected] equationsection | http://arxiv.org/abs/1707.09288v2 | {
"authors": [
"Ignatios Athanasiadis",
"Simon J. Wheeler",
"Peter Grassl"
],
"categories": [
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.mtrl-sci",
"published": "20170726131024",
"title": "Hydro-mechanical network modelling of particulate composites"
} |
Oleg SolovievAlias-free basis for sensorless adaptive optics Alias-Free Basis for Modal Sensorless Adaptive Optics Using the Second Moment of IntensityOleg Soloviev[DCSC, 3mE, Delft University of Technology, Mekelweg 2, 2628 CD Delft, the Netherands, [email protected]] December 30, 2023 ======================================================================================================================================== In theory of optical aberrations, an aberrated wavefront is represented by its coefficients in some orthogonal basis, for instance by Zernike polynomials. However, many wavefront measurement techniques implicitly approximate the gradient of the wavefront by the gradients of the basis functions. For a finite number of approximation terms, the transition from a basis to its gradient might introduce an aliasing error. To simplify the measurements, another set offunctions, an “optimal basis” with orthogonal gradients, is often introduced, for instance Lukosz-Braat polynomials. The article first shows that suchbases do not necessarily eliminate the aliasing error and secondly considers the problem of finding an alias-free basis on example of second-moment based indirect wavefront sensing methods. It demonstrates that for these methods any alias-free basis should be formed by functions simultaneously orthogonal in two dot-products and be composed of the eigenfunctions of the Laplace operator.The fitness of such alias-free basis for optical applications is analysed by means of numerical simulations on typical aberrations occurring in microscopy and astronomy.§ INTRODUCTION In optics, the quality of an optical system can be characterised by the rms of thewavefront aberration. Thus, in adaptive optics, the rms of the residual wavefront error is the measure of the correction:ϵ = ∫_A (ϕ() - φ_DM())^2,where ϕ and φ_DM denote the incoming and correction wavefronts, andisthe coordinate vector insideaperture A. On the other hand, wavefront sensors often provide information based on the gradient of the wavefronts, and control algorithmsuse a quantity related (maybe implicitly) to the rms of the gradient of residual aberration:ϵ_∇ = ∫_A ∇ϕ() - ∇φ_DM()^2.While without limitation on φ_DM, minimisation of both errorsgives equivalent up to a constant phase term results, this is not the case when φ_DM is confined to some subspace, for instance, spanned by the response functions of a deformable mirror. This article considers an abstract mathematical problem of finding such a basis that minimisation of both error given by <ref> over any subspace spanned by a finite subset of the basis functions provides equivalent results and discusses the advantages of using such bases in optical applications on example of indirect wavefront sensing method based on the second moment of intensity. The method is briefly recapitulated in the following section.§ INDIRECT WAVEFRONT SENSING In feedback adaptive optics (AO) systems, the goal of wavefront correction is to minimise the rms of the residual aberration <cit.>, which is measured by means of the wavefront sensing. In applications with (quasi-)static aberrations, such as microscopy, indirect (also referred to as wavefront sensor-less) methods provide some advantage in comparison with direct wavefront sensing and at present can be considered as a common approach to aberration sensing and correction in AO microscopy <cit.>. In these methods, the aberration is corrected by optimising some image quality metric as a function of the control signals applied to an adaptive element. The chosen metric shouldof course be related to the rms value of the residual aberration, so that optimisation of the metrics should minimise the wavefront error. In model-based indirect methods,the correction wavefront is modelled as a linearfunction of some parameters c = (c_1,…,c_N)^T, where N is the number of the degrees of freedom in the model; the parametersc are often (but not necessarily) linearly related to the control signals v of the adaptive element: v = Lc. In this case one effectively controls the adaptive element via modes formed from the response functions by columns of matrix L.If the image qualitymetric J attains its optimum value J_o at some optimal point c_o = (c_1,o,…,c_N,o)^T, itcan be considered as quadratic in the vicinity of c_o: J(c_1,…,c_N) ≈ J_o ±∑_i,j =1^N m_ij (c_i-c_i,o)(c_j - c_j,o), providedc-c_ois small,where m_ij are the elements of some positive semidefinite matrix M≽ 0. In this case, the metric can be optimised using one or more iterations consisting of P N +1 independent images each <cit.>, where P is some integer number, provided the coefficients m_ij of the quadratic form are known.The structure of matrix M is influenced by the choice of the metric andthe choice of the modes (that is by defining matrix L).The particular case of thediagonalmatrix is obviously the most advantageous, as it allows to adjust each of the modes independently, which increases the accuracy of the wavefront reconstruction and decreases the number of required iterations; it is achieved by careful selection of the metric function, or by selection of the modes that diagonalises the matrix m_ij:M = VSV^T, where S is some positive semidefinite diagonal matrix and the rows of V define the new modes. The shape and properties of V depend on the orthogonalisation procedure used:for instance, Gram-Schmidt orthogonalisation returns alower-triangular V; and in SVDV is an orthogonal matrix formed by the eigenvectors of M, with S is formed by the eigenvalues of M.The modal functions that diagonalise the metric Jareusually referred to as “optimal” <cit.> and can be obtained analytically, numericallyor empirically <cit.>. A special class of metrics is based on measuring the change of the second moment (SM) of the image. This method is applicable both to an image of a point source <cit.> and of an extended object <cit.>. This metric can be expressed via the squared gradient of the residual aberration; consequently it is quadratic also for larger aberrations and can be optimised in N+1 steps. It has been shown that the Lukosz-Braat (LB) polynomials are the optimal modes for the second-moment methods <cit.>. The LB polynomials are obtained from Zernike polynomials by Gram-Schmidtorthogonalisation of their gradients <cit.> and thus minimise the transverse ray aberration (the second moment of a PSF). This is an example of an analytically obtained optimal basis.For a low-order deformable mirror, the LB polynomials cannot be approximated well enough with the mirror response functions, andthe optimal modes are obtained by SVD orthogonalisation of their gradients <cit.>, or by SVD decomposition of an empirically obtained matrix M using <ref>. Modes obtained in such a way can be considered as numerical and empirical optimal bases respectively; this approach is more often used in practice. It was shown <cit.> that use of SVD modes is more advantageous over approximated LB modes for an adaptive element with a small number of actuators.The subject of this paper is the derivation of ananalyticaloptimal basis for the case when the number of degrees of freedom of an adaptive element is very large, such as for SLM or photo-addressed DM <cit.>.By “very large” or “practically infinite” we assume that the adaptive element is able to represent any analytical mode of interest accurately enough.We can mark the following differences with the case of a small number of degrees of freedom.Firstly, because of the increased dimensionality, it becomes unpractical to findthe optimal modes via SVDand they should be determined analytically — one needs to perform N^2 measurements to empirically obtain matrix M, and the results of this calibration procedure would be prone to errors, as a higher density of actuators usually results in decreased amplitude of individual response functions. Secondly, unlike the small-dimensional case, where all optimal bases are related to each other via a (small-dimensional) isometric transformation <cit.>, in a large-dimensional case, the corresponding transformation would result in a computation of large series, prone to the numerical errors; and anyway this procedure is effectively equivalent to just definingnew modes point-wise.At last, while in the case of a small number of actuators one can use all of them to obtain the best correction that can be achieved, in big-dimensional situations one needs to limit effective number of degrees of freedom to some smaller valueN'in order to avoid overexposure of the specimen; in general, one wants to keep N' as small as possible.It is natural to think thatany incoming aberration ϕcan be decomposed as a sum of twofunctions, one from the space that an adaptive element (or only those its modes that are used) can correct and the residual which cannot be corrected, so one cansimply ignore it:ϕ = ∑_n=1^N' f_n +∑_n>N' f_n = φ_DM + φ_r,where, to make the decomposition unique, the function φ_r is chosen in the least square-sense approximation, which is equivalent to the condition that φ_DM and φ_r are mutually orthogonal. Then, following this reasoning, matrix M is obtained and diagonalised only for the first N' basis terms. However, it is often forgotten that there are functions that cannot be corrected but which can stillbe “seen” by the metric and, consequently, there are functions with a non-zero correctable part φ_DM that will not be seen by the method and remain under-corrected. In this situation, the accuracy of the correction using only the first N' modes would not only depend on the statistical properties of the aberration, but also on the way the optimal basis was obtained.For instance, LB modes, obtained by sequential orthogonalisation of the gradients of the Zernike modes by Gram-Schmidt algorithm, introduce a systematic erroras is shown in Section <ref>.The rest of the article is organised as follows.First, we show on an example the problem of aliasing if suboptimal modes are used.Second, we show that second-moment metric infers a new dot product in the space of modes and show that diagonalisation of M is equivalent to orthogonalisation of the modal basis.Thenwe derive the condition for the aliasing-free basis.Finally, we present and discuss the results of numerical simulations using optimal modes on examples of typical aberration in microscopy and in astronomy. § ALIASING ERROR IN THE SECOND-MOMENT METHODS In this section we demonstrate how use of the first terms of a basis, although optimal in the sense of diagonalisation of matrix M, can result in asystematic aliasing error.Although the aliasing or coupling error is familiar to all using adaptive optics with direct or indirect wavefront sensing, we dwell a little bit on it to demonstrate that the error is caused completely by the choice of basis and can be avoided by using an “even more optimal” basis. We name the bases correspondingly as suboptimal and optimal (or aliasing-free) bases.In the examples, we use indirect wavefront sensing based onany image metric related to the computation of the second moment of the intensity in the focal plane.The concise description of such a metric for images of a point source is given in the next section. The same reasoning, however, can be applied and similar examples can be constructed for other metrics.§.§ Second-moment based wavefront sensor-less methodFor an image of a point source obtained with control signals v,let metric J(v) be defined as a second moment of its normalised intensityI(x,y) = I(x,y|v):J(v) = ∑_x,y I(x,y) (x^2 + y^2)/∑_x,y I(x,y) ,where the origin (x,y) = 0 is given by the intersection of the optical axis with the imaging plane.Under some natural limitations, the sums in the equation above can be considered as discrete approximations of two-dimensional integrals over ℝ^2, and by denoting as φ = φ(v) and ϕ the phasegenerated by the adaptive element andthat of the initial aberration, in the case of uniform illumination, metric J can be shown <cit.> to be linear dependent on the averaged square gradient of the phase of light field in the aperture:J(v) ≈ J_d + 1/4 π^ 2 ∫_^2P^2()∇ (ϕ- φ(v))^2 /∫_^2 P^2(),where = (ξ, η) is the coordinate vector in the pupil plane, P() is the pupil function, and J_d is the second moment of the diffraction-limited PSF.Assuming linear dependence of the correction phase on the control signalsφ (v) = ∑_n=1^N f_n() v_n,where f_n(),n =1, …, N are the response functions of the adaptive element,expanding the modulus square in the right hand side of <ref> and completing the full square, the metric J can now be written in the form of <ref> as J(v) ≈ J_d + m_o + (v -v_o)^T M (v -v_o),where m_o ≥ 0 is some constant dependent on φ, such that the minimum value J_o of <ref> is given byJ_o = J_d + m_o, and the elements m_i,j of the matrix M are given by the averaged inner product of the gradients of the response functions f_i and f_j:m_i,j = 1/4 π^ 2 ∫_^2P^2()(∇ f_i, ∇ f_j)/∫_^2 P^2() .If allelements m_i,j are known (either from the calibration procedure or by diagonalisation of M in a theoretical, numerical, or empirical way), there remain N+1 unknowns — m_o, v_o — in <ref>; they can be extracted using K ≥ N+1 measurementsJ_k = J(v_k),k=1,…, K of the metric in K test points v_k. Thus, in the case of diagonal M, the test points c_k,k=0,…, N can be chosen as c_k = (0,…, 1,…, 0),where 1 is present only on the k-th position. The calculation of the optimal point c_o from measurements J_k is then straightforward.§.§ Example of aliasing in suboptimal basisConsider the followinghypothetical example of wavefront correction with a deformable mirror able to reproduce any Zernike polynomial, and suppose we want to limit ourselves only to the correction of the first 6 polynomials[usually in microscopy, tip, tilt, and defocus (and of course piston) are discarded from the modal space as they correspond to the translation of the image in 3D space; here we kept them for the sake of simplicity—the example can be easily modified to include any other first N' Zernike modes as the modal space and a higher order mode as the aberration] using the SM-based method, that is, using metric J given by <ref>. Suppose also that we can calculate without error the second moment of PSF, i.e. there is no noise, and the domain of (x,y) (the camera size) is infinite.Lukosz-Braat (LB) polynomials diagonalise the matrix given by <ref> for a circular aperture for any N, so m_i,j = δ_ij,i,j = 1,…,∞, where δ _ij is the Kronecker symbol; consequently they are often used as a theoretical basis for SM-based methods. The first 6 LB polynomials span the modal space F given byF= ⟨1, x, y, x^2, xy, y^2⟩,where ⟨·⟩ stands for the linear span. This isthe same space as spanned by the first 6 Zernike polynomials,so any aberration that belongs to this space would be corrected by the mirror and SM method with zero error.In our example, let's take the incoming wavefront ϕ outside of space F, for instance a LB polynomial of a higher degreeB_4^0 ϕ = B_4^0 =6 r^4 - 8r^2+2. As LB basis also diagonalises the metric J for any number of the firstterms, the value of metric J will not change when applying any test aberration from the first 6 modes. Thus, the second moment methods will not see the aberration and leave it without change.On the other hand, in the sense of <ref>, ϕ contains a non-zero part φ_DM = -2 r^2 +1 that should be corrected by the mirror.Thus, B_4^0 is not seen by the SM method and will be under-corrected. Symmetrically, the spherical aberration Z_4^0 from the space {φ_r} ofuncorrectable terms of <ref> can be expressed asϕ' = 6 r^4 - 6 r^2 +1 = B_4^0 +(2r^2 -1),will be seen by the SM method as the defocus term Z_2^0 (see <ref>):ϕ≈φ' = 2 r^2-1.and will be over-corrected by applyingthe corresponding correction. This example can be considered as an illustration of aliasing, a manifestation of a “high-order” term Z_4^0 not belonging to the modal space F, as a non-zero “low-order”term Z_2^0 ∈ F, or equivalently, the presence of a non-zero correctable term in a aberration “not-seen” by the metric. In other words, the aliasing here was caused by the choice of space F of the first N' mirror responses, that was not orthogonalto a high-order function from the basis that diagonalises the metric (B_4^0).From the dot-product matrixof the LB polynomials (see second line of Fig. <ref>), it is clear that any subspace F formed by a finite number of LB modes (not necessarily the first ones) will have an infinite number of non-orthogonal to it LB modes.Moreover, the aliasing will remain for any other orthogonal basis in F (e.g. a basis obtained by SVD of the first N' modes with tip, tilt, and defocus excluded), as choice of basis does not change the space itself. Thus we can conclude that the nature of Zernike (or equivalently LB) polynomials does not allow the use of the decomposition of <ref> without introducing an aliasing error. From our example, with the space Fformed by the first N' modes of the LB basis, it is clear that the necessary condition for a basis diagonalising SM metric J to bealiasing-free is that the functions fromits tail should be orthogonal to all the functions from its start.This can be generalised further to conclude that bases that diagonalise the metric but of which the dot-product matrix is not block-diagonal are not aliasing-free and are thus sub-optimal. Therefore for use in indirect wavefront sensing methods, it would be more appropriate to call a basis “optimal” if it a) isformed by a set of orthonormal functions and b) diagonalises the metric. In this redefined optimal basis, decreasing the number of decomposition terms will only introduce a corresponding truncation error caused by the non-zero norm of φ_r; in a sub-optimal basis, anadditional aliasing error will also be present. The natural question is whether a defined in such a way optimal, aliasing-free, basis does exist for a given metric.The next section provides an example of a positive answer. §.§ Example of an aliasing-free basis It's relatively easy to find an example of an optimal basis for the second-moment metric and a square aperture. This is provided by the trigonometrical basis, defined on a unit square (x,y) ∈ [0,1] × [0,1] asf_k l () = ^2 π (k x + l y),k, l ∈ℤ,or its real-valued equivalent {sin (k x + l y), cos( k x + l y)}. The basis is orthogonalon the unit square, and it is easy to check that it diagonalises the matrix given by <ref> as its gradients are also orthogonal on the unit square, so the truncated Fourier series of the decomposition of ϕ in this basis can be used for alias-free indirect wavefront sensing.§ DERIVATION OF AN ALIASING-FREE BASIS FOR ANY APERTURE Now we're going to find an optimal basis for the SM metric fora circular aperture, for which boththe modesand their gradients should form two orthogonal sets on a unit disk. Moreover, we find a sufficient and necessary condition for a basis to be optimalfor second-moment based methodson any aperture. For this, consider the following mathematical formalisation of the problem.Consider the Hilbert space L = L_2[P], where P is the phase domain, that is, the pupil of the system, with the “standard” L_2 inner product defined as(f,g) = ∫_P f() g() ,and letF ⊂ L be a subspace, spanned by some N' functions or modes f_n(),n=1,…, N'F = ⟨ f_1, …, f_N'⟩.Let ϕ∈ L be some function[for the sake of brevity, from now on the dependence onwill be omitted if it is not causing any problems, and tensor notation will be used, so a variable with an upper letter index c^n denotes a column vector c^n = [c^1, …, c^N']^T, and a variable with a lower index denotes a row vector; a repeating upper and low index denotes summation] that is approximated by anelement φ of F:ϕ() ≈c^0 + φ() = c^0 + ∑_n=1^N' c^n f_n () ≡ c^0 + c^n f_n,where the piston term c^o is usually separated as it has no influence on the PSF.With ϕ representing the incoming wavefront, and f_1,…,f_N' representing the decomposition modes, themodal wavefront reconstructionmay now be represented as an optimisation problem in the space L_2[P]min_c^0, c^nϕ - c^0 - c^n f_n _2^2.Without loss of generality, we can consider the setf_n to be the first N' functions of a full and orthonormal basis f_n, n=1,…, ∞,(f_n, f_n') = δ_n,n', whereδ is Kronecker's symbol. Then the best approximation φof ϕ is given by truncation of itsdecomposition by the basis functionsϕ = ∑_n=1^∞ c^n f_n⇒φ = ∑_n=1^N' c^n f_n,where the decomposition coefficients c^n are obtained by the inner product of φ with the basis functions:c^n = (f^n,ϕ),and are known as the (generalised) Fourier coefficients.From the orthonormality of the basis, we get the Euclidean norm of the approximation error ε = ϕ - φ asε^2= ϕ^2 -φ^2 = ∑_n=N'+1^∞ (c_n)^2 = ϕ^2 -∑_n=1^N' (c_n)^2. From <ref> it follows that the second-moment-based methods approximate the gradient of the incoming wavefront with the gradients of the basis function, solving thus another optimisation problemmin_ c^n∇ϕ- c^n ∇ f_n _2^2.Here, the piston term c_0 is cancelled by the gradient operator ∇.Disregarding the piston term, the optimal (alias-free) basis should provide the same solutions for the problems <ref> and <ref>. Let us remove the piston term from both problems by redefining L as a factor space by the gradient kernel: L=L_2[P]/{ϕ:ϕ=const}. In the factor space L, the optimisation problem <ref> is equivalent to the decomposition ofLinto the direct sum of F and its orthogonal complementation O= F^⊥:L = F ⊕ O, whereφ∈ F and ε∈ O are the orthogonal projections of ϕ on the modal space and its orthogonal complementation, andϕ^2 = φ^2 + ε^2.Let us also introduce another inner product in L, defined as the “standard” inner product[the inner product (∇ f, ∇ g) is defined here for the vector-functions from L× L in a natural way: (∇ f, ∇ g) =(∂ f/∂ x, ∂ g/∂ x) + (∂ f/∂ y, ∂ g/∂ y)] of the gradients:(f,g)_∇def=(∇ f, ∇ g) = ∫_P ∇ f()·∇ g() ,where a · b denotes the “standard” inner product in ℝ^2. One can easily verify that in the factor space L, relation (<ref>) satisfies all properties of an inner product,which we will call the “gradient-dot product”, and therefore we can define a corresponding “gradient-norm” asf_∇ = √((f,f)_∇)≡√(∫_P ∇ f^2 ). With the gradient-norm, the minimisation problem of <ref> can be written as min_c^nϕ - c^n f_n_∇,and solving this problem is equivalent to finding the orthogonal complementation of Fwith respect to the gradient-dot product: L = F ⊕_∇ O_∇.Obviously, as <ref> and <ref> use different norms to minimise the residue, their solutions do not necessarilymatch.It is easy to prove (see Fig. <ref>) that they would be the same if and only if the two orthogonal complementations of F are the same, that isO=O_∇ .Indeed, if for any ϕ its projections on F in both inner products match, i.e. φ= φ ', then consequently its approximation errors match too, i.e. ε = ε'. Then for any vector x∈ O_∇, considering its approximation errors, one hasx= ε ' = ε∈ O, and thus O_∇⊆ O. In the same way,O ⊆ O_∇ and thus O_∇ = O.Vice versa, suppose that O_∇ = O. Then from Eqs.(<ref>) and (<ref>) and from the definition of the direct sum, we get that φ = φ '.This explains why optimisation of the Zernike basis by changing to LB or SVD modes may introduce an aliasing error. By performing this optimisation, we re-orthogonalise the basis modes in F with respect to the dot-gradient product and thus also implicitly introduce a new inner product, which changes the orthogonal complementation from O to O_∇. For instance, no first N terms of the Zernike modes considered in the example above can form an optimal basis for the second-moment-based moments.Noll <cit.>has calculated the coefficients of decomposition of the x- and y-components of the gradient of Zernike polynomials through themselves, andfrom his calculation it follows that(Z_n^0,Z_n+2^0)_∇≠ 0∀ n,nis even.Hence, if n is the maximal order of spherical aberration from F, then Z_n+2^0 ∉ O_∇. But as Z_n+2^0 ∈ O, this means O ≠ O_∇.Let us call the basisf_n, n=1,…,∞ in L “optimal” if its elements are both orthogonal and gradient-orthogonal:(f_i, f_j) = (f_i, f_j)_∇ = 0if i≠ j.For the modal space F formed by any N' functions of the basis F= ⟨ f_i_1,…,f_i_N'⟩, condition (<ref>) is obviously satisfied, and thus limiting the number of decomposition modes of the optimal basis will not introduce an aliasing error. In this basis, the solution to the optimisation problem (<ref>) is given in a way similar to <ref>:c'_n = (f_n,ϕ)_∇/(f_n,f_n)_∇. Now we are going to find a necessary condition for a basis to be optimalfor an arbitraryaperture. Suppose F has matching orthogonal complements O and O'. Then for any vector ϕ, its orthogonal projection on F should match in both inner products_F (ϕ) = _F,∇ (ϕ).From this it follows that the L_2-scalar product and the gradient-dot product of any test function ϕ and any modal vector f_n should be proportional to each other, or more accurately, if k_n is the gradient-norm of a basis vector f_n, then(ϕ, f_n) = 1/k_n^2 (ϕ, f_n)_∇.Let us take a Dirac δas a test function, ϕ() = δ( -_0) for some _0 ∈ P.By definition of the gradient-dot product <ref> and by Stokes theorem, we getf_n(_0) = ( δ( -_0), f_n) = 1/k_n^2 ( δ( -_0), f_n)_∇ =1/k_n^2 ( ∇δ( -_0), ∇ f_n) = - 1/k_n^2 (δ( -_0), ∇^2 f_n)= -1/k_n^2Δ f_n (_0) ∀_0 ∈ P,that is to say, f_n should be the eigenfunction of the Laplace operator, and is given by the solution of the Helmholtz equation in the aperture P:Δ f_n + k_n^2 f_n = 0for some k_n ∈. This equations comprises the necessary condition for {f_n} to form an optimal basis.As the eigenfunctions of the Laplace operator satisfying to the homogeneous boundary condition form an orthogonal basis, we have just proved that any optimal modal space for the gradient operator is given by the solutions to <ref> with some homogeneous boundary conditions. The trigonometric basis for the square aperture of <ref> is a particular case of such asolution to <ref>, with periodic boundary conditions. For a circular aperture, this is a classical problem of mathematical physics, with the analytical solutionbounded at =0 phase given in the polar coordinates = (r,θ) byf_m,n(r, θ) = J_m(k_m,n r) 0pt1cossin (m θ),where J_m(r) are the Bessel functions of the first kind, and k_m,n some constants depending on the boundary conditions. It's interesting to note that this set of functions, referred as the Bessel circular functions or the membrane vibration modes, has already popped up recently in the literature<cit.> as being more suitable thanZernike polynomials for some applications, but, to our knowledge, was not yet considered as optimal for wavefront sensor-less methods.§ NUMERICAL SIMULATION In this section we compare the accuracy of aberration correction using sub-optimal and optimal baseswith a numerical simulation. It is not evident whether a basis satisfying the optimality criteria would suit a particular application, because optimality as defined in this paper means only that the result of two optimisation problems would be the same and no aliasing will be introduced by using the gradient norm.In other words, the basis might be optimal for a given wavefront sensing method, but not for representing a typical aberration of an application.The magnitude of the aliasing error is also defined by the statistical properties of the aberration andby the speed of convergence of the generalised Fourier series.For instance, the eigenfunctions defined with the Dirichlet boundary conditions seem a priori to be less suitable foraberrations with non-constant values on the boundary. As test cases, we have used a) wavefront aberrations produced by mouse oocyte cell as an example of specimen induced aberration in microscopy <cit.> and b) wavefront aberrations produced by Kolmogorov's turbulence, covering two important applications of AO[see Ref. for the fitness analysis for ophthalmic applications]. Although the iterative nature of the indirect wavefront sensing methods makes them less suitable for thecase of the atmospheric turbulence, we have included this example for comparison purposes and to get an impression of the extent to which the improvement given by the optimal modes depends on the wavefront statistics. The oocyte phase screens were obtained by the procedure described in Ref. with the same parameters as its authors used for the results demonstrated in their Figure 5.The Kolmogorov phase screens were calculated by the method described in Ref. with 20 phase screens for the ratios D/r_0 = 25/ iwith integer i in the range from 1 to 25.The typical phase screens for both test cases are shown in the left columns of <ref>. The screens are shown wrapped for the presentation purposes only; unwrapped phase screens were used for approximations. 500 random phase screens for each of the cases were generated; several aberration-free phase screens were generated by the method of Ref. ; they were discarded, resulting in a total of 459 phase screens for the oocyte-case.For each of the phase screens, we compared the numerical results of its approximationby the gradients of the first 10 orders (= 65 terms) of thea) Zernike polynomials (denoted in the plots as Z); b) LB polynomials(LB); c) SVD modes obtained from the first 4 orders of the Zernike modes with respect to the gradient-dot product (SML); d) SVD modes obtained fromall 10 orders of the Zernike modes with respect to the gradient-dot product (SM), and eigenmodes of the Laplacian satisfying either to d) Neumann boundary conditions (OMN) or e) Dirichlet boundary conditions (OMD)[the results for the OMD case are not present in this paper, but are available in the supplementary material <cit.>]. Each of the basis functions and aberrations was discreetized on a 128 × 128 grid, with a circular aperture of a 125 pixels diameter (see <ref>).The Zernike polynomials were calculated using their analytical formulae and normalised with respect to their ℓ_2 norm on the aperture.Their gradients were approximated by first-order finite differences inside the aperture; the values on the aperture boundary were zeroed.The Lukosz-Braat polynomials were calculated numerically through Cholesky decomposition of the gradient-dot product matrix M'_Z of the Zernike modes (becausethe gradients are not defined on the boundary, this provided more accurate results for the chosen resolution than calculationusing analytical formulae).The SVD modes were obtained by SVD decomposition of M'_Z.The optimal modes were obtained in Mathematica software with thefunction with the corresponding boundary condition. All modes were, if necessary, first renormalised with respect to the aperture and then reordered by increasing their gradient-norm. The dot product and the gradient dot product matrices M and M' were calculated numerically for the discreetized bases for illustration purposes (see right column of <ref>); the matrices allow to see the key difference between seemingly similar basis function shapes.Please note how SVD modes diagonalise both matrices only for the orders from which they were calculated (the top left minor formed by the red lines in the matrix plots). For each of the phase screens and each of the bases, we have calculated [a)] * its rms error by approximating it by the first N' Zernike terms, * the rms error of its approximation by the gradients of the first N' terms of the basis, * the Strehl ratio, and * the second moment of the PSF of the remaining aberration (normalised to that of the diffraction-limited PSF). The Strehl ratio was defined as the maximum intensity of the central pixel, normalised to the diffraction-limited maximum. This definition results in a specific dip of the plots for the oocyte-produced small aberrations, like in the top rows of <ref>: for instance, correction of tilt only would move the brightest pixel further awayfrom the image centre. The results for 5 representative aberrations of various strengths for each set are shown in Figs. <ref> and <ref>. Figure <ref> presents the box-plot for the relative rms error for the ensembles of the phase screens; it is shown only for the first third of the modes for better visibility. From the results shown in Fig. <ref>, it is clear that theoptimal modes obtained from solution of Helmholtz equation with Neumann boundary conditions (denoted as OMN) provide on average better results for smaller dimension of the modal space (first 4 orders of Zernike, denoted as SML), and similar approximation quality to that of the numerically calculated SVD modes obtained from the first 10 orders of Zernike polynomials (SM). This result can be explained by the fact that for large enough N, the part of the aberration not covered by the span of the basis function is small, and correspondingly the aliasing error introduced by it is small as well.§ DISCUSSION AND CONCLUSIONSWhile the results of <ref> describe the best analytical basis for SM-based methods, this basis can be difficult to implement in practice because of alignment errors, necessity of exact calibration of the adaptive element and so on, so SVD modes are used commonly. Numerical results of <ref> show that singular modes obtained from the large number of Zernike polynomials look similar to the optimal modes (<ref>), which can be seen as the limiting case of the SVD-ing with the number of modes going to infinity.This is also confirmed by the similarity of the results (see <ref>, cases SM and OMN).The optimal basis for a given aperture is not unique (for instance, both OMN and OMD are optimal),and thus the question of a proper optimal basis for a given application remains open.Similarity of SVD and optimal bases suggests that an approximation of the optimal basis can be used.For instance, for a piecewise smooth aberration like that of the oocyte-induced aberrations, SVD orthogonalisation of Gaussian radial base functions might give better results. Moreover, as one can see by the first row of <ref>, for a sparse aberration present only at one edge of the aperture and a small number of correction modes, the Lukosz-Braat functions give better results. This can be explained by the specific shape of the aberration, which has 70 per cent of its power uniformly contributed by the first 30 Zernike polynomials.An important conclusion on the practical use of SM-based wavefront sensorless methods can be drawn from <ref>. To minimise the aliasing effect, the response functions ofthe adaptive element should be a good approximation of some optimal basis, i.e. approximately satisfy <ref>. Since the response functions of a membrane deformable mirror, a bi-morph mirror, and a free-edge mirror with push-pull actuatorssatisfy either Poisson or biharmonic equations with a piece-wise constant right-hand part <cit.>, none of them, nor any basis obtained by orthogonalisation of their gradients, can form a true optimal basis, and thus the use of low-order deformable mirrors for correction with a second-moment based method will suffer from aliasing effects.Again, some approximation to the optimal basis can still be obtained with these types of mirrors with a large number of actuators, e.g. a photo-controlled deformable membrane mirror <cit.>. For a low-order adaptive element, the layout of the actuators should be optimised in order to get response functions closer to the optimal modes (and not to Zernike modes, for instance).To conclude, in this paper we have derived the necessary condition for an analytically defined basis to be optimal, that is, aliasing-free, for wavefront sensorless methods based on the second moment of intensity, namely that each function of the basis should satisfy the Helmholtz equation, and we have provided examples of such a basis for square and circular apertures.The analytical optimal basis can be considered as a limiting case of a numerically computed optimal basis via SVD of the gradient-dot product when the number of basis functions is sufficiently large.The proposed analytical modes can be useful when controlling an adaptive element with a large number of actuators, as probe modes (starting point) for building an empirical optimal basis, and when designing low-order adaptive elements for SM-based methods.§ ACKNOWLEDGEMENTS The author is grateful to G. Vdovin, M. Verhaegen, and D. Wilding (TU Delft) for their attention to the work and valuable comments and to D. Soloviev for proofreading the manuscript.plain | http://arxiv.org/abs/1707.08489v2 | {
"authors": [
"Oleg Soloviev"
],
"categories": [
"physics.app-ph"
],
"primary_category": "physics.app-ph",
"published": "20170726152006",
"title": "Alias-Free Basis for Modal Sensorless Adaptive Optics Using the Second Moment of Intensity"
} |
[NO \title GIVEN] [NO \author GIVEN] December 30, 2023 ====================== We address the task of entity-relationship (E-R) retrieval, i.e, given a query characterizing attributes of two or more entities and relationships between them, retrieve the relevant tuples of related entities. Answering E-R queries requires gathering and joining evidence from multiple unstructured documents that mention the given entities and specified relationships. In this work, we consider entity and relationships of any type, i.e, characterized by context terms instead of pre-defined types or relationships. Unlike real documents, these entity tuples do not have direct and explicit term-based representations. Therefore, we propose a novel IR-centric approach for E-R retrieval, that builds on the basic early fusion design pattern for object retrieval, to provide extensible entity-relationship representations, suitable for complex, multi-relationships queries. We performed experiments with Wikipedia articles as entity representations combined with relationships extracted from ClueWeb-09-B with FACC1 entity linking. We obtained promising results using 3 different test sets comprising 469 E-R queries which indicates this strategy as a flexible baseline for further experimentation with entity-relationship retrieval from unstructured documents.Unlike real documents, these entity tuples do not have direct and explicit term-based representations. Therefore,We address the task of entity-relationship (E-R) retrieval, i.e, given a query characterizing types of two or more entities and relationships between them, retrieve the relevant tuples of related entities. Answering E-R queries requires gathering and joining evidence from multiple unstructured documents. In this work, we consider entity and relationships of any type, i.e, characterized by context terms instead of pre-defined types or relationships. We propose a novel IR-centric approach for E-R retrieval, that builds on the basic early fusion design pattern for object retrieval, to provide extensible entity-relationship representations, suitable for complex, multi-relationships queries. We performed experiments with Wikipedia articles as entity representations combined with relationships extracted from ClueWeb-09-B with FACC1 entity linking. We obtained promising results using 3 different query collections comprising 469 E-R queries.§ INTRODUCTIONIn recent years, we have seen increased interest in using online information sources to find concise and precise information about specific issues, events, and entities rather than retrieving and reading entire documents and web pages. Modern search engines are now presenting entity cards, summarizing entity properties and related entities, to answer entity-bearing queries directly in the search engine result page. Examples of such queries are “Who founded Intel?" and “Works by Charles Rennie Mackintosh". Existing strategies for entity search can be divided in IR-centric and Semantic-Web-based approaches. The former usually rely on statistical language models to match and rank co-occurring terms in the proximity of the target entity <cit.>. The latter consists in creating a SPARQL query and using it over a structured knowledge base to retrieve relevant RDF triples <cit.>. Neither of these paradigms provide good support for entity-relationship (E-R) retrieval, i.e., searching for multiple unknown entities and relationships connecting them. Contrary to traditional entity queries, E-R queries expect tuples of connected entities as answers. For instance, “Ethnic groups by country" can be answered by tuples <ethnic group, country>, while “Companies founded by the creator of Star Wars" is expecting tuples of the format <company, George Lucas>. In essence, an E-R query can be decomposed into a set of sub-queries that specify types of entities and types of relationships between entities.Recent work in E-R search followed a Semantic-Web-based approach by extending SPARQL and creating an extended knowledge graph <cit.>. However, it is not always convenient to rely on a structured knowledge graph with pre-defined and constraining entity types. For instance, search over transient information sources, such as social media <cit.> or online news <cit.>, require more flexible approaches. We hypothesize that it should be possible to generalize the term dependence models to represent entity-relationships and achieve effective E-R retrieval without entity type restrictions. We propose a novel IR-centric approach using fusion-based design patterns for E-R retrieval from unstructured texts. We make the first step in that direction by presenting an early fusion strategy that consists in creating meta-documents for entities and entity-pairs (relationships) and then apply standard retrieval models.In order to leverage information about entities and relationships in a corpus, it is necessary to create a representation of entity related information that is amenable to E-R search. In our approach we focus on sentence level information about entities although it can be applied to more complex methods for text segmentation. We use Wikipedia entity articles and entity-pairs occurrences from ClueWeb-09-B data set with FACC1 text annotations that refer to entities found in the text, including the variances of their surface forms. Each entity is designated by its unique ID and for each unique entity instance we created entity documents comprising a collection of sentences that contain the entity. These context documents are indexed, comprising the entity index. The same is done by creating entity-pair documents and the entity-pair index. These two indices enable us to execute E-R queries using an early fusion strategy with two different retrieval models, Language Models and BM25. The approach was tested on a reasonably large-scale scenario, involving 4.1 million unique entities and 71.7 M of entity pairs.As far as we know this is the largest experiment in E-R retrieval, considering the size of the query sets (469 E-R queries) and the data collection, expecting significant contributions to this line of research:* Generalize the problem of entity-relationship search to cover entity types and entity-relationships represented by any attribute and predicate, respectively, rather than a pre-defined set.* Propose an indexing method that supports a retrieval approach to that problem.* Propose a novel fusion-based strategy that builds on the basic early fusion design pattern for object retrieval <cit.> to provide extensible entity-relationship representations, suitable for complex, multi-relations queries. * Provide results of experiments at scale, with a comprehensive set of queries and corpora.§ RELATED WORK Li et al. <cit.> were the first to study relationship queries for structured querying entities over Wikipedia text with multiple predicates. This work used a query language with typed variables, for both entities and entity pairs, that integrates text conditions. First it computes individual predicates and then aggregates multiple predicate scores into a result score. The proposed method to score predicates relies on redundant co-occurrence contexts.Yahya et al. <cit.> defined relationship queries as SPARQL-like subject-predicate-object (SPO) queries joined by one or more relationships. They cast this problem into a structured query language (SPARQL) and extended it to support textual phrases for each of the SPO arguments. Therefore it allows to combine both structured SPARQL-like triples and text simultaneously. In the scope of relational databases, keyword-based graph search has been widely studied, including ranking <cit.>. However, these approaches do not consider full documents as graph nodes and are limited to structured data. While searching over structured data is precise it can be limited in various respects. To increase the recall when no results are returned and enable prioritization of results when there are too many, Elbassuoni et al. <cit.> propose a language-model for ranking results. Similarly, the models like EntityRank by Cheng et al. <cit.> and Shallow Semantic Queries by Li et al. <cit.>, relax the predicate definitions in the structured queries and, instead, implement proximity operators to bind the instances across entity types. Yahya et al. <cit.> propose algorithms for application of a set of relaxation rules that yield higher recall.Web documents contain term information that can be used to apply pattern heuristics and statistical analysis often used to infer entities as investigated by <cit.>, <cit.> and <cit.>. In fact, early work by Conrad and Utt <cit.> proposes a method that retrieves entities located in the proximity of a given keyword. They show that a fixed-size window around proper-names can be effective for supporting search for people and finding relationship among entities. Similar considerations of the co-occurrence statistics have been used to identify salient terminology, i.e. keyword to include in the document index <cit.>.Existing approaches to the problem of entity-relationship (E-R) search are limited by pre-defined sets of both entity and relationship types. In this work, we generalize the problem to allow the search for entities and relationships without any restriction to a given set and we propose an IR-centric approach to address it.§ ENTITY-RELATIONSHIP QUERIES E-R queries aim to obtain a ordered list of entity tuples T_E= <E_i, E_i+1,..., E_n> as a result. Contrary to entity search queries where the expected result is a ranked list of single entities, results of E-R queries should contain two or more entities. For instance, the complex information need “Silicon Valley companies founded by Harvard graduates” expects entity-pairs (2-tuples) <company, founder> as results. In turn, “European football clubs in which a Brazilian player won a trophy" expects triples (3-tuples) <club, player, trophy> as results. Each pair of entities E_i, E_i+1 in an entity tuple is connected with a relationship R(E_i,E_i+1). A complex information need can be expressed in a relational format, which is decomposed into a set of sub-queries that specify types of entities E and types of relationships R(E_i,E_i+1) between entities. For each relationship query there is one query for each entity involved in the relationship. Thus a E-R query Q that expects 2-tuples, is mapped into a triple of queries (Q^E_i, Q^R_i,i+1, Q^E_i+1), where Q^E_i and Q^E_i+1 are the entity types for E_i and E_i+1 respectively, and Q^R_i,i+1 is a relationship type describing R(E_i,E_i+1). For instance, “football players who dated top models” with answers such as <Cristiano Ronaldo, Irina Shayk>) is represented as three queries Q^E_i={football players}, Q^R_i,i+1={dated}, Q^E_i+1={top models}. Consequently, we can formalize that a query Q contains a set of sub-queries Q^E = {Q^E_1, Q^E_2, ..., Q^E_n} and a set of sub-queries Q^R = {Q^R_1,2, Q^R_2,3, ..., Q^R_n-2,n-1}. Automatic mapping of terms from a natural language information need Q to queries Q^E_i or Q^R_i,i+1 is out of the scope of this work and can be seen as a problem of query understanding <cit.>. We assume that the information needs are decomposed into constituent queries either by processing the original query Q or by user input through an interface that enforces this structure Q = {Q^E_i, Q^R_i,i+1, Q^E_i+1}.§ EARLY FUSION E-R retrieval requires collecting evidence for both entities and relationships that can be spread across multiple documents. Therefore, it is not possible to create direct term-based representations. Documents serve as bridges between entities, relationships and queries. We propose an early fusion strategy specific to E-R retrieval that is inspired on the early fusion design pattern for object retrieval <cit.>. Therefore, our design pattern can be thought as creating a meta-document D^R_i,i+1 for each pair of entities (relationship) that co-occur close together in raw documents and a meta-document D^E_i for each entity, similar to Model 1 of <cit.>. In our approach we focus on sentence level information about entities and relationships although the design pattern can be applied to more complex segmentations of text (e.g. dependency parsing). We rely on Entity Linking methods for disambiguating and assigning unique identifiers to entity mentions on raw documents D. We collect entity contexts across the raw document collection and index them in the entity index. The same is done by collecting and indexing entity pair contexts in the relationship index.We define the (pseudo) frequency of a term t for an entity meta-document D^E_i as follows: f(t,D^E_i) =∑_j=1^n f(t,E_i,D_j) w(E_i,D_j) where n is the total number of raw documents in the collection,f(t,E_i,D_j) is the term frequency in the context of the entity E_i in a raw document D_j.w(E_i,D_j) is the entity-document association weight that corresponds to the weight of the document D_j in the mentions of the entity E_i across the raw document collection. Similarly, the term (pseudo) frequency of a term t for a relationship meta-document D^R_i,i+1 is defined as follows: f(t,D^R_i,i+1) =∑_j=1^n f(t,R_i,i+1,D_j) w(R_i,i+1,D_j) where f(t,R_i,i+1,D_j is the term frequency in the context of the pair of entity mentions corresponding to the relationship R_i,i+1 in a raw document D_j and w(R_i,i+1,D_j) is the relationship-document association weight. In this work we use binary associations weights indicating the presence/absence of an entity mention in a raw document, as well as for a relationship. However, other weight methods can be used.The relevance score for an entity tuple T_E can then be calculated by summing the score of individual entity meta-documents and relationship meta-documents using standard retrieval models. Formally, the relevance score of an entity tuple T_E given a query Q is calculated by summing individual relationship and entity relevance scores for each Q^R_i,i+1 and Q^E_i in Q. We use the terminology |Q| to denote the number of sub-queries in a query Q. We define the score for a tuple T_E given a query Q as follows: score(T_E, Q) =∑_i=1^|Q|-1 score(D^R_i,i+1, Q^R_i,i+1)+∑_i=1^|Q| score(D^E_i, Q^E_i) w(E_i, R_i,i+1) where score(D^R_i,i+1, Q^R_i,i+1) represents the retrieval score resulting of the match of the query terms of a relationship (sub-)query Q^R_i,i+1 and a relationship (entity-pair) meta-document D^R_i,i+1. The same applies to the retrieval score score(D^E_i, Q^E_i) which corresponds to the result of the match of an entity (sub-)query Q^E_i with a entity meta-document D^E_i. We use a binary association weight for w(E_i, R_i,i+1) which represents the presence of a relevant entity E_i to a sub-query Q^E_i in a relationship R_i,i+1 relevant to a sub-query Q^R_i,i+1. This entity-relationship association weight is the building block that guarantees that two entities relevant to (sub-)queries Q^E that are also part of a relationship relevant to a (sub-)query Q^R will be ranked higher than tuples where just one or none of the entities are relevant to the entity (sub-)queries Q^E.For computing both score(D^R_i,i+1, Q^R_i,i+1) and score(D^E_i, Q^E_i) any retrieval model can be used. In this work we run experiments using Dirichlet smoothing Language Models (LM) and BM25. Considering Dirichlet smoothing unigram Language Models (LM) the scores can be computed as follows:score_LM(D^R_i,i+1, Q^R_i,i+1) = ∑_t^Q^R_i,i+1log ( f(t,D^R_i,i+1) +f(t,C^R)/|C^R|μ^R/|D^R_i,i+1| + μ^R )score_LM(D^E_i, Q^E_i) = ∑_t^Q^E_ilog (f(t,D^E_i)+f(t,C^E)/|C^E|μ^E /|D^E_i| + μ^E ) where t is a term of a (sub-)query Q^E_i or Q^R_i,i+1, f(t,D^E_i) and f(t,D^R_i,i+1) are the (pseudo) frequencies defined in equations <ref> and <ref>. The collection frequencies f(t,C^E), f(t,C^R) represent the frequency of the term t in either the entity index C^E or in the relationship index C^R. |D^E_i| and|D^R_i,i+1| represent the total number of terms in a meta-document while |C^R| and |C^E| represent the total number of terms in a collection of meta-documents. Finally, μ^E and μ^R are the Dirichlet prior for smoothing which generally corresponds to the average document length in a collection. Using BM25, the score is computed as summation over query terms, as follows: score_BM25(D^R_i,i+1, Q^R_i,i+1) = ∑_t^Q^R_i,i+1 f(t,D^R_i,i+1)(K_1 + 1)/f(t,D^R_i,i+1) + K_1 (1 - b + b |D^R_i,i+1|/avg(D^R)) IDF(t)score_BM25(D^E_i, Q^E_i) = ∑_t^Q^E_i f(t,D^E_i) (K_1 + 1)/f(t,D^E_i) + K_1 (1 - b + b |D^E_i|/avg(|D^E|)) IDF(t) where IDF(t) is computed as logN - n(t) + 0.5/n(t) + 0.5 with N as the number of meta-documents on the respective collection and n(t) the number of meta-documents where the term occurs. |D^E_i| and |D^R_i,i+1| are the total number of terms in a meta-document, while avg(|D^E|) and avg(D^R) are the average meta-document lengths. K_1 and b are free parameters usually chosen as 1.2 and 0.75, in the absence of specific optimization. § EXPERIMENTAL SETUP§.§ Test Collections We ran experiments with a total of 469 E-R queries aiming for 2-tuples of entities as results. We leave experimentation with longer E-R queries (e.g. 3-tuples) for future work. Relevance judgments consist of pairs of entities linked to Wikipedia.Query sets for E-R retrieval are scarse. Generally entity retrieval query sets are not relationship-centric <cit.>. To the best of our knowledge there are only 3 test collections specifically created for E-R retrieval: ERQ <cit.>, COMPLEX <cit.> and RELink <cit.>.Neither ERQ nor COMPLEX provide complete relevance judgments and consequently, we manually evaluated each answer in our experiments.ERQ consists of 28 queries that were adapted from INEX17 and OWN28 initiatives. Twenty two of the queries express relationships, but already have one entity instance named and fixed in the query (e.g. “Find Eagles songs”). Only 6 queries ask for pairs of unknown entities, such as “Find films starring Robert De Niro and please tell directors of these films.”. COMPLEX queries were created with a semi-automatic approach. For a specific domain in a knowledge graph, a pivot entity is selected based on prior domain popularity. A chain of 2-4 entities connected to the entity is created based on a number of facts connecting to the pivot table. A set of different chains from several domains was given to human editors to formulate E-R queries answered by the entities in each chain. The query set contains 70 queries from which we removed 10 that expect 3-tuples of entities. COMPLEX consists of pure relationship-centric queries for unknown pairs of entities, such as “Currency of the country whose president is James Mancham “Kings of the city which led the Peloponnesian League.”and “Who starred in a movie directed by Hal Ashby?”.RELink queries and relevance judgments were also created with a semi-automatic approach. A sample of relational tables from Wikipedia was used as input to human editors for manually creating E-R queries. Columns from selected tables represent entity types and the table structure implies one or more relationships among the entities. Relevance judgments are automatically collected from each table. RELink comprises 600 queries aiming 2-tuples and 3-tuples of entities from which we use the subset of 381 queries aiming for pairs of related entities as results.§.§ Data and Indexing We aim to answer E-R queries without specific or pre-defined entity or relationship types. Therefore we use unstructured texts mentioning entities and relationships between entities to create our indices. We use a dump of English Wikipedia from October 2016 and the ClueWeb-09-B[<http://www.lemurproject.org/clueweb09/>] collection combined with FACC1<cit.> text span annotations with links to Wikipedia entities (via Freebase). The entity linking precision and recall in FACC1 are estimated at 85% and 70-85%, respectively <cit.>. For our experiments we create two main indices: one for entity extractions and one for entity pairs (relationships) extractions. For a given Wikipedia article representing an entity we index each sentence and consider it as an entity occurrence extraction in the entity index. The Wikipedia dump used contains 4.1M entities. We use ClueWeb-09-B corpus with FACC1 annotations to extract relationship occurrences using an Open Information Extraction method like <cit.>. We look for co-occurring entities in the same sentence of ClueWeb-09-B and we extract the separating string, i.e., the context of the relationship connecting them. We obtained 418M entity pairs extractions representing 71M unique entity-relationships. We ran our experiments using Lucene and made use of GroupingSearch for grouping extractions by entity and entity pair on query time. §.§ Retrieval Method We adopted a two stage retrieval approach. First, queries Q^E_i,Q^E_i+1 are submitted against the entity index and Q^R_i,i+1 is submitted against the entity-pair index. Initial sets of top 20K results grouped by entity or entity-pairs, respectively, are retrieved using Lucene's default search settings. Second, the score functions of the specific retrieval model are calculated for each set, using an in-house implementation. This process is easily parallelized. The final ranking score for each entity-pair is then computed using the early fusion strategy equation for score(T_E, Q). We do not optimize the Dirichlet priors μ^E and μ^R in language models and set them equal to the average entity and relationships extractions length, respectively. The same happens with K_1 and b in BM25, set to default values of 1.2 and 0.75, respectively. Evaluation scores are reported on the top 100 entity-pair results.§ RESULTSWe present the results of our experiments in Table <ref>. We report scores of four different retrieval metrics: Mean Average Precision at 100 results (MAP), precision at 10 (P@10),normalized discounted cumulative gain at 10 (NDCG@10) and mean reciprocal rank (MRR).The first observation is concerning the retrieval model (LM vs BM25). On ERQ, LM shows higher MAP and MRR while BM25 has higher scores for metrics at top 10 results (P@10 and NDCG@10). Although results of both retrieval models are similar, LM outperforms BM25 for every metric on COMPLEX query collection. BM25 has higher MAP on RELink but it is lower on the remaining metrics. The second observation is concerned with the RELink results which are far lower for both retrieval models on all metrics. The RELink collection is by far the largest collection from the 3, comprising a total of 381 queries. It contains several queries regarding dates. For instance, the query “Find australian films of 1981 and their directors.” returns several entity-pairs comprising australian films and directors of those films but not from 1981. The most common relationship query Q^R in this collection is “located in” which is a very frequent relationship string in our entity-pair index. We hypothesize that returning 20k entity-pairs on the first passage might result insufficient for RELink as it reduces the search space. In the future, we will further experiment with higher number of results.§ CONCLUDING REMARKS Work reported in this paper is concerned with expanding the scope of entity-relationship search methods to enable search over large corpora with flexible entity types and complex relationships. We have presented an early fusion strategy for fusion-based E-R retrieval. We anticipate that such strategy can be used as flexible baseline for further experimentation. For the sake of simplicity and clarity, we have reported on the basic E-R retrieval comprising a single relationship between two entities. In future work, we will report experiments with multiple relationships, as well as, an alternative late fusion strategy for E-R retrieval.unsrt | http://arxiv.org/abs/1707.09075v2 | {
"authors": [
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"Natasa Milic-Frayling",
"Eduarda Mendes Rodrigues",
"Carlos Soares"
],
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"published": "20170727233306",
"title": "Early Fusion Strategy for Entity-Relationship Retrieval"
} |
assumeAssumption ExampleExample RemarkRemark[section] CorollaryCorollary[section] DefinitionDefinition ProblemProblemTheoremTheorem[section] PropositionProposition[section] LemmaLemma[section] ConjectureConjecture[section] AssumptionAssumption | http://arxiv.org/abs/1707.08421v1 | {
"authors": [
"Wei Liu",
"Jie Huang"
],
"categories": [
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"primary_category": "math.OC",
"published": "20170726130935",
"title": "Cooperative Global Robust Output Regulation for a Class of Nonlinear Multi-Agent Systems by Distributed Event-Triggered Control"
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Yusuke Aso [email protected]]Yusuke Aso Department of Astronomy, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan Subaru Telescope, National Astronomical Observatory of Japan, 650 North A'ohoku Place, Hilo, HI 96720, USA Academia Sinica Institute of Astronomy and Astrophysics, P.O. Box 23-141, Taipei 10617, Taiwan Subaru Telescope, National Astronomical Observatory of Japan, 650 North A'ohoku Place, Hilo, HI 96720, USA Academia Sinica Institute of Astronomy and Astrophysics, P.O. Box 23-141, Taipei 10617, Taiwan Center for Computational Sciences, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8571, Japan Department of Astronomy, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan Department of Earth and Planetary Sciences, Faculty of Sciences Kyushu University, Fukuoka 812-8581, Japan Chile Observatory, National Astronomical Observatory of Japan, Osawa 2-21-1, Mitaka, Tokyo 181-8588, Japan Nobeyama Radio Observatory, Nobeyama, Minamimaki, Minamisaku, Nagano 384-1305, Japan SOKENDAI, Department of Astronomical Science, Graduate University for Advanced Studies Department of Physics and Astronomy, Graduate School of Science and Engineering, Kagoshima University, 1-21-35 Korimoto, Kagoshima, Kagoshima 890-0065, Japan Academia Sinica Institute of Astronomy and Astrophysics, P.O. Box 23-141, Taipei 10617, Taiwan Department of Earth and Space Science, Osaka University, Toyonaka, Osaka 560-0043, Japan National Astronomical Observatory of Japan, Osawa, 2-21-1, Mitaka, Tokyo 181-8588, Japan European Southern Observatory, Karl-Schwarzschild-Str. 2, D-85748 Garching, GermanyWe have newly observed the Class 0/I protostar L1527 IRS using the Atacama Large Millimeter/submillimeter Array (ALMA) during its Cycle 1 in 220 GHz dust continuum and C^18O (J=2-1) line emissions with a ∼ 2 times higher angular resolution (∼ 05) and ∼ 4 times better sensitivity than our ALMA Cycle 0 observations. Continuum emission shows elongation perpendicular to the associated outflow, with a deconvolved size of 0 53× 0 15. C^18O emission shows similar elongation, indicating that both emissions trace the disk and the flattened envelope surrounding the protostar. The velocity gradient of the C^18O emission along the elongation due to rotation of the disk/envelope system is re-analyzed, identifying Keplerian rotation proportional to r^-0.5 more clearly than the Cycle 0 observations. The Keplerian-disk radius and the dynamical stellar mass are kinematically estimated to be ∼ 74 AU and ∼ 0.45, respectively. The continuum visibility is fitted by models without any annulus averaging, revealing that the disk is in hydrostatic equilibrium. The best-fit model also suggests a density jump by a factor of ∼ 5 between the disk and the envelope, suggesting that disks around protostars can be geometrically distinguishable from the envelope from a viewpoint of density contrast. Importantly, the disk radius geometrically identified with the density jump is consistent with the radius kinematically estimated. Possible origin of the density jump due to the mass accretion from the envelope to the disk is discussed. C^18O observations can be reproduced by the same geometrical structures derived from the dust observations, with possible C^18O freeze-out and localized C^18O desorption.§ INTRODUCTION T Tauri stars are widely known to be associated with protoplanetary disks or T Tauri disks, which often show Keplerian rotation, as has been shown by interferometric observations in the last two decades <cit.>. Class 0/I protostars have also been, on the other hand, considered to be associated with protostellar disks, which are supposed to be precursors of T Tauri disks. Details of protostellar disks around protostars, such as their structures and dynamics, however, are poorly characterized because protostellar disks are deeply embedded in protostellar envelopes. The question of whether disks and envelopes in protostellar systems can be distinguished from each other by detailed observations of protostellar disks is therefore critical. One promising way to achieve this goal is to distinguish them kinematically by inspecting their rotational motion, i.e., envelopes that often have infall motion with conserved angular momentum are expected to have rotation proportional to r^-1, while disks are expected to have Keplerian rotation proportional to r^-0.5. Initial attempts with such an approach have been successfully made recently, identifying a couple of disks with Keplerian rotation around protostars <cit.>. Further attempts have been made more recently with higher angular resolutions (∼ 100 AU) and sensitivity, enabling us to identify even smaller, less bright disks around younger protostars <cit.>. Once envelopes and disks are kinematically distinguished, disk radii can also be kinematically estimated. Another important aspect to identify Keplerian rotation in disks around protostars is that it allows us to estimate dynamical masses of central protostars directly with no assumption. Such direct estimation of the dynamical mass enables us to examine whether or not infall motions around protostars are free fall, as has been assumed so far. L1527 IRS <cit.>, L1551 IRS 5 <cit.>, and TMC-1A <cit.> have infall motions that appear significantly slower than free fall velocities yielded by their dynamical masses, while L1489 IRS shows inflow at free fall velocity toward its Keplerian disk <cit.>.These previous studies indicate that protostellar disks around protostars are kinematically similar to T Tauri disks in the sense that both disks show Keplerian rotation. It is not, however, clear whether protostellar disks are structurally similar to T Tauri disks as well. Physical structures of T Tauri disks, such as radial dependences of surface density and temperature have been well investigated, suggesting that their radial power-law profiles of surface density and temperature are well described as Σ∝ r^-p with p∼ 1 regardless of exponential tails and T∝ r^-q with q∼ 0.5, respectively. Their scale height described as H∝ r^h was also investigated, and was found to be in hydrostatic equilibrium (HSEQ) with a power-law index of h=1.25.On the other hand, similar studies to investigate physical structures of protostellar disks have not been performed yet except for very limited caseswith radial and vertical structures for the Butterfly Star <cit.>, L1527 IRS <cit.>, TMC-1A <cit.>, four protostars in the NGC 1333 star forming region, and VLA 1623 <cit.> and without vertical structures for TMC-1A, TMC1, TMR1, L1536 <cit.>, and seven protostars in Perseus molecular cloud <cit.>. Understanding physical structures of protostellar disks would be important to see whether disks and their surrounding envelopes can be geometrically distinguished from each other. If this is the case, we can have two independent ways (kinematically and geometrically) to identify disks around protostars. In addition, investigating physical structures of protostellar disks can help us to understand the disk formation process and also the process of evolution into T Tauri disks. In particular recent observations of T Tauri disks have revealed irregular structures such as spirals, gaps, and central holes, which might be related to planet formation. Tantalizing ring structures are discovered even in the disk around the class I/ young star HL Tau. It would be very important to understand when and how such structures are formed in these disks, and to answer this question, it is required to investigate structures of protostellar disks, which would be precursors of disks with irregular structures.L1527 IRS (IRAS 04365+2557) is one of the youngest protostars, whose disk has been studied. L1527 IRS, located in one of the closest star-forming regions, the Taurus Molecular Cloud (d=140 pc), has bolometric luminosity L_ bol=2.0 L_⊙ and bolometric temperature T_ bol=44 K <cit.>, indicating that L1527 IRS is a relatively young protostar. The systemic velocity of L1527 IRS in the local standard of rest (LSR) frame was estimatedto be V_ LSR∼ 5.7 from C^18O J=1-0 observations with Nobeyama 45 m single-dish telescope <cit.>, while N_2H^+ J=1-0 observations with Five College Radio Astronomy Observatory (FCRAO) 14 m and Institut de Radioastronomie Millimetrique (IRAM) 30 m single-dish telescopes, estimated it to be V_ LSR∼5.9 <cit.>. We adopt V_ LSR=5.8 for the systemic velocity of L1527 IRS in this paper, which is reasonable as will be shown later.On a ∼ 30000 AU scale, a bipolar outflow associated with this source was detected in the east-west direction by FCRAO single-dish observations in ^12CO J=1-0 molecular line emission <cit.> and by James Clerk Maxwell Telescope (JCMT) single-dish observations in ^12CO J=3-2 molecular line emission <cit.>. Their results show that the blue and red robes of the outflows are on the eastern and western sides, respectively. On the other hand, inner parts (∼ 8000 AU scale) of the outflow mapped with Nobeyama Millimeter Array (NMA) in ^12CO J=1-0 show the opposite distribution, i.e., stronger blueshifted emission on the western side and stronger redshifted emission on the eastern side<cit.>. Mid-infrared observations toward L1527 IRS with Spitzer Space Telescope shows bright bipolar scattered light nebulae along the outflow axis in the ∼ 20000 AU scale <cit.>. They fitted a protostellar envelope model to near- and mid-infrared scattered light images and spectral energy distribution (SED). As a result the inclination angle of the envelope around L1527 IRS was estimated to be i=85^∘, where i=90^∘ means the edge-on configuration. In addition <cit.> found that the western side is closer to observers. This inclination angle is consistent with a disk-like structure of dust highly elongated along the north-south direction, which was spatially resolved for the first time in 7-mm continuum emission by the Very Large Array (VLA) <cit.>. By expanding the studies by <cit.>, <cit.> fitted a model composed of an envelope and a disk to (sub)millimeter continuum emissions and visibilities observed with SMA and CARMA as well as infrared images and SED. Their best-fitting model suggests a highly flared disk structure (H∝ R^1.3, H=48 AU at R=100 AU) with a radius of 125 AU. This study has geometrically distinguished the protostellar disk and envelope around L1527 IRS, although they were not kinematically distinguished from one another. The first interferometric observations of the envelope surrounding L1527 IRS in molecular line emission were reported by <cit.>, identifying an edge-on flattened envelope elongated perpendicularly to the associated outflow, in their C^18O J=1-0 map obtained with NMA at an angular resolution of ∼ 6. It was found that kinematics of the envelope can be explained with dynamical infall motion (∼ 0.3 ) and slower rotation (∼ 0.05) at 2000 AU. Its mass infalling rate was also estimated to be Ṁ∼ 1× 10^-6 yr^-1.Higher-resolution (∼ 1) observations using Combined Array for Research in Millimeter Astronomy (CARMA) in ^13CO J=2-1 line were carried out toward this source by <cit.>. They measured emission offsets from the central protostar at each channel and fitted the position-velocity data with a kinematical model using the LIne Modeling Engine (LIME) by assuming Keplerian rotation. According to their best-fit result, the mass of L1527 IRS was estimated to be M_*=0.19± 0.04 and the disk radius was also estimated to be 150 AU. It should be noted, however, that no other kinds of rotation, such as the one conserving its angular momentum, were compared with the observations in the work. They attempted later to compare their rotation curve with rotation laws conserving angular momentum, which did not change their original conclusion <cit.>. In order to investigate the rotational velocity around L1527 IRS without assuming Keplerian rotation, a radial profile of the rotational velocity V_ rot was measured by <cit.> from their SMA observations in C^18O J=2-1 line with the angular resolution of 4 2 × 2 5. In their analysis, the rotation profiles are derived from Position-Velocity (PV) diagrams cutting along a line perpendicularly to the outflow axis. The rotation profile of L1527 IRS was measured at r≳ 140 AU to be V_ rot∝ r^-1.0± 0.2, which is clearly different from Keplerian rotation V_ rot∝ r^-1/2. Further investigation of the rotation profile around L1527 IRS was performed with much higher sensitivity as well as higher angular resolution provided by Atacama Large Millimeter/submillimeter Array (ALMA) <cit.>. The rotation profile obtained in C^18O J=2-1 at a resolution of ∼ 09 mostly shows velocity inversely proportional to the radius being consistent with the results obtained by <cit.>, while it also suggests a possibility that the profile at ≲ 54 AU can be interpreted as Keplerian rotation with a central stellar mass of ∼ 0.3. In addition, infall velocity in the envelope is found to be slower than the free fall velocity yielded by the expected central stellar mass <cit.>.In this paper we report new ALMA Cycle 1 observations of L1527 IRS in C^18O (J=2-1) and 220 GHz continuum, with a ∼ 2 times higher angular resolution and a ∼ 4 times higher sensitivity as compared with our previous ALMA cycle 0 observations, which allow us to give a much better constraint on the rotation profile of the disks and the envelope, and also their geometrical structures. Our observations and data reduction are described in Section <ref>. In Section <ref>, we present the continuum and molecular-line results. In Section <ref>, we analyze rotation velocity measured by the C^18O line and perform χ ^2 fitting to explain the continuum visibility using a model. In Section <ref>, we investigate the validity and consistency of the model that reproduces the observations the best. We present a summary of the results and our interpretation in Section <ref>.§ ALMA OBSERVATIONS AND DATA REDUCTIONWe observed our target, L1527 IRS, during Cycle 1 using ALMA on 2014 July 20. The observations were composed of two tracks in the same day with a separation of ∼ 40 minutes. Each track was ∼ 30 minutes including overhead. J0510+1800 was observed as the passband, gain, and flux calibrator for the former track and J0423-013 was observed as the flux calibrator for the latter track. Thirty-four antennas were used in the first track, while one antenna was flagged in the latter track. The antenna configuration covers projected baseline length from 17 to 648 m (13-474 kλ in uv-distance at the frequency of C^18O J=2-1). This minimum baseline resolves out more than 50% of the flux when a structure is extended more than 7 1 <cit.>, corresponding to ∼990 AU at a distance of L1527 IRS. The coordinates of the map center during the observations were α (J2000)=04^ h39^ m5390, δ (J2000)=26^∘03 1000. C^18O J=2-1 line and 220 GHz continuum emission in Band 6 were observed for 6.9+6.7=14 minutes (on source). To achieve high velocity resolution for molecular line observations, we configured the correlator in Frequency Division Mode for two spectral windows. Each spectral window has 3840 channels covering 234 MHz bandwidth. Emission-free channels in the lower side band are used to make the continuum map centered at 220 GHz. The total bandwidth of the continuum map is ∼234 MHz. We performed self-calibration for the continuum observations using tasks (clean, gaincal, and applycal) in Common Astronomy Software Applications (CASA), and the obtained calibration table for the continuum observations was applied to the C^18O observations. The self-calibration has improved the rms noise level of the continuum map by a factor of 2-3, while the noise level of the C^18O map has been improved by less than a few percent. The noise level of the C^18O map was measured in emission-free channels. All the mapping process was carried out with CASA.Because the original map center during the observations was not coincident with the continuum peak position estimated from 2D Gaussian fitting in the uv-domain by 054, the phase center of the observed visibilities was shifted from the original phase center with fixvis in CASA, making the map center of the resultant maps in this paper the same as the continuum peak position.The visibilities were Fourier transformed and CLEANed. In this process we adopted superuniform weighting with npixel=2 and binned two frequency channels; the resultant frequency resolution in this paper is 122 kHz, corresponding to 0.17in the velocity resolution at the frequency of C^18O J=2-1. We set a 12× 12 area centered on the map center as a CLEAN box with a threshold of 3σ. The synthesized beam sizes of the CLEANed maps are 0 50× 0 40 for the C^18O line, and 047× 037 for the continuum emission. The parameters of our observations mentioned above and others are summarized in Table <ref>.c|cc Summary of the ALMA observational parameters Date 2c2014.Jul.20Target 2cL1527 IRSCoordinate center 2cR.A. (J2000)=4^ h39^ m53^ s.9 2cDec. (J2000)=26^∘03 10 0Projected baseline length 2c17.4 - 647.6 mPrimary beam 2c286Passband calibrator 2cJ0510+1800Flux calibrator 2cJ0423-013, J0510+1800Gain calibrator 2cJ0510+1800 Continuum C^18O J=2-1 Frequency 219.564200 219.560358Synthesized beam (P.A.) 0 47 × 0 37 (-0.4^∘) 0 50× 0 40 (3.1^∘)Velocity resolution 234 MHz 0.17 1σ 0.22.6 § RESULTS§.§ 220 GHz ContinuumFigure <ref> shows the 220 GHz continuum emission in L1527 IRS observed with ALMA. Strong compact continuum emission is detected. The emission is clearly elongated in the north-south direction and shows weak extensions to the northwest and the southeast. The 6σ contour in Figure <ref> shows a full width of ∼ 2=280 AU along the north-south direction. Its deconvolved size derived from a 2D Gaussian fitting is 531± 2mas× 150±2mas,P.A.=1.5^∘± 0.2^∘. This major-axis direction is almost perpendicular to the direction of the associated outflow, indicating that the continuum emission traces a dust disk and/or a flattened dust envelope around L1527 IRS. Compared with the synthesized beam size (0 47× 0 37,P.A.=-0.4^∘), the major axis of the emission is longer than the beam size and thus spatially resolved, which is consistent with the previous observations at a higher angular resolution <cit.><cit.>. The aspect ratio, 0.28, is a half of (i.e. thinner than) the ratio reported by <cit.> with lower angular resolutions than this work. The peak position is also measured from the Gaussian fitting to be α (2000)=04^ h39^ m53 88, δ (2000)=+26^∘03 09 55, which is consistent with previous measurements <cit.>. We define this peak position and the major-axis direction as the central protostellar position of L1527 IRS and the orientation angle of its dust disk/envelope respectively in this paper. The peak intensity and the total flux density of the emission derived from the Gaussian fitting are 101.4± 0.2 and 164.6± 0.5, respectively, while the total flux density is 176 when measured in the whole region of Figure <ref>. By assuming that the dust continuum emission is optically thin and dust temperature is isothermal, total mass can be calculated with the total flux density <cit.>. The total fluxes derived above correspond to a mass of M_ gas∼ 0.013 by assuming a dust opacity of κ(220GHz)=0.031cm^2g^-1 <cit.>, a dust temperature of 30 K <cit.>, and a standard gas to dust mass ratio, g/d, of 100.§.§ C^18O J=2-1The C^18O J=2-1 emission was detected above 3σ level at the relative velocity range from -3.3 to 3.2 in the LSR frame with respect to the systemic velocity V_ LSR=5.8. Figure <ref> shows the total integrated intensity (moment 0) map in white contours and the intensity-weighted-mean velocity (moment 1) map in color; both are derived from the above velocity range with 3σ cutoff. The moment 0 map overall shows an elongated structure perpendicular to the outflow axis, centered at the protostellar position. In more detail, lower contours (∼ 3-6σ) show extensions to north-northeast, north-northwest, south-southeast, and south-southwest. The moment 0 map also shows two local peaks on the northern and southern sides of the central protostar with a separation of ∼ 1. This double peak is due to a “continuum subtraction artifact”; the continuum emission was subtracted even at channels where the C^18O emission has low contrast with respect to the continuum emission and is resolved out by the interferometer. Subtraction of the continuum thus results in negative intensity at the protostellar position. Regardless of the double peak, the map was fitted with single 2D Gaussian to measure the overall structure of the C^18O emission; a deconvolved size of the C^18O emission is estimated to be 2 17± 0 04 × 0 88± 0 02, with P.A.=-1.8^∘± 0.7^∘. Peak integrated intensity and total flux measured in the whole region of Figure <ref> are 0.20 and 2.2Jy. The moment 1 map shows a velocity gradient in the north-south direction, which is perpendicular to the outflow axis. The morphology of C^18O emission indicates that it traces a flattened gas envelope and/or a gas disk around L1527 IRS and thus the velocity gradient seen in the C^18O emission is mainly due to their rotation, as already suggested by <cit.> and <cit.>. Because the C^18O emission shows a more complicated structure than the continuum emission, we assume the orientation angle of the gas disk/envelope to be the same as that of the dust disk/envelope in this paper.Figure <ref> shows channel maps of the C^18O emission, which enable us to investigate velocity structures in more detail. In higher blue- and redshifted velocities (|V|≳ 1.6), emissions show overall circular shapes and their sizes at 3σ level are smaller than ∼ 1 5. The emission peaks are located on the southern side in the blueshifted range while on the northern side in the redshifted range, making a velocity gradient from the south to the north as seen in Figure <ref>. In a middle velocity range (0.4≲ |V|≲ 1.5), more complicated structures can be seen. For example, at -1.15, -0.98, 0.85, 1.02 , the emissions appear to composed of a strong compact (∼ 1) structure close to the protostar and a more extended (>2) structure, resulting in a plateau structure. The extended emissions are located mainly on the southern side in the blueshifted range while mainly on the northern side in the redshifted range. Furthermore some blueshifted channels (-0.65 and -0.48) show an extension from the protostar to the northwest. In the other lower velocities, emissions are strongly resolved out and negative emission can be seen from 0.02 to 0.35. This redshifted negative emission is due to a continuum subtraction artifact with an extended infalling envelope around L1527 IRS as <cit.> confirmed using a infalling envelope+Keplerian disk model with radiative transfer calculation. The higher angular resolution than <cit.> can make the negative emission deeper and result in a double peak in Figure <ref> that did not appear in <cit.>.Other recent observations reported that spatial thickness of the envelope decreases outward from a radius of ∼ 150 AU by a factor of two in CCH (N=4-3, J=9/2-5/2, F=5-4 and 4-3) line emission <cit.>. No such structure, however, can be confirmed in our results in C^18O emission, which is considered to be a better tracer of the overall column density and H_2 gas distribution. The CCH emission has a higher critical density than the C^18O emission by three orders of magnitude and thus traces only dense regions in the envelope.In the moment 1 map and channel maps, L1527 IRS shows a velocity gradient from the south to the north. The PV diagrams along the major and minor axes are considered to represent a velocity gradient due to rotation and radial motion, respectively, toward disk-like structures as discussed in previous work. Figure <ref>a and b show PV diagrams of the C^18O emission cutting along the major and minor axes, respectively. The overall velocity gradient from the south to the north can be confirmed in the PV diagram along the major axis. In more detail, the so-called “spin up” rotation can also be seen in |V|≳ 1.5, that is, an emission peak at a velocity channel is closer to the central position at higher velocity. We will analyze the dependence of rotational velocity on radial distance from the central position in Section <ref>. In 0.6≲ |V|≲ 1.4, a strong compact emission and a more extended emission appear to be superposed, which corresponds to the plateau structures seen in the channel maps (Figure <ref>). In the PV diagram along the minor axis, there are four strong peaks in the western redshifted and blueshifted components, and the eastern blueshifted and redshifted ones in |V|≲ 1.5 while the emission is mainly concentrated on the central position in the higher velocity range. The western blueshifted component extends to higher velocities than the eastern blueshifted one, and also the eastern redshifted component extends to higher velocities than the western redshifted one. These extensions can be interpreted as a small velocity gradient from the west to the east, which was detected in observations in CS J=5-4 line emission and considered to be due to infalling motion in the protostellar envelope by <cit.>. § ANALYSIS§.§ Rotation Profile In the previous section we identified rotation in the C^18O gaseous component tracing either a flattened envelope, disk, or both. In this section the radial profile of the rotation will be investigated with the PV diagram along the major axis (Figure <ref>) so as to characterize the nature of the observed rotation.The method used in this section to obtain the radial profile of rotational velocity is based on the analyses presented by <cit.> and also explained by <cit.> in detail. The representative position at each velocity channel of the PV diagram along the major axis is measured as the intensity-weighted 1D mean position, x_m(v)=∫ xI(x,v)dx /∫ I(x,v)dx. Pixels having intensity more than 5σ are used to calculate the sum. The error bar of each representative position is also derived by considering propagation of errors. The derived representative positions are overlaid with error bars on the PV diagram along the major axis (Figure <ref>) and also plotted in a log R-log V plane (Figure <ref>). The abscissa of Figure <ref> is calculated from the offset position in the PV diagram by assuming that the distance of L1527 IRS is 140 pc. Figure <ref> shows a clear negative correlation that the rotational velocity is higher at the position closer to the central protostar, i.e., differential rotation. Furthermore the log R-log V diagram in Figure <ref> exhibits two different linear regimes with a break radius of ∼ 60 AU. The data points in the log R-log V plane are, therefore, fitted by a double power-law function with four free parameters: inner and outer power-law indices p_ in and p_ out, respectively, and a break point (R_b,V_b) <cit.>. The best-fit parameter set is (R_b,V_b,p_ in,p_ out)=(56± 2AU, 2.31± 0.07,0.50± 0.05, 1.22± 0.04), giving a reasonable reduced χ ^2 of 1.6. For comparison, χ ^2 fitting with a single power function is also performed, where V_b is fixed at 2.31. The best-fit parameter set for this case is (R_b,p)=(49.5± 0.3AU,0.88± 0.01), giving reduced χ ^2=4.0. These reduced χ^2 suggest that the radial profile of rotational velocity is characterized by the double power function better than any single power function. In addition, to examine whether the LSR velocity of L1527 IRS we adopt, 5.8, is reasonable, we also carried out fitting including the systemic velocity as another free parameter, which was fixed to be 5.8 above, confirming that the adopted systemic velocity is the most reasonable to fit the log R-log V diagram.The best-fit inner power-law index is almost equal to Keplerian rotation law (p=1/2), suggesting that the inner/higher-velocity component of the C^18O emission traces a Keplerian disk. In fact, the Keplerian rotation was marginally detected in <cit.>.On the other hand, the best-fit outer power-law index is roughly equal to that of rotation conserving its angular momentum (p=1), which is steeper than the Keplerian rotation law, and thus suggests that the rotation in the envelope around the disk cannot support material against the gravity yielded by the central protostar. This is consistent with the fact that the envelope is in infalling motion, as was reported by <cit.>.The best-fit result is quite consistent with that obtained in our previous Cycle 0 observations of L1527 IRS <cit.>, while the higher angular resolution and sensitivity of the Cycle 1 observations has enabled us to sample twice as many data points within the break radius in the log R-log V diagram as the previous work, making the break in the rotation profile more definite with more precise measurements of the radius of the break and the inner power-law index. The radius of the identified Keplerian disk can be estimated from the best-fit break radius. Note that low angular resolutions along the minor axis of disks cause emission within the radius to be measured at a given velocity channel and the emission moves the representative position inward at the channel, as pointed out by <cit.>. This underestimation strongly affects in the case of edge-on configurations, such as L1527 IRS, because such configurations make the angular resolution even lower along the minor axis. We thus take into account the underestimation. In their notation, the correction factor is a constant, 0.760, when Keplerian disk radius R_o is smaller than ≲ 480 AU with the inclination angle i=85^∘ and the angular resolution θ∼ 0 5=70 AU. With this correction factor taken into account, the Keplerian disk radius is estimated to be ∼ 74 AU. From the best-fit break velocity together with this disk radius, the central protostellar mass M_* of L1527 IRS and a specific angular momentum j at the outermost radius of the Keplerian disk can also be estimated to be M_*∼ 0.45 and j∼ 8.3× 10^-4 pc, respectively, where the inclination angle is assumed to be i=85^∘ <cit.>. §.§ Structures of the Keplerian DiskAs shown in the previous section, a Keplerian disk has been kinematically identified by the C^18O results. This disk around the protostar L1527 IRS seems to be kinematically quite similar to those around T Tauri stars <cit.>. A tantalizing question is whether this disk is also geometrically similar to those around T Tauri stars. Because this disk is almost edge-on, it is also possible to investigate the vertical structures of the disk. In this subsection geometrical structures of the Keplerian disk are investigated.§.§.§ Continuum Visibility distribution To investigate the disk structures, we have performed direct model fitting to the observed continuum visibility data, which is free from any non-linear effects associated with interferometric imaging. Figure <ref> shows distributions of the continuum visibilities in three panels, where red and blue points denote the same groups; the red points are located near the major axis (± 15^∘ on the uv-plane) while the blue points are located near the minor axis.Note that all the visibilities of each baseline in each track for ∼30 minutes are averaged in these plots. That is why any trajectory due to Earth's rotation is not drawn on the uv-plane (Figure <ref>a). It should be stressed that although all the visibilities of each baseline are averaged, no further azimuthal average has been done in our analysis as is obvious in Figure <ref>a. This is because information on structures that are not spherically symmetric such as disks is missed with azimuthally averaged visibilities unless the disks are face-on, as explained below in more detail.Figure <ref>b exhibits a trend that the visibility amplitudes are higher at shorter uv-distances. The total flux density 176 measured in the image space (Figure <ref>) appears consistent with an amplitude at zero uv-distance, which can be derived from visual extrapolation of the amplitude distribution. Figure <ref>b also exhibits the data points appearing to scatter more widely at longer uv-distances. This is clearly not due to the error of each data point, which is ∼ 2mJy but due to the structures of the continuum emission. In more detail, blue and red points have the highest and lowest amplitudes at each uv-distance, respectively. Because visibility is derived by Fourier transforming an image, these distributions of the blue and red points indicate that structures of the continuum emission are largest along the major axis and smallest along the minor axis in the image domain, which is consistent with the image of the continuum emission shown in Figure <ref>. Similarly the scattering of the green points between the blue and red points is due to structures along directions at different azimuthal angles. In addition, the data points near the major axis (red points) are also compared with two simple functions in Figure <ref>b: Gaussian and power-law profiles. Best-fit profiles are 0.16Jy exp (-4ln 2 (β /234m)^2) and 0.13Jy (β /100m)^-0.48, respectively, where β denotes the uv-distance. The power-law profile cannot explain the observations at all. Although the Gaussian profile matches the observations better, this profile is not necessarily realistic for circumstellar disk structures <cit.> and also the comparison shows systematic deviation; the points are higher than the Gaussian profiles in <100 m and >300 m while they are lower in ∼ 200 m. Figure <ref>c exhibits that most phases of the visibility are smaller than ≲ 5^∘, which corresponds to ≲ 0 04 where uv-distance is larger than 100 m. This indicates that emission is centered at the protostellar position at most spatial frequencies. Red points and blue points appear to be well mixed in Figure <ref>c, which means that the distribution of phases in the azimuthal direction is roughly uniform. As Figure <ref>b clearly demonstrates, the analysis of visibility without azimuthal average in a uv-plane is quite powerful for investigating spatially resolved not-spherically symmetric structures such as disks except for face-on cases.No such analysis with sufficient signal-to-noise ratio has been done in previous studies. Note that a few studies attempted to use 2D distributions in model fitting <cit.>. Those studies, however, showed only azimuthally averaged visibilities deprojected by considering inclination angles, but lacked the signal-to-noise ratio to perform comparable exploration to the 2D visibilities, making it impossible to evaluate how good their fittings were in 2D uv-space. Exceedingly high sensitivity as well as high angular resolution of ALMA allows us to perform such data analyses.§.§.§ Model fitting Our analysis of the disk structures is performed by χ^2 fitting of models to the continuum visibilities shown in Figure <ref>b. It should be noted that the full size of the continuum emission at 6σ level in Figure <ref>, ∼ 280 AU (see Section <ref>) [The 6σ size is referred here to see the extension of the whole continuum emission, which is much larger than the FWHM derived from Gaussian fitting.] is twice the disk size expected from the radius kinematically estimated in Section <ref>. This suggests that the continuum emission arises not only from the disk but also from the envelope. Hence our models should include envelope structures as well as disk structures. Because the envelope around L1527 IRS shows a flattened morphology, the model we use in this section is based on a standard disk model <cit.> but modified to express a flattened dust envelope as well as a dust disk, as described below. We used the code described in <cit.>. The model includes 12 parameters summarized in Table <ref>. The radial dependence of temperature T(R) and scale height H(R) are described as T(R)=T_1(R/1AU)^-q and H(R)=H_1(R/1AU)^h, respectively. This means that the scale height in our model is not assumed to be in HSEQ. To express dust disk and dust envelope structures, the radial dependence of surface density Σ (R) is described by a combination of inner and outer profiles formulated asΣ (R)=(2-p)M_ disk/2π( R_ out^2-p -R_ in^2-p) R^-p×{[1(R≤ R_ out); S_ damp(R>R_ out) ]. ,where M_ disk is a disk mass (mass within R_ out) determined from g/d=100 and S_ damp is a damping factor of the surface density for the outer dust envelope. The model has no envelope when S_ damp is zero, while the model has no density jump when S_ damp is unity. R_ out is the outer radius of the disk defined as the boundary between the disk and the envelope. In our model, mass density distribution ρ(R,z) is determined from the scale height H(R) and the surface density Σ (R) as Σ/(√(2π)H)exp (-z^2/2H^2) within the outermost radius of 1000 AU. Our model adopts the same power-law index of density for both disk and envelope, which is consistent with theoretical simulations by <cit.>. Some of the parameters, including the power-law index of the temperature distribution, are fixed as shown in Table <ref>. Radio observations are considered to be not sensitive to temperature distribution because most observed continuum emissions are optically thin, which makes it hard to constrain temperature distribution by mm-continuum observations. Therefore, we fixed the temperature distribution in our model, referring a total luminosity derived in infrared wavelengths <cit.>, which are more sensitive to temperature than radio observations. In addition, because our angular resolution is not high enough to resolve vertical structures of the disk, the radial temperature profile we adopted is vertically isothermal. We confirmed that our profile provides representative temperature of the 2D temperature distribution derived by <cit.> using a self-consistent radiative transfer model <cit.> at each radius on 10-100 AU scales. The inclination angle of the dust disk/envelope is fixed at i=85^∘ and the eastern side is on the near side for the observers <cit.>. The other six quantities are free parameters (M_ disk,R_ out,p,S_ damp,H_1,h). When radiative transfer were solved in 3D space to produce a model image, the following condition and quantities were assumed: local thermodynamic equilibrium (LTE), g/d=100, and a dust opacity of 0.031cm^2g^-1 calculated from a opacity coefficient of κ (850 μ m)=0.035cm^2g^-1 and an opacity index of β =0.25 <cit.>.After a model image was calculated from the radiative transfer, model visibility was obtained by synthetic observations through the CASA tasks simobserve and listvis. In the synthetic observations, the phase center was set at the center of the model image and the orientation (P.A.) was assumed to be the same as that of the observed continuum emission. Following the two observational tracks, we performed the synthetic observations without artificial noise with the same antenna configurations as the observations. Then model visibilities were derived at the same points on the uv-plane as the observations. Using the model visibility and the observed continuum visibility, reduced χ ^2 was calculated to evaluate the validity of each model. Without azimuthally averaging visibilities, all 1089 data points were used to calculate the reduced χ ^2 defined asχ^2_ν=1/σ ^2(2N_ data-N_ par-1)∑ _i[ (Re V_i^ obs- Re V_i^ mod)^2.+. (Im V_i^ obs- Im V_i^ mod)^2],where σ, V_i^ obs, V_i^ mod, N_ data, and N_ par are the standard deviation of noise in the observed visibility amplitude, the observed visibility, model visibility, the number of data points, and the number of free parameters, respectively; N_ data=1089 and N_ par=6 as mentioned above. N_ data is multiplied by two because each visibility includes two independent values, real and imaginary parts. Using the distribution of reduced χ ^2 in the parameter space, the uncertainty of each parameter is defined as the range of the parameter where the reduced χ ^2 is below the minimum plus one (=6.6) when all parameters are varied simultaneously. We also used the Markov Chain Monte Carlo method to find the minimum χ ^2 efficiently.Figure <ref>a represents a comparison of the observed continuum visibility with our best-fit model showing that the best-fit model overall reproduces the observations. The reduced χ ^2 of the best-fit model is 5.6, which corresponds to a residual of ∼ 2.4σ on average in the uv-space. It is also confirmed that the best-fit model can reproduce the observations in an image space, as shown in Figure <ref>. Note that the model image in Figure <ref>a was not made through the synthetic observation using CASA but was simply made based on the best-fit parameters in Table <ref> with convolution using the synthesized beam of our observations. This is because the synthetic observations using CASA produce a beam that is slightly different from the actual observations. Synthetic observation is not very crucial for this comparison, but convolution with the beam exactly same as the synthesized beam used in the actual observations is more crucial because the model image is relatively compact as compared to the synthesized beam. The residual shown in Figure <ref>b was obtained by subtracting the best-fit model, in the image space, from the observations, indicating that almost no significant residual can be seen in the image space. This comparison demonstrates that when the residual in the uv-space is small, the image-space residual derived from the analysis above is also small.The parameters of the best-fit model are summarized in Table <ref>. The damping factor S_ damp=0.19^+0.03_-0.09, which is not zero but significantly smaller than unity, suggests that the observed continuum emission arises from both a dust disk and a dust envelope with a significant jump of the column density between them. A much larger or smaller value than 0.19 cannot explain the observed visibilities as shown in Figure <ref>b and <ref>c. The former and the latter show models with S_ damp=0 and 1, respectively, where the other parameters are the same as those of the best-fit model. In Figure <ref>b and <ref>c visibility amplitude of models at longer uv-distance (≳ 300 m) appears to be similar to the observations in both cases, whereas visibility amplitude of the models with S_ damp=0 and 1 are lower and higher than the observations, respectively, at shorter uv-distance (≲ 300 m). One might wonder whether a jump of dust opacity, as well as surface density, could also explain the observations. It is observationally confirmed, however, that the opacity index β does not change on scales of 100 to 1000 AU, based on observed dependence of spectral index on uv-distance <cit.>, and possible uncertainty of Δβ, ∼ 0.2, adds only ∼ 8% to the relative uncertainty of S_ damp. Further discussion on the significant jump of the surface density between the dust disk and envelope will be presented in Section <ref>. The best-fit value of R_ out=84^+16_-24 AU is close to the Keplerian disk radius kinematically derived from the C^18O results; the discrepancy between the two values are within their errors. This result suggests that the dust disk identified geometrically as density contrast by this model fitting is consistent with the kinematically identified gaseous Keplerian disk. The power-law index of the surface density, p=1.7^+0.1_-0.3, is a bit steeper than the typical value for T Tauri disks, ∼ 1.0 <cit.> while a similarly steep p has been found toward a few other protostars as well <cit.>. The power-law index of the scale height, h=1.2^+0.1_-0.1, corresponds to that for HSEQ (h=1.25) within the error range, where the temperature distribution is assumed to be vertically isothermal. To examine whether the best-fit model is indeed in HSEQ, the scale height of the disk at a certain radius can be compared directly with that in HSEQ. The scale height of the best-fit model at R=R_ out is calculated to be ∼ 20 AU while the scale height of a disk in HSEQ is estimated to be ∼ 15 AU at the same radius when the central stellar mass is M_*=0.45 as kinematically estimated from the rotation profile (see Section <ref>), the temperature is 44 K as estimated from the temperature profile we used in the model, and a mean molecular weight is 2.37 m_ H. This comparison suggests that the disk around L1527 IRS is most probably in HSEQ. Note that the scale height in HSEQ also depends on vertical temperature distribution; it can be higher if temperature is higher in upper layers of disks. <cit.> suggested a highly flared disk around L1527 IRS, which has a scale height of 38 AU at R_ out. We consider that their estimated scale height is higher than ours because their observations included infrared wavelength, which may be traced by scattered light making the disk geometrically thick in appearance, or because smaller grains traced by infrared wavelength are in upper layers than larger grains traced by mm wavelength. In addition, the scale height and the index derived by our best-fit model are relatively large and steep, respectively, when compared with protoplanetary disks in Ophiuchus star forming region <cit.>.The best-fit model provides us a comparison between the disk around L1527 IRS and T Tauri disks from a geometrical point of view. The disk around L1527 IRS has a mass and a radius similar to those of T Tauri disks. On the other hand, the power-law index of the disk surface density, the scale height, and the power-law index of the scale height are possibly steeper, larger, and steeper, respectively, than those of T Tauri disks.ccccccc Fixed and Free Parameters of the Model FittingFixed i R_in T_1 q κ (220GHz) g/d 85^∘ 0.1 AU 403.5 K 0.5 0.031cm^2g^-1 100Free M_ disk R_ out p S_ damp H_1 hBest 1.3^+0.3_-0.4× 10^-2 84^+16_-24 AU 1.7^+0.1_-0.3 0.19^+0.03_-0.09 0.11^+0.02_-0.03 AU 1.2^+0.1_-0.1§ DISCUSSION§.§ Possible Origin of the Surface Density JumpA jump of the surface density between the envelope and the Keplerian disk around L1527 IRS has been found by our model fitting to the continuum visibility. In fact, such a density jump can be qualitatively confirmed in numerical simulations of disk evolution <cit.> and a similar density jump by a factor of ∼ 8 is suggested to reproduce continuum emission arising from the disk around HH 212 <cit.>. Furthermore it is important to note that the disk radius of L1527 IRS geometrically estimated with the density jump is fairly consistent with the radius kinematically estimated, suggesting that disks and envelopes around protostars may be not only kinematically but also geometrically distinguishable from the viewpoint of density contrast. The results also suggest that the density jump may be physically related to kinematical transition from infalling motions to Keplerian rotation. In this section, the possible origin of the surface density jump is quantitatively discussed. Surface density of the disk at the boundary ∼ 84 AU can be calculated from our best-fit model to be Σ _ disk= 0.42gcm^-2 with Equation <ref> and the best-fit parameters shown in Table <ref>. This surface density is within the typical range derived for ClassYSO disks in Taurus <cit.> and Ophiuchus <cit.>. On the other hand, volume density of the envelope at the boundary can be calculated from the best-fit model to be ρ_ env= 1.0× 10^-16gcm^-3, which corresponds to the number density of n_ env=2.5× 10^7cm^-3. For embedded young stars in the Taurus-Auriga molecular cloud, a typical density distribution of protostellar envelopes can be described as ∼ (0.3-10)× 10^-13gcm^-3 (R/1AU)^-3/2 <cit.>, which provides (0.4-13)× 10^-16gcm^-3, corresponding to (0.1-3.5)× 10^7cm^-3, at the boundary, R_ out. This indicates that the envelope density in the best-fit model is also reasonable when compared with other observations. The envelope density is also consistent with that in <cit.> at their boundary radius 125 AU.Because the disk around L1527 IRS is considered to still grow with mass accretion from the envelope, as discussed in <cit.>, a possible origin of the density jump may be mass accretion from the envelope to the disk.First, density in the Keplerian disk is higher than that in the infalling envelope when radial motion in the disk is not as fast as that in the envelope, which is reasonable because gravity and centrifugal force are balanced in the disk while they are not balanced in the envelope. This difference of mass infall rates makes mass build up in the disk, resulting in disk growth. Secondly, to explain the factor S_ damp=0.19 quantitatively, we consider isothermal shock due to the mass accretion between the infalling envelope and the Keplerian disk. The gravity of a central protostar causes material in the envelope to infall dynamically and thus the material has radial infall velocity u_r (R) as a function of radius. When ρ _ env, ρ _ disk, and c_s indicate envelope density, disk density, and sound speed, respectively, the isothermal shock condition is S_ damp=ρ _ env/ρ _ disk=c_s^2/u_r^2. The sound speed at the boundary can be calculated from our best-fit model to be c_s=0.39. Regarding the infall velocity, we assume u_r to be the product of a constant coefficient α and free fall velocity, i.e., u_r=α√(2GM_*/R_ out), as was discussed by <cit.>. By using M_*=0.45 and R_ out=84 AU, this infall velocity can be calculated to be 3.1α. In order to explain S_ damp=0.19, α should be ∼ 0.3, which is consistent with the range of α (0.25-0.5) <cit.> found. This quantitative discussion suggests that mass accretion from the envelope to the disk is a possible origin of the density jump we found. §.§ Structures of the C^18O gas disk The previous sections have discussed the structures of the disk and the envelope around L1527 IRS based on the continuum observations tracing the disk and the envelope. C^18O emission, on the other hand, also traces them, as was discussed in <cit.>. Because it can be reasonably assumed that gas and dust are well coupled and mixed in the protostellar phase in contrast to the T Tauri phase where gas and dust can be decoupled because of grain growth, it is important to examine whether the structures of the disk and the envelope revealed in the dust observations can also be valid for those traced in C^18O emission.In order to answer the question mentioned above, in this section, models for C^18O are constructed based on the best-fit dust model derived from the fitting to the continuum visibility shown in Section <ref>, and models are compared with the C^18O observations. The models of C^18O have the same profiles of the surface density, temperature, and scale height as those of the best-fit dust model derived from the fitting to the continuum visibility shown in Section <ref>. The surface density profile has a jump at a radius of 84 AU, which is the boundary between the disk and the envelope, as was suggested by the best-fit dust model. In addition to these constraints on structures, C^18O models also require velocity fields, which cannot be constrained by the continuum observations. For velocity fields, we assume that C^18O models follow the radial profile of the rotation derived from the C^18O observations in Section <ref>, and that of infall introduced in Section <ref>. Note that we adopt 74 AU as the radius where the rotation profile has a break even though the geometrical boundary between the disk and the envelope is set at 84 AU in radius, as mentioned above. These geometrical and kinematical structures of the C^18O models are fixed in the following discussion. Importantly the C^18O models still depend on the fractional abundance of C^18O relative to H_2, X( C^18O), as discussed later.In order to compare models and observations, C^18O data cubes are calculated from the models described above. It should be noted that the C^18O emission obviously traces more extended structures arising from outer parts of the envelope, as compared with the dust emission. Because structures of the disk and the envelope adopted for C^18O model are based on the continuum emission detected within a radius of ∼ 1 (see Figure <ref>), comparisons between C^18O models and observations should be made only within a radius of 1. According to the C^18O velocity channel maps shown in Figure <ref>, the C^18O emission arises within a radius of ∼ 1 when |V_ LSR|>2.0, and only C^18O emission having these LSR velocities is discussed in comparisons between the models and the observations. When C^18O data cubes are calculated from the models, radiative transfer, including both dust and C^18O opacities, are also solved in 3D and velocity space, and then dust continuum emission is subtracted to derive the final model cubes. The model data cubes calculated from the radiative transfer are convolved with a Gaussian beam having the same major and minor axes, and orientation as the synthesized beam of our observations. A moment 0 map is made from this convolved data cube to compare with the observations. Even though visibilities are not compared between models and observations here, we can still judge how good each model is based on this comparison using moment 0 maps, as demonstrated in the model fitting for the continuum data in Section <ref>.Figure <ref>a shows a comparison of moment 0 maps between a model and the observations. In this model, a constant C^18O abundance of 4× 10^-8 is adopted as a nominal value. In this case, significant residuals at 9σ level are left, as shown in Figure <ref>b. Inner regions show negative residuals while outer regions show positive residual, demonstrating that the model C^18O is too strong in inner regions while it is too weak in outer regions, as compared with the observations. Note that neither a higher nor a lower constant value than 4.0× 10^-8 of the C^18O abundance improves the model. For instance, the value in interstellar medium (ISM), 5.0× 10^-7 <cit.>, provides more negative residuals than Figure <ref>b. Although there are a couple of factors to change the C^18O intensity in the model, the abundance of C^18O is the only one that changes the intensity of C^18O if we still remain the same physical structures of the disk in the model, i.e., in order to make the C^18O intensity weaker in inner regions and stronger in outer regions in the model, the C^18O abundance might be lower in inner regions and higher in outer regions. For instance, if the C^18O abundance in outer regions (R>84 AU) is the same as the one in ISM, 5.0× 10^-7 <cit.>, and the abundance in the inner regions (R<84 AU) decreases by a factor of ∼ 20 due to the freeze-out of C^18O molecules, the observations can be explained by the model. T Tauri disks are, however, usually considered to have temperature profiles having higher temperature in inner regions <cit.>, and the disk around L1527 IRS is also considered to have such a temperature profile as <cit.> suggested. With such a temperature profile, it would be difficult for the disk to have a lower C^18O abundance in the inner regions because of the molecular freeze-out.One possible C^18O abundance distribution that can explain the observations is the one with a local enhancement of the C^18O molecule, as has been suggested for the SO molecular abundance around L1527 IRS by <cit.>; they suggested that the SO abundance is locally enhanced around L1527 IRS because of accretion shocks making the dust temperature sufficiently high for SO molecules frozen out on dust grains to be desorbed. An example of such a C^18O abundance distribution is the ISM abundance <cit.> at 80≤ R≤88 AU and a lower constant abundance of 2.8× 10^-8 elsewhere. Figure <ref> shows the comparison between the model with this C^18O abundance distribution and the observations, suggesting that the model with this C^18O abundance distribution can reproduce the observations with reasonably small residual.The reason of the lower C^18O abundance in the inner disk region is not clear. According to the temperature distribution of L1527 IRS by <cit.>, the midplane temperature becomes lower than 30 K at r≳ 100 AU, sublimation temperature of CO on dense conditions <cit.>. Thus CO freeze-out could be present only at the outer region. In general, CO molecules indeed could not be frozen-out so easily in protostellar disks as in T Tauri disks because surrounding envelopes heat up such embedded disks <cit.>, although the degree of this heating effect depends on a couple of factors, such as density distribution.There are two other possibilities to explain the observed C^18O abundance decrease in the inner disk region. One is that dust can be optically thick in inner regions and hides part of the C^18O emission as indicated by our best-fit model shown in Section <ref>. The other possibility is that chemistry on the warm and dense conditions. On such a condition CO can be converted into more complex molecules such as CO_2 and organic molecules. Indeed, recent CO observations of protoplanetary disks with ALMA have been reporting similar decrease of CO abundance, which is attributed to such a chemical effect <cit.>. The CO conversion is also reported in protostellar phases as well based on single-dish and interferometric observations <cit.>. Although it is difficult to give a strong constraint on the width of the local enhancement and the lower C^18O abundance in the frozen-out or converted region in models with the current observations, the density and temperature structures derived from the continuum observations can also reproduce the C^18O observations with a radial abundance profile of C^18O with a local enhancement like the one discussed above. Future observations at a higher angular resolution can give a better constraint on the C^18O radial abundance profile.§ CONCLUSIONSWe have observed the Class 0/I protostar L1527 IRS in the Taurus star-forming region with ALMA during its Cycle 1 in 220 GHz continuum and C^18O J=2-1 line emissions to probe the detailed structures of the disk and the envelope around L1527 IRS.The 220 GHz continuum emission spatially resolved with an angular resolution of ∼ 05 × 0 4 shows a similar elongated structure in the north-south direction. Its deconvolved size is estimated from a 2D Gaussian fitting to be ∼ 053 × 0 15, showing a significantly thinner structure than those previously reported on the same target.The C^18O J=2-1 emission overall shows an elongated structure in the north-south direction with its velocity gradient mainly along the same direction. The integrated intensity map shows a double peak with the central star located between the peaks, due to a continuum subtraction artifact. The elongation of the continuum as well as C^18O clearly indicates that these emissions trace the disk/envelope system around L1527 IRS and the velocity gradient along the elongation is naturally considered to be due to rotation of the system, as was previously suggested.The radial profile of rotational velocity of the disk/envelope system obtained from the position-velocity diagram of the C^18O emission cutting along the major axis of the continuum emission was fitted with a double power-law, providing the best-fit result with a power-law index for the inner/higher-velocity (p_ in) of 0.50 and that for the outer/lower-velocity component (p_ out) of 1.22. This analysis clearly suggests the existence of a Keplerian disk around L1527 IRS, with a radius kinematically estimated to be ∼ 74 AU. The dynamical mass of the central protostar is estimated to be ∼ 0.45.In order to investigate structures of the disk/envelope system, χ ^2 model fitting to the continuum visibility without any annulus averaging have been performed, revealing a density jump between the disk and the envelope, with a factor of ∼ 5 higher density on the disk side. The disk radius geometrically identified as the density jump is consistent with the Keplerian disk radius kinematically estimated, suggesting that the density jump may be related to the kinematical transformation from infalling motions to Keplerian motions. One possible case to form such a density jump is isothermal shock due to mass accretion at the boundary between the envelope and the disk. If this is the case, to form the density jump with a factor of ∼ 5 requires the infall velocity in the envelope to be ∼ 0.3 times slower than the free fall velocity yielded by the central stellar mass. In addition to the density jump, it was found that the disk is roughly in hydrostatic equilibrium. The geometrical structures of the disk found from the χ^2 model fitting to the continuum visibility can also reproduce the C^18O observations as well, if C^18O freeze-out, conversion, and localized desorption possibly occurring within ∼ 1 from the central star are taken into account. This paper makes use of the following ALMA data: ADS/JAO.ALMA2012.1.00647.S (P.I. N. Ohashi). ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), NSC and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. We thank all the ALMA staff making our observations successful. We also thank the anonymous referee, who gave us invaluable comments to improve the paper. Data analysis were in part carried out on common use data analysis computer system at the Astronomy Data Center, ADC, of the National Astronomical Observatory of Japan. S.T. acknowledges a grant from the Ministry of Science and Technology (MOST) of Taiwan (MOST 102-2119-M-001-012-MY3), and JSPS KAKENHI Grant Number JP16H07086, in support of this work. Y.A. is supported by the Subaru Telescope Internship Program. ALMACASA, MIRIAD, IDL aasjournal | http://arxiv.org/abs/1707.08697v1 | {
"authors": [
"Yusuke Aso",
"Nagayoshi Ohashi",
"Yuri Aikawa",
"Masahiro N. Machida",
"Kazuya Saigo",
"Masao Saito",
"Shigehisa Takakuwa",
"Kengo Tomida",
"Kohji Tomisaka",
"Hsi-Wei Yen"
],
"categories": [
"astro-ph.SR",
"astro-ph.GA"
],
"primary_category": "astro-ph.SR",
"published": "20170727034355",
"title": "ALMA Observations of the Protostar L1527 IRS: Probing Details of the Disk and the Envelope Structures"
} |
[email protected] Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, New York 11973, USA Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439, USAAdvanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439, USA School of Physical Sciences, Dublin City University, Dublin 9, Ireland Department of Condensed Matter Physics, Charles University, Ke Karlovu 3, Prague 12116, Czech Republic Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Collaborative Innovation Center of Quantum Matter, Beijing, China [email protected] Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, [email protected] Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, New York 11973, USAWe report on the tuning of magnetic interactions in superlattices composed of single and bilayer SrIrO_3 inter-spaced with SrTiO_3. Magnetic scattering shows predominately c-axis antiferromagnetic orientation of the magnetic moments for the bilayer justifying these systems as viable artificial analogues of the bulk Ruddlesden-Popper series iridates.Magnon gaps are observed in both superlattices, with the magnitude of the gap in the bilayer being reduced to nearly half that in its bulk structural analogue, Sr_3Ir_2O_7. We assign this to modifications in the anisotropic exchange driven by bending of the c-axis Ir-O-Ir bond and subsequent local environment changes, as detected by x-ray diffraction and modeled using spin wave theory. These findings explain how even subtle structural modulations driven by heterostructuring in iridates are leveraged by spin orbit coupling to drive large changes in the magnetic interactions.Magnetism in artificial Ruddlesden-Popper iridates leveraged by structural distortions M. P. M. Dean December 30, 2023 ======================================================================================§ INTRODUCTION Recent years have seen iridates, compounds composed of active Ir 5d orbitals in oxygen octahedra, emerge as an important new class of strongly correlated materials <cit.>. The combination of crystal field interactions and strong spin-orbit coupling generatesnarrow bands leading to insulating antiferromagnetic ground states that arise from modest values of the Coulomb repulsion U <cit.>. Many appealing structural and electronic analogies between iridates and lighter 3d-electron based cuprates have been identified <cit.>. One crucial difference, however, is that iridates host spin-orbit coupled J_eff=1/2 magnetic moments, which have a more intricate coupling to orbital distortions than pure spin S=1/2 moments <cit.>. This is borne out in observations: different magnetic ground states appear in iridates composed of similar Ir-O octahedra when they are subtly distorted or interspaced with different atoms. Sr_2IrO_4, hosting isolated IrO_2 layers, forms an ab-plane canted antiferromagnetic state <cit.>, while Sr_3Ir_2O_7, hosting isolated IrO_2 bi-layers, has c-axis collinear antiferromagnetic ordering <cit.>.Towards understanding how the structural modulations tailor magnetic ground states, resonant inelastic x-ray scattering (RIXS) has been extremely successful in quantifying the magnetic interactions present in iridate crystals <cit.>. For example, a large (92 meV) spin gap was measured in , reflecting the substantial interlayer anisotropic coupling that causes a “dimensionality driven spin flop transition" with respect to<cit.>. Recently, artificial layered iridates in analogy to Ruddlesden-Popper iridates were realized via alternating layers of nSrIrO_3 and SrTiO_3 (nSIO/1STO), which showed a metal-insulator transition as a function of n, closely mirroring their bulk analogues<cit.>. However, an ab-plane canted antiferromagnetic state was argued to be maintained for n≤3, suggesting that the spin flop transition is suppressed and breaking the analogy to bulk crystals<cit.>. To achieve this change in the ground state, anisotropic coupling between the two Ir layers in 2SIO/1STO, which favors the c-axis antiferromagnetic state, would need to be substantially modified compared to<cit.>. Further, this behavior is disputed by density functional theory (DFT) predictions that find c-axis collinear magnetism in 2SIO/1STO <cit.>. Taken together, these conflicting results point to the need for direct observation of the exchange coupling and the resulting magnetic structure to unravel the impact of heterostructuring on the behavior of these proposed artificial analogues to the Ruddlesden-Popper iridates. In this work, we directly probe the magnetic behavior of nSIO/1STO and extend the sensitivity of Ir L_3 RIXS to quantify the interactions that stabilize this state. We find a c-axis antiferromagnetic ground state in 2SIO/1STO, in contrast with an earlier report <cit.>, demonstrating that the magnetic ground state mimics bulk . In both 1SIO/1STO and 2SIO/1STO, the magnetic excitation spectrum shows a clear dispersion with a magnon gap of 55 meV in n=2, substantially reduced to about half that in bulk<cit.>. Based on modeling the magnetic dispersion, the predominately c-axis moments in 2SIO/1STO are stabilized by the anisotropic coupling between Ir-O planes as seen in . However, the lowering of the magnon gap evidences a significant reduction in the tetragonal distortion of the octahedra, while for 1SIO/1STO the gap size is similar to bulk<cit.>. The source of the modulation of the tetragonal distortion was determined to be substantial bending of the c-axis Ir-O-Ir bonds alongside changes in the local environment beyond the octahedra. This work establishes these artificial structures as true analogues to the Ruddlesden-Popper iridates, with the caveat that changes in the magnetic ground states are highly susceptible to subtle structural distortions <cit.>.These distortions push the heterostructure towards a quantum critical point between the ab-plane and c-axis antiferromagnets, exemplifying how magnetic states in iridates can be transformed in a tractable manner owing to their strong spin-orbit coupling. § EXPERIMENTAL DETAILSSuperlattices (SLs) of form [nSIO/1STO]× m with n = 1, 2 and m = 60, 30, respectively, were grown with pulsed laser deposition using methods described in Ref. <cit.>, as depicted in Fig. <ref>(a). High sample quality was verified by x-ray diffraction, x-ray magnetic circular dichroism, transport and magnetometry measurements (Fig. S1-S3 of <cit.>), consistent with previous studies <cit.>. Resonant elastic x-ray scattering (REXS), RIXS, and non-resonant diffraction data were taken at the 6-ID-B, 27-ID-B, and 33-BM-C beamlines of the Advanced Photon Source at Argonne National Laboratory. The RIXS energy resolution was 35 meV, full width half maximum. Further details are available in the Supplemental Material <cit.>. § CRYSTAL AND MAGNETIC STRUCTUREPrevious investigations of nSIO/1STO with n = 1, 2, 3 displayed a net ferromagnetic moment for all samples, which was taken as evidence for the stabilization of canted ab-plane magnetic moments as in<cit.>. Although there is a strong consensus that this is valid for 1SIO/1STO, the result for 2SIO/1STO is more controversial as itbreaks the analogy between n=2 and , which has purely c-axis collinear antiferromagnetism implying no spontaneous net moment <cit.>. Theory also predicted c-axis moments and posited that the observed net ferromagnetic moment comes from oxygen vacancies <cit.>. Establishing the true magnetic ground state is of high importance towards extracting the magnetic exchange parameters that ultimately dictate the overall magnetic behavior of these heterostructures.In view of this controversy, we directly measured the spin ordering direction using azimuthal REXS scans, as was done in<cit.>. This dependence is shown for 2SIO/1STO in Fig. <ref>(b), left panel[For the magnetic reflection the SL structure is used for r. l. u, with a≈ b ≈ c/3.].The calculated azimuthal dependence for c-axis oriented antiferromagnetic moments, shown as the grey line, matches the data well and establishes predominately c-axis moments <cit.>. To further emphasize this distinction, we also show the magnetic Bragg peaks where the maximum intensity for both cases is expected (-90^∘), and also where no intensity for in-plane moments is expected (7^∘), Fig. <ref> (b), right panel. Clearly, a magnetic peak persists with integrated intensity that matches that expected for c-axis orient moments (∼ 30%). These results then agree with theoretical predictions, showing the 2SIO/1STO SL maintains the same magnetic ground state as , strengthening the analogy to Ruddlesden-Popper series iridates <cit.>. Armed with the correct ground state, the magnetic excitation spectrum can be correctly interpreted, allowing the probing of the magnetic exchange couplings to unravel the true impact of artificial heterostructuring.To directly probe the magnetic interactions, we utilize RIXS to map the magnetic dispersion of the established magnetic ground state. Although Ir L_3-edge RIXS has been applied extensively to iridate crystals and thin films, a full 2D magnetic dispersion curve has never been characterized on SL heterostructures due to the relatively large (5 μm) x-ray penetration depth at the Ir L_3 edge <cit.>. This challenge was overcome by growingrelatively thick SLs (60 IrO_2 planes) and working near grazing incidence (1^∘) <cit.>. Raw RIXS spectra for the SLs are displayed in Fig. <ref>. Each spectra displays a high energy feature around 0.75 eV energy loss, corresponding to both an intra-t_2g orbital excitation and the e-h continuum <cit.>. A sharp peak arises at zero energy due to elastic scattering, along with a small phonon feature at around 40 meV. Finally, a dispersive feature from 50 to 140 meV is seen in all spectra, and is identified as the magnon excitation [Only one magnetic excitation was observed for both samples, despite the presence of both optical and acoustic modes for the bilayer. This is due to the intensity dependence of each mode as discussed in section IV. ], with the higher energy tail including multimagnon excitations <cit.>. The spectra were fit using a combination of peaks in a similar approach to that used previously (see supplemental materials)<cit.>. Examples of fits along the nodal direction for each sample are displayed in Fig. <ref>(a). From these one can extract the energy, width, and integrated intensity of the magnetic excitation, Figs. <ref>(b). The intensity peaks at the magnetic ordering wave vectorand the energy loss is within the bandwidth seen forand , corroborating our assignment of the feature as a magnetic type excitation <cit.>. From the extracted magnon dispersion, some important observations are immediately clear: (i) both SLs have nearly identical dispersion around theandpoints with maxima of ∼ 120 and 150 meV, respectively, (ii) both samples show magnon gaps. In the case of the 1SIO/1STO, the size of the gap is not well defined, being ∼11-36 meV, due to the worse reciprocal space resolution, 0.46 Å^-1 (0.073 r.l.u.) <cit.>. For 2SIO/1STO, a mask was used to improve the resolution to 0.12 Å^-1 (0.018 r.l.u.) for theand (0, 0) Q-points. Here, a larger gap is much better defined as between 50 and 60 meV at<cit.>. Compared with the dispersion for , Fig. <ref>(b), the overlap is very robust everywhere except at these minima <cit.>.We analyzed the origin of this anomalous behavior using linear spin wave theory [It should be noted, forthe dispersion and magnon gap can also be described in terms of a bond-operator approach utilizing spin dimers along the c-axis <cit.>, not considered here for several reasons. In this case, the SLs lies firmly within the antiferromagnetic order regime as clearly evidenced by the smaller magnon gaps, whereas the dimer model requires gaps ≥ 90 meV. Finally, the sizable total moments observed, on the order of , also conflict with a dimer picture in this case <cit.>.] applied to the Hamiltonian described in Ref. <cit.>, with ab-plane canted and c-axis collinear ground states for the single and bilayer SLs, respectively <cit.>. Importantly, the canted nature of the moments in 1SIO/1STO gives a dispersion relation that fundamentally differs from the out-of-plane Néel state <cit.>. As was the case for , the nine magnetic couplings can be parameterized in terms of: (i) the tetragonal distortion θ, defined by tan2θ = 2√(2)λ/λ - 2Δ, with spin-orbit coupling λ and tetragonal splitting Δ, (ii) η=J_H/U, with Hund's coupling J_H and Coulomb repulsion U, (iii) the octahedral rotation angle α. The form of the exchange couplings in terms of these parameters is described in the supplemental (J^'_ab, J^''_ab, andJ^'_c are treated as free fit parameters) <cit.>. Concerning (ii), η = 0.24 was established forand is unlikely to change significantly, leaving the tetragonal distortion and rotation angles to explain the observed dispersion and magnon gaps [As shown in Fig. <ref>(b), the phase transition is not sensitive to η in this regime, nor is the dispersion fit.]. Regarding α,andboth feature large in-plane rotations (α = 12^∘ and 11^∘ respectively), but no tilts (rotations about a/b axes that bend the c-axis bond) <cit.>. Bulk-like SIO films, on the other hand, show substantial tilts and rotations implying that similar effects may be present in SLs <cit.>. We consequently tested for the presence of octehedral tilts and rotations by scanning the half order Bragg peaks locations. While an exact structural solution of the SLs is unfeasible due to the complex orthorhombic structure of SIO <cit.>, published methods allow us to associate different half order reflections with different antiphase distortions <cit.>. We measured several reflections for the n=1 and 2 samples and illustrate the important behavior in Fig. <ref>(a) <cit.>. The (1/2, 3/2, 3/2) reflection (left panel) arises from a combination of rotations and tilts; whereas the (1/2, 1/2, 3/2) reflection (right panel) comes from only tilts. Both peaks are of similar magnitude for n=2, but the tilt-peak is suppressed by an order of magnitude in n=1. This data suggests that both SLs have similar rotations of ∼ 8^∘. In contrast with the nearly straight c-axis bonds seen in<cit.>, n=2 likely hosts tilting of a similar size (of order 8^∘), while n=1 has small, but finite tilts[In fitting the magnetic dispersion, values of α in the range 5-15^∘ are consistent with the data and thus we fix α = 8^∘]. Most importantly, the presence of tilting generates the ab-plane ferromagnetic moment in 2SIO/1STO, observed experimentally <cit.>, through canting the c-axis antiferromagnetic moments, resolving the conflict between previous experimental interpretations and theory <cit.>. § TUNING MAGNETIC GROUND STATES Having established the approximate rotation angles of both samples as α=8^∘, we can fit the dispersion of the SLs, as discussed above, displayed for 2SIO/1STO in Fig. <ref>(b) left panel <cit.>. For 1SIO/1STO with gap ∼11-36 meV (compared to ∼25 meV for ), we find θ=0.221-0.247π. Within this range, spin wave theory can model well the dispersion throughout reciprocal space <cit.>. For 2SIO/1STO, fitting with θ=0.225±0.009 πadequately reproduces the gaps and dispersion. For the bilayer, optical and acoustic modes are present, but only one mode is observed due to the Q-dependence of their intensities, discussed in the supplemental <cit.>. The similar theta values, which do not reflect the differences seen between bulkand , are initially surprising due to the less distorted octahedra observed for both SIO and(≤ 2%, ∼ 8% for ) <cit.>. However, the Ir 5d orbitals are rather extended spatially and couple strongly to next nearest neighbors, breaking the local symmetry <cit.>. The similar θ of the SLs further corroborates this, pointing to the SL structure as the dominant determinant of θ.Extracted values of the exchange couplings for each of the SLs are shown in Table <ref> <cit.>. For 1SIO/1STO, the values are similar to those found in both doped and undoped , owing to the relatively small change in θ between the SL and(∼ 0.01π) <cit.>. This indicates the 1SIO/1STO provides another magnetic analogue to cuprates, similar to that found in , but with the higher tunability afforded by heterostructuring <cit.>. Comparing 2SIO/1STO with , on the other hand, the changes are quite substantial, owing to the much more significant shift of θ (0.035π) <cit.>. Here, the ratio J_ab/J_c is half that found in , a reasonable change in light of the more uniform octahedra expected in the SL (due to the ability of tilting to circumvent distortion) [Within the dimer model, this ratio is strongly inverted with much larger J_c and smaller J_ab <cit.>.]. Finally, from fitting the smaller magnon gap,a 28% decrease in the pseudodipolar anisotropic coupling Γ_c is observed, which is chiefly responsible for stabilizing the c-axis magnetic ground state. To investigate the stability of the observed magnetic phases, we map the SLs on the classical (θ,η) magnetic phase diagram in Fig. <ref>(b) alongsideand<cit.>. Intriguingly, both SLs lie close to their respective phase transitions. This is especially significant for 2SIO/1STO, where the material lies closer to the phase transition than its bulk analogue, within 0.02π or ∼ 10 meV tetragonal splitting change. The fact that such a shift happens despite the similar structures of 2SIO/1STO andshows how relatively small distortions can strongly modify the interactions of these SLs.Further bending of the c-axis bond can then be expected to drive the system closer to, and eventually through, a quantum critical point between the ab-plane canted and c-axis collinear antiferromagnets. This could be accomplished through applying epitaxial tensile strain, by changing the substrate, or by substituting Sr with Ca. Based on the calculated change in the crystal field of strained films of , applied strain of only a few percent could be enough to drive 2SIO/1SIO to the quantum critical point <cit.>. In this way, strong spin-orbit coupling provides a means to exploit small structural distortions to stabilize large changes in the magnetic ground state.§ CONCLUSIONIn conclusion, we demonstrate that 2SIO/1STO has predominantly c-axis antiferromagnetic moments, establishing the nSIO/1STO SL series as viable artificial analogues to the Ruddlesden-Popper crystals. We furthermore reconcile previous contradictory reports by identifying finite octahedral tilting that generates a net canted moment <cit.>. Hard x-ray RIXS is shown to be sufficiently sensitive to probe the magnetic interactions that stabilize the observed ground state for the first time. For the bilayer 2SIO/1STO, the magnon gap is significantly smaller than that observed inand spin-wave based modelling shows that this material is closer to a phase transition between different ground states. Heterostructuring iridates and probing their magnetic interactions with RIXS thus shows how spin orbit coupling can leverage small structural distortions to alter magnetic interactions with potential to realize quantum critical artificial Ruddlesden-Popper phases. We would like to acknowledge helpful discussions with G. Jackeli, G. Khaliullin, Wei-Guo Yin, and Yilin Wang and experimental assistance from C. Rouleau, Z. Gai, and J. K. Keum. This material is based upon work supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Early Career Award Program under Award No. 1047478. Work at Brookhaven National Laboratory was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-SC0012704. J.L. acknowledges the support by the Science Alliance Joint Directed Research & Development Program and the Transdisciplinary Academy Program at the University of Tennessee. J. L. also acknowledges support by the DOD-DARPA under Grant No. HR0011-16-1-0005. A portion of the fabrication and characterization was conducted at the Center for Nanophase Materials Sciences, which is a DOE Office of Science User Facility. Use of the Advanced Photon Source, an Office of Science User Facility operated for the U.S. DOE, OS by Argonne National Laboratory, was supported by the U.S. DOE under Contract No. DE-AC02-06CH11357. X. Liu acknowledges support by MOST (Grant No.2015CB921302) and CAS (Grant No. XDB07020200). | http://arxiv.org/abs/1707.08910v1 | {
"authors": [
"D. Meyers",
"Yue Cao",
"G. Fabbris",
"Neil J. Robinson",
"Lin Hao",
"C. Frederick",
"N. Traynor",
"J. Yang",
"Jiaqi Lin",
"M. H. Upton",
"D. Casa",
"Jong-Woo Kim",
"T. Gog",
"E. Karapetrova",
"Yongseong Choi",
"D. Haskel",
"P. J. Ryan",
"Lukas Horak",
"X. Liu",
"Jian Liu",
"M. P. M. Dean"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170727154528",
"title": "Magnetism in artificial Ruddlesden-Popper iridates leveraged by structural distortions"
} |
1.5ptazuki IkedaDepartment of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan [email protected]============================================================================================================================== Recent advances in the Langlands program shed light on a vast area of modern mathematics from an unconventional viewpoint, including number theory, gauge theory, representation, knot theory and etc. By applying to physics, these novel perspectives endow with a unified account of the (integer/ fractional) quantum Hall effect. The plateaus of the Hall conductance are described by Hecke eigensheaves of the geometric Langlands correspondence. Especially, the particle-vortex duality, which is explained by S-duality of Chern-Simons theory, corresponds to the Langlands duality in Wilson and Hecke operators.Moreover the Langlands duality in the quantum group associated with the Hamiltonian describes fractal energy spectrum structure, know as Hofstadter's butterfly. These results suggest that the Langlands program has many physically realistic meanings. § INTRODUCTION Ramanujan's finding on automorphic forms is crucial to modern number theory. Around 1916, he calculated the expansion coefficients a_n of the following infinite series. q∏_n=1^∞ (1-q^n)^2(1-q^11n)^2=∑_n=1a_nq^n.About 40 years later, Eichker proved that there is a profound correspondence between the automorphic form above and the elliptic curve defined on ℚy^2+y=x^3-x.Astonishingly, for b_p=p+1-# (points of (<ref>) mod p), the equationa_p=b_pis true for any prime p. It is the Langlands program <cit.> that connects these dualities from general viewpoints of mathematics. In the Langlands program, the correspondence between automorphic forms and elliptic curves are all about the correspondence between the eigenvalues of Hecke operators and Frobenius operators, namely a_p and b_p are eigenvalues of Hecke and Frobenius operators respectively. The Langlands program can be interpreted geometrically <cit.>, and the geometric Langlands correspondence foresees many nontrivial aspects of gauge theories. From a perspective of geometry, in a simple case, this is achieved by considering a gauge theory on a Riemann surface, and a Hecke operator modifies the given principle bundle at a singular point so that the 1st Chern-number jumps at the singularity, as a vortex or a monopole operator do, and a Frobenius operator is analogous to a holomony operator, like a Wilson loop <cit.>. The aim of this article is to endow it with physical meaning. In a seminal piece of research made by A. Kapustin and E. Witten <cit.>, they predict the electric magnetic duality and mirror symmetry are intimately related to the geometric Langlands correspondence. While there are many relevant works <cit.>, what would follow in view of non supersymmetric physics had not been known. In this work, we address the quantum Hall effect and enjoy the panoptic picture of the quantum Hall effect drawn as a natural consequence of the Langlands program. One can seek a cue from the Langlands/GNO dual group to understand the connection with the quantum Hall effect and the geometric Langlands duality. In electric-magnetic duality, Dirac monopoles and Dirac's quantization condition explain the dual group ^LG of a general Lie group G from a perspective of physics <cit.>. The quantization condition of the Hall conductance is understand in a similar manner. More concretely, what we address in this article is* Integer quantum Hall effect (IQHE)* Fractional quantum Hall effect (FQHE)* Hofstadter's butterflyFirst of all, what is common in these three is that they are described by the S-duality like picture, as often discussed in gauge theory. They may look different at first sight, however, the Langlands philosophy connects them all eventually. The two dimensional IQHE shows the typical Hall conductance σ_xy classified by integers (figure <ref>).σ_xy is given by the sum of the Chern numbers[In this article, we will often call the 1st Chern number of a U(1)-bundle as its Chern number since we focus on two-dimensional physics. When we say Chern numbers, it means we consider many U(1)-bundles.] of the U(1)-bundles, associated with the energy bands below the Fermi level, on the Brillouin zone (BZ) <cit.>. The plateau regions clearly explains the existence of flat connections forming 𝒟-modules, which turn out to be Hecke eigensheves. And quantum jumps in the conductivity are naturally described by the Hecke modification. The Hall conductance σ which accompanies the Chern-Simons action and q-parameter accommodates σ in such a way that q=exp(σ). The duality of Chern-Simons theory induces flipping q→ ^Lq=exp(-1/σ) which suggests the IQHE-FQHE duality. This is how the Langlands/S-duality unites the IQHE and the FQHE. Secondly, it is important to recall that the Hamiltonian of the IQHE is written by the quantum group 𝒰_q(sl_2) and S-duality maps it to the dual quantum group with the dual ^Lq-parameter. Flipping q→ ^Lq corresponds to the particle-vortex duality. And we explain that Hofstadter's butterfly is captured in this way. A better understanding on the IQHE will also enhance comprehension of general topological insulators (TI). Roughly speaking, TI's are extension of the IQHE to general gauge theories and they are classified in a similar manner <cit.>. And analogous idea of the particle-vortex duality can be seen in general ways, including supper conductors <cit.>. While the QHE and TI's are well known systematically, geometric Langlands correspondence for general cases remains a conjecture. Hence knowledge on the QHE and TI's will endow the Langlands program with hints for being developed. This piece is orchestrated as follows. In section 2, we address the IQHE and the FQHE from a perspective of the geometric Langlands correspondence. In section 3, we describe the Langlands duality in the quantum group 𝒰_q(sl_2) and build a connection with the geometric Langladns duality based on Chern-Simons theory. Hofstadter's butterfly appears on this way. *Acknowledgement I thank Kentaro Nomura for useful conversations on the quantum Hall effect and topological insulators. I am grateful to Sergei Gukov for various comments. § INTEGER QUANTUM HALL EFFECT AND LANGLANDS PROGRAM §.§ General Setup Throughout this article we consider the IQHE on Laughlin's type of geometry <cit.>, namely a square lattice inthe uniform magnetic flux ϕ perpendicular to the system which has a period L_y∈ℤ in the y direction. We assume ϕ is rational P/Q, where P and Q are mutually prime integers, then the Hall conductance σ_xy is quantized and there are Q energy bands. To consider energy bands, we prefer to work on a generic tight-binding Hamiltonian H=∑_m,n(c_m+1,n^† c_m,ne^iθ^x_m,n+c_m,n+1^† c_m,ne^iθ^y_m,n+h.c.), where c_m,n (c_m,n^†) is the annihilation (creation) operator at (m,n) site. If we choose the Landau gauge (θ^x_m,n,θ^y_m,n)=(0,2π mϕ), the Shcrödinger equation becomes Ψ_m+1,n+Ψ_m-1,n+e^i2π mϕΨ_m,n+1+e^-i2π mϕΨ_m,n-1=EΨ_m,n.We write Ψ_m,n=e^ik_ynψ_m(k_y) (0≤ k_y≤ 2π) since the system is periodic in the y direction. Under the assumption that L_y is sufficiently large, the wave number k_y=2π l/L_y (l∈ℤ) is usually regarded as a continuous parameter. Moreover the system has a period Q in the x direction because of the rational flux ϕ=P/Q, therefore Bloch's theorem allows us to write the wave function as ψ_m(k_y)=e^i2π mk_xu_m(k_x, k_y) (0≤ k_x≤ 2π/Q), where u_m is periodic u_m+Q=u_m. In view of the wavenumber space or the Brillouin zone, the system has two periods 2π/Q and 2π in the k_x and k_y directions respectively, hence we identify the BZ with a torus T_BZ^2 by gluing its boundary.There are Q-energy bands[One should be careful not to confuse the band and site indexes. Their total numbers are the same.] and each of them is a U(1)-bundle on T^2_BZ.A U(1)-connection for the j-th bundle is given by the Berry connection A^j=-i∑_m=1^Q(u_m^j^†∂_k_xu_m^jdk_x+u_m^j^†∂_k_yu_m^jdk_y), where u_m^j is the Bloch function for the j-th energy band and normalized |u^j|^2=∑_m=1^Qu_m^j^† u_m^j=1. §.§ Hall ConductanceOne of the reasons for the quantized Hall conductance σ_xy can described by the Chern numbers of the fiber bundles. Let σ_xy^j be the Hall conductance of the j-th energy band. The well-known formula σ^j_xy=e^2/h∫_T^2_BZd^2k/2π(∂ A^j_y/∂ k_x-∂ A^j_x/∂ k_y) tells that the Hall conductance is given by the Chern number of the j-th U(1)-bundle ℒ^j. If the Berry connection a^j is holomorphic on entire T^2_BZ, then the Stokes theorem implies σ^j_xy=0. So for σ^j_xy being nontrivial, a^j must have a singular point on T^2_BZ. At such a point, the Bloch function u^j(k) vanishes and the Chern number corresponds to vorticity of the function <cit.>. If there exist many singular points, σ^j_xy is given by the total vorticity. This viewpoint is important for the Langlands correspondence, especially for the Hecke modifications of bundles.Let us generalize the statement above. Suppose the Fermi energy E_F lies in the M-th gap, and we write M states of bands below E_F by u=(u^1,⋯, u^M). This multiplet forms a U(M)-bundle ℰ over T^2_BZ, whose Berry connection is given by A=-iTr(u^† du). Then the Hall conductance σ_xy is given by the Chern number c(ℰ)=1/2π∫_T^2_BZTr(dA). The famous TKNN formula <cit.> tells that the total Hall conductance σ_xy is given by the sum of Chern numbers c(ℒ_j)=1/2π∫_T^2_BZ dA^j associated with all energy bands below E_F: σ_xy=e^2/h∑_j=1^Mc(ℒ_j). §.§ Geometric Langlands CorrespondenceThe geometric Langlands correspondence is a branch of the Langlands program. There are a lot of surveys and the conjecture is partly proven <cit.>.The GL_1=ℂ^× case[The compactification of GL_1=ℂ^×=ℂ∖{0} is U(1).] is the simplest and established. We will focus on this case for a while. A readable introduction is <cit.> whose part II will help us greatly. Let X be a compact Riemann surface. We consider a holonomy representation ρ:π_1(X)→ GL_1. A famous mathematical theorem[This is true for any smooth manifold M and any representation ρ:π_1(M)→ G, where G is an arbitrary Lie group.] guaranties a bijection between the set Loc_1(X) of isomorphism classes of flat GL_1-bundles on X and the set of conjugacy classes of the holomony representations. Hence, one can attach a flat connection for a given representation ρ. (This is the same trick we define a Wilson loop.) An element of Loc_1(X) is called a local system. Now we introduce another character of the geometric Langlands correspondence. We denote by Pic(X) the set of isomorphism classes of holomorphic line bundles on X, which classifies the line bundles by their 1st Chern classes:Pic(X)=_d=0{ℒ∈Pic(X):d=∫_Xc_1(ℒ)}. For a given x∈ X, we consider the map, called a Hecke operator (functor), h_x:Pic(X) →Pic(X)ℒ↦ℒ(x),where ℒ(x) is the line bundle whose sections are sections of ℒ which may vanish at x. Under h_x, the Chern number of ℒ jumps by 1 (c(ℒ(x))=c(ℒ)+1). One can consider a more general modification of ℒ to ℒ' at N-tuple of points (x_i),i=1,⋯, N so that c(ℒ')=c(ℒ)+N. What the geometric Langlands correspondence expects is that for a given flat GL_1-bundle ℰ on X, there exist a unique 𝒟-module ℱ_ℰ on Pic(X) associated with the modification h_x. This correspondence is proven by P. Deligne. The general conjecture of the Langlands correspondence for a Lie group G can be stated as follows.We denote by ^LG the Langlands dual group of G. If G=GL_1, then its dual is isomorphic to GL_1. The set Loc_^LG(X) of local systems is again identified with the set of conjugacy classes of representations ρ:π_1(X)→^LG. And Pic(X) is generalized to the moduli stack Bun_G(X) of principle G-bundles on X. So the geometric Langlands correspondence implies that for a given flat ^LG-bundle ℰ on X, there is a unique 𝒟-module ℱ_ℰ, called a Hecke eigensheaf, defined on Bun_G(X) associated with the Hecke modification. §.§ Hecke Eigensheaf, Landau Level, and Anderson Localization In this section we give a physical explanation about Hecke eigensheaves. The sections 4.3∼4.5 in <cit.> will be helpful for more information. For this purpose, we physically interpret a sheaf. We are interested in a sheaf 𝒮=(𝒮,π, B) whose fiber 𝒮_p=π^-1(p), p∈ B is a vector space. The dimension of fibers may differ at points. A standard example is the skyscraper sheaf 𝒪(x), which is a sheaf supported at a single point x∈ B. How will they come into play in our story? First of all, sections of our sheaf are wave functions and the base space is the Riemann (complex energy) surface <cit.>. In comparison with the Landau levels, our sheaf can be visualizable as follows. On the left side of the figure <ref>, the blank zones between the Landau levels show there are no eigenstates of the Hamiltonian, and the only states living in the Landau levels contribute to the quantum Hall effect. On the right side, the eigenstates forms bundles on the complex energy surface and the other states vanish. So the set of "complexified" Landau levels can be recognized as a sheaf ⋃_i𝒪(E_i). Now we are ready to explain Hecke eigensheavs. Note that each of the Landau levels in the figure <ref> is linear, therefore only one-dimensional momentum k_y is a good quantum number. So one can define the only one-dimensional Berry connection A_k_y. Hence we may regard it as a flat connection[This flat connection, which is a Berry connection of U(1)-bundle, is a different one we used for a Wilson loop W: π_1(T^2_BZ)→^LU(1).] on T^2_BZ, by setting A_k_x=0. A Hecke eigensheaf on Pic^0(X)={ℒ∈Pic(X):0=∫_Xℒ} is a 𝒟-module of such flat connections. Of course this is not the whole story. Actual energy bands are "wavy" as shown with pictures in <cit.>, and the wavy parts possess nontrivial Chern numbers. This is the mechanism of Hecke modifications ℒ→ℒ(x)=ℒ⊗𝒪(x), x∈ T^2_BZ. Moreover the existence of plateaus in the figure <ref> can be described by Hecke eigensheaves as shown in the figure below. If there exist impurity potentials in the system, wave functions localize around the potentials (in the real space). This phenomenon is called the Anderson localization <cit.>. As a result, each of the Landau levels becomes wide. However, the localized wave functions do not carry non trivial Chern numbers and only the extended wave function living in the original Landau level contributes to the Hall conductance <cit.>. This is why the Hall conductance has plateaus. In the language of the geometric Langlands correspondence, this can be explained by saying that the Berry connections associated with those localized wave functions are flat, and hence they form a 𝒟-module.§.§ Duality, Hecke operator, and K-theoretic viewAlgebraically, a vortex operator for G is defined by a homomorphism ϱ:U(1)→ G, which is classified by highest weights of the dual group ^LG up to conjugation.We write it in the most general way as ϱ:e^iα→diag(e^im_1α,⋯,e^im_Mα), where ^Lw=(m_1,⋯,m_M) is an M-plet of integers with m_1≥⋯≥ m_M, which is a highest weight of ^LG=U(M). As we have already seen, we obtain decomposition of the U(M)-bundle ℰ into the sum of line bundles ⊕_i=1^Mℒ_i. Let k_i be a singular point of ℒ_i. The vortex operator V(^Lw) acts on ℒ_j as V(^Lw):ℒ_i→ℒ_j⊗𝒪(k_j)^m_j, where m_j is vorticity at k_j. In other words, it changes the Chern number c(ℒ_j) by m_j. The total Hall conductance σ_xy is the total Chern numbers of this system σ_xy=∑_j=1^Mσ^j_xy, which is the total vorticity of the system, in other words. By the way, the classification of vortex operators or Hecke operators is exactly the same as that of effective Hamiltonians. So far we have neglected contribution from conduction bands, and from now we suppose there are N conduction bands and M valence bands. So this system has U(M+N) gauge group in general. Then effective Hamiltonians of the quantum Hall system is classified by the Grassmannian Gr_M, M+N=U(M+N)/U(M)× U(N) from the K-theoretic perspectives <cit.>.In terms of the geometric Langlands correspondence, the Hecke operators are classified as follows. Let ℳ,ℳ'∈Bun_G be principle G=U(M+N) bundles on a Riemann surface X such that ℳ⊂ℳ' and ℳ'/ℳ≃𝒪(x)^M, where x∈ X. As discussed, the Hecke operators modify the G-bundle ℳ to ℳ' at this singular point x, and it is known the space of such modifications is parametrized by points in Gr_M,M+N. (One may discover extra value in mathematical explanations <cit.> or in physical explanations <cit.>). §.§ Chern-Simons Theory, S-duality and Mirror SymmetryQuantum Hall Effect has common description based on Chern-Simons theory, therefore it is meaningful to give some comments on the relation with the Langlands duality. We consider 2+1-dimensional system which is parametrized by x=(x^0,x^1,x^2), where x^0 stands for the time-direction and x^1,x^2 represent the space-directions. We may regard our system is product of ℝ and a torus T^2 since our physics on the two dimensional space we have considered so far is periodic in the x and y directions respectively. Let A be the background gauge filed of electromagnetism. The integer Hall conductance σ_xy=k/2π is described by the Chern-Simons action S_CS=∫ d^3xk/4πϵ^μνρA_μ∂_ν A_ρ,whose U(1)-current is J^μ=∂ S/∂ A_μ=kϵ^μνρ∂_ν A_ρ.Especially this is nothing but the Hall current if one takes μ=x^1, and the level k corresponds to the bulk Hall conductance. The duality in quantum Hall effect that acts on the filling fraction ν can be understood as S-duality that acts on the inverse level ħ = 1/k, which needs to be analytically continued away from integer values in order for the quality to be meaningful. When level k is integer, the Langlands/S-duality formally maps q = exp (2 π i k) to ^Lq =exp (2 π i / k). Continuing q away from roots of unity is naturally accommodated in the complex Chern-Simons theory, which now indeed enjoys Langlands/S-duality <cit.>. This duality describes the particle vortex duality <cit.> and the geometric Langladns correspondence is easily understood. As we discussed in section <ref>, the geometric Langlands correspondence states the duality between a Wilson loop and a Hecke operator. The Hecke operator corresponds to the vortex operator, which picks up the Chern number associated with the bulk Hall conductivity. To find the corresponding Wilson loop, we consider an anyon, which is a quasi-particle with magnetic flux.The Landau level filling factor is defined by ν=The number of electrons in the system/The number of flux quanta passing through the system.So ν is equivalent to the ratio of the number of electrons n_e and flux quanta ϕ per placket. If ν=1/k there are k flux quanta per electron. We may simply regard it as ϕ=k and n_e=1. The integer quantum Hall effect ν=k case can be regarded as (ϕ,n_e)=(1,k) or (ϕ,n_e)=(1/k, 1). If one prefers the former perspective, the duality (ν, ϕ, n_e)→(1/ν,n_e, ϕ) is similar to electric-magnetic duality as we see below soon. When one chooses the latter, the duality ν→1/ν along ϕ→ 1/ϕ makes sense. This perspective is crucial for the duality in Hofstadter's butterfly as we discuss latter. We first investigate the case where ν is an integer (the Integer Hall effect) and will treat the fractional case latter. The modular group SL(2,ℤ) acts on the complex Hall conductivity σ=σ_xx+iσ_xy in such a way thatS :σ→-1/σT :σ→σ+1 So, with respect to q=exp(2πσ) and ^Lq=exp(-2π/σ), the particle-vortex duality at σ_xx=0 in the integer quantum Hall system (σ_xy=ν=k) simply reads to q→^Lq. Especially, on the plateau regions, where gauge connection is flat, the equation σ_xx=0 holds <cit.> and the S-duality agrees with the Langlands duality of Wilson and Hecke operators. Let ν=k, which means an unit flux ϕ_0 is attached to an electron. The corresponding Wilson loop appears as an Aharonov-Bohm (AB) phase. Namely, the vector filed α which generates imaginary and negligibly thin magnetic flux attached to each electron should satisfy ∇×α=0, however an electron moving around ϕ_0 feels the vector potential and gains the phase exp(i2πϕ_0). From this picture, another way to understand aHecke operator, which is dual to the Wilson loop, is obtained as follows: the flux ϕ_0 moving around n_e=k electrons picking up the dual AB phase exp(i2π kϕ_0). This corresponds to the Hecke operator we discussed before as the vortex operator which pics up the Chern number associated with the Berry curvature. When we consider the vorticity, the location of the vortex depends on our gauge choice, however the former perspective of the AB-phase allows us to look the system in a gauge invariant way. By the way, this S-duality can be understood as so-called the particle-vortex duality and is consistent with the physical understanding of the geometric Langlands correspondence as electric-magnetic duality. By S-duality ν→1/ν, the Wilson loop and the Hecke operators are exchanged and the Wilson loop has the phase exp(i2π kϕ_0). Therefore the geometric Langlands correspondence suggests the duality of the IQHE and the FQHE. The geometric Langlands correspondence is accompanied by the Lang- lands dual group ^LG of a given gauge group G. In our case G=U(1) and hence ^LG=U(1). The ν=1/k Laughlin state have an emergent U(1) gauge field a, which is a global symmetry and dual to the background U(1) gauge filed A. We begin with the Lagrangian ℒ[a]= -k/4πϵ^μνρa_μ∂_ν a_ρThis Lagrangian can be generally written as ℒ_CS=∑_i,jk_i,j/4πA_i∧ dA_j,k_i,j is known as the K-matrix. The S-duality is achieved by adding an off-diagonal part 1/2πϵ^μνρA_μ∂_ν a_ρ to the Lagrangian (<ref>)ℒ[a]→ℒ'[A]=ℒ[a]+1/2πA∧ da.Integrating out a from ℒ'[A], we obtain ℒ[A]=-1/4π kϵ^μνρA_μ∂_ν A_ρ.Immediately, we can see the Hall conductance σ_xy is proportional to 1/k.So to understand the IQHE-FQHE duality, consider the actionwith Lagrange's multiplier γ associated with Bianchi's identity ϵ^μνρ∂_μ F_νρ=0S=∫ d^3x-1/4F_μνF^μν+1/2γϵ^μνρ∂_μ F_νρ,where F_μν=∂_μ A_ν-∂_ν A_μ. Suppose the boundary term banishes and integrate it with respect to F, we have ∂^μγ=1/2ϵ^μνρF_νρ. Substituting it into the original Lagrangian, we obtain S=∫ d^3x1/2ϵ^μνρ∂_μF_νρ=2π∫_S^21/2πF∈ 2πℤ,where S^2 is a two-sphere existing at the spacial infinity of ℝ^2,1. The form (<ref>) is nothing but the Chern number associated with the integer Hall conductance.The Langlands/S-duality is summarized as S: G=U(1) → ^LG=U(1) ℒ[a] →ℒ'[A]=ℒ[a]+1/2πA∧ a ν=k →ν=1/k Mirror symmetry and Chern-Simons theory is discussed in <cit.>, and here we leave a short summary which connects our perspective above. A central architecture in the study of Langlands/S-duality is Hitchin's moduli space ℳ(G,C), which is the classical phase space of (G) Chern-Simons theory on a 3-manifold with a boundary Riemann surface C. If one consider the complexification of G which we denote by G_ℂ (e.g. G=SU(2), G_ℂ=SL(2,ℂ)), then ℳ(G,C) is equivalent to the space ℳ(G_ℂ, C) of flat G_ℂ connections on C. The S-duality maps ℳ_flat(G_ℂ,C) to ℳ_flat(^LG_ℂ,C) and indeed they are a pair of mirror symmetry <cit.>. The particle-vortex duality can be described by the Landau-Ginzburg theory and, in supersymmetric situations, its analogue is referred to as mirror symmetry. Therefore, as long as we focus on the Langlands correspondence, the S-duality picture we have discussed in this work would be essentially same as the S(or T)-duality picture discussed in <cit.> as mirror symmetry. § HOFSTADTER'S BUTTERFLY AND LANGLANDS DUALITY OF QUANTUM GROUPSNow we see that the Langlnads/S-duality also enhance our understanding on a different aspect of the quantum Hall effect. As we discussed, ν is equivalent to the ratio of the number of electrons n_e and flux quanta ϕ per placket. If ν=1/k there are k flux quanta per electron. We identify it with ϕ=k and n_e=1. The integer quantum Hall effect ν=k case can be regarded as (ϕ,n_e)=(1/k, 1). The duality (ν,ϕ)→ (1/ν,1/ϕ) is crucial for the duality in Hofstadter's butterfly. On the q-parameter level, it states the duality between theories with q and ^Lq. As a well-known subject, the Hamiltonian (<ref>) of the integer quantum Hall effect can be written by use of the quantum group 𝒰_q(sl_2) <cit.> and the Langlands duality of the quantum group endows with novel perspective on its fractal energy spectrum structure, known as Hofstadter's butterfly <cit.>. It is known that this fractal spectrum is generated by the maps (ϕ,E) → (ϕ+1, E)(ϕ,E) → (1/ϕ, f(E)),where f is a some function. In <cit.>, this duality is described by a quantum geometric viewpoint and they relate the butterfly with the energy spectrum of relativistic Toda lattice. Tough the duality works for general ϕ, we consider rational ϕ=P/Q to consider the butterfly (P and Q are co-prime). Now, our q-parameter is q=exp(i2πϕ). The duality is understood by the formulaP_ϕ(E)=P_1/ϕ(E), E=f(E). A generic tight binding Hamiltonian we are interested in is H=∑_m,n(c_m+1,n^† c_m,ne^iA^x_m,n+R^2c_m,n+1^† c_m,ne^iA^y_m,n+h.c.), where c_m,n (c_m,n^†) is the annihilation (creation) operator at (m,n) site. When we choose the Landau gauge A^x_m,n=0, A^y_m,n=2π mϕ, it can be written asH=T_x+T_x^†+R^2(T_y+T^†_y),where we choose a Q-dimensional representation ρ_Q of 𝒰_q(sl_2)={K^±1,X^±} with q=e^i2π P/Q so thatT_x =e^ik_xρ_Q(X^+), T_y=e^ik_yρ_Q(K) ρ_Q(X^+) =[ 0 1 0 ⋯ 0; ⋮ ⋱ ⋱ ⋱; ⋮ ⋱ ⋱ 0; 0 ⋱ 1; 1 0 ⋯ ⋯ 0 ] , ρ_Q(K)=diag(q,q^2,⋯,q^Q)These operators T_x and T_y are non commutative because of the Aharonov-Bohm phase for an electron moving around the flux:T_xT_y=qT_yT_x. The energy spectrum consists of eigenvalues of this Hamiltonian, which is described by the Chambers relation <cit.>(H(k,R)-E)=P_ϕ(E,R)+h(k,R), k=(k_x,k_y)where P_ϕ(E,R) is a polynomial and h(k,R)=2(-1)^Q-1(cos(Qk_x)+R^2Qcos(Qk_y)). The energy spectrum displayed in Fig. <ref> satisfies the equation P_ϕ(E,R)=0 under the mid band point condition h(k_0,R)=0, where k_0=(π/2Q,π/2Q). Hence the anticipated formula P_P/Q(E,R)=P_Q/P(E,R) implies the equivalence of the Q-dimensional representation (<ref>) of 𝒰_q(sl_2) and the P-dimensional representation of 𝒰_^Lq(sl_2), where ^Lq=e^i2π/ϕ and 𝒰_^Lq(sl_2) is the Langlands dual quantum group of 𝒰_q(sl_2) <cit.>. We write this duality map by S:(𝒰_q(sl_2), H)→ (𝒰_^Lq(sl_2), H),where the dual Hamiltonian H is given by the following P× P matrix of the form H=T_x+T_x^†+R^2(T_y+T^†_y),where T_x=e^ik_xρ_P(X) and T_y=e^ik_yρ_P(Y). Since we expect the correspondence of the characteristic polynomials (H-E)=(H-E), we find R=R^1/ϕ by comparing order of R and R in h(k,R) and h(k,R). This is consistent with the Langlands duality of quantum groups explained by the interpolating quantum group 𝒰_q,t(sl_2) <cit.>, which is parametrized by arbitrary nonzero complex values q,t and generated by X^±,K^±1,K^±1 such thatKX^± =q^±2X^± K, KX^±=t^±2 X^±K,[X^+,X^-] =KK-(KK)^-1/qt-(qt)^-1.The interpolating property of 𝒰_q,t(sl_2) appears as 𝒰_q,1(sl_2)/{K=1}≃𝒰_q(sl_2), 𝒰_1,t(sl_2)/{K=1}≃𝒰_t(sl_2).By definition, 𝒰_q,t(sl_2) is equivalent to the usual quantum group 𝒰_ϱ(sl_2) with generators X^±, KK and the parameter ϱ=qt. Taking q=e^i2π P/Q and t= ^Lq=e^i2π Q/P, we find ϱ=q ^Lq=e^i2π(P/Q+Q/P) is symmetric under exchanging P and Q. The Langlands duality of quantum groups states that any irreducible representation of 𝒰_q(sl_2) would be t-deformed uniquely to a representation of 𝒰_q,t(sl_2) in such a way that its specialization at q=1 gives a representation of 𝒰_t(sl_2). The easiest case is P=1 and Q=2. A two-dimensional representation of 𝒰_q(sl_2) is dual to a one-dimensional representation of 𝒰_^Lq(sl_2), which is equivalent to P_1/2(E,R)=P_2/1(E,R). Generically, we observe that a Q-dimensional representation of 𝒰_q(sl_2) and a P-dimensional representation of 𝒰_^Lq(sl_2) are dual. This explain the formula P_P/Q(E,R)=P_Q/P(E,R). To connect ϕ and ν, we consider a two dimensional square system of size L× L and suppose there are N_e electrons. For the case ϕ=1/Q, there are Q energy bands and each of them contains L^2/Q stats since there are L^2 one-particle states in total. On the other hand, magnetic flux per placket is 1/Q and the total magnetic flux is N_ϕ=L^2/Q, which corresponds to the degeneracy of each of the Landau levels. By definition ν=N_e/N_ϕ=N_e/L^2/Q∝1/ϕ. So flipping ν→ 1/ν is essentially equivalent to mapping 1/ϕ→ϕ. This is gives a more direct connection to the geometric Langlands duality associated with the Chern-Simons theory and the Langlands duality of the quantum group.§ FRACTIONAL QUANTUM HALL EFFECT AND LANGLANDS PROGRAM§.§ Frobenius OperatorLet q be prime and z be an element of ℂ. The Frobenius operator is defined by Frob_q(z)=z^q. The Landau level filling factor ν=1/q means that q magnetic flux is attached to an electron. And the Frobenius operator comes into the FQHE as the Laughlin's wave function Ψ_ν=1/q=∏_i<j(z_i-z_j)^qexp(-1/4l^2∑_i=1^N_e|z_i|^2),where z_i stand for the coordinates of electrons.Namely, the factor q of Frob_q corresponds to 1/ν. The corresponding Wilson loop is given by the Aharonov-Bohm phase. §.§ Knot, Chern-Simons, and Topological Quantum ComputationKnots are objects in 3-space, however at first sight the Jones polynomial seems to be irrelevant to the background space. To interpret the Jones polynomial from three-dimensional geometry was a problem proposed by Atiyah very long ago, and it was solved by Witten <cit.>. Here we give a short introduction to this matter.We consider a principle bundle (E, π, ^LG, M) on an oriented 3-manifold M without boundary, where ^LG is the Langlands dual group of a compact simple Lie group G (we may take it to be simply connected if necessary). The Chern-Simons action is given byS_CS(A)=1/4π∫_MTr(A∧ A+2/3A∧ A∧ A),where A is a connection one-form (or a gauge field) on E. The partition function is defined by performing the Feynman path integral on the space 𝒰 of connections:Z_k(M)=1/vol∫_𝒰DAexp[ikS_SC(A)],where k∈ℤ is the level of the theory. A knot K in M enters our story if we consider a Wilson loop of A.We pic up an irreducible representation ^LR of ^LG and then the Wilson loop is given by W(K,^LR)=Tr_^LRPexp(-∫_KA). For simplicity we consider the case G=SU(2) and ^LR is its fundamental representation. Then the vacuum expectation value of the Wilson loop Z_k(M=S^3, K, ^LR)=1/vol∫_𝒰DAexp[ikS_SC(A)]W(K,^LR)corresponds to the Jones polynomial evaluated at q=exp(2π i/k+2). § FINALENow we conclude this article with some comments. We successfully understand the Langlands philosophy in terms of the integer quantum Hall system. Our discussions can be summarized in the following dictionary: Langlands Program ↔Quantum Hall effect Geometric Langlands ↔IQHE/FQHE Quantum groups' duality ↔Hofstadter's butterfly This article is the first penguin for exploring topological insulators from a viewpoint of the Langlands program, and doors for further adventures are always open. We naively expect that many similar phenomena observed or expected in generic topological insulators will be addressed in the same way as we discussed. Moreover the theory of Anderson's localization, which distinguishes metal-insulator transitions, is one of the essential and cross-cutting issues in topological insulators. The key ingredients are the topological terms (the Wess-Zumino-Witten (WZW) terms) associated with the non linear sigma models <cit.>. And the geometric Langlands correspondence manifests power for investigating the WZW model <cit.>. This suggests that mathematical background of topological insulators would be much more fruitful than what it had been believed.It is interesting to build another connection to the work done by Kapustin and Witten <cit.>, where 𝒩=4 super Yang-Mills theory is essential to explain the geometric Langlands correspondence via mirror symmetry and S-duality. We can seek for a likely scenario in string theoretical approaches to the quantum Hall effect (and topological insulators) <cit.>, in which the two-dimensional quantum Hall effect is described by using the D3-brane, on which the 𝒩=4 supper symmetry does live <cit.>. Moreover a generic topological insulator can be explained by the corresponding D-brane configuration, hence it may attract a general interest to build more strong connections among the geometric Langlands, topological insulators, and the supper symmetric theory. Finally, 3-dimensional understanding on Jones polynomials, which describes knots, is given by the Chern-Simons theory <cit.>. Categorufication of the Jones polynomial is known as Khovanov homology <cit.>. Its physical interpretation is also proposed by Gukov, Vafa, and Schwarz <cit.> in the context of topological string theory.More recently its connection with gauge theory has been considered<cit.> and it is predicted that the Langlands correspondence based on the 𝒩=4 super Yang-Mills theory plays a fundamental role to understand Khovanov homology. It is a challenging open problem to find a physically realistic explanation about the conjecture, since knot theory and Jones polynomial endow the FQHE with fundamental description of anyons.utphys | http://arxiv.org/abs/1708.00419v2 | {
"authors": [
"Kazuki Ikeda"
],
"categories": [
"cond-mat.mes-hall",
"hep-th",
"math-ph",
"math.MP"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170727154527",
"title": "Quantum Hall Effect and Langlands Program"
} |
Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USADriven-dissipative systems define a broad class of non-equilibrium systems where an external drive (e.g. laser) competes with a dissipative environment.The steady state of dynamics is generically distinct from a thermal state characteristic of equilibrium. As a representative example, a driven-dissipative system with a continuous symmetry is generically disordered in two dimensions in contrast with the well-known algebraic order in equilibrium XY phases. In this paper, we study a 2D driven-dissipative model of weakly interacting bosons with a continuous U(1) symmetry. Our aim is two-fold: First, we show that an effectively equilibrium XY phase emerges despite the driven nature of the model, and that it is protected by a natural ℤ_2 symmetry of the dynamics. Second, we argue that this phase is unstable against symmetry-breaking perturbations as well as static disorder, whose mechanism in most cases has no analog in equilibrium. In the language of renormalization group theory, we find that, outside equilibrium, there are more relevant directions away from the XY phase. Fragile fate of driven-dissipative XY phase in two dimensions Mohammad F. Maghrebi December 30, 2023 ============================================================= § INTRODUCTION A time-dependent drive continuously pumps energy into a driven system, and eventually heats it up to infinite temperature. On the other hand, a driven system coupled to a dissipative bath approaches a nontrivial non-equilibrium steady state due the competition between dissipation and external drive. In many-body driven-dissipative systems, the steady state of dynamics may exhibit new, and inherently nonequilibrium, phases. The latter, however, pose a fundamental challenge to our understanding of phases of matter. Non-equilibrium systems are, almost by definition, less constrained than their equilibrium counterparts. This implies that, away from equilibrium, dynamics and fluctuations can explore a larger “phase space”. It is then natural to expect non-equilibrium phases that are not accessible in equilibrium. The converse of this statement could also be true in the sense that a generic equilibrium phase may be non-generic far from equilibrium. A representative example is a driven-dissipative model with U(1) symmetry in low dimensions. This model is particularly relevant to driven-dissipative condensates consisting of exciton polaritons in semiconductor quantum wells <cit.>. It has been argued that such driven-dissipative Bose systems in two dimensions cannot exhibit algebraic order, characteristic of the equilibrium XY model, unless they are strongly anisotropic <cit.>. This is partly due to the emergence of the Kardar-Parisi-Zhang (KPZ) equation that describes a broad range of driven classical phenomena <cit.>. This manuscript makes a case for the emergence of the XY phase in driven-dissipative systems on the basis of symmetry. We present a case study of a 2D driven-dissipative bosonic model with U(1) symmetry which nevertheless gives rise to an XY phase. We further argue that this is due to the underlying symmetries of the model including an additional ℤ_2 symmetry.Despite the emergence of the XY phase, the model is shown to be generically unstable to symmetry-breaking perturbations as well as static disorder. We shall argue that, while U(1)-symmetry breaking perturbations find a description similar to those in equilibrium, perturbations of the ℤ_2 symmetry as well as static disorder are of a genuinely non-equilibrium nature (see Fig. <ref>).The structure of this paper is as follows. In Sec. <ref>, we introduce the 2D driven-dissipative model of weakly interacting bosons, and argue on the basis of the Keldysh functional integral that an effectively classical equilibrium XY phase emerges. In Sec. <ref>, we undertake a detailed study of the symmetries of the model and the way they constrain the emergent thermodynamic phase. We further discuss perturbations away from symmetries as well as static disorder. Finally, in Sec. <ref>, we summarize our results and discuss future directions.§ MODELWe consider a driven-dissipative model of weakly interacting bosons on a square lattice in two dimensions. This model is inspired by the spin model introduced in Ref. <cit.> and its subsequent treatment in Ref. <cit.> where spins were mapped to bosons. To define the model, we start from the quantum master equation∂_t ρ =-i [Ĥ, ρ] +∑_j (L̂_j ρL̂_j^† - 1/2L̂_j^†L̂_j ρ- 1/2ρL̂_j^†L̂_j ).The first term on the right-hand side gives the usual coherent evolution via the Hamiltonian Ĥ. The dissipation is subsumed in the second term characterized by the Lindblad operators L_js that describe the incoherent processes. We take the Hamiltonian asĤ= J∑_⟨ i j⟩(â_i â_j+â_i^†â_j^†) + U∑_j â_j^†â_j^†â_j â_j.The first term in the Hamiltonian describes anomalous hopping between nearest neighbors, while the second term is the on-site interaction.[For notational convenience, J is defined two times that of Refs. <cit.>.] We can also consider a “chemical-potential” term ∼∑_j a_j^† a_j in the Hamiltonian; the latter, however, does not alter our main conclusions, and will be discussed from the point of view of symmetry in Sec. <ref>. Furthermore, we consider weakly interacting bosons where the the interaction (U) can be treated perturbatively. Finally, the incoherent dynamics is given by a single-particle lossL̂_j= √(Γ) â_j .The Hamiltonian and the Lindblad operators should be understood in a rotating frame—determined by the frequency of the external drive—after making the rotating wave approximation. The latter is an excellent approximation provided that the drive frequency is much larger compared to other energy scales. We shall not provide a microscopic time-dependent model[The underlying time-dependent model is not unique, and its explicit form is constrained by experimental feasibility rather than physical principles.]; however, we argue that the driven nature of the dynamics is inherent in the quantum master equation. To this end, note that there is a competition between the Hamiltonian and dissipative dynamics. While the dissipation via L_js favors a state with no particles, or a vacuum, the Hamiltonian produces pairs of particles out of the vacuum state. The competition between the two gives rise to a steady state at long times with a finite density of particles. This feature has no analog in equilibrium, and is the defining character of driven-dissipative models.An important property of the model introduced here is that it possesses a U(1) symmetry. To see this, let us consider the checkerboard sublattices A and B of the square lattice.The quantum master equation is invariant under the following staggered U(1) transformationâ_j∈ A→ e^iθâ_j∈ A,â_j∈ B→ e^-iθâ_j ∈ B,where bosons on the two sublattices are “rotated” in opposite directions. This is to ensure that the anomalous hopping in the Hamiltonian remains invariant; all the other terms in the master equation (including the Lindblad terms) are acting on a single site, and respect the symmetry as well. It is then natural to ask whether the continuous U(1) symmetry is broken in the steady state. A mean-field analysis would be a first step to this end (for the spin analog of this model, see Ref. <cit.>). However, mean-field-type treatments are at best incomplete since they ignore fluctuations that are crucial to finding the fate of ordered phases in low dimensions. Furthermore, in a nonequilibrium setting, there is even a larger phase space available to dynamics and fluctuations. Instead, we shall follow a field-theory treatment based on the Keldysh formalism. §.§ Overview of Keldysh formalism and previous resultsThe Keldysh formalism adapts the functional-integral techniques to density matrices where two time contours/branches represent the evolution of the bra and ket states in the density matrix.In transitioning to the functional integral, the operator â_j is mapped to the fields a_j,±(t) with the subscripts ± representing the two branches. The Keldysh functional integral gives a weighted sum (integral) over all configurations of a_j,±(t). The weight associated with each configuration is given by the Keldysh action S_K[a_j,±(t)], the form of which is directly determined from the quantum master equation (<ref>). In a coherent-state representation, the Keldysh action can be cast as <cit.>S_K=∫_t[∑_j (a_j,+^* i∂_t a_j,+ -a_j,-^* i∂_t a_j,-) -iL].L contains information about dynamics, and is given byL= -i(H_+-H_-) + ∑_j[ L_j,+L_j,-^* -1/2( L_j,+^*L_j,++L_j,-^*L_j,-)]where H_± as well as L_± contain fields on the ± contour only. Clearly, the first line of this equation captures the unitary dynamics, while the second line describes the dissipative dynamics. The particular form of various terms are determined by the simple rule that a term of the form ÔρÔ' in the quantum master equation translates to O_+ O'_- in the action <cit.>.It is often more convenient to work in the Keldysh basis defined as <cit.>a_j,cl=a_j,++ a_j,-/√(2), a_j,q=a_j,+- a_j,-/√(2).This basis is more convenient in separating out the mean value (represented by a_cl) from the fluctuations around it (due to both a_cl and a_q which may be nevertheless of different nature). Next, we provide a summary of the previous results obtained in the context of a spin model <cit.> to the extent that it is relevant to our discussion. Along the way, we also give an overview of the by-now standard techniques and methods. Motivated by the staggered U(1) symmetry (<ref>), we allow the order parameter to be different on the two sublattices, but assume that it is uniform within each sublattice. With this assumption, one can take the continuum limit of the lattice model. Following Ref. <cit.>, we define the bosonic operators on the two sublattices A and B in the continuum asâ_j∈ A⟶â(), â_j∈ B⟶b̂().(With a slight abuse of notation, we have now used â() to denote the bosonic operators corresponding to the sublattice A.) The corresponding quantum and classical fields associated with the operators â() and b̂() should be identified as a_cl/q(t,) and b_cl/q(t,). Subsequently, the Keldysh action can be written as a functional of these fields. It was pointed out in Ref. <cit.> that one can make the transformationψ_cl/q(t, ) =∓[e^± iπ/4 b_cl/q(t, )+e^∓ iπ/4 a_cl/q^*(t,)],χ_cl/q(t,) =e^∓ iπ/4 b_cl/q(t,)+e^± iπ/4 a_cl/q^*(t,),to bring the Keldysh action into a more transparent form at or near the critical point to be further discussed below. This transformation casts the quadratic part of the Keldysh Lagrangian density (the integrand of the space-time integral in the action) asL_K^(2)= 1/2 {ψ_q^*[-∂_t+J∇^2 -r] ψ_cl + c.c. +iΓ |ψ_q|^2+ χ_q^*[-∂_t -R] χ_cl + c.c. +iΓ |χ_q|^2 },with the constantsr=Γ/2-4J,R= Γ/2+4J.Importantly, the constant r can be tuned to zero, or criticality, while R is always finite. Indeed we have used this fact to drop the gradient term acting on χ_cl at long wavelengths. It should be then clear that the critical behavior is captured by ψ_cl/q, while χ_cl/q are non-critical. At the quadratic level, the two fields are decoupled, and χ_cl/q can be simply dropped; however, interaction mixes the critical and non-critical fields together. We shall not reproduce the interaction terms in the new basis, and refer the interested reader to Ref. <cit.>. Integrating out χ_cl/q produces an effective interaction term of the formL_K^ int=-u/2(|ψ_cl|^2 ψ_clψ_q^*+ c.c.).The (real) coefficient u ∼ U^2 /Jis obtained via a second-order perturbation theory in the vicinity of the critical point r=0 or J= Γ/8. Of course, a perturbative treatment is justified in the limit of weak coupling U≪ J. We remark that there are various nonlinear terms generated in the second-order perturbation theory; however, a simple scaling analysis renders nonlinear terms with higher powers of the quantum field ψ_q irrelevant in the sense of renormalization group (RG) theory. A first step of perturbative RG is to determine scaling dimensions of the fields at the Gaussian fixed point corresponding to the quadratic part of the action.Demanding that the latter should be scale-invariant at the critical pointunder the transformation → b and t → b^2 t (relative scaling of space and time coordinates follows from the diffusive nature of the dynamics) requires ψ_cl→ b^0 ψ and ψ_q→ b^2 ψ_q in two dimensions. The corresponding scaling dimensions are then [ψ_cl]=0 and [ψ_q]=2. The difference in the scaling dimensions is a consequence of the fact that ψ_q is “gapped” in the sense that the action contains the term Γ |ψ_q|^2 whose coefficient, unlike that of rψ_q^* ψ_cl+ c.c., cannot be tuned to zero. The relevance of nonlinear terms at the Gaussian fixed point is determined by their RG flow; generally, terms containing higher powers of fields of a larger scaling dimension will be less relevant.Putting together the quadratic terms in the first line of Eq. (<ref>) with the interaction term in Eq. (<ref>), we find the effective Keldysh Lagrangian density obtained by integrating out the non-critical fields. Incidentally, the latter can be written asL_K^ eff=1/2{ψ_q^*[-∂_t ψ_cl-δ H^ eff/δψ_cl^*]+ c.c. +i T^ eff |ψ_q|^2},where T^ eff=Γ, and the functional H^ eff≡ H^ eff[ψ_cl] is given byH^ eff[ψ]=∫_ J|∇ψ|^2+r|ψ|^2 +u |ψ|^4.(A function(al) of ψ should be always interpreted as a function(al) of both ψ and ψ^*.)The latter has the same form as the Landau-Ginzburg free energy for a complex-valued field ψ. The description in terms of the effective action and Hamiltonian can be equivalently cast as a stochastic equation,∂_t ψ=-δ H^ eff/δψ^*+ξ(t,),where ξ represents a stochastic noise that is correlated as ⟨ξ(t,)ξ^*(t',')⟩= 2T^ effδ(t-t')δ(-'). Using standard techniques <cit.>, one can show that the asymptotic steady state of the effective dynamics in Eq. (<ref>) is given by the probability distribution (ψ_cl→ψ)P[ψ]∼exp(- H^ eff[ψ]/T^ eff),which is nothing but a thermal distribution function.Despite the nonequilibrium dynamics at the microscopic scale, at long wavelengths, the model effectively behaves as if it is in equilibrium. Of course, the effective Hamiltonian (free energy) and temperature are not in any direct way related to those at the microscopic level. §.§ Emergence of XY phaseAn effective classical and equilibrium behavior opens up the sophisticated toolbox of statistical mechanics. In the context of the model considered here, one can immediately draw intuition from the classical XY model in two dimensions. In particular, vortices should be properly taken into account. To this end, let us take ψ_0=⟨ψ_cl⟩, and define K≡ 2J|ψ_0|^2/Γ which is commonly known as spin stiffness. It is a classical result due to Kosterlitz and Thouless that a quasi-long-range order with algebraic decay of correlations emerges when <cit.>K>2/π.In the opposite regime where this constraint is not satisfied, vortices proliferate destroying the algebraic order and leading to an exponential decay of correlations.For the spin model introduced in Ref. <cit.>, it was shown that the constraint (<ref>) cannot be satisfied <cit.>, and the XY phase will not be realized.This is because ψ_0 represents the expectation value of a spin operator which may be saturated, and consequently K∼|ψ_0|^2 would not be large enough. In contrast, in our bosonic model, |ψ_0| can be arbitrarily large in favor of the constraint (<ref>). The saddle-point approximation of Eq. (<ref>) yields|ψ_0|^2≈|r|/2u,when r<0. Recalling that u∼ U^2/J, the constraint (<ref>) is easily satisfied in the weak-coupling regime U≪ J.We thus conclude that an XY phase is realized in the driven-dissipative model of weakly interacting bosons introduced here.Our perturbative treatment still lacks an important discussion. We have used second-order perturbation theory to show that the resulting action takes a form that finds a description in terms of an effective free energy. It is important, however, to show that this is not an artifact of our approximation, but rather is protected by the symmetries of the model. This is particularly important in the context of a 2D driven-dissipative model with U(1) symmetry where the XY phase is shown to be generically unstable to KPZ-like physics <cit.>. In the next section, we discuss the symmetries of the model, and argue that the XY phase is indeed protected by these symmetries. Furthermore, we show that relaxing these symmetries generically tends to destroy the XY phase.§ ROLE OF SYMMETRYInaddition to the U(1) symmetry (Eq. (<ref>)), the model defined in the previous section has a ℤ_2 symmetry under sublattice exchange A ↔ B.Hence, in a unit cell consisting of two sites (one from each sublattice), we have an enlarged ℤ_2× U(1) symmetry. In the continuum, the ℤ_2 symmetry interchanges the fields, â() ↔b̂(), which constitutes a fundamental symmetry of the model beyond any approximation or perturbation scheme. The latter directly translates to a transformation in terms of the fields, a_cl/q(t,) ↔ b_cl/q(t,), as a symmetry of the Keldysh action S_K[a_cl/q,b_cl/q]. In the basis of the fields ψ_cl/q and χ_cl/q defined in Eq. (<ref>), this symmetry simply reads as complex conjugation,ψ_cl/q(t,)↔ψ^*_cl/q(t,),χ_cl/q(t,)↔χ^*_cl/q(t,).Being a fundamental symmetry of the model, this transformation should also be a symmetry of the effective Keldysh action S^ eff_K[ψ_cl/q] obtained by integrating out the non-critical fields χ_cl/q,S^ eff_K[ψ_cl^*,ψ_q^*]= S^ eff_K[ψ_cl,ψ_q].This equation imposes a strong constraint on the form of the Keldysh action. To fully exploit it, we also note that a general Keldysh action S_K[a_cl, a_q] (with a_cl/q representing all the fields with quantum numbers suppressed) always satisfies S^*_K[a_cl, a_q]=- S_K[a_cl, -a_q]. This equation follows from the causal structure of the Keldysh action, and ensures that (G^R)^†=G^A and (G^K)^†=-G^K where G^R,A,K are retarded, advanced, and Keldysh Green's functions, respectively. It then follows that ( S_K[a_cl, a_q]) is odd in a_q while ( S_K[a_cl, a_q]) is even in a_q. For future reference, we specialize the causality structure to the effective Keldysh action,( S^ eff_K[ψ_cl, ψ_q])^*=- S^ eff_K[ψ_cl, -ψ_q]. Using the ℤ_2× U(1) symmetry, we next expand the Keldysh action in classical and quantum fields, and only keep spatial and time derivatives to the lowest order. With the exception of the classical field, this is well justified due to the corresponding scaling dimensions ([∂_t]=2, [∂_]=1, and [ψ_q]=2). However, a similar scaling argument fails for the classical field since [ψ_cl]=0 at the critical point. Moreover, we need to consider a parameter regime at a finite distance away from the critical point where Eq. (<ref>) is satisfied and the classical field assumes a finite value. We will present a more general argument below; for now, we simply expand the action in both classical and quantum fields as[The expansion starts at the linear order in quantum field since S_K[a_cl,a_q=0]=0 as a general property of the Keldysh action <cit.>.]S^ eff_K=∫_t, ψ_q^*[-Z∂_tψ_cl -J̃∇^2 ψ_cl-∂ V/∂ψ_cl^*]+ c.c.+i Γ̃|ψ_q|^2+⋯.V(ψ_cl) is a function of the modulus |ψ_cl|, and can be expanded as V(ψ)=r̃|ψ|^2+ũ |ψ|^4+⋯. Note that the action is at most quadratic in the quantum field, but possibly contains higher-order terms in the classical field. Also it is written in a way that explicitly respects the U(1) symmetry. Furthermore, the complex conjugation in the first line of the action is to ensure the reality of ψ_q-odd terms that follows from Eq. (<ref>). In general, the coefficients Z, J̃, r̃, ũ,⋯ can be complex-valued (Γ̃ has to be real on the basis of Eq. (<ref>)). However, the symmetry constraint in Eq. (<ref>) ensures that all the coefficients are real. Therefore, the action can be written in a form consistent with Eqs. (<ref>, <ref>) with a (real-valued) Hamiltonian, which is what we wanted to show.It is instructive to present a more general argument within the XY phase. Let us first represent the classical and quantum fields asψ_cl=√(ρ_0+π)e^iθ, ψ_q=ζ e^iθ,where ρ_0≡ |ψ_0|^2=⟨|ψ_cl|^2⟩ is the average density in the ordered phase, π characterizes density fluctuations, and ζ=ζ_1+iζ_2 is a complex field representing the quantum field. Here we have followed the notation in Ref. <cit.> in factoring out a common phase factor from both classical and quantum fields. The symmetry constraint (<ref>) in the new basis readsS_K^ eff[π, θ,ζ_1,ζ_2]= S_K^ eff[π, -θ,ζ_1,-ζ_2].Moreover, the U(1) symmetry requires the action to be invariant under θ→θ + const. With these constraints on the form of the Keldysh action together with Eq. (<ref>), we can write the most general Keldysh action as[Again, the expansion starts at the linear order in ζ representing the quantum field. A linear term in ζ_1 is also allowed, but nevertheless can be absorbed in a redefinition of π, or equivalently a renormalization of the density ρ_0.]S_K^ eff=∫_t, ζ_2(- Z'∂_t θ-J'∇^2 θ)+J”ζ_1 (∇θ)^2 -u'ζ_1 π+i Γ'ζ_1^2+iΓ”ζ_2^2+⋯,where the ellipses represent irrelevant terms that contain higher powers of spatial and time derivatives or the fields π, ζ_1 and ζ_2 due to the corresponding scaling dimensions ([π]=[ζ_1]=[ζ_2]=2). All the coefficients (Z',K',K”,u',Γ', Γ”) in the action are real as a consequence of the causality structure in Eq. (<ref>). We note that, unlike Eq. (<ref>), we have not made an expansion in powers of ψ_cl which in the ordered phase can be possibly large. For the special case of the action in Eq. (<ref>), we have Z'∼√(ρ_0) Z, K'∼ K”∼√(ρ_0)J̃, u'∼√(ρ_0)u, and Γ'=Γ”=Γ̃. Now note that the integration over π in Eq. (<ref>) gives a delta function that sets ζ_1=0, and the Keldysh action can be simply written asS_K^ eff=∫_t,ζ_2(- Z'∂_t θ-K'∇^2 θ)+iΓ”ζ_2^2,where irrelevant terms are simply dropped. It is then straightforward to see that the steady state is given as an effective thermal distribution with the partition function∫ Dθexp[-K/2∫(∇θ)^2],corresponding to the XY Hamiltonian with the spin stiffness K=K'Z'/Γ”.Having shown that a description in terms of the XY Hamiltonian is guaranteed by the symmetries of the model, we next turn our attention to symmetry-breaking perturbations. We shall see that the XY phase is generically unstable to such perturbations (see Fig. <ref>). §.§ Perturbing U(1) symmetryThere are a number of ways that U(1) symmetry can be explicitly broken. A representative example is nearest-neighbor hopping,ΔH= α∑_⟨ i j⟩â_i^†â_j+ H.c.Since neighboring sites belong to different sublattices, the U(1) symmetry in Eq. (<ref>) is explicitly broken. In the spin analog of Ref. <cit.>, this amounts to having J_x ≈ -J_y with a slightly different |J_x| and |J_y| of the corresponding XX and YY interactions. In the continuum, we have ΔH∝α∫_â^† () b̂()+ H.c.+⋯, and the corresponding term in the action reads Δ S∝α∫_t, a_q^* b_cl+a_cl^*b_q+ c.c.+⋯ with the ellipses indicating the less relevant terms. Writing the latter in the basis of ψ_cl/q and χ_cl/q and integrating out the non-critical fields χ_cl/q, we find, to leading order in U/J,Δ S_K^ eff∝α U/J∫_t, [ψ_q^*(ψ_cl^*+|ψ_cl|^2ψ_cl^* + ψ_cl^3)+ c.c.].All the terms reported here explicitly break the U(1) symmetry, while those that simply renormalize terms already present in the non-perturbed action as well as the less relevant terms are omitted. Also, we have not kept track of relative coefficients (of order 1) between different terms. The first term under the integral in the effective action (ψ_q^* ψ_c^*+ c.c.) can be cast as a correction to the effective Hamiltonian in Eq. (<ref>) asΔ H^eff_1∝α U/J∫_ψ^2 +(ψ^*)^2,with the subscript denoting the corresponding term. This expression is nothing but a perturbation of the XY model familiar in the context of statistical physics. In the ordered phase, ψ_cl≈√(ρ_0) e^iθ, the correction to the effective Hamiltonian becomes Δ H^eff_1∝αρ_0 U/J∫_cos (2θ).Generally, a cosine perturbation of the from cos(pθ) is irrelevant if K< p^2/(8π) <cit.>. However, the latter cannot be satisfied due to the condition (<ref>), and thus the cosine perturbation grows under RG pinning the value of θ. The XY phase and its characteristic algebraic order will be then destroyed at long wavelengths.On the other hand, the second and the third terms under the integral in Eq. (<ref>) cannot be derived from a Hamiltonian.[A term in the Keldysh action of the form ψ_q^* f(ψ_cl) + c.c. can be cast as ψ_q^* δ H/δψ_cl^* + c.c.—in a fashion similar to Eq. (<ref>)—if the function f(ψ) satisfies the condition ∂ f/∂ψ=∂ f^*/∂ψ^*.]To treat these terms, we resort to the density-phase representation of Eq. (<ref>). In this representation, the U(1) perturbation leads to a correction to the action of the form (relative and overall coefficients are neglected)Δ S_K^ eff∝∫_t,ζ_1cos(2θ)+ζ_2sin (2θ).This action contains the most relevant perturbations that also respect the ℤ_2 symmetry (Eq. (<ref>)). Now the first term under the integral simply drops since the functional integration over π sets ζ_1=0. The second term perturbs the U(1) symmetry, but can be similarly cast as a correction to the effective Hamiltonian as in Eq. (<ref>).In short, a perturbation of the U(1) symmetry of the form considered in this section can be simply considered as a perturbation of the XY Hamiltonian, which, using standard techniques, can be shown to destroy the XY phase at long wavelengths. §.§ Perturbing ℤ_2 symmetryIn this section, we study the consequences of breaking the ℤ_2 sublattice symmetry. Naively, this symmetry can be broken by adding to the Hamiltonian a staggered chemical potential (with different chemical potentials on the two sublattices),μ_A∑_j∈ Aâ_j^†â_j+ μ_B ∑_j∈ Bâ_j^†â_j.However, one can remove the asymmetry by exploiting a gauge transformation via the unitary operatorÛ(t)=exp[-iμ t(∑_j∈ Aâ_j^†â_j-∑_j∈ Bâ_j^†â_j)].In an appropriate rotating frame with μ=(μ_A-μ_B)/2, the above perturbation can be brought into a form with μ_A'=μ_B'. The latter satisfies all the symmetries (ℤ_2× U(1) as well as the translation symmetry) of the model, and thusonly slightly renormalizes the effective Hamiltonian. It is instructive to view this argument in the basis of the Keldysh action. The corresponding correction to the effective Keldysh action is given byΔ S_K^ eff = iμ_A-μ_B/4∫_t,(ψ_q^* ψ_cl- c.c.).Naively, this term breaks the ℤ_2 symmetry since it is not invariant under ψ_cl,q↔ψ_cl,q^*. However, the stochastic equation that follows from the Keldysh action reads (ψ_cl→ψ)∂_t ψ=-δ H_ eff/δψ^* +i μ_A-μ_B/2ψ+ξ(t,).Making the transformation ψ→ψ e^it(μ_A-μ_B)/2, we recover the unperturbed form of the stochastic equation that can be equivalently described by an effective Hamiltonian, consistent with the gauge transformation in Eq. (<ref>). To explicitly break the ℤ_2 sublattice symmetry, we take the decay rates to be slightly different on the two sublattices asL̂_j∈ A=√(Γ_A) â_j, L̂_j∈ B=√(Γ_B) â_j,with Γ_A/B= Γ±ΔΓ. We stress that any generic perturbation of the ℤ_2 symmetry should also lead to the same conclusions. Writing the corresponding Keldysh action and integrating out the non-critical fields, we find an effective Keldysh action, to the first nontrivial order in U/J, asΔ S_K^ eff∝iΔΓ U/J∫_t, (ψ_q^*ψ_cl+ψ_q^*|ψ_cl|^2ψ_cl - c.c.).(We have not kept track of relative coefficients.) The first term under the integral can be gauged away by going to a rotating frame similar to Eq. (<ref>); however, the second term cannot be dealt with in a similar fashion. Indeed adding the latter to the non-perturbed Keldysh action, we find a renormalized interaction term ∼ u_ renψ_q |ψ_cl|^2 ψ +c.c. with a complex-valued coefficient u_ ren=u'_ ren+iu”_ ren. As shown in Refs. <cit.>, this feature generically leads to the KPZ equation, and takes us outside the effective equilibrium description.The emergence of the KPZ equation can be generally argued on the basis of symmetry. In the absence of the ℤ_2 symmetry, many new terms are allowed in the Keldysh action. One such term is ζ_2 (∇θ)^2, a term that was previously disallowed due to the symmetry under the simultaneous transformation ζ_2→ -ζ_2 and θ→ -θ.[Other generated terms, ζ_1 ∇^2θ, ζ_2 π, and ζ_1 ζ_2, also lead to the same qualitative behavior.] The inclusion of the latter term in Eq. (<ref>) leads to the KPZ equation∂_tθ=J∇^2θ +λ(∇θ)^2+η(t,),where η represents a (real-valued) stochastic noise that is correlated as ⟨η(t,)η(t',')⟩= (Γ/√(ρ_0)) δ(t-t')δ(-'). The coefficient of the nonlinear term (λ) vanishes with the perturbation (∼ΔΓ) away from the ℤ_2 symmetry. Notice that this term cannot be derived from a Hamiltonian in a similar fashion as Eq. (<ref>). At the Gaussian fixed point (ignoring the compact nature of θ), the scaling dimension [θ]=0, and the new term in the KPZ equation is marginal <cit.>. To a higher order in perturbation theory, the latter term can be shown to be marginally relevant, and leads to a stretched-exponential decay of the correlation function and the destruction of the XY phase <cit.>. The interested reader is referred to Refs. <cit.> for more details on the emergence of the KPZ equation in driven-dissipative condensates. §.§ Random disorderIn this section, we consider the effect of disorder on the behavior of our model. In a disordered system, translation symmetry is broken at a microscopic level, which nevertheless is restored by an ensemble average over disorder configurations. A generic example is a disordered chemical potentialΔH=∑_j μ_jâ_j^†â_j,where μ_j on each site is a static random variable drawn from a Gaussian distribution. Unlike a staggered chemical potential (see the previous subsection), a disordered chemical potential cannot be gauged away. Carrying out the same steps of writing the perturbation in the continuum, and integrating out noncritical fields, we find a correction to the effective action, to the zeroth order in U/J, asΔ S_K^ eff =i/2∫_t,υ(ψ_q^* ψ_cl- c.c.),where υ≡υ() is correlated as ⟨υ()υ(')⟩=ϰδ(-') with ϰ the disorder strength. Disorder superficially breaks the ℤ_2 symmetry (Eq. (<ref>)); however, the integral over the Gaussian distribution restores this symmetry (in the same way that translation symmetry is restored in a disordered system). This can be more precisely formulated as a modified symmetry under the transformation in Eq. (<ref>) together with υ() → -υ(). It is more convenient to cast the above equation in the density-phase representation of Eq. (<ref>) to find Δ S_K^ eff =√(ρ_0)∫_t,ζ_2 υ. This term satisfies the ℤ_2 symmetry which, in this representation, is defined as the symmetry under θ→ -θ, ζ_2→-ζ_2, and υ→-υ as a close analog of Eq. (<ref>). Indeed this is the only relevant correction to the action that involves υ. This follows from the scaling dimension of static disorder, [υ]=1, determined from its Gaussian distribution.[Note that the disorder average can be performed at the level of the Keldysh functional integral since the partition function is normalized to Z=1 by construction. This allows us to directly compare the terms in the action to the disorder distribution <cit.>.] Putting all the relevant terms together in the Keldysh action after integrating out π and setting ζ_1=0, one finds S_K^ eff =∫_t,√(ρ_0)ζ_2(-∂_tθ+J∇^2θ+υ)+i Γ/2ζ_2^2.Importantly the term depending on υ cannot be cast as the functional derivative of a proper potential term (a naive guess υθ does not respect the gauge freedom θ→θ+ const.). It is instructive to write the corresponding Langevin equation,∂_t θ=J∇^2θ+υ()+η(t,).Note that η represents white noise, while υ denotes delta-function-correlated static disorder. Clearly, the latter cannot be gauged away by going to a rotating frame due to its spatial dependence. Furthermore, static disorder, being perfectly correlated in time, should be expected to dominate over white noise. This is indeed the case, and is easily seen on the basis of scaling analysis. A convenient way to see this is to obtain the disorder-averaged Keldysh action by integrating over υ,S_K^ eff = ∫_t,[√(ρ_0)ζ_2(-∂_tθ+J∇^2θ) + i Γ/2ζ_2^2] + iρ_0ϰ/2∫_t, t',ζ_2(t,)ζ_2(t',),where the double time integral in the last line runs from -∞ to +∞ for both t and t'. With [ζ_2]=2 at the XY fixed point described by the first line of this equation, a simple power-counting analysis reveals that ϰ grows under RG asdϰ/dl=2ϰ.Therefore, static disorder takes the system into a disordered phase at long wavelengths. We stress that the instability to static disorder discussed here is a purely non-equilibrium phenomenon. Alternatively, imagine that the effect of disorder could be absorbed in a correction to the effective Hamiltonian of the form Δ H^ eff∼∫_υ̃|ψ|^2 with υ̃() a static random potential; in fact, the disorder potential in Eq. (<ref>) produces such a correction at a higher order in U/J.Nevertheless, various terms generated from the disordered effective Hamiltonian can be shown to either vanish or become irrelevant in the sense of RG. Crucially, it is the non-equilibrium nature of the disorder potential that is responsible for the destruction of the XY phase.§ SUMMARY AND OUTLOOKIn this paper, we have considered a driven-dissipative model of weakly interacting bosons with U(1) symmetry in two dimensions. We have shown that an effectively classical equilibrium XY phase emerges as the steady state despite the driven nature of the model. The emergence of the XY phase has been argued on the basis of an additional ℤ_2 symmetry due to the sublattice exchange of the lattice model. Various perturbations of symmetry as well as static disorder have been considered, against which the XY phase is shown to be unstable. It is further argued that ℤ_2 symmetry-breaking perturbations as well as static disorder are genuinely of nonequilibrium nature, perturbing the XY phase in directions that are not accessible in equilibrium. More generally, nonequilibrium systems allow for new types of dynamics and fluctuations, which should be properly taken into account in order to determine the nature of phases and phase transitions in the thermodynamic limit. A natural question for future study is the fate of this model in the limit of strong coupling (large U). The perturbative arguments presented in this manuscript are not directly applicable in this limit. It is also interesting to consider other types of symmetry (O(n) symmetry, for example), and to compare and contrast the emergent behavior in and out of equilibrium on the basis of symmetry. It would be worthwhile to identify additional symmetries, if any, that constrain the corresponding driven-dissipative models to exhibit an effectively equilibrium behavior.§ ACKNOWLEDGEMENTSWe acknowledge Sergey V. Syzranov for a critical reading of the manuscript. The author acknowledges start-up funding from Michigan State University. 18 fxundefined [1]ifx#1fnum [1]#1firstoftwosecondoftwo fx [1]#1firstoftwosecondoftwonoop [0]secondoftworef[1]@startlink#1@href href[1]#1@endlink anitize@url [0]` 12`$12`&12`#12`1̂2`_12`%12 startlink[1] endlink[0]rl [1]href #1 @bib@innerbibempty[Kasprzak et al.(2006)Kasprzak, Richard, Kundermann, Baas, Jeambrun, Keeling, Marchetti, Szymańska, André, Staehli, Savona, Littlewood, Deveaud, andDang]Kasprzak06 author author J. Kasprzak, author M. Richard, author S. Kundermann, author A. Baas, author P. Jeambrun, author J. M. J. Keeling, author F. M. Marchetti, author M. H. Szymańska, author R. André, author J. L. Staehli, author V. Savona, author P. B. 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"authors": [
"Mohammad F. Maghrebi"
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"primary_category": "cond-mat.quant-gas",
"published": "20170726200001",
"title": "Fragile fate of driven-dissipative XY phase in two dimensions"
} |
equation*endequation* | http://arxiv.org/abs/1707.08884v2 | {
"authors": [
"Steven A Balbus"
],
"categories": [
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"published": "20170727143523",
"title": "The general relativistic thin disc evolution equation"
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[email protected] Department of Computer Science and Engineering, Indian Institute of Technology, KharagpurDepartment of Radiation Oncology, Tata Medical Center, Kolkata Department of Nuclear Medicine, Tata Medical Center, Kolkata Department of Medical Physics, Tata Medical Center, Kolkata[corra]Corresponding authorDeformable registration of spatiotemporal Cone-Beam Computed Tomography (CBCT) images taken sequentially during the radiation treatment course yields a deformation field for a pair of images. The Jacobian of this field at any voxel provides a measure of the expansion or contraction of a unit volume. We analyze the Jacobian at different sections of the tumor volumes obtained from delineation done by radiation oncologists for lung cancer patients. The delineations across the temporal sequence are compared post registration to compute tumor areas namely, unchanged (U), newly grown (G), and reduced (R) that have undergone changes. These three regions of the tumor are considered for statistical analysis. In addition, statistics of non-tumor (N) regions are taken into consideration. Sequential CBCT images of 29 patients were used in studying the distribution of Jacobian in these four different regions, along with a test set of 16 patients. Statistical tests performed over the dataset consisting of first three weeks of treatment suggest that, means of the Jacobian in the regions follow a particular order.Although, this observation is apparent when applied to the distribution over the whole population, it is found that the ordering deviates for many individual cases. We propose a hypothesis to classify patients who have had partial response (PR). Early prediction of the response was studied using only three weeks of data. The early prediction of response of treatment was supported by a Fisher's test with odds ratio of 5.13 and a p-value of 0.043.Deformable Image Registration Jacobian Cone-Beam Computed Tomography(CBCT) Statistical analysis § INTRODUCTIONLung cancer patients undergo a six week fractionated course of radiotherapy as part oftheir curative treatment. Radiation therapy planning requires identifying the tumor based on a pre-treatment planning computed tomography(CT) taken in treatment position followed by localization of the tumor by radiation oncologists. During the process nearby “organs at risk” are also identified to ensure that radiation induced damage is avoided. This localized volume is then planned to receive a specified (often 60Gy) dose of radiation in thirty fractions over a period of six weeks. Modern radiation therapy machines can acquire a cone beam CT scan(CBCT) image on a daily or more commonly, a weekly basis prior to delivery of the radiation to negate any positional changes in the identified tumor. Such image guided radiation therapy is often the standard of care for the curative therapy of Lung cancers. Adaptive Radiation Therapy (ART) requires oncologists to adapt the radiation volume based on spatio-temporal changes in the tumor, and has the potential to personalize radiation therapy based on such changes. Registration of such images therefore remain a key area of concern and could not only help identify the target accurately, but could also potentially prognosticate the outcomes in such patients.From a big dataset of radically treated lung cancer patients in a tertiary care center, we randomly selected their pre-treatment CBCT and on-treatment serial CBCT images. Weekly CBCTs of the patients are delineated by experienced radiologists for approval, and treatment. Each of the CBCTs are first delineated by an expert radiologist, followed by correction and approval of a second senior radiologist. Finally, the approved delineations are used by the radiation planning team to position the patient for radiotherapy.The work studies the statistical analysis of deformation fields obtained by registering pre-treatment CBCT and serial on-treatment CBCT images of lung cancer patients who underwent radiotherapy. Using only the first three weeks of treatment as data, we aim to predict the response of the patient to treatment before the therapy completes. To ensure clinical relevance, we used the actual target volumes delineated by the clinical oncologist on pre-treatment CBCT for treatment and thereafter on CBCT to assess the changes in the targets. The delineations were used to categorize the tumor into different regions. We study the Jacobian statistic on the deformation fields in these regions and study how it varies across patients. We observe that patients satisfying the proposed hypothesis have better response to treatment than those who do not satisfy the hypothesis. Using this hypothesis, we are able to classify patients into PR and non-PR categories, and support the hypothesis using Fisher's association test. §.§ Related work Computational Anatomy <cit.> is a study of anatomy using mathematical and computational models. The deformable template that exhibits diffeomorphism, a bijection between anatomical coordinates is sometimes used in such a study. In <cit.>, landmarks around the area of interest were used to track changes, where the Jacobian maps showed the direction of growth in ventral and dorsal parts of the hippocampus of a mouse. Deformation fields obtained after deformable image registration were earlier used in <cit.> for characterizing different regions of the lung, based on the motion properties like directional change, volume change and nature of change. In another study <cit.>, the deformation fields were used to track changes in tissue volumes. Different regions of the lung were segregated into blocks, and average Jacobian in these regions was observed to report lung activity and for tracking change in volume of tissues. In <cit.>, anatomical changes in the brain were observed to be different across spatial scales using the Helmholtz decomposition of the deformation field. A difference of Gaussians (DOG) operator was applied on the irrotational component of the decomposition to identify the areas of maximal volume change. However, this method was applied only to brain MR images.In another work <cit.>, an average template was generated using registration of brain MR images. Statistical analysis on the deformation tensors with respect to this average template was performed to find anomalies in brain. Longitudinal analysis of spatiotemporal data to measure the growth of hippocampus in the brain is reported in <cit.>. Here, differences between growth trajectories are used to estimate a mean growth behavior of a population. This is compared with subjects to identify delay in growth for children diagnosed with autism. Biological growth using deformation fields is mathematical modeled in <cit.>. The model is called a Growth by Random Iterated Diffeomorphisms (GRID). This model uses random seeds with radial deformations around it, to capture growth based deformations. In <cit.> and <cit.>, the growth variables in GRID model are directly captured from image data. All the aforementioned approaches use Jacobian for measuring growth and decay from images using nonlinear registration.§.§ ContributionThis work presents a statistical analysis of deformation fields obtained by registering CBCT images of lung cancer patients who underwent radiotherapy. In our work, we also make use of delineation of tumor in CBCT images by radiologists. This is the first work of its kind on a dataset annotated by radiologists, and studying the behavior across population. The delineations were used to categorize the tumor into different regions. We study the Jacobian statistic on the deformation fields in these regions, and study how it varies across patients. Patients satisfying our proposed hypothesis were found to have better response to treatment. The results of the classification suggest that early prediction is feasible. § METHOD §.§ Image RegistrationGiven two images S, T ∈ℝ^3, the goal of image registration is to find a transformation g⃗ : ℝ^3 ↦ℝ^3 that maps/aligns S onto T. The approach of computing both the forward (registering S to T) and reverse (registering T to S) transformations is termed as bidirectional symmetric registration. This process involves computation of inverse of the deformation field.A symmetric log-domain based nonlinear image registration technique was proposed in <cit.>. The deformation field obtained using this method is diffeomorphic and the true inverse can be computed efficiently at a very low cost. Hence this method is suitable for computational anatomy. This technique, however, is based on the sum of squared differences (SSD) in the intensities of the image. This technique is not quite applicable in our scenario, on account of the high amount of noise present in CBCT images. The noise in CBCT images is due to several artifacts arising from usage of low energy beams to produce the image. Further improvement in the results of the registration technique is reportedin <cit.>, where the symmetric Local Correlation Coefficient (LCC) provided a robust similarity measure. The technique was found to give smooth deformation fields owing to the regularization imposed on the total energy. We use this technique to perform non-linear image registration. The deformation field that warps S to T is denoted as ϕ⃗ = z⃗ - g⃗(z⃗), where z⃗ is the set of points in S that are mapped to corresponding points z⃗ - g⃗(z⃗) in T with displacement field g⃗(z⃗). Such a deformation ϕ⃗ yields proper alignment of the two images. Trilinear interpolation was used to compute the warping of deformation field at non-integer coordinates. The deformation field obtained using this method is diffeomorphic and the inverse can be computed efficiently at a very low cost. Moreover, this method guarantees invertibility of the deformation and is invertible. Hence this method is suitable for computational anatomy. §.§ Analysis of deformation fields The deformation index for interpreting the information in a deformation field is the determinant of Jacobian, commonly referred in the literature as simply Jacobian. The Jacobian of a deformation field ϕ is defined as:J(ϕ⃗(z⃗)) = [ ∂ϕ_1(z⃗)/∂ z_1 ∂ϕ_1(z⃗)/∂ z_2 ∂ϕ_1(z⃗)/∂ z_3 ; ∂ϕ_2(z⃗)/∂ z_1 ∂ϕ_2(z⃗)/∂ z_2 ∂ϕ_2(z⃗)/∂ z_3 ; ∂ϕ_3(z⃗)/∂ z_1 ∂ϕ_3(z⃗)/∂ z_2 ∂ϕ_3(z⃗)/∂ z_3 ] The determinant of J(ϕ⃗(z⃗)), denoted as |J(ϕ⃗(z⃗))| or in short J , can also be expressed in terms of its eigen values. The Jacobian (J) measures the local volume change with respect to that of a unit cube. This can be visualized in Fig. <ref>. J varies between 0 and ∞. When J = 1, it means there is no change in volume. A value of J < 1 denotes, net contraction; and J > 1 denotes, net expansion. We computed the Jacobian in different areas of the 3D image and calculated the variation in the Jacobian statistic. The categorization can be visualized in Fig. <ref>. The Jacobian is computed individually in the following regions: * Deformed Tumor Region in week i (T_i): Set of tumor voxels when the image is deformed.* Tumor Region in week i (T^'_i): Set of tumor voxels when the image is undeformed.* Deformed Non-Tumor Region in week i (N_i): Set of non-tumor voxels computed as negation of the set T_i.* Non-Tumor Region in week i (N^'_i): Set of non-tumor voxels computed as negation of the set T^'_i.* Unchanged Region in week i (U_i): The set of tumor voxels that is the intersection of T_i and T_i+1 i.e, U_i = T_i ∩ T^'_i+1.* Reduced Region in week i (R_i): The set of tumor voxels that is defined as, R_i = T_i ∖ T^'_i+1.* Newly grown Region in week i (G_i): The set of tumor voxels that is defined as, G_i = T^'_i+1∖ T_i. Applying the discussed model on real images, several slices are shown in Fig. <ref>. We study the deformation fields in these regions. §.§ Distribution of JacobianStatistical tests on the Jacobian are performed for the entire dataset on observing the distribution of J in regions of different categories as described in Section. <ref>. We assume normality of the dataset in all the regions due to the very large number of samples. The number of samples in R, G, U, and N regions are approximately, 1.8, 1.5, 3.5 and 169 million, respectively.The confidence intervals around the mean can be estimated as [x̅± 1.96 σ/√n], for a confidence of 95%. We also computed the confidence intervals for the different regions using bootstrap method. In bootstrapping, the samples are resampled to compute the confidence intervals. The bootstrap confidence intervals and the mean for a population of 29 patients are shown in the Table. <ref>. It can be observed that the intervals are very narrow. §.§ Two-sided t-testThe two-sided t-test was performed under the null hypothesis that two independent distributions have identical expected values. The t-test was done with an assumption that the two populations have unknown identical variances. The test gives a p-value that explains the level of significance achieved. The p-value gives the probability of achieving a result equal to or more extreme than what was observed, assuming the null hypothesis is true. A very low p-value leads to the rejection of null hypothesis. The p-value was found to be close to zero for all the pair of regions. The t-statistic reveals which of the two samples have higher or lower expected values.Samples were collected from the different regions of the tumor and the test was conducted on the population of 29 patients. Let us denote the mean(μ) and standard deviation(σ) in the regions with the respective subscripts, where T corresponds to tumor regions, N corresponds to non-tumor regions, U corresponds to unchanged regions, R corresponds to reduced regions, and G corresponds to newly grown regions, respectively. The t-statistic values are interpreted such that, when the alternative hypothesis is μ_X ≥μ_Y, then large positive values of t lead to rejection of the null hypothesis, where X and Y are defined as any of the four regions i.e, X,Y ∈R,U,G,N| X ≠ Y. Alternatively, when the t-statistic values are large and negative, then μ_X ≤μ_Y.Fig. <ref> shows the box-plot for the population in different regions. The t-statistic values are shown in Table. <ref>. Here, X corresponds to the first column and Y corresponds to the first row. It can be seen that μ_R ≤μ_G, μ_R ≤μ_U, and μ_G ≤μ_U, as the t-statistic values are large and negative. Similarly, μ_R ≥μ_N, μ_G ≥μ_N, and μ_U ≥μ_N, respectively, due to large positive values of the t-statistic.By arranging the t-statistic values on a real line, we observe that μ_N ≤μ_R ≤μ_G ≤μ_U for the population. These results corroborate with the intuition also, as the R region must be more aggressive than the N region. R region has recently reduced to non-tumor. Further, the G region has recently manifested as tumor, hence we expect it to be aggressive; however, lower than the U region which has currently not undergone any change. The above observation also is in conformation with the fact that G and U are relatively more aggressive than R and N regions.§.§ Ordering of Jacobian for individual patientsRadiology Response Evaluation Criteria in Solid Tumors(RECIST) <cit.> is used to assess the imaging of each patient and analyze response to radiation treatment. Patients who have undergone treatment are categorized according to this criteria according to their response. The response of a patient is checked by radiation oncologists typically three months after completion of radiotherapy. A follow up CT scan is performed to check for progression. Based on the tumor size and metastasis, the patient is classified into one the categories as follows: * Complete Response (CR): Disappearance of all target lesions.* Partial Response (PR): At least a 30% decrease in the sum of the longest diameter(LD) of target lesions, taking as reference the baseline sum LD.* Stable Disease (SD): Neither sufficient shrinkage to qualify for PR nor sufficient increase to qualify for PD, taking as reference the smallest sum LD since the treatment started (% change between 30% decrease and 20% increase).* Progressive Disease (PD): At least a 20% increase in the sum of the LD of target lesions, taking as reference the smallest sum LD recorded since the treatment started or the appearance of one or more new lesions.* Distant Progression (DP): The cancer is spreading from the original (primary) tumor in the prostate to lymph nodes or distant organs such as the bones, liver and lungs.§.§.§ Ordering hypothesisWe perform the t-test for each of the patients to record the ordering of the means in different regions. The hypothesis to classify each patient is explained in Algorithm. <ref>. The aim is to correctly classify partial response(PR) patients using the hypothesis. It is to be noted that, by definition, a CR qualifies as a PR but a PR does not qualify as a CR. Therefore, in our analysis PR category of classification refers to patients with response as either PR or CR, while the non-PR category refers to patients with response neither as PR nor as CR. We perform the t-test for each patient in the population and test set. The ordering from t-test is modified in the hypothesis for classifying patients into PR and non-PR categories. It is designed based on the following logic: By definition, we know that the expansion in G and U should be greater than that of R, because of more tumorous activity in these regions. Therefore, we hypothesize that μR≤{μ_G,μ_U}. This hypothesis does not consider the ordering between G and U. Additionally we add another condition based on the tumor activity in the R region. Intuitively, since R region should result in net reduction of the volume, the mean of Jacobian μ_R should be less than 1.0. It can be observed from Fig. <ref>, that the mean of patients for PR is less than 1.0, while that of NPR is close to 1.0.Fig. <ref> shows the block diagram of the classification process. Our data consists of the radiation response of each patient in terms of the RECIST criteria. Table. <ref> shows the corresponding radiation response(RX_Response) for each patient. Patients whose response to radiation could not be determined by doctors are labeled as “NA". Based on this hypothesis, we label each patient in the “Classification" column of Table. <ref> in <ref>. We compare the predicted and the actual response(RX_Response) to summarize into Table. <ref> and Table. <ref>, while excluding the patients whose response was “NA". Overall, we find that 12 out of 21 patients are correctly classified as PR, and 13 out of 17 are correctly classified as non-PR patients. Fisher's exact test for association between PR and the hypothesis was performed on the contingency table Table. <ref> and Table. <ref>, which yields an odds ratio(OR) and a p-value, to measure the association between two categories PR and non-PR. The null hypothesis is that the occurrence of PR and non-PR are equally likely. The odds ratio(OR) measures the ratio of odds of occurrence of an event to the odds of an event not occurring. The contingency table Table. <ref> corresponding to complete six weeks data yielded an OR and p-value of 4.33 and 0.051, respectively.Using only the first three weeks data, we find that 11 out of 21 patients are correctly classified as PR, and 14 out of 17 are correctly classified as non-PR patients. From Table. <ref>, for the first three weeks data, the obtained result for OR and p-value were 5.13 and 0.043, respectively. Assuming a critical p-value of 0.10 as threshold, we can reject the null hypothesis that there is similarity in occurrence between PR and non-PR categories. The OR of 5.13 represents that it is 5.13 times more likely that PR happens when the hypothesis is satisfied than non-PR when the hypothesis is satisfied. This analysis suggests that the hypothesis is satisfied for patients with response as PR during the first three weeks of the treatment also. This early prediction can help doctors take necessary actions for improving the response of the patient to treatment. The accuracy, precision and recall were found to nearly the same on the three weeks data as shown in Table. <ref>. The level of significance is close to being significant considering the full six weeks and first three weeks of the dataset.§ CONCLUSIONIn this work we discussed analysis of Jacobian obtained from the deformation fields of registering spatiotemporal data of CBCT images for patients who underwent radiotherapy. Using clinical radiation oncologists delineation over the first three weeks of treatment duration, we analyzed the behavior of Jacobian in four regions of each delineated tumor. We observed very narrow variation of the Jacobian in the population in each of the regions. Two-sided t-test was performed to identify the ordering of mean Jacobian for the population in the four regions as described above. Based on the ordering obtained from the population, we proposed a hypothesis for classification of each patient within the population and test data into PR and non-PR classes. This hypothesis was used to segregate patients into those satisfying the hypothesis and not satisfying the hypothesis. We observed that patients satisfying the proposed hypothesis had better RECIST response to radiation treatment. Significant association between the proposed hypothesis and better response was confirmed by using Fisher's test.This early prediction has significant impact in providing radiologists necessary feedback regarding the response of patient to the treatment. Early prediction before the stipulated radiation course of 6 weeks gives enough time for doctors to take alternate actions like increased radiation dose, or alternative treatment approaches like chemotherapy, surgery. Using this result, the images of new patients who have undergone treatment can be processed using the discussed work flow, to qualitatively predict the response of a treatment. Further analysis can be performed to predict quantitative measures that can indicate prognosis for the patient.§ ACKNOWLEDGMENTThis work is carried out under the MHRD sponsored project entitled as “Predicting Cancer Treatment outcomes of lung and colo-rectal cancer by modeling and analysis of anatomic and metabolic images". We would additionally like to thank, Partha Sen and Gaurav Goswami for helping us procure the data and validating the image registrations.§ REFERENCES § DETAILS OF THE DATASET | http://arxiv.org/abs/1707.08719v1 | {
"authors": [
"Bijju Kranthi Veduruparthi",
"Jayanta Mukherjee",
"Partha Pratim Das",
"Mandira Saha",
"Raj Kumar Shrimali",
"Sanjoy Chatterjee",
"Soumendranath Ray",
"Sriram Prasath"
],
"categories": [
"stat.AP"
],
"primary_category": "stat.AP",
"published": "20170727065127",
"title": "Analysis of Deformation Fields in Spatio-temporal CBCT images of lungs for radiotherapy patients"
} |
Why there is no an existence theorem for a convex polytope with prescribed directions and perimeters of the faces? Victor Alexandrov Originally submitted: July 25, 2017. Corrected: November 25, 2017 ================================================================================================================================== We choose some special unit vectorsn_1,…,n_5 in ℝ^3 and denote by ℒ⊂ℝ^5the set of all points (L_1,…,L_5)∈ℝ^5with the following property: there exists a compact convex polytope P⊂ℝ^3such that the vectors n_1,…,n_5(and no other vector) are unit outward normals to the faces ofP and the perimeter of the face with the outward normaln_k is equal to L_k for all k=1,…,5. Our main result reads that ℒ is nota locally-analytic set, i. e., we prove that, for some point(L_1,…,L_5)∈ℒ,it is not possible to find a neighborhood U⊂ℝ^5and an analytic set A⊂ℝ^5 such thatℒ∩ U=A∩ U.We interpret this result as an obstacle for finding anexistence theorem for a compact convex polytope with prescribeddirections and perimeters of the faces. Mathematics Subject Classification (2010): 52B10; 51M20. Key words:Euclidean space, convex polyhedron, perimeter of a face, analytic set. 1. Introduction and the statement of the main result. In 1897, Hermann Minkowski proved the following uniquenesstheorem: Theorem 1 (H. Minkowski, <cit.> and <cit.>). A convex polytope is uniquely determined, up to translations, by the directions and the areas of its faces.Here and below a convex polytope is the convex hull of a finitenumber of points. Bythe direction of a face, we mean the direction of theoutward normal to the face. Theorem 1 has numerous applications and generalizations.In order to discuss some of them, we will use the followingnotation.Let P be a compact convex polytope in ℝ^3 and n∈ℝ^3 be a unit vector.By P^n we denote the intersection of P and its support plane with the outward normal n.Note that P^n is either a vertex, or an edge,or a face of P. Accordingly, we say that P^nhas dimension 0, 1, or 2.In 1937, A.D. Alexandrov proved several generalizations ofthe aboveuniqueness theorem of Minkowski, including the followingTheorem 2 (A.D. Alexandrov, <cit.> and <cit.>).Let P_1 and P_2 be convex polytopes in ℝ^3. Then one of the following mutually exclusive possibilities realizes:(i) P_1 is obtained from P_2 by a parallel translation;(ii) there exist k=1,2 and a unit vectorn∈ℝ^3 such that P_k^n has dimension 2 and, for sometranslation T:ℝ^3→ℝ^3, the formulaT(P_j^n)⊊ P_k^nholds true, where j∈{1,2}{k}.Note that the formulaT(P_j^n)⊊ P_k^nmeans that the face P_j^n can beembedded inside the face P_k^n by translationT as a proper subset.For more details about Theorems 1 and 2, the reader is referred to <cit.>.For us, it is important that Theorem 1 is a special case of Theorem 2. In fact, suppose the conditions of Theorem 1 are fulfilled.Then, using the notation of Theorem 2, we observe that,for every unit vector n such that P_k^nhas dimension 2, P_j^n also has dimension 2and its area is equal to the area of P_k^n.Hence, there is no translation T such that T(P_j^n)⊊ P_k^n.This means that the possibility (ii) in Theorem 2 is not realized. Therefore, the possibility (i) in Theorem 2 is realized and Theorem 1 is a consequence of Theorem 2.In fact, Theorem 2 has many other consequences, including the following Theorem 3 (A.D. Alexandrov, <cit.>). A convex polytope in ℝ^3 is uniquelydetermined, up to translations, by the directions and the perimeters of its faces.For the sake of completeness, we mention that a direct analogof Theorem 1 is valid in ℝ^d for all d⩾ 4; a direct analogof Theorem 2 is not valid in ℝ^d for every d⩾ 4; in ℝ^3,a refinement of Theorem 2was found by G.Yu. Panina <cit.> in 2008.We explained above that uniqueness Theorems 1 and 3 are similarto each other and both follow from Theorem 2.In the rest part of this sectionwe explain the differencethat appears when we are interested in existence resultscorresponding to uniqueness Theorems 1 and 3.In 1897, Hermann Minkowski also proved the following existencetheorem: Theorem 4 (H. Minkowski, <cit.> and <cit.>). Let unit vectorsn_1,…,n_m in ℝ^3 and real numbers F_1,…,F_m satisfy the following conditions: (i) n_1,…,n_mare not coplanar and no two of them coincide with each other;(ii) F_k is positive for every k=1,…,m;(iii) ∑_k=1^m F_k n_k=0.Then there exists a convex polytopeP⊂ℝ^3 such thatn_1,…,n_m (and no other vector) are outward face normals for P and F_k is the area ofthe face with outward normal n_k for every k=1,…,m.For the sake of completeness, we mention that a directanalog of Theorem 4 is valid in ℝ^d for all d⩾ 4. For more details about Theorem 4, the reader is referredto <cit.>.Recall that a set A⊂ℝ^d is said to be algebraicif A={x∈ℝ^d: p(x)=0} for some polynomial p:ℝ^d→ℝ,and A is said to be locally-algebraic if, for every x∈ A,there is a neighborhood U⊂ℝ^d and an algebraic setA_0⊂ℝ^d such that A∩ U=A_0∩ U. For a given set {n_1,…,n_m}of vectors satisfying the condition (i) of Theorem 4,denote by ℱ(n_1,…,n_m) the set of all points (F_1,…,F_m)∈ℝ^msuch that there exists a convex polytope P⊂ℝ^3 for which n_1,…,n_m (and noother vector) are the outward face normals for P, and F_kis the area of the face with the outward normaln_k for every k=1,…,m. The set ℱ(n_1,…,n_m)⊂ℝ^m can be referred to as a natural configuration space of convexpolytopes (treated up to translations) with prescribed set{n_1,…,n_m}of outward unit normals when a polytope is determined by the areas F_1,…,F_m of its faces.From Theorem 4, it follows immediately thatthe set ℱ(n_1,…,n_m)⊂ℝ^m is locally-algebraic forevery set {n_1,…,n_m}of vectors satisfying the condition (i) of Theorem 4. In fact, we can define the algebraic set A_0 as the zero set of the quadratic polynomial ∑_j=1^3(∑_k=1^m(n_k,e_j) F_k)^2,where (n_k,e_j) stands for the standard scalar product in ℝ^3 and{e_1, e_2, e_3} is the standard orthonormal basis in ℝ^3.Recall that a set B⊂ℝ^d is said to be analyticif B={x∈ℝ^d: φ(x)=0} for somereal-analytic function φ:ℝ^d→ℝ,and B is said to be locally-analytic if, for every x∈ B,there is a neighborhood U⊂ℝ^d and an analyticset B_0⊂ℝ^d such that B∩ U=B_0∩ U.Obviously, every locally-algebraic set is locally analytic.Let unit vectors n_1,…,n_5 in ℝ^3 be defined by the formulasn_1=(0,0,-1),n_2= (1/√(2),0,1/√(2)),n_3=(-1/√(2),0,1/√(2)), n_4= (0,1/√(2),1/√(2)),n_5=(0,-1/√(2),1/√(2)).For convenience of the reader, the vectors n_1,…,n_5 are shown schematically in Figure 1.By ℒ=ℒ(n_1,…,n_5)⊂ℝ^5we denote the set of all points (L_1,…,L_5)∈ℝ^5with the following property: there exists a convex polytope P⊂ℝ^3 such thatthe vectors n_1,…,n_5 (and noother vector) are the unit outward normals to the faces of P,and L_k is the perimeter of the face with the outward normaln_k for every k=1,…,5. The set ℒ(n_1,…,n_5)⊂ℝ^5 can be referred to as a natural configuration space of convexpolytopes (treated up to translations) with prescribed set{n_1,…,n_5}of outward unit normals, when a polytope is determined by the perimeters L_1,…,L_5 of its faces.The main result of this article reads as follows:Theorem 5.Let the vectors n_1,…,n_5be given by the formulas (1). Then the setℒ(n_1,…,n_5)⊂ℝ^5is not locally-analytic.From our point of view, Theorem 5 explains why a generalexistence theorem is not known whichdetermines a convex polytope in ℝ^3 via unitnormals and perimeters of its faces.The reason is that no analytic condition, similar tothe condition (iii) in Theorem 4, does exist.2. Auxiliary constructions and preliminary results. Let P⊂ℝ^3 be a convex polytope such thatthe vectors n_1,…,n_5defined by the formulas (1) (and no other vector) are theunit outward normals to the faces of P. For k=1,…,5, denote by π_k the 2-dimensional plane in ℝ^3 containing the face of P with theoutward normal n_k.The straight lines π_1∩π_2 and π_1∩π_3 are parallel to the vector e_2=(0,1,0), and the straight lines π_1∩π_4 and π_1∩π_5 are parallel to the vector e_1=(1,0,0). Hence, the face P∩π_1 is a rectangle. Computing the angles between the vectors n_1 and n_k for k=2,…, 5, we conclude that the dihedral angle attached to any edge of the face P∩π_1is equal to π/4. Now it is clear that the polytope P can be of one of the three types schematically shown in Figure 2.In order to be more specific, we put by definition A=π_1∩π_2∩π_5, B=π_1∩π_2∩π_4, C=π_1∩π_3∩π_4, and D=π_1∩π_3∩π_5. Denote by 2x the length of the straight line segment AB, and by 2y the length of the straight line segment BC. We say that the polytope P is of Type I, if x<y; is of Type II, if x=y; and is of Type III, if x>y. Polytopes P of Types I–III are shown schematicallyin Figure 2. Denote byℒ_I(n_1,…,n_5)(respectively, byℒ_II(n_1,…,n_5) and ℒ_III(n_1,…,n_5)) the set of all points (L_1,…,L_5)∈ℝ^5 such that there exists a convex polytope P⊂ℝ^3of Type I (respectively, of Type II or Type III)such that the vectors n_1,…,n_5(and no other vector) are outward unit normals to the faces of P, and L_k is the perimeter of the face of P with the outward normal n_k for every k=1,…, 5. Below, we use also the following notationv_I = (2, -(3+2√(3)), -(3+2√(3)), 5, 5), v_II = (2(√(3)-1), 1,1,1,1), v_III = (2,5,5,-(3+2√(3)), -(3+2√(3))).Note that in this paper the notation AB can as well denote the straight line segment and its length.Lemma 1.Let the set {n_1,…,n_5}of unit vectors in ℝ^3 be defined by the formulas (1). Then the following three statements are equivalent to each other:(i) (L_1,…,L_5)∈ℒ_I(n_1,…,n_5);(ii) L_1=(2√(3)-3)L_2+L_4, L_2=L_3, L_4=L_5, L_4>L_2>0;(iii) (L_1,…,L_5)=αv_I +βv_II for some α, β∈ℝsuch that β>(3+2√(3))α>0.Proof : Suppose the statement (i) of Lemma 1 holds true. In addition to the notation introduced above in Section 2, letE=π_2∩π_4∩π_5, F=π_3∩π_4∩π_5, G be the base of the perpendicular dropped from E on theedge AB,H be the base of the perpendicular dropped from E on the face ABCD,and K be the base of the perpendicular dropped from E onthe edge BC, see Figure 3. Using this notation, we obtain easily BG = GH=EH=BK=x,BH = x√(2),BE = x√(3),EF = BC-2 BK=2y-2x,L_1 = 2AB+2BC=4x+4y,L_2 = L_3=AB+2BE=2(1+√(3))x,L_4 = L_5=BC+EF+2BE=2(√(3)-1)x+4y.Eliminating x and y from the last three formulas, we get L_1=(2√(3)-3)L_2+L_4. Since the relations L_2=L_3, L_4=L_5, and L_4>L_2>0 are obvious for every polytope P of Type I, we conclude that the statement (i) implies the statement (ii).Now suppose the statement (ii) of Lemma 1 holds true. We, first, find the numbers x and y that satisfy therelations (2)–(4) and, second, construct a convexpolyhedron P of Type I for which L_1,…, L_5 arethe perimeters of the faces.In accordance with (3), we put by definitionx=L_2(√(3)-1)/4. Since L_2>0, x>0. In accordance with (4), we put by definition y=L_2(√(3)-2)/4+L_4/4. Since L_1=(2√(3)-3)L_2+L_4, the condition (2)is satisfied: 4x+4y=(2√(3)-3)L_2+L_4=L_1. Using the inequalities L_4>L_2>0, we obtain 4y=L_4+(√(3)-2)L_2>L_2+(√(3)-2)L_2=(√(3)-1)L_2>0. Hence, there exists a rectangle ABCD in ℝ^3such that the edge AB is parallel to the vectore_2=(0,1,0) and its length is equal to 2x, and the edge BC is parallel to the vectore_1=(1,0,0) and its length is equal to 2y.Denote by π_1 the plane that contains ABCD.Obviously, π_1 is perpendicular to n_1. Denote by π_2 the plane perpendicular to n_2 and containing AB. Denote by π_3 the plane perpendicular to n_3 and containing CD. Denote by π_4 the plane perpendicular to n_4 and containing BC. And denote by π_5 the plane perpendicular to n_5 and containing AD. The five planes π_1,…,π_5 determine a compactconvex polyhedron with outward normalsn_1,…,n_5. Denote it by P.According to the statement (ii), L_4>L_2. Hence, 4y=L_4+(√(3)-2)L_2>L_2+(√(3)-2)L_2=(√(3)-1)L_2=4x. Thus, P∈ℒ_I and the statement (ii) implies the statement (i).So, we have proved that the statements (i) and (ii) are equivalent to each other.In order to prove that the statements (ii) and (iii) are equivalent to each other, we observe that the equationsL_1=(2√(3)-3)L_2+L_4, L_2=L_3, and L_4=L_5 from the statement (ii) define a 2-dimensional plane inℝ^5. Denote this plane by λ_I. Moreover, the vectors v_I and v_II constitute an orthogonal basisin λ_I. This means that every vector (L_1,…,L_5)∈λ_I can be uniquely written in the formαv_I+βv_II. Direct calculations show that the inequalities L_4>L_2>0 from the statement (ii) are equivalentto the inequalities β>(3+2√(3))α>0 from the statement (iii). □Lemma 2.Let the set {n_1,…,n_5}of unit vectors in ℝ^3 be defined by the formulas (1). Then the following three statements are equivalent to each other:(i) (L_1,…,L_5)∈ℒ_II(n_1,…,n_5);(ii) L_1=2(√(3)-1)L_2, L_2=L_3=L_4=L_5>0;(iii) (L_1,…,L_5)=γv_II for some γ∈ℝsuch that γ>0.Proof is left to the reader. It can be obtained by arguments similar to those used above in the proof of Lemma 1.But in fact, it is sufficient to observe that Lemma 2 is the limit case of Lemma 1 as L_4 approaches L_2.Lemma 3.Let the set {n_1,…,n_5}of unit vectors in ℝ^3 be defined by the formulas (1). Then the following three statements are equivalent to each other:(i) (L_1,…,L_5)∈ℒ_III(n_1,…,n_5);(ii) L_1=L_2+(2√(3)-3)L_4, L_2=L_3, L_4=L_5, L_2>L_4>0;(iii) (L_1,…,L_5)=δv_III +εv_II for some δ, ε∈ℝsuch that ε>(3+2√(3))δ>0.Proof is left to the reader. It can be obtained by arguments similar to those used above in the proof of Lemma 1.But in fact, it is sufficient to observe that if we rotatea polytope of Type III around the vectore_3=(0,0,1) to the angle π/2, we get a polytope of Type I and can apply Lemma 1 to it. In the proof of Lemma 1, we denoted by λ_Ithe 2-dimensional subspace in ℝ^5 which is spanned by the vectorsv_I and v_II. Now we denote by λ_IIthe 1-dimensional subspace in ℝ^5spanned by v_II and denote by λ_IIIthe 2-dimensional subspace spanned by v_II and v_III. Lemma 4.λ_II=λ_I∩λ_III.Proof : Each subspace λ_I, λ_II, and λ_III contains v_II. Hence, (λ_I∩λ_III)⩾ 1.On the other hand, λ_I=λ_III=2.Hence, (λ_I∩λ_III) is equal to either 1or 2. Suppose (λ_I∩λ_III)=2. Then λ_I=λ_III. Hence, the vectors v_I, v_II,and v_III are linearly dependant. But this is not the case because the 3×3 minorcomposed of the first, third and fifth columns of the matrix[ v_I;v_II; v_III ] = [2 -(3+2√(3)) -(3+2√(3))55;2(√(3)-1)1111;255 -(3+2√(3)) -(3+2√(3)) ]is non-zero. Hence, (λ_I∩λ_III)=1, and λ_II=λ_I∩λ_III. □3. Half-branches of analytic sets and the proof of Theorem 5.Let A be a one-dimensional analytic set, and x∈ A.For every sufficiently small open ball U with center x,A∩ (U{x}) has a finite number ofconnected components A_1,…,A_k such that x belongs to the closure of A_j for every j=1,…, k. These A_j are called the half-branches of Acentered at x. It is known that the number of half-branchesof a one-dimensional analytic set centered at a point is even,see, e. g. <cit.>.For completeness, we mention that a comprehensive exposition of a similar result foralgebraic sets of dimension 1 may be found in <cit.>.Proof of Theorem 5 : Let Λ be the straight line in ℝ^5 defined by the formula Λ={x∈ℝ^5| x= v_II+tv_Ifor some t∈ℝ}.Our proof is by contradiction. Suppose the setℒ(n_1,…,n_5)⊂ℝ^5is locally-analytic. Then Λ∩ℒ(n_1,…,n_5) is also locally-analytic. Moreover, it is one-dimensional, contains the point v_II, and has only one half-branch centered at v_II. Let us explain the last statements in more details. From the definition of polytopes of Types I–III we know thatℒ(n_1,…,n_5)= ℒ_I(n_1,…,n_5)∪ℒ_II(n_1,…,n_5)∪ℒ_III(n_1,…,n_5).From Lemma 1 we know thatℒ_I(n_1,…,n_5) is an angle on the 2-dimensional plane λ_I⊂ℝ^5. From Lemma 3 we know thatℒ_III(n_1,…,n_5) is an angle on the 2-dimensional plane λ_III⊂ℝ^5. These angles are glued together along the ray ℒ_II(n_1,…,n_5) (see Lemma 2), and no 2-dimensional plane contains the both of them (see Lemma 4).The line Λ lies in the plane λ_I and passes through the point v_II. Hence, for every sufficiently small open ballU⊂ℝ^5 with center v_II, U∩Λ∩ℒ(n_1,…,n_5)= {v_II}∪(U∩Λ∩ℒ_I(n_1,…,n_5)).This formula means that we may obtain U∩Λ∩ℒ(n_1,…,n_5) in the following way: first, we divide the straight line Λ into two rays by the point v_II; then we observe that only one of these rays has at least one common point with the angle ℒ_I(n_1,…,n_5)and select that ray; at last, we intersect the ray selected with U. From this description, it is clear thatU∩Λ∩ℒ(n_1,…,n_5) is the half-branch of the locally-analytic setℒ(n_1,…,n_5)⊂ℝ^5 centered at v_II.Moreover, this is the only half-branch centered at v_II.This contradicts to the fact that the number of half-branchesof a locally-analytic set centered at a point is even, see <cit.>. □Remark : The proof of Theorem 5 provides us with a new, more technical, answer to thequestion of the title of this article. As a part of the proof of Theorem 5, we demonstrated that the setℒ(n_1,…,n_5)⊂ℝ^5 is not convex. In Section 1, we mentioned that ℒ(n_1,…,n_5) can be considered as a natural configuration space of convexpolytopes (treated up to translations) with prescribedoutward unit normals and perimeters of its faces. The reader, familiar with the proof of Theorem 4 given in <cit.>,may remember that convexity of the analogous `natural configuration space' ℱ(n_1,…,n_m)⊂ℝ^m plays an important role in that proof.8Al37Alexandrov, A.D.: An elementary proof of the Minkowski and some other theorems on convex polyhedra (in Russian). Izv. Akad. Nauk SSSR, Ser. Mat. No. 4, 597–606 (1937).JFM 63.1234.02Al96Alexandrov, A.D.: Selected works. Part 1: Selected scientific papers. Gordon and Breach Publishers, Amsterdam (1996). MR1629804, Zbl 0960.01035Al05Alexandrov, A.D.: Convex polyhedra.Springer, Berlin (2005). MR2127379, Zbl 1067.52011BCR8Bochnak, J.; Coste, M.; Roy, M.-F.: Real algebraic geometry. Springer, Berlin (1998). MR1659509, Zbl 0912.14023Mi97Minkowski, H.: Allgemeine Lehrsätze über die convexen Polyeder.Gött. Nachr. 198–219 (1897). JFM 28.0427.01Mi11Minkowski, H.: Gesammelte Abhandlungen von Hermann Minkowski. Band I. Teubner, Leipzig (1911). JFM 42.0023.03Pa08Panina, G.: A.D. Alexandrov's uniqueness theorem for convex polytopes and its refinements. Beitr. Algebra Geom. 49, No. 1, 59–70 (2008). MR2410564, Zbl 1145.52007Su71Sullivan, D.: Combinatorial invariants of analytic spaces.Proc. Liverpool Singularities-Sympos. I,Dept. Pure Math. Univ. Liverpool 1969–1970, 165–168 (1971). MR0339241, Zbl 0227.32005 Victor AlexandrovSobolev Institute of MathematicsKoptyug ave., 4Novosibirsk, 630090, RussiaandDepartment of PhysicsNovosibirsk State UniversityPirogov str., 2Novosibirsk, 630090, Russiae-mail: [email protected] | http://arxiv.org/abs/1707.08288v2 | {
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"title": "Why there is no an existence theorem for a convex polytope with prescribed directions and perimeters of the faces?"
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http://arxiv.org/abs/1707.08830v1 | {
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thefnmarkfootnotetext fnsymbolroman [NO \title GIVEN] [NO \author GIVEN] December 30, 2023 ======================In this paper we use a recent version of the Ruelle-Perron-FrobeniusTheorem to compute, in terms of the maximal eigendata of the Ruelleoperator,the pressure derivative of translation invariant spin systems taking values on a general compact metric space.On this setting the absence of metastable states for continuous potentials on one-dimensional one-sided lattice is proved. We apply our results, to show that the pressureof an essentially one-dimensionalHeisenberg-type model, on the lattice ℕ×ℤ,is Fréchet differentiable, on a suitable Banach space. Additionally,exponential decay of the two-point function, for this model,is obtained for any positive temperature.Mathematics Subject Classification: 37D35. Keywords:Thermodynamic Formalism, pressure derivative, transfer operator.§ INTRODUCTIONThe Ruelle operatorwas introduced by David Ruelle in the seminalpaper<cit.>, in order toprovetheexistence and uniqueness of the Gibbs measuresfor some long-range Statistical Mechanics models in theone-dimensional lattice. Ever since the Ruelle operatorhas become a standard toolin a varietyof mathematical fields, for instance, in Dynamical Systems,and other branches of Mathematics and Mathematical Physics. The Ruelle operator was generalized in severaldirections and its generalizations are commonlycalled transfer operators.Transfer operatorsappear in IFS theory, Harmonic Analysis andC^*-algebras, see for instance<cit.> respectively.In Dynamical Systems the existenceof Markov Partitions allows one to conjugate uniform hyperbolic maps on compactdifferentiable manifolds with the shift map in the Bernoulli space. For more details see, for example, <cit.> and references therein. A field in which the Ruelle operator formalismhas also been proved useful isthe Multifractal Analysis. Bowen, in the seminal work <cit.>, has established a relationship between theHausdorff dimensionof certain fractal sets andtopological pressure,for more details see <cit.>and also the introductory texts <cit.>. The classical Thermodynamic Formalism wasoriginally developedin the Bernoulli space M^ℕ, with M being a finite alphabet, see <cit.>.The motivation to consider more general alphabets from thedynamical system point of view is given in <cit.>, where proposed models withinfinite alphabet M=ℕ are used todescribe somenon-uniformly hyperbolic maps,for instance, the Manneville-Pomeau maps. Unbounded alphabets as general standard Borel spaces, which includescompact and non-compact, are considered in details in <cit.>. In <cit.> the authors considered the alphabet M=S^1 and a Ruelle operator formalism is developed.Subsequently, in <cit.>, this formalism was extended to general compact metric alphabets. Those alphabets do not fit in the classical theory, since the number of preimagesunder the shift map may be not countable. To circumvent thisproblem the authors considered an a priori measure μ defined on M, and so a generalized Ruelle operator can be defined and a version of the Ruelle-Perron-Frobenius Theorem is proved.In this general setting concepts of entropy and pressure are also introduced.A variational principle is obtained in <cit.>. The authors also show that theirtheory can also be used to recovery some results of Thermodynamic Formalism for countablealphabets, by taking the one-point compactification of ℕ,and choosing a suitable a priori measure μ.In Classical Statistical Mechanics uncountable alphabetsshows up, for example, in the so-called O(n) models with n≥ 2. Theseare modelson a d-dimensional lattice for which a vectorin the (n-1)-dimensionalsphere is assignedto every lattice siteand the vectorsatadjacent sites interactferromagneticallyvia their inner product. More specifically, let n≥ 1 be an integer and let G=(V(G), E(G)) a finite graph.A configurationof the Spin O(n) model on G is anassignment σ:V(G)→ S^n-1,we denote by Ω:=(S^n-1)^V(G)thespace of configurations. Atinverse temperature β∈ (0, ∞),configurations are randomly chosen fromthe probability measure μ_G,n, βgiven by dμ_G,n, β(σ):=1Z_G, n, βexp ( β∑_u,v ∈ E(G)σ_u·σ_v)dσ,where σ_u·σ_v denotes the inner productin ℝ^n of the assignments σ_u and σ_v,Z_G, n, β=∫_Ωexp(β∑_u,v ∈ E(G)σ_u·σ_v)dσ and dσ is the uniform probability measureon Ω.Special cases of these models have names of their own:when n=0 this model is the Self Avoiding Walking (SAW); when n=1,this model isthe Ising model, when n=2,the model iscalled the XY model; finally for n=3 this model is calledthe Heisenberg model, see <cit.> for details. In <cit.> the authors generalize the previousRuelle-Perron-Frobenius Theorem for a more general class of potentials,satisfying what the authors called weak andstrong Walters conditions,which in turn are natural generalizations for the classicalWalters condition. The regularity properties of the pressure functional are studiedand its analyticity is proved on the space of the Höldercontinuous potentials.An exponential decay rate for correlations are obtained, in the case wherethe Ruelle operator exhibits the spectral gap property. An example, derived from the long-range Ising model, of a potential in the Walters class wherethe associated Ruelle operator has no spectral gap isgiven. One of the main results of this work provide anexplicit expression for the derivativeof the pressure P:C^α(Ω)→ℝ,where Ω≡ M^ℕ and M is a general compact metric space. To be more precise, we show that1 P^'(f)φ=∫_Ωφ h_fdν_f,where h_f and ν_f are eigenfunctionandeigenmeasure of the associated Ruelle operator.The proof follows closely the one givenin <cit.>, where the expression (<ref>) is obtainedin the context of finite alphabet. We also prove the existence of the limitP(f)=lim_n→∞1nlogℒ_f^n 1(x)in the uniform sense, for any continuous potential f. We would liketo point outthat the existence of this limit in this setting has beenproved in<cit.>.We give a new proof of this fact here for two reasons:first, it is different from the proof presented in <cit.>and we believe it is more flexible to be adapted to other contexts; second, some pieces of it are used to compute the pressurederivative. In the last section we apply our results in cases where M=(S^2)^ℤendowed with a suitable a priori DLR-Gibbs measure.We introduce a Heisenberg-type model on the lattice ℕ×ℤ,depending on a real parameter α, anduse the Ruelle operator to obtain differentiability of the pressure andexponential decay rate for the two-point function for any positive temperature. § BASIC DEFINITIONS AND RESULTS In this section we set up the notation and presentsome preliminaries results.Let M=(M,d) be a compact metric space, equippedwith a Borel probability measure μ:ℬ(M)→ [0,1],having the whole space M as its support.In this paper, the set of positive integers is denoted by ℕ. We shall denote by Ω=M^ℕthe set of all sequences x=(x_1,x_2,… ), wherex_i ∈ M, for all i∈ℕ.We denote the left shift mapping by σ:Ω→Ω, which is givenby σ(x_1,x_2,…)=(x_2,x_3,…). We consider the metric d_Ω on Ω given byd_Ω (x,y) = ∑_n=1^∞1/2^nd(x_n,y_n).The metric d_Ω induces the product topology and therefore it follows fromTychonoff's theorem that(Ω,d_Ω) is a compact metric space.The space of allcontinuous real functions C(Ω, ℝ) is denoted simply by C(Ω) and will be endowed with the norm ·_0 definedby f_0=sup_x∈Ω |f(x)|, which in turn isa Banach space.For any fixed 0< α< 1 we denote by C^α(Ω)the space of all α-Hölder continuous functions,that is, the set of all functions f:Ω→ℝ satisfying Hol_α(f) = sup_x,y∈Ω: x≠ y|f(x)-f(y)|d_Ω(x,y)^α <+∞.We equip C^α(Ω), 0< α< 1, with the norm given by f_α= f_0+ Hol(f). We recall that(C^α(Ω),·_α) is a Banach space for any 0< α< 1. Our potentials will be elements of C(Ω) and, in order to have a well defined Ruelle operatorwhen (M,d) is a general compact metric space, we needto consider an a priori measure which is simply aBorel probability measure μ:ℬ(M)→ℝ, where ℬ(M) denotes the Borel σ-algebraof M. For many of the most popular choices of an uncountable space M, there is a natural a priori measure μ.Throughout this paper the a priori measure μis supposed to have the whole space M as its support.The Ruelle operatorℒ_f: C(Ω) → C(Ω) is the mapping sending φ to ℒ_f(φ)defined for any x∈Ω by the expressionℒ_f(φ)(x) = ∫_M e^f(ax)φ(ax)dμ(a),where ax denotes the sequenceax=(a, x_1, x_2, …)∈Ω. The classical Ruelle operator can be recovered on this setting by considering M={0,1,…,n} andthea priori measureμ as the normalized counting measure. Let (M,d) be a compact metric space, μ aBorel probability measure on M having full support andf a potential in C^α(Ω),where 0<α<1. Then ℒ_f: C^α(Ω) → C^α(Ω) has a simple positive eigenvalue of maximal modulus λ_f,and there are a strictly positive function h_fand a Borel probability measure ν_f on Ω such that, (i) ℒ_f h_f=λ_f h_f, ℒ^*_fν_f=λ_fν_f;(ii) the remainder of the spectrum ofℒ_f: C^α(Ω) → C^α(Ω) is contained ina disc of radius strictly smaller than λ_f; (iii) for all continuous functions φ∈ C(Ω) we havelim_n→∞λ_f^-nℒ^n_fφ-h_f∫φ dν_f_0 = 0.See <cit.> for the case M=S^1 and <cit.>for a general compact metric space. Following the references <cit.> we definethe entropy of a shift invariant measure and the pressure of the potential f, respectively, as followsh(ν) = inf_f ∈ C^α(Ω){-∫_Ωf dν+ logλ_f} and P(f) = sup_ν∈ℳ_σ{h(ν)+∫_Ωfdν},where ℳ_σ is the set of all shift invariant Borelprobability measures.For each f ∈ C^α(Ω) we have for allx∈Ω that P(f) = lim_n→∞1/nlog[ ℒ_f^n( 1)(x)] = logλ_f =sup_ν∈ℳ_σ{h(ν)+∫_Ωfdν}.Moreover thesupremum is attained by m_f=h_f ν_f. See <cit.> Corollary 1. The function defined by C^α(Ω)∋ f↦ P(f)∈ℝis a real analytic function.See <cit.> for a proof.Let f∈ C^α(Ω) be a Hölder continuous potential, ν_f be the measure given by Theorem <ref> and φ, ψ∈ C^α(Ω). For each n∈ℤ we define the correlation functionC_φ,ψ,m_f(n) = ∫ (φ∘σ^n)ψdm_f -∫φ dm_f ∫ψdm_f.We have that the above correlation functionhas exponential decay.More precisely we have the following proposition:For each n∈ℕ let C(n) denote the correlation functiondefined by (<ref>). Then there exist constants K>0 and 0<τ<1 such that |C(n)|≤ K τ^n.The proof when M is finite is given in <cit.>. Due toTheorem <ref> this proof can be easily adapted to the case where M is compact metric space.We will include the details here for completeness. Before prove the above theorem, we present two auxiliary lemmas.Let f∈ C^α(Ω), and ∂ D the boundary of a disc D withcenter in λ_f, then the spectral projection π_f≡π_ℒ_f=∫_∂ D(λ I-ℒ_f)^-1dλ.is given byπ_f(φ)=(∫φdν_f) · h_f.Let be φ,ψ∈ C^α(Ω) then ℒ_f^n(φ∘σ^n ·ψ· h_f) = φℒ_f^n( ψ h_f).The proof of both lemmas arestraightforward computation so they will be omitted.Proof ofProposition <ref>. Since m_f=h_f dν_f it follows from the definition of the correlation function that|C_φ,ψ,m_f (n)|= |∫ (φ∘σ^n)ψ h_fdν_f -∫φh_fdν_f∫ψ h_fdν_f|.Notice that(ℒ^*_f)^nν_f = λ_f^nν_fand therefore the rhs above is equal to | ∫λ_f^-nℒ_f^n((φ∘σ^n)ψ h_f) dν_f - ∫φh_fdν_f∫ψ h_f dν_f|.By using the Lemma <ref> and performing simplealgebraic computations we get |C_φ,ψ,m_f (n)| ≤( ∫|φ|dν_f) λ^-n_fℒ_f^n ( ψ h_f-h_f∫ψ h_fdν_f)_0. By Theorem <ref> we know that the spectrum ofℒ_f: C^α(Ω)→ C^α(Ω)consists in a simple eigenvalue λ_f>0 and a subsetof a disc of radius strictly smaller than λ_f. Setτ=sup{|z|; |z|<1 and z·λ_f∈Spec(ℒ_f)}.The existence of the spectral gap guarantees that τ<1.Let π_fthe spectral projection associated to eigenvalue λ_f, then,the spectral radius of the operatorℒ_f(I-π_f) is exactly τ·λ_f.Since the commutator [ℒ_f,π_f]=0, we get ∀ n ∈ℕ that [ℒ_f(I-π_f)]^n = ℒ_f^n(I-π_f). From the spectral radius formula it follows that for each choice of τ>τthere is n_0≡ n_0(τ)∈ℕso that for all n≥ n_0 we have ℒ_f^n(φ-π_fφ)_0 ≤λ_f^nτ^n φ_0, ∀φ∈C^α(Ω). Therefore there is a constantC( τ)>0such that for every n≥ 1ℒ_f^n(φ-π_fφ)_0 ≤C( τ)λ_f^n τ^n φ_0 ∀φ∈C^α(Ω).By using the Lemma <ref>and the above upper boundin the inequality (<ref>) we obtain |C_φ,ψ,m_f(n)|≤( ∫ |φ|dν_f) C τ^n ψ h_f_0 ≤ C(τ)h_f_0( ∫ |φ|dν_f) ψ_0τ^n. * § MAIN RESULTSProposition <ref> ensures for any Hölder potential f that the limit P(f) = lim_n→∞ n^-1log[ ℒ_f^n( 1)(x)] always exist and is independent of x∈Ω.In what follows, we will extend this result for all continous potentials f∈ C(Ω). This is indeed a surprising result and it does not have a counter part on one-dimensional two-sided lattices,due to the existence of metastable states discovered by Sewell in <cit.>. Before present the proof of this fact,we want to explain what is the mechanism behind the absence of metastablestates for one-dimensional systems on the lattice ℕ. The absence of metastable states for continuous potentials and finite state space M, on one-dimensional one-sided lattices, as far as we knownwas first proved by RicardoMañé in <cit.>, but apparently he did not realize it. A generalization of this result for M being a compact metric space appearsin <cit.> and again no mention to metastable states is made in this paper.The proof of this result presented in <cit.> is completely different from ours and we believe that one presented here is more suitable to be adapted to other context.An alternative explanation of this fact can be found in <cit.> Remark 2.2, and following the first author of <cit.> the first person to realize this fact was Aernout van Enter. Let (ℬ,·) and (ℬ,|||·||| ) the classicalBanach spaces of interactions defined as in <cit.>.In case of free boundary conditions, we have for any Φ∈ℬ and a finite volume Λ_n⊂ℤ that |P_Λ_n(Φ)-P_Λ_n(Ψ)| ≤ |||Φ-Ψ||| and therefore the finite volume pressure with free boundary conditionsis 1-Lipschitz functionfrom (ℬ,|||·|||) to ℝ.On the lattice ℤ, when boundary conditions are consideredthe best we can prove is|τ_nP_Λ_n(Φ)-τ_nP_Λ_n(Ψ)| ≤Φ-Ψ, for interactions Φ,Ψ∈ℬ. From this we have that finite volume pressure with boundary conditionsis 1-Lipschitz function from the smaller Banach space(ℬ,·) to ℝ. The last inequality can not beimproved for general boundary conditions and interactions in the big Banach space (ℬ,|||·|||), becauseas we will see in the proof of Theorem <ref>it would imply the independence of the boundaryconditions of the infinite volume pressure which is a contradiction withSewell's theorem. The mechanism that prevents existence of metastablestates for continuous potential on the lattice ℕ is thepossibility of proving that the analogous of the finite volume pressure with boundary conditions on the lattice ℕ is indeed 1-Lipschitz function. To be more precise.For each continuous potentialf∈ C(Ω) there is a real number P(f) such that lim_n→∞1nlogℒ_f^n 1 -P(f)_0 =0.It is sufficient to prove that Φ_n:C(Ω) → C(Ω), given by Φ_n(f)(x)=1/nlogℒ_f^n 1(x)converges to a Lipschitz continuous function Φ:C(Ω) → C(Ω)in the following sense Φ_n(f)-Φ(f)_0→ 0, when n→∞. Indeed, by Proposition <ref>, for any fixed 0<α< 1 we haveΦ(C^α(Ω))⊂⟨ 1⟩, where ⟨ 1 ⟩ denotes the subspace generated by the constant functions in C(Ω). Since C^α(Ω) is a dense subset of (C(Ω),·_0) and Φ is Lipschitz, thenΦ(C(Ω))=Φ(C^α(Ω))⊂⟨ 1 ⟩. In order to deduce the convergence of (Φ_n)_n∈ℕ, it is more convenient to identifyΦ_n:C(Ω)→ C(Ω) with the function Φ_n:C(Ω)×Ω→ℝ, given by Φ_n(f,x)=(1/n)logℒ_f^n 1(x). For any fixed x∈Ω,follows from the Dominated Convergence Theorem that the Fréchet derivative ofΦ_n:C(Ω)×Ω→ℝ,evaluatedat f and computedin φ, is given by∂/∂ fΦ_n(f,x)·φ = 1/n∫_M^n(S_nφ)( x)exp(S_nf)( x)dμ()/ℒ_f^n 1(x),where (S_nφ)(x)≡∑_j=0^n-1φ∘σ^j(x). Clearly, n^-1S_nφ_0≤φ_0, and therefore we have the following estimate |∂/∂ fΦ_n(f,x)·φ|= |∫_M^n1/n(S_nφ)( x)exp(S_n f)( x)dμ()/ℒ_f^n 1(x)| ≤∫_M^n|1/n(S_nφ)( x)exp(S_nf)( x) | dμ()/|∫_M^nexp(S_nf)( x)dμ()|≤φ_0,for any f∈ C(Ω) independently of x∈Ω. Taking the supremum in (<ref>) over x∈Ω we obtain ∂/∂ fΦ_n(f,·)·φ_0 ≤φ_0,for alln∈ℕ. The above inequalityallows to concludethat∂/∂ fΦ_n(f)≤ 1,where · means the operator norm.Fix f and f̃ in C(Ω)and define for eachn∈ℕ the map Φ̂_n(t)=Φ_n(α(t),x),where α(t)=t f+(1-t)f̃ with 0≤ t≤ 1. Obviously, Φ̂_n is a differentiable map whenseen asa map from [0,1] toℝ and we have that|Φ̂_n(1)-Φ̂_n(0)|=|d/dtΦ̂_n(t̂)(1-0)|for somet̂∈ (0,1).Using the above estimative of the Fréchet derivative norm we have that |Φ_n(f,x)-Φ_n(f̃,x)| = |∂/∂ fΦ_n(f,x)(f-f̃)|≤f-f̃_0.As an outcome for any fixed f∈ C(Ω) the sequence(Φ_n(f))_n∈ℕisuniformly equicontinuous. Moreover, sup_n∈ℕΦ_n(f)_0<∞.Indeed, from inequality (<ref>), the triangular inequalityand existence of the limitlim_n→∞Φ_n( 1), it follows that|Φ_n(f)|=|Φ_n(f)-Φ_n( 1)|+|Φ_n( 1)| ≤f- 1_∞ +|Φ_n( 1)|≤f- 1_∞ +sup_n∈ℕ|Φ_n( 1)|≡ M(f). Now, we are ableto apply theArzelà-Ascoli's Theoremto obtain asubsequence (Φ_n_k(f))_k∈ℕ,whichconverges to a function Φ(f)∈ C(Ω).We now show that Φ_n(f)→Φ(f), when n→∞. Let ε>0andg∈ C^α(Ω) such that f-g_0<ε. Choose n_k and n sufficiently large so that the inequalities Φ_n_k(f)- Φ(f)_0<ε, Φ_n(g)-Φ_n_k(g)_0<ε and Φ_n_k(g)-Φ_n_k(f)_0<ε are satisfied. For these choices ofg,n andn_k, we have by the triangular inequality and inequality (<ref>) thatΦ_n(f)- Φ(f)_0 ≤Φ_n(f)-Φ_n_k(f)_0 + Φ_n_k(f)- Φ(f)_0< Φ_n(f)-Φ_n_k(f)_0 + ε≤Φ_n(f)-Φ_n(g)_0 + Φ_n(g)-Φ_n_k(f)_0 + ε<Φ_n(g)-Φ_n_k(f)_0 + 2ε≤Φ_n(g)-Φ_n_k(g)_0 + Φ_n_k(g)-Φ_n_k(f)_0 + 2ε< 4ε,thus proving the desired convergence. To finish the proof it is enough to observethat the inequality <ref> implies that Φ is a Lipschitz continuous function. For each fixed0<α<1 and f∈ C^α(Ω)theFréchet derivative of the pressurefunctionalP:C^α(Ω)→ℝ is given byP'(f)φ=∫φ h_fdν_fFor eachfixed f ∈C^α(Ω), 0<α< 1, there existsthe limitlim_n→∞1/nℒ_f^n(S_nφ)/ℒ^n_f 1 - ∫φ h_fdν_f_0 = 0for everyφ∈ C(Ω), andthe convergence is uniform. A straightforward calculation shows that 1/nℒ_f^n(S_nφ)/ℒ^n_f 1 - ∫φ h_fdν_f_0 = λ_f^n/ℒ^n_f 11/nλ_f^-nℒ_f^n(S_nφ) - ∫φ h_fdν_f_0≤sup_n∈ℕλ_f^n/ℒ^n_f 1_01/nλ_f^-nℒ_f^n(S_nφ)- (λ_f^-nℒ^n_f 1)∫φ h_fdν_f_0.Therefore to get the desired result it issufficient to show that: (a)sup_nλ_f^n/ℒ^n_f 1_0 is finite; (b) (λ_f^-nℒ^n_f 1)∫φ h_fdν_f converges to h_f∫φ h_fdν_f; (c)1/nλ_f^-nℒ_f^n(S_nφ) converges to h_f∫φ h_fdν_f. The first two itemsare immediate consequences of the Ruelle-Perron-Frobenius Theorem. Indeed,theconvergence λ_f^-nℒ_f^n1·_0⟶h_f, immediatelygive that (λ_f^-nℒ^n_f 1) ∫φ h_fdν_f·_0⟶ h_f∫φ h_fdν_f.Sinceh_f is a continuous strictly positive function,it follows from compactness of Ω thath_f is bounded awayfrom zero,and consequentlyλ_f^n/ℒ_f^n 1·_0⟶ 1/h_f.Onceh_fis strictly positive 1/h_f is also positive andbounded away from zero, which givesthatsup_nλ_f^n/ℒ^n_f 1_0<∞.The third expressionin(c)isharder to analyze than the previoustwo, sowe will splitthe analysisin three claims.Claim 1.For all φ∈ C(Ω) and n∈ℕ we havethat,λ_f^-nℒ_f^n(S_nφ) = ∑_j=0^n-1λ_f^-(n-j)ℒ_f^n-j (φλ_f^-jℒ^j_f 1). We first observe that, ℒ_f^n(φ∘σ^n) = φℒ^n_f 1,whichis an easy consequence of the definition of theRuelle operator. From that andthe linearity of the Ruelle operator it follows,ℒ_f^n(S_nφ)= ℒ_f^n(φ)+ℒ_f^n-1(ℒ_f(φ∘σ))+ …+ℒ_f(ℒ_f^n-1φ∘σ^n-1)= ℒ_f^n(φ)+ℒ_f^n-1(φℒ_f 1)+ …+ℒ_f(φℒ^n-1_f 1) = ∑_j=0^n-1ℒ_f^n-j(φℒ^j_f 1) finishing the proof. Fromthe linearity of the Ruelle operatorwe easily getλ_f^-nℒ_f^n(S_nφ) = ∑_j=0^n-1λ_f^-(n-j)ℒ_f^n-j (φλ_f^-jℒ^j_f 1).Claim2. For eachφ∈ C(Ω)we have lim_n→∞1/n( λ_f^-nℒ_f^n(S_nφ)-∑_j=0^n-1λ_f^-(n-j)ℒ_f^n-jφ h_f) _0 = 0.To verify (<ref>)we use(<ref>)to obtain the following estimate,1/n{λ_f^-nℒ_f^n(S_nφ) - ∑_j=0^n-1λ_f^-(n-j)ℒ_f^(n-j)φ h_f }_0 = 1/n∑_j=0^n-1λ_f^-(n-j)ℒ_f^n-j(φλ_f^-jℒ^j_f 1-φ h_f)_0≤const./n ∑_j=0^n-1φλ_f^-jℒ^j_f 1-φ h_f_0.The last term in the above inequality convergesto zero, when n→∞, because itis a Cesàro summation associated tothe sequenceφλ_f^-jℒ^j_f 1-φ h_f, which converges tozero in the uniform norm bytheRuelle-Perron-Frobenius Theorem, so the claim is proved. Claim 3. For eachφ∈ C(Ω) there exists the following limitlim_n→∞1/n∑_j=0^n-1λ_f^-(n-j)ℒ_f^n-jφ h_f- h_f∫φ h_fdν_f_0=0.Define A_n,j:=λ_f^-(n-j)ℒ_f^n-jφ h_fandB:=h_f∫φ h_fdν_f.In one hand,bytheRuelle-Perron-Frobenius Theorem,we must have for any fixed j∈ℕ that lim_n→∞A_n,j-B_0=0. On the other hand, we have by the triangular inequalityand the convergence in the Cesàro sense that 1n∑_j=0^n-1A_n,j-B_0≤1n∑_j=0^n-1A_n,j-B_0⟶ 0,whenn→∞, finishing the proof ofClaim 3. Therefore the proof of the Lemma is established. Proof of Theorem <ref>Fix x∈Ω and definefunctionΦ_n:C^α(Ω)→ℝ as Φ_n(f)≡1/nlog(ℒ_f^n 1)(x). As we have seen in the proof of Theorem <ref> theFréchet derivative ofΦ_n at f evaluated inφ∈ C^α(Ω) isgivenbyΦ_n'(f)φ = 1/nℒ_f^n(S_nφ)/ℒ^n_f 1,Sincewe havethe analyticity of the pressure functional inC^α(Ω) (Theorem <ref>) it follows fromthe previousLemma that P'(f)φ = lim_n→∞Φ_n'(f)φ = ∫φ h_fdν_f.§ A HEISENBERG TYPE MODEL The aim of this section is tointroduce a Heisenberg type model on the half-space ℕ×ℤ,prove absence of phase transition and exponential decay of correlationsfor this model.The construction of this model is split in two steps.Firts step. We consider(S^2)^ℤ as the configuration space. Atinverse temperature β∈ (0, ∞),the configurationsare randomly chosen accordingto the following probabilitiesmeasures μ_n, β dμ_n, β(σ):=1Z_n, βexp ( β∑_i,j∈Λ_nσ_i·σ_j )dσ,where Λ_n denotes the symmetric interval ofintegers [-n,n], σ_i·σ_j denotes the inner product in ℝ^3 ofthe first neighbors σ_i and σ_j, Z_n, β=∫_(S^2)^Λ_nexp ( β∑_i,jσ_i·σ_j)dσand dσ is the uniform probability measureon (S^2)^Λ_n.Let measure ν̂ bethe unique accumulation point of the sequence of probability measures given by (<ref>), see <cit.> for details. The measure ν̂ will be used as the a priori measurein the second step.Second step. Now we introduce a Heisenberg type model.We begin with the compact metric space (S^2)^ℤ, where S^2 is the2-dimensional unit sphere inℝ^3, as our alphabet.Now the configuration space is the Cartesian productΩ =((S^2)^ℤ)^ℕ,that is, a configuration is a pointσ=(σ(1),σ(2), ⋯)∈Ω, where each σ(i) is of the formσ(i)=(…,σ_(i,-2),σ_(i,-1),σ_(i,0), σ_(i,1), …),and each σ_(i,j)∈ S^2. We denote by · the ℝ^3 Euclidean norm, and v· w the inner product of two elements of ℝ^3. Fix a summable ferromagnetic translation invariant interactionJ on ℤ, that is,a function J:ℤ→ (0,∞) and assume thatJ(n)= e^-|n|α, for some α>0. Of course, we have ∑_n∈ℤ J(n)<∞. Now we consider the potential f:Ω→ℝ given by f(σ)= ∑_ n∈ℤ J(n) σ_(1,n)·σ_(2,n).Note that this potential has only first nearest neighbors interactions.Thepotential f given by (<ref>) is actuallyan α-Hölder continuous function.Indeed, |f(σ)-f(ω)|≤ |∑_n∈ℤJ(n)σ_(1,n)· (σ_(2,n)-ω_(2,n))| + |∑_n∈ℤJ(n)ω_(2,n)· (σ_(1,n)-ω_(1,n))| ≤∑_n∈ℤJ(n)σ_(2,n)-ω_(2,n)+∑_n∈ℤJ(n)σ_(1,n)-ω_(1,n)From the very definition of the distance we haved(σ,ω) ≥12^n∑_j∈ℤ12^|j|σ_(n,j)-ω_(n,j)≥12^n+|j|σ_(n,j)-ω_(n,j).By using (<ref>) and (<ref>) we get that |f(σ)-f(ω)|d(σ,ω)^α≤ K_1∑_n∈ℕJ(n)2^α n + K_2∑_n∈ℕJ(n)2^α n.Since for all n∈ℕ we haveJ(n)2^α n≤exp(-n(α(1-log 2))),and the constant α (1-log 2) is positive, follows that the series ∑_n∈ℕJ(n)2^α n isconvergent. Therefore, f is an α-Hölder continuous function.From Theorem <ref>, we have that there is aunique probability measure ν_f so thatℒ_f^*ν_f=λ_fν_f. This probabilitymeasure, following <cit.>, is a unique DLR-Gibbs measure associatedto a quasilocal specification associated to f, see <cit.> for theconstruction of this specification. By observing that the horizontal interactions, in our model, goes fast to zero in the y-directionis naturally to expect that the model is essentially a one-dimensional model. This feature allow us to obtain the following result Let β f be a potential, where f is given by(<ref>) and the inverse temperature β∈ (0,∞). Consider the a priori measure ν̂ on (S^2)^ℤ,constructed from (<ref>). Then for any fixed β>0 and m∈ℤthere are positive constantsK(β) and c(β) such that for all n∈ℕ we have∫_Ω (σ_(1,m)·σ_(n+1,m)) dν_β f≤K(β)e^-c(β)n.Furthermore, the pressure functional is differentiable at β f andits derivative is given by expression (<ref>). Fix m∈ℤ and letσ_(n,m)≡(σ_(n,m)^x,σ_(n,m)^y, σ_(n,m)^z). Consider the following continuous potentials φ^u, ξ^u given byφ^u(σ) = σ_(1,m)^uandξ^u(σ) = σ_(1,m)^u/h_β f(σ),u=x,y,z.Note that ∫_Ωξdm_β f = ∫_Ωσ_(1,m)^udν_β f(σ)= 0, u=x,y,z,where the last equality comes from the O(3)-invariance ofthe eingemeasure. Since h_β f(σ)=h_β f(-σ), see <cit.>, it follows again from theO(3)-invariance of ν_f that ∫_Ωσ_(1,m)^udm_β f=∫_Ωσ_(1,m)^u h_f(σ) dν_β f= 0,u=x,y,z.Therefore ∫_Ω (σ_(1,m)^uσ_(n+1,m)^u) dν_β f = ∫ (φ^u∘σ^n)ξ^u dm_β f= C_φ^u,ξ^u,m_β f(n) = O(e^-c(β)n),u=x,y,z. By summing (<ref>) with u=x,y,z we get the claimed exponential decay. The last statement follows from the fact that (S^2)^ℤ is a compact metric space,ν̂ is a full support a priori measure in (S^2)^ℤ and β f is an α-Hölder. Therefore the Proposition <ref> applies andcorollary follows.§ ACKNOWLEDGMENTSThe author would like to thankLeandro Cioletti, Artur Lopes and Andréia Avelar for fruitfull discussions and comments.Dillo 83le A. T. Baraviera, L. M. Cioletti, A. O. Lopes, J. Mohr and R. R. Souza.On the general one-dimensional XY Model:positive and zero temperature, selection and non-selection.Rev. Math. Phys., v. 23, p. 1063–1113, 2011.Baladi V. Baladi.Positive Transfer Operators and Decay of Correlations. World Scientific Publishing Co., 2000.Barreira L. Barreira .Thermodynamic Formalism and Applications to Dimension Theory. Birkhauser, 2010.bo R. Bowen.Equilibrium states and the ergodic theory of Anosov diffeomorphisms.Lecture Notes in Mathematics, v. 470,Springer, 1994.boo R. Bowen.Hausdorff dimensions of quasicircles. IHES Publ. Math., v. 50, p. 11–25, 1977. CO81 M. Cassandro and E. Olivieri. Renormalization group and analyticity inone dimension: a proof of Dobrushin's theorem. Comm. Math. Phys.,v. 80, p. 255–269, 1981.CER17L. Cioletti, A.C.D. van Enter, and R. Ruviaro.The Double Transpose of the Ruelle Operator. ArXiv e-print:1710.03841, p. 1–19, 2017. CL14 L. Cioletti and A. Lopes. Interactions, Specifications, DLR probabilities and theRuelle Operator in the One-Dimensional Lattice. Disc. Cont. Dyn. Sys.-A, Vol 37, n 12, p. 6139–6152, 2017. CL16 L. Cioletti and A. Lopes. Ruelle Operator for Continuous Potentials and DLR-Gibbs Measures. arXiv:1608.03881, 2016. CL17 L. Cioletti and A. Lopes. Correlation Inequalities and Monotonicity Properties of the Ruelle Operator.arXiv:1703.06126, 2017.CL-SIL L. Cioletti and E. A. Silva. Spectral properties of the Ruelle operator on the Walters class over compact spaces. Nonlinearity v. 29, p. 2253–2278, 2016. Ellis R. Ellis Entropy, Large Deviation and Statistical Mechanics. Springer, 2005. Excel-Lopes R. Exel, A. Lopes.C^*-Algebras, approximately proper equivalence relations and thermodynamic formalism. Erg.Theo. and Dyn. Syst.,v. 24(4), p. 1051–1082, 2004.GJ-Book J. Glimm and A. Jaffe. Quantum physics: A functional integral point of view. Second edition, Springer-Verlag, New York, 1987. Geogii88H.-O. 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Soc.,v. 236, p. 121–153, 1978. | http://arxiv.org/abs/1707.09072v3 | {
"authors": [
"Eduardo A. Silva"
],
"categories": [
"math-ph",
"math.MP",
"37D35"
],
"primary_category": "math-ph",
"published": "20170727231111",
"title": "Pressure Derivative on Uncountable Alphabet Setting: a Ruelle Operator Approach"
} |
A Jointly Learned Deep Architecture for Facial Attribute Analysis and Face Detection in the Wild Keke He, Yanwei Fu, Xiangyang XueFudan University {kkhe15, yanweifu, xyxue}@fudan.edu.cnDecember 30, 2023 =================================================================================================Facial attribute analysis in the real world scenario is very challenging mainly because of complex face variations. Existing works of analyzing face attributes are mostly based on the cropped and aligned face images. However, this result in the capability of attribute prediction heavily relies on the preprocessing of face detector. To address this problem, we present a novel jointly learned deep architecture for both facial attribute analysis and face detection. Our framework can process the natural images in the wild and our experiments on CelebA and LFWA datasets clearly show that the state-of-the-art performance is obtained.§ INTRODUCTIONThe problem of automatically analyzing the facial attributes received increasing attention recently <cit.>. Face attributes may potentially benefit a number of real-world applications, such as face alignment<cit.>, head pose estimation <cit.> and face verification <cit.>. Analyzing facial attributes still remains challenging in real-world scenarios. For example, most existing works <cit.> predict the facial attributes on well-cropped face images. In an extension to real-world scenarios, those works have to utilize the face detector to localize the bounding box of human faces before facial attribute analysis. Such a pipeline of face detection followed by facial attribute prediction is nevertheless undesirable in real-world facial attribute prediction due to two reasons. Firstly, with the preprocessing of face detector, the capability of facial attribute prediction has to heavily relies on the results of face detection. Unfortunately in an uncontrolled setting <cit.>, the face is likely to have large out-of-plane tilting, occlusion and illumination variations, which may affect the algorithms of face detection and facial attribute analysis simultaneously. For example, the Viola-Jones face detector <cit.> works well for near-frontal faces, but less effective for faces in the pose far from frontal views. Second, face detection is heterogeneous but subtly correlated with facial attribute prediction, and vice versa <cit.>. Efficiently and effectively exploiting this correlation may help in both detecting faces and predicting facial attributes.We propose an end-to-end deep architecture to jointly learning to detect faces and analyze facial attributes. Essentially, the two tasks share the same bottom layers (feature map in Fig. <ref>) in the architecture to alleviate the problems of face variations aforementioned. For instance, one face region of one image is failed to be detected as a face due to the partial occlusion, and yet the network can detect the existence of some facial attributes such as “eyeglasses”, and “smiling”. The face region can still be used to optimize our network and in turn help supervise the learning of the face detection part, to improve the performance of face detection part.In this work, the deep learning architecture is proposed for both face detection and facial attribute analysis. The network structure is illustrated in Fig. <ref>. Given an entire image, our architecture will firstly pass the image with convolutional layers (conv1– conv5) and max pooling layers to produce the conv feature map for face region proposals.The region of interest (RoI) pooling layer and face region proposal layer are two layers introduced to facilitate the jointly learning of our architecture. A region of interest (RoI) pooling layer pools each face proposal on the conv5 feature map into a fixed-length feature vector which is further processed by two fully connected layers (fc6 and fc7). Built on the fc7 layer, each individual task of facial attribute analysis and face detection are jointly optimized. By virtue of the face detection subnet, our architecture can directly predict the facial attributes from the whole images, rather than using the well-cropped images as previous work <cit.>. Extensive experiments on benchmark CelebA <cit.> and LFWA <cit.> datasets demonstrate that our method outperforms state-of-the-art alternatives. Contribution. Our main contribution is to propose a novel deep architecture of jointly learning of face detection and facial attribute tasks, which is capable of utilizing both tasks to better optimize the shared network and thus improve the performance of both tasks. Two layers – region of interest (RoI) pooling layer and face region proposal layer are introduced to enable learning these two tasks simultaneously. More importantly, we can predict the facial attributes in the wild and the input images do not need to be cropped and aligned as the standard practice in <cit.>. Finally, our joint deep architecture achieves state-of-the-art result on the biggest benchmark dataset for facial attribute analysis– CelebA dataset <cit.> and LFWA <cit.> datasets.§ RELATED WORK Facial attribute analysis. It was first studied by Kumar et al. <cit.>. In term of different visual features and distinctive learning paradigm, the facial attribute analysis has been developed into three categories: (1) the methods <cit.> of using hand-crafted visual features, such as SIFT <cit.> and LBP <cit.>; (2) the methods of utilizing the recent deep features <cit.>; and (3) multi-task methods of learning facial attribute <cit.>. We here highlight the differences between our architecture and these previous works. Liu et al. <cit.> cascaded three deep networks pre-trained for facial attribute prediction. In contrast, we show that the tasks of face detection and facial attribute prediction are highly correlated and our jointly learning architecture can improve both tasks. Rudd et al. <cit.> introduced a mixed objective optimization network which utilizes distribution of attribute labels to learn each task. Abdulnabi et al. <cit.> proposed a multi-task CNN model sharing of visual knowledge between tasks for facial attribute analysis. Comparing with <cit.>, we focus on jointly learning the face detection and facial attribute analysis; and our model can predict facial attributes on the images in the wild <cit.> i.e. without cropped and aligned, as illustrated in Fig. <ref> and Fig. <ref>.Face detection. The work of Viola and Jones <cit.> made face detection usable in many real world applications. The cascaded classifiers were built on Harr-like features to detect human faces. The deformable part model (DPM) <cit.> on top of HOG feature is a general object detector and can also be used for face detection <cit.>. Recent advances of deep learning architectures also inspire another category of methods for face detection. Yang <cit.> used the fully convolutional networks (FCN) to generate the heat map of facial parts for producing face proposals. In contrast to these works, our face detection task gets benefit from not only the well-designed architecture (in Fig. <ref>), but also the jointly learning process with facial attribute prediction. Further inspired by Fast-RCNN <cit.>, we take the face detection as a special case of the general semi-rigid object detection. More specifically, given an image, our face detection will try to answer two questions: (1) whether this patch contains faces or not? i.e. face score task in Fig. <ref>. (2) can we detect the bounding box of faces if there is any face? i.e. face bounding box task in Fig. <ref>. The jointly optimizing these two sub-tasks will better solve face detection. Multi-task learning. Our framework can be categorized as multi-task learning <cit.>, which shares the information and explores the similarity of related tasks on the same data. The multi-task learning can facilitate a wide range of tasks and applications, including but not limited to action recognition <cit.>, information retrieval <cit.>, facial landmark detection <cit.>, and facial attribute prediction <cit.>. The recent technical report <cit.> also combine face detection with the tasks of locating face landmarks and recognizing gender. However, unlike our facial attribute prediction on the images in the wild, their work still has to firstly detect face regions for the facial landmarks and predicting gender. § OUR DEEP ARCHITECTUREIn this section, we firstly overview our network in Sec. <ref>, and then the tasks are defined in Sec. <ref>. The Face region proposal and RoI pooling layers are explained accordingly in Sec. <ref> and Sec. <ref>. Finally, we utilize the network to solve the facial attribute prediction and face detection in Sec. <ref>.§.§ Overview Figure <ref> shows our framework for jointly face detection and attribute analysis. Our architecture takes an entire image as input and a set of face bounding boxes as labels for training. The whole network firstly processes the image with several convolutional layers (conv1– conv5), and max pooling layers to produce a conv feature map for face region proposal. For feature map of each proposed face, we employ a region of interest (RoI) pooling layer to pool it into a fixed-length feature vector. Each feature vector is further processed by two fully connected layers (fc6 and fc7) and thus used for the tasks of facial attribute analysis and face detection. On top layers, our architecture has face detection branch and facial attribute branch. Our network follows the art and design of VGG-16 <cit.>. Particularly, the kernel size, stride and the number of filters in convolutional layers (conv1– conv5) and the two fully connected layers (fc6 and fc7) are exactly the same as the corresponding layers in VGG-16 architecture. §.§ Task formulation Suppose we have the labelled source training dataset 𝒟_s={𝐈,𝐚,𝐋} with N training instances and M attributes. 𝐈 denotes the patches of training images and 𝐋 denotes the labels. We use the 𝐋 matrix to both denote whether an image patch contains the face, and whether a facial attribute exists in the image patch. Particularly, for the i-th image patch 𝐈_i (i=1,⋯,N), we use 𝐋_i⋆=0 to indicate that this image patch does not contain a human face. If 𝐈_i is a face image patch, we use 𝐋_ij=+1 to denote the existence of j-th facial attribute 𝐚_j (j=1,⋯,M); 𝐋_ij=-1 otherwise. As illustrated in Fig. <ref>, our network extracts the image patch to a 4096-dim feature vector, and it is denoted as f(𝐈;Θ), where Θ is the parameter set of the deep architecture. We have the prediction tasks of facial attribute, face score and face bounding box. Each task has their own parameters on the last layer. Facial attribute prediction. To predict the attributes of the image 𝐈^*, we need to learn a function 𝐋_i𝐚^*=Ψ(𝐈^*) to predict the facial attribute 𝐚. We thus have the predicting function Ψ=[ψ_i]_i=1,⋯,M, and ψ_i(𝐈):ℝ^4096→{ +1,-1}; specifically, we consider Ψ(x)=W_a^Tx, where W_a⊆ℝ^4096×40 for all 40 attributes. Face score. This task aims to predict the score whether an image patch is face. Essentially, we need to learn Φ(x):ℝ^4096→{ +1,-1}, and Φ(x)=W_s^Tx, where W_a⊆ℝ^4096×2 for binary prediction task. Face bounding box. We regress the bounding box of faces by Ω(x):ℝ^4096→ℝ^4, and we configure the form as Ω(x)=W_b^Tx, and W_b⊆ℝ^4096×4 for the new bounding box. §.§ Face region proposal layerInspired by the work of object detection <cit.>, our network has the branch of face region proposal. Specifically, this branch takes an image of any size as input and outputs a set of rectangular face proposals. To generate the facial region proposal, we slide the branch over the convolutional feature map after the five convolutional layers (conv1– conv5) in Fig. <ref>. The extracted conv5 features are further fed into two sibling fully connected layers, i.e. face bounding box (bbox) and face score layer. Face bounding box layer employs the smooth L_1 loss (defined in <cit.>) enables a regressor to predict the facial bounding box. Face score layer utilizes the softmax loss to indicate whether the bounding box is a face. For each image, we selected top-300 face region proposals in term of the face scores computed. §.§ The RoI pooling layerThe RoI pooling layer can convert the feature maps of face region proposal (with the size of h× w) to a fixed spatial extent of H× W, which facilitates the further process. Here, h,w,H,W are heights and widths of each rectangle region of feature map in Fig. <ref>. For the varying size of input feature patches, we vary the size of filters of pooling layer with the sub-window of approximate size h/H× w/W. The RoI pooling technique is adopted from the work of object detection <cit.> which inspired by the SPPnets <cit.>. §.§ Facial attribute analysis and face detectionThe fixed-length feature vector extracted from RoI Pooling layer is further fed into two fully connected layers (fc6 and fc7) in order to enable the tasks of facial attribute analysis. Particularly, built on the layers of fc6 and fc7, the multi-task network structure is employed to analyze each attribute individually. This can be modeled as the minimization of the expected loss over all the training instances which is {Θ,W_a,W_s,W_b}=argmin ℒ(𝐈;Θ,W_a,W_s,W_b) where ℒ(𝐈;Θ,W_a,W_s,W_b) is the loss function of jointly learning; and we have ℒ(𝐈;Θ,W_a,W_s,W_b) =λ_1ℒ_a(Φ(f(𝐈)))+λ_2ℒ_s(Ψ(f(𝐈))) +λ_3ℒ_b(Ω(f(𝐈)))here the loss function ℒ_a(Φ(f(𝐈))) on facial attribute prediction is the mean square error loss of all attributes; we use softmax loss ℒ_s(Ψ(f(𝐈))) for face score task, and finally the smooth L_1- loss ℒ_b(Ω(f(𝐈))) <cit.> is employed to regress the face bounding box.We denote the shared parameters of deep architectures as Θ which are jointly optimized by all these three tasks. Since these three tasks are highly related, learning in such a way can not only greatly reduce the prediction error of each individual task, but also accelerate the convergence rate of learning the whole network.For the testing image 𝐈_k, the predicted result 𝐋̂_kj of the j-th facial attribute 𝐚_j (j=1,⋯,M) is thresholded by 𝐋̂_kj= 1ψ(𝐈_k)>τ-1ψ(𝐈_k)≤τ where τ is the threshold parameter. § EXPERIMENTS§.§ Datasets and settings We conduct the experiments on the CelebA dataset <cit.> and LFWA dataset <cit.>.CelebA <cit.> contains approximately 200k images of 10k identities. Each image is annotated with 5 landmarks (two eyes, the nose tips, the mouth corners) and binary labels of 40 attributes. To make a fair comparison with the other facial attribute methods, the standard split is used here: the first 160k images are used for training, 20k images for validation and remaining 20k for testing. CelebA provides two types of training images, i.e., aligned and cropped face images and raw images as shown in the first and second row of Fig. <ref> respectively. We use the raw images for training our joint deep architecture. Since there is no ground-truth bounding box label for faces in raw images, the face detector <cit.> is employed here to help generate the face bounding box for training images. We use the implementation of dlib toolbox <cit.> for generating labels. Note that face bounding box generation is only required in the training stagesfor synthesizing labels. At testing phrase, raw images are directly input for both facial attribute analysis and face detection. LFWA <cit.> is constructed based on face datasets LFW <cit.>. It contains approximately 13143 images of 10k identities. Following <cit.>, 50% of the images for training, and the other 50% are used for testing. LFWA has 40 binary facial attributes, the same as CelebA. We also generate face bounding box for LFWA to train our joint learning network. Evaluation metrics. We take the attribute prediction as classification tasks and thus mean accuracy can be computed. Particularly, we evaluate the performance as comparable to <cit.> by the mean error which is defined as mean error=1-mean accuracy. Implementation and Parameter settings. The τ is set as 0 in Eq (<ref>); and square error loss is used in Eq (<ref>). We empirically set λ_1=λ_2=1,λ_3=2 in Eq (<ref>). The convolutional layers and fully connected layers are initialized by the 16 layer VGG network <cit.> individually. We use the open source deep learning framework Caffe <cit.> to implement our structure. A single end-to-end model is used for all the testing. We employ the stochastic gradient descent to train our network. Dropout is used for fully connected layers and the ratio is set to 0.5. For training CelebA dataset, with initial learning rate 0.001, and gradually decreased by 1/10 at 100k, 150k iterations, the total training iterations are 180k. For training LFWA dataset, as there are only limited training images - 6263, We fine-tune from pre-trained CelebA model, with initial learning rate 0.0001, and decreased by 1/10 at 40k iterations, the total training iterations is 60k. Once trained, our framework can predict the facial attributes and detect faces on the images in the wild for any testing images. Running cost. Our facial model get converged with 180k iterations and it takes 29 hours on CelebA with one NVIDIA TITANX GPU.On LFWA dataset, Our facial model get converged with 60k iterations and takes 9 hours. For training all the model, and it takes around 4 GB GPU memory. §.§ Competitors.Our model is compared against state-of-the-art methods and several baselines.Particularly, (1) FaceTracer <cit.> is one of the best methods with the hand-crafted features. The features used including HOG <cit.> and color histograms of facial regions of interested to train SVM classifier for predicting facial attributes. (2) LNets+ANet <cit.> migrates two deep CNN face localization networks to one deep CNN network for facial attribute classification. (3) Walk and Learn <cit.> learns good representations for facial attributes by exploiting videos and contextual data (geo-location and weather) as the person walks. (4) Moon <cit.> is a mixed objective optimization multi-task network to learn all facial attributes, achieves the best result on CelebA dataset. (5) Cropped model is a variant of our model without using the face detection branch. The raw image is also used as the input and we crop the faces from the images by Dlib toolkit to train facial attribute models. The processed face images are utilized to train the model. (6) Aligned model uses the aligned and cropped images provided by CelebA and LFWA to train facial attribute model, which is used by most of the state-of-the-art methods. To make a fair comparison, these two baseline models are initialized by 16 layer VGG network; same as the proposed structure.§.§ Comparison with the State of the ArtWe compared our model with state-of-the-art methods: FaceTracer <cit.>, LNets+ANet <cit.>, Walk and Learn <cit.>, Moon <cit.> and two baselines. The results on the testing split of CelebA and LFWA are reported in Tab. <ref>, Fig. <ref> and Fig. <ref>, respectively.Note that please refer to the supplementary material for the full comparison results on each attribute. Comparing with all the competitors, we draw the following conclusions. Our model beats all the other methods by the mean error on CelebA and LFWA dataset. As we can see from Fig. <ref>, our approach obtains the mean error 8.41% which is the lowest among all the competitors, and it outperforms all the other methods on CelebA dataset. And on LFWA dataset as shown in Fig. <ref>, our mean error is only 13.13% which is the lowest among all the competitors. This validates the efficacy of our joint learning architecture. Particularly, We compare the classification error in each individual attribute in Fig. <ref> and Fig. <ref>. We note that on more than half among the total 40 attributes, our framework is significantly better than the other competitors, since our jointly learning architecture can efficiently leverage the information between face detection and facial attribute prediction. Particularly, comparing with the other works, all our attribute prediction tasks share the same deep learning architecture, i.e. conv1-conv5, feature map and fc6, fc7 layers as illustrated in Fig. <ref>). These shared structures implicitly model the correlations between each attribute task. Furthermore, on gender attribute prediction, our framework can achieve 1.7% error on CelebA; in contrast, the error of HyperFace <cit.> is 3.0%. That indicates that our results can get 1.3% improvement over that of HyperFace <cit.>. Our model can process the images in the wild. Our face region proposal and RoI pooling layers are flexible enough to directly process the images in the wild. This thus better demonstrates the effectiveness of our models. Specifically, our model not only achieves the best performance on the benchmark dataset – CelebA and LFWA, but also our tasks of facial attribute prediction do not need to align and crop the facial images as have done in many previous work <cit.>. Additionally, unlike the work of Walk and Learn <cit.>, our model is not trained by the external data; and yet still obtains better results than Walk and Learn <cit.>. Our model is very efficient in term of jointly learning to detect faces and predict facial attributes. Our joint learning results have greatly improved over those of two baseline models – Cropped, Aligned models. These two methods are yet another two naive baselines of directly learning the facial attribute tasks. Particularly, on CelebA dataset, our model beats the two methods on the classification error of 27 attributes (totally 40 facial attributes) as compared in Fig. <ref>. Figure <ref> shows that our joint model hit better results than the two baseline models on all the 40 attributes. This reveals attribute prediction tasks get benefit from the face detection task. Also note that unlike LNets+ANet <cit.> using two networks to localize the face and another one network to extract features, we can train our model by a single end-to-end network here since the shared architectures (conv1 – conv5) can explicitly model the important facial parts, again thanks to the face detection subnet.Qualitative results. The important facial parts extracted by the shared feature map are visualized in Fig. <ref>. In particular, four groups of images are shown; and the feature map has higher activation on the regions of human faces. The first 3 columns are success cases, even thought the images have large pose (Column 2) or very high occlusion (Column 3). We also list some failure examples in Column 4, which are some extreme cases. Some of them are caused by ambiguous or too small view (e.g. the cat face is also similar to human's). The feature map reveals our model pay attention on the important facial parts which helps to analyze facial attributes.§.§ Results of Face Detection Our face detection is compared against the face detector of dlib toolbox <cit.>, which is an implementation of the generic object detector <cit.> on face images. On CelebA dataset, there are only 5 landmarks of faces, but no labeled ground-truth bounding box of faces for raw images. We generate the ground truth bounding box by 5 landmark coordinates for the test set. If the IOU (Intersection over Union) of predicted bounding box and the ground truth bounding box is larger than 0.5, we assume the prediction is correct. We compare the two detectors on the testing split of CelebA.The results are shown in Fig. <ref>. We compare the Precision-Recall curve for two methods. We find that both the Dlib detector and our face detector can achieve very high face detection on the CelebA test set as shown in the left subfigure of Fig. <ref>, which shows the efficacy of both detectors of solving the task of face detection. Nevertheless, our face detector still beats the Dlib detector by a relatively large margin. The AUC values of Dlib detector and ours are 0.938 and 0.982 respectively. To better show the difference, we highlight the up-right corner of PR curve in the right subfigure as shown in Fig. <ref>. This result further proves that our jointly learned architecture can make the face detection and face attribute tasks help each other via sharing the same parameters of the deep network. This shows that attribute learning does help for face detection also it provides more detailed information about the face. § CONCLUSION In this paper, we propose a novel joint deep architecture for facial attribute prediction and face detection. Different from the previous pipeline of face detection followed by facial attribute prediction, our architecture takes an entire image as input, enables both face detection and facial attribute analysis. The proposed architecture can not only exploit the correlation of face detection and face attribute prediction, but also boost both tasks. The experimental results on CelebA and LFWA datasets show the efficacy of proposed methods over the other state-of-the-art methods.ieee | http://arxiv.org/abs/1707.08705v1 | {
"authors": [
"Keke He",
"Yanwei Fu",
"Xiangyang Xue"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170727044542",
"title": "A Jointly Learned Deep Architecture for Facial Attribute Analysis and Face Detection in the Wild"
} |
[pages=1-last]paper.pdf | http://arxiv.org/abs/1707.08551v3 | {
"authors": [
"Hao Dong",
"Akara Supratak",
"Luo Mai",
"Fangde Liu",
"Axel Oehmichen",
"Simiao Yu",
"Yike Guo"
],
"categories": [
"cs.LG",
"cs.DC",
"stat.ML"
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"primary_category": "cs.LG",
"published": "20170726172949",
"title": "TensorLayer: A Versatile Library for Efficient Deep Learning Development"
} |
[NO \title GIVEN] [NO \author GIVEN] December 30, 2023 ====================== Computational efficiency demands discretised, hierarchically organised, and individually adaptive time-step sizes (known as the block-step scheme) for the time integration of N-body models. However, most existing N-body codes adapt individual step sizes in a way that violates time symmetry (and symplecticity), resulting in artificial secular dissipation (and often secular growth of energy errors). Using single-orbit integrations, I investigate various possibilities to reduce or eliminate irreversibility from the time stepping scheme. Significant improvements over the standard approach are possible at little extra effort. However, in order to reduce irreversible step-size changes to negligible amounts, such as suitable for long-term integrations of planetary systems, more computational effort is needed, while exact time reversibility appears elusive for discretised individual step sizes. gravitation — methods: numerical — celestial mechanics§ INTRODUCTIONAstrophysical N-body problems typically have a large range of dynamical time scales with factors of 10^2-4 between the shortest and longest orbital times. Consequently, instead of using time steps of (fixed or varying) global size, contemporary N-body algorithms advance each particle with time steps of a size which is individually adapted along its trajectory <cit.>. There are two components to such an individual time-stepping method: a time-step function and a time-stepping scheme. The time-step function (or criterion) returns an appropriate step size given the instantaneous state of a particle's trajectory. The time-stepping scheme, on the other hand, is a method that adapts the individual particle step sizes to follow these time-step criteria as best as possible.This study is concerned solely with the second ingredient, the time-stepping scheme. There are two important conditions for such a scheme: (1) it should not hinder computational efficiency and (2) it should be time reversible (and/or support symplectic time integration[A symplectic integrator advances thetrajectories by a canonical map. An equivalent statement is that the JacobianJ⃗ = ∂ξ(t+h)/∂ξ(t)between the initial state ξ={x⃗,p⃗} of the system and that advanced by step size h satisfies J⃗^T·Ω⃗·J⃗=J⃗ with the symplectic matrixΩ⃗ =[0⃗ -I⃗;I⃗ -0⃗ ].As a consequence, the geometric structure of phase space, most first integrals, and the Poincaré invariants are preserved and the energy error tends to be bounded, but see footnote <ref>.]). This latter condition is important to avoid artificial numerical dissipation <cit.>. In order to meet the first condition, all contemporary astrophysical N-body methods for large N employ the block-step method, where particle time-step sizes are discretised and hierarchically synchronised <cit.>. The original motivation for this scheme was the reduction in the number of predictions of particle positions, which are required for the computation of the forces on other particles. Moreover, with modern gravity solvers the simultaneous computation of allgravitational force between N particles requires only 𝒪(Nln N) <cit.> or (fewer than) 𝒪(N) operations <cit.>, instead of𝒪(N^2) for a direct force summation, which allows considerable efficiency saving from the synchronisation. It appears that the block-step is the only possibility to achieve these savings and yet allow for individual time-step sizes. It is therefore mandatory to use this method. However, (as far as I am aware) none of the contemporary N-body codes employing the block-step is time reversible or symplectic, and only little effort has been made towards that goal <cit.>. One problem is that integrating the mutual forces between any two particles with different step sizes for either particle cannot be reconciled with a canonical map and hencesymplecticity. Note, however, that the block-step method itself does not destroy symplecticity[Any dependence of h on ξ alters the Jacobian (<ref>) and destroys symplecticity <cit.>. However, when using discrete step sizes (as with the block-step scheme), the function h(ξ) is piece-wise constant with jump discontinuities. Thus, the neighbourhood of almost all trajectories use the same h and the integration remains symplectic (Tremaine, private communication, 2016). The exceptions (trajectories hitting the jumps in h) occupy a volume of measure zero in phase space and are irrelevant.].(, see their Fig. 5) demonstrate this with an example where the Poincaré invariant is conserved to within round-off error using the block-step scheme.Symplecticity can be restored by integrating the force between any two particles with the same time-step size (either that of the faster or slower of the two particles), resulting in exact conservation of total momentum <cit.>. However, such schemes are only reversible as long as the time-step adaption process (the application of the jumps discussed in footnote <ref>) is. Since the N-body dynamics is strictly reversible, irreversibility of the numerical integration method tends to result in artificial dissipation. As a consequence, the energy error (for example) is not guaranteed to be bounded, irrespective of whether the integrator is symplectic[Symplecticity by itself does not imply time reversibility. A simple counter example is the second-order accurate leapfrog integrator followed by a positional offset proportional to h^4. This is a symplectic, second-order accurate integrator, yet is not reversible and suffers from significant energy drift.] or not. This consideration suggests that reversibility of the integration scheme is more important than symplecticity. Unfortunately, so far no reversible yet efficient block-step-based time-stepping scheme is known. In fact, most practitioners determine the step size by the time-step function evaluated at the start of the time step <cit.>. This simple forward method violates time symmetry whenever the step size is adapted and is well known to destroy long-term stability <cit.>. Thus, the most wanted ingredient for reversible N-body integration is a reversible method for adapting the individual particle step sizes given some time-step function.The goal of this study is to consider ways to improve this situation. If h is not discretised, but continuous, the situation is much simpler and various adaptive time stepping methods have been proposed <cit.>. Here, I adapt several of these to discrete h and the block-step and study them in the context of single-orbit integrations. This constitutes a first test that any such method must pass before being considered for the full N-body problem. In particular, I do not toy with the time-step function or the underlying time integrator.The paper is organised as follows. In section <ref> the problem is stated more formally, while sections <ref> and <ref> present two different approaches of improving reversibility. Section <ref> presents a third approach, which is more accurate but also more computationally expensive and hence more suitable for long-term integration of planetary systems. Finally, sections <ref> and <ref>discuss the findings and conclude.§ STATEMENT OF THE PROBLEM Let ξ denote the state of a particle orbit and ϕ_h a self-adjoint (time-symmetric) integrator used for individual time steps (i.e. the composition ϕ_-h∘ϕ_h obtains the identity map). Then the state of the integrated orbit after n steps of sizes h_i isξ_n = ϕ_h_n-1/2 ξ_n-1 = ϕ_h_n-1/2∘…∘ϕ_h_1/2 ξ_0,where h_n+1/2=t_n+1-t_n. Thus, the step sizes naturally have half-integer indices to satisfy time symmetry. Since ϕ_h is reversible, so is the combined map (<ref>) iff the individual step sizes h are reversibly adapted. With the block-step scheme individual step sizes are discretised ash = 2^-r h_maxwith integer rung r≥0 and arranged hierarchically, as shown in Fig. <ref>. As a consequence of the hierarchical structure, a change to higher rung (δ r>0: shorter step size) is always possible, but a change to lower rung (δ r<0: longer steps size) only if the synchronisation requirement is met (which is every 2^-δ r steps). Furthermore, the change in rung is usually limited to |δ r|≤1, corresponding to changing h by a factor two either way. These block-step synchronisation constraints imply that given a desired step size τ, there exist at any time t_n a unique block-step maximum step sizeh_block(τ,t_n) ≤τof the form (<ref>). The dependence of h_block on the actual simulation time t_n originates only from the block-step synchronisation constraint, i.e. the fact that doubling the step size (reducing the rung) is not always possible. In view of Noether's theorems, this dependence of the integration method on absolute time suggests that the energy error may never be fully bounded with the block-step method. However, one would expect this time dependence to affect the energy not in a systematic but rather pseudo-random way. In fact, when experimentally dropping the block-step synchronisation constraint in single-orbit integrations, I found if anything a deterioration of the energy conservation. In the notation used in the remainder of this paper, the time-dependence of h_block is suppressed for brevity.If the integrator ϕ_h requires a force at the end (and start) of each time step, then the force computations are synchronised not only between particles with the same rung but between particles of all rungs larger than the smallest active rung at the given time (see also Fig. <ref>). This is the main reason why most astrophysical N-body methods for collision-less dynamics use the kick-drift-kick (rather than the drift-kick-drift) version of the leapfrog integratorϕ_h = ϕ^K_h/2∘ϕ^D_h∘ϕ^K_h/2with kick and drift operatorsϕ^K_h∘(t, x⃗, v⃗) = (t, x⃗, v⃗-h∇⃗Φ(x⃗,t)), ϕ^D_h∘(t, x⃗, v⃗) = (t+h, x⃗+hv⃗, v⃗),where x⃗ and v⃗ denote position and velocity, respectively. All tests in this study are performed using this integrator, but most conclusions regarding the effect of the time-step adaptation schemes are also valid for other integrators, including those of higher order.The statement of the problem then is:∙adapt h reversibly and such that h≈ T(ξ),where T(ξ) denotes the time-step function. This problem is non-trivial because the states ξ, and hence T(ξ), are not synchronised with the step sizes h, as indicated by the half-integer indices for the latter. Thus, in order to solve the problem, some form of synchronisation of relation (<ref>) is required, and different synchronisation attempts result in different stepping schemes.§.§ What is wrong with the simple forward method?As mentioned in the introduction, the state of the art for adapting individual step sizes is simplyh_n+1/2 = h_block(T_n) with T_n≡ T(ξ_n),when the step size matches (block-step allowing) its optimal value at the start of the step. This simply ignores the synchronisation problem in equation (<ref>). As a consequence, any time dependence of T result in an O(h) synchronisation error in equation (<ref>). The forward method gives the adaptation schemeδ r = {[ -1 ifh_n-1/2≤ 12 T_n and block-step allows,;0 ifh_n-1/2≤ T_n,; +1 4lotherwise. ].Here, I have limited rung changes to |δ r|≤1, which is common practice and which is assumed in the remainder of this study unless otherwise stated.For two orbits integrated with this scheme, Fig. <ref> plots the time evolution of radius, h, T, and the energy over one orbital period. The time-step function used isT(ξ) = η√(r^3/GM(r))with η=0.02 and M(r) the mass enclosed at radius r (of course, such a time-step function is not directly available in N-body simulations, but reasonable approximations are, e.g. ). Evidently, the energy varies considerably over one orbit. Such oscillations of energy are characteristic of symplectic integrators. If using a (sufficiently small) fixed step size, the error made on approach to peri-centre is exactly undone on the way out again. With the adaptive method, shorter time steps near peri-centre reduce the energy error there and improve the overall trajectory accuracy, while avoiding unnecessarily short steps for most of the orbit. However, the error is not exactly symmetric w.r.t. peri-centre. Instead, with the forward adaptation method, the integration accuracy is on average higher on the outward than on the inward part of the orbit and a residual error remains. For an integrator of order n (n=2 for the leapfrog used here), the residual error is proportional to h^n+1∝η^n+1 per irreversible step-size change. As a consequence, the energy error grows over the long term.This can be seen in Fig. <ref>, which plotsfor an integration over 10^4 periods the ratio of the long-term energy error over the short-term energy error (maximum error over a single orbit as shown in Fig. <ref>) for η=0.02 (black) and 0.01 (red). Evidently, δ E_long/|δ E_short| grows roughly linearly at a rate ∝η. Thus, unless η is chosen exceedingly small the long-term error (for these eccentric orbits) is soon dominated by the adverse effects of irreversible step-size changes. This result is independent of the order of the underlying integrator ϕ_h.§.§ Counting stepping errors The departure of the energy error from exactly symmetrical behaviour over each orbit, and hence the net energy error and long-term drift, are a consequence of each step-size change being irreversible, which results in deviations of the step size h from the condition (<ref>). While the forward scheme does not avoid such deviations, one can detect them a posteriori.There are four types of deviations, depending on the sign of Ṫ and whether h was too large or too small. One may estimate the ideal step size as T_n+1/2 if known (i.e. for a longer step when in fact two short steps have been taken), otherwise as √(T_nT_n+1). Then a step is too long if h_n+1/2>√(T_nT_n+1) and two adjacent steps of the same size h are too short if a larger step was possible and 2h<T_middle.The forward scheme (<ref>) only makes two types of errors: too long steps at Ṫ<0 and too short steps at Ṫ>0. Both tend to result in an energy error of the same sign (which is different for the two orbits considered in Fig. <ref>).Errors of the other types (too long steps at Ṫ>0 or too short steps at Ṫ<0) did not occur, but would have resulted in energy errors of the opposite sign. I define the net rate of stepping errors as difference in the number of errors of different sign divided by the total number of step changes:[ R_err≡[number of too long steps atṪ<0; + number of too short steps atṪ>0; -number of too long steps atṪ>0; - number of too short steps atṪ<0] /; 2ltotal number of step-size changes, ]where a pair of adjacent too short steps is only counted once. For the forward scheme, the chance for a stepping error is only 50%, in agreement with the results from the orbit integrations (indicated in Fig. <ref>). Of course, R_err is a rough measure, as it only accounts for the typical sign but not the actual size of the errors. In reversed time, Ṫ changes sign and hence also R_err. Therefore, a reversible scheme should obtain R_err=0, even though it may not always chose the ideal step size. Such a scheme should have long-term energy errors comparable to the short-term energy error. Since the short-term error is ∝ h^n∝η^n for an integrator of order n, the ratio of long-term errors owed to irreversible step-size changes to the short-term error scales like η R_err. For the forward method, R_err∼0.5 independent of η, and therefore δ E_long/|δ E_short|∝η, in agreement with Fig. <ref>. Of the two orbits presented, the Kepler orbit has three times as many step-size changes than the cusp orbit, but owing to the bimodal structure of the error (bottom left panel of Fig. <ref>), some of the resulting energy errors cancel, unlike the situation for the cusp orbit. Therefore, the net effect on the ratio δ E_long/|δ E_short| happens to be similar for these two orbits. §.§ How can the problem be solved?§.§.§ Reducing or eliminating stepping errorsIf one can reduce or even eliminate stepping errors, or at least their net rate such that errors of opposite sign largely cancel, the integration will be near-reversible, even if no attempt is made at constructing an exactly reversible scheme.One possibility is to extrapolate the time-step function into the future. In section <ref>, I consider such a method which has synchronisation error 𝒪(h^3), i.e. as good as the leapfrog's trajectory error, but still incurs stepping errors.A more rigorous approach is to attempt to eliminate stepping errors by trying different h and then chosing those satisfying the time-step criteria. Applying this approach to each particle separately ignores the interdependence of particle orbits and time-step functions, but is close enough to reversibility for most practical purposes (see section <ref>). However, this approach requires twice as many force computations as actually used. This overhead can be reduced (but not eliminated) when combined with the extrapolation method, see section <ref>. §.§.§ Aiming for reversibility When instead trying to obtain truly reversible schemes, past and future (step sizes) must be treated symmetrically, which severely restricts our hands in how to use our knowledge about the past. In order to avoid computationally expensive iterations, a reversible method must be explicit, i.e. the future step size must be obtained from the current value of the time-step function and the past step size in a reversible way. Since h is discretised, the choice for the future is always between a long step (either equal to or twice the previous step size) and two shorter steps. If the decision for the next step size is based solely on the first of these shorter steps, time symmetry is violated, while the second shorter step is beyond the horizon of prediction (at the moment of the decision). Therefore, in order to maintain time symmetry, the decision must be based solely on the merit of the long step: an explicit scheme must prefer longer steps. Attempts to obtain such schemes are presented in sections <ref> and <ref>. § EXTRAPOLATING AND ITERATING For the situation of a continuous step size h, formally exact time symmetry can be obtained by matching h to some mean of the time-step function T at the start and end of the step: [ Matching h instead to the time-step function evaluated in the middle of the step h_n+1/2 = T_n+1/2≡ T(ϕ_1/2h_n+1/2ξ_n) fails to obtain exact time symmetry, because in general ϕ_h/2≠ϕ_-h/2∘ϕ_h, such that T_n+1/2 obtained in the forward and backward directions differ.]h_n+1/2=μ(T_n,T_n+1) with T_n+1≡ T(ϕ_h_n+1/2∘ξ_n).Here, μ(x,y) denotes a general mean, satisfying μ(x,y)=μ(y,x) and min{x,y}≤μ(x,y)≤max{x,y}, for example the arithmetic meanh_n+1/2=12(T_n+T_n+1)<cit.>, the geometric meanh_n+1/2=√(T_nT_n+1),or the harmonic mean. Equation (<ref>) is a non-trivial implicit relation for h_n+1/2, which requires an iterative approach for an exact solution.For h discretised with the block-step, equation (<ref>) is naturally replaced by h_n+1/2=h_block(μ(T_n,T_n+1)),when the step size can only take a few discrete values, such that a finite number of iterations suffices for full convergence. However, owing to the interdependence of the particle trajectories, an exact solution requires iterating not just over the trajectory of each particle individually, but over the N-body trajectory of all particles combined. While this appears to give a reversible scheme <cit.>, it requires an enormous computational effort both in time and memory, and is completely unpractical. §.§ Extrapolating the time-step function One possibility to avoid such iterations is to estimate T_n+1 and obtain an approximately reversible scheme. The lowest-order approximation is simply T_n+1≈ T_n, when equation (<ref>) results in the forward scheme (<ref>). At the next order, one can use the previous value of T and estimate√(T_nT_n+1)≈ T_n (T_n/T_n-1)^2 h_n+1/2/2 h_n-1/2,which amounts to linear extrapolation of ln(T) in time. When inserted into equation (<ref>) this still gives an implicit relation for h_n+1/2, but with a trivial dependence. For the block-step with |δ r|≤1, h_n+1/2 can take only three allowed values and one can easily solve this relation to obtain the schemeδ r = {[ -1 if h_n-1/2 T_n-1≤ 12T_n^2 and block-step allows,;0 if h_n-1/2^2 T_n-1≤ T_n^3,; +1 otherwise. ].Strictly, these conditions are not unique if h_n-1/2≥2T_n. While this should never occur for appropriate time-step functions, it can be easily resolved by testing for longer steps first, say.Equation (<ref>) has an error ∝ h^2 (^2ln(T)/ t^2), but since even-order time derivatives are time symmetric, the synchronisation error of scheme (<ref>) is 𝒪(h^3) as opposed to 𝒪(h) for the forward scheme.Fig. <ref> shows the ratio of long- to short-term energy error over 10^4 periods with this scheme for exactly the same two orbits already considered above with the naive forward scheme. The net rate R_err of stepping errors is much smaller than for the forward scheme (<ref>), representing a significant improvement. As a result, the long-term and orbital energy variations are comparable in stark contrast to the situation for the forward scheme (see Fig. <ref>). Another significant improvement over the forward method is that R_err decreases with η. As a consequence, the ratio of irreversibility-induced long-term energy errors to the orbital energy error decreases faster than ∝η.In this context, it is worth noting that not all of the measured long-term energy error is due to irreversible step-size changes (it also occurs for integrations without any violation of time symmetry, such as for the Kepler orbit integrated with η=0.01 in Fig. <ref> below). As already discussed in section <ref>, Noether's theorems and the dependence of h_block(τ) on absolute time suggest such pseudo-random behaviour of the long-term energy error. Alternatively, one may interpret this as a random walk caused by an incommensurability between period and time stepping (the cusp orbit exhibits some quasi-periodic behaviour of the error, clearly visible at η=0.01 in Figs. <ref>-<ref>, <ref> & <ref>, indicating a resonance between period and time stepping). §.§ A try-and-reject schemeThe extrapolation of the time-step function is only advisable if it is well-behaved, i.e. does not change much on time scales of itself, implying |Ṫ|≪1. This is not necessarily the case in N-body simulations, where fluctuations may occur due to close and insufficiently softened encounters as well as force approximation errors.Therefore, I now consider an alternative approach to deal with the implicit relations. If one ignores the violation of time symmetry resulting from the mutual dependence of the particle trajectories, one can consider each particle trajectory separately. Since there are only few discrete values allowed for h_n+1/2, one can, instead of iterating, simply try them one after the other, starting with the shortest allowed step.To this end, the position at t_n+12h_n-1/2, corresponding to halving the step size, is predicted first and the time-step function T at that point computed. If h_n-1/2>T the step size h_n+1/2=h_n-1/2 is too long (in the sense of equation <ref>) and instead the shorter step must be used, the first part of which has already be done by predicting the position. Otherwise, the step is continued to size h_n+1/2=h_n-1/2 at the end of which the procedure is repeated (subject to block-step synchronisation constraints).In order to continue the step, the kick-drift-kick leapfrog can be re-phrased as a predictor-corrector scheme. The predictor simply predicts position and velocity into the future assuming a constant acceleration a⃗=a⃗_beg equal to that at the startof the step. The prediction operation is associative and second-order accurate in position. A full time step consists of any number of predictions followed by a force computation (obtaining a⃗_end) and the correction stepv⃗←v⃗ + 12 h (a⃗_end-a⃗_beg).In practice, this scheme requires storage for a⃗_beg and h, the accumulated length of the step so far (which is incremented by the predictor). The scheme for a full time step for one particle is then as follows. (i) Increment the rung (r← r+1) to try a shorter step. (ii) Predict ξ until the rung r is block-step synchronised. (iii) Calculate T and the force (obtaining a⃗_end)[Strictly speaking, a⃗_end is only required in (v), but since T typically depends on the gravitational potential and/or acceleration, the efficiency saving by evaluating a⃗_end only when needed is not available.]. (iv) If 2h≤ T and if rung r-1 is block-step synchronised at time t+h, reject the step (discarding the force computation just made), decrement the rung (r← r-1), and recurse with point (ii). (v) Otherwise, finish the step by applying the correction (<ref>), then reset a⃗_beg←a⃗_end and h←0. This scheme is similar in spirit to that proposed by <cit.>, with the main difference that their scheme tests for longer steps first, while I test for shorter steps first. I should also note that this scheme violates time symmetry not only because of the mutual dependence of particle step sizes, but even when applied to a single orbit in a fixed potential. This is because the predictor is only time symmetric between the start and end of a step, but not with respect to intermediate times. The position predicted for the middle of a step (at unchanged step size) differs between forward and backward steps by 18h^2(a⃗_end-a⃗_beg), see also footnote <ref>. Consequently, the time-step function T evaluated at these positionsin (iii) differs too and with it possibly the decision about the step size in (iv).However, these violations of time symmetry are not obviously systematic and do not introduce a coherent arrow of time. Hence, no significant systematic energy drift is to be expected. Fig. <ref> shows that for the same two orbits used before the ratio of long- to short-term energy error is slightly smaller than for the extrapolation scheme form the previous sub-section.§.§ Combining extrapolation with try-and-rejectThe try-and-reject scheme is rather wasteful, as it requires (on average) twice as many force computations as it actually uses for the orbit integration. However, by combining extrapolation of the time-step function with the rejection idea, one may avoid to always try for a shorter step. One can parameterise the ignorance about the precise future values for T by a parameter λ≤1 and assume thatT_n+1/2≥ T_n λ^2 h_n+1/2/2 h_n-1/2,which corresponds to the assumption that Ṫ≳ln(λ). With this supposition, one can replace step (i) in the algorithm given in the previous sub-section by r ← r + δ r withδ r = {[ -1 if h_n-1/2≤12T_n λand block-step allows,; -0 if h_n-1/2^2 ≤ T_n^2λ,; -1 otherwise. ].For λ=0.8, Fig. <ref> shows the evolution of the ratio of long- to short-term energy error for the same orbits as considered before. The long-term error is similar to that for the try-and-reject scheme before (Fig. <ref>). For these orbits, the ratio of force calculations to time steps is down from 2 for the pure try-and-reject scheme to 1.19 and 1.36 for Kepler and cusp orbit, respectively. § ATTEMPTING EXPLICIT REVERSIBILITY The schemes discussed in the previous section attempt to obtain T_n+1/2, i.e. synchronise the right-hand side of equation (<ref>). Instead, the schemes considered in this section attempt to synchronise the left-hand side of equation (<ref>). For continuous step sizes (unconstrained by the blockstep scheme), this can be accomplished by swapping the roles of h and T in the implicit relation (<ref>), yielding <cit.>μ(h_n-1/2,h_n+1/2) = T_n,which is an explicit relation for h_n+1/2, i.e. requires no iterations for an exact solution. Since in reversed time h_n-1/2 and h_n+1/2 are swapped while T_n remains unaltered, this scheme is reversible.proposed the harmonic mean, 2/μ=1/x+1/y. I will use the geometric mean μ^2=xy, which of the one-parameter family of averages[ μ^α = 12[x^α+y^α]for α≠0,; lnμ = 12[ln x+ln y] for α=0 ]is the only one that always obtains a unique positive h_n+1/2 for any given T_n, h_n-1/2>0. In the remainder of this study, I stick to the geometric mean, but I experimented with various α without finding α=0 inferior.The resulting integration schemes for continuous h give long-term energy errors consistent with zero for the two test orbits used before (not shown). It seems expedient, therefore, to try to transfer this idea to the block-step scheme.For the further quantification of (the lack of) reversibility, I introduce the net rate of irreversible step-size changes in close analogy to the net rate of stepping errors:[ R_irr≡[ number of steps with h>h_back at Ṫ<0; + number of steps with h<h_back at Ṫ>0; - number of steps with h>h_back at Ṫ>0; - number of steps with h<h_back at Ṫ<0] /; 4lnumber of step-size changes. ]Here, h_back is the step size which the same scheme would have taken when applied after the step of size h in the backward direction. Like stepping errors, irreversibilities come in four flavours, depending on the sign of Ṫ and the relation between h and h_back. Note that R_irr is an exact measure, unlike R_err of equation (<ref>), which involved an estimation for the correct step size. §.§ Reversible adaptation of a continuous time step variableA direct porting of equation (<ref>) to discrete step sizes obtains a scheme only slightly better than the forward scheme, as described in the appendix. The problem is that for certain intermediate values of T the discrete step size h flips between two values bracketing T. These flips are reversible only if step-size changes are unlimited, which allows the possibility of arbitrary long steps.To avoid too long steps, one must not allow this flipping of h. This can be achieved by following the initial scheme (<ref>) with an auxiliary continuous variable τ instead of h and seth_n+1/2 = h_block(τ_n+1/2).However, implementing this directly via τ_n+1/2=T_n^2/τ_n-1/2(for μ the geometric mean) fails. The problem is that τ oscillates around T with amplitude increasing at each change in step size[ This increasing oscillation amplitude seems at odds with the reversible nature of the method. However, this can be understood in analogy to linear instability where time symmetry implies that each growing mode is accompanied by a decaying mode of no practical significance.]. If h remains unchanged, the relation (<ref>) places ln(τ_n-1/2), ln(T_n) and ln(τ_n+1/2) on a line when plotted against time (rather than step index). So, to avoid the amplification of oscillations when h changes, one may simply retain that linear relation, which givesτ_n+1/2=T_n(T_n/τ_n-1/2)^h_n+1/2/h_n-1/2instead of (<ref>). Together with equation (<ref>), this is an implicit relation for h_n+1/2 and τ_n+1/2 given the previous values and T_n. Again, solving these relations is straightforward, obtaining the schemeδ r = {[ -1 if h_n-1/2τ_n-1/2^2≤ 12 T_n^3 and block-step allows,;0 if h_n-1/2τ_n-1/2≤ T_n^2,; +1 4lotherwise. ].The resulting ratio of long- to short-term energy error for the two test orbits is shown in Fig. <ref> when started at apo-centre and Fig. <ref> when started half way between apo- and peri-centre. In the latter case, one sees a significant difference between an orbit integration started with τ_1/2=T_0 or τ_1/2=√(T_0T_1) (and h_1/2=h_block(τ_1/2) in both cases). This difference can be understood by a significant number of irreversible steps (as reported via R_irr).Fig. <ref> shows the mechanics of irreversible step-size changes with this scheme. They an occur for both signs of Ṫ and originate from an integration/extrapolation error of τ, which causes the value for τ in the middle of a long step to be different in the backward direction, if that step was not taken. The chance for such integration errors and hence irreversible step-size changes is greatly enhanced if the auxiliary quantity τ oscillates around the time-step function (as it does to some extent in Fig. <ref>). If τ_n-1/2=T_n-1/2, relation (<ref>) corresponds to a linear extrapolation of ln(τ)=ln(T) in time. But if τ_n-1/2≠ T_n-1/2 or if ln(T) has curvature, this extrapolation goes slightly wrong and causes step-to-step oscillations of τ around T, like those detected byfor non-discrete h. This explains why carefully choosing the initial τ greatly reduces the number of irreversible steps (and with it the energy error) reported in the bottom panels of Fig. <ref>.§.§ Can the situation be improved?However, this reduction of irreversible steps through careful choice of the initial τ works only as long as the time-step function is sufficiently smooth. This is not necessarily the case in N-body simulations, where T(ξ) may fluctuate.In order to reduce the number of irreversible step-size changes, one may devise techniques to suppress these oscillations, even though this inevitably introduces another type of irreversibility. One technique is to re-align τ=T when log_2τ is close to a half-integer, i.e. as far away from any step-size changes as possible. The resulting test orbit integrations give comparable results to those obtained with a careful choice of the first step size (Fig. <ref>, bottom panels).Another idea is to avoid irreversible step-size changes by means of the try-and-reject method of section <ref> (which violates time reversibility in a more subtle and less harmful way than irreversible steps). Because one cannot detect the violation of reversibility in time to avoid it, the only possibility is to try avoiding stepping errors. Then, instead to prefer the longer step in case of ambiguity (as per the argument of section <ref>), one tries the shorter step but rejects it if the longer step is found acceptable. However, this introduces a new type of irreversible rung change, because not all stepping errors of the scheme (<ref>) are irreversible. § INTEGRATING THE STEP SIZE The time stepping schemes considered in the previous two sections attempt to synchronise the right- and left-hand sides of h=T (equation <ref>), respectively. I now consider schemes which instead synchronise a differential form, likeΔ h^-1 = Δ T^-1or Δln(h) = Δln(T). The left-hand and right-hand sides of these relations can be expressed as finite difference and differential, respectively. For non-discrete step sizes, this obtains the schemes 1/h_n+1/2 - 1/h_n-1/2 = - Ṫ_n/T_n<cit.> orln h_n+1/2 - ln h_n-1/2 = Ṫ_n. According toand my own experiments, these schemes perform very well for continuous step sizes (no block-step): even for chaotic orbits h closely follows T. Most importantly, this scheme avoids step-by-step oscillations of h, and hence is unlikely to suffer from any significant number of irreversible step-size changes, when adapted to the block-step.The price to pay is the need to compute the time derivative Ṫ of the time-step function. This is usually not efficiently possible in large-N simulations, but is a reasonable option for small-N methods. Since the block-step does not allow incremental changes to h, one cannot use equations (<ref>) directly, but must use an auxiliary variable τ. One must then also account for the discrete step over which τ is integrated, giving 1/τ_n+1/2 = 1/τ_n-1/2 - h_n-1/2+h_n+1/2/2Ṫ_n/T_n^2, lnτ_n+1/2 = lnτ_n-1/2+ h_n-1/2+h_n+1/2/2Ṫ_n/T_nin place of equations (<ref>).Combined with equation (<ref>), one again has trivially solvable implicit relations for h_n+1/2. When restricting rung changes to |δ r|≤1 as usual, this obtains the stepping schemesδ r= {[ -1 if2/τ_n-1/2 -3d/T_n≤1/h_n-1/2 and block-step allows,;0 if 1/τ_n-1/2 -d/T_n≤1/h_n-1/2; +1 4lotherwise, ].δ r= {[ -1 if ρ_n-1/2-3/2(d/ln2)≤r_n-1/2-1 and block-step allows,; -0 if ρ_n-1/2-3/2(d/ln2)≤ r_n-1/2,; +1 4lotherwise. ]. where d ≡ h_n-1/2(Ṫ_n/T_n), while ρ≡-log_2(τ/h_max) is a continuous rung variable replacing τ. To initialise τ, one must integrate the first half time step, i.e.1/τ_1/2 = 1/T_0 - h_1/2Ṫ_0/2 T_0^2, τ_1/2 =T_0 exp(h_1/2Ṫ_0/2T_0). Fig. <ref> shows the run of the ratio of long- to short-term energy error for this scheme and the two test orbits. There are still very few irreversible step-size changes, which can occur for either sign of Ṫ. These are caused by the same basic mechanism that is also responsible for irreversible step-size changes with the scheme (<ref>), see Fig. <ref>. However, because the time integration of τ is more accurate and avoids oscillations, the schemes (<ref>) only suffer very occasionally fromirreversible rung changes. For the e=0.9 Kepler orbit, for example, no irreversible step-size changed occurred for η=0.01 over 10^4 orbits (the small-amplitude trend of the long-term energy for this orbit must be owed to other sources, see also the last paragraph of Section <ref>).One can differentiate the relation between h and T once more and numerically integrate a second-order differential equation for τ, for example using the kick-drift-kick method. This obtains τ_n+1reversibly predicted from τ_n and τ̇_n and enables the time-step condition h_n+1/2≤μ(τ_n,τ_n+1). Such an approach is conceptually very similar to a zonal time-step function T=T(x⃗), discussed in section <ref> below. It is quiestionable, however, whether this significantly reduces the chances of irreversible step-size changes, since the integration schemes (<ref>) are already second-order accurate. § DISCUSSION The usage of individual adaptive time-step sizes in N-body simulations can result in enormous efficiency savings. Without these, hardly any of the many computer simulations of stellar dynamics, large-scale structure formation, and galaxy formation would have been possible. However, this technique still suffers from fundamental limitations in that almost all contemporary implementations violate time symmetry at the basic level, by setting the step size equal to its desired value at the beginning of each step (the forward method). Such violations are well known to affect the long-term stability of the simulations, resulting in artificial dissipative behaviour, which is often (but not necessarily) revealed by a secular drift of the total energy.§.§ Can exact reversibility be achieved? Adapting the size h of each time step reversibly is non-trivial because thetime-step function T(ξ), which provides the ideal value for h given the state ξ of the trajectory, can only be evaluated at the start of each step to inform the choice of its size h. Thus, T and h are not naturally synchronised and to achieve time reversibility some form of synchronisation is necessary.This problem of synchronising h and T also exists in the case of a single continuous step size (unconstrained by the block-step scheme). This simpler situation has been well studied and several solutions have been proposed and demonstrated to give good results <cit.>. These methods essentially integrate the step size itself in a reversible way.However, porting these methods to the block-step scheme and at the same time retaining reversibility appears impossible (at least when efficiency is retained. So how does the block step render the problem so much more difficult?§.§.§ Discreteness and the horizon of predictability The effect of the block-step scheme is to discretise the step sizes h. Without such discretisation synchronisation between T and h can be achieved by exactly solving an explicit reversible adaption condition, such as equations (<ref>) or (<ref>). When discretising h, this approach is no longer viable, as none of the discrete values will solve the reversibility condition exactly.Instead, one has to follow an explicit and reversible adaption condition with an auxiliary continuous variable τ, which in turn is then used to inform the choice for the discretised h. Complications arise because the time evolution of τ depends on the actual discrete step sizes h used, which creates a mutual dependency between h and τ and hence an implicit relation for h. Because only a finite number of discrete values are allowed for h, this implicit relation can be solved without iterations, using the longest allowed step in case of ambiguity.The reason for preferring the longest allowed step is that the merit of the alternative, a pair of two shorter steps, cannot be assessed without violating time symmetry, since the second short step is still in the future and beyond the horizon of predictability.The integration of the continuous step-size variable τ is inevitably subject to errors and, as a consequence, different values are obtained by different sequences of steps. Consider the situation depicted in Fig <ref>. If two short steps are taken between t_0 and t_2. then the value τ_for for τ(t_1) predicted at time t_0 in the forward direction differs (in general) from τ_back, the value predicted for the same quantity at time t_2 in the backward direction.If this difference is that between taking the long step from t_0 to t_2 or two shorter steps, an irreversible step-size change results. Thus, if two short steps are taken instead of one long one, then the value τ_back for the alternative longer step is beyond the horizon of predictability at the moment τ_for is computed. This mechanism only produces irreversibly step-size changes with h_for < h_back but of either sign of Ṫ. This is exactly what I find for the schemes (<ref>) and (<ref>): all irreversible step-size changes are of this kind and their frequency is substantially larger when removing the block-step's hierarchical synchronisation (but still requiring discretised step sizes). §.§.§ The case of zonal time-step functions A zonal time-step function T=T(x⃗) depends only on the particle position. This is useful in simulations of near-equilibrium galactic dynamics, when the local orbital time is well described by a simple function of radius. The discrete step sizes allowed by the block-step then correspond to radial zones <cit.>.For zonal time-step functions (and using the kick-drift-kick leapfrog) it seems that exact reversibility can be obtained with the block-step, because the value T_n+1 can be readily computed from the information known at time t_n. , for example, uses for h_n+1/2 the minimum of the discrete step sizes required at t_n and t_n+1. However, the trajectory is subject to integration errors and hence so is the step size, such that the problem discussed in the previous section still pertains, as explained in Fig. <ref>, and even for zonal time-step functions exact reversibility is elusive.§.§ Applicability to contemporary N-body methodsThe importance of the adverse effects of irreversible step-size adaptation varies considerably between different applications of the N-body method and so does the necessity to and potential benefit of applying any of the methods considered in this study. One potentially serious problem is the noise level, as quantified by |Ṫ|, of any practical time-step-function implementation, which for most of the adaptation schemes will result in poorer reversibility and hence long-term stability than for single-orbit integrations. §.§.§ Simulations of collision-less dynamicsIn contemporary N-body simulations of collision-less stellar dynamics (galaxy formation and interactions, as well as large-scale structure formation) the adverse effects of irreversible step-size adaptation from the naive forward scheme are presumably tolerable, i.e. the resulting errors remain small over the duration of the simulations (though this has not been rigorously validated), for two reasons. First, such simulations only cover ∼10^2-3 dynamical times, many fewer than the test orbit integrations presented here.Second, the time-step function most commonly used in such simulations, T=η√(ϵ/|a⃗|), where ϵ is the softening length and a⃗ the acceleration, varies only weakly over typical orbits. (In fact, this time-step function is sub-optimal, since for the vast majority of orbits it gives unnecessarily short time steps, seefor a better alternative.) Moreover, in situations where ϵ is smaller than the inter-particle separation close encounters will result in |Ṫ|≪̸1. For most of the stepping schemes discussed, this inevitably leads to a degradation of the performance compared to the simple test-orbit integrations considered in this study. Thus, if a scheme with considerably better reversibility than the state-of-the-art forward method is required, some form of adaptive force softening appears desirable. §.§.§ Simulations of collisional dynamicsSimulations of star clusters, on the other hand, span usually ∼10^4-6 dynamical times, when adverse effects from violating time symmetry may well become important. However, such simulations are usually done very carefully, typically by controlling or monitoring the total energy error and hence guarding against artificial secular dissipation. In this case, the total net energy error may still be dominated by the adverse effects of the step-size adaptation. If this is so, then a more careful scheme for adapting the step sizes may improve the efficiency of collisional N-body simulations. The problem of fluctuating time-step functions is likely much less of a problem than for collision-less dynamics, simply because shorter time steps are chosen for enhanced accuracy. This directly reduces |Ṫ| and with it the rate of irreversible step changes for all of the adaptation schemes considered in this study and in contrast to the forward method. §.§.§ Simulations of planetary systemsFinally, N-body simulations of planetary systems span ∼10^8-12 dynamical times, enough for even mild secular dissipation to accumulate. In order to avoid this, such integrations are typically done with a single global time-step size (which may be adapted).Of all the block-step based adaptation schemes considered in this study, only those of Section <ref> appear at all suitable for integrations of planetary systems. These techniques avoid oscillations of the auxiliary continuous step-size variable τ and only suffer from the apparently unavoidable irreversibilities of the type described in Figs. <ref>&<ref> and Section <ref>.In order to assess how the rate of these irreversible step-size changes depends on the intended integration accuracy, i.e. on the parameter η of the time-step function (<ref>), I ran some integrations of the e=0.9 Kepler orbit for 10^6 orbits (100 times longer than in Fig. <ref>), see Fig. <ref>. Encouragingly, the net-rate R_irr of irreversible step-size changes (but also their total number) decrease like η^2 (the same scaling as for the energy error). This is because decreasing η increases the integration accuracy not only of the trajectory, but also of the continuous step-size variable τ, and hence reduces the chances of irreversible step-size changes.As a consequence, the ratio of the irreversibility-driven long-term energy error over the short-term energy error decreases like ∼η^3 with decreasing η, independent of the order of the integrator (which determines the short-term energy error). As a consequence, the effects of irreversibilities on the long-term energy error are hardly relevant for sufficiently small η. Indeed, there is no visible correlation between the instances of irreversible step-size changes in the bottom panel of Fig. <ref> (indicated by thin red lines on top and bottom) and the run of the long-term energy error. This suggests that the methods of Section <ref> may well be useful in long-term integrations of planetary system dynamics. § CONCLUSION My attempts to improve on the simple forward method for adapting individual particle step sizes were met with varied success. When merely trying to reduce deviations form reversibility (section <ref>), progress is possible either by extrapolating the step size or by a try-and-reject approach, which however comes at increased computational costs. Porting these methods to N-body simulations requires a well-behaved time-step function (without short-term fluctuations), which may require adaptive force softening.The situation is more complicated when attempting explicit reversibility. Adapting the scheme of <cit.> to the block-scheme step (section <ref>) obtains methods that suffer from oscillations of the step size or of an auxiliary continuous step-size variable. These oscillations are already present in the original method of , but are much more problematic with the block-step scheme, where they cause irreversible step-size changes. More promising is the idea, pursued in section <ref>, to integrate a continuous step-size variable, analogous to the method of <cit.> for non-discrete h. This avoids the oscillations and obtains near-reversibity, but requires the computation of the time derivative of the time-step function. Such a method may well be useful in long-term integration of planetary systems, when reversibility is much more important than in any other N-body model.It appears that no practical (explicit or nearly explicit) scheme exists to adapt individual block-step discretised particle step sizes exactly reversibly. As outlined in sub-section <ref>, the ultimate reason appears to be the discretisation of the step size itself. § ACKNOWLEDGEMENTSI thank David M. Hernandez for a critical reading and Scott Tremaine for useful conversations on symplecticity and time symmetry. WD is partly supported by STFC grant ST/N000757/1.mnras§ A FLIPPING SCHEME The obvious way to adapt the adaptation method (<ref>) to discrete step sizes is (with μ the geometric mean) to seth_n+1/2 = h_block(T_n^2/h_n-1/2),which gives the schemeδ r = {[ -1 ifh_n-1/2≤1√(2) T_n and block-step allows,;0 ifh_n-1/2≤ T_n,; +1 4lotherwise. ].The only difference to the forward scheme (<ref>) is the factor 1/√(2) instead of 1/2 in the first clause. Fig. <ref> shows the ratio of long- to short-term energy error for this scheme. Obviously, this is only a little better than the forward scheme (<ref>). So obviously constraining the time steps to follow the block-step breaks the reversibility of this method, but how?Fig. <ref> plots the run of radius, h, T, and energy over the first orbit in the cusp model. One sees first that the step sizes flip between two values if 2^1/2-r≤(T/h_max)≤2^1-r for some rung r. By itself this flipping is reversible. However, there are ten such flips between steps 59 and 88 on the way to peri-centre, but only nine between steps 158 and 185 on the way out again – a violation of time symmetry. Upon closer inspection, one finds that the last flip on the in-falling part of the orbit (at step 88) is not reversible: in reversed time the scheme (<ref>) would not chose to change h=2^-7 to h=2^-6 at that moment.This is a consequence of the restriction to changes in h by at most a factor two. If allowing larger rung changes, the algorithm jumps from h=2^-6 to h=2^-8 at step 88 and in reversed time jumps back. Unfortunately, enabling |δ r|>1 allows the occasional much too long time step with disastrous consequences for the integration accuracy. Using different limits for δ r and/or functions μ(x,y) can give some improvement over the reported attempt, but did not yield a reliable method significantly better than the simple forward scheme (<ref>). Moreover, the flipping of step sizes is inefficient for N-body force solvers, since in some places half the particles will have rung r and the other r+1, switching every other step.The scheme (<ref>) only produces irreversible step-size adaptations of the first two types in equation (<ref>), which give energy errors of the same sign but opposite to the second two types. | http://arxiv.org/abs/1707.09069v1 | {
"authors": [
"Walter Dehnen"
],
"categories": [
"astro-ph.IM"
],
"primary_category": "astro-ph.IM",
"published": "20170727230642",
"title": "Towards time symmetric N-body integration"
} |
Towards de Sitter from 10D Hsi-Wei Yen December 30, 2023 ==========================Robots operating alongside humans in diverse, stochastic environments must be able to accurately interpret natural language commands. These instructions often fall into one of two categories: those that specify a goal condition or target state, and those that specify explicit actions, or how to perform a given task. Recent approaches have used reward functions as a semantic representation of goal-based commands, which allows for the use of a state-of-the-art planner to find a policy for the given task. However, these reward functions cannot be directly used to represent action-oriented commands.We introduce a new hybrid approach, the Deep Recurrent Action-Goal Grounding Network (DRAGGN), for task grounding and execution that handles natural language from either category as input, and generalizes to unseen environments. Our robot-simulation results demonstrate that a system successfully interpreting both goal-oriented and action-oriented task specifications brings us closer to robust natural language understanding for human-robot interaction. § INTRODUCTION Natural language affords a convenient choice for delivering instructions to robots, as it offers flexibility, familiarity, and does not require users to have knowledge of low-level programming. In the context of grounding natural language instructions to tasks, human-robot instructions can be interpreted as either high-level goal specifications or low-level instructions for the robot to execute.Goal-oriented commands define a particular target state specifying where a robot should end up, whereas action-oriented commands specify a particular sequence of actions to be executed. For example, a human instructing a robot to “go to the kitchen” outlines a goal condition to check if the robot is in the kitchen.Alternatively, a human providing the command “take three steps to the left” defines a trajectory for the robot to execute. We need to consider both forms of commands to understand the full space of natural language that humans may use to communicate their intent to robots. While humans also combine commands of both types into a single instruction, we make the simplifying assumption that a command belongs entirely to a single type and leave the task of handling mixtures and compositions to future work.Existing approaches can be broadly divided into one of two regimes. Goal-based approaches like <cit.> and <cit.> leverage some intermediate task representation and then automatically find a low-level trajectory to achieve the goal using a planner. Other approaches, in the action-oriented regime, directly infer action sequences <cit.> from the syntactic or semantic parse structure of natural language. However, these approaches can be computationally intractable for large state-action spaces or use ad-hoc methods to execute high-level language rather than relying on a planner. Furthermore, these methods are unable to adapt to dynamic changes in the environment; for example, consider an environment in which the wind, or some other force moves an object that a robot has been tasked with picking. Action sequence based approaches would fail to handle this without additional user input, while goal-based approaches would be able to re-plan on the fly, and complete the task.To address the issue of dealing with both goal-oriented and action-oriented commands, we present a new language grounding framework that, given a natural language command, is capable of inferring the latent command type. Recent approaches leveraging deep neural networks have formulated the language grounding problemas sequence-to-sequence learning or multi-label classification <cit.>. Inspired by the recent success of neural networks to model programs that are highly compositional and sequential in nature, we present the Deep Recurrent Action/Goal Grounding Network (DRAGGN) framework, derived from the the Neural Programmer-Interpreter (NPI) of <cit.> and outlined in Section <ref>. We introduce two instances of DRAGGN models, each with slightly different architectures. The first, the Joint-DRAGGN (J-DRAGGN) is defined in Section <ref>, while the second, the Independent-DRAGGN (I-DRAGGN) is defined in Section <ref>. § RELATED WORK There has been a broad and diverse set of work examining how best to interpret and execute natural language instructions on a robot platform <cit.>. <cit.> produce policies using language and expert trajectories based rewards, which allow for planning within a stochastic environment along with re-planning in case of failure. <cit.> instead grounds language to trajectories satisfying the language specification. <cit.> chose to ground language to constraints given to an external planner, which is a much smaller space to perform inference over than trajectories.<cit.> formulate language grounding as a machine translation problem, treating propositional logic functions as both a machine language and reward function.Reward functions or cost functions can allow richer descriptions of trajectories than plain constraints, as they can describe preferential paths. Additionally, <cit.> simplify the problem from one of machine translation to multi-class classification, learning a deep neural network to map arbitrary natural language instructions to the corresponding reward function.Informing our distinction between action sequences and goal state representation is the division presented by <cit.>, who posited that natural language can be interpreted as both a goal state specification and an action specification. Rather than producing both from each language command, our DRAGGN framework makes the simplifying assumption that only one representation captures the semantics of the language; additionally, our framework does not require a manually pre-specified grammar.Recently, deep neural networks have found widespread success and application to a wide array of problems dealing with natural language <cit.>. Unsurprisingly, there have been some initial steps taken towards applying neural networks to language grounding problems. <cit.> uses a recurrent neural network (RNN) with long short-term memory (LSTM) cells <cit.> to learn sequence-to-sequence mappings between natural language and robot actions. This model augments the standard sequence-to-sequence architecture by learning parameters that represent latent alignments between natural language tokens and robot actions. <cit.> used an RNN-based model to produce grounded reward functions at multiple levels of an Abstract Markov Decision Process hierarchy <cit.>, varying the abstraction level with the level of abstraction used in natural language. Our DRAGGN framework is closely related to the Neural Programmer-Interpreter (NPI) <cit.>. The original NPI model is a controller trained via supervised learning to interpret and learn when to call specific programs/subprograms, which arguments to pass into the currently active program, and when to terminate execution of the current program. We draw a parallel between inferred NPI programs and our method of predicting either lifted reward functions or action trajectories. § PROBLEM SETTINGWe consider the problem of mapping from natural language to robot actions within the context of Markov decision processes. A Markov decision process (MDP) is a five-tuple ⟨𝒮, 𝒜, 𝒯, ℛ, γ⟩defining a state space 𝒮, action space 𝒜, state transition probabilities 𝒯, reward function ℛ, and discount factor γ <cit.>. An MDP solver produces a policy that maps from states to actions in order to maximize the total expected discounted reward. While reward functions are flexible and expressive enough for a wide variety of task specifications, they are a brittle choice for specifying an exact sequence of actions, as enumerating every possible action sequence as a reward function (i.e. a specific reward function for the sequence Up 3, Down 2) can quickly become intractable. This paper introduces models that can produce desired behavior by inferring either reward functions or primitive actions. We assume that all available actions 𝒜 and the full space of potential reward functions (i.e., the full space of possible tasks) are known a priori. When a reward function is predicted by the model, an MDP planner is applied to derive the resultant policy (see system pipeline Figure <ref>).We focus our evaluation of all models on the the Cleanup World mobile-manipulator domain <cit.>. The Cleanup World domain consists of an agent in a 2-D world with uniquely colored rooms and movable objects. A domain instance is shown in Figure <ref>. The domain itself is implemented as an object-oriented Markov decision process (OO-MDP) where states are denoted entirely by collections of objects, with each object having its own identifier, type, and set of attributes <cit.>. Domain objects include rooms and interactable objects (e.g a chair, basket, etc.) all of which have location and color attributes. Propositional logic functions can be used to identify relevant pieces of an OO-MDP state and their attributes; as in <cit.> and <cit.>, we treat these propositional functions as reward functions. In Figure <ref>, the goal-oriented command “take the chair to the green room” may be represented with the reward function blockInRoom block0 room1, wherethe blockInRoom propositional function checks if the location attribute of block0 is contained in room1.§ APPROACHWe now outline the pipeline that converts natural language input to robot behavior. We begin by first defining the semantic task representation used by our grounding models that comes directly from the OO-MDP propositional functions of the domain. Next, we examine our novel DRAGGN framework for language grounding and, in particular, address the separate paths taken by action-oriented and goal-oriented commands through the system as seen in Figure <ref>. Finally, we discuss two different implementations of the DRAGGN framework that make different assumptions about the relationship between tasks and constraints. Specifically, we introduce the Joint-DRAGGN (J-DRAGGN), that assumes a probabilistic dependence between tasks (i.e. goUp) and the corresponding arguments (i.e. 5 steps) based on a natural language instruction, and the Independent-DRAGGN (I-DRAGGN) that treats tasks and arguments as independent given a natural language instruction. §.§ Semantic Representation In order to map arbitrary natural language instructions to either action trajectories or goal conditions, we require a compact but sufficiently expressive semantic representation for both. To this end, we define the callable unit, which takes the form of a single-argument function. These functions are paired with binding arguments whose possible values depend on the callable unit type. As in <cit.> and <cit.>, our approach generates reward function templates, or lifted reward functions, for goal-oriented tasks along with environment-specific constraints. Once these templates and constraints are resolved to get a grounded reward function, the associated goal-oriented tasks can be solved by an off-the-shelf planner thereby improving transfer and generalization capabilities.Goal-oriented callable units (lifted reward functions) are paired with binding arguments that specify properties of environment entities that must be satisfied in order to achieve the goal. These binding arguments are later resolved by the Grounding Module (see Section <ref>) to produce grounded reward functions (OO-MDP propositional logic functions) that are handled by an MDP planner. Action-oriented callable units directly correspond to the primitive actions available to the robot and are paired with binding arguments defining the number of sequential executions of that action. The full set of callable units along with requisite binding arguments is shown in Table <ref>.§.§ Deep Recurrent Action/Goal Grounding Network (DRAGGN)While the Single-RNN model of <cit.> is effective, it cannot model the compositional argument structure of language. A unit-argument pair not observed at training time will not be predicted from input data, even if the constituent pieces were observed separately. Additionally, the Single-RNN model requires every possible unit-argument pair to be enumerated, to form the output space. As the environment grows to include more objects with richer attributes, this output space becomes intractable.To resolve this, we introduce the Deep Recurrent Action/Goal Grounding Network (DRAGGN) framework. Unlike previous approaches, the DRAGGN framework maps natural language instructions to separate distributions over callable units and (possibly multiple) binding constraints, generating either action sequences or goal conditions. By treating callable units and binding arguments as separate entities, we circumvent the combinatorial dependence on the size of the domain. This unit-argument separation is inspired by the Neural Programmer-Interpreter (NPI) of <cit.>. The callable units output by DRAGGN are analogous to the subprograms output by NPI. Additionally, both NPI and DRAGGN allow for subprograms/callable units with an arbitrary number of arguments (by adding a corresponding number of Binding Argument Networks, as shown at the top right of Figure <ref>, each with its own output space).We assume that each natural language instruction can be represented by a single unit-argument pair with only one argument. Consequently, in our experiments, we assume that sentences specifying sequences of commands have been segmented, and each segment is given to the model one at a time. The limitation to a single argument only arises because of the domain's simplicity; as mentioned above, it is straightforward to extend our models to handle extra arguments by adding extra Binding Argument Networks.To formalize the DRAGGN objective, consider a natural language instruction l. Our goal is to find the callable unit ĉ and binding arguments â that maximize the following joint probability:ĉ, 𝐚̂ = max_c, 𝐚(c, 𝐚| l) Depending on the assumptions made about the relationship between callable units c and binding arguments 𝐚, we can decompose the above objective in two ways: preserving the dependence between the two, and learning the relationship between the units and arguments jointly, and treating the two as independent. These two decompositions result in the Joint-DRAGGN and Independent-DRAGGN models respectively.Given the training dataset of natural language and the space of unit-argument pairs, we train our DRAGGN models end-to-end by minimizing the sum of the cross-entropy losses between the predicted distributions and true labels for each separate distribution (i.e. over callable units and binding arguments). At inference time, we first choose the callable unit with the highest probability given the natural language instruction. We then choose the binding argument(s) with highest probability from the set of valid arguments. The validity of a binding argument given a callable unit is given a priori, by the specific environment, rather than being learned at training time. Our models were trained using Adam <cit.>, for 125 epochs, with a batch size of 16, and a learning rate of 0.0001. §.§ Joint DRAGGN (J-DRAGGN)The Joint DRAGGN (J-DRAGGN) models the joint probability in Equation <ref>, coupled via the shared RNN state in the DRAGGN Core (as depicted in Figure <ref>), but selects the optimizer sequentially, as follows:ĉ, 𝐚̂ = max_c, 𝐚(c, 𝐚| l) ≈max_𝐚 [ max_c(c, 𝐚| l) ] We first encode the constituent words of our natural language segment into fixed-size embedding vectors. From there, the sequence of word embeddings is fed through an RNN denoted by the DRAGGN Core[We use the gated recurrent unit (GRU) as our RNN cell, because of its effectiveness in natural language processing tasks, such as machine translation <cit.>, while requiring fewer parameters than the LSTM cell <cit.>.]. After processing the entire segment, the current gated recurrent unit (GRU) hidden state is then treated as a representative vector for the entire natural language segment. This single hidden core vector is then passed to both the Callable Unit Network and the Binding Argument Network, allowing for both networks to be trained jointly, enforcing a dependence between the two.The Callable Unit Network is a two-layer feed-forward network using rectified linear unit (ReLU) activation. It takes the DRAGGN Core output vector as input to produce a softmax probability distribution over all possible callable units. The Binding Argument Network is a separate network with an identical architecture and takes the same input, but instead produces a probability distribution over all possible binding arguments. The two models do not need to share the same architecture; for example, callable units with multiple arguments require multiple different argument networks, one for each possible binding constraint. §.§ Independent DRAGGN (I-DRAGGN)The Independent DRAGGN (I-DRAGGN), contrary to the Joint DRAGGN, decomposes the objective from Equation <ref> by treating callable units and binding arguments as being independent, given the original natural language instruction. More precisely, the I-DRAGGN objective is:ĉ, 𝐚̂ = max_c, 𝐚(c | l)(𝐚| l)The I-DRAGGN network architecture is shown in Figure <ref>. Beyond the difference in objective functions, there is another key difference between the I-DRAGGN and J-DRAGGN architectures. Rather than encoding the constituent words of the natural language instruction once, and feeding the resulting embeddings through a DRAGGN Core to generate a shared core vector, the I-DRAGGN model embeds and encodes the natural language instruction twice, using two separate embedding matrices and GRUs, one each for the callable unit and binding argument. In this way, the I-DRAGGN model encapsulates two disjoint neural networks, each with their own individual parameter sets that are trained independently. The latter half of each individual network (the Callable Unit Network and Binding Argument Network) remains the same as that of the J-DRAGGN. §.§ Grounding ModuleIf a goal-oriented callable unit is returned (i.e. a lifted reward function), we require an additional step of completing the reward function with environment-specific variables. As described in <cit.>, we use a Grounding Module to perform this step. The Grounding Module maps the inferred callable unit and binding argument(s) to a final grounded reward function that can be passed to an MDP planner. In our implementation, the Grounding Module is a lookup table mapping specific binding arguments to room ID tokens. A more advanced implementation of the Grounding Module would be required in order to handle domains with non-unique binding arguments (e.g. resolving between multiple objects with overlapping attributes). § EXPERIMENTSWe assess the effectiveness of both our J-DRAGGN and I-DRAGGN models via instruction grounding accuracy for robot navigation and mobile-manipulation tasks. As a baseline, we compare against the state-of-the-art Single-RNN model introduced by <cit.>.§.§ Procedure To conduct our evaluation, we use the dataset of natural language commands for the single instance of Cleanup World domain seen in Figure <ref>, from <cit.>. In the user study, Amazon Mechanical Turk users were presented with trajectory demonstrations of a robot completing various navigation and object manipulation tasks. Users were prompted to provide natural language commands that they believed would have generated the observed behavior. Since the original dataset was compiled for analyzing the hierarchical nature of language, we were easily able to filter the commands down to only those using high-level goal specifications and low-level trajectory specifications. This resulted in a dataset of 3734 natural language commands total.To produce a dataset of action-specifying callable units, experts annotated low-level trajectory specifications from the <cit.> dataset. For example, the command “Down three paces, then up two paces, finally left four paces” was segmented into “down three spaces,” “then up two paces,”“finally left four paces,” and was given a corresponding execution trace of goDown 3, goUp 2, goLeft 4. The existing set of grounded reward functions in the dataset were converted to callable units and binding arguments. Examples of both types of language are presented in Table <ref> with their corresponding callable unit and binding arguments.To fully show the capabilities of our model, we tested on two separate versions of the dataset. The first is the standard dataset, consisting of a 90-10 split of the collected action-oriented and goal-oriented commands We also evaluated our models on an “unseen” dataset, which consists of a specific train-test split that evaluates how well models can predict previously unseen action sequence combinations. For example, in this dataset the training data might consist only of action sequences of the form goUp 3, and goDown 4, while the test data would only consist of the “unseen” action sequence goUp 4. Note that in both datasets, we assume that the test environment is configured the same as the train environment. §.§ Results Language grounding accuracies for our two DRAGGN models, as well as the baseline Single-RNN, are presented in Table <ref>. All three models received the same set of training data, consisting of 2660 low-level action-oriented segments and 693 high-level goal-based sentences. All together, there are 17 unique combinations action-oriented callable units and respective binding arguments, and 6 unique combinations of goal-oriented callable units and binding arguments present in the data. Then, we evaluated all three models on the same set of held-out data, which consisted of 295 low-level segments and 86 high-level sentences.In aggregate, the models that use callable units for both action- and goal-based language grounding demonstrate superior performance to the Single-RNN baseline, largely due to their ability to generalize, and output combinations unseen at train time. We break down the performance on each task in the following three sections.§.§ Action Prediction We evaluate the performance of our models on low-level language that directly specifies an action trajectory. An instruction is correctly grounded if the output trajectory specification corresponds to the ground-truth action sequence. To ensure fairness, we augment the output space of Single-RNN to include all distinct action trajectories found in the training data (an additional 17 classes, as mentioned previously). All models perform generally well on this task, with Single-RNN correctly identifying the correct action callable unit on 95.8% of test samples, while both DRAGGN models slightly outperform with on 96.6% and 97.0% respectively.§.§ Goal PredictionIn addition to the action-oriented results, we evaluate the ability for each model to ground goal-based commands. An instruction is correctly grounded if the output of the grounding module corresponds to the ground-truth (grounded) reward function.In our domain, all models predict the correct grounded reward function with an accuracy of 84.9% or higher, with the Single-RNN and J-DRAGGN models being too close to call. §.§ Unseen Action Prediction The Single-RNN baseline model is completely unable to produce unit-argument pairs that were never seen during training, whereas both DRAGGN models demonstrate some capacity for generalization. The I-DRAGGN model in particular demonstrates a strong understanding of each token within the original natural language utterances which, in large part, comes from the separate embedding spaces maintained for callable units and binding constraints respectively.§ DISCUSSIONOur experiments show that the DRAGGN models have a clear advantage over the existing state-of-the-art in grounding action-oriented language. Furthermore, due to the factored nature of the output, I-DRAGGN generalizes well to unseen combinations of callable units and binding arguments.Nevertheless, I-DRAGGN did not perform as well as Single-RNN and J-DRAGGN on goal-oriented language. This is possibly due to the small number of goal types in the dataset and the strong overlap in goal-oriented language. Whereas the Single-RNN and J-DRAGGN architectures may experience some positive transfer of information (due to the shared parameters in each of the two models), the I-DRAGGN model does not because of its assumed independence between callable units and binding arguments. This ability to allow for positive information transfer suggests that J-DRAGGN would perform best in environments where there is a strong overlap in the instructional language, with a relatively smaller but complex set of possible action sequences and goal conditions.On action-oriented language, J-DRAGGN has grounding accuracy of around 20.2% while I-DRAGGN achieves a near-perfect 97.0%. Since J-DRAGGN only encodes the input language instruction once, the resulting vector representation is forced to characterize both callable unit and binding argument features. While this can result in positive information transfer and improve grounding accuracy in some cases (e.g. goal-based language), this enforced correlation heavily biases the model towards predicting combinations it has seen before. By learning separate representations for callable units and binding arguments,I-DRAGGN is able to generalize significantly better. This suggests that I-DRAGGN would perform best in situations where the instructional language consists of many disjoint words and phrases.While our results demonstrate that the DRAGGN framework is effective, more experimentation is needed to fully explore the possibilities and weaknesses of such models. One of the shortcomings in the DRAGGN models is the need for segmented data. We found that all evaluated models were unable to handle long, compositional instructions, such as “Go up three steps, then down two steps, then left five steps”. Handling conjunctions of low-level commands requires extending our model to learn how to perform segmentation, or producing sequences of callable units and arguments. § CONCLUSIONIn this paper, we presented the Deep Recurrent Action/Goal Grounding Network (DRAGGN), a hybrid approach that grounds natural language commands to either action sequences or goal conditions, depending on the language. We presented two separate neural network architectures that can accomplish this task, both of which factor the output space according to the compositional structure of our semantic representation. We show that overall the DRAGGN models significantly outperform the existing state of the art. Most notably, we show that the DRAGGN models are capable of generalizing to action sequences unseen during training time.Despite these successes, there are still open challenges with grounding language to novel, unseen environment configurations. Furthermore, we hope to extend our models to handleinstructions that are a mixture of goal-oriented and action-oriented language, as well as to long, sequential commands. An instruction such as “go to the blue room, but avoid going through the red hallway” does not map to either an action sequence or a traditional, Markovian reward function. We believe new tools and approaches will need to be developed to handle such instructions, in order to handle the diversity and complexity of human natural language. § ACKNOWLEDGEMENTS This material is based upon work supported by the National Science Foundation under grant number IIS-1637614 and the National Aeronautics and Space Administration under grant number NNX16AR61G.Lawson L.S. Wong was supported by a Croucher Foundation Fellowship. acl_natbib | http://arxiv.org/abs/1707.08668v1 | {
"authors": [
"Siddharth Karamcheti",
"Edward C. Williams",
"Dilip Arumugam",
"Mina Rhee",
"Nakul Gopalan",
"Lawson L. S. Wong",
"Stefanie Tellex"
],
"categories": [
"cs.AI",
"cs.CL"
],
"primary_category": "cs.AI",
"published": "20170726235729",
"title": "A Tale of Two DRAGGNs: A Hybrid Approach for Interpreting Action-Oriented and Goal-Oriented Instructions"
} |
Pseudo-spin Skyrmions in the Phase Diagram of Cuprate Superconductors C. Pépin December 30, 2023 ===================================================================== We prove a generalization of the Fujita-Kawamata-Zuo semi-positivity Theorem <cit.> for filtered regular meromorphic Higgs bundles and tame harmonic bundles. Our approach gives a new proof in the cases already considered by these authors. We give also an application to the geometry of smooth quasi-projective complex varieties admitting a semisimple complex local system with infinite monodromy.§ STATEMENT OF THE MAIN RESULTS If X is a smooth projective complex variety and D = ∪_i ∈ I D_i ⊂ X is a simple normal crossing divisor, recall that a filtered regular meromorphic Higgs bundle (, θ) on (X,D) consists of a parabolic bundletogether with a Higgs field θ which is logarithmic with respect to the parabolic structure along D, meaning that for any parabolic weight α∈^I we haveθ :^α^α⊗Ω^1_X(log D).See section <ref> for a reminder. We denote by ^♢ the vector bundleinwith parabolic weight (0, ⋯, 0) ∈^I. Our first result is a generalization of Fujita-Kawamata-Zuo semi-positivity Theorem <cit.>, see Remarks <ref> for a discussion of former results. Let X be a smooth projective complex variety,be an ample line bundle on X and D ⊂ X be a simple normal crossing divisor. Let (, θ) be a filtered regular meromorphic Higgs bundle on (X,D). Assume that (, θ) is μ_-polystable with vanishing parabolic Chern classes. Ifis a subsheaf of ^♢ contained in the kernel of θ, then its dual ^⋆ is weakly positive. The somewhat technical definition of a weakly positive torsion-free coherent sheaf, which is a generalisation of the notion of pseudo-effective line bundles for higher-rank sheaves, is recalled in section <ref>. We will show more precisely that the restricted base locus _-(^⋆) of ^⋆ is contained in the union of D and the locus in X whereis not a locally split subsheaf of ^♢, cf. Theorem <ref>.In the special case where the parabolic filtration is trivial, one has the following more precise result: Let X be a smooth projective complex variety,be an ample line bundle on X and D ⊂ X be a simple normal crossing divisor. Let (, θ) be a μ_-polystable logarithmic Higgs bundle on (X,D) with vanishing Chern classes. Ifis a locally split subsheaf ofcontained in the kernel of θ, then its dual ^⋆ is nef.Let X,and D as before, and set j : U := X \ D ↪ X the inclusion. Given a μ_-polystable filtered regular meromorphic Higgs bundles (, θ) on (X,D) with vanishing parabolic Chern classes, it follows from the works of Simpson <cit.>, Biquard <cit.> and Mochizuki <cit.> that there exists an essentially unique tame pluriharmonic metric h on (^♢_|U, θ) whose asymptotic behaviour is controlled by the parabolic vector bundle . The triple (^♢_|U, θ , h) defines a tame harmonic bundle on U, and ^♢ can be recovered from this data as the subsheaf of j_*(^♢_|U) whose sections have sub-polynomial growth with respect to h. Using this, we will see that Theorem <ref> is a consequence of the following result: Let X be a complex manifold, and D ⊂ X be a simple normal crossing divisor. Set U := X \ D and j : U ↪ X the inclusion. Let (, θ , h) be a tame harmonic bundle on U, andbe a subsheaf of . The hermitian metric h induces a (possibly singular) hermitian metric h_ on . Denote by ^♢ the subsheaf of j_* whose sections have sub-polynomial growth with respect to h_. Then ^♢ is a torsion-free coherent sheaf on X. Moreover, ifis contained in the kernel of θ, then there exists a unique singular hermitian metric with semi-negative curvature on ^♢ whose restriction to U is h_.For the notion of singular hermitian metric on a torsion-free coherent sheaf, the reader is referred to section <ref>. Theorem <ref> follows then from the computation of the Lelong numbers of the induced singular hermitian metric on the dual sheaf ^⋆, see Proposition <ref>.An important example of tame harmonic bundle is given by the following construction. Let X be a complex manifold, and D ⊂ X be a simple normal crossing divisor. Let (, ∇ ,^, h) be a complex polarized variation of Hodge structures (-PVHS) on U := X \ D, and denote by (, θ) the associated Higgs bundle on U, i.e. := _ and θ := _∇. If h_ denotes the positive-definite hermitian metric oninduced by h, then the triple (, θ , h_) defines a tame harmonic bundle on U, cf. section <ref>. As a direct consequence of Theorem <ref>, we get the Let X be a complex manifold, and D ⊂ X be a simple normal crossing divisor. Let (, ∇ ,^, h) be a log -PVHS on (X,D), and denote by (, θ) the corresponding log-Higgs bundle. Assume that the eigenvalues of the residues of (, ∇) along the irreducible components of D, which are known to be real numbers, are non-negative.Ifis a subsheaf ofcontained in the kernel of the Higgs field θ, then the (possibly singular) hermitian metric h_ on _|U induced by h_ extends uniquely as a singular hermitian metric onwith semi-negative curvature.In the situation of Theorem <ref>, if moreover X is projective, then the dual ^⋆ ofis weakly positive. * In the situation of Corollary <ref>, if one assumes moreover that the residues of (, ∇) are nilpotent and thatis a locally split subsheaf ofcontained in the kernel of the Higgs field θ, then one can apply Theorem <ref> to get that its dual ^⋆ is nef. This special case is already known, cf. <cit.> and <cit.>. See also <cit.> for a different proof. From this, one can easily deduce Corollary <ref> in the special case where the eigenvalues of the residues are rational numbers, as explained for example in <cit.>. * With the notations of Theorem <ref>, let p be the biggest integer such that ^p=. It follows from Griffiths’ transversality that Gr^p_ = ^p+1 is a subbundle ofcontained in the kernel of the Higgs field, so that its dual is weakly positive (note that we make no assumption on the monodromy at infinity). When the monodromy at infinity is unipotent, (Gr^p_ )^⋆ is canonically isomorphic to the lowest piece of the Hodge filtration of the dual log -PVHS. Therefore, Theorem <ref> implies that the lowest piece of the Hodge filtration of a log -PVHS with unipotent monodromy at infinity is nef, as previously shown for real polarized variation of Hodge structures by Fujita <cit.> and Zucker <cit.> for curves, and by Kawamata <cit.>, Fujino-Fujisawa <cit.> and Fujino-Fujisawa-Saito <cit.> in higher dimensions. An algebraic proof of this last result appears in <cit.>. Theorem <ref> has some consequences for the geometry of smooth quasi-projective complex varieties admiting a complex local system with infinite monodromy that we now discuss. In analogy with <cit.> we say that a complex local system L on a smooth quasi-projective complex variety X is generically large when there exists countably many closed subvarieties D_i ⊊ X such that for every smooth quasi-projective complex variety Z equipped with a proper map f : ZX satisfying f(Z) ⊄∪ D_i the pullback local system f^∗ L is non-trivial. This condition is less exceptional that one might first thinks, since conjecturally at least any semisimple complex local system is virtually the pull-back of a generically large local system.Let X be a smooth projective complex variety and D ⊂ X be a simple normal crossing divisor. If there exists a generically large semisimple complex local system with discrete monodromy on X \ D, then the logarithmic cotangent bundle Ω^1_X(log D) of (X,D) is weakly positive. Under the same assumptions, if the local system underlies a complex polarized variation of Hodge structures, then it is known that (X,D) is of log-general type, cf. <cit.>. More generally, we conjecture that (X,D) is of log-general type if there exists a generically large semisimple complex local system on X \ D whose monodromy representation is Zariski-dense in a semisimple complex algebraic group, as already known when D = ∅ <cit.>.§ A REVIEW OF THE NON-ABELIAN HODGE CORRESPONDENCE FOR QUASI-PROJECTIVE VARIETIES In this section, we give a brief review of the non-abelian Hodge correspondence for smooth quasi-projective complex varieties, which is one of the key input of our approach. References include <cit.>. §.§ Parabolic bundlesWe follow <cit.> and <cit.>, however be aware that our terminology differs slightly from theirs. Let X be a complex manifold and D = ∪_i ∈ I D_i ⊂ X be a simple normal crossing divisor. A parabolic sheafon (X,D) is a meromorphic bundleon (X,D), i.e. a torsion-free coherent _X[*D]-module, endowed with a collection of torsion-free coherent _X-submodules ^α indexed by multi-indices α = (α_i)_i ∈ I with α_i ∈, satisfying the following conditions:* (the filtration is exhaustive and decreasing) = ∪_α^α and ^α↪^β whenever α≥β (i.e. α_i ≥β_i for all i),* (normalization/support) ^α + δ^i = ^α(- D_i), where δ^i denotes the multi-index δ^i_i =1 and δ^i_j =0 for j ≠ i,* (semicontinuity) for any given α there exists c > 0 such that for any multi-index ϵ with 0 ≤ϵ_i < c we have ^α - ϵ = ^α. We denote by ^♢ the subsheaf ofwith parabolic weight (0, ⋯, 0) ∈^I. Fixing c ∈, it follows from the axioms that a parabolic sheaf is determined by the ^α for the collection of jumping indices α with c ≤α_i < c +1 for any c ∈, and this collection is finite when D has a finite number of irreducible components.We say thatis locally abelian if in a Zariski neighborhood of any point x ∈ X,is isomorphic to a direct sum of parabolic line bundles (i.e. parabolic sheaves which are locally-free of rank 1). A parabolic bundle on (X,D) is a parabolic sheaf wich is locally abelian. In particular, all the ^α are locally-free. The parabolic structure is called trivial when all coefficients of the jumping indices are integers.There is a notion of parabolic Chern classes for parabolic bundles (we refer the reader to <cit.> and <cit.> for the quite complicated formulas). Whenis a parabolic bundle with trivial parabolic structure, then its parabolic Chern classes coincide with those of ^♢. §.§ Prolongation according to growth conditionsLet X be an n-dimensional complex manifold, and let D= ∪_ i ∈ ID_i be a simple normal crossing divisor. Let P be a point of X, and let D_1, ⋯, D_l be the irreducible components of D containing P. An admissible coordinate system around P is a pair (U, ψ), where: * U is an open subset of X containing P,* ψ is a holomorphic isomorphism U ≃Δ^n = { (z_1, ⋯ , z_n) ∈^n | |z_i| < 1 } such that ψ(P) = (0, ⋯ , 0) and ψ(D_i) = {z_i = 0 } for any i = 1, ⋯ , l.Letbe a holomorphic vector bundle on X \ D equipped with a hermitian metric h. Let α = (α_i)_ i ∈ I∈^I be a tuple of real numbers. Let U be an open subset of X, and s be an element of ^0(U \ D, ). We say that the order of growth of s is bigger than α if for any point P ∈ U, any admissible coordinate system (U, ψ) around P and any positive real number ϵ >0, there exists a positive constant C such that the following inequality holds on U \ D:‖ s(z) ‖_h ≤C ·∏_i=1^l |z_i|^α_i - ϵWe denote by ^α the subsheaf of j_* of sections with order of growth bigger than α. The ^α, α∈^I, form a decreasing filtration of the subsheaf of j_*. [Moderate hermitian metric on a holomorphic vector bundle] Let X be a complex manifold, D ⊂ X be a simple normal crossing divisor, andbe a holomorphic bundle on X \ D. Set j: X \ D ↪ X the inclusion. A hermitian metric h onis called moderate if the subsheaf ∪_α∈^I^α of j_* consisting of sections with moderate growth near D is a meromorphic sheaf (i.e. a torsion-free coherent _X[*D]-module), and for every α∈^I the subsheaf ^α of j_* is a coherent _X-module. When the metric is moderate, one verifies immediately that the collection of torsion-free coherent sheaves ^α form a parabolic sheaf on the pair (X,D). §.§ λ-Connection bundles [λ-Connection] Let λ∈. Let X be a complex manifold andbe a locally-free _X-module of finite rank. A λ-connection onis a -linear map D^λ : ⊗__XΩ_X^1 which satisfies the twisted Leibniz rule:D^λ (f · s) = f · D^λ (s) + λ· df ∧ s, where f and s are sections of _X andrespectively.One verifies immediately that a λ-connection D^λ induces for every integer p ≥ 0 a -linear map D^λ : ⊗__XΩ_X^p ⊗__XΩ_X^p+1. A λ-connection D^λ is called flat or integrable if D^λ∘ D^λ =0. [Higgs bundle] A pair (, θ) consisting of a locally-free _X-module of finite rankequipped with a flat λ-connection θ is called a Higgs bundle when λ = 1. [Logarithmic λ-connection] Let X be a complex manifold and D ⊂ X be a normal crossing divisor. A logarithmic λ-connection bundle on (X,D) is a pair (, D^λ) consisting of a locally-free _X-module of finite ranktogether with a -linear map D^λ : ⊗__XΩ_X^1( log D) which satisfies the twisted Leibniz rule. [Meromorphic λ-connection] Let X be a complex manifold and D ⊂ X be a normal crossing divisor. A filtered regular meromorphic λ-connection bundle on (X,D) is a pair (, D^λ) consisting of a parabolic bundletogether with a -linear map D^λ : ⊗__X[*D]Ω_X^1[*D] which satisfies the twisted Leibniz rule, and such that for any parabolic weight α the λ-connection D^λ induces a structure of logarithmic λ-connection bundle on (^α,D^λ). The notion of flatness generalizes to the logarithmic and the meromorphic settings. In particular, by specializing to λ = 0 we get the notions of logarithmic Higgs bundles and filtered regular meromorphic Higgs bundles on (X,D).[Residues of a logarithmic λ-connection] Let X be a complex manifold, D ⊂ X be a normal crossing divisor and (, D^λ) be a logarithmic λ-connection bundle on (X,D). For any irreducible component D_k of D, there is an associated Poincaré residue _X-linear map R_k : Ω^1_X(log D) _D_k. The map (R_k ⊗id) ∘ D^λ induces an _D_k-linear endomorphism res_D_k(D^λ) ∈(_|D_k) which is called the residue of D^λ along D_k.If X is a projective manifold equipped with an ample line bundle , there is a corresponding notion of μ_-stability for logarithmic λ-connection bundles and filtered regular meromorphic λ-connection bundles on (X,D), cf. <cit.>. When (, D^λ) is a filtered regular meromorphic λ-connection bundle on (X,D) such that the parabolic structure onis trivial, then its μ_-stability is equivalent to the μ_-stability of the logarithmic λ-connection bundle (^♢, D^λ). §.§ Pluriharmonic metricsLet (, θ) be a Higgs bundle on a complex manifold X, and h_ be a hermitian metric on . Let E := ^∞_X ⊗__X be the smooth vector bundle underlying , and ∂_ be the operator defining the holomorphic structure on . Let ∇^u = ∂_ + ∂_ be the Chern connection associated to the hermitian metric h_ on the holomorphic bundle , and θ^⋆ be the adjoint of θ with respect to h_. By definition, the metric h_ is said pluriharmonic if the operator ∇ := ∂_ + ∂_ + θ + θ^⋆ is integrable, i.e. if the differentiable form ∇^2 ∈ A^2((E)) is zero. In that case, the holomorphic bundle ℰ := (E,∂_E + θ^⋆ ) equipped with the connection ∇ defines a flat bundle.Conversely, let (ℰ, ∇) be a flat bundle on a complex manifold X, and denote by E the smooth vector bundle underlying ℰ. The choice of a hermitian metric h_ℰ on ℰ induces a canonical decomposition ∇ = ∇^u + Ψ, where ∇^u is a unitary connection on ℰ with respect to h_ℰ and Ψ is autoadjoint for h_ℰ. Both decompose in turn in their components of type (1, 0) and (0, 1):∇^u = ∂_ + ∂_, Ψ = θ + θ^⋆. By definition, the metric h_ℰ is said pluriharmonic if the operator D^'': =∂_ + θ is integrable, i.e. if the differentiable form (D^'')^2 ∈ A^2((E)) is zero. In that case, the holomorphic bundle := (E,∂_E) endowed with the one-form θ∈ A^1((E)) defines a Higgs bundle.More generally, there is a notion of pluriharmonicity for metrics on a holomorphic vector bundle equipped with a flat λ-connection, see <cit.>. §.§ Tame harmonic bundles [Harmonic bundle] A harmonic bundle on a complex manifold is a Higgs bundle endowed with a pluriharmonic metric, or equivalently a flat bundle equipped with a pluriharmonic metric. [Tame harmonic bundle] Let X be a complex manifold and D ⊂ X a simple normal crossing divisor. A harmonic bundle (, θ, h) on U := X \ D is called tame if there exists a logarithmic Higgs bundle on (X,D) extending (, θ).In that case, the characteristic polynomial of the Higgs field θ, which is a polynomial whose coefficients are holomorphic symmetric differential forms on U, extends to X as an element of ( Ω^1_X(log D)) [T]. This property turns out to be sufficient to show that a harmonic bundle (, θ, h) on U := X \ D is tame. Note that the element of ( Ω^1_X(log D)) [T] does not depend on the extension, because its restriction to U is fixed. In particular, the eigenvalues of the residues are independent of the extension.Let X be a complex manifold and D ⊂ X be a simple normal crossing divisor. If (ℰ, ∇, h) (resp. (, θ, h)) is a tame harmonic bundle on X \ D, then the metric h is moderate and the corresponding filtration according to the order of growth on ℰ (resp. ) defines a parabolic bundle on (X,D).§.§ Deligne-Manin filtrationLet X be a complex manifold and D = ∪_I D_i⊂ X be a simple normal crossing divisor. Let (ℰ, ∇) be a meromorphic bundle on (X,D) equipped with a regular meromorphic connection. For every α∈^I, let ℰ^α be the unique locally-free _X-module of finite rank contained in ℰ such that ∇ induces a connection with logarithmic singularities ∇ : ℰ^αℰ^α⊗Ω^1_X( log D) such that the real part of the eigenvalues of the residue of ∇ along D_i belongs to [ α_i , α_i + 1 ), cf. <cit.>. We call this filtration Deligne-Manin filtration. The meromorphic bundle ℰ equipped with this filtration defines a parabolic bundle ℰ, and (ℰ, ∇) is a filtered regular meromorphic connection bundle on (X,D).Let now (ℰ, ∇, h) be a tame harmonic bundle on U := X \ D. There exists a unique meromorphic bundle on (X,D) equipped with a regular meromorphic connection extending (ℰ, ∇) (the so-called Deligne's extension). When equipped with the Deligne-Manin filtration introduced above, this defines a canonical filtered regular meromorphic connection bundle (ℰ^DM, ∇) on (X,D) which extends (ℰ, ∇). We have also the filtered regular meromorphic connection bundle (ℰ^h, ∇) on (X,D) defined using the filtration according to the order of growth with respect to h, cf. Theorem <ref>. The following result is well-known and is a direct consequence of the table in <cit.>.Let X be a complex manifold and D⊂ X be a simple normal crossing divisor. Let (ℰ, ∇, h) be a tame harmonic bundle on X \ D, and (, θ) be the corresponding Higgs bundle. Then the following properties are equivalent: * The eigenvalues of the residues of θ along the irreducible components of D are purely imaginary.* The filtered regular meromorphic connection bundles (ℰ^DM, ∇) and(ℰ^h, ∇) are canonically isomorphic. §.§ The non-abelian Hodge correspondence in the non-compact case Let X be a smooth irreducible complex projective variety equipped with an ample line bundle , and D be a simple normal crossing divisor of X.Fix λ∈. Let (, D^λ) be a flat filtered regular meromorphic λ-connection bundle on (X,D). Then the following conditions are equivalent: * (, D^λ) is μ_-polystable with vanishing first and second parabolic Chern classes.* There exists a pluriharmonic metric on (_|X \ D,D^λ) adapted to the parabolic structure.Such a metric is unique up to obvious ambiguity. In particular, this induces a natural equivalence of categories between the category of μ_-polystable flat filtered regular meromorphic connection bundles on (X,D) with vanishing first and second parabolic Chern classes and the category of μ_-polystable filtered regular meromorphic Higgs bundles on (X,D) with vanishing first and second parabolic Chern classes. This equivalence preserves tensor products, direct sums and duals.§ SINGULAR HERMITIAN METRICS ON TORSION-FREE SHEAVESWe recall here the basic definitions concerning singular hermitian metrics on torsion-free sheaves, after <cit.>. We follow very closely the presentation of <cit.>, to which the reader is referred for more details.§.§ Singular hermitian inner products A singular hermitian inner product on a finite-dimensional complex vector space V is a function ‖·‖ : V [0, + ∞ ] with the following properties: * ‖λ· v ‖ = |λ| ·‖ v ‖ for every λ∈\{0} and every v ∈ V, and ‖ 0 ‖ = 0.* ‖ v + w ‖≤‖ v ‖ + ‖ w ‖ for every v, w ∈ V, where by convention ∞≤∞.* ‖ v + w ‖^2 + ‖ v - w ‖^2 = 2 ·‖ v ‖^2 + 2 ·‖ w ‖^2 for every v, w ∈ V.Let V_0 (resp. V_fin) be the subset of V of vectors with zero (resp. finite) norm. It follows easily from the axioms that both V_0 and V_fin are linear subspaces of V. By definition, ‖·‖ is said positive definite if V_0 = 0 and finite if V_fin = V. A singular hermitian inner product ‖·‖ on V induces canonically a singular hermitian inner product ‖·‖^⋆ on its dual V^⋆ = _(V, ) by setting‖ f ‖^⋆ := sup{|f(v)|/‖ v ‖ | v ∈ V with ‖ v ‖≠ 0 } for any linear form f ∈ V^⋆, with the understanding that a fraction with denominator + ∞ is equal to 0. (If V_0 = V, then we define ‖ f ‖^⋆= 0 for f = 0, and ‖ f ‖^⋆= + ∞ otherwise.) Note however that in general there is no induced singular hermitian inner product on V.Observe that the dual of a finite singular hermitian inner product is positive definite, and conversely. §.§ Singular hermitian metrics on vector bundlesLet X be a connected complex manifold, and letbe a non-zero holomorphic vector bundle on X.(compare <cit.> and <cit.>) A singular hermitian metric onis a function h that associates to every point x ∈ X a singular hermitian inner product ‖·‖_h,x : _x[0, + ∞ ] on the complex vector space _x, subject to the following two conditions: * h is finite and positive definite almost everywhere, meaning that for all x outside a set of measure zero, ‖·‖_h,x is a hermitian inner product on _x.* h is measurable, meaning that the function‖ s ‖_h,x : U[0, + ∞ ], x ↦‖ s(x) ‖_h,xis measurable whenever U ⊂ X is open and s ∈^0(U, ). A singular hermitian metric h onhas semi-negative curvature if the function x ↦log‖ s(x) ‖_h,x is plurisubharmonic for every local section s ∈^0(U, ). It has semi-positive curvature if its dual metric h^⋆ on ^⋆ has semi-negative curvature. A priori, a singular hermitian metric h is only defined almost everywhere, because its coefficients are measurable functions. However, if h has semi-positive or semi-negative curvature, then the singular hermitian inner product ‖·‖_h,x is unambiguously defined at each point of X. Note also that, if h is a singular hermitian metric with semi-negative curvature, then the singular hermitian inner product ‖·‖_h,x is finite at each point of X. As a converse, we have the: Letbe a non-zero holomorphic vector bundle on a connected complex manifold X. Let U be an open dense subset of X, and h be a singular hermitian metric with semi-negative curvature on _|U. The following two assertions are equivalent: * h is a restriction to U of a singular hermitian metric with semi-negative curvature on ,* the function x ↦log‖ s(x) ‖_h,x is locally bounded from above in the neighborhood of every point of X for every local section s ∈^0(U, ).Moreover, the metric extending h is unique when it exists. This is a direct application of Riemann extension theorem for plurisubharmonic functions.§.§ Singular hermitian metrics on torsion-free sheavesLet X be a connected complex manifold, and letbe a non-zero torsion-free coherent sheaf on X. Let X_⊂ X denote the maximal open subset whereis locally free, so that X \ X_ is a closed analytic subset of codimension at least 2.A singular hermitian metric onis a singular hermitian metric h on the holomorphic vector bundle _| X_. We say that such a metric has semi-positive curvature if the pair (E, h) has semi-positive curvature. § WEAKLY POSITIVE TORSION-FREE SHEAVES For the reader convenience, we recall in this section the notion of weak positivity for torsion-free coherent sheaves, due to Viehweg and later refined by Nakayama. Let X be a complex quasi-projective scheme. A coherent sheafon X is globally generated at a point x ∈ X if the natural map ^0(X, ) ⊗__X is surjective at x.[Nakayama,<cit.>] Letbe a torsion-free coherent sheaf on a smooth projective complex varietyX. We say thatis weakly positive at a point x ∈ X if for every ample invertible sheafon X and every positive integer α> 0 there exists an integer β > 0 such that S^α·β⊗__X^β is globally generated at x.Here the notation S^k stands for the reflexive hull of the sheaf S^k, i.e. S^k = i_* ( S^k i^*) where i : X_↪ X is the inclusion of the maximal open subset whereis locally free. (Restricted base locus) Letbe a torsion-free sheaf on a smooth projective complex varietyX. The restricted base locus _-() ofis the subset of X of points at whichis weakly positive. It is a priori only a union of closed subvarieties of X. A torsion-free coherent sheafis weakly positive in the sense of Viehweg if there exists a dense open subset U ⊂ X such thatis weakly positive at every x ∈ U, or equivalently when _-()is not Zariski-dense. Note that ifis locally free, thenis nef if and only if it is weakly positive at every x ∈ X.The following result gives an analytical criterion to show that a sheaf is weakly positive.Let X be a smooth projective variety, and letbe a torsion-free coherent sheaf on X equipped with a singular hermitian metric h with semi-positive curvature. Ifis locally free at x and ‖·‖_h,x is finite (hence it is a positive-definite hermitian inner product), thenis weakly positive at x .§ PROOF OF THEOREM <REF> Let X be a complex manifold, and D ⊂ X be a simple normal crossing divisor. Set U := X \ D and j : U ↪ X the inclusion. Let (, θ , h) be a tame harmonic bundle on U, andbe a subsheaf of . The hermitian metric h induces a (possibly singular) hermitian metric h_ on .Let ^♢ be the subsheaf of j_* whose sections have sub-polynomial growth with respect to h_. Equivalently, ^♢ := j_* ∩^♢, where^♢ denotes the subsheaf of j_* whose sections have sub-polynomial growth with respect to h_. As ^♢ is a locally-free sheaf of finite rank (cf. Theorem <ref>), it follows that ^♢ is a torsion-free coherent sheaf. Assume now thatis contained in the kernel of θ. It follows from the next lemma that h_ has semi-negative curvature. Let (, θ , h) be a harmonic bundle on a complex manifold X. If s ∈^0(X, ) is a section ofwhich satisfies θ(s) =0, then the functionlog‖ s ‖_h : X[- ∞, + ∞ ), x ↦log‖ s(x) ‖_his plurisubharmonic. Denoting by ∂_ + ∂_ the Chern connection associated to the hermitian metric h on the holomorphic bundleand by θ^⋆ the adjoint of θ with respect to h, we know by assumption that the connection ∇ := ∂_ + ∂_ + θ + θ^⋆ on the ^∞-vector bundle ^∞_X ⊗__X is flat. From this, it follows that the curvature Θ := (∂_ + ∂_ )^2 of (, h) satisfies the formula:Θ + θ∧θ^⋆ + θ^⋆∧θ = 0.On the other hand, if s is a section of , we have thati ·∂∂‖ s ‖_h^2 = -i · (Θ s, s)_h + i ·‖∂_ (s) ‖_h^2(see for example <cit.>). If moreover θ(s) = 0, it follows thati ·∂∂‖ s ‖_h^2 ≥ -i · (Θ s, s)_h= - i · (θ(s), θ(s))_h+ i · (θ^⋆(s), θ^⋆(s))_h = i · (θ^⋆(s), θ^⋆(s))_h ≥ 0,hence the function ‖ s ‖_h : X[0, + ∞ ), x ↦‖ s(x) ‖_h is plurisubharmonic.Now, recall that given a non-negative valued function v, log v is plurisubharmonic if and only if v · e^2 ·Re (q) is plurisubharmonic for every polynomial q (we learned this observation from <cit.>). The preceding computation shows that for every polynomial q the function x ↦‖ s(x) ‖_h^2 · e^2 ·Re (q(x)) = ‖ s(x) · e^ q(x)‖_h^2 is plurisubharmonic, because s · e^q is also a holomorphic section ofwhich satisfies θ(s · e^q) = e^q ·θ(s) =0. This completes the proof. Let us now prove that there exists a unique singular hermitian metric with semi-negative curvature on ^♢ whose restriction to U is h_. In view of Lemma <ref>, this will follow from theLet X be a complex manifold, D ⊂ X be a simple normal crossing divisor and (, θ,h) be a tame harmonic bundle on U := X \ D.If s is a section of ^♢ whose restriction to U satisfies θ(s_|U ) =0, then the function‖ s_|U‖_h : U[0, + ∞ ), x ↦‖ s_|U (x) ‖_h is locally bounded in the neighborhood of every point in X.Clearly it is sufficient to treat the case X = Δ^n and D = ∪_i=1^l D_i, where D_i = {z =(z_1, ⋯, z_n) ∈Δ^n | z_i = 0 }. We will show the stronger statement: sup_z∈ U‖ s_|U (z) ‖_h ≤sup_z∈∂ X‖ s_|U (z) ‖_h, where ∂ X := (∂Δ)^n is a product of circles. We claim that it is sufficient to treat the case the case n =1. Indeed, to bound ‖ s_|U (z) ‖_h at a point z∈ U, by successive applications of the case n =1, we get the inequalities: ‖ s_|U ((z_1, ⋯, z_n)) ‖_h ≤sup_α_1 ∈∂Δ‖ s_|U ((α_1, ⋯, z_n)) ‖_h ≤sup_α_1 ∈∂Δ, α_2 ∈∂Δ‖ s_|U ((α_1, α_2, ⋯, z_n)) ‖_h⋯≤sup_α∈ (∂Δ)^n‖ s_|U ((α_1, α_2, ⋯, α_n)) ‖_h . It remains to prove the case n =1. We will first show as a consequence of Simpson's norm estimates that the function ‖ s_|U‖_h : U[0, + ∞ ), x ↦‖ s_|U (x) ‖_h is locally bounded. It follows that the function log‖ s_|U‖_h extends as a plurisubharmonic function to Δ, and we conclude using the maximum principle.Finally, we have reduced the proof to the case where X = Δ and D = { 0 }. Let R := (^♢) be the residue of θ at 0. It is a -linear endomorphism of the complex vector space ^♢_|0. Let ^♢_|0 = ⊕_λ∈ (^♢_|0)_λ be the decomposition in generalized eigenspaces of R. On each piece (^♢_|0)_λ the nilpotent endomorphism R - λ· defines a weight filtration W_k ( (^♢_|0)_λ) indexed by integers k, thanks to the well-known: Let N be a nilpotent endomorphism of a finite dimensional complex vector space V. There exists a unique finite increasing filtration W_k (V) of V such that: * N (W_k ) ⊂ W_k-2 for every k,* N^k induces an isomorphism Gr_k^W VGr_-k^W V for every k ≥ 0.The sum of the weight filtrations for different eigenvalues λ yields in turn a filtration of ^♢_|0 that we still denote by W_k. As proved by Schmid <cit.> for variations of Hodge structures and Simpson <cit.> for general tame harmonic bundles, this filtration controls the growths of the sections of ^♢:If s be a section of ^♢ defined on Δ, then s(0) ∈ W_k if and only if there exists a constant C>0 such that the following estimates holds (cf. <cit.>):‖ s(z) ‖_h ≤ C ·(- log |z|)^k/2 .Let s be a section of ^♢ defined on Δ such that θ(s) =0. It follows that s(0) belongs to the kernel of (^♢). In particular, it belongs to the generalized eigenspace where (^♢) is nilpotent. Moreover, by the lemma below, it belongs to W_0 and by the norm estimates it follows that ‖ s ‖_h is bounded on Δ^⋆. Let N be a nilpotent endomorphism of a finite dimensional complex vector space V, and let W_k (V) be the corresponding filtration of V. Then ( N) ⊂ W_0 (V).§ PROOFS OF THEOREM <REF> AND THEOREM <REF> Let X be a smooth projective complex variety,be an ample line bundle on X and D ⊂ X be a simple normal crossing divisor. Let (, θ) be a filtered regular meromorphic Higgs bundle on (X,D), which is μ_-polystable with vanishing parabolic Chern classes. Let finallybe a subsheaf of ^♢ contained in the kernel of θ. Set U := X \ D and j : U ↪ X the inclusion. By Theorem <ref>, there exists an essentially unique pluriharmonic metric h on (^♢_|U, θ) adapted to the parabolic structure. The triple (^♢_|U, θ, h) defines a tame harmonic bundle on U, and ^♢ is equal to the subsheaf of j_*(^♢_|U) whose sections have sub-polynomial growth with respect to h. Observe also thatbeing torsion-free, the canonical map j_* (_|U) ∩^♢ is an injective map of sheaves and an isomorphism when restricted to U. By Theorem <ref> the (possibly singular) hermitian metric induced by h on _|U extends uniquely as a singular hermitian metric with semi-negative curvature on j_* (_|U) ∩^♢. By restriction we obtain a singular hermitian metric with semi-negative curvature onthat we denote h_. This metric induces in turn a singular hermitian metric with semi-positive curvature h_^⋆ on ^⋆. One can then apply Theorem <ref> to conclude that its dual ^⋆ is weakly positive. In fact one obtains the more precise Let X be a smooth projective complex variety,be an ample line bundle on X and D ⊂ X be a simple normal crossing divisor. Let (, θ) be a filtered regular meromorphic Higgs bundle on (X,D). Assume that (, θ) is μ_-polystable with vanishing parabolic Chern classes. Ifis a subsheaf of ^♢ contained in the kernel of θ, then its dual ^⋆ is weakly positive in the sense of Viehweg, and its restricted base locus _-(^⋆) is contained in the union of D and the locus in X whereis not a locally split subsheaf of ^♢.Assume from now on thatis a locally split subsheaf of . Let π : (^⋆)X be the projective scheme over X associated to ^⋆, i.e. (^⋆) := Proj__X ((^⋆)). Let _^⋆(1) be the tautological line bundle on (^⋆), so that there is an exact sequence π^* ^⋆_^⋆(1)0. The singular hermitian metric h_^⋆ with semi-positive curvature on ^⋆ induces canonically a singular hermitian metric h__^⋆(1) with semi-positive curvature on _^⋆(1).The Lelong numbers of the metric h__^⋆(1) are smaller than 1. In particular, the multiplier ideal (h__^⋆(1)) is trivial. If moreover the parabolic structure onis trivial, then all Lelong numbers are zero (see below for the definition of Lelong numbers). [compare <cit.>] On a complex manifold X, letbe a line bundle equipped with a singular hermitian metric h of semi-positive curvature. The Lelong number of h at the point x ∈ X is defined by ν(h, x) := lim inf_zxlog‖ s(z) ‖_h/- log |z - x|, where s is a holomorphic section ofdefined in a neighborhood of x with s(x) ≠ 0. One verifies that it is a well-defined non-negative real number which does not depend on the local section s, cf. loc. cit.From the exact sequences of hermitian holomorphic vector bundles π^* ^⋆_^⋆(1)0 and (^♢)^⋆^⋆ 0 one sees that the order of growth of π^* ^⋆ with respect to π^* h_^⋆ is bigger or equal than the order of growth of (^♢)^⋆ with respect to h^⋆. Let us show that the latter is bigger than -1. First note that (^♢)^⋆ is canonically isomorphic to (^⋆)^> -1, where ^⋆ denotes the dual of the parabolic bundleand (^⋆)^> -1 denotes the subsheaf of the parabolic bundle ^⋆ corresponding to the weight (- 1 + ϵ, ⋯, -1 + ϵ) for ϵ >0 sufficiently small. This follows immediately from the definition of the dual filtration: λ∈ (^⋆)^α if and only if λ ( ^β) ⊂_X^α + β for all β. On the other hand, the filtration according to the order of growth is easily seen to be compatible with taking duals, hence (^⋆)^> -1 is the subsheaf of j_* (_|U^⋆) of sections whose order of growth with respect to h^⋆ is bigger than -1 along every irreducible component of D. From this discussion, it follows that the Lelong numbers of the metric h__^⋆(1) are smaller than 1. In particular, all holomorphic sections of _^⋆(1) are locally L^2 with respect to h__^⋆(1) (cf. <cit.>), meaning that the multiplier ideal (h__^⋆(1)) is trivial. When the parabolic structure onis trivial, one sees that (^♢)^⋆ is canonically isomorphic to (^⋆)^♢, hence the Lelong numbers of the metric h__^⋆(1) are all zero. Finally, Theorem <ref> is a consequence of the preceding Proposition <ref> and the Let X be a smooth projective variety andbe a line bundle on X equipped with a singular metric h with semi-positive curvature. If the Lelong numbers of (, h) satisfy ν(h, x) = 0 for all x ∈ X but a countable set, thenis nef.Follows from <cit.> and<cit.>.§ VARIATIONS OF HODGE STRUCTURES AND PROOF OF THEOREM <REF> We begin by recalling some definitions. Let V be a complex vector space of finite dimension. A (complex polarized) Hodge structure (of weight zero) on V is the data of a non-degenerate hermitian form h and a h-orthogonal decomposition V = ⊕_p ∈ V^p such that the restriction of h to V^p is positive definite for p even and negative definite for p odd.The associated Hodge filtration is the decreasing filtration F on V defined byF^p := ⊕_ q ≥ p V^q. Let X be a complex manifold. A variation of polarized complex Hodge structures (-PVHS) on X is the data of a holomorphic vector bundle ℰ equipped with an integrable connection ∇, a ∇-flat non-degenerate hermitian form h and for all x ∈ X a decomposition of the fibre ℰ_x = ⊕_p ∈ℰ^p satisfying the following axioms:* for all x ∈ X, the decomposition ℰ_x = ⊕_p ∈ℰ_x^p defines a Hodge structure polarized by h_x,* the Hodge filtrationvaries holomorphically with x, * (Griffiths' transversality) ∇( ^p) ⊂^p-1⊗__XΩ_X^1 for all p.One obtains a positive-definite hermitian metric h_H on ℰ from h by imposing that for all x ∈ X the decomposition ℰ_x= ⊕_p ∈ℰ_x^p is h_H-orthogonal and setting h_H := (-1)^p · h on ℰ_x^p. We call h_H the Hodge metric.[Log -PVHS] Let X be a complex manifold, and D ⊂ X be a simple normal crossing divisor.A log complex polarized variation of Hodge structure (log -PVHS) on (X,D) consists of the following data: * A holomorphic vector bundle ℰ on X endowed with a connection ∇ with logarithmic singularities along D,* An exhaustive decreasing filtrationon ℰ by holomorphic subbundles (the Hodge filtration), satisfying Griffiths transversality∇^p ⊂^p-1⊗Ω^1_X(log D), * a ∇-flat non-degenerate hermitian form h on ℰ_|X \ D,such that (ℰ_| X \ D, ∇, ^_|X \ D, h) is a -PVHS on X \ D. Note that by setting:= _ℰ and θ:= _∇ we get a logarithmic Higgs bundle on (X,D), called the associated logarithmic Higgs bundle. The following result explains the link between -PVHS and harmonic bundles. If (ℰ, ∇ ,^, h) is a -PVHS on a complex manifold, then the triplet (ℰ, ∇, h_H) forms a harmonic bundle, where h_H denotes the Hodge metric on ℰ. The corresponding Higgs bundle (, θ) is given by := _ℰ and θ:= _∇. Let X be a complex manifold, and D = ∪_i ∈ I D_i ⊂ X be a simple normal crossing divisor. Let (ℰ, ∇ ,^, h) be a -PVHS on U := X\ D. First note that the harmonic bundle (ℰ, ∇, h_H) is tame. Indeed, if ( = ⊕_p ∈^p, θ) is the associated Higgs bundle, then one sees imediately that θ is nilpotent, and as a consequence its characteristic polynomial T^ U extends clearly to X as an element of (Ω_X^1(log D)) [T ].There exists a unique meromorphic bundle on (X,D) equipped with a regular meromorphic connection extending (ℰ, ∇) (the so-called Deligne's extension). When equipped with the Deligne-Manin filtration (cf. section <ref>), this defines a canonical filtered regular meromorphic connection bundle (ℰ^DM, ∇) on (X,D) which extends (ℰ, ∇). As θ is nilpotent, the eigenvalues of its residues along the D_i are zero, hence by Lemma <ref> (ℰ^DM, ∇)is canonically isomorphic to the filtered regular meromorphic connection bundle (ℰ^h, ∇) on (X,D) defined using the filtration according to the order of growth with respect to h. We denote by ℰ^α the elements of this filtration. We can extend the Hodge filtration to every ℰ^α by setting:^p ℰ^α:= ℰ^α∩ j_* ^p.For every p, we have an exact sequence of locally-free sheaves on U:0 ^p+1ℰ^pℰ Gr_^pℰ = ^p0. The Hodge metric on ℰ induces canonical hermitian metrics on these locally-free sheaves. From this it follows that for every α and p one get an exact sequence of sheaves on X:0 ^p+1ℰ^α^pℰ^α(^p)^α. Consequently, there is a canonical injective map of sheaves g_α : Gr_ℰ^α^α for every α. Note however that it is not clear a priori that these maps are surjective (when the residues of (ℰ, ∇) are nilpotent, it is a consequence of the work of Schmid <cit.>). Even if this fact is not strictly needed in our discussion, we will give a proof for the sake of completeness. First note that the preceding maps g_α are isomorphisms if and only if the induced maps on the determinants g_α :(Gr_ℰ^α)(^α) are isomorphisms. But Gr_ℰ^α is canonically isomorphic to ℰ^α and the filtration according to growth conditions is compatible with taking determinant, hence this reduces the problem to show that the canonical maps (ℰ)^α ()^α are isomorphisms for every α. This is clear because in restriction to U the map (ℰ)() is an isomorphism of hermitian holomorphic vector bundles. Therefore we have proved that the maps g_α : Gr_ℰ^α^α are isomorphisms of sheaves (in particular, the ^pℰ^α form a filtration of ℰ^α by locally split subsheaves).We can now give a proof of the Theorem <ref>. With the notations of the statement, it follows from the discussion above that there exists an injective map of sheaves 0 ^♢. Moreover,is in the kernel of the Higgs field. We get the result by applying Theorem <ref> to this particular situation. § PROOF OF THEOREM <REF>In this section we prove Theorem <ref> as a direct consequence of the two following propositions. Let X be a smooth projective complex variety, D ⊂ X be a simple normal crossing divisor and (, θ, h) be a tame harmonic bundle on U := X \ D. If the map of _U-modules ϕ : T_U () associated to θ∈Ω^1_U(()) is injective, then the logarithmic cotangent bundle Ω^1_X(log D) of (X,D) is weakly positive.Let (, θ) = (^h, θ) be the filtered regular meromorphic Higgs bundle on (X,D) associated to (, θ, h) using the filtration according to the order of growth with respect to h, cf. Theorem <ref>. The parabolic bundle () inherits a canonical structure of filtered regular meromorphic Higgs bundle, whose Higgs field Θ is defined by(Θ_s( Ψ ))(v) = θ_s ( Ψ (v)) - Ψ( θ_s (v))for Ψ a local holomorphic section of (), v a local holomorphic section ofand s a local holomorphic section of T_U. Here we denote by θ_s the contraction of θ with s, alias ϕ(s) (and similarly for Θ_s). The composition of ϕ : T_U () with the Higgs field Θ : () Ω^1_U ⊗_𝒪_U() is zero: if s and t are local holomorphic sections of T_U and v is a local holomorphic section of , then the condition θ∧θ = 0 implies (Θ_s( θ_t ))(v) = θ_s ( θ_t (v)) - θ_t( θ_s (v)) = 0,where ϕ(t) = θ_t ∈(). As a consequence, if we denote by ⊂ ()^♢ the image of ϕ : T_X(- log D)()^♢, thenis contained in the kernel of the Higgs field Θ :()^♢Ω^1_X(log D) ⊗_𝒪_X()^♢.Ifis an ample line bundle on X, then by Theorem <ref> the filtered regular meromorphic Higgs bundle (, θ) is μ_-polystable with vanishing parabolic Chern classes, and the same is true for ((), Θ). The subsheaf ⊂()^♢ is contained in the kernel of Θ, hence by Theorem <ref> its dual ^⋆ is weakly positive. Since the map T_X(- log D) is generically an isomorphism by assumption, the same is true for the dual map ^⋆Ω^1_X(log D). It follows that Ω^1_X(log D) is weakly positive. Let X be a smooth projective complex variety and D ⊂ X be a simple normal crossing divisor. A tame harmonic bundle (ℰ, ∇, h) ≡ (, θ, h) is called purely imaginary if the eigenvalues of the residues of θ along the irreducible components of D are purely imaginary. It is equivalent to ask that the filtered regular meromorphic connection bundles (ℰ^DM, ∇) and (ℰ^h, ∇) are canonically isomorphic, cf. Lemma <ref>. Recall that a complex local system on X \ D is semisimple if and only if the corresponding connection bundle comes from a purely imaginary tame harmonic bundles on X \ D <cit.>. Also, if X and Y are smooth projective complex varieties, D_X and D_Y are simple normal crossing divisors of X and Y respectively and f : XY is a morphism such that f^-1(D_Y) ⊂ D_X, thenfor any tame harmonic bundle (,θ ,h) on Y \ D_Y, its pullback f^*(,θ ,h) is a tame harmonic bundle on X \ D_X. Moreover, if (,θ ,h) is pure imaginary, then f^* (,θ ,h) is also pure imaginary, cf. <cit.> . Let X be a smooth projective complex variety, D ⊂ X be a simple normal crossing divisor and (, θ, h) be a pure imaginary tame harmonic bundle on U := X \ D. If the associated semisimple complex local system on U is generically large and has discrete monodromy, then the map of _U-modules ϕ : T_U () associated to θ∈Ω^1_U(()) is injective. Before giving the proof of the proposition, let us prove the following easy lemma. Let (, θ, h) be a pure imaginary tame harmonic bundle on a smooth quasi-projective complex variety U := X \ D, and let L be the associated semisimple complex local system on U. Then the monodromy of L is finite if and only if it is discrete and the Higgs field θ is zero. If the monodromy of L is finite, then the pullback of L to a finite étale cover of U is trivial. Therefore the pullback of the Higgs field by a finite étale cover is zero, hence θ is zero itself. For the converse, first observe that given a harmonic bundle (, θ, h) on a complex manifold, the Higgs field θ is zero if and only if the pluriharmonic metric h is parallel for the Chern connection of (,h). Therefore, the monodromy being unitary and discrete, it has to be finite.By assumption, there exist countably many closed subvarieties D_i ⊊ U such that for every smooth quasi-projective complex variety Z equipped with a proper map f : ZU satisfying f(Z) ⊄∪ D_i the pullback local system f^∗ L is non-trivial (and has in fact infinite monodromy). Let also Y ⊊ U be the closed subvariety where the cokernel of the map of _U-modules ϕ : T_U () is not locally free. Fix x ∈ X \ (∪ D_i ∪ Y) and let us show that ϕ (s) ≠ 0 for all s ∈ T_X(x) \{ 0 }. Let C ⊂ U be a complete intersection curve containing x with T_C(x) = · s ⊂T_X(x), let ν : C̃ C be its normalization and f : C̃ X be the composition of ν with the inclusion C ⊂ X. Since f(C̃) = C ⊄∪ D_i, the monodromy of the local system f^∗ L is infinite and discrete. By the preceding Lemma, it follows that the Higgs field of the tame harmonic bundle f^*(,θ ,h) is nonzero. In other words, the composition T_Cf^∗ T_Uf^∗() is nonzero. Since in restriction to a neighborhood of x ∈ C the cokernel of the map T_Cf^∗() is locally free, the image of v is necessarily nonzero, i.e. ϕ(s) ≠ 0.alphaYohan Brunebarbe, Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Schweiz E-mail address: | http://arxiv.org/abs/1707.08495v1 | {
"authors": [
"Yohan Brunebarbe"
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"categories": [
"math.AG"
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"primary_category": "math.AG",
"published": "20170726152742",
"title": "Semi-positivity from Higgs bundles"
} |
Corresponding author [email protected] Istituto dei Sistemi Complessi, CNR, Via dei Taurini 19, 00185 Rome, Italy and INFN “Tor Vergata” Dipartimento di Fisica and INFN, Università di Roma “Tor Vergata”, Via Ricerca Scientifica 1, 00133 Roma, Italy Dipartimento di Fisica and INFN, Università di Torino,Via P. Giuria 1, 10125 Torino, ItalyDipartimento di Fisica and INFN, Università di Roma “Tor Vergata”, Via Ricerca Scientifica 1, 00133 Roma, ItalyThe irreversible turbulent energy cascade epitomizes strongly non-equilibrium systems.At the level of single fluid particles, time irreversibility is revealed by the asymmetry of the rate of kinetic energy change, the Lagrangian power, whose moments display a power-law dependence on the Reynolds number, as recently shown by Xu et al. [H Xu et al, Proc. Natl. Acad. Sci. U.S.A. 111, 7558 (2014)].Here Lagrangian power statistics are rationalized within the multifractal model of turbulence, whose predictions are shown to agree with numerical and empirical data.Multifractal predictions are also tested, for very large Reynolds numbers, in dynamical models of the turbulent cascade, obtaining remarkably good agreement for statistical quantities insensitive to the asymmetry and, remarkably, deviations for those probing the asymmetry. These findings raise fundamental questions concerning time irreversibility in the infinite-Reynolds-number limit of the Navier-Stokes equations. 05.70.Ln,47.27.-i,47.27.ebTime irreversibility and multifractality of power along single particle trajectories in turbulence[Version accepted for publication (postprint) on Phys. Rev. Fluids 2, 104604 – Published 27 October 2017] Massimo De Pietro=========================================================================================================================================================================================================== § INTRODUCTION In nature, the majority of the processes involving energy flow occur in nonequilibrium conditions from the molecular scale of biology <cit.> to astrophysics <cit.>. Understanding such nonequilibrium processes is of great interest at both fundamental and applied levels, from small-scale technology <cit.> to climate dynamics <cit.>. A key aspect of nonequilibrium systems is the behavior of fluctuations that markedly differ from equilibrium ones.As for the latter, detailed balance establishes equiprobability of forward and backward transitions between any two states, a statistical manifestation of time reversibility <cit.>, while, irreversibility of nonequilibrium processes breaks detailed balance.In three-dimensional (3D) turbulence, a prototype of very far-from-equilibrium systems, detailed balance breaks in a fundamental way <cit.>: It is more probable to transfer energy from large to small scales than its reverse.Indeed, in statistically stationary turbulence, energy, supplied at scale L at rate ϵ (≈ U_L^3/L, U_L being the root mean square single-point velocity), is transferred with a constant flux approximately equal to ϵ up to the scale η, where it is dissipated at the same rate ϵ, even for vanishing viscosity (ν→ 0) <cit.>. As a result, time reversibility, formally broken by the viscous term, is not restored for ν→ 0 <cit.>.Time irreversibility is unveiled by the asymmetry of two-point statistical observables. In particular, the constancy of the energy flux directly implies, in the Eulerian frame, a non vanishing third moment of longitudinal velocity difference between two points at distance r (the 4/5 law <cit.>) and, in the Lagrangian frame, a faster separation of particle pairs backward than forward in time <cit.>.Remarkably, time irreversibility has been recently discovered at the level of single-particle statistics <cit.> that is not a priori sensitive to the existence of a nonzero energy flux. This opens important challenges also at applied levels for stochastic modelization of single-particle transport, e.g., in turbulent environmental flows <cit.>.Both experimental and numerical data revealed that the temporal dynamics of Lagrangian kinetic energy E(t)=1/2v^2(t), where v(t)= u( x(t),t) is the Lagrangian velocity along a particle trajectory x(t), is characterized by events where E(t) grows slower than itdecreases.Such flight-crash events result in the asymmetry of distribution of the Lagrangian power, p(t)=Ė= v(t)· a(t) (a≡v̇=∂_tu+ u·∇ u being the fluid particle acceleration). While in stationary conditions the mean power vanishes ⟨ p⟩=0, the third moment is increasingly negative with the Taylor scale Reynolds number Re_λ≈ (U_LL/ν)^1/2≈ T_L/τ_η measuring the ratio between the timescales of energy injection T_L and dissipation τ_η, which easily exceeds 10^3 in the laboratory.In particular, it was found that ⟨ p^3⟩/ϵ^3 ∼ -Re_λ^2 <cit.> and ⟨ p^2⟩/ϵ^2 ∼Re_λ^4/3. Interestingly, the Re_λ dependence deviates from the dimensional prediction based on Kolmogorov phenomenology <cit.> ⟨ p^q⟩/ϵ^q ∝Re_λ^q/2, signaling that the Lagrangian power is strongly intermittent as exemplified by its spatial distribution and the strong non-Gaussian tails of the probability distribution function of p (Fig. <ref>).From a theoretical point of view, the above scaling behavior of the power with Re_λ implies that the skewness of the probability density function (PDF) of p, S = ⟨ p^3⟩ /⟨ p^2⟩ ^3/2 , is constant, suggesting thattime irreversibility is robust and persists even in the limit Re_λ→∞.It is important to stress that one might use different dimensionless measures of the symmetry breaking, e.g., S̃ = ⟨ p^3⟩ /⟨ |p|^3⟩ , which directly probes the ratio between the symmetric and asymmetric contributions to the PDF. In the presence of anomalous scaling S and S̃ can have a different Re_λ dependence, as highlighted for the problem of statistical recovery of isotropy <cit.>.The aim of our work is twofold. First, we use direct numerical simulations (DNSs) of 3D Navier-Stokes equations (NSEs) to quantify the degree of recovery of time reversibility along single-particle trajectories using different definitions as discussed above. Second, we show that it is possible to extend the multifractal formalism (MF) <cit.> to predict the scaling of the absolute value of the Lagrangian power statistics.Moreover, in order to explore a wider range of Reynolds numbers, we also investigate the equivalent of the Lagrangian power statistics in shell models <cit.>.The rest of the paper is organized as follows. Section <ref> is devoted to a brief review of the multifractal formalism for fully developed turbulence and the predictions for the statisticsof the Lagrangian power. In Sec. <ref> we compare thesepredictions with the results obtained from direct numerical simulations of the Navier-Stokes equations and from a shell model of turbulence. Section <ref> is devoted to a summary and conclusions. The Appendix reports some details of the numerical simulations. § THEORETICAL PREDICTIONS BY THE MULTIFRACTAL MODEL We start by recalling the MF for the Eulerian statistics <cit.>. The basic idea is to replace the global scale invariance in the manner of Kolmogorovwith a local scale invariance, by assuming that spatial velocity increments δ_r u over a distance r≪ L are characterized by a range of scaling exponents h∈ℐ≡(h_m,h_M), i.e., δ_r u∼ u_L (r/L)^h.Eulerian structure functions ⟨ (δ_r u)^q⟩ are obtained by integrating over h∈ℐ and the large-scale velocity u_L statistics 𝒫(u_L), which can be assumed to be independent of h. The MF assumes the exponent h to be realized on a fractal set of dimension D(h), so the probability to observe a particular value of h, for r≪ L, is 𝒫_h(r) ∼ (r/L)^3-D(h). Hence, we find ⟨ (δ_r u)^q⟩∼⟨ u_L^q⟩∫_h∈ℐ dh (r/L)^hq+3-D(h)∼⟨ u_L^q⟩ (r/L)^ζ_q, where a saddle-point approximation for r≪ L givesζ_q= inf_h∈ℐ{hq+3-D(h)} .For the MF to be predictive, D(h) should be derived from the NSE, which is out of reach. One can, however, use the measured exponents ζ_p and, by inverting (<ref>), derive an empirical D(h).Here, following <cit.>, we useD(h)= 3-d_0-d(h)[ln(d(h)/d_0) -1] ,with d(h)=3(1/9-h)/lnβ and d_0=2/[3(1-β)] corresponding, via (<ref>), to ζ_q=q/9+(2/3)(1-β^q/3)/(1-β), which, for β=0.6, fits measured exponents fairly well <cit.>.The MF has been extended from Eulerian to Lagrangian velocity increments <cit.>.The idea is that temporal velocity differences δ_τ v over a time lag τ, along fluid particle trajectories, can be connected to equal time spatial velocity differences δ_r u by assuming that the largest contribution to δ_τ v comes from eddies at a scale r such that τ∼ r/δ_r u.This implies δ_τ v ∼δ_r u, withτ∼ T_L (r/L)^1-h,where T_L=L/u_L.By combining Eq. (<ref>) and the D(h) obtained from Eulerian statistics, one can derive a prediction for Lagrangian structure functions, which has been found to agree with experimental and DNS data <cit.>. The MF can be used also for describing the statistics of theacceleration a along fluid elements <cit.>. The acceleration can be estimated by assuming a ∼δ_τ_η v/τ_η .According to the MF, the dissipative scale fluctuates as η∼ (ν L^h/u_L)^1/(1+h)<cit.>, which leads, via (<ref>), toτ_η∼ T (ν/Lu_L)^(1-h)/(1+h) .Substituting (<ref>) in (<ref>) yields the acceleration conditioned on given values of h and u_L:a ∼ν^(2h-1)/(1+h) u_L^3/(1+h) L^-3h/(1+h) .Equation (<ref>) has been successfully used to predict the acceleration variance <cit.> and PDF <cit.>.We now use (<ref>) to predict the scaling behavior of the Lagrangian power moments with Re_λ.These can be estimated as ⟨ p^q ⟩∼⟨ (au_L)^q⟩∼∫ du_L 𝒫(u_L) ∫_h∈ℐ dh 𝒫_h(τ_η) (au_L)^q with 𝒫_h(τ_η)=(τ_η/T)^[3-D(h)]/(1-h).Using (<ref>) with ν=U_LL Re_λ^2 (with U_L^2=⟨ u_L^2 ⟩), we have⟨ p^q⟩/ϵ^q∼∫dṽ𝒫(ṽ)∫_h∈ℐdhṽ^[4q+h-3+D(h)]/(1+h)Re_λ^2 [(1-2h)q-3+D(h)]/(1+h) ,with ṽ=u_L/U_L [Possible divergences in ṽ→ 0 should not be a concern as the MF cannot be trusted for small velocities].In the limit Re_λ→∞, a saddle point approximation of the integral (<ref>) yields, up to a multiplicative constant (depending on the large scale statistics), ⟨ p^q⟩/ϵ^q∼Re_λ^α(q) withα(q)=sup_h{2(1-2h)q-3+D(h)/1+h} .§ COMPARISON WITH NUMERICAL SIMULATIONS To test the MF predictions (<ref>) we use two sets of DNS of homogeneous isotropic turbulence on cubic lattices of sizes from 128^3 up to 2048^3, with Re_λ up to 540, obtained with two different forcings (see the Appendix for details). In particular, to probe both the symmetric and asymmetric components of the Lagrangian power statistics, we study the nondimensional moments𝒮_q=⟨ |p|^q⟩/ϵ^q, 𝒜_q=⟨ p|p|^q-1⟩/ϵ^q ,where the latter vanishes for a symmetric (time-reversible) PDF. In Fig. <ref> we show the second-and third-order moments of (<ref>) as a function of Re_λ.We observe that (i) the MF prediction (<ref>) is in excellent agreement with the scaling of 𝒮_q (see also Fig. <ref>) and (ii) the asymmetry probing moments 𝒜_q are negative, confirming the existence of the time-symmetry breaking, and scale with exponents compatible with those of 𝒮_q. This implies that time reversibility is not recovered even for Re_λ→∞.Actually, irreversibility is independent of Re_λ if measured in terms of the homogeneous asymmetry ratio S̃=𝒜_q/𝒮_q, while if quantified in terms of the standard skewness S, it grows as Re_λ^χ with χ=α(3)-(3/2) α(2) ≃ 0.35 due to anomalous scaling.In the inset of Fig. <ref> we compare S with S̃. Evaluating (<ref>) with D(h) given by (<ref>), we obtain α(2)≈ 1.17 and α(3)≈ 2.10, which are close to the 4/3 and 2 reported in <cit.>. We remark that the authors of <cit.> explained the observed exponents by assuming that the dominating events are those for which the particle travels a distance r ∼ U_Lτ in a frozenlike turbulent velocity field, so that δ_τ_η v ∼ (ϵτ_η U_L)^1/3. Hence, for the acceleration (<ref>) one has a ∼ U_L^1/3ϵ^1/3τ_η^-2/3, which, using the dimensional prediction τ_η=(ν/ϵ)^1/2, ends up in p∼ U_L a∼ U_L^4/3ϵ^2/3ν^-1/3∼ϵRe_λ^2/3. This argument provides only a linear approximation 2q/3 for α(q), while the multifractal model is able to describe its nonlinear dependence on q. In Fig. <ref> we show the whole set of exponents for both 𝒜_q and 𝒮_q as observed in DNS data andcompare them with the prediction (<ref>). It is worth noticing that the MF provides an excellent prediction for the statistics of p also in 1D compressible turbulence, i.e., in the Burgers equation, studied in <cit.>. Here, out of a smooth (h=1) velocity field, the statistically dominant structures are shocks (h=0). The velocity statistics is thus bifractal with D(1)=1 and D(0)=0 <cit.>.Adapting (<ref>) to one dimension and noticing that Re∝Re_λ^2, we have ⟨ p^q⟩∼Re^α_1D(q) with α_1D(q)=sup_h{[(1-2h)q-1+D(h)]/(1+h)}, which for Burgers means α_1D(q)=q-1, in agreement with the results of <cit.>.To further investigate the scaling behavior of the symmetric and asymmetric components of the power statistics in a wider range ofReynolds numbers and with higher statistics, in the following we study Lagrangian powerwithin the framework of shell models of turbulence <cit.>. Shell models are dynamical systems built to reproduce the basic phenomenology of the energy cascade on a discrete set of scales, r_n=k_n^-1=L2^-n (n=0,…,N), which allow us to reach high Reynolds numbers.For each scale r_n, the velocity fluctuation is represented by a single complex variable u_n, which evolves according to the differential equation <cit.>u̇_n=ik_n(u_n+2u^*_n+1-1/4u_n+1u^*_n-1 +1/8u_n-1u_n-2)-ν k_n^2 u_n+f_nwhose structure is a cartoon of the 3D NSE in Fourier space but for the nonlinear term that restricts the interactions to neighboring shells, as justified by the idea localness of the energy cascade <cit.>.Energy is injected with rate ϵ=⟨∑_n Re{f_n u_n^*}⟩. See the Appendix for details on forcing and simulations. As shown in <cit.>, this model displays anomalous scaling for the velocity structure functions, ⟨ |u_n|^q⟩∼ k_n^-ζ_q, with exponents remarkably close to those observed in turbulence and in very good agreement with the MF prediction (<ref>).Following <cit.>, we model the Lagrangian velocity along a fluid particle as the sum of the real part of velocity fluctuations at all shells v(t)≡∑_n=1^NRe{u_n}.Analogously, we define the Lagrangian acceleration a≡∑_n=1^NRe{u̇_n} and power p(t)=v(t)a(t).In Figs. <ref>(a)and <ref>(b) we show the moments 𝒮_q and 𝒜_q for q=2,3 obtained from the shell model. The symmetric ones 𝒮_q perfectly agree with the multifractal prediction obtained using the same D(h), i.e., (<ref>) for β=0.6, which fits the Eulerian statistics. The asymmetry-sensitive moments 𝒜_q are negative (for q>1), as in Navier-Stokes turbulence, and display a power-law dependence on Re_λ with a different scaling respect to their symmetric analogs. In particular, as summarized in Fig. <ref>(c), we observe smaller exponents with respect to the MF up to q=4. Rephrased in terms of the skewness, these findings mean that the time asymmetry becomes weaker and weaker with increasing Reynolds numbers if measured in terms of S̃ [Fig. <ref>(c) inset], as distinct from what was observed for the NSE (Fig. <ref> inset). The standard skewness S, on the other hand, is still an increasing function of Re_λ though with an exponent smaller than the MF prediction α(3)-(3/2)α(2), because 𝒜_3 has a shallower slopethan the multifractal one.§ CONCLUSIONS We have shown that the multifractal formalism predicts the scaling behavior of the Lagrangian power moments in excellent agreement with DNS data and with previous results on the Burgers equation. In the range of explored Re_λ, we have found that symmetric and antisymmetric moments share the same scaling exponents, and therefore the MF is able to reproduce both statistics.It is worth stressing that the effectiveness of the MF in describing the scaling of 𝒜_q is not obviousas the MF, in principle, bears no information on statistical asymmetries [See Sect.8.5.4 in <cit.> for a discussion.]. By analyzing the Lagrangian power statistics in a shell model of turbulence, at Reynolds numbers much higher than those achievable in DNS, we found that symmetric and antisymmetric moments possess two different sets of exponents.While the former are still well described by the MF formalism, the latter, in the range of q explored, are smaller.As a consequence, the ratios 𝒜_q/𝒮_q in the shell model decrease with Re_λ. However, we observe that the mismatch between the two sets of scaling is compatible with the assumption that 𝒜_q ∼𝒮_q ⟨sign(p)⟩, i.e., that the main effect is given by a cancellation exponent introduced by the scaling of sgn(p). Our findings raise the question whether the apparent similar scaling among symmetric and asymmetric components in the NSE is robust for large Reynolds numbers or a sort of recovery of time symmetry would be observed also in Navier-Stokes turbulence as for shell models. We conclude by mentioning another interesting open question. In <cit.> it was found that the Lagrangian power statistics is asymmetric also in statistically stationary 2D turbulence in the presence of an inverse cascade. Like in three dimensions, the third moment is negative and its magnitude grows with the separation between the timescale of dissipation by friction (at large scale) and of energy injection (at small scale), which is a measure of Re_λ for the inverse cascade range. Moreover, the scaling exponents are quantitatively close to the 3D ones. This raises the question on the origin of the scaling in 2two dimensions that cannot be rationalized within the MF, since the inverse cascade is not intermittent <cit.>. Likely, to answer the question one needs a better understanding of the influence of the physics at and below the forcing scale on the 2D Lagrangian power. We thank F. Bonaccorso for computational support.We acknowledge support from the COST Action MP1305 “Flowing Matter.”L.B.and M.D.P. acknowledge funding from ERC under the EU 7^th FrameworkProgramme, ERC Grant Agreement No. 339032. G.B. acknowledgeCineca within the INFN-Cineca agreement INF17fldturb.* § DETAILS ON THE NUMERICAL SIMULATIONS §.§ Direct Numerical SimulationsWe performed two sets of DNSs at differentresolutions and Reynolds numbers with two different forcing schemes. The values of the parameters characterizing all the simulations are shown in Table <ref>. In all cases we integrated the Navier-Stokes equations∂_tu+u·∇ u≡ a= - ∇ P + νΔ u+f ,for the incompressible velocity field u( x,t) with a fully parallel pseudo-spectral code, fully dealiased with 2/3 rule <cit.>, in a cubic boxof size ℒ=2 π with periodic boundary conditions. In (<ref>) P represents the pressure and ν is the kinematic viscosity of the fluid.For the set of runs DNS1 we used a Sawford-type stochastic forcing, involving the solution of the stochastic differential equations <cit.>d f̃_i = ã_i(t) dt ,d ã_i = -a_1 ã_i(t) dt -a_2 f̃_i(t) dt + a_3 dW_i(t),where a_1 = 1/τ_f, a_2 = (1/8)/τ_f^2, a_3 = √(2 a_1 a_2), and dW_i(t) = r √(dt) is an increment of a Wiener process (r is a random Gaussian number with ⟨ r ⟩ = 0 and ⟨ r^2 ⟩ = 1). The forcing f(k,t) in Fourier space is thenf(k,t) = i k× [i k× (0.16 k^-4/3f̃)]for k ∈ [k_f,min, k_f,max]0for k ∉ [k_f,min, k_f,max] .Time integration is performed by a second-order Adams-Basforth scheme with exact integration of the linear dissipative term <cit.>. For the set of runs DNS2 we use a deterministic forcing acting on a spherical shell of wavenumbers in Fourier space0 < | k| ≤ k_f, wherek_f=1.5 with imposed energy input rate ε <cit.>.In Fourier space the forcing readsf(k,t) =εu( k,t) /[2 E_f(t)]for k ∈ [k_f,min, k_f,max]0for k ∉ [k_f,min, k_f,max] .where E_f(t)=∑_k=0^k_f E(k,t), and E(k,t) is the energy spectrum at time t. This forcing guarantees the constancy of the energy injection rate. Notice that Eq. (<ref>) explicitly breaks the time-reversal symmetry; however, owing to the universality properties of turbulence with respect to the forcing, we expect this effect to be negligible as compared to the energy cascade.Time integration is performed by a second-order Runge-Kutta midpoint method with exact integration of the linear dissipative term <cit.>.Simulations have a resolution N sufficient to resolve the dissipative scale with k_maxη≃ 1.7 (k_max=N/3).We have checked in the simulations that the velocity field is statistically isotropic with a probability density function (for each component) close to a Gaussian.Simulations are performed for several large-scale eddy turnover times T, after an initial transient to reach the turbulent state, in order to generate independent velocity fields in stationary conditions.From the velocity fieldsthe acceleration field is then computed by evaluating the right hand side of (<ref>) and the power field is obtained as p= u· a.§.§ Simulations of the shell model As for the shell model (<ref>), simulations have been performed by fixing the number of shells N=30 and varying the viscosity ν in the range [3.16 × 10^-4 , 3.16 × 10^-8]. For each value of ν we performed ten independent realizations lasting approximately 10^6 T_L each.Time integration is performed using a fourth-order Runge-Kutta scheme with exact integration of the linear term.Forcing is stochastic and acts only on the first shell f_n=fδ_n,1.The stochastic forcing is obtained by choosing f=F(f^R+if^I) with F=1 andḟ^α =-1/τ_ff^α + √(2)/τ_fθ^α(t) , θ̇^α =-1/τ_fθ^α + √(2/τ_f)η^α(t) ,where η^α is a zero mean Gaussian variable with correlation ⟨η^α(t)η^β(t')⟩ = δ_αβδ(t-t'). 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Sawford, title title Reynolds number effects in Lagrangian stochastic models of turbulent dispersion,@noopjournal journal Phys. Fluids Avolume 3, pages 1577–1586 (year 1991)NoStop [Canuto and Quarteroni(2006)]canuto2006spectral author author C. Canuto and author A. Quarteroni, @nooptitle Spectral Methods(publisher Wiley Online Library, New York, year 2006)NoStop [Lamorgese et al.(2005)Lamorgese, Caughey, and Pope]lamorgese2005direct author author A. G. Lamorgese, author D. A. Caughey, and author S. B. Pope, title title Direct numerical simulation of homogeneous turbulence with hyperviscosity, @noopjournal journal Phys. Fluids volume 17, pages 015106 (year 2005)NoStop [Boyd(2001)]boyd2001chebyshev author author J. P. Boyd, @nooptitle Chebyshev and Fourier Spectral Methods (publisher Courier, North Chelmsford, year 2001)NoStop | http://arxiv.org/abs/1707.08837v3 | {
"authors": [
"Massimo Cencini",
"Luca Biferale",
"Guido Boffetta",
"Massimo De Pietro"
],
"categories": [
"physics.flu-dyn",
"cond-mat.stat-mech"
],
"primary_category": "physics.flu-dyn",
"published": "20170727123802",
"title": "Time irreversibility and multifractality of power along single particle trajectories in turbulence"
} |
UNIVERSITY OF IOANNINA Searching for Localized Black-Holesolutions in Brane-World models byTheodoros Nakas A thesis submitted in partial fulfilment for the Master's degree in the< g r a p h i c s >PHYSICS DEPARTMENTSCHOOL OF NATURAL SCIENCESUNIVERSITY OF IOANNINAGREECEJULY 2017roman“Every atom in your body came from a star that exploded. And, the atoms in your left hand probably came from a different star than your right hand. It really is the most poetic thing I know about physics: You are all stardust. You couldn't be here if stars hadn't exploded, because the elements - the carbon, nitrogen, oxygen, iron, all the things that matter for evolution and for life - weren't created at the beginning of time. They were created in the nuclear furnaces of stars, and the only way for them to get into your body is if those stars were kind enough to explode. So, forget Jesus. The stars died so that you could be here today." Lawrence M. Krauss Abstract tocchapterAbstractIn the context of this thesis, the question that is going to occupy us, is the existence of a 5-dimensional braneworld black hole solution that is localized close to the 3-brane and has the properties of a regular 4-dimensional one. For this purpose, the 4-dimensional part of the complete 5-dimensional spacetime is considered to be a generalized Vaidya metric, in the context of which, the mass parameter m is allowed to vary with respect to time, while it is also allowed to have both y and r dependence. The dependence on the r-coordinate essentially means that our black hole solution can deviate from the conventional Schwarzschild solution. Additionally, the dependence on the y-coordinate leads to a non-trivial profile of the black hole along the extra dimension. In order to justify physically the existence of such general mass parameter, we consider the case of two scalar fields ϕ(v,r,y), χ(v,r,y) which interact with each other and they are also non-minimally coupled to gravity via a general coupling function f(ϕ,χ). In all the cases that were investigated in the context of this particular scenario, the result for the existence of a viable 5-dimensional localized black hole solution was negative, a result that causes concern about the compatibility of brane-world models with basic predictions of General Theory of Relativity.plain Acknowledgements tocchapterAcknowledgements First and foremost, I would like to thank my parents; without their love and support this thesis would never have been started. I would also like to thank my supervisor, Panagiota Kanti, not only for the guidance, help and support, but also for providing me the opportunity to work with her and learn more about the enchanting field of General Relativity. I am very thankful to Charalambos Kolasis as well. His passion for mathematics and his clear proofs in every theorem or every problem that he mentioned, helped me understand the way of reasoning as an undergraduate student. Last but not least, I would like to say a big thank you to Ilias for letting me evaluate my mathematica codes on his “monster" computer and Georgia, Konstantinos and my friends for helping me maintain my sanity. This journey would not have been so enjoyable without you (cliché but also true).plaintocchapterContentsarabicCHAPTER: BASIC NOTATIONtocchapterBasic Notation* The signature of the metric tensor (g_μν) that is going to be used in the context of this thesis is chosen to be: (-,+,+,…,+). Therefore, a flat and 4-D (four-dimensional) space-time has the following line-element.ds^2=-(cdt)^2+(dx)^2+(dy)^2+(dz)^2 * Upper-case Latin indices M,N,… will denote bulk coordinates, thus for a 6-D space-time they will take the values 0,1,2,…,5. Greek indices μ,ν,… will be used for brane coordinates, hence they will take the values 0,1,2,3. Finally, lower-case Latin indices a,b,… will denote the three spatial coordinates taking the values 1,2,3.* Most of the times, we are going to use natural units or Planck units in order mathematical expressions to be less complicated, i.e. c=ħ=1 or c=ħ=G=1 respectively.* The partial derivative of a function f=f(x^0,x^1,x^2,x^3) with respect to x^μ can be expressed asf/ x^μ=_μ f=f_,μ * The covariant derivative of a tensor quantity with respect to x^ is denoted asT^μν_;or ∇_ T^μνwhere we arbitrarily chose the contravariant tensor T^μν to present the notation. Of course, the same notation holds for any kind of tensor. CHAPTER: INTRODUCTIONThe fundamental similarity of people throughout the history of humankind is curiosity. Since ancient times, this special characteristic has made humans wonder about nature, the origin of the universe and consequently the origin of life. The questions that immediately come to mind are: How did the universe begin? Which are the fundamental building blocks of the universe? Are there extra dimensions?These questions are still open and they may remain open for many more years. Our understanding about the universe is limited. It is worth mentioning that everything that is possible to be observed in the night sky with a naked eye or by using telescopes adds up to about 4.9% of the entire universe, the other 95.1% of the universe is made of dark matter <cit.> (26.8%) and dark energy <cit.> (68.3%)[The name dark is related to the fact that they do not interact electromagnetically. Therefore, they are invisible to the entire electromagnetic spectrum.]. Almost nothing is known about dark matter and dark energy, but it is certainly known that both of them interact gravitationally. Dark matter behaves like ordinary matter (in terms of gravitation) but it is not luminous, its existence and properties are deduced by astrophysical and cosmological measurements i.e. galaxy rotation curves, gravitational lensing, cosmic microwave background (CMB), etc. On the other hand, dark energy is associated with the accelerated expansion of the universe, hence it is needed to act repulsively. The most popular method and historically the first one that was formulated in order to describe dark energy, is via the cosmological constant, which can be identified to a perfect fluid of constant energy density and negative pressure that fills the entire universe homogeneously[An alternative method to describe dark energy is via quintessence <cit.>, a scalar field that it is allowed to be both space and time varying.]. The evidence of the existence of dark energy is indicated also by astrophysical and cosmological observations, i.e. high redshifted observations from supernovae, CMB, Large-scale structures, observational Hubble constant data (OHD), etc. So far we have discussed about what we do not know, thus, it is now reasonable to discuss about what we actually know about the universe. Everything that is known about the universe nowadays can be provided by two theories. The first one in chronological order is General Theory of Relativity (GTR or GR) <cit.> and the second one is the Standard Model (SM) <cit.> of particle physics. GR constitutes the modern theory of gravity and it was formulated by Albert Einstein in 1915 <cit.>. The mathematical framework of the theory is Differential Geometry and, in this context, gravity emerges as a geometric property of spacetime[More precisely, the curvature of the spacetime and the gravitational field are the two sides of the same coin.], which is a four dimensional (4-D) manifold. GR provides a clear description and explanation for a number of astronomical and cosmological observations. However, only a year after its publication, Karl Schwarzschild showed that GR breaks down in high energies by proving that the spherically symmetric vacuum solution of the gravitational field equations[In 1923, George David Birkhoff proved that any spherically symmetric solution of the vacuum gravitational field equations must be static and asymptotically flat, which means that the spacetime outside of a spherical, non-rotating body must be described by the Schwarzschild metric.] leads to a singularity at r=0 <cit.>. For obvious reasons, infinities (or singularities) are undesirable features for a physical theory. Consequently, GR cannot be considered as the ultimate theory of gravity. When quantum effects become important GR fails to describe them. R0.4< g r a p h i c s >Particle interactions as described by Standard Model.SM on the other hand, constitutes the theory of the quantum world. It successfully describes the remaining three of the four known fundamental forces of nature (electromagnetism, weak interaction and strong interaction) and additionally accommodates all the known elementary particles (see Figure 1.1). SM is the most experimentally tested theory and its predictions are confirmed with extreme accuracy. Although its phenomenal success to explain the vast majority of processes in particle physics, SM is still not capable of being the fundamental theory that describes all four fundamental forces (including gravity). Moreover, SM does not provide any insight for the nature of dark energy and dark matter, which as we already mentioned are the biggest mysteries nowadays. It is crystal clear from the aforementioned problems that a new and more fundamental theory is needed, or a deeper understanding of the already established ideas. Such a theory should encompass both GR and SM, namely a theory of everything, but it is not necessary to be based on either of them. However, this hypothetical theory should reproduce both GR and SM in the appropriate limit. Currently, the most promising theory of unification is string theory, which replaces elementary particles with one-dimensional strings, instead of being accounted for as point-like objects. The fatal flaw of string theory is its complexity; it requires at least 10 dimensions in order to be consistent and also it is beyond experimental verification because of the extremely high energies that are needed for such a purpose. R0.4< g r a p h i c s >The 5-D spacetime in Kaluza Klein theory.Historically, the first theory that introduced extra dimensions was Kaluza-Klein theory. The primary idea came from Theodor Kaluza (1921) <cit.>, who extended general theory of relativity by allowing the existence of another spatial dimension in addition to the four dimensional spacetime fabric of GR. Furthermore, he constrained the five-dimensional metric tensor by demanding none of its components to be dependent on the extra dimension. This condition is also known as the “cylinder condition". His motivation for a higher dimensional spacetime was the unification of electromagnetism and gravity. The contribution of Oscar Klein (1926) <cit.> was his quantum interpretation of Kaluza's theory. He hypothesized that the extra dimension is curled up and tiny (see Figure 1.2) in order to explain the cylinder condition and also evaluated the scale of the extra dimension by taking into consideration the quantum nature of the electric charge.plainIn the last twenty years, two additional extra dimensional theories was added to the literature, ADD model (1998) <cit.>and RS models (1999) <cit.>. Both models <cit.> were motivated by the Hierarchy Problem[Hierarchy problem is the vast discrepancy between the electroweak scale m_EW∼ 1TeV and Planck scale m_P∼ 10^19 GeV. (m_EW/m_P∼ 10^-16. m_P=√(ħ c/G), while M_P=m_P/√(8π)∼ 10^18 GeV).] and also both of them were based on the concept of braneworld, which simply indicates that Standard Model particles are confined on our 4-D spacetime (3-brane) while gravity can freely propagate in the bulk, i.e the entire (4+n)-dimensional spacetime, where n is the number of the extra dimensions. Particularly, the ADD model allows the existence of n compact extra spatial dimensions of the same radius R. Thus, the total number of dimensions that gravity encounters are (4+n). Subsequently, the fundamental Planck scale corresponds to the higher dimensional one, namely M_P(4+n), while the Planck scale M_P which we experience is an effective one. Hence, it is possible the fundamental Planck scale M_P(4+n) to be equal to the electroweak scale m_EW and simultaneously the effective Planck scale to maintain its huge value. As it is indicated in <cit.>, the gravitational potential between two masses m_1 and m_2 which are separated by a distance r≪ R in (4+n)-dimensions is given by the following expression V(r)∼m_1 m_2/M_P(4+n)^n+21/r^n+1 Equation (<ref>) can be derived by the gravitational Gauss's law in (4+n) dimensions <cit.>. Assuming now that r≫ R we obtain V(r)∼m_1m_2/M^n+2_P(4+n)R^n1/r Therefore, the effective Planck scale should fulfil the following equationM_P^2∼ M_P(4+n)^n+2R^n Solving the above equation with respect to R and setting M_P(4+n)=m_EW, we obtainR∼ 10^30/n-19(1 TeV/m_EW)^1+2/n m m_EW∼ 1 TeVR∼ 10^30/n-19 m Equation (<ref>) associates the number of the extra dimensions n with the size of their radius R. It is obvious that for n=1 and R∼ 10^11 m Newton's gravitational law would differ from the law that we are all used to. Deviations from the conventional Newton's law have not been measured, thus, n≥ 2. The upper bound for the size of the extra dimensions is R∼ 10^-4 m <cit.> and it results for n=2. The figures below depict schematically the two different types of topologies of the extra dimensions that was taken into consideration in <cit.>. plainADD model is elegant from a mathematical point of view, but only seemingly solves the Hierarchy Problem. In particular, ADD model simply replaces the problem of hierarchy with that of the size of the extra dimensions R, because now it is requisite to explain the reason why R can be so much larger than the length 10^-19 m, which is associated with the fundamental Planck scale M_P(4+n)=m_EW∼ 1 TeV. Although it is difficult to measure gravitational deviations of the Newton's law in sub-millimeter distances, the fact that the fundamental gravitational scale M_P(4+n) can be equal to 1 TeV gives us the opportunity to detect (indirectly) the existence of extra dimensions in collider experiments through the formation of tiny black holes from highly energetic particles <cit.>.plainOn the other extreme, the RS models <cit.> -which were published only one year after the ADD model- examine the case of only but one curved extra dimension in the bulk. The special characteristic of these models is that now the 3-brane itself possesses tension, therefore it interacts gravitationally with the bulk. Particularly, the first RS model (RS1) assumes the existence of two 3-branes in the bulk and achieves to generate the electroweak scale from the Planck scale through an exponential hierarchy, which arises purely from the geometry of the 5-D spacetime. RS2 model assumes the existence of only one 3-brane embedded in an infinitely long extra dimension but nevertheless manages to reproduce 4-dimensional gravity on the brane. Both RS models will be discussed in Chapter 2, thus, any further detail will be postponed for later.§ MOTIVATION AND THESIS OUTLINEIn the context of GR, all the information that is needed to describe a black hole comes only from three classical parameters: mass (M), electric charge (Q) and angular momentum (J). It is impossible to distinguish two black holes if they are characterized by the same aforementioned parameters and these parameters have the same value. This is an incredible characteristic of the 4-D black holes and it is rarely encountered in other objects in nature. Given these parameters, there are only four different black hole solutions[Schwarzschild metric (M≠ 0, Q=0, J=0) <cit.>, Reissner-Nordström metric (M≠ 0, Q≠ 0, J=0) <cit.>, Kerr metric (M≠ 0, Q=0, J≠ 0) <cit.>, Kerr-Newman metric (M≠ 0, Q≠ 0, J≠ 0) <cit.> (See also [bhs]Appendix A).] <cit.> that can be derived by the gravitational field equations of GR; this particular statement is also known as the no-hair theorem <cit.>. On the other hand, braneworld black holes are not so easily manageable and most importantly there is not a corresponding no-hair theorem for higher dimensional spacetimes. Therefore, the “families" of higher dimensional black holes solutions are not yet known. This thesis is entirely focused on braneworld black holes and more specifically on the existence of localized black hole solutions in an RS2-type braneworld model.The outline of the thesis is as follows. Chapter 2 -as it was mentioned previously- constitutes a detailed analysis of the RS models. The existence of black string solutions in the context of RS models and the difficulties in finding a localized 5-D black hole solution on our 3-brane are reviewed in Chapter 3. In the same Chapter, the theoretical framework of the thesis, namely the geometrical background that the thesis is based on, and the scalar field theory (two non-minimally-coupled and interacting scalar fields with a general coupling to the Ricci scalar) are also presented. This scalar field theory constitutes a more general theory compared to the scalar field theory that is presented in <cit.>. This extra degree of freedom -that the additional scalar field offers- may lead to a solution to the problem of localizing a 5-dimensional black hole on a 3-brane. Finally, in Chapter 3, the field equations of this particular ansatz are derived. Subsequently, the various cases[Different cases vary on the spacetime coordinates from which the scalar fields and the coupling function depend on.] which are studied in this particular scenario are presented in a series of paragraphs of increasing complexity in Chapter 4. Finally, our results and our conclusions are discussed in Chapter 5. plainCHAPTER: RANDALL-SUNDRUM BRANE-WORLD MODELSIn their first model <cit.> (RS1), Lisa Randall and Raman Sundrum introduced a compact extra dimension, which is finite and bounded by two 3-branes. This specific type of spacetime enriched with some additional properties, which are going to be discussed in the following section, manages to address the hierarchy problem. Amazingly, they proved that the 4-D gravity can also be recovered on the brane even if the extra dimension has infinite size, this constitutes the RS2 model <cit.>. Let us now proceed to the analysis of these models.§ RS1 MODELThe RS model postulates the existence of one extra dimension which is compactified on a circle S^1 (one-dimensional sphere) and also possesses a ℤ_2 symmetry, which simply means that the points (x^μ,y) and (x^μ,-y) are identified. Hence, the extra dimension is an S^1/ℤ_2 orbifold. As it is illustrated in the figure below, this type of compactification contains two fixed points, at y=0 and y=π r_c≡ L. The range of y is from -L to L, but the metric is completely specified by the values in the range 0≤ y≤ L. These fixed points of the extra dimension (y=0 and y=L) host the two 3-branes, which essentially are two separated 4-D worlds.The action of the model is as follows:S=S_grav+S_1+S_2 S_grav=∫ d^4x∫_-L^Ldy √(-g^(5))(R/2κ_(5)-Λ_5) S_1=∫ d^4x√(-g_1) (_1-_1) S_2=∫ d^4x√(-g_2) (_2-_2)where=[g_MN(x^,y)] g_1=[g^1_μν(x^)] g_2=[g^2_μν(x^)]The determinants g_1 and g_2 are derived by the metrics g^1_μν and g_μν^2 of the two 3-branes which are located at y=0 and y=L respectively. R is the 5-D Ricci scalar, Λ_5 is the higher dimensional cosmological constant and κ_(5)=8π G_(5)=M_P(5)^-3 where M_P(5) is the fundamental Planck scale of the 5-D spacetime. The quantities _1 and _2 represent the energy densities of the 3-branes, while _1 and _2 are the Lagrangians on each 3-brane.The variation of Eq.(<ref>) with respect to the components of the 5-D metric tensor g_MN provides us with the field equations (see [var]Appendix F for more details about the variation of a general action). The field equations, which are depicted below, were deduced under the assumption that _1=_2=0. This particular assumption was made in order to determine the geometrical background of the model. The field equations have the following form:√(-) G_MN=-κ_(5)[√(-) g_MN Λ_5. +√(-g_1) _1 g^1_μν ^μ_M ^ν_N (y).+√(-g_2) _2 g^2_μν ^μ_M ^ν_N (y-L)]In order to continue the analysis of the model it is necessary to introduce an appropriate metric for such a setup. A property that we need to impose on the 5-D metric is to respect the Poincaré symmetry on the two 3-branes. This property comes naturally from the fact that the 4-D induced metrics on the 3-branes should describe the real world and therefore they should respect the same symmetries as the physical world. The general form of a five-dimensional line-element that includes an extra dimension with S^1/ℤ_2 compactification and also respects Poincaré symmetry has the following formds^2=e^2A(y)η_μνdx^μ dx^ν+dy^2 The function A(y) is called warp factor and it will be evaluated subsequently by the gravitational field equations of this theory. It is obvious that the metric tensor that results from this particular line-element is of the form(g_MN)=([ -e^2A(y)0000;0e^2A(y)000;00e^2A(y)00;000e^2A(y)0;00001 ]) g_MN=e^2A(y)η_μν_g_μν^μ_M^ν_N+^4_M^4_N=g_μν^μ_M^ν_N+^4_M^4_N (g^MN)=( [ -e^-2 A(y)0000;0e^-2 A(y)000;00e^-2 A(y)00;000e^-2 A(y)0;00001;]) g^MN=e^-2A(y)η^μν_g^μν_μ^M_ν^N+_4^M_4^N=g^μν_μ^M_ν^N+_4^M_4^NConsequently, the induced metric tensors on the 3-branes at the points y=0 and y=L are given byg^1_μν=e^2A(0)η_μν g^2_μν=e^2A(L)η_μνThe non-zero and two-times covariant components of the Einstein tensor G_MN of the aforementioned ansatz are depicted below:{[ G_00=G_tt=-3e^2A(y)[2A'^2(y)+A”(y)];; G_11=G_22=G_33=3e^2A(y)[2A'^2(y)+A”(y)];;G_44=G_yy=6A'^2(y) ]}{[ G_μν=3[2A'^2(y)+A”(y)]g_μν; ;G_44=6A'^2(y) ]}Combining equations (<ref>), (<ref>) and (<ref>) for M=N=4 we get:6A'^2(y)=-κ_(5)Λ_5 A'^2(y)=-Λ_5/6while for M=μ and N=ν we obtain:6A'^2(y)+3A”(y)=-Λ_5- _1 (y)- _2 (y-L)(<ref>) 3A”(y)=-[_1 (y)+_2 (y-L)] A”(y)=-/3[_1 (y)+_2 (y-L)]The five-dimensional cosmological constant Λ_5 must be negative in order to have a real solution for the function A(y). A negative 5-D cosmological constant affects decisively the geometry of the spacetime between the two 3-branes. Particularly, the bulk spacetime of this scenario leads to an anti-de Sitter spacetime between the two branes, which is also denoted as AdS_5. Subsequently, we define the following constant k^2≡ -Λ_5/6 The specific function of A(y) can be easily deduced from Eq.(<ref>). Combining equations (<ref>) and (<ref>), we get A'^2(y)=k^2 A'(y)=± k A(y)=± kyA(y)=-k|y| The reason that we kept the minus sign in Eq.(<ref>) will be understood later. In addition, it is necessary to preserve the orbifold symmetry S_1/ℤ_2 for the extra dimension y. Hence, we are obliged to express the function A(y) in terms of |y|. Substituting Eq.(<ref>) into Eq.(<ref>) we obtain the geometrical background of the RS model: ds^2=e^-2k|y|η_μνdx^μ dx^ν+dy^2 We can relate the constants _1 and _2 with k by evaluating the second derivative of A(y) as given by Eq.(<ref>) and then equate the result with Eq.(<ref>). A'(y)=-k(|y|)'=-k sgn(y)=-k[H(y)-H(-y)] where H(y)={[ 0, -L≤ y< 0; ; 1, 0≤ y ≤ L ]}, H(-y)={[ 1, -L≤ y< 0; ; 0, 0≤ y ≤ L ]}A”(y)=-k[H'(y)-H'(-y)]=-2k[(y)-(y-L)] where{ H'(y)=(y)-(y-L) H'(-y)=[-(y-L)]-(-y)=-(y)+(y-L) } (<ref>)(<ref>)-/3[_1 (y)+_2 (y-L)]=-2k[(y)-(y-L)] _1=-_2=6k/ Let us now focus on the 3-brane at y=L and include the Higgs field in the 4-dimensional action. We are going to evaluate the Vacuum Expectation Value (VEV) of the Higgs field on the brane which determines the physical masses in the Standard Model. The action will have the form:S_H=∫ d^4x∫_-L^L dy√(-)[g^MND_M H^† D_N H-(H^† H-v_0^2)^2](y-L) S_H=∫ d^4x√(-g_2)[g_2^μν D_μ H^† D_ν H-(|H|^2-v_0^2)^2][√(-g_2)=e^4A(L)=e^-4kL]g_2^μν=e^-2A(L)η^μν=e^2kLη^μν S_H=∫ d^4x e^-4kL[e^2kLη^μνD_μ H^† D_ν H-(|H|^2-v_0^2)^2]S_H=∫ d^4x[η^μνD_μH̃^† D_νH̃-(|H̃|^2-v^2)^2] In order to obtain a canonically normalized action we definedH̃≡ e^-kLH v≡ e^-kLv_0The action of Eq.(<ref>) depicts the ordinary action of the Higgs field. The corresponding VEV of the renormalized Higgs scalar H̃ is v and is given by Eq.(<ref>), while the Higgs scalar H is the bare Higgs with VEV v_0. As we mentioned earlier, the VEV of the Higgs field determines all the mass parameters in the context of SM, thus we can safely conclude that m=e^-kLm_0 where m constitutes the physical mass as it is measured on the 3-brane at y=L. Eq.(<ref>) is a simple and “powerful" result. It can be considered as a necessary condition for the solution of the hierarchy problem, because it does not demand a huge discrepancy between parameters k and L in order to achieve that. It is easy to verify the previous statement by setting m_0 equal to the Planck mass M_P∼ 10^18 GeV and m equal to the electroweak scale m_EW∼ 1 TeV. Then:10^3 GeV=e^-kL 10^18 GeV e^-kL=10^-15 kL=15ln(10) kL≈ 35In order to confidently state that the type of exponential suppression of Eq.(<ref>) successfully addresses the hierarchy problem, it is also necessary to examine the dependence of the effective scale (4-D scale) of gravity on the size of the extra dimension y. For this purpose, we need to perturb the four-dimensional part of the five-dimensional line-element of Eq.(<ref>) and then extract the 4-D gravitational action from the original 5-D action which is shown at Eq.(<ref>). The form of the perturbative line-element is given by the following relation:ds^2 =e^-2k|y|[η_μν+h_μν(x)]dx^μ dx^ν+dy^2=e^-2k|y|g^per_μνdx^μ dx^ν+dy^2where |h_μν|≪ 1. Using the metric that follows from Eq.(<ref>) into Eq.(<ref>) and focusing on the term which includes the Ricci scalar, we obtain S_eff= ∫ d^4x ∫_-L^Ldy e^-2k|y|/2√(-g_per) R^4D(g^per_μν) The S_eff should be also equal to S_eff=∫ d^4x √(-g_per) R^4D/2κ where κ=8π G=M_P^-2. Consequently, by equating the last two relations we obtain the following expression for the effective scale of gravity.1/κ=1/∫_-L^L dy e^-2k|y| M_P^2=M_P(5)^3(∫_-L^0 dy e^2ky+∫_0^Ldy e^-2ky) M_P^2=M_P(5)^3(1/2k-e^-2kL/2k-e^-2kL/2k+1/2k) M_P^2=M_P(5)^3/k(1-e^-2kL)M_P^2=M_P(5)^3/k(1-e^-2kL) Substituting now the value of the product kL given by Eq.(<ref>) into Eq.(<ref>) we get M_P^2=M_P(5)^3/k(1-e^-70)≃M_P(5)^3/k It is clear from Eq.(<ref>) and Eq.(<ref>) that gravity is essentially independent of the size of the extra dimension. Surprisingly, even if we infinitely extend the length L of the extra dimension in Eq.(<ref>), the four-dimensional Planck scale M_P remains finite. This particular observation was the central point of the RS2 model which is going to be analyzed in the section that follows. Conclusively, it was shown that in the context of the RS model the hierarchy problem has an extremely simple and clear solution. Simultaneously, the RS model does not introduce new huge hierarchies (in contrast with the ADD model) between its fundamental parameters (k,or M_P(5), v_0, L or r_c). The only constraint that is required by the model is that kL≈ 35. Of course, a stabilizing mechanism (Goldberger-Wise mechanism <cit.>) must be included in the model for this purpose but this is beyond the context of the present analysis.§ RS2 MODELIn their second paper, Lisa Randall and Raman Sundrum proved that is in fact possible to extend the length L of the extra dimension to an infinite value and nevertheless get an effectively four-dimensional gravity. The verification of the previous statement is derived from the fact that the 5-D graviton is localized near the 3-brane at y=0. The setup of the RS2 model is similar to the RS1 model with the difference that the 3-brane at y=L is practically removed from the original picture, by taking L to infinity. Thus, the action of the RS2 model isS=S_grav+S_1 S_grav=∫ d^4x∫ dy √(-)(R/2-Λ_5) S_1=∫ d^4x√(-g_1)(_1-_1)The line-element of the RS2 model is given by Eq.(<ref>) as well. Let us now proceed to the derivation of the graviton modes. Gravitons correspond to small fluctuations in the spacetime “fabric". Therefore, in the context of RS2 model we have ds^2=e^-2k|y|[η_μν+h_μν(x,y)]dx^μ dx^ν+dy^2 In the last equation it was chosen h_M4=0. It is always possible to find a set of coordinates with this particular property in the region of the 3-brane. We now seek a new variable z for the description of the extra dimension in order to construct a metric tensor which will be more convenient for future calculations. For this purpose, we demand from the new variable z to satisfy the following relation: dy^2≡ e^-2k|y|dz^2 Performing the integration of the previous equation we find k|z|=e^k|y|-1 where we chose the integration constant appropriately in order to get z=0 when y=0. Eq.(<ref>) leads to e^-2k|y|=1/(k|z|+1)^2 Combining equations (<ref>) and (<ref>) we obtainds^2=1/(k|z|+1)^2{[η_μν+h_μν(x,z)]dx^μ dx^ν+dz^2}h_4M=0 ds^2=e^2A(z)[η_MN+h_MN(x,z)]dx^Mdx^Nds^2=e^2A(z)g̅_MN(x,z)dx^Mdx^Nwheree^2A(z)≡1/(k|z|+1)^2g̅_MN(x,z)≡η_MN+h_MN(x,z)From Eq.(<ref>) we getA(z)=-ln(k|z|+1) A'(z)=-k sgn(z)/k|z|+1=-k[H(z)-H(-z)]/k|z|+1 A”(z)=-2k(z)/k|z|+1+k^2/(k|z|+1)^2Under the above transformation of the extra dimension (from y to z), Eq.(<ref>) is also going to be transformed into S_grav=∫ d^4x∫_-L_z^L_z dz√(-)(R/2-Λ_5) where L_z=(e^kL-1)/k. The field equations which are yielded by the variation of Eq.(<ref>) with respect to the components of the 5-D metric are depicted below (_1 in Eq.(<ref>) is set to be 0).√(-) G_MN=-[√(-) g_MN Λ_5+√(-g_1) g^1_μν ^μ_M ^ν_N (z) _1] whereg^1_μν (z)=e^2A(z)(η_μν+h_μν) (z) g_1 (z)= e^-2A(z) (z)Hence, the combination of equations (<ref>)-(<ref>) results to G_MN=-[g_MN Λ_5+e^A(z)(η_μν+h_μν)^μ_M ^ν_N (z) _1] The calculation of the Einstein tensor components G_MN using “brute force" is a difficult task in this model. Hence, we are going to use a conformal transformation[For more details about conformal transformations see <cit.>.] in order to obtain the components of the Einstein tensor more easily. Particularly, we mention (without proof) that if g̃_MN is the conformally transformed metric of the metric g_MN and the two metrics are connected through the relation g̃_MN=Ω^2(x,z) g_MN then the corresponding components of the Einstein tensor are connected as follows: G̃_MN=G_MN+D-2/2Ω^2[4Ω_,MΩ_,N+(D-5)Ω_,KΩ^,K g_MN]-D-2/Ω(Ω_;MN-g_MN□Ω) where D indicates the total number of spacetime's dimensions. The adjustment of equations (<ref>) and (<ref>) in our case is straightforward. We simply execute the following substitutions: {g̃_MN g_MN, Ω(x,z) e^A(z), g_MNg̅_MN and D=5}. Hence, we obtain the expression:G_MN=G̅_MN+3[_M A _N A-∇_M∇_NA+g̅_MN(□ A+_LA ^LA)] G_MN=G̅_MN+3[_M A _N A -_M_NA+^L_MN_LA+g̅_MN(_L^LA+^L_LK^KA+_LA ^LA)]For all the subsequent calculations in this section, the terms which contain fluctuations (h_MN) of order higher than the first will be neglected. Additionally, it is possible and extremely convenient to perform appropriate coordinate transformations[A complete analysis of the legitimacy and derivation of this particular gauge is presented in <cit.>. See [gnc]Appendix B as well.] in which the fluctuations satisfy the following properties: {[ h_4M=0; ;h=h^μ_μ=0; ; _μ h^μ_ν=0 ]} In this gauge and with the use of equations (<ref>) and (<ref>) it is quite trivial to show that the Christoffel symbols ^L_MN and the components of Einstein tensor G̅_MN are expressed through the following equations:^L_MN=1/2(_Nh^L_M+_Mh^L_N-^Lh_MN) G̅_MN=-1/2_L^Lh_MN Using equations (<ref>)-(<ref>) we are led to {[G_44=6A'^2;;G_4μ=0;; G_μν=-1/2_L^Lh_μν-3/2 _4h_μν A'+3(η_μν+h_μν)(A”+A'^2) ]}The combination of equations (<ref>) and (<ref>) for M=N=4 results to Eq.(<ref>), namely6A'^2=-Λ_5 g_44=-Λ_5 e^2A(z)(η_44_1+h_44_0)(<ref>) A'^2(z)=k^2e^2A(z) A(z)=-ln(k|z|+1)where we fixed again the integration constant appropriately (the reason has already been discussed in the sentence before equation (<ref>)). Correspondingly, for M=μ and N=ν we have: -1/2_L^Lh_μν-3/2 _4h_μν A'+3(η_μν+h_μν)(A”+A'^2)=-(η_μν+h_μν)[e^2AΛ_5+e^A(z) _1] It is not hard to combine equations (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>) in order to prove the following expressions (these are going to simplify Eq.(<ref>) afterwards): -Λ_5 e^2A(z)=-Λ_5/6 6 e^2A(z)=6 k^2 e^2A(z)=6A'^2(z) - _1 e^A(z) (z)=-6 k e^A(z) (z)=3[A”(z)-A'^2(z)] Consequently, from equations (<ref>)-(<ref>) we get _L^Lh_μν+3 _4h_μν A'=0 We now rescale the fluctuations h_μν as follows. h_μν e^ A(z)h_μν Then, Eq.(<ref>) takes the form: h_μν[(3+^2)A'^2+ A”]+(2+3)A' _4h_μν+_L^Lh_μν=0 It is now obvious that we can nullify the term which contains the quantity _4h_μν by choosing =-3/2. Furthermore, we perform a Kaluza-Klein decomposition on the fluctuations h_μν(x,z): h_μν(x,z)=∑_n=0^∞ h_μν^n(x)ψ_n(z) where {[ h^n_μν(x)=e^i p_nx=e^i p^n_ x^; ; _^ h_μν^n(x)=m_n^2 h_μν^n(x) ]}Substituting equations (<ref>) and (<ref>) into Eq.(<ref>) and setting =-3/2, we obtain_L^L[∑_n=0^∞ h^n_μν(x)ψ_n(z)]-(3/2A”+9/4A'^2)∑_n=0^∞ h^n_μν(x)ψ_n(z)=0 _4^4[∑_n=0^∞ h^n_μν(x)ψ_n(z)]+_^[∑_n=0^∞ h^n_μν(x)ψ_n(z)]-(3/2A”+9/4A'^2)∑_n=0^∞ h^n_μν(x)ψ_n(z)=0 ∑_n=0^∞{h^n_μν(x)[ψ_n”(z)+m_n^2 ψ_n(z)-(3/2A”+9/4A'^2)ψ_n(z)]}=0-ψ_n”(z)+V(z) ψ_n(z)=m_n^2 ψ_n(z) whereV(z)=3/2A”+9/4A'^2=-3k(z)/k|z|+1+15k^2/4(k|z|+1)^2=15k^2/4(k|z|+1)^2-3k(z) The boundary condition at z=0 can be found as follows:(<ref>)∫_0^-^0^+dz[-ψ_n”(z)+V(z)ψ(z)]=∫_0^-^0^+dz m_n^2 ψ_n(z) -ψ_n'(0^+)+ψ_n'(0^-)-3kψ_n(0)=0[ψ'_n(z)=-ψ'_n(-z)]ψ_n(z)=ψ_n(-z)ψ'_n(0^+)=-3/2 k ψ_n(0) Although we have focused on the RS2 model, we may develop a unified analysis for the study of gravitons in both RS models. Thus, if we reintroduce the second brane at y=L=π r_c or at z=(e^kπ r_c-1)/k≡ L_z (as it is indicated by Eq.(<ref>)), then the potential of Eq.(<ref>) will be modified as follows and a new boundary condition at z=L_z will be added as well: V_new(z)=15k^2/4(k|z|+1)^2-3k[(z)-(z-L_z)]/k|z|+1 The boundary condition at z=L_z can be deduced with the use of equations (<ref>) and (<ref>).∫_L_z^-^L_z^+dz[-ψ_n”(z)+V_new(z)ψ_n(z)]=∫_L_z^-^L_z^+dz m_n^2 ψ_n(z) -ψ_n'(L_z^+)+ψ_n(L_z^-)+3kψ_n(L_z)/k|L_z|+1=0ψ_n(L_z^+)=-ψ_n(L_z^-)ψ_n'(L_z)=-3kψ_n(L_z)/2(kL_z+1)§.§ Kaluza-Klein Modes The zero-mode ψ_0(z) for m_0=0 can be easily evaluated from Eq.(<ref>). Thus, it is-ψ”_0(z)+[3/2A”(z)+9/4A'^2(z)]ψ_0(z)=0ψ_0(z)=N_0 e^3/2A(z)(<ref>)ψ_0(z)=N_0(k|z|+1)^-3/2 where N_0 is a normalization constant which is going to be evaluated followingly.∫_-L_z^L_zdz |ψ_0(z)|^2=1 (N_0)^2∫^L_z_-L_zdz(k|z|+1)^-3=2(N_0)^2∫_0^L_zdz(kz+1)^-3=1(N_0)^2[-1/k(kL_z+1)^2+1/k]=1kL_z=e^kL-1≫ 1(N_0)^21/k=1 N_0=√(k) From equations (<ref>) and (<ref>) we are led to ψ_0(z)=1/k(|z|+1/k)^-3/2 The figure above depicts the plot of the potential V(z) as it is given by equation (<ref>), where it was used the approximation (z)∼(1/a√(π))e^-(z/a)^2 with a=0.05. This particular approximation helps us to visualize better the behaviour of the potential. The boundary conditions are both satisfied. The figure in the next page depicts the plot of the graviton zero-mode at the region of the 3-brane which is located at z=0.The Kaluza-Klein (KK) modes for n>0 can be provided from the general equation (<ref>). ψ_n”(z)+[m_n^2-15k^2/4(k|z|+1)^2]ψ_n(z)=0 while for z=0 and z=L_z equations (<ref>) and (<ref>) should be satisfied respectively. The above differential equation has the following general solution: ψ_n(z)=(|z|+1/k)^1/2{a_n J_2[m_n(|z|+1/k)]+b_nY_2[m_n(|z|+1/k)]} where a_n, b_n are constant coefficients and J_2(z), Y_2(z) are the Bessel functions of the first and second kind respectively. The derivative of the function ψ_n(z) is given byψ_n'(z)= m_n(|z|+1/k)^1/2{a_n J_1[m_n(|z|+1/k)]+b_nY_1[m_n(|z|+1/k)]}-3/2(|z|+1/k)^-1/2{a_n J_2[m_n(|z|+1/k)]+b_nY_2[m_n(|z|+1/k)]}We can relate the constants a_n and b_n by simply applying the boundary condition at z=0. Thus, from equations (<ref>), (<ref>) and (<ref>) we get: a_n=-b_nY_1(m_n/k)/J_1(m_n/k) Hence, it is: ψ_n(z)=N_n(|z|+1/k)^1/2{-Y_1(m_n/k)J_2[m_n(|z|+1/k)]+J_1(m_n/k)Y_2[m_n(|z|+1/k)]} where we defined N_n=b_n/J_1(m_n/k) Since the massless graviton mode is accompanied by a tower of massive Kaluza-Klein states, one may wonder about the gravitational potential that an observer on the visible brane will feel. In order to answer this, we will now consider each RS model separately.The limit m_n/k≫ 1Combining equations (<ref>), (<ref>), (<ref>) and (<ref>) we geta_n J_1(m_n/k)+b_nY_1(m_n/k)=0 a_n J_1[m_n(L_z+1/k)]+b_nY_1[m_n(L_z+1/k)]=0It is well-known that a homogeneous system of equations in order to have a non-trivial solution it is necessary to have a vanishing determinant. Thus, for the previous system of equations we demand that J_1(m_n/k)Y_1[m_n(L_z+1/k)]-J_1[m_n(L_z+1/k)]Y_1(m_n/k)=0kL_z=e^kL-1J_1(m_n/k)Y_1(m_n/ke^kL)-J_1(m_n/ke^kL)Y_1(m_n/k)=0In this limit, the large value of the quantity m_n/k results to the following expressions for the Bessel functions: {[ J_1(m_n/k)∼√(2k/π m_n)cos(m_n/k-3π/4), J_1(m_n/ke^kL)∼√(2k/π m_ne^-kL)cos(m_n/ke^kL-3π/4); ; Y_1(m_n/k)∼√(2k/π m_n)sin(m_n/k-3π/4), Y_1(m_n/ke^kL)∼√(2k/π m_ne^-kL)sin(m_n/ke^kL-3π/4) ]} Therefore, Eq.(<ref>) takes the formcos(m_n/k-3π/4)sin(m_n/ke^kL-3π/4)-cos(m_n/ke^kL-3π/4)sin(m_n/k-3π/4)=0 sin[m_n/k(e^kL-1)]=0kL_z=e^kL-1sin(m_n L_z)=0m_n=sπ/L_z,s=1,2,… Correspondingly, Eq.(<ref>) is given byψ_n(z)∼ N_n√(2k/π m_n)√(|z|+1/k) {-sin(m_n/k-3π/4)J_2[m_n(|z|+1/k)]..+cos(m_n/k-3π/4)Y_2[m_n(|z|+1/k)]}m_n(|z|+1/k)≫ 1 ψ_n(z)∼ N_n2√(k)/π m_n {-sin(m_n/k-3π/4)cos[m_n(|z|+1/k)-5π/4]..+cos(m_n/k-3π/4)sin[m_n(|z|+1/k)-5π/4]}ψ_n(z)∼ N_n2√(k)/π m_nsin(m_n|z|-π/2) where we have used the following expressions for the Bessel function at the limit m_n(|z|+1/k)≫ 1. (|z|+1/k)^1/2J_2[m_n(|z|+1/k)]∼√(2/π m_n)cos[m_n(|z|+1/k)-5π/4] (|z|+1/k)^1/2Y_2[m_n(|z|+1/k)]∼√(2/π m_n)sin[m_n(|z|+1/k)-5π/4] It is now trivial to evaluate the normalization constant N_n.∫_-L_z^L_zdz ψ_n^2(z)=1 N_n^24k/π^2m_n^2∫_-L_z^L_zdz sin^2(m_n|z|-π/2)=1N_n^24k/π^2m_n^2 2 ∫_0^L_zdz sin^2(m_nz-π/2)_L_z/2 (for L_z≫ 1)=1N_n=π m_n/2√(kL_z)(<ref>)(<ref>)ψ_n(z)∼1/√(L_z)sin(m_n|z|-π/2) For the higher graviton modes to be very heavy, as assumed above, we should have a very small inter-brane distance according to Eq. (<ref>). Then, this case applies to the RS1 model. Each graviton mode will contribute to the gravitational potential on the brane through a Yukawa-like potential, thusU(r) ∝∑_n G_(4) m_1 m_2 e^-m_n r/rBut since all the KK masses are very heavy, it is only the zero graviton that will create a 1/r potential while the contributions of the higher modes will be very much suppressed.The limit m_n/k≪ 1At the approximation of m_n/k≪ 1, it is J_1(m_n/k)∼m_n/2k, Y_1(m_n/k)∼ -2k/π m_n Hence, we can safely state that -Y_1(m_n/k)≫ J_1(m_n/k). Subsequently, Eq.(<ref>) results to the following simple requirement in order to have a vanishing determinant: J_1(m_n/k e^kL)=0m_n=k e^-kL R^J_1_n≃R^J_1_n/L_z where R^J_1_n are the roots of the function J_1(x), namely J_1(R^J_1_n)=0. The masses m_n of the massive graviton modes constitute the graviton spectrum (or Kaluza Klein spectrum). The substitution of the approximate relations from equation (<ref>) into (<ref>) leads toψ_n(z)∼ N_n(|z|+1/k)^1/2{2k/π m_nJ_2[m_n(|z|+1/k)]+m_n/2kY_2[m_n(|z|+1/k)]}N_n N_nm_n/2kψ_n(z)∼ N_n(|z|+1/k)^1/2{4k^2/π m_n^2J_2[m_n(|z|+1/k)]+Y_2[m_n(|z|+1/k)]}The normalization constant N_n can be determined by the integral∫_-L_z^L_zdz ψ_n^2(z)=1The fact that L_z∞ allows us to perform the calculation of N_n in the approximation of m_n|z|≫ 1. Using also the fact that m_n/k≪ 1k/m_n≫ 1, we can ignore the second term of Eq.(<ref>) as negligible. Thus, the substitution of Eq.(<ref>) into Eq.(<ref>) leads to ψ_n(z)∼ N_n4k^2/π m_n^2√(2/π m_n)cos(m_n|z|-5π/4) Thus, from Eq.(<ref>) we have:∫_-L_z^L_zdz ψ_n^2(z)=1 N^2_n 32k^4/π^3m^5_n∫^L_z_-L_zcos^2(m_n|z|-5π/4)=1N^2_n 32k^4/π^3m^5_n 2 ∫^L_z_0cos^2(m_nz-5π/4)_L_z/2(for L_z≫ 1)=1N_n=π^3/2m_n^5/2/4k^2√(2L_z) Consequently, the graviton modes ψ_n(z) for n>0 and m_n|z|≫ 1 are given byψ_n(z)=1/√(L_z) cos(m_n|z|-5π/4)In this case, the KK masses are assumed to be small and this can be realised only for a large inter-brane distance. Therefore, this analysis applies to the RS1 model, where resolving the hierarchy problem is not an objective any more, or to the RS2 model. Especially, in the latter case where L∞, even the higher graviton modes become massless. It is thus even more important to verify if the Newtonian limit of gravity is indeed recovered on the visible brane. Here, the sum over the KK graviton states changes to an integral that must be carefully evaluated - we refer the interested reader to <cit.> for more details on this. At the end, we present the result from <cit.> which is related to the gravitational potential that is generated by a massive object with mass M on the 3-brane at z=0. The energy-momentum tensor for a point-mass at rest on the brane at r⃗=0 is the following:T_μν=M^(3)(r⃗) ^0_μ ^0_νFor r=|r⃗|≫ 1/k we get: h_μν=2G_(5)kM/r[(1+1/3k^2r^2)η_μν+(2+1/k^2r^2)^0_μ^0_ν] Consequently, from equations (<ref>) and (<ref>) we obtainh_00=2G_(4)M/r(1+2/3k^2r^2) h_ij=2G_(4)M/r(1+1/3k^2r^2)η_ijThe gravitational potential V(r) is given by h_00 as follows: V(r)=1/2h_00=G_(4)M/r(1+2/3k^2r^2)V(r)∼G_(4)M/r Obviously, the desirable behaviour of an effective 4-dimensional gravity at the brane is produced by the model as kr≫ 1. On the other hand, if we consider the case of kr≪ 1, then the gravitational potential is proportional to 1/r^2, i.e. V(r)∝ 1/r^2. This simply means that at short distances -compared to the scale k which is also associated with the AdS curvature- gravity becomes 5-dimensional. The possibility to observe deviations from the Newton's law of gravitation depends entirely on the value of k.CHAPTER: LOCALIZATION OF A 5-DIMENSIONAL BRANE-WORLD BLACK HOLE The preceding analysis of RS models made clear that there is indeed a possibility to live at the boundaries of a higher dimensional cosmos (specifically a five-dimensional one) and the effective gravitational interaction between massive objects on our four-dimensional brane can be the same (in some limit) as the gravitational interaction which is provided by the established and well-tested gravitational theory of General Relativity. Hence, it is absolutely natural to wonder about the behaviour of 5-dimensional black holes in the context of RS models.The first attempt to find a black hole solution in the aforementioned scenario was in <cit.>. The line-element which was considered is of the following form: ds^2=e^2A(y)[-(1-2M/r)dt^2+(1-2M/r)^-1dr^2+r^2(dθ^2+sin^2θ dφ^2)]+dy^2 where M is a constant quantity and represents the mass of the black hole. As stated in the previous chapter, the coordinate y is used for the extra dimension and A(y) denotes the warp factor. For A(y)=-k|y| we get the warp factor of the RS model. It is quite obvious that for y=0 the induced 4-dimensional line-element is identified with the well-known Schwarzschild spacetime geometry. However, a 5-dimensional observer will not be able to associate the complete 5-dimensional line-element of Eq.(<ref>) with a regular black hole. The previous statement can be easily verified by the Kretschmann scalar K≡ R_MNKLR^MNKL that emanates fromthe line-element (<ref>). K≡ R_MNKLR^MNKL=8 [2 A”(y)^2+5 A'(y)^4+4 A'(y)^2 A”(y)+6 M^2 e^-4 A(y)/r^6] The last term of Eq.(<ref>) reveals the existence of a singularity at r=0 which extends from -∞ to +∞ along the extra dimension y. Hence, the line-element of Eq.(<ref>) generates a black string rather than a black hole. Of course, for any descending function A(y) the last term of Eq.(<ref>) becomes even more problematic, because when y∞ the quantity e^-4A(y) goes to infinity as well. This particular behaviour is in complete contrast to the purpose of the RS model, which intends to keep gravity localized near the brane at y=0. There is a plethora of attempts in the literature that try to derive a 5-dimensional black hole which is localized close to the brane (we mention some of them: <cit.>. Additionally, there are numerical solutions which describe both mini and large brane-world black holes in the context of RS models and beyond (i.e. 6-dimensional black holes) <cit.>. However, an analytical solution which can be written in closed form has not been found, thus, the investigation of such a solution is still incomplete. Finally, it is important to refer that in <cit.> black hole solutions were found which are localized on a 2-brane embedded in a (3+1)-dimensional bulk. Therefore, the motivation to search for a corresponding solution for the higher dimensional problem (3-brane in a (4+1)-dimensional bulk) is completely justified. This Chapter and the rest of the thesis is entirely focused on the examination of the existence of a localized black hole solution in the context of a brane-world model which is similar to the RS2 model. The word “similar" was used because the warp factor A(y) is allowed to differ from the expression -k|y| (but it is necessary to be a descending function, as it is in the RS models) in order to increase the chances of finding a localized 5-dimensional black hole solution. In the following section we present the geometrical background that is going to be used in the framework of this thesis.§ THE GEOMETRY OF THE 5-DIMENSIONAL SPACETIMEWe assume that the 4-dimensional part of the general 5-dimensional geometrical background is a Vaidya metric[See [vaid]Appendix D for more information about Vaidya metric.]. We also consider that mass is a function of three variables m=m(v,r,y) (motivated by <cit.>). The non-trivial dependence of the mass from the extra dimension y is needed in order to counter the anomalous effect of the quantity e^-4A(y) in the last term of equation (<ref>). An appropriate mass function that decreases faster than e^-4A(y) increases and has also the necessary r-dependence in order to describe a black hole, would solve the localization problem. The drawback of this non-trivial assumption of the mass parameter is that it demands the existence of a bulk matter distribution in order to get consistent field equations. The bulk matter distribution that it is going to be considered in the context of this thesis will be presented later in this Chapter. Thus, the line element is of the form ds^2=e^2A(y)[-(1-2m(v,r,y)/r)dv^2+2dvdr+r^2(dθ^2+sin^2θd^2)]+dy^2 It is easily deducible that the covariant components of the metric tensor in matrix form are: (g_MN)=( [ -e^2 A(y)(1-2 m(u,r,y)/r)e^2 A(y) 0 0 0;e^2 A(y) 0 0 0 0; 0 0e^2 A(y) r^2 0 0; 0 0 0 e^2 A(y) r^2 sin ^2(θ ) 0; 0 0 0 0 1; ]) Thus, the contravariant components of the metric tensor are of the following form: (g^MN)=( [ 0 e^-2 A(y) 0 0 0; e^-2 A(y) e^-2 A(y)(1-2 m(u,r,y)/r) 0 0 0; 0 0 e^-2 A(y)/r^2 0 0; 0 0 0 e^-2 A(y)/r^2sin^2θ 0; 0 0 0 0 1; ])For the metric ansatz (<ref>), the quantities that are invariant and also contain the complete information of the 5-dimensional curvature, are R=g_MNR^MN, R_MNR^MN and R_ABCDR^ABCD. As it is mentioned earlier, an acceptable solution for the mass function m=m(v,r,y) should have an appropriate dependence on the y-coordinate in order to expunge any singularities in the bulk. Hence, the desirable solution should yield to a regular Anti-de Sitter space-time at a finite distance from the brane and a 5-dimensional black hole localized on the brane, otherwise the solution is rejected. The Christoffel symbols can be evaluated with the use of equations (<ref>), (<ref>) and the following relation: ^K_MN=1/2g^KL(g_ML,N+g_NL,M-g_MN,L) In the table that follows are depicted the non-zero Christoffel symbols. [box=]equation [^0_00=m-r _r m/r^2^0_04=^0_40=A'^0_22=-r;; ^0_33=-r sin^2θ ^1_00=r^2 _v m-(r-2 m) (r _r m-m)/r^3 ^1_01=^1_10=r _rm-m/r^2;; ^1_04=^1_40=_ym/r^1_14=^1_41=A' ^1_22=2 m-r;; ^1_33=(2 m-r)sin^2θ ^2_12=^2_21=1/r^2_24=^2_42=A';; ^2_33=-sinθcosθ ^3_13=^3_31=1/r ^3_23=^3_32=θ;;^3_34=^3_43=A'^4_00=e^2 A (A' (r-2 m)-_ym)/r ^4_01=^4_10=-e^2 A A';; ^4_22=-r^2 e^2 A A' ^4_33=-r^2A' e^2 A sin^2θ ]Subsequently, we present the 5-dimensional curvature invariant quantities. [box=]equation R=R_MNg^MN=-20 A'^2-8 A”+2 e^-2 A/r (_r^2m+2_rm/r)[box=]equation [ R_MNR^MN=80A'^2+64A'^2 A”+20A”^2-4 e^-2 A/r(_r^2m+2_rm/r)(4A'^2+A”);+2e^-4A/r^2[(_r^2m)^2+4 (_rm)^2/r^2] ][box=]equation [ R^ABCDR_ABCD=40A'^4+32A'^2A”+16A”^2+48e^-4Am^2/r^6-8e^-2AA'^2/r(_r^2m+2_rm/r);; +4e^-4A/r^2[(_r^2m)^2+4m/r^2(_r^2m-4_rm/r)-4_rm_r^2m/r+8(_r m)^2/r] ] The non-zero covariant components of the Einstein tensor G_MN are:[box=]equation [ G_00=-e^2 A [3 (r-2 m) (2 A'^2+A”)+4 A' _ym+_y^2m] r^2+2 r_vm +2(r-2 m)_r m/r^3;; G_01=G_10=3 e^2 A (2 A'^2+A”)-2 _rm/r^2;; G_04=G_40=_ym+r_r_y m/r^2;;G_22=r [3 e^2 A r (2 A'^2+A”)-_r^2m];;G_33=r sin^2θ[3 e^2 A r (2 A'^2+A”)-_r^2m];;G_44=6 A'^2-e^-2 A (2 _rm+r_r^2 m)/r^2 ]while the mixed components of the Einstein tensor G^M_N=g^MKG_KN are the following: [box=]equation [ G^0_0=6 A'^2+3 A”-2 e^-2 A _rm/r^2; ; G^1_0=-4 r A' _ym+r _y^2m-2 e^-2 A _vm/r^2; ; G^1_1=6 A'^2+3 A”-2 e^-2 A_r m/r^2; ; G^1_4=e^-2 A (_ym+r _r_ym)/r^2; ; G^2_2=6 A'^2+3 A”-e^-2 A_r^2 m/r; ; G^3_3=6 A'^2+3 A”-e^-2 A _r^2m/r; ;G^4_0=_ym+r _r_ym/r^2; ;G^4_4=6 A'^2-e^-2 A (2 _rm+r_r^2 m)/r^2 ]A detailed presentation of all the geometrical quantities i.e. Christoffel symbols, Riemann tensor, Ricci tensor and Einstein tensor, takes place in [geom]Appendix E. Moreover, in the same Chapter of Appendices, one can find the proper mathematica commands in order to verify the validity of all the aforementioned quantities. In the following section, we introduce the field theory model in the context of which the existence of a viable solution to the localization problem of a 5-dimensional black hole will be examined.§ TWO NON-MINIMALLY COUPLED AND INTERACTING SCALAR FIELDS We consider the most general case of two interacting scalar fields ϕ,χ which are not minimally coupled to gravity. These scalar fields can freely propagate into the bulk. Subsequently, the action of this model is the following: S=∫ d^4xdy√(-)[f(ϕ,χ)/2R-1/2(∇ϕ)^2-1/2(∇χ)^2-V(ϕ,χ)-Λ_B] where =[(g_MN)]. The scalar fields ϕ, χ are both functions of the variables (v,r,y) and Λ_B denotes the cosmological constant of the bulk, it is the same constant as Λ_5 which was used in Chapter 2. The field equations of this model can be derived by varying the action with respect to the metric tensor and also with respect to the fields ϕ,χ. The steps that are going to be followed for the variation of the above action are the same as the steps which are followed in [var]Appendix F, where it is thoroughly presented the variation of a single scalar field which is non-minimally coupled to gravity. Thus, we have:S=0 =∫ d^4xdy (√(-))[f(ϕ,χ)/2R-1/2(∇ϕ)^2-1/2(∇χ)^2-V(ϕ,χ)-Λ_B]+∫ d^4xdy√(-){f(ϕ,χ)/2 R-1/2[(∇ϕ)^2]-1/2[(∇χ)^2]}Moreover, it is [(∇ϕ)^2]=(∇^Mϕ∇_Mϕ)=(g^MN∇_Mϕ∇_Nϕ)=∇_Mϕ∇_Nϕg^MN Using equations (<ref>), (<ref>), (<ref>) and (<ref>) into (<ref>) we get:0 =∫ d^4xdy√(-)g^MN{-1/2g_MN[f(ϕ,χ)/2R-1/2(∇ϕ)^2-1/2(∇χ)^2-V(ϕ,χ)-Λ_B]..+1/2f(ϕ,χ)R_MN-1/2∇_M∇_Nf(ϕ,χ)+1/2g_MN□ f(ϕ,χ)-1/2∇_Mϕ∇_Nϕ-1/2∇_Mχ∇_Nχ} 0 =f(ϕ,χ)/(R_MN-1/2g_MNR)+g_MN[(∇ϕ)^2/2+(∇χ)^2/2+V(ϕ,χ)]+g_MNΛ_B-∇_Mϕ∇_Nϕ-∇_Mχ∇_Nχ-∇_M∇_Nf(ϕ,χ)+g_MN□ f(ϕ,χ) f(ϕ,χ)(R_MN-1/2g_MNR)=f(ϕ,χ) G_MN=(T_MN-g_MNΛ_B) where T_MN=∇_Mϕ∇_Nϕ+∇_Mχ∇_Nχ-g_MN[(∇ϕ)^2/2+(∇χ)^2/2+V]+∇_M∇_Nf-g_MN□ f The above field equations are more convenient for calculations when they are expressed in terms of mixed tensor components. Hence, we have [box=]equation f(ϕ,χ) G^M_N=T^M_N-^M_NΛ_Bwhere we have absorbed the constantinside the function f(ϕ,χ) and [box=]equation [ T^M_N=∇^Mϕ∇_Nϕ+∇^Mχ∇_Nχ-^M_N[(∇ϕ)^2/2+(∇χ)^2/2+V]+∇^M∇_Nf-^M_N□f; ;=^Mϕ_Nϕ+^Mχ_Nχ-^M_N[(ϕ)^2/2+(χ)^2/2+V]+∇^M∇_Nf-^M_N□f ]The variation of the action (<ref>) with respect to the fields ϕ, χ (as it is done in the second section of [var]Appendix F) provides us with two additional equations that should be satisfied in order to have an acceptable solution for the mass function and the scalar fields. In complete analogy to the variation method of section F.2, it is straightforward to derive the following expressions: [box=]equation √(-)(1/2f/ϕR-V/ϕ)=-_M(√(-) g^MN_Nϕ)[box=]equation √(-)(1/2f/χR-V/χ)=-_M(√(-) g^MN_Nχ) In order to derive the independent field equations that are resulting from relations (<ref>) and (<ref>) we firstly need to evaluate the mixed components of the energy-momentum tensor T^M_N. In [enmom]Appendix G there is a comprehensive evaluation of the components of the energy-momentum tensor given by Eq.(<ref>), thus, it is redundant to repeat any of these calculations. Subsequently, the non-zero components of the energy-momentum tensor are depicted in their compact form. [box=]align T^0_1 =e^-2A[(_1ϕ)^2+(_1χ)^2+_1^2f]T^1_0 =e^-2A[(_0ϕ)^2+(_0χ)^2+_0^2f+(1-2m/r)(_1_0ϕ+_1χ_0χ+_1_0f). . +_1(m/r)_0f-_0(m/r)_1f+e^2A_4(m/r)_4f]T^4_0 =_4ϕ_0ϕ+_4χ_0χ+_4_0f-A'_0f-_4m/r_1fT^0_4 =e^-2A(_1ϕ_4ϕ+_1χ_4χ+_1_4f-A'_1f)T^4_1 =e^2AT^0_4T^1_4 =e^-2AT^4_0+(1-2m/r)T^0_4T^0_0 =e^-2A[_1ϕ_0ϕ+_1χ_0χ+_1_0f-_1(m/r)_1f]+A'_4f+-□fT^1_1 =T^0_0+(1-2m/r)T^0_1T^2_2 =e^-2A/r[_0f+(1-2m/r)_1f]+A'_4f+-□fT^3_3 =T^2_2T^4_4 =(_4ϕ)^2+(_4χ)^2+^2_4f+-□f =-e^-2A/2{2(_1ϕ_0ϕ+_1χ_0χ)+(1-2m/r)[(_1ϕ)^2+(_1χ)^2]} -1/2[(_4ϕ)^2+(_4χ)^2]-V(ϕ,χ)□f =e^-2A_0_1f+e^-2A/r^2_1[r^2_0f+r^2(1-2m/r)_1f]+e^-4A_4(e^4A_4f)§ THE FIELD EQUATIONS OF THE THEORYHaving in our disposal both the components of the Einstein tensor from equation (<ref>) and the components of the energy-momentum tensor given by equation (<ref>), the independent field equations can immediately ensue with the use of equation (<ref>). It is clear that the independent field equations are the following. Equation (^0_1):(<ref>)[(<ref>)](<ref>)(_1ϕ)^2+(_1χ)^2+_1^2f=0 Substituting _1^2f from equation (<ref>) into (<ref>) we get(1+_ϕ^2f)(_1ϕ)^2+(1+_χ^2f)(_1χ)^2+2_χ_ϕ f_1χ_1ϕ+_ϕ f_1^2ϕ+_χ f_1^2χ=0Equation (^1_0): (<ref>)[(<ref>)](<ref>)(_0ϕ)^2+(_0χ)^2+_0^2f+(1-2m/r)(_1ϕ_0ϕ+_1χ_0χ+_1_0f)+_1(m/r)_0f-_0(m/r)_1f+e^2A_4(m/r)_4f=f[2/r^2_0m-e^2A/r(_4^2m+4A'_4m)]The substitution of quantities _0^2f and _1_0f from equations (<ref>) and (<ref>) respectively into equation (<ref>) yields toe^-2A{(1+_ϕ^2f)(_0ϕ)^2+(1+_χ^2f)(_0χ)^2+2_χ_ϕ f_0ϕ_0χ+_ϕ f_0^2ϕ+_χ f_0^2χ. +(1-2m/r)[(1+_ϕ^2f)_1ϕ_0ϕ+(1+_χ^2f)_1χ_0χ+_ϕ_χ f(_1ϕ_0χ+_1χ_0ϕ)+_ϕ f_1_0ϕ+_χ f_1_0χ] .+(_1m/r-m/r^2)(_ϕ f_0ϕ+_χ f_0χ)-_0m/r(_ϕ f_1ϕ+_χ f_1χ)+e^2A_4m/r(_ϕ f_4ϕ+_χ f_4χ)}==f[2/r^2_0m-e^2A/r(_4^2m+4A'_4m)]Equation (^4_0):(<ref>)[(<ref>)](<ref>)_4ϕ_0ϕ+_4χ_0χ+_4_0f-A'_0f-_4m/r_1f=f/r(_4m/r+_1_4m) The last equation in its extended form is:(1+_ϕ^2f)_4ϕ_0ϕ+(1+_χ^2f)_4χ_0χ+_χ_ϕ f(_4χ_0ϕ+_4ϕ_0χ)+_ϕ f_4_0ϕ+_χ f_4_0χ-A'_0f-_4m/r(_ϕ f_1ϕ+_χ f_1χ)=f/r(_4m/r+_1_4m)Equation (^0_4):(<ref>)[(<ref>)](<ref>)_1ϕ_4ϕ+_1χ_4χ+_1_4f-A'_1f=0 Using equation (<ref>) into equation (<ref>) we obtain(1+_ϕ^2f)_1ϕ_4ϕ+(1+_χ^2f)_1χ_4χ+_ϕ_χ f(_1χ_4ϕ+_1ϕ_4χ)+_ϕ f_1_4ϕ +_χ f_1_4χ-A'(_ϕ f_1ϕ+_χ f_1χ)=0 Equation (^0_0): (<ref>)[(<ref>)](<ref>)e^-2A[_1ϕ_0ϕ+_1χ_0χ+_1_0f-_1(m/r)_1f]+A'_4f+-□ f-Λ_B==f(6A'^2+3A”-2e^-2A/r^2_1m)Equation (<ref>) is equivalent toe^-2A[(1+_ϕ^2f)_1ϕ_0ϕ+(1+_χ^2f)_1χ_0χ+_χ_ϕ f(_1χ_0ϕ+_1ϕ_0χ)+_ϕ f_1_0ϕ+_χ f_1_0χ. .+_ϕ f_1ϕ+_χ f_1χ/r(m/r-_1m)]+A'(_ϕ f_4ϕ+_χ f_4χ)+-□ f-Λ_B==f(6A'^2+3A”-2e^-2A/r^2_1m) Equation (^2_2): (<ref>)[(<ref>)](<ref>)e^-2A/r[_0f+(1-2m/r)_1f]+A'_4f+-□ f-Λ_B==f(6A'^2+3A”-e^-2A/r_1^2m)Expanding the partial derivatives of the function f=f(ϕ,χ) we gete^-2A/r[(_ϕ f_0ϕ+_χ f_0χ)+(1-2m/r)(_ϕ f_1ϕ+_χ f_1χ)]+A'(_ϕ f_4ϕ+_χ f_4χ)+-□ f-Λ_B=f(6A'^2+3A”-e^-2A/r_1^2m) Equation (^4_4):(<ref>)[(<ref>)](<ref>)(_4ϕ)^2+(_4χ)^2+^2_4f+-□ f-Λ_B=f(6A'^2-e^-2A/r_1^2m-2e^-2A/r^2_1m)The combination of equations (<ref>) and (<ref>) gives(1+_ϕ^2f)(_4ϕ)^2+(1+_χ^2f)(_4χ)^2+2_χ_ϕ f_4χ_4ϕ+_ϕ f_4^2ϕ+_χ f_4^2χ+-□ f-Λ_B==f(6A'^2-e^-2A/r_1^2m-2e^-2A/r^2_1m)Instead of using equations (<ref>), (<ref>), (<ref>) and the corresponding extended equations (<ref>), (<ref>) and (<ref>), which contain the termsand □ f that add extra complexity, we can be exempted from these terms by combining the aforementioned equations with each other.Equation (^0_0)-Equation (^2_2):Subtracting equation (<ref>) from (<ref>) we get:r(_1ϕ_0ϕ+_1χ_0χ+_1_0f)-_0f-_1f(_1m+1-3m/r)=f(_1^2m-2/r_1m) Correspondingly, from equations (<ref>) and (<ref>) one obtainsr[(1+_ϕ^2f)_1ϕ_0ϕ+(1+_χ^2f)_1χ_0χ+_ϕ_χ f(_1χ_0ϕ+_1ϕ_0χ)+_ϕ f_1_0ϕ+_χ f_1_0χ] -(_ϕ f_0ϕ+_χ f_0χ)-(_1m+1-3m/r)(_ϕ f_1ϕ+_χ f_1χ)=f(_1^2m-2/r_1m) Equation (^0_0)-Equation (^4_4):In the same way, subtracting equation (<ref>) from (<ref>) and (<ref>) from (<ref>) we respectively have:e^-2A[_1ϕ_0ϕ+_1χ_0χ+_1_0f-_1(m/r)_1f]+A'_4f-(_4ϕ)^2-(_4χ)^2-_4^2f==f(3A”+e^-2A/r_1^2m) e^-2A[(1+_ϕ^2f)_1ϕ_0ϕ+(1+_χ^2f)_1χ_0χ+_ϕ_χ f(_1χ_0ϕ+_1ϕ_0χ)+_ϕ f_1_0ϕ+_χ f_1_0χ-_1(m/r)(_ϕ f_1ϕ+_χ f_1χ)]+A'(_ϕ f_4ϕ+_χ f_4χ)-(1+_ϕ^2f)(_4ϕ)^2-(1+_χ^2f)(_4χ)^2-2_ϕ_χ f_4ϕ_4χ-_ϕ f_4^2ϕ-_χ f_4^2χ=f(3A”+e^-2A/r_1^2m) Equation (^2_2)-Equation (^4_4):Finally, the last field equation that has no dependence on the quantitiesand □ f can be provided by equations (<ref>) and (<ref>) or equations (<ref>) and (<ref>). We respectively obtain e^-2A/r[_0f+(1-2m/r)_1f]+A'_4f-(_4ϕ)^2-(_4χ)^2-_4^2f=f(3A”+2e^-2A/r^2_1m) e^-2A/r[_ϕ f_0ϕ+_χ f_0χ+(1-2m/r)(_ϕ f_1ϕ+_χ f_1χ)]+A'(_ϕ f_4ϕ+_χ f_4χ)-(1+_ϕ^2f)(_4ϕ)^2-(1+_χ^2f)(_4χ)^2-2_ϕ_χ f_4ϕ_4χ-_ϕ f_4^2ϕ-_χ f_4^2χ=f(3A”+e^-2A/r_1^2m) Of course, equation (<ref>) is not independent from equations (<ref>) and (<ref>), the subtraction of Eq.(<ref>) from (<ref>) gives Eq.(<ref>). Correspondingly, the subtraction of Eq.(<ref>) from Eq.(<ref>) gives Eq.(<ref>). The reason that equations (<ref>) and (<ref>) were presented is that in some of the cases -which are going to be investigated in the next Chapter- these equations might be easier to be solved or they might provide more directly useful information about the mass function m(v,r,y). In the context of the scalar field theory model, which was introduced at the beginning of this section, a solution to the localization problem -except from an appropriate expression for the mass function- requires also appropriate expressions for the scalar fields. A solution of the field equations that achieves to produce a mass function m(v,r,y) which describes a 5-dimensional black hole and it is also localized close to the 3-brane at y=0, with the cost of producing functions for the scalar fields ϕ(v,r,y), χ(v,r,y) that have an infinite value at y∞, cannot be accepted. Subsequently, we sum up the field equations. The field equations in their compact form are given by equations (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>) (the last equation is not independent), while the field equations in their extended form are given by equations (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>) (of course, the last equation is also not independent). [box=]gather Compact Form(_rϕ)^2+(_rχ)^2+_r^2f=0(_vϕ)^2+(_vχ)^2+_v^2f+(1-2m/r)(_rϕ_vϕ+_rχ_vχ+_r_vf)+_r(m/r)_vf -_v(m/r)_rf+e^2A_y(m/r)_yf=f[2/r^2_vm-e^2A/r(_y^2m+4A'_ym)] _yϕ_vϕ+_yχ_vχ+_y_vf-A'_vf-_ym/r_rf=f/r(_ym/r+_r_ym)_rϕ_yϕ+_rχ_yχ+_r_yf-A'_rf=0(_yϕ)^2+(_yχ)^2+^2_yf+-□f-Λ_B=f(6A'^2-e^-2A/r_r^2m-2e^-2A/r^2_rm)r(_rϕ_vϕ+_rχ_vχ+_r_vf)-_vf-_rf(_rm+1-3m/r)=f(_r^2m-2/r_rm) e^-2A[_rϕ_vϕ+_rχ_vχ+_r_vf-_r(m/r)_rf]+A'_yf-(_yϕ)^2-(_yχ)^2-_y^2f=f(3A”+e^-2A/r_r^2m) e^-2A/r[_vf+(1-2m/r)_rf]+A'_yf-(_yϕ)^2-(_yχ)^2 -_y^2f=f(3A”+2e^-2A/r^2_rm)Last but not least, there are also the following two equations for the scalar fields that are necessary to be satisfied: [box=]equation √(-)(1/2f/ϕR-V/ϕ)=-_M(√(-) g^MN_Nϕ)[box=]equation √(-)(1/2f/χR-V/χ)=-_M(√(-) g^MN_Nχ) The last two equations are going to bother us only in the case that we find appropriate functions for the quantities A(y), m(v,r,y), ϕ(v,r,y) and χ(v,r,y) which satisfy equations (<ref>)-(<ref>). In this hypothetical scenario it would be necessary to verify if these two equations are satisfied as well.CHAPTER: SOLVING THE FIELD EQUATIONS In this Chapter, we will try to solve the field equations, meaning that we will try to determine from the field equations the mass function m=m(v,r,y) and then the functions of the scalar fields ϕ(v,r,y), χ(v,r,y). As we already mentioned in the previous Chapter, the mass function should have a suitable dependence on the extra dimension y in order to be able to constitute a 5-dimensional black hole that is localized on our 4-dimensional 3-brane (namely our universe). In order to achieve that, we are going to use equations (<ref>)-(<ref>). The aforementioned field equations resulted from the assumption that the scalar fields ϕ, χ and consequently the coupling function f=f(ϕ,χ) depend on the coordinates (v,r,y). However, it is possible to consider simpler cases where one or both of the scalar fields depend on just one or two of the (v,r,y) coordinates. It is also possible to consider cases in which either _ϕ f or _χ f equals to zero but not both of them simultaneously, because it is important to preserve the non-minimal coupling. As will be clear from the next pages of this Chapter, it is extremely difficult to find a suitable solution of the field equations within the framework of our field theory which could yield to a mass function m=m(v,r,y) that has the desirable dependence on the extra dimension y. On the contrary, in most of these cases the field equations are not consistent with our assumptions.§ ALL POSSIBLE CASESWe now present all the possible cases that were mentioned previously, starting from the simplest cases and ending to the most complicated ones. *Both scalar fields depend on one coordinate[ 1){ϕ=ϕ(v), χ=χ(v)}; ; 2){ϕ=ϕ(v), χ=χ(r)}; ; 3){ϕ=ϕ(v), χ=χ(y)}; ; 4){ϕ=ϕ(r), χ=χ(r)}; ; 5){ϕ=ϕ(r), χ=χ(y)}; ; 6){ϕ=ϕ(y), χ=χ(y)}; ;]The field equations are manifestly symmetrical under the exchange of ϕ and χ. Therefore, the case {ϕ=ϕ(v), χ=χ(r)} is the same as {ϕ=ϕ(r), χ=χ(v)}. This property reduces significantly the number of independent cases. *One scalar field depends on two coordinates and the other one depends on one [7){ϕ=ϕ(v,r), χ=χ(v)};;8){ϕ=ϕ(v,r), χ=χ(r)};;9){ϕ=ϕ(v,r), χ=χ(y)};; 10){ϕ=ϕ(v,y), χ=χ(v)};; 11){ϕ=ϕ(v,y), χ=χ(r)};; 12){ϕ=ϕ(v,y), χ=χ(y)};; 13){ϕ=ϕ(r,y), χ=χ(v)};; 14){ϕ=ϕ(r,y), χ=χ(r)};; 15){ϕ=ϕ(r,y), χ=χ(y)};; ]*Both scalar fields depend on two coordinates [ 16){ϕ=ϕ(v,r), χ=χ(v,r)};; 17){ϕ=ϕ(v,r), χ=χ(v,y)};; 18){ϕ=ϕ(v,r), χ=χ(r,y)};; 19){ϕ=ϕ(v,y), χ=χ(v,y)};; 20){ϕ=ϕ(v,y), χ=χ(r,y)};; 21){ϕ=ϕ(r,y), χ=χ(r,y)};; ] *One scalar field depends on all three coordinates and the other one depends on one [ 22){ϕ=ϕ(v,r,y), χ=χ(v)};; 23){ϕ=ϕ(v,r,y), χ=χ(r)};; 24){ϕ=ϕ(v,r,y), χ=χ(y)};; ] *One scalar field depends on all three coordinates and the other one depends on two [ 25){ϕ=ϕ(v,r,y), χ=χ(v,r)};; 26){ϕ=ϕ(v,r,y), χ=χ(v,y)};; 27){ϕ=ϕ(v,r,y), χ=χ(r,y)};; ]*Both scalar fields depend on all three coordinates 28){ϕ=ϕ(v,r,y), χ=χ(v,r,y)} We henceforth start our quest for a valid solution of the field equations examining one by one the cases that were presented previously. The examination of the aforementioned cases will be split in two large categories. The first category includes the cases from 1 to 21 which can be studied (and excluded as it is shownbelow) without further assumptions. The cases from 22 to 28 belong to the second category in which it is necessary to introduce an expression for the coupling function f(ϕ,χ) in order to be able to proceed to the solution of the field equations. In these last cases, if we do not fix the coupling function f(ϕ,χ) the field equations are unapproachable.§ EXPLICITLY REJECTED CASES1){ϕ=ϕ(v), χ=χ(v), f=f(ϕ,χ)=f(v)}(<ref>) 0=f(3A”+e^-2A/r_r^2m)_r^2m=-3A”e^2A r_rm=-3/2A”e^2Ar^2+m_0(v,y)(<ref>)-_vf=f(_r^2m-2/r_rm)f≠0 -_vf/f=_r^2m-2/r_rm(<ref>) -_vf/f=-3A”e^2Ar-2/r(-3/2A”e^2Ar^2+m_0(v,y)) -_vf/f=-3A”e^2Ar+3A”e^2Ar-2/rm_0(v,y)_vf/f=2/rm_0(v,y)Obviously, the last equation is totally inconsistent, the left hand side (LHS) depends only on the v-coordinate while the right hand side (RHS) depends on the coordinates (v,r,y). Even if we demand _ym_0=0 in order to eliminate the y-dependence, we definitely cannot cancel the factor 1/r in the RHS. Therefore, this case is rejected.2){ϕ=ϕ(v), χ=χ(r), f=f(ϕ,χ)=f(v,r)} (<ref>) -A'_rf=0 A'_rf=0{[ A'=_yA=0; or; _rf=_χ f_rχ=0_rχ≠0_χ f=0 ]}From equation (<ref>) it is easily deducible that this case is rejected as well. The possibility A'(y)=0 is immediately rejected, while the constraint _χ f=0 leads to _rχ=0 if we use equation (<ref>). Hence, none of the possibilities (A'=0 or _χ f=0) can be valid because they contradict with our assumptions about the functions A(y) and χ(r).3){ϕ=ϕ(v), χ=χ(y), f=f(ϕ,χ)=f(v,y)} (<ref>) A'_yf-(_yχ)^2-_y^2f=f(3A”+e^-2A/r_r^2m) The LHS of equation (<ref>) has (v,y)-dependence, while the RHS of the same equation is depended on (v,r,y), hence, we are led to the following constraint:_r[f(3A”+e^-2A/r_r^2m)]=0_rf_0(3A”+e^-2A/r_r^2m)+fe^-2A_r(_r^2m/r)=0f≠0_r(_r^2m/r)=0_r^3m/r-_r^2m/r^2=0_r^3m-_r^2m/r=0 The differential equation (<ref>) can be solved easily assuming that the function of mass m=m(v,r,y) can be written as a power series expansion with respect to r-coordinate, containing either positive or negative powers of r. The same assumption has been used as well in <cit.>. Thus, it is m=m(v,r,y)=∑_n a_n(v,y)r^n The combination of equations (<ref>) and (<ref>) lead to∑_n n(n-1)(n-2)a_n(v,y)r^n-3-∑_n n(n-1)a_n(v,y)r^n-3=0 ∑_nn(n-1)(n-3)a_n(v,y)r^n-3=0{[a_n=0∀ n≠{0,1,3};; a_0,a_1,a_3arbitraryfunctions ]} m(v,r,y)=a_0(v,y)+a_1(v,y)r+a_3(v,y)r^3 (<ref>)-_vf=f(_r^2m-2/r_rm)f≠0-_vf/f=_r^2m-2/r_rm(<ref>) -_vf/f=6a_3(v,y)r-2/r[a_1(v,y)+3a_3(v,y)r^2]=6a_3(v,y)r-6a_3(v,y)r-2/ra_1(v,y)_vf/f=2/ra_1(v,y)Similarly to the case 1, the LHS of equation (<ref>) depends on (v,y)-coordinates while the RHS has the factor 1/r which cannot be cancelled, therefore this case is inconsistent as well.4){ϕ=ϕ(r), χ=χ(r), f=f(ϕ,χ)=f(r)}(<ref>)-A'_rf=0 A'_rf=0{[ A'=_yA=0; or;_rf=0 ]}In this case, we are also led to a contradiction to our primary assumption. Either A'(y)=0 or _r f=0 cannot be true. Especially, the constraint _r f=0 implies that the coupling function f(ϕ,χ) is completely independent from both scalar fields ϕ and χ, thus, the non-minimal coupling of the scalar fields to gravity is entirely vanished. Therefore, this case is also excluded from the list of possible solutions.5){ϕ=ϕ(r), χ=χ(y), f=f(ϕ,χ)=f(r,y)}(<ref>) (_rϕ)^2+_r^2f=0_r^2f=-(_rϕ)^2(_rf)/ r=-(_rϕ)^2 ∫ dr(_rf)/ r=-∫ (_rϕ)^2dr_rf-f_0(y)=-∫ (_rϕ)^2dr_rf=-∫ (_rϕ)^2dr+f_0(y) (<ref>)_y(_rf)-A'_rf=0(<ref>)_y[-∫ (_rϕ)^2dr+f_0(y)]-A'[-∫ (_rϕ)^2dr+f_0(y)]=0 -_y[∫(_rϕ)^2dr]_0+_yf_0+A'∫(_rϕ)^2dr-A'f_0=0A'f_0-_yf_0=A'∫(_rϕ)^2dr The LHS of equation (<ref>) depends only on y-coordinate, while the RHS depends on (r,y)-coordinates. Hence, the constraint that is derived is the following: _r[A'∫ (_rϕ)^2dr]=0 A'(_rϕ)^2=0{[ A'=_yA=0; or;_rϕ=0 ]} According to our assumptions, both functions A' and _rϕ cannot be zero, so none of the constraints which are demanded by equation (<ref>) is able to be fulfilled. Neither _ϕ f=0 nor _χ f=0 are able to be assumed in the context of this case. The constraint _ϕ f=0 leads to _rϕ=0 through equation (<ref>) and _χ f=0 leads to the same analysis and the same negative result as the original case 5 with the only difference that in the case of _χ f=0 f_0 is a constant and not a function of y.6){ϕ=ϕ(y), χ=χ(y), f=f(ϕ,χ)=f(y)}(<ref>) 0=f(_r^2m-2/r_rm)f≠0_r^2m-2/r_rm=0(_rm)/ r=2/r_rm1/_rm(_rm)/ r=2/r[ln(_rm)]/ r=2/r∫ dr[ln(_rm)]/ r=∫2/rdrln(_rm)-m_0(v,y)=2ln r_rm=e^m_0(v,y)+ln r^2e^m_0(v,y) m_0(v,y)_rm=m_0(v,y)r^2∫ dr m/ r=m_0(v,y)∫ r^2drm(v,r,y)=m_0(v,y)r^3/3+m_1(v,y) (<ref>) 0=f/r(_ym/r+_r_ym)f≠0_ym/r+_r_ym=0(<ref>)_ym_0r^2/3+_ym_1+_ym_0r^2=0 2r^2_ym_0+_ym_1=0{[ _ym_0=0 m_0=m_0(v);and; _ym_1=0 m_1=m_1(v) ]} (<ref>)(<ref>)m=m(v,r)=m_0(v)r^3/3+m_1(v)It is clear from equation (<ref>) that in order to satisfy simultaneously both field equations (<ref>) and (<ref>), we are led to a mass function which does not depend on the y-coordinate. A mass function which is y-independent is impossible to describe a localized black hole. Therefore, this case is rejected without second thought. 7){ϕ=ϕ(v,r), χ=χ(v), f=f(ϕ,χ)=f(v,r)}(<ref>)-A'_rf=0{[ A'=_yA=0; or; _rf=0_ϕ f_rϕ=0_ϕ f=0 ]}Of course, A'(y)≠ 0. Thus, if _ϕ f=0 then equation (<ref>) leads to _r ϕ=0. Hence, we reject this case as well.8){ϕ=ϕ(v,r), χ=χ(r), f=f(ϕ,χ)=f(v,r)}(<ref>)-A'_rf=0{[ A'=_yA=0; or; _rf=0_ϕ f=_χ f=0 ]}Clearly, this case is also excluded.9){ϕ=ϕ(v,r), χ=χ(y), f=f(ϕ,χ)=f(v,r,y)}(<ref>) (_rϕ)^2+_r^2f=0_r^2f=-(_rϕ)^2_rf=-∫(_rϕ)^2dr+f_0(v,y) (<ref>)_y_rf-A'_rf=0(<ref>)_y[-∫(_rϕ)^2dr+f_0]-A'[-∫(_rϕ)^2dr+f_0]=0 -_y∫(_rϕ)^2dr_0+_yf_0+A'∫(_rϕ)^2dr-A'f_0=0A'f_0-_yf_0=A'∫(_rϕ)^2dr The LHS of equation (<ref>) depends on (v,y)-coordinates while the RHS depends on (v,r,y)-coordinates. Hence, equation (<ref>) leads to the following constraint. _r[A'∫(_rϕ)^2dr]=0 A'(_rϕ)^2=0{[ A'=_yA=0; or;_rϕ=0 ]} Equation (<ref>) is not possible to be satisfied because it contradicts with the original assumptions. We cannot assume _ϕ f=0 because equation (<ref>) results to _rϕ=0 as well. Moreover, _χ f=0 is not helpful either, because the only difference with the previous analysis is that f_0(v,y) f_0(v).10){ϕ=ϕ(v,y), χ=χ(v), f=f(ϕ,χ)=f(v,y)}(<ref>)A'_yf-(_yϕ)^2-_y^2f_(v,y)-dependent=f(3A”+e^-2A/r_r^2m)_(v,r,y)-dependent_r(RHS)=0 _r[f(3A”+e^-2A/r_r^2m)]=0 fe^-2A_r(_r^2m/r)=0f≠0_r(_r^2m/r)=0_r^3m-_r^2m/r=0 Equation (<ref>) is identical to equation (<ref>). Hence, combining (<ref>) with (<ref>) as we did in case 3, we obtain the mass function which is given by equation (<ref>). Subsequently, substituting equation (<ref>) into (<ref>) we are led to the same result as in case 3, namely equation (<ref>) which is inconsistent in this case as well.11){ϕ=ϕ(v,y), χ=χ(r), f=f(ϕ,χ)=f(v,r,y)}(<ref>) (_rχ)^2+_r^2f=0_r^2f=-(_rχ)^2_rf=-∫ (_rχ)^2dr+f_0(v,y) (<ref>)_r_yf-A'_rf=0(<ref>)_y[-∫ (_rχ)^2dr+f_0(v,y)]-A'[-∫ (_rχ)^2dr+f_0(v,y)]=0 _yf_0+A'∫ (_rχ)^2dr-A'f_0=0A'f_0-_yf_0_(v,y)-dependent=A'∫ (_rχ)^2dr_(r,y)-dependent_r(RHS)=0_r[A'∫ (_rχ)^2dr]=0 A'(_rχ)^2=0{[ A'=_yA=0; or;_rχ=0 ]} None of the two choices of equation (<ref>) can be satisfied. In addition, demanding either the constraint _ϕ f=0 or _χ f=0, nothing changes. Both sub-cases lead to the inconsistent result _rχ=0 as well.12){ϕ=ϕ(v,y), χ=χ(y), f=f(ϕ,χ)=f(v,y)} Using equations (<ref>), (<ref>) and the expansion of the mass function given by equation (<ref>) we obtain the same differential equations and constraints as in case 10. Therefore, this case results to an inconsistency as well.13){ϕ=ϕ(r,y), χ=χ(v), f=f(ϕ,χ)=f(v,r,y)}(<ref>) (_rϕ)^2+_r^2f=0_r^2f=-(_rϕ)^2_rf=-∫(_rϕ)^2dr+B(v,y)f(v,r,y)=-∫[∫(_rϕ)^2dr]dr+B(v,y)r+C(v,y) (<ref>)_yϕ_rϕ+_r_yf-A'_rf=0(<ref>) _rϕ_yϕ+_y[-∫(_rϕ)^2dr+B(v,y)]-A'[-∫(_rϕ)^2dr+B(v,y)]=0 _rϕ_yϕ-2∫_rϕ _y_rϕ dr+_yB+A'∫(_rϕ)^2dr-A'B=0 A'B-_yB_(v,y)-dependent=_rϕ_yϕ+∫_rϕ(A'_rϕ-2_y_rϕ)dr_(r,y)-dependent As it is indicated by the last equation (<ref>), its LHS depends on (v,y)-coordinates, while its RHS depends on (r,y)-coordinates. Consequently, we can derive two new constrains by demanding the partial derivative of the LHS with respect to v to be zero and likewise the partial derivative of the RHS with respect to r to be zero. Thus, we have: (<ref>)_v(LHS)=0_v(A'B-_yB)=0_v_yB=A'_vB (<ref>) _r(RHS)=0_r[_rϕ_yϕ+∫_rϕ(A'_rϕ-2_y_rϕ)dr]=0_r^2ϕ_yϕ+_rϕ_r_yϕ+_rϕ(A'_rϕ-2_r_yϕ)=0_r^2ϕ_yϕ-_rϕ_r_yϕ+A'(_rϕ)^2=0-(_rϕ_r_yϕ-_r^2ϕ_yϕ)+A'(_rϕ)^2=0_rϕ≠0-_rϕ_r_yϕ-_r^2ϕ_yϕ/(_rϕ)^2(_rϕ)^2+A'(_rϕ)^2=0-_r(_yϕ/_rϕ)(_rϕ)^2+A'(_rϕ)^2=0 A'=_r(_yϕ/_rϕ)_yϕ/_rϕ=A'r+F(y)_yϕ=_rϕ[A'(y)r+F(y)](<ref>) _y_vf-A'_vf-_ym/r_rf=f/r(_ym/r+_r_ym)_y_vf-A'_vf=f_ym/r^2+f_r_ym/r+_rf_ym/r(<ref>) r_y_vB+_y_vC-A'(r_vB+_vC)=f_ym/r^2+_r(f_ym)/r(<ref>)rA'_vB+_v_yC-rA'_vB-A'_vC=f_ym/r^2+_r(f_ym)/r _v_yC-A'_vC_(v,y)-dependent=f_ym/r^2+_r(f_ym)/r_(v,r,y)-dependent_r(RHS)=0_r[f_ym/r^2+_r(f_ym)/r]=0 _r(f_ym)/r^2-2f_ym/r^3+_r^2(f_ym)/r-_r(f_ym)/r^2=0_r^2(f_ym)-2f_ym/r^2=0 The last differential equation, with respect to the function f_ym, leads us to seek out the solution in the following form:f_ym=∑_n b_n(v,y)r^n Substituting equation (<ref>) into equation (<ref>) we get ∑_n n(n-1)b_n(v,y)r^n-2-2∑_n b_n(v,y)r^n-2=0 ∑_n (n^2-n-2)_roots: {-1,2}b_n(v,y)r^n-2=0{[b_n=0∀ n≠{-1,2}; ; b_-1,b_2arbitraryfunctions ]} f_ym=b_-1(v,y)/r+b_2(v,y)r^2[b_-1(v,y) D(v,y)]b_2(v,y) E(v,y)f_ym=D(v,y)/r+E(v,y)r^2[we need the dependence on the extra dimension]_ym≠ 0 f(v,r,y)=1/_y[m(v,r,y)][D(v,y)/r+E(v,y)r^2](<ref>)_vf=-_v_ym/(_ym)^2(D/r+E r^2)+1/_ym(_vD/r+_vE r^2)(<ref>)_vf=_vB r+_vC (<ref>),(<ref>) _vB r+_vC=-_v_ym/(_ym)^2(D/r+E r^2)+1/_ym(_vD/r+_vE r^2) r(_ym)^2(r_vB+_vC)=-_v_ym(D+E r^3)+_ym(_vD+_vE r^3)The combination of equations (<ref>) and (<ref>) leads to the relation r(∑_n_ya_n r^n)^2(r_vB+_vC)=-(∑_n_v_ya_n r^n)(D+E r^3)+(∑_n_ya_n r^n)(_vD+_vE r^3)(∑_n_ya_n r^n+1)^2_vB+r(∑_n_ya_n r^n)^2_vC=∑_n_v(D/_ya_n)(_ya_n)^2r^n+∑_n_v(E/_ya_n)(_ya_n)^2r^n+3 Equation (<ref>) can be mathematically consistent if and only if each one of its terms vanish. Concentrating our attention on the LHS, we can immediately deduce that in order to nullify both terms, it should be either {_ya_n=0∀ n} or {_vB=0 and _vC=0}. If {_ya_n=0∀ n}, then we lose the desirable dependence of the mass function on the extra dimension. On the other hand, if {_vB=0 and _vC=0} then equation (<ref>) becomes f=-∫[∫(_rϕ)^2dr]dr+rB(y)+C(y)_vf=0_χ f_≠0_vχ=0_vχ=0 which is inconsistent with our assumption about the field χ=χ(v). Therefore, this case also fails to offer us a solution to the problem. If we consider _ϕ f=0 then equation (<ref>) gives _rϕ=0, which is inconsistent with the assumption about ϕ=ϕ(r,y). Finally, we consider the sub-case in which _χ f=0. Hence, we have: _χ f=0_vf=0(<ref>)_v B=_C=0 Moreover, the substitution of equation (<ref>) into (<ref>) leads to 0=f_ym/r^2+_r(f_ym)/r_r(f_ym)/f_ym=-1/rf_ym=D(v,y)/r From equation (<ref>) and (<ref>) we get-_rf(_rm+1-3m/r)=f(_r^2m-2_rm/r)(<ref>) D(_r_ym r+_ym)/(r_ym)^2(_rm+1-3m/r)=D/r_ym(_r^2m-2_rm/r)(r_r_ym+_ym)(_rm+1-3m/r)=r_ym(_r^2m-2_rm/r)(<ref>) ∑_n(_ya_n)(n+1)r^n(1+∑_ℓ a_ℓ(ℓ-3)r^ℓ-1)=∑_n(_ya_n)r^n∑_ℓ a_ℓ[ℓ(ℓ-1)-2ℓ]_ℓ(ℓ-3)r^ℓ-1 ∑_n(_ya_n)r^n[(n+1)+∑_ℓ a_ℓ(n+1)(ℓ-3)r^ℓ-1-∑_ℓ a_ℓℓ(ℓ-3)r^ℓ-1] ∑_n(_ya_n)r^n[(n+1)+∑_ℓ a_ℓ(ℓ-3)(n+1-ℓ)r^ℓ-1]=0We demand that _ya_n≠0 ∀ n, thus equation (<ref>) can only hold if n,ℓ take the values 1 and 3. Consequently, we obtain:∑_n=1,3(_ya_n)r^n[(n+1)+a_1(-2)(n+2)]=0(_ya_1)r(2-6a_1)+(_ya_3)r^3(4-10a_1)=0{[ a_1=1/3; and; a_1=5/2 ]} rejectedIf on the other hand, we assume that n,ℓ take only the value 1. Then we have:(_ya_1)r(2-6a_1)=0 a_1=1/3_ya_1=0which is also an undesirable result.14){ϕ=ϕ(r,y), χ=χ(r), f=f(ϕ,χ)=f(r,y)}(<ref>)-_ym/r_rf=f/r(_ym/r+_r_ym)f_ym/r^2+_r(f_ym)/r=0_r(f_ym)=-f_ym/r[_ym≠ 0]f≠01/f_ym(f_ym)/ r=-1/r[ln(f_ym)]/ r=-1/rln(f_ym)=-ln r+B(v,y)e^B(v,y) C(v,y) f_ym=C(v,y)/r_ym≠ 0f=f(r,y)=C(v,y)/_ym(v,r,y) rUsing now equation (<ref>) together with (<ref>) we are led to the same result as in the sub-case _χ f=0 of case 13, which was presented before.* _ϕ f=0: From equation (<ref>) we obtain:-A'_r f=0 -A'_χ f_rχ_≠ 0=0{[ A'=0_yA=0;or; _χ f=0_χ f=_ϕ f=0 ]}rejected * _χ f=0: Using the same equations and performing the same steps as in the original case, we can show that this sub-case provides us with exactly the same negative result as the initial case 14.15){ϕ=ϕ(r,y), χ=χ(y), f=f(ϕ,χ)=f(r,y)} In this case, similarly to the case 14, the coupling function f depends on (r,y)-coordinates. Therefore, it is straightforward to verify that equations (<ref>) and (<ref>) result exactly to the same differential equations as in case 14 and subsequently to the same inappropriate form of the mass function. Consequently, case 15 is rejected as well.16){ϕ=ϕ(v,r), χ=χ(v,r), f=f(ϕ,χ)=f(v,r)}(<ref>) -A'_rf=0{[ A'=_yA=0; or; _rf=0_ϕ f=_χ f=0 ]}Obviously, this is also not a viable solution to the problem.17){ϕ=ϕ(v,r), χ=χ(v,y), f=f(ϕ,χ)=f(v,r,y)}(<ref>) (_rϕ)^2+_r^2f=0_rf=-∫(_rϕ)^2 dr+B(v,y)f=-∫[∫(_rϕ)^2 dr]dr+r B(v,y)+C(v,y)(<ref>) _y_rf-A'_rf=0(<ref>)_y[-∫(_rϕ)^2 dr+B]-A'[-∫(_rϕ)^2 dr+B]=0 -2∫_rϕ _y_rϕ_0 dr+_yB+A'∫(_rϕ)^2 dr-A'B=0A'B-_yB_(v,y)-dependent=A'∫(_rϕ)^2 dr_(v,r,y)-dependent_r(RHS)=0_r[A'∫(_rϕ)^2 dr]=0 A'(_rϕ)^2=0{[A'=0;or; _rϕ=0 ]}None of the choices which are depicted above is in agreement with our assumptions, hence, in this case the field equation (<ref>) is inconsistent.* _χ f=0:(<ref>) -A'_r f=0{[A'=0;or; _r f=0_ϕ f _rϕ_≠ 0=0_ϕ f=0=_χ f ]}rejected * _ϕ f=0:(<ref>)-_v f=f(_r^2m-2/r_rm)f≠ 0-_vf/f_(v,y)-dependent=_r^2m-2/r_rm_(v,r,y)-dependent_r(RHS)=0 _r(_r^2m-2/r_rm)=0_r^3m-2/r_r^2m+2/r^2_rm=0(<ref>) ∑_n a_nn(n-1)(n-2)r^n-3-2∑_na_nn(n-1)r^n-3+2∑_na_nnr^n-3=0 ∑_na_n[n(n-1)(n-2)+2n-2n(n-1)]_n(n-2)(n-3)r^n-3=0∑_na_nn(n-2)(n-3)r^n-3=0 m(v,,r,y)=a_0(v,y)+a_2(v,y)r^2+a_3(v,y)r^3Combining equations (<ref>) and (<ref>) we have:_yχ_vχ+_y_vf-A'_vf=f/r(_ya_0/r+_ya_2 r+_ya_3 r^2+2r_ya_2+3r^2_ya_3) _yχ_vχ+_y_vf-A'_vf/f_(v,y)-dependent=_ya_0/r^2+3_ya_2+4r_ya_3_(v,r,y)-dependent[consistency]for {[ _ya_0=0; _ya_2=0; _ya_3=0 ]}_ym=0This sub-case should also be excluded, because with _ym=0 the localization is not possible. 18){ϕ=ϕ(v,r), χ=χ(r,y), f=f(ϕ,χ)=f(v,r,y)}(<ref>) (_rϕ)^2+(_rχ)^2+_r^2f=0_rf=-∫[(_rϕ)^2+(_rχ)^2]dr+B(v,y)f=-∫{∫[(_rϕ)^2+(_rχ)^2]dr}dr+rB(v,y)+C(v,y) (<ref>) _rχ_yχ+_y_rf-A'_rf=0_rχ_yχ+_y{-∫[(_rϕ)^2+(_rχ)^2]dr+B}-A'{-∫[(_rϕ)^2+(_rχ)^2]dr+B}=0_rχ_yχ-∫(2 _rϕ _y_rϕ_0+2 _rχ _y_rχ)dr+_yB+A'∫[(_rϕ)^2+(_rχ)^2]dr-A'B=0A'B-_yB_(v,y)-dependent=_rχ_yχ+∫[_rχ(A'_rχ-2 _y_rχ)+A'(_rϕ)^2]dr_(v,r,y)-dependent (<ref>) _r(RHS)=0_r{_rχ_yχ+∫[_rχ(A'_rχ-2 _y_rχ)+A'(_rϕ)^2]dr}=0_r^2χ _yχ+_rχ _r_yχ+A'(_rχ)^2-2 _rχ _y_rχ+A'(_rϕ)^2=0_r^2χ _yχ-_rχ _r_yχ+A'(_rχ)^2_(r,y)-dependent=-A'(_rϕ)^2_(v,r,y)-dependent_v(RHS)=0_v[A'(_rϕ)^2]=0 2A'_rϕ _v_rϕ=0[_rϕ≠0]A'≠0_v_rϕ=0_v[_rϕ(v,r)]=0_rϕ(v,r)=C_1(r)ϕ(v,r)=∫ C_1(r) dr+C_2(v)ϕ(v,r)=ϕ_1(r)+ϕ_2(v)(<ref>)_vf=[(2 _rϕ _v_rϕ_0+2 _rχ _v_rχ_0)dr]dr+r_vB+_vC_vf=r_vB+_vC (<ref>)(<ref>)A'B-_yB_(v,y)-dependent=_rχ_yχ+[_rχ(A'_rχ-2 _y_rχ)+A'(_rϕ)^2^r-dependent]dr_(r,y)-dependent_v(LHS)=0_v[A'B-_yB]=0A'_vB=_v_yB (<ref>) _y_vf-A'_vf-_ym/r_rf=f/r(_ym/r+_r_ym)(<ref>) r_y_vB+_y_vC-A'(r_vB+_vC)=f_ym/r^2+_r(f_ym)/r(<ref>)_y_vC-A'_vC_(v,y)-dependent=f_ym/r^2+_r(f_ym)/r_(v,r,y)-dependent(<ref>)_r(RHS)=0_r[f_ym/r^2+_r(f_ym)/r]=0_r(f_ym)/r^2-2f_ym/r^3+_r^2(f_ym)/r-_r(f_ym)/r^2=0_r^2(f_ym)-2/r^2(f_ym)=0Equation (<ref>) is identical to equation (<ref>) and equation (<ref>) is identical to (<ref>) of case 13; we can henceforth follow the same steps as in case 13 in order to show that this case should also be rejected.* _ϕ f=0:(<ref>)(_rϕ)^2+(_rχ)^2+_r^2f=0_rf=-∫[(_rϕ)^2+(_rχ)^2]dr+B(y)f(r,y)=-∫{∫[(_rϕ)^2+(_rχ)^2]dr}dr+r B(y)+C(y) (<ref>)_vf=0-∫(∫ 2_rϕ_v_rϕ dr)dr=0_v_rϕ=0ϕ(v,r)=ϕ_1(v)+ϕ_2(r)(<ref>) 0=f_ym/r^2+_r(f_ym)/r=0_ym≠0f(r,y)=D(v,y)/r _y[m(v,r,y)] where the following constraint is necessary to be satisfied: _vf=0_v(D/_ym)=0 (<ref>)e^-2A/r(1-2m/r)_rf+A'_yf-(_yχ)^2-_y^2f=f(3A”+2e^-2A/r^2_rm)_v( )-2e^-2A/r^2_vm _rf=f2e^-2A/r^2_v_rm-_vm _rf=f_v_rm (<ref>)[(<ref>),(<ref>)](<ref>)-_vm[-D(_r_ym r+_ym)/(r_ym)^2]=D/r_ym_v_rm_vm(r_r_ym+_ym)=r_ym _v_rm(∑_n(_va_n)r^n)(∑_ℓ(_ya_ℓ)(ℓ+1)r^ℓ)=(∑_n(_ya_n)r^n)(∑_ℓ(_va_ℓ) ℓ r^ℓ)∑_n,ℓ(_va_n)(_ya_ℓ)(ℓ-n+1)r^n+ℓ=0{_va_n=0 ∀ n}_vm=0Thus, the combination of equation (<ref>) and (<ref>) results to _vD=0 Hence, we obtain: (<ref>)[(<ref>)](<ref>)f(r,y)=D(y)/r _y[m(r,y)] (<ref>) r_rϕ_vϕ-_rf(_rm-3m/r+1)=f(_r^2m-2/r_rm)_v( )r_v_rϕ_0_vϕ+r_rϕ_v^2ϕ=0_v^2ϕ=0_v^2ϕ_1=0ϕ_1(v)= v+ξ whereand ξ are constants. _rϕ_vϕ=_rf/r(_rm-3m/r+1)+f/r(_r^2m-2/r_rm)(<ref>)(_vϕ)^2+(1-2m/r)_rϕ_vϕ =-e^-2A/r(_ym _yf+f_y^2m+4A' f_ym)=-e^-2A/r(e^4A_ym _yf+e^4Af_y^2m+4A'e^4Af_ym)=-e^-2A/r_y(e^4Af_ym)=-e^-2A/r_y(e^4AD/r)=-e^-2A/r^2_y(De^4A)Substituting equation (<ref>) into (<ref>) and using also equations (<ref>), (<ref>) and (<ref>), we obtain:^2+e^-2A/r^2_y(De^4A)+(1-2m/r)[-D(r _r_ym+_ym/(r_ym)^2)(_rm+1-3m/r). .+D/r^2_ym(_r^2m-2/r_rm)]=0 ^2+e^-2A/r^2_y(De^4A)+(1-2m/r)D/(r_ym)^2[-(r _r_ym+_ym)(_rm+1-3m/r). ._ym(_r^2m-2/r_rm)]=0^2(r_ym)^2+e^-2A_y(De^4A)(_ym)^2-(_r_ym r+_ym)(_rm+1-3m/r)+2m/r(_r_ym r+_ym)(_rm+1-3m/r) _ym(_r^2m-2/r_rm) -2m/r_ym(_r^2m-2/r_rm)=0 ∑_n,ℓ,k_ya_n{(n+1)+[(_ya_ℓ)(^2 r^2+e^-2A_y(e^4AD))+a_ℓ((n+1)(5-ℓ)r^-1+ℓ(ℓ-3)r^-2).. ..+2a_ℓ a_k(ℓ-3)((n+1)r^k-2-ℓ r^k-3)]r^ℓ}r^n=0Demanding _ya_n≠0 the above equation is impossible to be satisfied. Therefore, in order to have a consistent field equation we need to allow _ya_n=0 at least for some values of n, but this is catastrophic for the localization of the 5-dimensional black hole. * _χ f=0:(<ref>) (_rϕ)^2+(_rχ)^2+_r^2f=0_y( )2_rϕ_y_rϕ_0+2_rχ _y_rχ+_y_r^2f_0=0_y_rχ=0χ(r,y)=χ_1(r)+χ(y) (<ref>)_rχ_yχ-A'_rf=0_v( )-A'_v_rf=0_v_rf=0 f(v,r)=f_1(v)+f_2(r) (<ref>)_rχ_yχ-A'_rf=0χ_1'(r)χ_2'(y)-A'(y)f_2'(r)=0χ_2'(y)/A'(y)=f_2'(r)/χ_1'(r)= χ_2(y)= A(y)+κf_2(r)= χ_1(r)+κ̃ (<ref>)[(<ref>)]_v( )2_rϕ _v_rϕ=0_v_rϕ=0ϕ(v,r)=ϕ_1(v)+ϕ_2(r) (<ref>)-A'_vf_(v,y)-dependent=f_ym/r^2+_r(f_ym)/r_(v,r,y)-dependent_r(RHS)=0_r^2(f_ym)-2f_ym/r^2=0_ym≠0f(v,r)=1/_y[m(v,r,y)][B(v,y)/r+C(v,y) r^2](<ref>)_yf=0{[ _y(B/_ym)=0;; _y(C/_ym)=0 ]} _y(B/_ym)=0_B_ym-B_y^2m=0∑_n(_yB_ya_n-B_y^2a_n)r^n=0_yB/B=_y(_ya_n)/_ya_n_y[a_n(v,y)]/B(v,y)=b_n(v) Similarly, we have: _y[a_n(v,y)]/C(v,y)=c_n(v) Equation (<ref>) with the use of equations (<ref>) and (<ref>) gets the following form: f(v,r)=1/∑_n d_n(v)r^n+1+1/∑_n c_n(v)r^n-2 Finally, from equation (<ref>) the relation _v_rf=0 should be satisfied. Thus, it is:_v_rf=0(<ref>)∑_n d_n'(n+1)r^n/(∑_n d_nr^n)^2-2∑_n d_n(n+1)r^n∑_ℓ d_ℓ' r^ℓ+1/(∑_n d_nr^n+1)^3+∑_n c_n'(n-2)r^n-3/(∑_n c_nr^n-2)^2-2∑_n c_n(n-2)r^n-3∑_ℓ c_ℓ' r^ℓ-2/(∑_n c_nr^n-2)^3=0 {[ d_n'(v)=0 ∀ n; and; c_n'(v)=0 ∀ n ]}_vf=0 Therefore, this sub-case should be rejected as well. In the previous analysis we silently assumed that both B(v,y) and C(v,y) were not equal to zero. In case that one of these functions is zero, we are led to exactly the same result, namely _vf=0. This happens because even if we nullify one of the functions B(v,y) or C(v,y) in equation (<ref>), none of the following steps is going to be changed. 19){ϕ=ϕ(v,y), χ=χ(v,y), f=f(ϕ,χ)=f(v,y)}(<ref>)A'_yf-(_yϕ)^2-(_yχ)^2+_y^2f_(u,y)-dependent=f(3A”+e^-2A/r_r^2m)_(v,r,y)-dependent_r(RHS)=0fe^-2A_r(_r^2m/r)=0_r^3m-_r^2m/r=0m(v,r,y)=a_0(v,y)+a_1(v,y)r+a_3(v,y)r^3 (<ref>)-_vf=f(_r^2m-2/r_rm)(<ref>)-_vf=-f2a_1/r_v[f(v,y)]/f(v,y)=2a_1(v,y)/rIt is clear that the last equation cannot be true, because of the factor 1/r in the RHS. For the sub-cases _χ f=0 and _ϕ f=0 nothing changes. The previous equations have exactly the same form, thus, the sub-cases are rejected as well.20){ϕ=ϕ(v,y), χ=χ(r,y), f=f(ϕ,χ)=f(v,r,y)}(<ref>) (_rχ)^2+_r^2f=0_rf=-∫ (_rχ)^2dr+B(v,y)f=-∫[∫(_rχ)^2dr]dr+rB(v,y)+C(v,y) (<ref>)_rχ_yχ+_r_yf-A'_rf=0(<ref>) _rχ_rχ-2∫_rχ _y_rχ dr+B-A'[-∫(_rχ)^2dr+_yB]=0A'B-_yB_(v,y)-dependent=_rχ_yχ+∫_rχ(A'_rχ-2 _y_rχ)dr_(r,y)-dependent(<ref>)_v(LHS)=0_v(A'B-_yB)=0A'_vB=_v_yB (<ref>)_r(RHS)=0_r^2χ_yχ+_rχ _r_yχ+_rχ(A'_rχ-2 _r_yχ)=0 _r^2χ_yχ-_rχ _r_yχ+A'(_rχ)^2=0_rϕχ0-_r(_yχ/_rχ)(_rχ)^2+A'(_rχ)^2=0_r(_yχ/_rχ)=A'_yχ=_rχ[A' r+F(y)](<ref>) _vϕ _yϕ+_y_vf-A'_vf-_ym _rf/r=f/r(_ym/r+_r_ym)_vϕ _yϕ+r_y_vB+_y_vC-A'(r_vB+_vC)=f_ym/r^2+_r(f_ym)/r(<ref>)_vϕ _yϕ+_y_vC-A'_vC_(v,y)-dependent=f_ym/r^2+_r(f_ym)/r_(v,r,y)-dependent_r(RHS)=0_r[f_ym/r^2+_r(f_ym)/r]=0_r^2(f_ym)-2/r^2(f_ym)=0f(v,r,y)=1/_y[m(v,r,y)][D(v,y)/r+E(v,y)r^2]We have already shown in previous cases (see case 13) that the last related combined with equation _vf=r_vB+_vC and (<ref>) leads to an inconsistency. If we assume _ϕ f=0 then equation (<ref>) gives _rχ=0 which is not accepted. If on the other hand assume _χ f=0 then using equations (<ref>), (<ref>) and (<ref>) as before, we obtain again the relation (<ref>) (but now _vf=0). Finally, with the use of (<ref>) it is:-_rf(_rm+1-3m/r)=f(_r^2m-2/r_rm)Using equations (<ref>), (<ref>) into the previous expressions and after some algebra we get∑_n(_ya_n){D/r^2(n+1)+E r(n-2)+∑_ℓ a_ℓ(ℓ-3)[D/r^3r^ℓ(ℓ+n+1)+E r^ℓ(ℓ+n-2)]}=0In the above equation, if we demand _ya_n≠0 then it is also restrictive to nullify both functions D(y) and E(y) in order for the relation to be consistent. However, there is the possibility to fix n,ℓ=1 and E(y)=0. Then, we can calculate the value of the function D(y) that manages to make the LHS of the above expression to vanish. The problem in this sub-sub-case is that for n=1 the metric tensor does not describe a black hole. Hence, this complete case does not give a viable solution to the problem.21){ϕ=ϕ(r,y), χ=χ(r,y), f=f(ϕ,χ)=f(r,y)} The field equation (<ref>), in this case, leads to the same expression for the coupling function f(r,y) as in case 14, namely (<ref>). Additionally, if we combine equations (<ref>), (<ref>) and (<ref>), as it has already been done in case 13, we are led to the same negative results. The sub-cases _ϕ f=0 and _χ f=0 do not solve the problem either. They produce the same results as the original case 21. § F(Φ,Χ)=A Φ+B ΧThe rest of the cases are extremely difficult to be checked and even more difficult to be solved by using the field equations (<ref>)-(<ref>); the complexity of the differential equations that emerge by the field equations increases dramatically with each variable that we add into the fields. Hence, we are going to examine the behaviour of some special coupling functions f(ϕ,χ)=aϕ+bχ in each one of the remaining cases (22-28).22){ϕ=ϕ(v,r,y), χ=χ(v)}We consider the general case where a,b∈∧ a≠0∧ b≠0. (<ref>)(_rϕ)^2+a _r^2ϕ=0 -(_rϕ)/(_rϕ)^2=dr/a (_rϕ)^-1=r/a+B(v,y)_rϕ=1/r/a+B(v,y)_rϕ=a/r+a B(v,y)a B(v,y) B(v,y)_rϕ=a/r+B(v,y)ϕ(v,r,y)=aln[r+B(v,y)]+C(v,y) (<ref>) _rϕ _yϕ+a _r_yϕ-A'a _rϕ=0(<ref>)a/r+B(_yB/r+B+_yC)-a^2_yB/(r+B)^2-A' a^2/r+B=0a_yC-A' a^2/r+B=0_yC=a A'C(v,y)=a A(y)+D(v) (<ref>)(<ref>)ϕ(v,r,y)=a{ln[r+B(v,y)]+A(y)}+D(v)(<ref>) _yϕ_vϕ+a _y_vϕ-A'( a_vϕ+b_vχ)-_ym/ra _rϕ=f/r(_ym/r+_r_ym)(<ref>)(a_yB/r+B+aA')(a _vB/r+B+_vD)+a(_y_vB/r+B-a_vB_yB/(r+B)^2)-A'(a^2_vB/r+B+aD'+bχ')-_ym/ra^2/r+B=aϕ+bχ/r(_ym/r+_r_ym)×(r+B)aD'_yB+a^2_v_yB-bBA'χ'_(v,y)-dependent=(aϕ+bχ)(r+B)(_ym/r^2+_r_ym/r)+A'bχ'r+a^2_ym/r_(v,r,y)-dependentDemanding the consistency of the previous equation, we are led to demand _r(RHS)=0. Thus, it is:(<ref>)_r(RHS)=0A'bχ'+a^2(_r_ym/r-_ym/r^2)+a^2(_ym/r^2+_r_ym/r)+(aϕ+bχ)(_ym/r^2+_r_ym/r)+(aϕ+bχ)(r+B)(-2_ym/r^3+_r^2_ym/r)=0 A'bχ'+2a^2_r_ym/r+(aϕ+bχ)(_r_ym/r-_ym/r^2+_r^2_ym-2B_ym/r^3+B_r^2_ym/r)=0(<ref>) A'bχ'+2a^2∑_n(_ya_n)nr^n-2+(aϕ+bχ)[∑_n(_ya_n)nr^n-2-∑_n(_ya_n)r^n-2+. .+∑_n(_ya_n)n(n-1)r^n-2-2B∑_n(_ya_n)r^n-3+B∑_n(_ya_n)n(n-1)r^n-3]=0A'bχ'+∑_n(_ya_n){2a^2nr+(aϕ+bχ)(n+1)[(n-1)r+B(n-2)]}r^n-3Both for b=0 and b≠0 the above equation cannot have an overall value of zero, without allowing some of the function a_n(v,y) to be y-independent. This is easy to be understood by writing the equation in the following expanded form.A'bχ'+∑_n(_ya_n)[2a^2n+bχ(n+1)]r^n-2+Bbχ∑_n(_ya_n)(n+1)(n-2)r^n-3++aϕ∑_n(_ya_n)(n+1)(n-1)r^n-2+Baϕ∑_n(_ya_n)(n+1)(n-2)r^n-3=0Demanding finite number of values for n[The parameter n is directly connected to the behaviour of the mass function m(v,r,y) with respect to the variable r via equation (<ref>). Thus, the parameter n should take some specific values in order to lead to a mass function that describes a black hole, or even a modified black hole. In any case, we cannot allow n to take infinite number of values.], it is impossible to find a set of values that could allow the last two terms of equation (<ref>) to cancel each other (any other combination of terms in order to have a vanishing LHS either results to _rϕ=0 or to an inconsistency in the expression of ϕ with respect to r). Hence, the consistency of the equation can only be achieved by setting _ya_n=0 for some values of n. However, this would be devastating for the localization. Consequently, this case should also be excluded. Lastly, for a=0, equation (<ref>) results to _rϕ=0, meaning that we need to investigate the case {ϕ(v,y), χ=χ(v)}, but this case has already been excluded.23){ϕ=ϕ(v,r,y), χ=χ(r)}(<ref>)(_rϕ)^2+(χ')^2+a _r^2ϕ+b χ”=0 (_rϕ)^2+a _r^2ϕ=-(χ')^2-b χ”(_rϕ)^2+a _r^2ϕ=h(r)=-(χ')^2-b χ”With the use of mathematica or any other software that can solve differential equations, one can verify that the above differential equations (with respect to ϕ and χ) have extremely complicated solutions even in case that h(r)=r. One can also examine different functions for h(r) in order to be completely convinced. As an example we present the solutions of the differential equations with respect to the scalar fields ϕ, χ that emanate from the relation (<ref>) in case of h(r)=r.ϕ(v,r,y)= -_1^r a {√(-1/a)√(1/a) q^3/2[-c_1 J_-4/3(2/3√(-1/a)√(1/a) q^3/2)+c_1J_2/3(2/3√(-1/a)√(1/a)q^3/2)]}/2 q (c_1J_-1/3(2/3√(-1/a)√(1/a)q^3/2)+J_1/3(2/3√(-1/a)√(1/a)q^3/2))dq+_1^r a[2 √(-1/a)√(1/a) q^3/2J_2/3(2/3√(-1/a)√(1/a) q^3/2)+c_1 J_-1/3(2/3√(-1/a)√(1/a) q^3/2)]/2 q (c_1J_-1/3(2/3√(-1/a)√(1/a)q^3/2)+J_1/3(2/3√(-1/a)√(1/a)q^3/2))dq+c_2χ(r)=_1^rq^3/2[-c_1J_-4/3(-2 q^3/2/3 b)+c_1 J_2/3(-2q^3/2/3 b)-2 J_-2/3(-2 q^3/2/3b)]+b c_1 J_-1/3(-2 q^3/2/3 b)/2q (c_1 J_-1/3(-2 q^3/2/3b)+J_1/3(-2 q^3/2/3 b)) dq+c_2Thus, it would be practically impossible to use these complicated expression of the fields ϕ, χ into the other field equations in order to examine their consistency. Consequently, we are essentially obliged to consider only the cases in which the function h(r) is simply a constant (negative or positive) and zero. * h(r)=C=Q^2>0∧ a≠ 0∧ b≠0: (<ref>)(_rϕ)^2+a _r^2ϕ=Q^2=-(χ')^2-b χ”{[ (_rϕ)^2+a _r^2ϕ=Q^2;;(χ')^2+b χ”=-Q^2 ]} From equation (<ref>) we obtain the following functions for the scalar fields:ϕ(v,r,y)=a ln[cosh(Q/ar+B(v,y))]+D(v,y)χ(r)=b ln[cos(Q/br+E)]+FWe have assumed that C>0. In case that C<0, we would get the same expressions for the fields with the difference that now the hyperbolic cosine would describe the field χ, while the cosine would correspond to the field ϕ. The subsequent analysis in each case is the same.(<ref>)_rϕ_yϕ+a_r_yϕ-A'(a_rϕ+b_rχ)=0[(<ref>)](<ref>) Qtanh(Q/ar+B)[atanh(Q/ar+B)_yB+_yD]+aQ_yB/cosh^2(Q/ar+B)-A'aQtanh(Q/ar+B)+A'bQtan(Q/br+E)=0 a_yB+tanh(Q/ar+B)(_yD-aA')+A'btan(Q/br+E)=0{[ _yB=0; and; _yD-aA'=0; and;A'=0 ]}A'≠0rejected * h(r)=C=-Q^2<0∧ a≠ 0∧ b=0:(<ref>)(_r ϕ)^2+a _r^2ϕ=-Q^2=-(χ')^2{[ (_rϕ)^2+a _r^2ϕ=-Q^2; ; (χ')^2=Q^2 ]} From equation (<ref>) we get: ϕ(v,r,y)=a ln[cos(Q/ar+B(v,y))]+D(v,y) χ(r)=± Qr+E(<ref>)_rϕ_yϕ+a_r_yϕ-A'a_rϕ=0(<ref>)-Qtan(Q/ar+B)[-atan(Q/ar+B)_yB+_yD]-aQ_yB/cos^2(Q/ar+B)+A'aQtan(Q/ar+B)=0 -aQ_yB-Qtan(Q/ar+B)(_yD-aA')=0{[_yB=0B=B(v);and; _yD=aA'D(v,y)=aA(y)+F(v) ]}Thus, equations (<ref>) and (<ref>) lead to ϕ(v,r,y)=a ln[cos(Q/ar+B(v))]+a A(y)+F(v)(<ref>)_yϕ_vϕ+a_y_vϕ_0-A'a_vϕ=aϕ/r(_ym/r+_r_ym)+_ym/ra_rϕ aA'[-tan(Q/ar+B)_vB+_vF]-A'a[-atan(Q/ar+B)_vB+_vF]=aϕ_ym/r^2+_r(aϕ_ym)/raϕ_ym/r^2+_r(aϕ_ym)/r=0(ϕ_ym)/ϕ_ym=-dr/r_ym=G(v,y)/rϕThe substitution of equation (<ref>) into (<ref>) leads to _ym=G(v,y)r^-1{a ln[cos(Q/ar+B(v))]a A(y)+F(v)}^-1 An expansion of the form m(v,r,y)=∑_na_n(v,y)r^n and a finite number of values for the index n cannot produce the above equation for the mass function. Hence, this sub-case does not result to a viable solution for the localization problem. * h(r)=C=Q^2>0∧ a=0∧ b≠0:Using again equation (<ref>) in the same way as in the previous sub-case, we obtain: ϕ(v,r,y)=± Qr+B(v,y) χ(r)=b ln[cos(Q/br+D)]+E(<ref>)_rϕ_yϕ-A'b _rχ=0[(<ref>)](<ref>)± Q_yB+A'Qtan(Q/br+D)=0±_yB+A'tan(Q/br+D)=0{[ _yB=0; and;A'=0 ]}A'≠0rejected * h(r)=0∧ a≠ 0∧ b≠0:In this sub-case, it is not possible to assume either a=0 or b=0, because from equation (<ref>) we are led to _rϕ=0 or _rχ=0 respectively, but these results are inconsistent with our primary assumption for the fields. Consequently, from equation (<ref>) we get:(_rϕ)^2+a _r^2ϕ=0=(χ')^2+b χ”{[ (_rϕ)^2+a _r^2ϕ=0;; (χ')^2+b χ”=0 ]}The solutions that emanate from the above differential equations are: ϕ(v,r,y)=a ln[r+B(v,y)]+C(v,y) χ(r)=b ln(r+D)+E(<ref>)_rϕ_yϕ+a _r_yϕ-A'(a _rϕ+b _rχ)=0[(<ref>)](<ref>)a/r+B(a_yB/r+B+_yC)-a^2_yB/(r+B)^2-A'(a^2/r+B+b^2/r+D)=0a_yC/r+B-A'a^2/r+B-b^2A'/r+D=0_y[C(v,y)]-aA'(y)_(v,y)-dependent=A'(y)b^2/a r+B(v,y)/r+D_(v,r,y)-dependentDemanding the consistency of equation (<ref>), we get the following constraint. (<ref>)_r(RHS)=0_r(r+B(v,y)/r+D)=0B(v,y)=D Substituting equation (<ref>) into (<ref>), we obtain: _y[C(v,y)]-aA'(y)=A'(y)b^2/aC(v,y)=a^2+b^2/aA(y)+F(v) Hence, from equations (<ref>), (<ref>) and (<ref>) it is: ϕ(v,r,y)=a ln(r+D)+a^2+b^2/aA(y)+F(v)(<ref>)_yϕ_vϕ+a_y_vϕ_0-A'a _vϕ-_ym/r_rf=f/r(_ym/r+_r_ym)a^2+b^2/aA'F'-aA'F'_(v,y)-dependent=f_ym/r^2+_r(f_ym)/r_(v,r,y)-dependent_r(RHS)=0_r^2(f_ym)-2f_ym/r^2=0_ym≠0f(v,r,y)=1/_y[m(v,r,y)][G(v,y)/r+H(v,y)r^2]However, the coupling function f(v,r,y) is also given by the primary expression f=a ϕ+b χ. Thus, the substitution of equations (<ref>) and (<ref>) into the previous expression and then equating the result with the RHS of equation (<ref>), we obtain:(a^2+b^2)ln(r+D)+(a^2+b^2)A(y)+aF(v)+bE=1/_ym[G(v,y)/r+H(v,y)r^2]_ym=G(v,y)/r+H(v,y)r^2/(a^2+b^2)ln(r+D)+(a^2+b^2)A(y)+aF(v)+bE Similarly to sub-case (ii), the above equation cannot be consistent with the expansion of equation (<ref>) and the condition of having a finite number of values for the index n. Hence, this sub-case is also rejected.24){ϕ=ϕ(v,r,y), χ=χ(y)} (<ref>) (_rϕ)^2+a _r^2ϕ=0ϕ(v,r,y)=a ln[r+B(v,y)]+C(v,y) (<ref>)_rϕ_yϕ+a _r_yϕ-A'a _rϕ=0a/r+B(a_yB/r+B+_yC)-a^2_yB/(r+B)^2-a^2A'/r+B=0_yC=aA'C(v,y)=aA(y)+D(v)(<ref>)(<ref>)ϕ(v,r,y)=a ln[r+B(v,y)]+aA(y)+D(v)(<ref>)_yϕ_vϕ+a _v_yϕ-A'a _vϕ-_ym/ra _rϕ=aϕ+bχ/r(_ym/r+_r_ym)(<ref>)a _yB D'+a^2_v_yB_(v,y)-dependent=(aϕ+bχ)(r+B)(_ym/r+_r_ym)+a^2_ym/r_(v,r,y)-dependentThe RHS of equation (<ref>) is similar to the RHS of equation (<ref>) except for the missing term A'bχ'r. Thus, demanding _r(RHS)=0 of equation (<ref>) (as in case 22) and also using equation (<ref>), we get: ∑_n(_ya_n){2a^2nr+(aϕ+bχ)(n+1)[(n-1)r+B(n-2)]}r^n-3=0 The above equation cannot be consistent if we demand a finite number of values for n. The arguments here are the same as the arguments which were presented in case 22. 25){ϕ=ϕ(v,r,y), χ=χ(v,r)}(<ref>) (_rϕ)^2+(_rχ)^2+a _r^2ϕ+b _r^2χ=0 (_rϕ)^2+a _r^2ϕ=-(_rχ)^2-b _r^2χ(_rϕ)^2+a _r^2ϕ=h(v,r)=-(_rχ)^2-b _r^2χAs it was justified in case 23, the field equation are essentially unapproachable if we do not demand _r[h(v,r)]=0. Therefore, in this case, we are going to consider that h=h(v). In complete analogy to the case 23, four different sub-cases are going to be investigated in the context of the original case.* h(v,r)=[C(v)]^2>0∧ a≠ 0∧ b≠0:From equation (<ref>) we are led to the following functions: ϕ(v,r,y)=a ln[cosh(C(v)/ar+B(v,y))]+D(v,y) χ(v,r)=b ln[cos(C(v)/br+E(v))]+F(v) (<ref>)_rϕ_yϕ+a _r_yϕ-A'(a _rϕ+b _rχ)=0[(<ref>)](<ref>) a_yB+_yDtanh(C/ar+B)-A'atanh(C/ar+B)+A'btan(C/br+E)=0 {[ _yB=0; and; _yD-aA'=0; and;A'=0 ]}A'≠0rejected* h(v,r)=-[C(v)]^2<0∧ a≠ 0∧ b=0:In this sub-case equation (<ref>) leads to ϕ(v,r,y)=a ln[cos(C(v)/ar+B(v,y))]+D(v,y) χ(v,r)=± C(v)r+E(v) (<ref>)_rϕ_yϕ+a _r_yϕ-A'a_rϕ=0(<ref>) -a_yB-_yDtan(C/ar+B)+A'atan(C/ar+B)=0{[_yB=0B=B(v);and; _yD=aA'D(v,y)=aA(y)+F(v) ]}Thus, equations (<ref>) and (<ref>) give: ϕ(v,r,y)=a ln[cos(C(v)/ar+B(v))]+aA(y)+E(v) Consequently, we have:(<ref>)_yϕ_vϕ+a _v_yϕ_0-A'a_vϕ=aϕ_ym/r^2+_r(aϕ_ym)/r ϕ_ym/r+_r(ϕ_ym)=0ϕ_ym=G(v,y)/rϕ≠0_ym=G(v,y)/rϕ_ym=G(v,y)r^-1{a ln[cos(C(v)/ar+B(v))]+aA(y)+E(v)}^-1The reason that we reject this sub-case is exactly the same as in sub-case (ii) of case 23.One can easily verify that the remaining two sub-cases of case 25, namely h(v,r)=[C(v)]^2>0∧ a=0∧ b≠0 and h(v,r)=0∧ a≠ 0∧ b≠0, can be excluded as well. The reason is that the complete analysis of case 25 can be simply deduced by the analysis of case 23 if we just replace the constant Q with the function C(v) (To clarify this statement, compare the analysis of the previous two sub-cases -which were performed extensively- to the corresponding analysis of case 23). This particular substitution does not change the negative results that was found in each sub-case of case 23. Therefore, we proceed to the following case. 26){ϕ=ϕ(v,r,y), χ=χ(v,y)} (<ref>) (_rϕ)^2+a _r^2ϕ=0ϕ(v,r,y)=a ln[r+B(v,y)]+C(v,y) (<ref>)_rϕ_yϕ+a _r_yϕ-A'a _rϕ=0a/r+B(a_yB/r+B+_yC)-a^2_yB/(r+B)^2-a^2A'/r+B=0_yC=aA'C(v,y)=aA(y)+D(v)(<ref>)(<ref>)ϕ(v,r,y)=a ln[r+B(v,y)]+aA(y)+D(v)(<ref>)_yϕ_vϕ+_yχ_vχ+a _v_yϕ+b_v_yχ-A'(a _vϕ+b_vχ)=f_ym/r^2+_r(f_ym)/r(<ref>)a _yB D'+_yχ_vχ B+a^2_v_yB-A'bB_vχ_(v,y)-dependent==(aϕ+bχ)(r+B)(_ym/r^2+_r_ym/r)-_yχ_vχ r-b_y_vχ r+A'b_vχ r+a^2_ym/r_(v,r,y)-dependentThe RHS of equation (<ref>) is similar to the RHS of equation (<ref>) except from the additional terms -_yχ_vχ r and -b_y_vχ r. Thus, demanding _r(RHS)=0 of equation (<ref>) (as in case 22) and also using equation (<ref>), we get: A'b_vχ-_yχ_vχ-b_v_yχ+∑_n(_ya_n){2a^2nr+(aϕ+bχ)(n+1)[(n-1)r+B(n-2)]}r^n-3=0The above equation cannot be consistent if we demand a finite number of values for n. The arguments here are the same as the arguments which were presented in case 22 and also used in case 24. 27){ϕ=ϕ(v,r,y), χ=χ(r,y)}(<ref>) (_rϕ)^2+(_rχ)^2+a _r^2ϕ+b _r^2χ=0 (_rϕ)^2+a _r^2ϕ=-(_rχ)^2-b _r^2χ(_rϕ)^2+a _r^2ϕ=h(r,y)=-(_rχ)^2-b _r^2χThe same argument as it was presented in case 23 leads us to the following sub-cases: * h(r,y)=C(y)>0∧ a≠ 0∧ b≠0:From euqation (<ref>) we obtain: ϕ(v,r,y)=a ln[cosh(√(C(y))/ar+B(v,y))]+D(v,y) χ(r,y)=b ln[cos(√(C(y))/br+E(y))]+F(y) (<ref>)_rϕ_yϕ+_rχ_yχ+a_r_yϕ+b_r_yχ-A'(a_rϕ+b_rχ)=0[(<ref>)](<ref>) a_yB-bE'+tanh(√(C)/ar+B)(_yD+aC'/2C-aA')-tan(√(C)/br+E)(F'+bC'/2C-bA')=0{[ _yB=b/aE'B(v,y)=b/aE(y)+G(v);and;_yD-aA'+a/2_y[ln C(y)]=0D(v,y)=aA(y)-a/2ln[C(y)]+H(v);and; F'-bA'+b/2_y[ln C(y)]=0F(y)=bA(y)-b/2ln[C(y)]+J_const. ]}Thus, equations (<ref>) and (<ref>) take the following form: ϕ(v,r,y)=a ln[cosh(√(C(y))/ar+b/aE(y)+G(v))]+aA(y)-a/2ln[C(y)]+H(v) χ(r,y)=b ln[cos(√(C(y))/br+E(y))]+bA(y)-b/2ln[C(y)]+J (<ref>)_yϕ_vϕ+a_v_yϕ-A'a_vϕ=f_ym/r^2+_r(f_ym)/rf_ym/r^2+_r(f_ym)/r=arC'G'/2√(C)+abE'G'+tanh(√(C)/ar+b/aE+G)(C'H'r/2√(C)+bE'H'-aC'G'/2C)-aC'H'/2CMoreover, it is: f=aϕ+bχ=a^2 ln[cosh(√(C(y))/ar+b/aE(y)+G(v))]+b^2 ln[cos(√(C(y))/br+E(y))]+(a^2+b^2)A(y)-a^2+b^2/2ln[C(y)]+aH(v)+bJObviously, the substitution of equation (<ref>) into (<ref>) leads to a very complicated equation which is not possible to be satisfied using equation (<ref>) for the mass function and demanding n to take a finite number of values. The previous statement holds even in case of C'=0.* h(r,y)=C(y)=-[K(y)]^2<0∧ a≠ 0∧ b=0:Equation (<ref>) results to the following functions for the fields: ϕ(v,r,y)=a ln[cos(√(C(y))/ar+B(v,y))]+D(v,y) χ(r,y)=±√(C(y)) r+E(y) Using equation (<ref>) we obtain:_rϕ_yϕ+_rχ_yχ+a_r_yϕ-A'a_rϕ=0 -a_yB-tan(√(C)/ar+B)(_yD+aC'/2C-aA')± E'=0{[_yB=±E'/aB(v,y)=±E(y)/a+F(v); and; _yD-aA'+a/2_y[ln C(y)]=0D(v,y)=aA(y)-a/2ln[C(y)]+G(v) ]}Hence, we have: ϕ(v,r,y)=a ln[cos(√(C(y))/ar±E(y)/a+F(v))]+aA(y)-a/2ln[C(y)]+G(v) (<ref>)_yϕ_vϕ+a_y_vϕ-A'a_vϕ=f_ym/r^2+_r(f_ym)/rf_ym/r^2+_r(f_ym)/r=-a^2E'(C'r/2√(C)a±E'/a)-atan(√(C)/ar±E/a+F)(G'C'r/2√(C)a±G'E'/a-aF'C'/2C)-aC'G'/2C For _ym≠0, it is necessary to nullify the quantity _ym/r^2+_r_ym/r which is the factor of the term which includes the coupling function f in the LHS of the above equation. This term cannot be equated with another term, so it must vanish, otherwise the field equation is not consistent. Thus, we get: _ym/r^2+_r_ym/r=0m(v,r,y)=(v,y)/r For the same reason the quantity _r^2m-2_rm/r should also be zero in the RHS of equation (<ref>). Hence, it is: _r^2m-2/r_rm=0m(v,r,y)=κ(v,y)r^3+ξ(v,y) It is clear that equations (<ref>) and (<ref>) cannot be simultaneously true. Consequently, we should reject this sub-case as well, because it leads to an inconsistency.* h(r,y)=C(y)>0∧ a=0∧ b≠0:This sub-case is almost identical to the previous one, but the functions of ϕ and χ are alternated. Here, equation (<ref>) leads to ϕ(v,r,y)=±√(C(y)) r+B(v,y) χ(r,y)=bln[cos(√(C(y))/br+D(y))]+E(y) Following the same steps as in sub-case (ii), we obtain again equations (<ref>) and (<ref>). Therefore, this sub-case is also not capable of providing a viable solution to the localization problem. * h(r,y)=0∧ a≠ 0∧ b≠0:From equation (<ref>) we have: ϕ(v,r,y)=aln[r+B(v,y)]+C(v,y) χ(r,y)=bln[r+D(y)]+E(y)(<ref>)[[(<ref>)](<ref>)_yC-aA'_(v,y)-dependent=-(b/aE'-b^2/aA')r+B/r+D_(v,r,y)-dependent (<ref>)_r(RHS)=0B(v,y)=D(y)Thus: (<ref>)(<ref>)C(v,y)=a^2+b^2/aA(y)-b/aE(y)+F(v)(<ref>)[(<ref>)](<ref>)ϕ(v,r,y)=aln[r+D(y)]+a^2+b^2/aA(y)-b/aE(y)+F(v)Using equations (<ref>) and (<ref>) as they were used in sub-cases (ii) and (iii), we obtain again the equations (<ref>) and (<ref>). Hence, the case 27 is complete excluded. 28){ϕ=ϕ(v,r,y), χ=χ(v,r,y)} Finally, the most general case 28 can be investigated and excluded in exactly the same way as case 27. The only difference is that in this case, we have to replace the functions h(r,y) and C(y) with h(v,r,y) and C(v,y) respectively. CHAPTER: CONCLUSIONS AND DISCUSSION In summary, the study that was preceded in the framework of this thesis had the following structure. In Chapter 1 (Introduction) took place a concise presentation of the biggest mysteries in physics that remain unsolved until these days i.e. the nature of Dark Matter and Dark Energy and the existence of a Unified Theory. The attempts that were made in order to resolve these problems led to various extra-dimensional theories, in which String Theory plays a starring role. The Hierarchy Problem led to the formulation of the Randall-Sundrum models (RS1 and RS2), which were analyzed in detail in Chapter 2. Briefly, RS1 model manages to resolve the Hierarchy Problem by assuming the existence of a compact extra dimension, which is finite and also bounded by two 3-branes. However, in the context of the RS2 model, the extra dimension is allowed to be infinite and therefore the second brane is essentially removed from the model. The astonishing result of the RS2 model is that although we have an infinite extra dimension, the gravity on the remaining 3-brane is effectively 4-dimensional. Although very popular, these models face the problem of the absence of an analytic solution describing a regular, 5-dimensional black hole solution localised close to the brane. Therefore, in Chapter 3, the geometrical framework and the scalar field theory -which were used in the context of the thesis in an attempt to solve black hole localization problem- were presented and the field equations of our theory were derived. Subsequently, these field equations were used in Chapter 4 in order to find a localized black hole solution in the context of a generalized RS2 brane-world model. The main purpose of this thesis was to find a localized 5-dimensional black hole solution close to our 3-brane (or our 4-dimensional universe) by using an RS2-type geometrical background and a scalar field theory which consists of two scalar fields ϕ, χ that interact with each other and they are also non-minimally coupled to gravity. This complicated choice of a scalar field theory was made because simpler types of scalar field theories failed to provide a localized black hole solution. Therefore, we assumed the existence of an extra scalar field χ. Despite the additional degree of freedom that was provided by the second scalar field χ, we were not able to find a viable configuration thus the analytical solution to the localization problem remains still an open problem. Although we were able to exclude mathematically all considered cases, the complexity of the equations did not allow us to formulate a no-go theorem that would exclude altogether the existence of a viable configuration. Our negative result is only one in a series of failed analytical attempts over a period of almost 20 years, and this certainly creates well-founded concerns about the compatibility of brane-world models with the predictions of General Relativity. It is also crucial to indicate that some of the solutions rejected may comprise instead novel black-string solutions, whose study we will undertake in the near future. If this proves to be true, it will refuel the existing debate in the literature regarding the question of why black-string solutions are so much easier to find in the context of brane-world models compared to black-hole solutions. In the context of the scalar field theory that was considered in this thesis, there are some extra coupling functions that is possible to be examined, though they are more complicated: f(ϕ,χ)=a ϕ^2+b χ^2+c ϕ χf(ϕ,χ)=a ϕ^κ+b χ^ f(ϕ,χ)=e^a ϕ+b χExcept for the most complicated case where a≠0∧ b≠0∧ c≠0, it is possible to set a=0 and/or b=0 and/or c=0. The previous analysis for the coupling function f(ϕ,χ)=aϕ+bχ have made clear that the most valuable information about the form of the functions of the scalar fields ϕ and χ emanates from equation (<ref>). The reason is that compared to the other field equations, equation (<ref>) is the simplest one. If we therefore consider a coupling function, which is complicated enough to make the differential equation (<ref>) unsolvable, then we are doomed to give up the attempt to solve the localization problem. Consequently, one should be careful about the choice of the coupling function f(ϕ,χ). If the coupling function is too simple, then probably the localization of a 5-dimensional black hole would not be possible, because the simple cases have already been investigated and excluded. On the other hand, if the coupling function is too complicated, then, even if there is a solution to the problem, we might not be able to find it. After about 20 years of research in the direction of finding a closed-form, analytical 5-dimensional localized braneworld black hole solution, there is still work to do. Fortunately, there are numerical solutions to the problem, thus, the research for an analytical solution is not completely in vain. CHAPTER: BLACK HOLE SOLUTIONS IN GENERAL RELATIVITYSchwarzschild Solution (M≠ 0, Q=0, J=0): ds^2=-(1-r_S/r)(cdt)^2+(1-r_S/r)^-1dr^2+r^2(dθ^2+sin^2θ dφ^2) wherer_S=2GM/c^2 Reissner-Nordström Solution (M≠0, Q≠0, J=0): ds^2=-Δ (cdt)^2+Δ^-1 dr^2+r^2(dθ^2+sin^2θ φ^2) whereΔ=1-r_S/r+r_Q^2/r^2andr_Q^2=GQ^2/4πϵ_0c^4 Kerr Solution (M≠0, Q=0, J≠0): ds^2=-(1-r_S r/ρ^2)(cdt)^2-2 r_S r a sin^2θ/ρ^2cdtdφ+ρ^2/Δdr^2+ρ^2 dθ^2 +sin^2θ/ρ^2[(r^2+a^2)^2-a^2Δsin^2θ]dφ^2 wherea=J/McΔ=r^2-r_S r+a^2ρ^2=r^2+a^2cos^2θThe Kerr-Newman line element (M≠0,Q≠0,J≠0) results from Eq.(<ref>) by replacingr_S r with r_S r-r_Q^2.CHAPTER: GAUSSIAN NORMAL COORDINATESLet us consider a Lorentzian (n+1)-manifold ℳ with coordinates {x^M}={x^0,x^1…, x^n}, where M runs from 0 to n. A (d+1)-submanifold 𝒮 of the original (n+1)-dimensional manifold ℳ with intrinsic coordinates {y^μ}={y^0,y^1,…,y^d} (d<n) can be defined via the following set of n+1 parametric equations: x^M=x^M(y^0,y^1,…,x^d){[ x^0=x^0(y^0,y^1,…,x^d); x^1=x^1(y^0,y^1,…,x^d);⋮; x^n=x^n(y^0,y^1,…,x^d) ]} For d=n-1 we have a hypersurface. In this case, it is possible to define the hypersurface without using the above parametric equations. The way that this can be done, is the following: F(x^0,x^1,…,x^n)=c where c is a constant. In every point p of the hypersurface 𝒮there is a tangent plane T̅_p which can be thought as an n-dimensional subspace of the tangent plane T_p of the (n+1)-dimensional manifold ℳ. It is intuitively clear that there is always an (n+1)-vector n⃗∈ T_p (unique up to scaling) which is orthogonal to all vectors in T̅_p. This vector n⃗ is said to be normal to the hypersurface 𝒮. The unit normal vector n̂ in any point on 𝒮 is given by n̂=n^Me⃗_M=^M F/√(|g^AB_AF_BF|)e⃗_M In order to clarify the unitarity of n̂ one can perform the scalar product of n̂ with itself and substitute e⃗_M·e⃗_N for g_MN. We silently assumed that g_AB n^An^B≠ 0, in the opposite case we have a null-hypersurface and therefore is not possible to normalize the normal vector as we did in Eq.(<ref>). Let us now consider two events on the hypersurface 𝒮 which are separated by the n-vector dr⃗=dx^Ae⃗_A. These events are so close that we can safely assume that the vector dr⃗ lies on the tangent plane T̅_p. Hence, the scalar product between dr⃗ and n̂ is zero. n̂· dr⃗=0(n^Ae⃗_A)·(dx^Be⃗_B)=0 n^Adx^Bg_AB=n^A dx_A=n_A dx^A=0 The Gaussian Normal Coordinates for any non-null hypersurface can be constructed by using the following steps. Firstly, for each point p∈𝒮 we find the unique geodesic curve that passes by the point p and its tangent vector is n̂. Thereafter, we choose a coordinate system {x^0,x^1,…,x^n-1} on 𝒮 and then we characterize each point in a neighborhood of the hypersurface 𝒮 by using these coordinates and the parameter y which is along the geodesic curve that emanates from the point p∈𝒮. Therefore, it is always possible to find a local coordinate system {x^0,x^1,…,x^n,y} in a neighborhood of a point p∈𝒮 where a vector along coordinate y is perpendicular to the hypersurface 𝒮.CHAPTER: LINEARIZED GRAVITYLinearized gravity is simply an approximation which is used to describe weak gravitational fields. The spacetime in the context of this approximation is considered nearly flat, thus, the metric tensor is expressed by the following relation: g_MN=η_MN+h_MN(x^,y) where |h_MN|≪ 1 The use of capital Latin characters indicates that we take into account extra spatial dimensions. In this chapter particularly, we consider a total of (4+1)-dimensions {x^0,x^1,…,x^3,y}. In the calculations that follow we ignore as negligible any term that contains non-linear orders of h_MN. Consequently, the Christoffel symbols are calculated as follows:^L_MN=1/2g^LR(g_MR,N+g_NR,M-g_MN,R) ^L_MN=1/2(η^LR+h^LR)[_N(η_MR+h_MR)+_M(η_NR+h_NR)-_R(η_MN+h_MN)] ^L_MN=1/2η^LR(h_MR,N+h_NR,M-h_MN,R)=1/2(h^L_M,N+h^L_N,M-h_MN^,L)The components of the Riemann tensor can be computed very easily as well.R^L_MRN=^L_MN,R-^L_MR,N+^L_RK^K_MN_h^20-^L_NK^K_MR_h^20R^L_MRN=1/2(h^L_M,NR+h^L_N,MR-h_MN,R^L-h^L_M,RN-h^L_R,MN+h_MR,N^L) R^L_MRN=1/2(h^L_N,MR-h_MN,R^L-h^L_R,MN+h_MR,N^L)Using Eq.(<ref>) and contracting indices L and R we obtain the components of the Ricci tensor.R_MN=1/2(h^L_N,ML-h_MN,L^L-h^L_L,MN+h_ML,N^L) R_MN=1/2(-□ h_MN-h_,MN+h_ML,N^L+h^L_N,ML)whereh≡ h^L_L Subsequently, the Ricci scalar is going to be evaluated, which emanates directly from Eq.(<ref>).R=g^MNR_MN=η^MNR_MN+h^MNR_MN_h^20=1/2(-□ h-□ h+h^ML_,ML+ h^ML_,ML) R=-□ h+h^ML_,MLFinally, the components of the Einstein tensor are given byG_MN=R_MN-1/2R g_MN=R_MN-1/2R η_MNG_MN=1/2(-□ h_MN-h_,MN+h_ML,N^L+h^L_N,ML-η_MN□ h+η_MNh^RL_,RL)CHAPTER: FROM SCHWARZSCHILD TO VAIDYAFirst of all, we clarify that for all the mathematical expressions that are depicted below, the Planck units have been used. The Schwarzschild metric as the static and spherically symmetric solution to Einstein's equations of gravity is given by the following line element ds^2=-(1-2M/r)dt^2+(1-2M/r)^-1dr^2+r^2(dθ^2+sin^2θ dφ^2) Someone could switch to Eddington-Finkelstein coordinates in order to remove the undesirable coordinate singularity of the metric at r=2M. The new coordinates are using the null coordinate v with the following definition. t=v-r-2Mln(r/2M-1)dt=dv-(1-2M/r)^-1dr Subsequently, using equation (<ref>) into (<ref>) the line element takes the form ds^2=-(1-2M/r)dv^2+2dvdr+r^2(dθ^2+sin^2θ dφ^2) If we now extend the mass parameter M from a constant to a function of v we get the Vaidya metric. ds^2=-(1-2M(v)/r)dv^2+2dvdr+r^2(dθ^2+sin^2θ dφ^2)CHAPTER: COMPLETE INFORMATION ABOUT THE 5-D GEOMETRICAL BACKGROUND§ CHRISTOFFEL SYMBOLSChristoffel symbols can be calculated by equation ^K_MN=1/2g^KL(g_ML,N+g_NL,M-g_MN,L) Hence, combining equations (<ref>), (<ref>) and (<ref>), we obtain[box=]equation [^0_00=m-r _r m/r^2^0_04=^0_40=A'^0_22=-r;; ^0_33=-r sin^2θ ^1_00=r^2 _v m-(r-2 m) (r _r m-m)/r^3 ^1_01=^1_10=r _rm-m/r^2;; ^1_04=^1_40=_ym/r^1_14=^1_41=A' ^1_22=2 m-r;; ^1_33=(2 m-r)sin^2θ ^2_12=^2_21=1/r^2_24=^2_42=A';; ^2_33=-sinθcosθ ^3_13=^3_31=1/r ^3_23=^3_32=θ;;^3_34=^3_43=A'^4_00=e^2 A (A' (r-2 m)-_ym)/r ^4_01=^4_10=-e^2 A A';; ^4_22=-r^2 e^2 A A' ^4_33=-r^2A' e^2 A sin^2θ ] § RIEMANN TENSORRiemann tensor's components are defined by R^L_KMN=^L_NK,M-^L_MK,N+^L_MJ^J_NK-^L_NJ^J_MK Thus, having in our disposal the Christoffel symbols from equation (<ref>) it is possible to compute the components of the Riemann tensor, but also one would need an eternity to make calculations. Therefore, it will be presented a less time-consuming method that provides the Riemann tensor's components.First and foremost, the main ingredients of this method are the matrices _M and B_MN. * The _M matrices are resulting from Christoffel symbols ^K_ML as follows. The component ^K_ML is defined to be the element of the K-th row and L-th column of the matrix _M. So, there are 5 matrices _M of size 5× 5.* The B_MN matrices are resulting from the components of Riemann tensor R^L_KMN. The component R^L_KMN is defined to be the element of the L-th row and K-th column of the matrix B_MN. Thus, there are 25 matrices B_MN of size 5× 5.It is now obvious that making use of the previous definitions, equation (<ref>) can be written in the following form. B_MN=_N,M-_M,N+_M_N-_N_M It is necessary to mention that the matrices B_MN have the useful property B_MN=-B_NM. This property arise from the antisymmetry of Riemann tensor R^L_KMN=-R^L_KNM. Therefore, this property ensures us that B_MM=0. We can easily extract the Γ_M matrices from equation (<ref>). Hence, we have _0=( [ m-r _rm/r^2 0 0 0A'; r^2 _vm-(r-2 m) (r _rm-m)/r^3 r _rm-m/r^2 0 0 _ym/r; 0 0 0 0 0; 0 0 0 0 0; e^2 A[(r-2 m) A'-_ym]/r -e^2 A A' 0 0 0; ]) _1=( [ 0 0 0 0 0; r _rm-m/r^2 0 0 0A'; 0 0 1/r 0 0; 0 0 0 1/r 0; -e^2 A A' 0 0 0 0; ]) _2=( [ 0 0-r 0 0; 0 0 2 m-r 0 0; 0 1/r 0 0A'; 0 0 0 θ 0; 0 0 -e^2 A r^2 A' 0 0; ]) _3=( [000 -r sin ^2θ0;000(2 m-r) sin ^2θ0;000-cosθsinθ0;01/rθ0 A';000 -e^2 A r^2 sin ^2θA'0;]) _4=( [A' 0 0 0 0; _ym/rA' 0 0 0; 0 0A' 0 0; 0 0 0A' 0; 0 0 0 0 0; ])The combination of equations (<ref>)-(<ref>) leads to the following B_MN matrices. 𝐁_01=-𝐁_10 B_01=_1,0-_0,1+_0_1-_1_0B_01=( [ -e^2 A A'^2+2 (m-r _rm)/r^3+_r^2m/r 0 0 0 0; -(r-2 m) [r (e^2 A r^2 A'^2+2 _rm-r _r^2m)-2 m]/r^4 e^2 AA'^2-r^2_r^2m -2 r_rm +2 m/r^3 0 0 _ym-r _r_ym/r^2; 0 0 0 0 0; 0 0 0 0 0;e^2 A(r _r_ym-_ym)/r^2 0 0 0 0; ]) 𝐁_02=-𝐁_20 B_02=_2,0-_0,2+_0_2-_2_0B_02=( [ 0 0_rm-e^2 A A'^2 r^3+m/r 0 0; 0 0_vm-e^2 A r A' _ym 0 0; -e^2 A A' r^3[(r-2 m) A'-_ym] -r^2_vm +(r-2 m) (r_rm-m)/r^4e^2 A A'^2 r^3-r_rm +m/r^3 0 0-_ym/r^2; 0 0 0 0 0; 0 0e^2 A_ym 0 0; ]) 𝐁_03=-𝐁_30 B_03=_3,0-_0,3+_0_3-_3_0B_03=( [ 0 0 0-sin ^2θ(e^2 A A'^2 r^3-r_rm +m)/r 0; 0 0 0-sin ^2θ(e^2 A r A' _ym-_vm) 0; 0 0 0 0 0; -e^2 A A' r^3[(r-2 m) A'-_ym] -r^2_vm +(r-2 m) (r_rm-m)/r^4e^2 A A'^2 r^3-r_rm +m/r^3 0 0-_ym/r^2; 0 0 0 e^2 Asin ^2θ_ym 0; ]) 𝐁_04=-𝐁_40 B_04=_4,0-_0,4+_0_4-_4_0B_04=( [r _r_ym-_ym/r^2000 -A'^2-A”;(r-2 m) (r _r_ym-_ym)/r^3_ym-r _r_ym/r^200 -2A' _ym+_y^2m/r;00000;00000; e^2 A[-(r-2 m) (A'^2+A”)+2 A' _ym+_y^2m]/r e^2 A(A'^2+A”)000;])𝐁_12=-𝐁_21 B_12=_2,1-_1,2+_1_2-_2_1=( [00000;00 _rm-e^2 A A'^2 r^3+m/r00; e^2 A A'^2 r^3-r_rm +m/r^30000;00000;00000;]) 𝐁_13=-𝐁_31 B_13=_3,1-_1,3+_1_3-_3_1=( [00000;000 -sin ^2θ(e^2 A A'^2 r^3-r_rm +m)/r0;00000; e^2 A A'^2 r^3-r_rm +m/r^30000;00000;]) 𝐁_14=-𝐁_41 B_14=_4,1-_1,4+_1_4-_4_1=( [00000;0000 -A'^2-A”;00000;00000; e^2 A(A'^2+A”)0000;]) 𝐁_23=-𝐁_32 B_23=_3,2-_2,3+_2_3-_3_2=( [ 0 0 0 0 0; 0 0 0 0 0; 0 0 0 sin ^2θ(2 m-e^2 A r^3 A'^2)/r 0; 0 0e^2 A r^2 A'^2-2 m/r 0 0; 0 0 0 0 0; ]) 𝐁_24=-𝐁_42 B_24=_4,2-_2,4+_2_4-_4_2=( [ 0 0 0 0 0; 0 0-_ym 0 0; _ym/r^2 0 0 0-A'^2-A”; 0 0 0 0 0; 0 0 e^2 A r^2 (A'^2+A”) 0 0; ]) 𝐁_34=-𝐁_43 B_34=_4,3-_3,4+_3_4-_4_3=( [00000;000-sin ^2θ_ym0;00000;_ym/r^2000 -A'^2-A”;000 e^2 A r^2 sin ^2θ(A'^2+A”)0;]) Finally, knowing the matrices B_MN from equations (<ref>)-(<ref>) and using the fact that the L-th row and K-th column of the matrix B_MN gives the R^L_KMN component of the Riemann tensor, it is quite easy to extract the components of the Riemann tensor, all non-zero components are depicted below.[box=]equation [R^0_001=-R^0_010=2 (m-r _rm)/r^3+^2_rm/r-e^2 A A'^2 R^0_004=-R^0_040=r _r_ym-_ym/r^2; ;R^0_202=-R^0_220=_rm-m+r^3 e^2 A A'^2/rR^0_303=-R^0_330=-sin^2θ(-r _rm+m+r^3 e^2 A A'^2)/r; ;R^0_404=-R^0_440=-A”(y)-A'(y)^2 R^1_004=-R^1_040=(r-2 m) (r _r_ym-_ym)/r^3; ; R^1_001=-R^1_010=-(r-2 m) [r (2 _rm-r ^2_rm+r^2 e^2 A A'^2)-2 m]/r^4R^1_101=-R^1_110=e^2 A A'^2-r^2 _r^2m-2 r _rm+2 m/r^3; ; R^1_104=-R^1_140=_ym-r _r_ym/r^2R^1_202=-R^1_220=_vm-r e^2 A A' _ym; ;R^1_212=-R^1_221=_rm-m+r^3 e^2 A A'^2/rR^1_224=-R^1_242=-_ym; ; R^1_303=-R^1_330=-sin^2θ(r e^2 A A' _ym-_vm)R^1_313=-R^1_331=-sin^2θ(-r _rm+m+r^3 e^2 A A'^2)/r; ;R^1_334=-R^1_343=-sin^2θ_ym R^1_401=-R^1_410=_ym-r _r_ym/r^2; ; R^1_404=-R^1_440=-2 A' _ym+^2_ym/rR^1_414=-R^1_441=-A”-A'^2; ; R^2_002=-R^2_020=-r^3e^2 A A' [A' (r-2 m)-_ym]-r^2 _vm+(r-2 m) (r _rm-m)/r^4 R^2_012=-R^2_021=-r _rm+m+r^3 e^2 A A'^2/r^3; ; R^2_024=-R^2_042=_ym/r^2 R^2_102=-R^2_120=-r _rm+m+r^3 e^2 A A'^2/r^3; ;R^2_323=-R^2_332=sin^2θ(2 m-r^3 e^2 A A'^2)/rR^2_402=-R^2_420=-_ym/r^2; ;R^2_424=-R^2_442=-A”-A'^2 R^3_013=-R^3_031=-r _rm+m+r^3 e^2 A A'^2/r^3; ;R^3_003=-R^3_030=-r^3e^2 A A' [A' (r-2m)-_ym]-r^2 _vm+(r-2 m) (r _rm-m)/r^4 R^3_034=-R^3_043=_ym/r^2; ; R^3_103=-R^3_130=-r _rm+m+r^3 e^2 A A'^2/r^3R^3_223=-R^3_232=r^2 e^2 A A'^2-2 m/r; ;R^3_403=-R^3_430=-_ym/r^2R^3_434=-R^3_443=-A”-A'^2; ; R^4_001=-R^4_010=e^2 A (r _r_ym-_ym)/r^2 R^4_014=-R^4_041=e^2 A (A”+A'^2); ;R^4_004=-R^4_040=e^2 A [2 A'_ym+(A”+A'^2) (2m-r)+_y^2m]/r R^4_104=-R^4_140=e^2 A (A”+A'^2); ; R^4_202=-R^4_220=e^2 A _ym R^4_224=-R^4_242=r^2 e^2 A (A”+A'^2); ; R^4_303=-R^4_330=e^2 A sin^2θ_ym R^4_334=-R^4_343=r^2 e^2 A sin^2θ(A”+A'^2) ]§ RIEMANN SCALARRiemann scalar is defined as R^ABCDR_ABCD Therefore, it is necessary to calculate the four-times contravariant and four-times covariant components of Riemann tensor in order to be able to execute the above contraction. Contravariant and covariant components of Riemann tensor have some very useful properties that reduce the number of independent components. Hence, we present these properties below in order to use them afterwards. * R^ABCD=-R^BACD=-R^ABDC* R^ABCD=R^CDAB* R_ABCD=-R_BACD=-R_ABDC* R_ABCD=R_CDABKnowing the components R^L_KMN of the Riemann tensor, the four-times contravariant components are given by the equation R^ABCD=R^A_KMNg^KBg^MCg^ND and the non-zero four-times contravariant components are [ R^0101=-R^1001=-R^0110=R^1010=e^-6 A(e^2 A A'^2-r^2 _r^2m-2 r _rm+2 m/r^3); ;R^0114=-R^1014=-R^0141=R^1401=-R^4101=-R^1410=R^1041=R^4110=e^-4 A(r _r_ym-_ym)/r^2; ; R^0212=-R^2012=-R^0221=R^1202=-R^2102=-R^1220=R^2021=R^2120=-e^-6 A(-r _rm+m+r^3 e^2 A A'^2)/r^5; ; R^0313=-R^3013=-R^0331=R^1303=-R^3103=-R^1330=R^3031=R^3130=-e^-6 A ^2θ(-r _rm+m+r^3 e^2 A A'^2)/r^5; ; R^0414=-R^4014=-R^0441=R^1404=-R^4104=-R^1440=R^4041=R^4140=-e^-2 A(A”+A'^2); ; R^1212=-R^2112=-R^1221=R^2121=e^-6 A[(r-2 m) (_rm-m+r^3 e^2 A A'^2/r)+r (_vm-r e^2 A A'_ym)]/r^5; ; R^1224=-R^2124=-R^1242=R^2412=-R^4212=-R^2421=R^2142=R^4241=-e^-4 A_ym/r^4; ; R^1313=-R^3113=-R^1331=R^3131=e^-6 A ^2θ[r^2(_vm-r e^2 A A' _ym)-(r-2 m) (-r _rm+m+r^3 e^2A A'^2)]/r^6; ; R^1334=-R^3134=-R^1343=R^3413=-R^4313=-R^3431=R^3143=R^4331=-e^-4 A ^2θ_ym/r^4; ;R^2323=-R^3223=-R^2332=R^3232=e^-6 A ^2θ(2 m-r^3 e^2 AA'^2)/r^7; ; R^2424=-R^4224=-R^2442=R^4242=-e^-2 A(A”+A'^2)/r^2; ; R^3434=-R^4334=-R^3443=R^4343=-e^-2 A ^2θ(A”+A'^2)/r^2; ; R^1414=-R^4114=-R^1441=R^4141=-e^-2 A[2 A' _ym+(A”+A'^2) (r-2m)+_y^2m]/r; ;] The four-times covariant components are given by the equation R_ABCD=g_ALR^L_BCD and the non-zero four-times covariant components are [R_0101=-R_1001=-R_0110=R_1010=e^4 A A'^2-e^2 A[r (r_r^2m-2 _rm)+2 m]/r^3;;R_0104=-R_1004=-R_0140=R_0401=-R_4001=-R_0410=R_1040=R_4010=e^2 A(_ym-r _r_ym)/r^2;;R_0202=-R_2002=-R_0220=R_2020=e^2 A[(r-2 m) (-r _rm+m+r^3 e^2 A A'^2)/r^2-re^2 A A' _ym+_vm];; R_0212=-R_2012=-R_0221=R_1202=-R_2102=-R_1220=R_2021=R_2120=e^2 A(_rm-m+r^3 e^2 A A'^2/r);; R_0224=-R_2024=-R_0242=R_2402=-R_4202=-R_2420=R_2042=R_4220=-e^2 A_ym;; R_0303=-R_3003=-R_0330=R_3030=e^2 Asin ^2θ[(r-2 m) (-r _rm+m+r^3 e^2 A A'^2)/r^2-re^2 A A' _ym+_vm];; R_0313=-R_3013=-R_0331=R_1303=-R_3103=-R_1330=R_3031=R_3130=-e^2 Asin ^2θ(-r_rm+m+r^3 e^2 A A'^2)/r;;R_0334=-R_3034=-R_0343=R_3403=-R_4303=-R_3430=R_3043=R_4330=-e^2 Asin ^2θ_ym;; R_0404=-R_4004=-R_0440=R_4040=e^2 A[-2 A'_ym+(A”+A'^2) (r-2 m)-_y^2m]/r;; R_0414=-R_4014=-R_0441=R_1404=-R_4104=-R_1440=R_4041=R_4140=-e^2 A(A”+A'^2);;R_2323=-R_3223=-R_2332=R_3232=r e^2 Asin ^2θ(2 m-r^3 e^2 A A'^2);;R_2424=-R_4224=-R_2442=R_4242=-r^2e^2 A(A”+A'^2);; R_3434=-R_4334=-R_3443=R_4343=-r^2e^2 Asin ^2θ(A”+A'^2);; ] Possessing all non-zero contravariant and covariant components of Riemann tensor from equations (<ref>) and (<ref>) respectively, we are now able to calculate the Riemann scalar. R^ABCDR_ABCD = 4R^0101R_0101+8R^0212R_0212+8R^0313R_0313+8R^0414R_0414 +4R^2323R_2323+4R^2424R_2424+4R^3434R_3434 All the other combinations of contraction of the indices ABCD nullify either R^ABCD or R_ABCD. [box=]equation [ R^ABCDR_ABCD=40A'^4+32A'^2A”+16A”^2+48e^-4Am^2/r^6-8e^-2AA'^2/r(_r^2m+2_rm/r);; +4e^-4A/r^2[(_r^2m)^2+4m/r^2(_r^2m-4_rm/r)-4_rm_r^2m/r+8(_r m)^2/r] ]§ RICCI TENSOR AND RICCI SCALARSThe two-times covariant components of Ricci tensor is defined by R_MN=R^L_MLN=R^0_M0N+R^1_M1N+R^2_M2N+R^3_M3N+R^4_M4N Subsequently, the non-zero components of the Ricci tensor are [box=]equation [ R_00=e^2 A r [(r-2 m) (4 A'^2+A”)-4 A' _ym-_y^2m]-(r-2 m)_r^2m+2 _vm/r^2; ;R_01=R_10=_r^2m/r-e^2 A (4 A'^2+A”); ;R_04=R_40=_ym+r _r_ym/r^2; ; R_22=2 _rm-e^2 A r^2 (4 A'^2+A”); ;R_33= sin^2θ[2 _rm-e^2 A r^2 (4 A'^2+A”)]; ;R_44=-4 (A'^2+A”) ] There are two Ricci scalars that one can evaluate in order to extract information for the curvature of spacetime. The first one is defined as R=R_MNg^MN and the second one is defined as R_MNR^MN. It is obvious that for the evaluation of the latter scalar the two-times contravariant components of Ricci tensor are necessary. The following table depicts all the non-zero contravariant components of Ricci tensor. The evaluation is made by the following equation R^MN=R_ABg^AMg^BN and the in between analytical actions are skipped here as well. Therefore, we have [box=]equation [ R^01=R^10=e^-4 A [_r^2m/r-e^2 A (4 A'^2+A”)]; ; R^11=e^-4 A [-e^2 A r ((r-2m) (4 A'^2+A”)+4 A'_ym+_y^2m)+(r-2 m) _r^2m+2_vm]/r^2; ; R^14=R^41=e^-2 A (_ym+r _r_ym)/r^2; ;R^22=e^-4 A [2 _rm-e^2 A r^2 (4 A'^2+A”)]/r^4; ; R^33=e^-4 A ^2θ[2 _rm-e^2 A r^2 (4 A'^2+A”)]/r^4; ; R^44= -4 (A'^2+A”) ] We now have everything that is needed for the evaluation of the two aforementioned Ricci scalars. [box=]equation R=R_MNg^MN=-20 A'^2-8 A”+2 e^-2 A/r (_r^2m+2 _rm/r)[box=]equation R_MNR^MN=80A'^2+64A'^2 A”+20A”^2-4 e^-2 A/r(_r^2m+2_rm/r)(4A'^2+A”)+ 2e^-4A/r^2[(_r^2m)^2+4 (_rm)^2/r^2] § EINSTEIN TENSOREinstein tensor G_MN is defined through Ricci tensor R_MN and Ricci scalar R as follows G_MN=R_MN-1/2g_MNRThe combination of equations (<ref>),(<ref>) and (<ref>) can be done easily, thus, after a bit of algebra we are led to the following non-zero components of Einstein tensor. [box=]equation [ G_00=-e^2 A [3 (r-2 m) (2 A'^2+A”)+4 A' _ym+_y^2m] r^2+2 r_vm +2(r-2 m)_r m/r^3;; G_01=G_10=3 e^2 A (2 A'^2+A”)-2 _rm/r^2;; G_04=G_40=_ym+r_r_y m/r^2;;G_22=r [3 e^2 A r (2 A'^2+A”)-_r^2m];;G_33=r sin^2θ[3 e^2 A r (2 A'^2+A”)-_r^2m];;G_44=6 A'^2-e^-2 A (2 _rm+r_r^2 m)/r^2 ]The mixed components of Einstein tensor G^M_N are given by G^M_N=g^MAG_AN Equations (<ref>),(<ref>) and (<ref>) yield to [box=]equation [ G^0_0=6 A'^2+3 A”-2 e^-2 A _rm/r^2; ; G^1_0=-4 r A' _ym+r _y^2m-2 e^-2 A _vm/r^2; ; G^1_1=6 A'^2+3 A”-2 e^-2 A_r m/r^2; ; G^1_4=e^-2 A (_ym+r _r_ym)/r^2; ; G^2_2=6 A'^2+3 A”-e^-2 A_r^2 m/r; ; G^3_3=6 A'^2+3 A”-e^-2 A _r^2m/r; ;G^4_0=_ym+r _r_ym/r^2; ;G^4_4=6 A'^2-e^-2 A (2 _rm+r_r^2 m)/r^2 ] § MATHEMATICA CODEThe mathematica code that is used to verify all the above calculations is illustrated below.Definition of coordinates and metric tensorn=5;coord={v,r,θ ,φ ,y};g={{-Exp[2A[y]](1-2m[v,r,y]/r),Exp[2A[y]],0,0,0},{Exp[2A[y]],0,0,0,0}, {0,0,Exp[2A[y]]*r^∧2,0,0},{0,0,0,Exp[2A[y]]*r^∧2*Sin[θ ]^∧2,0},{0,0,0,0,1}};StringJoin[Characters[(g_MN)=]]MatrixForm[g]invg=Simplify[Inverse[g]];StringJoin[Characters[(g^MN)=]]MatrixForm[invg]Christoffel Symbols: ^𝐋_𝐌𝐍christoffel=FullSimplify[Table[(1/2)*Sum[(invg[[λ ,ρ ]])*(D[g[[μ ,ρ ]],coord[[ν ]]] +D[g[[ν ,ρ ]],coord[[μ ]]]-D[g[[μ ,ν ]],coord[[ρ ]]]),{ρ ,1,n}],{λ ,1,n},{μ ,1,n},{ν ,1,n}]];chr=Table[If[UnsameQ[christoffel[[λ ,μ ,ν ]],0],{ToString[Γ [λ -1,μ -1,ν -1]], christoffel[[λ ,μ ,ν ]]}],{λ ,1,n},{μ ,1,n},{ν ,1,n}];TableForm[Partition[DeleteCases[Flatten[chr],Null],2],TableSpacing→{2,2}]Riemann Tensor's Components:𝐑^𝐊_𝐋𝐌𝐍riemann=FullSimplify[Table[D[christoffel[[α ,μ ,σ ]],coord[[ρ ]]]-D[christoffel[[α ,μ ,ρ ]],coord[[σ ]]] +Sum[christoffel[[α ,ρ ,λ ]]*christoffel[[λ ,μ ,σ ]],{λ ,1,n}]-Sum[christoffel[[α ,σ ,λ ]] *christoffel[[λ ,μ ,ρ ]],{λ ,1,n}],{α ,1,n},{μ ,1,n},{ρ ,1,n},{σ ,1,n}]];rie=Table[If[UnsameQ[riemann[[α ,μ ,ρ ,σ ]],0],{ToString[R[α -1,μ -1,ρ -1,σ -1]], riemann[[α ,μ ,ρ ,σ ]]}],{α ,1,n},{μ ,1,n},{ρ ,1,n},{σ ,1,n}];TableForm[Partition[DeleteCases[Flatten[rie],Null],2],TableSpacing→{2,2}]Contravariant Components of Riemann Tensor:𝐑^𝐀𝐁𝐂𝐃riemanncon=FullSimplify[Table[Sum[Sum[Sum[riemann[[μ ,α ,β ,γ ]]*invg[[α ,ν ]],{α ,1,n}]* invg[[β ,κ ]],{β ,1,n}]*invg[[γ ,λ ]],{γ ,1,n}],{μ ,1,n},{ν ,1,n},{κ ,1,n},{λ ,1,n}]];riecon=Table[If[UnsameQ[riemanncon[[α ,μ ,ρ ,σ ]],0],{ToString[Rcon[α -1,μ -1,ρ -1,σ -1]], riemanncon[[α ,μ ,ρ ,σ ]]}],{α ,1,n},{μ ,1,n},{ρ ,1,n},{σ ,1,n}];TableForm[Partition[DeleteCases[Flatten[riecon],Null],2],TableSpacing→{2,2}]Covariant Components of Riemann Tensor:𝐑_𝐀𝐁𝐂𝐃riemanncov=FullSimplify[Table[Sum[g[[μ ,δ ]]*riemann[[δ ,ν ,κ ,λ ]], {δ ,1,n}],{μ ,1,n},{ν ,1,n},{κ ,1,n},{λ ,1,n}]];riecov=Table[If[UnsameQ[riemanncov[[α ,μ ,ρ ,σ ]],0],{ToString[Rcov[α -1,μ -1,ρ -1,σ -1]], riemanncov[[α ,μ ,ρ ,σ ]]}],{α ,1,n},{μ ,1,n},{ρ ,1,n},{σ ,1,n}];TableForm[Partition[DeleteCases[Flatten[riecov],Null],2],TableSpacing→{2,2}]Covariant Components of Ricci Tensor:𝐑_𝐌𝐍ricci=FullSimplify[Table[Sum[riemann[[μ ,α ,μ ,β ]],{μ ,1,n}],{α ,1,n},{β ,1,n}]];StringJoin[Characters[(R_MN)=]]MatrixForm[ricci]Contravariant Components of Ricci Tensor:𝐑^𝐌𝐍riccicon=FullSimplify[Table[Sum[ricci[[α ,β ]]*invg[[α ,μ ]]*invg[[β ,ν ]],{α ,1,n},{β ,1,n}],{μ ,1,n}, {ν ,1,n}]];StringJoin[Characters[(R^MN)=]]MatrixForm[riccicon]Ricci and Riemann Scalarsscalarricci=FullSimplify[Sum[invg[[μ ,ν ]]*ricci[[μ ,ν ]],{μ ,1,n},{ν ,1,n}]];StringJoin[Characters[R_MNg^MN=]].scalarricciscalarricci2=FullSimplify[Sum[ricci[[μ ,ν ]]*riccicon[[μ ,ν ]],{μ ,1,n},{ν ,1,n}]];StringJoin[Characters[R_MNR^MN=]] .scalarricci2StringJoin[Characters[R_ABCDR^ABCD=]] .FullSimplify[Sum[riemanncov[[μ ,ν ,κ ,λ ]]* riemanncon[[μ ,ν ,κ ,λ ]],{μ ,1,n},{ν ,1,n},{κ ,1,n},{λ ,1,n}]]Covariant Components of Einstein Tensor: 𝐆_𝐌𝐍einstein=FullSimplify[Table[ricci[[μ ,ν ]]-(1/2)g[[μ ,ν ]] *scalarricci,{μ ,1,n},{ν ,1,n}]];StringJoin[Characters[(G_MN)=]]MatrixForm[einstein]Mixed Components of Einstein Tensor: 𝐆^𝐌_𝐍einsteinUD=FullSimplify[Table[Sum[invg[[μ ,λ ]]*einstein[[λ ,ν ]],{λ ,1,n}],{μ ,1,n},{ν ,1,n}]];StringJoin[Characters[(G_N^M)=]]MatrixForm[einsteinUD] CHAPTER: NON-MINIMAL COUPLING AND VARIATION OF THE ACTION§ VARIATION WITH RESPECT TO THE METRIC TENSOR We consider the following general action for a non-minimally coupled scalar field. S=∫ d^4x√(-g)[f(Φ)/2κ R-1/2_μΦ^μΦ-V(Φ)-Λ_B]+∫ d^4x√(-g) _m where _m describes matter and/or radiation. It is obvious from the above action that we are restricted in a 4-D spacetime. Every calculation and proof in this section is going to be done in 4-D spacetime. However, the generalization for extra spatial dimensions is instantaneous, the only thing that changes in the final field equations are the indices. For a 4-D spacetime the indices are Greek letters while for (4+n)-D spacetime the indices are capital Latin letters.In order to deduce the field equations of this general theory, we apply the principle of least action to Eq.(<ref>). We vary the action with respect to the metric tensor. Thus, we have:S=0 =∫ d^4x{(√(-g))[f(Φ)/2κ R-1/2_μΦ^μΦ-V(Φ)-Λ_B]+√(-g)[f(Φ)/2κ R-1/2(_μΦ^μΦ)]}+∫ d^4x (√(-g) _m)0 =∫ d^4x{(√(-g))[f(Φ)/2κ R-1/2_μΦ^μΦ-V(Φ)-Λ_B]+√(-g)[f(Φ)/2κ R-1/2(_μΦ_νΦ) g^μν]}+∫ d^4x (√(-g) _m)We will calculate one by one the varying terms of the previous equation. First of all, we will prove the Jacobi's formulad/dt{det[A(t)]}=tr{adj[A(t)]dA(t)/dt} which is necessary in order to derive the desirable field equations. We will prove Eq.(<ref>) in two steps. At first, we prove a preliminary lemma.Let A and B be a pair of square matrices of the same dimension n. Then∑_i=1^n ∑_j=1^n A_ijB_ij=tr(A^TB) (AB)_jk =∑_i=1^n A_jiB_ik (A^TB)_jk =∑_i=1^n A^T_jiB_ik=∑_i=1^n A_ijB_ik tr(A^TB) =∑_j=1^n (A^TB)_jj=∑_j=1^n∑_i=1^n A_ijB_ij=∑_i=1^n∑_j=1^nA_ijB_ij (Jacobi's formula):d[det(A)]=tr[adj(A)dA]Laplace's formula (or cofactor expansion) for the determinant of a matrix A can be stated asdet(A)=∑^n_j=1 A_ij(-1)^(i+j)M_ij=∑^n_j=1 A_ijC_ij=∑^n_j=1 A_ij[adj(A)]_ji=∑^n_j=1 A_ij[adj^T(A)]_ij where M_ij is the i, j minor matrix of A, that is, the determinant of the (n-1)×(n-1) matrix that results from deleting the i-th row and j-th column of A. The summation is performed over some arbitrary row i of the matrix. The i, j cofactor of A is the scalar C_ij defined by C_ij≡(-1)^i+jM_ij. We note as well that the adjugate matrix of A is the transpose of the cofactor matrix C of A, which means that adj(A)≡ C^T [adj(A)]_ij=C_ji. The determinant of A can also be considered to be a function of elements of A:det(A)=F(A_11,A_12,...,A_21,...,A_nn)so that, by the chain rule, its differential isd[det(A)]=∑_i,j∂ F/∂ A_ijdA_ij=∑_i,j∂ det(A)/∂ A_ijdA_ij ∂ det(A)/∂ A_ij =∂/∂ A_ij(∑_k A_ik[adj^T(A)]_ik)=∑_k∂(A_ik[adj^T(A)]_ik)/∂ A_ij=∑_k∂ A_ik/∂ A_ij__kj[adj^T(A)]_ik+∑_k A_ik∂ [adj^T(A)]_ik/∂ A_ij_=0=∑_kδ_kj[adj^T(A)]_ik=[adj^T(A)]_ijHence, we haved[det(A)]=∑_i,j∂ det(A)/∂ A_ijdA_ij=∑_i,j [adj^T(A)]_ijdA_ij=tr[adj^TdA]=tr[adj(A)dA]Consequently, Eq.(<ref>) is proved.We are now ready to calculate the variations of Eq.(<ref>). * δ (√(-g)):δ (√(-g))=1/21/√(-g)δ (-g)=-1/21/√(-g)δ g It is important to explain the following notation g≡ det(g_μν) where (g_μν) constitutes the matrix which depicts the components of metric tensor g_μν. From equations (<ref>) and (<ref>) we obtainδ g=δ[det(g_μν)]=∑_,β[adj^T(g_μν)]_β δ g_βIn matrix calculus it is well known that A^-1=1/det(A)adj(A), hence we have(g_μν)^-1=(g^μν)=1/det(g_μν)adj(g_μν)adj(g_μν)=g(g^μν)=adj^T(g_μν)⇒ [adj^T(g_μν)]_β=g g^βThus, from Eq.(<ref>) and Eq.(<ref>) we getδ g=∑_,βg g^βδ g_β⇒δ g=g g^μνδ g_μνIt also obvious that δ(g_μνg^νλ) = δ(δ^λ_μ)=0 ⇒ δ(g_μν)g^νλ = -g_μνδ (g^νλ)⇒δ g^ρσ=-g^ρμg^σνδ g_μν ,δ g_αβ=-g_αμg_βνδ g^μν Combining now equations (<ref>) and (<ref>) we haveδ g=g g^μνδ g_μν=-g g_αβδ g^αβTherefore, we can now easily evaluate the quantity (√(-g)). δ (√(-g))=-1/21/√(-g)g g^μνδ g_μν=1/21/√(-g)g g_αβδ g^αβδ (√(-g))=√(-g)/2 g^μνδ g_μν=-√(-g)/2 g_αβδ g^αβ * δ R:In order to calculate δ R we need to prove first the Palatini's identity:δ R_μν=(δΓ^λ_μν)_;λ-(δΓ^λ_λμ)_;ν=∇_(δΓ^λ_μν)-∇_ν(δΓ^λ_λμ)R_μν=Γ^λ_μν ,λ-Γ^λ_μλ ,ν+Γ^α_μνΓ^λ_αλ-Γ^λ_ανΓ^α_μλ⇒ δ R_μν=δΓ^λ_μν ,λ -δΓ^λ_μλ,ν+δ (Γ^α_μν)Γ^λ_αλ+Γ^α_μνδ (Γ^λ_αλ)-δ (Γ^λ_αν)Γ^α_μλ-Γ^λ_ανδ (Γ^α_μλ)Christoffel symbols of first kid _μν is defined by Γ_μνσ ≡g_μλΓ^λ_νσ=1/2g_μg^ρ(g_νρ,+g_ρ,ν-g_ν,ρ)=1/2^ρ_μ(g_νρ,+g_ρ,ν-g_ν,ρ)= 1/2(g_μν ,σ+g_μσ ,ν-g_νσ ,μ) where we have used the expression of Christoffel symbols of second kid (or simply Christoffel symbols) which are well known. Using Eq.(<ref>) we haveδ(Γ^λ_νσ) =δ(g^λμΓ_μνσ)=δ(g^λμ)Γ_μνσ+g^λμδΓ_μνσ=-g^λαg^μβδ (g_αβ)Γ_μνσ+g^λμδΓ_μνσ=-g^λαδ (g_αβ)Γ^β_νσ+g^λμ δ[1/2(g_μν ,σ+g_μσ ,ν-g_νσ ,μ)] =-g^λμδ (g_μβ)Γ^β_νσ+g^λμ1/2(δ g_μν ,σ+δ g_μσ ,ν-δ g_νσ ,μ) =1/2g^λμ(δ g_μν ,σ+δ g_μσ ,ν-δ g_νσ ,μ-2Γ^β_νσδ g_μβ) =1/2g^λμ(δ g_μν ,σ+δ g_μσ ,ν-δ g_νσ ,μ-2Γ^β_νσδ g_μβ-Γ^β_νμδ g_σβ+Γ^β_νμδ g_σβ-Γ^β_σμδ g_νβ+Γ^β_σμδ g_νβ) =1/2g^λμ[(δ g_μν ,σ-Γ^β_σμδ g_νβ-Γ^β_νσδ g_μβ)+(δ g_μσ ,ν-Γ^β_νμδ g_σβ-Γ^β_νσδ g_μβ)-(δ g_νσ ,μ-Γ^β_νμδ g_σβ-Γ^β_σμδ g_νβ)] =1/2g^μ(∇_ g_μν+∇_ν g_μ-∇_μ g_ν) It is clear that elements like g_μν and g_μν; are tensors. Thus, from Eq.(<ref>) we can easily deduce that ^_μν is also a tensor, because ^_μν is expressed as a linear combination of tensors. Moreover, the quantity (^_μν)_;-(^_μ)_;ν constitutes a tensor. Performing the expansion of the expression (^_μν)_;-(^_μ)_;ν we obtain(δΓ^λ_μν)_;λ-(δΓ^λ_μλ)_;ν= +(δΓ^λ_μν ,λ+Γ^λ_λρδΓ^ρ_μν-Γ^ρ_μλδΓ^λ_ρν-Γ^ρ_νλδΓ^λ_ρμ) -(δΓ^λ_μλ ,ν+Γ^λ_νρδΓ^ρ_μλ-Γ^ρ_μνδΓ^λ_ρλ-Γ^ρ_νλδΓ^λ_ρμ) = +(δΓ^λ_μν ,λ+Γ^λ_λρδΓ^ρ_μν-Γ^ρ_μλδΓ^λ_ρν) -(δΓ^λ_μλ ,ν+Γ^λ_νρδΓ^ρ_μλ-Γ^ρ_μνδΓ^λ_ρλ)The right hand side of Eq.(<ref>) is identical to the right hand side of Eq.(<ref>), so the left hand sides should be equal to each other as well. Hence, we derived the desirable Palatini's identity.The calculation of the quantity R using the Eq.(<ref>) is very simple.δ R =δ (g^μνR_μν)=(δ g^μν)R_μν+g^μνδ R_μν=(δ g^μν)R_μν+g^μν(∇_δΓ^λ_μν-∇_νδΓ^λ_μλ)We have shown that δΓ^λ_νσ =1/2g^μ(∇_ g_μν+∇_ν g_μ-∇_μ g_ν)=1/2g^ρ(∇_ g_ρν+∇_ν g_ρ-∇_ρ g_ν)Using Eq.(<ref>) we rename the indexto μ. Furthermore, using the symmetry ^_νμ=^_μν we get δΓ^λ_μν=1/2g^ρ(∇_μ g_ρν+∇_ν g_ρμ-∇_ρ g_νμ) Contracting the indexwith ν we have δΓ^λ_μ=δΓ^λ_μ = 1/2g^ρ(∇_μ g_ρ+∇_ g_ρμ-∇_ρ g_μ)= 1/2g^ρ∇_μ g_ρ+1/2(g^ρ∇_ g_ρμ-g^ρ∇_ρ g_μ)= 1/2g^ρ∇_μ g_ρ+1/2(g^ρ∇_ρ g_μ-g^ρ∇_ρ g_μ_0)= 1/2g^ρ∇_μ g_ρ We combine equations (<ref>), (<ref>) and (<ref>). Thus, we get R_μν = ∇_[1/2g^ρ(∇_μ g_ρν+∇_ν g_ρμ-∇_ρ g_νμ)]-∇_ν(1/2g^ρ∇_μ g_ρ)= 1/2g^ρ(∇_∇_μ g_ρν+∇_∇_ν g_ρμ-∇_∇_ρ g_νμ)-1/2g^ρ∇_ν∇_μ g_ρ= 1/2g^ρ(∇_∇_μ g_ρν+∇_∇_ν g_ρμ-∇_∇_ρ g_νμ-∇_ν∇_μ g_ρ) Previously, we used the property ∇_ g^μν=0. The proof of this property is presented below.∇_ g^μν =g^μν_,+^μ_ρ g^ρν+^ν_ρ g^ρμ=g^μν_,+1/2g^μκ(g_κ,ρ+g_ρκ,-g_ρ,κ)g^ρν+1/2g^νκ(g_κ,ρ+g_ρκ,-g_ρ,κ)g^ρμ=g^μν_,+1/2g^μκ(g_κ,ρ+g_ρκ,-g_ρ,κ)g^ρν+1/2g^νρ(g_ρ,κ+g_κρ,-g_κ,ρ)g^κμ=g^μν_,+g^μκ g_κρ, g^ρν=g^μν_,+(g^μκ g_κρ)_,_^μ_ρ,=0 g^ρν-g^μκ_, g_κρ g^ρν__κ^ν=g^μν_,-g^μν_,=0 g^μν R_μν = 1/2g^μνg^ρ(∇_∇_μ g_ρν+∇_∇_ν g_ρμ-∇_∇_ρ g_νμ-∇_ν∇_μ g_ρ)= 1/2(g^μνg^ρ∇_∇_μ g_ρν+g^μνg^ρ∇_∇_ν g_ρμ-g^μνg^ρ∇_∇_ρ g_νμ-g^μνg^ρ∇_ν∇_μ g_ρ)= 1/2(g^νμg^ρ∇_∇_ν g_ρμ+g^μνg^ρ∇_∇_ν g_ρμ-2g^μνg^ρ∇_∇_ρ g_νμ)= 1/2(2g^ρg^νμ∇_∇_ν g_ρμ-2g^μνg^ρ∇_∇_ρ g_νμ)= g^ρg^μν∇_∇_μ g_ρν-g^μνg^ρ∇_∇_ρ_□ g_μν=∇^ρ∇^ν g_ρν-g^μν□ g_μν Using Eq.(<ref>) into Eq.(<ref>) we obtain g^μν R_μν = g^ρg^μν∇_∇_μ(-g_ρg_νβ g^β)-g^μνg^ρ∇_∇_ρ(-g_μg_νβ g^β)= -g^ρg^μνg_ρg_νβ∇_∇_μ g^β+g^μνg^ρg_μg_νβ∇_∇_ρ g^β= -^_^μ_β∇_∇_μ g^β+g^ρ^ν_g_νβ∇_∇_ρ g^β= -∇_∇_β g^β+g_βg^ρ∇_∇_ρ_□ g^β= -∇_∇_β g^β+g_β□ g^β Putting together Eq.(<ref>) and Eq.(<ref>) we have g^μν R_μν=∇^μ∇^ν g_μν-g^μν□ g_μν=-∇_μ∇_ν g^μν+g_μν□ g^μν From Eq.(<ref>) and Eq.(<ref>) we conclude thatδ R=δ (g^μν)R_μν-∇_μ∇_ν g^μν+g_μν□ g^μν * (√(-g)_m):The energy-momentum tensor is defined by T^(m)_μν≡-2/√(-g)(√(-g)_m)/ g^μν Thus, we have (√(-g)_m)=-1/2 T^(m)_μν√(-g)g^μν Replacing the right hand sides from equations (<ref>), (<ref>), (<ref>) into Eq.(<ref>). We get 0=-1/2∫ d^4x √(-g) g_μνg^μν[f(Φ)/2κ R-1/2_Φ^Φ-V(Φ)-Λ_B] +1/2∫ d^4x√(-g) {1/κ[ g^μν f(Φ)R_μν-f(Φ)∇_μ∇_νg^μν+g_μνf(Φ)□g^μν]- g^μν_μΦ_νΦ} -1/2∫ d^4x T^(m)_μν√(-g)g^μν 0= ∫ d^4x √(-g)g^μν{g_μν[-f(Φ)/2κ R+1/2_Φ^Φ+V(Φ)+Λ_B]+f(Φ)/κR_μν-_μΦ_νΦ-T^(m)_μν} +1/κ∫ d^4x√(-g) (-f(Φ)∇_μ∇_νg^μν+g_μνf(Φ)□g^μν)The integral in the second line of Eq.(<ref>) can be modified to a more useful one, but first it is necessary to prove one general property that will help us doing the modification.∇_μ A^μ=_μ A^μ+^μ_μA^ ^μ_μ =1/2g^μρ(g_ρ,μ+g_μρ,-g_μ,ρ)=1/2g^μρg_ρ,μ+1/2g^μρg_μρ,-1/2g^μρg_μ,ρ=1/2g^μρg_ρ,μ+1/2g^μρg_μρ,-1/2g^ρμg_ρ,μ=1/2g^μρg_μρ,1/√(-g)√(-g)/ x^ =1/2g g/ x^=1/2g^-1 g/ x^=1/2ε_μ_0…μ_3 g^0μ_0⋯ g^3μ_3/ x^(ε^ν_0…ν_3 g_0ν_0⋯ g_3ν_3)=1/2ε_μ_0…μ_3ε^ν_0…ν_3 g^0μ_0⋯ g^3μ_3(g_0ν_0,⋯ g_3ν_3+⋯+g_0ν_0⋯ g_3ν_3,)=1/2|[ _μ_0^ν_0 _μ_0^ν_1 _μ_0^ν_2 _μ_0^ν_3; _μ_1^ν_0 _μ_1^ν_1 _μ_1^ν_2 _μ_1^ν_3; _μ_2^ν_0 _μ_2^ν_1 _μ_2^ν_2 _μ_2^ν_3; _μ_3^ν_0 _μ_3^ν_1 _μ_3^ν_2 _μ_3^ν_3 ]|g^0μ_0⋯ g^3μ_3(g_0ν_0,⋯ g_3ν_3+⋯+g_0ν_0⋯ g_3ν_3,)=1/2_μ_0^ν_0 _μ_1^ν_1 _μ_2^ν_2 _μ_3^ν_3 g^0μ_0⋯ g^3μ_3(g_0ν_0,⋯ g_3ν_3+⋯+g_0ν_0⋯ g_3ν_3,)=1/2g^0ν_0 g^1ν_1 g^2ν_2 g^3ν_3(g_0ν_0,⋯ g_3ν_3+⋯+g_0ν_0⋯ g_3ν_3,)=1/2(g^0ν_0g_0ν_0,+g^1ν_1g_1ν_1,+g^2ν_2g_2ν_2,+g^3ν_3g_3ν_3,)=1/2(g^0ρg_0ρ,+g^1ρg_1ρ,+g^2ρg_2ρ,+g^3ρg_3ρ,)=1/2g^μρg_μρ, Combining equations (<ref>), (<ref>), (<ref>) we get∇_μ A^μ=_μ A^μ+1/√(-g)√(-g)/ x^A^=1/√(-g)_μ(√(-g) A^μ)Let us now modify the integral of the last term of Eq.(<ref>) by using Eq.(<ref>), Eq.(<ref>) and the fact that the variation of the metric g^μν vanishes at infinity.∫ d^4x√(-g) (g_μνf(Φ)□g^μν-f(Φ)∇_μ∇_νg^μν)= ∫ d^4x√(-g)(g_μν f g^ρ∇_ρ∇_ g^μν-f ∇_μ∇_ν g^μν)= ∫ d^4x√(-g)[f ∇_ρ∇_(g_μν g^ρ g^μν)-∇_μ(f ∇_νg^μν)+(∇_μ f)∇_ν g^μν]= ∫ d^4x√(-g)[f ∇_ρ∇_(g_μν g^ρ g^μν)+(∇_μ f)∇_ν g^μν]-∫ d^4x _μ(√(-g) f ∇_ν g^μν)_0= ∫ d^4x√(-g){∇_ρ[f ∇_(g_μν g^ρ g^μν)]-(∇_ρ f)∇_(g_μν g^ρ g^μν)+∇_ν( g^μν∇_μ f)- g^μν∇_μ∇_ν f }= ∫ d^4x√(-g)[-(∇_ρ f)∇_(g_μν g^ρ g^μν)- g^μν∇_μ∇_ν f ]+∫ d^4x _ρ[√(-g) f ∇_(g_μν g^ρ g^μν)]_0+∫ d^4x _ν(√(-g)g^μν∇_μ f)_0= ∫ d^4x√(-g)[-∇_(g_μν g^ρ g^μν∇_ρ f)+g_μν g^ρ g^μν∇_∇_ f- g^μν∇_μ∇_ν f]= ∫ d^4x√(-g)g^μν(g_μν□ f-∇_μ∇_ν f)-∫ d^4x _(√(-g) g_μν g^ρ g^μν∇_ρ f)_0∫ d^4x√(-g)(g_μνf(Φ)□g^μν-f(Φ)∇_μ∇_ν g^μν)=∫ d^4x√(-g)g^μν[g_μν□ f(Φ)-∇_μ∇_ν f(Φ)]The combination of equations (<ref>) and (<ref>) leads to 0 =∫ d^4x √(-g)g^μν{g_μν[-f(Φ)/2κ R+1/2_Φ^Φ+V(Φ)+Λ_B]..+1/κ[f(Φ)R_μν-∇_μ∇_ν f(Φ)+g_μν□ f(Φ)]-_μΦ_νΦ-T^(m)_μν}0 =-g_μν[f(Φ)/2κ R-1/2_Φ^Φ-V(Φ)-Λ_B]+1/κ[f(Φ)R_μν-∇_μ∇_ν f(Φ)+g_μν□ f(Φ)]-_μΦ_νΦ-T^(m)_μν(T^(m)_μν-g_μνΛ_B)=f(Φ)/κ[R_μν-1/2g_μνR]+g_μν[_Φ^Φ/2+V(Φ)] +1/κ[-∇_μ∇_ν f(Φ)+g_μν□ f(Φ)]-_μΦ_νΦWe can define now the following tensor-T^(Φ)_μν=g_μν[_Φ^Φ/2+V(Φ)]+1/κ[-∇_μ∇_ν f(Φ)+g_μν□ f(Φ)]-_μΦ_νΦT^(Φ)_μν=_μΦ_νΦ-g_μν[_Φ^Φ/2+V(Φ)]+1/κ[∇_μ∇_ν f(Φ)-g_μν□ f(Φ)] It is known that the covariant derivative of a scalar ∇_μΦ equals to the simple derivative _μΦ. Thus, in the above equation □≡ g_β∇^∇^β=∇_∇^=∇^2 and _Φ^Φ=∇_Φ∇^Φ=(∇Φ)^2. Hence, we can write Eq.(<ref>) as T^(Φ)_μν=∇_μΦ∇_νΦ-g_μν[(∇Φ)^2/2+V(Φ)]+1/κ[∇_μ∇_ν f(Φ)-g_μν∇^2 f(Φ)] The combination of equations (<ref>) and (<ref>) yields to(T^(m)_μν-g_μνΛ_B)=f(Φ)/κ[R_μν-1/2g_μνR]-T^(Φ)_μνκ(T^(m)_μν+T^(Φ)_μν-g_μνΛ_B)=f(Φ)(R_μν-1/2g_μνR)Finally, we present the above equations in the case of (4+n)-dimensional spacetime. As we already mentioned the only difference is that Greek indices become capital Latin indices. Moreover, we should have in mind that in this case κκ_(4+n) and Λ_B is the higher dimensional cosmological constant. Therefore, it is [box=]equation T_MN^(Φ)=∇_M Φ∇_NΦ-g_MN[(∇Φ)^2/2+V(Φ)]+1/κ_(4+n)[∇_M ∇_N f(Φ)-g_MN∇^2 f(Φ)]where now it is[box=]equation (∇Φ)^2=∇_KΦ∇^KΦ, □=∇^2=∇_K∇^K The field equations are [box=]equation κ_(4+n)(T^(m)_MN+T^(Φ)_MN-g_MNΛ_B)=f(Φ)(R_MN-1/2g_MNR) § VARIATION WITH RESPECT TO THE SCALAR FIELDBy varying the action of equation (<ref>) with respect to the scalar field Φ and _μΦ, we obtain a new equation that relates functions f(Φ) and V(Φ) to the field Φ and its derivatives. The procedure that one should follow is depicted below. Firstly, we write the action (<ref>) in the following form in order to be more convenient for evaluation. S=∫ d^4x√(-g) _tot where _tot=f(Φ)/2κR-1/2_μΦ^μΦ-V(Φ)-Λ_B+_m The variation of the action (<ref>) with respect to Φ and _μΦ will provide us the Euler-Lagrange equation.S=0=∫ d^4x (√(-g) _tot)=∫ d^4x[(√(-g) _tot)/ΦΦ+(√(-g) _tot)/(_μΦ)(_μΦ)](_μΦ)=_μ(Φ) 0 =∫ d^4x [(√(-g) _tot)/ΦΦ+(√(-g) _tot)/(_μΦ)_μ(Φ)] =∫ d^4x [(√(-g) _tot)/ΦΦ+_μ((√(-g) _tot)/(_μΦ)Φ)-_μ((√(-g) _tot)/(_μΦ))Φ] However ∫ d^4x _μ((√(-g) _tot)/(_μΦ)Φ)=0because at the limits of the integration Φ=0. Thus, combining equations (<ref>) and (<ref>) we get0=∫ d^4x [(√(-g) _tot)/Φ-_μ((√(-g) _tot)/(_μΦ))]Φ(√(-g) _tot)/Φ=_μ[(√(-g) _tot)/(_μΦ)]Eq.(<ref>) constitutes the Euler-Lagrange equation. Substituting now the quantity _tot from Eq.(<ref>) into Eq.(<ref>) we obtain√(-g) /Φ[f(Φ)/2κR-V(Φ)]=_μ[√(-g) /(_μΦ)(-1/2_νΦ^νΦ)] √(-g)(1/2κdf/dΦR-dV/dΦ)=-1/2_μ[√(-g) /(_μΦ)(g^ρν_νΦ_ρΦ)] √(-g)(1/2κdf/dΦR-dV/dΦ)=-1/2_μ[√(-g)( g^ρν^μ_ν_ρΦ+g^ρν_νΦ^μ_ρ)] √(-g)(1/2κdf/dΦR-dV/dΦ)=-1/2_μ[√(-g)( g^ρμ_ρΦ+g^μν_νΦ)_2g^μρ_ρΦ][box=]equation √(-g)(1/2κdf/dΦR-dV/dΦ)=-_μ(√(-g) g^μρ_ρΦ) CHAPTER: ENERGY-MOMENTUM TENSOR'S COMPONENTSFirstly, the non-diagonal and non-zero components of energy-momentum tensor T^M_N are going to be calculated. Subsequently, the diagonal components will be calculated and finally the zero-valuedcomponents will be presented. 𝐓^0_1:(<ref>)[N=1]M=0 T^0_1 =^0ϕ_1ϕ+^0χ_1χ+∇^0∇_1f=g^0K_Kϕ_1ϕ+g^0K_Kχ_1χ+g^0K∇_K∇_1f=g^01(_1ϕ)^2+g^01(_1χ)^2+g^01∇_1^2f=g^01_e^-2A[(_1ϕ)^2+(_1χ)^2+∇_1^2f]=e^-2A[(_1ϕ)^2+(_1χ)^2+∇_1^2f] ∇_1^2f=∇_1(∇_1f)=∇_1(_1f)=_1(_1f)-^L_11_0_Lf=_1^2f_1^2f =_1(_ϕ f_1ϕ+_χ f_1χ)=_ϕ^2f(_1ϕ)^2+_χ_ϕ f_1χ_1ϕ+_ϕ f_1^2ϕ+_χ^2f(_1χ)^2+_ϕ_χ f_1ϕ_1χ+_χ f_1^2χ=_ϕ^2f(_1ϕ)^2+2_χ_ϕ f_1χ_1ϕ+_ϕ f_1^2ϕ+_χ^2f(_1χ)^2+_χ f_1^2χCombining equations (<ref>)-(<ref>), we obtain T^0_1=e^-2A[(_1ϕ)^2+(_1χ)^2+_1^2f] or T^0_1=e^-2A[(1+_ϕ^2f)(_1ϕ)^2+(1+_χ^2f)(_1χ)^2+2_χ_ϕ f_1χ_1ϕ+_ϕ f_1^2ϕ+_χ f_1^2χ]𝐓^1_0:(<ref>)[N=0]M=1 T^1_0 =^1ϕ_0ϕ+^1χ_0χ+∇^1∇_0f=g^1K_Kϕ_0ϕ+g^1K_Kχ_0χ+g^1K∇_K∇_0f=g^10(_0ϕ)^2+g^11_1ϕ_0ϕ+g^10(_0χ)^2+g^11_1χ_0χ+g^10∇_0^2f+g^11∇_1∇_0f=g^10_e^-2A[(_0ϕ)^2+(_0χ)^2+∇_0^2f]+g^11_e^-2A(1-2m/r)[_1ϕ_0ϕ+_1χ_0χ+∇_1∇_0f]T^1_0=e^-2A{[(_0ϕ)^2+(_0χ)^2+∇_0^2f]+(1-2m/r)[_1ϕ_0ϕ+_1χ_0χ+∇_1∇_0f]}∇_0∇_0f =_0^2f-^L_00_Lf=_0^2f-^0_00_0f-^1_00_1f-^4_00_4f=_0^2f-(m/r^2-_1m/r)_0f-(_0m/r-_1m/r+m/r^2+2m_1m/r^2-2m^2/r^3)_1f-e^2A(A'-2mA'/r-_4m/r)_4f ∇_1∇_0f =_1_0f-^L_10_Lf=_1_0f-^1_10_1f-^4_10_4f=_1_0f-(_1m/r-m/r^2)_1f+e^2AA'_4f _0^2f =_0(_ϕ f_0ϕ+_χ f_0χ)=_ϕ^2f(_0ϕ)^2+_χ_ϕ f_0χ_0ϕ+_ϕ f_0^2ϕ+_χ^2f(_0χ)^2+_ϕ_χ f_0ϕ_0χ+_χ f_0^2χ=_ϕ^2f(_0ϕ)^2+2_χ_ϕ f_0χ_0ϕ+_χ^2f(_0χ)^2+_ϕ f_0^2ϕ+_χ f_0^2χ _1_0f =_1(_ϕ f_0ϕ+_χ f_0χ)=_ϕ^2f_1ϕ_0ϕ+_χ_ϕ f_1χ_0ϕ+_ϕ f_1_0ϕ+_χ^2f_1χ_0χ+_ϕ_χ f_1ϕ_0χ+_χ f_1_0χ=_ϕ^2f_1ϕ_0ϕ+_χ^2f_1χ_0χ+_χ_ϕ f(_1χ_0ϕ+_1ϕ_0χ)+_ϕ f_1_0ϕ+_χ f_1_0χFrom equations (<ref>)-(<ref>) we haveT^1_0 =e^-2A[(_0ϕ)^2+(_0χ)^2+_0^2f+(1-2m/r)(_1_0ϕ+_1χ_0χ+_1_0f).. +_1(m/r)_0f-_0(m/r)_1f+e^2A_4(m/r)_4f]orT^1_0 =e^-2A{(1+_ϕ^2f)(_0ϕ)^2+(1+_χ^2f)(_0χ)^2+2_χ_ϕ f_0ϕ_0χ+_ϕ f_0^2ϕ+_χ f_0^2χ.+(1-2m/r)[(1+_ϕ^2f)_1ϕ_0ϕ+(1+_χ^2f)_1χ_0χ+_ϕ_χ f(_1ϕ_0χ+_1χ_0ϕ)+_ϕ f_1_0ϕ+_χ f_1_0χ]. +(_1m/r-m/r^2)(_ϕ f_0ϕ+_χ f_0χ)-_0m/r(_ϕ f_1ϕ+_χ f_1χ)+e^2A_4m/r(_ϕ f_4ϕ+_χ f_4χ)}plain𝐓^4_0:(<ref>)[N=0]M=4T^4_0 =^4ϕ_0ϕ+^4χ_0χ+∇^4∇_0f=g^4K(_Kϕ_0ϕ+_Kχ_0χ+∇_K∇_0f)=g^44_1(_4ϕ_0ϕ+_4χ_0χ+∇_4∇_0f)=_4ϕ_0ϕ+_4χ_0χ+∇_4∇_0f ∇_4∇_0f =_4_0f-^L_40_Lf=_4_0f-^0_40_0f-^1_40_1f=_4_0f-A'_0f-_4m/r_1f _4_0f =_4(_ϕ f_0ϕ+_χ f_0χ)=_ϕ^2f_4ϕ_0ϕ+_χ_ϕ f_4χ_0ϕ+_ϕ f_4_0ϕ+_χ^2f_4χ_0χ+_ϕ_χ f_4ϕ_0χ+_χ f_4_0χ=_ϕ^2f_4ϕ_0ϕ+_χ^2f_4χ_0χ+_ϕ_χ f(_4χ_0ϕ+_4ϕ_0χ)+_ϕ f_4_0ϕ+_χ f_4_0χEquations (<ref>)-(<ref>) yield to T^4_0=_4ϕ_0ϕ+_4χ_0χ+_4_0f-A'_0f-_4m/r_1f orT^4_0=(1+_ϕ^2f)_4ϕ_0ϕ +(1+_χ^2f)_4χ_0χ+_χ_ϕ f(_4χ_0ϕ+_4ϕ_0χ)+_ϕ f_4_0ϕ+_χ f_4_0χ-A'_0f-_4m/r_1fplain 𝐓^0_4:(<ref>)[N=4]M=0T^0_4 =^0ϕ_4ϕ+^0χ_4χ+∇^0∇_4f=g^0K(_Kϕ_4ϕ+_Kχ_4χ+∇_K∇_4f)=g^01_e^-2A(_1ϕ_4ϕ+_1χ_4χ+∇_1∇_4f)=e^-2A(_1ϕ_4ϕ+_1χ_4χ+∇_1∇_4f) ∇_1∇_4f =_1_4f-^L_14_Lf=_1_4f-^1_14_1f=_1_4f-A'_1f _1_4f= =_1(_ϕ f_4ϕ+_χ f_4χ)=_ϕ^2f_1ϕ_4ϕ+_χ_ϕ f_1χ_4ϕ+_ϕ f_1_4ϕ+_χ^2f_1χ_4χ+_ϕ_χ f_1ϕ_4χ+_χ f_1_4χ=_ϕ^2f_1ϕ_4ϕ+_χ^2f_1χ_4χ+_ϕ_χ f(_1χ_4ϕ+_1ϕ_4χ)+_ϕ f_1_4ϕ+_χ f_1_4χ Inserting the expressions of the quantities ∇_1∇_4f and _1_4f from equations (<ref>) and (<ref>) respectively, into equation (<ref>) we get T^0_4=e^-2A(_1ϕ_4ϕ+_1χ_4χ+_1_4f-A'_1f) orT^0_4=e^-2A[(1+_ϕ^2f)_1ϕ_4ϕ +(1+_χ^2f)_1χ_4χ+_ϕ_χ f(_1χ_4ϕ+_1ϕ_4χ) +_ϕ f_1_4ϕ+_χ f_1_4χ-A'(_ϕ f_1ϕ+_χ f_1χ)]𝐓^4_1:(<ref>)[N=1]M=4T^4_1 =^4ϕ_1ϕ+^4χ_1χ+∇^4∇_1f=g^4K(_Kϕ_1ϕ+_Kχ_1χ+∇_K∇_1f)=g^44_1(_4ϕ_1ϕ+_4χ_1χ+∇_4∇_1f)=_4ϕ_1ϕ+_4χ_1χ+∇_4∇_1fIt is obvious from equations (<ref>) and (<ref>) thatT^4_1=e^2AT^0_4𝐓^1_4:(<ref>)[N=4]M=1T^1_4 =^1ϕ_4ϕ+^1χ_4χ+∇^1∇_4f=g^1K(_Kϕ_4ϕ+_Kχ_4χ+∇_4∇_1f)=g^10_e^-2A(_0ϕ_4ϕ+_0χ_4χ+∇_0∇_4f)+g^11_e^-2A(1-2m/r)(_1ϕ_4ϕ+_1χ_4χ+∇_1∇_4f)=e^-2A(_0ϕ_4ϕ+_0χ_4χ+∇_0∇_4f)+e^-2A(1-2m/r)(_1ϕ_4ϕ+_1χ_4χ+∇_1∇_4f)From equations (<ref>), (<ref>) and (<ref>) we deduce that T^1_4=e^-2AT^4_0+(1-2m/r)T^0_4 plain 𝐓^0_0:(<ref>)[N=0]M=0T^0_0 =^0ϕ_0ϕ+^0χ_0χ+∇^0∇_0f+-□ f=g^01(_1ϕ_0ϕ+_1χ_0χ+∇_1∇_0f)+-□ f=e^-2A(_1ϕ_0ϕ+_1χ_0χ+∇_1∇_0f)+-□ fwhere≡ -(ϕ)^2+(χ)^2/2-V(ϕ,χ)=-^Lϕ_Lϕ+^Lχ_Lχ/2-V(ϕ,χ)□ f≡∇^2f=∇^K∇_KfWe are going to calculate firstly the quantitiesand □ f.=-1/2(^Lϕ_Lϕ+^Lχ_Lχ)-V(ϕ,χ)=-1/2(^0ϕ_0ϕ+^1ϕ_1ϕ+^4ϕ_4ϕ+^0χ_0χ+^1χ_1χ+^4χ_4χ)-V(ϕ,χ)=-1/2(g^0K_Kϕ_0ϕ+g^1K_Kϕ_1ϕ+g^4K_Kϕ_4ϕ+g^0K_Kχ_0χ+g^1K_Kχ_1χ+g^4K_Kχ_4χ)-V(ϕ,χ)=-1/2{2g^01_e^-2A(_1ϕ_0ϕ+_1χ_0χ)+g^11_e^-2A(1-2m/r)[(_1ϕ)^2+(_1χ)^2]+g^44_1[(_4ϕ)^2+(_4χ)^2]}-V(ϕ,χ)=-e^-2A/2{2(_1ϕ_0ϕ+_1χ_0χ)+(1-2m/r)[(_1ϕ)^2+(_1χ)^2]}-1/2[(_4ϕ)^2+(_4χ)^2]-V(ϕ,χ) =-e^-2A/2{2(_1ϕ_0ϕ+_1χ_0χ)+(1-2m/r)[(_1ϕ)^2+(_1χ)^2]}-1/2[(_4ϕ)^2+(_4χ)^2]-V(ϕ,χ)□ f =∇^K∇_Kf=g^KL∇_L∇_Kf=2g^01∇_0∇_1f+g^11∇_1∇_1f+g^22∇_2∇_2f+g^33∇_3∇_3f+g^44∇_4∇_4f ∇_1∇_1f and ∇_0∇_1f are known from equations (<ref>) and (<ref>) respectively. ∇_2∇_2f =_2^2f_0-^L_22_Lf=-^0_22_0f-^1_22_1f-^4_22_4f=r_0f-(2m-r)_1f+r^2e^2AA'_4f ∇_3∇_3f =_3^2f_0-^L_33_Lf=-^0_33_0f-^1_33_1f-^4_33_4f=[r_0f-(2m-r)_1f+r^2e^2AA'_4f]sin^2θ=(∇_2∇_2f)sin^2θ ∇_4∇_4f =_4_4f-^L_44_0_Lf=_4^2fCombining equations (<ref>), (<ref>), (<ref>)-(<ref>) we obtain□ f =2g^01∇_0∇_1f+g^11∇_1∇_1f+g^22∇_2∇_2f+g^33∇_3∇_3f+g^44∇_4∇_4f=2e^-2A[_0_1f+(m/r^2-_1m/r)_1f+e^2AA'_4f]+e^-2A(1-2m/r)_1^2f+g^22∇_2∇_2f+g^22/sin^2θ(∇_2∇_2f)sin^2θ_2g^22∇_2∇_2f+_4^2f=2e^-2A_0_1f+e^-2A2(m/r^2-_1m/r)__1(1-2m/r)_1f+2A'_4f+e^-2A(1-2m/r)_1^2f+2e^-2A/r^2[r_0f-(2m-r)_1f+r^2e^2AA'_4f]+_4^2f=e^-2A_0_1f+e^-2A_1(_0f)+e^-2A_1[(1-2m/r)_1f]+2A'_4f+e^-2A/r^22r__1(r^2)_0f+2e^-2A/r_e^-2A/r^2_1(r^2)(1-2m/r)_1f+2A'_4f+_4^2f=e^-2A_0_1f+e^-2A/r^2r^2_1(_0f)+e^-2A/r^2r^2_1[(1-2m/r)_1f]+4A'_4f+e^-2A/r^2_1(r^2)_0f+e^-2A/r^2_1(r^2)[(1-2m/r)_1f]+_4^2f=e^-2A_0_1f+e^-2A/r^2_1[r^2_0f+r^2(1-2m/r)_1f]+e^-4A_4(e^4A_4f) □ f=e^-2A_0_1f+e^-2A/r^2_1[r^2_0f+r^2(1-2m/r)_1f]+e^-4A_4(e^4A_4f)Substituting ∇_1∇_0f from equation (<ref>) into (<ref>) we get plainT^0_0=e^-2A[_1ϕ_0ϕ+_1χ_0χ+_1_0f+_1f/r(m/r-_1m)]+A'_4f+-□ fT^0_0=e^-2A[_1ϕ_0ϕ+_1χ_0χ+_1_0f-_1(m/r)_1f]+A'_4f+-□ fUsing equation (<ref>) as well, we obtainT^0_0 =e^-2A[(1+_ϕ^2f)_1ϕ_0ϕ+(1+_χ^2f)_1χ_0χ+_χ_ϕ f(_1χ_0ϕ+_1ϕ_0χ)+_ϕ f_1_0ϕ. +._χ f_1_0χ+_ϕ f_1ϕ+_χ f_1χ/r(m/r-_1m)]+A'(_ϕ f_4ϕ+_χ f_4χ)+-□ fplain 𝐓^1_1:(<ref>)[N=1]M=1T^1_1 =^1ϕ_1ϕ+^1χ_1χ+∇^1∇_1f+-□ f=g^1K(_Kϕ_1ϕ+_Kχ_1χ+∇_K∇_1f)+-□ f=g^10_e^-2A(_0ϕ_1ϕ+_0χ_1χ+∇_0∇_1f)+g^11_e^-2A(1-2m/r)[(_1ϕ)^2+(_1χ)^2+∇_1^2f]+-□ fEquations (<ref>) and (<ref>) combined with (<ref>) give T^1_1=T^0_0+(1-2m/r)T^0_1 𝐓^2_2:(<ref>)[N=2]M=2T^2_2 =^2ϕ_2ϕ+^2χ_2χ+∇^2∇_2f+-□ f=g^2K(_Kϕ_2ϕ+_Kχ_2χ+∇_K∇_2f)+-□ f=g^22_e^-2A/r^2(_2ϕ_2ϕ_0+_2χ_2χ_0+∇_2∇_2f)+-□ f=e^-2A/r^2∇_2∇_2f+-□ f (<ref>)(<ref>)T^2_2 =e^-2A/r^2[r_0f-(2m-r)_1f+r^2e^2AA'_4f]+-□ f=e^-2A/r[_0f+(1-2m/r)_1f]+A'_4f+-□ fThus, we have T^2_2=e^-2A/r[_0f+(1-2m/r)_1f]+A'_4f+-□ f orT^2_2 =e^-2A/r[(_ϕ f_0ϕ+_χ f_0χ)+(1-2m/r)(_ϕ f_1ϕ+_χ f_1χ)]+A'(_ϕ f_4ϕ+_χ f_4χ)+-□ f𝐓^3_3:(<ref>)[N=3]M=3T^3_3 =^3ϕ_3ϕ+^3χ_3χ+∇^3∇_3f--□ f=g^3K(_Kϕ_3ϕ+_Kχ_3χ+∇_K∇_3f)--□ f=g^33_e^-2A/r^2sin^2θ(_3ϕ_3ϕ_0+_3χ_3χ_0+∇_3∇_3f)--□ f=e^-2A/r^2sin^2θ∇_3∇_3f+-□ f (<ref>)(<ref>)T^3_3 =e^-2A/r^2sin^2θ(∇_2∇_2f)sin^2θ--□ f=e^-2A/r^2∇_2∇_2f+-□ fT^3_3=T^2_2 plain 𝐓^4_4:(<ref>)[N=4]M=4T^4_4 =^4ϕ_4ϕ+^4χ_4χ+∇^4∇_4f+-□ f=g^4K(_Kϕ_4ϕ+_Kχ_4χ+∇_K∇_4f)+-□ f=g^44_1[(_4ϕ)^2+(_4χ)^2+∇_4∇_4f]+-□ f=(_4ϕ)^2+(_4χ)^2+∇_4∇_4f+-□ f The combination of equations (<ref>) and (<ref>) gives T^4_4=(_4ϕ)^2+(_4χ)^2+^2_4f+-□ f Moreover, it is_4^2f =_4(_ϕ f_4ϕ+_χ f_4χ)=_ϕ^2f(_4ϕ)^2+_χ_ϕ f_4χ_4ϕ+_ϕ f_4^2ϕ+_χ^2f(_4χ)^2+_ϕ_χ f_4ϕ_4χ+_χ f_4^2χ=_ϕ^2f(_4ϕ)^2+_χ^2f(_4χ)^2+2_χ_ϕ f_4χ_4ϕ+_ϕ f_4^2ϕ+_χ f_4^2χTherefore, equations (<ref>) and (<ref>) are combined to give T^4_4=(1+_ϕ^2f)(_4ϕ)^2+(1+_χ^2f)(_4χ)^2+2_χ_ϕ f_4χ_4ϕ+_ϕ f_4^2ϕ+_χ f_4^2χ+-□ f𝐓^0_2:(<ref>)[N=2]M=0 T^0_2 =^0ϕ_2ϕ+^0χ_2χ+∇^0∇_2f=g^0K_Kϕ_2ϕ_0+g^0K_Kχ_2χ_0+g^0K∇_K∇_2f=g^01_e^-2A∇_1∇_2f=e^-2A∇_1∇_2f ∇_1∇_2f=(_1_2f_0-^L_12_Lf)=-^L_12_Lf=-^2_12_2f_0=0Thus, equations (<ref>) and (<ref>) giveT^0_2=0 𝐓^0_3:(<ref>)[N=3]M=0 T^0_3 =^0ϕ_3ϕ+^0χ_3χ+∇^0∇_3f=g^0K_Kϕ_3ϕ_0+g^0K_Kχ_3χ_0+g^0K∇_K∇_3f=g^01_e^-2A∇_1∇_3f=e^-2A∇_1∇_3f ∇_1∇_3f=(_1_3f_0-^L_13_Lf)=-^L_13_Lf=-^3_13_3f_0=0Equations (<ref>) and (<ref>) yield toT^0_3=0 plain 𝐓^1_2:(<ref>)[N=2]M=1T^1_2 =^1ϕ_2ϕ_0+^1χ_2χ_0+∇^1∇_2f=g^1K∇_K∇_2f=g^10∇_0∇_2f+g^11∇_1∇_2f=e^-2A∇_0∇_2f+e^-2A(1-2m/r)∇_1∇_2f ∇_0∇_2f=_0_2f_0-^L_02_0_Lf=0Combining equations (<ref>), (<ref>), (<ref>) we getT^1_2=0 𝐓^1_3:(<ref>)[N=3]M=1T^1_3 =^1ϕ_3ϕ_0+^1χ_3χ_0+∇^1∇_3f=g^1K∇_K∇_3f=g^10∇_0∇_3f+g^11∇_1∇_3f=e^-2A∇_0∇_3f+e^-2A(1-2m/r)∇_1∇_3f ∇_0∇_3f=_0_3f_0-^L_03_0_Lf=0From equations (<ref>) and (<ref>) into (<ref>) we obtainT^1_3=0 𝐓^2_0:(<ref>)[N=0]M=2T^2_0=^2ϕ_0_0ϕ+^2χ_0_0χ+∇^2∇_0f=g^2K∇_K∇_0f=g^22∇_2∇_0f_0 (<ref>)=0 T^2_0=0 𝐓^2_1:(<ref>)[N=1]M=2T^2_1=^2ϕ_0_1ϕ+^2χ_0_1χ+∇^2∇_1f=g^2K∇_K∇_1f=g^22∇_2∇_1f_0 (<ref>)=0 T^2_1=0 plain 𝐓^2_3:(<ref>)[N=3]M=2T^2_3=^2ϕ_0_3ϕ+^2χ_0_3χ+∇^2∇_3f=g^2K∇_K∇_3f=g^22∇_2∇_3f∇_2∇_3f=_2_3f_0-^L_23_Lf=-^3_23_3f_0=0From equations (<ref>) and (<ref>) it is obvious thatT^2_3=0 𝐓^2_4:(<ref>)[N=4]M=2T^2_4=^2ϕ_0_4ϕ+^2χ_0_4χ+∇^2∇_4f=g^2K∇_K∇_4f=g^22∇_2∇_4f∇_2∇_4f=_2_4f_0-^L_24_Lf=-^2_24_2f_0=0Equations (<ref>) and (<ref>) are combined to giveT^2_4=0 𝐓^3_0:(<ref>)[N=0]M=3T^3_0=^3ϕ_0_0ϕ+^3χ_0_0χ+∇^3∇_0f=g^3K∇_K∇_0f=g^33∇_3∇_0f_0 (<ref>) T^3_0=0 𝐓^3_1:(<ref>)[N=1]M=3T^3_1=^3ϕ_0_1ϕ+^3χ_0_1χ+∇^3∇_1f=g^3K∇_K∇_1f=g^33∇_3∇_1f_0 (<ref>) T^3_1=0𝐓^3_2:(<ref>)[N=2]M=3T^3_2=^3ϕ_0_2ϕ+^3χ_0_2χ+∇^3∇_2f=g^3K∇_K∇_2f=g^33∇_3∇_2f_0 (<ref>) T^3_2=0 𝐓^3_4:(<ref>)[N=4]M=3T^3_4=^3ϕ_0_4ϕ+^3χ_0_4χ+∇^3∇_4f=g^3K∇_K∇_4f=g^33∇_3∇_4f∇_3∇_4f=_3_4f_0-^L_34_Lf=-^3_34_3f_0=0The combination of equations (<ref>) and (<ref>) yield to T^3_4=0 plain 𝐓^4_2:(<ref>)[N=2]M=4T^4_2=^4ϕ_0_2ϕ+^4χ_0_2χ+∇^4∇_2f=g^4K∇_K∇_2f=g^44∇_4∇_2f_0 (<ref>) T^4_2=0 𝐓^4_3:(<ref>)[N=3]M=4T^4_2=^4ϕ_0_3ϕ+^4χ_0_3χ+∇^4∇_3f=g^4K∇_K∇_3f=g^44∇_4∇_3f_0 (<ref>)T^4_3=0* unsrttocchapterReferences | http://arxiv.org/abs/1707.08351v1 | {
"authors": [
"Theodoros Nakas"
],
"categories": [
"gr-qc"
],
"primary_category": "gr-qc",
"published": "20170726100428",
"title": "Searching for Localized Black-Hole solutions in Brane-World models"
} |
A.N. Frumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy of Sciences, 31 Leninsky Prospect, 119071 Moscow, RussiaA.N. Frumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy of Sciences, 31 Leninsky Prospect, 119071 Moscow, Russia A.N. Frumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy of Sciences, 31 Leninsky Prospect, 119071 Moscow, Russia Institute of Mechanics, M.V. Lomonosov Moscow State University, 119991 Moscow, Russia[Corresponding author: ][email protected] A.N. Frumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy of Sciences, 31 Leninsky Prospect, 119071 Moscow, Russia Department of Physics, M.V. Lomonosov Moscow State University, 119991 Moscow, Russia DWI - Leibniz Institute for Interactive Materials, Forckenbeckstraße 50, 52056 Aachen, Germany The hydrodynamics of liquid flowing past gas sectors of unidirectional superhydrophobic surfaces is revisited. Attention is focussed on the local slip boundary condition at the liquid-gas interface, which is equivalent to the effect of a gas cavity on liquid flow. The system is characterized by a large viscosity contrast between liquid and gas, μ/μ_g ≫ 1. We interpret earlier results, namely the dependence of the local slip length on the flow direction, in terms of a tensorial local slip boundary condition and relate the eigenvalues of the local slip length tensor to the texture parameters, such as the width of the groove, δ, and the local depth of the groove, e(y, α). The latter varies in the direction y, orthogonal to the orientation of stripes, and depends on the bevel angle of groove's edges, π/2 - α, at the point, where three phases meet. Our theory demonstrates that when grooves are sufficiently deep their eigenvalues of the local slip length tensor depend only on μ/μ_g, δ, and α, but not on the depth. The eigenvalues of the local slip length of shallow grooves depend on μ/μ_g and e(y, α), although the contribution of the bevel angle is moderate. In order to assess the validity of our theory we propose a novel approach to solve the two-phase hydrodynamic problem, which significantly facilitates and accelerates calculations compared to conventional numerical schemes. The numerical results show that our simple analytical description obtained for limiting cases of deep and shallow grooves remains valid for various unidirectional textures.Boundary conditions at the gas sectors of superhydrophobic grooves. Olga I. Vinogradova December 30, 2023 ===================================================================§ INTRODUCTIONAnisotropic superhydrophobic (SH) surfaces have raised a considerable interest over the recent years. Such surfaces in the Cassie state, i.e., where the texture is filled with gas, can induce exceptional lubricating properties <cit.> and generate secondary flows transverse to the direction of the applied pressure gradient <cit.>. These are important for a variety of applications that involve a manipulation of liquids at the small scale and can be used to separate particles <cit.> and enhance their mixing rate <cit.>in microfluidic devices. During past decade the quantitative understanding of liquid flow past SH anisotropic surfaces was significantly expanded. However, many fundamental issues still remain unresolved.To quantify the drag reduction and transverse hydrodynamic phenomenaassociated withSH surfaces with given area of gas and solid fractions it is convenient to construct the effective slip boundary condition applied at the imaginary homogeneous surface <cit.>, which mimics the real one and is generally a tensor <cit.>. Once eigenvalues of the slip-length tensor, which depend on the local slip lengths at the solid and gas areas, are determined, they can be used to solve various hydrodynamic problems. To calculate these eigenvalues, SH surface is usually modeled as a perfectly smooth with patterns of local boundary conditions at solid and gas sectors.It is widely accepted that one can safely impose no-slip at the solid area, i.e., neglect slippage of liquid past smooth solid hydrophobic areas, which is justified provided the nanometric slip is small compared to parameters of the texture <cit.>. For gas sectors of SH surfaces the situation is much less clear. Prior work often appliedshear-free boundary conditions at the flat gas areas <cit.>. In this idealized description both a meniscus curvature and a viscous dissipation in the gas phase, which could affect the local slippage, are fully neglected.Several groups have recently studied the effect of a meniscus curvature on the friction properties of SH surfaces <cit.>. Most of these studies neglected a viscous dissipation in the gas, by focussing on the connection of the meniscus protrusion angle and effective slip length (but note that no attempts have been made to connect the meniscus curvature and a local slip at the gas area). It has been generally concluded that if the protrusion angle is ±π/6 or smaller the effective slip of the SH surface does not differ significantly from expected in the case of the flat interface, so that the model of a flat meniscus can always serve as a decent first-order approximation.There is some literature describing attempts to provide a satisfactory theory of a local slip length, which take into account a viscous dissipation in the gas phase (and as far as we know, all these studies have modeled the liquid-gas interface as flat). We mention below what we believe are the most relevant contributions. To account for a dissipation in gas it is necessary to solve Stokes equations both for the liquid and for the gas phases (Fig. <ref>(a)), by imposing boundary conditionsz=0:𝐮= 𝐮_g,μ∂𝐮_τ∂ z=μ_g∂𝐮_gτ∂ z,where 𝐮 and μ are the velocity and the dynamic viscosity of the liquid, and 𝐮_g and μ _g are those of the gas, 𝐮_τ=(u_x,u_y)is the tangential velocity.Thisproblem has been resolved numerically for rectangular grooves <cit.>. It is however advantageous toreplace the two-phase approach, by a single-phase problemwith spatially dependent partial slip boundary conditions <cit.>. For unidirectional (1D) surfaces they are normally imposed asz=0:𝐮_τ-b(y)∂𝐮_τ∂ z=0,where b(y) is the local scalar slip length, which is varying in one direction only.That the gas flow can be indeed excluded from the analysis being equivalently replaced by b(y) is by no means obvious. Early work has suggested that the effect of gas-filled cavities is equivalent to the introduction of a slip length, proportional to their depth if shallow and to their spacing if deep <cit.>. To model the trapped gas effects theoretically a semi-analytical method, which predicted a non-uniform local slip length distribution across the liquid-gas interface, has been proposed <cit.>. During a last few years several authors havediscussed that the local slip depends on the flow direction <cit.>. Although they did not fully recognize it, their results are equivalent to a tensorial generalization ofEq.(<ref>) z=0:𝐮_τ-𝐛∂𝐮_τ∂ z=0,where 𝐛=b{i,j} is the second-rank local slip length tensor, which isrepresentedby symmetric,positive definite2× 2 matrix diagonalized by a rotation with respect to the alignment of the SH grooves.Recent work has elucidateda mechanism which transplants the flow in the gas to a local slip boundary condition at the liquid-gas interface <cit.>. This study has concluded that thenon-uniform local slip length of a shallow texture is defined by the viscosity contrast and local thickness of a gas cavity, similarly to infinite systems <cit.>. In contrast, the local slip length of a deep texture has been shown to be fully controlled by the dissipation at the edge of the groove, i.e. the point where three phases meet, but not by the texture depth as has sometimes been invoked for explaining the extreme local slip. These results led to simple formulas describing liquid slippage at the trapped gas interface of 1D grooves of width δ and constant depth e^* <cit.> b_,⊥≃μμ_gδβ_,⊥,whereb_,⊥are eigenvalues of the slip length tensor, 𝐛, and β_,⊥ are eigenvalues of the tensorial slip coefficient, β. The latter become linear in e^*/δ, when e^*/δ is small and saturate at large e^*/δ. The validity of this ansatz for rectangular grooves has been confirmed numerically <cit.> and, although indirectly, experimentally <cit.>. Previous investigationshave addressed the question of local slip at the gas areas of rectangular grooves with a constant depth only. We are unaware of any previous work that has quantified liquid slippage at gas areas of more general 1D SH surfaces. In this paper, we explore grooves with beveled edges and a non-uniform depth, which varies with y and depends on the bevel angle, π/2 - α (see Fig. <ref>(b)). An obvious practical advantage of such a relief is that the manufactured grooves become mechanically more stable against bending compared to rectangular ones. This is especially important for dilute textures of large δ, which induce larger slip. It is therefore very timely to understand important consequences of a bevel angle for a generation of liquid slippage at the gas areas of such grooves. Here we present theoretical arguments, which allow one to relate b_,⊥ to e(y, α). Our results show that b_,⊥ are not really sensitive to a bevel angle when SH grooves are shallow and weakly slipping, but the large local slip at deep SH grooves is controlledby their width and bevel angle only. These two parameters could be used to tune the large slip at the gas areas of any grooved SH surface and constrain its attainable upper value. Our paper is organized as follows. In Sec. II we describe our model and justify the choice of model surfaces. Sec. III gives a brief summary of our theoretical results for b_,⊥ obtained in some limiting situations. In Sec. IV we describe a numerical method developed here to compute local slip length profiles. To solve numerically the two-phase hydrodynamic problem we consider separately flows in liquid and gas phases, which is in turn done by using different computational techniques. Our results are discussed in Sec. V, and we conclude in Sec. VI. Details of our asymptotic analysis can be found in Appendix <ref>. § MODEL We considercreeping flow past 1D SH textures of period Land gas area fraction ϕ. The coordinate axis x is parallel to the grooves; the cross-plane coordinates are denoted by y and z. The width of the groove is denoted as δ=L ϕ. The bevel angle of the grooves is π/2 - α, where an angle α≤π/2 is defined relative to the vertical, and the depth of the grooves, e(y, α), is varying in only one direction(see Fig. <ref>(b)). Since we consider SH textures with air trapped in the grooves in a contact with water, theratio of liquid and gas viscosities is typically μ / μ_g ≃ 50, which is much larger than unity. Our results apply to a situation where the capillary and Reynolds numbers are sufficiently small, so that the liquid-gas interface does not deform, but not to the opposite case, where significant deformation of this interface is expected <cit.>.The texture material is characterized by the (unique) Young angle Θ (above π/2) measured taking the horizontal as a reference. Displacing a contact line to the sharp edge of a bevel angle π/2 - α we would observe an apparent variability of the contact angle from Θ to π/2 - α + Θ since the line becomes pinned (seeFig. <ref>). Using simple geometry one can then conclude that in the Cassie statepermitted protrusion angles of a meniscus are confined between π/2 + α - Θ and π - Θ, so that one of these (many) possibleprotrusion angles, which would be observed in practice, should be determined by a pressure drop between the liquid and the gas phases. Sincepositive protrusion angles (convex meniscus) are expected only when an external pressure is applied to the gas phase <cit.>, we can exclude this artificial for SH surfaces case from our consideration. Now simple estimates suggest that forΘ = 2 π /3, typical for SH texture materials, and when α = 0, bounds, which constrain attainable protrusion angles are 0 (flat meniscus) and - π /6 (concave meniscus). Tighter bounds for finite α further constrain the attainable protrusion angle. These angles are too small to significantly reduce the effective slip <cit.>. Therefore, to highlight the effect of viscous dissipation on the slippage in simpler terms we here assume that liquid-gas interface to be flat, which implies that pressure in gas is equal to that in liquid. Such a situation occurs when trapped by texture gas is in contact with the atmosphere <cit.>.Our aim is to investigate how the local depth e(y,α)/δ modifies the slippage of liquid past gas areas. We will provide some general theoretical arguments and results valid for any shape of the SH grooves, and also some simulation results for representative SH grooves. As an initial illustration of our theoretical and computational approach we will consider atrapezoidal surface, where α and the maximal depth, e^*, can be varied independently and in the very large rangee(y, α) =y α, y ≤ e^* tanαe^*, e^* tanα < y ≤δ - e^* tanα(δ - y) α,δ - e^* tanα < yWe should like to mention that motivated by a recent experiment <cit.> there have been already attempts to determine an effective slip past trapezoidal SH grooves <cit.>, but that work has simply assumed the trapezoidal shape of a local scalar slip and no attempt has been made to properly connect it with viscous dissipation in the confined gas.Another type of SH surfaces we explore here are grooves bounded by arcs of circles of radii δ/(2 cosα). These arc-shaped grooves do not have areas of a constant depth, and the e(y, α) profile does not contain sectors where dependence on α and δ disappearse(y, α)=δ/2(√(1/cos^2α-(2y/δ-1)^2)-tanα).Their maximal depth, e^∗, is attained at the midplane, y/δ=0.5, and is given bye^∗ = δ (1 - sinα)/2 cosαEq.(<ref>) shows that arc-shaped grooves can never be really deep since e^∗/δ cannot exceed 0.5. It will be therefore instructive to compare their local slip with that of trapezoidal textures of the same δ and e^∗.§ THEORY Our aim is to calculate the eigenvalues of the local slip length tensor, 𝐛, which can be found from the solution of the two-phase problem for the longitudinal(fastest) and transverse(slowest) flow directions by imposing conditions, Eq.(<ref>)b_∥=.u_x∂ u_x/ ∂ z|_z=0≡μμ_g.u_g,x∂ u_g,x/ ∂ z|_z=0, b_⊥=.u_y∂ u_y/ ∂ z|_z=0≡μμ_g.u_g,yδ∂ u_g,y/ ∂ z|_z=0. However, since all properties of β are inherited by the local slip-length tensor, 𝐛, below we will focus more on calculations of scalar non-uniform eigenvalues, β_∥,⊥ <cit.>. In the general case they can be calculated only numerically. However, for some limiting cases explicit expressions canbe obtained.The solution for local slip lengths can be found analytically close to the edge of the grooves, y/δ≪ 1, and the details of our analysis are given inAppendix <ref>. Here we highlight only the main results.Our theory predicts that in the vicinity of groove edges, the eigenvalues of local slip length of any 1D texture augment from zero as b_∥≃μ/μ_g2y/tan(π/4+α/2), b_⊥≃b_∥/4.In what follows that at small y/δ the slip coefficient grows as β_∥,⊥≃β'_∥,⊥yδ,by having slopesβ'_∥≃2tan(π/4+α/2),β'_⊥≃12tan(π/4+α/2).Note that when α = 0, Eqs.(<ref>) predict β'_∥≃ 2 and β'_⊥≃ 1/2. If α = π/2, i.e. there are no gas sectors, β'_∥,⊥≃ 0. We also remark that the linearity of the Stokes equations implies that near the second edge of the groove, y/δ = 1, we haveβ_∥,⊥≃β'_∥,⊥(1 - yδ).It has been suggested before that local slip length of a shallow groove is defined only by the viscosity contrast and local thickness of a thin lubricating films <cit.>, similarly to infinite systems <cit.>, but also depends strongly on the flow direction. Since in our case the local thickness depends on α the early results <cit.> can be generalized asb_∥≃μμ_g e(y, α), b_⊥≃b_∥/4,and by using Eq.(<ref>) we can immediately formulate equations for eigenvalues of a slip coefficient in this limitβ_∥≃ e(y, α)/δ,β_⊥≃β_∥/4. Although Eqs.(<ref>) and (<ref>) are valid for any groove depth, in the case of weakly slipping shallow groves the contribution of dissipation in the area y/δ≪ 1 should be small compared to the central part of the gas areas, so that β_∥,⊥are controlled by the groove depth as predicted by Eq.(<ref>). However, for strongly slipping deep grooves β_∥,⊥ should depend mainly on α since their maximal values are constrained by β'_∥,⊥/2. This suggests that when grooves are deep, β_∥,⊥ are not sensitive to the shape of their bottom and should saturate at some e(y, α)/δ. Coming back to Eq.(<ref>) we conclude that local slip length at the gas areas of sufficiently deep grooves is defined by the viscosity contrast, their width, and bevel angle, but not by their depth. This implies that δ and α are the only two parameters that could be used to tune the large local slip at SH surfaces. An important remark would be that they also constrain its attainable upper value, and we can conclude that for rectangular grooves b_∥ should be inevitably below μδ/μ_g, and that b_⊥ is always smaller than μδ/4μ_g. For grooves of the same δ with beveled edges these upper (and in fact unattainable) bounds will be even smaller.§ COMPUTATION OF THE TWO-PHASE FLOW NEAR SUPERHYDROPHOBIC GROOVES. A precise discussion of the flow in the vicinity of a SH surface requires a numerical solution of the self-consistent two-phase boundary problem, which is normally done by using finite element <cit.> or boundary element <cit.> methods. Here we suggest a simple alternative approach, which is easier to implement. We start by noting that Eq.(<ref>) does not contain any parameters associated with the flow of liquid. This implies that if Eq.(<ref>) is correct, then β_∥,⊥ are universal characteristics of the groove geometry only. However, applying the same β_∥,⊥ as boundary conditions yields different velocities at the gas-liquid interface, u^int_x,y(y)=u_x,y|_z=0, since the solution of the Stokes equations in liquid depends onμ/μ_g and ϕ. This suggests that one can construct a simple iterative scheme to solve two-phase problem. We apply a lattice-Boltzmann method (LBM) <cit.> to simulate gas flow. It is robust, easy to implement, and allows one to measure velocity 𝐮_g,τ and shear rate ∂𝐮_g,τ/ ∂ z at the interface independently.Webegin with a solution for u^int_x,y obtained for an isolated perfect slip stripe, i.e. in the limit of small ϕ <cit.>u_x,y^int/u^max_x,y = √(1-(2y/δ-1)^2),and impose it as a boundary condition to calculate a flow in the gas phase. The eigenvalues of the local slip coefficient can then be calculated using Eqs.(<ref>) and(<ref>). Note that it is also convenient to use a more accurate formula u_x,y^int/u_x,y^max=cosh ^-1[ cos( π y) /cos( πϕ /2) ] /cosh ^-1[ 1/cos( πϕ /2) ], which is valid for an arbitrary ϕ <cit.>, and that this velocity profile is very close to given by Eq.(<ref>) when ϕ≤ 0.5.These computed eigenvalues specify boundary conditions for liquid flow. We then solve Stokes equations in liquid numerically by using a method based on Fourier series <cit.>. The solution satisfying these new partial slip conditions leads in turn to a better approximation for the velocity at the gas-liquid interface. At the next iteration we use this new velocity profile instead of Eq.(<ref>) to compute more accurate values of ∂𝐮_g,τ/ ∂ z and β_∥,⊥, and so on, until an exact solution is obtained.We use D3Q19 implementation of LBM with the unit length set by the lattice step a, the time-step Δ t, and mass density ρ_0=1. The kinematic viscosity of the fluid isdefined through the relaxation time scale τ as ν=(2τ-1)/6 and kept constant in all the simulations ν=0.03. The simulated system is built in such way that the upper boundary represents a liquid-gas interface (z=0 in Fig. <ref>(b)) and the groove walls are perpendicular to yz-plane. For each simulation run the height of the system N_z is chosen equal to the maximal depth of the groove. To provide sufficient resolution of groove shapes we use a simulation domain of the size N_y = δ = 126 a, N_x = 8 a, and N_z = 12 a - 122 a. To verify the results we have repeated separate runswith 2 times and 4 times larger space resolution and have shown that the maximal error due to discretization does not exceed 1%. At the liquid-gas interface we set 𝐮_g|_z=0=(u^int_x,0,0) for longitudinal and 𝐮_g|_z=0=(0,u^int_y,0) for transverse grooves with u_x,y^int/u^max_x,y given by Eq.(<ref>). The maximum velocity u^max_x,y at y=δ/2 is taken equal to 10^-3 a/Δ t. At the gas-solid interface, given by z=-e(y), where e(y)is defined by Eq.(<ref>) or Eq.(<ref>), we impose no-slip boundary conditions. For trapezoidal grooves we have varied N_z/N_y=e/δ from 0.1 to 0.97 and α from 0 to π/6. For arc-shaped grooves the same values of α define N_z=e^∗ (see Eq.(<ref>)). Finally, the periodic boundary conditions have been set along the x-axis. All simulations are made with an open-source package ESPResSo <cit.>. Toassess the validity and convergence of the approachwecompute the interface velocity, u_x,y^int/u^max_x,y, and the local slip coefficients, β_∥,⊥, at fixedϕ=0.5, α=3 π/32, and e^*/δ=0.97. Fig. <ref> shows the interface velocity profiles and β_∥,⊥ obtained after two iteration steps. We conclude that both longitudinal and transverse velocity at the gas sector nearly coincide with predictions ofEq.(<ref>) taken as an initial approximation, but we remark and stress that at the second iteration step u^max_x,y becomes quite different from used as an initial guess.We also see that β_∥,⊥ computed at the first and the second iteration steps are practically the same. Therefore, in reality our iteration procedure converges extremely fast, so that below we use two iteration steps only. Finally, to calculate the effective slip lengths for given computed local slip length profilesb_eff^,⊥=.⟨ u_x,y⟩⟨∂_z u_x,y⟩|_z=0we solve the Stokes equation in liquid numerically by using the Fourier series method described before in <cit.>. In these calculations we vary ϕ in the range from 0.1 to 0.9. § RESULTS AND DISCUSSIONWe begin by studying the flow field in the gas phase of the grooves. For a longitudinal configuration the gas flows in x-direction only, and its velocity monotonously decays from gas-liquid interface to the bottom of the groove. The streamlines for a transverse case form a single eddy, and the locus of its center depends slightly on the shape of the groove and on α. Fig. <ref> shows typical streamlines for a transverse flow in gas computed for trapezoidal grooves and arc-shaped grooves of α=0 and α=3π/32. The other parameters of a trapezoidal relief are taken the same as used in Fig. <ref>. Note that with these parameters the trapezoidal grooves are roughly twice deeper than arc-shaped. Thus, if α=0, Eq.(<ref>) gives e^∗/δ = 1/2 (cf. e^*/δ = 0.97). Altogether the simulation results show that the velocity field in the gas phase is not a unique function of α and that it generally depends on the relief of grooves and their depth. We note, however, that when the grooves are sufficiently deep, the gas in them is almost stagnant near a bottom, so that increasing the depth further should not change the liquid flow past the SH surface. The computed flow field in the gas phase allows one to immediately deduceβ_∥,⊥. Fig. <ref> shows β_∥,⊥ for trapezoidal grooves ofa fixed α=3π/32 and several depths e^*/δ. We see that for shallow grooves (e^*/δ≪ 1)the β_∥,⊥ profiles can be approximated by trapezoids with the central region of a constant slip β_∥≃ e/δ and β_⊥≃ e/4 δ given by Eqs.(<ref>), and linear regions near edges with local slip coefficients described by Eqs.(<ref>) and (<ref>). For relatively deep grooves, e^*/δ = O(1), β_∥,⊥ profiles practically converge into single curves and are no longer dependent on e^*/δ. We also see that the crossover between the regimes of shallow and deep grooves takes place at intermediate e^*/δ, where β_∥,⊥ are controlled both by e^*/δ and α. To illustrate this we have included in Fig. <ref> the β_∥,⊥ curves computed for arc-shaped grooves of the same α=3π/32, which implies their e^∗/δ≃ 0.37. Remarkably, the β_∥,⊥ profiles are very close to computed for trapezoidal grooves of e^*/δ = 0.35. Some small discrepancies in the shapes of β_∥,⊥ obtained for two types of textures indicate that in the crossover regime the shape of the grooves contributes to a slip length profile, but this should be seen as a second-order correction only. Now we focus on the role of α and study first only deep, e^*/δ=0.97, and shallow, e^*/δ=0.094, trapezoidal groovesThe computed β_⊥ are displayed in Fig. <ref>(a). One can see that in the case of shallow textures β_⊥ generally does not depend on αand is consistent with Eq.(<ref>), but note that there are deviations from Eq.(<ref>) in the vicinity of the edges. For deep trapezoids it decreases with α. These trends fairly agree with the predictions of Eqs.(<ref>) and Eqs.(<ref>). We can now compare β_⊥ for deep trapezoidal grooves with that of arc-shaped grooves with the same bevel angle. The results of calculations are included to Fig. <ref>(a). The data show that at the central part of the gas sector the arc-shaped grooves induce smaller β_⊥ than trapezoidal ones, which simply reflects the fact that they are not deep enough. Indeed, with our values of α, their maximal depth e^∗/δ varies from 0.5 down to 0.32 indicating the crossover regime. However, in the vicinity of the groove edges β_⊥ for both textures appears to be the same. To examine this more closely, the dataobtained with two values of α in the edge region is plotted in Fig. <ref>(b), and we see that in the vicinity of the point where three phases meet β_⊥ computed for different grooves indeed coincide. Altogether these results do confirm that β'_⊥ is determined by α only.Fig. <ref> includes β'_∥,⊥ curves computed fordeep, e^*/δ=0.97, and shallow, e^*/δ=0.094, trapezoidal grooves and arc-shaped grooves. The calculations are made using several α in the range from 0 to π/6, which implies that e^∗/δ varies from 0.5 down to 0.29.The general conclusion from this plot is that β'_∥,⊥ does not depend on the groove shape and depth being a function of α only, so that our results fully verify Eq.(<ref>).We now turn to the effective slip lengths, b_eff^,⊥, and will try to understand if such a two-phase problem can indeed be reduced to a one-phase problem, and whether a SH surface can be modeled as a flat one with patterns of stripes with piecewise constant apparent local slip lengths b^c_∥,⊥. We first fit our numerical data for b_eff^,⊥ to the known formulae <cit.>:[b_eff^≃Lπln[(πϕ/2)]1+Lπ b^c_∥ln[(πϕ/2)+tan(πϕ/2)],; b_eff^⊥≃L2πln[(πϕ/2)]1+L2π b^c_⊥ln[(πϕ/2)+tan(πϕ/2)], ]taking b^c_∥,⊥ as fitting parameters, and then deduce β^c_∥,⊥, which can be defined as <cit.>β^c_∥,⊥ = μ_g μb^c_∥,⊥/δWe stress that with such a definition the eigenvalues of apparent slip coefficients, β^c_∥,⊥, of trapezoidal grooves depend only on α and e^*/δ, but not on y/δ. The curves for β ^c_∥,⊥, computed for several α are plotted Fig. <ref>. For all α these functions saturate already at e^*/δ≥ 1, thereby imposing constraints on the attainable b_eff^,⊥. Also included in Fig. <ref> are β ^c_∥,⊥, for arc-shaped texture of the same α. In this case e(y, α)/δ is nonuniform throughout the cross-section, and we make no attempt to calculate its average or effective value at a given α. Instead, Fig. <ref> is intended to indicate the range of β ^c_∥,⊥, that is expected for arc-shaped grooves, so that in this case we simply plot data as a function of e^∗/δ. One can see that data for arc-shapedgrooves nearly coincide with results for trapezoidal grooves at α = π/8,but at smaller angles there is some discrepancy, especially when α=0 and for a transverse case. The discrepancy is always in the direction of smaller β ^c_∥,⊥, than found for trapezoidal grooves of e=e^∗, indicating that effects of the groove shape become discernible at intermediate depths. We stress, however, that both for trapezoidal and arc-shaped grooves the values of β ^c_∥,⊥, are close and show the same trends. So our results for two types of grooves provide a good sense of the possible local slip of 1D texture of any shape. These resultscan be used to predict upper attainable local and effective slip lengths of complex grooves if their bevel angle and maximal depth are known. Indeed, Fig. <ref> allows one to immediately evaluate β ^c_∥,⊥, and the apparent local slip lengths can be found by using Eq.(<ref>). Once they are known, the eigenvalues of the effective length tensor can be calculated analytically by usingEqs.(<ref>). § CONCLUSION We have analyzed liquid slippage at gas areas of 1D SH surfaces and developed an asymptotic theory, which led to explicit expressions for the longitudinal and transverse slip lengths near the edge of the groove, i.e. the point, where three phases meet. The theory predicts that the local slip lengths in the vicinity of this point always increase linearly with the slope determined solely by a bevel angle of grooves. We have also shown that at a given viscosity contrast the eigenvalues of local slip tensor of strongly slipping deep grooves are fully determined by their width and the value of their bevel angle, but not by their depth as sometimes invoked for explaining the extreme local slip. Thus, it is not necessary to deal with very deep grooves to get a largest possible local slip at the gas areas. However, the eigenvalues of a local slip tensor of weakly slipping shallow grooves are not really sensitive to the bevel angle and are determined mostly by their depth.Altogether, our study shows that for a given width, SH grooves with beveled edges are less efficient than rectangular ones for drag reduction purposes. However, the use of grooves with beveled edges appears as a good compromise between the positive effect of their stability against bending and a moderate reduction of the slippage effect due to a bevel angle. A very large local slip length can be induced by using wide grooves with beveled edges, which would be often impossible for wide rectangular grooves due to their bending instability.To check the validity of our theory, we have proposed an approach to solve numerically the two-phase hydrodynamic problem by considering separately flows in liquid and gas phases, which can in turn be done by using different techniques. Our method significantly facilitates and accelerates calculations compared to classical two-phase numerical schemes. Generally, the numerical results have fully confirmed the theory for limiting cases. They have also clarified that at intermediate depths a particular shape of 1D SH surface modifies the local slip profiles, but only slightly.Our strategy and computational approach can be extended to a situation of a sufficiently curved meniscus. Another fruitful direction could be to consider more complex 2D textures, which include various pillars and holes. Thus, they may guide the design of textured surfaces with superlubricating potential in microfluidic devices, tribology, and more. It would be also interesting to revisit recent analysis of an effective slip of SH surfaces and its various implications since our results suggest that instead of a piecewise constant slip length at the gas areas, a local tensorial slip should be employed to obtain more accurate eigenvalues of the effective slip length tensor.Finally, we mention that our approach can be immediately applied to compute local slip lengths of grooves filled by immiscible liquid of low or high viscosity.This research was partly supported by the the Russian Foundation for Basic Research (grants No. 15-01-03069). § LOCAL SLIP LENGTHS NEAR THE EDGE OF THE GROOVE. Here we obtain the solutions of the Stokes equations and explicit formulas for eigenvalues of the local slip length in the vicinity of the groove edge ( y,z)=(0,0). Following <cit.> we use polar coordinates (r,θ), so that y/δ=-rcosθ , z/δ=rsinθ (see Fig.<ref>), and focus on the case r≪ 1. We will show that in this situation the eigenvalues of the local slip length, b_∥,⊥ augment linearly with r, are proportional to the viscosity ratio when it is large, μ/μ_g≫ 1, and decrease with α. In the case of longitudinal grooves the velocity has only one component 𝐮=(u,0,0) and the Stokes equations reduce to the Laplace equation, Δ u=0. The general solution implies a power-law dependence of velocities on the distance:u_l,g = r^λ_∥[ a_l,gsin( λ_∥θ) +c_l,gcos( λ_∥θ) ],where unknown exponent λ_∥ and coefficients a_l,a_g,c_l,c_g can be found by imposing boundary conditions. These are defined in the usual way. Namely, we use the no-slip boundary conditions at the solid wallsu_l(r,0)=0,u_g(r,θ_w)=0where θ_w=3π /2-α, and at the liquid-gas interface, θ =π, we imposeu_l=u_g,μ∂ _θu_l=μ _g∂ _θu_g.Applying these boundary conditions, we obtain a linear homogeneous systemc_l=0, a_gsin( λ_∥θ_w ) +c_gcos( λ_∥θ_w ) =0, (a_l-a_g)sin( λ_∥π)+ (c_l-c_g)cos( λ_∥π)=0, (μ a_l-μ _ga_g)cos( λ_∥π)- (μ c_l-μ _gc_g)sin( λ_∥π)=0,which allows us to derive an equation for λ_∥ by eliminating a_l, a_g, c_g,tan[ λ_∥(3 π/2-α)] =( μ-μ_g) tan( λ_∥π) /μ+μ_gtan^2( λ_∥π) .This result implies that the value of λ_∥ is determined by μ/μ_g and α.By expanding (<ref>) into a Taylor series in μ_g/μ≪ 1 we getλ_∥ =1/2-μ_g/μtan(π/4+α/2)π+O(μ_g^2μ^2 ) .The longitudinal slip length profile near the edge of the groove can be then calculated as b_∥ = r δ u_x(r, π)∂ _θ u_x(r, π )=-r δtan( λ_∥π) λ_∥.We note that the last equality has been obtained by using Eq.(<ref>). By using Eqs.(<ref>) and (<ref>) we can then derive for μ/μ_g≫ 1b_∥≃μ/μ_g2y/tan(π/4+α/2). In the case of transverse grooves, the solution of Stokes equations can be expressed in terms of a streamfunction ψ which satisfies the biharmonic equation, Δ ^2ψ =0. A general solutionin the liquid phase has the following form <cit.>:[ψ_l (r,θ ) =r^λ_⊥[ a_lsin (λ_⊥θ )+g_lsin ((λ_⊥ -2)θ ).; +. c_lcos (λ_⊥θ )+h_lcos ((λ_⊥ -2)θ )], ]and the radial and the angular components of the velocity areu_lr(r,θ )=∂ _θψr, u_lθ(r,θ )=-∂ _rψ .The same equations are valid for a streamfunction ψ _g and velocities u_gr,u_gθ, in the gas phase, with coefficients a_g, g_g, c_g and h_g in Eq.(<ref>). To find the eight unknown coefficients and λ_⊥ we again apply the boundary conditions at the solid walls and at the liquid-gas interface. The no-slip conditions at the wall, u_r=u_θ=0 at θ=0 and θ=θ_w allow us to formulate four equations of the system:c_l+h_l=0, λ_⊥ a_l+(λ_⊥ -2)g_l=0, a_gsin (λ_⊥θ_w )+g_gsin ((λ_⊥ -2)θ_w )+c_gcos (λ_⊥θ_w )+h_gcos ((λ_⊥ -2)θ_w )=0, λ_⊥ a_gcos (λ_⊥θ_w )+(λ_⊥ -2)g_gcos ((λ_⊥ -2)θ_w )-λ_⊥ c_gsin (λ_⊥θ_w )-(λ_⊥ -2)h_gsin ((λ_⊥ -2)θ_w )=0. The conditions at the interface, θ=π, namely ofthe impermeability, u_lθ=u_gθ=0, and of the continuity of the tangent velocity, u_lr=u_gr, and the shear stress, μ∂ _θu_lr=μ _g∂ _θu_gr, give( a_l+g_l) sin (λ_⊥π )+( c_l+h_l) cos (λ_⊥π ) =0, ( a_g+g_g) sin (λ_⊥π )+( c_g+h_g) cos (λ_⊥π ) =0, [ (a_l-a_g)λ_⊥+(g_l-g_g)(λ_⊥-2)] cos (λ_⊥π )-[ (c_l-c_g)λ_⊥+(h_l-h_g)(λ_⊥-2)] sin (λ_⊥π )=0, [ (μ a_l-μ _ga_g)λ_⊥ ^2+(μ g_l-μ _gg_g)(λ_⊥-2)^2] sin (λ_⊥π )+[ (μ c_l-μ _gc_g)λ_⊥ ^2+(μ h_l-μ _gh_g)(λ_⊥ -2)^2] cos (λ_⊥π )=0.The solution of a system of Eqs.(<ref>)-(<ref>) cannot be reduced to a single equation for λ_⊥, as it has been done in the longitudinal case (and led to λ_∥ described by Eq.(<ref>)). However, it is possible to obtain an asymptotic solutions when μ/μ_g ≫ 1:λ_⊥ =3/2-μ_g/μ2tan(π/4+α/2)π+O(μ_g^2μ^2 ).The local transverse slip length can be calculatedas b_⊥ = rδ u_lr(r, π)∂ _θ u_lr(r, π )=r δtan( λ_⊥π) 2(λ_⊥-1),where the last equality is obtained by using Eqs.(<ref>) and (<ref>). This immediately givesb_⊥≃μμ _gy2tan(π/4+α/2)=b_∥4. | http://arxiv.org/abs/1707.08511v4 | {
"authors": [
"Alexander L. Dubov",
"Tatiana V. Nizkaya",
"Evgeny S. Asmolov",
"Olga I. Vinogradova"
],
"categories": [
"physics.flu-dyn"
],
"primary_category": "physics.flu-dyn",
"published": "20170726155932",
"title": "Boundary conditions at the gas sectors of superhydrophobic grooves"
} |
Spectral Properties]On the First Eigenvalue of the Degenerate p-Laplace Operator in Non-Convex DomainsIn this paper we obtain lower estimates of the first non-trivialeigenvalues of thedegenerate p-Laplace operator, p>2, in a large class of non-convex domains.This study is based on applications of the geometric theory of compositionoperators on Sobolev spaces that permits us to estimates constants of Poincaré-Sobolev inequalities and as an application toderive lowerestimates of the first non-trivialeigenvalues for the Alhfors domains (i.e. to quasidiscs). This class ofdomains includes some snowflakes type domains with fractal boundaries. [ V. Gol'dshtein, V. Pchelintsev, A. Ukhlov=============================================Key words and phrases: elliptic equations, Sobolev spaces, quasiconformal mappings.2010 Mathematics Subject Classification: 35P15, 46E35, 30C65.§ INTRODUCTION In this article we consider the Neumann eigenvalue problem for the two-dimensional degenerate p-Laplace operator (p>2)Δ_p u=div(|∇ u|^p-2∇ u). This operator arises in study of vibrations of nonelastic membranes <cit.>. The weak statement of the frequencies problem for the vibrations of a nonelastic membrane is equivalent to the follows spectral problem: to find eigenvalues μ_p andeigenfunctions u ∈ W^1_p(Ω) for the following variational problem∬ _Ω (|∇ u(x,y)|^p-2∇ u(x,y) ·∇ v(x,y)) dxdy = μ_p ∬ _Ω |u(x,y)|^p-2 u(x,y) v(x,y) dxdy,p>2, for all v ∈ W^1_p(Ω), Ω⊂ℝ^2. The problem of estimates ofμ_p^(1)(Ω) is one of mainly interesting problems of the modern geometric analysis and its applications to the continuummechanics.The classical upper estimate for the first nontrivial Neumann eigenvalue of the Laplace operator μ_2^(1)(Ω)≤μ_2^(1)(Ω^∗)=p^2_n/2/R^2_∗was proved by Szegö <cit.> for simply connected planar domains and by Weinberger<cit.> fordomains in ℝ^n. In this inequality p_n/2 denotes the first positive zero of the function (t^1-n/2J_n/2(t))', and Ω^∗ is an n-ball of the same n-volume as Ω with R_∗ as its radius. In particular, if n=2, we have p_1=j_1,1'≈1.84118 where j_1,1' denotes the first positive zero of the derivative of the Bessel function J_1. The upper estimates of the Laplace eigenvalues with the help of different techniques were intensively studied in the recent decades, see, for example, <cit.>.The usual approach to lower estimates of the Laplace eigenvalues is based on the integral representations machinery inconvex domains. On this base lower estimates of first non-trivial eigenvalues for convex domains were given in terms of Euclidean diameter of the domains (see, for example, <cit.>). But Nikodim type examples <cit.> show that in non-convex domains μ_p^(1)(Ω) any estimates in terms of Euclidean diameters are not relevant.We suggested in our previous works another type of estimates of in terms of integrals of conformal derivatives that can be reformulated in terms of hyperbolic radii of domains. So, we can say that hyperbolic geometry represents a natural language for the spectral properties of the Laplace operator. The integrals of conformal derivatives are not simple for analytical estimates, but if domains allow quasiconformal reflections <cit.> we can simplify the problem and obtain the lower estimates of the principal frequency μ_p^(1)(Ω) in terms of "quasiconformal" geometry of domains.The main result of the paper is:0.2cm Theorem A. Let Ω⊂ℝ^2 be a K-quasidisc. Thenμ_p^(1)(Ω)≥M_p(K)/|Ω|^p/2=M^∗_p(K)/R_∗^p,where R_∗ is a radius of a disc Ω^∗ of the same area as Ω and M^∗_p(K)=M_p(K)π^-p/2. 0.2cmThe quantity M_p(K) in Theorem A depends on p and a quasiconformity coefficient K only:M_p(K)= π^p/2/2^p-2K^2exp(-K^2 π^2(2+ π^2)^2/2log3) × inf_2<α< α*inf_1≤ q≤ 2{(1-δ/1/2-δ)^(δ-1)p C_α^-2} , C_α=10^6/[(α -1)(1- ν(α))]^1/α, where δ=1/q-(α-2)/pα, α*=min(K^2/K^2-1, γ*), where γ* is the unique solution of the equationν(α):=10^4 α((α -2)/(α -1))(24π^2K^2)^α=1. The functionν(α) is a monotone increasing function. Hence for any α<α* the number (1- ν(α))>0 and C_α>0.Recall that K-quasidiscs are images of the unit discs under K-quasiconformal homeomorphisms of the plane ℝ^2. This class includes all Lipschitz simply connected domains but also includes a class of fractal domains (for example, so-called Rohde snowflakes <cit.>). Hausdorff dimensions of quasidiscs boundary can be any number of [1,2). Theorem A is based on the following theorem, which characterizes the Neumann eigenvalues in the terms of conformal derivatives. In previous works we introduced a concept of conformal α-regular domains. Let φ : 𝔻→Ω be a conformal mapping from the unit disc 𝔻 onto bounded domain Ω. The domain Ω is a conformal α-regular for some ∞≥α >2 if φ' | L_α(𝔻)<∞.0.2cm Theorem B. Let φ : 𝔻→Ω be a conformal mapping from the unit disc 𝔻 onto conformal α-regular domain Ω. Then for any p>2 the following inequality holds:1/μ_p^(1)(Ω)≤ C_p · |Ω|^p-2/2·φ' | L_α(𝔻)^2,where C_p=2^p π^α-2/α-p/2inf_q ∈ [1, 2](1-δ/1/2-δ)^(1-δ)p, δ=1/q-α-2/pα.0.2cm As an application of this result we obtain lower estimates of μ_p^(1)(Ω) in the domains bounded by an epicycloid of (n-1) cusps, which are non-convex domains. Example. For n ∈ℕ, the diffeomorphism φ(z)=z+1/nz^n,z=x+iy,is conformal and maps the unit disc 𝔻 onto the domain Ω_n bounded by an epicycloid of (n-1) cusps, inscribed in the circle |w|=(n+1)/n. Since Ω_n is a conformal ∞-regular domain, then we have1/μ_p^(1)(Ω_n)≤2^p+2(n+1/n)^p-2inf_q ∈ [1, 2](1-δ/1/2-δ)^(1-δ)p ,where δ=1/q-1/p. The Theorem B is based on the existence of the composition operator in Sobolev spacesφ^∗: L^1_p(Ω)→ L^1_q(𝔻),q<p,with the norm φ^∗≤ K_p,q(𝔻). In the case of conformal mappings φ : 𝔻→Ω and 1≤ q≤ 2< p<∞, we haveφ^∗≤ K_p,q(𝔻)= (∬_𝔻 |φ'(x,y)|^(p-2)q/p-q dxdy)^p-q/pq≤ |Ω|^p-2/2p·π^2-q/2q. Therefore we can distinguish three different cases of estimates of the composition operators norm if they are generated by conformal mappings. The first case p=q=2 corresponds to the classical Laplace operator. Conformal mappings induces isometries of spaces L^1_2(Ω) and L^1_2(𝔻) and as result we obtain estimates of a first non-trivial eigenvalue with the help ofLebesgue norms of conformal derivatives in spaces L_α(𝔻) <cit.> for α-regular domains Ω. The case p<2 corresponds to singular p-Laplace operators, then (<ref>) is the singular integral and its convergence and estimates of first non-trivial eigenvalues depends on Brennan's Conjecture <cit.> for composition operators. The case p>2 corresponds to degenerate p-Laplace operators, the integral (<ref>) is finite for conformal α-regular domains andthe inverse Hölder inequality permit us to estimate this integral for quasidiscs. The proposed approach permits us also to obtain spectral estimates of degenerate p-Laplace Neumann operator in quasidiscs (Theorem A) in terms of quasiconformal geometry. Theorem A will be illustrated by estimates of the first non-trivial eigenvalue of degenerate p-Laplace operator instar-shaped and spiral-shaped domains. Reformulating the notion of quasidiscs in terms of Ahlfors's three-point condition we obtain Theorem C that gives estimates of the first non-trivial eigenvalues in terms of bounded turning condition. As a consequence we obtain the spectral estimates in snowflake like domains.Recall one more time that a domain Ω is called a conformal α-regular domain <cit.> if φ'∈ L_α(𝔻) for some α>2. The degree α does not depends on choice of a conformal mapping φ:𝔻→Ω(by the Riemann Mapping Theorem) and depends on the hyperbolic metric on Ω only. A domain Ω is a conformal regular domain if it is an α-regular domain for some α>2. Note that any C^2-smooth simply connected bounded domain is ∞-regular (see, for example, <cit.>). A problem of exact constants in (r,q)-Poincaré-Sobolev inequalities is an open problem even in the unit disc.We can use only existing rough estimates of such constants in the case of convex domains <cit.>.Theorem B can be reformulated in terms of hyperbolic radii R(φ(z),Ω) <cit.> of domains.0.2cm Theorem B*. Let φ : 𝔻→Ω be a conformal mapping from the unit disc 𝔻 onto conformal α-regular domain Ω. Then for any p>2 the following inequality holds:1/μ_p^(1)(Ω)≤ C_p · |Ω|^p-2/2(∬_𝔻(R(φ(z),Ω)/1-|z|^2)^α dxdy)^2/α,where C_p=2^p π^α-2/α-p/2inf_q ∈ [1, 2](1-δ/1/2-δ)^(1-δ)p, δ=1/q-α-2/pα. 0.2cm Few words about our machinery that is basedon the geometric theory of composition operators <cit.> and its applications to the Sobolev type embedding theorems <cit.>. The following diagram roughly illustrates the main idea: [W^1_p(Ω)φ^*⟶W^1_q(𝔻); 1c↓ 1c↓;L_s(Ω) (φ^-1)^*⟵ L_r(𝔻). ] Here the operator φ^* defined by the composition rule φ^*(f)=f ∘φ is a bounded composition operator on Sobolev spaces induced by a homeomorphism φ of Ω and 𝔻 and the operator (φ^-1)^* defined by the composition rule (φ^-1)^*(f)=f ∘φ^-1 is a bounded composition operator on Lebesgue spaces. This method allows to transfer Poincaré-Sobolev inequalitiesfrom regular domains (for example, from the unit disc 𝔻) to Ω.0.2cm In the recent works we studied composition operators on Sobolev spaces defined on planar domains in connection with the conformal mappings theory <cit.>. This connection leads to weighted Sobolevembeddings <cit.> with the universal conformal weights. Another application of conformal composition operators was given in <cit.> where thespectral stability problem for conformal regular domains was considered.§ COMPOSITION OPERATORS IN Α-REGULAR DOMAINS Let Ω be a domain in the Euclidean plane ℝ^2. For any 1≤ p<∞ we consider the Lebesgue space L_p(Ω) of measurable functions f: Ω→ℝ equipped with the following norm:f| L_p(Ω)=(∬ _Ω|f(x,y)|^pdxdy)^1/p<∞. We consider the Sobolev space W^1_p(Ω), 1≤ p<∞, as a Banach space of locally integrable weakly differentiable functions f:Ω→ℝ equipped with the following norm: f| W^1_p(Ω)=(∬ _Ω|f(x,y)|^pdxdy)^1/p+ (∬ _Ω|∇ f(x,y)|^pdxdy)^1/p.Recall that the Sobolev space W^1_p(Ω) coincides with the closure of the space of smooth functions C^∞(Ω) in the norm of W^1_p(Ω). We consider also the homogeneous seminormed Sobolev space L^1_p(Ω), 1≤ p<∞, of locally integrable weakly differentiable functions f:Ω→ℝ equipped with the following seminorm: f| L^1_p(Ω)=(∬ _Ω|∇ f(x,y)|^pdxdy)^1/p. By the standard definition functions of L^1_p(Ω) are defined only up to a set of measure zero, but they can be redefined quasieverywhere i. e. up to a set of p-capacity zero. Indeed, every function f∈ L^1_p(Ω) has a unique quasicontinuous representation f̃∈ L^1_p(Ω). A function f̃ is termed quasicontinuous if for any ε >0 there is an openset U_ε such that the p-capacity of U_ε is less than ε and on the set Ω∖ U_ε the functionf̃ is continuous (see, for example <cit.>).Let φ:Ω→Ω be weakly differentiable in Ω. The mapping φ is the mapping of finite distortion if |Dφ(z)|=0 for almost all x∈ Z={z∈Ω : J(x,φ)=0}.Let φ:Ω→Ω be a homeomorphism. Then φ is called a mapping of bounded (p,q)-distortion <cit.>, if φ∈ W^1_1,(Ω), has finite distortion, andK_p,q(Ω)=(∬_Ω(|φ'(x,y)|^p/|J_φ(x,y)|)^q/p-q dxdy)^p-q/pq<∞. Classes of mappings of bounded (p,q)-distortion are closely connected with composition operators on Sobolev spaces.Let Ω and Ω be domains in ℝ^2. We say that a diffeomorphism φ:Ω→Ω induces a bounded composition operator φ^∗:L^1_p(Ω)→ L^1_q(Ω), 1≤ q≤ p≤∞,by the composition rule φ^∗(f)=f∘φ, if the composition φ^∗(f)∈ L^1_q(Ω) is defined quasi-everywhere in Ω and there exists a constant K_p,q(Ω)<∞ such that φ^∗(f)| L^1_q(Ω)≤ K_p,q(Ω)f| L^1_p(Ω)for any function f∈ L^1_p(Ω) <cit.>.The following theorem gives an analytic description of composition operators on Sobolev spaces:<cit.> A homeomorphism φ:Ω→Ω between two domains Ω and Ω induces a bounded composition operator φ^∗:L^1_p(Ω)→ L^1_q(Ω), 1≤ q< p<∞, if and only if φ has finite distortion and is a mapping of bounded (p,q)-distortion. The norm of the composition operator φ^*≤ K_p,q(Ω).Now we establish a connection between conformal α-regular domains and the composition operators on Sobolev spaces. Let Ω⊂ℝ^2 be a simply connected domain. Then Ω is a conformal α-regular domain if and only if any conformal mapping φ: 𝔻→Ω generates a bounded composition operatorφ^*:L_p^1(Ω) → L_q^1(𝔻)for any p ∈ (2 , +∞) and q=pα /(p+ α -2).By Theorem <ref>K_p,q(𝔻)=(∬_𝔻(|φ'(x,y)|^p/J_φ(x,y))^q/p-q dxdy)^p-q/pq<∞if and only if a homeomorphism φ: 𝔻→Ω has finite distortion and induces a bounded composition operatorφ^*: L_p^1(Ω) → L_q^1(𝔻),1≤ q<p<∞. Let Ω is a conformal α-regular domain. Since conformal mappings have finite distortion then for any conformal mapping φ: 𝔻→Ω ∬_𝔻 |φ'(x,y)|^α dxdy< ∞for someα >2. Using the conformal equality |φ'(x,y)|^2= J_φ(x,y)>0, we obtainK_p,q^pq/p-q(𝔻)= ∬_𝔻(|φ'(x,y)|^p/J_φ(x,y))^q/p-q dxdy = ∬_𝔻 |φ'(x,y)|^(p-2)q/p-q dxdy = ∬_𝔻 |φ'(x,y)|^α dxdy< ∞for α=(p-2)q/(p-q). Hence we have a bounded composition operatorφ^*: L_p^1(Ω) → L_q^1(𝔻)for any p ∈ (2 , +∞) and q=pα/(p+α -2).Let us check that q<p. Because p>2 we have that p+α -2> α >2 and so α /(p+α -2) <1. Hence we obtainq<p.Suppose that the composition operatorφ^*: L_p^1(Ω) → L_q^1(𝔻)is bounded for any p ∈ (2 , +∞) and q=pα/(p+α -2). Then ∬_𝔻 |φ'(x,y)|^α dxdy= ∬_𝔻 |φ'(x,y)|^(p-2)q/p-q dxdy = ∬_𝔻(|φ'(x,y)|^p/J_φ(x,y))^q/p-q dxdy= K_p,q^pq/p-q(𝔻) < ∞.If Ω⊂ℝ^2 is a conformal α-regular domain, then by the Sobolev embedding theorem φ belongs to theHölder class H^γ(𝔻), γ=(α-2)/α. Hence, the class of conformal regular domains allows description in terms of γ-hyperbolic boundary condition <cit.>:ρ_Ω≤1/γlog(z_0,∂Ω)/(z,∂Ω)+C_0,z=(x,y),where ρ_Ω is the hyperbolic metric in Ω.Note, that if Ω is a conformal α-regular domain, then it is a domain with γ-hyperbolic boundary condition for γ=(α-2)/α. Inverse, if Ω is a domain with γ-hyperbolic boundary condition, then Ω is a conformal regular domain for some α, but calculation of γ in terms ofα is a non solved problem. For our study we need the exact value of α.Theorem <ref> implies:Let Ω⊂ℝ^2 be a simply connected domain. Then Ω satisfies a γ-hyperbolic boundary condition if and only if any conformal mapping φ: 𝔻→Ω generates a bounded composition operatorφ^*:L_p^1(Ω) → L_q^1(𝔻)for any p ∈ (2 , +∞) and some q=q(p,γ)>2. We define the geodesic diameter _G(Ω) of a domain Ω⊂ℝ^n as_G(Ω)=sup_x,y∈Ω_Ω(x,y).Here _Ω(x,y) is the intrinsic geodesic distance:_Ω(x,y)=inf_γ∈Ω∫_0^1 |γ'(t)| dtwhere infimum is taken over all rectifiable curves γ∈Ω such that γ(0)=x and γ(1)=y.Using <cit.> and Corollary <ref> we obtain a simple necessary geometric condition for domains with γ-hyperbolic boundary condition.Let Ω⊂ℝ^2 be a simply connected domain. IfΩ satisfies a γ-hyperbolic boundary condition, then Ω has a finite geodesic diameter. Note, that this theorem gives a simple proof that "maze-like" domain <cit.> does not satisfies the γ-hyperbolic boundary condition, because this domain obviously has infinite geodesic diameter.§ POINCARÉ-SOBOLEV INEQUALITIES Two-weight Poincaré-Sobolev inequalities. Let Ω⊂ℝ^2 be a planar domain and let h : Ω→ℝ be a real valued locally integrable function such that h(x)>0 a. e. in Ω. We consider the weighted Lebesgue space L_p(Ω,h), 1≤ p<∞ is the spaceof measurable functions f: Ω→ℝwith the finite normf | L_p(Ω,h):= (∬_Ω|f(x)|^ph(x,y) dxdy )^1/p< ∞.It is a Banach space for the norm f | L_p(Ω,h).In the following theorem we obtain the estimate of the norm of the composition operator on Sobolev spaces in any simply connected domain with finite measure.Let Ω⊂ℝ^2 be a simply connected domain with finite measure. Thenconformal mapping φ: 𝔻→Ω generates a bounded composition operatorφ^*:L_p^1(Ω) → L_q^1(𝔻)for any p ∈ (2 , +∞) and q ∈ [1, 2]. By Theorem <ref> a homeomorphism φ: 𝔻→Ω induces a bounded composition operatorφ^∗:L^1_p(𝔻)→ L^1_q(Ω), 1≤ q<p<∞,if and only if φ∈ W_1,^1(Ω), has finite distortion andK_p,q(𝔻)=(∬_𝔻(|φ'(x,y)|^p/J_φ(x,y))^q/p-q dxdy)^p-q/pq<∞.Because φ is a conformal mapping, then φ have finite distortion. Using the conformal equality |φ'(x,y)|^2= J_φ(x,y)>0 we obtainK_p,q(𝔻) = (∬_𝔻(|φ'(x,y)|^p/J_φ(x,y))^q/p-q dxdy)^p-q/pq= (∬_𝔻 |φ'(x,y)|^(p-2)q/p-q dxdy)^p-q/pq.Note that if q≤2 then the quantity (p-2)q/(p-q)≤2. Hence applying Hölder inequality to the last integral we get(∬_𝔻 |φ'(x,y)|^(p-2)q/p-q dxdy)^p-q/pq ≤[(∬_𝔻 |φ'(x,y)|^2 dxdy)^(p-2)q/2(p-q)(∬_𝔻 dxdy)^(2-q)p/2(p-q)]^p-q/pq.By the condition of the theorem, the domain Ω is a simply connected with finite measure thereforeK_p,q(𝔻) ≤ |Ω|^p-2/2p·π^2-q/2q< ∞.We proved that a composition operatorφ^*: L_p^1(Ω) → L_q^1(𝔻)is bounded for any p ∈ (2 , +∞) and q ∈ [1, 2].On the base of this theorem we prove existence of universal two-weight Poincaré-Sobolev inequalities in any simply connected domain Ω⊂ℝ^2 with finite measure. Let Ω⊂ℝ^2 be a simply connected domain with finite measure and h(u,v) =J_φ^-1(u,v) isthe conformal weight defined by a conformal mappingφ : 𝔻→Ω. Then for every function f ∈ W^1_p(Ω), p>2, the inequalityinf_c ∈ℝ(∬_Ω |f(u,v)-c|^rh(u,v) dudv)^1/r≤ B_r,p(Ω,h) (∬_Ω |∇ f(u,v)|^p dudv)^1/pholds for any r ≥ 1 with the constantB_r,p(Ω,h) ≤inf_q ∈ [1, 2]{B_r,q(𝔻) ·π^2-q/2q}· |Ω|^p-2/2p.Here B_r,q(𝔻) is the best constant in the (non-weighted) Poincaré-Sobolev inequality for the unit disc 𝔻⊂ℝ^2. Let r ≥ 1. By the Riemann Mapping Theorem there exists a conformal mapping φ : 𝔻→Ω. Denote by h(u,v): =J_φ^-1(u,v) the conformal weight in Ω.Using the change of variable formula for conformal mapping, the classical Poincaré-Sobolev inequality for the unit disc 𝔻⊂ℝ^2 inf_c ∈ℝ(∬_𝔻 |g(x,y)-c|^r dxdy)^1/r≤ B_r,q(𝔻) (∬_𝔻 |∇ g(x,y)|^q dxdy)^1/qand Theorem <ref>, we get for a smooth function g∈ W^1_p (Ω)inf_c ∈ℝ(∬_Ω |f(u,v)-c|^rh(u,v) dudv)^1/r = inf_c ∈ℝ(∬_Ω |f(u,v)-c|^rJ_φ^-1(u,v) dudv)^1/r= inf_c ∈ℝ(∬_𝔻 |g(x,y)-c|^r dxdy)^1/r≤ B_r,q(𝔻) (∬_𝔻 |∇ g(x,y)|^q dxdy)^1/q ≤ B_r,q(𝔻) ·π^2-q/2q· |Ω|^p-2/2p(∬_Ω |∇ f(u,v)|^p dudv)^1/p. Approximating an arbitrary function f ∈ W^1_p(Ω) by smooth functions we haveinf_c ∈ℝ(∬_Ω |f(u,v)-c|^rh(u,v) dudv)^1/r≤ B_r,p(Ω,h) (∬_Ω |∇ f(u,v)|^p dudv)^1/pwith the constantB_r,p(Ω,h) ≤inf_q ∈ [1, 2]{B_r,q(𝔻) ·π^2-q/2q}· |Ω|^p-2/2p.The property of the conformal α-regularity implies the integrability of aJacobian of conformal mappings and therefore for any conformal α-regular domain we have the embedding of weighted Lebesgue spaces L_r(Ω,h) into non-weighted Lebesgue spaces L_s(Ω) for s=α -2/αr: Let Ω be a conformal α-regular domain.Then for any function f ∈ L_r(Ω,h), α / (α - 2) ≤ r < ∞, the inequalityf | L_s(Ω)≤(∬_𝔻|φ'(x,y)|^α dxdy )^2/α·1/s ||f | L_r(Ω,h)||holds for s=α -2/αr.Because Ω is a conformal α-regular domain then for any conformal mapping φ: 𝔻→Ω we have(∬_𝔻 J_φ^r/r-s(x,y) dxdy)^r-s/rs = (∬_𝔻|φ'(x,y)|^α dxdy)^2/α·1/s< +∞,for s=α -2/αr.Using the change of variable formula for conformal mappings, Hölder's inequality with exponents (r,rs/(r-s)) and the conformal weight h(u,v):=J_φ^-1(u,v), we getf | L_s(Ω)= (∬_Ω |f(u,v)|^s dudv)^1/s =(∬_Ω |f(u,v)|^s J_φ^-1^s/r(u,v) J_φ^-1^-s/r(u,v) dudv)^1/s ≤(∬_Ω |f(u,v)|^r J_φ^-1(u,v) dudv)^1/r(∬_Ω J_φ^-1^- s/r-s(u,v) dudv)^r-s/rs ≤(∬_Ω |f(u,v)|^r h(u,v) dudv)^1/r(∬_𝔻 J_φ^r/r-s(x,y) dxdy)^r-s/rs= (∬_Ω |f(u,v)|^r h(u,v) dudv)^1/r(∬_𝔻|φ'(x,y)|^α dxdy)^2/α·1/s.The following theorem gives the upper estimate of the Poincaré-Sobolev constant as an application of Theorem <ref> and Lemma <ref>: Let Ω⊂ℝ^2 be a conformal α-regular domain. Then for any function f ∈ W^1_p(Ω), p>2, the Poincaré-Sobolev inequalityinf_c ∈ℝ(∬_Ω |f(u,v)-c|^s dudv)^1/s≤ B_s,p(Ω) (∬_Ω |∇ f(u,v)|^p dudv)^1/pholds for any s ≥ 1 with the constantB_s,p(Ω) ≤(∬_𝔻|φ'(x,y)|^α dxdy)^2/α·1/s B_r,p(Ω, h)≤inf_q ∈ [1, 2]{B_α s/α-2,q(𝔻) ·π^2-q/2q}· |Ω|^p-2/2p·φ' | L_α(𝔻)^2/s. Let f ∈ W^1_p(Ω), p>2. Then by Theorem <ref> and Lemma <ref> we get inf_c ∈ℝ(∬_Ω |f(u,v)-c|^s dudv)^1/s ≤(∬_𝔻|φ'(x,y)|^α dxdy)^2/α·1/sinf_c ∈ℝ(∬_Ω |f(u,v)-c|^rh(u,v) dudv)^1/r ≤ B_r,p(Ω, h) (∬_𝔻|φ'(x,y)|^α dxdy)^2/α·1/s(∬_Ω |∇ f(u,v)|^p dudv)^1/pfor s ≥ 1.Because by Lemma <ref> s=α -2/αr and by Theorem <ref> r ≥ 1, then s ≥ 1. By the generalized version of Rellich-Kondrachov compactness theorem (see, for example, <cit.> or <cit.>) and the (r,p)–Sobolev-Poincaré inequality for r>p follows thatthe embedding operatori: W^1_p(Ω) ↪ L_p(Ω)is compact in conformal α-regular domains.Hence, the first non-trivial Neumann eigenvalue μ_p^(1)(Ω) can be characterized <cit.> asμ_p^(1)(Ω)=min{∬_Ω |∇ u(x,y)|^p dxdy/∬_Ω |u(x,y)|^p dxdy : u ∈ W_p^1(Ω) ∖{0},∬_Ω |u|^p-2u dxdy=0 }. Moreover, μ_p^(1)(Ω)^-1/p is the best constant B_p,p(Ω) ( see, for example, <cit.>) in the following Poincaré-Sobolev inequalityinf_c ∈ℝ(∬_Ω |f(x,y)-c|^p dxdy)^1/p≤ B_p,p(Ω) (∬_Ω |∇ f(x,y)|^p dxdy)^1/p,f ∈ W_p^1(Ω). Theorem <ref> in case s=p implies the lower estimates of the first non-trivial eigenvalue of the degenerate p-Laplace Neumann operator in conformal α-regular domains Ω⊂ℝ^2 with finite measure.0.2cm Theorem B. Let φ : 𝔻→Ω be a conformal mapping from the unit disc 𝔻 onto conformal α-regular domain Ω. Then for any p>2 the following inequality holds1/μ_p^(1)(Ω)≤ C_p · |Ω|^p-2/2·φ' | L_α(𝔻)^2,where C_p=2^p π^α-2/α-p/2inf_q ∈ [1, 2](1-δ/1/2-δ)^(1-δ)p, δ=1/q-α-2/pα.0.2cm In case conformal ∞-regular domains we have the following assertion:Let φ : 𝔻→Ω be a conformal mapping from the unit disc 𝔻 onto conformal ∞-regular domain Ω.Then for any p>2 the following inequality holds1/μ_p^(1)(Ω)≤ C_p · |Ω|^p-2/2·φ' | L_∞(𝔻)^2,where C_p=2^p π^1-p/2inf_q ∈ [1, 2](1-δ/1/2-δ)^(1-δ)p, δ=1/q-1/p. 0.2cm As examples, we consider the domains bounded by an epicycloid. Since the domains bounded by an epicycloid are conformal ∞-regular domains, we can apply Corollary <ref>, i.e.: For n ∈ℕ, the diffeomorphism φ(z)=z+1/nz^n,z=x+iy,is conformal and maps the unit disc 𝔻 onto the domain Ω_n bounded by an epicycloid of (n-1) cusps, inscribed in the circle |w|=(n+1)/n. The image of φ for n=2, n=5 and n=8 is illustrated in Figure 2.Now we estimate the norm of the complex derivative φ' in L_∞(𝔻) and the area of domain Ω_n. A straightforward calculation yieldsφ' | L_∞(𝔻)=_|z|≤ 1(|1+z^n-1|)≤ 2and |Ω_n| ≤π(n+1/n)^2.Then by Corollary <ref> we have 1/μ_p^(1)(Ω_n)≤2^p+2(n+1/n)^p-2inf_q ∈ [1, 2](1-δ/1/2-δ)^(1-δ)p ,where δ=1/q-1/p.§ SPECTRAL ESTIMATES IN QUASIDISCSRecall that a domain Ω⊂ℝ^2 is called a K-quasidisc if it is the image of the unit disc 𝔻 under a K-quasiconformal mapping of the plane ℝ^2 onto itself. Note that quasidiscs represent large class domains including fractal type domains like snowflakes.In this section we obtain estimates of integrals of conformal derivatives inquasidiscs.Follow <cit.> a homeomorphism φ:Ω→Ω' between planar domains is called K-quasiconformal if it preserves orientation, belongs to the Sobolev class W_2,^1(Ω) and its directional derivatives ∂_ξ satisfy the distortion inequalitymax_ξ|∂_ξφ|≤ Kmin_ξ|∂_ξφ| a.e. in Ω . For any planar K-quasiconformal homeomorphism φ:Ω→Ω' the following sharp result is known: J(z,φ)∈ L_p,(Ω) for any 1 ≤ p<K/K-1 (<cit.>). Hence for any conformal mapping φ:𝔻→Ω of the unit disc 𝔻 onto a K-quasidisc Ω its derivatives φ'∈ L_p(𝔻) for any1≤ p<2K^2/K^2-1 <cit.>.Using integrability of conformal derivatives on the base of the weak inverse Hölder inequality and the measure doubling condition <cit.> we obtain an estimate of the constant in the inverse Hölder inequality for Jacobians of quasiconformal mappings. The following theorem was proved but not formulated in <cit.>.0.2cmLet φ:ℝ^2 →ℝ^2 be a K-quasiconformal mapping. Then for every disc 𝔻⊂ℝ^2 and for any 1<κ<K/K-1 the inverse Hölder inequality(∬_𝔻 |J_φ(x,y)|^κ dxdy )^1/κ≤C_κ^2 K π^1/κ-1/4exp{K π^2(2+ π^2)^2/2log3}∬_𝔻 |J_φ(x,y)| dxdy.holds. HereC_κ=10^6/[(2κ -1)(1- ν)]^1/2κ, ν = 10^8 κ2κ -2/2κ -1(24π^2K)^2κ<1.If Ω is a K-quasidisc, then a conformal mapping φ: 𝔻→Ω allows K^2-quasiconformal reflection <cit.>. Hence, by Theorem <ref> we obtain the following integral estimates of complex derivatives of conformal mapping φ:𝔻→Ω of the unit disc onto a K-quasidisc Ω: Let Ω⊂ℝ^2 be a K-quasidisc and φ:𝔻→Ω be a conformal mapping. Suppose that2<λ<2K^2/K^2-1. Then (∬_𝔻 |φ'(x,y)|^λ dxdy )^1/λ≤C_λ K π^2-λ/2 λ/2exp{K^2 π^2(2+ π^2)^2/4log3}· |Ω|^1/2.whereC_λ=10^6/[(λ -1)(1- ν)]^1/λ, ν = 10^4 λλ -2/λ -1(24π^2K^2)^λ<1.Combining Theorem B and Corollary <ref> we obtain spectral estimates of the degenerate p-Laplace operator with the Neumann boundary condition:0.2cm Theorem A. Let Ω⊂ℝ^2 be a K-quasidisc. Thenμ_p^(1)(Ω)≥M_p(K)/|Ω|^p/2=M^∗_p(K)/R_∗^p,where R_∗ is a radius of a disc Ω^∗ of the same area as Ω and M^∗_p(K)=M_p(K)π^-p/2. 0.2cm Quasidiscs are conformal α-regular domains for 2<α<2K^2/K^2-1 <cit.>. Then by Theorem B for any 2<α<2K^2/K^2-1 we have1/μ_p^(1)(Ω)≤ C_p · |Ω|^p-2/2·φ' | L_α(𝔻)^2,where C_p=2^p π^α-2/α-p/2inf_q ∈ [1, 2](1-δ/1/2-δ)^(1-δ)p, δ=1/q-α-2/pα.Now we estimate integral from the right-hand side of this inequality. According to Corollary <ref> we obtainφ' | L_α(𝔻)^2= (∬_𝔻 |φ'(x,y)|^α dxdy)^1/α· 2 ≤C_α^2 K^2 π^2/α-1/4exp{K^2 π^2(2+ π^2)^2/2log3}· |Ω|. Combining inequalities (<ref>) and (<ref>)we get the required inequality.As an application of Theorem A we obtain the lower estimates of the first non-trivial eigenvalue on theNeumann eigenvalue problem for the degenerate p-Laplace operator in the star-shaped and spiral-shaped domains.0.2cmStar-shaped domains. We say that a domain Ω^* is β-star-shaped (with respect to z_0=0) if the function φ(z), φ(0)=0, conformally maps a unit disc 𝔻 onto Ω^* and the condition satisfies <cit.>:|z φ^'(z)/φ(z)| ≤βπ/2,0 ≤β <1,|z|<1. In <cit.> proved the following:the boundary of the β-star-shaped domain Ω^* is a K-quasicircle with K= ^2(1-β)π/4.Then by Theorem A we have1/μ_p^(1)(Ω^*)≤inf_α∈(2,2/1-tan ^4(1-β) π/4)inf_q ∈ [1, 2](1-δ/1/2-δ)^(1-δ)p ×2^p-2 C_α^2^4(1-β)π/4/π^p/2exp{π^2(2+ π^2)^2^4(1-β)π/4/2log3}·|Ω^*|^p/2, where δ=1/q-(α-2)/pα,C_α=10^6/[(α -1)(1- ν)]^1/α, ν = 10^4 αα -2/α -1(24π^2^4(1-β)π/4)^α<1. 0.2cmSpiral-shaped domains. We say that a domain Ω_s is β-spiral-shaped (with respect to z_0=0) if the function φ(z), φ(0)=0, conformally maps a unit disc 𝔻 onto Ω_s and the condition satisfies <cit.>:| e^i γz φ^'(z)/φ(z)| ≤βπ/2,0 ≤β <1,|γ|<βπ/2,|z|<1. In <cit.> proved the following: the boundary of the β-spiral-shaped domain Ω_s is a K-quasicircle with K= ^2(1-β)π/4.Then by Theorem A we have1/μ_p^(1)(Ω_s)≤inf_α∈(2,2/1-tan ^4(1-β) π/4)inf_q ∈ [1, 2](1-δ/1/2-δ)^(1-δ)p ×2^p-2 C_α^2^4(1-β)π/4/π^p/2exp{π^2(2+ π^2)^2^4(1-β)π/4/2log3}·|Ω_s|^p/2, where δ=1/q-(α-2)/pα,C_α=10^6/[(α -1)(1- ν)]^1/α, ν = 10^4 αα -2/α -1(24π^2^4(1-β)π/4)^α<1. 0.2cmIn order to obtain the lower estimates of the first non-trivial eigenvalue on theNeumann eigenvalue problem for the degenerate p-Laplace operator in fractal type domains we use description of quasidiscs in the terms of the Ahlfors's 3-point condition: a Jordan curve Γ satisfies the Ahlfors's 3-point condition: there exists a constant C such that|ζ_3-ζ_1| ≤ C|ζ_2-ζ_1|,C ≥ 1for any three points on Γ, where ζ_3 is between ζ_1 and ζ_2.In <cit.> was proved (Theorem 5.1) that if a domain Ω is bounded by Jordan curve Γ satisfies the Ahlfors's 3-point condition, then a conformal mapping φ:𝔻→Ω allows a K^2-quasiconformal extension φ̃:ℝ^2→ℝ^2 withK< 1/2^10exp{(1+e^2 πC^5)^2}. Using the estimate (<ref>) for the quasiconformal coefficient in Theorem A, we obtain lower estimates of the first non-trivial eigenvalues in domains satisfy the Ahlfors's 3-point condition.0.2cm Theorem C. Let a domain Ω⊂ℝ^2 is bounded by a Jordan curve Γ satisfies the Ahlfors's 3-point condition. Then1/μ_p^(1)(Ω)≤inf_q ∈ [1, 2](1-δ/1/2-δ)^(1-δ)p ×2^p-22 C_α^2 e^2(1+e^2πC^5)^2/π^p/2exp{π^2(2+ π^2)^2e^2(1+e^2πC^5)^2/2^21log3}· |Ω|^p/2 holds for 2<α<min(2K^2/K^2-1, γ*), where δ=1/q-(α-2)/pα,γ* is the unique solution of the equation ν(α):=10^4 αα -2/α -1(24π^2K^2)^α=1and C_α=10^6/[(α -1)(1- ν(α))]^1/α.0.2cmUsing this theorem we obtain lower estimates of μ_p^(1) for snowflakes.Rohde snowflake. In <cit.> S. Rohde constructed a collection S of snowflake type planar curves with the intriguing property that each planar quasicircle is bi-Lipschitz equivalent to some curve in S.Rohde's catalog is S:= ⋃_1/4 ≤ t < 1/2 S_t where t is a snowflake parameter. Each curve in S_t is built in a manner reminiscent of the construction of the von Koch snowflake. Thus, each S ∈ S_t is the limit of a sequence S^n of polygons where S^n+1 is obtained from S^n by using the replacement rule illustrated in Figure 3:for each of the 4^n edges E of S^n we have two choices, either we replace E with the four line segments obtained by dividing E into four arcs of equal diameter, or we replace E by a similarity copy of the polygonal arcA_t pictured at the top right of Figure 3. In both cases E is replaced by four new segments, each of these with diameter (1/4) diam(E) in the first case or with diameter t diam(E) in the second case. The second type of replacement is done so that the "tip" of the replacement arc points into the exterior of S^n. This iterative process starts with S^1 being the unit square, and the snowflake parameter, thus the polygon arcA_t, is fixed throughout the construction.The sequence S^n of polygons converges, in the Hausdorff metric, to a planar quasicircle S that we call a Rohde snowflake constructed with snowflake parameter t. Then S_t is the collection of all Rohde snowflakes that can be constructed with snowflake parameter t.In <cit.> established that each Rohde snowflake S in S_t is C-bounded turning withC=C(t)=16/1-2t,1/4 ≤ t < 1/2. A planar curve Γ satisfies the C-bounded turning, C ≥ 1, if for each pair of points x, y, on Γ, the smaller diameter subarc Γ[x,y] of Γ that joins x, y satisfiesdiam(Γ[x,y]) ≤ C|x-y|.The C-bounded turning condition (<ref>) is equivalent the Ahlfors's 3-point condition (<ref>) with the same constant C <cit.>.According to Theorem C and by a known fact that any L-bi-Lipschitz planar homeomorphism is K-quasiconformal with K=L^2 we obtain the following lower estimates of the first non-trivial eigenvalue of the degeneratep-Laplace Neumann operator in domains type a Rohde snowflakes: 0.2cm Let S_t⊂ℝ^2, 1/4 ≤ t < 1/2, be the Rohde snowflake. Then the following inequality1/μ_p^(1)(S_t) ≤inf_q ∈ [1, 2](1-δ/1/2-δ)^(1-δ)p×2^p-22 C_α^2 e^4(1+e^4π(16/(1-2t))^5)^2/π^p/2×exp{π^2(2+ π^2)^2e^4(1+e^4π(16/(1-2t))^5)^2/2^21log3}· |S_t|^p/2 holds for 2<α<2K^2/K^2-1, where δ=1/q-(α-2)/pα,C_α=10^6/[(α -1)(1- ν)]^1/α, ν = 10^4 αα -2/α -1(3π^2/2^17e^4(1+e^2π(16/(1-2t))^5)^2)^α<1. 0.2cm AhlL. 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Anal. 5 (1956) 633–636.0.3cmDepartment of Mathematics, Ben-Gurion University of the Negev, P.O.Box 653, Beer Sheva, 8410501, IsraelE-mail address: [email protected] Department of Higher Mathematics and Mathematical Physics, Tomsk Polytechnic University, 634050 Tomsk, Lenin Ave. 30, Russia;Department of General Mathematics, Tomsk State University, 634050 Tomsk, Lenin Ave. 36, Russia Current address: Department of Mathematics, Ben-Gurion University of the Negev, P.O.Box 653,Beer Sheva, 8410501, Israel E-mail address: [email protected] Department of Mathematics, Ben-Gurion University of the Negev, P.O.Box 653, Beer Sheva, 8410501, Israel E-mail address: [email protected] | http://arxiv.org/abs/1707.08867v2 | {
"authors": [
"V. Gol'dshtein",
"V. Pchelintsev",
"A. Ukhlov"
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"published": "20170727135731",
"title": "On the First Eigenvalue of the Degenerate $p$-Laplace Operator in Non-Convex Domains"
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Analysis of Deformation Fields in Spatio-temporal CBCT images of lungs for radiotherapy patients Sriram Prasath Received: date / Accepted: date ================================================================================================A new MHD solver, based on thespectral/hp element framework, is presented in this paper. The velocity and electric potential quasi-static MHD model is used. The Hartmann flow in plane channel and its stability, the Hartmann flow in rectangular duct, and the stability of Hunt's flow are explored as examples.Exponential convergence is achieved and the resulting numerical values were found to have an accuracy up to 10^-12 for the state flows compared to an exact solution, and 10^-5 for the stability eigenvalues compared to independent numerical results. § INTRODUCTION It is well known that high-order methods have good computational properties, fast convergence, small errors, and the most compact data representation. For many problems in hydrodynamics, high-order methods are necessary. Such problems include the time-dependent simulation of transient flow regimes and the investigation of hydrodynamic stability. Of course, turbulent flows can be investigated using low accuracy schemes (achieving low precision results), but in most cases problems in channels of hydrodynamic stabilityrequire the use of spectral methods. The classical Orr-Sommerfield equation has a small parameter 1/Re at the highest derivative which causes rapidly oscillating solutions. The first numerical calculation of eigenvalues for this equation <cit.> used a high-order finite difference scheme. Later, Orszag <cit.>achieved more accurate results and explained the convenience of using high-order methods for problems of flow stability. A recent review of flow stability in complex geometries and the advantages and disadvantages of high-order methods can be found in <cit.>.§ PROBLEM FORMULATION Consider a flow of incompressible, electrically conducting liquid in the presence of an imposed magnetic field. We suppose that Re_m ≪ Re. In this case a magnetic field generated by the fluid movement does not affect the imposed magnetic field. This is correct for most engineering applications <cit.>.It is now possible to write the Navier-Stokes equation in the form∂v/∂t+( v∇)v = -1/ρ∇ p + νΔv + F(v,H),div v = 0,where v is the velocity, p is the pressure, ν is the viscosity, ρ is the density, F is the magnetic force, and H is the magnitude of the imposed magnetic field.Ohm's law is:j = σ( -∇φ+v×H),where j is the density of electirc current, φ is the electric potential, and σ is the conductivity. Using the law of conservation of electric charge (div j = 0), it is possible to derive the equation for electric potential as:Δφ = ∇(v×H). The system (<ref>) can be written in the form: ∂v/∂t+( v∇)v = -∇ p + 1/ReΔv + St ( -∇φ+v×H)×H ,div v = 0,Δφ = ∇(v×H),where Re = L_0V_0/ν is the Reynolds number, St=σ H_0^2 L_0/ρ V_0 is the magnetic interaction parameter (Stuart number), and L_0, V_0, and H_0 represent the scales of length, velocity and magnitude of the imposed magnetic field, respectively. The system (<ref>) is also known as the MHD system in quasi-static approximation in electric potential form. This system is widely used in theoretical investigations and accurately approximate many cases of liquid metal flows (for example, see the appropriate discussion and reference in <cit.>).The boundary condition for velocity at walls have the form:v = 0,and the boundary condition for the electric potential at perfectly conducting walls is:φ = const.The boundary condition for insulating walls is:∂φ/∂n = 0. § NUMERICAL TECHNIQUE OVERVIEW Our new MHD solver has been developed on the basis of an open source spectral/hp element framework<cit.>. The incompressible Navier-Stokes solver () from the framework has been taken as the source for the MHD solver. uses the velocity correction scheme as described in <cit.>, assuming the time grid t_0, t_1, …,t_n-1,t_n,t_n+1. Using a first-order difference scheme, it is possible to define intermediate velocity ṽ by the equation:ṽ-v_n/δ t = -(v_n ∇)v_n + F_n+St · v_n×( -∇φ_n+v_n×H),where F is a force acting of a fluid. At this stage, thesoftware allows force to be imposed which acts on the flow. The electrical potential is found by solving the equation Δφ_n = ∇(v_n×H) and, after this, the magnetic force in (<ref>) is calculated. We define the second intermediate velocity as:ṽ̂-ṽ/δ t = - ∇ p_n+1.The Poisson equationΔ p_n+1 = ∇(ṽ/δ t)is immediately derived using div ṽ̂ = 0. Thus, at this stage, the divergence-free condition is approximately satisfied. The boundary conditions for pressure are discussed in <cit.>. The last step of the velocity correction procedure is the equation:(Δ - Re/δ t)v_n+1 = - Re/δ tṽ + Re ∇ p_n+1,wich allows us to find the next time-step velocity v_n+1.can use first, second and third order schemes. § THE HARTMANN FLOW IN A PLANE CHANNEL Figure <ref> illustrates a flow in a plane channel. Two parallel infinite planes are installed at points y = ± 1. The liquid between the planes flows under a constant pressure gradient ∇ p in a x direction. The magnetic field H is perpendicular to the planes. This is the Hartmann flow in the plane channel. According to <cit.> the solution of (<ref>) is u(y)/u(0)=cosh (M) - cosh (M y)/cosh (M) - 1where M = √(St · Re), u(0) is a centreline velocity. Velocity graphs are shown in Figure <ref> for M=10,100,10000.Figure <ref> shows a 2D mesh for numeric calculations of the flow. In Figure <ref> one can see large gradients of velocity near the walls in cases of large M and we should take special attention to this part of the flow. It is possible to increase accuracy by mesh concentration near the walls where there are large velocity gradients. This is the h-type solution refinement. The high-order method can increase accuracy by increasing the polynomials' order of an approximation, this is p-refinement. For the flow calculations we will combine both methods by using the order of approximation p from 5 to 25 and mesh condensation near the walls with a coefficient β (β=1 for a uniform grid).When it is supposed that the flow is two-dimensional, all functions are independent from z-coordinate and v_z=0. This proposition leads to the equation Δφ = 0,this means that boundary conditions for φ are not required. The boundary conditions for velocity and pressure arev = 0,∂ p/∂ n = 0at walls, ∂v/∂ n = 0, ∂ p/∂ n = 0at inflow and outflow. In Table <ref>, maximum deviations from the exact solution are presented at M=10∼ 10^4 for different orders of polynomial approximation p. The state flow (<ref>) is calculated as a time-dependent flow with zero initial conditions. Table <ref> includes the running time of the solver on an AMD Ryzen Threadripper 1920X machine with 12 threads. § FLOW IN RECTANGULAR DUCTConsider a steady flow in a rectangular duct. A sketch of this flow is shown in Figure <ref>. The rectangle in Figure <ref> is the cross section of the channel and a uniform magnetic field is applied vertically. Liquid flows under a constant pressure gradient, perpendicular to the plane of the diagram. This flow was investigated analytically in <cit.>.For flow computations in the rectangular duct, the authors use a mesh shown in Figure <ref>.software allows us to set up 3D problems where the homogeneous direction is z, the number of Fourier modes is 2(the minimal possible value).The velocity convergence at points (0.95,0.0) for M=10^3 and (0.98,0.0) for M=10^4 is presented in Table <ref> for the case of perfectly electro-conducting walls. The table includes the mesh concentration coefficient β and velocity values for different p from 5 to 25, the time of calculation and memory usage, for an AMD Ryzen Threadripper 1920X machine with 12 threads. The velocity graph for M=10^3, M=10^4 is shown in Figure <ref>. § THE STABILITY PROBLEMFor a stability analysis let us decompose velocity, pressure and electric potential to formv = U+v, φ = φ_0+φ,p= p_0 +p,where U, φ_0, and p_0 are values of a steady-state flow and v, φ and p are small disturbances. The system (<ref>) becomes linear:∂v/∂t+( U∇)v+( v∇)U = -∇ p + 1/ReΔv + St ( -∇φ+v×H)×H ,div v = 0,Δφ = ∇(v×H). From equations (<ref>) a linear operator A can be constructed:v(x,y,z,T) = A(T)v(x,y,z,0) = λ(T) v(x,y,z,0),where T is a time interval. The linear operator A(T) is constructed numerically by the splitting procedure in the same way as in the case ofnonlinear equations (<ref>). In order to find an eigenvalue λ(T), it is convenient to construct a Krylov subspaceK_n(A,v_0) = span{v_0, A(T) v_0,A(T)^2 v_0, …,A(T)^n-1v_0 },where A(T)^i v_0 is obtained by direct calculation v_1 = A(T)v_0, v_2 = A(T)v_1, …. Further eigenvalue calculations are carried out by standard numerical algebraic techniques, such as the Arnoldi method. The eigenvalues are obtained byin the form:λ(T) = m · e^θ i,and if m>1 then the flow is unstable. The time-independent growth is σ = ln(m)/T and the time-independent frequency is ω=θ/T. § STABILITY OF THE HARTMANN FLOWIn Section <ref>, the Hartmann flow was considered. Now, we will explore the stability of this flow. We take a Hartmann 2D flow disturbance (<ref>) in the form:{v,φ,p } = q(x,y) e^iα(x-C t),where q(x,y)is the amplitude of disturbance, α=2π/γ is the wave-vector, γ is the wavelength, C=X+iY is the phase velocity of disturbance, α X=ω is the frequency, α Y = σ is the growth of disturbance. When σ≤ 0, it means that the flow is stable. The disturbance form (<ref>) is widely used in hydrodynamics stability analysis and leads to the eigenvalue problem, equivalent to (<ref>). As reference data, we take Takashima critical values <cit.> for 2D disturbances. In Table<ref>, growths and frequencies of the disturbances are given for several cases. The values of Re and Mare taken from the article <cit.>, and these are critical values of Hartmann flow. The computational grid is shown in Figure <ref>; nx and ny are the number of cells in the horizontal and vertical directions. The length L of the grid is set up by using a critical wave-vectorα_c = 2π/L. Boundary conditions at inlet and outlet are periodical. In Table <ref> complete coincidence is observed with the reference data from <cit.> and convergence of the eigenvalues is achieved up to 10^-7. Additionally, the table includes the time and memory usage (for the AMD Phenom FX-8150 processor with 8 threads).§ STABILITY OF HUNT'S FLOWConsider a steady flow in the rectangular duct (Figure <ref>), where horizontal walls are perfectly electrically conducting and vertical walls are perfectly electrically insulating. The flow was investigated in <cit.> and it is known as the Hunt's flow. A mesh for base flow and stability calculations is shown at Figure <ref>. Figure <ref> presents a graph of velocity over a line y=0 at M=10, 10^2 10^3. In Table <ref> our calculated eigenvalues are compared with reference values from article <cit.>; the time of calculation and the memory usage are presented. It is possible to see numerical convergence by increasing the order p and match with the reference values from <cit.> up to 10^-5, excluding the case M=10^3.§ CONCLUSIONThis article presents the spectral/hp element solver for MHD problems based on theframework. The solver also makes it possible to investigate the stability of such flows. In order to demonstrate the solver's capacity, several examples were considered: the Hartmann flow in a plane channel and its stability, the Hartmann flow in a rectangular duct, and the stability ofHunt's flow. For the flows, it is easy to find steady-state solutions analytically, and these results were used as the reference test solutions. It was found that the margin of error decreases exponentially with an increasing degree of approximating polynomials, an accuracy 10^-12 can be achieved. To estimate the costs of computer time and memory, these data were listed in the tables for several cases. The computational costs for the stability calculations are large. The first reason for this is the fact that a non-stationary algorithm was used, which allowed use of the non-stationary solver with small adaptations. To obtain eigenvalues with high accuracy, we should set a small time step. The second reason is that we are considering the test examples in 2D for the Hartmann flow and 3D for flows in duct. Usually, these problems can be reduced to the more simple cases described in <cit.>, which can be investigated with much lower computational costs. In this article we demonstrated the accuracy of the method using the well investigated examples.In general, our numerical technique is intended for complex geometry flows where such simplifications are not possible.mhd | http://arxiv.org/abs/1707.08957v4 | {
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"Anatoly M. Sagalakov"
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"published": "20170727055627",
"title": "A spectral/hp element MHD solver"
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The tangent number T_2n+1 is equal to the number of increasing labelled complete binary trees with 2n+1 vertices. This combinatorial interpretation immediately proves that T_2n+1 is divisible by 2^n. However, a stronger divisibility property is known in the studies of Bernoulli and Genocchi numbers, namely, the divisibility of (n+1)T_2n+1 by 2^2n. The traditional proofs of this fact need significant calculations. In the present paper, we provide a combinatorial proof of the latter divisibility by using the hook length formula for trees. Furthermore, our method is extended to k-ary trees, leading to a new generalization of the Genocchi numbers. The Caccioppoli UltrafunctionsVieri Benci, Luigi Carlo Bersellicorresponding author, and Carlo Romano Grisanti====================================================================================§ INTRODUCTIONThe tangent numbers[Some mathematical literature uses a slightly different notation where tan x is written T_1 x + T_2 x^3/3! + T_3 x^5/5! + ⋯(See <cit.>)] (T_2n+1)_n≥ 0 appear in the Taylor expansion of tan(x):tan x = ∑_n≥ 0 T_2n+1x^2n+1/(2n+1)!.It is known that the tangent number T_2n+1 is equal to the number of all alternating permutations of length 2n+1 (see <cit.>). Also, T_2n+1 counts the number of increasing labelled complete binary treeswith 2n+1 vertices. This combinatorial interpretation immediately impliesthat T_2n+1 is divisible by 2^n. However, a stronger divisibility property is known related to the study of Bernoulli and Genocchi numbers <cit.>, as stated in the following theorem.The number (n+1)T_2n+1 is divisible by 2^2n, and the quotient is an odd number.The quotient is called Genocchi number and denoted by G_2n+2:=(n+1)T_2n+1/2^2n. Letg(x):=∑_n≥ 0G_2n+2x^2n+2/(2n+2)!be the exponential generating function for the Genocchi numbers. Then, (<ref>) is equivalent tog(x)=xtanx/2. The initial values of the tangent and Genocchi numbers are listed below:n0 1 2 3 4 5 6T_2n+11 2 16 272 7936 353792 22368256 G_2n+21 1 3 17 155 2073 38227 The fact that the Genocchi numbers are odd integers is traditionally proved by using the von Staudt-Clausen theorem on Bernoulli numbers and the little Fermat theorem<cit.>. Barsky <cit.> gave a different proof by using the Laplace transform. To the best of the authors' knowledge, no simple combinatorial proof has been derived yet and it is the purpose of this paper to provide one.Our approach is based on the geometry of the so-called leaf-labelled tree and the fact that the hook length h_v of such a tree is always an odd integer (see Sections <ref> and <ref>).In Section <ref> we consider the k-ary trees instead of the binary trees and obtain a new generalization of the Genocchi numbers. For each integer k≥ 2, let L_kn+1^(k) be the number of increasing labelled complete k-ary trees with kn+1 vertices.Thus, L^(k)_kn+1 will appear to be a natural generalizationof the tangent number. The general result is stated next. (a) For each integer k≥ 2, the integer (k^2 n-kn+k)! L^(k)_kn+1/ (kn+1)!is divisible by (k!)^kn+1. (b) Moreover, the quotientM^(k)_k^2 n-kn+k:= (k^2 n-kn+k)!L^(k)_kn+1/(k!)^kn+1(kn+1)!≡ 1k,k=p, 1p^2,k=p^t, t≥ 2, 0k,otherwise,where n≥ 1 and p is a prime number. We canrealize that Theorem <ref> is a direct generalization of Theorem <ref>, if we restate the problem in terms of generating functions. Let ϕ^(k)(x) and ψ^(k)(x) denote the exponential generating functions forL^(k)_kn+1 and M^(k)_k^2n-kn+k, respectively, that is,ϕ^(k)(x) =∑_n≥ 0L^(k)_kn+1x^kn+1/(kn+1)!;ψ^(k)(x) =∑_n≥ 0M^(k)_k^2n-kn+kx^k^2n-kn+k/(k^2n-kn+k)!.If k is clear from the context, the superscript (k) will be omitted. Thus, we will writeL_kn+1:=L^(k)_kn+1, M_k^2 n-kn+k := M^(k)_k^2 n-kn+k, ϕ(x):=ϕ^(k)(x), ψ(x):=ϕ^(k)(x). From Theorem <ref> we haveϕ'(x)=1+ϕ^k(x); ψ(x)=x ·ϕ(x^k-1/k!).The last relation becomes the well-known formula (<ref>) when k=2. Several generalizations of the Genocchi numbers have been studied in recent decades. They are based onthe Gandhi polynomials <cit.>, Seidel triangles <cit.>, continued fractions <cit.>, combinatorial models <cit.>, etc. Our generalization seems to be the first extension dealing withthe divisibility of (n+1)T_2n+1 by 2^2n. It also raises the following open problems.Problem 1. Find a proof of Theorem <ref> à la Carlitz, or à la Barsky. Problem 2. Find the Gandhi polynomials, Seidel triangles, continued fractions and a combinatorial model for the new generalization of Genocchi numbers M_k^2n-kn+k à la Dumont. Problem 3. Evaluate m_n:=M_k^2n-kn+k k for k=p^t, where p is a prime number and t≥ 3. It seems that the sequence (m_n)_n≥ 0 is always periodic for any p and t.Computer calculation has provided the initial values:(m_n)_n≥ 0 = (1,1,5,5,1,1,5,5,⋯) fork=2^3, (m_n)_n≥ 0 = (1,1,10,1,1,10,1,1,10⋯) fork=3^3,(m_n)_n≥ 0 = (1,1,126,376,126,1,1,126,376,126,⋯) fork=5^4, (m_n)_n≥ 0 = (1,1,13,5,9,9,5,13,1,1,13,5,9,9,5,13,⋯) fork=2^4. § INCREASING LABELLED BINARY TREES In this section we recall some basic notions onincreasing labelled binary trees. Consider the set 𝒯(n) of all (unlabelled) binary trees with n vertices. For each t∈𝒯(n) let ℒ(t) denote the set of all increasing labelled binary trees of shape t,obtained from t by labeling its n vertices with {1,2,…,n} in such a way that the label of each vertex is less than that of its descendants. For each vertex v of t, the hook length of v, denoted by h_v(t) or h_v, is the number of descendants of v (including v). The hook length formula (<cit.>) claims that the number of increasing labelled binary trees of shape t is equal to n! divided by the product of the h_v's (v∈ t)#ℒ(t)=n!/∏_v∈ t h_v. Let 𝒮(2n+1) denote the set of all complete binary trees s with 2n+1 vertices, which are defined to be the binary trees such that the two subtrees of each vertex are, either both empty, or both non-empty. For example, there are five complete binary trees with 2n+1=7 vertices, labelled by their hook lengths in Fig. 1.beginfig(1, "1.5mm"); setLegoUnit([3,3]) #showgrid([0,0], [20,14])# show grid r=0.15 rtext=r+0.5# distance between text and the pointdef ShowPoint(ptL, labelL, dir, fill=True): [circle(p[z-1], r, fill=fill) for z in ptL] [label(p[ptL[z]-1], labelL[z], dist=[rtext, rtext], dist_direction=dir) for z in range(len(ptL))]dist=[1,1,1,1] p=btree([6,4,7,2,5,1,3], [4,8], dist=dist, dot="fill", dotradius=r) ShowPoint([6,7,5,3], [1,1,1,1], 270) ShowPoint([4,2,1], [3,5,7],135) label(addpt(p[0],[0,1.6]), "s_1")p=btree([4,2,6,5,7,1,3], [8.2,8], dist=dist, dot="fill", dotradius=r) ShowPoint([4,6,7], [1,1,1], 270) ShowPoint([2,1], [5,7],135) ShowPoint([5,3], [3,1],45) label(addpt(p[0],[0,1.6]), "s_2")p=btree([2,1,6,4,7,3,5], [12.4,8], dist=dist, dot="fill", dotradius=r) ShowPoint([5,6,7], [1,1,1], 270) ShowPoint([4,2,1], [3,1,7],135) ShowPoint([3], [5],45) label(addpt(p[0],[0,1.6]), "s_3")p=btree([2,1,4,3,6,5,7], [16.6,8], dist=dist, dot="fill", dotradius=r) ShowPoint([2,4,6,7], [1,1,1,1], 270) ShowPoint([5,3,1], [3,5,7],45) label(addpt(p[0],[0,1.6]), "s_4")p=btree([4,2,5,1,6,3,7], [22.8,8], dist=[1.2, 0.8], dot="fill", dotradius=r) ShowPoint([5,4,6,7], [1,1,1,1], 270) ShowPoint([1,2], [7,3],135) ShowPoint([3], [3],45) label(addpt(p[0],[0,1.6]), "s_5")endfig();< g r a p h i c s >Fig. 1. Complete binary trees with 7 vertices We now define an equivalence relation on 𝒮(2n+1), called pivoting. A basic pivotingis an exchange of the two subtrees of a non-leaf vertex v. For s_1, s_2∈𝒮(2n+1), if s_1 can be changed to s_2 by a finite sequence of basic pivotings, we write s_1∼ s_2. It's routine to check that ∼ is an equivalence relation. Let 𝒮̅(2n+1) = 𝒮(2n+1)/∼. Since s_1∼ s_2 implies that #ℒ(s_1)=#ℒ(s_2), we define #ℒ(s̅)=#ℒ(s) for s∈s̅. ThenT_2n+1 = ∑_s̅∈𝒮̅(2n+1) T(s̅),whereT(s̅)=∑_s ∈s̅#ℒ(s) = #s̅×#ℒ(s̅).For example, consider 𝒮(7) (see Fig. 1), we have[shapes_1s_2s_3s_4s_5;∏_v h_v3· 5· 73· 5· 73· 5· 73· 5· 73· 3· 7; n!/∏_v h_v 48 48 48 48 80 ]Trees s_1, s_2, s_3 and s_4 belong to the same equivalence class s_1, while s_5 is in another equivalence class s_5. Thus T(s_1)=4× 48=192, T(s_5)=80 and T_7=T(s_1)+T(s_5)=272.The pivoting can also be viewed as anequivalence relation on the set ∪_s∈s̅ℒ(s), that is, all increasing labelled trees of shape s with s∈s̅. Since the number of non-leaf vertices is n in s, there are exactly 2^n labelled trees in each equivalence class. Hence, T(s̅) is divisible by 2^n. Take again the example above, T(s_1)/2^3=24, T(s_5)/2^3=10, and T_7/2^3 = 24+10=34.This is not enough to derive that 2^2n| (n+1)T_2n+1. However, the above process leads us to reconsider the question in each equivalence class. We can show that the divisibility actually holds in each s̅, as stated below. For each s̅∈𝒮(2n+1), the integer (n+1)T(s̅) is divisible by 2^2n. Let G(s̅):= (n+1)T(s̅)/2^2n. Proposition <ref> implies that G(s̅) is an integer. By (<ref>) and (<ref>),G_2n+2 = ∑_s̅∈𝒮̅(2n+1) G(s̅).We give an example here and present the proof in the next section.For n=4, there are three equivalence classes. beginfig(2, "1.6mm"); setLegoUnit([3,3]) #showgrid([0,0], [20,14])# show grid r=0.15 rtext=r+0.5# distance between text and the pointdef ShowPoint(ptL, labelL, dir, fill=True): [circle(p[z-1], r, fill=fill) for z in ptL] [label(p[ptL[z]-1], labelL[z], dist=[rtext, rtext], dist_direction=dir) for z in range(len(ptL))]dist=[1,1,1,1] p=btree([8,6,9,4,7,2,5,1,3], [4,8], dist=dist, dot="fill", dotradius=r) ShowPoint([8,9,7,5,3], [1,1,1,1,1], 270) ShowPoint([6,4,2,1], [3,5,7,9],135) label(addpt(p[0],[0,1.6]), "s_1∈s_1")dist=[1.6,1.6,1,1] p=btree([6,4,7,2,8,5,9,1,3], [11,8], dist=dist, dot="fill", dotradius=r) ShowPoint([8,9,7,6,3], [1,1,1,1,1], 270) ShowPoint([4,2,1], [3,7,9],135) ShowPoint([5], [3],45) label(addpt(p[0],[0,1.6]), "s_2∈s_2") dist=[1.6,1,1,1] p=btree([6,4,7,2,5,1, 8,3,9], [18,8], dist=dist, dot="fill", dotradius=r) ShowPoint([8,9,7,6,5], [1,1,1,1,1], 270) ShowPoint([4,2,1], [3,5,9],135) ShowPoint([3], [3],45) label(addpt(p[0],[0,1.6]), "s_3∈s_3") endfig();< g r a p h i c s >Fig. 2. Three equivalence classes for n=4 In this case, Proposition <ref> and relation (<ref>) can be verified by the following table.s̅ #s̅ ∏ h_v #ℒ(s̅) T(s̅) G(s̅) s_1 8 3· 5· 7· 9 384 3072 60 s_2 2 3· 3· 7· 9 640 1280 25 s_3 4 3· 3· 5· 9 896 3584 70 sum 147936 155 § COMBINATORIAL PROOF OF THEOREM <REF>Let n be a nonnegative integer and s̅∈𝒮̅(2n+1) be an equivalence class in the set of increasing labelled complete binary trees. The key of the proof is the fact that the hook length h_v is always an odd integer. For each complete binary tree s, we denote the product of all hook lengths by H(s)=∏_v∈ s h_v. Also, let H(s̅)=H(s) for s∈s̅, since all trees in the equivalence class s̅ share the same product of all hook lengths. For each complete binary tree s, the product of all hook lengths H(s) is an odd integer. By Lemma <ref>, Proposition <ref> has the following equivalent form. For each s̅∈𝒮̅(2n+1), the integer (2n+2)H(s̅)T(s̅) is divisible by 2^2n. By identities (<ref>) and (<ref>) we have(2n+2)H(s̅)T(s̅) =(2n+2)H(s̅)×#s̅×#ℒ(s̅) =(2n+2)×#s̅× (2n+1)! =(2n+2)!×#s̅. Suppose that s is a complete binary tree with 2n+1 vertices, then s has n+1 leaves. Let s^+ be the complete binary tree with 4n+3 vertices obtained from s by replacing each leaf of s by the complete binary tree with 3 vertices. So s^+ has 2n+2 leaves. Let ℒ^+(s^+) be the set of all leaf-labelled trees of shape s^+,obtained from s^+ by labeling its 2n+2 leaves with {1,2,…, 2n+2}. It is clear that #ℒ^+(s^+)=(2n+2)!. By (<ref>) we have the following combinatorial interpretation:For each s̅∈𝒮̅(2n+1), the number of all leaf-labelled trees of shape s^+ such that s∈s̅ is equal to (2n+2)H(s̅)T(s̅). This time we take the pivoting for an equivalence equation on the set of leaf-labelled trees ∪_s∈s̅ℒ^+(s^+). Since a leaf-labelled tree s^+ has 2n+1 non-leaf vertices, and each non-trivial sequence of pivotings will make a difference on the labels of leaves, every equivalence class contains 2^2n+1 elements. Hence, we can conclude that (2n+2)H(s̅)T(s̅) is divisible by 2^2n+1. For example, in Fig. 3, we reproduce a labelled tree with 9 vertices and a leaf-labelled tree with 19 vertices. There are 4 non-leaf vertices in the labelled tree and the 9 non-leaf vertices in the leaf-labelled tree, as indicated by the fat dot symbol “∙”. Comparing with the traditional combinatorial model,our method increases the number of non-leaf vertices. Consequently, we establish a stronger divisibility property. beginfig(3, "1.6mm"); setLegoUnit([3,3]) #showgrid([0,0], [20,14])# show grid dist=[1.6, 1.6, 1] r=0.15 rtext=r+0.5# distance between text and the pointdef ShowPoint(ptL, labelL, dir, fill=True): [circle(p[z-1], r, fill=fill) for z in ptL] [label(p[ptL[z]-1], labelL[z], dist=[rtext, rtext], dist_direction=dir) for z in range(len(ptL))]p=btree([6,4,7,2,8,5,9,1,3],pt=[7,0], dist=dist, dot="frame", dotradius=r, labeled=False) ShowPoint([1,2,4], [1,2,4], 135, fill=True) ShowPoint([6,7,8,9,3], [8,5,7,9,3], 270, fill=False) ShowPoint([5], [6], 60, fill=True) label(addpt(p[0],[0,2.2]), "Labeled tree") label(addpt(p[0],[0,1.4]), "n=4 non-leaf vertices") # third tree is composed by 2 trees, because dist is not equal: (9–4) small dist=[2.2, 2, 1, 0.5] pa=[19,0] p=btree([10,6,11,4,12,7,13,2,14,8,15,5,16,9,17,1,3],pt=pa, dist=dist, dot="frame", dotradius=r) [circle(p[z-1], r, fill=True) for z in [1,2,3,4,5,6,7,8,9]] ShowPoint([10,11,12,13,14,15,16,17], [5,8,2,6,1,7,10,3], 270, fill=False)p=btree([2,1,3], pt=p[2], dist=[dist[3]], dot="frame", dotradius=r) ShowPoint([2,3], [9,4], 270, fill=False) label(addpt(pa,[-0.8,2.2]), "Leaf-labelled tree") label(addpt(pa,[-0.8,1.4]), "2n+1=9 non-leaf vertices") endfig();< g r a p h i c s >Fig. 3. Trees, non-leaf vertices and divisibilities For proving Theorem <ref>, it remains to show that G_2n+2=∑ G(s̅) is an odd number. Since H(s̅) is odd, we need only to prove that the weighted Genocchi numberf(n)=∑_s̅∈𝒮̅(2n+1) H(s̅)G(s̅)is odd.For example, in Fig. 2., G_10=G(s_1)+G(s_2)+G(s_3)=60+25+70=155, andf(4) = H(s_1)G(s_1)+H(s_2)G(s_2)+H(s_3)G(s_3)=3·5·7·9·60 +3·3·7·9·25+3·3·5·9·70 =(3·5·7)^2· 9.The weighted Genocchi number f(n) is more convenient for us to study, since it has an explicit simple expression. Let f(n) be the weighted Genocchi number defined in (<ref>). Then,f(n)=(1· 3 · 5 · 7 ⋯ (2n-1))^2 · (2n+1)=(2n-1)!!· (2n+1)!!. We successively havef(n) = ∑_s̅ H(s̅)G(s̅)= ∑_s̅H(s̅) (n+1) T(s̅)/2^2n= ∑_s̅(2n+2)! ×#s̅/2^2n+1= (2n+2)!/2^2n+1∑_s̅#s̅= (2n+2)!/2^2n+1·#𝒮(2n+1). While #𝒮(2n+1) equals to the Catalan number C_n, we can calculate thatf(n) = (2n+2)!/2^2n+1· C_n= (2n+2)!/2^2n+1·1/n+12nn= (2n-1)!!· (2n+1)!!.FromTheorem <ref>, the weighted Genocchi number f(n) is an odd number. Therefore, the normal Genocchi number G_2n+2 is also odd. This achieves the proof of Theorem <ref>. § GENERALIZATIONS TO K-ARY TREES In this section we assume that k≥ 2 is an integer. Recall the hook length formula for binary trees described in Section 2. For general rooted trees t (see <cit.>), we also have #ℒ(t)=n!/∏_v∈ t h_v,where ℒ(t) denote the set of all increasing labelled trees of shape t. Let L_kn+1 be the number of increasing labelled complete k-ary trees with kn+1 vertices. Then,L_kn+1=∑_n_1+⋯+n_k=n-1knkn_1+1, ⋯, kn_k+1L_kn_1+1⋯ L_kn_k+1.Equivalently, the exponential generating function ϕ(x) for L_kn+1ϕ(x)=∑_n≥ 0L_kn+1x^kn+1/(kn+1)!is the solution of the differential equationϕ'(x)=1+ϕ^k(x)such that ϕ(0)=0.Let ψ(x) be the exponential generating function for M_k^2 n-kn+k which is defined in Theorem <ref>,ψ(x):= ∑_n≥ 0M_k^2 n-kn+kx^k^2n-kn+k/(k^2n-kn+k)!.Thenψ(x)=x ·ϕ(x^k-1/k!). From identities (<ref>) and (<ref>), Theorem <ref> can be restated in the form of power series and differential equations:Letψ(x) be a power series satisfying the following differential equationxψ'(x)-ψ(x)=k-1/k!(x^k+ψ^k(x)),with ψ(0)=0. Then, for each n≥ 1, the coefficient of x^k^2n-kn+k/(k^2n-kn+k)! in ψ(x) is an integer. Moreover, it is congruent to (i)1k, if k=p; (ii) 1 p^2, if k=p^t with t≥ 2; (iii) 0k, otherwise. When k=2, L_2n+1 is just the tangent number T_2n+1 and M_2n+2 is the Genocchi number G_2n+2. For k=3 and 4, the initial values of L_kn+1 and M_k^2 n-kn+k are reproduced below: n L_3n+1 M_6n+3 0 1 1 1 670 2 540 500500 3 184680 43001959000 4 157600080 21100495466050000 5 270419925600 39781831724228093500000 Table for k=3 n L_4n+1 M_12n+4 0 1 1 1 24 525525 2 32256 10258577044340625 3 285272064 42645955937142729593062265625 4 8967114326016 6992644904557760596067178252404694486328125Table for k=4Now we define an equivalence relation (k-pivoting) on the set of all (unlabelled) complete k-ary trees ℛ(kn+1). A basic k-pivotingis a rearrangement of the k subtrees of a non-leaf vertex v. Let r_1, r_2 be two complete k-ary trees, if r_1 can be changed to r_2 by a finite sequence of basic k-pivotings, we write r_1∼ r_2. Hence the set of all complete k-ary trees can be partitioned into several equivalence classes. Let ℛ̅(kn+1) = ℛ(kn+1)/∼, define #ℒ(r̅) = #ℒ(r) for r ∈r̅, then we have∑_r̅∈ℛ̅(kn+1) L(r̅)= L_kn+1,whereL(r̅)=∑_r∈r̅#ℒ(r) = #r̅×#ℒ(r̅). Similar to the case of the tangent numbers, this equivalence relation implies that L(r̅) is divisible by (k!)^n. There is still a stronger divisibility, stated as below: For each r̅∈ℛ̅(kn+1), the number (k^2 n-kn+k)!L(r̅)/(kn+1)! is divisible by (k!)^kn+1.First, we show that the coefficient (k^2 n-kn+k)!/(kn+1)! is divisible by (k-1)!^kn+1. In fact, (k^2 n-kn+k)!/(kn+1)!· (k-1)!^kn+1 =(k^2n-kn+k)·∏_i=1^kn+1i(k-1)-1k-2. It remains to provek^kn+1|(k^2 n-kn+k)! L(r̅)/(kn+1)!· (k-1)!^kn+1. For each vertex v in a complete k-ary tree r, we observe that the hook length h_v satisfies h_v≡ 1 k. Thus,H(r̅)=∏_v∈ rh_v≡ 1 k.Consequently, relation (<ref>) is equivalent to k^kn+1|(k^2 n-kn+k)! L(r̅)H(r̅)/(kn+1)!· (k-1)!^kn+1,which can be rewrittenas (k!)^kn+1| (k^2 n-kn+k)! ×L(r̅)H(r̅)/(kn+1)!. We will prove this divisibility using the following combinatorial model. Let r be a complete k-ary tree with kn+1 vertices. It is easy to show that r has (k-1)n+1 leaves. Replacing all leaves of r by the complete k-ary tree with k+1 vertices, we get a new tree with k^2 n-kn+k leaves, denoted by r^+. Let ℒ^+(r^+) be the set of all leaf-labelled tree of shape r^+, obtained from r^+ by labeling all the leaves with 1,2,…, k^2 n-kn+k. It is clear that #ℒ^+(r^+)=(k^2 n-kn+k)!. On the other hand, by the hook length formula we haveL(r̅)H(r̅)/(kn+1)! = H(r̅)×#r̅×#ℒ(r)/(kn+1)! = #r̅.Thus, the right-hand side of (<ref>) is equal to (k^2 n-kn+k)! ×#r̅, that is, the number of all leaf-labelled trees of shape r^+ such that r∈r̅.Translate the k-pivoting to the set of all leaf-labelled trees of shape r^+ such that r∈r̅. It is easy to check that the k-pivoting is still an equivalence relation. Since a leaf-labelled tree has kn+1 non-leaf vertices, there are (k!)^kn+1 leaf-labelled trees in each equivalence class, which implies that the right-hand side of (<ref>) is divisible by (k!)^kn+1. The following two lemmas will be used for proving Theorem <ref>. Suppose that p is prime number. For each positive integer k, let α(k) be the highest power of p dividing k! and β(k) be the sum of all digits of k in base p. Then,α(k)=∑_i≥ 1⌊k/p^i⌋ =k-β(k)/p-1.For the proof of Lemma <ref>, see <cit.>. Let p≥ 3 be a prime number, then(pk+1)(pk+2)⋯(pk+p-1)≡ (p-1)! p^2. The left-hand side of (<ref>) is equal to(pk)^p-1e_0+⋯+(pk)^2e_p-3+(pk)e_p-2+e_p-1≡ (pk)e_p-2+(p-1)!p^2, where e_j:=e_j(1, 2, ⋯, p-1) are the elementary symmetric functions. See <cit.>. Sincee_p-2=(p-1)!∑_ii^-1≡ (p-1)!∑_ii≡(p-1)!p(p-1)/2≡ 0 p,equality (<ref>) is true. We are ready to prove Theorem <ref>. The first part (a) is an immediate consequence of Lemma <ref> and (<ref>). Let n≥ 1, we construct the following weighted function f(n)=∑_r̅∈ℛ̅(kn+1)H(r̅)M(r̅),whereM(r̅)=(k^2 n-kn+k)!L(r̅)/(k!)^kn+1(kn+1)!.Since H(r̅)≡ 1 k, we have f(n)≡∑_r̅∈ℛ̅(kn+1)M(r̅) = M_k^2 n-kn+k k.Thus, we only need to calculate f(n).f(n) = ∑_r̅ H(r̅)M(r̅)= ∑_r̅H(r̅) × (k^2 n-kn+k)!L(r̅)/(k!)^kn+1(kn+1)!= ∑_r̅(k^2 n-kn+k)! ×#r̅/(k!)^kn+1= (k^2 n-kn+k)!/(k!)^kn+1 C_k(n), whereC_k(n) is the number of all (unlabelled) complete k-ary trees, that is equal to the Fuss-Catalan number <cit.>C_k(n)=(kn)!/n!(kn-n+1)!.Consequently,f(n) = (k^2 n-kn+k)!/(k!)^kn-n+1(kn-n+1)!·(kn)!/(k!)^nn!= ∏_i=0^kn-nik+k-1k-1×∏_j=0^n-1jk+k-1k-1. For proving the second part (b), there are three cases to be considered depending on the value of k.(b1) k=p is a prime integer. We have ip+p-1p-1=(ip+1)(ip+2)⋯(ip+p-1)/1× 2×⋯× (p-1)≡ 1p. Thus f(n)≡ 1p by identity (<ref>). (b2) k=p^t(t≥ 2) where p is a prime integer. If p≥ 3, by Lemma <ref>, we haveip^t+p^t-1p^t-1 =∏_s=0^p^t-1-1(ip^t+sp+1)⋯(ip^t+sp+p-1)/(sp+1)⋯(sp+p-1)·∏_s=1^p^t-1-1ip^t+sp/sp≡[(p-1)!/(p-1)!]^p^t-1·ip^t-1+p^t-1-1p^t-1-1p^2≡ip^t-1+p^t-1-1p^t-1-1p^2≡⋯≡ip+p-1p-1p^2 = (ip+1)(ip+2)⋯(ip+p-1)/1× 2×⋯× (p-1)≡ 1p^2.Thus f(n)≡ 1p^2 for k=p^t with p≥ 3 and t≥ 2.Now suppose p=2 and k=2^t (t≥ 2). We havei2^t+2^t-12^t-1 =∏_s=0^2^t-1-1i· 2^t+2s+1/2s+1·∏_s=1^2^t-1-1i· 2^t+2s/2s = ∏_s=0^2^t-2-1(i· 2^t+4s+1)(i· 2^t +4s+3)/(4s+1)(4s+3)·∏_s=1^2^t-1-1i· 2^t-1+s/s≡(-1/-1)^2^t-2·i· 2^t-1+2^t-1-12^t-1-14≡i· 2^t-1+2^t-1-12^t-1-14≡⋯≡i· 2+2-12-14 = 2i+1.Therefore, by identity (<ref>), we can check thatf(n)≡∏_i=0^(2^t-1)n(2i+1)×∏_j=0^n-1(2j+1) ≡ 14.(b3) Suppose that k has more than one prime factors. We want to prove f(n)≡ 0 k. Let p be a prime factor of k,and write k=b p^m with b≥ 2 and p∤ b. Notice that f(n)| f(n+1) by identity (<ref>). Thus, it suffices to show thatf(1)= (k^2)!/(k!)^k+1≡ 0p^m,which is equivalent toα(b^2 p^2m) - (bp^m+1) α(b p^m) ≥ m.By Legendre's formula (<ref>), the left-hand side of (<ref>)is equal toΔ = 1/p-1( b^2 p^2m - β(b^2) -(b p^m+1) (bp^m -β(b)) ) = 1/p-1( β(b) - β(b^2) +b p^m β(b) -b p^m ).Since β(b^2) ≤ b β(b) and β(b)≥ 2,b≥ 2, we haveΔ ≥1/p-1( (bp^m -b +1)β(b)-b p^m )≥1/p-1( b(p^m -2) +2)≥1/p-1( 2p^m -2 )≥ m.This completes the proof. Acknowledgments. The first author would like to thank Zhi-Ying Wen for inviting me to Tsinghua University where the paper was finalized. 10Andre1879 D. André. Développement de x and tan x. C. R. Math. Acad. Sci. Paris, 88:965–979, 1879.Aval2008 Jean-Christophe Aval. Multivariate Fuss-Catalan numbers. Discrete Math., 308(20):4660–4669, 2008.Barsky1980 D. Barsky. Congruences pour les nombres de Genocchi de deuxième espèce. Groupe d'études d'analyse ultramétrique, Paris, 34:1–13, 1980-81.Carlitz1960 L. Carlitz. The Staudt-Clausen theorem. Math. Mag., 34:131–146, 1960/1961.Carlitz1971 L. Carlitz. A conjecture concerning Genocchi numbers. Norske Vid. Selsk. Skr. (Trondheim), (9):4, 1971.Dickson1919 Leonard Eugene Dickson. History of the theory of numbers. Vol. I: Divisibility and primality. Chelsea Publishing Co., New York, 1966.Domaratzki2004 Michael Domaratzki. Combinatorial interpretations of a generalization of the Genocchi numbers. J. Integer Seq., 7(3):Article 04.3.6, 11, 2004.DumontRand1994 Dominique Dumont and Arthur Randrianarivony. Dérangements et nombres de Genocchi. Discrete Math., 132(1-3):37–49, 1994.Euler1755 Leonhard Euler. Institutiones calculi differentialis, Chap. 7, volume 10 of Opera Mathematica 1, 1913. 1755.FoataHan2008 Dominique Foata and Guo-Niu Han. Principes de combinatoire classique. (on line), 2008. (Cours et exercices corrigés). Niveau master de mathématiques.HanZeng1999den Guo-Niu Han and Jiang Zeng. q-polynômes de Gandhi et statistique de Denert. Discrete Math., 205(1-3):119–143, 1999.Knuth1998Vol3 Donald E. Knuth. The art of computer programming. Vol. 3. Addison-Wesley, Reading, MA, 1998. Sorting and searching, Second edition.KnuthBuckholtz1967 Donald E. Knuth and Thomas J. Buckholtz. Computation of tangent, Euler, and Bernoulli numbers. Math. Comp., 21:663–688, 1967.Macdonald1995 I. G. Macdonald. Symmetric functions and Hall polynomials. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, second edition, 1995. With contributions by A. Zelevinsky, Oxford Science Publications.Nielsen1923 Niels Nielsen. Traité élémentaire des nombres de Bernoulli. Gauthier-Villars, Paris, 1923.RiordanStein1973 John Riordan and Paul R. Stein. Proof of a conjecture on Genocchi numbers. Discrete Math., 5:381–388, 1973.Viennot1982 Gérard Viennot. Interprétations combinatoires des nombres d'Euler et de Genocchi. In Seminar on Number Theory, 1981/1982, pages 94, Exp. No. 11. Univ. Bordeaux I, Talence, 1982.ZengZhou2006 Jiang Zeng and Jin Zhou. A q-analog of the Seidel generation of Genocchi numbers. European J. Combin., 27(3):364–381, 2006. | http://arxiv.org/abs/1707.08882v2 | {
"authors": [
"Guo-Niu Han",
"Jing-Yi Liu"
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"published": "20170727143138",
"title": "Combinatorial proofs of some properties of tangent and Genocchi numbers"
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Ashok Kumar Verma [email protected]]Ashok K. Verma Department of Earth, Planetary, and Space Sciences, University of California, Los Angeles, CA 90095, USA0000-0001-9798-1797]Jean-Luc Margot Department of Earth, Planetary, and Space Sciences, University of California, Los Angeles, CA 90095, USA Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA0000-0001-8834-9423]Adam H. Greenberg Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA We evaluated the prospects of quantifying the parameterized post-Newtonian parameter β and solar quadrupole moment J_2⊙ with observations of near-Earth asteroids with large orbital precession rates (9 to 27 arcsec century^-1).We considered existing optical and radar astrometry, as well as radar astrometry that can realistically be obtained with the Arecibo planetary radar in the next five years.Our sensitivity calculations relied on a traditional covariance analysis and Monte Carlo simulations.We found that independent estimates of β and J_2⊙ can be obtained with precisions of 6×10^-4 and 3×10^-8, respectively.Because we assumed rather conservative observational uncertainties, as is the usual practice when reporting radar astrometry, it is likely that the actual precision will be closer to 2×10^-4 and 10^-8, respectively.A purely dynamical determination of solar oblateness with asteroid radar astronomy may therefore rival the helioseismology determination. § INTRODUCTIONThe parameterized post-Newtonian (PPN) formalism is a useful framework for testing metric theories of gravity <cit.>.It consists of 10 dimensionless parameters that describe the general properties of the metric.In general relativity (GR), only 2 of the 10 parameters are non-zero. They are known as the Eddington-Robertson-Schiff parameters γ and β.γ represents the amount of curvature produced by a unit mass, and β represents the amount of nonlinearity in the superposition law for gravity.Several techniques have been used to place observational bounds on these parameters <cit.>, including observations of the bending and delay of light by spacecraft tracking <cit.> or Very Long Baseline Interferometry <cit.>, and fitting of ephemerides to observations of planetary positions <cit.>.In GR, γ and β are equal to one.Doppler tracking of the Cassini spacecraft has shown that γ does not differ from one by more than 2 × 10^-5 <cit.>.Ephemeris-based studies prior to 2009 indicated that β - 1 does not differ from zero by more than 10^-4 <cit.>.More recently, the availability of precise ranging data from the MESSENGER Mercury orbiter <cit.> enabled improved estimates <cit.>.Here, we evaluate the prospect of asteroid orbit precession measurements to place more stringent bounds on β.We consider Earth-based radar observations of near-Earth asteroids with perihelion shifts larger than 10 arcsec century^-1.Orbital precession can also be caused by the nonuniformity of the gravity field that results from the oblate shape of the Sun.The solar oblateness is characterized by the solar quadrupole moment, J_2⊙ <cit.>.Simultaneous estimation of β and J_2⊙ requires that the precessional effects due to GR and to the Sun's oblateness be disentangled.Fortunately, GR is a purely central effect, whereas the oblateness-induced precession has an inclination dependence.The two effects also have a different distance dependence <cit.>.As a result, observations of a small sample of near-Earth asteroids with a variety of semi-major axes and inclinations (Table <ref>) can in principle be used to estimate β and J_2⊙ <cit.>.Current estimates of the solar quadrupole moment are typically derived on the basis of interior models of the Sun constrained by helioseismology data <cit.>.The current best value from the helioseismology literature is J_2⊙ = (2.2 ± 0.1) × 10^-7 <cit.>.Dynamical estimates that do not rely on fits to helioseismology data yield similar values of J_2⊙ = 2.3± 0.25 × 10^-7 <cit.> and J_2⊙ = 2.25± 0.09 × 10^-7 <cit.>. High-precision dynamical estimates are important to validate our understanding of the interior structure of the Sun.Our simulations of the determination of β and J_2⊙ using a variety of asteroid orbits suggest that independent values of β and J_2⊙ can be obtained with satisfactory precision: with the traditionally conservative assignment of radar uncertainties, β can be constrained at the 6×10^-4 level and J_2⊙ can be constrained at the 3×10^-8 level.With uncertainties that more closely reflect measurement errors, this precision may be improved by a factor of ∼3. (Section <ref>).The outline of this paper is as follows.In Section <ref>, we describe our choice of target asteroids.In Section <ref>, we discuss the estimation of asteroid orbits with optical and radar measurements.Our dynamical model and data reduction procedures are described in Section <ref> and <ref>, respectively. Orbit determination results are presented in Section <ref>. Simulations of the determination of β and J_2⊙ are described in Section <ref>.§ TARGET ASTEROIDS The per-orbit secular advance in the angular position of the perihelion is given by <cit.>δω = 6π GM_⊙/a(1-e^2)c^2[(2-β+2γ)/3] + 6π/2R^2_⊙(1-3/2 sin^2i)/a^2(1-e^2)^2J_2⊙,where ω is the argument of perihelion, GM_⊙ is the Sun's gravitational parameter, R_⊙ is the radius of the Sun, c is the speed of light, and a, e, and i are the semi-major axis, eccentricity, and orbital inclination (with respect to the solar equator) of a planetary body, respectively.Because both GR and solar oblateness affect perihelion precession, estimates of β and J_2⊙ are highly correlated and it is desirable to track a variety of solar system bodies with a range of a, e, i values to disentangle the two effects.Our selection of target asteroids follows the method of <cit.>.We select asteroids with both large perihelion shift values and favorable observing conditions with radar (Table <ref> and Figure <ref>).This sample of asteroid orbits includes a wide range of semi-major axes, eccentricities, and inclinations, which are advantageous when simultaneously solving for β and J_2⊙.The predicted rates of perihelion advance, δ̇ω̇, shown in Figure <ref> and Table <ref> were computed assuming γ = β = 1 and J_2⊙=2.2 × 10^-7. § METHODSWe first determined nominal trajectories for asteroids in our sample with astrometric (i.e., positional) data, both optical and radar (Table <ref>). The process involved three steps: (1) numerical integration of each asteroid's orbit and calculation of partial derivatives of the equations of motion with respect to the solve-for parameters (i.e., the six components of the state vectors), (2) evaluation ofsimulated optical and radar observables and computation of their partial derivatives with respect to the solve-for parameters, and (3) least-squares adjustments to the solve-forparameters.We used the Mission Operations and Navigation Toolkit Environment (MONTE) software <cit.> for orbit determination and parameter estimation. MONTE is an astrodynamics computing platform developed by NASA's Jet Propulsion Laboratory (JPL). MONTE is used for spacecraft navigation and trajectory design.MONTE has also been used for a variety of scientific purposes, including gravity analysis <cit.> and ephemeris generation <cit.>.§.§ Dynamical modelMONTE uses a variable-step Adams-Bashforth method to numerically integrate the equations of motion and corresponding partial derivatives. Our dynamical model includes gravitational forces from the Sun, 8 planets, and 21 minor planets with well-determined masses <cit.>, general relativistic effects, and perturbations due to the oblateness of the Sun.In addition to these forces, we have also modeled the nongravitational Yarkovsky orbital drift.Perihelion advance due to GR and solar oblateness does not affect the value of the semi-major axis, but Yarkovsky drift does.This nongravitational effect has been shown to affect the semi-major axes of small bodies due to the anisotropic re-emission of absorbed sunlight <cit.>. The change in semi-major axis with time due to Yarkovsky orbital drift, ⟨ da/dt ⟩, was estimated for all target asteroids with the method of <cit.>.The values ranged in amplitude between 4 and 50 au/My, which is plausible for kilometer-sized bodies.Only one target (1566 Icarus) is common between our target list and the 42 Yarkovsky detections of <cit.>, and only one target (1999 MN) is common between our target list and the 21 Yarkovsky detections of <cit.>.In both cases, our Yarkovsky drift estimates are consistent with and better constrained than prior work.To initialize the integration process, we used a priori state vectors extracted from the Minor Planet Center (MPC) database <cit.>. §.§ Existing optical and radar astrometry We used both optical and radar astrometry to determine the nominal trajectory of each asteroid. Optical measurements provide positional information on the plane of the sky.They are typically expressed as right ascension (R.A.) and declination (decl.) in the equatorial frame of epoch J2000.0.We downloaded optical astrometry from the MPC <cit.>.We debiased optical astrometry and assigned data weights according to the algorithm recommended by <cit.>.Radar astrometry consists of round-trip light time, a measurement that can provide the asteroid-observer distance, and Doppler shift, a measurement that can provide the line-of-sight velocity of the asteroid with respect to the observer. Radar measurements have fractional uncertainties as small as 10^-8.The addition of radar astrometry can decrease orbital element uncertainties by orders of magnitude compared to an optical-only orbit solution <cit.>.However, the number of radar measurements is typically small compared to the number of optical observations (Table <ref>).We processed a total of 12,102 optical measurements (R.A. and decl. pairs obtained at 6051 epochs), as well as 56 range and 17 Doppler measurements that have been published. §.§ Orbit determination for nominal trajectories In order to compute nominal asteroid trajectories, we computed the expected values of the observables and their partial derivatives with respect to initial state vectors.We calculated weighted residuals by subtracting computed measurements (C) from observed measurements (O) and dividing the result by the corresponding observational uncertainty (σ).We adjusted initial state vectors with an iterative least-squares techniques that minimized the sum of squares of weighted residuals. Because there are 9 targets and 6 orbital elements per asteroid in the nominal situation (γ = 1, β =1,J_2⊙ = 2.2 × 10^-7), we adjusted a total of 54 parameters. We defined outliers as measurements with weighted residuals in excess of three.We identified and rejected 127 epochs with outliers in the optical astrometry.There were no outliers in the radar astrometry.We obtained a measure of the quality of the fit at each iteration by computing the dimensionless rms of the weighted residuals:RMS = √(1/N∑_i=1^N( O_i-C_i/σ_i)^2),where N is the number of observations, O_i is the ith observation, C_i is the ith computed measurement, and σ_i is the observational uncertainty associated with the ith observation.We stopped the iterative process when the change in the RMS of the weighted residuals between two successive iterations was less than 0.01%. RMS residuals smaller than one indicate solutions that provide good fits to the observations (Table <ref>).§.§ Anticipated radar astrometry The objectives of this study are to evaluate the precision with which PPN parameter β and solar quadrupole moment J_2⊙ can be determined from orbital fits constrained by existing and anticipated optical and radar astrometry. To quantify the effect of anticipated radar astrometry on the determination of these parameters,we simulated all existing optical and radar astrometry (Table <ref>) and a number of anticipated Arecibo Observatory range measurements (Table <ref>) with the nominal asteroid trajectories described above. We did not attempt to simulate the effect of additional optical astrometry, which is expected to improve the overall quality of the fits, albeit not as powerfully as radar astrometry <cit.>. To supplement the published astrometry with realistic anticipated values, we used the epochs of closest approach to Earth when the asteroids are detectable with the Arecibo radar (Table <ref>). On the basis of prior experience, we assumed that two to four independent data points would be collected at each future apparition.For apparitions in the past (identified in bold in Table <ref>), we used the number of data points that were actually obtained.In total, we simulated 61 independent range measurements in addition to the 56 published values.For each realization in our simulations, we added noise to the observations by randomly drawing from a Gaussian distribution with zero mean and standard deviation equal to the observational uncertainty.Observational uncertainties for observations in the future were assigned according to signal-to-noise ratio (S/N) and experience, with values ranging between 30 and 900 m. Uncertainties for observations in the past mirrored the actual measurement uncertainties adopted by the observer for these data points. §.§ Orbit determination with estimation of β and J_2⊙ We assigned solve-for parameters to one of two categories: local and global.Local parameters are specific to each asteroid, i.e., the 6 orbital elements or initial state vector (total of 9 × 6 = 54 parameters), whereas global parameters are common to all asteroids, i.e., β and J_2⊙.We jointly solved for these 56 parameters.We used two independent approaches to evaluate the precision in the determination of global parameters β and J_2⊙.First, we used a traditional covariance analysis (Section <ref>) as described in <cit.>. Second, we performed Monte Carlo simulations (Section <ref>) to verify the results of the covariance analysis.§ RESULTS§.§ Covariance analysis A covariance analysis is a powerful technique that can be used to evaluate the precision of solve-for parameters.First, simulated, noise-free measurements and their partial derivatives are computed on the basis of nominal trajectories.A least-squares estimation is then performed, where the estimates logically converge on the nominal values.In the process, the associated covariance matrix is produced. The expected precision of the estimated parameters is then inferred by examining the covariance matrix. The square roots of the diagonal elements provide the one-standard-deviation formal uncertainties.After globalfits of 56 parameters, we obtained the following formal uncertainties:σ_β = 5.6 × 10^-4, σ_J_2⊙ =2.7 × 10^-8,with a correlation coefficient of -0.72.The parameters remain correlated because both GR and solar oblateness contribute to perihelion precession.However, the range of asteroid orbital parameters (Table <ref>) helps reduce the correlation coefficient. Consideration of the Lense-Thirring effect for the Sunincreases our σ_β and σ_J_2⊙ estimates by 0.2% and 4%, respectively. The expected formal uncertainty on J_2⊙ with direct dynamical measurement of asteroids is 2.7 times the uncertainty based onfits to helioseismology data <cit.>. For β, the expected formal uncertainty is about twice the uncertainty obtained with pre-MESSENGER planetary ephemerides <cit.>, ∼7 times the uncertainty obtained with post-MESSENGER planetary ephemerides <cit.>, and ∼14 times the uncertainty obtained with MESSENGER range data <cit.> The formal uncertainties scale linearly with the uncertainties assigned to the measurements.It is often the case that radar observers assign conservative uncertainties, as evidenced by RMS residuals or reduced chi-square metrics that are almost always smaller than unity and most often <0.3 (Table <ref>).Therefore, we anticipate that the actual precision may be improved by a factor of ∼3, and the dynamical determination of J_2⊙ may be as precise as the helioseismology determination.In order to investigate the benefit of future observations, we also performed covariance analyses under the assumption that observations would stop at the end of 2017, 2019, or 2021, as opposed to 2022 in our nominal scenario.The results were σ_β,2017 = 9.6× 10^-4, σ_β,2019 = 7.6× 10^-4, σ_β,2021 = 7.5× 10^-4 and σ_J_2⊙,2017 = 1.9× 10^-7, σ_J_2⊙,2019 = 4.2× 10^-8, σ_J_2⊙,2021 = 3.8× 10^-8.§.§ Monte Carlo simulationsMore robust results can be obtained by performing end-to-end simulations that approximate the actual measurement and estimation process.In these analyses, integration of the trajectories and estimation of the parameters are conducted as described in Section <ref> with two variations.First, we chose initial values of the solve-for parameters that are not identical to their nominal values.For instance, the initial positions and velocities of all asteroids were changed by 10 km and 0.1 ms^-1 in each direction, respectively.Likewise, initial values for β and J_2⊙ were changedby 4 × 10^-4 and 5 × 10^-8, which is approximately five times the uncertainty of recent estimates. Second, we polluted the simulated measurements with independent noise realizations as described in Section <ref>.We performed 500 Monte Carlo simulations. After convergence of the least-squares estimation, we compared the estimated values of solve-for parameters with their nominal values, which produced error estimates. To arrive at an estimate of the uncertainties, we can fit Gaussian distributions to the histograms of error estimates, or we can compute the covariance matrix, as follows:cov(p_i, p_j) = 1/N-1∑_k=1^N(p_i^k - p_i^n)(p_j^k - p_j^n),where N is the total number of simulations, p_i^n is the nominal value of the ith parameter (β=1, J_2⊙ = 2.2 × 10^-7), and p_i^k is the estimated value of the ith parameter from the kth simulation of observations. We used Equation (<ref>) and estimated the formal uncertainties in the solve-for parameters by computing the square root of diagonal elements.We foundσ_β = 7.4 × 10^-4, σ_J_2⊙ =3.7 × 10^-8,with a correlation coefficient of -0.81.These values confirm the covariance analysis results. § CONCLUSIONS A modest observing campaign requiring 50-60 hours of Arecibo telescope time over the next five years can provide about 20 range measurements of asteroids whose orbits exhibit large perihelion shift rates.The Arecibo Planetary Radar facility is required for these measurements because its sensitivity is ∼20 times better than that of other radar systems <cit.>, allowing detection of asteroids that are not detectable elsewhere.The Arecibo measurements will complement existing optical and radar astrometry and enable joint orbital solutions with β and J_2⊙ as adjustable parameters.Independent, purely dynamical determinations of both parameters are important because they place bounds on theories of gravity and the interior structure the of Sun, respectively.Our simulation results likely under-estimated actual precision for two reasons. First, we did not attempt to simulate the impact of future optical astrometry nor improvements to the accuracy of star catalogs.Both of these effects will inevitably improve the quality of the orbital determinations.Second, we assumed, based on historical evidence, that radar observers assign fairly conservative uncertainties to their measurements, which often underestimate the precision of the measurements by a factor of ∼3 (Table <ref>).As a result, we anticipate that the uncertainties of the final estimates will be close toσ_β∼ 2 × 10^-4, σ_J_2⊙∼ 10^-8. § ACKNOWLEDGMENTS A.K.V., J.L.M., and A.H.G. were supported in part by the NASA Planetary Astronomy program under grant NNX12AG34G.J.L.M. and A.H.G. were supported in part by NSF Planetary Astronomy program AST-0929830 and AST-1109772.This work was enabled in part by the Mission Operations and Navigation Toolkit Environment (MONTE).MONTE is developed at the Jet Propulsion Laboratory, which is operated by Caltech under contract with NASA.MONTE v124 <cit.> | http://arxiv.org/abs/1707.08675v3 | {
"authors": [
"Ashok K. Verma",
"Jean-Luc Margot",
"Adam H. Greenberg"
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"categories": [
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"physics.space-ph"
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"primary_category": "astro-ph.EP",
"published": "20170727004925",
"title": "Prospects of dynamical determination of General Relativity parameter beta and solar quadrupole moment J2 with asteroid radar astronomy"
} |
Edge-Fault-Tolerant Single-Source Spanners via Best (or Good) Swap Edges Davide Bilò et al. Dipartimento di Scienze Umanistiche e Sociali, University of Sassari, Italy [email protected] Sasso Science Institute, L'Aquila,Italy. [email protected] di Ingegneria dell'Impresa, University of Rome “Tor Vergata”,Italy. [email protected] of Computer Science, ETH Zürich, Switzerland. [email protected] di Ingegneria e Scienze dell'Informazione e Matematica, University of L'Aquila, Italy. [email protected] di Analisi dei Sistemi ed Informatica, CNR, Rome, Italy.Effective Edge-Fault-Tolerant Single-Source Spanners via Best (or Good) Swap Edges Davide Bilò1 Feliciano Colella2 Luciano Gualà3 Stefano Leucci4 Guido Proietti5,6 December 30, 2023 ==================================================================================== Computing all best swap edges (ABSE) of a spanning tree T of a given n-vertex and m-edge undirected and weighted graph G means to select, for each edge e of T, a corresponding non-tree edge f, in such a way that the tree obtained by replacing e with f enjoys some optimality criterion (which is naturally defined according to some objective function originally addressed by T). Solving efficiently an ABSE problem is by now a classic algorithmic issue, since it conveys a very successful way of coping with a (transient) edge failure in tree-based communication networks: just replace the failing edge with its respective swap edge, so as that the connectivity is promptly reestablished by minimizing the rerouting and set-up costs. In this paper, we solve the ABSE problem for the case in which T is a single-source shortest-path tree of G, and our two selected swap criteria aim to minimize either the maximum or the average stretch in the swap tree of all the paths emanating from the source. Having these criteria in mind, the obtained structures can then be reviewed as edge-fault-tolerant single-source spanners. For them, we propose two efficient algorithms running in O(m n+n^2 log n) and O(m n logα(m,n)) time, respectively, and we show that the guaranteed (either maximum or average, respectively) stretch factor is equal to 3, and this is tight. Moreover, for the maximum stretch, we also propose an almost linear O(m logα(m,n)) time algorithm computing a set of good swap edges, each of which will guarantee a relative approximation factor on the maximum stretch of 3/2 (tight) as opposed to that provided by the corresponding BSE. Surprisingly, no previous results were known for these two very natural swap problems. § INTRODUCTIONNowadays there is an increasing demand for an efficient and resilient information exchange in communication networks. This means to design on one hand a logical structure onto a given communication infrastructure, which optimizes some sought routing protocol in the absence of failures, and on the other hand, to make such a structure resistant against possible link/node malfunctioning, by suitably adding to it a set of redundant links, which will enter into operation as soon as a failure takes place.More formally, the depicted situation can be modeled as follows: the underlying communication network is an n-vertex and m-edge undirected input graph G=(V(G),E(G),w), with positive real edge weights defined by w, the logical (or primary) structure is a (spanning) subgraph H of G, and finally the additional links is a set of edges A in E(G) ∖ E(H). Under normal circumstances, communication takes place on H, by following a certain protocol, but as soon as an edge in H fails, then one or more edges in A come into play, and the communication protocol is suitably adjusted.In particular, if the primary structure is a (spanning) tree of G, then a very effective way of defining the set of additional edges is the following: with each tree edge, say e, we associate a so-called best swap edge, namely a non-tree edge that will replace e once it (transiently) fails, in such a way that the resulting swap tree enjoys some nice property in terms of the currently implemented communication protocol. By doing in this way, rerouting and set-up costs will be minimized, in general, and the quality of the post-failure service remains guaranteed. Then, an all best swap edges (ABSE) problem is that of finding efficiently (in term of time complexity) a best swap edge for each tree edge.Due to their fault-tolerance application counterpart, ABSE problems received a large attention by the algorithmic community. In such a framework, a key role has beenplayed by the Shortest-Path Tree (SPT) structure, which is commonly used for implementing efficiently the broadcasting communication primitive. Indeed, it is was shown already in <cit.> that an effective post-swap broadcast protocol can be put in place just after the original SPT undergoes an edge failure. Not surprisingly then, several ABSE problems w.r.t. an SPT have been studied in the literature, for many different swap criteria.Previous work on swapping in an SPT. Since an SPT enjoys several optimality criteria when looking at distances from the source, say s, several papers have analyzed the problem in various respects. However, most of the efforts focused on the minimization w.r.t. the following two swap criterion: the maximum/average distance from s to any node which remained disconnected from s after a failure. The currently fastest solutions for these two ABSE problems run in O(m logα(m,n)) time <cit.> and O(mα(n,n) log^2 n) time <cit.>, respectively. Moreover, it has been shown that in the swap tree the maximum (resp., average) distance of the disconnected nodes from s is at most twice (resp., triple) that of the new optimum SPT <cit.>, and these bounds are tight.Other interesting swap criteria which have been analyzed include the minimization of the maximum increase (before and after the failure) of the distance from s, and the minimization of the distance from s to the root of the subtree that gets disconnected after the failure <cit.>. Besides the centralized setting, all these swap problems have been studied also in a distributed framework (e.g., see <cit.>).On the other hand, no results are known for the case in which one is willing to select a BSE with the goal of minimizing either the maximum or the average stretch from the source s of the disconnected nodes, where the stretch of a node is measured as the ratio between its distance from s in the swap tree and in a new optimum SPT. This is very surprising, since they are (especially the former one) the universally accepted criterion leading to the design of a spanner, i.e., a sparse subgraph preserving shortest paths (between pairs of vertices of interest) in a graph (also in the presence of failures). In this paper, we aim to fill this gap, by providing efficient solutions exactly for these two swap criteria. Our results. Let us denote by and the ABSE problem w.r.t. the maximum and the average stretch swap criterion, respectively. For such problems, we devise two efficient algorithms running in O(m n +n^2 log n) and O(m n logα(m,n)) time, respectively. Notice that both solutions incorporate the running time for computing all the replacement shortest paths from the source after the failure of every edge of the SPT, as provided in <cit.>, whose computation essentially dominates in an asymptotic sense the time complexity. Our two solutions are based on independent ideas, as described in the following:* for the problem, we develop a centroid decomposition of the SPT, and we exploit a distance property that has to be enjoyed by a BSE w.r.t. a nested and log-depth hierarchy of centroids, which will be defined by the subtree detached from the source after the currently analyzed edge failure. A further simple filtering trick on the set of potential swap edges will allow to reduce them from O(m) to O(n), thus returning the promised O(n^2 log n) time.* for the problem, we instead suitably combine a set of linearly-computable (at every edge fault) information, that essentially will allow to describe in O(1) time the quality of a swap edge. This procedure is in principle not obvious, since to compute the average stretch we need to know, for each swap edge, the O(n) distances to all the nodes in the detached subtree.Again, by filtering on the set of potential swap edges, we will get an O(n^2) running time, which will be absorbed by the all-replacement paths time complexity. Concerning the quality of the corresponding swap trees,we instead show that the guaranteed (either maximum or average, respectively) stretch factor w.r.t. the paths emanating from the source (in the surviving graph) is equal to 3, and this is tight. By using a different terminology, our structures can then be revised as edge-fault-tolerant single-source 3-spanners, and we qualified them as effective since they can be computed quickly, are very sparse, provide a very simple alternative post-failure routing, and finally have a small (either maximum or average) stretch.Although the proposed solutions are quite efficient, their running time can become prohibitive for large and dense input graphs, since in this case they would amount to a time cubic in the number of vertices. Unfortunately, it turns out that their improvement is unlikely to be achieved, unless one could avoid the explicit recomputation of all post-failure distances from the source. To circumvent this problem, we then adopt a different approach, which by the way finds application for the (most relevant) max-stretch measure only: we renounce to optimality in the detection of a BSE, in return of a substantial improvement (in the order of a linear factor in n) in the runtime. More precisely, for such a measure, we will compute in an almost linear O(m logα(m,n)) time a set of good swap edges (GSE), each of which will guarantee a relative approximation factor on the maximum stretch of 3/2 (tight) as opposed to that provided by the corresponding BSE. Moreover, a GSE will still guarantee an absolute maximum stretch factor w.r.t. the paths emanating from the source (in the surviving graph) equal to 3 (tight).Besides that, we also point out another important feature concerned with the computation in a distributed setting of all our good swap edges. Indeed, in <cit.> it was shown that they can be computed in an asynchronous message passing system in essentially optimal ideal time,[This is the time obtained with the ideal assumption that the communication time of each message to a neighboring process takes constant time, as in the synchronous model.] space usage, and message complexity, as opposed to the recomputation of all the corresponding BSE, for which no efficient solution is currently available. Other related results. Besides swap-based approaches, an SPT can be made edge-fault-tolerant by further enriching the set of additional edges, so that the obtained structure has almost-shortest paths emanating from the source, once an edge fails. The currently best trade off between the size of the set of additional edges and the quality of the resulting paths emanating from s is provided in <cit.>, where the authors showed that for any arbitrary constant ε>0, one can compute in polynomial time a slightly superlinear (in n, and depending on ε) number of additional edges in such a way that the resulting structure retains (1+ε)-stretched post-failure paths from the source.For the sake of completeness, we also quickly recall the main results concerned with ABSE problems. For the minimum spanning tree (MST), a BSE is of course one minimizing the cost of the swap tree, i.e., a swap edge of minimum cost. This problem is also known as the MST sensitivity analysis problem, and can be solved in O(mlogα(m,n)) time <cit.>. Concerning the minimum diameter spanning tree, a BSE is instead one minimizing the diameter of the swap tree <cit.>, and the best solution runs in O(m logα(m,n)) time <cit.>. Regarding the minimum routing-cost spanning tree, a BSE is clearly one minimizing the all-to-all routing cost of the swap tree <cit.>, and the fastest solutions for solving this problem has a running time of O(m 2^O(α(n,n))log^2 n) <cit.>. Finally, for a tree spanner, a BSE is one minimizing the maximum stretch w.r.t. the all pair distances, and the fastest solution to date run in O(m^2 logα(m,n)) time <cit.>.To conclude, we point out that the general problem of designing fault-tolerant spanners for the all-to-all case has been extensively studied in the literature, and we refer the interested reader to <cit.> and the references therein.§ PROBLEM DEFINITIONLet G = (V(G), E(G), w) be a 2-edge-connected, edge-weighted, and undirected graph with cost function w : E(G) →ℝ^+. We denote by n and m the number of vertices and edges of G, respectively. If X ⊆ V, let E(X) be the set of edges incident to at least one vertex in X. Given an edge e ∈ E(G), we will denote by G-e the graph obtained from G by removing edge e. Similarly, given a vertex v ∈ V(G), we will denote by G-v the graph obtained from G by removing vertex v and all its incident edges. Let T be an SPT of G rooted at s ∈ V(G). Given an edge e ∈ E(T), we let C(e) be the set of all the swap edges for e, i.e., all edges in E(G) ∖{ e } whose endpoints lie in two different connected components of T-e, and let C(e,X) be the set of all the swap edge for e incident to a vertex in X ⊆ V(G). For any e ∈ E(T) and f ∈ C(e), let T_e/f denote the swap tree obtained from T by replacing e with f. Let T_v = (V(T_v), E(T_v)) be the subtree of T rooted at v ∈ V(G). Given a pair of vertices u,v ∈ V(G), we denote by d_G(u,v) the distance between u and v in G. Moreover, for a swap edge f=(x,y), we assume that the first appearing endvertex is the one closest to the source, and we may denote by w(x,y) its weight. We define the stretch factor of y w.r.t. s,T,G as σ_G(T,y) = d_T(s,y)/d_G(s,y).Given an SPT T of G, the problem is that of finding, for each edge e=(a,b) ∈ E(T), a swap edge f^* such that: f^* ∈min_f ∈ C(e){μ(f):= max_v∈ V(T_b)σ_G-e(T_e/f,v) }. Similarly, the problem is that of finding, for each edge e=(a,b) ∈ E(T), a swap edge f^* such that:f^* ∈min_f ∈ C(e){λ(f):=1/|V(T_b)|∑_v∈ V(T_b)σ_G-e(T_e/f,v) }. We will call μ(f) (resp., λ(f)) the max-(resp., avg-)stretch of f w.r.t. e.§ AN ALGORITHM FOR In this section we will show an efficient algorithm to solve the problem in O(mn + n^2 log n) time. Notice that a brute-force approach would require O(mn^2) time, given by the O(n) time which is needed to evaluate the quality of each of the O(m) swap edges, for each of the n-1 edges of T. Our algorithm will run through n-1 phases, each returning in O(m+nlog n) time a BSE for a failing edge of T, as described in the following.Let us fix e=(a,b) as the failing edge. First, we compute in O(m+ n log n) time all the distances in G-e from s. Then, we filter the O(m) potential swap edges to O(n), i.e., at most one for each node v in T_b. Such a filtering is simply obtained by selecting, out of all edges f=(x,v) ∈ C(e, { v}), the one minimizing the measure d_G(s,x)+w(f). Indeed, it is easy to see that the max-stretch of such selected swap edge is never worse than that of every other swap edge in C(e). This filtering phase will cost O(m) total time. As a consequence, we will henceforth assume that |C(e)| = O(n).Then, out of the obtained O(n) swap edges for e, we further restrict our attention to a subset of O(log n) candidates as BSE, which are computed as follows. Let Λ denote a generic subtree of T_b, and assume that initially Λ=T_b. First of all, we compute in O(|V(Λ)|) time a centroid c of Λ, namely a node whose removal from Λ splits Λ in a forest F of subtrees, each having at most |V(Λ)|/2 nodes <cit.>; then, out of all the swap edges, we select a candidate edge f minimizing the distance from s to c in T_e/f, i.e.,f ∈min_(x',v') ∈ C(e){d_T(s,x')+w(x',v')+d_T(v',c)};then, we compute a critical node z for the selected swap edge f, i.e.,z ∈max_z' ∈ V(T_b)σ_G-e(T_e/f,z').We now select a suitable subtree Λ' of the forest F, and we pass to the selection of the next candidate BSE by recursing on Λ', until |V(Λ')|=1. More precisely, Λ' is the first tree of F containing the first vertex of V(Λ) that is encountered by following the path in T from z towards c (see Figure <ref>).Due to the property of the centroid, the number of recursions will be O(log |V(T_b)|)= O(log n), as promised, each costing O(n) time. Moreover, at least one of the candidate edges will be a BSE for e, and hence it suffices to choose the edge minimizing the maximum stretch among the corresponding O(log n) candidate edges. This step is done within the recursive procedure by comparing the current candidate edge f with the best candidate resulting from the nested recursive calls.A more formal description of each phase is shown in Algorithm <ref>. In the following we prove the correctness of our algorithm.Let e=(a,b) be a failing edge, and let Λ be a subtree of T_b. Given a vertex c ∈ V(Λ), let f ∈min_ (x',v') ∈ C(e){ d_T(s,v') + w(x',v') + d_T(v', c) } and let z be a critical node for f. Let F be the forest obtained by removing the edges incident to c from Λ, and let Λ' be the tree of F containing the first vertex of the path from z to c in T that is also in V(Λ). For any swap edge f' ∈ C(e,V(Λ)), if μ(f') < μ(f) then f' ∈ C(e, V(Λ')). Let f=(x, v) and f' = (x', v'). We show that if v' ∈ V(Λ) ∖ V(Λ') then μ(f') ≥μ(f) (see also Figure <ref>). Indeed: μ(f')≥σ_G-e(T_e/f', z) = d_T_e/f'(s,z)/d_G-e(s, z) =d_T(s,x') + w(f') + d_T(v',c) + d_T(c,z) /d_G-e(s, z)≥ d_T(s,x) + w(f) + d_T(v,c) + d_T(c,z) /d_G-e(s, z)≥ d_T_e/f(s,z) /d_G-e(s, z) = σ_G-e(T_e/f, z)= μ(f),where we used the equality d_T(v',z) = d_T(v',c) + d_T(c,z), which follows from the fact that the path from v' to z in T must traverse c as v' and z are in two different trees of F. If C(e, V(Λ)) contains a BSE for e then (e, Λ) returns a BSE for e. First of all notice that Algorithm <ref> only returns edges in C(e). We prove the claim by induction on |V(Λ)|. If |V(Λ)|=1 and C(e, V(Λ)) contains a BSE f^* for e, then let f be the edge of C(e) returned by Algorithm <ref> and let V(Λ)={c}. By choice of f, for every v ∈ V(T_b),d_T_e/f(s,v)≤ d_T_e/f(s,c)+d_T(c,v)=d_T_e/f^*(s,c)+d_T(c,v) = d_T_e/f^*(s,v),from which we derive that μ(f) = μ(f^*), and the claim follows. If |V(Λ)| > 1 and C(e, V(Λ)) contains a BSE for e, we distinguish two cases depending on whether the edge f computed by Algorithm <ref> is a BSE for e or not. If that is the case, then μ(f) ≤μ(f”)∀ f”∈ C(e) and the algorithm correctly returns f. Otherwise, by Lemma <ref>, any edge f' ∈ C(e, V(Λ)) such that μ(f') < μ(f) must belong to C(e, V(Λ')). It follows that Λ' contains a BSE for e and since 1 ≤ |V(Λ')| < |V(Λ)| we have, by inductive hypothesis, that the edge f' returned by (G, e, Λ') is a BSE for e. Clearly μ(f') < μ(f) and hence Algorithm <ref> correctly returns f'. Since each invocation of Algorithm <ref> requires O(n) time, Lemma <ref> together with the previous discussions allows us to state the main theorem of this section: There exists an algorithm that solves the problem in O(mn + n^2 log n) time.§ AN ALGORITHM FOR In this section we show how the problem can be solved efficiently in O(mn logα (m,n)) time. Our approach first of all, in a preprocessing phase, computes in O(m n logα(m,n)) time all the replacement shortest paths from the source after the failure of every edge of T <cit.>. Then, the algorithm will run through n-1 phases, each returning in O(m) time a BSE for a failing edge of T, as described in the following. Thus, the overall time complexity will be dominated by the preprocessing step.Let us fix e=(a,b)∈ E(T) as the failing edge of T. The idea is to show that, after a O(n) preprocessing time, we can compute the avg-stretch λ(f) of any f in constant time. This immediately implies that we can compute a BSE for e by looking at all O(m) swap edges for e.Let U=V(T_b) and let y be a node in U, we define:M(y)=∑_v ∈ Ud_T(y,v)/d_G-e(s,v)andQ = ∑_v ∈ U1/d_G-e(s,v).Let f=(x,y) be a candidate swap edge incident in y ∈ U. The avg-stretch of f can be rewritten as: λ(f)=∑_v ∈ Ud_T(s,x)+w(f)+d_T(y,v)/d_G-e(s,v) = (d_T(s,x)+w(f))Q + M(y). Hence, the avg-stretch of f can be computed in O(1) time, once Q and M(y) are available in constant time. Observe that Q does not depend on y and can be computed in O(n) time. The rest of this section is devoted to show how to compute M(y) for every y ∈ U in O(n) overall time.§.§.§ Computing M(y) for all y ∈ U.Let y e y' be two nodes in U such that y is a child of y' in T. Moreover, let U_y=V(T_y), and let Q_y = ∑_v ∈ U_y1/d_G-e(s,y). Hence, we can rewrite M(y) and M(y') as follows: M(y) = ∑_v ∈ U_yd_T(y,v)/d_G-e(s,v) + ∑_v ∈ U-U_yw(y,y')+d_T(y',v)/d_G-e(s,v)andM(y') = ∑_v ∈ U_yw(y,y')+d_T(y,v)/d_G-e(s,v) + ∑_v ∈ U-U_yd_T(y',v)/d_G-e(s,v).Therefore, we have:M(y) = M(y') + w(y,y') (- Q_y + (Q - Q_y)) = M(y') + w(y,y') (Q - 2Q_y).The above equation implies that M(y) can be computed in O(1) time, once we have computed M(y'), Q and Q_y. As a consequence, we can compute all the M(y)'s as follows. First, we compute Q_y for every y∈ D in O(n) overall time by means of a postorder visit of T_b. Notice also that Q=Q_b. Then, we compute M(b) explicitly in O(n) time. Finally, we compute all the other M(y)'s by performing a preorder visit of T_b. When we visit a node y, we compute M(y) in constant time using (<ref>). Thus, the visit will take O(n) time. We have proved the following:There exists an algorithm that solves the problem in O(mn logα (m,n)) time. § AN APPROXIMATE SOLUTION FOR In this section we show that for the max-stretch measure we can compute in an almost linear O(m logα(m,n)) time, a set of good swap edges (GSE), each of which guarantees a relative approximation factor on the maximum stretch of 3/2 (tight), as opposed to that provided by the corresponding BSE. Moreover, as shown in the next section, each GSE still guarantees an absolute maximum stretch factor w.r.t. the paths emanating from the source (in the surviving graph) equal to 3 (tight).Let e be a failing edge in T, letg=(x,y) ∈min_(x',v') ∈ C(e){d_T(s,x')+w(x',v')},and, finally, let f=(x',y') be a best swap edge for e w.r.t. . Then, μ(g)/μ(f) ≤ 3/2. Let z be the critical node for the good swap edge g, and let t (resp., t') denote the least common ancestor in T between y' and z (resp., y' and y). Let D=d_T(s,x)+w(x,y)=d_G-e(s,y). By choice of g, it holds that d_G-e(s,z) ≥ D and d_G-e(s,y' ) ≥ D. We divide the proof into the following two cases, as depicted in Figure <ref>: either (1) t is an ancestor of t' in T, or (2) t' is an ancestor of t in T. Let A,B,C denote the distance in T between y and t', t' and t, t and z, respectively.§.§.§ Case 1Since t is an ancestor of t' (left side of Figure <ref>), we have that d_T_e/f(s,y) ≥ D+A and we can write:σ_G-e(T_e/f,y) ≥D+A/d_G-e(s,y) = D+A/D≥D+A/d_G-e(s,z), and similarly σ_G-e(T_e/f,z) ≥D+B+C/d_G-e(s,z). Moreover, by the definition of μ(·) we have that μ(f) ≥max{σ_G-e(T_e/f,y), σ_G-e(T_e/f,z)}. The previous inequalities together imply:μ(g)/μ(f)≤σ_G-e(T_e/g,z)/max{σ_G-e(T_e/f,y), σ_G-e(T_e/f,z)}≤A+B+C+D/D+max{ A, B+C }. Now we divide the proof into two subcases, depending on whether B+C ≥ A or B+C < A. Observe that D ≥ d_G(s,y) ≥ A. If B+C ≥ A, then (<ref>) becomes:μ(g)/μ(f)≤A+B+C+D/B+C+D = 1 + A/B+C+D≤ 1 + A/2A = 3/2,otherwise, if B+C < A, then(<ref>) becomes:μ(g)/μ(f)≤A+B+C+D/A+D < 2A+D/A+D = 1 + A/A+D≤ 1 + A/2A = 3/2. §.§.§ Case 2Assume now that t' is an ancestor of t (right side of Figure <ref>). Since μ(f) ≥σ_G-e(T_e/f,y) ≥d_G-e(s,y')+A+B/d_G-e(s,y)=d_G-e(s,y')+A+B/D,we have that:μ(g)/μ(f) ≤A+B+C+D/d_G-e(s,z)·D/d_G-e(s,y')+A+B≤A+B+C+D/d_G-e(s,z)·D/A+B+Dand since d_G-e(s,z) ≥ d_G(s,z) ≥ C, and recalling that d_G-e(s,z) ≥ D, we have:μ(g)/μ(f)≤A+B+C+D/A+B+D·D/max{ C,D } = ( 1 + C/A+B+D) ·D/max{ C,D }.Moreover, notice that also the following holds:μ(g)/μ(f) ≤μ(g)/σ_G-e(T_e/f,z)≤A+B+C+D/d_G-e(s,z)·d_G-e(s,z)/d_G-e(s,y') + d_T(y',t)+C≤A+B+C+D/C+D = 1+ A+B/C+D.We divide the proof into the following two subcases, depending on whether D ≥ C or D < C. In the first subcase, i.e., D ≥ C, we have that (<ref>) becomes μ(g)/μ(f)≤ 1 + C/A+B+D, and hence, by combining this inequality with (<ref>), we obtain:μ(g)/μ(f) ≤ 1+ min{C/A+B+D, A+B/C+D}≤ 1+ min{C/A+B+C, A+B/2C}≤ 1+1/2=3/2.In the second subcase, i.e., D < C, (<ref>) becomes:μ(g)/μ(f)≤( 1 + C/A+B+D) ·D/C≤D/C + D/A+B+D < 1 + D/A+B+D,and hence, by combining (<ref>) and (<ref>), we have that:μ(g)/μ(f) ≤ 1+ min{D/A+B+D, A+B/C+D}≤ 1+ min{D/A+B+D, A+B/2D}≤ 1+1/2=3/2,from which the claim follows. Given the result of Lemma <ref>, we can derive an efficient algorithm to compute all the GSE for . More precisely, in <cit.> it was shown how to find them in O(mα(m,n)) time. Essentially, the approach used in <cit.> was based on a reduction to the SPT sensitivity analysis problem <cit.>. However, in <cit.> it was proposed a faster solution to such a problem, running in O(mlogα(m,n)) time. Thus, we can provide the followingThere exists a 3/2-approximation algorithm that solves the problem in O(m logα (m,n)) time.We conclude this section with a tight example which shows that the analysis provided in Lemma <ref> is tight (see Figure <ref>).§ QUALITY ANALYSISAs for previous studies on swap edges, it is interesting now to see how the tree obtained from swapping a failing edge e=(a,b) with its BSE f compares with a true SPT of G-e. According to our swap criteria, we will then analyze the lower and upper bounds of the max- and avg-stretch of f, i.e., μ(f) and λ(f), respectively.As already observed in the introduction, it is well-known <cit.> that for the swap edge, say g, which belongs to the shortest path in G-e between s and the root of the detached subtree T_b, we have that for any v ∈ V(T_b), σ_G-e(T_e/g,v) ≤ 3. This immediately implies that μ(g),λ(g) ≤ 3, namely μ(f),λ(f) ≤ 3. These bounds happen to be tight, as shown in Figure <ref>.Let us now analyze the lower and upper bounds of the max-stretch of a good swap edge g, i.e., μ(g), as defined in the previous section. First of all, once again it was proven in <cit.> that for any v ∈ V(T_b), σ_G-e(T_e/g,v) ≤ 3, which implies that μ(g) ≤ 3. Moreover, the example shown in Figure <ref> can be used to verify that this bound is tight. § CONCLUSIONSIn this paper we have studied two natural SPT swap problems, aiming to minimize, after the failure of any edge of the given SPT, either the maximum or the average stretch factor induced by a swap edge. We have first proposed two efficient algorithms to solve both problems. Then, aiming to the design of faster algorithms, we developed for the maximum-stretch measure an almost linear algorithm guaranteeing a 3/2-approximation w.r.t. the optimum.Concerning future research directions, the most important open problem remains that of finding a linear-size edge-fault-tolerant SPT with a (maximum) stretch factor w.r.t. the root better than 3, or to prove that this is unfeasible. Another interesting open problem is that of improving the running time of our exact solutions. Notice that both our exact algorithms pass through the computation of all the post-failure single-source distances, and if we could avoid that we would get faster solutions. At a first glance, this sounds very hard, since the stretches are heavily dependant on post-failure distances, but, at least in principle, one could exploit some monotonicity property among swap edges that could allow to skip such a bottleneck. Besides that, it would be nice to design a fast approximation algorithm for the average-stretch measure. Apparently, in this case it is not easy to adopt an approach based on good swap edges as for the maximum-stretch case, since swap edges optimizing other reasonable swap criteria (e.g., minimizing the distance towards the root of the detached subtree, or minimizing the distance towards a detached node) are easily seen to produce an approximation ratio of 3 as opposed to a BSE. A candidate solution may be that of selecting a BSE w.r.t. the sum-of-distances criterium, which can be solved in almost linear time <cit.>, but for which we are currently unable to provide a corresponding comparative analysis.Finally, we mention that a concrete task which will be pursued is that of conducting an extensive experimental analysis of the true performances of our algorithms, to check whether for real-world instances the obtained stretches are sensibly better or not w.r.t. the theoretical bounds.plain | http://arxiv.org/abs/1707.08861v1 | {
"authors": [
"Davide Bilò",
"Feliciano Colella",
"Luciano Gualà",
"Stefano Leucci",
"Guido Proietti"
],
"categories": [
"cs.DS"
],
"primary_category": "cs.DS",
"published": "20170727133513",
"title": "Effective Edge-Fault-Tolerant Single-Source Spanners via Best (or Good) Swap Edges"
} |
Polarizability Extraction for Waveguide-Fed Metasurfaces David Smith July 14, 2017 ========================================================We use the method of Laplace transformation to determine the dynamics of a wave packet that passes a barrier by tunneling. We investigate the transmitted wave packet and find that it can be resolved into a sequence of subsequent wave packages. This result sheds new light on the Hartman effect for the tunneling time and gives a possible explanation for anexperimental result obtained by Spielmann et. al. § INTRODUCTION There are several definitionsof tunneling times (<cit.>,<cit.>,<cit.>,<cit.>,<cit.>,<cit.>,<cit.>,<cit.>)and the discussion about their meaning is stillongoing (<cit.>,<cit.>,<cit.>,<cit.>). Recently,experiments with atoms stimulated by ultrashort,infrared laser pulses, the so-called attoclock experiments, reinforced the interest in the prediction of tunneling times (<cit.>,<cit.>,<cit.>).In this article weinvestigate the dynamics of a wavepacket that tunnels through a barrier. The tunneling time we determine is theso-called group delay or phase time (<cit.>,<cit.>). It can be characterized as the time intervalbetween the moment the peak of a freely evolving incomingwave packet would reach the barrier and the arrival of the peak of the transmittedwave packet at the end of the barrier.One of the oldest results for this tunneling time was obtained by Mac Coll(<cit.>) in 1932 by the application of the stationary phase method. He concluded that there was ”no appreciable delay in the transmission of the packet through the barrier”. These calculations were laterrefined by Hartmann (<cit.>,<cit.>),who found a finite delay time. For thicker barriers this delay time becomes independent of the thickness of the barrier and tends to a fixed value which is known as the Hartmann effect. This saturationwas also found for other definitions of tunneling time (<cit.>), whereas in the framework of fractional quantum mechanics a decreasing of thetunneling time with barrier width was obtained(<cit.>).A formal analogy between the Schrödinger and theHelmholtz equation (see for instance <cit.>)made it possible to test predictions for the tunneling time with optical experiments.The results confirmed the saturation of the tunneling time (<cit.>,<cit.>, <cit.>,see <cit.> for more references )though a more detailed discrepancy between experiment and calculations remained open in <cit.>. Since the Hartmann effect seems to threaten Einstein causality, because superluminal velocities can be infered from transmission times saturating with barrier width ,an objection was made that only the modes above the barrier energy may contribute to the transmitted wave packet(<cit.>, <cit.>). But it was argued in (<cit.>)with reference to numerical calculations and experimental results that this effect can not be used as an explanation of the barrier tunneling phenomenon(see also <cit.> for a more detailed discussion and further references).The results of the attoclock experiments with strong laser fields that lower the Coulomb potentialcan be best explained by using atunneling time probability amplitude constructed with Feynnman path integrals (<cit.>, <cit.>), whereas several otherdefinitions of tunneling times do not agree with the data.At present there exists no unifiedformalism for the calculation of tunneling times for different experimental settings. In the case of wavepacket tunneling through static barriers the phase time, keeping track of the peak is the best candidate for the tunneling time, and in good agreement with the experimental results(<cit.>,<cit.>, <cit.>). The phase time is in general obtained by representing the solution of the time-dependent Schrödinger equation as the integral over stationary solutions for different energies.Under the assumption that the energy distribution of the initial wave packet is sufficiently peaked the application of thestationary phase method yields a time for the arrival of the peak of the wavepacket at the end of the barrier. In this article we use the method of Laplace transformation <cit.> to determine the wavepacket dynamics.Instead of immediately applying an approximation for oscillatory integrals, we first obtain exact solutions for the wavefunction at each side and insidethe barrier, which we then simplify assuming that the initial wavepacket is sufficiently peaked in the momentumrepresentation. Since we are free to choose explicitely the initial wavepacket in position space (which we do not have to construct as superposition of eigenstates), it is technically much easier to start with a wavepacket located exclusively at the left side of the barrier.Furthermore we give estimates for the consequences of the applied approximations and derive a consistency condition in the case of a special class of initial wavepackets.We find that what comes out of the barrier is not asingle wave packet, but infinitely many, one after the other. The time each of thesewavepackets needsfor the tunnelingis completely independent of the thickness of the barrier. But for thicker barriers, the later wavepackets are much more attenuated and so there remains a single transit time that equals the result Hartmann obtained asthe upper limit for the tunneling time. However in our approach the tunneling decreases before it reaches the limiting value for thick barriers. This is in accordance with the results of the optical experiments performed by Spielmann et al. <cit.>.Bernardini (<cit.>) investigated the transmission of wavepackets with energies above the height of the barrier. He also finds that there is not only one reflected and one transmitted wave packet, but many of them that leave the barrier. This result was obtained bythe decomposition of the transmission and reflection coefficients of the stationary solution ininfinite series.Our result is the counterpartin the tunneling regime. § THE SOLUTION OF THE TIME-DEPENDENT SCHRÖDINGER EQUATION A finite barrier is described by the potential[V(x)=0 x ≤ 0;V(x)=V 0 < x < d;V(x)=0 x ≥ d . ] The dynamics of a wave packet ψ(x,0)is determined by the Schrödinger equation-ħ^2/2 m∂^2 ψ(x,t)/∂ x^2= iħ ∂ψ(x,t)/∂ tx < 0, -ħ^2/2 m∂^2 ψ(x,t)/∂ x^2+V ψ(x,t)= iħ ∂ψ(x,t)/∂ t 0 < x < d,-ħ^2/2 m∂^2 ψ(x,t)/∂ x^2= iħ ∂ψ(x,t)/∂ tx > d, where ψ(x,t) is supposed to be continuously differentiable everywhere and square integrable .We will further assume that the initial wave packets are located at the left side of the barrierψ(x,0)=0x ≥ 0 .We apply the method of Laplace transformation, already introduced in (<cit.>). The Laplace transformed wavepacket φ(x,s) φ(x,s) = ℒ(ψ(x,t)) = ∫_0^∞ψ(x,t)e^-s t dt.obeys the transformed equations-ħ^2/2 m∂^2 φ(x,s)/∂ x^2 φ(x,s) = i ħsφ(x,s) - i ħ ψ(x,0)x < 0,-ħ^2/2 m∂^2 φ(x,s)/∂ x^2 + Vφ(x,s) = i ħsφ(x,s) - i ħ ψ(x,0)0 < x < d,-ħ^2/2 m∂^2 φ(x,s)/∂ x^2 φ(x,s) = i ħsφ(x,s) - i ħ ψ(x,0) x > d. The solution of (<ref>) is determined by the method of variation of constants:φ(x,s) =√(m/2 i s ħ){u_1(x,s)∫_-d^xu_2(y,s)ψ(y,0) dy-. .u_2(x,s)∫_-d^xu_1(y,s) ψ(y,0) dy}+ α(s) u_1(x,s) + β(s) u_2(x,s) x < 0 φ(x,s) = γ(s) u_3(x,s) + δ(s) u_4(x,s)0 <x < dφ(x,s) = μ(s) u_1(x,s) + ν(s) u_2(x,s)x > d.The functions u_1(x,s),u_2(x,s),u_3(x,s),u_4(x,s) are the solutions of the homogeneous equations corresponding to (<ref>)u_1(x,s)=e^i√(2 m s i/ħ) xu_2(x,s)=e^-i √(2 m s i/ħ) x0 <x < du_4(x,s)=e^-i √(2 m s i/ħ-2 m V/ħ^2) x u_3(x,s)=e^i√(2 m s i/ħ-2 m V/ħ^2) xx <0,x > d.Since φ(x,s) must vanish for x→±∞, we findα(s)=0 ,ν(s)=0 . If we evaluate φ(x,s) and its first derivative at x=0 and x=d, we obtain, imposing continuous differentiability β(s)=√(m/2 i ħ s)(I_1-I_2) +√(m/2 i ħ s)·2I_2/1+√(1-V/i ħ s)·ρ(s)-e^-2 i d √(2 m s i/ħ-2 m V/ħ^2)/ρ^2(s)-e^-2 i d √(2 m s i/ħ-2 m V/ħ^2)γ(s)= -√(m/2 i ħ s)·2I_2/1+√(1-V/i ħ s)·e^-2 i d √(2 m s i/ħ-2 m V/ħ^2)/ρ^2(s)-e^-2 i d √(2 m s i/ħ-2 m V/ħ^2)δ(s)=√(m/2 i ħ s)·2I_2 ρ(s)/1+√(1-V/i ħ s)·1/ρ^2(s)-e^-2 i d √(2 m s i/ħ-2 m V/ħ^2) ,μ(s)=√(m/2 i ħ s)·2I_2/1+√(1-V/i ħ s) e^- i d √(2 m s i/ħ)·1-ρ(s)/ρ^2(s)-e^-2 i d √(2 m s i/ħ-2 m V/ħ^2) e^-2 i d √(2 m s i/ħ-2 m V/ħ^2) ,where we have introduced the abbreviationsI_1=∫_-∞^0e^i √(2 m s i/ħ)yψ(y,0)dy ,I_2=∫_-∞^0e^-i √(2 m s i/ħ)yψ(y,0)dyand also usedρ(s)=2/ 1+√(1-V/ħ s i)-1 . Inserting this result into (<ref>) and applying theseries expansion 1/1-ρ^2(s)e^2 i d √(2 m s i/ħ-2 m V/ħ^2)=∑_k=0^∞(ρ(s))^2 ke^2 i d k√(2 m s i/ħ-2 m V/ħ^2)yields φ(x,s)=√(m/2 s i ħ){∫_-∞^0 e^i√(2 m s i/ħ) |x-y|ψ(y,0) dy+∫_-∞^0 e^-i√(2 m s i/ħ) (x+y)ψ(y,0) dy ·ρ(s)} + √(m/2 s i ħ)∑_l=0^∞∫_-∞^0 e^2 i d (l+1)√(2 m s i/ħ-2 m V/ħ^2) -i (x+y)√(2 m si/ħ)ψ(y,0)dy · (ρ^2(s)-1)ρ^2l+1(s)x < 0 φ(x,s)=√(m/2 s i ħ)∑_l=0^∞∫_-∞^0 e^ i(2 d l+x)√(2 m s i/ħ-2 m V/ħ^2) -i y √(2 m si/ħ)ψ(y,0)dy · (ρ(s)+1)ρ^2l(s) -√(m/2 s i ħ)∑_l=0^∞∫_-∞^0 e^ i(2 d (l+1)-x)√(2 m s i/ħ-2 m V/ħ^2) -i y√(2 m si/ħ)ψ(y,0)dy · (ρ(s)+1)ρ^2l+1(s)0 ≤ x ≤ d,φ(x,s)=√(m/2 s i ħ)∑_l=0^∞∫_-∞^0 e^i d (2l+1)√(2 m s i/ħ-2 m V/ħ^2) +i (x-d-y)√(2 m si/ħ)ψ(y,0)dy · (-ρ^2(s)+1)ρ^2l(s)x > d. We introduce the shifted momentum representation f(Q,p)=1/√(2 πħ)∫_-∞^∞ψ(x+Q,0) e^-i p x/ħ dx = e^i p Q/ħf(p) ,where f(0,p) = f(p)denotes the representation of the wave function in momentum space.Applying the abbreviations a_l(t)=ℒ^-1{ρ^2 l+1(s)(1-ρ^2(s))} ,b_l(t)=ℒ^-1{ρ^2 l(s)(1+ρ(s))} c_l(t)=ℒ^-1{ρ^2 l+1(s)(1+ρ(s))} ,g_l(t)=ℒ^-1{ρ^2 l(s)(1-ρ^2(s))} ,we find proceeding as for the asymmetric square well in <cit.> for the inverse Laplace transform of (<ref>)ψ(x,t)= κ/√(2 π ti)∫_-∞^0e^i (x-y)^2 κ ^2/2 tψ(y,0) dy + ∫_0^tκ/√(2 π (t-τ)i)∫_-∞^0e^i (x+y)^2 κ ^2/2 (t-τ)ψ(y,0) r(τ)dydτ + ∑_l=0^∞∫_0^t∫_-∞^∞ K(2d( l+1),p,t-τ) f(-x,p)dp · a_l(τ)dτ x < 0, ψ(x,t)= ∑_l=0^∞∫_0^t∫_-∞^∞ K( 2 d l+x),p,t-τ) f(p)dp · b_l(τ)dτ + ∑_l=0^∞∫_0^t∫_-∞^∞ K(d (2l+1)-x),p,t-τ) f(p)dp · c_l(τ)dτ0 < x < d, ψ(x,t)= ∑_l=0^∞∫_0^t∫_-∞^∞ K(d(2 l+1),p,t-τ) f(x-d,p)dp · g_l(τ)dτx > d, where K(x,p,t) is defined byK(x,p,t)= 1/2 √(2 πħ) e^- i V t/ħ-i t q^2/2 ħ m{e^-i x q/ħ[-i √(2 m i/ħt)x/2-i √(it/2 ħ m) q]+e^i x q/ħ[-i √(2 m i/ħt)x/2+i √(it/2 ħ m) q] } ,and we used κ= √(m/ħ) and q=√(p^2-2 m V). § TUNNELING OF WAVE PACKETS In order to investigate the tunneling process, weconsidera wave packet that is represented in momentum space byf(p)=e^-i p x_0/ħ F(p-p_0) , where F(p) fulfills∫_-∞^∞ F^*(p) pF(p) dp=0,∫_-∞^∞F^*(p)F'(p) dp=0 .. The expectation values of position and momentum are then given by⟨x̂⟩=x_0 , ⟨p̂⟩=p_0 .Weassume that the wavepacket isconcentrated within a region 0 < p <p_max <√(2 m V). so that we can use the approximationf(p) ≈ 0p > p_maxp < p_min .Moreover it should be sufficiently peaked around the momentum expectation value to justify the approximationF(p-p_0) (p-p_0)≈ 0 .Finally the difference √(2 m V)-p_m should be big enough to ensure 1/√(1-p^2/2 m V) = O(1) p_min ≤p ≤ p_max .In Appendix <ref> it is shown that these assumptions about the momentum distribution are compatible with the requirement (<ref>) for ψ(x,0).We start with the solution for x>d (<ref>). If we rewrite K(x,p,t)for q^2=p^2-2 m V < 0 and use X_l≡ d(2l+1) , we find(see section 4.4 in <cit.> for details) K(X_l,p,t)=U_0+U_1-U_2 , U_0=e^-i V t/ħ1/√(2 πħ)e^-i t q^2/2 ħ m+i X_l q/ħ =1/√(2 πħ)e^- i p^2 t/2 m ħ-X_l√(2 m V-p^2)/ħ ,U_1=κ/√(2 π ti)1/√(2 πħ) e^-i V t/ħ∫_0^∞e^i (X_l+u)^2 κ ^2/2 t e^i √(2 m V-p^2) u/ħ du , U_2=κ/√(2 π ti)1/√(2 πħ) e^-i V t/ħ∫_0^∞e^i (X_l-u)^2 κ ^2/2 t e^i √(2 m V-p^2) u/ħdu .Inserting κ/√(2 π ti)e^i (X_l± u)^2 κ ^2/2 t= ∫ _-∞^∞1/2 πħ e^-i q^2 t/2 m ħ+i q X_l/ħ±i q u/ħdq ,we obtainU_1-U_2=1/(2 πħ)^3/2e^-i V t/ħ∫_-∞^∞ e^-i q^2 t/2 m ħ+i q X_l/ħ2 i ħ q/2 m V-p^2+q^2dq.We find forU_1-U_2, that contains no oscillatory part ∫_-∞^∞(U_1-U_2)f(x-d,p)dp≈ 0 , since ∫_-∞^∞f(p) p^n e^i p (x-d)/ħ dp=(-i ħ∂/∂ x )^nψ(x-d,0)=0x > d , n=0,1,2,N , and so the contribution to (<ref>) will be proportional toΔ p^N+1 ,if f(p) decays rapidly enough so that the integrals ∫_-∞^∞f(p) p^n dpexist up to n=N+1. If ψ(x) is infinitely differentiable forx>d, th eleft hand side of (<ref>) vanishes identically. So we conclude that for wavepackets that are sufficiently smooth for x > d , U_0 is the only relevant contribution.We obtain for the wavefunction for x > d ψ(x,t) ≈∑_l=0^∞∫_0^t1/√(2 πħ)e^-X_l√(2 m V-p_0^2)/ħ∫_-∞^∞e^- i p^2 (t-τ)/2 m ħe^i p (x-d)/ħf(p) dp · g_l(τ)dτ . This result contains the free time evolutionof the initial wave packet that is shifted by d to the right. Proceeding as for the asymmetric square well(see <cit.>) we can approximate theconvolution integral by∫_0^t1/√(2 πħ)∫_-∞^∞e^- i p^2 (t-τ)/2 m ħe^i p (x-d)/ħf(p) dp · g_l(τ)dτ≈∫_0^∞1/√(2 πħ)∫_-∞^∞e^- i p^2 (t-τ)/2 m ħe^i p (x-d)/ħf(p) dp · g_l(τ)dτ≈1/√(2 πħ)∫_-∞^∞e^- i p^2 t/2 m ħe^i p (x-d)/ħ R(p)^2l (1-R(p)^2)f(p) dp≈ R(p_0)^2l (1-R(p_0)^2) ·1/√(2 πħ)∫_-∞^∞e^- i p^2 t/2 m ħe^i p (x-d)/ħf(p) dp,where R(p) is given by R(p)=-1 + 2 k-2 √(k(k-1)) k=p^2/2 m V .Here the assumptions about the concentration of the wave packet (<ref>, <ref>) justifies putting R(p) beforethe integral. Evaluating the sum in (<ref>), we obtainψ(x,t) ≈∑_l=0^∞∫_0^t1/√(2 πħ)e^-X_l√(2 m V-p_0^2)/ħ∫_-∞^∞e^- i p^2 (t-τ)/2 m ħe^i p (x-d)/ħf(p) dp · g_l(τ)dτ≈ e^- d √(2 m V-p_0^2)/ħ/1-e^-2 d √(2 m V-p_0^2)/ħR^2(p_0)(1-R^2(p_0))1/√(2 πħ)∫_-∞^∞e^- i p^2 t/2 m ħe^i p (x-d)/ħf(p) dp.For the neglected part of the time integral (<ref>)Δ_l≡∫_t^∞1/√(2 πħ)∫_-∞^∞e^- i p^2 (t-τ)/2 m ħe^i p (x-d)/ħf(p) dp · g_l(τ)dτ ,we find the estimates (see Appendix <ref>)|Δ_l|∼ O(l)·O([2 ħ/V t]^1/4)Since ∑_l=0^∞e^-X_l√(2 m V-p_0^2)/ħΔ_l∼e^d √(2 m V-p_0^2)/ħ/(-1+e^2 d √(2 m V-p_0^2)/ħ)^2·O([2 ħ/V t]^1/4)the approximation for ψ(x,t) given by (<ref>) can be used for t ≫ħ/V. The wavepacket <ref> describes a free evolving wavepacket that started at t=0 at the position x_0+ d. It leaves thebarrier without any time delay and is instantaneously transmitted through the barrier attenuatedby a factor of the magnitudee^ -d √(2 m V-p_0^2)/ħ .So within our approximations we find that the tunneling time is zero, where we assume in accordance with <cit.>that the transmission begins when a freely evolvingwave packet starting at t=0 at the position x_0 would have arrived at the barrier, or equivalently when a classical particle starting at t=0, x_0=0 has reached the barrier. It is not practicable to follow the peak of the original wavepacket until theentrance of the tunnel since this peak will be deformed by oscillations during the reflection process(<cit.>,<cit.>) and also the position expectation value might undergo a slight attenuation before the tunnel as it was shown for infinite walls (<cit.>). Performing the inverse Laplace transform for x<0 and 0 < x < d(<ref>,<ref>)and making the same approximationsas in the previous case we find for the wavepacket to the left of and within the barrier: ψ (x,t)≈κ/√(2 π ti)∫_-∞^0e^i (x-y)^2 κ ^2/2 tψ(y,0) dy + 1/√(2 πħ)∫_-∞^∞e^- i p^2 t/2 m ħe^-i p x/ħ R(p)f(p) dp + ∑_l=0^∞e^-2 d (l+1) √(2 m V-p_0^2)/ħ1/√(2 πħ)∫_-∞^∞e^- i p^2 t/2 m ħe^-i p x/ħ R(p)^2 l +1(R(p)^2-1)f(p) dp ≈κ/√(2 π ti)∫_-∞^0e^i (x-y)^2 κ ^2/2 tψ(y,0) dy +{1+e^- 2 d √(2 m V-p_0^2)/ħ/1-e^-2 d √(2 m V-p_0^2)/ħR(p_0)^2· (R^2(p_0)-1)} R(p_0)1/√(2 πħ)∫_-∞^∞e^- i p^2 t/2 m ħe^-i p x/ħ f(p) dpx < 0, ψ (x,t)≈∑_l=0^∞e^-(2 d l+x) √(2 m V-p_0^2)/ħ1/√(2 πħ)∫_-∞^∞e^- i p^2 t/2 m ħ R(p)^2 l(R(p)+1)f(p) dp ≈ -∑_l=0^∞e^-(2 d (l+1)+x) √(2 m V-p_0^2)/ħ1/√(2 πħ)∫_-∞^∞e^- i p^2 t/2 m ħR(p)^2 l+1(R(p)+1)f(p) dp ≈{1/1-e^-2 d √(2 m V-p_0^2)/ħR(p_0)^2- e^- 2( d-x) √(2 m V-p_0^2)/ħ/1-e^-2 d √(2 m V-p_0^2)/ħR(p_0)^2·R(p_0)} · (R(p_0)+1) e^- x√(2 m V-p_0^2)/ħ· 1/√(2 πħ)∫_-∞^∞e^- i p^2 t/2 m ħ f(p) dp0 < x < d. The solution within the barrier differs from the solution of the potential step (see <cit.>) only by a time-independent factor ψ_step(x,t)≈ (1+R(p_0)) e^-√(2 m V-p_0^2) x/ħ1/√(2 πħ)∫_-∞^∞e^- i p^2 t/2 m ħ f(p) dp . So we see that the time it needs until the barrier is (approximately) empty again, is independent of the thickness of the barrier. The solution on the left side consists of the incoming and the reflected wave packet (<ref>). In contrast to the potential step the reflected wavepacket experiences a permanent attenuation, since a part of the wavefunction has tunneled through the barrier. After the reflection process the wavefunction consists of a reflected and a transmitted wavepacket (<ref>) only. An explicit calculation yields | {1+e^- 2 d √(2 m V-p_0^2)/ħ/1-e^-2 d √(2 m V-p_0^2)/ħR(p_0)^2· (R^2(p_0)-1)} R(p_0) |^2+ |e^- d √(2 m V-p_0^2)/ħ/1-e^-2 d √(2 m V-p_0^2)/ħR^2(p_0)(1-R^2(p_0))|^2 = 1, which confirms that the integral over the probability density is conserved and our approximations are consistent. § THE TUNNELING TIME Within our approximations we found the tunneling time to be zero,since our solution (<ref>) indicates that the wavepacket leaves the tunnel, shifted by d to the right.Here we have assumed,that all functions of R(p) can be pulled out of the integral (<ref>), andthereforethey do not influence thedynamics of the wave packet. If we take into account the first order contributions of R(p),we find a small, but finite tunneling time.We will from now on assume that the initial wave function is an uncorrelated function of the formf(p)=G(p)e^-i p x_0/ħ ,where G(p) is a real function that yields a momentum expectation value p_0. The position expectation value is then given by x_0.The impact of an additional factor Z(p) on the initial wavepacket f(p) is twofoldZ(p)f(p)=z(p)e^i μ (p) f(p) , z(p)=|Z(p)| . The probability density in momentum space is only affected by the absolute value z(p). If we restrict ourselves to first order contributions in p-p_0, z(p) ≈ z(p_0)+ z_1(p-p_0) we find that we can neglect the momentum shift:N^2≡∫_-∞^∞|f(p)|^2 z(p)^2 dp≈z(p_0)^2∫_-∞^∞|f(p)|^2 dp =z(p_0)^2 ⟨p̂⟩ =N^-2∫_-∞^∞p|f(p)|^2 z(p)^2 dp ≈ p_0+2 (z(p_0 ))^-1 z_1∫_-∞^∞|f(p)|^2 p(p-p_0) dp ≈ p_0 . Using also a linear approximationfor μ'(p) μ'(p) ≈μ_1+μ_2(p-p_0)we find for the position expectation value x_0 = N^-2∫_-∞^∞ z(p) f^*(p)e^-i μ (p)(i ħ∂/∂ p) z(p) f(p)e^i μ (p) dp = N^-2 x_0∫_-∞^∞G(p)^2 z(p)^2 dp-N^-2ħ∫_-∞^∞G(p)^2 z(p)^2 μ'(p)dp ≈x_0-ħμ_1 . Therefore the phase of the functionsR(p)^2l (1-R(p)^2)in (<ref>) yields a shift of the position of each particular wave packet constituting(<ref>)ψ(x,t) ≈∑_l=0^∞1/√(2 πħ)e^-X_l√(2 m V-p_0^2)/ħ∫_-∞^∞e^- i p^2 t/2 m ħe^i p (x-d)/ħ R(p)^2l (1-R(p)^2)f(p) dp. We findArg[1-R(p)^2]= Arctan[-1+2 l/2 √(k-k^2)] ≈Arctan[-1+2 k_0/2 √(k_0-k_0^2)]+ 2/√(2 m V-p^2_0)·(p-p_0) Arg[R(p)]= -Arccos[1-2 k] ≈-Arccos[1-2 k_0]+ 2/√(2 m V-p^2_0)·(p-p_0),where we have used the definitionsk=p^2/2 m V , k_0=p_0^2/2 m V .So we see that the lth term of (<ref>) will experience a phase shift of 2 (1+2l) (p-p_0)/√(2 m V-p^2_0) ,corresponding to a translation byδ x_l = 2 (1+2l) ħ/√(2 m V-p^2_0)=2 (1+2l) ħ/p_0√(k_0/1-k_0) to the left. So the delay time will beT_l=2 (1+2l) m ħ/p_0√(2 m V-p^2_0)instead of zero.Each term is attenuated by a factor of the magnitudee^ -X_l√(2 m V-p_0^2)/ħ .For thick barriers, ifd √(2 m V-p_0^2)/ħ≫ 1 , the first term will dominate the sum (<ref>), and the tunneling time will be given byT_0=2 m ħ/√(2 m V-p^2_0)p_0 This is exact the time Hartmann <cit.>found as upper limit for the tunnelingtime through thick barriers. If the barrier gets thinner the other wavepackets for l>0 will also come into play (see figure <ref>). Each of them leaves the barrier at a different time T_l. But since they appear very shortly after each other they may appear as one smeared out wavepacket with a delay time bigger than T_0. Note that apart from the absolute and relative magnitude of the wavepackets, all characteristic quantities of tunneling are independent of the width d. In the case of the wavefunction pictured in figure <ref> where the uncertainties are given by Δ x≈√(a/2) ,Δ p ≈√(ħ/2 a) ,the ratio between the distance of the centre of the wavepackets and the position uncertainty the determines the distinguishability between the wavepackets readsδ x_l+1-δ x_l/Δ x= 4 ħ/p_0Δ x√(k_0/1-k_0)≈8 Δ p/p_0√(k_0/1-k_0) .§ DISCUSSION AND CONCLUSIONS Our result for thetunneling time (<ref>) is not an exact reproduction of Hartmann's result (<cit.>,<cit.>) which predicts an increasing tunneling time with the thickness of the barrierbefore saturationtakes place. However Spielmann et al. found a decreasing tunneling time in their experiments with electromagnetic waves propagating through photonic band gap materials (see <cit.> fig. 3, <cit.> fig 1 ). This qualitative behaviour is in good agreement with our results that also predict a decreasing tunneling time since for thicker barriers the later wave packets are more and more attenuated.For thicker barriers the conclusion of both calculations is that the tunneling time for sufficiently peaked wave packets is given by (<ref>). This is as far interesting as the results were obtained by completely different methods. Moreover we ensured in our calculations that the initial wavepacket is only located at the left side of the barrier (<ref>) which is not clearly guaranteed by Hartmann's approach. So our result makes sure that the Hartmann time (<ref>) is not some relic of the parts ofthe initial wavepacket that were at the right hand side of the barrier from the beginning. We did not take into account the parts of the initial wavepacket with energiesnear or greater than the critical energy of standard transmission through the barrier (see <ref>). In Appendix <ref> we have derived a consistency condition for thisapproximation for the case of a special class of initial wavepackets. So this is a further counterexample to the idea thatonly energies greater than the barrier height contribute to tunneling (<cit.>).We also found out that the approximate solution within a finite barrier differs from the solution within the potential step only by a time-independent factor (<ref>,<ref>) which also indicates that important dynamical properties are independent of the thickness of the barrier. It would be especially interesting if this is also true for more general tunneling processes as the tunneling out of a potential well that could model radioactive decay or tunneling out of atoms as provided by the attoclock experiment<cit.>. Moreover an application of the method of Laplace transformation to relativistic wave equations would yield a picture of the reflection and tunneling processes in the relativistic case. § ACKNOWLEDGMENTS We thank the referees for their comments and suggestions that helped to improve this article. § ESTIMATION OF THE CONVOLUTION INTEGRAL According to <cit.>, the inverse Laplace transform of ρ(s)^l is given by ℒ^-1( ρ(s)^l)=l/i^l tJ_l[V t/2 ħ]e^-i V t/2 ħ . Therefore we findg_l(t)=ℒ^-1(ρ^(2 l)(s)(1-ρ^2(s)))= l/i^(2l) tJ_2l[V t/2 ħ]e^-i V t/2 ħ- l/i^(2(l+1)) tJ_2(l+1)[V t/2 ħ]e^-i V t/2 ħ . For an integral of the formu(l,t)= ∫_t^∞ e^∓ i p Q/ħ e^-i p^2 (t-τ)/2m ħ e^-i V τ/2 ħl/i^lτ J_l[V τ/2 ħ] dτ ,we get the following estimate |u(l,t)|=| ∫_t^∞e^i p^2 τ/2m ħ-i V τ/2 ħl/τ J_l[V t/2 ħ] dτ| ≤∫_t V/2 ħ^∞|l J_l(y)/y| dy ≤(∫_t V/2 ħ^∞l/y^1+2ϵ dy )^1/2·(∫_0^∞(J_l(y))^2/y^1-2ϵ)^1/2 =l 1/√(2 ϵ)(2 ħ/V t)^ϵ·( 2^2 ϵΓ [1-2 ϵ]Γ[l+ϵ]/2 (Γ[1-ϵ])^2 Γ[1+l-ϵ])^1/2 ≤ l 1/√(2 ϵ)(2 ħ/V t)^ϵ·( 2^2 ϵΓ [1-2 ϵ]/2 (Γ[1-ϵ])^2 )^1/2 0 < ϵ < 1/2 ,where we have applied the Schwarz inequality and the integral formula (<cit.>)∫_0^∞(J_l(y))^2/y^1-2ϵ= 2^2 ϵΓ [1-2 ϵ]Γ[l+ϵ]/2 (Γ[1-ϵ])^2 Γ[1+l-ϵ] . So we conclude setting ϵ=1/4 for Δ_l (<ref>)|Δ_l|∼ O(l)·O([2 ħ/V t]^1/4) § DECREASING BEHAVIOUR IN MOMENTUM SPACE OF FUNCTIONS WITH COMPACT SUPPORT AND THE EXAMINATION OF THE USED ASSUMPTIONS ABOUT ORDERS OF MAGNITUDE BASED ON AN EXAMPLEAccording to the Palay Wiener theorem (<cit.>), functions with compact support in position space can not be restrictedto a finite interval in momentum space as well.Nevertheless the concentration of those functions in momentum space around theirexpectation value can be shownto be prescribed by the Fourier transform of agenericreference function. We choose an appropriate wavepacket with a referencefunction of Gaussian shape and derive the conditions that justify (<ref>,<ref>,<ref>). We also show that we can use the reference function for the evaluation of (<ref>) as we did for the example presented in figure <ref>. We start with a normalized wavefuntion ψ(x) with position expectation value x_0 thatis assumed to be zero for |x-x_0| ≥ B. Let ψ_0(x) be a generic reference function that fulfills[ψ_0(x)=ψ(x) |x-x_0| ≤ B;ψ_0(x)=-δψ(x)|x-x_0| > B, ]where δψ is zero for |x-x_0| ≤ B. Then ψ(x) is represented by the sumψ(x)=ψ_0(x)+δψ (x)If the norm of δψ is given by ϵ we find for the reference function ⟨ψ_0 |ψ_0⟩ =1+ϵ^2 .The representation of ψ(x) in momentum space reads f(p)=f_0(p)+δ f(p)where f_0(p) and δ f(p) are the representationsψ_0(x) and δψ(x) in momentum space.Using the Schwarz inequality and taking into account that the Fourier transform preserves the L_2 norm weget for the integral over the momentum density in the region outside the interval (p_min,p_max) ∫_-∞^p_min f^*(p)f(p)dp + ∫_p_max^∞ f^*(p)f(p)dp≤∫_-∞^p_min f_0^*(p)f_0(p) dp+ ∫_p_max^∞ f_0^*(p)f_0(p) dp +ϵ^2 + 2 √(1+ϵ^2)ϵ . In order to provide an explicit example, we choose the initial wavepacket with the position and momentum expectation values x_0,p_0ψ (x)=N^-1/2e^-i p_0 x/ħe^-(x - x_0)^2/ 2 a[-L + x - x_0/√(a)]^2[L + x - x_0/√(a)]^2|x-x_0| < L √(a)ψ(x)=0 |x-x_0| ≥ L √(a) , where the normalization constant Nis given by N= √(a)/16 e^-L^2 (2 L (-105 + 50 L^2 - 20 L^4 + 8 L^6) + e^L^2 (105 + 8 L^2 (-15 + 9 L^2 - 4 L^4 + 2 L^6)) √(π)[L]) .The factors multiplied to the Gaussian ensure that ψ(x) is continuously differentiable at x=x_0± L √(a).The position and momentum uncertainty are determined by(Δ x) ^2= a (2 L (-945 + 210 L^2 - 52 L^4 + 8 L^6) + e^L^2 (945 - 840 L^2 + 360 L^4 - 96 L^6 + 16 L^8) √(π)[L])/ 4 L (-105 + 50 L^2 - 20 L^4 + 8 L^6) + 2 e^L^2 (105 - 120 L^2 + 72 L^4 - 32 L^6 + 16 L^8) √(π)[L](Δ p) ^2 = ħ^2 2 L (-225 + 18 L^2 + 12 L^4 + 8 L^6) + e^L^2 (225 + 8 L^2 (-21 + 5 L^2 + 4 L^4 + 2 L^6)) √(π)[L])/2 a (2 L (-105 + 50 L^2 - 20 L^4 + 8 L^6) + e^L^2 (105 - 120 L^2 + 72 L^4 - 32 L^6 + 16 L^8) √(π)[L]) ,wherelim_L →∞(Δ x) ^2 =a/2 Δ x ^2≤a/2 L > 2 lim_L →∞(Δ p) ^2 =ħ^2 /2 a Δ p ^2≤ 1.2 ħ^2/a L > 2 So we see that it is possible to choose the momentum uncertainty sufficiently small to justify (<ref>). We take asreference function ψ_0(x) the extension of (<ref>) to the whole real line. We find for the norm of the difference function δψ according to (<ref>)ϵ^2 = 2 L (105 - 50 L^2 + 20 L^4 - 8 L^6) + e^L^2 (105 + 8 L^2 (-15 + 9 L^2 - 4 L^4 + 2 L^6)) √(π)[L] / 2 L (-105 + 50 L^2 - 20 L^4 + 8 L^6) + e^L^2 (105 + 8 L^2 (-15 + 9 L^2 - 4 L^4 + 2 L^6)) √(π)[L] , where lim_L →∞ϵ =√(24/√(π))e^-L^2/2L^-9/2ϵ≤√(24/√(π))e^-L^2/2L^-9/2 L > 3For the further calculations we introduce the dimensionless quantitiesP=√(a)/ħ p , P_0=√(a)/ħ p_0 , X=x/√(a) ,X_0=x_0/√(a) , D=d/√(a) .We find for the momentum representation of the reference function ψ_0(x) f(p)=a^1/4/√(h)4e^-(1/2) (P - P_0) (P - P_0 + 2 I X_0) (3 + L^4 + 2 L^2 (-1 + (P - P_0)^2) - 6 (P - P_0)^2 + (P - P_0)^4)/√( 2 e^-L^2L (-105 + 50 L^2 - 20 L^4 + 8 L^6) +(105 + 8 L^2 (-15 + 9 L^2 - 4 L^4 + 2 L^6)) √(π)[L]) .Assuming that the interval (p_min,p_max) is symmetric around p_0 > 0 we find for the probability outside the interval𝒫_rest≡∫_-∞^p_min f_0^*(p)f_0(p)+∫_p_max^∞ f_0^*(p)f_0(p) dp ={2 e^-P_0^2(1+(K-2)K)(-1 + K) P_0 (-39 + 72 L^2 - 88 L^4 + 32 L^6 +. .. 2 (-1 + K)^2 (83 + 24 L^2 (-3 + L^2)) P_0^2 +4 (-1 + K)^4 (-17 + 8 L^2) P_0^4 + 8 (-1 + K)^6 P_0^6 ) +.(105 +8 L^2 (-15 + 9 L^2 - 4 L^4 + 2 L^6)) √(π)[P_0 (K-1)] }·(Abs[ 2 e^-L^2L (-105 + 50 L^2 - 20 L^4 + 8 L^6) + (105 + 8 L^2 (-15 + 9 L^2 - 4 L^4 + 2 L^6)) √(π)[L]] )^-1 ,where K ≡ p_max/p_0.According to (<ref>) the application of the approximation (<ref>) means that we neglect a portion of the probabilityof the magnitude ϵ^2 + 𝒫_Rest. Since the wavepackets that leave the barrier are of the magnitudee^ -d(2l+1) √(2 m V-p_0^2)/ħ=e^-(2l+1) D P_0√(1-k_0/k_0) .our results are relevant compared to the neglected parts if the condition D P_0 (2l+1)√(1-k_0/k_0)≲[ϵ+𝒫_Rest] is fulfilled.For the initial wavfunction evaluated in fig. <ref> with the parameters P_0=10, k_0=1/2, X_0=-20, L=20and with the choice K=p_max/p_0=1.4 ,(<ref>) readsD (2l+1)· 10≲18 . So the wavepackets up toD (2l+1)≤ 1.8 meet <ref>. Moreover the set of parameters fulfills the requirements of (<ref>), since1/√(1-p_max^2/2 m V)= 1/√(1-p_max^2/p_0^2)≈ 7 . For the evaluation of the first three terms of (<ref>) in fig.<ref> we have used f_0(p) instead of f(p): This is justified since for our choice of parameters (<ref>)ϵ≪ 1 . Moreover the obtained position shift (<ref>)δ x_l = 2 (1+2l) ħ/√(2 m V-p^2_0)=2 (1+2l)√(a)/P_0√(k_0/1-k_0)is bigger than the correction of the positions expectation value x_0 caused by δψ. We find δ x_ref≡⟨ψ_0x ψ_0⟩ - ⟨ψx ψ⟩ = ⟨δψx δψ⟩ = x_0·2 L (105 - 50 L^2 + 20 L^4 - 8 L^6) + e^L^2 (105 + 8 L^2 (-15 + 9 L^2 - 4 L^4 + 2 L^6)) √(π)[L]/ 2 L (-105 + 50 L^2 - 20 L^4 + 8 L^6) + e^L^2 (105 + 8 L^2 (-15 + 9 L^2 - 4 L^4 + 2 L^6)) √(π)[L] .Since | δ x_ref| ≤|x_0|·24/√(π)e^-L^2L^-9 L > 2 ,the condition δ x_ref ≪ δ_l is fulfilled if24/√(π)e^-L^2L^-9 ≪ 2(1+2l) P_0^-1|X_0^-1| √(k_0/1-k_0) , which is the case for the choice of parameters (<ref>).99 HaugeE.H.Hauge,J.A. Stovneng (1989): Tunneling Times: A critical review, Rev.Mod.Phys. 61, 917 LandauerR.Landauer , Th.Martin (1994): Barrier interaction time in tunneling, Rev.Mod.Phys.66, 217 Winful H.Winful (2006): Tunneling time, the Hartmann effect and superluminality:A proposed resolution of an old paradox, Phys. Rep.436,1-69Olkhovsky V. Olkhovsky et. al. (2004): Unified analysis of photonand particle tunneling Phys. Rep. 398, 133-178Razavy M.Razavy: Quantum theory of Tunneling, World Scientific Singapore 2003 GalaponE.Galapon (2012): Only above barrier energy components contribute to barrier traversal time, Phys.Rev.Lett.108,170402Landsmann A. Landsman, U.Keller (2014: Tunneling time in strong field ionization, J.Phys.B 47, 204024Landsmann1 A. Landsman, U.Keller (2015: Attosecond science and the tunneling time problem, Phys.Rep.547,1-24 Wang Z.S.Wang et.al.(2004: Quantum tunneling time, Phys.Rev. A 69, 052108 Cheng Y. Cheng (2010): On the tunneling time of arbitrary continuous potentials and the Hartmann effect, Chin.Phys.B, 19,No.11, 117305 Pfeiffer A.Pfeiffer et. at. (2012): Attoclock reveals natural coordinates of thelaser-induced tunneling current flow in atoms, Nature Physics Vol. 8, 76 Eckle P.Eckle et.al. (2008): Science Vol.322 1525 Landsmann2 A.Landsman (2014): Ultrafast resolution Hartmann T. Hartmann(1962): Tunneling of a Wave Packet, J. Appl. Phys. 33, 3427 of tunneling delay time, Optica Vol.1,No.3,343 MacColl L.A.MacColl(1932): Note on the Transmission and Reflection of Wave Packets by Potential Barriers, Phys.Rev. 40/ May 1932/621-626Hasan M. Hasan, B.Mandal (2018): Phys.Lett. A382, 248-252SteinbergA. Steinberg et. al. (1993): Measurement of single-photon tunneling time, Phys.Rev.Lett. 71/5,708Spielmann Ch. Spielmann, R. Szipös, A.Stingl, F. Krausz (1994) : Tunneling of optical pulses throug photonic band gaps, Phys.Rev. Lett. 73, 2308 Longhi S.Longhi et. al (2001):Superluminal optical pulse propagation, Phys. Rev. E 64055602 Chiao R.Y. Chiao et. al.(1993): Faster than light?, Scientific American 269, 52 Winful1 H.G. Winful (2003): Mechanism for 'superluminal' tunneling, Nature 424,638 Methode N.Riahi (2017): Solving the time-dependent Schrödinger equation via Laplace transform, Quantum Stud.: Math. Found.4/2, 103-126Bernardini A.E. Bernardini (2009): Stationary phase method and delay times for relativisticand non-relativistic tunneling particles, Ann.of Phys. 324, 1303-1339 Pereyra P.Pereyra (2000): Closed formulas for tunneling time in superlattices, Phys. Rev.Lett. 84/8, 1772 LosTunnel V.F.Los,M.V. Los (2012) : A time-dependent exact solution for wave-packet scattering at a rectangular barrier, J.Phys.A: Math.Theor. 45,095302BounceM.A.Doncheski et. al. (1999): Anatomy of a quantum 'bounce',Eur.J.Phys.20, 29-37Erdely A.Erdely: Tables of Integral Transforms, McGraw-Hill Book Company, 1954Magnus W.Magnus, F.Oberhettinger, R.P.Soni: Formulas and Theorems for the Special Functions of Mathematical Physics, Springer-VerlagZa A.Zayed: Handbook of Function and Generalized Function Transformations, CRC Press 1996 | http://arxiv.org/abs/1707.08917v3 | {
"authors": [
"Natascha Riahi"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170727155658",
"title": "Analysis of wavepacket tunneling with the method of Laplace transformation"
} |
α β̱ ε̧ δ̣ ϵ ϕ γ θ κ̨ łλ μ ν ψ ∂ ρ̊ σ τ ῠ φ̌ ω ξ η ζ Δ Γ Θ̋ ŁΛ Φ Ψ Σødiag Spin SOØ O SU U Sp SL trM_ PlInt. J. Mod. Phys. Mod. Phys. Lett. Nucl. Phys. Phys. Lett. Phys. Rev. Phys. Rev. Lett. Prog. Theor. Phys. Z. Phys. df BHinf inf evap eq smM_ Pl GeVm_ stop 0.7cm1.35cm Minimal Non-Abelian Supersymmetric Twin Higgs 1.2cm Marcin Badziak^1,2,3 and Keisuke Harigaya^2,3 0.4cm ^1 Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, ul. Pasteura 5, PL–02–093 Warsaw, Poland ^2 Department of Physics, University of California, Berkeley, California 94720, USA ^3 Theoretical Physics Group,Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA 1.5cm We propose a minimal supersymmetric Twin Higgs model that can accommodate tuning of the electroweak scale for heavy stops better than 10% with high mediation scales of supersymmetry breaking. A crucial ingredient of this model is a new SU(2)_X gauge symmetry which provides a D-term potential that generates a large SU(4) invariant coupling for the Higgs sector and only small set of particles charged under SU(2)_X, which allows the model to be perturbative around the Planck scale. The new gauge interaction drives the top yukawa coupling small at higher energy scales, which also reduces the tuning. § INTRODUCTION Supersymmetry (SUSY) provides one of the most promising solutions to the hierarchy problem of the Standard Model (SM) <cit.>. However, the lack of finding of SUSY partners casts serious doubts on whether SUSY can still naturally explain the electroweak (EW) scale. Fine-tuning of the EW scale in minimal SUSY models implied by the LHC searches was recently quantified in refs. <cit.>, which demonstrated that the current limits on stop and gluino masses exclude regions with fine-tuning better than 10%, even if a very low mediation scale of the SUSY breaking of 100 TeV is assumed.[The fine-tuning may be improved if the higgsino mass is not tied to the Higgs mass squared, see e.g. refs. <cit.>. In such a case higgsino could be heavier leading to compressed spectra for which the lower bounds on stops, and gluino are much weaker.]The fine-tuning quicklygets worse for larger mediation scales due to longer RG running of the soft Higgs mass.This isindication of the little hierarchy problem.A possible remedy tothe little hierarchy problem is offered by Twin Higgs mechanism <cit.>. In the scenario, the Higgs is a pseudo-Nambu-Goldstone boson of a global SU(4) symmetry emerging from ℤ_2 symmetry exchanging the SM with its mirror (or twin) copy. We refer to <cit.> for composite Twin Higgs models, and <cit.> for cosmological aspects of Twin Higgs scenario. Early realisations ofSUSY UV completion of Twin Higgs scenario <cit.>, which generate an SU(4) invariant quartic term with an F-term potential of a heavy singlet superfield, are not able to significantly reduce fine-tuning as compared to non-Twin SUSY models <cit.>.It was only very recently that SUSY Twin Higgs models were proposed in which tuning at the level of 10% is possible byintroducing either hard ℤ_2 symmetry breaking in the F-term model <cit.> or a new U(1)_X gauge symmetry whose D-term potential provides a large SU(4) invariant quartic term <cit.>. It should be, however, emphasised that the tuning at the level of 10% can be obtained in these models only for a low mediation scale or a low Landau pole scale. In the F-term model of ref. <cit.> a fine-tuning penalty for a larger mediation scale and hence a longer RG running is severe because the large SU(4) invariant coupling induces growth of the top yukawa coupling at higher energy scales. In the D-term model the RG effect of the gauge coupling g_X of the new interaction is to reduce the top yukawa coupling, and the effect of a higher mediation scale is not as severe as the one for the F-term model. However, the RG running of the U(1)_X gauge coupling is fastand hence the Landau pole scale of g_X is as low as 10^5-10^6 GeV for values of g_X that are large enough to guarantee approximate SU(4) symmetry of the Higgs potential. While such a low mediation scale or a low Landau pole scale is in principle possible, it strongly limits possible schemes of the mediation of the SUSY breaking and UV completions above the Landau pole scale.In the present work, we point out that the Landau pole scale and the mediation scale of the D-term model can be much higherif the SU(4) invariant term is generated by a D-term potential of a new non-abelian gauge symmetry. We construct a consistent model with SU(2)_X gauge symmetry with small number of flavors charged under this symmetry. The new gauge interaction drives the top yukawa coupling small at higher energy scales, which also helps obtain the EW scale more naturally. As a result, the tuning of the EW scale for 2 TeV stops and gluino can be at the level of 5-10% for mediation scales as high as 10^9-10^13 GeV. One can keep perturbativity up to around the Planck scale with tuning better than 5% (for low mediation scales).The model allows for moderate tuning better than few percent with the mediation scale around the Planck scale. If the gluino mass is a Dirac one, the tuning may be as good as 10%, which realizes a natural SUSY with a gravity mediation. § A SUSY D-TERM TWIN HIGGS WITH AN SU(2) GAUGE SYMMETRY In this section we present a SUSY D-term Twin Higgs model <cit.> where the D-term potential of a new SU(2)_X gauge symmetry generates the SU(4) invariant quartic coupling. We assume a ℤ_2 symmetry exchanging the SM with its mirror copy, and denote mirror objects with supersctripts '. The matter content of the model is shown in Table <ref>. In addition to the SU(3)_c× SU(2)_L× U(1)_Y gauge symmetry and its mirror counterpart, we introduce an SU(2)_X gauge symmetry which is neutral under the ℤ_2 symmetry. We embed an up-type Higgs H_u into a bi-fundamental of SU(2)_L× SU(2)_X, H, and its mirror partner H_u' into that of SU(2)'_L× SU(2)_X, H'.As we will see later, the D-term potential of SU(2)_X is responsible for the SU(4) invariant quartic coupling of H_u and H_u'. The SU(2)_X symmetry is broken by the vacuum expectation value (VEV) of a pair of SU(2)_X fundamental S and S̅. Except for S and S̅ all matter fields have their mirror partner.The right-handed top quark is embedded into Q̅_R and allow for a large enough top yukawa coupling through the superpotential term HQ̅_R Q_3, where Q_3 is the third generation quark doublet. E̅ is necessary in order to cancel the U(1)_Y`-SU(2)_X^2 anomaly. The VEV of ϕ_u is responsible for the masses of the up and charm quarks. Q_1,2,3, u̅_1,2, e̅_1,2,3, d̅_1,2,3 and L_1,2,3 are usual MSSM fields. To cancel the gauge anomaly ofSU(3)_c^2 `-U(1)_Y and U(1)_Y^3 originating from the extra up-type right handed quark in Q̅_R and two extra right-handed leptons in E̅, we introduce U and E_1,2. There are three up-type Higgses in H and ϕ_u, so we need to introduce three down-type Higgsses ϕ_d1,2,3. Their VEVs are responsible for the masses of down-type quarks and charged leptons.§.§ SU(2)_X symmetry breakingWe introduce a singlet chiral field Z and the superpotential coupling W = κ Z (S S̅ - M^2).We assume that the soft masses of S and S̅ are the same, V_ soft = m_S^2 (|S|^2 + |S̅|^2).Otherwise, the magnitude of the VEVs of S and S̅ are different from each other, and give large soft masses to the Higgs doublets through the D-term potential. The VEVs of S and S̅ are given by S = [ 0; v_S ], S̅ = [ v_S; 0 ], v_S = √(M^2 - m_S^2 / κ^2).The constraint on the T (ρ) parameter requires that v_S ≳2.9 TeV in the limit of large tanβ and neglecting the effect of mixing between the SM and the mirror Higgses, see Appendix <ref> for a derivation of this constraint and more precise formula. The masses of the SU(2)_X gauge bosons are given by m_X^2 = g_X^2 v_S^2. After integrating out massive particles with a mass as large as v_S, the potential of H and H' is given by 1/8g_X^2 ∑_i=1,2,3(H^†σ^iH +H'^†σ^iH')^2( 1 - ϵ^2 ), ϵ^2 = m_X^2/2m_S^2 + m_X^2.In the SUSY limit, m_S^2 =0, the D-term potential vanishes. In terms of the model parameters M,m_S, κ, g_X, ϵ^2 is given by ϵ^2 = g_X^2 (m_S^2 - κ^2 M^2)/g_X^2 (m_S^2 - κ^2 M^2) - 2 κ^2 m_S^2In the limit where κ≪ g_X, ϵ^2=1 and hence the D-term potential decouples. In order to obtain a large D-term potential, it is preferable that κ is as large g_X.To estimate the maximal possible value of κ, we solve the renormalization group equation of g_X and κ, d/ dlnμ g_X = g_X^3/16π^2 1 + 21/16π^2 g_X^2 - 1/8π^2κ^2/1-g_X^2/4π^2,d/ dlnμκ = κ/16π^2 (4 κ^2 - 3 g_X^2),from a high energy scale M_* towards low energy scales, with a boundary condition at M_* of g_X = κ≃ 2π. M_* can be identified with the Landau pole scale.The running of g_X and κ is shown in Fig. <ref>, which shows that κ≃ g_X much below M_*. We obtain the same conclusion as long as κ(M_*)1. For κ≃ g_X, ϵ^2 is ϵ^2 ≃g_X^2 M^2 - m_S^2/g_X^2 M^2 + m_S^2We may obtain a sufficiently small ϵ^2, say ϵ^20.2, for m_S^20.6 g_X^2 M^2.Notice also that for ϵ^2<1 there is a threshold correction to the soft Higgs mass which is proportional to a new gauge bosons mass squared:(δ m_H_u^2)_X= 3 g_X^2/64 π^2 m_X^2 ln(ϵ^-2),which may be a source of tuning of the EW scale. The same threshold correction is present also for the right-handed stop soft mass squared m_U_3^2. §.§ SU(4) invariant quartic coupling and μ terms We give masses to H = (H_1,H_2)^T and ϕ_u by pairing them with ϕ_d,1,2,3 through the superpotential terms, W = λ_1 ϕ_d,1 H S + λ_2 ϕ_d,2 HS̅ + m ϕ_u ϕ_d,3The pairs (H_1,ϕ_d,1), (H_2,ϕ_d,2) and (ϕ_u,ϕ_d,3) obtain masses of λ_1 v_S, λ_2 v_S and m, respectively. We assume that λ_2 v_S, m1 TeV and neglect (H_2,ϕ_d,2) and (ϕ_u,ϕ_d,3) for the dynamics of the electroweak symmetry breaking. We identify H_1 and ϕ_d,1 with H_u and H_d in the Higgs sector of the standard SUSY model. The μ parameter is given by μ=λ_1 v_S.The SU(4) invariant quartic coupling of (H_u,H_u') is given by V =g_X^2/8 (1-ϵ^2) (|H_u|^2 + |H_u'|^2)^2. As we will see, the VEV of ϕ_u is responsible for the masses of the up and charm quarks, and the neutrinos. To give a VEV to ϕ_u, we introduce a coupling W = δ m ϕ_u ϕ_d,1.Through the F term potential of ϕ_d,1, ϕ_u obtains a tadpole term after H_u obtains its VEV, which induces a non-zero VEV of ϕ_u.Through the coupling λ_2 (> λ_1), m_H_u^2 receive a quantum correction from m_S^2, Δ m_H_u^2≃ - λ_2^2/8π^2m_S^2 L = - (600 GeV)^2 ( λ_2/0.3)^2 m_S^2/(6 TeV)^2L/ ln10^4where L denotes a log-enhancement through an RGE. As long as λ_2 0.4, this contribution is always smaller than that from stops and/or the threshold correction from X, and hence we neglect it. Note, however, that even larger values of λ_2 may be possible without introducing tuning if the mediation scale of SUSY breaking is relatively low and/or m_S^2 runs to smaller values at higher energies.Note that the Z_2 symmetry S ↔S̅ is explicitly broken by the above superpotential couplings. Even if we assume the Z_2 symmetry of the soft masses of S and S̅, we expect a quantum correction to a mass difference of them, Δ m_S^2 ≡ m_S^2 - m_S̅^2 ≃λ_2^2/8π^2m_S^2 L.This leads a asymmetric VEV of S and S̅, which give m_H_u^2 through the D-term potential, m_H_u^2≃ -ϵ^2/2Δ m_S^2,which is always smaller than the direct one-loop quantum correction in Eq. (<ref>). It is also possible to maintain the Z_2 symmetry. Instead of the coupling in Eqs. (<ref>) and (<ref>), we introduce W = H ( λ_1 ϕ_d,1 + λ_3 ϕ_d,3 )( S + S̅) + λ_2Hϕ_d,2 ( S - S̅)+ ϕ_u ( m_1ϕ_d,1 + m_3 ϕ_d,3)Here we have assumed that ϕ_d,2 is odd under the Z_2 symmetry. After S and S̅ obtain their VEVs, the mass terms become W = v_S (λ_1 ϕ_d,1 + λ_3 ϕ_d,3 ) (H_1 - H_2) + λ_2 v_S ϕ_d,2 (H_1 + H_2) + ϕ_u ( m_1ϕ_d,1 + m_3 ϕ_d,3).We assume that λ_2 v_S, m_i1 TeV. Then (H_1 + H_2) /√(2) and ϕ_u obtain a large mass paired withϕ_d,2 and a linear combination of ϕ_d,1 and ϕ_d,3, respectively, andare irrelevant for the dynamics of the electroweak symmetry breaking. H_u ≡ (H_1 - H_2)/√(2) obtains a mass of O(λ_2,3 v_S) paired with another linear combination of ϕ_d,1 and ϕ_d,3 which we call H_d. §.§ Masses of matter particles We first consider a case where the Z_2 symmetry S↔S̅ is explicitly broken. A large enough top yukawa coupling is obtained by the superpotential W = y_tHQ̅_R Q_3 →y_t(H_2Q̅_R,1 - H_1 Q̅_R,2) Q_3,where Q̅_R = ( Q̅_R,1 , Q̅_R,2)^T. We give a large mass to Q̅_R,1 by introducing a coupling W = y Q̅_R U S,and identify Q̅_R,2 with a right-handed top quark u̅_3.The yukawa couplings of the up and charm quarks originates from the couplings with ϕ_u, W = y_u,ijϕ_u Q_i u̅_̅j̅.The left-handed neutrino masses are obtained in a similar manner once right-handed neutrinos are introduced. The yukawa couplings of the down-type quarks and the charged leptons is given by couplings with ϕ_d,i, W = y_d,ijkϕ_d,i Q_j d̅_k + y_e,ijkϕ_d,i L_j e̅_k. The extra SU(2)_X charged particle E̅ obtains its mass paired with E_1,2 through the SU(2)_X symmetry breaking, W= E̅ (y_E,1 E_1 + y_E,2 E_2 ) S +E̅ (y̅_E,1 E_1 + y̅_E,2 E_2 ) S̅.Next we consider a case where the Z_2 symmetry is maintained. The top yukawa coupling is obtained by the superpotential W = y_tHQ̅_R Q_3 →y_t H_u 1/√(2)(Q̅_R,1 +Q̅_R,2) Q_3.One linear combination of Q̅_R,1 and Q̅_R,2 obtains a Dirac mass term paired with U, W = y Q̅_R U (S + S̅) → y v_S 1/√(2) ( Q̅_R,1 - Q̅_R,2 )U.We identify the massless combination (Q̅_R,1 +Q̅_R,2 )/√(2)≡u̅_3 as a right-handed top quark. The extra SU(2)_X charged particle E̅ obtains its mass paired with E_1,2 through the coupling, W= E̅ (y_E,1 E_1 + y_E,2 E_2 ) S +E̅ (y_E,1 E_1 - y_E,2 E_2 ) S̅.Here we assume that E_2 is odd under the Z_2 symmetry S ↔S̅, so that all particles in E̅ and E_1,2 obtains their masses.So far we have assumed that a linear combination of Q̅_R,1 and Q̅_R,2 obtains a large mass paired with U. It is also possible to identify the linear combination with the right-handed charm quark. In such a model U and u̅_2 are not necessary. The mass of the right-handed scharm is predicted to be as large as that of the right-handed stop. This choice is beneficial for a high mediation scale, as it makes the SU(3)_c and U(1)_Y coupling constants relatively smaller, reducing the fine-tuning from the gluino and the bino.§ FINE-TUNING OF THE ELECTROWEAK SCALE Let us now discuss fine-tuning of the EW scale in the model. We quantify the degree of fine-tuning by introducing the measure <cit.>, Δ_v ≡Δ_f ×Δ_v/f,whereΔ_v/f = 1/2( f^2/v^2 -2), Δ_f = max_i ( |∂ ln f^2/∂ ln x_i(Λ)|, 1 ) .Here f ≡√(v^2 + v^'2) is the decay constant of the spontaneous SU(4) breaking. Δ_v/f measures the fine-tuning to obtain v < f via explicit soft ℤ_2 symmetry breaking. Δ_f measures the fine-tuning to obtain the scale f from the soft SUSY breaking which is analogous to the fine-tuning to obtain the electroweak scale from the soft SUSY breaking in the MSSM. x_i(Λ) are the parameters of the theory evaluated at the mediation scale of the SUSY breaking Λ. We include the important seven parameters, m_H_u^2, m_Q_3^2, m_u̅_3^2, M_1^2, M_2^2, M_3^2 and μ^2. To evaluate Δ_f we solve the renormalization group equations (RGEs) of parameters betweenand Λ. We assume that the right-handed charm quark is also embedded in Q̅_R. Betweenand m_X we solve MSSM RGEs at the one-loop level appropriately modifying the beta function of m_Q_3^2.At a scale m_X we perform matching by including the threshold correction (<ref>) to m_H_u^2 and m_U_3^2.Above m_X we solve the RGEs (that include the effects of non-MSSM states) at least at the one-loop level. The RGEs of the gauge couplings are solved at the two-loop level, but set, for simplicity, κ=0.[Non-zero κ slightly slows down the running of g_X but the impact on Δ_v and the scale ofthe Landau pole is negligible. ] The yukawa couplings other than the top yukawa are neglected.As clearly seen from eqs. (<ref>)-(<ref>),for a given value of f there is a lower bound on Δ_v of Δ_v/f. f/v is constrained by the Higgs coupling measurements <cit.> to be at least 2.3<cit.>. The latter value has been obtained neglecting invisible decays of the Higgs to mirror particles, which are generically non-negligible, so in our numerical analysis we use less extremal value off=3 v. Nevertheless, the tuning is quite independent of this choice (unless f is so large that Δ_v/f determines Δ_v). In fig. <ref> we present contours of Δ_v assuming low and high mediation scales of SUSY breaking Λ. [In the figure we shade the parameter region where the Landau pole scale of the gauge coupling g_X is above Λ. It is also possible that the SUSY breaking is mediated above the Landau pole scale, but we cannot calculate the fine-tuning measure unless we specify the description of the model above the Landau pole scale.] Here and hereafter, the stop mass m_ stop and the gluino mass M_3 refer to the values at the TeV scale. For Λ=100 tuning at the level of 10% can be obtained for the stop masses as large as 3 TeV, as seen from the upper left panel. An important constraint on the parameter space is provided by the Higgs mass measurement <cit.>. In order to assess the impact of this constraint we compute the Higgs mass following closely the procedure described in ref. <cit.>. The blue bands show the parameter region with m_h = 125 ± 3 GeV, where the error is a theoretical one. It can be seen from the upper right panel of fig. <ref> that this constraint prefers rather light stop unless tanβ is small enough.Since we are most interested in stop masses that easily avoid current or even potential future LHC constraint we set for the low scale mediation case tanβ=2.5 whichimplies the stop masses in the range between about 1.5 and 3 TeV. This range narrows to between 1.7 and 2 TeV if one demands tuning better than 10%. Interestingly, tuning is minimised for intermediate values of the stop masses which is a consequence of some cancellation between the thresholdcorrection from X and corrections from stops and gluino to m_H_u^2. In this region the value of |m_H_u^2| at the mediation scale is somewhat suppressed.For lighter stops (which can be compatible with the Higgs mass constraint for larger tanβ) the tuning is dominated by the threshold correctionwhich implies tuning at the level of few percent.It should be noted that fine-tuning of the EW scale is minimized at some intermediate value of g_X of about 1.5-2 even though perturbativity constraint allows for g_X as large as about 2.5. This is because for appropriately large g_X the tuning is dominated by the threshold correction to m_H_u^2 from the new gauge bosons. Since the latter must be rather heavy for large g_X due to EW precision constraints, the threshold correction dominates for g_X≳2 and the tuning gets worse with increasing g_X in spite of larger SU(4) invariant coupling. In fact, for very large value of g_X there is essentially no tuning of the EW scale from stops and gluino but the overall tuning is at the level of few percent. In the region of large g_X, where the threshold correction dominates the fine-tuning, larger values of ϵ lead to smaller tuning. On the other hand, for smaller g_X, when the threshold correction is subdominant, it is preferred to have smaller ϵ to suppress correctionsfrom stops and gluino by larger SU(4) invariant coupling. It is interesting to compare the fine-tuning of the present model to that in the model where an SU(4) invariant coupling originates from a non-decoupling D-term of U(1)_X gauge symmetry proposed in ref. <cit.>. For the stop mass below about 1 TeV, the U(1)_X is less tuned with tuning even better than 20%. This is because the threshold correction from the X gauge bosonsin the U(1)_X case is three times smaller than in the case of SU(2)_X. As the stop mass increases the tuning in the U(1)_X model gets worse and already for 2 TeV stops the tuning in the SU(2)_X model becomes better than in theU(1)_X model due to larger SU(4) invariant coupling which suppresses the correction from stops.The biggest advantage of the SU(2)_X model is that RGE running of g_X is relatively slow so the Landau pole scale, for given g_X, is much higher than in theU(1)_X model.For example in the case of Λ=10^16 GeV presented in the lower panels of fig. <ref>, values of g_X up to about 1.2 are possible without the Landau pole below Λ.In the previously proposed SUSY Twin Higgs models it is was not possible to keep perturbativity up to such high scale. We see from fig. <ref> that for Λ=10^16 GeV the fine-tuningbetter than few % can be obtained for the stop masses as large as 2 TeV. This is obviously worse than in the low-scale mediation case discussed before but for high-scale mediation there are more possible mechanisms of the mediation of the SUSY breaking. The fine-tuning is also much better than in the MSSMwith high-scale mediation. This is due to suppression of the corrections from stops and gluino (which dominates tuning for high mediation scales) bythe SU(4) invariant coupling but also because a large value of g_X efficiently drives the top yukawa coupling to smaller values at higher scales.Dependence of fine-tuning on the mediation scale for 2 TeV stops is presented in fig. <ref>.We see that moderate tuning of few percent can be obtained for high mediation scales. For high mediation scales the tuning is dominated by the correction from the gluino so the tuning crucially depends on the gluino mass limits. It was recently emphasised in ref. <cit.>that one should convert running soft masses to pole masses when assessing the impact of experimental constraints on naturalness of SUSY models. It was shown that the loop corrections <cit.> from 2 TeV squarks increase the gluino pole mass by 10% as compared to the soft mass.For heavier 1st/2nd generation of squarks, as experimentally preferred, the correction may be much larger e.g. 20% for 10 TeV squarks. In the left panel of fig. <ref> we fix the soft gluino mass to 2 TeV which easily satisfies the LHC constraints even for moderate loop corrections from squarks <cit.>. In such a case, 5% tuning is possible with the mediation scale, being below the Landau pole scale, as high as𝒪(10^12) GeV. For M_3=2.5 TeV, presented in the right panel, for which the gluino is definitely outside of the LHC reach <cit.>, mediation scale of order 𝒪(10^10) GeV can still allow for better than 5% tuning. Notice also a sharp increase in tuning when the mediation scale approaches the Planck scale. This originates from the fact that U(1)_Y gauge coupling constant runs rather fast due to many new states carrying hypercharge and eventually enters non-perturbative regime around the Planck scale. In consequence, bino strongly dominates fine-tuning when the mediation scale is close to the Landau pole for U(1)_Y.The fine-tuning for high mediation scales is even better if the gluino obtains its mass paired with an adjoint chiral superfield by a supersoft operator,due to the absence of the log-enhanced correction to m_H_u^2 <cit.>. The soft stop mass and the higgs mass are dominantly generated by the threshold correction around the gluino mass, m_ stop^2 ≃ 1/16M_3^2, m_ H_u^2 ≃ 3y_t(M_3)^2 /4π^2 m_ stop lnM_3/m_ stop .In non-Twin models the fine-tuning may be at a few % level even if the stop mass is as large as 2 TeV, which is further improved by the Twin-Higgs mechanism. The contour of Δ_v assuming the Dirac gluino is shown in fig. <ref>. For the stop mass of 2 TeV, 𝒪(10)% tuning is possible even if the mediation scale is as high as 10^16 GeV. Note that in Dirac gluino models the large log enhancement of the quantum correction to the Higgs mass squared is already absent. Thus the improvement of the fine-tuning by the Twin higgs mechanism simply originates from a large SU(4) invariant coupling. For g_X =1 - 1.5, the improvement is by a factor of 2-4. In some UV completions of the Dirac gluino, the fine-tuning may be worse and at the O(1)% level <cit.>. For example in gauge mediated models, a tachyonic soft mass term of the adjoint chiral superfield larger than the Dirac gluino mass is often generated. See ref. <cit.> for a pedagogical discussion.To prevent the instability of the adjoint field one needs to cancel the tachyonic mass by additional large soft mass or a supersymmetric mass of the adjoint, which leads to fine-tuning. See ref. <cit.> for a gauge mediated model free from this problem. In gravity mediated model the tachyonic mass is not necessarily larger than the Dirac gluino mass. Our D-term model, together with the Dirac gluino, realizes the natural SUSY even for the gravity mediation. The wino and the bino masses are also bounded from above by naturalness. The constraint is stronger than that in the MSSM as we add extra SU(2)_L and/or U(1)_Y charged fields which makes the corresponding gauge couplings and gaugino masses growing faster with the renormalization scale. Fine-tuning from bino and wino may be very large especially for high mediation scales. In the left panel of fig. <ref> we fix Λ=10^17 GeV and present contours of fine-tuning in the plane M_1-M_2. We see that bino as light as 700 GeV induces tuning at the level of 1 % for this mediation scale. The tuning from wino is slightly smaller but still 1 TeV wino results in about 1 % tuning. From the comparison of figs. <ref> and <ref> we see that in order not to increase tuning by more than a factor of two, bino (wino) must be lighter than about 400 (600) GeV. Thus, one generally expects all neutralinos to be light and the LSP to be a mixture of bino, wino and higgsino. Assuming majorana gluino, the impact of wino and bino on the tuning is less pronounce but in order not to increase tuning by more than a factor of two their masses are still expected to be below about 1 TeV, cf. figs. <ref> and <ref>. For smaller mediation scales the tuning from bino and wino is milder. The tuning from bino is subdominant unless Λ≳10^16 GeV. In the right panel of fig. <ref> we present tuning from wino as a function of the mediation scale. We see that even for small mediation scale wino mass should generally be below 1 TeV in order not to dominate tuning. The bounds on the masses is avoided if the wino and the bino also obtain Dirac masses.Interestingly, with an additional SU(2)_L adjoint paired with wino, the SU(2)_L gauge coupling constant also blows up around the Planck scale. In the above analysis we have ignored the contribution to the RGE running of m_H_u^2 proportional to the SU(2)_X gauge coupling constant. As long as the SU(2)_X gaugino mass is suppressed, one-loop contributions are negligible. At the two loop level, there is a contribution, d/ dlnμ m_H_u^2 ⊃3 g_X^4/256π^4∑_im_i^2,where m_i^2 is a soft mass squared of a SU(2)_X fundamental. Although this is a two-loop effect, the largeness of m_S^2 required to obtain a large non-decoupling SU(4) invariant quartic and the largeness ofg_X around the Landau pole scale can make thiscontribution non-negligible. In the left panel of fig. <ref>, we show the fine-tuning including this two-loop effect to m_H_u^2, with m_S^2 fixed at the value determined by eq. (<ref>), while ignoring contribution from other SU(2)_X charged fields. The fine-tuning gets worse than the case ignoring the two-loop effect, especially when the mediation scale is close to the Landau pole scale, while it remains the same if the mediation scaleis much smaller than the Landau pole scale. We note, however, that the two-loop effect strongly depends on the boundary condition of soft masses at the mediation scale and might be much smaller. For example, if m_S^2 = - m_E̅^2 at the boundary, the two-loop effect is suppressed. This special boundary condition should be explained by a UV completion of our model. It is also possible that m_S^2 at the UV boundary is much smaller than around the m_X scale, and is generated through the RG running. Actually couplings with the fields Z and/or E_1,2 can generate a non-zero and positive m_S^2, if the soft masses of them are negative. For example, in the right panel of fig. <ref> we show tuning under the assumption that m_S^2 vanishes at the mediation scale and gets renormalized to appropriately large value determined by eq. (<ref>) at the EW scale via the interaction (<ref>) (by suitable choice of the soft mass for Z) with κ =0.3 at the UV boundary. In this case the impact of m_S^2 on tuning is rather small unless the mediation scale is very close to the Landau pole scale.Even though naturalness does not require sparticles to be within discovery reach of the LHC (perhaps except for wino if the mediation scale is high enough) it does require that the twin Higgs boson is relatively light. The mass of the twin Higgs boson is well approximated by 2√(λ)f with λ being the SU(4) invariant coupling. Both λ and f are constrained from above by naturalness. For example, demanding better than 10% tuning f must be below about 4.5. This upper bound on f is quite generic for Twin Higgs models unless hard ℤ_2 breaking is non-negligible. The upper bound on λ is specific for this model and is set by the requirement of not too large threshold correction from X gauge bosons. We find that betterthan 10% tuning requires λ≲0.5 which, together with the upper bound on f, leads to the upper bound on the twin Higgs boson mass of about 1 TeV. The twin Higgs tends to be lighter for a larger Landau pole scale.For recent studies of the phenomenologyof the twin Higgs boson we refer thereader to refs. <cit.>. It is also noteworthy that in this model MSSM-like Higgs bosons and their mirror counterparts are not required to be light by naturalness because H_d is not charged under SU(2)_X. § SUMMARY We proposed a new SUSY Twin Higgs model in which an SU(4) invariant quartic term originates from a D-term potential of a new SU(2)_X gauge symmetry. The choice of the non-abelian gauge symmetry, together with a minimal number of flavors charged under SU(2)_X, makes the running of the new gauge coupling constant rather slow allowing for a large SU(4) invariant quartic term without generating a low-scale Landau pole. The Twin Higgs mechanism, together with the negative contribution from the new gauge coupling to the RG running of the top yukawa coupling, allows for tuning of the EW scale better than 10% for high mediation scales up to 𝒪(10^9) GeV even for sparticle spectra that may be outside of the ultimate LHC reach. If the gluino obtains a Dirac mass term, tuning of 10% is possible even ifthe mediation scale is around the Planck scale. The model may be tested at the LHC by searching for a twin Higgs boson whose mass is bounded from above by naturalness and is anti-correlated with the Landau Pole scale. In parts of parameter space with tuning better than 10% the twin Higgs boson is expected to be lighter than about 1 TeV. All electroweakinos are expected to be rather light, with masses in the sub-TeV region, especially if the mediation scale of SUSY breaking is high.§ ACKNOWLEDGMENTSThis work has been partially supported by National Science Centre, Poland, under research grant DEC-2014/15/B/ST2/02157, by the Office of High Energy Physics of the U.S. Department of Energy under Contract DE-AC02-05CH11231, and by the National Science Foundation under grant PHY-1316783. MB acknowledges support from the PolishMinistry of Science and Higher Education through its programme Mobility Plus (decision no. 1266/MOB/IV/2015/0).The work of KH was in part performed at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611.§ ELECTROWEAK PRECISION MEASUREMENTS We use the so-called S,T,U parametrization <cit.> to constrain the parameter space of our model. We follow the method presented in <cit.>, where the observables shown in Table <ref> are used to constrain S,T,U. We take U=0 and show the constraint on (S,T) in Fig. <ref>. The Higgs multiplet H is charged under SU(2)_X. After the electroweak symmetry breaking scale, Z boson mixes with the SU(2)_X gauge bosons. The mixing breaks the custodial symmetry and we expect a severe constraint from the electroweak precision measurement. After integrating out the SU(2)_X gauge bosons, we obtain the effective dimension 6 operator, L_ eff = g_X^2/8 m_X^2( H_u^† D_μ H_u - (D_μ H_u)^† H_u )^2.This generates a non-zero T parameter, T=1/2αc_W^2/g_2^2g_X^2 m_Z^2/ m_X^2sin^2β .The dependence on tanβ originates from subdominant H_d component of the Higgs. S and U parameters are negligibly small. 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The Kazdan-Warner equation on graphs] The Kazdan-Warner equation oncanonically compactifiablegraphs M. Keller]Matthias Keller M. Keller, Institut für Mathematik, Universität Potsdam 14476Potsdam, Germany [email protected] M. Schwarz]Michael Schwarz M. Schwarz,Institut für Mathematik, Universität Potsdam 14476Potsdam, Germany [email protected] We study the Kazdan-Warner equation on canonically compactifiablegraphs. These graphs are distinguished as analytic properties of Laplacians on these graphs carry a strong resemblance to Laplacians on open pre-compact manifolds.[ [ May 29, 2018 ================§ INTRODUCTIONIn recent years various topics in the analysis on graphs received a lot of attention. While the main focus was put so far mainly on the study of linear equations such as the heat equation, the Poisson equation or fundamental topics in spectral theory, see e.g. <cit.>, there is an upcoming interest in non-linear problems. On the onehand this concerns non-linear operators such as the p-Laplacian, <cit.>, and on the other hand non-linear equations such as the Kazdan-Warner equation <cit.>. A major goal is often to unveil the analogies in the continuum, i.e., Riemannian manifolds, and the discrete setting, i.e. graphs. The deeper reason of these analogies stems from the fact that both the Laplacian in the continuum and in the discrete are generators of so called Dirichlet forms or from another perspective they are generators of certain Markov processes, see <cit.>. A major challenge in the discrete is often to find the correct geometric notions to even formulate analogous statements. The Kazdan-Warner equation arises from the basic geometric question which functions are potential curvatures of 2-dimensional manifolds. Kazdan and Warner studied this questionon compact manifolds <cit.> and on open manifolds that are diffeomorphic to an open set in a compactmanifold <cit.>. From there on there is an enormous amount of work on this topic and we refer here only to <cit.>Compactness in the discrete setting is obviously equivalent to finiteness of the graph. For this case a very satisfying answer was foundby Grigor'yan/Lin/Yang <cit.>, see also <cit.>. On the other hand it is not a priori clear what are graph analogues topre-compact open manifolds. This is the starting point of our paper. We propose a class called canonically compactifiable graphs. These infinite graphs were recently introduced in <cit.>, see also <cit.>, and the analysis of Laplacians onthese graphs resembles a lot of features from the analysis of Laplacians on bounded domains or pre-compact open manifolds with sufficiently nice boundaries. For example the Dirichlet problem on the Royden boundary is uniquely solvable and the spectrum of the Laplacian is purely discrete in the case of finite measure. Here, for the Kazdan-Warner equation, we prove analogous results as in the finite case for canonically compactifiable graphs, and, indeed, our results include finite graphs as a special case.Let us recall the original geometric motivation of the Kazdan-Warner equation.This traces back to an equation on a 2-dimensional Riemannian manifold M with finite volume and two conformal metrics g andg=e^2g for some smooth function . Specifically denoting the corresponding Gaussian curvatures by K and K this gives rise to the equationΔ_g=K-K e^2,with the Laplace-Beltrami operator Δ_g with respect to the metric g. Provided there is a solution ψ to the equation Δ_gψ=K-K with K being the averaged curvature and letting u=2(-ψ) this translates into the equationΔ_g u=2K-(2Ke^2ψ)e^u.Hence, the question whether a prescribed function K is a curvature function under a conformal transformation is equivalent to the question whether the equation above has a solution u. Of course, from the geometric picture one already expects certain restrictions on K depending on K as for example thatK is positive somewhere whenK>0 or that K changes sign if K≠ 0 in the flat case K =0 (at least when M is compact). Although there are already several very promising notions of curvatures proposed on graphs, see e.g. <cit.>, it still seems to be out of reach to study this problem in terms of one of these curvature notions. However, looking at this equation from an analytic perspectiveis interesting in its own right. This reduces to the study of an equationLu=-c+he^uwhere L is the graph Laplacian, c is a constant, h is a function, and u is a function in the domain of L, see Section <ref> for the precise definitions.As stated above we study this equation on so called canonically compactifiable graphs and under the assumption that the graph has finite measure. One way of characterizing canonically compactifiable graphs is that functions of finite energy are already bounded, see <cit.> (while for this work it is sufficent that all finite energy functions in ℓ^2 are bounded).Our results can besummarized as follows in dependence of the sign of c and where h denotes the average of h: c=0 : There exists a solution if and only ifh<0 and h changes sign. c>0 : There exists a solution if and only if h is positive somewhere. c<0 : If there exists a solution then h<0 and if h<0 then there is -∞< c(h)<0 such that there is a solution for c(h)<c<0.We only assume h ≠0 to be square summable. Moreover,we consider the so-called Neumann-Laplacian which is the self-adjoint operator associated to the quadratic form on all ℓ^2-functions of finite energy. Solutions are then requested to be in theoperator domain of the Neumann-Laplacian. So, we are indeed talking about strong solutions of the equation.From this it is clear that while the basic ideas of our proofs owe to <cit.>, theyneed substantially more care on the operator theoretic level. Furthermore, in contrast to the case of manifolds we do not have a theory of elliptic regularity in the discrete case and no chain rule.The paper is structured as follows. In the next section we introduce the setting and in particular canonically compactifiable graphs. We furthermore provide basic tools such as a Sobolev embedding theorem, a Poincar inequality and a Trudinger Moser inequality. In Section <ref> we characterize existence of solutions in the case c=0 and in Section <ref> we study the case c>0. Finally, in Section <ref> we give a necessary and sufficient criterion on the existence of solutions in the case c<0.In this paper C denotes an arbitrary positive constant which might change from line to line. If we want to indicate that a constant C depends on a parameter β we write C=C(β). Note added: After this work was completed we learned about the recent preprint of Ge/Jiang <cit.>. This paper studies the Kazdan-Warner equation on infinite graphs, however with different geometric assumptions on the graph and on the function h. It seems that these assumptions are independent from the assumptions of this paper and the techniques used there are totally different from ours.§ THE LAPLACIAN ON CANONICALLY COMPACTIFIABLE GRAPHS In this section we introduce canonically compactifiable graphs over discrete measure spaces. These graphs, which can be seen as discrete analogues to open pre-compact manifolds,were studied systematically in <cit.>, see also <cit.>. The definition is here slightly weakened by taking also a finite measure into consideration. We then introduce several tools that are needed for the proof. The availability of these tools such as the Sobolev embedding theorem, a Poincar inequality and a Trudinger Moser inequality, which hold trivially in the finite setting, see <cit.>,owes here to the assumption on the graph being canonically compactifiable and of finite measure.Let X be a countable[Here, countable means finite or countably infinite, although we are of course mainly interested in the countably infinite case] set. A symmetric function b:X× X→ [0,∞) that vanishes on the diagonal and satisfies ∑_y∈ X b(x,y)<∞for every x∈ X is called a graph on X.A graph is called connected if for every x,y∈ X there are x_1,…, x_n∈ X such that b(x,x_1)>0, b(x_1,x_2)>0,…, b(x_n,y)>0. We equip the set X with the discrete topology, so that every measure m on (the Borel-σ-field of)X is given by afunction, also denoted by m, from X to [0,∞], via m(A):=∑_x∈ A m(x). Throughout this paper we assume thatm(x)>0 for all x∈ X and that m is a finite measure, m(X)<∞,and call (X,m) a finite discrete measure space. Given a measure m on X we defineas the space of real valued functions on X which are p-square-summable with respect to m, 1≤ p<∞ and ℓ^∞(X) the space of bounded functions. Denote the corresponding norm ·_p by f_p:=(∑_x∈ X|f(x)|^pm(x))^1/pFurthermore, we denote the dual pairing of ℓ^p and ℓ^q for p,q∈[1,∞] with 1/p +1/q=1 by⟨·,·⟩.Denote by C(X) the space of real valued functions. We define a quadratic form :C(X)→ [0,∞] via(u)=1/2∑_x,y∈ X b(x,y)(u(x)-u(y))^2.Let:={u∈ C(X)|(u)<∞}.Via polarizationgives rise to a semi-inner product onalso denoted by . It is easy to see thatis Markovian, i.e.(0∨ u∧ 1)≤(u),u∈ . By Fatou's lemma Q is lower semi-continuous and, therefore,the restrictionofto:=∩ is closed and, hence, a Dirichlet form which is often referred to as the Dirichlet form with Neumann boundary conditions or the Neumann form. We denote the form norm by·_:=((·)+·_2^2)^1/2.Furthermore, we let (f):=∞ for f∈ℓ^2(X,m)∖ D(). There is a self-adjoint operator, the Neumann Laplacian, L in correspondence withgiven by D()={u∈| there isf∈ such that (u,v)=⟨ f,v⟩ holds for everyv∈}, u=f.From this definition it is clear that the constant functions are in D(L) in the case of finite measure. It turns out, <cit.>,by the virtue of Green's formulaacts asu(x)=1/m(x)∑_y∈ X b(x,y)(u(x)-u(y)),u∈ D(). The goal of this paper is to study the so called Kazdan-Warner equation for functions u∈ D()u=-c+he^u, for some given function h∈ℓ^2(X,m), h≠ 0, and a constant c. We will study this equation on what we call canonically compactifiable graphs over (X,m).A connected graph b over a finite discrete measure space (X,m) is called canonically compactifiable if⊆ℓ^∞(X). The notion of canonically compactifiable graphs b over a discrete set X (instead of ameasure space (X,m)) was introduced in <cit.> via the condition ⊆ℓ^∞(X). For these graphs the Royden compactification (which arises as space of maximal ideals of the commutiative Banach-algebra ofclosed with respect to ·_∞) turns out to have particular nice properties. For example the Dirichlet problem with respect to the Royden boundary is uniquely solvable. Furthermore,various results in<cit.> suggest that canonically compactifiable stand in parallel to bounded domains in ^d. Next, we present two embedding results. Let b be a connected, canonically compactifiable graph over (X,m) with m(X)<∞. Then, there is a C>0 such thatu_∞≤ Cu_, u∈. The statement follows from the closed graph theorem. Let b be a connected, canonically compactifiable graph over (X,m) with m(X)<∞. Then, (,·_) embeds compactly into . In particular, the operatorhas purely discrete spectrum. For the semigroupe^-t, t≥0, of we have by the definition of canonically compactifiable graphs and finiteness of the measure m e^-tLℓ^2(X,m)⊆ D()⊆ D()⊆ℓ^∞(X)⊆ℓ^2(X,m). Thus, e^-tcan be viewed as an operator from ℓ^2(X,m) to ℓ^∞(X) composed with the embedding ℓ^∞(X)→ℓ^2(X,m) and is, therefore, a Hilbert-Schmidt operator by the factorization principle, see <cit.>. However, compactness of e^-tL is equivalent to the compactness of the embedding of D() into ℓ^2(X,m) which is equivalent to discreteness of the spectrum of . Let b be a connected, canonically compactifiable graph over (X,m)with m(X)<∞. Then, ()=span{1}. Furthermore,from {1} ^⊥ to {1}^⊥ is invertible. The first part of the statement carries over from <cit.>. The second part now follows from the fact that by the Sobolov embedding above,has purely discrete spectrum and is, therefore, bijective on {1}^⊥.In the case of finite measure m we obviously haveℓ^∞(X)⊆ℓ^p(X,m)⊆ℓ^1(X,m), p∈[1,∞),and, hence,for u∈ℓ^p(X,m) we can defineu:=1/m(X)⟨ u,1⟩.Next, we come to a Poincaré-inequality on canonically compactifiable graphs.Let b be a connected, canonically compactifiable graph over (X,m)with m(X)<∞. Then, there is C>0 such that u-u_2^2≤ C(u), u∈. Supposethere is a sequence (u_n) insuch that u_n=0,(u_n)→ 0 and u_n_2=1. Since (u_n) is bounded with respect to ·_it contains an ℓ^2-convergent subsequence by the Sobolev embedding, Proposition <ref>. Without loss of generality we assumethat (u_n) itself is ℓ^2-convergent to a limitu. Obviously, u_2=1 and due to the finiteness of m, we have u =0 sinceℓ^2-convergence induces ℓ^1-convergence (by the closed graph theorem).Sinceis lower semicontinuous with respect to -convergence, we conclude(u)≤lim inf_n→∞(u_n)=0and, therefore u must be constant. Thus, u=0 implies u=0 which however contradicts u_2=1. Hence theinequality holds for every u ∈ with u=0. For general u∈ observe that u-u=0 and ( u-u) =( u).Next, we discuss the following Trudinger-Moser inequality. Let b be a connected, canonically compactifiable graph over(X,m) with m(X)<∞. Then, for every β∈ there is a constant C>0such that ⟨ e^β (u-u)^2,1⟩≤ m(X)e^C|β|(u) for all u∈.In particular, for all β∈ there is C=C(β) such that⟨ e^β u^2,1⟩≤ C(β) for all u∈ with (u)=1 and u=0.The inequality holds for β≤ 0 since m(X)<∞. Let β>0. By Proposition <ref> and the Poincaréinequality, Proposition <ref>,(u(x)-u)^2≤u-u_∞^2≤ Cu-u_^2=C((u)+u-u_2)≤ C (u). Hence, the statement follows. Next, we show that for canonically compactifiable graphs with finite measure the form domain is invariant under composition with continuously differentiable functions. Let b be a connected, canonically compactifiable graph over (X,m) with m(X)<∞. Let u∈and f:→ be continuously differentiable. Then, f∘ u∈. Let u∈⊆ℓ^∞(X). Since both f and f' are continuous, we get that f∘ u,f'∘ u∈ℓ^∞(X)⊆ℓ^2(X).Let C:= f'|_[-u_∞,u_∞]_∞^2. Foranyx,y∈ X we infer by the mean value theoremthat there exists a point ξ between u(x) and u(y) such that (f∘ u(x)-f∘ u(y))^2=f'(ξ)^2(u(x)-u(y))^2<C(u(x)-u(y))^2 Thus, (f∘ u)≤ C (u)<∞ and, hence, f∘ u∈.§ THE CASE C=0 Let b be a canonically compactifiable graph over (X,m) such that m(X)<∞ and 0≠ h∈ℓ^2(X,m). Then, there is an u∈ D() withu=he^uif and only if h<0 and h changes sign. Suppose u∈ D() is a solution. Then, u∈ℓ^∞(X) since the graph is graph canonically compactifiable and, hence,he^u∈ℓ^2(X,m). Furthermore, m(X)<∞ also implies 1∈ and as u is a solution, we obtain⟨ he^u,1⟩=⟨ u,1⟩=(u,1)=0.Hence, h must change sign, as h≢0 by assumption. Then,since e^-u∈ by Lemma <ref>, wecompute⟨ h,1⟩=⟨ he^u,e^-u⟩=⟨ u, e^-u⟩=(u,e^-u)<0, where the last inequality follows by definition ofand the fact that t↦ e^-t is monotone decreasing. Furthermore,the inequality is strict since u can not be a constant if h changes its sign, or otherwise we would have 0=Lu=he^u and, therefore, h≡ 0 which contradicts our assumption.Let us turn to the other direction. So, assume h∈ℓ^2(X,m) is such thath<0 and there is x_0∈ X such thath(x_0)>0.Define the setB={v∈|⟨ he^v,1⟩= ⟨ v,1⟩=0}.The strategy of the proof is to show that Q admits a minimizer on B which turns out to be a solution up to an additive constant. As a first step we show that B is non empty. Claim 1: The set B is not empty. Define v_0=1_x_0. Then, tv_0∈ for every t∈ and a simple calculation shows ⟨ he^tv_0,1⟩=(e^t-1)h(x_0)m(x_0)+⟨ h,1⟩.Thus,F:[0,∞)→, t↦⟨ he^tv_0,1⟩is a continuous map with F(0)<0 and F(t)>0 for large t. Hence, there is t_0 such that F(t_0)=0. Then, v=t_0v_0-t_0m(x_0)/ m(X)∈ and v satisfies⟨ v,1⟩=0,and ⟨ he^v,1⟩ = e^-t_0m(x_0) /m(X)⟨ he^t_0v_0,1⟩=e^-t_0 m(x_0)/m(X)F(t_0)=0.Thus, we get v∈ B.Claim 2: There is a minimizer ofinB. Let (w_n) be a sequence in B such that (w_n)→inf_w∈ B(w). Using the Poincaré inequality, Proposition <ref>, we infer that (w_n) is a bounded sequence with respect to ·_. Hence, (w_n) has an -convergent subsequence by the Sobolev Embedding, Proposition <ref>, and we assume without loss of generalitythat (w_n) itself is -convergent.We denote the limit by w.We infer w∈ D() bythe lower semi-continuity of , i.e., (w)≤lim inf_n→∞(w_n)=inf_w∈ B(w). Next we show w∈ B.By Proposition <ref> we have sup_n∈w_n_∞<∞. Hence, by the finiteness of m and Lebesgue's dominated convergence theorem we observe⟨ w,1⟩=lim_n→∞⟨ w_n,1⟩=0.Furthermore, we have |he^w-he^w_n| ≤ |h|(e^w_∞+e^sup_n∈w_n_∞) and another application of Lebesgue'sdominated convergence theorem yields ⟨ he^w,1⟩=lim_n→∞⟨ he^w_n,1⟩=0. This shows w∈ B and, therefore, inf_w∈ B(w)≤(w). This finishes the proof of the claim.Let w be a minimizer ofon B.Then, the map →, f↦(f) has a minimum for f=w under the restrictions⟨ he^f,1⟩=0⟨ f,1⟩=0.Weapply theLagrange multiplier theoremin Banach spaces, c.f. <cit.>.For this, we calculate the Frchet derivatives of the mapsQ :D()→,f↦(f)F :D()→,f↦⟨ f,1⟩H :D()→,f↦⟨ he^f,1⟩. Claim 3: The maps Q , FandH are continuously Frchet differentiable with Frchet derivatives at f∈ D()given by D Q[f](·)=2(f,·), D F[f](·)= ⟨· ,1⟩, D H[f](·)=⟨· he^f,1⟩. For the derivative of Q we calculate |(f+g)-(f)-2(f,g)|/g_ =(g)/g_≤g_^2/g_→ 0, for g_→ 0. By the Cauchy-Schwarz-inequality theFrchet derivative DQ[f](·) =2Q(f,·) is continuous. The statement for F followseasily as F is linear. Finally, for the derivative of H we estimate using that e^g-1-g_∞/g_≤ C∑_k=2^∞g_∞^k/k!/g_∞ ≤ C∑_k=1^∞g_∞^k/k!= C( e^g_∞-1)→ 0forg_→0 since ·_-convergence implies ·_∞-convergence by Proposition <ref>, and |⟨ he^f+g,1⟩-⟨ he^f,1⟩-⟨ hge^f,1⟩|/g_ = |⟨ he^f(e^g-1-g),1⟩|/g_ →0 To see that the Fréchet-derivative of H is continuous, we estimate by the Cauchy-Schwarz-inequality and ·_2≤·_ |DH[f-g]()|= |⟨φ h(e^f-e^g),1⟩| =|⟨ h(e^f-e^g),φ⟩|≤h(e^f-e^g)_2φ_ This proves the continuity of the Frchet derivative of H.Let w be a minimizer of Q on B whose existence was proven above.The map(DF[w],DH[w]): D()→^2, f↦ (⟨ f,1⟩,⟨ f he^w,1⟩)is surjective, since there are x_0,x_1∈ X with h(x_0)>0, h(x_1)<0 by assumption, and 1_x_0,1_x_1∈, and, thus, DF[w](1_x_0), DF[w](1_x_1)>0 and DH[w](1_x_0)>0, DH[w](1_x_0)<0.Thus, we can apply the Lagrange multiplier theorem in Banach spaces and infer that there are λ,μ∈ such that 2(w,g)=λ⟨ ghe^w,1⟩+μ⟨ g,1⟩= ⟨λ he^w +μ,g⟩holds for every g∈, where λ he^w+μ∈ as m is finite and w∈ D() is bounded, since the graph is canonically compactifiable. Therefore, by definition, w∈ D() andw=λ/2 he^w+μ/2. Furthermore, by 1∈ we deduce0=2(w, 1)=λ⟨ he^w,1⟩ +μ m(X). Since w∈ B we see ⟨ he^w,1⟩=0 and, thus, μ=0.Finally, we show λ>0. It is easy to see that λ can not be zero, since otherwise we would infer w≡ 0 and the constant zero is not in B.Thus, we have 0>2(w,e^-w)=λ⟨ h,1⟩, where the first inequality can be seen by an easy calculation. Since by assumption ⟨ h,1⟩=h m(X)<0, we infer λ>0. Therefore, we get λ/2 =e^σ for some σ∈ and the function u=w+σ∈ D() is a solution. This concludes the proof. § THE CASE C>0 Let b be a canonically compactifiable graph over (X,m) such that m(X)<∞. Let c>0 and 0≠ h∈ℓ^2(X,m). Then, there is an u∈ D() withu=he^u-cif and only if h is positive somewhere. Suppose u∈ D() is a solution. Then, by0=(u,1)=⟨ u, 1 ⟩=⟨ -c,1 ⟩+⟨ he^u,1 ⟩we infercm(X)=⟨ he^u,1 ⟩and, since c>0, the function h must be positive somewhere. Now suppose that h is positive somewhere. DefineB={v∈|⟨ he^v-c,1 ⟩ =0} and J:D()→, J(v):=1/2(v)+cm(X)v. The strategy of the proof is similar to the proof of Theorem <ref> above. We show that J assumes a minimum on B which turns out to be a solution. To this end we first show that B is non-empty and J is bounded below on B. Afterwards we show that J assumes a minimum and then we show that this minimum is a solution by the virtue of Lagrange multipliers. Claim 1: The set B is not empty. Let x_0∈ X be such that h(x_0)>0 and define v_t,l=t1_x_0-l for t,l∈. We get v_t,l∈ for every t,l∈. Define F:^2→, F(t,l):=⟨ he^v_t,l,1 ⟩=(e^t-1)e^-lh(x_0)m(x_0)+e^-l⟨ h,1 ⟩ whichis obviously continuous. Furthermore, for fixed t we infer F(t,l)→ 0 for l→∞ and for fixed l we infer F(t,l)→∞ for t→∞. Since the continuous image of connected sets is connected, we conclude that there are t_0,l_0∈ such that F(t_0,l_0)=cm(X) and, hence, v_t_0,l_0∈ B. Next we show that J is bounded from below on B. Claim 2: There is a constant C such that J(v)≥1/4(v)+C, v∈ B. For v∈ B we infer e^v=cm(X)/⟨ he^v-v,1 ⟩ from the calculation ⟨ he^v-v,1 ⟩=e^-v⟨ he^v,1 ⟩=e^-vcm(X). Therefore,J(v) =1/2(v)+cm(X)v=1/2(v)+cm(X)log(e^v)=1/2(v)+cm(X)log(cm(X))-cm(X)log(⟨ he^v-v,1 ⟩). The idea is now to control the third term on the right hand side by (v). To this end weapply the Cauchy-Schwarz inequality, the basic inequality 2st≤ 2ε s^2+2/εt^2 for s=(v)^1/2, t=(v-v)/(v)^1/2 and ε>0, and, the Trudinger-Moser inequality, Proposition <ref> for the function u=(v-v)/ (v)= (v-v)/(v-v) and β=2/ε ⟨ he^v-v,1 ⟩^2≤h_2^2⟨ e^2(v-v),1 ⟩≤h_2^2⟨ e^2ε(v)+2(v-v)^2/ε(v),1 ⟩ ≤h_2^2Ce^2ε(v). Therefore, by the computation for J(v) above we obtain withε=1/(4cm(X)) J(v) ≥1/2(v)+cm(X)log(cm(X))-cm(X) log(Ch_2 e^ε(v)) = 1/4(v)+cm(X)log(cm(X)/Ch_2)= 1/4(v)+C. Claim 3: There is a minimizer of J in B. Let u_n be a minimizing sequence in B such that J(u_n)→inf_w∈ BJ(w)>-∞. This yields by Claim 2 and bythe Poincaré inequality, Proposition <ref>, that the sequence (u_k-u_k) is a bounded sequence in (,·_). By the definition of Ju_k=1/m(X)(J(u_k)-(u_k))we infer that the sequence (u_k) is bounded in . Hence, by u_k_≤u_k-u_k_+u_k_=u_k-u_k_+u_km(X)^1/2the sequence (u_k) is bounded in (,·_). Bythe weak compactness of closed balls in Hilbert spaces (u_k) has a ·_ weakly convergent subsequencewhich is without loss of generality (u_k) itself. By the Sobolev Embedding, Proposition <ref>, (u_k) converges to an u in ℓ^2(X,m) which is in D() as well. Furthermore, by Proposition <ref> we infer sup_n∈u_k_∞<∞ and, hence, u is bounded as well. By the inequality |he^u-he^u_n| ≤ |h|(e^u_∞+e^sup_n∈u_n_∞) we can apply Lebesgue'sdominated convergence theorem to obtain ⟨ he^u,1 ⟩=lim_n→∞⟨ he^u_n,1 ⟩=cm(X). Therefore u∈ B and, by the lower semi-continuity of , we inferJ(u)≤lim inf_n→∞J(u_n)=inf_v∈ BJ(v)and we conclude that u is a minimizer. Let u∈ B be a minimizer of J on B. Then the map J has a minimum for u∈ D() under the restriction given by the map I: D()→ I(f):=⟨ he^f-c,1 ⟩=0. So, our next goal is to apply the theorem about Lagrange multipliers in Banach spaces, c.f. <cit.>[Theorem 43.D]. For this we need to calculate the Frchet derivatives of J and I. Claim 4: The maps J and Iare continuously Frchet differentiable and have Frchet derivatives at f∈ D() given by DJ[f](·)=(f,·)+c⟨·,1 ⟩, DI[f](·)=⟨· he^f,1 ⟩. With the notation from the proofof Theorem <ref> the maps J and I can be represented as J=1/2Q+cF I= H-cF. Hence, the statement follows from Claim 3 of the proof of Theorem <ref>. It is easy to see that the map DI[f](·):→ is surjective. Hence, we can apply <cit.>[Theorem 43.D] and infer the existence of λ∈ such that (u,f)+c⟨ f,1 ⟩=λ⟨ fhe^u,1 ⟩holds for every f∈ and the minimizer u. Hence,(u,f)=⟨ -c+λ he^u,f ⟩for every f∈. Note that λ he^u-c∈ as u is bounded and m is finite. Therefore, by definition, u∈ D() andu=λ he^u-c. By 1∈ we infer0=(u, 1)=λ⟨ he^u,1 ⟩ - cm(X). Since u∈ B we see ⟨ he^u,1 ⟩=cm(X) and, thus, λ=1. This concludes the proof. § THE CASE C<0In this section we treat the case c<0 and prove the following theorem. Let b be a connected, canonically compactifiable graph over (X,m) and m(X)<∞. Let0≠ h∈ℓ^2(X,m) andc>0. Then, the following holds: (1) If there is u∈ D() with u=he^u-c, then h<0. (2)If h<0, then there exists a constant -∞≤ c_-(h)<0 such that the equation u=he^u-c has a solution for all 0>c>c_-(h) and no solution for every c<c_-(h). If h≤0, then c_-(h)=-∞. While the proof of (1) follows by a direct calculation, the proof of (2) involves the construction of a solution via lower and upper solutions. In particular, we call a function u_0∈ D() a lower solution ifu_0≤ he^u_0-c, and we call a function u_1∈ D() an upper solution if u_1≥ he^u_1-c.We introduce for a function k:X→ (0,∞) the bilinear form+k on ℓ^2(X,m) via the form sumD(+k)=D()∩ℓ^2(X,km)(+k)(f,g)=(f,g)+⟨ kf,g⟩, f,g∈ D(+k).An immediate consequence of Fatou's lemma (and the fact that D()=D∩ℓ^2(X,m)) yields that +k is lower semicontiuous and, hence,closed. We denote the associated non-negative, selfadjoint operator on ℓ^2(X,m) by +k which acts as(+k )f(x)= f(x)+ k(x)f(x), f∈ D(+k), x∈ X.We show next that for canonically compactifiable graphs over(X,m) with m(X)<∞ the domains of the formsand +k and the corresponding operators coincide whenever k∈ℓ^2(X,m). Let b be acanonically compactifiable graph over (X,m), m(X)<∞ and k∈ℓ^2(X,m) with k>0. Then, D(+k)= D(). Moreover, the operator L+k is bijective.Let f∈.We estimate by using Hlder's inequality and Proposition <ref>⟨ kf,f⟩≤k_1f_∞^2≤ C k_1(f).Since k∈ℓ^2(X,m), we infer k_1<∞ asℓ^2(X,m)⊆ℓ^1(X,m) in the case of finite measure. Hence, f∈ℓ^2(X,km) and, therefore,D()⊆ℓ^2(X,km).We concludeD(+k)=D()∩ℓ^2(X,km) =D(). To show the equality of the operator domains let f,g∈ℓ^2(X,m).Let G_0=(+1)^-1 and G_k=((+k)+1)^-1. Furthermore, with slight abuse of notation wedenote the operator of multiplication by k on ℓ^2(X,m) by k. SinceG_kℓ^2(X,m)⊆ D(+k)=D( )⊆ℓ^∞(X)we havek G_kf∈ℓ^2(X,m)and we calculate⟨ G_0k G_kf,g⟩=⟨ k G_kf,G_0g⟩= (+k)(G_kf,G_0g)-(G_kf,G_0g)Using the facts that (+k)(G_kf,G_0g)+⟨G_kf,G_0g⟩=⟨ f,G_0g⟩ and(G_kf,G_0g)+⟨G_kf,G_0g⟩=⟨ G_kf,g⟩ we infer⟨ G_0k G_kf,g⟩ =⟨ G_kf,g⟩ -⟨ f,G_0g⟩ = ⟨ (G_k-G_0)f,g⟩.Since g was chosen arbitrarily in ℓ^2(X,m) we have for all f∈ℓ^2(X,m)G_kf=G_0k G_kf +G_0fSince G_k is surjective on D(+k) and G_0ℓ^2(X,m)⊆ D() we infer the statement D(+k)⊆ D().Theother inclusion D()⊆ D(+k) follows analogously.Finally, we show that L+k is bijective. Note, that L+k has compact resolvent by the equality G_kf=G_0k G_kf +G_0f, f∈ℓ^2(X,m), shown above and since G_0 is compact by Proposition <ref>. It is left to show that (L+k)={0}. Let u∈(L+k). Then, we have0=⟨ (L+k)u,u⟩=(Q+k)(u,u)≥⟨ k u,u⟩and, since k> 0, we infer u≡ 0. Furthermore, we need the following maximum principle Let b be a graph over (X,m) and k∈ℓ^2(X,m), k≥0. Let u∈ D(), u not constant, such that the inequality (+k) u≤ 0 holds. Then, we get u≤ 0. Byu_±=(± u)∨ 0∈ D(+k) one has0≥⟨ ( +k)u, u_+⟩=(+k)(u,u_+)=(+k)(u_+,u_+)-(+k)(u_+,u_-).Hence, (+k)(u_+,u_+)≤(+k)(u_+,u_-)≤ 0,where the latter inequality holds by (+k)(|u|)≤ (+k)(u) as k>0 and Q is a Dirichlet form. Thus u_+≡ 0 since k>0 by assumption. With these preparations weshow that the existence of suitable lower and upper solutions imply the existence of a solution. Let b be a connected, canonically compactifiable graph over (X,m), m(X)<∞ and let c<0, 0≠h∈ℓ^2(X). If there is an upper solution u_1 and a lower solution u_0 such that u_1≥ u_0, then there is a solution u∈ D() with u_1≥ u≥ u_0 and u=he^u-c. Taking the upper solution u_1∈ D() and h∈ℓ^2(X), we let k=(1∨(-h))e^u_1.Since the graph is canonically compactifiable, and thus D()⊆ℓ^∞(X), we have e^u_1∈ℓ^∞(X) and, therefore, k≥ e^inf u_1>0 k∈ℓ^2(X,m).Hence, the operator+k is bijective by Lemma <ref>.For u∈ C(X) and c from our assumption we let the right hand side of the Kazdhan-Warner equation be denoted byf_u:X→, f_u=he^u-c .If u∈ℓ^∞(X,m), in particular if u∈ D(+k)=D(), we have f_u∈.Starting with u_1 we inductively defineu_j+1=(+k)^-1(f_u_j+ku_j)∈, j≥ 1.We next show u_0≤…≤ u_j+1≤ u_j≤…≤ u_1 by induction. For j=2we calculate(+k) (u_2-u_1)= f_u_1+ku_1- u_1-ku_1= f_u_1- u_1≤ 0as u_1 is an upper solution. Using the maximum principle, Lemma <ref>, we infer u_2-u_1≤ 0. For j≥ 2assume u_j≤ u_j-1≤…≤ u_1. We infer (+k) (u_j+1-u_j) =f_u_j-f_u_j-1+ku_j-ku_j-1=h·(e^u_j-e^u_j-1)+k·(u_j-u_j-1)=(he^ξ+k)·(u_j-u_j-1)≤ 0, where the function ξ:X→ withu_j-1≥ξ≥ u_j is given by the mean value theorem. Here, the last inequality follows from the definition of k. Note that u_1≥ξ by the induction hypothesis. Thus, the maximum principle, Lemma <ref>, yields u_j+1≥ u_j. Next we show u_j≥ u_0 for every j≥1. The case j=1 follows by the assumptionu_1≥ u_0. Supposeu_j≥ u_0 for some j≥1. We calculate (+k) (u_j+1-u_0) =f_u_j+ku_j- u_0-ku_0≥ h·(e^u_j-e^u_0)+k·(u_j-u_0)=(he^ξ+k)·(u_j-u_0)≥ 0, where the function ξ:X→ withu_j≥ξ≥ u_0 is given by the mean value theorem. Thus, using the maximimum principle we infer u_0≤ u_j. Hence, the sequence u_j is pointwise monotonically decreasingand, thus, there is a pointwise limit u:X→ with u_0≤ u≤ u_1. Using the finiteness of m and Lebesgue's dominated convergence theorem, we infer u∈ and u_j-u_2→ 0, j→∞. On the other hand we have u_j∈ D() by D(+k) =D(), Lemma <ref>. Hence, (+k) u_j=f_u_j-1+ku_j-1 yields u_j=f_u_j-1+ku_j-1-k u_j→ f_uinby Lebesgue'sdominatedconvergence theorem. The closedness of the operatoryields u∈ D(). By definition of f_u=he^u-c we conclude the equality u=he^u-c. By the lemma above it suffices to present a lower and an upper solution u_0 and u_1 with u_0≤ u_1 to prove Theorem <ref> (2). The next lemma shows the existence of a lower solutions in general and upper solutions forc <0 sufficiently large. Let b be a connected, canonically compactifiable graph over (X,m), m(X)<∞ and let h∈ℓ^2(X,m) be such that h <0. (a) For all c<0, there is a lower solutionu_0∈D(). (b)There is c_-(h) <0 such that there existsan upper solution in u_1∈ D() for c_-(h)<c<0. If furthermore h≤ 0, then c_-(h) =-∞. Furthermore, u_0 and u_1 can be chosen such that u_0≤ u_1. Recall that the operatoris bijective on the ℓ^2 functions which are orthogonal to the constants, Proposition <ref>. Wedenote this inverse with slight abuse of notation by ^-1. Then ^-1maps the orthogonal complement of the constants into D()⊆ D()⊆ℓ^∞(X) and we define v_α,β=-α^-1(h_--h_-) - βfor α,β∈ where f_± denotes the positive and negative part of a functionf, i.e.,f_±=(± f)∨ 0.Hence, v_α,β∈ D()⊆ℓ^∞ (X) and v_α,β→-∞ uniformly for β→∞and α∈ fixed. We estimate -h ≤ h_- to get v_α,β-he ^v_α,β+c ≤-α h_-+αh_-+h_-e^v_α,β+c=(e^v_α,β-α)h_-+αh_-+c. Choosing 0≤α_0≤ |c|/h_- and β _0 sufficiently large we obtain the statement (a) with u_0=v__0,β_0. To show (b) let Nbe such that h_N=h∨(- N ) still satisfies h_N<0. Furthermore,let C:=^-1(h_N-h_N) _∞and define for α≥0 v_α=α(^-1(h_N-h_N)- C ). We observe that -2α C≤ v_α≤ 0. We calculate (v_α+logα)-he ^v_α+log ≥ (v_α+logα)-h_Ne ^v_α+log=(α- e^v_α+logα)h_N-αh_N=α(1- e^v_α)h_N-αh_N. Next, we applythe mean value theorem and infer that there is a function ξ:X→ [-2α C,0 ] with 1-e^v_α=-e^ξ v_α, use that 0≥ v_α≥ -2α C and denote by h_N,- the negative part of h_N to deduce… =-α e^ξ v_α h_N-αh_N≥α e^ξ v_α h_N,--αh_N≥ -2α^2e^2α CCN-αh_N. Since h_N <0 there is α_0 such that C_1:=-2α_0^2e^2α_0 CCN-α_0h_N>0 Hence, for 0> c_-(h)> -C_1 we infer that u_1=v__0+log_0 is an upper solution for 0>c>c_-(h). This is the first statement of (b). Now, assume h≤0.For α≥0 define v_α=α(^-1(h-h)-C ) with C:=^-1(h-h) _∞. Then, v_≤ 0 and we deduce ( v_α+log)-he ^v_α+log= (1-e^v_)h-h≥-h sinceα >0, v_≤0, h≤ 0. So, for given c<0let _c≥ 0 be such that -_ch≥ -c. Then, u_1=v__c+log_c is an upper solution. To see that we can choose u_0≤ u_ 1 we observe that u_0,u_1∈ D()⊆ℓ^∞(X) and β_0 in the definition of u_0 can be chosen arbitrary large. (1): Let u be a solution. Then we have ⟨ h,1⟩= ⟨ he^u,e^-u⟩ = ⟨ u,e^-u⟩+c⟨ 1,e^-u⟩=(u,e^-u)+c⟨ 1,e^-u⟩<0, where the last inequality follows since t↦ e^-t is monotone decreasing and the definition of . (2): This follows fromLemma <ref> and Lemma <ref>. alpha | http://arxiv.org/abs/1707.08318v1 | {
"authors": [
"Matthias Keller",
"Michael Schwarz"
],
"categories": [
"math.AP",
"math.FA",
"math.MG"
],
"primary_category": "math.AP",
"published": "20170726083338",
"title": "The Kazdan-Warner equation on canonically compactifiable graphs"
} |
In this article, we study the thermodynamic behavior of anisotropic shape (rod and disk) nanoparticle within the block copolymer matrix by using self-consistent field theory (SCFT) simulation. In particular, we introduce various defect structures of block copolymers to precisely control the location of anisotropic particles. Different from the previous studies using spherical nanoparticles within the block copolymer model defects, anisotropic particles are aligned with preferred orientation near the defect center due to the combined effects of stretching and interfacial energy of block copolymers. Our results are important for precise controlling of anisotropic nanoparticle arrays for designing various functional nano materials. § INTRODUCTION In recent years, ordered array of nanoparticles (NPs) by controlled assembly has received great attention due to the inter-particle plasmonic and electric coupling between NPs <cit.>. Long-range ordering of NPs can open the possibility of new functional materials for numerous applications such as SERS (Surface Enhanced Raman Scattering) spectroscopy <cit.>,memory devices <cit.>, and solar cells <cit.>. In addition to the 0-dimensional spherical NPs, 1-dimensional nanorods show tunable plasmonic properties depending on their geometrical arrangement <cit.>. Furthermore, 2-dimensional nanodisks such as graphene <cit.> and molybdenum disulfide <cit.> quantum dots have been investigated recently due to their tunable band gap for numerous possible electronic applications. For the next generation NP-based device, precise localization of NP over multiple length scales is the most critical issue. Block copolymers (BCPs) have been regarded as attractive materials for the scaffold for NP array due to their ability to self-assemble into variety of structures at nanometer length scale <cit.>. By Blending NPs and BCPs, well ordered NPs are found within the self-assembled BCP matrix. It is well known that the ligand chemistry <cit.> and size <cit.>of NP determine the location of NPs within the BCP matrix due to their enthalpic and entropic contributions to the blends respectively. In addition to the ligand chemistry and size of NP, conformation entropy (stretching energy) of BCPs is another important factor for the localization of NPs within the BCP matrix. For example, higher spring constant by using stiffer comb block of coil-comb BCPs helps the alignment of spherical NPs into 3d structures <cit.>, as well as the arrangement of nanorods <cit.>. Defect structures of BCP are another great example of the role of conformation entropy of BCP to localization of NPs. NPs are aggregated at curved grain boundary <cit.>, curved pattern by chemoepitaxy <cit.>, and dislocation between monolayer and double-layer of thin film <cit.> of BCPs to minimize the conformation entropy loss. In this manner, localization of NPs at specific location would be possible, if one can design artificial defect structure of BCP at desired locations. Templated assisted self-assembly of BCPs is a promising method to generate controlled "defect" structure of BCPs. It is well known that control the lattice constant and shape of hydrogen silsesquioxane (HSQ) nano post array by e-beam lithography helps the formation of desired structure of highly stretched BCPs <cit.>. Self-consistent field theory (SCFT) simulation study of templated assisted self-assembly of BCPs shows the similar result described by experiments <cit.>. Recently, blends of NPs and BCPs with various defect structures by templated assisted self-assembly are studied by using SCFT simulation <cit.>. Free energy plots of the blends system show that the spheric NP is trapped at the defect center for every defect shape. However, when the shape of the particle deviates from the perfect sphere, we can expect that there will be favorable orientation of those particles for the different shape of BCP defect. Therefore, theoretical investigation of anisotropic shape of NP within the model defects of BCPs will be interesting problem for the both fundamental and engineering aspects.In this article, we study blends of anisotropic (rod and disk) NPs and BCPs using SCFT simulation method with "model defect" structures of BCPs. In particular, we investigate the free energy of blends as a function of configuration of anisotropic NP near the centers of X, T, and Y shape defects of BCPs. We expect that our result will open a possibility of new way to precise control of anisotropic NP at the desired location to design new functional materials. § COMPUTATION METHOD §.§ Self-Consistent Field TheoryIn this work, we simulate block copolymers (BCPs) and anisotropic (rod and disk) nanoparticles (NPs) using Self-Consistent Field Theory (SCFT) simulation. Cavity function <cit.> is introduced to describe the external particle (NPs) by excluding polymer density where the cavity function is defined. In this work, we expand this method to three dimensional system for better description of the blends of BCPs and NPs.The effective hamiltonian used in this work including the cavity function ρ_ext is given by,H[ Ω_+, Ω_-, {Ω_ext}] = -CV ln Q[ Ω_+, Ω_-, {Ω_ext} ] - iC ∫ d rΩ_+( 1 - ρ_ext/ρ_0)+ C/χ N∫ d rΩ_-^2where C = ρ_0R_g^3/ N corresponds to the dimensionless concentration, ρ_0 is the monomer concentration, N is the degree of polymerization, and R_g is radius of gyration of an ideal copolymer. Ω_+ is interpreted as a fluctuating pressure field which couples with summation of densities of polymer A and B (ρ_+ = ρ_A + ρ_B) to enforce the incompressibility condition for the local polymer density (ρ_+ + ρ_ext = ρ_A + ρ_B + ρ_ext = ρ_0). On the other hand, Ω_- is interpreted as an exchange potential field that couples with local density differences between the two blocks (ρ_- = ρ_A - ρ_B). The set of field {Ω_ext} corresponds to the set of the external field imposed on the system by the rod or disk NP which contains affinity of NPs to both blocks. The cavity function ρ_ext describes the density of the rod or disk NP within the polymer matrix and will be discussed at next section in detail.Q[ Ω_+, Ω_-, {Ω_ext} ] is the single polymer partition function, and can be calculated as Q[ Ω ] = 1/V∫d r q(r,1,Ω) where q(r,s, Ω), called the propagator, satisfies a diffusion-like equation given by, ∂/∂ sq(r,s,Ω) = ∇^2q(r,s,Ω)- Ω(r,s)q(r,s,Ω) having initial condition q(r,0,Ω) = 1. The time variable in the diffusion-like equation s corresponds to the position of a polymer segment (s = 0 and s = 1 correspond to both ends of a single polymer). Therefore, the set of fields Ω(r,s) is a function of type of polymer (A and B) described as, Ω(r,s)= {[ iΩ_+(r) - Ω_-(r)- γ_Aρ_ext(r) / ρ_0,0 < s ≤ f;iΩ_+(r) + Ω_-(r)- γ_Bρ_ext(r) / ρ_0, f < s ≤ 1 ]where f is the volume fraction of the A block in the block copolymer. γ_A and γ_B are the effective affinities of the nanoparticle for each block A and B. A positive value of the effective affinity yields an attractive force with the corresponding block of the BCPs. The local polymer densities ϕ_A and ϕ_B are calculated by integrating the proper propagators obtained from the diffusion-like equation.ϕ_A(r,Ω)= 1/Q∫_0^f ds q^†(r,1-s,Ω) q(r,s,Ω) ϕ_B(r,Ω)= 1/Q∫_f^1 ds q^†(r,1-s,Ω) q(r,s,Ω) We introduce the functionq^†(r,s,Ω) which is equivalent to the functionq(r,s,Ω) but the propagation along the chain starts from the B ends of the polymer. We use a Lattice Boltzman method recently developed to solve the diffusion-like equation for the propagator q(r,s,Ω), and the program has been optimized for GPU parallel computation <cit.>. To calculate the full partition function Z = ∫ DΩ e^H[Ω], we have to integrate the over all possible configuration of the field Ω. However, the SCFT of BCPs is based on the approximation of full partition function as a single partition function at the mean field (saddle point) configuration of the field Ω* described by Z ≈ e^H[Ω^*]. To find the mean field solution for Ω_+^* and Ω_-^* that satisfy the minimization of the effective Hamiltonian condition ∂ H / ∂Ω_+,- |_Ω_+,-^*= 0, a Langevin dynamic scheme is used to update Ω_+^* and Ω_-^*. The computational procedure to find the mean field solution is executed as described below. First, we update the Ω_- field with a small Gaussian real noise to escape from metastable states, then calculate local polymer densities ϕ_A and ϕ_B. Next, we update the Ω_+ field until the system satisfies the local incompressibility condition at every grid point. We repeat these two steps until the system reaches the equilibrium where the effective Hamiltonian is stabilized. We run 10^6 iterations (each iteration contains to a single update of Ω_- and full update of corresponding Ω_+) per simulation point until the system reaches equilibrium in this work. §.§ Description of anisotropic nanoparticles The cavity functions ρ_ext for rod and disk NPs are modified here to account for the anisotropy of NPs in comparison to the isotropic cavity function for spheric NP previously described <cit.>. For the rod NP, ρ_ext-rod is a Gaussian function defined along the axis of the rod of length L-D and standard deviation 0.5D.ρ_ext-rod(r) = ρ_0exp( -|r - r_0|^2/2(0.5D)^2 )where r_0 is the closest point along the finite rod axis to r. In this way, we generate a soft rod NP of length L and diameter D. For the disk NP, ρ_ext-disk is a Gaussian function defined in plane of a disc of diameter D-T with standard deviation 0.5T.ρ_ext-dis(r) = ρ_0exp( -|r - r_0|^2/2(0.5T)^2 )where r_0 is the closest point in the disk plane to r. In this way, we can generate a soft disk NP with diameter D and thickness T.To simulate such complex NP, an extremely refined grid is required to describe the NP correctly. Since the computational cost for a SCFT simulations directly scales with the number of chosen grid of simulation box, grid refinement of the entire simulation box is not an ideal solution. Instead, local grid refinement is required where the complex NPs exist to obtain better calculation accuracy and faster simulation execution. Such refinement is not possible for the traditional pseudo-spectral (PS) method for the diffusion-like equation solver because the simulation box has to be periodic. To solve this problem, we recently adapted the Lattice Boltzmann Method (LBM) to solve the diffusion-like equation used in SCFT simulations <cit.>. In this study, the refined grid region is defined as a three dimensional box having as 2.0R_g× 2.0R_g× 2.0R_g to contain all field information of ρ_ext which represents the rod and disk NP. Refinement of the fine grid is four times higher than coarse grid to reduce energy fluctuation of the molecule resulting from 3d rotation to the order of 10^-6.We set χ N to 20 with f = 0.7 where the minority block (B-block) forms hexagonally closed-packed cylinders in a bulk system. Each desired defect is designed by using the same concept as the templated-assembly of BCPs. By fixing the Ω_- field at desired locations, we can model the preferential wetting of one block of the BCP at certain grid points <cit.>. Then we optimized the system dimensions L_x, L_y, L_z to minimize the free energy of the system with a desired defect shape. Using this method, we could generate X, T, and Y shape defects. For each defect, we plot the free energy of the system as a function of 3d rotation angle of director vector of rod and disk NP. We designed the NP to be highly attractive to the B-block but repulsive to the A-block by setting γ_A to -40 and γ_B to +40. § RESULTS AND DISCUSSION§.§ Case of normal cylinder without defect Before we study the mixture of rod and disk NPs with BCPs containing defects, we first simulated the system having a rod or disk NP within the minority block region of the defect-free cylinder phase of BCPs. First, we simulated the rod molecule with length L and diameter D (See Figure 1-a). The red arrow represents the director vector of the rod molecule which is rotated by an angle θ from the axis perpendicular to the cylinder axis (See Figure 1-b). In Figure 2, we plot the free energy of the system as a function of θ, L, and D.From Figure 2, the rod NP has the highest free energy when the director vector is perpendicular to the cylinder axis (θ = 0^ ∘). If the rod is aligned perpendicular to the cylinder axis, the interface between the majority and minority blocks should be curved toward the director vector. This results in an increase of both the stretching energy and interfacial energy. However, if the rod molecule is aligned parallel to the cylinder axis (θ = 90^ ∘), the occupation of the rod component in the vicinity of the polymer ends is maximized. This results in the minimization of both the stretching energy and the interfacial energy of the block copolymer, and the free energy shows a corresponding minimum for every value of L and D of the rod NP at θ = 90^ ∘. Also, this minimization of the free energy is amplified if we increase the rod length L and diameter D because more rod volume will occupy the cylinder axis. We can clearly see this feature in Figure 2. When we increase L and D, the free energy has a lower minimum at θ = 90^ ∘. A disk NP having D as diameter and T as thickness was also simulated within the defect-free cylinder matrix (See Figure 3a). The red arrow represents the director vector of the disk which is rotated by an angle θ with respect to the axis perpendicular to the cylinder axis (See Figure 3-b). In Figure 4, we plot the free energy of the system as a function of θ, D, and T. In contrast to the rod case, the free energy minimum is achieved when the director vector of the disk is perpendicular to the cylinder axis (θ =0^ ∘). This configuration of the disk NP maximizes the occupation of the disk at the axis of the cylinder where the polymer ends exist, minimizing the stretching energy of the block copolymer. Intuitively, we also can derive that the larger and thicker disk decreases the free energy more by reducing θ to 0^ ∘. We can clearly see this feature in Figure 4. When we increase D and T, the free energy has a lower minimum at θ = 0^ ∘.§.§ Case of X-shape defect From the defect-free cylinder case, we established that a free energy minimum is realized when most of the volume of the molecule occupies the cylinder axis where the ends of BCPs exist. However, when a defect is introduced to the system, the free energy well is constructed at the defect center. The stiffness of the free energy is different when we move the external particle in different directions from the defect center and we found that it is due to the different curvature of interface of BCPs which isotropic (spheric) NP is faced at <cit.>. Therefore, different from spheric NP case, we can expect that the anisotropic feature of the rod and disk NPs results in preferred orientations of their director vectors near the defect center. From this section, we study the free energy landscape as a function of 3D rotation of director vector of rod and disk NPs at the center of three model defects.The first case considered is the rod and disk NPs within the X-shape defects. Since the size of NP has been relevant to the magnitude but not the tendency of free energy landscape from defect-free cylinder case, we only consider the single size of rod and disk NP ( D = 0.4R_g, L = 1.2R_g for rod and D = 1.2R_g, T = 0.4R_g for disk). In Figure 5,we show the rod and disk NPs are located at the center of X-shape defect structure of BCPs. Red arrow in Figure 5 represents the director vector of each NP and we plot the free energy as a function of director vector at the defect center in Figure 6 for rod and Figure 7 for disk respectively.From Figure 6-a, the free energy maximum is achieved when the director vector of the rod is perpendicular to the two cylinder axes where the chain stretching and interfacial area is maximized. For rod NP, free energy gets lower when we rotate the director vector toward the plane which contains both cylinder axes of BCPs as seen in Figure 6-b and 6-c. However, aligning the rod to one of two cylinder axes (Figure 6-c) results lower free energy compared to the configuration where the director vector is aligned toward the interface of BCPs (Figure 6-b) due to the extra increment of interfacial area of BCPs. Therefore, we can conclude that the alignment of the rod to x or y axis is equilibrated configuration of the rod within the X-shape defect of BCPs. For the disk NP, different from the rod case, free energy is maximize when the director vector of the disk shares the same plane with two axes of BCP cylinders due to the large interfacial area stretching to z-direction(Figure 7-b and 7-c). When the director vector of the disk is perpendicular to the two axes of cylinders where the volume of the disk shares both cylinder axes (Figure 7-a), the minimization of free energy is achieved from the minimized stretching of polymer chains and interfacial area. §.§ Case of T-shape defect The T-shape defect of BCPs is also simulated with the rod and disk NPs. Similar to the X-shape defect case, we rotate the director vector of the rod and disk at the defect center of T-shape BCP cylinders as shown in Figure 8-a and 8-b. In Figure 9, we plot the free energy the rod within the T-shape defect as a function of director vector. We also plot the free energy of the disk NP within the T-shape defect of BCPs in Figure 10. For the X-shape defect, the rod maximize the volume occupied along the cylinder axis of BCPs by aligning with either the x or y axis. However, for the T-shape defect, if the rod is aligned toward the axial direction of the cylinder aligned along the y axis as shown in Figure 9-c, the rod molecule will experience more curved interface of BCPs. Therefore, the director vector of the rod prefers to be aligned with the x axis at the defect center in a more energetically favorable configuration (Figure 9-b). The free energy maximum is achieved when the director vector of the rod is perpendicular to two cylinder axes of cylinders at the defect center as shown in Figure 9-a. On the other hand, when the director vector of the disk NP is perpendicular to both cylinder axes (Figure 10-a), the free energy minimum is achieved because the disk shares two axes of cylinders at this configuration. As director vector of the disk rotate toward to the axial direction of cylinders (Figure 10-b and 10-c), the free energy is increased due to the stretching of polymers and interfacial area. Among those configurations, free energy is maximized when the director vector of the disk aligned toward x axis as shown in FIgure 10-b since the disk has faced highly curved interface of BCPs.§.§ Case of Y-shape defect Finally, we simulate the rod and disk NPs with Y-shape defects. As shown in Figure 11, we rotate each director vector of rod and disk at the center of Y-shape defect structure. We also plot the free energy landscape as a function of the director vector of the rod and disk in Figure 12 and Figure 13 respectively.Similar to the other defect structures of BCPs, the free energy maximum is achieved when the director vector of the rod is perpendicular to the three cylinder axes as shown in Figure 12-a. Since two other cylinders (aligned to the (+x,+y) direction and (-x,+y) direction) provide a highly curved interface of two blocks compare to the cylinder aligned to the (-y) direction, the extra stretching energy from highly curved interface prohibits the rod from aligning with those two axial directions. As a result, rod aligned to y-axis has the lowest free energy than any other configuration of this system as shown in Figure 12-b and Figure 12-c. As expected, for the disk NP, the free energy minimum is achieved when the director vector is aligned perpendicular to the axial directions of all cylinders (Figure 13-a). The free energy maxima is achieved when the director vector of the disk aligned toward the (-y) direction because the disk directly faces the highest curved interface of BCPs at this configuration as shown in Figure 13-c.§ CONCLUSIONS In this study, we investigated the behavior of rod and disk NP under the confinement of various defect structures within a BCP matrix. Self-Consistent Field Theory (SCFT) simulations with cavity function for NPs were used to describe the rod and disk NPs as an extra field excluding the surrounding polymer density. By solving the diffusion-like equation with the Lattice Boltzmann Method (LBM), we could use the local grid refinement to simulate the rod and disk NP with higher accuracy than possible with the pseudo-spectral method.We first plotted the free energy of the rod and disk NP as a function of rotation angle within a defect-free cylinder phase of the BCPs. For the rod, the free energy minimum is achieved when the director vector is aligned with the cylinder axis. In this configuration, the rod molecule maximizes the volume occupancy of the cylinder axis where the polymer ends exist to minimize the stretching energy of the BCPs. This is also applied to the disk NP case, where the director vector of the disk should be perpendicular to the axial direction of cylinder for the maximum volume occupancy of the cylinder axis. For this configuration, the system reaches the free energy minimum. Based on the result from defect-free cylinder case, we also studied the rod and disk NPs confined within the X, T, and Y-shape defects of the BCPs. Unlike the defect-free cylinder case, here the curved interface of the BCPs add an extra stretching energy to the system. As a result, a higher energy barrier exists near the defect center when the NP approaches the highly curved interfaces. For the anisotropic NPs, this determines the specific configuration of the director vector of NP within the defect structure of the BCPs. For the X-shape defect case, the director vector of the rod is aligned with one of two cylinders at the defect center (cylinder junction) due to the symmetry of the X-shape defect. The director vector of the rod in both the T-shape and Y-shape defect is aligned with the x axis (Figure 9-b) and y axis (Figure 12-c) respectively to minimize the extra chain stretching from the curved interface. For the disk NP, the director vector is perpendicular to the defect plane which shares every axial direction of cylinders to minimize the free energy for all types of defects. In this configuration, the disk NP can occupy all cylinder axes equally to maximize the volume coverage by the disk itself. Since the chain stretching energy is important factor for the preferred orientation of NPs, we expect that BCPs contain semi-flexible polymer will enhance the confinement effect. There results give a better understanding of the effect of confinement on the preferred configuration of complex and anisotropic NPs blended with BCPs. 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Rees products of posets and equivariantgamma-positivity] Some applications of Rees products of posets toequivariant gamma-positivityDepartment of Mathematics National and Kapodistrian University of AthensPanepistimioupolis15784 Athens, Greece [email protected] words and phrases.Rees product, poset homology, group action,Schur gamma-positivity, local face module. The Rees product of partially ordered sets wasintroduced by Björner and Welker. Using the theoryof lexicographic shellability, Linusson, Shareshianand Wachs proved formulas, of significance in the theoryof gamma-positivity, for the dimension of the homology of the Rees product of a graded poset P with a certaint-analogue of the chain of the same length as P.Equivariant generalizations of these formulas areproven in this paper, when a group of automorphisms acts on P, and are applied to establish theSchur gamma-positivity of certain symmetric functionsarising in algebraic and geometric combinatorics. [ Christos A. Athanasiadis May 1, 2019 ============================§ INTRODUCTIONThe Rees product P ∗ Q of two partially orderedsets (posets, for short) was introduced and studied byBjörner and Welker <cit.> as a combinatorialanalogue of the Rees construction in commutativealgebra (a precise definition of P ∗ Q can befound in Section <ref>). The connection ofthe Rees product of posets to enumerativecombinatorics was hinted in <cit.>,where it was conjectured that the dimension of thehomology of the Rees product of the truncatedBoolean algebra B_n {∅} of rankn-1 with an n-element chain equals the number ofpermutations of [n] := {1, 2,…,n} withoutfixed points. This statement was generalized inseveral ways in <cit.>, using enumerative andrepresentation theoretic methods, andin <cit.>, using the theory of lexicographicshellability. One of the results of <cit.> proves formulas <cit.> for the dimension of thehomology of the Rees product of an EL-shellable posetP with a contractible poset which generalizesthe chain of the same rank as P. This paper providesan equivariant analogue of this result which seemsto have enough applications on its own to be ofindependent interest. To state it, let P be afinite bounded poset, with minimum element 0̂and maximum element 1̂, which is graded ofrank n+1, with rank function ρ: P →{0, 1,…,n+1} (for basic terminology on posets,see <cit.>). Fix a fieldandlet G be a finite group whichacts on P by order preserving bijections. Then, Gdefines a permutation representation α_P(S)overfor every S ⊆ [n], induced bythe action of G on the set of maximal chains ofthe rank-selected subposetP_S={ x ∈ P: ρ(x) ∈ S } ∪ {0̂, 1̂} of P. Following Stanley's seminal work <cit.>,we may consider the virtual G-representation β_P(S)=∑_T ⊆ S (-1)^|S-T| α_P(T), defined equivalently by the equations α_P(T)=∑_S ⊆ Tβ_P(S) for T ⊆ [n]. The dimensions of α_P(S)and β_P(S) are important enumerative invariantsof P, known as the entries of its flag f-vector and flag h-vector, respectively. When P is Cohen–Macaulayover , β_P(S) is isomorphic to thenon-virtual G-representation H_|S|-1(P̅_S; ) induced on the top homology group ofP̅_S := P_S {0̂, 1̂} <cit.>. As discussed and illustratedin various situations in <cit.>, the decompositionof β_P(S) as a direct sum ofirreducible G-representations often leads to veryinteresting refinements of the flag h-vector of P.As in references <cit.>, we writeβ(P̅) in place of β_P([n]) and note, as just mentioned, that this G-representation isisomorphic to H_n-1 (P̅; ) ifP is Cohen–Macaulay over . Wedenote by T_t,n the poset whose Hasse diagramis a complete t-ary tree of height n, rooted atthe minimum element. We denote by P^-, P_-and P̅ the poset obtained from P byremoving its maximum element, or minimum element,or both, respectively, and recallfrom <cit.> (see also Section <ref>)that the action of G on P induces actions on P^- ∗ T_t,n and P̅∗ T_t,n-1as well. We also write [a, b] := {a,a+1,…,b} for integers a ≤ b and denoteby (Θ) the set of all subsets, calledstable, of Θ⊆ which do notcontain two consecutive integers. The followingresult reduces to <cit.>,proven in <cit.> under additionalshellability assumptions on P, in the specialcase of a trivial action.Let G be a finite group acting on a finite boundedgraded poset P of rank n+1 by order preservingbijections. Then, β ((P^- ∗ T_t,n)_-) ≅_G∑_S ∈ ([n-1])β_P ([n]S)t^|S| (1+t)^n-2|S|+ ∑_S ∈ ([n-2])β_P([n-1]S)t^|S|+1 (1+t)^n-1-2|S| and β (P̅∗ T_t,n-1) ≅_G∑_S ∈ ([2, n-2])β_P([n-1]S)t^|S|+1 (1+t)^n-2-2|S|+ ∑_S ∈ ([2, n-1])β_P ([n]S)t^|S| (1+t)^n-1-2|S| for every positive integer t, where ≅_Gstands for isomorphism of G-representations. If P is Cohen–Macaulay over , then theleft-hand sides of (<ref>) and (<ref>)may be replaced by the G-representationsH_n-1 ((P^- ∗ T_t,n)_-; )and H_n-1 (P̅∗ T_t,n-1;), respectively, and all representations whichappear in these formulas are non-virtual. Several applications of <cit.>to γ-positivity appear in <cit.><cit.> and are summarized in<cit.>. Theorem <ref>has non-trivial applications to Schurγ-positivity, which we now briefly discuss.A polynomial in t with real coefficients is saidto be γ-positive if for some m ∈, it can be written as a nonnegative linearcombination of the binomials t^i (1+t)^m-2i for0 ≤ i ≤ m/2. Clearly, all such polynomialshave symmetric and unimodal coefficients. Twosymmetric function identities due to Gessel(unpublished), stated without proofin <cit.> <cit.>,can be written in the form1-t/E(; tz) - tE(; z) = 1 +∑_n ≥ 2 z^n∑_k=0^⌊ (n-2)/2 ⌋ξ_n,k() t^k+1 (1 + t)^n-2k-2 and (1-t) E(; tz)/E(; tz) - tE(; z) =1 +∑_n ≥ 1 z^n∑_k = 0^⌊ (n-1)/2 ⌋γ_n,k() t^k+1 (1 + t)^n-1-2k, where E(; z) = ∑_n ≥ 0 e_n() z^nis the generating function for the elementarysymmetric functions in = (x_1, x_2,…)and the ξ_n,k() and γ_n,k()are Schur-positive symmetric functions, whosecoefficients in the Schur basis can be explicitly described (see Corollary <ref>). Thecoefficients of z^n in the right-hand sides ofEquations (<ref>) and (<ref>) areSchur γ-positive symmetric functions, inthe sense that their coefficients in the Schurbasis are γ-positive polynomials in t with the same center of symmetry. Their Schurγ-positivity refines the γ-positivity of derangement and Eulerian polynomials, respectively; see <cit.> for more information.We will show (see Section <ref>) thatGessel's identitiescan in fact be derived from the special case ofTheorem <ref> in which P^- is theBoolean algebra B_n, endowed with the naturalsymmetric group action. Moreover, applying thetheorem when P^- is a natural signed analogueof B_n, endowed with a hyperoctahedral groupaction, we obtain new identities of the formE(; tz) - t E(; z)/E(; tz) E(; tz) - tE(; z) E(; z) =∑_n ≥ 0 z^n∑_k=0^⌊ n/2 ⌋ξ^+_n,k(, ) t^k (1 + t)^n-2k,E(; z) - E(; tz)/E(; tz) E(; tz) - tE(; z) E(; z) =∑_n ≥ 1 z^n ∑_k=0^⌊ (n-1)/2 ⌋ξ^-_n,k(, ) t^k (1 + t)^n-1-2k,E(; z) E(; tz)( E(; tz) - t E(; z) )/E(; tz) E(; tz) - tE(; z) E(; z) =∑_n ≥ 0 z^n∑_k=0^⌊ n/2 ⌋γ^+_n,k (, ) t^k (1 + t)^n-2k and t E(; z) E(; tz)( E(; z) - E(; tz) )/E(; tz) E(; tz) - tE(; z) E(; z) = ∑_n ≥ 1 z^n ∑_k=1^⌊ (n+1)/2 ⌋γ^-_n,k(, ) t^k (1 + t)^n+1-2k,where the ξ^±_n,k(, ) andγ^±_n,k(, ) are Schur-positivesymmetric functions in the sets of variables =(x_1, x_2,…) and = (y_1, y_2,…)separately. Their Schur positivity refinesthe γ-positivity of type B analogues orvariants of derangement and Eulerian polynomials;this is explained andgeneralized in the sequel <cit.> to thispaper. Note that for = 0, the left-hand sideof Equation (<ref>) specializes to thatof (<ref>) (withreplaced by )and the sum of the left-hand sides ofEquations (<ref>) and (<ref>) specializes to that of (<ref>) (again with replaced by ). Various combinatorial and algebraic-geometricinterpretations of the left-hand sides ofEquations (<ref>) and (<ref>) arediscussed in <cit.><cit.> <cit.>. For instance, by <cit.>, the coefficient of z^n in the left-hand side of (<ref>)can be interpreted as the Frobenius characteristicof the symmetric group representation on the localface module of the barycentric subdivision of the(n-1)-dimensional simplex, twisted by the signrepresentation. Thus, the Schur γ-positivity of this coefficient, manifested byEquation (<ref>), is an instance of thelocal equivariant Gal phenomenon, as discussed in<cit.>. Section <ref>shows that another instance of this phenomenonfollows from the specialization = ofEquation (<ref>). Similarly, setting=0 to (<ref>) yields anotheridentity, recently proven by Shareshian and Wachs(see Proposition 3.3 and Theorem 3.4 in <cit.>)in order to establish the equivariant Gal phenomenonfor the symmetric group action on the n-dimensionalstellohedron and Section <ref> combinesEquation (<ref>) with (<ref>) toestablish the same phenomenon for the hyperoctahedral group action on its associated Coxeter complex. Furtherapplications of Theorem <ref> are givenin <cit.>. It would be interesting to finddirect combinatorial proofs ofEquations (<ref>) – (<ref>) andto generalize other known interpretations of theleft-hand sides of Equations (<ref>)and (<ref>) to those of (<ref>) –(<ref>).Outline. The proof of Theorem <ref>is given in Section <ref>, after therelevant background and definitions are explained inSection <ref>. This proof is fairlyelementary and different from thatof <cit.>. Section <ref>derives Equations (<ref>) – (<ref>)from Theorem <ref> and provides explicitcombinatorial interpretations, in terms of standardYoung (bi)tableaux and their descents, for theSchur-positive symmetric functions which appearthere. Sections <ref> and <ref>provide the promised applications ofEquations (<ref>) and (<ref>) tothe equivariant γ-positivity of the symmetricgroup representation on the local face module ofa certain triangulation of the simplex and thehyperoctahedral group representation on the cohomology of the projective toric variety associated to theCoxeter complex of type B.§ PRELIMINARIESThis section briefly records definitions andbackground on posets, group representations and(quasi)symmetric functions. For basic notions andmore information on these topics, the reader isreferred to the sources <cit.> <cit.> <cit.><cit.> <cit.>. Thesymmetric group of permutations of the set [n]is denoted by _n and the cardinality of afinite set S by |S|.Group actions on posets and Rees products. All groups and posets considered here are assumedto be finite. Homological notions for posets alwaysrefer to those of their order complex;see <cit.>. A poset P has thestructure of a G-posetif the group G acts on P by order preservingbijections. Then, G induces a representation onevery reduced homology group H_i (P;), for every field . Suppose that P is a G-poset with minimumelement 0̂ and maximum element 1̂.Sundaram <cit.> (see also<cit.>) established theisomorphism of G-representations ⊕_k ≥ 0 (-1)^k ⊕_x ∈ P/GH_k-2 ((0̂, x); )↑^G_G_x≅_G0. Here P/G stands for a complete set of G-orbitrepresentatives, (0̂, x) denotes the openinterval of elements of P lying strictly between0̂ and x, G_x is the stabilizer ofx and ↑ denotes induction. Moreover, H_k-2 ((0̂, x); ) isunderstood to be the trivial representation1_G_x if x = 0̂ and k=0, or xcovers 0̂ and k = 1. The Lefschetz character of a finiteG-poset P (over the field ) is definedas the virtual G-representation L(P; G):=⊕_k ≥ 0(-1)^kH_k (P; ). Note that L(P; G) = (-1)^r H_r(P; ), if P is Cohen–Macaulay over of rank r. Given finite graded posets P and Q with rankfunctions ρ_P and ρ_Q, respectively,their Rees product is defined in <cit.> as P ∗ Q = { (p, q) ∈ P× Q: ρ_P (p) ≥ρ_Q (q) }, withpartial order defined by setting (p_1, q_1)≼ (p_2, q_2) if all of the followingconditions are satisfied: =0pt∙ p_1 ≼ p_2 holds in P, ∙ q_1 ≼ q_2 holds in Q and ∙ ρ_P (p_2) - ρ_P (p_1) ≥ρ_Q(q_2) - ρ_Q (q_1). Equivalently, (p_1, q_1) is covered by (p_2, q_2)in P ∗ Q if and only if (a) p_1 is covered byp_2 in P; and (b) either q_1 = q_2, or q_1 iscovered by q_2 in Q. We note that the Rees product P ∗ Q is graded with rank function given byρ(p,q) = ρ_P(p) for (p,q) ∈ P ∗ Q, andthat if P is a G-poset, then so is P ∗ Qwith the G-action defined by setting g · (p,q) = (g · p, q) for g ∈ G and (p, q) ∈P ∗ Q.A bounded graded G-poset P, with maximum element 1̂, is said to be G-uniform<cit.> if the following hold: =0pt∙ the intervals [x, 1̂] and [y, 1̂] inP are isomorphic for all x, y ∈ P of the samerank, ∙ the stabilizers G_x and G_y are isomorphic forall x, y ∈ P of the same rank, and ∙ there is an isomorphism between [x, 1̂] and[y, 1̂] that intertwines the actions of G_xand G_y, for all x, y ∈ P of the same rank. Given a sequence of groups G = (G_0,G_1,…,G_n), a sequence of posets (P_0,P_1,…,P_n) is said to be G-uniform<cit.> if=0pt∙ P_k is G_k-uniform of rank k for all k, ∙ G_k is isomorphic to the stabilizer (G_n)_x for every x ∈ P_n of rank n-k and all k,and ∙ there is an isomorphism between P_k and theinterval [x, 1̂] in P_n that intertwinesthe actions of G_k and (G_n)_x for every x∈ P_n of rank n-k and all k. Under these assumptions, Shareshian and Wachs<cit.> established theisomorphism of G-representations 1_G_n⊕ ⊕_k=0^nW_k(P_n;G_n) [k+1]_t L((P_n-k∗T_t,n-k)_-; G_n-k) ↑^G_n_G_n-k≅_G0 for every positive integer t, where W_k(P_n;G_n)is the number of G_n-orbits of elements of P_nof rank k and [k+1]_t := 1 + t + ⋯ + t^k.The Boolean algebra B_n consists of all subsetsof [n], partially ordered by inclusion. When endowedwith the standard _n-action, it becomes aprototypical example of an _n-uniform poset.Every element x ∈ B_n of rank k is a set ofcardinality k; its stabilizer (_n)_x = {w ∈_n: w x = x} is isomorphic to the Youngsubgroup _k ×_n-k of _n, whichcan be defined as the stalilizer of [k]. The sequence(B_0, B_1,…,B_n) can easily be verified to be(G_0, G_1,…,G_n)-uniform for G_k := _k×_n-k.Permutations, Young tableaux and symmetric functions. Our notation concerning these topicsfollows mostly that of <cit.><cit.> <cit.>. In particular, the set of standard Young tableauxof shape λ is denoted by (λ),the descent set { i ∈ [n-1]: w(i) > w(i+1) }of a permutation w ∈_n is denoted by (w) and that of a tableau Q ∈(λ), consisting of those entries i forwhich i+1 appears in Q in a lower row than i, is denoted by (Q). We recall that theRobinson–Schensted correspondence is abijection from thesymmetric group _n to the set of pairs(, Q) of standard Young tableaux of the sameshape and size n. This correspondence has the property <cit.> that (w)= (Q(w)) and (w^-1) = ((w)),where ((w), Q(w)) is the pair of tableauxassociated to w ∈_n.We will consider symmetric functions in theindeterminates = (x_1, x_2,…) over thecomplex field . We denote by E(; z) :=∑_n ≥ 0 e_n() z^n and H(; z) :=∑_n ≥ 0 h_n() z^n the generatingfunctions for the elementary and completehomogeneous symmetric functions, respectively,and recall the identity E(; z) H(; -z) = 1.The (Frobenius) characteristic map, a -linearisomorphism of fundamental importance from thespace of virtual _n-representations to thatof homogeneous symmetric functions of degree n, will be denoted by . This map sends theirreducible _n-representation corresponding to λ⊢ n to the Schur functions_λ() associated to λ and thus it sends non-virtual _n-representations toSchur-positive symmetric functions. The fundamental quasisymmetric functionassociated to S ⊆ [n-1] is defined asF_n, S ()= ∑_1 ≤ i_1 ≤ i_2 ≤⋯≤ i_n j ∈ S⇒ i_j < i_j+1 x_i_1 x_i_2⋯ x_i_n. The following well known expansion<cit.>s_λ()=∑_Q ∈ (λ)F_n, (Q) () will be used in Section <ref>.For the applications given there, we need the analogues of these concepts in therepresentation theory of the hyperoctahedral groupof signed permutations of the set [n], denotedhere by _n. We will keep this discussion rather brief and refer to <cit.> formore information.A bipartition of a positive integer n,written (λ, μ) ⊢ n, is any pair(λ, μ) of integer partitions of totalsum n. A standard Young bitableau ofshape (λ, μ) ⊢ n is any pair Q= (Q^+, Q^-) of Young tableaux such thatQ^+ has shape λ, Q^- has shape μ and every element of [n] appears exactly once as an entry of Q^+ or Q^-. The tableaux Q^+ and Q^- are called theparts of Q and the number n is itssize. The Robinson–Schenstedcorrespondence of type B, as describedin <cit.> (seealso <cit.>), is abijection from the group _n to the setof pairs (, Q) of standard Young bitableauxof the same shape and size n. The (Frobenius) characteristic map for thehyperoctahedral group, denoted by _,is a -linear isomorphism from the space ofvirtual _n-representations to that ofhomogeneous symmetric functions of degree n inthe sets of indeterminates = (x_1, x_2,…)and = (y_1, y_2,…) separately; see, forinstance, <cit.>. The map _ sends the irreducible _n-representationcorresponding to (λ, μ) ⊢ n to thefunction s_λ() s_μ() and thus it sends non-virtual _n-representations toSchur-positive functions, meaning nonnegativeinteger linear combinations of the functionss_λ() s_μ(). The following basicproperties of _ will be useful inSection <ref>: =1pt∙ _(1__n) = h_n(), where 1__n is the trivial _n-representation, ∙ _ (σ⊗τ↑^_n__k ×_n-k) =_(σ) ·_ (τ) for allrepresentations σ and τ of _kand _n-k, respectively, where _k×_n-k is the Young subgroup of _n consisting of all signed permutations which preserve the set {± 1, ± 2,…,± k},∙ _ (↑^_n__nρ) =(ρ) (, ) for every _n-representationρ, where _n ⊂_n is the standard embedding. We denote by E(, ; z) := ∑_n ≥ 0 e_n (, ) z^n and H(, ; z) := ∑_n ≥ 0h_n(, ) z^n the generating function for theelementary and complete homogeneous, respectively, symmetric functions in the variables (, ) =(x_1, x_2,…,y_1, y_2,…) and note thatE(, ; z) = E(; z) E(; z), since e_n (, ) = ∑_k=0^n e_k() e_n-k(),and similarly that H(, ; z) = H(; z)H(; z).The signed descent set<cit.> <cit.> of w ∈_n, denoted (w), records the positionsof the increasing (in absolute value) runs ofconstant sign in the sequence (w(1),w(2),…,w(n)). Formally, we may define(w) as the pair ((w), ε),where ε = (ε_1,ε_2,…,ε_n) ∈{-, +}^nis the sign vector with ith coordinate equal tothe sign of w(i) and (w) consists of theindices i ∈ [n-1] for which either ε_i= + and ε_i+1 = -, or else ε_i = ε_i+1 and |w(i)| >|w(i+1)| (this definition is slightly differentfrom, but equivalent to, the ones given in<cit.>). The fundamentalquasisymmetric function associated to w, introduced by Poirier <cit.> in a moregeneral setting, is defined asF_(w) (, )= ∑_i_1 ≤ i_2 ≤⋯≤ i_n j ∈(w)⇒i_j < i_j+1 z_i_1 z_i_2⋯ z_i_n,where z_i_j = x_i_j if ε_j = +, andz_i_j = y_i_j if ε_j = -. Given astandard Young bitableau Q of size n, one definesthe signed descent set (Q) as the pair((Q), ε), where ε =(ε_1, ε_2,…,ε_n)∈{-, +}^n is the sign vector with ithcoordinate equal to the sign of the part of Q inwhich i appears and (Q) is the set of indicesi ∈ [n-1] for which either ε_i = +and ε_i+1 = -, or elseε_i= ε_i+1 and i+1 appears in Q in alower row than i. The function F_s(Q) (,) is then defined by the right-hand side ofEquation (<ref>), with w replaced by Q; see <cit.>. The analogue s_λ() s_μ()= ∑_Q ∈(λ,μ) F_s(Q) (, )of the expansion (<ref>) holds(<cit.>) and theRobinson–Schensted correspondence of type B hasthe properties that (w) = (Q^B(w)) and(w^-1) = (^B(w)), where (^B(w),Q^B(w)) is the pair of bitableaux associated to w∈_n; see <cit.>. § PROOF OF THEOREM <REF>This section proves Theorem <ref> usingonly the definition of Rees product and the definingequation (<ref>) of the representationsβ_P(S). For S = {s_1, s_2,…,s_k}⊆ [n] with s_1 < s_2 < ⋯ < s_k weset φ_t(S) := [s_1 + 1]_t [s_2 - s_1 + 1]_t⋯ [s_k - s_k-1 + 1]_tψ_t(S) := [s_1]_t [s_2 - s_1 + 1]_t ⋯[s_k - s_k-1 + 1]_t where, as mentioned already, [m]_t = 1 + t + ⋯ +t^m-1 for positive integers m.Let G be a finite group, P be a finite boundedgraded G-poset of rank n+1, as inTheorem <ref>, and Q, R be the posetsdefined by Q̅ = (P^- ∗ T_t,n)_- andR̅ = P̅∗ T_t,n-1. Then, α_Q (S)≅_Gφ_t(S)α_P (S)α_R (S)≅_Gψ_t(S)α_P (S) for all positive integers t and S ⊆ [n].Let S = {s_1, s_2,…,s_k}⊆ [n] withs_1 < s_2 < ⋯ < s_k and let ρ: T_t,n→ be the rank function of T_t,n. By thedefinition of Rees product, the maximal chains inQ_S have the form 0̂ ≺(p_1, τ_1)≺(p_2, τ_2)≺ ⋯ ≺(p_k, τ_k)≺ 1̂ where 0̂≺ p_1 ≺ p_2 ≺⋯≺ p_k ≺1̂ is a maximal chain inP_S and τ_1 ≼τ_2 ≼⋯≼τ_k is a multichain in T_t,n suchthat=0pt∙ 0 ≤ρ(τ_1) ≤ s_1 and ∙ ρ(τ_j) - ρ(τ_j-1) ≤ s_j - s_j-1for 2 ≤ j ≤ k. Let m_t(S) be the number of these multichains.Since the elements of G act on (<ref>) byfixing the τ_j and acting on the correspondingmaximal chain of P_S, we have α_Q(S)≅_G m_t(S)α_P(S). To choose such amultichain τ_1 ≼τ_2 ≼⋯≼τ_k, we need to specify the sequencei_1 ≤ i_2 ≤⋯≤ i_k of ranks of itselements so that i_j - i_j-1≤ s_j - s_j-1for 1 ≤ j ≤ k, where i_0 := s_0 := 0, andchoose its maximum element τ_k in t^i_kpossible ways. Summing over all such sequences,we get m_t (S)=∑_(i_1, i_2,…,i_k)t^i_k =∑_0 ≤ a_j ≤ s_j -s_j-1 t^a_1 + a_2 + ⋯ + a_k =φ_t (S)and the result for α_Q(S) follows. The same argument applies to α_R(S); one simply has toswitch the condition for the rank of τ_1 to 0≤ρ(τ_1) ≤ s_1 - 1.The proof of the following technical lemma will begiven after that of Theorem <ref>.For every S ⊆ [n] we have ∑_S ⊆ T ⊆ [n] (-1)^n-|T| φ_t(T)= 0,if [n]S is not stable,t^r(1+t)^n-2r,if [n]S isstable and n ∈ S,t^r(1+t)^n+1-2r,if [n]S isstable and n ∉S and ∑_S ⊆ T ⊆ [n] (-1)^n-|T| ψ_t(T)= 0,if 1 ∉S,0,if [n]S is not stable,t^r(1+t)^n-1-2r,if [n]S isstable and 1, n ∈ S,t^r(1+t)^n-2r,if [n]S isstable, 1 ∈ S and n ∉S, where r = n - |S|. Proof of Theorem <ref>. UsingEquations (<ref>) and (<ref>),as well as Lemma <ref>, we compute that β_Q ([n]) =∑_T ⊆ [n](-1)^n-|T| α_Q (T)≅_G∑_T ⊆ [n] (-1)^n-|T| φ_t(T)α_P (T) =∑_T ⊆ [n] (-1)^n-|T| φ_t(T)∑_S ⊆ Tβ_P(S) =∑_S ⊆ [n]β_P(S)∑_S ⊆ T ⊆ [n] (-1)^n-|T| φ_t(T) and find similarly that β_R ([n])≅_G∑_S ⊆ [n]β_P(S) ∑_S ⊆ T ⊆ [n] (-1)^n-|T| ψ_t(T). The proof follows from these formulas andLemma <ref>.For the last statement of the theorem one has to notethat, as a consequence of <cit.>,if P is Cohen–Macaulay over , then so are theRees products P^-∗ T_t,n and P̅∗T_t,n-1. Proof of Lemma <ref>. Let us denote by χ_t(S) the left-hand side of (<ref>) and write S = {s_1, s_2,…,s_k}, with 1 ≤s_1 < s_2 < ⋯ < s_k ≤ n. By definition, wehaveχ_t(S)=χ_t(s_1)χ_t(s_2 - s_1) ⋯χ_t(s_k - s_k-1)ω_t(n - s_k), whereχ_t(n) :=∑_n ∈ T ⊆ [n] (-1)^n-|T| φ_t(T) ω_t(n) :=∑_T ⊆ [n] (-1)^n-|T| φ_t(T) for n ≥ 1 and ω_t(0) := 1. We claim thatχ_t(n)= 1+t,if n = 1,t,if n = 2,0,if n ≥ 3 and ω_t(n)= 1,if n = 0,t,if n = 1,0,if n ≥ 2. Equation (<ref>) is a direct consequenceof (<ref>) and this claim. To verify theclaim, note that the defining equation forχ_t(n) can be rewritten asχ_t(n)=∑_(a_1, a_2,…,a_k)n (-1)^n-k[a_1 + 1]_t [a_2 + 1]_t ⋯ [a_k + 1]_t, where the sum ranges over all sequences(compositions) (a_1, a_2,…,a_k) of positiveintegers summing to n. We leave it as a simplecombinatorial exercise for the interested reader to show (for instance, by standard generatingfunction methods) that χ_t(n) = 0 forn ≥ 3. The claim follows from this fact andthe obvious recurrence ω_t(n) = χ_t(n) -ω_t(n-1), valid for n ≥ 1.Finally, note that Equation (<ref>) isequivalent to (<ref>) in the case 1 ∈S. Otherwise, the terms in the left-hand sidecan be partitioned into pairs of terms,corresponding to pairs {T, T ∪{1}} ofsubsets with 1 ∉T, canceling each other. This shows that the left-hand side vanishes. § SYMMETRIC FUNCTION IDENTITIESThis section derives Equations (<ref>)– (<ref>) from Theorem <ref>(Corollaries <ref>, <ref>and <ref>) and interprets combinatorially the Schur-positive symmetric functions whichappear there. We first explain why Gessel'sidentities are special cases of this theorem.The set of ascents of a permutation w ∈_n is defined as (w) := [n-1] (w) and, similarly, we have ():= [n-1] () for every standardYoung tableauof size n. Let us recallthe fact (used in the following proof) thatthe reduced homology groups of a poset with aminimum or maximum element vanish. Equations (<ref>) and (<ref>) arevalid for the functionsξ_n,k ()=∑_λ⊢ nc_λ,k· s_λ ()=∑_w F_n,(w)() and γ_n,k ()=∑_λ⊢ nd_λ,k· s_λ ()=∑_w F_n,(w)(), where c_λ,k (respectively,d_λ,k ) stands for the number oftableaux ∈(λ) for which ()∈([2,n-2]) (respectively,() ∈([n-2]) ) has kelements and, similarly, w ∈_n runs throughall permutations for which (w^-1) ∈([2,n-2]) (respectively, (w^-1)∈([n-2]) ) has k elements.We will apply Theorem <ref> when P^- is theBoolean lattice B_n, considered as an _n-posetas in Example <ref>. On the one hand, we havethe equality 1 +∑_n ≥ 2 ( H_n-1 ((B_n {∅}) ∗ T_t,n-1; ) ) z^n=1-t/E(; tz) - tE(; z) which, although not explicitly stated in <cit.>, follows as in the proof of its special case t=1<cit.>. On the other hand, since B_n has a maximum element, the second summand in the right-hand side of Equation (<ref>) vanishes and hence this equation gives ( H_n-1 ((B_n {∅}) ∗ T_t,n-1; ) )= ∑_S ∈ ([2,n-2])( β_B_n ([n-1]S) )t^|S|+1(1+t)^n-2-2|S| for n ≥ 2. The representations β_B_n (S) for S ⊆ [n-1] are known to satisfy (see,for instance, <cit.>)( β_B_n (S) )=∑_λ⊢ n c_λ,S· s_λ (), where c_λ,S is the number of standardYoung tableaux of shape λ and descent setequal to S. Combining the previous threeequalities yields the first equality inEquation (<ref>). The second equalityfollows from the first by expanding s_λ()according to Equation (<ref>) to get ξ_n,k ()=∑_λ⊢ n∑_∑_Q ∈ (λ) F_n, (Q) () where, in the inner sum,runs through alltableaux in (λ) for which () ∈_([2,n-2]) has k elements, and then usingthe Robinson–Schensted correspondence and itsstandard properties (w) = (Q(w)) and(w^-1) = ((w)) to replace thesummations with one running over elements of _n, as in the statement of the corollary. The proof of (<ref>) is entirelysimilar; one has to use Equation (<ref>)instead of (<ref>), as well as the equality 1 +∑_n ≥ 1 ( H_n-1 ((B_n ∗ T_t,n)_-; ) ) z^n=(1-t) E(; tz)/E(; tz) - tE(; z). The latter follows from the proof of Equation (3.3)in <cit.>, where the left-hand sideis equal to -F_t(-z), in the notation used in thatproof. The coefficient of z^4 in the left-hand sidesof Equations (<ref>) and (<ref>)equals =3pt∙ e_4() (t + t^2 + t^3) + e_2()^2 t^2,and ∙ e_4() (t + t^2 + t^3 + t^4) + e_1()e_3() (t^2 + t^3) + e_2()^2 (t^2 + t^3), respectively. These expressions may be rewritten as =3pt∙ s_(1,1,1,1) () t(1+t)^2 + s_(2,1,1) ()t^2 + s_(2,2) () t^2, and ∙ s_(1,1,1,1) () t(1+t)^3 + 2 s_(2,1,1)() t^2(1+t) + s_(2,2) () t^2(1+t), respectively, and hence ξ_4,0 () =s_(1,1,1,1) (), ξ_4,1 () = s_(2,1,1)() + s_(2,2) (), γ_4,0 () =s_(1,1,1,1) () and γ_4,1 () = 2s_(2,1,1) () + s_(2,2) (). We leave it tothe reader to verify that these formulas agree withCorollary <ref>. We now focus on the identities (<ref>) – (<ref>). We will applyTheorem <ref> to the collection sB_nof all subsets of {1, 2,…,n}∪{ -1,-2,…,-n} which do not contain {i, -i}for any index i, partially ordered byinclusion. This signed analogue of the Booleanalgebra B_n is a graded poset of rank n,having the empty set as its minimum element, onwhich the hyperoctahedral group _n acts in the obvious way <cit.>, turningit into a _n-poset. It is isomorphic to theposet of faces (including the empty one) of theboundary complex of the n-dimensionalcross-polytope and hence it is Cohen–Macaulayoverand any field. The left-hand sides ofEquations (<ref>) and (<ref>)for P^- = sB_n will be computed using themethods of <cit.>. Consider the n-element chain C_n = {0,1,…,n-1}, with the usual total order.Following <cit.>, we denote by I_j(B_n) theorder ideal of elements of (B_n {∅}) ∗ C_n which are strictly less than ([n], j).Then I_j(B_n) is an _n-poset for every j ∈C_n and one of the main results of <cit.> (see <cit.> <cit.>)states that 1 +∑_n ≥ 1z^n∑_j=0^n-1t^j(H_n-2 (I_j(B_n); ) )=(1-t) E(; z)/E(; tz) - tE(; z). For the _n-poset sB_n we have 1 +∑_n ≥ 1 _( H_n-1 ((sB_n {∅})∗ T_t,n-1; ) ) z^n =(1-t) E(; z)/E(; tz) E(; tz) - tE(; z) E(; z). Following the reasoning in the proofof <cit.>, we apply (<ref>) to the Cohen–Macaulay _n-poset obtained from(sB_n {∅}) ∗ T_t,n-1 by adding a minimum and a maximum element. For 0 ≤ j < k ≤n, there are exactly t^j _n-orbits of elementsx of rank k in this poset with second coordinateof rank j in T_t,n-1 and for each one of these,the open interval (0̂, x) is isomorphic to I_j(B_k)and the stabilizer of x is isomorphic to _k×_n-k. We conclude that H_n-1 ((sB_n {∅}) ∗ T_t,n-1; )≅__n ⊕_k=0^n (-1)^n-k ⊕_j=0^k-1t^j (H_k-2 (I_j(B_k); ) ⊗ 1__n-k)↑^_n__k ×_n-k. Applying the map _ andusing the transitivity ↑^_n__k×_n-k ≅__n ↑^_k×_n-k__k ×_n-k ↑^_n__k ×_n-k ofinduction and properties of _ discussed inSection <ref>, the right-hand side becomes ∑_k=0^n (-1)^n-k ∑_j=0^k-1t^j(H_k-2 (I_j(B_k); ) )(, ) · h_n-k(). Thus, the left-hand side of Equation (<ref>) is equal to H(; -z) ·( 1 +∑_n ≥ 1z^n∑_j=0^n-1t^j(H_n-2 (I_j(B_n); ) )(, )) and the result follows from Equation (<ref>)and the identities E(; z) H(; -z) = 1 andE(, ; z) = E(; z) E(; z). Recall the definition of the sets (w) and() for a signed permutation w ∈_nand standard Young bitableauof size n,respectively, from Section <ref>. Following <cit.>, we define the typeB descent set of = (^+, ^-) as_B() = () ∪{n}, if n appears in ^+, and _B() = () otherwise. The complement of _B() in the set [n] iscalled the type B ascent set ofandis denoted by _B(). Similarly, we definethe type B descent set of w ∈_nas _B(w) = (w) ∪{n}, if w(n) is positive, and _B(w) = (w) otherwise. Thecomplement of _B(w) in the set [n] iscalled the type B ascent set of w andis denoted by _B(w). The sets _B(w) and _B() depend only on the signed descent sets_B(w) and _B(), respectively, and <cit.>, mentioned at theend of Section <ref>, implies that _B(w)= _B(Q^B(w)) and _B(w^-1) = _B (^B(w)) for every w ∈_n.Equations (<ref>) and (<ref>) arevalid for the functionsξ^+_n,k (, )=∑_(λ, μ) ⊢ nc^+_(λ,μ),k· s_λ () s_μ()=∑_w F_(w)(, ) and ξ^-_n,k (, )=∑_(λ, μ) ⊢ nc^-_(λ,μ),k· s_λ () s_μ()=∑_w F_(w)(, ), where c^+_(λ,μ),k (respectively,c^-_(λ,μ),k ) stands for the numberof bitableaux ∈(λ,μ) for which_B() ∈([2,n]) has k elements andcontains (respectively, does notcontain) n and where, similarly, w ∈_n runs through all signed permutations for which_B(w^-1) ∈([2,n]) has kelements and contains (respectively, does notcontain) n.We apply the second part of Theorem <ref>for P^- = sB_n, thought of as a _n-poset.The representations β_P (S) for S ⊆[n] were computed in this casein <cit.>, which implies that_( β_B_n (S) )=∑_(λ,μ) ⊢ n c_(λ,μ),S· s_λ () s_μ() for S ⊆ [n], where c_(λ,μ),Sis the number of standard Young bitableauxofshape (λ,μ) such that _B() = S.Switching the roles ofandandcombining this result with the second part ofTheorem <ref> andProposition <ref> we get(1-t) E(; z)/E(; tz) E(; tz) - tE(; z) E(; z)=∑_n ≥ 0 z^n∑_k=0^⌊ n/2 ⌋ξ^+_n,k(, ) t^k (1 + t)^n-2k+ ∑_n ≥ 1 z^n ∑_k=0^⌊ (n-1)/2 ⌋ξ^-_n,k(, ) t^k (1 + t)^n-1-2k, where the ξ^±_n,k(, ) are givenby the first equalities in (<ref>)and (<ref>). We now note that theleft-hand side of Equation (<ref>) is equal to the sum of the left-hand sides, say Ξ^+(, , t; z) and Ξ^-(, , t;z), of Equations (<ref>)and (<ref>). Since, as one can readilyverify, Ξ^+(, , t; z) is left invariant under replacing t with 1/t and z with tz, while Ξ^-(, , t; z) is multiplied by t after these substitutions, the coefficient of z^n in Ξ^+(, , t; z) (respectively, Ξ^-(, , t; z)) is a symmetricpolynomial in t with center of symmetry n/2(respectively, (n-1)/2) for every n ∈. Since the corresponding properties are clear for the coefficient of z^n in the two summands in the right-hand side of Equation (<ref>)and because of the uniqueness of the decomposition of a polynomial f(t) as a sum of two symmetric polynomials with centers of symmetry n/2 and(n-1)/2 (see <cit.>), weconclude that (<ref>)and (<ref>) follow from the singleequation (<ref>). The second equalities in (<ref>)and (<ref>) follow by expandings_λ() s_μ() according toEquation (<ref>) and then using theRobinson–Schensted correspondence of type B andits properties (w) = (Q^B(w)) and_B(w^-1) = _B(^B(w)), exactly asin the proof of Corollary <ref>.The coefficient of z^2 in the left-hand sideof Equations (<ref>) and (<ref>)equals =3pt∙ e_1() e_1() t + e_2() t = s_(1) () s_(1)() t + s_(1,1)() t, and ∙ e_2() (1 + t) = s_(1,1)() (1 + t), respectively and hence ξ^+_2,0 (, ) =0, ξ^+_2,1 (, ) = s_(1)() s_(1)()+ s_(1,1)() and ξ^-_2,0 (, ) =s_(1,1)(), in agreement withCorollary <ref>. For the _n-poset sB_n we have 1 +∑_n ≥ 1 _( H_n-1((sB_n ∗ T_t,n)_-; ) ) z^n=(1-t) E(; z) E(; tz) E(; tz)/E(; tz) E(; tz) - tE(; z) E(; z). Following the reasoning in the proofof <cit.>, we setL_n (, ; t):=_(L((sB_n ∗ T_t,n)_-; _n) ), where L(P; G) denotes the Lefschetz character ofthe G-poset P over(seeSection <ref>). Since (sB_n ∗T_t,n)_- is Cohen–Macaulay overof rankn-1, we have_( H_n-1 ((sB_n ∗ T_t,n)_-; ) )=(-1)^n-1L_n(, ; t). Thus, the left-hand side of (<ref>) isequal to -∑_n ≥ 0 L_n(, ; t) (-z)^n.The sequence of posets (sB_0, sB_1,…,sB_n) caneasily be verified to be (_0 ×_n, _1×_n-1,…,_n ×_0)-uniform(see Section <ref>). Moreover, there is a single _n-orbit of elements of sB_n of rank k for each k ∈{0, 1,…,n}. Thus, applying (<ref>) to thissequence gives 1__n⊕ ⊕_k=0^n [k + 1]_t L((sB_n-k∗ T_t,n-k)_-; _n-k×_k) ↑^_n__n-k×_k≅__n 0. Applying the characteristic map _, as inthe proof of Proposition <ref>, gives ∑_k=0^n [k + 1]_t h_k(, ) L_n-k (, ; t)=- h_n (). Standard manipulation with generating functions,as in the proof of <cit.>,results in the formula ∑_n ≥ 0 L_n(, ; t) z^n=- H(; z)/∑_n ≥ 0[n+1]_th_n(, ) z^n =-(1-t) H(; z)/H(, ; z) - t H(, ; tz).The proof now follows by switching z to -z andusing the identities E(; z) H(; -z) = 1 andE(, ; z) = E(; z) E(; z). Equations (<ref>) and (<ref>)are valid for the functionsγ^+_n,k (, )=∑_(λ, μ) ⊢ nd^+_(λ,μ),k· s_λ () s_μ()=∑_w F_(w)(, ) and γ^-_n,k (, )=∑_(λ, μ) ⊢ nd^-_(λ,μ),k· s_λ () s_μ()=∑_w F_(w)(, ), where d^+_(λ,μ),k (respectively,d^-_(λ,μ),k ) is thenumber of bitableaux ∈(λ,μ) forwhich _B() ∈([n]) has kelements and does not contain (respectively, contains) n and, similarly, w ∈_nruns through all signed permutations for which_B(w^-1) ∈([n]) has k elementsand does not contain (respectively, contains) n.This statement follows by the same reasoning as inthe proof of Corollary <ref>, provided oneappeals to the first part of Theorem <ref>and Proposition <ref> instead. The coefficient of z^2 in the left-hand sideof Equations (<ref>) and (<ref>)equals =3pt∙ e_2() (1 + t + t^2) + e_1()^2 t +e_1() e_1() t = s_(1,1)() (1 + t)^2 + s_(2) () t + s_(1)() s_(1)()t, ∙ e_1() e_1() (t + t^2) + e_2()(t + t^2) = s_(1) () s_(1)() t(1+t) +s_(1,1)() t(1+t),respectively and hence we have γ^+_2,0 (,) = s_(1,1)(), γ^+_2,1 (, ) =s_(2)() + s_(1)() s_(1)() andγ^-_2,1 (, ) = s_(1)() s_(1) () + s_(2)(), in agreement withCorollary <ref>.§ AN INSTANCE OF THE LOCAL EQUIVARIANT GALPHENOMENONThis section uses Equation (<ref>) toverify an equivariant analogue of Gal'sconjecture <cit.> for the local face moduleof a certain triangulation of the simplex withinteresting combinatorial properties. Backgroundand any undefined terminology on simplicialcomplexes can be found in <cit.>.To explain the setup, let V_n = {ε_1,ε_2,…,ε_n } be the set of unit coordinate vectors in ^n andΣ_n be the geometric simplex on the vertexset V_n. Consider a triangulation Γ ofΣ_n (meaning, a geometric simplicialcomplex which subdivides Σ_n) withvertex set V_Γ and the polynomial ringS = [x_v: v ∈ V_Γ] in indeterminateswhich are in one-to-one correspondence with thevertices of Γ. The face ring<cit.> of Γ is definedas the quotient ring [Γ] = S/I_Γ, where I_Γ is the ideal of S generated bythe square-free monomials which correspond to the non-faces of Γ. Following<cit.>, we consider the linear forms θ_i=∑_v ∈ V_Γ⟨ v, ε_i ⟩ x_v for i ∈ [n], where ⟨ ,⟩is the standard inner product on ^n, anddenote by Θ the ideal in [Γ]generated by θ_1,θ_2,…,θ_n.This sequence is a special linear system ofparameters for [Γ], in the sense of<cit.>. As a result, thequotient ring (Γ) = [Γ] / Θis a finite dimensional, graded -vector spaceand so is the local face module L_V_n(Γ), defined<cit.> as the image in(Γ) of the ideal of [Γ]generated by the square-free monomials whichcorrespond to the faces of Γ lying in the interior of Σ_n. The Hilbert polynomials∑_i=0^n _ ((Γ))_i t^i and ∑_i=0^n _ (L_V_n(Γ))_i t^iof (Γ) and L_V_n (Γ) are twoimportant enumerative invariants of Γ, namely the h-polynomial <cit.> and the local h-polynomial<cit.> <cit.>, respectively.Suppose that G is a subgroup of theautomorphism group _n of Σ_n which acts simplicially on Γ. Then, G acts onthe polynomial ring S and (as discussed on<cit.>) leaves the -linearspan of θ_1, θ_2,…,θ_ninvariant. Therefore, G acts on thegraded -vector spaces (Γ)and L_V_n (Γ) as well and thepolynomials ∑_i=0^n ((Γ))_i t^iand ∑_i=0^n (L_V_n(Γ))_i t^i, whosecoefficients lie in the representation ring of G, are equivariant generalizations of theh-polynomial and local h-polynomial ofΓ, respectively. The pair (Γ, G) is said (see also <cit.>) tosatisfy the local equivariant Galphenomenon if∑_i=0^n (L_V_n(Γ))_i t^i=∑_k=0^⌊ n/2 ⌋ M_k t^k(1+t)^n-2k for some non-virtual G-representations M_k. This is an analogue for local face modules ofthe equivariant Gal phenomenon, formulated byShareshian and Wachs <cit.> for group actions on (flag) triangulations of spheres as an equivariant version of Gal's conjecture <cit.>. For trivial actions on flag triangulations of simplices,the validity of the local equivariant Galphenomenon was conjectured in <cit.> andhas been verified in many special cases; see<cit.> <cit.> and references therein. Although it would be too optimistic to expect that the local equivariant Gal phenomenon holdsfor all group actions on flag triangulations ofΣ_n, the case G = _n deserves specialattention. We then use the notation ( (Γ), t ) :=∑_i=0^n( (Γ) )_it^i,( L_V_n(Γ), t ) :=∑_i=0^n( L_V_n(Γ) )_it^i. For the (first) barycentric subdivision ofΣ_n we have the following result of Stanley.(<cit.>)For the _n-action on the barycentricsubdivision Γ_n of the simplex Σ_n,we have 1 +∑_n ≥ 1 ( L_V_n(Γ_n), t ) z^n=1 - t/H(; tz) - tH(; z). Combining this result with Gessel's identity(<ref>) gives( L_V_n(Γ_n), t )=∑_k=0^⌊ (n-2)/2 ⌋ω ξ_n,k() t^k+1 (1 + t)^n-2k-2, where ω is the standard involution onsymmetric functions exchanging e_λ()and h_λ() for every λ, whenceit follows that (Γ_n, _n) satisfies thelocal equivariant Gal phenomenon for every n.The combinatorics of the barycentric subdivisionΓ_n is related to the symmetric group _n. We now consider a triangulation K_n of the simplexΣ_n, studied in <cit.> (seealso <cit.><cit.>) and shown on the right ofFigure <ref> for n=3, the combinatorics ofwhich is related to the hyperoctahedral group _n. The triangulation K_n can be defined as thebarycentric subdivision of the standard cubicalsubdivision of Σ_n, shown on the left ofFigure <ref> for n=3, whose faces are ininclusion-preserving bijection with the nonemptyclosed intervals in the truncated Boolean latticeB_n {∅}. Thus, the faces ofK_n correspond bijectively to chains of nonemptyclosed intervals in B_n {∅}and _n acts simplicially on K_n in theobvious way. As a simplicial complex, K_n can bethought of as a `half Coxeter complex' for _n.For the _n-action on K_n we have 1 +∑_n ≥ 1 ( (K_n), t ) z^n=H(; z) ( H(; tz) - tH(; z) )/H(; tz)^2 - tH(; z)^2 and 1 +∑_n ≥ 1 ( L_V_n(K_n), t ) z^n=H(; tz) - tH(; z)/H(; tz)^2 - tH(; z)^2. Moreover, the pair (K_n, _n) satisfies thelocal equivariant Gal phenomenon for every n. The proof relies on methods developed byStembridge <cit.> to study representations ofWeyl groups on the cohomology of the toric varieties associated to Coxeter complexes. To prepare for it,we recall that the h-polynomial of a simplicial complex Δ of dimension n-1 is defined ash(Δ, t)=∑_i=0^n f_i-1(Δ) t^i (1-t)^n-i, where f_i(Δ) stands for the number ofi-dimensional faces of Δ. Consider a pair (Γ, G), consisting of atriangulation Γ of Σ_n and asubgroup G of _n acting on Γ, asdiscussed earlier. Following <cit.>, we call the action of Gon Γ proper if w fixes allvertices of every face F ∈Δ which isfixed by w, for every w ∈ G. Note thatgroup actions, such as the _n-actions onΓ_n and K_n, on the order complex(simplicial complex of chains) of a poset whichare induced by an action on the poset itself,are always proper. Under this assumption, theset Γ^w of faces of Γ which arefixed by w forms an induced subcomplex ofΓ, for every w ∈ G.Although Stembridge <cit.> deals withtriangulations of spheres, rather than simplices,his methods apply to our setting and his Theorem 1.4, combined with the considerations of Section 6 in <cit.>, imply that ( (Γ), t )=1/n!∑_w ∈_nh(Γ^w, t)/(1 - t)^1+(Γ^w) ∏_i ≥ 1(1 - t^λ_i(w)) p_λ_i(w) () for every proper _n-action on Γ, whereλ_1(w) ≥λ_2(w) ≥⋯ are thesizes of the cycles of w ∈_n and p_k() is a power sum symmetric function.Proof of Proposition <ref>. To proveEquation (<ref>), we follow the analogouscomputation in <cit.> for thebarycentric subdivision of the boundary complex ofthe simplex. We first note that (K_n)^w is combinatorially isomorphic toK_c(w) for every w ∈_n, where c(w) isthe number of cycles of w. Furthermore, it wasshown in <cit.> that, in thenotation of Section <ref>, h(K_n, t) is the `half _n-Eulerian polynomial' B^+_n(t)=∑_w ∈^+_n t^|_B(w)|,where ^+_n consists of the signed permutations w ∈_n with negative first coordinate. These remarks and Equation (<ref>) implythat( (K_n), t ) z^n= ∑_λ = (λ_1, λ_2,…) ⊢ n m^-1_λ B^+_ℓ(λ)(t)/(1 - t)^ℓ(λ) ∏_i ≥ 1(1 - t^λ_i) p_λ_i ()z^λ_i, where n!/m_λ is the cardinality of theconjugacy class of _n which corresponds toλ⊢ n and ℓ(λ) is the numberof parts of λ. The polynomials B^+_n(t) are known (see, for instance,<cit.>) to satisfyB^+_n(t)/(1 - t)^n =∑_k ≥ 0( (2k+1)^n - (2k)^n ) t^kand hence, we may rewrite the previous formula as( (K_n), t ) z^n= ∑_k ≥ 0t^k ∑_λ = (λ_1, λ_2,…) ⊢ n m^-1_λ( (2k+1)^ℓ(λ) - (2k)^ℓ(λ)) ∏_i ≥ 1(1 - t^λ_i) p_λ_i ()z^λ_i . Summing over all n ≥ 1 and using the standardidentities H(; z)=∑_λ m^-1_λ p_λ () z^|λ| =exp(∑_n ≥ 1 p_n() z^n / n ) just as in the proof of <cit.> (one considers the p_n as algebraicallyindependent indeterminates and replaces first each p_n with (2k+1) (1 - t^n)p_n, then with (2k)(1 - t^n)p_n), we conclude that1 +∑_n ≥ 1 ( (K_n), t ) z^n = 1 +∑_k ≥ 0t^k( H(; z)^2k+1/H(; tz)^2k+1- H(; z)^2k/H(; tz)^2k)= 1 +( H(; z)/H(; tz)- 1 ) ( 1 - tH(; z)^2/H(; tz)^2)^-1=H(; z) ( H(; tz) - tH(; z) )/H(; tz)^2 - tH(; z)^2 and the proof of (<ref>) follows. To prove (<ref>), it suffices to observe that 1 +∑_n ≥ 1 ( L_V_n(K_n), t ) z^n=E(, -z) ( 1 +∑_n ≥ 1 ( (K_n), t ) z^n ). The latter follows exactly as the correspondingidentity for the barycentric subdivisionΓ_n, shown in the proof of<cit.>. Finally, from Equations (<ref>) and (<ref>) we deduce that ( L_V_n(K_n), t )=∑_k=0^⌊ n/2 ⌋ω ξ^+_n,k (, ) t^k (1 + t)^n-2k. This expression, Corollary <ref> and thewell known fact that s_λ() s_μ()is Schur-positive for all partitions λ, μ imply that ( L_V_n(K_n), t ) isSchur γ-positive for every n, as claimed in the last statement of the proposition. § AN INSTANCE OF THE EQUIVARIANT GALPHENOMENONA very interesting group action on a simplicialcomplex is that of a finite Coxeter group W onits Coxeter complex <cit.>. When Wis crystallographic, this action induces a gradedW-representation on the (even degree) cohomology ofthe associated projective toric variety which has beenstudied by Procesi <cit.>,Stanley <cit.>, Dolgachev andLunts <cit.>, Stembridge <cit.> and Lehrer <cit.>, among others. The graded dimension of this representation is equal to the W-Eulerianpolynomial. Its equivariant γ-positivity is a consequence of a variant of Equation (<ref>) in the case of the symmetric group _n; see<cit.> <cit.>. In thecase of the hyperoctahedral group _n, by<cit.> or <cit.>,the Frobenius characteristic of this graded_n-representation is equal to the coefficient ofz^n in(1-t) H(; z) H(; tz)/H(, ; tz) - tH(, ; z) . The following statement (and its proof) shows that theequivariant γ-positivity of this gradedrepresentation is a consequence of the results ofSection <ref> and confirms another instanceof the equivariant Gal phenomenon of Shareshian andWachs. As discussed in <cit.>, it isreasonable to expect that the same holds for theaction of any finite crystallographic Coxeter groupW on its Coxeter complex; that would provide anatural equivariant analogue to the γ-positivityof W-Eulerian polynomials <cit.>.The coefficient of z^n in (<ref>)is Schur γ-positive for every n ∈.Using Equations (<ref>) with (<ref>) we find that(1-t) E(; z) E(; tz)/E(, ; tz) - tE(, ; tz)=E(; z) E(; tz)(E(; tz) - tE(; z))/E(; tz)E(; tz) - tE(; z)E(; tz)·1-t/E(; tz) - tE(; z)=(∑_n ≥ 0 z^n∑_i=0^⌊ n/2 ⌋γ^+_n,i (, ) t^i (1 + t)^n-2i) ·( 1 +∑_n ≥ 2 z^n∑_j=1^⌊ n/2 ⌋ξ_n,j-1() t^j (1 + t)^n-2j) =∑_n ≥ 0 z^n ∑_k+ℓ=n∑_i, j γ^+_k,i (, ) ξ_ℓ,j-1()t^i+j (1+t)^n-2i-2j , where we have set ξ_0, -1() := 1 and ξ_1, -1 () := 0. Since Schur-positivity is preservedby sums, products and the standard involution on symmetricfunctions, this computation implies that the coefficient ofz^n in (<ref>) is Schur γ-positivefor every n ∈ and the proof follows.We have shown that (1-t) H(; z) H(; tz)/H(, ; tz) - tH(, ; z) =1+∑_n ≥ 1 z^n ∑_i=0^⌊ n/2 ⌋γ^B_n,i(,) t^i (1 + t)^n-2i for some Schur-positive symmetric functions γ^B_n,i (, ). It is an open problem to find an explicit combinatorial interpretation of the coefficient c^B_(λ, μ), i of s_λ() s_μ() inγ^B_n,i(, ), for (λ, μ) ⊢ n. Comparing the graded dimensions of the_n-representations whose Frobenius characteristic is given by the two sides of Equation (<ref>) weget B_n(t)=∑_(λ,μ) ⊢ nn|λ| f^λ f^μ∑_i=0^⌊ n/2 ⌋ c^B_(λ,μ), i t^i (1 + t)^n-2i , where B_n(t) := ∑_w ∈_n t^|_B(w)| is the_n-Eulerian polynomial and f^λ stands for thenumber of standard Young tableaux of shape λ. Thus,a solution to this problem would provide a refinement of theknown γ-positivity of _n-Eulerian polynomials;see <cit.>. Acknowledgements.The author wishes to thank Eric Katz and Kalle Karu for some helpful e-discussions about the localequivariant Gal phenomenon and the anonymous referees for their useful comments.99AAER17 R.M. Adin, C.A. Athanasiadis, S. Elizalde andY. Roichman, Character formulas and descents for thehyperoctahedral group, Adv. in Appl. Math. 87 (2017), 128–169.Ath12 C.A. Athanasiadis, Flag subdivisions and γ-vectors, Pacific J. Math. 259 (2012), 257–278.Ath14 C.A. Athanasiadis, Edgewise subdivisions, local h-polynomialsand excedances in the wreath productℤ_r ≀𝔖_n, SIAM J. Discrete Math. 28 (2014), 1479–1492.Ath16a C.A. Athanasiadis, A survey of subdivisions and local h-vectors, in The Mathematical Legacy of Richard P. Stanley (P. Hersh, T. Lam, P. Pylyavskyy, V. Reiner, eds.), Amer. Math. Society, Providence, RI, 2016, pp. 39–51.Ath17 C.A. Athanasiadis, Gamma-positivity in combinatorics and geometry, Sém. Lothar. Combin. 77 (2018),Article B77i, 64pp (electronic).Ath1x C.A. Athanasiadis, Binomial Eulerian polynomials for coloredpermutations, arXiv:1812.00434.Bj84 A. Björner, Some combinatorial and algebraic propertiesof Coxeter complexes and Tits buildings, Adv. Math. 52 (1984), 173–212.BW05 A. Björner and V. Welker, Segre and Rees products of posets, withring-theoretic applications, J. Pure Appl. Algebra 198 (2005), 43–55.DL94 I. Dolgachev and V. Lunts, A character formula for the representation ofa Weyl in the cohomology of the associated toricvariety, J. Algebra 168 (1994), 741–772.Ga05 S.R. Gal, Real root conjecture fails for five- andhigher-dimensional spheres, Discrete Comput. Geom. 34 (2005), 269–284.Le08 G.I. Lehrer, Rational points and Coxeter group actionson the cohomology of toric varieties, Ann. Inst. Fourier (Grenoble) 58 (2008), 671–688.LSW12 S. Linusson, J. Shareshian and M.L. 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Wachs, From poset topology to q-Eulerian polynomials to Stanley's chromatic symmetric functions, in The Mathematical Legacy of Richard P. Stanley (P. Hersh, T. Lam, P. Pylyavskyy and V. Reiner, eds.), pp. 301–321, Amer. Math. Society, Providence, RI, 2016.SW17 J. Shareshian and M.L. Wachs, Gamma-positivity of variations of Eulerian polynomials, arXiv: 1702.06666v3.Sta82 R.P. Stanley,Some aspects of groups acting on finite posets,J. Combin. Theory Series A 32 (1982), 132–161.Sta89 R.P. Stanley, Log-concave and unimodal sequences in algebra,combinatorics, and geometry, in Graph Theory and its Applications: Eastand West, Annals of the New York Academy of Sciences 576, New York Acad. Sci., New York, 1989,pp. 500–535.Sta92 R.P. Stanley, Subdivisions and local h-vectors, J. Amer. Math. Soc. 5 (1992), 805–851.StaCCA R.P. Stanley, Combinatorics and Commutative Algebra, Progress in Mathematics 41, second edition, Birkhäuser, Boston, 1996.StaEC1 R.P. Stanley, Enumerative Combinatorics, vol. 1, Cambridge Studies in Advanced Mathematics 49, Cambridge University Press, second edition,Cambridge, 2011.StaEC2 R.P. Stanley, Enumerative Combinatorics, vol. 2, Cambridge Studies in Advanced Mathematics 62, Cambridge University Press, Cambridge, 1999.Ste92 J.R. Stembridge,The projective representations of the hyperoctahedral group,J. Algebra 145 (1992), 396–453.Ste94 J.R. Stembridge, Some permutation representations of Weylgroups associated with the cohomology of toricvarieties, Adv. Math. 106 (1994), 244–301.Su94 S. Sundaram,The homology representations of the symmetricgroup on Cohen–Macaulay subposets of the partitionlattice,Adv. Math 104 (1994), 225–296.Wa07 M. Wachs, Poset Topology: Tools and Applications,in Geometric Combinatorics(E. Miller, V. Reiner and B. Sturmfels, eds.), IAS/Park City Mathematics Series 13, pp. 497–615, Amer. Math. Society, Providence,RI, 2007. | http://arxiv.org/abs/1707.08297v3 | {
"authors": [
"Christos A. Athanasiadis"
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"primary_category": "math.CO",
"published": "20170726063517",
"title": "Some applications of Rees products of posets to equivariant gamma-positivity"
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On a possibility of baryonic exoticaPresentedat the2nd Jagiellonian Symposium on Fundamental and Applied Subatomic Physics, June 3-11, 2017, Kraków, Poland. Michał PraszałowiczM. Smoluchowski Institute of Physics, Jagiellonian University, ul. S. Łojasiewicza 11, 30-348 Kraków, Poland.================================================================================================================================================================ Models based on chiral symmetry predict pentaquarks that have relatively low masses.We briefly review both theoreticaland experimental status of exotica in the light sector. Next,shall show how to extend chiral models to baryons with one heavy quark and show that one expects exotica also in this case. Finally, weinterpret recently discovered by the LHCb Collaboration five Ω^*_c resonances in terms of regular and exotic excitations of the ground state Ω_c. 14.20.Lq, 12.38.Lg, 12.39.Hg§ INTRODUCTION: CHIRAL QUARK-SOLITON MODELThe Chiral Quark-Soliton Model (χQSM) is based on an old argument by Witten <cit.>, which says that in theN_c →∞ limit(N_c stands for number of colors), N_c relativistic valence quarks generate chiral mean fields represented by a distortion of a Dirac sea that in turninteract with the valence quarks themselves (fora review see Ref.<cit.>).In this way, a self-consistent configuration called a soliton is formed. In Fig. <ref> (a) we plotschematic pattern of light quark energy levels corresponding to this scenario. It is assumed that the mean fields exhibit so called hedgehog symmetry, which means thatneither quark spin (S_q) nor quark isospin (T_q)are "good" quantum numbers. Instead a grand spin K=S_q+T_q is a "good" quantum number. The lowest valence level has K^P=0^+.In order to project out spin and isospin one has to rotate the the soliton, both in flavor and configuration spaces. These rotations are then quantized semiclassically and the collective Hamiltonianis computed.The model predicts rotationalbaryon spectra that satisfy the following selection rules:* allowed SU(3) representations must contain states with hypercharge Y^'=N_c/3,* the isospin T^' of the states with Y^' =N_c/3 couples with the soliton spin J to a singlet: T^'+J=0.In the case of light positive parity baryons the lowest allowed representations are 8 of spin 1/2, 10 of spin 3/2, and also exotic 10 of spin 1/2 with the lightest state corresponding to the putative Θ^+(1540). They are shown in Fig. <ref>.Chiral models in general predict that pentaquarks are light<cit.> and – in some specific models – narrow <cit.>.Afterthe first enthusiastic announcements of the discovery of pentaquarks in 2003by LEPS <cit.> and DIANA <cit.> collaborations, the experimental evidence for the light exotica has been questioned (see e.g. <cit.>). Nevertheless, both DIANA <cit.> and LEPS <cit.> upheld theiroriginal claims after performing higher statistics analyses. The report on exotic Ξ states (see Fig. <ref>) by NA49 <cit.> from 2004, to the best of my knowledge, has not been questioned so far,however the confirmation is still strongly needed.Another piece of information on 10 comes from the η photo-production off the nucleon.Different experiments confirm the narrow structure at the c.m.s. energy W ∼1.68 GeV observed in the case of the neutron, whereas no structure is observed on the proton (see Fig. 27 in the latest report by CBELSA/TAPS Collaboration <cit.>andreferences therein). The natural interpretation of this "neutron puzzle" was proposed already in 2003 in Ref. <cit.>. There one assumes that the narrow excitation at W ∼1.68 GeV corresponds to the non-exotic penta-nucleon resonance belonging to 10. Indeed, the SU(3) symmetry forbids photo-excitation of the proton member of 10, while the analogous transition on the neutron is possible. This is due to the fact that photon is an SU(3)U-spin singlet, and the U-spin symmetry is exact in the SU(3) symmetric limit. An alternative interpretation is based on a partial wave analysis in terms ofthe Bonn-Gatchina approach <cit.>.There is an ongoing dispute on the interpretation of the "neutron puzzle" (for the latest arguments see Ref. <cit.>).§ HEAVY BARYONS IN THE CHIRAL QUARK-SOLITON MODELIn a recent paper <cit.> following <cit.> we have extended the χQSM to baryons involving one heavy quark. In this case the valence level is occupied by N_c-1 light quarks (see Fig <ref> (b)) that couple with a heavy quark Q to form a color singlet. The lowest allowed SU(3) representations are shown in Fig. <ref>. They correspond to the soliton in representation in 3 of spin 0andto 6 of spin 1. Therefore, the baryons constructedfrom such a soliton and a heavy quark form an SU(3) antitriplet of spin 1/2 and two sextetsof spin 1/2 and 3/2 that are subject to a hyper-fine splitting. The next allowed representation of the rotational excitations corresponds to the exotic 15 of spin 0 or spin 1 <cit.>. The spin 1 soliton haslower mass and when it couples with a heavy quark, it forms spin 1/2 or 3/2 exoticmultiplets that should be hyper-fine split similarly to the ground state sextets by ∼ 70 MeV.The rotational states described abovecorrespond to positive parity and are clearly seen in the data <cit.>. Negative parity states are generated by soliton configurations with one lightquark excited to the valence level from the Dirac sea (Fig. <ref> (c)).The selection rules for excited quark solitons can besummarized as follows <cit.>: * allowed SU(3) representations must contain states with hypercharge Y^'=(N_c-1)/3,* the isospin T^' of the states with Y^' =(N_c-1)/3 couples with the soliton spin J as follows: T^'+J=K, where K is the grand spin of the excited level. The first allowed SU(3) representation for one quark excited soliton is again 3, Fig. <ref>, with T^'=0, whichfor K=1 is quantized as spin 1. The coupling of a heavy quark results in two hyperfine split antitriplets that are indeed seen in the data <cit.>. The hyperfine splitting parameter is in this case κ^'/m_c∼ 30 MeV. Next possibility is flavor 6 with T^'=1, which may couple with K=1 to J=0,1 and 2 resulting in 5 hyperfine split heavy sextets: two 1/2^-, two 3/2^- and one 5/2^- (see Tab. <ref>).§ POSSIBLE INTERPRETATION OF THE LHCB Ω_C^0 RESONANCESIn a very recent paper the LHCb Collaboration announced five Ω^0_c states with masses in the range of 3 - 3.2 GeV <cit.>. The simplest possibility would be to associate them with the five sextets describedat the end of Sect. <ref>. We have shown, however, in <cit.> that this scenario fails, as can be seen fromTable <ref>.In the second scenario proposed in <cit.>, we have interpreted three LHCb states as quark excitations of the ground state sextets, shown in Fig. <ref> as vertical lines. Two remaining sextetexcitations have higher mass and are above the threshold for the decays into charm mesons. They can be, therefore, wide and the branching ratio to Ξ^+_c +𝐾^- final state may be small. This would explain why they are not seen by LHCb. On the other hand, two remaining Ω_c^0 peaks are in this scenario interpreted as rotational excitations corresponding to the exotic 15. As such, they are isospin triplets and should decaynot only to Ξ^+_c + 𝐾^- but also toΞ_c^0+𝐾^- or Ξ_c^+ + 𝐾̅^0and Ω_c+π final states. This scenario is,therefore, very easy to confirm or falsify. Moreover, they are very narrow with widths around 1 MeV,and the χQSM provides a mechanism that suppresses pentaquark decays both in the light sector andin the present approach to heavy baryons <cit.>. Summarizing, let us stress that despite many "null findings" there is still an experimental support forlight and narrow pentaquarks. Using the ideas of the χQSM, we have proposed an interpretationof recently discovered Ω_c^0 states in terms of quark and and rotational excitations of theground state charmed baryons, the latter corresponding to the pentaquarks.§ ACKNOWLEDGEMENSThis note is based on Refs. <cit.> where more complete list of references can be found. I would like to thank H.C. Kim, M.V. Polyakov and G.S. Yang for a fruitful collaboration. 99 Witten:1979khE. Witten,Nucl. Phys. B 160, 57 (1979) andNucl. Phys. B 223 (1983) 422 and Nucl. Phys. B 223 (1983) 433. Christov:1995vmC. V. Christov, A. Blotz, H.-Ch. Kim, P. Pobylitsa, T. Watabe, T. Meissner, E. Ruiz Arriola and K. Goeke,Prog. Part. Nucl. Phys. 37 (1996) 91. Praszalowicz:2003ik M. Praszalowicz,Phys. Lett. B 575 (2003) 234.Diakonov:1997mm D. Diakonov, V. Petrov and M. V. Polyakov,Z. Phys. A 359 (1997) 305.Nakano:2003qx T. Nakano et al. [LEPS Collaboration],Phys. Rev. Lett.91 (2003) 012002. Barmin:2003vv V. V. Barmin et al. [DIANA Collaboration],Phys. Atom. Nucl.66 (2003) 1715[Yad. Fiz.66 (2003) 1763]. Hicks:2012zz K. H. Hicks,Eur. Phys. J. H 37 (2012) 1. Barmin:2013lva V. V. Barmin et al. [DIANA Collaboration],Phys. Rev. C 89 (2014) no.4,045204. Nakano:2017fui T. Nakano [LEPS and LEPS2 Collaborations],JPS Conf. Proc.13 (2017) 010007.Alt:2003vb C. Alt et al. [NA49 Collaboration],Phys. Rev. Lett.92 (2004) 042003.Witthauer:2017pcy L. Witthauer et al. [CBELSA/TAPS Collaboration],Eur. Phys. J. A 53 (2017) no.3,58. Polyakov:2003dx M. V. Polyakov and A. Rathke,Eur. Phys. J. A 18 (2003) 691. Anisovich:2017xqg A. V. Anisovich, V. Burkert, E. Klempt, V. A. Nikonov, A. V. Sarantsev and U. Thoma,Phys. Rev. C 95 (2017) no.3,035211.Kuznetsov:2017qmo V. Kuznetsov et al.,JETP Lett.105 (2017) 625.Yang:2016qdz G. S. Yang, H. C. Kim, M. V. Polyakov and M. Praszalowicz,Phys. Rev. D 94 (2016) 071502. Diakonov:2010tf D. Diakonov,arXiv:1003.2157 [hep-ph].Kim:2017jpx H. C. Kim, M. V. Polyakov and M. Praszalowicz,Phys. Rev. D 96 (2017) no.1,014009. Petrov:2016vvl V. Petrov,Acta Phys. Pol. B 47 (2016) 59.Aaij:2017nav R. Aaij et al. [LHCb Collaboration],Phys. Rev. Lett.118 (2017) no.18,182001. | http://arxiv.org/abs/1707.08474v2 | {
"authors": [
"Michal Praszalowicz"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170726144535",
"title": "On a possibility of baryonic exotica"
} |
APS/123-QEDPhotonic Research Centre, Department of Electronic and Information Engineering, the Hong Kong Polytechnic University, Hung Hom, Hong Kong SAR, China The Hong Kong Polytechnic University Shenzhen Research Institute, Shenzhen, 518057, China School of Engineering, Fraser Noble Building, University of Aberdeen, Aberdeen AB24 3UE, UK Department of Applied Mathematics, University of Washington, Seattle, WA 98195-2420, USA [email protected] Photonic Research Centre, Department of Electronic and Information Engineering, the Hong Kong Polytechnic University, Hung Hom, Hong Kong SAR, China State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, P.O. Box 72, 100876 Beijing, China Photonic Research Centre, Department of Electronic and Information Engineering, the Hong Kong Polytechnic University, Hung Hom, Hong Kong SAR, China Photonic Research Centre, Department of Electronic and Information Engineering, the Hong Kong Polytechnic University, Hung Hom, Hong Kong SAR, China [email protected] Photonic Research Centre, Department of Electronic and Information Engineering, the Hong Kong Polytechnic University, Hung Hom, Hong Kong SAR, China The Hong Kong Polytechnic University Shenzhen Research Institute, Shenzhen, 518057, China High frequency fluctuation in the optical signal generated in Fourier-Domain Mode Locked fiber laser (FDML-FL), which is the major problem and degrades the laser performance, is not yet fully analyzed or studied. The basic theory which is causing this high frequency fluctuation is required to clearly understand its dynamics and to control it for various applications. In this letter, by analyzing the signal and system dynamics of FDML-FL, we theoretically demonstrate that the high frequency fluctuation is induced by the intrinsic instability of frequency offset of the signal in cavity with nonlinear gain and spectral filter. Unlike the instabilities observed in other laser cavities this instability is very unique to FDML-FL as the central frequency of the optical signal continuously shifts away from the center frequency of the filter due to the effects like dispersion and/or nonlinearity. This instability is none other than the Eckhaus instability reported and well studied in fluid dynamics governed by real Ginzburg-Landau equation.Eckhaus Instability in the Fourier-Domain Mode Locked Fiber Laser Cavity P. K. A. Wai December 30, 2023 ========================================================================Nonlinear systems either naturally existing or manmade exhibit fascinating dynamics and have various applications in different fields. Development of appropriate and complete theoretical models of these nonlinear systems play vital role in understanding their dynamics and to control them for various applications. Nonlinear systems are governed either by nonlinear ordinary differential equations or nonlinear partial differential equations (NPDEs). Some NPDE governs more than one kind of systems. Complex Ginzburg Landau equation (CGLE) is one such very famous NPDE which governs nonlinear dynamical systems from different fields like fluids, optics, superconductivity and Bose-Einstein Condensate. The derivatives of CGLE like real Ginzburg Landau equation (RGLE) and the family of nonlinear Schrödinger equation (NLSE) also widely appear as system model equations across different fields. As a system dynamical equation governs more than one type of systems it became very common to utilize the results, studies and analyzes obtained in one field to the corresponding other fields systems governed by the same model equation. This helps in understanding the dynamics of the systems governed by the same dynamical equation at a faster pace and also to make further appropriate modifications and/or to achieve complete theoretical model.In this work we consider the Fourier-Domain Mode Locked fiber laser (FDML-FL) cavity which is a long cavity wavelength swept laser source and has a very important application in optical coherent tomography (OCT) <cit.>. FDML-FL was experimentally demonstrated for the first time in 2005 <cit.>. To avoid the rebuilding of laser signal from the spontaneous emission, which intrinsically limits the sweeping speed of wavelength swept lasers, a long fiber delay line was introduced into the cavity of FDML-FL to buffer the entire wavelength sweeping signal. With this kind of FDML-FL cavity, the wavelength sweeping speed can be enhanced by one to two orders to MHz level <cit.>. Although FDML-FL has been successfully deployed in the OCT systems as the swept source, the performance of the FDML-FL is limited by its large instantaneous linewidth <cit.>, which is in fact the high frequency fluctuations of the signal waveform. In order to improve the performance and to understand the dynamics of the FDML-FL, in this letter using an appropriate theoretical model we report the complete working mechanism and the intrinsic reason of such fluctuations in the FDML-FL cavity.In an FDML-FL, the fiber dispersion degrades the quality of the signal due to the mismatch between the filter sweeping period and the different round trip time of different parts of the sweeping signal in the cavity <cit.>. The linewidth of the signal is further increased by the nonlinearity of the fiber and the linewidth enhancement factor of the semiconductor optical amplifier (SOA) <cit.>. To investigate the dynamics of the signal in the FDML-FL cavity, a theoretical model in a co-moving frame of the filter has been proposed as <cit.>∂_zu=g(u,ω_s)(1-iα)u-σ (ω _s)u-a( i∂_t )u +iD_2ω _s^2(t)u+iD_3ω _s^3(t)u+iγ| u |^2u-iD_2∂ _t^2u, where u is the amplitude in filter frame defined as u=Aexp(i∫^tω_s(t')dt'), ω_s is the instantaneous center frequency of the sweeping filter and A is the complex amplitude of the signal in lab frame. This model includes the effects such as dispersion, nonlinearity, linewidth enhancement factor and frequency filtering. Equation (<ref>) is widely used in the numerical simulation of FDML-FL. Through numerical simulations of the FDML-FL, there are always high frequency fluctuations in the signal waveform<cit.>. These fluctuations appear predominantly on the desired signal and hence usually the waveform is artificially smoothened over a long period of time scale <cit.>. To understand the intrinsic reason for this like noise fluctuation, we use a simplified model. Utilizing the Wentzel–Kramers–Brillouin (WKB) analysis <cit.>, we find that the nonlinear phase shift and in-band dispersion are playing minor role and those terms can be neglected in the first order approximation <cit.>. The large linewidth enhancement factor, which is a special effect of SOAs, can also be neglected in a general laser cavity. Also neglecting the wavelength dependence of the gain and considering a typical Gaussian spectral filter with bandwidth B, Eq. (<ref>) becomes∂_zu=gu-σ u+12B^-2∂ _t^2u+iD_2ω _s^2(t)u+iD_3ω _s^3(t)u, where g(u)=g_0/(1+|u|^2/I_sat) is the saturated gain. It should be noted that such fast response gain saturation model is valid not only in the case with a fast recovery gain, but also in the cavity with a slow recovery gain and a fast response nonlinear loss element such as nonlinear optical loop mirror, nonlinear polarization rotation device. The simulation results of the simplified model of Eq. (<ref>) capture most of the signal dynamics of the FDML-FL system, especially the high frequency intensity fluctuation of the waveform. Now, the gain saturation factor is expressed in Taylor series of |u|^2 and keeping only the first order term, Eq. (<ref>) reduces to∂_zu=(g_0-σ)u-g_0I_sat^-1|u|^2u+12B^-2∂_t^2u +iD_2ω _s^2(t)u+iD_3ω _s^3(t)u, which can be normalized to∂_ZU=U-|U|^2U+∂ _T^2 U+iϵ^-1C(ϵ T)U, where C(ϵ T)=ϵ[ S_2Ω _s^2(T)+S_3Ω _s^3(T) ] is the combined phase term caused by the dispersion and S_2, S_3 are the normalized dispersion coefficients. The time scaling factor ϵ is defined as the inverse of the round trip time of the laser cavity. Equation (<ref>) is a RGLE with a chirp phase term C contributed by the dispersion in the FDML-FL cavity. In an ideal cavity without dispersion, Eq. (<ref>) will reduce to a standard RGLE as∂_ZU=U-| U |^2U+∂ _T^2U. RGLE has been extensively studied in fluid dynamics. Most importantly, a set of stationary solutions are available for the system governed by the RGLE which are single frequency continuous waves. The stationary solution with normalized angular frequency Ω isU=√(1-Ω ^2)e^-iΩ T, which are nontrivial in the frequency region |Ω|<1. But the stationary solutions are unstable when Ω^2>1/3, which is known as Eckhaus instability. Eckhaus instability was first discussed in the modeling of convection in fluidic systems governed by RGLE in 1960s <cit.>. After a very short time, the interest on the instability has moved to the system described by the CGLE, which is the modulation instability, where the Eckhaus instability can be treated as a reduced case with zero imaginary terms <cit.>. As a reduced form of CGLE, NLSE has attracted more attention than the RGLE because of the existence of the analytical solitary solution, especially after soliton was reported in optical fibers by Hasegawa and Tappert <cit.>. With NLSE, the modulation instability of optical continuous wave has been very well studied. In a system with dissipative and gain elements, such as a laser cavity, the more general system equation, CGLE is adapted to model the nonlinear pulse dynamics in the cavity <cit.>. But in all those studies, the nonlinear phase shift and dispersion were considered as the dominant effects in pulse shaping and stable propagation. Although RGLE is seldom used to describe an optical system, Eckhaus instability has also been discussed when considering the spatial effects in lasers <cit.>. In an one dimensional cavity, where spatial effects are not considered, the Eckhaus instability never appeared as a dominating effect.In the laser cavities described by Eq. (<ref>), there are no stationary solutions because of the existence of a single nonzero phase term. Only when an initial field is given, the evolution of the signal can be solved for Eq. (<ref>). To solve Eq. (<ref>), we introduce an amplitude-phase form U̅=Uexp [-iϵ^-1C(ϵ T)Z] and split the temporal dynamics by fast time t_1=T and slow time t_2=ϵ T, then the zero-th order governing equation of U̅ can be obtained as∂_ZU̅=U̅-| U̅|^2U̅+∂ _t_1^2U̅-C'^2Z^2U̅+2iC'Z∂_t_1U̅, where C'=∂_ϵ TC. If a slowing varying signal is considered, the evolution of the signal along Z governed by Eq. (<ref>) can be written asU̅(Z)=exp ( -iΩ̅_0(t_2)t_1 )/√(I_0^-1e^Q(0)-Q(Z)+2∫_0^Ze^Q(x)-Q(Z)dx), where I_0 and Ω̅_0(t_2) are the intensity and instantaneous frequency ofU̅ at Z=0, and Q(Z)=2∫_0^Z[ 1-(Ω̅_0(t_2)-C'x)^2 ]dx. The stability of the solution (<ref>) can be investigated through linear stability analysis. A perturbation a(t_1,t_2) is applied to the solution at Z = Z_0 as W(Z_0)=U̅(Z_0)(1+a). The evolution of the perturbed solution is assumed asW(Z,t_1,t_2)=U̅(Z,t_1,t_2)[1+e^Λ (Z,t_2)a(t_1,t_2)], where ∂_ZΛ =λ (Z,t_2) indicates the growing speed of the perturbation. If λ (Z>Z_0,t_2)>0, the perturbation at time point t_2 gets continuously amplified and the solution becomes unstable. By substituting Eq. (<ref>) into Eq. (<ref>), and using the solution described by Eq. (<ref>), where ∂_t_1U̅=-iΩ̅_0(t_2)U̅, the governing equation ofthe perturbation is written asλ a=-I( a+a^*)+∂ _t_1^2a+2i Ω_i∂_t_1a, where I is the intensity and Ω_i=C'Z-Ω̅_0(t_2) is the instantaneous frequency of U. Considering the coupling between the conjugated fields, the field is written asa(t_1,t_2)=α_k(t_2)exp (-ikt_1)+β_k(t_2)exp (ikt_1), where k>0 is the mode number of the perturbation. Substituting Eq. (<ref>) into Eq. (<ref>), it is easy to find thatα_k and β_k have nonzero solutions only when| I+k^2+2kΩ_i+λI+λ I+λI+k^2-2kΩ_i+λ |=0, which has solutions k=0, or λ =2Ω _i^2-I-0.5k^2 for k>0. Clearly, λ will have negative values for all k>0 modes only when 2Ω _i^2-I<0, which is the criterion of the stability of the solution. With the solution described by Eq. (<ref>), the stability condition isλ_max=2( C'Z-Ω̅_0)^2 -[ I_0^-1e^Q(0)-Q(Z)+2∫_0^Ze^Q(x)-Q(Z)dx]^-1<0. As the simplest case, where C'=0, solution (<ref>)reduces to a quasi-stationary solutionU̅_0=√(1-Ω̅_0^2(t_2))exp[ -iΩ̅_0(t_2)t_1]. Also, Q is simplified to Q(Z)=2Z[ 1-Ω̅_0^2(t_2) ], and stability criterion of the eigenvalue changes to λ_max=3Ω̅_0^2(t_2)-1<0, which is exactly the condition for Eckhaus instability. Especially when Ω̅_0(t_2) is a constant, the solution becomes the stationary solution of Eq. (<ref>). Such stationary solution can be found in laser cavities modeled by RGLE. A nonzero relative frequency offset Ω̅_0 can be introduced either by a frequency shifter, or a fast tuning of the spectral filter in the cavity. The stability of the stationary signal depends on the offset frequency Ω̅_0, which can be divided into three regions, Ω̅_0^2<1/3, 1/3≤Ω̅_0^2≤ 1 and Ω̅_0^2>1. The frequencies Ω̅_0^2=1/3 and Ω̅_0^2=1 stand for the critical point to trigger the Eckhaus instability and the threshold frequency of positive net gain, respectively.Figure <ref> shows the spectra of signals with frequencies in the three different regions. In Fig. <ref>(a), the frequency Ω̅_0=0.5, which is in the stable region of Ω̅_0^2<1/3, the amplitude of the signal is -0.866 dB. The solution is stable and the sidebands have not grown at Z=50000. When Ω̅_0 increases to 0.6, which has passed the critical point Ω̅_0^2=1/3, as shown in Fig. <ref>(b), sidebands are generated on both sides of the signal at Z=3000. In this region, although the signal is unstable, the original signal decays very slowly until the sidebands grow to relatively large value. To explain the mechanism of the growth of sidebands and frequency switching, we show the spectral evolution of the signal in Fig. <ref>(d). Note that the signal switching occurs in a very short distance where higher order sidebands are also excited. Eventually, the sideband at the low frequency side, which experiences higher net gain, becomes the dominant mode and replaces the original single frequency signal. The new signal is stable as it is in the region of Ω̅_0^2<1/3. When Ω̅_0 is further increased to Ω̅_0^2>1, as shown in Fig. <ref>(c), the solution defined in Eq. (<ref>) does not exist anymore and the single frequency signal gets quickly attenuated. At the same time, new signal builds up from noise.In the FDML-FL cavity, the dispersion will introduce a time varying chirp to the signal. Before considering the dynamic chirp caused by the C'Z term, we first consider a signal with a stationary sinusoidal chirp profile Ω̅_0(t_2)=Ω̅_c-Ω̅_m×cos (2πt_2) in a cavity with C'=0, where Ω̅_c=0.4 and Ω̅_m=0.2 are the center and amplitude of the frequency modulation. Since Ω̅_0(t_2) is fully within the region of Ω̅_0^2<1, the solution Eq. (<ref>) is still valid. The satisfaction of stable criterion λ_max=3Ω̅_0^2(t_2)-1<0 depends on the value of Ω̅_0(t_2) at each individual temporal point t_2. Here, the maximum value of Ω̅_0^2(t_2) is 0.36 and larger than 1/3, the portion of the signal with t_2∈(0.4235,0.5765) falls within the region of Ω̅_0^2>1/3 and becomes unstable. Figure <ref>shows the spectrograms of the signal at different Z. The spectrograms are generated with a moving Chebyshev gating function applied to the signal in the time domain.At Z=5000, distinct sidebands have already formed at the extremum points of the frequency Ω̅_0. From Z=5000 to 7000, higher order sidebands are quickly generated and widely spread out on the temporal waveform. After Z=7000, a new signal with a frequency lower than the original signal is generated and becomes dominant in the section as the Eckhaus instability got triggered. Eventually, the signal in the whole unstable section is replaced by thus formed new signal in the stable region and the higher order sidebands got totally suppressed in the unstable region. During this entire dynamics, the signal initially in the stable region remains unaffected. From the evolution of the solution (<ref>) shown in Figs. <ref>–<ref>, it is clear that the Eckhaus instability plays a vital role in the dynamics of the signal with either a single frequency offset or a stationary frequency modulation. The portion of the signal with frequencies in the regionΩ̅_0^2>1/3 becomes unstable and replaced by a new signal in the stable region. But in a realistic FDML-FL, the chirp profile of the signal is not fixed but varying continuously along the propagation, which corresponds to nonzero C'. When C' is nonzero, the system dynamics becomes more complex since the solution should be given by Eq. (<ref>). From Ω_i=C'Z-Ω̅_0, if C' is nonzero, Ω_i will monotonically increase or decrease with the increase of Z. Such monotonic variation of Ω_i will inevitably push the signal to unstable region. The evolutions of the intensity of the solution Eq. (<ref>) and λ_max described by Eq. (<ref>) are shown in Fig. <ref>. The intensity and λ_max are plotted against the instantaneous frequency Ω_i since it is proportional to Z when C' ≠ 0, and identical to Ω̅_0 when C'=0. For the curves of C' ≠ 0, the initial signal at Z=0 are assumed to have Ω̅_0=0 and I_0=1. The dashed curves with C'=0 are for the solution Eq. (<ref>) and the corresponding stability factor. Figure <ref>(a) shows that for a given value of Ω_i, the intensity of the signal will increase with the increasing of C' especially for higher values of Ω_i. When C'<0.01, the difference between the dynamically varying chirped solution and the stationary chirped solution indicated by the dashed curve is very small especially in the stable region. In contrast, the intensity curves of higher values of C' deviates well away from the dashed curve. Besides the deviation of the intensity trace, the stability condition is also affected by the nonzero C', as shown in Fig. <ref>(b). When C' increases from 0 to 2, the critical point of the instability has been pushed from Ω_i=0.577 to Ω_i =0.677. When C'<0.01, the critical point is almost fixed to 0.577 which is accordant to the Eckhaus instability. To investigate the signal dynamics and the instability in FDML-FL cavities, we consider a practical FDML-FL cavity with a length of 1 km and a round trip time of 5 μs. The dispersion coefficients of the fiber are D_2=-1 ps^2/km and D_3=0.02 ps^3/km. A sweeping Gaussian filter with a bandwidth of B=0.04 ps^-1 is driven by a sinusoidal signal ω_s(t)=ω_mcos (2πf_0t) with repetition rate f_0=200 kHz and sweeping frequency range 2ω_m=80 ps^-1. The gain and loss coefficients of the cavity are g_0=2.5 km^-1 and σ=0.5 km^-1, respectively. The saturation power of the gain element is 1 mW. Then the normalized parameters are U=25u, Z=2z, Ω =12.5ω, t_1=0.08t, ϵ =2.5× 10^-6, t_2=ϵ t_1, andC(t_2)=-1.6×10^-4×[cos^2(2π t_2)-0.8 cos^3(2π t_2) ],C'(t_2)=10^-4× 3.2πsin (4π t_2) [1-1.2cos (2π t_2)], where the maximum of |C'| is 0.0019. With these normalized parameters, the dynamics of the signal can be simulated using Eq. (<ref>).We start the simulation of Eq. (<ref>) from a CW signal with Ω̅_0=0. During the propagation along Z, the frequency shift accumulates and continuously increases the swing range of the curve on the spectrogram, as shown in Fig. <ref>. In the range of Z<300, the spectrogram of the signal is always smooth since the entire signal is confined within the stable region of Ω^2<1/3. Once the peak points of the signal cross the threshold points of Ω^2=1/3 at Z ≈ 303, sidebands start to grow in the area just outside the stable region but it is too low to be observed during early stages. At Z=400, the sidebands are clearly visible. As the unstable portion of the signal continuously expands, more and more higher order sidebands are also generated. At the same time, new signal generated in the stable region are as well frequency shifted towards the unstable region by the effect of dispersion. Eventually, the new signal will exit the stable region and suffer from its own Eckhaus instability. This mechanism gets repeated for the generation of new signals in the stable region, getting frequency shifted to the unstable region because of the dispersion effect and thus vanishing due to Eckhaus instability, severely distort the laser signal. After many cycles inside the cavity, e.g. at Z=20000, the signal almost in the entire region becomes very noisy. Most of the signal energy gets distributed near the boundary of the stable region, which means the signal experiences a very high loss when passing through the filter. The frequency of the signal is completely dispersed even for the part of the signal located at the same side of the filter. This is a great limitation to the instantaneous linewidth of FDML-FL. It should be noted that the direction of frequency shift is determined by the sign of C'. Thus the signals with different sign of C' cluster on different sides of the filter. In this letter, we studied the Eckhaus instability in FDML-FL. We found that the FDML-FL can be modeled by a RGLE with a frequency shifting term, which is due to the dispersion of the fiber. The fast recovering gain saturation provides a nonlinear loss to the signal. We have derived the analytical solution for the system equation and analyzed its stability. In FDML-FL, the dispersion introduces a continuous frequency shift C' to the signal, which will unavoidably push the frequency outside the stable region of Ω^2<1/3. If C' is large, the stable region on frequency domain will be slightly enlarged. By considering practical parameter values, we numerically showed the repeatedly triggering of Eckhaus instability in FDML-FL cavities by the endless frequency shifting. Such mechanism is the root cause for the high frequency fluctuations of the signal that limits the signal quality of the FDML-FL. We acknowledge the support of Research Grant Council of Hong Kong SAR (PolyU5263/13E, PolyU152144/15E), The Hong Kong Polytechnic University (1-ZVGB), National Science Foundation of China (NSFC) (61475131) and Shenzhen Science and Technology Innovation Commission (JCYJ20160331141313917). | http://arxiv.org/abs/1707.08304v1 | {
"authors": [
"Feng Li",
"K. Nakkeeran",
"J. Nathan Kutz",
"Jinhui Yuan",
"Zhe Kang",
"Xianting Zhang",
"P. K. A. Wai"
],
"categories": [
"physics.optics"
],
"primary_category": "physics.optics",
"published": "20170726070138",
"title": "Eckhaus Instability in the Fourier-Domain Mode Locked Fiber Laser Cavity"
} |
Physik-Institut, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, SwitzerlandSwiss Light Source, Paul Scherrer Institut, CH-5232 Villigen PSI, SwitzerlandPhysik-Institut, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland Physik-Institut, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, SwitzerlandDepartment of Physics and Astronomy, Uppsala University, SE-75121 Uppsala, Sweden KTH Royal Institute of Technology, Materials Physics, SE-164 40 Kista, Stockholm, Sweden KTH Royal Institute of Technology, Materials Physics, SE-164 40 Kista, Stockholm, Sweden Physik-Institut, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland Physik-Institut, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland Physik-Institut, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, SwitzerlandInstitute of Physics, École Polytechnique Fedérale de Lausanne (EPFL), Lausanne CH-1015, SwitzerlandPhysik-Institut, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen PSI, SwitzerlandSwiss Light Source, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen PSI, SwitzerlandDiamond Light Source, Harwell Campus, Didcot OX11 0DE, UK.Diamond Light Source, Harwell Campus, Didcot OX11 0DE, UK. Department of Advanced Materials, University of Tokyo, Kashiwa 277-8561, Japan Department of Advanced Materials, University of Tokyo, Kashiwa 277-8561, Japan Department of Advanced Materials, University of Tokyo, Kashiwa 277-8561, JapanH. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, United KingdomH. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, United Kingdom Department of Physics, Hokkaido University - Sapporo 060-0810,JapanDepartment of Physics, Hokkaido University - Sapporo 060-0810,Japan Department of Applied Sciences, Muroran Institute of Technology,Muroran 050-8585, Japan Department of Physics, Hokkaido University - Sapporo 060-0810,Japan Physik-Institut, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland Physik-Institut, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland Direct Observation of Orbital Hybridisation in a Cuprate SuperconductorJ. Chang December 30, 2023 ======================================================================== The minimal ingredients to explain the essential physics oflayered copper-oxide (cuprates) materials remains heavily debated.Effective low-energy single-band models of the copper-oxygen orbitals are widely usedbecause there exists no strong experimental evidence supporting multi-band structures. Here we report angle-resolved photoelectron spectroscopy experiments onLa-based cuprates that provide direct observation of a two-band structure. This electronic structure, qualitatively consistent with density functional theory,is parametrised by a two-orbital ( and ) tight-binding model.We quantify the orbital hybridisationwhichprovidesan explanation for the Fermi surface topology and the proximity of the van-Hove singularity to the Fermi level. Our analysis leads to a unification of electronic hopping parameters for single-layer cuprates and we conclude that hybridisation, restraining d-wave pairing, is an important optimisation element for superconductivity. Identifying the factors that limit the transition temperatureof high-temperature cuprate superconductivity is a crucial step towards revealing the design principles underlying the pairing mechanism <cit.>.It may also provide an explanation for the dramatic variation ofacross theknown single-layer compounds <cit.>. Although superconductivity is certainlypromoted within the copper-oxide layers, the apical oxygen position may play an important role in defining the transition temperature <cit.>. The CuO_6 octahedron lifts the degeneracy of the nine copper 3d-electrons and generates fully occupied t_2g and 3/4-filled e_g states <cit.>. With increasing apical oxygen distanceto the CuO_2 plane, the e_g states split to create a 1/2-filledband.The distancethus defines whether single or two-band models are most appropriate to describe the low energy band structure. It has also been predicted thatinfluencesin at least two different ways.First, the distancecontrols the charge transfer gap between the oxygen and copper site which, in turn, suppresses superconductivity <cit.>.Second, Fermi level-hybridisation, depending on , reduces the pairing strength <cit.>. Experimental evidence, however, points in opposite directions. Generally, single layer materials with largerhave indeed a larger <cit.>. However, STM studies of Bi-based cuprates suggest an anti-correlation betweenand <cit.>. In the quest to disentangle these causal relation betweenand , it is imperative to experimentallyreveal the orbital character of the cuprate band structure.The comparably short apical oxygen distancemakes an ideal candidate for such a study. Experimentally, however, it is challenging to determine the orbital character of the states near the Fermi energy (). In fact, thebandhas never been identified directly byangle-resolved photoelectron spectroscopy (ARPES) experiments. A large majority of ARPES studies have focused on the pseudogap, superconducting gap and quasiparticle self-energy properties in near vicinity to the Fermi level <cit.>. An exception to this trend are studies of the so-called waterfall structure <cit.> that lead to the observation of band structures below theband <cit.>. However, the origin and hence orbital character of these bands was never addressed. Resonant inelastic x-ray scattering has been used to probe excitations between orbital d-levels. In this fashion, insight about the position of , , ,states with respect tohas been obtained <cit.>. Although difficult to disentangle, it has been argued that for LSCO thelevel is found above ,, <cit.>.To date, a comprehensive study of themomentum dependence is missing and therefore the coupling between theandbands has not been revealed. X-ray absorption spectroscopy (XAS) experiments, sensitive to the unoccupied states, concluded only marginal hybridisation ofandstates in (LSCO) <cit.>. Therefore, the role of -hybridisation remains ambiguous <cit.>.Here we provide direct ultra-violet and soft-xray ARPES measurements of theband inLa-based single layer compounds.Theband is located about 1 eV below the Fermi level at the Brillouin zone (BZ) corners.From these corners, theband is dispersing downwards along the nodal and anti-nodal directions, consistent with density functional theory (DFT) calculations.The experimental and DFT band structure, including onlyandorbitals, is parametrised using a two-orbital tight-binding model<cit.>. The presence of theband close to the Fermi level allows to describe the Fermi surface topology for all single layer compounds (includingand ) with similar hopping parameters for theorbital. This unification of electronic parameters implies that the main difference betweensingle layer cuprates originates fromthe hybridisation betweenandorbitals. The significantly increased hybridisation in La-based cuprates pushes the van-Hove singularity close to the Fermi level.This explains why the Fermi surface differs from other single layer compounds.We directly quantifythe orbital hybridisation that plays a sabotaging role for superconductivity.Results Material choices: Different dopings ofspanning from x=0.12 to 0.23 in additionto an overdoped compound ofwith x=0.21 have been studied.These compounds represent different crystal structures: low-temperature orthorhombic (LTO), low-temperature tetragonal (LTT) and the high-temperature tetragonal (HTT).Our results are verysimilar across all crystal structures and dopings (see Supplementary Fig. 1).To keep the comparisonto band structure calculations simple, this paper focuses on results obtained inthe tetragonal phase of overdopedwith x=0.23. Electronic band structure: A raw ARPES energy distribution map (EDM), along the nodal direction, is displayed in Fig. <ref>a.Near , the widely studied nodal quasiparticle dispersion with predominately character is observed <cit.>.This band reveals the previously reported electron-like Fermi surface of , x=0.23<cit.> (Fig. <ref>b), the universal nodal Fermi velocity v_F≈1.5 eVÅ<cit.> and a band dispersion kink around 70 meV<cit.>.The main observation reported here is the second band dispersion at ∼1 eV below the Fermi level E_F (see Fig. <ref> and <ref>) and a hybridisation gap splitting the two (see Fig. <ref>).This second band – visible in both raw momentum distribution curves (MDC)and constant energy maps (CEM) – disperses downwards away from the BZ-corners.Since a pronounced k_z-dependence is observedfor this band structure (see Figs. <ref> and <ref>) a trivial surface state can be excluded .Subtracting a background intensity profile (see Supplementary Fig. 2) is a standard method that enhances visualisation of this second band structure.In fact, using soft x-rays (160 eV - 600 eV),at least two additional bands (β and γ) are found below thedominatedband crossing the Fermi level.Here, focus is set entirely on the β band dispersionclosest to thedominated band.This band is clearly observed at the BZ corners (see Figs. 1–3). The complete in-plane (k_x,k_y) and out-of-plane (k_z) band dispersion is presented in Fig. <ref>. Orbital band characters: To gain insight into the orbital character of these bands, a comparison with a DFT band structurecalculation (see methods section) of La_2CuO_4is shown in Fig. <ref>. The e_g states ( and ) are generally found above the t_2g bands (, , and ).The overall agreement between the experiment andthe DFT calculation (see Supplementary Fig. 3) thus suggests that the two bands nearest to the Fermi level are composedpredominately ofandorbitals. This conclusion can also be reached by pureexperimental arguments. Photoemission matrix element selection rules contain information about the orbital band character.They can be probed in a particular experimental setup where a mirror-plane is defined by the incident light and the electron analyser slit <cit.>.With respect to this plane the electromagnetic light fieldhas odd (even) parity for() polarisation (see Supplementary Fig. 4).Orienting the mirror plane along the nodal direction (cut 1 in Fig. <ref>), theand() orbitals have even (odd) parity. For a final-state with even parity, selection rules <cit.> dictate that theand -derived bands should appear (vanish) in the() polarisation channel and vice versa for . Due to their orientation in real-space, the and orbitals are not expected to show a strict switching behavior along the nodal direction <cit.>. As shown in Fig. <ref>f-g, two bands (α and γ) appear with -polarised light while for -polarised light bands β and γ' are observed. Band α which crossesis assigned towhile band γ has to originate from / orbitals asand -derived states are fully suppressed for -polarised light. In the EDM, recorded with -polarised light, band (β) at ∼ 1 eV binding energy and again a band (γ') at ∼ 1.6 eV is observed. From the orbital shape, a smaller k_z dispersion is expected forand -derived bands than for those fromorbitals. As the β band exhibits a significant k_z dispersion (see Fig. <ref>), much larger than observed for theband, we conclude that it is ofcharacter. The γ' band which isvery close to the γ band istherefore ofcharacter. Interestingly, this -derived band has stronger in-plane than out-of-plane dispersion, suggesting that there is a significanthopping to in-plane p_x and p_y oxygen orbitals. Therefore the assumption thatthestates are probed uniquely through the apical oxygen p_z orbital <cit.> has to be taken with caution.DiscussionMost minimal models aiming to describe the cuprate physics start with an approximatelyhalf-filledsingleband ona two-dimensional square lattice. Experimentally, different band structures have been observedacross single-layer cuprate compounds. The Fermi surface topology ofis, for example, less rounded compared to(Bi2201),(Tl2201), and(Hg1201).Within a single-band tight-binding model the rounded Fermi surface shape of the single layer compounds Hg1201 and Tl2201 isdescribed by setting r=(|t_α'|+|t_α”|)/t_α∼0.4<cit.>, where t_α, t_α', and t”_α are nearest, next nearest and next-next nearest neighbour hopping parameters (see Table <ref> and Supplementary Fig. 4).Forwith more flat Fermi surface sections,significantly lower values of r have been reported. For example, for overdoped ,r ∼0.2 was found <cit.>.The single-band premise thus leads to varying hopping parameters across the cuprate families,stimulating the empirical observation that T_c^max roughly scales with t_α' <cit.>. This, however, is in direct contrast to t-J models that predict the opposite correlation <cit.>. Thus the single-band structure applied broadly to all single-layer cuprates lead to conclusions that challenge conventional theoretical approaches. The observation of theband calls for a re-evaluation of the electronic structure in La-based cuprates using a two-orbital tight-binding model (see methods section). Crucially, there is a hybridisation term Ψ(k)=2 t_αβ[cos( k_x )-cos(k_y)] between theandorbitals, where t_αβ is a hopping parameter that characterises the strength of orbital hybridisation.In principle, one may attempt to describe the two observed bandsindependently by taking t_αβ=0. However, the problem then returns to the single band description with the above mentioned contradictions.Furthermore, t_αβ=0 implies a band crossing in the antinodal direction that is not observed experimentally (see Fig. <ref>). In fact, from theavoided band crossing one can directly estimate t_αβ≈200 meV.As dictated by the different eigenvalues of the orbitals under mirror symmetry, the hybridisation term Ψ(k) vanishes on the nodal lines k_x=± k_y (see inset of Fig. 3). Hence the pureandorbital band character is expected along these nodal lines. The hybridisation Ψ(k) is largest in the anti-nodal region, pushing the van-Hove singularity of the upper band close to the Fermi energy and in case of overdopedacross the Fermi level. In addition to the hybridisation parameter t_αβ and the chemical potential μ, six free parametersenter the tight-binding model that yields the entire band structure (white lines in Figs. <ref> and <ref>). Nearest and next-nearest in-plane hopping parameters between(t_α, t_α') and(t_β,t_β') orbitals are introduced to capture theFermi surface topology and in-planeband dispersion (see Supplementary Fig. 4). The k_z dispersion is described by nearest and next-nearest out-of-plane hoppings (t_β z, t'_β z) of theorbital.The fourhopping parameters and the chemical potential μ are determined from the experimental band structurealong the nodal direction where Ψ(k)=0. Furthermore, theα and β band dispersion in the anti-nodal region and the Fermi surface topology provide the parameters t_α, t_α' and t_αβ. Our analysis reveals a finite band coupling-t_αβ=0.21 eV resulting in a strong anti-nodal orbital hybridisation (see Fig. <ref> and Table <ref>). Compared to the single-band parametrisation <cit.> a significantly larger value r ∼ -0.32is foundand hence a unification of t_α'/t_α ratios for all single-layer compounds is achieved. Finally, we discuss the implication of orbital hybridisation for superconductivity and pseudogap physics. First, we notice that a pronounced pseudogap is found in the anti-nodal regionofwith x=0.21 – consistent with transport experiments <cit.> (see Supplementary Fig. 5).The fact that t_αβ of La_1.59Eu_0.2Sr_0.21CuO_4 is similar tot_αβ of LSCO suggests that the pseudogap is not suppressed by the-hybridisation.To this end, a comparison to the 1/4-filled e_g systemEu_2-xSr_xNiO_4 with x=1.1 is interesting <cit.>. This material has the same two-orbital band structure with protection against hybridisation alongthe nodal lines. Both theandbands are crossing the Fermi level, producing two Fermi surface sheets <cit.>.Despite an even stronger-admixture of thederived band ad-wave-like pseudogap has been reported <cit.>. The pseudogap physicsthus seems to be unaffected by the orbital hybridisation. It has been argued that orbital hybridisation – of the kind reported here – is unfavourable for superconducting pairing<cit.>. It thus provides an explanation for thevarying T_c^max across single layercuprate materials. Although other mechanisms, controlled by the apical oxygen distance, (e.g. variation of the copper-oxygen charge transfer gap <cit.>) are not excluded our results demonstrate that orbital hybridisation exists and is an important control parameter for superconductivity.Acknowledgements:D.S., D.D., L.D., and J.C. acknowledge support by the Swiss National Science Foundation. Further, Y.S. and M.M. are supported by the Swedish Research Council (VR) through a project (BIFROST, dnr.2016-06955). This work was performed at the SIS<cit.>, ADRESS <cit.> and I05 beamlines at the Swiss Light Source and at the Diamond Light Source. A.M.C. wishes to thank the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1066293, for hosting during some stages of this work.We acknowledge Diamond Light Source for access to beamline I05 (proposal SI10550-1) that contributed to the results presented hereand thank all the beamline staff for technical support. Authors contributions: SP, TT, HT, TK, NM, MO, OJL,and SMH grew and prepared single crystals.CEM, DS, LD, MH, DD, CGF, KH, JC, NP, MS, OT, MK, VS, TS, PD, MH, MM, and YS prepared and carried out the ARPES experiment. CEM, KH, JC performed the data analysis. CEM carried out the DFT calculations andAMC, CEM, TN developed the tight-binding model. All authors contributed to the manuscript.Methods Sample characterisation: High-quality single crystals of , x=0.12, 0.23, and , x=0.21, were grown by thefloating-zone technique.The samples were characterised by SQUID magnetisation <cit.> to determine superconducting transition temperatures (T_c=27 K, 24 K and 14 K).For the crystal structure, the experimental lattice parameters are a=b=3.78 Åand c=2c'=13.2 Å<cit.>. ARPES experiments: Ultraviolet and soft x-ray ARPES experiments were carried out at the SISand ADRESS beam-lines at the Swiss Light Source and at the I05 beamline at Diamond Light Source. Samples were pre-aligned ex situ using a x-ray LAUE instrument and cleavedin situ – at base temperature (10 - 20 K) and ultra high vacuum (≤5·10^-11 mbar) – employing a top-post technique or cleaving device<cit.>.Ultraviolet (soft x-ray <cit.>) ARPES spectra were recorded using a SCIENTA R4000 (SPECS PHOIBOS-150) electron analyser with horizontal (vertical) slit setting. All data was recorded at the cleaving temperature 10-20 K. To visualize the -dominated band, we subtracted in Figs. <ref>f,g and Figs. <ref>-<ref> the backgroundthat was obtained by taking the minimum intensity of the MDC at each binding energy.Tight-binding model:A two-orbital tight-binding model Hamiltonian with symmetry-allowed hopping terms is employed to isolate and characterise the extent of orbital hybridisation of the observed band structure <cit.>. For compactness of the momentum-space Hamiltonian matrix representation, we introduce the vectors Q^κ_ = ( a, κ b,0)^𝖳,R^κ_1,κ_2 =(κ_1 a , κ_1κ_2b ,c )^𝖳 /2,T^κ_1,κ_2_1= (3κ_1 a , κ_1κ_2b ,c )^𝖳/2,T^κ_1,κ_2_2= (κ_1 a ,3 κ_1κ_2b ,c )^𝖳/2, whereκ, κ_1, and κ_2 take values ±1 as defined by sums in the Hamiltonian and 𝖳 denotes vector transposition. Neglecting the electron spin (spin orbit coupling is not considered) the momentum-space tight-binding Hamiltonian, H(k), at a particular momentum k = (k_x, k_y, k_z ) is then given by H(k) = [ [ M^x^2-y^2(k) Ψ(k); Ψ(k) M^z^2(k) ]] in the basis (c_k, x^2 - y^2, c_k, z^2)^⊤,where the operator c_k,α annihilates an electron with momentum k in an e_g-orbital d_α, with α∈{ x^2-y^2, z^2 }.The diagonal matrix entriesare given byM^x^2-y^2(k) =2t_α[ cos( k_x a) + cos( k_y b ) ] + μ+ ∑_κ=±1 2 t_α' cos(Q^κ_·k) + 2t_α”[ cos( 2k_x a) + cos( 2k_y b ) ], andM^z^2(k) = 2 t_β[ cos ( k_x a) + cos ( k_y b) ]-μ+∑_κ = ± 12 t_β'cos( Q^κ_·k)+∑_κ_1,2 = ± 1[ 2 t_β zcos( R_^κ_1,κ_2·k)+ ∑_i=1,2 2 t'_β zcos(T^κ_1,κ_2_i ·k) ], which describe the intra-orbital hopping forandd_z^2 orbitals, respectively. The inter-orbital nearest-neighbour hopping term is given byΨ(k)= 2 t_αβ[ cos( k_x a ) - cos( k_y b ) ].In the above, μ determines the chemical potential.The hopping parameters ,andcharacterise nearest neighbor (NN), next-nearest neighbour (NNN) and next-next-nearest neighbour (NNNN) intra-orbital in-plane hopping betweenorbitals.andcharacterise NN and NNN intra-orbital in-plane hopping betweenorbitals, whileandcharacterise NN and NNN intra-orbital out-of-plane hopping betweenorbitals, respectively (see Supplementary Fig. 3). Finally, the hopping parametercharacterises NN inter-orbital in-plane hopping.Note that in our model,intraorbital hopping terms described by the vectors (Eqs. <ref>) are neglected as theseare expected to be weak compared to those of theorbital.This is due to the fact that the inter-plane hopping is mostly mediated by hopping between apical oxygen p_z orbitals, which in turn only hybridize with theorbitals, not with theorbitals. Such an argument highlights that the tight-binding model is not written in atomic orbital degrees of freedom, but in Wannier orbitals, which are formed from the Cu d orbitals and the ligand oxygen p orbitals. As follows from symmetry considerations and is discussed in Ref. HirofumiPRB12, the Cuorbital together with the apical oxygen p_z orbital forms a Wannier orbital withsymmetry, while the Cuorbital together with the four neighboring p_σ orbitals of the in-plane oxygen forms a Wannier orbital withsymmetry. One should thus think of this tight-binding model as written in terms of these Wannier orbitals, thus implicitly containing superexchange hopping via the ligand oxygen p orbitals. Additionally we stress that all hopping parameterseffectively include the oxygen orbitals.Diagonalising Hamiltonian (<ref>), we find two bandsε_±(k) =1 2[ M^x^2-y^2(k) +M^z^2(k) ] ±12√([M^x^2-y^2(k ) - M^z^2(k )]^2 + 4 Ψ^2 (k ) ) and make the following observations: along thek_x=± k_y lines, Ψ(k) vanishes and hence no orbital mixing appears in the nodal directions.The reason for this absence of mixing lies in the different mirror eigenvalues of the two orbitals involved. Hence it is not an artifact of the finite range of hopping processes included in our model. The parameters of the tight-binding model are determined by fitting the experimental bandstructure and are provided in Table <ref>.DFT calculations:Density functional theory (DFT) calculations were performed for La_2CuO_4 in the tetragonal space group I4/mmm, No. 139, found in the overdoped regime ofusing the WIEN2K package <cit.>.Atomic positions are those inferredfrom neutron diffraction measurements<cit.> for x=0.225.In the calculation, the Kohn-Sham equation is solved self-consistently by using a full-potential linear augmented plane wave (LAPW) method. The self consistent field calculation converged properly for a uniform k-space grid in the irreducible BZ. The exchange-correlation term is treated within the generalized gradient approximation (GGA) in the parametrization of Perdew, Burke and Enzerhof (PBE)<cit.>. 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"authors": [
"Christian E. Matt",
"D. Sutter",
"A. M. Cook",
"Y. Sassa",
"M. Mansson",
"O. Tjernberg",
"L. Das",
"M. Horio",
"D. Destraz",
"C. G. Fatuzzo",
"K. Hauser",
"M. Shi",
"M. Kobayashi",
"V. Strocov",
"P. Dudin",
"M. Hoesch",
"S. Pyon",
"T. Takayama",
"H. Takagi",
"O. J. Lipscombe",
"S. M. Hayden",
"T. Kurosawa",
"N. Momono",
"M. Oda",
"T. Neupert",
"Johan Chang"
],
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"published": "20170726152223",
"title": "Direct Observation of Orbital Hybridisation in a Cuprate Superconductor"
} |
Supermetric Search [================== We describe a machine learning approach for the 2017 shared task on Native Language Identification (NLI). The proposed approach combines several kernels using multiple kernel learning. While most of our kernels are based on character p-grams (also known as n-grams) extracted from essays or speech transcripts, we also use a kernel based on i-vectors, a low-dimensional representation of audio recordings, provided by the shared task organizers. For the learning stage, we choose Kernel Discriminant Analysis (KDA) over Kernel Ridge Regression (KRR), because the former classifier obtains better results than the latter one on the development set. In our previous work, we have used a similar machine learning approach to achieve state-of-the-art NLI results. The goal of this paper is to demonstrate that our shallow and simple approach based on string kernels (with minor improvements) can pass the test of time and reach state-of-the-art performance in the 2017 NLI shared task, despite the recent advances in natural language processing. We participated in all three tracks, in which the competitors were allowed to use only the essays (essay track), only the speech transcripts (speech track), or both (fusion track). Using only the data provided by the organizers for training our models, we have reached a macro F_1 score of 86.95% in the closed essay track, a macro F_1 score of 87.55% in the closed speech track, and a macro F_1 score of 93.19% in the closed fusion track. With these scores, our team (UnibucKernel) ranked in the first group of teams in all three tracks, while attaining the best scores in the speech and the fusion tracks.§ INTRODUCTIONNative Language Identification (NLI) is the task of identifying the native language (L1) of a person, based on a sample of text or speech they have produced in a language (L2) other than their mother tongue. This is an interesting sub-task in forensic linguistic applications such as plagiarism detection and authorship identification, where the native language of an author is just one piece of the puzzle <cit.>. NLI can also play a key role in second language acquisition (SLA) applications where NLI techniques are used to identify language transfer patterns that help teachers and students focus feedback and learning on particular areas of interest <cit.>. In 2013, tetreault-blanchard-cahill:2013:BEA8 organized the first NLI shared task, providing the participants written essays of non-native English learners. In 2016, the Computational Paralinguistics Challenge <cit.> included a shared task on NLI based on the spoken response of non-native English speakers. The 2017 NLI shared task <cit.> attempts to combine these approaches by including a written response (essay) and a spoken response (speech transcript and i-vector acoustic features) for each subject. Our team (UnibucKernel) participated in all three tracks proposed by the organizers of the 2017 NLI shared task, in which the competitors were allowed to use only the essays (closed essay track), only the speech transcripts (closed speech track), or both modalities (closed fusion track).Our approach in each track combines two or more kernels using multiple kernel learning. The first kernel that we considered is the p-grams presence bits kernel[We computed the p-grams presence bits kernel using thecode available at http://string-kernels.herokuapp.com.], which takes into account only the presence of p-grams instead of their frequency. The second kernel is the (histogram) intersection string kernel[We computed the intersection string kernel using the code available at http://string-kernels.herokuapp.com.], which was first used in a text mining task by ionescu-popescu-cahill-EMNLP-2014. While these kernels are based on character p-grams extracted from essays or speech transcrips, we also use an RBF kernel <cit.> based on i-vectors <cit.>, a low-dimensional representation of audio recordings, made available by the 2017 NLI shared task organizers <cit.>. We have also considered squared RBF kernel versions of the string kernels and the kernel based on i-vectors. We have taken into consideration two kernel classifiers <cit.> for the learning task, namely Kernel Ridge Regression (KRR) and Kernel Discriminant Analysis (KDA).In a set of preliminary experiments performed on the development set, we found that KDA gives better results than KRR, which is consistent with our previous work <cit.>. Therefore, we decided to submit results using just the KDA classifier. We have also tuned the range of p-grams for the string kernels. Using only the data provided by the organizers for training our models, we have reached a weighted F_1 score of 86.95% in the essay track, a weighted F_1 score of 87.55% in the speech track, and a weighted F_1 score of 93.19% in the fusion track. The first time we used string kernels for NLI, we placed third in the 2013 NLI shared task <cit.>. In 2014, we improved our method and reached state-of-the-art performance <cit.>. More recently, we have shown that our method is language independent and robust to topic bias <cit.>. However, with all the improvements since 2013, our method remained a simple and shallow approach. In spite of its simplicity, the aim of this paper is to demonstrate that our approach can still achieve state-of-the-art NLI results, 4 years after its conception.The paper is organized as follows. Related work on native language identification and string kernels is presented in Section <ref>. Section <ref> presents the kernels that we used in our approach. The learning methods used in the experiments are described in Section <ref>. Details about the NLI experiments are provided in Section <ref>. Finally, we draw conclusions and discuss future work in Section <ref>.§ RELATED WORK §.§ Native Language IdentificationAs defined in the introduction, the goal of automatic native language identification (NLI) is to determine the native language of a language learner, based on a piece of writing or speech in a foreign language. Most research has focused on identifying the native language of English language learners, though there have been some efforts recently to identify the native language of writing in other languages, such as Chinese <cit.> or Arabic <cit.>.The first work to study automated NLI was that of Tomokiyo-2001. In their study, a Naïve Bayes model is trained to distinguish speech transcripts produced by native versus non-native English speakers. A few years later, a second study on NLI appeared <cit.>. In their work, Jarvis-2004 tried to determine how well a Discriminant Analysis classifier could predict the L1 language of nearly five hundred English learners from different backgrounds. To make the task more challenging, they included pairs of closely related L1 languages, such as Portuguese and Spanish. The seminal paper by koppel:2005:LNCS introduced some of the best-performing features for the NLI task: character, word and part-of-speech n-grams along with features inspired by the work in the area of second language acquisition such as spelling and grammatical errors. In general, most approaches to NLI have used multi-way classification with SVM or similar models along with a range of linguistic features. The book of jarvis-2012 presents some of the state-of-the-art approaches used up until 2012. Being the first book of its kind, it focuses on the automated detection of L2 language-use patterns that are specific to different L1 backgrounds, with the help of text classification methods. Additionally, the book presents methodological tools to empirically test language transfer hypotheses, with the aim of explaining how the languages that a person knows interact in the mind.In 2013, tetreault-blanchard-cahill:2013:BEA8 organized the first shared task in the field. This allowed researchers to compare approaches for the first time on a specifically designed NLI corpus that was much larger than previously available data sets. In the shared task, 29 teams submitted results for the test set, and one of the most successful aspects of the competition was that it drew submissions from teams working in a variety of research fields. The submitted systems utilized a wide range of machine learning approaches, combined with several innovative feature contributions. The best performing system in the closed task achieved an overall accuracy of 83.6% on the 11-way classification of the test set, although there was no significant difference between the top teams. Since the 2013 NLI shared task, several systems <cit.> have reported results above the top scoring system of the 2013 NLI shared task.Another interesting linguistic interpretation of native language identification data was only recently addressed, specifically the analysis of second language usage patterns caused by native language interference. Usually, language transfer is studied by Second Language Acquisition researchers using manual tools. Language transfer analysis based on automated native language identification methods has been the approach of jarvis-2012. Swanson-EACL-2014 also define a computational methodology that produces a ranked list of syntactic patterns that are correlated with language transfer. Their methodology allows the detection of fairly obvious language transfer effects, without being able to detect underused patterns. The first work to address the automatic extraction of underused and overused features on a per native language basis is that of malmasi-EMNLP-2014. The work of ionescu-popescu-cahill-COLI-2016 also addressed the automatic extraction of underused and overused features captured by character p-grams. §.§ String Kernels In recent years, methods of handling text at the character level have demonstrated impressive performance levels in various text analysis tasks <cit.>. String kernels are a common form of using information at the character level. They are a particular case of the more general convolution kernels <cit.>. LodhiSSCW02 used string kernels for document categorization with very good results. String kernels were also successfully used in authorship identification <cit.>. For example, the system described by PopescuG12 ranked first in most problems and overall in the PAN 2012 Traditional Authorship Attribution tasks. More recently, various blended string kernels reached state-of-the-art accuracy rates for native language identification <cit.> and Arabic dialect identification <cit.>. String kernels have also been used for sentiment analysis in various languages <cit.> and in cross-domain settings <cit.>.§ KERNELS FOR NATIVE LANGUAGE IDENTIFICATION §.§ String Kernels The kernel function captures the intuitive notion of similarity between objects in a specific domain and can be any function defined on the respective domain that is symmetric and positive definite. For strings, many such kernel functions exist with various applications in computational biology and computational linguistics <cit.>. String kernels embed the texts in a very large feature space, given by all the substrings of length p, and leave it to the learning algorithm to select important features for the specific task, by highly weighting these features.The first kernel that we use in the NLI experiments is the character p-grams presence bits kernel. The feature map defined by this kernel associates to each string a vector of dimension |Σ|^p containing the presence bits of all its substrings of length p (p-grams). Formally, for two strings over an alphabet Σ, s,t ∈Σ^*, the character p-grams presence bits kernel is defined as:k^0/1_p(s,t)=∑_v ∈Σ^p_v(s) ·_v(t),where _v(s) is 1 if string v occurs as a substring in s, and 0 otherwise.The second kernel that we employ is the intersection string kernel introduced in <cit.>. The intersection string kernel is defined as follows:k^∩_p(s,t)=∑_v ∈Σ^pmin{_v(s), _v(t) } ,where _v(s) is the number of occurrences of string v as a substring in s. Further details about the string kernels for NLI are given in <cit.>. The efficient algorithm for computing the string kernels is presented in <cit.>.Data normalization helps to improve machine learning performance for various applications. Since the value range of raw data can have large variation, classifier objective functions will not work properly without normalization.After normalization, each feature has an approximately equal contribution to the similarity between two samples. To ensure a fair comparison of strings of different lengths, normalized versions of the p-grams presence bits kernel and the intersection kernel are being used:k̂^0/1_p(s,t) =k^0/1_p(s,t)/√(k^0/1_p(s,s) · k^0/1_p(t,t)), k̂^∩_p(s,t) =k^∩_p(s,t)/√(k^∩_p(s,s) · k^∩_p(t,t)).Taking into account p-grams of different lengths and summing up the corresponding kernels, new kernels, termed blended spectrum kernels, can be obtained. We have used various blended spectrum kernels in the experiments in order to find the best combination. Inspired by the success of Radu-Andrei-ADI-2017 in using a squared RBF kernel based on i-vectors for Arabic dialect identification, we have also tried out squared RBF versions of the above kernels. We first compute the standard RBF kernels as follows:k̅^0/1_p(s,t) = exp ( -1 - k̂^0/1_p(s,t)/ 2 σ^2), k̅^∩_p(s,t) = exp ( -1 - k̂^∩_p(s,t)/ 2 σ^2). We then interpret the RBF kernel matrix as a feature matrix, and apply the dot product to obtain a linear kernel for this new representation:K̅ = K · K'. The resulted squared RBF kernels are denoted by (k̅^0/1_p)^2 and (k̅^∩_p)^2, respectively. §.§ Kernel based on Acoustic Features For the speech and the fusion tracks, we also build a kernel from the i-vectors provided by the organizers <cit.>. The i-vector approach <cit.> is a powerful speech modeling technique that comprises all the updates happening during the adaptation of a Gaussian mixture model (GMM) mean components to a given utterance. The provided i-vectors have 800 dimensions. In order to build a kernel from the i-vectors, we first normalize the i-vectors using the L_2-norm, then we compute the euclidean distance between each pair of i-vectors. We next employ the RBF kernel <cit.> to transform the distance into a similarity measure:k̂^i-vec(x, y) = exp (- √(∑^m_j=1(x_j - y_j)^2)/ 2 σ^2),where x and y are two i-vectors and m represents the size of the two i-vectors, 800 in our case. For optimal results, we have tuned the parameter σ in a set of preliminary experiments. We also interpret the resulted similarity matrix as a feature matrix, and we compute the product between the matrix and its transpose to obtain the squared RBF kernel based on i-vectors, denoted by (k̅^i-vec)^2.§ LEARNING METHODSKernel-based learning algorithms work by embedding the data into a Hilbert feature space and by searching for linear relations in that space. The embedding is performed implicitly, by specifying the inner product between each pair of points rather than by giving their coordinates explicitly. More precisely, a kernel matrix that contains the pairwise similarities between every pair of training samples is used in the learning stage to assign a vector of weights to the training samples. Various kernel methods differ in the way they learn to separate the samples. In the case of binary classification problems, kernel-based learning algorithms look for a discriminant function, a function that assigns +1 to examples belonging to one class and -1 to examples belonging to the other class. In the NLI experiments, we employed the Kernel Ridge Regression (KRR) binary classifier. Kernel Ridge Regression selects the vector of weights that simultaneously has small empirical error and small norm in the Reproducing Kernel Hilbert Space generated by the kernel function. KRR is a binary classifier, but native language identification is usually a multi-class classification problem. There are many approaches for combining binary classifiers to solve multi-class problems. Typically, the multi-class problem is broken down into multiple binary classification problems using common decomposition schemes such as: one-versus-all and one-versus-one. We considered the one-versus-all scheme for our NLI task. There are also kernel methods that take the multi-class nature of the problem directly into account, for instance Kernel Discriminant Analysis. The KDA classifier is sometimes able to improve accuracy by avoiding the masking problem <cit.>.More details about the kernel classifiers employed for NLI are discussed in <cit.>.§ EXPERIMENTS §.§ Data Set The corpus provided for the 2017 NLI shared task contains 13,200 multi-modal samples produces by speakers of the following 11 languages: Arabic, Chinese, French, German, Hindi, Italian, Japanese, Korean, Spanish, Telugu and Turkish. The samples are split into 11,000 for training, 1100 for development and 1100 for testing. The distribution of samples per prompt (topic) per native language is balanced. Each sample is composed of an essay and an audio recording of a non-native English learner. For privacy reasons, the shared task organizers were not able to provide the original audio recordings. Instead, they provided a speech transcript and an i-vector representation derived from the audio file. §.§ Parameter and System Choices In our approach, we treat essays or speech transcripts as strings. Because the approach works at the character level, there is no need to split the texts into words, or to do any NLP-specific processing before computing the string kernels. Hence, we apply string kernels on the raw text samples, disregarding the tokenized version of the samples. The only editing done to the texts was the replacing of sequences of consecutive space characters (space, tab, and so on) with a single space character. This normalization was needed in order to prevent the artificial increase or decrease of the similarity between texts, as a result of different spacing.We used the development set for tuning the parameters of our approach. Although we have some intuition from our previous work <cit.> about the optimal range of p-grams that can be used for NLI from essays, we decided to carry out preliminary experiments in order to confirm our intuition. We also carried out preliminary experiments to determine the optimal range of p-grams to be used for speech transcripts, a different kind of representation that captures other features of the non-native English speakers. We fixed the learning method to KDA based on the presence bits kernel and we evaluated all the p-grams in the range 3-9. For essays, we found that p-grams in the range 5-9 work best, which confirms our previous results on raw text documents reported in <cit.>. For speech transcripts, we found that longer p-grams are not helpful. Thus, the optimal range of p-grams is 5-7. In order to decide which classifier gives higher accuracy rates, we carried out some preliminary experiments using only the essays. The KRR and the KDA classifiers are compared in Table <ref>. We observe that KDA yields better results for both the blended p-grams presence bits kernel (k̂^0/1_5-9) and the blended p-grams intersection kernel (k̂^∩_5-9). Therefore, we employ KDA for the subsequent experiments. An interesting remark is that we also obtained better performance with KDA instead of KRR for the English L2, in our previous work <cit.>.After fixing the classifier and the range of p-grams for each modality, we conducted further experiments to establish what type of kernel works better, namely the blended p-grams presence bits kernel, the blended p-grams intersection kernel, or the kernel based on i-vectors. We also included squared RBF versions of these kernels. Since these different kernel representations are obtained either from essays, speech transcripts or from low-level audio features, a good approach for improving the performance is combining the kernels. When multiple kernels are combined, the features are actually embedded in a higher-dimensional space. As a consequence, the search space of linear patterns grows, which helps the classifier in selecting a better discriminant function. The most natural way of combining two or more kernels is to sum them up. Summing up kernels or kernel matrices is equivalent to feature vector concatenation. The kernels were evaluated alone and in various combinations, by employing KDA for the learning task. All the results obtained on the development set are given in Table <ref>.The empirical results presented in Table <ref> reveal several interesting patterns of the proposed methods. On the essay development set, the presence bits kernel gives slightly better results than the intersection kernel. The combined kernels yield better performance than each of the individual components, which is remarkably consistent with our previous works <cit.>. For each kernel, we obtain an improvement of up to 1% by using the squared RBF version. The best performance on the essay development set (85.55%) is obtained by sum of the squared RBF presence bits kernel and the squared RBF intersection kernel. On the speech track, the results are fairly similar among the string kernels, but the kernel based on i-vectors definitely stands out. Indeed, the best individual kernel is the kernel based on i-vectors with an accuracy of 81.64%. By contrast, the best individual string kernel is the squared RBF intersection kernel, which yields an accuracy of 59.82%. Thus, it seems that the character p-grams extracted from speech transcripts do not provide enough information to accurately distinguish the native languages. On the other hand, the i-vector representation extracted from audio recordings is much more suitable for the NLI task. Interestingly, we obtain consistently better results when we combine the kernels based on i-vectors with one or both of the string kernels. The best performance on the speech development set (85.45%) is obtained by sum of the squared RBF presence bits kernel, the squared RBF intersection kernel and the squared RBF kernel based on i-vectors. The top accuracy levels on the essay and speech development sets are remarkably close. Nevertheless, when we fuse the features captured by the kernels constructed for the two modalities, we obtain considerably better results. This suggests that essays and speech provide complementary information, boosting the accuracy of the KDA classifier by more than 6% on the fusion development set. It is important to note that we tried to fuse the kernel combinations that provided the best performance on the essay and the speech development sets, while keeping the original and the squared RBF versions separated. We also tried out a combination that does not include the intersection string kernel, an idea that seems to improve the performance. Actually, the best performance on the fusion development set (92.09%) is obtained by sum of the presence bits kernel (k̂^0/1_5-9) computed from essays, the presence bits kernel (k̂^0/1_5-7) computed from speech transcripts, and the kernel based on i-vectors (k̂^i-vec). In each track, we submitted the top two kernel combinations for the final test evaluation. §.§ Results The results on the test set are presented in Table <ref>. Although we tuned our approach to optimize the accuracy rate, the official evaluation metric for the NLI task is the macro F_1 score. Therefore, we have reported both the accuracy rate and the macro F_1 score in Table <ref>. Both kernel combinations submitted to the essay track obtain equally good results (86.95%). For the speech and the fusion tracks, the squared RBF kernels reach slightly lower performance than the original kernels. The best submission to the speech track is the KDA classifier based on the sum of the presence bits kernel (k̂^0/1_5-7) and the kernel based on i-vectors (k̂^i-vec), a combination that reaches a macro F_1 score of 87.55%. These two kernels are also included in the sum of kernels that gives our top performance in the fusion track (93.19%). Along with the two kernels, the best submission to the fusion track also includes the presence bits kernel (k̂^0/1_5-9) computed from essays. An interesting remark is that the results on the test set (Table <ref>) are generally more than 1% better than the results on the development set (Table <ref>), perhaps because we have included the development samples in the training set for the final test evaluation. The organizers have grouped the teams based on statistically significant differences between each team's best submission, calculated using McNemar's test with an alpha value of 0.05. The macro F_1 score of 86.95% places us in the first group of methods in the essay track, although we reach only the sixth best performance within the group. Remarkably, we also rank in the first group of methods in the speech and the fusion tracks, while also reaching the best performance in each of these two tracks. It is important to note that UnibucKernel is the only team ranked in first group of teams in each and every track of the 2017 NLI shared task, indicating that our shallow and simple approach is still state-of-the-art in the field.To better visualize our results, we have included the confusion matrices for our best runs in each track. The confusion matrix presented in Figure <ref> shows that our approach for the essay track has a higher misclassification rate for Telugu, Hindi and Korean, while the confusion matrix shown in Figure <ref> indicates that our approach for the speech track has a higher misclassification rate for Hindi, Telugu and Arabic. Finally, the confusion matrix illustrated in Figure <ref>, shows that we are able to obtain the highest correct classification rate for each and every L1 language (with respect to the other two confusion matrices) by fusing the essay and speech information. While there are no more than two misclassified samples for Chinese, Japanese, Spanish and German, our fusion-based approach still has some trouble in distinguishing Hindi and Telugu. Another interesting remark is that 5 native Arabic speakers are wrongly classified as French, perhaps because these Arabic speakers are from Maghreb, a region in which French arrived as a colonial language. As many people in this region speak French as a second language, it seems that our system gets confused by the mixed Arabic (L1) and French (L2) language transfer patterns that are observable in English (L3).§ CONCLUSION AND FUTURE WORKIn this paper, we have described our approach based on learning with multiple kernels for the 2017 NLI shared task <cit.>. Our approach attained generally good results, consistent with those reported in our previous works <cit.>. Indeed, our team (UnibucKernel) ranked in the first group of teams in all three tracks, while reaching the best marco F_1 scores in the speech (87.55%) and the fusion (93.19%) tracks. As we are the only team that ranked in first group of teams in each and every track of the 2017 NLI shared task, we consider that our approach has passed the test of time in native language identification.Although we refrained from including other types of features in order to keep our approach shallow and simple, and to prove that we can achieve state-of-the-art results using character p-grams alone, we will consider combining string kernels with other features in future work.§ ACKNOWLEDGMENTS This research is supported by University of Bucharest, Faculty of Mathematics and Computer Science, through the 2017 Mobility Fund. emnlp_natbib | http://arxiv.org/abs/1707.08349v2 | {
"authors": [
"Radu Tudor Ionescu",
"Marius Popescu"
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"published": "20170726100340",
"title": "Can string kernels pass the test of time in Native Language Identification?"
} |
Transition to chaos in the kinetic model] Transition to Chaos in the Kinetic Model of Cellulose Hydrolysis Under Enzyme Biosynthesis ControlBogolyubov Institute for Theoretical Physics, Nat. Acad. of Sci. of Ukraine 14b, Metrolohichna Str., Kiev 03680, Ukraine [email protected] [email protected] In the paper the kinetic model of the biochemical process of cellulose hydrolysis with cell application is presented. The model includes enzyme biosynthesis control and is open conditions it represents the dynamical system in the preturbulent regime. The limit cycle and its five consequencebifurcations of the doubling-period type are found. Also the limit regime of the system - the strange attractor - is presented. [ V.P. Gachok July 14, 2017 =================§ INTRODUCTION In the paper we present the dynamical system which has a direct interpretation in the biochemical process of the cellulose hydrolysis [1-3]. The factors present in the system such as the biosynthesis of cellobiase, the repression of biosynthesis by glucose, the cell lysis and the conditions of the input ofinitial substrate and the output of cells and connected to it the inactivation of cellobiase create the living conditions with complicated regimes in the kineticsof the system. The constructed biosystem is found to develop periodic oscillatory regimes including the chaotic regime, the strange attractor. In chemical kinetics the dynamical system with a strange attractor was studies by Rösler <cit.>.§ THE DESCRIPTION OF THE MODELWe consider the kinetics of hydrolysis of cellulose C under action of the enzyme E whose activity is controlled by the biosynthesis in themicroorganisms X. The end product is glucose G. This kinetics is described by the following 4-dimensional system of differential equations being the simplification of the cellulose hydrolysis model proposed in <cit.>: dC/dt=a - l E/(1+E)C/(1+C+G), dG/dt=2l E/(1+E)C/(1+C+G)-m GX/(1+X+G), dX/dt=m_1 GX/(1+X+G)-m_0X, dE/dt=E_0C/(1+C)N/(N+G)-eE, where C,G,X,E are the dimensionless concentrations of the corresponding reagents. The parameter a, according to the first equation, represents the input of the cellulose G in the system and terms fX and eE, according to the third and the fourth equations, characterize the output of the microorganisms X and the enzyme E (the dissipation). The system shows that the growth of substrate Cincludes the enzyme biosynthesis E which, consequently, leads to the productionof G and the growth of X. In turn the growth of X increases the biosynthesis of E. At large E and X the level of C and G decreases. Further the output leads to small concentrations of E and X and again due to the input C they start to increase and the process repeats. The existence of these feedback controls leads to the complicated oscillations. § COMPUTER RESULTS We fix the parameters, except f, as follows: a=1,l=200,m=400,m_1=230, E_0=1100, N=0.05, e=20. We investigate the phase portrait of the system changing the parameter f in the bounds 43 ≤ f ≤ 43.6. In this case the system has the stationaryfocus in the phase space. However, the stable focus is not a global attractor. The system with the above f has one more attractor. It was studied by computer simulations. The Cauchy datawere: G_0=0.9, G_0=0.24,X_0=0.64,E_0=0.8. The period was found with the application of the Poincaré map. This initial state of the system develops to the limit cycle. In growth of f we found that at the points f_1=43.39, f_2=43.53, f_3=43.562, f_4=43.5667, f_5=43.5677, the double period cycle occurs. The following increase of f leads to the chaotic regime, the strange attractor. Thus we have constructed the scenario ofthe following bifurcations and its limit, the strange attractor. The results are represented in the phase plane (C,G) in the Fig. <ref> - <ref>.991 V.P. Gachok, ITP-81-102R, Kiev (1981)2 V.P. Gachok, ITP-83-33E, Kiev (1983)3 V.P. Gachok, A.S. Zhokhin, ITP-83-54R, Kiev (1983)4 O. Rössler, Bull.Math.Biol.,v39 (1977) 2755 Th. Schulmeister, studia biophysica, v 105 (1985) 5 | http://arxiv.org/abs/1707.08914v1 | {
"authors": [
"A. S. Zhokhin",
"V. P. Gachok"
],
"categories": [
"q-bio.CB",
"nlin.AO"
],
"primary_category": "q-bio.CB",
"published": "20170727155254",
"title": "Transition to Chaos in the Kinetic Model of Cellulose Hydrolysis Under Enzyme Biosynthesis Control"
} |
Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, D-22607 Hamburg, Germany Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, D-22607 Hamburg, Germany Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, D-22607 Hamburg, Germany National Research Tomsk Polytechnic University (TPU), pr. Lenina 30, 634050 Tomsk, Russia SLAC National Accelerator Laboratory, 2575 Sand Hill Rd., Menlo Park, 94025 CA SLAC National Accelerator Laboratory, 2575 Sand Hill Rd., Menlo Park, 94025 CA SLAC National Accelerator Laboratory, 2575 Sand Hill Rd., Menlo Park, 94025 CA Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, D-22607 Hamburg, Germany Present address: European XFEL GmbH, Holzkoppel 4, D-22869 Schenefeld, Germany Van 't Hoff Laboratory for Physical and Colloid Chemistry, Debye Institute for Nanomaterial Science, Utrecht University, Padualaan 8, 3584 CH Utrecht, The Netherlands Present address: Division of Physical Chemistry, Department of Chemistry, Lund University, 22100 Lund, Sweden. SLAC National Accelerator Laboratory, 2575 Sand Hill Rd., Menlo Park, 94025 CA Present address: U.S. Department of Energy Brookhaven National Laboratory, 53 Bell Avenue, Upton, NY 11973-5000, USA SLAC National Accelerator Laboratory, 2575 Sand Hill Rd., Menlo Park, 94025 CA Present address: European XFEL GmbH, Holzkoppel 4, D-22869 Schenefeld, Germany SLAC National Accelerator Laboratory, 2575 Sand Hill Rd., Menlo Park, 94025 CA Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, D-22607 Hamburg, Germany European XFEL GmbH, Holzkoppel 4, D-22869 Schenefeld, Germany University of California San Diego, 9500 Gilman Dr., La Jolla, California 92093, USAVan 't Hoff Laboratory for Physical and Colloid Chemistry, Debye Institute for Nanomaterial Science, Utrecht University, Padualaan 8, 3584 CH Utrecht, The Netherlands Laboratory of Physical Chemistry, Department of Chemical Engineering and Chemistry, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, Netherlands[Corresponding author: ][email protected] Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, D-22607 Hamburg, Germany National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoe shosse 31, 115409 Moscow, RussiaWe demonstrate experimentally Hanbury Brown and Twiss (HBT) interferometry at hard X-ray Free Electron Laser (XFEL) on sample diffraction patterns. This is different from the traditional approach when HBT interferometry requires direct beam measurements in absence of the sample. HBT analysis was carried out on Bragg peaks from the colloidal crystals measured at Linac Coherent Light Source (LCLS). We observed nearly perfect (90%) spatial coherence and the pulse duration on the order of 11 fs for the monochromatized beam that is significantly shorter than expected from the electron bunch measurements. 41.60.Cr, 42.25.Kb, 42.50.Ar, 42.55.VcDiffraction based Hanbury Brown and Twiss interferometry performed at a hard x-ray free-electron laser I. A. Vartanyants December 30, 2023 ======================================================================================================X-ray free-electron lasers (XFELs) provide extremely bright and highly coherent x-ray radiation with femtosecond pulse duration. They find extensive applications in the wide range of scientific fields: structural biology <cit.>, solid density plasma <cit.>, matter under extreme conditions <cit.>, ultrafast photochemistry <cit.>, atomic physics <cit.> and many others. XFEL coherence properties often significantly affect its experimental performance. Several methods were employed to study spatial and temporal coherence, such as double pinholes <cit.>, Michelson type interferometry <cit.>, speckle contrast analysis <cit.>, and Hanbury Brown and Twiss (HBT) interferometry <cit.> (see for review <cit.>).One important aspect that makes XFELs substantially different from all other existing x-ray sources, is the degeneracy parameter, or average number of x-ray photons in one state. If for present high-brilliance synchrotron sources this value is about 10^-2, for the XFEL sources it can reach such high values as 10^10 <cit.>. This makes XFEL sources similar to optical lasers, and implies possibility of non-linear and quantum optics experiments, as was first suggested by Glauber <cit.>. This area in the FEL science is just on its early stage of development <cit.>. At the core of the quantum optics experiments stays HBT interferometry <cit.>. Since its first demonstration it was used, for example, to analyze nuclear scattering experiments <cit.>, to probe Bose-Einstein condensates <cit.> or to study effects of interaction on HBT interferometry <cit.>. HBT interferometry is especially well suited to study statistical behavior of XFELs due to their extremely short pulse duration. It allows to extract both the spatial and temporal XFEL coherence properties <cit.> as well as statistical information about the secondary beams and positional jitter <cit.>.The basic idea of the HBT interferometry <cit.> is to determine statistical properties of radiation from the normalized second-order intensity-intensity correlation function g^(2)(𝐫_1, 𝐫_2) = ⟨ I(𝐫_1) I(𝐫_2) ⟩/⟨ I(𝐫_1)⟩⟨ I(𝐫_2) ⟩ obtained by measuring the coincident response of two detectors at separated positions 𝐫_1 and 𝐫_2 (see for review <cit.>). In Eq. (<ref>), I(𝐫_1), I(𝐫_2) are the intensities of the wave field and the averaging denoted by brackets <...> is performed over a large ensemble of different realizations of the wave field, or different pulses in the case of XFEL radiation.In this Letter we present results of HBT interferometry performed on the Bragg peaks originating from the scattering on colloidal crystals. Due to a small beam size and large sample-detector distance instead of the conventional sharp Bragg peaks a comparably broad intensity distribution around each Bragg peak position is measured. Importantly, this intensity distribution depends not only on the crystal structure, but also on the incident pulse profile. Statistical changes of the XFEL beam structure from pulse to pulse lead to corresponding changes in the observed Bragg peaks intensity distribution. Therefore, fluctuating behavior of the Bragg peak intensity contains information about the statistical properties of the incident radiation typical for self-amplified spontaneous emission (SASE) XFELs <cit.>. This allowed us to extract information on statistical properties of the XFEL radiation during diffraction experiment on colloidal crystals. The measurements were performed at Linac Coherent Light Source (LCLS) in Stanford, USA at the x-ray pump probe (XPP) beamline <cit.>. LCLS was tuned to produce pulses with 3.3-3.7 mJ pulse energy, bunch charge 0.18 nC, and pulse repetition rate 120 Hz. An expected pulse duration from electron bunch measurements was about 41-43 fs. The double-crystal diamond (111) monochromator at LCLS with the thicknesses of the monochromator crystals 100 μm and 300 μm split the primary x-ray beam into a pink (transmitted) and monochromatic (diffracted) branches (see Fig. <ref>(a)). We used the monochromatic branch with the photon energy of 8 keV (1.55 Å) and relative energy bandwidth of 4.4 · 10^-5 <cit.>. Compound refractive lenses (CRLs) focused the beam at the sample position down to 50 μm full width at half maximum (FWHM). The number of photons in the focus was about 10^9 ph/pulse, and the experiment was performed in non-destructive mode [This was confirmed by comparing diffraction patterns in the beginning and the end of the run.]. Colloidal crystal sample was positioned vertically, perpendicular to the incoming XFEL pulse in the transmission diffraction geometry (see Fig. <ref>(a)). Series of x-ray diffraction patterns were recorded using the CSPAD megapixel x-ray detector positioned at the distance L = 10 m downstream from the sample consisting of 32 silicon sensors with pixel size of 110 x 110 μm^2 covering an area of approximately 17 x 17 cm^2 (see for experimental details Ref. <cit.>).Colloidal crystal films were prepared from the polystyrene (PS) using the vertical deposition method <cit.>. The film consisted of 30-40 monolayers of spherical particles. Two samples were investigated: PS colloidal crystals with a sphere diameter of 160 ± 3 nm (sample 1) and 420 ± 9 nm (sample 2).Examples of diffraction patterns measured from these crystals are shown in Fig. <ref> (b) and (c). Bragg peaks with the six-fold symmetry originating from the hexagonal colloidal crystal structure are clearly visible in this figure <cit.>. Intensity distributions around the Bragg peak 4 for sample 2 for three different incident pulses at the same position of the sample are shown in Fig. <ref>. It is well seen from this figure that Bragg peak profiles for each pulse have a complicated internal structure with additional sub-peaks. These sub-peaks have the same position from pulse to pulse but their relative intensity varies.Projection on the horizontal axis of the same Bragg peak intensities for three selected pulses as well as an average projected intensity for all pulses is shown in Fig. <ref>(d).In our experimental geometry we were in Fresnel scattering conditions (Fresnel number 1.7). It can be shown (see Appendix for details) that in general case of Fresnel scattering the intensity-intensity correlation function at a selected Bragg peak is given by an expression g^(2)(𝐐, 𝐐')= 1 + ζ_2(σ_ω)| μ(𝐐, 𝐐')|^2. Here vector 𝐐 is related to a radius vector 𝐫, measured from the center of the diffraction peak, by the relation 𝐐=k𝐫/L, where k=2π/λ and λ is the wavelength [In the following we perform evaluation in 𝐫-space.]. The contrast function ζ_2(σ_ω) introduced in Eq. (<ref>) is strongly dependent on the radiation bandwidth σ_ω. The spectral degree of coherence μ(𝐐, 𝐐') in Eq. (<ref>) is defined as μ(𝐐, 𝐐') = J(𝐐, 𝐐')/√(⟨ I(𝐐) ⟩)√(⟨ I(𝐐') ⟩), where J(𝐐, 𝐐') is the mutual intensity function (MIF) determined at the detector position. It is directly related to the statistical properties of the incident beam at the sample position by a two-dimensional Fourier transform | J(𝐐, 𝐐') |= |∬ e^-i(𝐐'𝐫'-𝐐𝐫) J_in(𝐫, 𝐫') 𝐫𝐫'|. Here J_in(𝐫, 𝐫') = ⟨ E_in^(𝐫') E_in(𝐫) ⟩ is the MIF function of the incoming field at the sample position, where E_in(𝐫)is the complex amplitude of the incident beam.It is important to note that the contrast function ζ_2(σ_ω) in Eq. (<ref>) is defined by the monochromator settings and its value is preserved between the sample and detector positions. This allows to connect pulse duration of the beam to coherence time of the monochromatized beam incident at the sample. The functional dependence of the intensity-intensity correlation function as given by Eq. (<ref>) allows to study both the spatial and temporal statistical properties of the XFEL radiation by the HBT interferometry. For correlation analysis we considered four Bragg peaks not obscured by detector gaps for each crystal (see Fig. <ref> (b-c)). In order to exclude the influence of the electron energy jitter, only patterns corresponding to pulses with the electron energies close to the mean value were selected (in total about 50,000, see Appendix for details). Average intensities of the Bragg peaks marked as 4 in Fig. <ref>(b-c) for both crystals are presented in Fig. <ref> (a-b). Projections of the selected Bragg peaks on the horizontal and vertical axes were then correlated and corresponding intensity-intensity correlation functions are presented in (Fig. <ref> (c-f)).The comparison of the intensity profiles with the intensity-intensity correlation functions reveals an interesting feature. While intensity profiles are not smooth and contain several sub-peaks reflecting non-perfect structure of the colloidal crystals, correlation functions are almost flat in a wide central region and then drop down fast to a background level, forming a square type shape (see Appendix for details). This is different from our earlier measurements at FELs <cit.>, when intensity-intensity correlation functions had been gradually decreasing with the distance between correlated positions.We were able to reproduce this form of intensity-intensity correlation functions in simulations (see Fig. <ref>). Two factors contribute to it: coherence length larger than the beam size and additional detector noise. If a fluctuating background is present on the detector, it limits the field of view of the correlation function to the area where the intensity around the Bragg peak is larger than the detector noise. If the coherence length of the incident beam is at least a few times larger than the size of the beam, it leads to a relatively flat intensity-intensity correlation function. We were also able to reproduce an appearance of a small area of higher contrast observed in Fig. <ref>(e). It can be simulated using the model of secondary beams introduced in <cit.>. A weak secondary beam (10% of the primary beam intensity) introduced in the vertical direction (see for its characteristics Appendix) leads to a similar feature in the intensity-intensity correlation function as observed in the experiment (see Fig. <ref>(b)). The fact that the models based on the assumption of a chaotic source describe well the behavior of the intensity-intensity correlation function supports an assumption that LCLS as a SASE XFEL can be considered as a rather chaotic source (compare with <cit.>). Our experimental results also allowed us to determine the degree of spatial coherence of LCLS radiation for hard x-rays. Performing similar analysis as in Refs. <cit.> we determined that the spatial degree of coherence on average for both samples and different peaks for each direction (horizontal and vertical) was about 0.90 ± 0.05 (see Appendix for details). This gave us an estimate of 81% of global transverse coherence of the full beam, which is in a good agreement with our previous observations <cit.>.We now explore the temporal properties of the beam. The contrast ζ_2(σ_ω) introduced in Eq. (<ref>) can be determined from the values of the intensity-intensity correlation function along the main diagonal g^(2)(x, x) (Fig. <ref>). In our experiment it wasapproximately 0.40 ± 0.05 and did not change significantly for different crystals and Bragg peaks (exact numbers for each crystal and peak can be found in Appendix). This suggests that the influence of the crystal structure variations on our results is insignificant. Assuming Gaussian Schell-model pulsed source <cit.> (see Appendix for details) the contrast function ζ_2(D_ω) can be expressed as ζ_2(σ_ω) = 1/√(1 + 4(T_rmsσ_ω)^2) , where T_rms is an effective pulse duration (r.m.s.) before the monochromator and σ_ω is the r.m.s. value of the monochromator bandwidth. It is important to point out that the effective pulse duration is extracted only from the part of the beam passing the monochromator. As such, it can be significantly shorter than that of non- monochromatized beam (see below). In derivation of equation (<ref>) it was assumed that the spectral width of the incoming radiation is much broader than the monochromator bandwidth and spectral coherence width. These conditions are well satisfied for the LCLS x-ray beam parameters and monocromator used in the experiment.Inversion of equation (<ref>) gives for the FWHM of the pulse duration (see Appendix for details)T = 2.355/2σ_ω√(1/[ζ_2(σ_ω)]^2-1) . Using the measured value of the contrast function ζ_2(σ_ω), we can estimate that for our experiment the pulse duration lies in the range of 11-12 fs. These values were significantly shorter than initially expected (about 41 fs) from the electron bunch measurements. To verify our findings we determined pulse duration by a different approach based on the mode analysis of the radiation as suggested inRef. <cit.>. According to this approach an average number of modes of radiation M is inversely proportional to the normalized dispersion of the energy distribution, that in our case coincide with the contrast function defined in Eq. (<ref>) M=1/[ζ_2(σ_ω)].We determined the number of modes by fitting intensity distribution at one of the Bragg peaks by Gamma distribution <cit.> (see Appendix for details). As a result, the number of longitudinal modes was M ≈ 2.3±0.1 and reproducible between different runs. Substituting this number in Eq. (<ref>) gives for the pulse duration 11.5 ± 0.5 fs in an excellent agreement to previously determined values from the HBT interferometry. Similar inconsistency factor of about three between the expected and observed pulse duration has been observed earlier in another LCLS experiment <cit.>.To explain the difference between thus obtained values of the pulse duration with the results of the electron bunch measurements several factors should be taken into account. An estimate of the pulse duration from the electron bunch measurements is mainly based on the longitudinal size of the electron beam as an FEL lasing medium. The electron beam size generally limits the maximum emitted hard x-ray pulse duration. The FEL gain is very sensitive to various electron beam properties, such as beam emittance, electron current and energy spread, and orbit alignment inside an undulator. These properties vary along the electron beam, which may result in a relatively short core, providing significantly better gain, compared to the rest of the beam. Another possible explanation may be the filtering of the bunch with the strong chirp by the high-resolution monochromator <cit.>. It was proven experimentally with cross-correlation measurements <cit.>, that 150 pC 50 fs long electron beam may radiate only 14 fs long 8.5 keV beam, which is comparable with our observations. The coherence time τ_c can be estimated from the bandwidth of the monochromator according to Ref. <cit.> as τ_c = √(π)/σ_ω.The obtained value is about 7.5 fs, which is only slightly shorter than the pulse duration. Therefore, x-ray pulses were effectively longitudinally coherent during the experiment.In summary, we have performed HBT interferometry at the Bragg peaks originating from the colloidal crystals measured at LCLS. This technique allowed us to extract information about spatial coherence and temporal properties of the incident beam directly from the diffraction patterns without additional equipment or specially dedicated measurements. We have determined a high degree of spatial coherence of the full XFEL beam that was about 81% which concord with our previous measurements. We have also observed a coherence length much larger than the beam size. We have obtained pulse durations of 11-12 fs, which are significantly shorter than expected in the operation regime of the LCLS used in our experiment. We also estimated coherence time for high-resolution monochromator used in our experiment and obtained the value of 7.5 fs that is just slightly below the pulse duration. That means that LCLS pulses in our experiment were not only spatially but also temporally coherent close to Fourier limited pulses.Our approach is quite general and is not limited to the analysis of the diffraction patterns originating from colloidal crystals. Any other crystalline sample can be used provided Bragg peaks are sufficiently broad to allow HBT measurement. This can be accommodated, for example, by the larger sample detector distance, or implementing a set of CRLs in the beam diffracted from the sample.Our measurements have demonstrated high degree of spatial coherence of the FEL radiation that could potentially lead to completely new avenue in the field of quantum optics. Such quantum optics experiments as exploration of non-classical states of light <cit.>, super-resolution experiments <cit.>, quantum imaging experiments <cit.> or ghost imaging experiments <cit.> may become possible at the hard XFEL sources in the near future. Finally, we foresee that HBT interferometry will become an important diagnostics and analytic tool at the XFEL sources.Portions of this research were carried out on the XPP Instrument at the LCLS at the SLAC National Accelerator Laboratory. LCLS is an Office of Science User Facility operated for the U.S. Department of Energy Office of Science by Stanford University. Use of the Linac Coherent Light Source (LCLS), SLAC National Accelerator Laboratory, is supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Contract No. DE-AC02-76SF00515. We thank Edgar Weckert, Evgeny Saldin, Dina Sheyfer and Gerhard Grübel for useful discussions. This work was partially supported by the Virtual Institute VH-VI-403 of the Helmholtz Association.§ APPENDIX I. INTENSITY-INTENSITY CORRELATION FUNCTIONS OF A RADIATION FIELD SCATTERED FROM A CRYSTAL We will consider a quasi-monochromatic x-ray beam E_in(𝐬) incident on a colloidal crystal in a shape of a thin slab of material (see Fig.<ref>).An exit surface wave from such a crystal can be written as E_ESW(𝐬) = O(𝐬)E_in(𝐬), where O(𝐬) is the so-called object function and 𝐬 is the two-dimensional (2D) vector in transverse direction to the incoming beam at the position of the sample. For a thin slab of material an object function can be expressed through refractive index n(𝐬,z) as <cit.> O(𝐬) = e^iφ(𝐬) , where φ(𝐬) = k∫_0^d(𝐬)(n(𝐬,z)-1)z is the phase difference due to refraction. Here d(𝐬) is the crystal thickness at the position 𝐬, k=2π/λ is the wave number and λ is the wavelength. At x-ray wavelength refractive index can be expressed as <cit.> n(𝐬,z) = 1-δ(𝐬,z)+iβ(𝐬,z), where δ(𝐬,z) is the real part of refractive index responsible for refraction and β(𝐬,z) is the imaginary part responsible for absorption. Neglecting absorption and taking into account known relation between the real part of refractive index and electron density of the crystal <cit.> δ(𝐬,z) = λ r_eρ(𝐬,z)/k, where r_e is the classical electron radius, we obtain for the phase in the object function in Eq. (<ref>) φ(𝐬) = - λ r_e∫_0^d(𝐬)ρ(𝐬,z)z. Taking into account that projection of a crystalline electron density is a periodic function we obtain that the object function in Eq. (<ref>) is also 2D periodic function.To determine distribution of the wavefield at the detector position we will propagate the exit surface wave to that position by performing convolution with the free space propagator P_L(𝐫) E_d(𝐫) = ∫ E_ESW(𝐬)P_L(𝐫-𝐬)𝐬 = ∫ O(𝐬) P_L(𝐫-𝐬) E_in(𝐬) 𝐬 , where 𝐫 is the 2D coordinate at the detector position and L is the sample-detector distance. Propagator in Eq. (<ref>) has a known form P_L(𝐫-𝐬) = 1/i λ Lexp[ik(𝐫-𝐬)^2/2L].Taking now into account that the object function is a 2D periodic function it can be expanded into Fourier series as O(𝐬) = ∑_𝐡 O_𝐡 e^i𝐡·𝐬 , where 𝐡 is the 2D reciprocal space vector and O_𝐡 = 1/V ∫ O(𝐬) e^-i 𝐡·𝐬𝐬 are the Fourier coefficients of the expansion. Substituting now this expansion in Eq. (<ref>) and considering scattering in the vicinity of a selected Bragg peak 𝐡 we obtain E_h(𝐬) = O_𝐡∫ e^i𝐡·𝐬 P_L(𝐫-𝐬) E_in(𝐬) 𝐬 .Using the far-field (D^2/(λ L) ≫ 1, where D is the size of the beam at the sample position) expression of the propagator P_L(𝐫-𝐬) ≃ exp(-i𝐪_𝐫·𝐬), where 𝐪_𝐫 = k 𝐫/L we obtain from Eq. (<ref>) E_h^FF(𝐐) = O_𝐡∫ e^-i 𝐐·𝐬E_in(𝐬) 𝐬 , where 𝐐 = 𝐪_𝐫 - 𝐡 is the momentum transfer vector calculated from the reciprocal space vector 𝐡. For the intensity of the scattered field in the far-field we have I_h^FF(𝐐) = |E_h(𝐐) |^2 = |O_𝐡|^2 ∫∫ e^-i 𝐐· (𝐬-𝐬') E_in^*(𝐬') E_in(𝐬) 𝐬𝐬' . In Fresnel (near-field) regime we can not use expansion expression for the propagator and we have for the scattered amplitude E_h^NF(𝐐) = O_𝐡 e^iϕ_u∫ e^-i 𝐐·𝐬E_in(𝐬) 𝐬 , where we introduced the phase ϕ_r=k𝐫^2/(2L) and defined a new amplitude E_in(𝐬) = e^i (k/2L)𝐬^2 E_in(𝐬). For intensity in the near-field we have I_h^NF(𝐐) = |E_h^NF(𝐐) |^2 = |O_𝐡|^2 ∫∫ e^-i 𝐐· (𝐬-𝐬')E_in^*(𝐬') E_in(𝐬) 𝐬𝐬' . As we can see expressions for the scattered intensities around selected Bragg peak coincide in the far-ield and near-field conditions with the change of the incoming wavefield expression to one given in Eq. (<ref>). As soon as the difference between two cases is in the constant phase factor it would not influence statistical characteristics of the scattered field. In the following we will use far-field expression (<ref>) keeping in mind that Fresnel conditions can be matched by the substitution given in Eq. (<ref>).We will now evaluate intensity-intensity correlation function at the detector position in the vicinity of the selected Bragg reflection 𝐡 g^(2)(𝐐, 𝐐') = ⟨ I(𝐐) I(𝐐') ⟩/⟨ I(𝐐)⟩⟨ I(𝐐') ⟩ , where momentum transfer vectors 𝐐 and 𝐐' are centered at reflection 𝐡 and related to the spatial coordinates at the detector position by 𝐐 = k 𝐮/L and 𝐐' = k 𝐮'/L. Averaging here is denoted by the brackets <...> and is performed over many realizations of the field. Substituting here expression for the intensity (<ref>) we have for the nominator ⟨ I(𝐐) I(𝐐') ⟩ = |O_𝐡|^4 e^-i𝐐· (𝐬-𝐬') - 𝐐'· (𝐬”-𝐬”')⟨ E_in^(𝐬') E_in(𝐬) E_in^(𝐬”') E_in(𝐬”)⟩𝐬𝐬'𝐬”𝐬”' .Assuming that the incoming radiation obeys Gaussian statistics we can use Gaussian moment theorem ⟨ E_in^(𝐬') E_in(𝐬) E_in^(𝐬”') E_in(𝐬”)⟩ = ⟨ E_in^(𝐬') E_in(𝐬)⟩⟨ E_in^(𝐬”') E_in(𝐬”)⟩ + + ⟨ E_in^(𝐬') E_in(𝐬”)⟩⟨ E_in^(𝐬”') E_in(𝐬)⟩ .Substituting now this expression in Eq. (<ref>) we obtain for the nominator ⟨ I(𝐐) I(𝐐') ⟩ = ⟨ I(𝐐) ⟩⟨ I(𝐐') ⟩ + | J(𝐐, 𝐐') |^2, where |J(𝐐, 𝐐')| is the absolute value of the mutual intensity function (MIF) defined at the detector position and related to the MIF of the incoming field J_in(𝐬, 𝐬') = ⟨ E_in^(𝐬') E_in(𝐬) ⟩ by the following relation | J(𝐐, 𝐐') |^2= |∬ e^-i(𝐐'𝐬'-𝐐𝐬) J_in(𝐬, 𝐬') 𝐬𝐬'|^2. Finally, we have for the normalized intensity-intensity correlation function (<ref>) g^(2)(𝐐, 𝐐') = ⟨ I(𝐐) I(𝐐') ⟩/⟨ I(𝐐)⟩⟨ I(𝐐') ⟩ = 1+| μ(𝐐, 𝐐')|^2, where μ(𝐐, 𝐐') = J(𝐐, 𝐐')/√(⟨ I(𝐐) ⟩)√(⟨ I(𝐐') ⟩) is the normalized spectral degree of coherence.Taking now into account that we have a finite bandwidth of radiation incoming from the monochromator we have for the intensity-intensity correlation function g^(2)(𝐐, 𝐐')= 1 + ζ_2(σ_ω)| μ(𝐐, 𝐐')|^2, where ζ_2(σ_ω) is the contrast function which strongly depends on the radiation bandwidth σ_ω. We will now evaluate this function inthe next section. § APPENDIX II. DETERMINATION OF THE PULSE DURATION FROM THE INTENSITY INTERFEROMETRY In the HBT interferometry the contrast function ζ_2(σ_ω) for a cross-spectral pure chaotic radiation can be defined as <cit.> ζ_2(σ_ω) = ∬_-∞^∞ |T(ω_1)|^2|T(ω_2)|^2|W(ω_1,ω_2)|^2 ω_1 ω_2/[∫_-∞^∞|T(ω)|^2S(ω)ω]^2 , where |T(ω)|^2 is the monochromator transmission function, W(ω_1,ω_2) is the cross spectral density function in the spectral domain, and S(ω) = W(ω,ω) is the spectral density function.We will assume in the following that monochromator transmission function is described by a Gaussian function with the r.m.s. width σ_ω |T(ω)|^2 = exp[ ω^2/2σ_ω^2] and pulsed x-ray radiation incoming on the monochromator can be approximated by a Gaussian Schell-model beam giving for the cross spectral density function <cit.> W(ω_1,ω_2)= W_0exp[-(ω_1-ω_0)^2+(ω_2-ω_0)^2/4Ω^2-(ω_1-ω_2)^2/2Ω_c^2], where W_0 is the normalization constant. Here ω_0 is the central pulse frequency, Ω is the spectral width, and Ω_c is the spectral coherence width. It can be shown <cit.> that these parameters can be related to the r.m.s. values of the pulse duration T_rms and coherence time T_c of the pulse before monochromator as <cit.> Ω^2 = 1/T_c^2 + 1/4T_rms^2 ;Ω_c = T_c/T_rmsΩ .Now substituting Eqs. (<ref> - <ref>) into the expression for the contrast function Eq. (<ref>) and performing integration we obtain ζ_2(σ_ω) = 2C/√(4A^2-B^2) , where A = 1/2σ_ω^2 + 1/2Ω^2 + 1/Ω_c^2 ; B = 2/Ω_c^2 ; C = 1/2σ_ω^2 + 1/2Ω^2 . This is the general expression for the contrast function for arbitrary values of all frequencies introduced in this expression. Now taking into account that in the conditions of our experiment at LCLS the monochromator bandwidth σ_ω and spectral coherence width Ω_c were much narrower than the spectral width Ω (σ_ω, Ω_c ≪Ω) we obtain for parameters (<ref>) the following approximate expression A ≃1/2σ_ω^2+ 1/Ω_c^2 ; B = 2/Ω_c^2 ; C ≃1/2σ_ω^2 . Substituting these values in expression (<ref>) we obtain for the contrast function ζ_2(ω) = Ω_c/√(Ω_c^2+4σ_ω^2) = 1/√(1+4(σ_ω/Ω_c)^2) . Taking now into account that in the conditions of our experiment at LCLS coherence time of radiation before the monochromator was much shorter than the pulse duration (T_c ≪ T_rms) we obtain from Eqs. (<ref>) for the pulse duration T_rms≃1/Ω_c . Substituting this expression in Eq. (<ref>) we obtain for the contrast function the following relation ζ_2(ω) = Ω_c/√(Ω_c^2+4σ_ω^2) = 1/√(1+4(T_rmsσ_ω)^2) . that was used in the main text of the manuscript for the analysis. In two limiting cases T_rmsσ_ω≪ 1 and T_rmsσ_ω≫ 1 we obtain from equation (<ref>) for the contrast function: ζ_2(σ_ω) ≃ 1-2 (T_rmsσ_ω)^2 in the first case and ζ_2(σ_ω) ≃ 1/[2(T_rmsσ_ω)] in the second. The first case corresponds to nearly Fourier limited radiation and the second one to rather incoherent (in time-domain) radiation. Expression (<ref>) can be inverted to determine pulse duration of the x-ray radiation before the monochromator. For the FWHM of the pulse duration we finally have T =2.355 T_rms = 2.355/2σ_ω√(1/[ζ_2(σ_ω)]^2-1) .§ APPENDIX III. ADDITIONAL EXPERIMENTAL RESULTS Here we present additional figures demonstrating our experimental results. Projections of the averaged intensity on the both horizontal and vertical axes (shown in Fig. <ref>) reveal presence of the small subpeaks due to the defect structure of the colloidal crystal. Cross sections of the intensity-intensity correlation function along the white line in Fig. 3 in the main text (see Fig. <ref>) demonstrate a relatively flat region in the center and a steep slope after that. § APPENDIX IV. SIMULATION OF THE INTENSITY-INTENSITY CORRELATION FUNCTIONS The model used for simulations in this work was first introduced in Ref. <cit.>. In this model, the X-ray beam is assumed to consist of several statistically independent Gaussian Schell-model beams with the total complex field amplitude E_Σ(𝐫,ω)= ∑_i=1^N E_i(𝐫,ω) , where E_i(𝐫,ω) is a complex amplitude of a single beam. Since all beams are statistically independent, the total spectral cross-correlation function W^(2)_Σ(𝐫_1,ω_1,𝐫_2,ω_2) and spectral density S_Σ(𝐫,ω) can be expressed as W^(2)_Σ(𝐫_1,ω_1,𝐫_2,ω_2)=∑_i=1^NJ_i(𝐫_1, 𝐫_2)W_i(ω_1, ω_2) , S_Σ(𝐫,ω)=∑_i=1^NI_i(𝐫)S_i(ω) . Intensity-intensity cross-correlation function is than calculated as obtained in Ref. <cit.> g_Σ^(2)(𝐫_1,𝐫_2) =1+ +∑_i,j=1^NJ_i(𝐫_1,𝐫_2)J^*_j(𝐫_1,𝐫_2)∬_-∞^∞|T(ω_1)|^2|T(ω_2)|^2W_i(ω_1, ω_2)W^*_j(ω_1, ω_2)ω_1 ω_2/∑_k, l = 1^NI_k(𝐫_1)I_l(𝐫_2)∫_-∞^∞|T(ω_1)|^2 S_k(ω_1)ω_1∫_-∞^∞|T(ω_2)|^2 S_l(ω_2) ω_2 . The model for simulating the fluctuating detector background was also introduced in Ref. <cit.>. The total intensity can be represented as I(x) = I_0(x) +I_B(x), where I_0(x) is the intensity of the beam and I_B(x) is the background intensity. The background signal is assumed to be statistically independent from the beam intensity fluctuations. It is then possible to express the normalized intensity-intensity correlation function modified by fluctuating background as g^(2)(x_1, x_2) = ⟨ I_0(x_1) I_0(x_2) ⟩ + ⟨ I_B(x_1) I_B(x_2) ⟩ + ⟨ I_0(x_1)⟩⟨ I_B(x_2) ⟩ + ⟨ I_B(x_1)⟩⟨ I_0(x_2) ⟩/(⟨ I_0(x_1)⟩+⟨ I_B(x_1)⟩)(⟨ I_0(x_2) ⟩+⟨ I_B(x_2)⟩), where the ensemble average ⟨ I_0(x_1) I_0(x_2) ⟩ = g_Σ^(2)(x_1,x_2)·⟨ I_0(x_1)⟩⟨ I_0(x_2)⟩. The background average intensity and intensity-intensity correlation function are assumed to have the form ⟨ I_B(x) ⟩= C , ⟨ I_B(x_1) I_B(x_2) ⟩= C^2(1+Aδ_x_1,x_2), where C<<max(I_0) and the background signal is therefore not significant in the center of the beam. The final expression that was used for modeling as it follows from Eqs. (<ref> - (<ref>)) has the form g^(2)(x_1, x_2) = g_Σ^(2)(x_1,x_2)⟨ I_Σ(x_1)⟩⟨ I_Σ(x_2)⟩ + C^2(1+Aδ_x_1,x_2) + C ⟨ I_Σ(x_1)⟩ + C ⟨ I_Σ(x_2) ⟩/(⟨ I_Σ(x_1)⟩+C)(⟨ I_Σ(x_2) ⟩+C).In simulations we used two models (see Table <ref>): one in the horizontal direction consisting of a single beam with the size (r.m.s.) 1.6 mm and coherence length 10 mm and second one in the vertical direction consisting of two beams shifted by 1.5 mm and with the relative intensity of 10%. The background level was considered to be 2% of the total intensity in both cases and parameter A in Eq.(<ref>) was taken as A=0.125. All further details of all parameters in both models are listed in Table <ref>.§ APPENDIX V. SPATIAL DEGREE OF COHERENCE AND CONTRAST VALUES Our experimental results also allowed us to determine the degree of spatial coherence ζ_S of LCLS radiation for hard x-rays. Similar to our previous work <cit.>, we obtained this value by applying the following relation ζ_S = ∫| W(𝐫_1, 𝐫_2)|^2 d𝐫_1 d𝐫_2/(∫⟨ I(𝐫) ⟩ d𝐫)^2 = ∫| μ (𝐫_1, 𝐫_2)|^2 ⟨ I(𝐫_1) ⟩⟨ I(𝐫_2) ⟩ d𝐫_1 d𝐫_2/(∫⟨ I(𝐫) ⟩ d𝐫)^2 and substituting values of | μ (𝐫_1, 𝐫_2)| obtained from the HBT interferometry analysis. Performing this analysis we determined the spatial degree of coherence for each Bragg peak for both samples see Table <ref>.Evaluation of the contrast values was performed based on their values determined from the main diagonal of intensity-intensity correlation function g^(2)(x, x). As a final value the mean value of g^(2)(x,x) over the region of FWHM of the averaged Bragg peak intensity ⟨ I(x) ⟩ was considered. § APPENDIX VI. MODE ANALYSIS AND ELECTRON BUNCH ENERGY FILTERING Jitter in the energy of the electron bunch introduces additional problem for the analysis of the monochromator filtered radiation. If electron bunch energy of a pulse is significantly different from an average, the central wavelength of the pulse is too far removed from the monochromator transmittance band. In such a case pulse intensity after monochromator will be significantly reduced, affecting observed statistics. This is clearly observed in Fig. <ref>, where the distribution of pulse intensities and corresponding electron bunch energies is shown.Therefore, it is important to filter the collected pulses by bunch energy. The filtering was performed by choosing only the pulses for which E_el-⟨ E_el⟩ < σ_E_el/2, where E_el is the electron bunch energy and σ_E_el^2 is the energy dispersion (see Fig. <ref> for the region considered for the following analysis). Around 50,000 pulses were left in each run after the filtering. The difference in the intensity distribution because of filtering can be seen if Fig. <ref>, where the histogram of the inegrated intensity from the Bragg peak in the diffraction pattern before and after the electron bunch filtering is shown. The number of modes is clearly underestimated without filtering.The pulse duration was also determined by using the mode analysis of the radiation as suggested inRef. <cit.>. According to this approach an average number of modes of radiation M is inversely proportional to the normalized dispersion of the energy distribution, that in our case coincide with the contrast function defined in Eq. (<ref>) M=1/[ζ_2(σ_ω)]. Substituting this relation in Eq. (<ref>) we obtain for the pulse duration T = 2.355/2σ_ω√(M^2-1) . We determined the number of modes by fitting integrated intensity distribution at one of the Bragg peaks by Gamma distribution <cit.> (see Fig. <ref>). As a result, the number of longitudinal modes was M ≈ 2.3±0.1 and reproducible between different runs. Substituting this number in Eq. (<ref>) gives for the pulse duration 11.5 ± 0.5 fs. | http://arxiv.org/abs/1707.08828v1 | {
"authors": [
"O. Yu. Gorobtsov",
"N. Mukharamova",
"S. Lazarev",
"M. Chollet",
"D. Zhu",
"Y. Feng",
"R. P. Kurta",
"J. -M. Meijer",
"G. Williams",
"M. Sikorski",
"S. Song",
"D. Dzhigaev",
"S. Serkez",
"A. Singer",
"A. V. Petukhov",
"I. A. Vartanyants"
],
"categories": [
"physics.optics"
],
"primary_category": "physics.optics",
"published": "20170727121813",
"title": "Diffraction based Hanbury Brown and Twiss interferometry performed at a hard x-ray free-electron laser"
} |
LPCC]O. Lopezcorrespondingauthor [correspondingauthor]Corresponding author [email protected],HORIA]M. Pârlog IPNO]B. Borderie IPNO]M.F. Rivetfn [fn]Deceased LPCC]G. Lehaut IPNO,TAMU]G. Tabacaru IPNO]L. Tassan-got IJPAN]P. Pawłowski GANIL]E. Bonnet LPCC]R. Bougault GANIL]A. Chbihi NAP,IPNO]D. Dell'Aquila GANIL]J.D. Frankland IPNO,CNAM]E. Galichet LPCC,GANIL]D. Gruyer NAP]M. La Commara LPCC]N. Le Neindre LNS,NAP]I. Lombardo EAMEA,LPCC]L. Manduci GANIL,CENBG]P. Marini LPCC]J.C. Steckmeyer IPNO]G. Verde LPCC]E. Vient GANIL]J.P. Wieleczko [LPCC]Normandie Université, ENSICAEN, UNICAEN, CNRS/IN2P3, F-14000 Caen, France [HORIA]Horia Hulubei National Institute for R&D in Physics and Nuclear Engineering (IFIN-HH), P.O.BOX MG-6, RO-76900 Bucharest-Màgurele, Romania [IPNO]Institut de Physique Nucléaire, CNRS/IN2P3, Univ. Paris-Sud, Université Paris-Saclay, F-91406 Orsay cedex, France [TAMU]Cyclotron Institute, Texas A&M University, MS 3366 College Station, Texas 77843, USA [IJPAN]Institute of Nuclear Physics PAN, ul. Radzikowskiego 152, 31-342 Krakow, Poland [GANIL]GANIL, CEA-DSM/CNRS-IN2P3,B.P. 5027, F-14076 Caen cedex, France [CNAM]Conservatoire National des Arts et Métiers, F-75141 Paris Cedex 03, France [NAP]Dipartimento di Scienze Fisiche e Sezione INFN, Università di Napoli “Federico II”, I80126 Napoli, Italy [LNS]INFN - Laboratori Nazionali del Sud, Via S. Sofia 62, 95125 Catania, Italy [EAMEA,LPCC]EAMEA, CC19 F-50115 Cherbourg-Octeville Cedex, France [CENBG]CEN Bordeaux-Gradignan, Le Haut-Vigneau, F-33175 Gradignan Cedex, France Profiting from previous works done with the INDRA multidetector <cit.> on the description of the light response ℒ ofthe CsI(Tl) crystals to different impinging nuclei <cit.>, we propose an improved Δ E - ℒ identification-calibration procedure for Silicon-Cesium Iodide(Si-CsI) telescopes, namely an Advanced Mass Estimate (AME) method. AME is compared to theusual,simple visual analysis of the corresponding two-dimensional map of Δ E - E type, by using INDRA experimental data from nuclear reactions induced byheavy ions in the Fermi energy regime. We show that the capability of such telescopes to identify both the atomic Z and the mass A numbers oflight and heavy reaction products, can be quantitatively improved thanks to the proposed approach. This conclusion opens new possibilities to use INDRA for studying these reactions especially with radioactive beams. Indeed, the determination of the mass for charged reaction products becomes of paramountimportance to shed light on the role of the isospin degree of freedom in the nuclear equation of state <cit.> . § INTRODUCTION One of the present motivations for investigating heavy-ion collisions at intermediate energies consists of improving our understanding of the equation of state for nuclear matter with the isospin degree of freedom. The advent of new accelerators, providing high intensity radioactive beams will cover a broad range of isospin (N/Z) ratios. Jointly, new detection arrays like FAZIA <cit.>, which fully exploit pulse shape analysis from silicon detectors, are under construction to benefit from these future possibilities. Information on the isospin dependence of the nuclear EOS can then be obtained by properly choosing projectile-target colliding systems. To improve the present experimental capabilities in this framework, we present a newAdvanced Mass Estimate (AME) approach, based upon thetelescope technique for INDRA Silicon-CsI telescopes <cit.>. This approach will extend the isotopic identification to nuclear reaction products heavier than those commonly identified with standard Δ E - ℒtwo-dimensional correlations. Here, Δ E indicates the energy lost in the 1^st Silicon stage (Si) of the telescope and ℒ the scintillation light produced in the 2^nd stage, made by a CsI(Tl) scintillator crystal read by a photomultiplier and corresponds to the residual energy E = E_0 depositedbyenergetic charged reaction products. The main difficulties for identifyingthe mass number over a broad range of elements are related to the non-linear energy response of each of the two stages and, in particular, of the scintillator. Actually, the light response of the scintillator strongly depends on the reaction product identity (charge and mass), whichmakes difficult even thedetermination of the deposited energy. At present time, the isotopic identification is visually achieved only for light nuclei from hydrogen up to (roughly)carbon isotopes for most of the INDRA Si-CsI telescopes.For some specific telescopes with smaller thickness - 150 μ m instead of 300 μ m -, an increased gain has been used in order toimprove the energy resolution and hence the isotopic separation during the 5^th INDRA campaign performed at GANIL a few years ago. In doingso, the isotopic identification for these specific telescopes has been slightly augmented up to oxygen isotopes for the best cases.To improve and optimize information coming from INDRA Si-CsI telescopes as far as the mass number is concerned, we started from the pionneering works of Pârlog et al. <cit.> which provide anaccurate physical description of the light response produced by the CsI(Tl) crystals. In these articles, two formulas havebeen derived concerning the relation between the light signal ℒ, the atomic number Z, the mass number A and the incident energyE_0 of a reaction product detected by a CsI scintillator. The proposed method was then used and tested on data recorded withINDRA during the fifth campaign, with telescopes having as first stage 300 μ m or 150 μ m-thick Silicon detectors. These experimentaldata were obtained by bombarding ^112,124Sn targets with ^124,136Xe beams at 32A MeV and 45A MeV.The paper is organized as follows. In section (II), we recall the main results of references <cit.> concerning the role of quenching and knock-on electrons in scintillation light from the CsI(Tl) crystals and show the quality of the analytical description. Section (III) describes the Advanced Mass Estimate(AME) method and the comparisons with standard INDRA isotopic identification. In section (IV) determination and uncertainty on A are discussed. In Section (V) we present a summary of this work.§ QUENCHING AND KNOCK-ON ELECTRONS (Δ-RAYS) IN SCINTILLATION LIGHT OF CSI(TL) CRYSTALS Cesium iodide scintillators, CsI(Tl), doped with thallium at a level of 0,02 - 0,2 % molar concentration, are inorganic crystals where the scintillation light is produced by the activation (excitation) of the thallium atoms encountered by the carriers (electrons andholes) produced during the motion of the incoming charged product. The activationresults in an emission of light by the excited thalliumatoms in the green band at 550 nm. The differential scintillation light output dℒ/dE as a function of energy E is often describedby means of the Birks formula <cit.>: d ℒ/dE=𝒮1/1+𝒦ℬ(dE/dx),𝒮 being the scintillation efficiency and 𝒦ℬ the quenching coefficient. The differential light decreases asthe stopping power (dE/dx) increases; this is the so-called quenching effect, more pronounced for the heavier ions leading to high carrierconcentrations. Under the approximation (dE/dx) ∝ AZ^2/E, the integral over the variable E of the above equationprovides a simple formula for the total light response ℒ <cit.> as a function of the initial energy E_0 of the detected ion: ℒ(E_0) = ∫_0^E_0ℒ(E) dE = a_1 E_0[1-a_2AZ^2/E_0ln(1+1/a_2 AZ^2/E_0)], The gain coefficient a_1 includes both the scintillation efficiency andthe electronic chain contribution to the signal amplification. The quenching coefficient a_2 is mainly related to the prompt direct recombinationof part of the electrons and holes, which thus are not participating to the excitation of the activator atoms. The expressions (<ref>) and (<ref>) were used, with reasonable results <cit.> in the case of light charged particlesor Intermediate Mass Fragments (IMFs) of rather low energy per nucleon E/A, i.e. as long as the contribution to the light response of theknock-on electrons or δ-rays, escaping the fiducial volume of very high carrier concentration close to the trajectory of the particle/ion,remains unsignificant. Actually, above a certain energy per nucleon threshold e_δ=E_δ/A, the incident particle/ionstarts to generate these rapid electrons, which are characterized by a small stopping power. Consequently, the fraction ℱ(E) -firstly introduced by Meyer and Murray <cit.> -, of the energy dE deposited into a slice dx and carried off by the knock-on electronsis practically not affected by quenching. The δ-rays increase thus the light output and this should be necessarily taken into account atenergies higher than a few MeV/nucleon, especially for heavier ions.As it penetrates into a CsI crystal, an energetic charged particle/ion is gradually loosing its energy (from E_0 to 0) mainly by ionization- the electronic stopping power -, leading to the scintillation, but also, in a smaller extent, by interacting with the host lattice nuclei- the nuclear stopping power -, lost for the radiative transitions. Both stopping powers can be quantitatively predicted, e.g. by using Ziegler tables <cit.>appealing to the work of Lindhard et al. <cit.>. Within the INDRA collaboration, we use stopping power tables for heavy ions in solids from Northcliffe and Schilling at low energies <cit.> and from Hubert and Bimbot at highenergies <cit.>, both matched at 2.5 MeV/nucleon. They provide quite accurate results in the low and intermediate energy range, i.e. from few hundreds of keV/nucleonup to 100 MeV/nucleon, of interest here. More than a decade ago, Pârlog et al. <cit.> put in evidence the role of the two types of energy loss to thequenching and also found the dependence of the fraction ℱ(E) on the instantaneous velocity (or energy per nucleon E/A). They disentangledthe contributions of the carriers produced in the main particle track and of the δ-rays to the scintillation too. The authors quantified theseprocesses in a simple Recombination and Nuclear Quenching Model (RNQM) connecting the exact value of the total emitted light ℒ to both the electronic and nuclear infinitesimal stopping powers along the incident particle track via numerical integration <cit.>. The modelcontains Eq. (<ref>) as a particular case. Under well argued approximations, they derived a more friendly analytical formula relatingℒ to the quantities Z, A and E_0 <cit.>: ℒ(E_0) = a_1 E_0[1-a_2 AZ^2/E_0ln(1+1/a_2 AZ^2/E_0)+a_2 a_4 AZ^2/E_0ln(E_0+a_2 AZ^2/a_3A+a_2 AZ^2)],for an incident energy E_0 in the CsI(Tl) higher than the threshold E_δ at which the δ-rays start to be generated. Besides thecoefficients a_1, a_2, with the same physical signification as above in Eq. (<ref>), two others appear: the energy per nucleon a_3=e_δ, (a few MeV/nucleon)and a_4 = ℱ - the fraction (a few tenths of percents) of energy - they are carrying off, taken as a constant irrespective of current energy E alongthe particle path above E_δ. At low energy (E ≤ E_δ), ℱ = 0 and only the first termis present, then Eq. (<ref>) is reduced to Eq. (<ref>). These four parameters have then to be evaluated by using a number of suitable calibrationpoints by a fit procedure.The relation (<ref>) is purely analytical and can then be easily implemented for calibration purpose. It is less accurate than theexact treatement provided in RNQM <cit.> especially at low energy. One drawback is also the step function used for ℱ(ℰ),which jumps from 0 to a_4 at E = E_δ in order to allow the analytical integration over E. This introduces a discontinuity in the functionℒ(E) at this connection point, especially for very heavy fragments <cit.>. Nevertheless, it may be ad hoc improved by slightlyimproving the continuity of the fraction ℱ(E) around E_δ. In this work, we consider that the use of the analytical expressionwill only marginally affect the results, taking into account the intrinsic quality of the Silicon wafers and of the CsI crystals of the INDRAtelescopes, which does not secure the precision required to appreciate such discrepancies. Moreover, the total light ℒ emitted by theCsI(Tl) scintillators is not directly measured, but reconstructed, through the procedure described by Pârlog et al. <cit.>, starting from two components ofthe scintillation light measured by integrating the signal in the fast and slow time gates <cit.>. Nevertheless, for a more rigorous and accurate treatment, the use of the exactformulation of RNQM <cit.> is preferable when possible, for example with high-quality detectors such as FAZIA Si-CsI telescopes. This will be the subject of aforthcoming paper.As an example of the quality attained with our analytical description for the scintillation light in CsI(Tl) crystal, Fig. <ref> displays the energy-light correlationE_0-ℒ using Eq. (<ref>) superimposed on INDRA data concerning the system ^136Xe+^124Sn at 32A MeV, for a specific Si-CsItelescope. Each full/coloured line in Fig. <ref> corresponds to a given nucleus with an atomic number Z and a mass number A. We have chosen here to displayisotopic lines with A=2Z+1 for even-Z nuclei. We will see in the following that this mass assumption is quite reasonable for IMFs when considering theneutron (n)-rich system ^136Xe+^124Sn. For a given energy E_0- determined as shown in the next section -,the heavier the nucleus, the smaller the light value ℒ is;this is a direct consequence of the ratio nuclear/electronic stopping powers, and also of the quenching effect. Both quantities increase with thecharge and the mass of the fragment and decrease when E_0 increases. Additionaly, above a certain velocity,δ-electrons are generated, very efficient for light production. These are the reasons why the curvature of the different isotopic curvesshown in Fig. <ref> evolves toward a linear behavior at higher light/energy, here ℒ>600. It is worthwhile to mention that the δ-rays contribution to the lightis quite large, reaching20-50% for Z>20, as pointed out in Ref.<cit.> and must be definitely included in order to reproduce the experimental data.To obtain the results displayed in Fig. <ref>, we have used calibration points coming from secondary light beams stopped in CsI detectorsfrom Z=1 up to Z=5 together with punched through events in the Silicon layer when possible. In a twodimensional Δ E - ℒ plot, these points are close to the ordinate Δ E axis, i.e. to fragment energies slightly higher than that necessary to traversethe Silicon stage of the telescope and to reach the CsI(Tl) one with a quite small residual energy, sufficient however to be seen in thescintillator stage.In order to better appreciate the performances concerning the isotopic identification in INDRA CsI telescopes, we display in Fig. <ref> the correlation between theenergy and the CsI light signal (same as for Fig. <ref>) for 4 selected elements (carbon, fluorine, magnesium and sulphur), and for systems withdifferent neutron content: ^124Xe+^112Sn and ^136Xe+^124Sn at 32A MeV. We can observe a significative difference between the two systems concerning the neutron richnessof the produced fragments (higher masses for the (n)-rich system in blue) as one could expect from simple physical arguments. It is worthwhile to mention that this result requiresindeed a very good stability for the CsIlight response. This is done in INDRA by monitoring a laser pulse all along the data taking <cit.>. Thus, Fig. <ref> suggests that the CsIlight signal can help to discriminate the different isotopes, here at least up to Z=16 (sulphur). In the following, we will use this additional valuable information to improve theusual Δ E - ℒ identification method for heavier elements than typically done up to carbon or nitrogen. To illustrate the overall sensitivity of the Si-CsI(Tl) telescopes to the mass number, Fig. <ref> displays the Δ E - ℒ correlation bidimensional matrix of the 2^ndmodule (including a 300 μ m-thick Si) for the 1^st ring (2°≤θ≤ 3°) of INDRA, and for ^124Xe and ^136Xe projectiles on^124Sn at 32A MeV and 45A MeV bombarding energies. The bright/yellow spots, indicated by arrows on the borders of the geometricalloci for Z = 54 (two in the region of the 32A MeV incident energy and other two in that of 45A MeV one) correspond in both cases to the (n)-poor or (n)-richprojectiles, respectively. These findings indeed show the good sensitivity of the response of INDRA telescopes to the mass of the detected ejectile, thus calling for a deeper analysis of the experimental data as presented hereafter.§ ADVANCED MASS ESTIMATE (AME) IN INDRA SI-CSI TELESCOPESIn this section, we are going to present the new AME identification method in details.We use information given by the energy lost in the Silicon detector, Δ E, and the atomic number Z taken from the usual Δ E-ℒidentification method in a Si-CsI map (see Fig. <ref> for example). Doing so, we benefit from the previous identification works done for INDRA data : Zidentification in Si-CsI matrices by semi-automatic <cit.> or handmade grids and the careful calibration of the Silicon detector, by means ofα particle source and secondary beams stopped in this layer <cit.>. For heavy ions (Z>15), the Pulse Height Defect (PHD) in this detector canbe large <cit.> and has to be carefully evaluated. For INDRA, we use the elastic scattering of low-energy heavy ion beams (Ni and Ta at 6 AMeV)which are stopped in Silicon detectors. Traditionally, we quantify the PHD as a function of the atomic number, the energy of the particle and the quality of thedetector, according to Moulton formula <cit.>.For a given element characterized by its atomic number Z, the measured energy Δ E deposited in the first layer of the Si-CsI(Tl) telescope depends on the velocity,or the initial energy and the mass of the incident particleand, in principle, it can not provide by itself the two quantities without ambiguity. To perform consistently the isotopic identification in Si-CsI matrices, we thenassume a starting value A_0 for the mass number concerning one detected nucleus with its atomic number Z and, by constraining the energy loss Δ E in the Silicon stage at themeasured value, we compute both the total energy at the entrance of the Silicon stage and the residual energy E_0 deposited in the CsI(Tl) by using the above-mentioned range and energy loss tables <cit.>. This procedure imposesalso to accurately evaluate the thickness of the Δ E Silicon detector. The value of the scintillation light ℒ given by Eq. (<ref>) for the residual energy valueE_0 associated to this starting valueof A is then compared to the experimental light output ℒ_exp from the CsI(Tl). In order to determine the best 'theoretical' valueℒ(E_0), we iterate on mass number A (and consequently on the value of E_0) until we find the best agreement between the theoretical andexperimental values of the light, always compatible with the energy lost in the Silicon stage. It is worthwhile to mention that the mass number is aninteger and, as such, is varied by increment of one mass unit. At the end of the iteration, we get an integer mass number, giving the best agreement for the experimentally determined quantities Δ E and ℒ_exp as displayed in Fig. <ref>. This is the basis of the AdvancedMass Estimate (AME) method, which, by making use in a consistent way of the experimental quantities Δ E and ℒ_exp, brings a more accurate informationon both the mass and the residual energy (and consequently the total energy too).As one may guess from Fig. <ref>, the calibration for the Silicon detector should be as accurate as possible to perform the best isotopic identification.The formula given by <cit.> used to calculate the PHD does not depend on the ion mass. This is certainly an advantage as it simplifies our approach, but itmay become a drawback too. Even if the calibration of the Silicon stage can be considered to be rather accurate, we estimate that it represents at presenttime one of the known limitations for the extension of the identification method toward heavier nuclei (Z>30). Nevertheless, we will see in the following that it does not hamper very much the isotopic identification for such heavy products. In the next sections, we will estimate the performances of this new identification procedure. As a first step, we will benchmarkthe new method for light nuclei where isotopic identification is already achieved (1 ≤ Z ≤ 6-8) with traditional methods. In a second step, we will then get somequantitative values concerning the improvement for the isotopic identification of heavier nuclei, up to xenon isotopes in our case.§.§ Benchmark with the standard Δ E -ℒ method Using the AME method, we obtained isotopic distributions of light nuclei that have been compared to the ones obtained with the standard method (making use of semi-automatic <cit.> or handmade grids) for INDRA Si-CsI telescopes. Fig. <ref> displays the isotopic distributions obtained by the new AME method (filled histograms)and the standard Δ E-ℒ one, using standard grids (empty histograms), from lithium (Z=3) up to oxygen (Z=8) isotopes. The numbers indicate the isotope masses. Forthe meaning of the colours of these numbers : black or gray/red, see section (IV). The Particle IDentifier (PID) defined asPID=8Z+A, and allowing to separately observe the neighbouring elements, was chosen as abscissa for this representation. The modules incorporating silicon detectors of only 300 μ m thickness were kept for this representation. We observe an overall good agreement for the most probable isotopes, found as having themass number A = 2Z+1 as already discussed for Fig. <ref>. We also notice that the new method can still provide isotopic identification for less abundant species ((n)-rich and (n)-poor carbon to oxygen isotopes for example) since it does not use any visual recognition to build the grids for which a sufficiently large production cross-section is needed.This is clearly an improvement compared to the standard methods since it allows to recover the overall isotopic distributions for a given element Z, at least in this range of atomic numbers Z=3-8. This new feature is welcome for studying isospin effects as for example isotopic yieldsor isoscaling <cit.>. To complete the benchmark on light nuclei, we also present in Fig. <ref> theisotopic distributions obtained for the specific 150 μ m-thick Silicon detectors with a high gain, but for lower statistics. These ones allow to betterdiscriminate the isotopes for light IMFs (up to Z≈8) and constitute a more stringent test for the comparison. Actually, the mass distribution forthe carbon isotopes given by the standard method becomes now significantly larger, closer to that provided by the AME method, which recovers more exotic species.We also notice that even the yield for the most probable isotopes given by the two methods are sometimes not the same, due to the absence of grids for some telescopes where thevisual inspection does not permit to define properly the isotopes curves and boundaries. This is particularly true for Z=7-8. In Figs. <ref> and <ref>,the black numbers indicate the masses estimated with an uncertainty lower than one mass unit, while the grey/red ones,those affected by higher uncertainty. This specific point is developed in section (IV). §.§ Comparison with different isospin systems To extend and confirm the previous results, we checked the isotopic identification by means of the AME method for two systems with different isospins : ^124Xe+^112Snand ^136Xe+^124Sn at the same incident energy per nucleon of 45 AMeV without any event selection except here a common trigger multiplicity M=1.These latter are also part of the data extracted from the 5^th INDRA campaign performed at GANIL. We could reasonably expect an overproductionof (n)-rich isotopes in the case of the (n)-rich ^136Xe+^124Sn system, for light nuclear fragments. In thefollowing, we compared the isotopic distributions obtained for both systems, in order to see whether we observe any difference reflecting the possible differentproduction yields for a given element Z.Fig. <ref> shows the isotopic distributions from lithium to phosphorus isotopes, provided by Silicon-CsI telescopes. We display here only the results for the 300 μ m-thickSilicon ones. We can observe a global shift of the isotopic distribution toward more (n)-rich species for the (n)-rich system (^136Xe+^124Sn) as compared to the (n)-poor one(^124Xe+^112Sn). If we consider the most abundant isotope per element, it is ^7Li instead of ^6Li and ^17O instead of ^16O for example, together with the enhancementof very (n)-rich isotopes production for the (n)-rich system (^136Xe+^124Sn) as one could expect. This illustrates the fact that the isotopic distributions determined withAME are not an artifact of the method but they truly could be associated to the genuine (physical) isotopic distributions. § QUALIFYING THE ISOTOPIC IDENTIFICATION The isotopic identification can be further qualified by some specific operations. More precisely, we can provide a quite accurate estimate for the mass numberA even if the full isotopic resolution is not achieved. We remind that, knowing the thickness of the Silicon detector (300 μ m all alongthis section), the atomic number Z value of a detected fragment and the well determined Δ E, corrected for the PHD <cit.>, we canstart by proposing an atomic mass A_i number and compute the corresponding residual energy E_0i in the CsI stage using the energy loss tables. They areconnected to the calculated scintillation light ℒ_i = ℒ(E_0i,A_i) via the Eq.(<ref>). Then, the integer mass number is varied, by stepsof one unit, in order to minimize the quantity |ℒ_i-ℒ_exp|/ℒ_exp, in such a way that the measured Δ E value be reproducedtoo. After a few iterations, the best integer value A^* of A_i and the related value of E_0i are found, characterized by the shortest normalized distanced_i=|ℒ_i-ℒ_exp|/ℒ_exp between the calculated ℒ_i and experimental ℒ_exp light. Finally, to get a representative valueA_est for the estimated mass number in a single event (one experimental point in the Δ E - ℒ plot) at a given Δ E, we simply weightthe different A_i values by the inverse of d_i^2 as: A_est =1/Σ_i 1/d_i^2Σ_i A_i/d_i^2We shall exploit thus not only the mass number as an integer but directly the PID, defined above as: PID=8Z+A_est, by letting now the mass number A_estto be a real number. Of course, for an experimental light ℒ_exp, the main contributions to A_est are coming from the shortest distances d_i.Doing so, we can obtain an estimation concerning the uncertaintyΔ A by taking the absolute difference between the optimum value A^*, correspondingto the smallest distance d_i and the weighted value A_est obtained with the Eq. (<ref>):Δ A=|A^*-A_est| If the two values A_est and A^* are close enough (Δ A<0.5, so comprised in one unit range), we assume a full isotopic identification, whereas if Δ A ≥ 0.5,we have only a limited isotopic identification. This procedure can be therefore considered as a simple and easy way to qualify the isotopic identification.This is illustrated by the black and grey/red numbers on Figs. <ref> and <ref>. The black numbers refer to Δ < 0.5 whereas the grey/red ones to Δ A ≥ 0.5.In order to further evaluate the validity of the method, we have also used INDRA results for the four different systems: ^124,136Xe+^112,124Snat 32 AMeV. Several tests are then proposed in the following. First, we have looked to data at the most forward angles, from rings 1 - 5, i.e. 2°≤θ≤ 15°. These ones are obtained from the system ^136Xe+^112Sn at 32 AMeV, by requiring a trigger multiplicity (fired telescopes) M ≥ 1, in orderto select mostly quasi-elastic events. From Fig. <ref>, we could indeed notice that we recover as main contribution the quasi-projectile (Z ≈ 54) in the most forward rings.For such high Z values, the isotopes are not visually separated in the Δ E - ℒ matrix; in the framework of the standard method, a hypothesis on the mass has to bemade for finding their velocities starting from the measured energies deposited in the 1^st stage of a telescope. Fig. <ref> displays the correlation between themass and the velocity parallel to the beam, for different ejectiles and for different mass estimates. The yellow stars indicate the maximum number of entries.The two upper panels of Fig. <ref> display the A-V_z correlation for the standard case (usual Δ E - E method), where only the atomic number Z is determinedfrom the Δ E - ℒ plot. In the upper panel, for Z=54 we took as mass hypothesis the prediction A=120 for the evaporation attractor line (EAL) <cit.> (see below).It leads to value V_z≈8.5 cm/ns of velocity parallel to the beam direction. In the middle panel, the β-stability hypothesis is used , and the mass for xenon is set toA=129, richer in neutron, and V_z≈8.2 cm/ns. In both cases, the values of the atomic mass A and the parallel velocity are peaked quite far from the expectedelastic contribution, in the present case: A=136 and V_z=7.9 cm/ns represented by the filled circles on Fig. <ref>. This is due to the incorrect values of the mass number A,simply calculated from the atomic numbers Z via different hypotheses. Consequently, the corresponding parallel velocities V_z are also incorrect since they were computed by meansof these hypothetical A values. At variance, we can notice that applying the iterative AME method - lower panel of Fig. <ref> -, the plotted distribution presents atA≈133 and V_z≈8.0 cm/ns a maximum located much closer to the elastic contribution. We could therefore infer that the obtained results with AME are more validfor the (n)-rich projectiles even for these very heavy ions detected in the region of quasi-elastic events. We also found the same conclusion for the proton (p)-rich system^124Xe+^112Sn at 32A MeV.Now, if we look at the PID distributions in Fig. <ref>, we may stress also the differences. In the upper panel, when we are not using the scintillation lightto determine the mass number (usual method), we can get some isotopic identification up to Z=6-8. By contrast, as shown in the lower panel, thanks to the new AME method, we are now able to distinguish a fair isotopic identification up to at least Z ≈ 12-13 for which we have Δ A ≤ 0.5 as obtained from Eq. (<ref>) for the mostabundant isotopes. The corresponding improvement concerning the isotopic resolution is indeed obtained by taking into account the additional information from the CsI crystal. We could also qualify the accuracy of the isotopic identification for Z much higher than 12 by taking advantage once again of the elastic channel for both reactions.For a trigger multiplicity M=1, an angular range between 2° and 7° (rings 1-3), and by selecting only the xenon nuclei (Z=54),we obtain the isotopic distributions displayed in Fig. <ref>. These latters are centered around A ≈ 124 for ^124Xe(the mass of the projectile) and A ≈ 133 for ^136Xe (three mass units smaller than the projectile). For the ^136Xe data, due to its neutron richness, onecould expect a loss of few neutrons for the projectile even in very peripheral collisions, transforming thus the elastic contribution into a quasi-elastic one. The resultsare therefore compatible with physical arguments and with those shown in the lower panel of Fig. <ref> (mass - velocity correlation). The width of these isotopic distributionsreflect indeed the convolution of the physical isotopic distribution as well as the uncertainty on the determined mass. We can therefore reasonably deduce thatthe uncertainty Δ A=3 found in Fig. <ref> could represent an upper limit for the mass uncertainty. Finally, we display in Fig. <ref> the N-Z charts for the reaction products in the above-mentioned reactions, their masses being determined via the AMEprocedure. We have selected the trigger condition M ≥ 4, thus removing the major part of the quasi-elastic contribution presented in the previous Figs. <ref>-<ref>.The grey/coloured curves represent the evaporation attractor lines <cit.>, predicting the number of neutrons N as an integer functionof Z: the steeper/pink line, with N= 1.072Z+2.032 × 10^-3Z^2 recommended for Z < 50 and the more gentle slope/green line, withN= 1.045Z+3.57 × 10^-3Z^2 recommended for Z ≥ 50. The black curve indicates a 3^rd degree polynomial fit of the β-stability valley as the integer ofN= 1.2875+0.7622Z+1.3879 × 10^-2Z^2-5.4875 × 10^-5Z^3, with i.e. nuclei more (n)-rich than for the EAL lines. The Z-N charts in Fig. <ref> concern the forward detection angles: 2°≤θ≤ 45°(rings 1-9 of INDRA). These data seemto reflect mainly the ratio N/Z of the projectile and none of these hypotheses on the number N of neutrons, and consequently on A, is able to reproduce in average the results,especially for the (n)-rich ^136Xe projectiles (lower panels). This overall view pleads in favour of the AME procedure compared to a simple mass hypothesis.With the present method we can obtain a better calibration of very thick CsI(Tl) scintillators allowing at the same time the full detection of very energetic chargedreaction products and their mass determination with the best resolution. AME upgrades thus the 4 π INDRA array, designed to measure only the atomic number Z of the heavy nuclear fragments stemming from multifragmentation reactions, to a device able to estimate their mass A too, up to Z≈12-13 for an isotopic resolutionΔ A ≤ 0.5 and Z≈54 for Δ A ≤ 3, and this in a very compact geometry.§ CONCLUSIONWe have presented a method called Advanced Mass Estimate (AME), a new approach for isotopic identification in Si-CsI telescopes using the analytical formulationfor the CsI(Tl) light response provided in <cit.>. It includes explicitely the light quenching and the δ-rays contribution to the scintillation of the CsI(Tl)crystals. In this framework, we have shown that it is possible to use an iterative procedure to accurately calibrate the CsI detectors and, at the same time, toestimate the mass number A of the charged reaction products, besides the charge Z one, with a resolution better than the one previously achieved by standard techniques. This method allows to recover not only the isotopic distributions obtained by the usual visual techniques for Z=1-8, but it can also be extended to heavier nuclei up to Z≈12-13,with an uncertainty of one atomic mass unit for the telescopes of the INDRA array. In addition, from the comparison with experimental data, we have shownthat it is reasonably possible to estimate the atomic mass within 2-3 mass units up to xenon isotopes, if one is able to carefully evaluate the thickness and the pulse height defectin the Δ E silicon layer. We then consider that the qualityof INDRA Si-CsI experimental results can be dramatically improved by using the new AME method, and that is particularly well adapted to undergo analyseswith radioactive beams exploring a large N/Z domain. The AME method is not only suited for INDRA Si-CsI(Tl) telescopes but can be alsosuccessfully exploited with any charged particle array using the same kind of telescopes. Further studies concerning the implementation of the Recombination and Nuclear QuenchingModel with the exact treatement mentioned in the first section are currently in progress, by using high-quality data from FAZIA telescopes, and will be the subject ofa forthcoming paper.§ REFERENCES 0Pouthas J. Pouthas, et al., Nucl. Instr. and Meth. A 357 (1995) 418. Parlog1 M. Parlog, et al., Nucl. Instr. and Meth. A 482 (2002) 674. Parlog2 M. Parlog, et al., Nucl. Instr. and Meth. A 482 (2002) 693. WCI Dynamics and Thermodynamics with nuclear degrees of Freedom, Eur. Phys. J. A 30 (1) (2006), and references therein. EPJA50 Topical Issue on Nuclear Symmetry Energy, Eur. Phys. J. A 50 (2) (2014), and references therein. Bougault R. Bougault et al. (FAZIA collaboration), Eur. Phys. J. A50 (2014) 47. Carboni S. Carboni et al. (FAZIA collaboration), Nuclear Instruments and Methods in Physics Research A 664 (2012) 251. Birks J.B. Birks, The theory and practice of scintillation counters, Pergamon Press, Oxford (1964). Horn D. Horn, et al., Nucl. Instr. and Meth. A 320 (1992) 273. DeFilippo E. De Filippo, et al., Nucl. Instr. and Meth. A 342 (1994) 527. Meyer A. Meyer, R.B. Murray, Phys. Rev. 118 (1962) 98. Ziegler J. F. Ziegler, Handbook of Stopping Cross Sections for Energetic Ions in All Elements, vol. 5 of series "Stopping and Ranges of Ions in Matter", Pergamon Press, New York (1980). LSS J. Lindhard, M. Scharff and H.E Schiott, K. Dan. Vidensk. Selk. Mat. Fys. Medd. 33 (14) (1963) 1-42. Northcliffe L. C. Northcliffe and R. F. Schilling, Nuclear Data Tables A7 (1970) 233. Hubert F. Hubert, R. Bimbot, H. Gauvin, Atomic Data and Nuclear Data Tables 46 (1990) 1. Tassan-Got L. Tassan-Got, Nucl. Instr. and Meth. B 194 (2002) 503. Tabacaru G. Tabacaru, et al., Nucl. Instr. and Meth. A 428 (1999) 379. Moulton J.B. Moulton, J.E. Stephenson, R.P. Schmitt, G.J. Wozniak, Nucl. Instr. and Meth. 157 (1978) 325. Ademard G. Ademard et al. (INDRA collaboration), Eur. Phys. J. A50 (2014) 33. Charity R. J. Charity, Phys. Rev. C58 (1998) 1073. | http://arxiv.org/abs/1707.08863v1 | {
"authors": [
"O. Lopez",
"M. Parlog",
"B. Borderie",
"M. F. Rivet",
"G. Lehaut",
"G. Tabacaru",
"L. Tassan-got",
"P. Pawlowski",
"E. Bonnet",
"R. Bougault",
"A. Chbihi",
"D. Dell'Aquila",
"J. D. Frankland",
"E. Galichet",
"D. Gruyer",
"M. La Commara",
"N. Le Neindre",
"I. Lombardo",
"L. Manduci",
"P. Marini",
"J. C. Steckmeyer",
"G. Verde",
"E. Vient",
"J. P. Wieleczko"
],
"categories": [
"physics.ins-det",
"nucl-ex"
],
"primary_category": "physics.ins-det",
"published": "20170727134523",
"title": "Improving isotopic identification with \\emph{INDRA} Silicon-CsI(\\emph{Tl}) telescopes"
} |
(i)equationsection equationsection==thebibliography[1] #1.8 plain theoremTheorem[section] proposition[theorem]Proposition lemma[theorem]Lemma corollary[theorem]Corollary definition[theorem]DefinitionUscrmn boldUscrbn Uscr'177 Uscrmn<-6>rsfs5<6-8>rsfs7<8->rsfs10 Uscrbn<-6>rsfs5<6-8>rsfs7<8->rsfs10definition remark[theorem]Remark example[theorem]ExampleEquilibrium Returns with Transaction Costs[We are grateful to Michalis Anthropelos, Peter Bank, Paolo Guasoni, and Felix Kübler for stimulating discussions and detailed comments. Moreover, we thank an anonymous referee for his or her careful reading and pertinent remarks.] Bruno BouchardUniversité Paris-Dauphine, PSL, CNRS, UMR [7534], CEREMADE, 75016 Paris, France, email . Masaaki FukasawaOsaka University, Graduate School of Engineering Science, 1-3 Machikayama, Toyonaka, Osaka, Japan,email:and Tokyo Metropolitan University, Graduate School of Social Sciences. Support from KAKENHI Grant number 25245046 is gratefully acknowledged.Martin HerdegenUniversity of Warwick, Department of Statistics, Coventry, CV4 7AL, UK, email. Partly supported by the Swiss National Science Foundation (SNF) under grant 150101.Johannes Muhle-KarbeCarnegie Mellon University, Department of Mathematical Sciences, 5000 Forbes Avenue, Pittsburgh, PA 15213, USA, email . Parts of this paper were written while this author was visiting ETH Zürich; he is grateful to the Forschungsinstitut für Mathematik and H.M. Soner for their hospitality.December 30, 2023 ================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================= We study how trading costs are reflected in equilibrium returns. To this end, we develop a tractable continuous-time risk-sharing model, where heterogeneous mean-variance investors trade subject to a quadratic transaction cost. The corresponding equilibrium is characterized as the unique solution of a system of coupled but linear forward-backward stochastic differential equations. Explicit solutions are obtained in a number of concrete settings. The sluggishness of the frictional portfolios makes the corresponding equilibrium returns mean-reverting. Compared to the frictionless case, expected returns are higher if the more risk-averse agents are net sellers or if the asset supply expands over time.Mathematics Subject Classification (2010): 91G10, 91G80.JEL Classification: C68, D52, G11, G12.Keywords: equilibrium, transaction costs, FBSDEs.§ INTRODUCTION It is empirically well documented that asset returns depend on liquidity <cit.>. To understand the theoretical underpinning of such “liquidity premia”, we study a continuous-time risk-sharing equilibrium with transaction costs.[Liquidity premia with exogenous asset prices are studied by <cit.>, for example.] For tractability, we assume (local) mean-variance preferences and quadratic trading costs, levied on the agents' trading rates. Then, both the unique equilibrium return that clears the market and the corresponding optimal trading strategies can be characterized by a system of coupled but linear forward-backward stochastic differential equations (FBSDEs). These equations can be solved explicitly in terms of matrix power series, leading to closed-form expressions for the liquidity premia compared to the frictionless benchmark. If the risk aversions of all agents are homogenous and the asset supply remains constant over time, then the frictionless price dynamics still clear the market. As a consequence, illiquidity only affects trading strategies but not equilibrium prices in this case. By contrast, if the asset supply expands over time, positive liquidity premia are necessary to compensate the agents for the trading costs incurred when purchasing these additional shares. Nontrivial liquidity premia also arise with heterogenous preferences. Then, the more risk averse agents have a stronger motive to trade and therefore have to provide additional compensation to the less risk-averse ones. This leads to positive liquidity premia when the more risk averse agents are net sellers. With heterogenous preferences, illiquidity also makes expected returns mean reverting. This result does not depend on mean-reverting fundamentals, but is instead induced by the sluggishness of the frictional portfolios. With trading costs, allocations do not move directly to their stationary allocation but only gradually adjust over time, leading to autocorrelated return dynamics. For example, if endowment exposures have independent increments, then the liquidity premia have Ornstein-Uhlenbeck dynamics. If the agents exogenous trading needs are also mean-reverting, they enter the liquidity premium as a stochastic mean-reversion level. Illiquidity in turn determines the fluctuations of the actual equilibrium return around this value.From a mathematical perspective, our analysis is based on the study of systems of coupled but linear FBSDEs. Since their forward components are degenerate, general FBSDE theory as in <cit.> only yields local existence in this context. As we need global existence and uniqueness results, we provide a direct argument. Using the theory of primary matrix functions, we extend the univariate results of Bank, Soner, and Voß <cit.> to the multivariate settings needed to analyze the interaction of multiple agents trading several assets. In order to cover tractable stationary models as a special case, we also show how to extend this analysis to infinite time horizons under suitable transversality conditions. Related Literature Equilibrium models with transaction costs are notoriously intractable, because trading costs severely complicate the agents' individual optimization problems. Moreover, representative agents cannot be used to simplify the analysis since they abstract from the trades between the individual market participants.Accordingly, most of the literature on equilibrium asset pricing with transaction costs has focused either on numerical methods or on models with very particular simplifying assumptions. For example, <cit.> propose algorithms for the numerical approximation of equilibrium dynamics in discrete-time, finite-state models. In contrast, <cit.> obtain explicit formulas in continuous-time models but focus on settings with deterministic asset prices for tractability. Garleanu and Pedersen <cit.> solve for the equilibrium returns in a model with a single rational agent and noise traders. For exogenous mean-reverting demands, they also obtain mean-reverting returns like in our model.[Mean-reverting fundamentals also drive the mean-reverting dynamics in the overlapping-generations model with linear costs studied in <cit.>, for example.] Our more general results show that this effect persists even in the absence of mean-reverting fundamentals, as the sluggishness of optimal portfolios with transaction costs already suffices to generate this effect.A similar observation is made by Sannikov and Skrzypacz <cit.>. Like us, they study a model with several rational mean-variance investors. However, by making information about trading targets private, they also strive to endogenize the price impact. If trades are implemented by means of a “conditional double auction”, where each agent observes all others' supply and demand schedules, then linear, stationary equilibria can be characterized by a coupled system of algebraic equations. However, this system generally admits multiple solutions and these are not available in closed form except in the case of (almost) homogenous risk aversions.Outline of the paper This article is organized as follows.Section <ref> describes the model, both in its frictionless baseline version and with quadratic trading costs. In Section 3, we derive the frictionless equilibrium, before turning to individual optimality with transaction costs (and given exogenous returns) in Section <ref>. Section <ref> in turn contains our main results, on existence, uniqueness, and an explicit characterization of the equilibrium return, complemented by several examples. Section <ref> concludes. Appendix <ref> contains the existence and uniqueness results for linear FBSDEs that are used in Section <ref> and <ref>. Appendix B summarizes some material on primary matrix functions that is needed in Appendix <ref>. Notation Throughout, we fix a filtered probability space (Ω,ℱ,(ℱ_t)_t ∈T,P), where either T = [0, T] for T ∈ (0, ∞) (“finite time horizon") or T = [0, ∞) for T = +∞ (“infinite time horizon"). To treat models with a finite and infinite time horizon in a unified manner, we fix a constant δ≥ 0,[This will be the time-discount rate below; for infinite horizon models, it needs to be strictly positive.] and say that an ℝ^ℓ-valued progressively measurable process (X_t)_t ∈T belongs to L^p_δ, p ≥ 1, if E[∫_0^Te^-δ t‖ X_t‖^p dt] < ∞, where ‖·‖ is any norm on ℝ^ℓ. Likewise, an ℝ^ℓ-valued local martingale (M_t)_t ∈T belongs to M^p_δ, p ≥ 1, if E[‖∫_0^T e^-2 δ s d[M]_s ‖^p/2] < ∞. Here, ‖·‖ denotes any matrix norm on ℝ^ℓ×ℓ. § MODEL§.§ Financial Market We consider a financial market with 1+d assets. The first one is safe, and normalized to one for simplicity. The other d assets are risky,with dynamics driven by d-dimensionalBrownian motion(W_t)_t ∈T:dS_t=μ_t dt+σ dW_t.Here, the ℝ^d-valued expected return process (μ_t)_t∈T∈L^2_δ is to be determined in equilibrium, whereas the constant ℝ^d× d-valued volatility matrix σ is given exogenously. Throughout, we write Σ= σσ^⊤ and assume that this infinitesimal covariance matrix is nonsingular.Since our goal is to obtain a model with maximal tractability, it is natural to assume that the exogenous volatility matrix σ is constant.[If one instead assumes that the volatility follows some (sufficiently integrable) stochastic process (σ_t)_t ∈T, then the subsequent characterization of individually optimal strategies and equilibrium returns in terms of coupled but linear FBSDEs as in(<ref>)–(<ref>) still applies. However, the stochastic volatility then appears in the coefficients of this equation, so that the solution can no longer be characterized (semi-)explicitly in terms of matrix power series. Instead, a “backward stochastic Riccati differential equation” appears as a crucial new ingredient already in the one-dimensional models with exogenous price dynamics studied by <cit.>.] However, stochastic volatilities are bound to appear naturally in more general models where they are determined endogenously. Such extensions of the current setting are an important direction for further research. §.§ Endowments, Preferences, and Trading Costs A finite number of agents n=1,…,N receive (cumulative) random endowments (Y^n_t)_t ∈T with dynamicsdY^n_t=d A^n_t +(ζ^n_t)^⊤σ dW_t +dM_t^⊥,n,n=1,…,N.Here, the ℝ-valued adapted process (A^n_t)_t ∈T with E[∫_0^T e^-δ s d | A |_s] < ∞ denotes the finite variation component of Agent n's endowment; it may contain lump-sum payments as well as absolutely continuous cash-flows. The ℝ^d-valued process ζ^n ∈L^2_δ describes the exposure of the endowment to asset price shocks. Finally, the orthogonal ℝ-valued martingale M^⊥,n∈M^2_δ/2 models unhedgeable shocks. Without trading costs, the goal of Agent n is to choose an ℝ^d-valued progressively measurable trading strategy φ∈L^2_δ (the number of shares held in each risky asset) to maximize the (discounted) expected changes of her wealth, penalized for the quadratic variation of wealth changes as in, e.g., <cit.>:E[∫_0^T e^-δ t(φ _t^⊤ dS_t+dY^n_t- γ^n/2d⟨∫_0^·φ_s^⊤ dS_s+Y^n⟩_t )] =E[ ∫_0^T e^-δ t(φ_t^⊤μ_t - γ^n/2 (φ_t+ζ^n_t)^⊤Σ(φ_t+ζ^n_t) )dt + ∫_0^T e^-δ t(d A^n_t -γ^n/2 d⟨ M^⊥,n⟩_t ) ] →max!Here, γ^n>0 and δ≥ 0 are Agent n's risk aversion and the (common) discount rate, respectively. We assume without loss of generality thatγ^N = max(γ^1, …, γ^N),so that Agent N has the highest risk aversion among all agents. For simplicity, we also suppose that the initial stock position φ^n_0- of each agent is zero. A strictly positive discount rate allows to postpone the planning horizon indefinitely to obtain stationary infinite-horizon solutions as in <cit.>. In that case, φ∈L^2_δ is an appropriate transversality condition that ensures that the problem is well posed. The solution of the frictionless problem (<ref>) is readily determined by pointwise optimization asφ^n_t=Σ^-1μ_t/γ^n-ζ^n_t.The first term is the classical (myopic) Merton portfolio; the second is the mean-variance hedge for the replicable part of the endowment. As in <cit.> we now assume that trades incur costs proportional to the square of the order flow φ̇_t=d/dtφ_t.[The assumption of quadratic rather than proportional costs is made for tractability. However, buoyed by the results from the partial equilibrium literature, we expect the qualitative properties of our results to be robust across different small transaction costs, compare with the discussion in <cit.>.] This trading friction can either be interpreted as temporary price impact proportional to both trade size and trade speed, or as a (progressive) transaction tax or trading fee. For the first interpretation it is natural to assume that trades also move the prices of correlated securities (compare <cit.>), and each agent's trades also affect the others' execution prices. In contrast, a tax as in <cit.> or the fee charged by an exchange affects trades in each asset and by each agent separately. We focus on the second specification here, which simplifies the analysis by avoiding a coupling of the agents' optimization problems through common price impact. To wit, λ^m > 0, m=1,…,d, describes the quadratic costs levied separately on each agent's order flow for asset m and we denote by Λ∈ℝ^d × d the diagonal matrix with diagonal entries λ^1, …, λ^d.[More general specifications do no seem natural for the tax interpretation of the model. Note, however, that the mathematical analysis below only uses that Λ is symmetric and positive definite.] With this notation, Agent n's optimization problem then reads as follows: J^n(φ̇):=E[ ∫_0^T e^-δ t(φ_t^⊤μ_t - γ^n/2 (φ_t+ζ^n_t)^⊤Σ(φ_t+ζ^n_t) - φ̇_t^⊤Λφ̇_t dt)dt] +E[ ∫_0^Te^-δ t(d A^n_t -γ^n/2 d⟨ M^n⊥⟩_t) ] →max!In order to avoid infinite transaction costs, all trading rates (as well as the corresponding trading strategies themselves) naturally have to belong to L^2_δ.The goal now is to solve for the equilibrium excess return that matches the agents' (and, potentially, noise traders') supply and demand. A similar model with a single strategic agent and noise traders with a particular parametric demand is analyzed in <cit.>. Conversely, <cit.> study models of the above form without noise traders (and with exponential rather than mean-variance preferences). § FRICTIONLESS EQUILIBRIUM For later comparison to the frictional case, we first consider the model without trading costs. To clear the market, the expected return process (μ_t)_t ∈T needs to be chosen so that the demand of the strategic agents and the exogenous demand of a group of noise traders matches the total supply of zero at all times. To wit, modeling the noise trader demand by an exogenous process ψ∈L^2_δ , the clearing condition reads as0=φ^1_t+…+φ^N_t+ψ_t.(Alternatively, one can interpret -ψ as the exogenous supply of the risky assets.) In view of (<ref>), the frictionless equilibrium expected return therefore isμ_t=Σ(ζ^1_t+ζ^2_t+…+ζ^N_t-ψ_t)/1/γ^1+1/γ^2+…+1/γ^N.The interpretation is that the investment demand induced by the equilibrium return needs to offset the difference between the noise trading volume and the strategic agents' total hedging demand. Whence, the equilibrium return scales with the (exogenous) covariance matrix of the risky assets, relative to the total risk tolerance.In this simple model, equilibrium dynamics and strategies are known in closed form, rather than only being characterized via martingale representation <cit.> or BSDEs <cit.>. This makes the model an ideal point of departure for analyzing the impact of transaction costs on the equilibrium return.§ INDIVIDUAL OPTIMALITY WITH TRANSACTION COSTS As a first step towards our general equilibrium analysis in Section <ref>, we now consider each Agent n's individual optimization problem with transaction costs (<ref>), taking an expected return process μ∈L^2_δ as exogenously given. A multidimensional generalization of the calculus of variations argument of <cit.> leads to the following representation of the optimal strategy in terms of a coupled but linear system of forward-backward stochastic differential equations (henceforth FBSDEs): Let φ^n_t =Σ^-1μ_t/γ^n-ζ^n_t be the frictionless optimizer from (<ref>). Then the frictional optimization problem (<ref>)for Agent n has a unique solution, characterized by the following FBSDE:dφ^Λ, n_t=φ̇^Λ, n_t dt, φ^Λ, n_0 =0, dφ̇^Λ, n_t= dM^n_t+γ^n Λ^-1Σ/2(φ^Λ, n_t-φ^n_t)dt + δφ̇^Λ, n_t dt.Here, φ^Λ, n_t, φ̇^Λ, n_t ∈L^2_δ, and the ℝ^d-valued square-integrable martingale M^n needs to be determined as part of the solution. If T < ∞, the dynamics (<ref>) are complemented by the terminal condition[This means that agents stop trading near maturity, when there is not enough time left to recuperate the costs of further transactions. If T=∞, this terminal condition is replaced by the transversality conditions implicit in φ^Λ, n_t, φ̇^Λ, n_t ∈L^2_δ for δ>0.]φ̇^Λ, n_T = 0.For T=∞, Agent n's unique individually optimal strategy φ^Λ,n has the explicit representation (<ref>); the corresponding optimal trading rate φ̇^Λ,n is given in feedback form by (<ref>). For T<∞, the corresponding formulas are provided in (<ref>) and (<ref>), respectively.Since the goal functional (<ref>) is strictly convex, (<ref>)has a unique solution if and only if there exists a (unique) solution to the following first-order condition <cit.>:⟨ J'(φ̇),ϑ̇⟩=0, ϑ with ϑ_0 = 0and ϑ, ϑ̇∈L^2_δ.Here, the Gâteaux derivative of J in the direction ϑ̇ is given by⟨ J'(φ̇),ϑ̇⟩ = lim_ρ→ 0J(φ̇+ρϑ̇)-J(φ̇)/ρ=E[∫_0^T e^-δ t((μ^⊤_t-γ^n(φ_t+ζ^n_t)^⊤Σ) (∫_0^t ϑ̇_s ds)- 2(φ̇_t)^⊤Λϑ̇_t)dt].By Fubini's theorem,∫_0^T (e^-δ t(μ^⊤_t -γ^n(φ_t+ζ^n_t)^⊤Σ)(∫_0^t ϑ̇_s ds ))dt =∫_0^T (∫_s^T e^-δ t(μ^⊤_t -γ^n(φ_t+ζ^n_t)^⊤Σ) dt )ϑ̇_s ds.Together with the tower property of the conditional expectation, this allows to rewrite the first-order condition (<ref>) as0= E[∫_0^T ( E[∫_t^T e^-δ s(μ^⊤_s -γ^n(φ_s+ζ^n_s)^⊤Σ)ds|ℱ_t] - 2e^-δ t(φ̇)^⊤Λ)ϑ̇_t dt].Since this has to hold for any perturbation ϑ̇, (<ref>) has a (unique) solution φ̇^Λ, n if and only if φ̇^Λ, n_t= γ^n Λ^-1Σ/2 e^δ t E[∫_t^T e^-δ s(Σ^-1μ_s/γ^n -ζ^n_s -φ^Λ, n_s)ds |ℱ_t]has a a (unique) solution. Now, assume that (<ref>) has a (unique) solution φ̇^Λ, n. Note that if T < ∞, (<ref>) is satisfied. Define the square-integrable martingale M̃_t=γ^n Λ^-1Σ/2 E[∫_0^T e^-δ s(φ^n_s -φ^Λ, n_s)ds|ℱ_t], t ∈T. Integration by parts then allows to rewrite (<ref>) asdφ̇^Λ,n_t =e^δ t dM̃_t- γ^nΛ^-1Σ/2 (φ^n_t-φ^Λ, n_t)dt+δφ̇^Λ, n_t dt.Together with the definition dφ_t^Λ, n= φ̇_t^Λ, n dt, this yields the claimed FBSDE representation (<ref>).Conversely, assume that (<ref>) has a (unique) solution (φ^Λ, n, φ̇^Λ, n, M^n), where φ^Λ, n, φ̇^Λ, n∈L^2_δ and M^n is an ℝ^ℓ-valued martingale with finite second moments.First note that, for t ∈T with t < ∞, integration by parts givese^-δ tφ̇^Λ, n_t = φ̇^Λ, n_0 + ∫_0^t e^-δ s dM^n_s + ∫_0^t e^-δ sγ^n Λ^-1Σ/2(φ^Λ, n_s-φ^n_t)ds.Next, we claim thatφ̇^Λ, n_0 =- ∫_0^T e^-δ s dM^n_s - ∫_0^T e^-δ sγ^n Λ^-1Σ/2(φ^Λ, n_s-φ^n_t)ds.If T < ∞, this follows from (<ref>) for t = T together with the terminal condition (<ref>). If T = ∞, we argue as follows: since φ̇^Λ, n∈L^2_δ, there exists an increasing sequence (t_k)_k ∈ℕ with lim_k →∞ t_k = ∞ along which the left-hand side of (<ref>) converges a.s. to zero. Moreover, Proposition <ref>, the martingale convergence theorem and φ^Λ, n ,φ^n ∈L^2_δ show that the right-hand side of (<ref>) converges (along t_k) a.s. to φ̇^Λ, n_0 + ∫_0^∞ e^-δ s dM^n_s + ∫_0^∞ e^-δ sγ^n Λ^-1Σ/2(φ^Λ, n_s-φ^n_t)ds.Hence, (<ref>) holds also in this case.Inserting (<ref>) into (<ref>), taking conditional expectations and rearranging in turn yields (<ref>).It remains to show that the FBSDE (<ref>) has a (unique) solution (φ^Λ, n, φ̇^Λ, n, M^n). Since the matrix γ^n/2Λ^-1Σ has only positive eigenvalues (because it is the product of two symmetric positive definite matrices, cf. <cit.>), this follows from Theorem <ref> (for T=∞) or Theorem <ref> (for T<∞), respectively. § EQUILIBRIUM WITH TRANSACTION COSTS§.§ Equilibrium Returns We now use the above characterization of individually optimal strategies to determine the equilibrium return (μ_t)_t ∈T, for which the agents' individually optimal demands match the zero net supply of the risky asset at all times. As each agent's trading rate is now constrained to be absolutely continuous, the same needs to hold for the exogenous noise-trading volume: dψ_t=ψ̇_tdt, where dψ̇_t=μ^ψ_tdt+ dM_t^ψ for μ^ψ∈L^2_δ and a local martingale M^ψ. We also assume that ψ, ψ̇∈L^2_δ. The key ingredient for the equilibrium return is the solution of another system of coupled but linear FBSDEs: There exists a unique solution(φ^Λ,φ̇^Λ)=(φ^Λ,1,…,φ^Λ, N-1,φ̇^Λ,1,…,φ̇^Λ,N-1) of the following FBSDE:dφ^Λ_t=φ̇^Λ_t dt, φ_0=0,dφ̇^Λ_t =dM_t +(Bφ^Λ_t+δφ̇^Λ_t-Aζ_t+χ_t) dt,satisfying the terminal condition φ̇^Λ_T = 0 if T < ∞. Here, M is an ℝ^d(N-1)-valued martingale with finite second moments, ζ=((ζ^1)^⊤,…,(ζ^N)^⊤)^⊤,B = [ (γ^N-γ^1/N+γ^1) Λ^-1Σ/2 ⋯γ^N-γ^N-1/NΛ^-1Σ/2; ⋮ ⋱ ⋮;γ^N-γ^1/NΛ^-1Σ/2 ⋯ (γ^N-γ^N-1/N+γ^N-1) Λ^-1Σ/2 ]∈ℝ^d (N-1)× d (N-1),A =[ (γ^1/N-γ^1) Λ^-1Σ/2 ⋯γ^N-1/NΛ^-1Σ/2γ^N/NΛ^-1Σ/2; ⋮ ⋱ ⋮ ⋮;γ^1/NΛ^-1Σ/2 ⋯ (γ^N-1/N-γ^N-1) Λ^-1Σ/2γ^N/NΛ^-1Σ/2 ]∈ℝ^d(N-1)× d N,andχ_t=1/N((γ^NΛ^-1Σ/2ψ_t+δψ̇_t-μ^ψ_t)^⊤,…,(γ^NΛ^-1Σ/2ψ_t+δψ̇_t-μ^ψ_t)^⊤)^⊤∈ℝ^d(N-1). Lemma <ref> shows that all eigenvalues of the matrix B are real and positive; in particular, B is invertible. The assertion in turn follows from Theorem <ref> for T = ∞ and from Theorem <ref> for T < ∞ because ζ, χ∈L^2_δ. We can now state our main result: The unique frictional equilibrium return is μ^Λ_t =∑_n=1^N-1(γ^n-γ^N)Σ/Nφ^Λ,n_t+∑_n=1^N γ^nΣ/Nζ^n_t-γ_N Σ/Nψ_t+ 2Λ/N(μ^ψ_t-δψ̇_t).The corresponding individually optimal trading strategies of Agents n=1,…,N are φ^Λ,1,…,φ^Λ, N-1 from Lemma <ref> and φ^Λ,N=-∑_n=1^N-1φ^Λ,n-ψ.Let ν∈L^2_δ be any equilibrium return and denote by ϑ^Λ=(ϑ^Λ,1,…,ϑ^Λ,N) the corresponding individually optimal trading strategies. Then, market clearing implies that not only the positions of the agents but also their trading rates must sum to zero, 0=∑_n=1^N ϑ̇^Λ,n+ψ̇. Together with the FBSDEs (<ref>) describing each agent's optimal trading rate, it follows that0=dM_t +∑_n=1^N Λ^-1/2(γ^n Σϑ^Λ,n_t -(ν_t -γ^n Σζ^n_t) )dt+∑_n=1^N δϑ̇^Λ,n_t dt+dψ̇_t,for a local martingale M. Market clearing implies ϑ^Λ,N=-∑_n=1^N-1ϑ^Λ,n-ψ, and so this gives0=dM_t + Λ^-1/2(∑_n=1^N-1(γ^n-γ^N) Σϑ^Λ,n_t -∑_n=1^N (ν_t -γ^n Σζ^n_t)-γ_N Σψ_t )dt-δψ̇_t dt+μ_t^ψ dt + dM^ψ_t.Since any continuous local martingale of finite variation is constant, it follows thatν_t =∑_n=1^N-1(γ^n-γ^N)Σ/Nϑ^Λ,n_t+∑_n=1^N γ^nΣ/Nζ^n_t-γ_N Σ/Nψ_t+ 2Λ/N(μ^ψ_t-δψ̇_t).Plugging this expression for ν_t back into Agent n=1,…,N-1's individual optimality condition (<ref>), we deduce thatdϑ̇^Λ,n_t = dM^n_t+Λ^-1Σ/2(γ^nϑ^Λ,n_t +∑_m=1^N-1γ^N-γ^m/Nϑ^Λ,m_t+γ^nζ_t^n -∑_m=1^N γ^m/Nζ^m_t)dt+1/N(γ_NΛ^-1Σ/2ψ_t+δψ̇_t-μ^ψ_t)dt,n=1,…,N-1.Hence, (ϑ^Λ,1,…,ϑ^Λ,N-1,ϑ̇^Λ,1,…,ϑ̇^Λ,N-1) solves the FBSDE (<ref>) and therefore coincides with its unique solution from Lemma <ref>. Market clearing in turn shows ϑ^Λ,N=φ^Λ,N, and (<ref>) implies that the equilibrium return coincides with (<ref>). This establishes that if an equilibrium exists, then it has to be of the proposed form.To verify that the proposed returns process and trading strategies indeed form an equilibrium, we revert the above arguments. Market clearing holds by definition of φ^Λ,N, so it remains to check that φ^Λ,n is indeed optimal for agent n=1,…,N. To this end, it suffices to show that the individual optimality conditions (<ref>) are satisfied for n=1,…,N. After inserting the definitions of μ^Λ, one first realises that for n =1, …, N-1, (<ref>) coincides with the respective equation in (<ref>), and for n=N, this follows from market clearing. This completes the proof.§.§ Equilibrium Liquidity Premia Let us now discuss the equilibrium liquidity premia implied by Theorem <ref>, i.e., the differences between the frictional equilibrium returns (<ref>) and their frictionless counterparts (<ref>). To this end, denote by φ̅^n, n=1,…,N,the frictionless optimal strategy from (<ref>) for Agent n, corresponding to the frictionless equilibrium return (<ref>):φ̅^n_t = 1/γ_n (∑_m=1^N ζ^m_t - ψ_t )/∑_m=1^N 1/γ^m- ζ^n_t.With this notation, the frictionless equilibrium return μ can be written asμ_t = ∑_n=1^N γ_n Σ/N (φ̅^n_t + ζ^n_t).Now subtract (<ref>) from the frictional equilibrium return (<ref>), use that - ∑_n =1^N-1φ^Λ, n = φ^Λ_N + ψ_t by the frictional clearing condition, and note that ∑_n=1^N ( φ_t^Λ,n- φ̅_t^n) = (-ψ_t + ψ_t) = 0 by frictional and frictionless market clearing. This yields the following expression for the liquidity premium:LiPr_t := μ^Λ_t - μ_t= ∑_n=1^Nγ^nΣ/Nφ^Λ,n_t+∑_n=1^N γ^nΣ/Nζ^n_t + 2Λ/N(μ^ψ_t-δψ̇_t) - ∑_n=1^N γ_n Σ/N (φ̅^n_t + ζ_t) =Σ/N∑_n=1^N γ^n ( φ_t^Λ,n- φ̅_t^n) + 2Λ/N(μ^ψ_t-δψ̇_t) =Σ/N∑_n=1^N (γ^n -γ̅)( φ_t^Λ,n- φ̅_t^n) + 2Λ/N(μ^ψ_t-δψ̇_t),where γ̅= ∑_n =1^N γ_n/N denotes the average risk aversion of the strategic agents.Let us now interpret this result. A first observation is that if all agents are strategic and have the same risk aversion, then the frictionless equilibrium returns (<ref>) also clear the market with transaction costs: Suppose there are no noise traders and all strategic agents have the same risk aversion γ̅=γ^1=…= γ^N. Then there are no liquidity premia. A similar result has been established for exponential investors in the limit for small transaction costs by <cit.>. In the present quadratic context, this result holds true exactly. A result in the same spirit in a static model is <cit.>, where incompleteness also only affects strategies but not equilibrium prices for mean-variance investors with homogenous risk aversions. Another related result is <cit.>, where homogeneous risk aversion imply that the frictional equilibrium converges to the frictionless one as the horizon grows. However, this result no longer remains true in the presence of noise traders: Suppose that all strategic agents have the same risk aversion γ̅=γ^1=…= γ^N. Then: LiPr_t = 2Λ/N(μ^ψ_t-δψ̇_t). To illustrate the intuition behind this result, consider the simplest case where the noise traders simply sell at a constant rate, ψ̇<0. Put differently, the number of risky shares available for trading expands linearly. Then LiPr_t= -2Λ/Nδψ̇>0. This illustrates how market growth can lead to positive liquidity premia even for homogenous agents. If there is only one strategy agent (N =1), we are always in the setting of Corollary <ref>. An example is the model of Garleanu and Pedersen <cit.> with a single risky asset (d=1), a single strategic agent without random endowment (ζ=0) and exogenous noise traders, whose positions ψ_t are mean-reverting around a stochastic mean:[Several groups of noise traders with different mean positions as considered in <cit.> can be treated analogously.] dψ_t=κ_ψ(X_t-ψ_t)dt, where X is an Ornstein-Uhlenbeck process driven by a Brownian motion W^X: dX_t=-κ_X X_t dt+σ_X dW^X_t. In the notation from Section <ref>, we then have ψ̇_t=κ_ψ (X_t-ψ_t) and μ^ψ_t=-κ_ψκ_X X_t - κ_ψψ̇_t=-κ_ψκ_X X_t - κ^2_ψ (X_t-ψ_t) = κ_ψ^2 ψ_t -κ_ψ(κ_ψ+κ_X)X_t.We therefore recover the nontrivial liquidity premia of <cit.>, to which we also refer for a discussion of the corresponding comparative statics. To obtain nontrivial liquidity premia in a model with only strategic agents and a fixed supply of risky assets, one needs to consider agents with heterogeneous risk aversions. To ease notation and interpretation, suppose there are no noise traders (ψ=0). Then, the liquidity premium (<ref>) is simplifies toLiPr_t= Σ/N∑_n=1^N (γ^n - γ̅)( φ_t^Λ,n- φ̅_t^n).This means that the liquidity premium is the sample covariance between the vector (γ^1,…,γ^N) of risk aversions and the current deviations (φ^Λ,1_t-φ̅^1_t,…,φ^Λ,N_t-φ̅^N_t) between the agents' actual positions and their frictionless targets. Hence, the liquidity premium is positive if and only if sensitivity and excess exposure to risk are positively correlated, i.e., if the more risk averse agents hold larger risky positions than in the (efficient) frictionless equilibrium. Then, these agents will tend to be net sellers and, as their trading motive is stronger than for the net buyers, a positive liquidity premium is needed to clear the market. To shed further light on the dynamics of liquidity premia induced by heterogenous risk aversions, we now consider some concrete examples where the aggregate of the agents' endowments is zero as in <cit.>. First, we consider the simplest case where endowment exposures have independent, stationary increments: Let the time horizon be infinite and consider two strategic agents with risk aversions γ_1 < γ_2, discount rate δ>0, and endowment volatilities following arithmetic Brownian motions: ζ^1_t=at+N_t, ζ^2_t =-ζ^1_t, for an ℝ^d-valued Brownian motion N and a ∈ℝ^d. Then, the frictionless equilibrium return vanishes. The equilibrium return with transaction costs has Ornstein-Uhlenbeck dynamics: dμ^Λ_t =(√(γ_1+γ_2/2ΣΛ^-1/2+δ^2/4I_d)-δ/2I_d) (2γ_1-γ_2/γ_1+γ_2δΛ a-μ^Λ_t)dt +(γ_1-γ_2)Σ/2dN_t. The frictionless equilibrium return vanishes by (<ref>). In view of Theorem <ref> and since ζ^2 =-ζ^1, its frictional counterpart is given by μ^Λ_t= (γ^1-γ^2)Σ/2 (φ^Λ,1_t+ζ^1_t). ByLemma <ref> as well as Theorem <ref> (with B=γ_1+γ_2/2Λ^-1Σ/2 and ξ_t=-ζ^1_t) and the representation (<ref>) from its proof, Agent 1's optimal trading rate is φ̇^Λ,1_t=ξ̅^1_t- (√(Δ)- δ/2I_d)φ^Λ,1_t, where Δ = γ_1+γ_2/2Λ^-1Σ/2+δ^2/4I_d,ξ̅^1_t= -(√(Δ) -δ2I_d ) E[∫_t^∞(√(Δ) +δ2 I_d ) e^-(√(Δ)+δ/2I_d)(u-t)(a u+N_u) d u|ℱ_t] = -(√(Δ) - δ/2I_d) (ζ^1_t+(√(Δ)+δ2I_d)^-1a). Here, we have used an elementary integration and the martingale property of N for the last equality. Plugging this back into (<ref>) yields φ̇^Λ,1_t=-(√(Δ) - δ/2I_d)(φ^Λ,1_t + ζ^1_t+(√(Δ) + δ2I_d)^-1a). Inserting this into (<ref>) in turn leads to the asserted Ornstein-Uhlenbeck dynamics: d μ^Λ_t= (γ^1-γ^2)Σ2 (φ̇^Λ,1_t dt+dζ^1_t) = (γ^1-γ^2)Σ2((-(√(Δ) - δ2I_d)(φ^Λ,1_t + ζ^1_t+(√(Δ)+δ2I_d)^-1a) + a) dt+d N_t) = ((γ^1-γ^2)Σ2(√(Δ) - δ2I_d) (Δ -δ^24I_d)^-1δ a) -Σ(√(Δ) - δ2I_d) Σ^-1μ_t^Λ) dt + (γ^1-γ^2)Σ2 dN_t = (2γ^1-γ^2γ_1+γ_2Σ(√(Δ) - δ2I_d) Σ^-1δΛ a -Σ(√(Δ) - δ2I_d) Σ^-1μ_t^Λ) dt + (γ^1-γ^2)Σ2 dN_t = Σ(√(Δ) - δ2I_d) Σ^-1(2γ^1-γ^2γ_1+γ_2δΛ a- μ^Λ_t) dt + (γ^1-γ^2)Σ2 dN_t = (√(γ_1+γ_22ΣΛ^-12+δ^24I_d)-δ2I_d) (2γ_1-γ_2γ_1+γ_2δΛ a-μ^Λ_t)dt +(γ_1-γ_2)Σ2dN_t, where we have used Lemma <ref>(b) in the last equality. Let us briefly discuss the comparative statics of the above formula. In line with Corollary <ref>, the average liquidity premium 2γ_1-γ_2/γ_1+γ_2δΛ a and the corresponding volatility both vanish if the agents' risk aversions coincide. More generally, its size is proportional to the degree of heterogenity, measured by γ_1-γ_2/γ_1+γ_2, multiplied by the discount rate δ, the trading cost Λ, and the trend a of Agent 1's position. To understand the intuition behind this result, suppose that a < 0 so that Agent 1 is a net buyer and Agent 2 is a net seller. Since γ_1 < γ_2, Agent's 2 motive to sell dominates Agent's 1 motive to buy and hence an additional positive drift is required to clear the market with friction. Since these readjustments do not happen immediately but only gradually over time, the size of these effects is multiplied by the discount rate δ: a higher demand for immediacy forces the more risk averse agent to pay a larger premium.Even for correlated assets, the average liquidity premium only depends on the trading cost for the respective asset and the corresponding imbalance in Agent 1 and 2's demands. The linear scaling in the trading cost also shows that for small costs, the average value of the liquidity premium is much smaller than its standard deviations (which then scale with the square-root of the liquidity premium).Finally, note that liquidity premia are mean reverting here even though the agents' endowments are not stationary. The reason is the sluggishness of the agents' portfolios with transaction costs: the stronger trading need of the more risk-averse agent is not realized immediately but only gradually. This endogenously leads to autocorrelated returns like in the reduced-form models from the frictionless portfolio choice literature <cit.>. Next, we turn to a stationary model where endowment exposures are also mean-reverting as in <cit.>. This generates an additional state variable, that appears in the corresponding liquidity premia as a stochastic mean-reversion level: Let the time horizon be infinite and consider two strategic agents with risk aversions γ_1 < γ_2, discount rate δ>0, and endowment volatilities with Ornstein-Uhlenbeck dynamics: dζ^1_t=-κζ^1_t dt+dN_t,dζ^2_t =-dζ^1_t, ζ^1_0=ζ^2_0=0, for an ℝ^d-valued Brownian motion N and a positive-definite mean-reversion matrix κ∈ℝ^d × d. Then, the frictionless equilibrium return vanishes. The equilibrium return with transaction costs has Ornstein-Uhlenbeck-type dynamics with a stochastic mean-reversion level that is a constant multiple of the endowment levels:dμ^Λ_t=Σ(√(Δ) - δ/2I_d) Σ^-1((γ^1-γ^2) Σ/2(κ (√(Δ)+δ2 I_d +κ)^-1-(√(Δ)-δ2 I_d)^-1κ)ζ^1_t-μ^Λ_t)dt+(γ^1-γ^2)Σ/2dN_t,where Δ = γ_1+γ_2/2Λ^-1Σ/2+δ^2/4I_d. As in Corollary <ref>, the frictionless equilibrium return vanishes by (<ref>), its frictional counterpart is given by (<ref>), and Agent 1's optimal trading rate is (<ref>). The only change is the target process ξ̅^1, which can be computed as follows in the present context: ξ̅^1_t= -(√(Δ) -δ2I_d ) ∫_t^∞(√(Δ) +δ2 I_d ) e^-(√(Δ)+δ/2I_d)(u-t)E[ζ^1_u|ℱ_t] du = -(√(Δ) - δ2I_d) (√(Δ)+δ2I_d) (√(Δ)+δ2I_d+κ)^-1ζ^1_t. Here, we have used the expectation E[ζ^1_u|ℱ_t]=e^-κ(u-t)ζ^1_t of Ornstein-Uhlenbeck processes and an elementary integration for the last equality. Plugging this back into (<ref>) yields φ̇^Λ,1_t=-(√(Δ) - δ2I_d)(φ^Λ,1_t + (√(Δ)+δ2I_d) (√(Δ)+δ2I_d+κ)^-1ζ^1_t). Inserting this (<ref>) in turn leads to the asserted Ornstein-Uhlenbeck dynamics: d μ^Λ_t= (γ^1-γ^2)Σ2 (φ̇^Λ,1_t dt+dζ^1_t) = (γ^1-γ^2)Σ2((-(√(Δ) - δ2I_d)(φ^Λ,1_t + (√(Δ)+δ2I_d) (√(Δ)+δ2I_d+κ)^-1ζ^1_t)-κζ^1_t) dt+d N_t)=Σ(√(Δ) - δ2I_d) Σ^-1((γ^1-γ^2) Σ2(κ (√(Δ)+δ2 I_d +κ)^-1-(√(Δ)-δ2 I_d)^-1κ)ζ^1_t-μ^Λ_t)dt+(γ^1-γ^2)Σ2dN_t. For a single risky asset (d=1), the mean-reversion level of the liquidity premium in Corollary <ref> can be rewritten as (γ^2-γ^1) Σκ(δ+κ)/2(√(Δ)+δ2 I_d +κ)(√(Δ)-δ2 I_d)ζ^1_t =2 γ_2 - γ_1/γ^1+ γ_2Λκ (κ + δ) (1 - κ/√(Δ)+δ2 I_d +κ) ζ^1_t.Since γ_1 < γ_2, the coefficient of ζ^1_t in this expression is positive, so that the sign of the liquidity premium depends on Agent 1's risky exposure ζ^1_t. If ζ^1_t>0, mean-reversion implies that Agent 1's exposure will tend to decrease, so that this agent will want to buy back part of her negative hedging position in the risky asset. Conversely, Agent 2 will tend to sell risky shares. Since Agent 2 is more risk averse, her selling motive dominates and needs to be offset by an additional positive expected return to clear the market, in line with the sign of the above expression.§ CONCLUSION In this paper, we develop a tractable risk-sharing model that allows to study how trading costs are reflected in expected returns. In a continuous-time model populated by heterogenous mean-variance investors, we characterize the unique equilibrium by a system of coupled but linear FBSDEs. This system can be solved in terms of matrix power series, and leads to fully explicit equilibrium dynamics in a number of concrete settings.If all agents are homogenous, positive liquidity premia are obtained if the asset supply expands over time. For a fixed asset supply but heterogenous agents, the sign of the liquidity premia compared to the frictionless case is determined by the trading needs of the more risk averse agents. Since these have a stronger motive to trade, they need to compensate their more risk-tolerant counterparties accordingly. The sluggishness of illiquid portfolios also introduces autocorrelation into the corresponding equilibrium expected returns even for fundamentals with independent increments.Several extensions of the present model are intriguing directions for further research. One important direction concerns more general specifications of preferences (e.g., exponential rather than quadratic utilities) or trading costs (e.g., proportional instead of quadratic). Such variations are bound to destroy the linearity of the corresponding optimality conditions, but might still lead to tractable results in the small-cost limit similarly as for models with exogenous prices <cit.>. Another important direction for further research concerns extensions where the price volatility is no longer assumed to be exogenously given, but is instead determined as an output of the equilibrium. However, even the simplest versions of such models are also bound to lead to nonlinear FBSDEs. § EXISTENCE AND UNIQUENESS OF LINEAR FBSDES For the determination of both individually optimal trading strategies in Section <ref> and equilibrium returns in Section <ref>, this appendix develops existence and uniqueness results for systems of coupled but linear FBSDEs:[Due to the degeneracy of the forward component (<ref>), general FBSDE theory as in <cit.> only yields local existence. However, the direct arguments developed below allow to establish global existence and also lead to explicit representations of the solution in terms of matrix power series.]dφ_t=φ̇_t dt, φ_0=0,t ∈T, dφ̇_t =dM_t +B(φ_t-ξ_t)dt + δφ̇_t dt,t ∈T,where B ∈ℝ^ℓ×ℓ has only positive eigenvalues, δ≥ 0, and ξ∈L^2_δ. If T<∞, (<ref>) is complemented by the terminal condition φ̇_T = 0.If T = ∞, we assume that δ > 0 and the terminal condition is replaced by the transversality conditions implicit in φ, φ̇∈L^2_δ for δ>0. A solution of (<ref>–<ref>) is a triple (φ, φ̇, M) for which φ, φ̇∈L^2_δ and M is a martingale on T with finite second moments.We first consider the infinite time-horizon case. In this case, the linear FBSDEs (<ref>-<ref>) can be solved using matrix exponentials similarly as in <cit.>. To this end, we first establish a technical result stating that the martingale M appearing in the solution of the FBSDE (<ref>–<ref>) automatically belongs to M^2_δ: Let T = ∞. If (φ, φ̇, M) is a solution to the FBSDE (<ref>–<ref>), then M ∈M^2_δ.Let (φ, φ̇, M) be a solution to the FBSDE (<ref>–<ref>), where φ, φ̇∈L^2_δ and M is a martingale on [0, ∞) with finite second moments. Fix t ∈ (0, ∞). Then integration by parts yieldse^-δ tφ̇_t = φ̇_0 + ∫_0^t e^-δ s dM_s + ∫_0^t e^-δ s B (φ_s - ξ_s) ds.Let ‖·‖_2 be the Euclidean norm in ℝ^ℓ and ‖·‖_max the maximum norm on ℝ^ℓ×ℓ. Rearranging (<ref>) and using subsequently the elementary inequality (a + b + c)^2 ≤ 3(a^2 + b^2 + c^2) for a, b, c ∈ℝ, the elementary estimate ‖ A x ‖_2 ≤√(ℓ)‖ A ‖_max‖ x ‖_2, and Jensen's inequality gives‖∫_0^te^-δ s dM_s ‖_2^2≤ 3 (‖φ̇_0 ‖_2^2 + e^- 2 δ t‖φ̇_t ‖^2_2 + (∫_0^t e^-δ s√(ℓ)‖ B ‖_max‖φ_s-ξ_s ‖_2 ds)^2 ) ≤ 3 (‖φ̇_0 ‖_2^2 + e^- δ t‖φ̇_t ‖^2_2 + ℓ/δ‖ B ‖^2_max∫_0^t e^-δ s‖φ_s-ξ_s ‖^2_2 ds ).The definitions of the maximum and Euclidean norms, the estimate |[N^1, N^2 ]| ≤12 ([N^1 ]+ [N^2]) for real-valued local martingales N^1 and N^2, and Itô's isometry give, for t ∈ (0, ∞),E[‖∫_0^t e^-2 δ sd[M]_s ‖_max]= E[ max_i, j ∈{1, …, ℓ}∫_0^t e^-2 δ sd[M^i, M^j]_s ] ≤1/2 E[ max_i ∈{1, …, ℓ}∫_0^t e^-2 δ sd[M^i]_s + max_j ∈{1, …, ℓ}∫_0^t e^-2 δ sd[M^j]_s ] = E[ max_i ∈{1, …, ℓ}∫_0^t e^-2 δ sd[M^i]_s ] ≤ E[ ∑_i =1^ℓ∫_0^t e^-2 δ sd[M^i]_s ] = E[ ∑_i =1^ℓ(∫_0^t e^-δ sdM^i_s)^2 ] = E[‖∫_0^te^-δ s dM_s ‖_2^2].Since φ̇∈L^2_δ, there exists an increasing sequence (t_k)_k ∈ℕ with lim_k →∞ t_k =∞ such that lim_n →∞ E[e^- δ t_k‖φ̇_t_k‖^2_2 ] = 0.Monotone convergence, (<ref>-<ref>) and φ, ξ∈L^2_δ in turn yieldE[‖∫_0^∞ e^-2 δ sd[M]_s ‖_max]= lim_k →∞ E[‖∫_0^t_k e^-2 δ sd[M]_s ‖_max] ≤lim_k →∞ E[‖∫_0^t_ke^-δ s dM_s ‖_2^2] ≤ 3 ℓ(‖φ̇_0 ‖_2^2 + ℓ/δ‖ B ‖^2_max E[ ∫_0^∞ e^-δ s‖φ_s-ξ_s ‖^2_2 ds ] ) < ∞.Thus, M ∈M^2_δ as claimed. Suppose that T = ∞, δ >0, and the matrix B from (<ref>) has only positive eigenvalues. Set Δ = B + δ^2/4 I_ℓ. Then, the unique solution of the FBSDE (<ref>–<ref>) is given by φ_t= ∫_0^t (e^-(√(Δ) - δ/2 I_ℓ)(t-s)ξ̅_s )ds, where ξ̅_t=(√(Δ) - δ2 I_ℓ) E[∫_t^∞(√(Δ) + δ2 I_ℓ) e^-(√(Δ)+ δ/2 I_ℓ) (s-t)ξ_s ds | ℱ_t].Let (φ,φ̇, M) be a solution to the FBSDE (<ref>–<ref>) and define φ̃_t := e^-δ/2tφ_t. Using that φ̇̃̇_t= -δ2φ̃_t + e^-δ/2 tφ̇_t, d φ̇̃̇_t = -δ2φ̇̃̇_t dt + e^-δ/2 t d φ̇_t -δ2 e^-δ/2 tφ̇_t dt and the FBSDE (<ref>-<ref>) for (φ,φ̇), it follows that (φ̃,φ̇̃̇) solves the FBSDE dφ̃_t=φ̇̃̇_t dt, φ̃_0=0,t ∈T, dφ̇̃̇_t =dM̃_t +B(φ̃_t- ξ̃_t)dt + δ^24φ̃_t dt,t ∈T, where dM̃_t = e^-δ/2 t dM_t and ξ̃_t = e^-δ/2 tξ_t. In matrix notation, this equation can be rewritten as d(φ̃_t,φ̇̃̇_t)^⊤=C_1dM̃_t+C_2(φ̃_t,φ̇̃̇_t)^⊤ dt-C_3ξ̃_tdt, with C_1=[ 0; I_ℓ ],C_2=[ 0 I_ℓ; Δ 0 ],C_3=[ 0; B ]. Integration by parts shows d(e^-C_2 t(φ̃_t,φ̇̃̇_t)^⊤)=e^-C_2 tC_1dM̃_t-e^-C_2 tC_3ξ̃_t dt, and in turn e^-C_2 u[ φ̃_u; φ̇̃̇_u ]= e^-C_2 t[ φ̃_t; φ̇̃̇_t ]+∫_t^u e^-C_2 sC_1d M̃_s -∫_t^u e^-C_2 sC_3ξ̃_s ds,Multiplying (<ref>) by the matrixC̃_2=[ I_ℓ √(Δ)^-1;√(Δ) I_ℓ ]and settingH(t)=C̃_2 e^-C_2 t yields H(u) [ φ̃_u; φ̇̃̇_u ]= H(t)[ φ̃_t; φ̇̃̇_t ]+∫_t^u H(s)C_1d M̃_s -∫_t^u H̃(s) C_3ξ̃_s ds, It follows by induction thatC̃_2 (- C_2 t)^2n =[Δ^n Δ^n - 12; Δ^n + 12Δ^n ] t^2n,C̃_2 (- C_2 t)^2n+1 =-[ Δ^n+1/2 Δ^n; Δ^n+1 Δ^n+1/2 ] t^2n +1,n ≥ 0.Now, the power series for the exponential function allows to deduceH(t)=[ e^- √(Δ) t √(Δ)^-1 e^- √(Δ) t;√(Δ) e^- √(Δ) t e^- √(Δ) t ].Together with (<ref>), it follows that√(Δ) e^- √(Δ) uφ̃_u + e^- √(Δ) uφ̇̃̇_u= √(Δ)e^- √(Δ) tφ̃_t + e^- √(Δ) tφ̇̃̇_t+∫_t^u e^- √(Δ) s dM̃_s -∫_t^u e^- √(Δ) s B ξ̃_s ds,By the assumption that φ, φ̇∈L^1_δ⊂L^2_δ, (<ref>) and the fact that all eigenvalues of √(Δ) are greater or equal than δ/2 (because B has only nonnegative eigenvalues), there exists an increasing sequence (u_k)_k ∈ℕ with lim_k →∞ u_k = + ∞ along which the left-hand side of(<ref>) converges a.s. to zero. Moreover, since M ∈M^2_δ by Proposition <ref> and ξ̃∈L^2_δ, the martingale convergence theorem and monotone convergence (together with Jensen's inequality) – also using that all eigenvalues of √(Δ) are greater or equal than δ/2 – imply that for u →∞ (and a fortiori along (u_k)_k ∈ℕ) the right hand side of (<ref>) converges a.s. to√(Δ)e^- √(Δ) tφ̃_t + e^- √(Δ) tφ̇̃̇_t +∫_t^∞ e^- √(Δ) s dM̃_s -∫_t^∞ e^- √(Δ) s B ξ̃_s ds.Together, these two limits show that (<ref>) vanishes. Multiplying (<ref>)by e^√(Δ) t, rearranging, and taking conditional expectations (using again that all eigenvalues of √(Δ) are larger than or equal to δ/2) we obtain φ̇̃̇_t = E[∫_t^∞ e^-√(Δ) (s-t) B e^-δ/2 sξ_s ds| ℱ_t] - √(Δ)φ̃_t. Now, (<ref>) and rearranging give φ̇_t = E[∫_t^∞ e^-√(Δ) (s-t) B e^-δ/2 (s-t)ξ_s ds| ℱ_t] - (√(Δ) - δ2 I_ℓ) φ_t.Finally, since B commutes with e^-√(Δ) (s-t) (as B = (√(Δ) - δ/2 I_ℓ)(√(Δ) + δ/2 I_ℓ) and by Lemma <ref>(a)) it follows that φ̇_t = ξ̅_t- (√(Δ) - δ2 I_ℓ) φ_t. By the variations of constants formula, this linear (random) ODE has the unique solution (<ref>). If a solution of the FBSDE (<ref>-<ref>) exists, it therefore must be of the form (<ref>). It remains to verify that (<ref>) indeed solves the FBSDE (<ref>-<ref>).To this end, we first show that ξ̅∈L^2_δ. Indeed, denote by ‖·‖_2 both the Euclidean norm in ℝ^ℓ and the spectral norm in ℝ^ℓ×ℓ. Since all eigenvalues of B are positive, there is ε > 0 such that all eigenvalues of √(Δ) + δ2 I_ℓ are greater or equal that δ + ε. Hence, by Lemma <ref>(c) and the definition of the spectral norm, it follows that ‖ e^-(√(Δ) + δ2 I_ℓ)t‖_2 ≤ e^-(δ + ε) t,t ∈ [0, ∞). Thus, by the definition of ξ̅ in (<ref>), the fact that B = (√(Δ) - δ/2 I_ℓ)(√(Δ) + δ/2 I_ℓ), Jensen's inequality and Fubini's theorem, we obtain E[∫_0^∞ e^-δ t‖ξ̅_t ‖^2_2 d t]≤‖ B ‖^2_2/δ + ϵ∫_0^∞ e^-δ t∫_t^∞ e^-(δ + ε) (s-t) E[‖ξ_s ‖^2_2] ds dt ≤‖ B ‖^2_2/δ + ε∫_0^∞(∫_0^se^ε t dt ) e^-(δ + ε) s E[‖ξ_s ‖^2_2] ds ≤‖ B ‖^2_2/ϵ(δ + ε)∫_0^∞e^-δ s E[‖ξ_s ‖^2_2] ds < ∞. Next, we show that φ∈L^2_δ. Arguing similarly as above, we have ‖ e^-(√(Δ) - δ2 I_ℓ)t‖_2 ≤ e^-ε t,t ∈ [0, ∞). Thus, by the definition of φ in (<ref>), Jensen's inequality and Fubini's theorem and since ξ̅∈L^2_δ by the above arguments, we obtain E[∫_0^∞ e^-δ t‖φ_t ‖^2_2 d t]≤1/ε∫_0^∞ e^-δ t∫_0^t e^-ϵ (t -s) E[‖ξ̅_s ‖^2_2] ds dt ≤1/ϵ∫_0^∞(∫_s^∞e^ -(δ + ε) t dt ) e^ε s E[‖ξ̅_s ‖^2_2] ds = 1/ϵ(ϵδ + ϵ)∫_0^∞e^-δ s E[‖ξ̅_s ‖^2_2] ds < ∞.By definition, we have φ_0=0. Next, integration by parts shows that φ̇ satisfies the ODE (<ref>), and this yields φ̇∈L^2_δ (because φ, ξ̅∈L^2_δ). Define the ℝ^ℓ-valued square-integrable martingale (M̅_t)_t ∈ [0, ∞) by[Note that ∫_0^∞ e^-√(Δ) s B ξ̃_s ds is square integrable because ξ∈L^2_δ and all eigenvalues of √(Δ) are greater than δ/2.] M̅_t= E[∫_0^∞ e^-√(Δ) s B ξ̃_s ds| ℱ_t],where φ̃_t := e^-δ/2tφ_t and ξ̃_t := e^-δ/2tξ_t as before. Then multiplying (<ref>) by the matrix e^-(√(Δ) + δ/2I_ℓ)t and using (<ref>) as well as Δ - δ^2/4 I_ℓ = B gives, after some rearrangement, e^-√(Δ) tφ̇̃̇_t = M̅_t - ∫_0^t e^-√(Δ) s B ξ̃_s ds - √(Δ) e^-√(Δ) tφ̃_t. Taking differentials, we therefore obtain-√(Δ) e^-√(Δ) tφ̇̃̇_t dt + e^-√(Δ) t d φ̇̃̇_t = d M̅_t - e^-√(Δ) t B ξ̃_t dt -√(Δ) e^-√(Δ) tφ̇̃̇_t dt + Δ e^-√(Δ) tφ̃_t dt. Rearranging, multiplying by e^√(Δ) t and using that √(Δ) and e^√(Δ) t commute, it follows that dφ̇̃̇_t= e^√(Δ) t d M̅_t - B ξ̃_t dt + Δφ̃_t dt. Finally, again taking into account (<ref>) and defining the martingale M (which has finite second moments) by d M_t = e^(√(Δ) + δ/2 I_ℓ) t d M̅_t,M_0 = M̅_0, we obtain that φ from(<ref>) indeed satisfies (<ref>–<ref>). Let us briefly sketch the financial interpretation of the solution; cf. <cit.> for more details. In the context of individually optimal trading strategies (cf. Lemma <ref>), the ODE (<ref>) describes the optimal trading rate. It prescribes to trade with a constant relative speed √(Δ) - δ2 I_ℓ towards the target portfolio(√(Δ) - δ2 I_ℓ)^-1ξ̅_t = E[∫_t^∞(√(Δ) + δ2 I_ℓ) e^-(√(Δ)+ δ/2 I_ℓ) (s-t)ξ_s ds | ℱ_t]. In the context of Lemma <ref>, this is an average of the future values of the frictionless optimal trading strategy ξ, computed using an exponential discounting kernel. As the trading costs tend to zero, the discount rate tends to infinity, and the target portfolio approaches the current value of the frictionless optimizer, in line with the small-cost asymptotics of <cit.>.We now turn to the finite-horizon case. In order to satisfy the terminal condition φ̇_T=0, the exponentials from Theorem <ref> need to be replaced by appropriate hyperbolic functions in the one-dimensional case <cit.>. In the present multivariate context, this remains true if these hyperbolic functions are used to define the corresponding “primary matrix functions” in the sense of Definition <ref>. The first step to make this precise is the following auxiliary result, which is applied for Δ =B+δ^2/4I_ℓ in Theorem <ref> below:Let Δ∈ℝ^ℓ×ℓ. The matrix-valued functionG(t) = ∑_n = 0^∞1/(2n)!Δ^n (T - t)^2nis twice differentiable on ℝ with derivativeĠ(t)=-∑_n = 0^∞1/(2n+1)!Δ^n+1 (T - t)^2n+1,and solves the following ODE:G̈(t)=Δ G(t), G(T)=I_d Ġ(T)=0.Moreover, if the matrix Δ has onlypositive eigenvalues then, in the sense of Definition <ref>, G(t) = cosh(√(Δ) (T-t)), Ġ(t) = -√(Δ)sinh(√(Δ) (T-t));for δ≥ 0, the matrix Δ G(t) - δ/2Ġ(t) is invertible for any t ∈ [0, T] and, for any matrix norm ‖·‖,sup_t ∈ [0, T]‖(Δ G(t) - δ/2Ġ(t))^-1‖ < ∞. Note that ∑_n = 0^∞1/(2n)!‖Δ‖^n (T - t)^2n < ∞ for any matrix norm ‖·‖. Whence, G(t) is well defined for each t ∈ℝ. By twice differentiating term by term, and estimating the resulting power series in the same way, it is readily verified that G is twice continuously differentiable on ℝ, has the stated derivative, and is a solution of (<ref>).Suppose now that Δ has only positive eigenvalues and δ≥ 0. Then the first two additional claims follow from Definition <ref> via the fact that B = (√(B))^2 and the series representation of the smooth functions cosh and sinh. The final claim follows from Lemma <ref>(c) and (d) since, for fixed x ∈ (0,∞),inf_t ∈ [0, T] x cosh(x (T -t)) + δ2 x sinh(x (T -t)) ≥ x > 0.The unique solution of our FBSDE can now be characterized using the function G(t) from Lemma <ref> as follows:Suppose that T < ∞ and that the matrix Δ=B + δ^2/4 I_ℓ has only positive eigenvalues. Then, the unique solution of the FBSDE (<ref>-<ref>) with terminal condition (<ref>) is given by φ_t= ∫_0^t (e^-∫_s^t F(u)duξ̅_s )ds,where[Note that the inverses are well defined by Lemma <ref>.]F(t)=-(Δ G(t) -δ/2Ġ(t))^-1 B Ġ(t), ξ̅_t=(Δ G(t) -δ/2Ġ(t))^-1E[∫_t^T (Δ G(s) - δ/2Ġ(s) ) B e^-δ/2 (s-t)ξ_s ds| ℱ_t]. Let (φ,φ̇) be a solution of the FBSDE (<ref>–<ref>) with terminal condition (<ref>) and setφ̃_t := e^-δ/2tφ_t.Using thatφ̇̃̇_t= -δ2φ̃_t + e^-δ/2 tφ̇_t, d φ̇̃̇_t = -δ2φ̇̃̇_t dt + e^-δ/2 t d φ̇_t -δ2 e^-δ/2 tφ̇_t dtand the FBSDE (<ref>–<ref>) for (φ,φ̇) with (<ref>), it follows that (φ̃,φ̇̃̇) solves the FBSDEdφ̃_t=φ̇̃̇_t dt, φ̃_0=0,t ∈ [0, T]dφ̇̃̇_t =dM̃_t +B(φ̃_t- ξ̃_t)dt + δ^24φ̃_t dt,t ∈ [0, T],with terminal conditionφ̇̃̇_T=-δ2φ̃_T.Here dM̃_t =e^-δ/2 t dM_t is a square-integrable martingale (because M is) and ξ̃_t = e^-δ/2 tξ_t. In matrix notation, this equation can be rewritten asd(φ̃_t,φ̇̃̇_t)^⊤=C_1dM̃_t+C_2(φ̃_t,φ̇̃̇_t)^⊤ dt-C_3ξ̃_tdt,withC_1=[ 0; I_ℓ ],C_2=[ 0 I_ℓ; Δ 0 ],C_3=[ 0; B ].Integration by parts showsd(e^C_2(T-t)(φ̃_t,φ̇̃̇_t)^⊤)=e^C_2(T-t)C_1dM̃_t-e^C_2(T-t)C_3ξ̃_t dt,and in turn[ φ̃_T; φ̇̃̇_T ]= e^C_2(T-t)[ φ̃_t; φ̇̃̇_t ]+∫_t^T e^C_2(T-s)C_1d M̃_s -∫_t^T e^C_2(T-s)C_3ξ̃_s ds.Set H(t)=e^C_2(T-t) and note thatH(t)=[G(t) -Δ^-1Ġ(t); -Ġ(t)G(t) ],for the function G(t) from Lemma <ref>, as is readily verified by induction. Together with (<ref>), it follows thatφ̃_T= G(t) φ̃_t -Δ^-1Ġ(t) φ̇̃̇_t - ∫_t^T Δ^-1Ġ(s) dM̃_s + ∫_t^T Δ^-1Ġ(s)B ξ̃_s ds,φ̇̃̇_T= -Ġ(t) φ̃_t + G(t) φ̇̃̇_t +∫_t^T G(s) dM̃_s -∫_t^T G(s) B ξ̃_s ds.Since φ̇̃̇_T=-δ/2φ̃_T by (<ref>), this in turn yields0 =(δ2 G(t) - Ġ(t) ) φ̃_t + (-δ2Δ^-1Ġ(t) +G(t) ) φ̇̃̇_t+∫_t^T (-δ2Δ^-1Ġ(s) + G(s) ) dM̃_s + ∫_t^T (δ2Δ^-1Ġ(s) - G(s)) B ξ̃_s ds.Multiplying this equation by Δ and taking conditional expectations gives(Δ G(t) -δ2Ġ(t)) φ̇̃̇_t = E[∫_t^T (Δ G(s) - δ2Ġ(s) ) B ξ̃_s ds| ℱ_t] +(ΔĠ(t) -δ2Δ G(t)) φ̃_t.Now, using (<ref>) and rearranging, it follows that(Δ G(t) -δ2Ġ(t)) e^-δ/2 tφ̇_t = E[∫_t^T (Δ G(s) - δ2Ġ(s) ) B e^-δ/2 sξ_s ds| ℱ_t] +(Δ - δ^24 I_ℓ) Ġ(t) e^-δ/2 tφ_t.After multiplying with the inverse of (Δ G(t) -δ/2Ġ(t)) (which exists by Lemma <ref>) and using that Δ - δ^2/4 I_ℓ = B, this leads toφ̇_t = ξ̅_t-F(t)φ_t.By the variations of constants formula, this linear (random) ODE has the unique solution (<ref>). If a solution of the FBSDE (<ref>-<ref>) exists, it therefore must be of the form (<ref>).It remains to verify that (<ref>) indeed solves the FBSDE (<ref>-<ref>). First, note that ξ̅∈L^2_δ by the fact that ξ∈L^2_δ and the estimate (<ref>), and in turn φ∈L^2_δ. Moreover, by definition, we have φ_0=0. Next, integration by parts shows that φ̇ satisfies the ODE (<ref>), and this yields φ̇∈L^2_δ (because φ, ξ̅∈L^2_δ) and φ̇_T=0 (because G(T)=I and Ġ(T)=0). Define the ℝ^ℓ-valued square-integrable martingale (M̅_t)_t ∈ [0, T] byM̅_t= E[∫_0^T (Δ G(s) - δ2Ġ(s) ) B ξ̃_s ds | ℱ_t],where φ̃_t := e^-δ/2tφ_t and ξ̃_t := e^-δ/2tξ_t as before. Then, multiplying (<ref>) by (Δ G(t) -δ2Ġ(t)) and using (<ref>) as well as Δ - δ^2/4 I_ℓ = B gives, after some rearrangement,(Δ G(t) -δ2Ġ(t)) φ̇̃̇_t = M̅_t - ∫_0^t (Δ G(s) - δ2Ġ(s) ) B ξ̃_s ds +(ΔĠ(t) -δ2Δ G(t)) φ̃_t.Taking differentials, we therefore obtain(ΔĠ(t) -δ2G̈(t)) φ̇̃̇_t dt + (Δ G(t) -δ2Ġ(t)) d φ̇̃̇_t = d M̅_t - (Δ G(t) - δ2Ġ(t) ) B ξ̃_t dt +(ΔĠ(t) -δ2Δ G(t)) φ̇̃̇_t dt + (ΔG̈(t) -δ2ΔĠ(t)) φ̃_t dt.Using that G̈(t) = Δ G(t) by the ODE (<ref>) and taking into account that Δ commutes with both G(t) and Ġ(t) by Lemma <ref>(a), it follows that(Δ G(t) -δ2Ġ(t)) d φ̇̃̇_t= d M̅_t - (Δ G(t) - δ2Ġ(t) ) B ξ̃_t dt+ (Δ G(t) -δ2 G(t)) Δφ̃_t dt.Now, multiplying with the inverse of (Δ G(t) -δ/2Ġ(t)) (which exists by Lemma <ref>) and using that Δ= B + δ^2/4 I_ℓ, we obtaind φ̇̃̇_t= (Δ G(t) - δ2Ġ(t) )^-1 d M̅_t + B(φ̃_t- ξ̃_t)dt + δ^24φ̃_t dt.Finally, again taking into account (<ref>) and defining the square-integrable martingale M byd M_t = e^δ/2 t(Δ G(t) - δ2Ġ(t) )^-1 d M̅_t,M_0 = M̅_0,we obtain that φ from(<ref>) indeed satisfies the FBSDE dynamics (<ref>-<ref>) with terminal condition (<ref>). Let us again briefly comment on the financial interpretation of this result in the context of Lemma <ref>. The basic interpretation is the same as in the infinite-horizon case studied in Theorem <ref>. However, to account for the terminal condition that the trading speed needs to vanish, the optimal relative trading speed in (<ref>) is no longer constant. Instead, it interpolates between this terminal condition and the stationary long-run value from Theorem <ref>, that is approached if the time horizon is distant. Analogously, the exponential discounting kernel used to compute the target portfolio in Theorem <ref> is replaced by a more complex version here; compare <cit.> for a detailed discussion in the one-dimensional case. To apply Theorems <ref> and <ref> to characterize the equilibrium in Theorem <ref> it remains to verify that the matrix B appearing there only has real, positive eigenvalues, since this implies that the matrix Δ := B + δ^2/4 I_ℓ has only real eigenvalues greater than δ^2/4 ≥ 0. Let Λ∈ℝ^d × d be a diagonal matrix with positive entries λ^1, …, λ^d > 0,Σ∈ℝ^d × d a symmetric, positive definite matrix and γ^1,…,γ^N>0 with γ^N = max(γ^1,…,γ^N).Then, the matrixB= [ γ^N-γ^1/NΛ^-1Σ/2⋯ γ^N-γ^N-1/NΛ^-1Σ/2;⋮⋯⋮; γ^N-γ^1/NΛ^-1Σ/2⋯ γ^N-γ^N-1/NΛ^-1Σ/2 ] + [γ^1 Λ^-1Σ/2 0; ⋱ ;0γ^N-1Λ^-1Σ/2 ]∈ℝ^d (N-1)× d (N-1)has only real, positive eigenvalues. First, recall that two matrices that are similar have the same eigenvalues. Since the matrix Λ^-1Σ/2 has only positive eigenvalues (because it is the product of two symmetric positive definite matrices, cf. <cit.>), there is an invertible matrix P ∈ℝ^d× d and a diagonal matrix U ∈ℝ^d× d with positive diagonal entries u^1, …, u^d such that Λ^-1Σ/2 = P U P^-1. Now, define the matrixQ = [ P 0; ⋱; 0 P ]∈ℝ^d (N-1)× d (N-1).A direct computation shows that Q is invertible with inverseQ^-1 = [ P^-1 0; ⋱ ;0P^-1 ]∈ℝ^d (N-1)× d (N-1). Whence B is similar to B̅ := P^-1 B P, andB̅ = [ γ^N-γ^1/N U ⋯ γ^N-γ^N-1/N U; ⋮ ⋯ ⋮; γ^N-γ^1/N U ⋯ γ^N-γ^N-1/N U ] + [ γ^1 U 0; ⋱; 0 γ^N-1 U ].To prove that B̅ (and hence B) only has real and positive eigenvalues, we calculate the determinant of V(x) = xI_d (N-1) - B̅ for x ∈ℂ∖ (0, ∞) and show that (V(x)) ≠ 0. So let x ∈ℂ∖ (0, ∞). Denote by ℛ_d the commutative subring of all diagonal matrices in ℂ^d× d and let ℵ^1, …, ℵ^N-1, ℷ^1(x), …ℷ^N-1(x) ∈ℛ_d be given byℵ^n = - γ^N - γ^n/N U andℷ^n(x) = x I_d - γ^n U.With this notation, the ℝ^d(N-1) × d (N-1)-valued matrix V(x) can also be understood as an element of ℛ_d^(N-1)×(N-1) (the (N-1)×(N-1) matrices with elements from the diagonal matrices in ℝ^d × d) and we haveV(x) = [ℵ^1 +ℷ^1(x)ℵ^2⋯ℵ^n-2ℵ^N-1;ℵ^1ℵ^2 +ℷ^2(x)⋱⋮⋮;⋮⋱⋱⋱⋮;⋮⋮⋱ ℵ^N-2+ℷ^N-2(x)ℵ^N-1;ℵ^1ℵ^2⋯ℵ^N-2 ℵ^N-1 + ℷ^N-1(x) ].Now use that by <cit.>, (V (x)) = ( 𝔡𝔢𝔱(V(x))), where 𝔡𝔢𝔱: ℛ_d^(N-1)×(N-1)→ℛ_d is the determinant map on the commutative ring ℛ_d. By subtracting the last row (in ℛ_d) of V(x) from the other rows, the problem boils down to calculating the determinant of V̅(x) := [ ℷ^1(x)0⋯0 - ℷ^N-1(x);0 ℷ^2(x)⋱⋮⋮;⋮⋱⋱0⋮;0⋯0 ℷ^N-2(x) - ℷ^N-1(x);ℵ^1ℵ^2⋯ℵ^N-2 ℵ^N-1 + ℷ^N-1(x) ].As x ∈ℂ∖ (0, ∞) and for n ∈{1, …, N-1}, the eigenvalues of γ^n U are γ^n u^1, …, γ^n u^d ∈ (0, ∞), it follows that (ℷ^n (x)) ≠ 0 and hence ℷ^n (x) is invertible for each n. Now, subtracting ℵ^n (ℷ^n(x))^-1-times the n-th row from the last row for n = 1, …, N-2, the problem simplifies to calculating the determinant of V̂(x) := [ℷ^1(x) 0 0- ℷ^N-1(x); 0 ⋱ 0 ⋮; 0 ⋱ℷ^N-2(x)- ℷ^N-1(x); 0 ⋯ 0 ℷ^N-1(x)(I_d + ∑_n =1^N-1ℵ^n (ℷ^n(x))^-1) ].As a result:𝔡𝔢𝔱(V̂(x)) = (∏_n = 1^N-1ℷ^n(x) ) (I_d+ ∑_n =1^N-1ℵ^n (ℷ^n(x))^-1),and in turn(V(x)) = (𝔡𝔢𝔱(V̂(x)) ) = (∏_n = 1^N-1(ℷ^n(x))) (I_d + ∑_n =1^N-1ℵ^n (ℷ^n(x))^-1).It therefore remains to show that (I_d + ∑_n =1^N-1ℵ^n (ℷ^n(x))^-1) ≠ 0. As I_d + ∑_n =1^N-1ℵ^n (ℷ^n(x))^-1 is a diagonal matrix, we have(I_d + ∑_n =1^N-1ℵ^n (ℷ^n(x))^-1) = ∏_i = 1^d (1 + ∑_n =1^N-1α^n u^i 1/x - γ^n u^i),where α^n = -γ^N -γ^n/N, n ∈{1, …, N-1}. As γ^N = max(γ^1, …, γ^N), we have α^n ≤ 0 for each n ∈{1, …, N-1}. It suffices to show that for i ∈{1, …, d},1 + ∑_n =1^N-1α^n u^i 1/x - γ^n u^i≠ 0.Writing x = (x) + i(x) and expanding each fraction in (<ref>) by (x) - (x) i - γ^n u^i, we obtain1 + ∑_n =1^N-1α^n u^i 1/x - γ^n u^i = 1 + ∑_n =1^N-1α^n u^i (x) - γ^n u^i - (x) i/((x) - γ^n u^i)^2 + (x)^2= 1 + (x) ∑_n =1^N-1α^n u^i 1/((x) - γ^n u^i)^2 + (x)^2 + ∑_n =1^N-1α^n u^i - γ^n u^i/((x) - γ^n u^i)^2 + (x)^2 - (x) i∑_n =1^N-1α^n u^i 1/((x) - γ^n u^i)^2 + (x)^2= 1 + (x) c^i(x) + d^i(x) - (x) i c^i(x) = 1 +d^i(x) + c^i(x) x,where x is the complex conjugate of x andc^i(x)= ∑_n =1^N-1α^n u^i 1/((x) - γ^n u^i)^2 + (x)^2, d^i(x)= ∑_n =1^N-1α^n u^i - γ^n u^i/((x) - γ^n u^i)^2 + (x)^2.As each α^n is nonpositive, d^i(x) is nonnegative and c^i(x) is nonpositive. Combining this with x∈ℂ∖ (0, ∞), it follows that 1 + d^i(x)+ c^i(x) x≠ 0 so that all eigenvalues of the matrix B are indeed real and positive. § PRIMARY MATRIX FUNCTIONSIn this appendix, we collect some facts about matrix functions from the textbook <cit.> that are used in Appendix <ref>. First, we recall the definition of a (primary) matrix function:Let A ∈ℂ^ℓ×ℓ be a matrix with distinct eigenvalues λ_1, …, λ_m, m ≤ℓ. Denote by n_ithe algebraic multiplicity of λ_i, i ∈{1, …, m}.Let O be an open neighbourhood of λ_1, …, λ_m in ℂ and f: O →ℂ a function.[If A ∈ℝ^ℓ×ℓ and all eigenvalues of A are real, O can be taken as an open neighbourhood of λ_1, …, λ_m in ℝ, provided that f is also real valued.] (a) The function f is said to be defined on the spectrum of A if it is n_i-1-times differentiable at λ_i, i ∈{1, …, m}. (b) If f is defined on the spectrum of A, then the primary matrix function f(A) is defined byf(A) :=p(A),where p:ℂ→ℂ is the unique Hermite interpolating polynomial satisfying p^(k)(λ_i) = f^(k)(λ_i) for k ∈{0, …, n_i-1} and i ∈{1, …, s}.[If A, λ_1, …, λ_m, O and f are all real valued and f defined on the spectrum of A, the Hermite interpolating polynomial is also real valued.]As a prime example, note that the exponential function is defined on the spectrum of all matrices A ∈ℂ^ℓ×ℓ and exp(A) is just the matrix exponential. We recall some elementary properties of (primary) matrix functions:Let A ∈ℂ^ℓ×ℓ be a matrix with distinct eigenvalues λ_1, …, λ_m, m ≤ℓ and f: ℂ→ℂ a function defined on the spectrum of A. Then: (a) If P ∈ℂ^ℓ×ℓ commutes with A, then f(A) and P also commute. (b) If P ∈ℂ^ℓ×ℓ is invertible, then P f(A) P^-1 = f(P A P^-1). (c) The eigenvalues of f(A) are f(λ_1), …, f(λ_m). (d) f(A) is invertible if and only if f(λ_i) ≠ 0 for all i ∈{1, …, m}. Assertions (a), (b) and (c) are parts of <cit.>. Finally, (d) follows from (c) and the fact that f(A) is invertible if and only if zero is not an eigenvalue.Finally, we recall a result on the principal square root <cit.>: Let A ∈ℝ^ℓ×ℓ be a matrix whose eigenvalues are all real and positive. Then there exists a unique matrix P ∈ℝ^ℓ×ℓ with positive eigenvalues such that P^2 = A. It is given by the primary matrix function P = √(A) in the sense of Definition <ref>. abbrv | http://arxiv.org/abs/1707.08464v4 | {
"authors": [
"Bruno Bouchard",
"Masaaki Fukasawa",
"Martin Herdegen",
"Johannes Muhle-Karbe"
],
"categories": [
"q-fin.PM",
"math.PR"
],
"primary_category": "q-fin.PM",
"published": "20170726143233",
"title": "Equilibrium Returns with Transaction Costs"
} |
[@twocolumnfalseSpectral sequences of Type Ia supernovae. I. Connecting normal and sub-luminous SN Ia and the presence of unburned carbon [========================================================================================================================= numberstyleCurrent virtual reality (VR) training simulators of liver punctures often rely on static 3D patient data and use an unrealistic (sinusoidal) periodic animation of the respiratory movement. Existing methods for the animation of breathing motion support simple mathematical or patient-specific, estimated breathing models. However with personalized breathing models for each new patient, a heavily dose relevant or expensive 4D data acquisition is mandatory for keyframe-based motion modeling. Given the reference 4D data, first a model building stage using linear regression motion field modeling takes place. Then the methodology shown here allows the transfer of existing reference respiratory motion models of a 4D reference patient to a new static 3D patient. This goal is achieved by using non-linear inter-patient registration to warp one personalized 4D motion field model to new 3D patient data. This cost- and dose-saving new method is shown here visually in a qualitative proof-of-concept study. §.§ KeywordsVirtual Reality, Liver Puncture Training, 4D Motion Models, Inter-patient Registration of Motion Models]§ INTRODUCTION The virtual training and planning of minimally invasive surgical interventions with virtual reality simulators provides an intuitive, visuo-haptic user interface for the risk-sensitive learning and planning of interventions.The simulation of liver punctures has been an active research area for years <cit.>.Obviously first, the stereoscopic visualization of the anatomy of the virtual patient body is important <cit.>. Second, the haptic simulation of the opposing forces through the manual interaction, rendered by haptic input and output devices, with the patient is key <cit.>. Third in recent developments, the simulation of the appearance and forces of the patient's breathing motions is vital <cit.>.The previously known VR training simulators usually use time invariant 3D patient models. A puncture of the spinal canal can be simulated sufficiently plausibly by such models. In the thoracic and upper abdominal region, however, respiratory and cardiac movements are constantly present. In the diaphragm area at the bottom of the lungs just above the liver, breathing movement differences in the longitudinal z direction of up to 5 cm were measured <cit.>. Now for 4D animation, the necessary data consists of a single 3D CT data set and a mathematical or personalized animation model. Ouraim here is to incorporate these physiological-functional movements into realistic modeling in order to offer the user a more realistic visuo-haptic VR puncture simulation. This means also to take into account the intra- and intercycle variability (hysteresis, variable amplitude during inhalation / exhalation).A major interest and long term goal of virtual and augmented reality is the planning <cit.> and intra-operative navigation assistance <cit.>. However, in these works breathing motion is not incorporated or applicability limits by neglecting breathing motion in terms of minimal tumor size are given <cit.>. Published approaches from other groups <cit.> model only a sinusoidal respiratory motion without hysteresis and amplitude variation. First steps in the direction of a motion model building framework were taken by our group <cit.>. Accurate simulation of respiratory motion depending on surrogate signals is relevant e.g. in fractionated radiotherapy. However, since a patient-specific 4D volume data set is required for personalized breathing model building and its acquisition is associated with a high radiation dose with 4D-CT (≥ 20-30 mSv (eff.)), our approach is the transfer of existing 4D breathing models to new 3D patient data. For comparison, the average natural background radiation is approximately 2.1 mSv (eff.) per year[http://www.bfs.de/EN/topics/ion/environment/natural-radiation-exposure/natural-radiation-exposure_node.htmlIntercontinental flight max. 0.11 mSv (eff.)].On the other hand, there is no medical indication to acquire 4D CT data just for training purposes and model building from 4D MR data to be included is unjustifiable for cost reasons.In this paper, we present a feasibility study with first qualitative results for the transfer of an existing 4D breathing model<cit.> to static 3D patient data, in which only a 3D CT covering chest and upper abdomen at maximum inhalation is necessary (approximately 2- 13 mSv (eff.))[https://static.healthcare.siemens.com/siemens_hwem-hwem_ssxa_websites-context-root/wcm/idc/groups/public/@global/@imaging/@ct/documents/download/mdaw/mtm1/ edisp/ct_somatom_definition_as_brochure-00032845.pdfSiemens Somatom Definition AS].§ RECENT SOLUTIONThe existing solution requires a full 4D data set acquisition for each new patient. In <cit.>, concepts for a 3D VR simulator and efficient patient modeling for the training of different punctures (e.g. liver punctures) have already been presented, see Fig. <ref>. A Geomagic Phantom Premium 1.5 HighForce is used for the manual control and haptic force feedback of virtual surgical instruments.Nvidia shutter glasses and a stereoscopic display provide the plausible rendering of the simulation scene. This system uses time invariant 3D CT data sets as a basis for the patient model. In case of manual interaction with the model, tissue deformation due to acting forces of the instruments are represented by a direct visuo-haptic volume rendering method.New developments of VR simulators <cit.> allow a time-variant 4D-CT data set to be used in real time for the visualized patient instead of a static 3D CT data set. The respiratory movement can be visualized visuo-haptically as a keyframe model using interpolation or with a flexible linear regression based breathing model as described below.§ PROPOSED SOLUTIONThe new solution requires only a 3D data set acquisition for each new patient. §.§ Modeling of Breathing Motion Realistic, patient-specific modeling of breathing motion in <cit.>relies on a 4D CT data set covering one breathing cycle. It consists of N_phases phase 3D images indexed by j. Furthermore, a surrogate signal (for example, acquired by spirometry) to parametrize patient breathing in a low-dimensional manner is necessary.We use a measured spirometry signal v (t) [ml] and its temporal derivative in a composite surrogate signal: (v (t), v '(t))^T. This allows to describe different depths of breathing and the distinction between inhalation and exhalation (respiratory hysteresis). We assume linearity between signal and motion extracted from the 4D data. First, we use the 'sliding motion'-preserving approach from <cit.> for N_phases-1 intra-patient inter-phase image registrations to a selected reference phase j_ref:φ^pat4D_j=φargmin(D_NSSD[I^pat4D_j, I^pat4D_j_ref∘φ] +α_S · R_S(φ)),j∈{1,..,j_ref-1,j_ref+1,..,N_phases},where a distance measure D_NSSD (normalized sum of voxel-wise squared differences <cit.>) and a specialized regularization R_S establishes smooth voxel correspondences except in the pleural cavity where discontinuity is a wished feature <cit.>. Based on the results, the coefficients a ^ pat4D_1..3 are estimated as vector fields over the positions 𝐱. The personalized breathing model then can be stated as a linear multivariate regression <cit.>:φ̂^pat4D(𝐱,t)=a^pat4D_1(𝐱)· v(t) +a^pat4D_2(𝐱)· v'(t) +a^pat4D_3(𝐱), 𝐱∈Ω_pat4D.Thus, a patient's breathing state can be represented by a previously unseen breathing signal: Any point in time t corresponds to a shifted reference image I^pat4D_j_ref∘φ̂^pat4D(𝐱,t). Equipped with a real-time capable rendering technique via ray-casting with bent rays (see <cit.> for full technical details), the now time variant model-based animatable CT data I^pat4D_j_ref can be displayed in a new variant of the simulator and used for training. The rays are bent by the breathing motion model and this conveys the impression of an animated patient body, while being very time efficient (by space-leaping and early ray-termination) compared to deforming the full 3D data set for each time point and linear ray-casting afterwards <cit.>.§.§ Transfer of Existing Respiratory Models to new, static Patient Data Using the method described so far, personalized breathing models can be created, whose flexibility is sufficient to approximate the patients' breathing states, which are not seen in the observation phase of the model formation.However, the dose-relevant or expensive acquisition of at least one 4D data set has thus far been necessary for each patient.Therefore, here we pursue the idea to transfer a readily built 4D reference patient breathing model to new static patient data pat3D and to animate it in the VR simulator described in Sec. <ref>.For this purpose, it is necessary to correct for the anatomical differences between the reference patient with the image data I^pat4D_j_ref and the new patient image data I^pat3D_ref based on a similar breathing phase. This is achieved, for example, by a hold-breath scan (ref) in the maximum inhalation state, which corresponds to a certain phase j_ref in a standardized 4D acquisition protocol. A nonlinear inter-patient registration φ (𝐱): Ω_pat3D→Ω_pat4D with minimization of a relevant image distance D ensures the necessary compensation <cit.>:φ^pat3D → pat4D_j_ref=φargmin(D_SSD[I^pat3D_ref, I^pat4D_j_ref∘φ] +α_D· R_D(φ)), where a distance measure D_SSD (sum of squared voxel-wise differences) and a diffusive non-linear regularization R_D establishes smooth inter-patient voxel correspondences. On both sides, the breathing phase 3D image of maximum inhalation is selected as the reference phase (ref). The distance measurement can be selected according to the modality and quality of the image data. The transformation φ^pat3D → pat4D_j_ref, which is determined in the nonlinear inter-patient registration, can now be used to warp the intra-patient inter-phase deformations of the reference patient φ^pat4D_j as a plausible estimate φ ^ pat3D_j (j ∈{1, …, n }; ∘: right to left):φ^pat3D_j =( φ^pat3D → pat4D_j_ref)^-1∘φ^pat4D_j∘φ^pat3D → pat4D_j_ref.The approach for estimating the respiratory motion for the new patient can now be applied analogously to the reference patient (see Sec. <ref>). With a efficient regression method <cit.>, the breathing movement of virtual patient models, which are only based on a comparatively low dose of acquired 3D-CT data, can be plausibly approximated:φ̂^pat3D(𝐱,t)= a^pat3D_1(𝐱)· v(t) +a^pat3D_2(𝐱)· v'(t) +a^pat3D_3(𝐱), 𝐱∈Ω_pat3D.Optionally, simulated surrogate signals v (t)can be used for the 4D animation of 3D CT data. Simple alternatives are to use the surrogate signal of the reference patient or also a (scaled) signal of the new patient pat3D, which can simply be recorded with a spirometric measuring device without new image acquisition. § EXPERIMENTS AND RESULTS We performed a qualitative feasibility study, results are animated in the 4D VR training simulator <cit.>.For the 4D reference patient, a 4D-CT data set of the thorax and upper abdomen with 14 respiratory phases (512^2 × 462 voxel to 1^3 mm) and a spirometry signal v (t) were used (Fig. <ref>). The new patient is represented only by a static 3D CT data set (512^2 × 318 voxel to 1^3 mm).All volume image data was reduced to a size of 256^3 voxel due to the limited graphics memory of the GPU used (Nvidia GTX 680 with 3 GB RAM).According to Eq. <ref> we first perform the intra-patient inter-phase registrations to a chosen reference phase j_ref.The registrations from Eqs. <ref> and <ref> use weights α_S = 0.1 and α_D=1 for the regularizers R_S and R_D. In both registration processes, the phase with maximum inhalation is used as the reference respiratory phase j_ref and for the training of the breathing model.The respiratory signal used for model training is shown in Fig. <ref>, gray curve. We show the areas with plausible breathing simulation and use the unscaled respiratory signal of pat3D with larger variance to provoke artifacts (Fig. <ref>, blue curve). The model training according to Eqs. <ref> and <ref> is very efficient using matrix computations.We use manual expert segmentations of the liver and lungs, available for every phase of the 4D patient, to mainly assess the quality of the inter-patient registration in Eq. <ref>. Via the availabe inter-phase registrations φ^pat4D_j(Eq. <ref>) to the 4D reference phase, we first warp the phase segmentation masks accordingly. After applying the inter-patient registration to pat3D, we have the segmentation masks of pat4D in the space of the targeted 3D patient. Now for this patient, also a manual expert segmentation is availabe for comparison. Quantitatively, the DICE coefficients of the transferred segmentation masks (liver, lungs) can be given to classify the quality of the registration chain of the reference respiratory phases (single atlas approach).Qualitatively, we present sample images for four time instants and a movie.The mean DICE coefficients of the single-atlas registration of the liver and lung masks to the new static patient pat3D yield satisfying values of 0.86±0.12 and 0.96±0.09. Note the clearly different scan ranges of the data sets (Fig. <ref>a). The animation of the relevant structures is shown as an example in Fig. <ref>, using a variable real breathing signal of the target patient pat3D (Fig. <ref>b). In the puncture-relevant liver region, the patient's breathing states are simulated plausibly for the 4D reference patient (Fig. <ref>) and, more importantly, the 3D patient (Figs. <ref>, <ref>), to which the motion model of pat4D was transferred[https://goo.gl/DVVYzwDemo movie, click here]. § DISCUSSION, OUTLOOK AND CONCLUSIONFor interested readers, the basic techiques for 4D breathing motion models have been introduced in <cit.> by our group. However there, the motion model is restricted to the inside of the lungs and by design a mean motion model is built from several 4D patients. The mean motion model is artificial to some degree, more complex and timely to build. The method described here for the transfer of retrospectively modeled respiratory motion of one 4D reference patient to a new 3D patient data set is less complex and extends to a larger body area. It already allows the plausible animation of realistic respiratory movements in a 4D-VR-training-simulator with visuo-haptic interaction. Of course in the future, we want to build a mean motion model for the whole body section including (lower) lungs and the upper abdomen, too. In other studies, we found α_D=1 in Eq. <ref> robust (compromise between accuracy and smoothness) for inter-patient registration with large shape variations<cit.>. In Eq. <ref> for intra-patient inter-phase registration, we use α_S=0.1 to allow more flexibility for more accuracy as the shape variation between two phases of the same patient is much smaller <cit.>.We achieve qualitatively plausible results for the liver area in this feasibility study. In the upper thorax especially at the rib cage in neighborhood to the dark lungs stronger artifacts can occur (Fig. <ref>). They are due to problems in the inter-patient registration that is a necessary step for the transfer of the motion model. The non-linear deformation sometimes is prone to misaligned ribs. The same is true for the lower thorax with perforation first of the liver and then diaphragm (Fig. <ref>). Further optimization have to be carried out as artifacts can appear on the high contrast lung edge (diaphragm, ribs) with a small tidal volume. For liver punctures only, the artifacts of smeared ribs are minor as can be seen in Fig. <ref>.Summing up, the previous assumption from Sec. <ref> of a dose-relevant or expensive acquisition of a 4D-CT data set for each patient, can be mitigated for liver punctures by the presented transfer of an existing 4D breathing model.Future work will deal with the better adaptation and simulation of the breathing signal. Further topics are the optimization of the inter-patient registration and the construction of alternatively selectable mean 4D reference breathing models. As in <cit.>, the authors plan to perform usability studies with medical practitioners.To conclude, the method allows VR needle puncture training in the hepatic area of breathing virtual patients based on a low-risk and cheap 3D data acquisition for the new patient only. The requirement of a dose-relevant or expensive acquisition of a 4D CT data set for each new patient can be mitigated by the presented concept. Future work will include the reduction of artifacts and building mean reference motion models.§ ACKNOWLEDGEMENTSupport by grant: DFG HA 2355/11-2. wscg-alpha 0 | http://arxiv.org/abs/1707.08554v2 | {
"authors": [
"Andre Mastmeyer",
"Matthias Wilms",
"Heinz Handels"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170726173409",
"title": "Interpatient Respiratory Motion Model Transfer for Virtual Reality Simulations of Liver Punctures"
} |
Center for Emergent Matter Science, RIKEN, Wako-shi, Saitama, 351-0198, Japan Institute for Theoretical and Applied Electrodynamics, Russian Academy of Sciences, Moscow, 125412 Russia Center for Emergent Matter Science, RIKEN, Wako-shi, Saitama, 351-0198, Japan Institute for Theoretical and Applied Electrodynamics, Russian Academy of Sciences, Moscow, 125412 Russia Moscow Institute for Physics and Technology (State University), Dolgoprudnyi, 141700 Russia Center for Emergent Matter Science, RIKEN, Wako-shi, Saitama, 351-0198, Japan Moscow Institute for Physics and Technology (State University), Dolgoprudnyi, 141700 Russia Institute for Theoretical and Applied Electrodynamics, Russian Academy of Sciences, Moscow, 125412 Russia Dukhov Research Institute of Automatics, Moscow, 127055 Russia Center for Emergent Matter Science, RIKEN, Wako-shi, Saitama, 351-0198, Japan Department of Physics, University of Michigan, Ann Arbor, MI 48109-1040, USAWe theoretically study the effects of electron-electron interaction in twisted bilayer graphene in applied transverse dc electric field. When the twist angle is not very small, the electronic spectrum of the bilayer consists of four Dirac cones inherited from each graphene layer. Applied bias voltage leads to the appearance of two hole-like and two electron-like Fermi surface sheets with perfect nesting among electron and hole components. Such a band structure is unstable with respect to exciton band gap opening due to the screened Coulomb interaction. The exciton order parameter is accompanied by the spin-density-wave order. The value of the gap depends on the twist angle. More importantly, it can be controlled by applied bias voltage which opens new directions in manufacturing of different nanoscale devices. 73.22.Pr, 73.21.Ac Externally controlled band gap in twisted bilayer graphene Franco Nori December 30, 2023 ==========================================================Introduction— It is known that application of the bias voltage to AB stacked bilayer graphene opens a gap in its electronic spectrum <cit.>. This feature makes bilayer graphene promising for applications in electronics. Experiment shows, however, that, in many cases, the structure of the bilayer graphene samples is different from a simple AB stacking, and is characterized by a non-zero twist angle between layers <cit.>. Twisting makes the physics of the bilayer graphene more complicated and rich. For example, it leads to the appearance of the moiré patterns – alternating dark and bright regions seen in STM images <cit.>. For a countable set of twist angles the system has a superstructure with the period, which may coincide or be a commensurate with the moiré period <cit.>.Rotational misorientation affects also the electronic properties of the bilayer.Analysis in the single-particle approximation shows that one can distinguish three qualitatively different types of behavior of the spectrum at low energies. When the twist angle θ is close to commensurate value corresponding to the superstructure with considerably small size of the supercell, the spectrum has a gap at Fermi level <cit.>. This gap, however, is very sensitive to small variations of the twist angle, and is non-negligible only for a limited number of superstructures <cit.>. With the exception of those values, one can assume that when θ is greaterthan critical value θ_c∼1–2^∘, the electron spectrum has a linear dispersion and consists of four Dirac cones inherited from two graphene layers. For a commensurate structure, which we only study in this paper, the four Dirac points of two graphene layers are distributed between two non-equivalent corners of the superlattice Brillouin zone, forming the band structure with two doubly-degenerate Dirac cones in the corners of the superlattice Brillouin zone. The Fermi velocity of these Dirac cones, however, turns out to be θ-dependent: it decreases monotonically down to zero when θ goes to critical value <cit.>. Finally, when θ<θ_c, the spectrum at low energies is characterized by flat bands and the density of states has a peak at the Fermi level <cit.>.Despite the progress in understanding the electronic properties of twisted bilayer graphene, many issues still remain unclear. First of all this concerns many-body effects, since in the majority of cases, theorists are limited to a single-electron approximation <cit.>. In our work, we consider the effects of electron-electron interaction for superstructures with θ>θ_c and not too small size of the supercell, when the single-electron gap can be neglected. We also assume that the transverse electric field is applied to the bilayer. This lifts the double degeneracy of four Dirac cones of the system; two of them are shifted upwards in energy, and the other two – downwards. As a result, two hole-like and two electron-like Fermi surface sheets appear in the system with a perfect nesting between hole-like and electron-like components. This leads to Fermi surface instability with respect to exciton band gap formation due to electron-electron interaction. We show also that the exciton order parameter is accompanied by the spin-density-wave (SDW) order. The dependence of the gap on the twist angle and on the bias voltage is analyzed. It is shown that the gap can be effectively controlled by external field, which may be useful for applications.Geometry of twisted bilayer graphene— The geometry of the twisted bilayer graphene is described in details, in several papers <cit.>. Here we only present the basic properties necessary for further considerations. Graphene monolayer has a hexagonal crystal structure consisting of two triangular sublattices A and B. Carbon atoms in the layer 1 are located in positions 𝐫_𝐧^1A=𝐫_𝐧^1≡ n𝐚_1+m𝐚_2 and 𝐫_𝐧^1B=𝐫_𝐧^1+δ with 𝐧=(n, m) (n, m are integers), where 𝐚_1,2=a(√(3),∓1)/2 are the graphene lattice vectors (a=2.46Å) and δ=a(1/√(3),0). The positions of atoms in layer 2 are𝐫_𝐧^2B=𝐫_𝐧^2≡ d𝐞_z+n𝐚_1'+m𝐚_2' and 𝐫_𝐧^2A=𝐫_𝐧^2-δ', where 𝐚_1,2', δ'are vectors𝐚_1,2, δ rotated by angle θ, symbol 𝐞_z is the unit vector along the z-axis, and d=3.35Åis the interlayer distance. We also introduce vectors δ^iα≡𝐫_𝐧^iα-𝐫_𝐧^i, whichare independent on 𝐧. Limiting case θ=0 corresponds to the AB stacking. The superstructure exists for twist angles equalcosθ=3m_0^2+3m_0r+r^2/2/3m_0^2+3m_0r+r^2 ,where m_0 and r are co-prime positive integers. Superlattice vectors 𝐑_1,2 are linear combinations of 𝐚_1 and 𝐚_2 with integer-valued coefficients <cit.>. The magnitude of these vectors is <cit.> |𝐑_1,2|=rL/√(g), where L=a/[2sin(θ/2)] is the moiré period and g=1 if r≠3n, or g=3 if r=3n (n is integer). Thus, only superstructures with r=1 coincide with the moiré lattice. The number of graphene unit cells of each layer inside the supercell is N_sc=(3m_0^2+3m_0r+r^2)/g. Thus, the number of carbon atoms in the superlattice cell is equal to 4N_sc.We introduce 𝐛_1,2=2π(1/√(3),∓1)/a, which are the reciprocal lattice vectors of the layer 1, and 𝐛_1,2' for layer 2 (𝐛_1,2' are connected to 𝐛_1,2 by rotation on angle θ). Vectors G_1,2 are the reciprocal vectors for superlattice. All these vectors are related to each other according to the following formulas: 𝐛_1'=𝐛_1-r G_1 and 𝐛_2'=𝐛_2+r( G_1+ G_2) if r≠3n, or 𝐛_1'=𝐛_1+r( G_1+2 G_2)/3 and 𝐛_2'=𝐛_2-r(2 G_1+ G_2)/3, otherwise. Each graphene layer has two non-equivalent Dirac points located at corners of its Brillouin zone. Thus, the total number of Dirac points for the bilayer is four. The Brillouin zone of the superlattice has a shape of hexagon. It can be obtained by N_sc times folding of the Brillouin zone of the layer 1 or 2. As a result of this folding, Dirac points of each layer are translated to two non-equivalent Dirac points of the superlattice, 𝐊_1 and𝐊_2, located at corners of the reduced Brillouin zone. Thus, one can say that the Dirac points 𝐊_1,2 are doubly degenerate since each of them corresponds to two non-equivalent Dirac points of constituent layers. Points 𝐊_1,2 can be expressed via vectors G_1,2 as 𝐊_1=( G_1+2 G_2)/3 and 𝐊_2=(2 G_1+ G_2)/3.Model Hamiltonian— We start from the tight-binding model for the p_z electrons in twisted bilayer graphene. Electrons are assumed to interact via the screened Coulomb interaction. Formally, we writeH = ∑_i𝐧j𝐦αβσt(𝐫_𝐧^iα;𝐫_𝐦^jβ) d^†_𝐧iασd^†_𝐦jβσ+ V_b/2∑_𝐧(n_𝐧1-n_𝐧2)+ 1/2∑_i𝐧j𝐦αβσσ'd^†_𝐧iασd^†_𝐧iασ V(𝐫_𝐧^iα-𝐫_𝐦^jβ)d^†_𝐦jβσ'd^†_𝐦jβσ' ,where d^†_𝐧iασ and d^†_𝐧iασ are the creation and annihilation operators of the electron with spin projection σ, located at site 𝐧 in the layer i(=1,2) in the sublattice α(=A,B), and n_𝐧i=∑_ασd^†_𝐧iασd^†_𝐧iασ. The first term describes the intersite hopping. For intralayer hopping we consider only the nearest-neighbor term with amplitude -t, where t=2.57 eV. The interlayer hopping amplitudes are parameterized as described in Refs. <cit.>, with the largest interlayer hopping amplitude equal to t_0=0.4 eV. Second term describes the potential energy difference between layers due to the applied bias voltage V_b. Third term corresponds to the Coulomb interaction between electrons. The precise form of the function V(𝐫) will be discussed below.Let us first consider single-particle part of the Hamiltonian (<ref>) (first and second terms). We proceed to the momentum representation introducing electronic operatorsd^†_𝐩𝐆iασ=1/√( N)∑_𝐧 e^-i(𝐩+𝐆)𝐫_𝐧^id_𝐧iασ ,where N is the number of graphene unit cells in the sample in one layer, momentum 𝐩 lies in the first Brillouin zone of the superlattice, while 𝐆=n G_1+m G_2 is the reciprocal vector of the superlattice, lying in the first Brillouin zone of the ith layer. The number of such vectors 𝐆 is equal to N_sc for each graphene layer. In this representation the single-particle part of Hamiltonian (<ref>) can be written asH_0 = ∑_𝐩[∑_𝐆_1𝐆_2 ijαβσt̃_ij^αβ(𝐩+𝐆_1;𝐆_1-𝐆_2) d^†_𝐩𝐆_1iασd^†_𝐩𝐆_2jβσ.+ .V_b/2∑_𝐆ασ( d^†_𝐩𝐆1ασd^†_𝐩𝐆1ασ- d^†_𝐩𝐆2ασd^†_𝐩𝐆2ασ)],wheret̃_ij^αβ(𝐤;𝐆)=1/N_sc∑'_𝐧𝐦e^-i𝐤(𝐫_𝐧^i-𝐫_𝐦^j) e^-i𝐆𝐫_𝐦^jt(𝐫_𝐧^iα;𝐫_𝐦^jβ) .In the last formula, the summation over 𝐦 is performed over sites inside the zeroth supercell, while summation over 𝐧 is performed over all sites in the sample (or vice versa).For a given momentum 𝐩, Hamiltonian (<ref>) can be represented in the form of 4N_sc×4N_sc matrix. We diagonalize this matrix numerically, and calculate both the spectrum E_𝐩^(S) (S=1, 2, …, 4N_sc) and eigenvectors U^(S)_𝐩𝐆iα. Figure <ref> shows the band structure calculated for the sample with m_0=5, r=1 (θ≅6.009^∘), and for bias voltage V_b=0.15t. Only bands within the energy window |E|/t<0.5 are shown. Low-energy spectrum consists of four Dirac cones located in pairs near two Dirac points of the superlattice, 𝐊_1 and 𝐊_2. Bias voltage shifts apexes of two of these cones to positive energies, and two – to negative energies. The electron density corresponding to upper (lower) Dirac cones is concentrated mostly in layer 1 (layer 2), even though the interlayer hybridization tends to distribute it uniformly between the layers. When bias voltage is applied, the system acquires a Fermi surface, which consists of two closed curves (valleys) located near two Dirac points. These curves are nearly circular when the bias voltage is small enough, while trigonal warping reveals itself at larger V_b (see Fig. <ref>). The trigonal warping becomes more pronounced for superstructures with smaller θ.The most important feature of the Fermi surface shown in Fig. <ref> is its double degeneracy; in each valley, the Fermi surface curve is created by both the electron-like band corresponding to the lower Dirac cone, and the hole-like band belonging to the upper Dirac cone. In other words, we have a situation with perfect nesting of Fermi surface. This leads to the instability of a Fermi liquid state with respect to the formation of some type of ordering due to the electron-electron interaction.Consider now the interaction term of Hamiltonian (<ref>). In terms of electron operators d^†_𝐩𝐆iασ, Eq. (<ref>), the last term in Eq. (<ref>) takes the form1/2 N∑_iασ jβσ'∑_𝐩_1𝐩_2𝐩_1'𝐩_2'∑_𝐆_1𝐆_2𝐆_1'𝐆_2'∑_𝐆δ_𝐆, 𝐤_1+𝐤_2-𝐤_1'-𝐤_2'× d^†_𝐩_1𝐆_1iασd^†_𝐩_1'𝐆_1'iασV_iα;jβ(𝐤_1-𝐤_1';𝐆) d^†_𝐩_2𝐆_2jβσ'd^†_𝐩_2'𝐆_2'jβσ',where 𝐤_1,2=𝐩_1,2+𝐆_1,2, 𝐤_1,2'=𝐩_1,2'+𝐆_1,2', andV_iα;jβ(𝐤;𝐆)=1/N_sc∑'_𝐧𝐦 e^-i𝐤(𝐫_𝐧^i-𝐫_𝐦^j) e^-i𝐆𝐫_𝐦^jV(𝐫_𝐧^iα-𝐫_𝐦^jβ) .As before, in the last equation the summation over 𝐦 is performed over sites inside the zeroth supercell, while summation over 𝐧 is performed over all sites in the sample. For intralayer (i=j) interaction one can separate the summation on 𝐧 and 𝐦 by substitution 𝐧→𝐧+𝐦. As a result, we obtain V_iα;iβ(𝐤;𝐆)=(∑_𝐛^iδ_𝐛^i, 𝐆) ∑_𝐧e^-i𝐤𝐫_𝐧^iV(𝐫_𝐧^i+δ^iα-δ^iβ), where the first and second summations are performed over all reciprocal lattice vectors (𝐛^i) and all lattice sites of the layer i, correspondingly. Let us denote by 𝐤̃^i the `vector 𝐤 modulo 𝐛^i', that is, the vector lying in the first Brillouin zone of the layer i and coinciding with 𝐤 upon the translation on some reciprocal vector 𝐛^i. By definition (<ref>), we have V_iα;iβ(𝐤;𝐆)=V_iα;iβ(𝐤̃^i;𝐆). Below we will use the continuum (low-𝐤) approximation for V_iα;iβ(𝐤;𝐆), when one can substitute the summation over lattice sites by the 2D integration. As a result, we obtainV_iα;iβ(𝐤;𝐆)=1/ V_cV_ii(𝐤̃^i) e^i𝐤̃^i(δ^iα-δ^iβ)∑_𝐛^iδ_𝐛^i, 𝐆 ,where V_c=√(3)a^2/2 is the graphene's unit cell area, and V_ii(𝐤)=∫ d^2rV(𝐫)e^-i𝐤𝐫 is the Fourier transform of the function V(𝐫).We introduce also the Fourier transform for interlayer interactionas V_ij(𝐤)=∫ d^2rV(d𝐞_z+𝐫)e^-i𝐤𝐫 (i≠ j). Substituting this equation to Eq. (<ref>), one obtainsV_iα;jβ(𝐤;𝐆) = 1/ V_c∑_𝐛^iV_ij(𝐤+𝐛^i) e^i(𝐤+𝐛^i)(δ^iα-δ^iβ)× ∑_𝐛^jδ_𝐛^j, 𝐛^i+𝐆 .For functions V_ij(𝐤) we use expressions for the screened Coulomb potential in the form <cit.>:V_ij(𝐪)=v_𝐪/1+Π_𝐪v_𝐪( [ 1 e^-qd; e^-qd 1 ]) ,where bare Coulomb potential is v_𝐪=2π e^2/ϵ q. The permittivity of the substrate is ϵ and Π_𝐪≡-P(ω=0,𝐪), where P(ω,𝐪) is the polarization operator of the bilayer. Since interlayer potential V_ij(𝐪) decays exponentially with q and e^-|𝐛_1,2|d≈5×10^-5, one can take only one term in the sum over 𝐛^i in Eq. (<ref>), such that𝐤+𝐛^i=𝐤̃^i. Below we will use the long wave length approximation for the function Π_𝐪. In this case, it is independent on 𝐪 and equals to the density of states of the bilayer at the Fermi level. We calculate the latter quantity following procedure described below.Exciton order parameter— To go further, we introduce new electronic operators ψ_𝐩Sσ defined according to the relations:ψ_𝐩Sσ=∑_𝐆iαU^(S)*_𝐩𝐆iαd_𝐩𝐆iασ , d_𝐩𝐆iασ=∑_SU^(S)_𝐩𝐆iαψ_𝐩Sσ .In this representation,the single-particle part of the Hamiltonian (<ref>) is diagonal. The interaction term takes a form ∑ V(𝐩_1S_1…𝐩_4S_4) ψ^†_𝐩_1S_1σψ^†_𝐩_2S_2σψ^†_𝐩_3S_3σ'ψ^†_𝐩_4S_4σ', where the summation is performed over all indices in this expression, and V(𝐩_1S_1;𝐩_2S_2;𝐩_3S_3;𝐩_4S_4) is the convolution of the product of the function V_iα;jβ(𝐤;𝐆) and four wave functions U^(S)_𝐩𝐆iα.Let us denote the indices of two electron-like and two hole-like bands closest to the Fermi level by S^μ_+ and S^μ_-, correspondingly, where μ=1,2. The perfect nesting of hole-like and electron-like Fermi surfaces gives rise to a dielectric order parameter driven by the Coulomb interaction. In general, this order parameter is a superposition of the expectation values of the type ⟨ψ^†_𝐩Sσψ^†_𝐩S'σ'⟩, where S≠ S'. Here we assume that non-zero expectation values are only those, which couple the bands S^μ_+ and S^μ_- (see Fig. <ref>). Analysis shows also that corresponding order is of the SDW type. Assuming planar spin configuration, we obtain that non-zero expectation values are the followingη_𝐩μσ=⟨ψ^†_𝐩S^μ_+σψ^†_𝐩S^μ_-σ̅⟩= ⟨ψ^†_𝐩S^μ_-σ̅ψ^†_𝐩S^μ_+σ⟩^*,where σ̅ means `not σ'. Note that possible charge-density-wave order, corresponding to the non-zero η̃_𝐩μ=∑_σ⟨ψ^†_𝐩S^μ_+σψ^†_𝐩S^μ_-σ⟩ is energetically unfavorable in comparison to the SDW ones.We will study the model (<ref>) in the mean-field approximation. This scheme involves the replacement of the product of two operators O_1O_2→ O_1⟨ O_2⟩+⟨ O_1⟩ O_2-⟨ O_1⟩⟨ O_2⟩, where ⟨ O_1,2⟩∝η_𝐩μσ. As a result, the interaction term of the Hamiltonian becomes quadratic in ψ^†_𝐩Sσ operators. In addition, we truncate the total mean-field Hamiltonian keeping only bands S^μ_±. Moreover, in the interaction part of this Hamiltonian we keep only terms coupling the electron band S^μ_+ with the hole band S^μ_- with the same μ. As a result, the effective mean-field Hamiltonian becomesH_MF=∑_𝐩μΨ^†_𝐩μĤ_𝐩μΨ^†_𝐩μ+U_c ,where we introduce the following 4-component operator Ψ^†_𝐩μ=(ψ^†_𝐩S^μ_+↑,ψ^†_𝐩S^μ_-↑, ψ^†_𝐩S^μ_+↓,ψ^†_𝐩S^μ_-↓)^T. In equation (<ref>), Ĥ_𝐩μ is the 4×4 matrixĤ_𝐩μ=([ E^(S^μ_+)_𝐩 0 0-Δ_𝐩μ↑^*; 0 E^(S^μ_-)_𝐩-Δ_𝐩μ↓ 0; 0-Δ_𝐩μ↓^* E^(S^μ_+)_𝐩 0;-Δ_𝐩μ↑ 0 0 E^(S^μ_-)_𝐩 ]),where Δ_𝐩μσ is the order parameter having the formΔ_𝐩μσ=1/ N∑_𝐪ν[A_μν(𝐩;𝐪)η_𝐪νσ+ B_μν(𝐩;𝐪)η^*_𝐪νσ̅],whereA_μν(𝐩;𝐪)≡ V(𝐩S^μ_-;𝐪S^ν_-;𝐪S^ν_+;𝐩S^μ_+) and B_μν(𝐩;𝐪)≡ V(𝐩S^μ_-;𝐪S^ν_+;𝐪S^ν_-;𝐩S^μ_+). The termU_c in Eq. (<ref>) is the c-numberU_c = 1/2 N∑_𝐩𝐪μνσ[A_μν(𝐩;𝐪)η^*_𝐩μση^*_𝐪νσ+. .B_μν(𝐩;𝐪)η^*_𝐩μση^*_𝐪νσ̅+H.c.].The precise form of the functions A_μν(𝐩;𝐪) and B_μν(𝐩;𝐪) is the following:A_μν(𝐩;𝐪) = ∑_iα jβ∑_𝐆_1𝐆_2𝐆_1'𝐆_2' U^(S^μ_-)*_𝐩𝐆_1iαU^(S^ν_-)_𝐪𝐆_1'iαU^(S^ν_+)*_𝐪𝐆_2jβU^(S^μ_+)_𝐩𝐆_2'jβ× V_iα;jβ(𝐩-𝐪+𝐆_1-𝐆_1';𝐆_1+𝐆_2-𝐆_1'-𝐆_2'),B_μν(𝐩;𝐪) = ∑_iα jβ∑_𝐆_1𝐆_2𝐆_1'𝐆_2' U^(S^μ_-)*_𝐩𝐆_1iαU^(S^ν_+)_𝐪𝐆_1'iαU^(S^ν_-)*_𝐪𝐆_2jβU^(S^μ_+)_𝐩𝐆_2'jβ× V_iα;jβ(𝐩-𝐪+𝐆_1-𝐆_1';𝐆_1+𝐆_2-𝐆_1'-𝐆_2'). Minimizing the total energy at zero temperature and at half-filling, we obtain the system of equations for the order parameters:Δ_𝐩μσ=1/2 N∑_𝐪ν[A_μν(𝐩;𝐪)Δ_𝐪νσ/√(|Δ_𝐪νσ|^2+E^2_𝐪ν)+ B_μν(𝐩;𝐪)Δ^*_𝐪νσ̅/√(|Δ_𝐪νσ̅|^2+E^2_𝐪ν)],where E_𝐪μ=[E^(S^μ_+)_𝐪-E^(S^μ_-)_𝐪]/2.For a given superstructure and bias voltage we calculate the functions A_μν(𝐩;𝐪) and B_μν(𝐩;𝐪) numerically. Analysis shows, that with a good accuracy the following relations take place:A_μν(𝐩;𝐪) ≈ δ_μνA(𝐩;𝐪) ,B_μν(𝐩;𝐪) ≈ δ_μν̅B(𝐩;𝐪) ,where ν̅ means `not ν'. The deviations from these equalities do not exceed 1% for any superstructures considered. Note thatin the limit of uncoupled graphene layers, t_0=0, Eqs. (<ref>) become exact. This follows from the symmetry of the wavefunctions presented in the definition of the functionsA_μν and B_μν, Eqs. (<ref>): for uncoupled layers the electrons are localized either in layer 1 or 2.Further, by choosing appropriate phases of the wave functions U^(S^μ_±)_𝐩𝐆iα one can make A(𝐩;𝐪) and B(𝐩;𝐪) real-valued functions. In this case, one can choose Δ_𝐩μσ to be real-valued functions satisfying the relations Δ_𝐩μ↑=Δ_𝐩μ↓≡Δ_𝐩μ. The equation for the order parameters then becomesΔ_𝐩μ=1/2∫d^2q/v_BZ[A(𝐩;𝐪)Δ_𝐪μ/√(Δ_𝐪μ^2+E^2_𝐪μ)+ B(𝐩;𝐪)Δ_𝐪μ̅/√(Δ_𝐪μ̅^2+E^2_𝐪μ̅)],where v_BZ=8π^2/(a^2√(3)) is the area of the graphene's Brillouin zone, while the integration is performed over the Brillouin zone of the superlattice. In numerical calculations we take A(𝐩;𝐪)=[A_11(𝐩;𝐪)+A_22(𝐩;𝐪)]/2 and B(𝐩;𝐪)=[B_12(𝐩;𝐪)+B_21(𝐩;𝐪)]/2.Approximate solution, weak interaction limit— The main interest is the value of the function Δ_𝐩1 at the Fermi surface since it gives us the energy gap.Numerical analysis shows that if momenta 𝐩 and 𝐪 belong to different valleys, that is, they are located near different Dirac points, we have A(𝐩;𝐪)≈0 and B(𝐩;𝐪)≈0. This property makes it possible to consider the functions Δ_𝐩μ near each Dirac point, 𝐊_1 and 𝐊_2, independently. Let's take for example the 𝐊_1 point. The Fermi surface near each Dirac point is a closed curve having near-circular shape. Below we neglect the trigonal warping and approximate the Fermi surface by a circle with radius q_F^* calculated numerically by averaging over the Fermi surface. We assume also that Δ_𝐩μ are the step-like functions, describing by the equations Δ_𝐊_1+𝐩μ=Δ_μΘ(q_Λ-|p-q_F^*|), where the cutoff momentum q_Λ will be specified below. Thus, the region of integration in Eq. (<ref>) becomes a ring centered at the Dirac point 𝐊_1 and having radii q_F^*-q_Λ and q_F^*+q_Λ if q_Λ<q_F^*, or the circle with radius q_Λ, otherwise. In further approximation, we replace the functions A(𝐩;𝐪) and B(𝐩;𝐪) by constants A̅ and B̅ obtained by averaging ofA(𝐩;𝐪) and B(𝐩;𝐪) over the Fermi surface. We also approximate energies E_𝐪μ by linear functionsE_𝐊_1+𝐪1≈ v_F^*(q-q_F^*) ,E_𝐊_1+𝐪2≈ v_F^*(q+q_F^*) ,where the renormalized Fermi velocity is calculated numerically by averaging the function ∑_μ|∂ E_𝐊_1+𝐪μ∂𝐪|/2 over the Fermi surface. As a result, the system of equations for order parameters becomesΔ_1 = ∫_q_1^q_2dq[1/2qλ_AΔ_1/√(Δ_1^2+v_F^*2(q-q_F^*)^2)+1/2qλ_BΔ_2/√(Δ_2^2+v_F^*2(q+q_F^*)^2)], Δ_2 = ∫_q_1^q_2dq[1/2qλ_AΔ_2/√(Δ_2^2+v_F^*2(q+q_F^*)^2)+1/2qλ_BΔ_1/√(Δ_1^2+v_F^*2(q-q_F^*)^2)],where q_1=q_F^*-q_Λ, q_2=q_F^*+q_Λ if q_Λ<q_F^*, or q_1=0, q_2=q_Λ, otherwise, λ_A=2πA̅/v_BZ, and λ_B=2πB̅/v_BZ. We solve this system numerically.The magnitude of the order parameters Δ_μ depends on the values of A̅ and B̅, as well as on the cut-off momentum q_Λ. Our analysis shows that the main contribution to the functions A(𝐩;𝐪) and B(𝐩;𝐪) comes from the interlayer interaction. Following Refs. <cit.> we define q_Λ from the condition V_12(q_Λ)=V_12(0)/2.Assuming that q_Λd≪1 (which is correct for V_b≲ t_0 and e^2/ϵ v_F≲1), from Eqs. (<ref>) we obtain the estimate q_Λ≈2πΠ_0v_Fα, where Π_0 is the density of states of the bilayer at the Fermi level, α=e^2/ϵ v_F is the graphene's fine structure constant, and v_F=ta√(3)/2 is the Fermi velocity of the single layer graphene. Neglecting the trigonal warping, the density of states is expressed as Π_0≈4q_F^*/(π v_F^*). Thus, we obtain for q_Λq_Λ=8v_F/v_F^*α q_F^* . Approximate solution, strong interaction limit— The limit of weak coupling corresponds the case of α≪1. Simple estimates show, however, that for bilayer suspended in vacuum, ϵ=1, the parameter α≈2.6. When α increases, the cut-off momentum can exceed the size of the superlattice's Brillouin zone. In this case, we should increase the number of bands in our effective Hamiltonian. Simultaneously, we should increase, the number of order parameters Δ_𝐩μ, with μ now changing from 1 to some N>2. The rank of the matrix functions A_μν and B_μν becomes equal to N. This consideration can be substantially simplified if we considerthe added `high-energy' bands in the limit of decoupled (t_0=0) graphene layers. This approximation is justified, since for these bands we have |E_𝐩^(S)|≳ t_0. For decoupled layers, one can associate the band index S to the momentum 𝐩 lying in the Brillouin zone of the layer 1 or 2, that is, one can perform the band unfolding <cit.>. As a result, one can assume that the number of order parameters Δ_𝐩μ is still equal to 2, but the integration in Eqs. (<ref>) or (<ref>) is extended to the momenta exceeding the reciprocalunit cell of the superlattice. Applying this procedure, one must keep in mind that the two valleys should be still considered independently. To understand why this is so, let us consider the situation in the unfolded Brillouin zone from the beginning. Layer 1 has two non-equivalent Dirac points, 𝐊 and 𝐊'(=-𝐊). Rotation by twist angle θ transforms them into Dirac points of the layer 2, 𝐊_θ and 𝐊'_θ. The considered ordering corresponds to the formation of the electron-hole pair consisting of the electron with momentum 𝐊+𝐩 and the hole with momentum 𝐊_θ'+𝐩 (valley 1), and the electron with momentum 𝐊'+𝐩 and the hole with momentum 𝐊_θ+𝐩 (valley 2). [For superstructures with r≠3n. Otherwise, the electron and the hole have momenta 𝐊+𝐩 and 𝐊_θ+𝐩 in valley 1, and 𝐊'+𝐩 and 𝐊_θ'+𝐩 in valley 2.] It is clear, that the annihilation of such a pair in the valley 1 with simultaneous creation of the pair in the valley 2 is prohibited by the momentum conservation law. Thus, in our model, the intervalley scattering can be neglected [This picture is valid, of course, for E_Λ≡ v_Fq_Λ≲ t. At larger energies, the spectrum cannot be described by separate Dirac cones.].The procedure described above implies the knowledge of the energies E_𝐩μ at large momenta. We approximate E_𝐊_1+𝐩μ by Eqs. (<ref>), when p<q_0, where q_0=| G_1,2|/(2√(3)) is the radius of the circle centered at Dirac point 𝐊_1 and touching the edges of the reciprocalunit cell of the superlattice (see Fig. <ref>).At larger p we use the limit of decoupled layersE_𝐊_1+𝐩1≈ E_𝐊_1+𝐩2≈ v_Fp .With this accuracy we neglect the effect of the bias voltage at high energies, since V_b≲ t_0. As a result, approximate equation for the order parameters Δ_μ becomeΔ_1 = ∫_0^q_0dq1/2qλ_AΔ_1/√(Δ_1^2+v_F^*2(q-q_F^*)^2)+ ∫_q_0^q_Λdq1/2qλ_AΔ_1/√(Δ_1^2+v_F^2q^2)+ ∫_0^q_0dq1/2qλ_BΔ_2/√(Δ_2^2+v_F^*2(q+q_F^*)^2)+ ∫_q_0^q_Λdq1/2qλ_BΔ_2/√(Δ_2^2+v_F^2q^2) , Δ_2 = ∫_0^q_0dq1/2qλ_AΔ_2/√(Δ_2^2+v_F^*2(q+q_F^*)^2)+ ∫_q_0^q_Λdq1/2qλ_AΔ_2/√(Δ_2^2+v_F^2q^2)+ ∫_0^q_0dq1/2qλ_BΔ_1/√(Δ_1^2+v_F^*2(q-q_F^*)^2)+ ∫_q_0^q_Λdq1/2qλ_BΔ_1/√(Δ_1^2+v_F^2q^2) . Results and Discussion— We solve the system of equations (<ref>) [or (<ref>), if q_Λ<q_0] numerically for several superstructures with r=1 in a wide range of V_b and α. We found that the ratio Δ_2/Δ_1≳0.3 for any interaction strength considered and goes to 1 when α increases. Our main interest is the value of Δ_1 since it gives us the energy gap. The dependencies of the band gap Δ_1 on α for superstructures with m_0=2, 4, 6, 8, 10 (corresponding twist angles are θ≅17.7, 7.3, 5.1, 3.9, 3.2 degrees) are presented in Fig. <ref>(a). The gap strongly (exponentially) depends on the interaction strength for all superstructures. It is seen from this figure that the gap is considerably large only when α≳1, that is, when ϵ≲2.5. The ratio Δ_1/t∼10^-2 corresponds to the value of the gap about 300 K. Thus, according to our calculations, in order to observe the gap at room temperatures, the permittivity of the sample should not be large.The important result demonstrating in Fig. <ref>(a) is that for any α, the band gap is larger for superstructures with smaller twist angles (larger m_0). This is illustrated in Fig. <ref>(b), where we plot the dependence of Δ_1 on θ calculated at α=1.044 and V_b/t=0.037. We see that the band gap increases by about 4 orders of magnitude, when twist angle changes from θ≅17.7^∘ (m_0=2) down to θ≅3.2^∘ (m_0=10). Such a strong enhancement can be explained by the reduction of the Fermi velocity due to the interlayer hybridization. To illustrate this, let us consider the weak interaction limit. Assuming that q_λ≪ q_F^* and Δ_μ≪ V_b, one can solve the system of equations (<ref>) analytically. This givesΔ_1≈2v_Fq_F^*α e^-1/Λ+4α^*,where α^*=e^2/(ϵ v_F^*) is the `renormalized α' andΛ≈2πA̅q_F^*/v_BZv_F^* .Calculations show that good approximation for A̅ and, consequently, for Λ can be obtained if in Eq. (<ref>) for A_μν(𝐩;𝐪) we will use wave functions corresponding to the limit of decoupled layers, t_0=0. In this case, expressions for U^(S)_𝐩𝐆iα can be found analytically, and as a result, we obtainA̅≈⟨cos^2(φ/2)V_12(2q_F^*|sin(φ/2)|)⟩ ,where φ is the polar angle parameterizing points on the Fermi surface, while averaging is performed over this angle. The factor cos^2(φ/2) before potential V_12 is inherited from the wave functions. Substituting this expression into Eq. (<ref>), we obtainΛ≈1/8⟨cos^2(φ/2)/1+1/4α^*|sin(φ/2)|⟩.Deriving the latter equality we assumed that q_F^*d≪1, which is valid for all bias voltages considered. According to Eq. (<ref>), Λincreases when α^* increases. When α^*≪1, we have Λ∝α^*ln(1/α^*). Since α^* inversely proportional to v_F^*, parameter Λ and, consequently, Δ_1 increases when v_F^* decreases. This can explain the dependence of Δ_1 on θ since v_F^* decreases with the twist angle <cit.>. Numerical calculations show also, that actual value of A̅ is greater than estimate (<ref>). This can be explained by the fact that at finite interlayer hybridization, the quasiparticles are no longer localized in one particular layer. As a result, intralayer potential contributes to A̅, and this effect is stronger for smaller twist angles.It is seen from Eqs. (<ref>), that exponent in Eq. (<ref>) is independent on the bias voltage. This result correlates with that obtained in Refs. <cit.>. Such a behavior can be understood if we realize that parameter Λ is a product of the interaction strength A̅∝1/q_F^* and the density of states at the Fermi level ρ_0∝ q_F^*∝ V_b. Thus, only pre-exponential factor depends the bias voltage. Numerical analysis shows, that at small V_b, the band gap linearly depends on V_b for any α, while at strong interaction the function Δ_1(V_b) can show non-monotonous behavior at larger bias voltages [see the inset to Fig. <ref>(a)].The order parameter goes to zero when V_b→0 for any superstructures and for any strength of the interaction considered. At the same time, it is seen from Fig. <ref>(b), that Δ_1 increases very fast with the decrease of the twist angle. We did not analyze what happens at twist angles very close and below critical value (θ_c≅1.89^∘ in our model), but we can expect that for these θs the Coulomb interaction can stabilize some type of ordering even at zero bias voltage. This suggestion is confirmed by recent studies performed in Ref. <cit.>, where authors predict the SDW ground state for bilayers with θ<θ_c in the framework of the Hubbard model. Note that our `exciton-plus-SDW' order parameter is stabilized mainly by the interlayer interaction, and it would not occur (or would be strongly suppressed) for Hubbard interaction. Similar type of order has been considered in Ref. <cit.> devoted to the study of the AA-stacked bilayer graphene in the applied electric field. The detailed investigation of the ordering type for bilayers with the twist angles close to the critical value is the subject for future study.In conclusion, we studied the ground state of the twisted bilayer graphene when the bias voltage is applied to the system. We showed that the bias voltage forms two hole-like and two electron-like Fermi surfaces with perfect nesting. As a result of such a band structure, the screened Coulomb interaction stabilizes the exciton order parameter in the system. The exciton order parameter is accompanied by the spin-density-wave order. The value of the gap depends on the twist angle and on the applied voltage. The latter property is quite useful for different applications in electronics.Acknowledgments.— This work is partially supported by the Russian Foundation for Basic Research (Projects 17-02-00323, and 15-02-02128), JSPS-RFBR grant No 17-52-50023, the RIKEN iTHES Project, MURI Center for Dynamic Magneto-Optics via the AFOSR Award No. FA9550-14-1-0040, the Japan Society for the Promotion of Science (KAKENHI), the IMPACT program of JST, JSPS-RFBR grant No 17-52-50023, CREST grant No. JPMJCR1676, and the Sir John Templeton Foundation.apsrevlong_no_issn_url 19 natexlab#1#1bibnamefont#1#1bibfnamefont#1#1citenamefont#1#1[Rozhkov et al.(2016)Rozhkov, Sboychakov, Rakhmanov, and Nori]ourBLGreview2016 authorA. Rozhkov, authorA. Sboychakov, authorA. Rakhmanov, and authorF. Nori, “titleElectronic properties of graphene-based bilayer systems,” journalPhysics Reports volume648, pages1 (year2016).[Mele(2012)]MeleReview authorE. J. 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"authors": [
"A. O. Sboychakov",
"A. V. Rozhkov",
"A. L. Rakhmanov",
"Franco Nori"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170727143841",
"title": "Externally controlled band gap in twisted bilayer graphene"
} |
Institute of Theoretical Physics, Lanzhou University, Lanzhou 730000, China In this review, we give a brief introduction on the aspects of some extra dimension models and the five-dimensional thick brane models in extended theories of gravity. First, we briefly introduce the Kaluza-Klein theory, the domain wall model, the large extra dimension model, and the warped extra dimension models. Then some thick brane solutions in extended theories of gravity are reviewed. Finally, localization of bulk matter fields on thick branes is discussed. Introduction to Extra Dimensions and Thick Braneworlds [To commemorate the great contributions to theoretical physics in China made by Prof. Yi-Shi Duan.]Yu-Xiao Liu[[email protected]] December 30, 2023 =============================================================================================================================================================== § INTRODUCTIONThe concept of extra dimensions has been proposed for more than one hundred years. In 1914, a Finnish physicistGunnar Nordström first introduced an extra spatial dimension to unify the electromagnetic and gravitational fields <cit.>. It is known that Nordström's work is not successful because it appeared before Einstein's general relativity. A few years later, a German mathematics teacher Theodor Kaluza put forward a five-dimensional theory that tries to unify Einstein's general relativity and Maxwell's electromagnetism <cit.>. Subsequently, in 1926 the Swedish physicist Oskar Klein suggested that the extra spatial dimension should be “compactified": it is curled up on a circle with a microscopically small radius so that it cannot be directly observed in everyday physics <cit.>. This theory is referred to Kaluza-Klein (KK) theory. Since then extra dimensions have aroused intense interest and study from physicists. Specifically, KK theory is usually regarded as an important predecessor to string theory, which attempts to address a number of fundamental problems of physics.However, the major breakthrough of the research along phenomenological lines occurred at the end of the 20th century. In 1982, Keiichi Akama presented a picture that we live in a dynamically localized 3-brane in higher-dimensional space-time, which is also called “braneworld" in modern terminology <cit.>. In 1983, Valery Rubakov and Mikhail Shaposhnikov proposed an extra dimension model, i.e., the domain wall model <cit.>, which assumes that our observable universe is a domain wall in five-dimensional space-time. The most remarkable characteristic of the two models is that the extra dimensions are non-compact and infinite <cit.>, which is also the seed of the subsequent thick brane models with curved extra dimensions. In 1990, Ignatios Antoniadis examined the possibility of the existence of a large internal dimension at relatively low energies of the order of a few TeV <cit.>. Such a dimension is a general prediction of perturbative string theories and this scenario is consistent with perturbative unification up to the Planck scale. But what really triggered the revolution of the extra dimension theory is the work done by Nima Arkani-Hamed,Savas Dimopoulos, and Georgi Dvali (ADD) in 1998 <cit.>, which has provided an important solution to the gauge hierarchy problem. Since the extra dimensions in the ADD model are large (compared to the Planck scale) and compact (similar to KK theory), now it has been a paradigm of large extra dimension models. Ignatios Antoniadis, Nima Arkani-Hamed,Savas Dimopoulos, and Georgi Dvali (AADD) <cit.> gave the first string realization of low scale gravity and braneworld models, and pointed out the motivation of TeV strings from the stabilization of mass hierarchy and the graviton emission in the bulk.One year later, Lisa Randall and Raman Sundrum (RS) proposed that it is also possible to solve the gauge hierarchy problem by using a non-factorizable warped geometry <cit.>. This model is also called RS-1 model and has been a paradigm of warped extra dimension models now. One of the basic assumptions of the two models is that the Standard Model particles are trapped on a three-dimensional hypersurface or brane, while gravity propagates in the bulk. This type of model is also known as braneworld model. It is worth mentioning that Merab Gogberashvili also considered a similar scenario <cit.>. After the ADD model and RS-1 model, the study of extra dimensions enters a new epoch and some of the extended extra dimension models are also well known, such as the RS-2 model <cit.>, the Gregory-Rubakov-Sibiryakov (GRS) model <cit.>, the Dvali-Gabadadze-Porrati (DGP) model <cit.>, the thick brane models <cit.>, the universal extra dimension model <cit.>, etc. It should be noted that the universal extra dimension model <cit.> is a particular case of the proposal of TeV extra dimensions in the Standard Model <cit.>. Here, we list some review references and books. References <cit.> are very suitable for beginners. References <cit.> provide very detailed introductions to extra dimension theories, including phenomenologicalresearch. There are also some books <cit.>, which may be of great help to the readers who want to do some related research in this direction. In this review, we mainly focus on thick brane models. Some review papers for some thick brane models can be found in Refs. <cit.>.§ SOME EXTRA DIMENSION THEORIES In this section, we will give a brief review of some extra dimension theories, including the KK theory, the domain wall with a non-compact extra dimension, the braneworld with large extra dimensions, and the braneworld with a warped extra dimension.In this review, we use capital Latin letters (such as M, N, ⋯) and Greek letters (such as μ, ν,...) to represent higher-dimensional and four-dimensional indices, respectively. The coordinate of the five-dimensional space-time is denoted by x^M=(x^μ,y) with x^μ and ythe usual four-dimensional and the extra-dimensional coordinates, respectively. A five-dimensional quantity is described by a “sharp hat" (in this section). For example, R̂ indicates the scalar curvature of the higher-dimensional space-time. §.§ KK theory First of all, let us review KK theory <cit.>. It is the first unified field theory of Maxwell's electromagnetism theory and Einstein's general relativity built with the idea of an extra spatial dimension beyond the usual four of space and time. The three-dimensional space ishomogeneous and infinite and the fourth spatial dimension y is a compact circle with a radius R_ED (see Fig. <ref>). So, this model was also known as “cylinder world" <cit.> in the early days. This theory is a purely classical extension of general relativity to five dimensions. Therefore, it assumes that there is only gravity in the five-dimensional space-time and the four-dimensional electromagnetism and gravity can be obtained by dimensional reduction. KK theory is viewed as an important precursor to string theory. KK theory is described by the five-dimensional Einstein-Hilbert actionS_KK=1/2κ_5^2∫ d^4xdy√(-ĝ) R̂and the following metric ansatzds^2=ĝ_MNdx^M dx^N =e^2αϕ g_μνd x^μ d x^ν+ e^-4αϕ(dy+ A_μ d x^μ)^2,where the five-dimensional gravitational constant κ_5 is related to the five-dimensional Newton constant G_N^(5) and the five-dimensional mass scale M_* asκ_5^2=8 π G_N^(5) = 1/M_*^3,α is a parameter, and all these components g_μν, A_μ, ϕ are functions of x^μ only (the so-called cylinder condition). The components of the five-dimensional metric ĝ_MN areĝ_μν = e^2αϕg_μν+ e^-4αϕ A_μ A_ν, ĝ_μ5 = e^-4αϕ A_μ, ĝ_55 = e^-4αϕ.Note that, among the 15 components of ĝ_MN, 10 components are identified with the four-dimensional metric g_μν, four components with the electromagnetic vector potential A_μ, and one component with a scalar field called “radion" or “dilaton". Substituting the above metric into (<ref>) and integrating the extra dimension y yields the following four-dimensional effective actionS_KK=∫ d^4x√(-g)(1/2κ_4^2R-1/2g^μν▽_μϕ▽_νϕ- 1/4e^6αϕF_μνF^μν),where κ_4^2 = √(8 π G_N) = 1/M_Pl with M_Pl the four-dimensional Plack mass, R is the four-dimensional scalar curvature defined by the metric g_μν, ϕ is a dilaton field, and F_μν=∂_μA_ν-∂_νA_μ is the four-dimensional field strength of the vector field A_μ(x^λ). Thus, one obtains a four-dimensional scalar-vector-tensor theory (<ref>) from a five-dimensional pure gravity. This theory only contains gravity and electromagnetic fields when ϕ is a constant:S_KK=∫ d^4x√(-g)( 1/2κ_4^2R -1/4 F_μνF^μν).Thus, Einstein's general relativity and Maxwell's electromagnetic theory in four-dimensional space-time can be unified in the five-dimensional KK theory. The relation between the fundamental Planck scale M_* and the four-dimensional effective one M_Pl isM_Pl^2=(2π R_ED)M_*^3.Here, R_ED is the radius of the extra dimension. It is easy to see that when the radius of the extra dimension R_ED is large, one can get a four-dimensional effective Planck scale M_Pl from a small fundamental scale M_*. This characteristic inspired the later large extra dimension model that tries to solve the hierarchy problem. In 1926, in order to explain the cylinder condition, Oskar Klein gave this classical theory a quantum interpretation by introducing the hypothesis that the fifth dimension is curled up and microscopic <cit.>. He also calculated a scale for the fifth dimension based on the quantization of charge. As an early theory of extra dimensions, KK theory is not a completely self-consistent theory. Now let us consider the following translation in the fifth coordinatex^μ→ x'^μ=x^μ, y → y'=y+κξ(x),which leads to the gauge transformation of the electromagnetic vector potential A_μ:A_μ(x) → A'_μ(x)=A_μ(x)+ κ∂_μξ(x).The above coordinate translation (<ref>) also leads to the gauge transformation of each KK mode Φ^(n)(x) of a bulk scalar Φ(x,y)=∑_n=0^∞Φ^(n)(x)e^i ny/R_ED:Φ^(n)(x) →Φ'^n(x) =Φ^(n)(x) e^i nκξ(x)/R_ED ,which indicates that each scalar KK mode has chargeQ_n=n κ/ R_ED = ne,with the charge quanta e given bye=κ/ R_ED=√(16π G_N)/ R_ED=√(4πα)=√(4π/137).Thus, one will obtain a tiny scale of the extra dimension:R_ED∼ 10^-33m,which is not much larger than the Plank length l_Pl∼ 10^-35 m. Such a tiny scale means that detecting the extra dimension is almost hopeless. On the other hand, in KK theory, the mass spectrum of KK modes of a bulk field with mass M_0 is given bym_n = √(M_0^2 + n^2/R_ED)≃ n × 10^17GeV,where the electroweak scale parameter M_0 is neglected. So, the masses of the massive KK modes of a bulk field will be much larger than the order of TeV. It isdifficult to detect such heavy KK particles in the present and future experiments. Therefore, only the zero modes of the bulk fields are observable. However, it is not acceptable that the charges of the KK modes of the bulk fields must satisfy Q_n=ne, i.e., all zero modes that describe the observed particles are neutral. Therefore, the predictions of KK theory about four-dimensional particles are completely inconsistent with experiments. This is the main reason why KK theory was not taken seriously for nearly half a century.More details and related issues about KK theory can be found in the book <cit.> or Refs. <cit.> and the references therein. §.§ Non-compact extra dimension: domain wall modelIn 1982, Akama presented a picture that we live in a dynamically localized 3-brane in a higher-dimensional space-time <cit.>. As an example, it was considered that our four-dimensional space-time is localized on a 3-brane by the dynamics of the Nielsen-Olesen-type vortex in six-dimensional space-time. At low energies, matters and gravity are trapped in the 3-brane.In 1983, Rubakov and Shaposhnikov proposed a domain wall model in a five-dimensional Minkowski space-time <cit.>, which is completely different from KK theory. In this model, our four-dimensional universe is restricted to a domain wall formed by a bulk scalar field, and the extra dimension is non-compact and infinitely large (see Fig. 2). There is an effective potential well around the domain wall, which can trap the lower energy KK modes of a bulk fermion field, i.e., the four-dimensional fermions, on the domain wall. So in general, the Standard Model particles are localized on the domain wall due to the potential well and we can only feel a three-dimensional space. Only when the energy of a particle is higher than the edges of the potential well, can one detect the effects of the extra dimension.In the original domain wall model, Rubakov and Shaposhnikov considered the following ϕ^4 model of a scalar field in a five-dimensional Minkowski space-time <cit.>:L_DW=-1/2η^MN∂_M ϕ∂_N ϕ- k^2/2v^2(ϕ^2-v^2)^2,where v and k are positive parameters. The ϕ^4 model usually gives a double-well potential and the minima of the potential are located at ϕ=± v. The model has the following static domain wall solution:ϕ(y)=v tanh(k y).The energy density of the system along extra dimension with respect to a static observer u^M=(1,0,0,0,0) isρ(y)= T_MNu^M u^N = 1/2η^MN∂_M ϕ∂_N ϕ + k^2/2v^2(ϕ^2-v^2)^2=k^2 v^2 sech^4(k y) .The shapes of the scalar field (<ref>) and energy density (<ref>) are shown in Fig. <ref>. Next, we show that the zero mode of a bulk Dirac fermion Ψ coupling with the background scalar ϕ can be localized on the domain wall even though the extra dimension is infinite. Suppose that there is a Yukawa coupling between a five-dimensional fermion field Ψ and the background scalar field ϕ:L_Ψ=Ψ̅γ^M∂_MΨ-ηΨ̅ϕΨ,where η is the coupling parameter. The equation of motion is given by (γ^M∂_M-ηϕ) Ψ=0. Then with the KK decompositionΨ(x,y)=∑_nΨ_n(x,y)=∑_n[ψ_Ln(x)f_Ln(y) +ψ_Rn(x)f_Rn(y)],where ψ_Ln=-γ^5ψ_Ln and ψ_Rn=γ^5ψ_Rn are the left- and right-chiral components of the Dirac fermion field, respectively, one can obtain the four-dimensional Dirac equations for ψ_Ln,Rn(x):[ (γ^μ∂_μ- m_n)ψ_Ln(x)=0,; (γ^μ∂_μ -m_n)ψ_Rn(x)=0, ]and the equations of motion for the KK modes f_Ln,Rn(y):[-∂_y^2+V_L(y)]f_Ln(y)= m^2_nf_Ln(y), [-∂_y^2+V_R(y)]f_Rn(y)= m^2_nf_Rn(y),where m_n is the mass of the four-dimensional fermion and the effective potentials are given byV_L,R(y)= η^2ϕ^2(y) ∓η∂_zϕ(y) = η ^2 v^2 (tanh ^2(k y) ∓k/ηvsech^2(k y)).The shapes of the effective potentials are plotted in Fig. <ref>. It can be seen that whether there is a potential well for V_R with η>0 is determined by the ratio k/(η v). When k/(η v)<1, there is a potential well, which may trap some bound massive KK fermions.One can derive the zero modes of the left- and right-chiral fermion fields:f_L0,R0(y) ∝exp(∓η∫ϕ(y)dy) =cosh(ky)^∓η v/k.So, when η>0, the left-chiral fermion field could be localized on the domain wall (to localize the right-chiral Fermion field one needs η<0), which is the most prominent feature of the model. The above localized left-chiral fermion zero mode is plotted in Fig. <ref>. It is clear that, even though the extra dimension is infinite, the zero mode of the left-chiral fermion can be localized on the domain wall through the Yukawa coupling with the background scalar field, while the right-chiral one cannot. Therefore, the fermion zero modes, i.e., the massless four-dimensional fermions localized on the domain wall, can be used to mimic our matters. They propagate with the speed of light along the domain wall, but do not move along the extra dimension. Therefore, domain wall is also called braneworld. In the real world, fermions have masses. So, in realistic theories fermion zero modes should acquire small masses by some mechanism. For the case of weak coupling |η| < k/v, there is no bound massive fermion KK modes. However, there may be one or more bound massive KK modes on the wall if the coupling is large enough (η≫ k/v). Besides, there is a continuous part of the spectrum starting at m =η v. These continuous states correspond to five-dimensional fermions that can escape to |y|=∞. For the localization of gauge fields and more details about this domain wall model, one can refer to Refs. <cit.>.It should be noted that the idea of Akama, Rubakov and Shaposhnikov is important because it provides a way basically distinct from the “compactification" to hide the extra dimensions. However, this domain wall model has a fatal weakness. Since the extra dimension is flat and infinitely large, the zero mode of gravity cannot be localized on the domain wall. Obviously, if extra dimensions are infinite and flat, i.e., for a D-dimensional Minkowski space-time, the gravitational force between any two static massive particles would be F∼ 1/r^D-2 instead of the inverse square law. Because of this shortcoming, this flat domain wall model is not taken seriously for a long time. After lRandall and Sundrum proposed the RS-2 model, the Rubakov-Shaposhnikov domain wall model was reconsidered in a warped five-dimensional space-time <cit.> (the jargon for this kind of model is thick brane model), which will be introduced in Sections <ref> and <ref>.§.§ Large extra dimensions: ADD braneworld model After the Akama brane model and the Rubakov-Shaposhnikov domain wall model, the phenomenological lines of extra dimensions almost have no significant development for a long time, except for compactifications at the electroweak scale (see, e.g., Refs. <cit.>). Until 1998, Arkani-Hamed, Dimopoulos, and Dvali suggested ingeniously that the gauge hierarchy problem can be addressed by the large extra dimension model (also called as ADD braneworld model) <cit.>. Then extra dimension theories re-attracted attentions of theoretical physicists.Before introducing the ADD model, it is necessary to describe what the hierarchy problem is. It is usually expressed as the huge discrepancy between the gravitation and electroweak interactions. In quantum field theory, it has another expression, i.e., why the Higgs boson mass is so much lighter than the Planck scale. These two expressions are equivalent and we will briefly introduce the latter one. In the Standard Model, the physical mass μ and bare mass μ_0 of the Higgs boson satisfy the following relationship:μ^2=μ_0^2+δμ_0^2.Here, δμ_0^2∼Λ^2 is the loop correction of the bare mass and Λ is a truncation parameter. According to the effective field theory, Λ should be the energy scale up to which the Standard Model is valid. It is known that the physical mass of the Higgs boson is μ∼10^2GeV. Assuming that the new physics appears at the Planck scale M_Pl∼10^19GeV, Eq. (<ref>) cannot be satisfied unless there is an unnatural fine-tuning between the bare mass and the loop correction. The essential reason why bare mass of Higgs boson requires such a high-precision adjustment is because of the huge hierarchy between the weak scale M_EW≈246GeV and the Planck scale M_Pl∼10^19GeV. If the new physics appears at the low-energy scale (such as 1TeV), there is no serious fine-tuning problem. Next, we will see that the ADD model does solve the hierarchy problem by assuming that the new physics in the bulk appears at M_*∼ 1TeV.In the ADD model, the space-time is assumed to be (4+d)-dimensional and the corresponding action is given by <cit.>S_ADD=M_*^d+2/2∫ d^4+dx√(-ĝ)R̂.If these extra spatial dimensions have the same radius R_ED, then one can obtain the following relationshipM_Pl^2=M_*^d+2(2π R_ED)^d,whose physical meaning and calculation are similar to Eq. (<ref>). In the bulk space, the fundamental scale of gravity is no longer the Plank mass M_Pl but M_*. To avoid the emergence of hierarchy, one assumes M_*∼ 1TeV. Then the radius of the extra dimensions reads asR_ED=1/2π M_*(M_Pl/M̂)^2/d∼1/2π10^32/dTeV^-1∼ 10^-17×10^32/dcm.For d=1,the radius of the extra dimension should be as large as 10^13m in order to address hierarchy problem. Obviously, it is against the tests of the gravitational inverse-square law <cit.>, which constrain the radius of the extra dimensions to be less than sub-millimeter. Therefore, according to the present gravity experiments, the number of the extra dimensions in the ADD model should be more than two. Of course, if the sizes of these extra dimensions are different, the result would be complex. More importantly, if one assumes that other fields live in the bulk, then the radius of the extra dimensions should be much smaller (according to the recent experiments it should be less than 10^-18m or more) in order not to violate the experiments at Large Hadron Collider (LHC). Even at the end of the last century, according to the nuclear-related researches, it can be deduced that the radius of the extra dimensions should usually be much less than 1μm. For this reason, ADD proposed another supposition inspired by string theory: except the gravitational field, all the Standard Model particles are bounded on a four-dimensional hypersurface or brane by an unknown natural mechanism. This hypothesis is the main difference between the ADD model and the KK theory. It should be noted that although the ADD model can eliminate the hierarchy between the weak scale and the Planck scale based on the assumptions that the extra dimensions are large (as compared to the Planck length) and the Standard Model particles are localized on a brane, the ratio between the fundamental scale M_* and the scale corresponding to the size of the extra dimensions, R_ED, is not acceptable. From Eq. (<ref>), we haveM_*/1/R_ED∼1/2π(M_Pl/M_*)^2/d=1/2π10^32/d.If one requires that 1/R_ED and M_* are in the same order of magnitude, the number of extra dimensions should be 32 or so. Naturally, a question arises: why there are so many extra dimensions? Therefore, the ADD model does not really solve the hierarchy problem.Other issues in the ADD model (including the difference between the ADD model and the KK theory, the features of the KK states, how the KK states interact with the fields on the brane, etc.) are discussed in detail in Refs. <cit.>§.§ Warped extra dimension: RS braneworld models The common feature of the KK theory, the domain wall model, and the ADD model is that extra dimensions are flat. They have solved some problems but left some new problems. In this section, we will see that two new extra dimension models in curved space-time give different physical pictures for our world. §.§.§ RS-1 model Inspired by the ADD model, in 1999 Randall and Sundrum proposed a braneworld model with a warped extra dimension to address the hierarchy problem, which is now called the RS-1 model <cit.>. The basic assumptions of the model are listed as follows: * There is only one extra spatial dimension, which is compactified on an S^1/Z_2 orbifoldwith a radius R_ED (y∈ [-π R_ED,π R_ED]).* There are two branes at the fixed points y=0 (called the hidden brane or Planck brane or UV brane) and y=π R_ED (called the visible brane or TeV brane or IR brane) in the bulk. And all the Standard Model particles are bounded on the visible brane.* The form of the five-dimensional metric is supposed to beds^2=e^2A(y)η_μνdx^μdx^ν + dy^2 =e^2A(y)η_μνdx^μdx^ν + R_ED^2 dϕ^2,where the warp factor A(y) isa function of the extra dimension y=R_EDϕ only.* The bulk is a five-dimensional anti-de Sitter (AdS) space-time, i.e., there is only a negative cosmological constant in the bulk.Therefore, the total action of the RS-1 model consists of three parts <cit.>:S_RS-1 = S_gravity+S_vis+S_hid= ∫ d^4x ∫dy√(-ĝ)(M_*^3/2R̂ -Λ) +∫ d^4x √(g_vis)(L_vis-V_vis)+∫ dx^4 √(g_hid)(L_hid-V_hid). Through a series of calculations and simplifications, one can obtain the four-dimensional effective action of the RS-1 model:S_eff=M_Pl^2/2∫ d^4x √(-g)R.Here, R is the scalar curvature defined by the four-dimensional metric g_μν and M_Pl is similar to Eq (<ref>):M_Pl^2=M_*^3/k(1-e^-2kπ R_ED),where the parameter k has the dimension of mass. To avoid new hierarchy problem, one requires that the parameter k satisfies k/M_*∼ 1. When the value of k R_ED becomes large, the fundamental scale M_* and the Plank scale M_Pl will be the same order. Note that the RS-1 model assumes that the fundamental scale M_* is still equivalent to the Planck scale, which is very different from the ADD model. But how does one get the four-dimensional TeV scale for the weak interaction?The RS-1 model assumes that the Higgs boson is bounded on the visible brane and the four-dimensional effective mass of the Higgs boson is given bym_H=√(λ) e^-k R_EDπ v_0= e^-k R_EDπm̂_H,where λ is a dimensionless parameter, and v_0 is the vacuum expected value of the Higgs field in the five-dimensional space-time. The mass of the Higgs boson in the five-dimensional space-time is m̂_H = √(λ) v_0. To eliminate the hierarchy, the RS-1 model requires that the fundamental parameters M_*, k, and v_0 are all at the order of M_Pl. Although the fundamental mass of the Higgs boson m̂_H in the five-dimensional space-time is the truncation scale M_Pl, the effective physical mass on the four-dimensional brane could be “red-shifted" to the order of TeV as long as the radius of the extra dimension satisfies R_ED∼10/k. Noted that R_ED^-1 is also a fundamental parameter. Since the exponential function is introduced into Eq. (<ref>), the ratio of k to R_ED^-1 does not need to be too large to “red-shift" m̂_H to TeV. This is why we usually say that the RS-1 model solves the hierarchy problem without introducing new hierarchy.§.§.§ RS-2 modelAs mentioned earlier, the fundamental scale of the five-dimensional space-time and the effective Plank scale in the KK theory and ADD model satisfy Eq. (<ref>), which requires that the radius of extra dimensions is finite. However, because of the warped space-time, it can be seen from Eq. (<ref>) that the scale of the extra dimension may be infiniteif one forgets the hierarchy problem. Enlightened by the RS-1 model, Randall and Sundrum provided another braneworld model (called as the RS-2 model) <cit.>) to solve the puzzle left by the domain wall model: the localization of gravity on the brane with an infinite co-dimension. The focus of the RS-2 model is mainly on how to restore the four-dimensional gravity on a thin brane when the extra dimension is infinite. Roughly, compared to the RS-1 model, the RS-2 model has made the following changes: * We live on the Plank brane at y=0 (the Standard Model particles are bounded on this brane).* The TeV brane located at y=π R_ED is moved to infinity, i.e., R_ED→∞. So the KK spectrum in the RS-2 model is continuous.The metric of the RS-2 model can be obtained by taking the limit R_ED→∞ in the metric (<ref>). Similarly, the KK spectrum in the RS-2 model can also be reduced from that of the RS-1 model <cit.>:m_n≈(n+1/4)π k e^-π k R_ED.Obviously, since R_ED→∞, the KK spectrum is continuous. In general, the Newton's gravitational potential between two static massive particles on the brane is contributed by all the KK gravitons. In the RS-2 model, due to the presence of the continuous massive KK states, it is necessary to consider how these KK states affect the Newton's gravitational potential. Randall and Sundrum showed that the Newton's gravitational potential of two static particles with masses m_1 and m_2 and distance |x⃗| on the brane has the following form <cit.>:V(|x⃗|) ∼ m_1m_2/|x⃗|+∫_0^∞dm/km_1m_2e^-m|x⃗|/|x⃗|m/k∼ m_1m_2/|x⃗|(1+1/|x⃗|^2k^2).Here, the first term is contributed from the zero mode of graviton, which corresponds to the standard Newton's gravitational potential. The second one is the contribution of all the massive KK modes, and it is the correction to the Newton's gravitational potential. As the distance |x⃗| increases, the correction term decays quickly. The effect of the extra dimension appears at the scale of the Planck length. So Randall and Sundrum proved that even if there is an infinite extra dimension, as long as it warps in some way, one can still get an effective four-dimensional Newtonian gravity. § SOLUTIONS OF THICK BRANE MODELS IN EXTENDED THEORIES OF GRAVITYIn this section, we introduce some thick brane models. It is known from the RS-2 model that, if the extra dimension is warped, it is possible to realize the localization of the matter fields and gravitational fields on a domain wall or thin brane with an infinite extra dimension. In the RS-2 model, the thickness of the brane is neglected and so the brane is called as thin brane. However, a brane without thickness is idealistic and a real braneworld should have a thickness. Furthermore, it may be hard to find thin brane solutions in some higher-order derivative gravity theories, such as the f(R) theory. It is easy to guess that a brane could be dynamically generated by some background fields, such as one or more scalar fields. Therefore, by combining the domain wall model and the RS-2 model, theoretical physicists investigated the so-called thick braneworld models, where the brane solutions are smooth. In literature, most five-dimensional thick branes are generated by one or more scalar fields with kink-like and/or bump-like configurations <cit.>, but a few brane models are based on vector fields or spinor fields <cit.>. Higher-dimensional thick branes were also considered <cit.>. They are smooth generalizations of the RS-2 model. Note that these fields should not be thought of as the matter fields that are related with those in the standard model. They are the matter fields generating a brane. There are also some brane models without matter fields <cit.>. These branes are embedded in higher-dimensional space-times which are not necessarily AdS far from the branes. In this review, we mainly consider scalar-field-generated thick branes embedded in AdS space-time and these smooth branes appear as domain walls interpolating between various vacua of the scalar fields. Unlike the thin RS-2 model, such thick brane solutions do not have any matter fields living on the brane <cit.>. In fact, all matter fields are assumed as bulk fields in thick brane models and one needs to investigate the localization of these bulk fields on the branes, which is the subject of the next section. In this section, we mainly consider constructions and solutions of thick brane models in extended theories of gravity. The system is described by the actionS = ∫ d^5 x √(-g) [ 1/2κ_5^2ℒ_G+ℒ_M(g_MN,ϕ^I,∇_Mϕ^I)],where the five-dimensional gravitational constant κ_5 is related to the five-dimensional Newton constant G_N^(5) and the five-dimensional Planck mass scale M_* asκ_5^2=8 π G_N^(5) = 1/M_*^3.Sometimes one sets κ_5=1 for convenience. ℒ_G is the Lagrangian of gravity, and ℒ_M(g_MN,ϕ^I) is the Lagrangian of the matter fields that generate the thick brane. Note that, a five-dimensional quantity is no longer described by a “sharp hat" from now on for convenience. For the simplest case of general relativity and a canonical scalar field, we haveℒ_G =R, ℒ_M =-1/2g^MN∂_M ϕ∂_N ϕ - V(ϕ),for which the energy-momentum tensor isT_MN = ∂_M ϕ∂_N ϕ-g_MN( 1/2∂^P ϕ∂_P ϕ + V(ϕ)).In this review, we only consider static branes. The five-dimensional line-element which preserves four-dimensional Poincaré invariance is assumed asds^2=g_MN dx^M dx^N= e^2A ds^2_brane + dy^2,whereds^2_brane= g̃_μν(x^λ)dx^μ dx^νdescribes the geometry of the brane. Usually, we are concerned with three typical branes:ds^2_brane= {[η_μν dx^μ dx^νflat brane;; e^2Hx_3(-dt^2+dx_1^2+dx_2^2)+dx_3^2 AdS brane;;-dt^2+e^2Htdx^i dx^idS brane ]..For a static brane, the warp factor A and scalar fields ϕ^I are functions of the extra dimensional coordinate y or z only, and the non-vanishing components of the energy-momentum tensor (<ref>) areT_μν = -g_μν( 1/2g^55(∂_5ϕ)^2 + V(ϕ)), T_55 = 1/2 (∂_5ϕ)^2 - g_55V(ϕ).One can make a coordinate transformation dz=e^-Ady and rewrite the five-dimensional line-element asds^2=e^2A (ds^2_brane + dz^2),which is very useful in the derivation of the perturbation equations of gravity and localization equations of various bulk matter fields in the current and next sections. The dynamical field equations read asR_MN-1/2R g_MN = κ_5^2T_MN,□^(5)ϕ≡ g^MN∇_M ∇_N ϕ= V_ϕ,whose non-vanishing component equations in the (x^μ,y) and (x^μ,z) coordinates are3 ( ε H^2 e^-2 A- A”-2 A'^2) = κ_5^2(1/2ϕ'^2+V),6(-ε H^2 e^-2 A+ A'^2)= κ_5^2(1/2ϕ'^2-V), 4A'ϕ'+ϕ” = V_ϕ,and3 (ε H^2 - ∂_z^2 A- ∂_z A ^2) = κ_5^2(1/2(∂_zϕ)^2+e^2 A(z) V ), 6(-ε H^2 +∂_z A^2)= κ_5^2(1/2(∂_zϕ)^2-e^2 A(z) V ),e^-2A(3 ∂_z A ∂_z ϕ + ∂_z^2ϕ)= V_ϕ,respectively. Here, primes denotes the derivatives with respect to the extra dimensional coordinate y, and ε=1, -1, and 0 for de Sitter, AdS, and flat brane solutions, respectively. Note that only two of the above equations (<ref>)-(<ref>) (or (<ref>)-(<ref>)) are independent. Before going to extended theories of gravity, we introduce some brane solutions in GR. The first example of a flat brane was given in Ref. <cit.>:e^2A(y) = sech^4 v^2/9(k y) e^v^2/9sech^2(k y), ϕ(y) = v/κ_5tanh(k y), V(ϕ) = k^2/54 κ _5^2 v^2[ 27 v^4-18 v^2 (2 v^2+3) κ_5^2 ϕ ^2+3(8 v^2+9) κ_5^4 ϕ^4-4 κ _5^6 ϕ ^6 ].A de Sitter thick brane solution in a five-dimensional space-time for the potentialV(ϕ)=1+3α/2ακ_5^2 3 H^2(cos(κ_5 ϕ/v) )^2(1-α)was found in Ref. <cit.>:e^2A(z) = sech^2α(H z/α) , ϕ(z) = v/κ_5arcsin(tanh(H z/α)),where v =√(3α(1-α)), 0<α<1, H>0. The domain wall configuration with warped geometry is dynamically generates by the soliton scalar. At last, we list the AdS brane solution found in Ref. <cit.>:V(ϕ) = 3 k^2/2 κ _5^2(4-v^2 cosh ^2(κ _5 ϕ (y)/v)) , ϕ(y) = v/κ _5arcsinh(tan (ky) ),e^2A(y) = 3 H^2/k^2 (v^2-3)cos^2(ky).It is obvious to see that the thick brane is bounded in the interval y∈(-π/2k, π/2k).Gauge-invariant fluctuations of the branes were analyzed in Ref. <cit.>, where scalar, vector and tensor modes of the geometry were classified according to four-dimensional Lorentz transformations. It was shown that the tensor zero mode is localized on the brane, which ensures the four-dimensional Newton's law of gravitation, while the scalar and vector fluctuations have no normalizable zero mode and hence are not localized on the brane. In Ref. <cit.>, Herrera-Aguilar et al. considered the mass hierarchy problem and the corrections to Newton's law in thick branes with Poincaré symmetry both in the presence of a mass gap in the graviton spectrum and without it. There is a special class of theories with noncanonical fields, namely the K-field theory, which was first proposed to drive the inflation with generic initial conditions <cit.>. This kind of theory introduces a general noncanonical Lagrangian ℒ≡ℒ_M(X,ϕ), whereX=-1/2g^MN∂_M ϕ∂_N ϕ.For example, ℒ=ℱ(X)-V(ϕ) is a simple one. The flat thick braneworld models generated by K field were considered in Ref. <cit.>, and the solutions with perturbative procedure in Ref. <cit.>. In Ref. <cit.>, Christoph Adam et al. chose a (non-standard) kinetic term such that the resulting kink in the extra dimension is a compacton, both with and without gravitational backreaction. This is slightly different from the thick branes of Refs. 113-115 which are not compactons. Compactons have some peculiar features (e.g. linear fluctuations are restricted to within the compacton, and the spectrum of perturbations is purely discrete). It was shown that, even with gravity included, the brane solutions remain compactons and are linearly stable <cit.>. The exact solutions with specific model ℒ=X-α X^2 - V(ϕ) were given in Ref. <cit.>. The system of a flat thick braneworld consists of the equations -3∂_y^2 A= κ _5^2ℒ_X(∂_y ϕ)^2,6(∂_y A)^2= κ _5^2(ℒ+ℒ_X(∂_y ϕ)^2),and(∂_y ^2ϕ)(ℒ_X+2X ℒ_XX) +L_ϕ-2X ℒ_Xϕ=-4ℒ_X(∂_y ϕ)( ∂_y A).The work <cit.> developed a first-order formalism to solve the K-brane system by assuming∂_y A=-1/3W(ϕ).Using this assumption Eqs. (<ref>) can be written as W_ϕ= κ _5^2ℒ_X(∂_y ϕ)^2, 2/3W^2= κ _5^2(ℒ+ℒ_X(∂_y ϕ)^2).For small parameter α, one can get the analytic solutions. Reference <cit.> gives another approach which is able to get the exact solutions. The strategy is to assume∂_y A= -1/3[ W(ϕ)+α Y(ϕ) ], ∂_y ϕ = W_ϕ,where Y(ϕ) satisfies Y_ϕ=W_ϕ^3. One can choose the superpotential asW=k ϕ_0^2sin(ϕ/ϕ_0)to get the Sine-Gordon solution:ϕ = ϕ_0 arcsin(tanh(k y)),A = -(1+2/3 k^2 αϕ _0^2 )ln (cosh (k y)) -1/6 k^2 αϕ _0^2+1/6 k^2 αϕ _0^2 sech^2(k y),V =-k^2 ϕ_0^2/18(6+5 k^2 αϕ_0^2+k^2 αϕ_0^2 cos(2 ϕ/ϕ_0))^2 sin^2(ϕ/ϕ_0)+ 1/2 k^2 ϕ_0^2 cos^2(ϕ/ϕ_0)+3/4 k^4 αϕ_0^4 cos^2(ϕ/ϕ_0).The analysis of tensor and full linear perturbations can be found in Refs. <cit.>. The tensor mode is similar to the standard case, while the scalar mode has significant difference. The work <cit.> shows that the scalar perturbation mode cannot be canonically normalized in the conformally flat coordinate z, and needs another coordinate transformation. The scalar zero mode cannot be localized on the brane, provided that ℒ_X>0 and 1+2ℒ_XX X/ℒ_X>0.In Ref. <cit.>, a de Sitter tachyon thick braneworld was considered with the following action:S =∫ d^5 x √(-g) [ 1/2κ_5^2 R-Λ_5-V(ϕ)√(1+g^MN∂_M ϕ∂_N ϕ) ],where ϕ is a tachyonic bulk scalar field. It was shown that the four-dimensional gravity is localized on the brane, and it is separated by a continuum of massive KK modes by a mass gap. The corrections to Newton's law in this model decay exponentially. The stability of the de Sitter tachyon braneworld under the scalar sector of fluctuations for vanishing and negative bulk cosmological constant was also investigated <cit.>. Corrections to Coulomb's law and fermion field localization were computed in Ref. <cit.>. Next we will introduce the thick brane models in extended theories of gravity. §.§ Metric f(R) theory Among the large amount of proposals of extended theories of gravity, the f(R) theory <cit.> has received growing interests due to its unique advantage: it is the simplest modification with higher-derivative curvature invariants, which are required by renormalization. In addition, some other theories with curvature invariants like R_MN R^MN and R_MNPQ R^MNPQ (except the Guass-Bonnet term) would inevitably lead to Ostrogradski instability <cit.>. This makes the f(R) theory most likely the only tensor theory of gravity that allows higher derivatives.Now let us review the five-dimensional thick brane model in the metric f(R) theory coupled with a canonical scalar field with the Lagrangian (<ref>). The action is given byS_met=∫ d^5 x √(-g)[1/2κ_5^2 f(R) -1/2g^MN∂_M ϕ∂_N ϕ - V(ϕ)].The gravitational field equations read asf_R R_MN-1/2f g_MN- (∇_M∇_N -g_MN□^(5)) f_R= κ_5^2T_MN,□^(5)ϕ≡ g^MN∇_M ∇_N ϕ =V_ϕ,where f_R and V_ϕ are defined as f_R ≡df(R)/dR and V_ϕ≡dV(ϕ)/dϕ. We only consider flat branes generated by a canonical scalar field, for which the line-element is given byds^2=g_MN dx^M dx^N =e^2A(y)η_μνdx^μ dx^ν + dy^2 = e^2A(z)(η_μνdx^μ dx^ν + dz^2).Then the field equations (<ref>) and (<ref>) read asf+2f_R(4A'^2+A”) -6f'_RA'-2f”_R = κ_5^2(ϕ'^2+2V),-8f_R(A”+A'^2)+8f'_RA' -f= κ_5^2(ϕ'^2-2V), 4A'ϕ'+ϕ” = V_ϕ,where the primes represent derivatives with respect to the coordinate y. This is a system with fourth-order derivatives on the metric. In general, it would be extremely hard to solve these fourth-order non-linear differential equations analytically. However, it is widely believed that the f(R) theory is equivalent to the Brans-Dicke theory with the Brans-Dicke parameter ω_0=0. Hence, it would be more comfortable to operate in the Brans-Dicke theory, which contains only up to second-order derivatives. This is actually the strategy used in Ref. <cit.>. Some thick brane solutions in the higher-order frame were studied in Refs. <cit.>. However, they are not perfect since the solution in Ref. <cit.> has singularity while the solution in Ref. <cit.> is numerical.The first exact solution in higher-order frame was given in Ref. <cit.>, where the specific model withf(R)=R+γ R^2was considered. The solution of equations (<ref>)-(<ref>) comes from the observation that only two of these equations are independent because of the conservation of the energy-momentum tensor <cit.>. This implies that one can solve the equations by giving one of the three functions, namely, the warp factor e^A(y), the scalar field ϕ(y), and the scalar potential V(ϕ). The prior choice is to assume the solution of e^A(y) since the fourth-order derivatives only act on A(y). With such choice, one only needs to solve the second-order field equations. To get an asymptotically AdS_5 geometry, the warp factor can be assumed as <cit.>e^A(y)=sech(k y).Then the scalar field and scalar potential can be solved as <cit.>ϕ (y) = v tanh(k y),V(ϕ) = λ^(5)(ϕ^2-v^2)^2+ Λ_5/2κ _5^2,where the parameters are related byλ ^(5)=29/98κ _5^2 k^2, v=√(3/29)7/κ _5,Λ_5=-318/29k^2,γ = 3/232 k^2. This is a ϕ^4 type potential with the vacua located at ϕ=± v. It can be seen that the spatial boundaries y=±∞ are mapped to the minima of the scalar potential. For the same f(R) (<ref>), a general solution with the warp factor e^A(y)=sech^B (k y) was constructed in Ref. <cit.>. Bazeia et al. <cit.> considered the generalised model with some other polynomial and nonpolynomial potential solutions. They expanded the scalar field as ϕ(y)=ϕ_0(y)+αϕ_α(y), and then wrote the scalar potential and warp function as V(ϕ)=V_0(ϕ)+α V_α(ϕ) and A(y)=A_0(y)+α A_α(y), with α a small parameter. To the first order of α, the solution is <cit.>ϕ(y) = vtanh(ky)+α∑_n=0^3 C_ϕ ntanh^2n+1(ky),V(ϕ(y)) = ∑_n=0^3 C_V ntanh^2n(ky) +α∑_n=0^6 C_V n+4tanh^2n(ky), A(y) = C_A 1(sech^2(ky)+4lnsech(ky)) +α∑_n=2^4C_A nsech^2n(ky),where C_ϕ n, C_Vn, and C_An are some coefficients related to the parameters α, γ,and κ_5. They also considered the case of multiple scalar fields in Ref. <cit.>, where the first-order formalism method was developed.There are also some pure geometric thick f(R)-branes without any scalar field. Zhong and Liu <cit.> gave the solutions for triangular f(R) andpolynomial f(R). Here we list the solution for the latter <cit.>:f(R) = Λ_5+c_1 R-c_2 / k^2R^2+c_3 / k^4R^3,e^A(y) = cosh^-20(k y),where Λ_5 is the five-dimensional cosmological constant, and c_n are dimensionless constants. Furthermore, a class of exact solutions in D dimensions were found by L et al. <cit.>. One of solutions of the warp factor ise^A(y) = [ e^bDycosh^2 (k y) ]^2/α D.The corresponding f(R) has a complex form.The stability of the tensor perturbation of general background was analyzed in Ref. <cit.>. The perturbed metric isds^2=e^2A(z)[(η_μν+h_μν)dx^μ dx^ν+dz^2],where h_μν is a transverse-traceless tensor, namely, η^μνh_μν=0=∂_μ h^μ_ ν. By making the decompositionh_μν(x^ρ,z)=(a^-3/2f_R^-1/2)ϵ_μν(x^ρ)ψ(z),where a ≡ e^2A, one can derive that the KK mode ψ(z) of the tensor perturbation satisfies the following Schrödinger-like equation <cit.>:[-∂_z^2 +W(z)]ψ(z) =m^2ψ(z),where the effective potential is given byW(z)=3/4(∂_z a)^2/a^2 +3/2∂_z^2 a/a +3/2∂_z a ∂_z f_R/a f_R -1/4(∂_z f_R)^2/f_R^2 +1/2∂_z^2 f_R/f_R.One can check that this equation can be factorized as𝒦 𝒦^† ψ(z)=m^2ψ(z)with𝒦 = ∂ _z+3/2∂_z a/a+1/2∂_z f_R/f_R, 𝒦^† =-∂ _z+3/2∂_z a/a+1/2∂_z f_R/f_R,which ensures that there is no graviton mode with m^2<0. The graviton zero mode (the four-dimensional massless graviton) can be solved asψ_0∝ (a^3 f_R)^1/2.Note that to make sure ψ_0 is real, f_R should be positive. This also avoids the graviton ghost. The recovering of four-dimensional gravity on the brane requires the normalization of graviton zero mode, namely∫^+∞_-∞ (ψ_0)^2 dz<∞.For the solution given in Ref. <cit.>, this condition cancertainly be satisfied. Thus thefour-dimensional gravity can be obtained. Besides the bound graviton zero mode, there are continuous unbound massive graviton KK modes. They will have a contribution to Newton's law of gravitation at short distance. The structure of other f(R)-brane models given in Ref. <cit.> was analyzed in Ref. <cit.>, where the effective potential for the graviton KK modes may have a singular structure and there is a series of graviton resonant modes.There is a problem that should be mentioned here. The above analysis involved the tensor mode only. This is not completesince the full perturbations contain tensor, vector, and scalar modes. In the original RS-1 model <cit.>, the fluctuation of the extra dimension radius gives a scalar mode (radion). If the extra dimension is not stabilized, the radion will be massless, which will contribute a long range fifth force. This is undoubtedly unacceptable. If the Goldberger-Wise mechanism <cit.> is introduced, the extra dimension radius can be stabilized and the radion will becomes massive.In thick braneworld scenario constructed with a background scalar field, the radion-like scalar mode has a continuous mass spectrum. But there is still a massless radion-like scalar mode. So the recovering of four-dimensional gravity implies that the scalar zero mode should not be localized. For general relativity coupled with a scalar field, the scalar zero mode is not localized.However, the situation is completely different for the case of the f(R) gravity. The tensor and vector modes are similar to the case of general relativity while the scalar mode is very different. In the higher-order frame, the dynamical equation of the scalar mode would be fourth order, which implies that there are actually two scalar degrees of freedom. Recall that the f(R) theory is equivalent to the Brans-Dicke theory. Based on this fact, the scalar perturbations of the f(R) theory can be investigated in the frame work of scalar-tensor theory, and we will review this part in section <ref>.§.§ Palatini f(ℛ) theory It is well known that there are two different formalisms in f(R) theories of gravity, namely, the metric formalism and the Palatini formalism <cit.>. In Palatini formalism, metric and connection are two independent fundamental variables. In general relativity these two formalisms are completely equivalent, but usually they will lead to different predictions in modified theories of gravity. Different from the metric f(R) theory, the Palatini f(ℛ) theory will lead to a second-order system. In this subsection, we consider thick brane models in the Palatini f(R) theory. The action is given byS_=∫ d^D x √(-g)[1/2κ_D^2 f(ℛ(g,Γ))-1/2g^MN∂_M ϕ∂_N ϕ - V(ϕ)],and the metric is also described by (<ref>). Variation with respect to the metric g and the connection Γ yields two equations of motion:f_ℛℛ_M N-1/2fg_M N = κ_D^2 T_M N, ∇̃_A(√(-g)f_ℛg^M N) = 0,where ∇̃_A is compatible with the independent connection Γ. For the case of f(ℛ)=ℛ, the theory is equivalent to general relativity. It would be convenient to define an auxiliary metric q_MN by√(-q) q^M N≡√(-g)f_ℛg^M N.Now Γ^P_ MN and ℛ_MN(Γ) can be viewed as the connection and Ricci tensor constructed from the auxiliary metric q_MN, respectively. One can eliminate the independent connection from the field equations by using the relation (<ref>), and obtain the following equations concerned with the metric only:G_M N = κ_D^2 T_M N/f_ℛ-1/2g_M N(ℛ-f/f_ℛ)+1/f_ℛ(∇_M∇_N-g_M N∇_A∇^A)f_ℛ -D-1/(D-2)f_ℛ^2(∇_Mf_ℛ∇_Nf_ℛ-1/2g_M N∇_Af_ℛ∇^Af_ℛ).In addition, contracting Eq. (<ref>) with g^MN, one gets an algebraic equation of ℛ and T:f_ℛℛ-D/2f=κ_D^2 T.This implies that f(ℛ) is just an algebraic expression of T. From this point of view, we can see that Eq. (<ref>) clearly tells that the Palatini f(ℛ) theory modifies the matter sector of Einstein equations. Note that there are derivatives on f(ℛ) and f_ℛ(ℛ) and thus T. This structure would lead to surface singularity of stars <cit.>, and also give higher derivatives on the matter fields. This is a significant difference with the metric f(R) theory, which has higher derivatives on the metric. For more details about the Palatini f(ℛ) theory, see Refs. <cit.>. The thick braneworld model in five-dimensional space-time with a scalar field in the Palatini f(ℛ) theory was first considered in Ref. <cit.>. The authors considered the flat braneworld model andintroduced two methodsto solve the system. In the first method, they assumed the following relationsdϕ/dy = bcos(bϕ),f_ℛ = (1+ab^2cos^2(bϕ))^-3/4 ,from which one can get ϕ(y) and f(ℛ(y)), but it is hard to get the expression of f(ℛ). The solution for ϕ(y) and A(y) is <cit.>ϕ(y) = 1/barcsin(tanh(b^2 y)), A(y)= -A_0 +ln(𝒰)+2κ_5^2/27b^2𝒰^3 -κ_5^2/3b^2(1+2ab^2/3) [arctanh(1/𝒰)+arctan(𝒰) ],where 𝒰(y)=(1+ab^2 sech^2(b^2y))^1/4.They also used the perturbative method to consider the model f(ℛ)=ℛ+ϵℛ^n with ϵ a small parameter. To the first order of ϵ, the analytic solutions were obtained. The complete solutions were first obtained in Ref. <cit.>, and the tensor perturbation was also investigated therein. As mentioned above, the field equations (<ref>) contain second-order derivatives on the trace of the energy-momentum tensor. This means that the field equations (<ref>) contain third-order derivatives on the scalar field and it is not a convenient choice to solve the equations (<ref>).The work <cit.> gave a strategy which avoids solving the higher-derivative equations. Note that the original equations (<ref>) and (<ref>) are second order at most, hence are much easier to solve. For the assumption of the space-time metric (<ref>), the auxiliary metric q_MN is given byds̃^2=q_MN dx^M dx^N =u^2(y)η_μνdx^μdx^ν+u^2(y)/a^2(y)dy^2,where u(y)=a(y)f_ℛ^1/3. In terms of these variables, the field equations (<ref>) are reduced to(6u'^2/u^2-3a'/au'/u-3u”/u)f_ℛ = κ_5^2ϕ'^2, 5f_ℛ(a'/au'/u+u”/u)-2f_ℛu'^2/u^2+f = 2κ_5^2 V.The above two equations and the scalar field equation□^(5)ϕ= V_ϕconstitute the system to be solved. For the model f(ℛ)=ℛ+αℛ^2, one can assume the relation u(y)=c_1a^n(y) with n 0, then the system can be solved <cit.>:a(y) = sech^2/3(n-1)(ky),V(y) =v_1 sech^4(ky) + v_2 sech^2(ky) +Λ_5/2κ _5^2, ϕ(y) = ϕ_0 [i √(3) E(ik y, 2/3)-i √(3) F(ik y, 2/3)+ √(2+cosh(2k y)) tanh(k y)],where F(y,m) and E(y,m) are the incomplete elliptic integrals of the first and second kinds, respectively. With some analyses on the energy density the parameters are constrained to be α>0 and n<-2/3 or 0<n<1/6 or n>1. Note that the warp factor is infinite at boundary for 0<n<1/6 or n<-2/3. Usually, the scalar potential should be a function of the scalar field ϕ, but in this case it cannot be solved.Different from the background solutions, it would be more convenient to consider the gravity fluctuations with the equation (<ref>). The final equation is similar to the case of the metric f(R) theory<cit.>:𝒦 𝒦^† Ψ(z)=m^2 Ψ(z),where 𝒦= ∂ _z+ 3/2∂_zln a+1/2∂_zln f_ℛ and Ψ(z) is defined by h_μν(x^σ,z)=ε_μν(x^σ)(a^3 f_ℛ )^1/2Ψ(z). Again, f_ℛ should be positive to keep the graviton zero mode real and to avoid the graviton ghost. It can be easily checked that the solutions (<ref>) always support localized graviton zero mode Ψ(z(y)) ∝sech^n/n-1(ky). There is a new feature, which is different from the regular case. For n<-2/3, the effective potential is an infinitely deep potential well, for which all the states are bounded. However, the four-dimensional gravity can still recover, but with a tiny correction <cit.>. §.§ Eddington inspired Born-Infeld theoryThere is another Palatini gravity theory which is widely considered in recent literatures, namely, the Eddington inspired Born-Infeld (EiBI) gravity theory <cit.>. It is an extension of Eddington's gravity theory, and contains the matter Lagrangian absent in Eddington's theory. The thick braneworld model was considered in Refs. <cit.>. The action of the theory is <cit.>S(g,Γ,Φ)=1/κ_5^2 b∫ d^5x[√(-|g_MN+bR_MN(Γ)|). -. λ√(-|g_MN|)]+S_M(g,Φ),where b and λ are some parameters, and R_MN(Γ) is the Ricci tensor built from the independent connection Γ. As mentioned in the last subsection, the Palatini version of a gravitational theory abandons the priority of the metric, so the variation has to be made with respect to both of the metric and connection. The field equations are√(-|g_PQ+bR_PQ|)/√(-|g_PQ|)[(g_PQ+bR_PQ)^-1]^MN-λ g^MN = -κ_5 b T^MN, ∇̃_K (√(-q)q^MN) = 0,where q_MN≡ g_MN+bR_MN is the new introduced auxiliary metric which satisfies q_MNq^MP=δ^P_N, and the covariant derivative ∇̃ is compatible with this metric. The work <cit.> constructed a flat brane generated by a canonical scalar field. By using the auxiliary metric and assuming the relation ϕ'(y)=K a^2(y), the authors obtained an analytic domain wall solution:a(y) = sech^3/4(ky), ϕ(y) = ϕ_0 (i E(ky/2,2)+sech^1/2(ky) sinh(ky)),V(ϕ(y)) = 7√(21)/24bκ_5sech^3(ky) -λ/bκ_5,where k=2/√(21b). Note that the scalar potential is expressed in terms of the space-time coordinates. However, according to the scalar field equation one can still find that the spatial boundaries y=±∞ are mapped to the minimum of the scalar potential V(ϕ). Besides, it can be easily checked that the energy density localizes near the origin.The above solution was generalised in Ref. <cit.> by imposing some more general assumptions ϕ'(y)=K a^2n(y) and ϕ'(y)=K_1 a^2(y)(1-K_2 a^2(y)). Under these assumptions, some solutions with interesting features, like double kink solution, are allowed.References <cit.> investigated linear perturbations of the EiBI brane system. It was found that the tensor perturbation is stable for the above models <cit.> and the stability condition for the scalar perturbations of a known analytic domain wall solution with e^2A(y)= sech^3/2p is 0<p<√(8) <cit.>. Quasi-localization of gravitational fluctuations was also studied <cit.>. §.§ Scalar-tensor theoryThe scalar-tensor theory has long been considered as an alternative gravity theory that deviates from general relativity. It was first proposed from the inspiration of Mach's principle by Brans and Dicke <cit.> in 1961. Thin braneworld models in this theory were studied in Refs. <cit.>, and thick braneworld models in Refs. <cit.>. Here we only review the thick brane models.We consider the following action in Jordon frame:S=∫ d^5x √(-g)(1/2κ_5^2F(ϕ)R -1/2(∂ϕ)^2-V(ϕ) + ℒ_M(g_MN,ψ)),where F(ϕ) is a positive function in order to avoid the problem of antigravity. In Jordon frame the scalar field ϕ non-minimally couples to the Ricci scalar. In Einstein frame ϕ does not couple to the Ricci scalar, but couples to the matter sector. For the flat brane assumption (<ref>) and ℒ_M(g_MN,ψ)=0, the equations of motion are3F(4A'^2+A”)+7A'F'+F” = -2κ_5^2 V, 3F A”-A'F'+F” =-κ_5^2 ϕ'^2, κ_5^2 ϕ”+4 κ_5^2 A'ϕ'-2F_ϕ (2A”+5A'^2) = κ_5^2 V_ϕ .One of the analytic solutions with F(ϕ)=1-ακ_5^2 ϕ^2 was given in Ref. <cit.>:ϕ(y) = 1/κ_5√(3(1-6α)/α(1-2α))tanh(ky),A(y) = -(α^-1-6)lncosh(ky),V(ϕ) = k^2/6α[9-54α/(1-2α)κ_5^2+6(α(7+24α)-2)ϕ^2+α(1-2α)(3-16α)(4-12α)/1-6ακ_5^2ϕ^4],where the parameter α should satisfy 0<α<1/6. Reference <cit.> found another interesting solution:F(ϕ) = v^2/3+(1-v^2/3)cosh(√(3)κ_5/vϕ), ϕ(y) = v/κ_5arctan(sinh k y),A(y) = -lncosh(k y),V(ϕ) = v^2 k ^2/2κ_5^2cos^2(κ_5ϕ/v) +2 k ^2 /√(3)κ_5^2(3-v^2)sin(2κ_5ϕ/v) sinh(√(3)κ_5ϕ/v)-k ^2/κ_5^2(2v^2 + (6-2v^2) cosh(√(3)κ_5 ϕ/v)) sin^2(κ_5ϕ/v).To avoid antigravity, it requires F>0 and thus0<v^2<3cosh(√(3)π/2)/3cosh(√(3)π/2)-1.All of these solutions constitute of domain walls, which can localize gravity and some matter fields. Such kind of works can be found in Refs. <cit.>.The tensor perturbation and localization of gravity were investigated in Refs. <cit.>, and we will not review this part. What is more interesting is the scalar perturbations, which have obvious differences with that of general relativity. However, the more general cases are the models with multiple scalars. Hence it would be more meaningful to consider multiple scalars.The study of scalar perturbations of general relativity with multiple canonical scalars can be found in Refs. <cit.>.For superpotential models, it has been shown that there is no tachyon instability, and only odd scalars can avoid the localized scalar zero mode. In particular, for the double field theory, there is always a normalizable zero mode. The scalar perturbations of 𝒩 nonminimally coupled scalars were systematically studied in Ref. <cit.> in Einstein frame, in which the theory is just general relativity with nonminimaly coupled scalar fields. The action isS=∫ d^D x√(-g)[1/2κ_D^2R+P(𝒢_IJ,X^IJ,Φ^I)],where X^IJ=-1/2g^MN∂_MΦ^I∂_NΦ^J is the kinetic function, 𝒢_IJ is the field-space metric, and P is the Lagrangian of the scalars. Here, indices I,J,K,L,⋯ (=1,2,⋯,𝒩) denote 𝒩-dimensional field-space indices lowered or raised by the field-space metric 𝒢 or its inverse, while M,N,P,Q,⋯ run over D-dimensional ones of the space-time. Using the Arnowitt-Deser-Misner variables and some calculations in the flat gauge, the coupled equations of the independent 𝒩 scalar modes (δΦ^I=Q^I) can be obtained <cit.>:1/a^D-1𝒟_y(a^n-1𝒟_yQ_I)-1/a^2 m^2 Q_I-ℳ_I^J Q_J=0,whereℳ_IJ = V_;IJ- ℛ_IKJL u^K u^L+𝒰_IJ,𝒰_IJ = 2 (D-2)a^D-1𝒟_y(a^D-1/A'u_I u_J), u^I≡ ∂_yΦ^I_0.Here 𝒟_y=u^I𝒟_I with 𝒟_I the covariant derivative compatible with the field-space metric 𝒢_IJ, and ℛ_IKJL is the Riemann tensor constructed from the field space metric 𝒢_IJ. If the field space is one-dimensional, Eq. (<ref>) is an usualSchrödinger-like equation. However, for the field-space with multi-field, Eq. (<ref>) becomes a series of coupled equations. Generally, Eq. (<ref>) cannot be factorized, but there is an exception. If the background solution is obtained by using the superpotential method, then this equation can be factorized in a supersymmetric formalism:(-δ^I_J𝒟_y-Z^I_J+(D-1)δ^I_J W)(δ^J_K𝒟_y-Z^J_K ) Q^K =m^2/a^2Q^IwithZ^I_J=(D-2)(W^I_;J-W^I W_J/W).Here W=-A'(y) is the superpotential. The above equations mean that there is no tachyon instability since∫ dy e^(D-3)Am^2Q^IQ_I= ∫ dy a^D-1Q_I (-δ^I_J𝒟_y-Z^I_J+(D-1)Wδ^I_J ) (δ^J_K𝒟_y-Z^J_K) Q^K= ∫ dy a^D-1|𝒟_y Q^I-Z^I_J Q^J|^2≥0.The scalar zero modes satisfy𝒟_y Q^I-Z^I_J Q^J=0.Now it is clear that the zero mode solutions are totally determined by the background solutions. For singular scalar case, the zero mode solution isQ_I=u_I/A'.It cannot be localized on the brane for asymptotically AdS_5 domain wall solution. For multiple scalars case, it is more convenient to define the tetrad fields satisfyinge^i_I e^j_J δ_ij=𝒢_IJ, e^i_I e^I_j =δ^i_j,𝒟_y e^i_I=0,and make a decomposition Q_I=∑_i e_I^i Q_i(m^2,y)e^ip_μ x^μ with η^μνp_μ p_ν=-m^2. In the conformally flat coordinate z and in terms of the canonically normalized modes Q̃_i≡ a^(D-2)/2Q_i, one gets the coupled Schrödinger-like equations <cit.>-∂_z^2Q̃_i+[((D-2)^2/4(∂_zA)^2 -(D-2)/2∂_z^2A)δ^j_i+a^2ℳ^j_i ]Q̃_j =m^2 Q̃_i.Actually, if the potential matrix ℳ^j_i is positive definite then there is no localized zero mode. The localized states should satisfy∫_-∞^+∞dy a^D-3𝒢_IJQ^IQ^J<∞.If one separates the field space into the background trajectory direction and its orthogonal space, then the perturbed modes are Q_σ and Q⃗_s, and the localization condition can be expressed as∫_-∞^+∞dy a^D-3(Q⃗_s^2+Q_σ^2)<∞.For the double-scalar superpotential case, for instance, Q_σ=u^I/|u^I| Q_I and Q_s=𝒟_yσ^I/|𝒟_yσ^I| Q_I, and the zero modes are given byQ_s = e^(D-2)∫ dy W_ss,Q_σ = √(W')/W∫ dy ω W/√(W') Q_s,with W_ss=W_,IJs^I s^J. The superpotential background solutions would lead to a normalized zero mode. It means that we will have a massless scalar field on the brane. This result is conflicted with observations for the fifth dimension and is not acceptable <cit.>. As mentioned in section <ref>, the metric f(R) theory can be studied in the context of scalar-tensor theory, hence one of the application of the above analysis is the scalar perturbations of the metric f(R) theory. Obviously, the metric f(R) theory with a scalar field is equivalent to a scalar-tensor theory with two scalars in Jordon frame, or general relativity with two nonminimally coupled scalars in Einstein frame. Using the above results and considering the solution given in Ref. <cit.>, we can show that the scalar perturbations are stable and no massless scalar mode can be localized on the brane <cit.>.§.§ f(T) theory In this subsection we will give a brief review of the thick braneworld scenarios in the f(T) gravity theory. Since the f(T) theory is successful in explaining the acceleration of the universe <cit.>, it has been investigated widely (for examples, see Refs. <cit.> and therein). The braneworld models in the f(T) theory have been studied in Refs. <cit.>. First, the f(T) theory is the generalization of the teleparallel gravity, so we will review the basics of the teleparallel gravity briefly <cit.>. The tangent space of any point in the specetime with the coordinate x^M can be expanded based on the orthogonal basis which is formed by the vielbein fields e_A(x^M). In this subsection, the Capital Latin letters A,B,C,⋯=0,1,2,3,5 label the tangent space, while M, N, O, P, ⋯ still represent the five-dimensional space-time indices. Obviously, the vielbein fields e_A(x^M) are vectors in the tangent space, and their components in the coordinates of space-time are labeled as e_A^M. The relation between the metric and vielbein isg_MN=e^A_M e^B_N η_AB,where η_AB=diag(-1,1,1,1,1) is the Minkowski metric of the tangent space. From the relation (<ref>) we can gete_A^Me^A_N=δ^M_N, e_A^Me^B_M=δ^B_A.Theconnection Γ̃^P_MN (Weitzenböck connection) inthe teleparallel gravity is defined asΓ̃^P_MN≡e_A^P∂_Ne^A_M.The torsion tensor isT ^P_ MN=Γ̃^P_ MN-Γ̃^P_ NM.The contortion tensor K^P_MN is defined as thedifference between the Weitzenbök connection and Levi-Civita connectionK^P_ MN≡Γ̃^P_ MN-Γ^P_ MN =1/2(T^ P_M N+T_N M^ P-T^P_ MN).The Lagrangian of the teleparallel gravity in five dimensions can be written asL_T=-M_*^3/4e T≡-M_*^3/4eS_P^ MNT^P_ MN,wheree is the determinant of e^A_M, M_* is the fundamental Planck scale in five-dimensional space-time (we will set M_*=1 in the following) and we have defined the tensor S_P^ MN asS_P^ MN≡1/2(K^MN_P-δ^N_PT^QM_Q+δ^M_PT^QN_Q).The action of the f(T) theory isS=-M_*^3/4∫ d^5x e f(T)+∫ d^5x e ℒ_M,where f(T) is a function of the torsion scalar T and ℒ_M denotes the Lagrangian of matters. The equations of motion can be obtained by varying the action with respect to the vielbein fields:e^-1f_T g_NP∂_Q (e S_M^ PQ)+f_TTS_MN^ Q∂_QT-f_TΓ̃^P_ QMS_PN^Q+1/4g_MNf(T)=M_*^-3𝒯_MN.The flat thick brane solutions for the f(T) theory were obtained in Refs. <cit.>. In the case of f(T)=T+α T^n and ℒ_M=-1/2∂^M ϕ∂_Mϕ-V(ϕ), the authors of Ref. <cit.> found the following solution: A(y) = -2/3 v^1/n∫ dy tanh ^1/n-1(ky), ϕ(y) =v tanh^n/2(n-1)(k y),V(ϕ) = 2/n^2ϕ^2(ϕ^2-2n/n-B_nϕ^2n-2/n) -4/3(ϕ^4/n-B_n/nϕ^4), where B_n=(-1)^n3^1-n2^4n-4n(2n-1)α and k=4(n-1)√(B_n)/n^2, and the scalar potential has two global minima at ϕ=± v=±B^n/4(n-1)_n and a local minimum atϕ=ϕ_0=0. In Ref. <cit.>, two explicit analytical thick solutions were found. The first solution is for n=0 or n=1/2: e^2A(y) = cosh^-2b(k y),ϕ(y) = √(6b)M_*^3/2arctan(tanh(ky/2)), V(ϕ) = 3bk^2M_*^3/4[(1+4b)cos^2(2ϕ/√(6b)M_*^3/2) -4b]. The second one is for n=2:e^2A(y) = cosh^-2b(k y),ϕ(y) = √(3b )/2 M_*^3/2[i√(2) E(i k y; u )-i√(2) F(i k y;u )+tanh (k y) √(72 αb^2 k^2+u cosh (2 k y)+1)], V(ϕ(y)) = 3/4 b k^2 M_*^3 [144 αb^3 k^2 tanh ^4(k y)-4 b tanh ^2(k y) (18 αb k^2 sech^2(k y)+1)+sech^2(k y)], where u=1-72 αb^2 k^2 and b,k are positive parameters, and F(y;q) and E(y;q) are the first and second kind elliptic integrals, respectively. For the case n=2, it requires that 72 α k^2b^2 < 1 in order to insure the reality of the scalar field ϕ.The Lagrangian density of the matter was generalized to a generic form ℒ_M=X+ λ [(1+β X)^p-1]-V(ϕ) in Ref. <cit.>,where X=-1/2∂_Mϕ∂^Mϕ. The parameter β is positive and λ is real. The analytical solution was found for p=2, λ=-36α k^7, β=4/3k^-5: A(y) = M_*^3/k^3ln(sech(k^4/M_*^3y)), ϕ(y) = √(3/2)M_*^3/k^3/2arcsin(tanh(k^4/M_*^3y)),V(ϕ) = 3/8 k^2 (C_1 cos(4 √(2/3) k^3/2ϕ/M_*^3)+C_2cos(2 √(2/3) k^3/2ϕ/M_*^3)+C_3), where C_1=9 αk^2 (3 k^3+4 M_*^3), C_2=36 αk^5+k^3-144 αk^2 M_*^3+4 M_*^3 and C_3=9 αk^5+k^3+108 αk^2 M_*^3-4 M_*^3. Besides, the localization of four-dimensional gravity was studied in Ref. <cit.> by analyzing linear tensor perturbation of the vielbein. It was found that the graviton KK modes of the tensor perturbation satisfy the following equation(∂_z+𝒦)(-∂_z+ℋ)ψ=m^2ψ,where 𝒦=3/2∂_z A+12e^-2A((∂_z A )^3-∂_z^2 A∂_z A)f_TT/f_T, which means m^2≥0, so any analytical thick brane solutions for the f(T) theory are stable under the transverse-traceless tensor perturbation. The graviton zero mode has the following formψ_0=N_0e^3/2A+ 12 ∫ e^-2A((∂_z A )^3-∂_z^2 A∂_z A)f_TT/f_T dz,where N_0 is the normalization coefficient. It is easy to show that the zero mode of the graviton can be localized on the brane for the above mentioned solutions.Furthermore, there were some related work in other gravity theories, such as Weyl (pure geometrical) gravity <cit.> and critical gravity <cit.>. § LOCALIZATION OF BULK MATTER FIELDS In the last section we know that the graviton zero mode of the tensor perturbation can be localized on the brane embedded in a five-dimensional AdS space-time and the Newtonian potential can be restored. As mentioned in the last section, all matter fields should be in the bulk in thick brane scenario. The zero modes of various bulk matter fields confined on the brane denote the particles or fields in the Standard Model, while the massive KK modes indicate new particles beyond the Standard Model. So a natural question is that whether various bulk matter fields can be localized on such brane. In order not to contradict the present observations, the zero modes of various matter fields should be confined on the brane, while the massive KK modes can be localized on the brane or propagate along extra dimensions. These massive KK modes give us the possibility of probing extra dimensions through their interactions with particles in the Standard Model <cit.>. Localization and resonances of various bulk matter fields on a brane have been investigated in five-dimensional brane models <cit.> and six-dimensional ones <cit.>. In this section we will review some works about localization of bulk matters on thick branes with the following metricds^2=e^2A(z)(g̃_μν(x)dx^μ dx^ν+dz^2).§.§ Scalar fields We first consider the case of scalar fields. We denote a bulk scalar field as Φ(x,y) to distinguish with the background scalar field ϕ(y). A main result is that a free massless scalar field can be localized on the brane if the gravity can.The action of a free massless scalar field can be written asS_0=∫√(-g)d^5x (-1/2 g^MN∂_MΦ∂_NΦ).The equation of motion can be obtained by varying the above action with respect to the scalar field Φ as□^(5)Φ =1/√(-g)∂_M (√(-g)g^MN∂_NΦ)=0.Combined with the conformal metric (<ref>), Eq. (<ref>) can be rewritten as(∂_z^2+3(∂_z A)∂_z+g̃^μν∂_μ∂_ν)Φ=0.Then, we introduce the KK decomposition Φ(x^μ, y)=Σ_nφ_n(x^μ)χ_n(y). By substituting this decomposition into Eq. (<ref>) and making separation of variables, one can find that φ_n satisfies the four-dimensional Klein-Gordon equation (g̃^μν∂_μ∂_ν-m_n^2)φ_n=0, and the extra component χ_n satisfies the following equation of motion(∂_z^2+3(∂_zA)∂_z+m_n^2)χ_n=0.At last, by integrating over the extra dimension and using Eq.(<ref>) and the following normalization condition∫_-∞^∞ dz e^3Aχ_m(z)χ_n(z)=δ_mn,one can reduce the fundamental five-dimensional action (<ref>) of a free massless scalar field to the effective four-dimensional action of a massless (m_0=0) and a series of massive (m_n>0) scalar fields:S_0 =∑_n ∫d^4x √(-g̃) [-1/2g̃^μν∂_μφ_n ∂_νφ_n-1/2 m_n^2 φ_n^2]. The mass spectrum m_n of the KK modes is determined by the equation of motion (<ref>). In order to investigate the mass spectrum, we introduce the following field redefinitionχ̃_n(z)=e^3/2Aχ_n(z),with which Eq. (<ref>) turns to bethe following Schrödinger-like equation[-∂_z^2+V_0(z)]χ̃_n(z)=m_n^2χ̃_n(z),where m_n is the mass of the n-th KK excitation of the scalar field. The effective potential V_0(z) takes the following form:V_0(z)=3/2∂_z^2 A+9/4(∂_z A)^2.Note that, the effective potential only depends on the warp factor A, and has the same form as the case of graviton KK modes in general relativity <cit.>. That is, the scalar zero mode will be localizedon the brane on the condition that the gravity can be localized on the brane. The effective potential has different forms for different solutions of the warp factor A. We take an explicit form as example: A(y)=ln(sech(ky)). This typical solution of the warp factor in the conformal coordinate has the following form:A(z)=ln(1/1+k^2z^2),where k is a parameter. Then the effective potential reads asV_0(z)=3k^2(4k^2z^2-1)/(1+k^2z^2)^2.From the above expression we can see that the potential V approaches to zero when z approaches to infinity, and the value of the potential at z=0 is -3k^2. So the effective potential has the volcano shape. In this case, there is no mass gap between the zero mode and the massive KK modes, and the mass spectrum is continuous. Any massive KK mode cannot be localized on the brane since V(|y|→∞) →0. The solution of the zero mode with m_0^2=0 is given byχ̃_0 (z)=N_0e^3/2A=N_0/(1+k^2 z^2)^3/2,where N_0=(8k/3π)^1/2 is the normalization constant. This is the lowest energy eigenfunction for the Schrödinger-like equation (<ref>), which indicates that there is no KK modes with negative m^2. In fact, this equation can be rewritten as 𝒦 𝒦^† χ̃_n=m^2_nχ̃_n, with 𝒦=∂_z+3/2∂_z A, which ensures m^2≥ 0, namely, there is no tachyonic scalar mode. Besides this scalar zero mode, there exist other continuous massive KK modes. The effective potential and the zero mode are shown in Fig. <ref>.There are other shapes of the effective potential, such as the infinite deep well and the Pöschl–Teller potential <cit.>. For example, the effective potential of the KK modes of a massless scalar field on the de Sitter brane with A(z)=ln[H/bsech(Hz)] has the following Pöschl–Teller form <cit.>:V_0(z)=3/4H^2[3 -5sech^2(Hz)],for which there are two bound KK statesχ̃_0(z)= √(2H/π) sech^3/2(Hz),χ̃_1(z)= √(H) sech^3/2(Hz) sinh(Hz),and the mass spectrum of the bound states is given bym_n^2=n(3-n)H^2, n=0,1.The effective potential, the zero mode, and the mass spectrum are shown in Fig. <ref>. §.§ Vector fields In this subsection, we review the localization of a bulk U(1) gauge vector field on the brane in some five-dimensional thick brane models. We first consider the following five-dimensional action for a free bulk vector field:S_A=-1/4∫ d^5 x √(-g) F^MNF_MN,where F_MN=∂_M A_N-∂_N A_M is the five-dimensional field strength. For the conformal metric (<ref>), the equations of motion1/√(-g)∂_M(√(-g) g^M N g^R S F_NS) = 0can be written as the following component equations:1/√(-g̃)∂_ν(√(-g̃) g̃^νρg̃^μλF_ρλ) +g̃^μλe^-A∂_z (e^A F_5λ)= 0, ∂_μ(√(-g̃) g̃^μν F_ν 5) =0.The five-dimensional vector field A_M(x^λ,z) can be decomposed asA_M(x^λ,z)=∑_n a_M^(n)(x^λ)ρ_n(z). It can be seen that the action (<ref>) is invariant under the following gauge transformation:A_M(x^λ,z) →A_M(x^λ,z)=A_M(x^λ,z)+∂_MF(x^λ,z),orA_μ(x^λ,z) →A_μ(x^λ,z)=A_μ(x^λ,z)+∂_μF(x^λ,z), A_5(x^λ,z) →A_5(x^λ,z)=A_5(x^λ,z)+∂_zF(x^λ,z),where F(x^λ,z) is an arbitrary regular scalar function. One can check that the gauge A_5(x^λ,z)=0 is allowed with the above gauge transformation. For the KK theory with finite extra dimension, A_5(x^λ,z) and F(x^λ,z) should be periodic functions of the extra dimension. However, for the braneworld scenario with an infinite extra dimension, which is the case we are considering, there isno any constraint on A_M(x^λ,z) and F(x^λ,z). From the transformation (<ref>) and the KK decomposition (<ref>), one has <cit.>A_5(x^λ,z) →A_5(x^λ,z)= ∑_nA_5^(n)(x^λ,z) =∑_na_5^(n)(x^λ)ρ_n(z) = ∑_na_5^(n)(x^λ)ρ_n(z)+∂_zF(x^λ,z).Therefore, if one choosesF(x^λ,z) =∑_n F_5^(n)(x^λ,z)= -∑_na_5^(n)(x^λ)∫ρ_n(z) dz,then the fifth component A_5 vanishes:A_5(x^λ,z) = 0,which is just the gauge choice we will take. Note that, for the case of the KK theory one has ρ_n(z)=cos (nkz), which indicates that one can only takeF(x^λ,z) =∑_n F_5^(n)(x^λ,z)= -a_5^(0)(x^λ) -∑_n ≠ 0a_5^(n)(x^λ)∫ρ_n(z) dz,i.e., the zero mode F_5^(0) is a function of x^λ only. Therefore, one can only choose the gauge condition A_5^(n)=0 for massive KK modes (n ≠ 0) instead of A_5^(0)=0. Thus, one has no the gauge A_5(x^λ,z) = 0 in the KK theory.Now we choose the gauge A_5=0 and make the decomposition A_μ(x,z)=∑_n a^(n)_μ(x)α_n(z)e^-A/2. Then it is easy to find that the vector KK modes α_n(z) satisfies the following Schrödinger equation:[-∂^2_z +V_1(z) ]α_n(z)=m_n^2α_n(z),where the effective potential V_1(z) is given byV_1(z)=1/2∂_z^2 A+1/4(∂_z A)^2.The vector zero mode with m_0^2=0 can be solved asα_0 (z)=N_0e^A/2.By introducing the orthonormalization conditions∫^+∞_-∞ α_m(z)α_n(z)dz=δ_mn,the fundamental action (<ref>) can be reduced to the effective one of a massless (m_0=0) and a series of massive (m_n>0) four-dimensional vector fields:S_1 = ∑_n∫ d^4 x √(-g̃) ( - 1/4g̃^μαg̃^νβf^(n)_μνf^(n)_αβ- 1/2m_n^2 g̃^μνa^(n)_μa^(n)_ν),where f^(n)_μν = ∂_μ a^(n)_ν - ∂_ν a^(n)_μ is the four-dimensional field strength tensor. The mass spectrum m_n and localization of the vector KK modes are also determined by the Schrödinger equation (<ref>). For the RS-like solution with A(z)=ln(1/1+k^2z^2), the effective potential and vector zero mode are given byV_1(z) = (2k^2z^2-1)/(k^2z^2+1)^2 k^2, α_0 (z) = N_0/√(1+k^2z^2),which are shown in Fig. <ref>. Since∫^+∞_-∞ |α_0(z)|^2 dz = ∫^+∞_-∞N_0^2 e^A dz = N_0^2 ∫^+∞_-∞ dy = ∞, the vector zero mode α_0(z) = N_0 e^A/2 with arbitrary A(z), including (<ref>), cannot satisfy the normalization condition ∫^+∞_-∞ |α_0(z)|^2 dz =1, and hence cannot be localized on the brane. So one needs some localization mechanisms for a bulk vector field for such brane models.For the de Sitter brane model with A(z)=ln[H/bsech(Hz)], the effective potential (<ref>) for the vector KK modes has the following Pöschl–Teller form <cit.>:V_1(z)=H^2/4[1-3 sech^2(Hz)].The above potential has a minimum -H^2/2 at z=0 and a maximum H^2/4 at z=±∞, which ensures the presence of a mass gap in the spectrum. There is only one bound state, i.e., the vector zero mode that can be localized on the de Sitter brane:α_0(z)= √(H/π sech(Hz)).The effective potential and the vector zero mode for the de Sitter brane model are shown in Fig. <ref>. Besides, the zero mode of a free five-dimensional vector field can also be localized on the brane in some other special braneworld senarios, such as AdS branes <cit.>, Weyl Thick Branes <cit.>,two-field thick branes with an finite extra dimension <cit.>. Note that if a RS-like brane has more than three space dimensions, then the vector zero mode can also be localized on the brane. In order to localize the zero mode of a bulk vector field on a RS-like brane, some mechanisms were proposed. In the following, we give a brief review. Kinetic energy term coupling. Inspired by the effective coupling of neutral scalar field to electromagnetic field and by the Friedberg-Lee model for hadrons <cit.>, Chumbes, Hoff da Silva, and Hott <cit.> explored the coupling between the kinetic term of the vector field and the background scalar field ϕ to realize the localization of the vector field. The general action of the vector field nonminimally coupled with the background scalar field is given by <cit.>S_A=-1/4∫ d^5 x √(-g)G(ϕ)F^MNF_MN.The localization condition for the vector zero mode is that the following integrate is finite:∫_-∞^+∞G(ϕ)dy < ∞.For the solutions of the background scalar field ϕ = v tanh(ky) and ϕ = v arcsin(tanh(ky)), the corresponding coupling functions can be chosen asG(ϕ)= (1-ϕ ^2/v^2)^p/2 for ϕ = v tanh(ky), G(ϕ)= (1-sin^2(ϕ/v))^p/2 for ϕ =varcsin(tanh(ky)),where p is a positive constant. The above two couplings would result in the same function G(ϕ(y))=sech^p(ky), which insures the normalization and hence the localization of the vector zero mode. References <cit.> considered such localization mechanism for the Bloch branes <cit.> and found localized zero mode and quasi-localized massive KK modes of a bulk vector field.For some two-field braneworld models <cit.>, the above coupling can also be used to localize the vector field with G=e^τπ(y), where π(y) is one of the two background scalars ϕ(y) and π(y), and τ is the coupling constant (see Refs. <cit.>).Yukawa-like coupling. An alternative approach to solve the localization problem of the gauge field is to introduce the Yukawa-like coupling <cit.>, namely, consider the Stueckelberg-like gauge field action <cit.>:S_A=∫ d^5x √(-g){-1/4F^MNF_MN -1/2G(ϕ)(A_M-∂_M B) (A^M-∂^M B)},where B is a dynamical scalar field just like in the Stueckelberg field <cit.>, and G(ϕ) is the coupling function of the background scalar field ϕ. With the gauge transformation A_M →A_M+∂_M ξ, B → B+ ξ, the action (<ref>) keeps gauge invariant. Through varying the action S_A, and parameterizing the five-dimensional field A_M as A_M=( A_μ, A_5)=( A_μ+∂_μφ, A_5) like the way in Ref. <cit.>, one can obtain[□^(4)+ e^2A(∂^2_y+2A'∂_y-G)]A_ν = 0 ,∂_y(e^2Aλ)-e^2AGρ = 0 , e^2A□^(4)λ+e^4AG(ρ '-λ) = 0 ,e^2AG□^(4)ρ+∂_y[e^4AG(ρ '-λ)] = 0 ,with the two redefined gauge invariant scalar fields λ=A_5-φ and ρ=B-φ.Then, by decomposing the gauge field as followsA^μ(x,y)=∑_na_n^μ(x)α _n(y),one can reduce Eq. (<ref>) as[∂_y^2+ 2A'∂_y-G]α_n(y)=-e^-2Am^2_nα_n(y),where □^(4)a_n^μ(x)=m^2_n a_n^μ(x).In order to localize the gauge field on the brane, a proper function form of the coupling G(ϕ) should be chosen. The authors in Ref. <cit.> chose the following function:G_c_1,c_2[ϕ(y)]=c_1 A”(y)+c_2 [A'(y)]^2.For the solution of the warp factor A(y)=A_0-blog[cosh (ay)], the mass spectrum of the vector KK modes is continuous since the effective potential of the KK modes approaches to zero when |z|→∞. The vector zero mode α_0(y) turns out to beα_0(y)=k_0 e^ξ A(y),which can be normalizable.Other mechanisms. Furthermore, the geometrical coupling with the gauge field was introducedby Alencar et al. in Ref. <cit.>. Zhao et al. <cit.> assumed that the five-dimensional gauge field has a dynamical mass term, which is proportional to thefive-dimensional scalar curvature. Vaquera-Araujo et al. <cit.> added a brane-gauge coupling into the action. All these mechanisms are effective for the localization of a bulk vector field on the brane. §.§ Kalb-Ramond fieldsIn this subsection, we review the localization of a bulk Kalb-Ramond field on a thick brane. It is known that a Kalb-Ramond field is an antisymmetric tensor field with higher spins proposed in string theory. The Kalb-Ramond field (NS-NS B-field) appears, together with the metric tensor and dilaton, as a set of massless excitations of a closed string. The action for a charged particle moving in an electromagnetic potential is given by -q∫ dx^M A_M. While the action for a string coupled to a Kalb-Ramond field is -∫ dx^M dx^N B_MN. This term in the action implies that the fundamental string of string theory is a source of the NS-NS B-field, much like charged particles are sources of the electromagnetic field. The Kalb-Ramond field is also used to describe the torsion of space-time in Einstein-Cartan theory.The action of a free Kalb-Ramond field isS_KR = -∫ d^5x √(-g)H_MNLH^MNL,where H_MNL=∂_[MB_NL] is the field strength for the Kalb-Ramond field, and H^MNL=g^MOg^NPg^LQH_OPQ. The field equations for the Kalb-Ramond field with the conformal metric (<ref>) read as∂_μ ( √(-g)H^μαβ)+∂_z(√(-g) H^4αβ) =0,∂_μ ( √(-g)H^μ4β) =0.One can make a decompositionB^αβ(x^λ,z)=∑_n b̂^αβ_(n)(x^λ)U_n(z) e^-7A/2,and set the gauge B_α4=0. Then it is not difficult to find that the KK mode U_n(z) satisfies the following Schrödinger-like equation: ( -∂^2_z+ V_KR(z))U_n(z), =m_n^2 U_n(z)where the effective potential V_KR(z) is given byV_KR=1/4(∂_z A)^2-1/2∂_z^2 A.By introducing the orthonormality conditions for the KK modes∫ dz U_m(z)U_n(z)=δ_mn,one can reduce the fundamental action (<ref>) to the following four-dimensional oneS_KR =-∑_n∫ d^4 x √(-g̃) (ĥ^(n)μαβĥ_μαβ^(n)+1/3m_n^2b̂^(n)αβb̂_αβ^(n))with ĥ_μαβ^(n)=∂_[μb̂_αβ]^(n) the four-dimensional field strength tensor. The solution of the Kalb-Ramond zero mode reads asU_0(z)=e^-A,and its normalization condition is∫ dz|U_0(z)|^2= ∫ dze^-2A=∫ dye^-3A.It is clear that for the RS-like solution with e^A=sech(ky), the zero mode of a free bulk Kalb-Ramond field cannot be localized on the brane.Similar to the case of a vector field, one can also introduce a nonminimal coupling between the Kalb-Ramond field and the background scalar fields ϕ, π, ⋯:S_KR = -∫ d^5x √(-g)G(ϕ, π, ⋯) H_MNLH^MNL.The localization and resonances of such KR field have been investigated in Refs. <cit.>. For example, Ref. <cit.> considered G=e^ζπ in a two-field brane model, where the background scalar π is given by π(z)=b A(z). The effective potential V_KR(z) and the zero mode are given byV_KR(z)= (1-√(3b) ζ)^2/4(∂_z A)^2+√(3b) ζ-1/2∂_z^2 A, U_0(z)= e^(√(3b) ζ -1/2)A(z).The localization condition is ζ>1/√(3b) for b≥ 1 or ζ>(2-b)/√(3b) for 0<b< 1 <cit.>. The resonances of the KR field have also been investigated in Refs. <cit.>. Other related work can be found in Refs. <cit.>. We know that the scalar, vector, and Kalb-Ramond fields are the 0-form, 1-form, and 2-form fields, respectively.In fact, there are higher-form fields in a higher-dimensional space-time with dimension larger than four. In four-dimensional space-time, free q-form fields are equivalent to scalar or vector fields by a duality. In higher space-time, they correspond to new types of particles. Some early works for localization and Hodge duality of a q-form field were studied in Refs. <cit.>, where some gauges were chosen to make the localization mechanism simpler. However, these gauge choices only reflect parts of the whole localization informations, including the Hodge duality of the KK modes. Recently, new localization mechanism, Hodge duality, and mass spectrum of a bulk massless q-form field on codimension-one branes (p-branes) were investigated in Refs. <cit.> by using a new KK decomposition. There are two types of KK modes for the bulk q-form field: the q-form and (q-1)-form modes, which cannot be localized on the p-brane simultaneously. The Hodge duality in the bulk naturally becomes two dualities on the brane. Dualities in the bulk and on the brane are shown in Table <ref>. For the detail, see Ref. <cit.>.§.§ Fermion fields In this subsection, we review the localization of bulk Dirac fermion fields on thick branes. In order to localize a fermione, one usually needs to introduce some interactions between the fermion and the background fields. Note that the general covariant equations for fields with arbitrary spin were derived by Y.-S. Duan <cit.>, and the general covariant Dirac equation previously obtained by V. A. Fock and D. D. Ivanenko <cit.> is a special case of the general covariant equations.Here, we consider a general action of the Dirac fermion <cit.>S_1/2=∫ d^5x√(-g)[F_1Ψ̅Γ^MD_MΨ +λ F_2Ψ̅Ψ+ηΨ̅Γ^M(∂_M F_3)γ^5Ψ].Here the functions F_1, F_2, and F_3 are functions of the background scalar fields ϕ^I and/or the Ricci scalar R, and λ and η are the coupling constants. In five-dimensional space-time, a Dirac fermion field is a four-component spinor and the corresponding gamma matrices Γ^M in curved space-time satisfy {Γ^M,Γ^N}=2g^MN. The operator D_M=∂_M+ω_M and the spin connection ω_M is defined asω_M=1/4ω_M^M̅N̅Γ_M̅Γ_N̅withω_M^M̅N̅=1/2E^NM̅(∂_ME^N̅_N-∂_NE^N̅_M)-1/2E^NN̅(∂_ME^M̅_N-∂_NE^M̅_M)-1/2E^PM̅E^QN̅E^R̅_M(∂_P E_QR̅-∂_Q E_PR̅).Here the letters with barrier M̅, N̅,⋯ are the five-dimensional local Lorentz indices and the vielbein E^M_M̅ satisfies E^M_M̅E^N_N̅η^M̅N̅ = g^MN. The relation between the gamma matrices Γ^M and Γ^M̅=(Γ^μ̅,Γ^5̅)=(γ^μ̅,γ^5) is given by Γ^M=E^M_M̅Γ^M̅.For the metric (<ref>), the non-vanishing components of the spin connection (<ref>) areω_μ=1/2(∂_z A) γ_μγ_5+ω̂_μ, where ω̂_μ is derived from the four-dimensional metric g̃_μν(x^λ). The five-dimensional Dirac equation reads as[γ^μ∂_μ +ω̂_μ +γ^5(∂_z+2∂_z A) +ℱ(z)]Ψ=0,whereℱ(z)= λ e^A(z)F_2/F_1+ η∂_zF_3/F_1.We make the following chiral decomposition for the five-dimensional Dirac field ΨΨ(x,z)=e^-2A(z)∑_n[ψ_Ln(x)f_Ln(z)+ψ_Rn(x)f_Rn(z)],where ψ_Ln=-γ^5ψ_Ln and ψ_Rn=γ^5ψ_Rn are the left- and right-chiral components of the Dirac fermion field, respectively, and the four-dimensional Dirac fermion fields satisfy[ γ^μ(∂_μ +ω̂_μ)ψ_Ln(x)=m_nψ_Rn(x),; γ^μ(∂_μ +ω̂_μ)ψ_Rn(x)=m_nψ_Ln(x), ]where m_n is the mass of the four-dimensional fermion fields ψ_Ln(x) and ψ_Rn(x). Substituting Eqs. (<ref>) and (<ref>) into Eq. (<ref>) yields the coupling equations of the KK modes f_Ln,Rn:[ (∂_z -ℱ(z) )f_Ln=+m_n f_Rn,;; (∂_z +ℱ(z) )f_Rn=-m_n f_Ln. ]The above two equations can also be rewritten as the Schrödinger-like equations[-∂_z^2+V_L(z)]f_Ln = m^2_nf_Ln, [-∂_z^2+V_R(z)]f_Rn = m^2_nf_Rn,where the effective potentials are given byV_L,R(z)= ℱ^2(z) ±∂_zℱ(z).The Schrödinger-like equations (<ref>) and (<ref>) can be decomposed by using the supersymmetry quantum mechanics as[ 𝒦^†𝒦f_Ln=m^2_nf_Ln; 𝒦𝒦^†f_Rn=m^2_nf_Rn ]with the operator 𝒦=∂_z-ℱ(z), which insure that the mass square is non-negative, i.e., m_n^2 ≥ 0. The corresponding chiral zero modes can be solved based on Eq. (<ref>) with m_0=0:f_L0,R0∝ e^±∫dzℱ(z).By introducing the following orthonormality conditions for the KK modes f_Ln,Rn∫_-∞^+∞F_1f_Lmf_Lndz =∫_-∞^+∞F_1f_Rmf_Rndz=δ_mn, ∫_-∞^+∞F_1f_Lnf_Rndz=0,one can derive the effective action of the four-dimensional massless and massive Dirac fermions from the five-dimensional Dirac action (<ref>):S_eff=∑_n∫ d^4x√(-ĝ) ψ̅_n[ γ^μ(∂_μ+ω̂_μ)-m_n] ψ_n.The conditions (<ref>) can be used to check whether the fermion KK modes can be localized on the brane.We know that there are two types of fermion localization mechanisms. The first one is the Yukawa coupling (F_1=1, F_3=0) between fermions and the background scalar fields <cit.>, which does work when the background scalar fields are odd functions of the extra dimension. This form of coupling λ F_2(ϕ)Ψ̅Ψ between the kink scalar ϕ and bulk fermions can be regarded as the coupling between a soliton and fermions in Ref. <cit.>. The corresponding effective potentials are (<ref>)V_L,R(z)=(λ e^AF_2)^2±∂_z(λ e^AF_2),and the chiral zero modes readf_L0,R0∝ e^±λ∫dz e^A F_2(ϕ) =e^±λ∫dy F_2(ϕ).We consider the brane models with the solutione^A(y→±∞)→ e^∓ ky and ϕ(y→±∞) →± v.For the simplest Yukawa coupling with F_2=ϕ(y) and strong enough but negative coupling (λ<λ_0≡ -k/v), the left-chiral fermion zero modef_L0(y→±∞) → e^±λ v ysatisfies the normalization condition ∫_-∞^+∞ e^-A(y)|f_L0|^2 dy<∞, and hence can be localized on the brane. It is worth pointing out that the right-chiral fermion zero mode cannot be localized at the same time.In Ref. <cit.>, a “natural” ansatz for the Yukawa term F_2Ψ̅Ψ is proposed, where F_2 inherits its odd nature directly from the geometry shape of the warp factor e^A(z). In order to guarantee the localization of the left-chiral fermion zero mode, the authors taken F_2 as F_2(z)=M∂_ze^-A(z), which is not arbitrariness and is independent of the braneworld model. With this choice, the localization of gravity on the brane implies the localization of spin-1/2 fermions as well. If the background scalar field is an even function of the extra dimension, the Yukawa coupling mechanism will do not work, since the Z_2 reflection symmetry of the effective potentials for the fermion KK modes cannot be ensured <cit.>. In order to solve this problem, a new localization mechanism was presented in Ref. <cit.>. The coupling is given by ηΨ̅Γ^M∂_MF_3(ϕ)γ_5Ψ (F_1=1, F_2=0), which is used to describe the interaction between π-meson and nucleons in quantum field theory and is called as the derivative coupling.For the above two mechanisms, the localization of bulk fermions depends on the coupling between bulk fermions and background scalar fields. For thick brane models without background scalar fields, the previous two mechanisms do not work any more. For such models, one can adopt the coupling between the bulk fermion fields and the scalar curvature R of the background space-time <cit.>. The form of coupling is the same as the derivative coupling ℒ_int=δΨ̅γ_5Γ^M∂_MF_3(R)Ψ <cit.> since the scalar curvature R is an even function of the extra dimension. With the derivative geometrical coupling, the corresponding effective potentials (<ref>) and chiral zero modes becomeV_L,R(z)=(η∂_zF_3)^2±∂_z(η∂_zF_3),andf_L0,R0∝ e^±η∫dz ∂_z F_3=e^±η F_3,where F_3 is a function of the background scalar fields or the scalar curvature. The normalization conditions for the fermion zero modes are∫_-∞^+∞ e^± 2η F_3dz < ∞.It can be seen that one of the left- and right-chiral fermion zero modes can be localized on the brane with some suitable choice of the function F_3(ϕ,R) (see Refs. <cit.> for detail). Here we should note that for a volcano-like effective potential, all the massive KK modes can escape to the extra dimension and the massive fermion KK resonancesdo not have contributions to the effective action (<ref>) in four-dimensional space-time since the integral of the square of a massive KK mode is divergent along the extra dimension. Recently a new localization mechanism <cit.> was proposedby considering the non-minimal coupling between bulk fermions and background scalar fields (see (<ref>)). Obviously, the localization of a bulk fermion on a brane is related to the function F_1 and this function has remarkable impacts on the normalization of the continuous massive KK modes (<ref>). One can see that the continuous massive KK modes may have contributions to the four-dimensional effective fermion action (<ref>) if one chooses a proper function F_1 (see Ref. <cit.>).It is known that the shapes of the effective potential of the left- or right-chiral fermion KK mode can be classified as three types: volcano-like <cit.>, finite- square-well-like <cit.>, and harmonic-potential-like <cit.>. The corresponding spectra of the KK fermions are continuous, partially discrete and partially continuous, and discrete, respectively. For the volcano-like effective potential, all the massive KK modes can escape to the extra dimension, and one might obtain the fermion resonances by using the numerical methods presented in Refs. <cit.>. Inspired by the investigation of Ref. <cit.>, Almeida et al. investigated the issue of localization of a bulk fermion on a brane, and firstly suggested that large peaks in the distribution of the normalized squared wave function |f_L,R(0)|^2 as a function of m would reveal the existence of fermion resonant states <cit.>. However, this method is suitable only for even fermion resonances because f_L,R(0)=0 for any odd wavefunction. In order to find all fermion resonances, Liu et al. introduced the following relative probability <cit.>:P=∫_-|z_b|^|z_b||f_Ln,Rn(z)|^2 dz/∫_z_max^-z_max|f_Ln,Rn(z)|^2 dz,where z_max=10|z_b| and the parameter z_b could be chosen as the coordinate that corresponds to the maximum of the effective potential V_L or V_R, which is also approximately the width of the brane. Here |f_Ln,Rn(z)|^2 can be explained as the probability density at z.If the relative probability (<ref>) has a peak around m = m_n and this peak has a full width at half maximum, then the KK mode with mass m_n is a fermion resonant mode. The total number of the peaks that have full width at half maximum is the number of the resonant modes. For the case of the symmetric potentials, the wave functions f_Ln,Rn(z) are either even or odd. Hence, we can use the following boundary conditions to solve the differential equations(<ref>) and (<ref>) numerically <cit.>:f_Ln,Rn(0) = 0, f'_Ln,Rn(0)=1, for odd KK modes,f_Ln,Rn(0) = 1, f'_Ln,Rn(0)=0, for even KK modes.One can also obtain the corresponding lifetime τ of a fermion resonance by the width (Γ) at half maximum of the peak with τ=1/Γ <cit.>. Fermion resonances can also be obtained by using the transfer matrix method <cit.>. The localization and resonances of a bulk fermion have been investigated based on the Yukawa coupling mechanism <cit.> and the derivative coupling mechanism <cit.>. Here we only list the results of the probabilities P_L,R and the resonances of the left- and right-chiral KK fermions for the coupling with F_1=1, F_2=0, and F_3=ϕ^2ln[χ^2+ρ^2] in a multi-scalar-field flat thick brane model in Figs. <ref> and <ref>, respectively <cit.>. Before closing this subsection, we give some comments. Firstly, the mass spectra and lifetimes of the fermion resonances for both the left- and right-chiral fermions are the same <cit.>. Secondly, the derivative coupling mechanism <cit.> can also be used for the branes generated by odd scalar fields <cit.>. Thirdly, the localized fermion zero mode is always chiral.Besides the above mentioned fields, some other fields such as Gravitino Fields <cit.>, Elko Spinors <cit.>, and new fermions <cit.> were also investigated in the content of extra dimensions and braneworlds.§ CONCLUSIONIn this review, we have given a brief introduction on several important extra dimension models and the five-dimensional thick brane models in extended theories of gravity. After introducing the KK theory, domain wall model, large extra dimension model, and warped extra dimension models, we listed some thick brane solutions in extended theories of gravity, and reviewed localization of bulk matters on thick branes.These extra dimension and/or braneworld models have been investigated, developed, or cited in thousands of literatures. But the study of extra dimensions and braneworld is far more than that. For other noteworthy extra dimension theories and related topics (including string theory, AdS/CFT correspondence, universal extra dimensions, multiple time dimensions, etc), interested readers can refer to the review papers or books mentioned earlier in this review.In recent years, the study of extra dimensions has evolved from the early pure theory to the experimental stage <cit.>. Although there is no direct evidence that there are extra spatial dimensions, the idea of extra dimensions and braneworld could help us to understand the new physical phenomena and provide a candidate for explaining the past and new physical problems, which is one of the major motivations for people to study theories of extra dimensions. Of course, there are still some problems that have not been solved. Further researches (mainly for thick brane models) in the future may include but not limited to the following directions: * Find analytic solutions of thick brane in new theories and study linear fluctuations of the solutions.* Localization of matter fields and gravitational field in new theories.* Intersecting brane models <cit.> and other new models.* Physical effects of new particles in thick brane models in high-energy accelerators.* Applications of braneworld models in cosmology (including neutrinos, black holes, inflation, dark energy and dark matter, and gravitational waves, etc) <cit.>.* Evolution and formation of braneworlds <cit.>.* Localized black-hole solutions in braneworld models <cit.>. Finally, we note that this short review cannot introduce the relevant researches comprehensively and we try our best to list the most relevant papers. § ACKNOWLEDGEMENTS We thanks C. Adam, C. Almeida, I. Antoniadis, N. Barbosa-Cendejas, D. Bazeia, M. Cvetic, V. Dzhunushaliev, A. 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"authors": [
"Yu-Xiao Liu"
],
"categories": [
"hep-th",
"gr-qc"
],
"primary_category": "hep-th",
"published": "20170726170359",
"title": "Introduction to Extra Dimensions and Thick Braneworlds"
} |
headings Hamiltonian MechanicsUniversitat de Barcelona University of Groningen Consorci Sanitari de Terrassa Serious Games Application for Memory Training Using Egocentric Images Gabriel Oliveira-Barra1 Marc Bolaños1 Estefania Talavera1,2 Adrián Dueñas1 Olga Gelonch3 Maite Garolera3 December 30, 2023 ============================================================================================================= Mild cognitive impairment is the early stage of several neurodegenerative diseases, such as Alzheimer's. In this work, we address the use of lifelogging as a tool to obtain pictures from a patient's daily life from an egocentric point of view. We propose to use them in combination with serious games as a way to provide a non-pharmacological treatment to improve their quality of life. To do so, we introduce a novel computer vision technique that classifies rich and non rich egocentric images and uses them in serious games. We present results over a dataset composed by 10,997 images, recorded by 7 different users, achieving 79% of F1-score. Our model presents the first method used for automatic egocentric images selection applicable to serious games.§ INTRODUCTION r0.4 < g r a p h i c s > Person using the Narrative Clip camera.Dementia can result from different causes, the most common being Alzheimer’s disease (AD) <cit.>, and it is often preceded by a pre-dementia stage, known as Mild Cognitive Impairment (MCI), characterized by a cognitive decline greater than expected by an individual's age, but which does not interfere notably with their daily life activities <cit.>. Currently, medical specialists design and apply special activities that could serve as a treatment tool for cognitive capabilities enhancement. Even though, these activities are not specially designed for the patients, which limits their engagement in some cases. A possible alternative to the application of generic exercises would be the use of personalized images of the daily life of the patients acquired by lifelogging devices. Lifelogging consists of a user continuously recording their everyday experiences, typically via wearable sensors including accelerometers and cameras, among others. When the visual signal is the only one recorded, typically by a wearable camera, it is referred to as visual lifelogging <cit.>. This is a trend that is rapidly increasing thanks to advances in wearable technologies over recent years. Nowadays, wearable cameras are very small devices that can be worn all-day long and automatically record the everyday activities of the wearer in a passive fashion, from a first-person point of view. As an example, Fig. <ref> shows pictures taken by a person wearing such a camera.Recent studies have described wearable cameras or lifelogging technologies as useful devices for memory support for people with episodic memory impairment, such as the one present in MCI <cit.>. The design of new technologies to be applied on this field requires to take into account people capabilities, limitations, needs and the acceptance of the wearable devices, since it can directly affect the treatment. So far, some studies have deeply focus into the factors associated to the use of these devices <cit.>.§.§.§ Lifelogging and privacy:In terms of privacy, in 2011, the European Union agency ENISA evaluated the risks, threats and vulnerabilities of lifelogging applications with respect to central topics as privacy and trust issues. In their final report, they highlighted that lifelogging itself is still in its infancy but nevertheless will play an important role in the near future <cit.>. Therefore, they recommended further and extensive research in order to influence its evolution to “be better prepared to mitigate the risks and maximize the benefits of these technologies”. In addition, other researchers have also evaluated the possible ethical risks involved on using lifelogging devices on medical studies <cit.>. §.§.§ Serious games for MCI: Serious games (also known as games with a purpose) are digital applications specialized for purposes other than simply entertaining, such as informing, educating or enhancing physical and cognitive functions. Nowadays they are widely recognized as promising non-pharmacological tools to help assess and evaluate functional impairments of patients, as well as to aid with their treatment, stimulation, and rehabilitation <cit.>. Boosted by the publication of a Nature letter showing that video game training can enhance cognitive control in older adults <cit.>, there is now a growing interest in developing serious games specifically adapted to people with AD and related disorders. Preliminary evidence shows that serious games can successfully be employed to train physical and cognitive abilities in people with AD, MCI, and related disorders <cit.>. <cit.> performed a literature review of the experimental studies conducted to date on the use of serious games in neurodegenerative disorders and <cit.> studied recommendations for the use of serious games in people with AD and related disorders, reporting positive effects on several health-related capabilities of MCI patients such as voluntary motor control, cognitive functions like attention and memory or social and emotional functions. For instance they can improve their mood and increase their sociability, as well as reduce their depression. §.§.§ Our contribution:Different studies have proven the benefits of directly stimulating the working memory. Our contribution in this paper consists in using as stimuli the autobiographical images of the MCI patients that was acquired by the wearable cameras. By doing this, we intend to accomplish the goal of enhancing their motivation and at the same time treat them in a more functional and multimodal manner <cit.>. The application, which will allow the user to exercise either at the sanitary center or at home, will be composed by serious games where the patient has to observe a series of images and interact with them. Although the stimuli provided by egocentric images can be of greater importance than non-personal images, it is important to note both, that egocentric images are captured in an uncontrolled environment, and that wearable cameras usually have free motion that might cause most images to be blurry, dark or empty of semantic content. Considering this important limitations together with the limited capabilities of MCI patients, we propose the development of an egocentric rich images detection system intended to select only images with semantic and relevant content.Our hypothesis is that, by using personal daily life rich images, the motivation of the patient will increase, and as a consequence, the health-related benefits provided by the treatment.This paper is organized as follows. We describe the proposed serious game and model for rich images selection in Section <ref> and Section <ref>, respectively. In Section <ref>, we describes the experimental setup and show quantitative and qualitative evaluation. Finally, Section <ref> draws conclusions and outlines future works.§ PROPOSED SERIOUS GAME: "POSITION RECALL" MCI patients experiment problems in their working memory <cit.>, herefore, it is of high importance to do exercises for stimulating it. All this under the neuroplasticity paradigm, which has proven that it is possible to modify the brain capabilities and the hypothesis of "use it or lose it", which are the basis of the studies related to the cognitive stimulation of elderly people <cit.>. Thus, in this work, we introduce a serious game that we name as "Position Recall", which was designed by neuropsychologist of Consorci Sanitari de Terrassa for improving the working memory.The mechanics of this game follow this scheme: The first screen explains to the patient the instructions of the game and in the second the patient is informed that, before starting the game, there will be some practice examples that will serve to understand its logic. To start, the patient must select his preferred level of difficulty (Level 1, 2 or 3).* Level 1 shows 3 images of the patients' day during 8 seconds and they are asked to remember their positions. Immediately after they disappear, a single "target" image is shown and they are asked to select in what position it was placed. After some trials the number of images displayed are increased to 4 and then to 5. * Level 2 follows the same procedure as the 1st level, but the timespan between the moment where the images disappear and the target image is shown is increased. During this timespan, called latency time, a black screen is shown. * Level 3 follows the same procedure as the 2nd level, but now a distractor image is shown instead of a black screen during the latency time. The distractor image is also an image belonging to the patients' day. The reward system of the game are pointsthat are given after each level, and are calculated as 100 x number of correct answers. There are 10 trials per level translating into a maximum of 1000 points per level and maximum of 3000 points per game. Figures <ref> and <ref> show the mechanics of the developed game. The images to be shown during the serious games should be significant for the patient. We propose to use images that represent past moments of the user's life, i.e. from the egocentric photostreams recorded by the patient. On the following section, we describe the proposed model for rich images selection.§ WHAT DID I SEE? RICH IMAGES DETECTION The main factor for providing a meaningful image selection algorithm is the fact that the proposed serious games intend to work on cognitive and sentiment enhancement. Considering the free-motion and non-intentionality of the pictures taken by wearable cameras <cit.>, it is very important to provide a robust method for images selection.Two of the most important and basic factors that determine the memorability of an image <cit.> can be described as 1) the appearance of human faces, and 2) the appearance of characteristic and recognizable objects. In this paper, we focus on satisfying the second criterion by proposing an algorithm based on computer vision. Our proposal consists in a rich images detection algorithm, which intends to detect images with a high number of objects and variability and at the same time avoids images with low semantical content, understanding as rich any image that is neither blur, nor dark and that contains clearly visible non-occluded objects. In Fig. <ref> we show the general pipeline of our proposal.Our algorithm for rich images detection (consists in 1) objects detection: where the neural network named YOLO9000 <cit.> is applied in order to detect any existent object in the images and their associated confidences c_i. 2) the image is divided in a pyramidal structure of cells, 3) a set of richness-related features are extracted, 4) the extracted features are normalized and 5) a Random Forest Classifier (RFC) <cit.> is trained to distinguish the differences between rich or non-rich images. When extracting features, the image is divided in a pyramidal structure of cells with different sizes at each level. The set of extracted features are:* Numbers of objects the cell contains.* Variance of color in the cell.* Does the cell contain people?* Object Scale. Real number between 0 and 1.* Object Class. Class identifier that varies between 1 and 9418.* Object Confidence c_i.where all features are repeated for each cell and the last three kinds of features are repeated for each object appearing in the cells. The image cell divisions applied are 1x1, 2x2 and 3x3, the maximum of objects selected per cell are 5, 3 and 2, respectively and all objects are sorted by their confidence c_i before selection. If the number of objects is less than the maximum number are found, the feature value in that specific position is set to 0.The pyramidal division of the images helps us consider smaller objects at higher levels (more cells) and bigger objects at lower levels (less cells). Thus, both small and big objects will be considered for the final prediction.In order to define the feature "Does the cell contain people?" we manually selected a set of person-related objects detected by the employed object detection method.The concepts representing people that we selected are "person", "worker", "workman", "employee", "consumer", "groom" and "bride".§ RESULTSThis section describes the results obtained in a quantitative and qualitative form. We compare the results obtained by variations of the proposed method on a self-made dataset of rich images.§.§.§ Dataset: The dataset used for evaluating our model was acquired by the wearable camera Narrative Clip 2[<www.getnarrative.com>], which takes a picture every thirty seconds automatically. The camera was worn during 15 days by 7 different people. Considering that on average the camera takes 1,500 images per day, our dataset consists of 10,997 photographs. The resulting data was labeled by neuropsychologist experts on MCI cognition following the criteria that any rich image has to be 1) properly illuminated, 2) not blurry and 3) contain one or more objects that are not occluded. After this manual selection the acquired images where split in 6,399 rich images and 4,598 non-rich images.In Fig. <ref> we can see some examples of egocentric rich images and in Fig. <ref> non-rich images. We observe that rich images show people or recognizable places. However, non-rich images are meaningless or dark images (that can hardly be seen), including pictures of the sky, ceilings or floor.The resulting data was divided in training, validation, and test. Considering the pictures taken during the same day can be very similar, we proceeded to randomly separate the different days into the three different sets. First, the training set consists of 60% of the days, in this case 9. Second, 20% of the days, in this case 3, were defined as the validation set. Finally, the remaining 20% was used for the test set. §.§.§ Evaluation Metrics:In order to evaluate the different results and compare them to get the best one, we make use of the F1-score (or F-measure) metric:F1 = 2*1/1/precision + 1/recall = 2*precision * recall/precision + recallwhere precision is the quotient between the number of True Positives objects and the number of predicted positive elements; and recall is the quotient between the number of True Positives objects and the number of real positive elements.§.§.§ Quantitative Results: Currently, there are no previous works addressing the challenge we introduce in this work. Thus, in order to compare the performance of our proposed model, we have defined and compared several variations to our main pipeline (see results in Table <ref>). As an alternative to our proposed approach (1), we tested an alternative feature vector representation by means of using the (2) Word2Vec word embedding <cit.>. This word characterization is a 300-dimensional vector representation created by Google that represents words in space depending on their semantic meaning (i.e. words with similar definitions will be represented close in space). The Word2Vec representation was used in two ways. On the one hand was used for defining the set of concepts related to "person" in the feature described as "Does the cell contain people?". Thus, we computed the similarity between the word "person" and any other concept detected in the image by the object detection and the maximum similarity achieved was used as an alternative to a 0/1 representation. On the other hand, the feature described as "Object Class" was replaced by the 300-dimensions Word2Vec representation.In the test setting (3) we additionally applied a PCA dimensionality reduction to the Word2Vec representation. Finally, in (4) we used a Support Vector Machine (SVM) classifier instead of a Random Forest Classifier. We applied a Grid Search on the variables C and gamma for parameter selection over the validation set.In conclusion we can see that using an RFC classifier (1) obtains better results than SVM (4) and at the same time none of the Word2Vec representations (2) and (3) helped improving the base results. §.§.§ Qualitative Results: Examples of the selected images by the proposed algorithm are shown in Fig. <ref>. On one hand, we can observe that rich images (left) are clearer, without shadows and with people or focused objects, which allows the user to infer what is happening in the scene.On the other hand, non-rich images (right) are discarded since they are not illustrative and make difficult the scene interpretation. Images selected by the proposed model are rich in information and memory trigger. We can foresee that the proposed model cannot only be used for serious games images selection, but also as a tool for images selection for autobiographical memories creation.§ CONCLUSIONS In this work, we have introduced a novel type of wearable computing application, aimingto provide non pharmacological treatment for MCI patients and to improve their life quality. We discussed lifelogging pictures obtained from wearable cameras combined with serious games as a channel for personalized treatments. We also introduced and tested a novel computer vision technique to classify rich and non rich images obtained from first-person point of view. We obtain 79% F1-score, promising results that will be further studied. As future work, we will implement more serious games to be included in the application tool. Specialists will use it for MCI patients, aiming to prove the the memory reinforcement hypothesis introduced in this work, as well as the motivation experienced by the subjects increase when using personalized rich images and serious games. Furthermore, in <cit.>, positiveness from egocentric images was addressed.Moreover, we will go deeper on the analysis of users acceptance over the proposed technology, their willingness to use it, and the factors that determine their acceptance toward it. Further improvements of the methodology will be developed in order to obtain more accurate results. § ACKNOWLEDGEMENTSThis work was partially founded by Ministerio de Ciencia e Innovación of the Gobierno de España, through the research project TIN2015-66951-C2. SGR 1219, CERCA, ICREA Academia 2014, Grant 20141510 (Marató TV3) and Grant FPU15/01347. The funders had no role in the study design, data collection, analysis, and preparation of the manuscript. The authors gratefully acknowledge the support of NVIDIA Corporation with the donation of the Titan Xp GPU used for this research. splncs03 | http://arxiv.org/abs/1707.08821v1 | {
"authors": [
"Gabriel Oliveira-Barra",
"Marc Bolaños",
"Estefania Talavera",
"Adrián Dueñas",
"Olga Gelonch",
"Maite Garolera"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170727113626",
"title": "Serious Games Application for Memory Training Using Egocentric Images"
} |
http://arxiv.org/abs/1707.08895v2 | {
"authors": [
"Camilo Ulloa",
"Roberto E. Troncoso",
"Scott A. Bender",
"R. A. Duine",
"A. S. Nunez"
],
"categories": [
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"primary_category": "cond-mat.mes-hall",
"published": "20170727144654",
"title": "Piezospintronic effect in honeycomb antiferromagnets"
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|
Instituto de Física, Universidade Federal Fluminense, Av. Gal. Milton Tavares de Souza s/n, Gragoatá, 24210-346 Niterói,Rio de Janeiro, Brazil. Department of Physics, University of Guelph, Guelph, Ontario, N1G 2W1, CanadaAn extended test body moving in a curved spacetime does not typically follow a geodesic, because of forces that arise from couplings between its multipole moments and the ambient curvature. An illustration of this fact was provided by Wisdom, who showed that the motion of a quasi-rigid body undergoing cyclic changes of shape in a curved spacetime deviates,in general, from a geodesic. Wisdom's analysis, however, was recently challenged on the grounds that the body's motion should be described by the Mathisson-Papapetrou-Dixon equations, and that these predict geodesic motion for the kind of body considered by Wisdom. We attempt to shed some light on this matter by examining the motion of an internally-moving tripod in Schwarzschild spacetime, as viewed by a Fermi observer moving on a timelike geodesic. We find that the description of the motion depends sensitively on a choice of cycle for the tripod's internal motions, but also on a choice of“center of mass” for the tripod; a sensible (though not unique) prescription for this “center of mass” produces a motion that conforms with Wisdom's prediction: the tripod drifts away from the observer, even when they are given identical initial conditions. We suggest pathways of reconciliation between this conclusion and the null result that apparently follows from the Mathisson-Papapetrou-Dixon equations of motion.Swimming in spacetime: the view from a Fermi observer Eric Poisson December 30, 2023 =====================================================§ INTRODUCTION [Disclaimer: In this work we employ a constrained Lagrangian formalismin order to model a body undergoing cyclic changes of shape in a curved spacetime. However, this constrained Lagrangian formalism is notnecessarily consistent with motions generated by internal forces alone, and may exhibit undesirable features even in special relativity. For instance, from the discussion in Sec. <ref> it can be seen that if a body initially at rest with respect to some inertialobserver starts a cyclic motion, it may acquire a nonzero velocitywith respect to the same observer for some choices of cycles. It appears, therefore, that additional restrictions should be imposed on the cycle for it to be consistent with the operation of internalforces alone, otherwise we might be led to interpret as “swimming”something that is due to an unreasonable power of the agentresponsible for keeping the body's motion.Considerations along these lines, raised largely through a discussion withAmos Ori, cast some doubts on the applicability of the constrained Lagrangian formalism to study swimming in curved spacetimes.Nonetheless, we still believe the calculations presented here are technically correct, and might be instructive for future research on related problems.] It is a well-known fact that in general relativity, an extendedtest body may not move on a timelike geodesic, because of forces that typically arise from couplings between its multipole moments and theambient curvature. Wisdom provided a vivid illustration of this fact in a 2003 paper<cit.>, where he demonstrated that a testbody undergoingcyclic changes of shape — a swimmer — does notmove ona geodesic. Wisdom gave a concrete example of this effectby calculating the displacement of a tripod relative to a geodesicin Schwarzschild spacetime; he found the displacement to scale withthe spacetime curvature and the extent of the tripod's internal motions. The effect was shown to be small and uninterestingfrom a practical point of view, but it is nevertheless interesting as a matter of principle. The validity of Wisdom's analysis, however, was recently called toquestion by Silva, Matsas, and Vanzella <cit.>. The objection raised in their workrelies on the observation that the swimmer's motion ought to be governed by the Mathisson-Papapetrou-Dixon (MPD) equations <cit.>,Dp_α/dτ = 1/2 S^μν u^λ R_μνλα, DS_αβ/dτ = 2 p_[α u_β],where p_α is the body's momentum vector, S_αβ its spin tensor, u^α the tangent to the world line,R_αβγδ the Riemann tensor, and D/dτ indicates covariant differentiation with respect to proper time τ. These equations are meant to describe the motion of a generic extended body, in a pole-dipole approximation that neglects the influence of higher multipole moments. The MPD equations indicate that the force acting on an extended body should scale withthe curvature, as was observed in Ref. <cit.>, but after a deeper examination, the authors of Ref. <cit.> concludethat they are incompatible with the swimming motion revealed by Wisdom;according to Eq. (<ref>), the tripod should move on a geodesic. The authors attempt to rescue the phenomenon by suggestingthat the motion might scale with the covariant derivative of the Riemann tensor, instead of the Riemann tensor itself, but this suggestion is incompatible with Wisdom's findings. We take this opportunity to revisit Wisdom's original analysis, and tocalculate in a novel way the motion of an internally-moving tripod in Schwarzschild spacetime. Instead of describing the motion in the static frame of the Schwarzschild spacetime, as Wisdom did, we prefer to exploit a reference frame attached to a freely moving observer who follows a timelike geodesic. Observer and tripodare given identical initial positions and velocities in thespacetime, and the tripod's motion is measured relative to theobserver's rest frame. Our implementation of this idea relies onFermi normal coordinates, which allow us to write the metric in aconvenient locally-flat form. The Riemann tensor appears explicitly in the metric, and this clarifies its influence on the tripod's motion. And because the Fermi coordinates are constructed from geodesic segments that are everywhere orthogonal to the observer's world line, they come with a transparent geometrical meaning that helps clarify the description of the motion. We begin in Sec. <ref> with a review of Fermi normal coordinates attached to a radial, timelike geodesic in a static, spherically symmetric spacetime. In Sec. <ref> we formulate our precise model for the tripod, specify the cycle of itsinternal motions, construct its Lagrangian, and derive the equations of motion. A delicate matter that presents itself is the designation of an appropriate “center of mass” (CM) for the tripod, which we use to track the motion of the tripod as a whole. Our relativistic definition is based on the requirements that there should be no swimming in Minkowski spacetime, and thatthe CM should always be contained within the body for any cycle of internal motions. These requirements determine the CM position up to a constant shift, and completing the prescription with aspecification of this shift turns out to be an important aspect of ouranalysis, with a significant bearing on our conclusions.We integrate the equations of motion in Sec. <ref>.We begin with a discussion of motion in de Sitter spacetime, and show that the shift freedom in the CM definition can be exploited to remove a drift from geodesic motion that would otherwise be present. With the CM position fully specified by this prescription, we place the tripod in the Schwarzschild spacetime and show that the coupling between its internal motions and the spacetime curvature prevents it from following a geodesic. The tripod's world line is not a geodesic, but we find that it asymptotes to a geodesic when the tripod is allowed toapproach the central singularity of the Schwarzschild metric; the asymptotic geodesic is distinct from the reference geodesic of the freely-falling observer.In Sec. <ref> we propose two paths of reconciliation withthe MPD equations. In the first, we suggest that the MPD equations may not apply to the tripod, because their derivation relies crucially onenergy-momentum conservation. The tripod, on the other hand, does not conserve energy and momentum, because external agents are required to keep the tripod on its cycle, and these can supply the missing energy and momentum. We provide evidence to support this suggestion by generalizing the MPD equations to a constrained mechanical system, and showing that the resulting equations do differ from Eqs. (<ref>). The second path of reconciliation is a suggestion that the apparent incompatibility between Wisdom's swimming and the MPD equations might not be real, but the result of amisinterpretation of the equations. The main point is that Eqs. (<ref>) are empty of content until a relation between p_α and u_α is specified through the selection of a suitable “center of mass” for the extended body. Our considerations in Sec. <ref> remind us that this can be a very delicate matter. A simple extended body might motivate the introduction of a simple auxiliary condition such as p_α S^αβ = 0, which yields an explicit relation between p_α and u_α. But a tripod undergoing cyclical internal motions is not simple, and it is likely that the relation between p_α and u_α is far more complicated in this case, involving the details of the tripod's design. And until this relation is identified, the predictions of Eq. (<ref>) must remain ambiguous.§ FERMI COORDINATESThe motion of an extended body in any spacetime can be described from the point of view of an observer moving on a reference timelike geodesic γ, which is imagined to stay within the body's neighborhood. The coordinates that best capture this point of view are the Fermi normal coordinates (t, x^a), in which tmeasures proper time on γ, and the spatial coordinates x^a measure proper distance on spacelike geodesics that are orthogonal to γ. In relativistic units with c=1, used throughout the paper, the metric of the spacetime is given by g_00 = -1 + h_00 = -1 - R_0a0b(t)x^a x^b,g_0a = h_0a = -2/3 R_0bac(t)x^b x^c,g_ab = δ_ab + h_ab = δ_ab- 1/3 R_acbd(t)x^c x^d, up to cubic terms in the spatial coordinates; the components of the Riemann tensor are evaluated on γ and expressed as functions of proper time t. The metric is flat on γ, with avanishing connection, and the Fermi coordinates define the restframe of the observer moving on γ. (An introduction to Fermicoordinates can be found in Ref. <cit.>. Analternative is Sec. 9 of Ref. <cit.>.)We consider a static, spherically symmetric spacetime with metricds^2 = -fdT^2 + f^-1dR^2 + R^2 (dΘ^2+ sin^2ΘdΦ^2), in which f is a function of R. For concreteness below we shalltake f = 1 - 2GM/R, so that the metric is that of the Schwarzschild spacetime, which describes the geometry outside a spherical body of mass M. We will also consider the case of de Sitterspacetime, for which f = 1 - k^2 R^2, with k^2 proportional to the spacetime curvature. The reference timelike geodesic γ is chosen to be a radial world line with tangent vector u^α = [ u^T, u^R, u^Θ, u^Φ]= [ E/f, -√(E^2-f), 0, 0 ], where E, the dimensionless energy parameter, is a constant of the motion. The radial component u^R of the tangent vector is negative, which indicates that R decreases along γ; in the context of the Schwarzschild spacetime, the reference observer falls toward the central object of mass M.The vectors e^α_a are defined to be mutually orthogonal, orthogonal to u^α, and parallel transported on γ. It is easy to show that the sete^α_1= [ 0, 0, 0, -1/(Rsinθ) ],e^α_2= [ 0, 0, 1/R, 0 ],e^α_3= [ -√(E^2-f)/f, E, 0, 0 ] satisfies these requirements. The vector e^α_1 points in the direction of decreasing Φ, e^α_2 in the direction of increasing Θ, and e^α_3 is mostly aligned with the direction of increasing R. The triad forms a right-handed system, and the third direction is identified with the “up” direction. The components of the Riemann tensor in Fermi coordinates are equal to its projections in the tetrad formed by u^α and e^α_a. We have R_0a0b = R_μανβu^μ e^α_a u^ν e^β_b, R_0bac = R_μβαγu^μ e^β_b e^α_a e^γ_c, R_acbd = R_αγβδe^α_a e^γ_c e^β_b e^δ_d, and calculation yields the nonvanishing components R_0101 = R_0202 = - R_1313 = -R_2323 = A, R_0303 = B, R_1212 = C, where A := f'/2R, B:= 1/2 f”, C := 1-f/R^2, with a prime indicating differentiation with respect to R. Making the substitutions in Eq. (<ref>) produces the nonvanishing components of the metric perturbation h_αβ.In particular, h_0a = 0 for our choice of γ.§ TRIPOD MODELWe wish to determine the motion of a swimmer in the spacetimedescribed in Sec. <ref>. We adopt Wisdom's model<cit.>, in which the swimmer is given the shape of a tripod; refer to Fig. 3 of his paper. The tripod consists of a “head” of mass m_0 and three “feet” of equal mass m_1. The feet are attached to the head with massless struts. Each strut has a length ℓ(t) and makes an angle α(t) with the axis of symmetry.We construct the tripod's Lagrangian by adding the individual Lagrangians of the head and feet and incorporating the constraints enforced by the struts; we neglect the stresses in the struts.§.§ Newtonian descriptionWe begin with a Newtonian description of the tripod, in the absence of gravity. In an inertial frame (x, y, z), the coordinates of the tripod's head are denoted r_0 = (x_0, y_0, z_0). The coordinates of each foot are given by r_1 - r_0= ( √(3)/2ℓsinα,-1/2ℓsinα, -ℓcosα),r_2 - r_0= (0, ℓsinα, -ℓcosα),r_3 - r_0= ( -√(3)/2ℓsinα,-1/2ℓsinα, -ℓcosα). We adopt the position of the tripod's center of mass (CM), given by x = x_0,y = y_0, z = z_0 - 3m_1/m_0 + 3m_1ℓcosα, as generalized coordinates. A simple calculation then reveals that up to an irrelevant function of time, the tripod's LagrangianL := L_0 + L_1 + L_2 + L_3, withL_i = 1/2 m_i (ẋ_i^2 + ẏ_i^2 + ż_i^2),is given by L̅ := L/m_0 + 3m_1 = 1/2 (ẋ^2 + ẏ^2+ ż^2), in which an overdot indicates differentiation with respect to t.This Lagrangian is identical to that of a free particle, and we conclude that the tripod's CM will stay at rest if it begins at rest. There is no swimming in this Newtonian description. §.§ Relativistic description: flat spacetimeWe continue to ignore gravity, and consider the tripod's dynamics in special relativity. We actually consider an approximate description in which all speeds are assumed to be small compared to the speed of light, so that only the leading relativistic correction is incorporated in each particle's Lagrangian,L_i = 1/2 m_i v_i^2 + 1/8 m_i v_i^4 with v_i^2 = ẋ_i^2 + ẏ_i^2 + ż_i^2. We work in a Lorentz frame (t, x, y, z) and continue to relate the coordinates of the feet to those of the head by Eq. (<ref>). We make a relativistic adjustment to the CM variables, so that they are now given by x = x_0,y = y_0, z = z_0 - 3m_1/m_0 + 3m_1ℓcosα - δ z, where δ z shall be determined below.Making the substitutions in the tripod's Lagrangian, we observe that it has the structure L̅ = 1/2 (ẋ^2 + ẏ^2 + ż^2) + 1/8 (ẋ^2 + ẏ^2 + ż^2)^2 + a_1(t) ẋ^2 + a_2(t) ẏ^2 + a_3(t) ż^2+ a_4(t) ż, where each a_i depends on the time derivative ofℓcosα and ℓsinα; the function a_4(t) also implicates δż. Our prescription[There would be no need for such a prescription if we had access to a complete energy-momentum tensor for the tripod. The CM would then be defined in terms of this tensor. But we do not have such an object, because the external agents responsible for keeping the tripod on its cycle of internal motions are not explicitly accounted for in the model. There is therefore no energy-momentum tensor, no statement of energy-momentum conservation, and no definition of a CM.]for the CM adjustment δ z is based on the requirements that (i) the CM should stay within the body for any cycle of internal motions, and (ii) the CM should move uniformly on a straight path; in particular, the CM should stay at rest if it begins at rest in the adopted Lorentz frame. Now, the form of the Lagrangian in Eq. (<ref>) implies thatp_z := ∂L̅/∂ż = ż(1 + ⋯) + a_4(t), where the ellipsis represents relativistic corrections;p_z is a constant of the motion by virtue of the Euler-Lagrangeequation. It follows that p_z = a_4(0) when we impose the initial condition ż(0) = 0, taking the CM to be initially at rest. At later times we have that a_4(0) = ż(1 + ⋯) + a_4(t), and we find that ż≠ 0 unlessa_4(t) = a_4(0). In other words, a CM initially at rest will be moving at later times, in violation of our second requirement, unless we demand that a_4(t) be a constant. Because this function implicates δż, we find that the conditiona_4(t) = a_4(0) implies δż = 3 m_0 m_1/2(m_0 + 3m_1)^2{m_0-3m_1/m_0 + 3m_1[ d/dt(ℓcosα) ]^2+ [ d/dt(ℓsinα) ]^2 }d/dt (ℓcosα) + k, where k := a_4(0).To determine k we appeal to our first requirement. We observethat for a given choice of cyclic functions ℓ(t) and α(t), the right-hand side of Eq. (<ref>) will be a periodic function with (typically) a nonzero average, giving rise to a δ z that features periodic oscillations superposed to a linear growth. To kill the growth and ensure that the CM does not drift away from the body, we choose k in such a way that the average of δż vanishes. In this way, our requirements determine the CM completely, except for the remaining freedom to choose the initial condition δ z(0) when integrating Eq. (<ref>). With this prescription, there is no swimming in flat spacetime.§.§ Relativistic description: curved spacetimeWe next incorporate gravity by placing the tripod in the spacetime described in Sec. <ref>.The coordinates (x, y, z) now refer to the Fermi frame attached to a radial geodesic of the spacetime described by Eq. (<ref>), and t is proper timemeasured by an observer moving on this geodesic. A point mass m moving freely in the spacetime is described by the word line x^α = r^α(t), with r^0 = t. The coordinate velocities are v^α = dr^α/dt, with v^0 = 1. Taking into account that h_0a = 0, the particle's Lagrangian is L = -m ( -g_αβ v^α v^β)^1/2= -m ( 1 - v^2 - h_00 - h_ab v^a v^b )^1/2, in which v^2 := δ_ab v^a v^b. As we did previously, we assume that v ≪ 1 and expand L in powers of v^2, taking h_αβ to be of order v^2 and keeping L linear in the curvature (that is, neglecting terms quadratic in h_00). If we also discard the irrelevant constant term -m, the Lagrangian becomesL/m = 1/2 v^2 + 1/2 h_00 + 1/8 v^4 + 1/4 h_00 v^2 + 1/2 h_ab v^a v^b. We recognize the Newtonian kinetic energy 1/2 v^2 and its relativistic correction 1/8 v^4, the Newtonian potential energy 1/2 h_00, and the remaining terms provide additional relativistic corrections to the Lagrangian. Substitution of Eqs. (<ref>) and (<ref>) into Eq. (<ref>) gives the explicit expressionL/m = 1/2 v^2 - 1/2 A ( x^2 +y^2)- 1/2 Bz^2 + 1/8 v^4 - 1/12 A [ ( x^2+ y^2)(3 ẋ^2 + 3 ẏ^2+ ż^2 ) - 2z^2 (ẋ^2 + ẏ^2)+ 4(xẋ +yẏ )zż]- 1/4 Bz^2v^2- 1/6 C (y^2 ẋ^2 - 2xy ẋẏ+x^2 ẏ^2 ), where ( x, y, z) are the components of the spatial vector r^a, (ẋ, ẏ, ż) those of v^a, andv^2 := ẋ^2 + ẏ^2 + ż^2. To construct the tripod's Lagrangian we apply Eq. (<ref>) to the head and feet, and incorporate the constraints of Eq. (<ref>).[Because the coordinates are constructed from geodesic segments originating on γ, the struts responsible for enforcing the constraints are themselves very close to geodesic segments. They are not exactly geodesic, because the struts do not originate on γ but at the tripod's head, a short distance away.] The tripod's position is described by the CM variables x, y, and z, which are defined by Eq. (<ref>), with δ z assigned to be a solution to Eq. (<ref>). We recall from Sec. <ref> that z represents the geodesic distance between the CM and the reference observer, in the longitudinal direction (increasing R), as measured in the observer's rest frame. On the other hand, x and y measure the geodesic distance between CM and observer in the transverse (angular) directions. The CM is given the initial conditionsx = y = z = 0 = ẋ = ẏ = ż, so that it is initially moving with the reference observer. We wish to determine the CM's motion at later times.We simplify the tripod's Lagrangian by removing all terms thatdo not involve the generalized coordinates and velocities (that is, terms that are prescribed functions of t) and performing a transformation to cylindrical coordinates (ϱ, φ, z) defined by x = ϱcosφ and y = ϱsinφ.The final expression is rather long, and we shall not display it here.Inspection of the Lagrangian reveals that it is independent of φ, so that p_φ = ∂ L/∂φ is a constant of the motion. We also find that p_φ∝φ̇, so that p_φ = 0 = φ̇ at all times by virtue of the initial conditions. We are therefore free to discard all terms involving φ̇ from the Lagrangian, and restrict the phase space to (ϱ̇, ϱ, ż, z). To leading order in an expansion in powers of v^2, the Lagrangian is given by L̅ = 1/2( ż^2 - B z^2 )+ 1/2( ϱ̇^2 - A ϱ^2 )+O(ϵ), with O(ϵ) representing the relativistic corrections; A and B are given by Eq. (<ref>). At this order the equations of motion arez̈ = -B z + O(ϵ), ϱ̈ = -A ϱ + O(ϵ), and these are recognized as components of the geodesic deviation equation. The equations imply that with the initial conditionsz = ϱ = 0 = ż = ϱ̇, the solution shall be of the form z = O(ϵ) and ϱ = O(ϵ), with deviations from the reference geodesic coming entirely from the relativistic terms in the Lagrangian. This fact allows us to simplify the Lagrangian further, by eliminating terms that scale as ϵ^3 and higher powers of ϵ. We thus obtainL̅ = 1/2ż^2 - 1/2 B z^2+ (a z + b ż) A + (c z + d ż) B + k ż+ O(ϵ^3), with a:= m_0 m_1/(m_0+3m_1)^2[ ℓsinαd/dt (ℓcosα) - ℓcosαd/dt (ℓsinα) ]d/dt (ℓsinα),b:= m_0 m_1/(m_0+3m_1)^2ℓsinα[ℓcosαd/dt (ℓsinα) + 1/2ℓsinαd/dt (ℓsinα) ],c:= -δ z + 3m_0 m_1/2(m_0+3m_1)^2ℓcosα{m_0-3m_1/m_0+3m_1[ d/dt (ℓcosα) ]^2 + [ d/dt (ℓsinα) ]^2 }, d:=3m_0 m_1 (m_0-3m_1)/2(m_0+3m_1)^3(ℓcosα)^2 d/dt (ℓcosα),and where k is defined by Eq. (<ref>).The terms 1/2 (ϱ̇^2 - A ϱ^2) wereeliminated from the Lagrangian, because ϱ and ϱ̇ do not appear in the remaining terms of order ϵ^2. This part of the Lagrangian therefore decouples from the one displayed in Eq. (<ref>), and its form implies that the solution to the equations of motion with the stated initial conditionsis ϱ = O(ϵ^3). The simplification has therefore eliminated ϱ and ϱ̇ from the list of dynamical variables, and the effective Lagrangian now depends solely upon z and ż. In this simplified description, the tripod's internal motions produce a longitudinal displacement with respect to the reference geodesic, but no transverse displacement.The equations of motion that follow from the Lagrangian of Eq. (<ref>) can be cast in the first-order formż = p_z - bA - dB - k, ṗ_z = -B z + aA + cB, where p_z is the momentum conjugate to z, or in the second-order formz̈ + B z = F := (a-ḃ) A + (c-ḋ) B- b Ȧ - d Ḃ. As stated previously, the equations are to be integrated with the initial conditions z = 0 = ż, so that the tripod begins its journey on the reference geodesic.§.§ Tripod cycleWe consider two families of cycles for the tripod's internal motions. The first is described by ℓ(t) = 1/2ℓ_0 (3 - cosα), α(t) = 2π t/T + χ, in which ℓ oscillates and α is monotonic; χ is an arbitrary phase constant and T is the period. This cycletraces a cosine function in the α–ℓ plane, and the areaunder the curve is equal to 3πℓ_0. This family is particularlysimple, and it allows us to integrate Eq. (<ref>) analytically;we obtain δ z= δ z_0+ 3π^2 m_0 m_1 ℓ_0^3/160(m_0 + 3m_1)^3T^2[(1320m_0 - 1980m_1) (cosα-cosχ)- (380m_0 - 570m_1) (cos2α-cos2χ)+ (40m_0+480m_1) (cos3α-cos3χ)- 405m_1 (cos4α-cos4χ)+ 108m_1 (cos5α-cos5χ)- 10m_1 (cos6α-cos6χ) ], where δ z_0 = δ z(0) is a constant of integration. Inserting these expressions in Eqs. (<ref>) returnsa= π^2 m_0 m_1 ℓ_0^3/8 (m_0+3m_1)^2 T^2[ 37 - 129cosα + 74cos2α - 15cos3α + cos4α],b= -π m_0 m_1 ℓ_0^3/128 (m_0+3m_1)^2T^2[ 126sinα - sin2α - 351sin3α + 220sin4α- 45sin5α + 3sin6α], c= -δ z_0-3π^2 m_0 m_1 ℓ_0^3/160 (m_0+3m_1)^3T^2[(560m_0 + 270m_1) - (1320m_0 - 1980m_1) cosχ+ (380m_0 - 570m_1) cos2χ- (40m_0 + 480m_1) cos3χ+ 405m_1 cos4χ - 108m_1 cos5χ + 10m_1 cos6χ - (60m_0 + 3780m_1)cosα+ (180m_0 + 2220m_1)cos2α- (20m_0 + 1680m_1)cos3α+ 1005m_1 cos4α- 252m_1 cos5α + 20m_1 cos6α], d= -3π m_0 m_1 (m_0-3m_1) ℓ_0^3/128 (m_0+3m_1)^3T^2[ 138sinα - 149sin2α + 153sin3α - 76sin4α+ 15sin5α - sin6α].The second family of cycles is described by ℓ(t) = 1/2ℓ_0 [ 3 - cos(2π t/T + χ) ] α(t) = π/12[ 3 - cos(2 π t/T) ], in which ℓ oscillates between ℓ_0 and 2ℓ_0 in the course of a complete period, while α oscillates between π/6 and π/3. This cycle traces an ellipse in the α–ℓ plane,sweeping out a surface with area (π^2/24) ℓ_0 sinχ.We shall focus our attention mostly on the cycle of Eq. (<ref>), and take advantage of its simplicity. We shall also, however, present numerical results for the cycle of Eq. (<ref>). § TRIPOD MOTION: RESULTS §.§ de Sitter spacetime We begin with a discussion of the tripod's motion in de Sitter spacetime, for which f = 1-k^2 R^2 and the curvatures of Eq. (<ref>) are A = B = -4/(9 t_0^2), where t_0 := 2/(3k) is a conveniently defined cosmological time scale. The solution to Eq. (<ref>) with vanishing initial conditions isz(t) = 2/3 t_0∫_0^tg(t') sinh[2(t'-t)/3t_0] dt',where g := a - ḃ + c - ḋ. From Eq. (<ref>) we see that when ℓ and α are periodic, g(t) will also be a periodic function of t with meang_0 := T^-1∫_0^Tg(t)dt = T^-1∫_0^T[a(t)+c(t)]dt. In particular, for the cycle ofEq. (<ref>),g_0= -δ z_0 + π^2 m_0 m_1 ℓ_0^3/160 (m_0 + 3 m_1)^3 T^2[ -470(2 m_0 - 3 m_1) + 1980 (2 m_0 -3 m_1) cosχ- 570 (2 m_0 - 3 m_1) cos 2χ+120 (m_0 + 12 m_1) cos 3χ -1215 m_1 cos 4χ + 324 m_1 cos 5 χ - 30 m_1 cos 6 χ].When T ≪ t_0, which defines the regime of rapid cycles, the oscillatory terms in g(t) give a negligible contribution to the integral of Eq. (<ref>), and z(t) is well approximated by z(t) = g_0 [ 1 - cosh(2t/3t_0) ].Now, for an arbitrary choice of δ z_0, g_0 is nonzero and the geodesic observer sees the tripod drifting away exponentially. Such a drift is paradoxical, because de Sitter spacetime is maximally symmetric, and there can be no preferred direction for the tripod's motion relative to the reference geodesic. But we have the option to eliminate this drift by adjusting δ z_0 so that g_0 =0. For this specific choice of CM we have that z(t) = 0, which means that when the CM is placed initially on a geodesic in de Sitter spacetime, it will continue to move on this geodesic (at least in the regime of rapid cycles). This choice of CM gives us the expected geodesic motion for a tripod in de Sitter spacetime, andtherefore provides us with a sensible prescription for the determination of δ z_0. This prescription can be applied to any spacetime. The point remains, however, that geodesic motion reflects a choice of CM, and that an alternative choice would generically produce the drift described by Eq. (<ref>).§.§ Schwarzschild spacetime We next turn to Schwarzschild spacetime, for which f = 1 - 2GM/R;the tripod falls toward a spherical body of mass M. For γ we choose a marginally bound, radial geodesic with E = 1, so thatdR/dt = -√(1-f) = -√(2GM/R), which integrates to R(t) = [ 9/2 G M (t_0 - t)^2 ]^1/3, where t_0 is now the time at which R is formally equal to zero; recall that in our notation, t is proper time on γ. The curvatures of Eq. (<ref>) becomeA = 2/9(t_0-t)^2, B = -4/9(t_0-t)^2 for this choice of spacetime and reference geodesic. With B given by Eq. (<ref>), the solution to Eq. (<ref>) with vanishing initial conditions isz (t) = 3/5 (t_0-t)^-1/3∫_0^t (t_0-t')^4/3 F(t')dt' - 3/5 (t_0-t)^4/3∫_0^t (t_0-t')^-1/3 F(t')dt'. To evaluate the integrals we discard the terms involving Ȧ and Ḃ in F, which give negligible contributions until t approaches t_0. This gives F ≃ 2 g̃/[9(t_0-t)^2], with g̃ := a - ḃ - 2(c - ḋ). The function g̃(t) is periodic with meang̃_0 := T^-1∫_0^Tg̃(t)dt= T^-1∫_0^T[a(t)- 2 c(t)]dt, and for the cycle described by Eq. (<ref>), g̃_0:= -δ z_0+ π^2 m_0 m_1 ℓ_0^3/80(m_0+3m_1)^3[(2050m_0+1920m_1) - (3960m_0-5940m_1)cosχ + (1140m_0-1710m_1)cos 2χ- (120m_0+1440m_1)cos 3χ + 1215m_1 cos 4χ - 324m_1 cos 5χ+30m_1 cos 6χ]. Discarding the oscillations in g̃(t), because they givenegligible contributions to the integrals in the regime of rapid cycles,and keeping only the mean, Eq. (<ref>) becomesz(t) = -2/5g̃_0 [ 1 - (1-t/t_0)^-1/3]- 1/10g̃_0 [ 1 - (1-t/t_0)^4/3]. When t/t_0 is small, Eq. (<ref>) reduces to z ∼1/9g̃_0 (t/t_0)^2. On the other hand, z ∼2/5g̃_0 (1 - t/t_0)^-1/3 when t approaches t_0. Equation (<ref>) describes a drift relative to the reference geodesic, and the drift is proportional to g̃_0 given by Eq. (<ref>). Because g̃_0 depends on δ z_0, we see once again that the description of the tripod's motion dependssensitively on the choice of CM. An option would be to adjustδ z_0 so that g̃_0 = 0, and to eliminate the drift in Schwarzschild spacetime. This would make an alternative prescription for the complete determination of the CM, and adopting it for de Sitter spacetime would give g_0 ≠ 0 and a drift relative to the reference geodesic; we would recover the same paradox as described in Sec. <ref>. A more sensible prescription is the one adopted in Sec. <ref>, which gives no drift in de Sitter spacetime. With this original prescription, and for the cycle of Eq. (<ref>), we have that g̃_0 = 111 π^2 m_0 m_1 ℓ_0^3/8(m_0 + 3 m_1)^2 T^2,and there is a drift in Schwarzschild spacetime.In Fig. <ref> we show that Eq. (<ref>) with theg̃_0 of Eq. (<ref>) is indeed a good approximation to the exact solution of Eq. (<ref>) with initial conditionsz = 0 = ż at t=0, for the cycle of Eq. (<ref>) and aδ z_0 determined by the de Sitter condition g_0 = 0. A similar comparison for the cycle of Eq. (<ref>) is presented inFig. <ref>. We note that whether the CM lags behind or sprints forward relative to the geodesic observer depends on the specific choice of cyclic motion. The analytical expression of Eqs. (<ref>) and (<ref>) allows us to calculate the tripod's drift after one cycle of its internal motions. This is measured by z(T), which evaluates to1/9g̃_0 (T/t_0)^2 after taking into account that t_0 ≫ T.Substituting Eq. (<ref>) and relating t_0^-2 to the spacetime curvature through Eqs. (<ref>) and (<ref>), we write this as z(T) = 37/9πm_0 m_1/(m_0 + 3m_1)^2 ℓ_0^2 (3πℓ_0) GM/R^3. We observe that the drift scales with m_0 m_1/(m_0 + 3m_1)^2, ℓ_0^2,the area 3πℓ_0 of the α–ℓ plane swept out by thetripod's cycle, and with the curvature GM/R^3. All these ingredients are also featured in Wisdom's Eq. (20) <cit.>, and we therefore have recovered the essence of his result. We do not get a precise match — the numerical prefactor is different — because we adopt a different cycle for the tripod's internal motions, model the struts in a slightly different way, and work in the Fermi frame of the free-falling observer instead of the static frame of the Schwarzschild spacetime.As a final comment, it is interesting to note that the asymptotic relation(<ref>), as t approaches t_0, describes a geodesic ofthe Schwarzschild spacetime. Indeed, the homogeneous version of Eq. (<ref>), z̈ + Bz = 0, is a component of the geodesic deviation equation, and its general solution z_ geo(t) =1/5[ 4z(0) + 3 t_0 ż(0)] (1-t/t_0)^-1/3+ 1/5[ z(0) - 3 t_0 ż(0) ] (1-t/t_0)^4/3 describes a geodesic of the Schwarzschild spacetime that neighbors the reference geodesic γ. The first term dominates as t approaches t_0, and we see that with 4z(0) + 3 t_0 ż(0) = 2 g̃_0, z_ geo becomes asymptotically equal to z. The world linedescribed by Eq. (<ref>), therefore, is a geodesic with initial conditions constrained by Eq. (<ref>).This allows us to give an interpretation to the numerical results ofFigs. <ref> and <ref>.What we see is the tripod's CM gradually transiting to a new geodesic described by Eq. (<ref>) after being launched from the reference geodesic. The tripod is initially directed along γ, but the coupling between its internal motions and the spacetime curvature prevents it from following the reference geodesic. The motion is therefore nongeodesic for 0 < t < t_0, but it becomes increasingly geodesic as t → t_0, that is, as the tripod approaches the curvature singularity at R = 0. The approach to the singularity follows a geodesic, irrespective of the internal motions[We should note that the description of the motion refers to the Fermi frame introduced in Sec. <ref>. The metric is approximate, and its domain of validity becomes increasingly narrow as the curvature increases. The approach to the singularity must therefore be handled with care; t cannot be allowed to be too close to t_0.]. The same reasoning can be applied to de Sitter spacetime, when a CM definition such that g_0 ≠ 0 is adopted. In this case, the motion asymptotes to z ∼1/2 g_0 e^2t/3t_0when t ≫ t_0, and this describes a geodesic of de Sitter spacetimewith initial conditions constrained by 2 z(0) + 3t_0 ż(0) = 2g_0.§.§ ConclusionOur main conclusion in this section is that the motion of the tripod relative to the reference geodesic depends sensitively on the choice of internal cycle, but also on the choice of δ z_0, the initial relativistic adjustment to the CM. The prescription advanced in Sec. <ref> left this quantityundetermined, and the prescription must therefore be completed by a choice of δ z_0. A sensible (though not unique) option is to choose δ z_0 so that g_0 = 0, thereby ensuring that a tripod placed in de Sitter spacetime does not drift away from the initial geodesic; this choice is motivated by the absence of a preferred direction in a maximally symmetric spacetime. Adopting the same CM when the tripod is placed in the Schwarzschild spacetime gives rise to a drift, with an approximate description given by Eqs. (<ref>) and (<ref>), in accordance to Wisdom's original results <cit.>. § RECONCILIATION WITH THE MATHISSON-PAPAPETROU-DIXON EQUATIONSThe Mathisson-Papapetrou-Dixon (MPD) equations, Dp_α/dτ = 1/2 S^μν u^λ R_μνλα, DS_αβ/dτ = 2 p_[α u_β],are meant to govern the motion of a generic extended body, in a pole-dipole approximation that neglects higher multipole momentsbut keeps the momentum vector p_α and spin tensor S_αβ; the velocity vector u^α is tangent to the world line, and D/dτ denotes a covariant derivative with respect to proper time τ. Many derivations of these equations have been provided (see, for example, Refs. <cit.>), with the most comprehensive analysis supplied by Dixon <cit.>. The puzzle that concerns us in this section is that while the MPD equations wouldbe thought to adequately govern the motion of the tripod, the detailed examination carried out by Silva, Matsas, and Vanzella <cit.> indicates that Eqs. (<ref>) seemto be incompatible with the swimming motion described in Sec. <ref>. One reason to suspect that Eqs. (<ref>) may not apply to the tripod is that their derivation relies heavily on conservation of energy-momentum, as embodied by ∇_β T^αβ = 0, where T^αβ is the energy-momentum tensor of the extendedbody. (Alternative derivations depend on the existence of a Lagrangian for the extended body, and this automatically enforces energy-momentum conservation.) The tripod, on the other hand, is a constrained mechanical system that does not conserve energy and momentum: the external agents responsible for the internal motions must supply energy and momentum to keep the tripod on its cycle. This issue can be investigated by generalizing the Lagrangian-based derivation of Eqs. (<ref>) provided by Bailey andIsrael <cit.> to a constrained system. To showcase the modifications that result from the introduction of constraints, weconsider the illustrative case of two particles moving freely in spacetime, except for a holonomic constraint that keeps their separation equal to a prescribed vector field f^α. This is a covariant formulation of the type of tripod model introduced in Sec. <ref>, and generalization to any number of particles (like four, for an actual tripod) is immediate.Bailey and Israel place the two particles on physical world lines P_1 and P_2, but describe their motion in terms of a reference world line C that represents an arbitrary choiceof “center of mass”. The world lines are given an arbitrary parameter t, the tangent vector to C is u^α, and theseparation between C and P_A at fixed t is-σ^α_A, with A = 1, 2 labelling the particle. The Lagrangian L_A of each particle is the usual -m_A dτ_A/dt evaluated on P_A, which is rewritten in terms of fields on C; an explicit expression for the Lagrangian is given by their Eq. (76). The complete Lagrangian for the constrained system isL = L_1 + L_2 + L_ cons, withL_ cons = λ_α( σ^α_2- σ^α_1 + f^α), where λ_α(t) is a Lagrange multiplier, and f^α(t) is the prescribed separation between the particles. The dynamical variables are defined by p_α := ∂ L/∂ u^α, S_αβ :=2 σ_1 [α∂ L/∂σ̇_1^β] + 2 σ_2 [α∂ L/∂σ̇_2^β], where σ̇^α_A := D σ_A^α/dt is thecovariant derivative of σ^α_A along C. Equations of motion for p_α and S_αβ follow from the requirement that S := ∫ Ldt be stationary under arbitrary variations of P_A and C; the derivation also involves an identity deduced from the invariance of the action under an infinitesimal coordinate transformation. Variation with respect to P_A implicates the constraints, and Eq. (50) from<cit.> generalizes to D/dt∂ L_1/∂σ̇^A_1- ∂ L_1/∂σ^α_1 = -λ_α, D/dt∂ L_2/∂σ̇^A_2- ∂ L_2/∂σ^α_2 = +λ_α; these equations can be used to determine the Lagrange multiplier. Variation with respect to C reproducesEq. (47) from <cit.> without change (we set theelectric charge e to zero), and their Eq. (51) acquires new terms coming from L_ cons.Putting all this together, truncating the description of the motion tothe pole-dipole approximation, and setting t = τ =, we find that the equations of motion take the form ofDp_α/dτ = 1/2 S^μν u^λ R_μνλα, DS_αβ/dτ = 2 p_[α u_β]+ 2 λ_[α f_β],which differ from Eqs. (<ref>) by an additional torque term provided by the constraints. As expected, the original MPD equations do not apply to a constrained Lagrangian system. Because Eqs. (<ref>) are derived from a Lagrangian, they must be physically equivalent (after generalization to four particles, and approximation to weakly relativistic motion) to the equations of motion obtained in Sec. <ref>, which also follow from a Lagrangian. To establish the equivalence explicitly would bedifficult, because the two sets of equations implicate different variables, and because the constraints are implemented differently in each approach: in Sec. <ref> the constraints were solved to express the Lagrangian in terms of the CM variables, while in this section they are enforced with Lagrange multipliers. Anotherobstacle toward establishing the equivalence of the two formulations is that Eqs. (<ref>) remain empty of content until a relation between p_α and u_α is specified through the selectionof a suitable “center of mass”; as we saw back in Sec. <ref>, this can be a delicate matter. But in spite of these obstacles, we can be confident that Eqs. (<ref>) or (<ref>) and (<ref>) describe the same system, because the two sets of equations originate from the same Lagrangian. In this admittedly incomplete way, we can reconcile Wisdom's swimming with theMPD framework. It seems to us that the selection of a “center of mass” is probably the most critical aspect of the comparison between Wisdom's results<cit.> and the MPD framework; the additional terms in Eq. (<ref>) may well be incidental. Indeed, we could imagine formulating a complete tripod model that includes all the springs and rubber bands that are dynamically responsible for the internal motions. Such a model could be described in terms of a Lagrangian or a conserved energy-momentum tensor, and such a tripod would be expected to satisfy Eqs. (<ref>). But the selection of a “center of mass” for this model would be a delicate affair, and the precise relation between p_α and u_α might be complicated by the many details of the tripod's design. For example, the choice of “center of mass” that comes with the oft-used covariant spin supplementary condition, p_α S^αβ = 0, might be entirely inadequate for such an object. In such circumstances, the relation between Eqs. (<ref>) and the actual motion of the tripod might be very subtle and difficult to describe. This is another path of reconciliation. | http://arxiv.org/abs/1707.08870v2 | {
"authors": [
"Raissa F. P. Mendes",
"Eric Poisson"
],
"categories": [
"gr-qc"
],
"primary_category": "gr-qc",
"published": "20170727141409",
"title": "Swimming in spacetime: the view from a Fermi observer"
} |
Are Triggering Rates of Labquakes Universal? Inferring Triggering Rates From Incomplete Information.Jordi Baró, Jörn DavidsenComplexity Science Group, Department of Physics and Astronomy, University of Calgary, CanadaDecember 30, 2023 =============================================================================================================================The acoustic emission activity associated with recent rock fracture experiments under different conditions has indicated that some features of event-event triggering are independent of the details of the experiment and the materials used and are often even indistinguishable from tectonic earthquakes. While the event-event triggering rates or aftershock rates behave pretty much identical for all rock fracture experiments at short times, this is not the case for later times. Here, we discuss how these differences can be a consequence of the aftershock identification method used and show that the true aftershock rates might have two distinct regimes. Specifically, tests on a modified Epidemic Type Aftershock Sequence model show that the model rates cannot be correctly inferred at late times based on temporal information only if the activity rates or the branching ratio are high. We also discuss both the effect of the two distinct regimes in the aftershock rates and the effect of the background rate on the inter-event time distribution. Our findings should be applicable for inferring event-event triggering rates for many other types of triggering and branching processes as well. § EVENT-EVENT TRIGGERING & AFTERSHOCKSMany striking features of physical, geophysical, biological or social processes can be portrayed as patterns or clusters of localized events. Specific examples include magnetization processes <cit.>, martensitic transformations <cit.>, fracture processes <cit.>, natural or induced earthquakes <cit.>, solar flares <cit.>, extreme bursts in the solar wind <cit.>, the spread of infections <cit.>, extinctions of species <cit.>, neural spikes <cit.>, booms and bursts of markets and economies <cit.>, media coverage <cit.> – to name a few. A generic attribute in all these cases is that one event can trigger or somehow induce another one to occur – or possibly numerous further events. One of the most prominent examples of such event-event triggering are aftershocks <cit.>. Aftershock sequences are characterized by time-varying (local) event rates, which are often empirically found to approximately follow — across a wide range of scales and systems from friction and fracture to socio-economic systems <cit.> — the Omori-Utsu (OU) relation,r(t) = K/(t + c)^p≡1/τ (t / c + 1)^p ,first proposed for earthquakes <cit.>. Here, t measures the time after the triggering event, p is typically close to 1 (p≳ 1 if one only considers directly triggered events <cit.>) and τ≡c^p/K. K is typically found to increase with the energy of the trigger though the exact dependence of K, c and hence τ on different parameters is an active field of research <cit.>. The OU relation with p ≈ 0.7 has in particular been observed in acoustic emission (AE) experiments of rock fracture across a range of different materials and conditions <cit.>, denoting some sort of universality in the response of disordered materials under mechanical stress, an hypothesis already suggested by the scale invariance in other physical and statistical relations <cit.>. Yet, a p-value significantly lower than 1 implies that the number of events directly or indirectly triggered by a single event is infinite. Indeed, a more recent study of rock fracture experiments using a more reliable technique to identify triggered events has shown that there are significant deviations from the OU relation at late times with a steeper decay that ensures that the number of triggered events is finite <cit.>. Identifying the aftershocks is a general challenge for all triggering processes since a detailed or “fundamental” knowledge of the underlying microscopic dynamics and causal information is typically not available <cit.>. In the case of earthquakes, the most reliable methods to identify triggering relations use spatio-temporal correlations between events <cit.>. In the absence of spatial information — as it is the case for previous rock fracture experiments <cit.> — this is not an option and one has to rely onthe measurement of the whole activity rate after each event <cit.>. This technique can lead to a strong bias in the estimation of triggering rates in cases where either the number of triggered events or the background rate of events activated by other mechanisms is high, or both, as we show explicitly here. Specifically, in this paper we aim to quantify the bias of this technique and establish under which conditions it can serve as a reasonable estimator for the triggering rates. We test it against synthetic catalogs generated by a modified Epidemic-Type Aftershock Sequence (ETAS) model with a triggering rate characterized by two power laws, as observed in the most recent rock fracture experiments <cit.>.We address the possibility that the low experimental OU exponent (p values) and some other inconsistent results observed in the previous rock fracture experiments <cit.> are a consequence of the hidden complexity of the triggering process.First, we formulate triggering in terms of branching processes.The true direct and compound triggering rates can be estimated reliably only under certain conditions. Instead, one is often limited to less reliable estimators such as those based on the mean aftershock sequence rates (MASR) and the distribution of waiting times (DWT) to extract the properties of the triggering process. We discuss the limitations of such methods. Next, we introduce the modified ETAS model with the triggering rates characterized by two power laws as observed in rock fracture experiments with spatio-temporal information <cit.>. Considering that only magnitude and temporal information is available, we interpret the measured MASR in terms of direct and compound triggering rates, and identify the parameters giving rise to different power-law regimes in the DWT. Finally, we discuss whether these numerical results can provide an explanation for anomalous features in the experimental results. § EVENT-EVENT TRIGGERING REPRESENTED AS A BRANCHING PROCESS The study of systems exhibiting localized events and triggering between them can be cast in the language of point processes <cit.>. A stochastic point process is fully determined by a function called the intensity, which quantifies the probability of occurrence of an event of size M at time t and at location z⃗: μ(t,z⃗,M) := Prob{eventofsizeM at t,z⃗} dt dr⃗dM.The measured activity rate can vary over time due to an explicit temporal variation of external parameters, and/or as a consequence of previous activity. In the latter case, the intensity depends explicitly on the history of the point process (ℋ_t:={all eventsi ;t_i < t }).The exact intensity (μ(t,ℋ_t)) of a process exhibiting triggering involves in general acomplex contribution of the whole history ℋ_t. In a simplified approach, the contribution to the intensity of each past event can be linearized in a Hawkes self-exciting point process <cit.>: μ (t,𝐳,m | ℋ_t) = μ_0(t,m) + ∑_i ∈ℋ_tϕ(t-t_i,z⃗-z⃗_i,m|M_i)Stochastic Hawkes processes can be reinterpreted as the outcome of branching processes where each event is either a background event or has a single parent. Given a background event (G=0), a sequence of first generation events (G=I) can be triggered, after a time difference τ and at relative position r⃗ from the background event, according to an intensity factor represented by the triggering kernel ϕ(τ,r⃗,m|M_i). Each event in the first generation can itself trigger a sequence of second generation events (G=II) with the same relative kernel ϕ(τ,r⃗,m|M_i), and so on until a whole triggering cascade or tree is generated up to some n-th generation that does not trigger further events.The major physical constraint to the model is the stability of the branching process, requiring that the average number of events directly triggered from a single parent —computed as the average branching ratio: n_b= ⟨∫ dτ∫ dr⃗∫ dm∫ dm'ϕ(τ,r⃗,m'|m) ⟩— has to be lower than one. Triggering trees can be spatially and temporally overlapping, generating a complex triggering forest difficult to disentangle in practice.§.§ Direct and compound event-event triggering rates The linear Hawkes model is useful for both the development of forecasting tools <cit.> and the deep understanding of the fundamental physics behind avalanche processes <cit.>.Both purposes require a reliable estimation of the triggering kernel.If the actual pairwise parent-child relations are retrievable, or can be estimated from declustering techniques <cit.>, one can measure the direct (or bare <cit.>) triggering rates and use them as a good estimator for the triggering kernel: ϕ(τ,r⃗,m'|m).In some cases one has to deal with time series without spatial information, or situations where the spatial kernel is too spread to retain meaningful information. One is left with the marginal temporal point process with a triggering kernel ϕ(τ,m | m_0). The compound (or dressed <cit.>) triggering rates are the expected temporal activity during the span of a triggering cascade generated from a background event (the root of the tree) of magnitude m_0at t_0. The compound triggering rates can be computed from the direct triggering rates as:Φ_c(τ,m | m_0) := ∑_G=I, II, ...^∞Φ_G(τ,m | m_0)where τ=t-t_0 and Φ_G=I corresponds to the direct rates from the background event, Φ_G=II designates the rate of events triggered by all first generation events, etc.. Under certain conditions one can calculate the compound rates analytically, or at least find an approximate solution, as for the case exposed below. The magnitude of triggered events is usually assumed as an independent variable: ϕ(τ,m|m_0)= ρ(m)ϕ(τ|m_0), where ρ(m) is the distribution of magnitudes. In such cases, the intensity of the first generation (G=I), equivalent to the kernel from the triggering background event, can be written as:Φ_I(τ,m|m_0) ≡ρ(m)ϕ(τ|m_0)andeach one of the higher order generations contribute to the intensity as:Φ_G(τ,m|m_0) =ρ(m) ∫_m_c^∞ dm'∫_0^τ dt' Φ_G-1(t'|m_0) ϕ(τ-t'|m') ρ(m')where t' and m' are the occurrence time and magnitude of the events from the (G-1)'th generation originated by the mainshock (m_0,t_0), and m' is distributed according to ρ(m').If the kernel can be further separated between the temporaland productivity term: ϕ(τ|m'):= k(m')ϕ(τ), the branching ratio is simplified as: n_b= ∫k(m') ρ(m')dm', and each element is directly triggered by the previous generation:Φ_G(τ|m_0) = n_b ∫_0^τ dt'Φ_G-1(t',m_0) ϕ(τ-t')Substituting Eq. (<ref>) in the generic recurrence the triggering rates of generation G can be expressed as a series of convolution operations (*) as: [Φ_G(τ|m_0) =K(m_0) n_b^g(ϕ_1 * ϕ_2 *... * ϕ_g) ]This recurrent expression can be solved in the Laplace transformed space. Given ψ(s) := ℒ(ϕ(y)) the compound rates can be found as: Φ_c(τ|m_0) =K(m_0) ∑_g=1^∞ℒ^-1((n_bψ(s))^g)§.§ Mean aftershock sequence ratesIn the absence of any information regarding the precise topological structure of the branching process we can still try to estimate the triggering kernel under certain conditions. The branching nature of the model, and the independence between terms, impose Markovian correlations in the parent-child relationship. All triggering branches are independent and, thus, triggering trees can be considered independently of the generation of the parent without loss of generality. Under these premises, we can consider all triggering trees originated from any event as statistically equivalent.Thus, we can measurethe activity rates conditioned to the presence of a trigger or mainshock (t_0,m_0) — the mean aftershock sequences rates (MASR) — as the expected density of events: MASR(τ,m|m_0) := ⟨ρ(t-t_i=τ,m,M_i=m_0)⟩In general, the measurement of MASR corresponds to the compound rates plus the contribution of all events without a causal connection to the mainshock. If the branching ratio is low, we can consider this second contribution as the independent activity:MASR(τ|m_0) ≈Φ_c(τ|m_0)+ ⟨μ_0(t)+∫ dt'μ(t') ϕ(t-t'|m_0) ⟩≈Φ_c(τ|m_0) +⟨μ(t) ⟩Under certain conditions the MASR serve as a good approximation to the compound or even the direct rates. If the background activity is low enough to isolate independent triggering trees, the contribution of independent events is small and MASR(τ|m_0) ≈Φ_c(τ|m_0). Furthermore, if the branching ratio is low enough to neglect secondary triggering as a major contribution to the triggering rates, compound and direct rates are similar and hence Φ_c(τ|m_0) ≈ϕ(τ|m_0).§.§ The Distribution of Waiting Times Finally, we shall mention the more naive approach to study triggering based on the distribution of waiting times (DWT) or times between consecutive events.Since the DWT is a memoryless measurement, different point process (with and without correlations) can give rise to similar distributions. It is advised to use the DWT with caution when assessing the presence of triggering, and to use more reliable techniques instead <cit.>. Specifically, different phenomena can give rise to power-law regimes in the DWT. For a Poisson process, all events are independent and their DWT renders a decaying exponential with a characteristic rate (e.g. μ_0 in Eq. <ref>). Yet, if this rate varies over time, the compound DWT corresponds to a superposition of exponential distributions. If the rate starts from zero (μ_0(t=0)=0) and increases with time, we can always expand the temporal dependence of the background rate in a power series around the origin: lim_t→ 0^+μ_0(t) ≈ t^1/ξ. It can be shown that, for long waiting times, the DWT for this process of independent events will be: DWT(δ) dδ∝δ^-ξ-2 dδ <cit.>. On the other hand, a power-law DWT can also emerge from triggering. Typically, the presence of triggering is identified as an anomalous behavior in the distribution of short waiting times. Direct triggering rates decaying in time as a power-law r(t-t_i) ∝ (t-t_i)^-p —such as the Omori-Utsu relation in Eq. (<ref>)— return also a power-law with an exponent 2-1/p <cit.>. However, some experimental measurements also display discrepancies with this exact relation <cit.>. Finally, when power-law triggering processes coexist with a time-dependent background rate we can find a double power-law DWT fulfilling certain scaling relations <cit.>. § TRIGGERING MODELS FROM EMPIRICAL DATAOne of the most studied triggering processes are aftershock sequences in seismology. Seismic activity increases after major earthquakes, a phenomenon consistent with the idea that this event — often called a main shock — triggered other ones.In 1894 F. Omori realized that the rate of the triggered earthquakes —the aftershocks— after the 1891 Nobi main shock decayed in time following a power-law relation with exponent p≈ 1 <cit.>. The modified Omori-Utsu relation <cit.>, as stated in Eq. (<ref>), presents a good approximation for the activity rates after almost all major earthquakes in recent history. The size of an earthquake is quantified by the magnitude m, a logarithmic measure of the seismic moment released by the slip associated with an earthquake. In 1944 Gutenberg and Richter <cit.> established that the number of earthquakes above a certain magnitude m approximately behaves as N_>(m) ∼ 10^-b m, equivalent to a power-law distribution of the seismic moment or an exponential distribution of magnitudes:ρ(m)dm =bln (10) 10^-b (m-m_c) dmfor m_c≤ mwhere m_c corresponds to the magnitude of completeness of the given catalog. The value of b is close to unity and Eq. (<ref>) extends down to magnitudes as low as m=-4.4 <cit.>. Typically, magnitudes can be considered to be independent <cit.>. There exists, however, a well established relationship between the magnitude of main shocks and the number of their aftershocks N_AS. The productivity relation of aftershocks states that N_AS scales with the magnitude of the mainshock m_0 as <cit.>:N_AS(m_0)∝10^αm_0This relation implies that the parameter K in the OU relation (<ref>) is not a constant and instead scales with the magnitude of the mainshock as:K(m_0) = k 10^α m_0 and, in this case, α≡α. Yet, recent studies of earthquake catalogs indicate that the productivity relation and the Omori-Utsu relation might need to be augmented and α≠α <cit.>. §.§ The Epidemic-Type Aftershock Sequence (ETAS) model of earthquakes The three statistical relations of seismology stated in Eqs. (<ref>,<ref>,<ref>) can be used to define a branching process [Notice that the exponent 1+θ and the p in the OU relation (<ref>) measured from the compound rates may differ <cit.>, as can be derived from Eq. (<ref>).], commonly known as the Epidemic Type Aftershock (ETAS) Model <cit.>, where all the explicit dependences can be separated as:ϕ(m,τ,𝐫|m_0)= ρ(m)K(m_0)ϕ_T(τ)ϕ_R(𝐫)with {[ρ(m) = b ln (10) 10^b(m_c-m);K(m_0) =k 10^α m_0;ϕ_T(τ) =θC^θ(C+τ)^-1-θ ]. where m_0 is the magnitude of the mainshock and τ=t-t_0. The average branching ratio is given by: n_b=kb/b-α10^α m_c. In the branching process approach, the number of triggered events is sampled as a Poisson variable with rate K(m_0).§.§ The modified ETAS model for rock fracture Many physical processes exhibit statistical features similar to those summarized in Eqs. (<ref>, <ref>, <ref>) for seismicity. Specifically, the ETAS model describes remarkably well some aspects of the temporal sequences of acoustic emission (AE) events recorded during the failure of rocks and porous materials under compression <cit.>. However, the full spatio-temporal triggering cascades, which have only become accessible very recently, in AE experiments of rock fracture reveal a more complex triggering kernel than the standard ETAS model <cit.>.While the event magnitudes appear to be independent (ϕ(m,τ|m_0)= ρ(m)ϕ(τ|m_0)), empirical evidence suggests that there is a characteristic time associated with the triggering rates that scales with the magnitude of the mainshock. Similar behavior has also been observed very recently in earthquake catalogs from Southern California <cit.>. Thus, the term ϕ(τ|m_0) cannot be separated in independent terms K(m_0)ϕ(τ). Specifically, the following scaling form has been observed:ϕ(τ| m_0)∼ 10^α m_0ϕ(10^-α_τ m_0τ) with α_τ=0.5 <cit.>.Furthermore, the short time regime, which is constant in the standard ETAS model, is better fitted by a generic power law with an exponent 0≤ p_1 < 1. In summary, the scaling function exhibits a transition from this power-law regime ϕ(x)∼ x^-p_1 below a characteristic value x_c of rescaled time, towards the standard ϕ(x)∼ x^-p_2 with p_2>1.In order to implement a modified ETAS model able to reproduce the empirical observations, we need to impose a branching ratio n_b≤ 1 for stability reasons, and set p_1<1 and p_2>1 to be integrable over the whole temporal domain. We define the positive parameters θ_1:=1-p_1 and θ_2:=p_2-1 for convenience. By normalizing the kernel and imposing the continuity constrain at x_±=x_c the scaling relation has the explicit form:1/x_cϕ(x/x_c)d x = dx/x_c(θ_1 θ_2/θ_2 + θ_1) {[(x_c/x)^1-θ_1 for 0 < x/x_c≤ 1; ;(x_c/x)^1+θ_2 for x/x_c≥ 1;].Considering the same K(m_0) used in Eq. (<ref>) and the definition of the productivity law stated in Eq. (<ref>), the total number of events triggered by a mainshock of magnitude m_0 now scales with the compound productivity exponent α=α+α_τ. Given the temporal and mainshock-magnitude dependent kernel ϕ(τ|m_0) the scaling function ϕ(x/x_c) from Eq. (<ref>) can be retrieved as:ϕ(10^-α_τ m_0τ/x_c)= x_c/k 10^-α m_0ϕ(τ|m_0).This is the form we focus on in the remainder of the paper, especially in the figures. Given the intensity of the process μ(t, ℋ_t) from Eq. (<ref>), the terms k, x, x_c and μ_0 involve a temporal scale. Here, we select ⟨μ_0(t) ⟩ as our time unit. Thus, we express the parameter x_c in units of the mean background rate: [x_c] = ⟨μ_0(t) ⟩ ^-1. We implement the ETAS model with the modified temporal kernel as defined in Eq. (<ref>) and performed simulations with the parameters estimated from the empirical data <cit.>: θ_1=0.25, θ_2=0.7 and a transition point τ_c=10^α_τ m_0x_c. The magnitudes of the events are generated from Eq. (<ref>) with b=2.0 and m_c=3.25 [Please note that the magnitudes of AE events in the lab experiments are defined in a different way than for earthquakes and not directly comparable. Hence the difference in scales and b-values.]. The productivity exponent of the kernel is set to α=0.55 and α_τ=0.5. Thus the average number of events generated by a mainshock is ⟨ N_AS(m_0)⟩ = k 10^1.05 m_0. Combining Eqs. (<ref>) and (<ref>), the explicit dependence of the productivity term on the branching ratio n_b reads:k(n_b)=∫ dm_0∫dτϕ(τ| m_0) = n_b(1-α-α_τ/b)10^-(α+α_τ)m_c . We generated sequences of 10^5 background events for different values of x_c and n_b. To highlight our main findings, we focus on two specific examples in the following: Low branching ratio with n_b=0.2 and high branching ration with n_b=0.95. The ratio between the transition point and average activity of unrelated events determines whether the second power-law regime can be observed in the mean aftershock sequence rates or not. For example, the transition point x_c is unobservable when the triggering rates ϕ(x_c) fall below the background rate μ_0.Considering the parameters of our simulations, this happens forx_c=θ_1θ_2/θ_1+θ_2= 5/300 ∼ 0.0167 in our reduced units. We simulate the model with values of x_c=1, 5/300, 10^-8 such that the transition point is found above, around and below the background level. In order to evaluate the effect of time-independent vs time-dependent background rates, we impose μ_0(t) ∼ t^σ-1 by sampling the background events from a cumulative distribution: CDF(t_i)=t^σ. Thus, a constant rate is sampled for σ=1 and quadratic increasing rate for σ=3, resembling the smooth increase of the rate observed at the beginning of AE experiments <cit.>. Fig. <ref> shows the measured bare rates for all simulations to verify the scaling relation (<ref>). Each line represents an average over all parent events with magnitude m < M_i < m + Δ m in each simulation. Indeed, the numerical results reproduce the expected relation from Eq. (<ref>), represented by the wide grey curve.The compound triggering rates are also invariant with respect to τ_c=10^-α_τ m_0x_c, which controls the temporal scale of the triggering with respect to the background rate, but it is sensitive to the productivity (given by ∫ϕ(τ| m_0) dm_0), as stated in Eq. (<ref>), and, thus, depends implicitly on n_b. The top panels of Fig. <ref> show the average of the compound rates in mainshock magnitude windows of Δ m = 0.5, measured in the numerical simulations. The curves are scaled according to Eq. (<ref>) in order to identify the deviations from the direct rates. For low branching ratio (n_b=0.2) thecompound rates are almost indistinguishable from the direct rates. This is not the case for higher branching ratios (n_b=0.95) where we identify an exceedance of activity starting at τ∼τ_c and extending to higher values.Eq. (<ref>) is only valid if the dependence on mainshock magnitude can be separated from the temporal kernel. In the modified ETAS model, the coupling imposed in the temporal scale τ_c(m_0) prevents this analytical approach. Yet, if we limit our analysis to the short-time power-law regime only (below τ_c), we can at least provide an explanation for the exceedance point observed for n_b=0.95 in Fig. <ref>.If the triggering kernel consisted of a single power-law regime with exponent θ_1>0 at all time-scales τ→∞, the compound rates would always diverge and increase exponentially fast above a certain characteristic time τ^*. When the branching ratio is high, the characteristic time τ^* is reached before the transition time τ_c towards the fast decaying regime. Since we now only consider the term: ϕ(τ/τ_c) = (τ/τ_c)^θ_1-1, we can separate the productivity from the pure temporal kernel and, following Eq. (<ref>), obtain the resulting rate for τ≤τ_c:Φ_c (τ/τ_c)= k/x_c 10^α m_0θ_1θ_2/θ_2+θ_1Γ(θ_1) ∑_g=1^∞1/Γ (g θ_1)(n_b(τ/τ_c)^θ_1)^gwith Γ(β):= ∫_0^∞ t^β-1exp(-t)dt. In Fig. <ref>.a,b the numerical solution from Eq. (<ref>) (computed up to g=50) is plotted as a guide to the eye, revealing the trend to the deviation from Eq. (<ref>) for high branching ratios. The solution for any value of θ_2>0 and n_b<1 above the transition point τ=τ_cshould not differ significantly from the power-law decay discussed in Ref. <cit.>.§ INFERRING TRIGGERING RATES IN THE MODIFIED ETAS MODEL Without spatial information, the catalogs generated from a point process are given in a sequence of events t_i,m_i and the direct and compound triggering rates cannot be measured directly. Instead, we have to rely on the measurement of mean aftershock sequence rates (MASR) to infer the triggering kernel. In this section we present the results for synthetic catalogs generated from the modified ETAS model (Eq. (<ref>)), and compare them with the actual direct and compound triggering rates. Due to the increase in the aftershock rates with the magnitude of the mainshock as expressed by the productivity law, the overall intensity of the process as defined in Eq. (<ref>) is typically dominated by an earlier large event or a more recent smaller one. To take advantage of this, we evaluate the mean aftershock sequence rates (MASR) as the activity after any event of any magnitude M_i until the next event j with magnitude M_j>M_i, in hopes to obtain long triggering sequences with reliable information. To obtain sufficient statistics, we average the MASR in mainshock magnitude windows (m < M_i < m + Δ m). We must normalize the sequences by the measurement range of each sequence: the time until an event is found with M_j>M_i, equivalent to the complementary cumulative distribution of waiting times CCDF(δ, M_k>m). In Fig. <ref>.c,d the MASR measured in simulations are compared to the compound triggering rates (Φ_c (τ, m)) with the addition of the time-averaged rate ⟨μ (t) ⟩, presenting an approximation to the final expression of Eq. (<ref>). Although the approximation is especially well-suited for low branching ratios, at n_b∼ 1, this assumption overestimates the contribution of unrelated events ⟨μ(t) ⟩ at late times. This is due to taking the time average since⟨μ (t) ⟩ can significantly vary over time. Since triggered events are considered as mainshocks in the MASR, the rate of events triggered by independent branches within the same triggering tree are non-negligible. Although not directly triggered, this type of activity occurs predominantly at early times after the mainshock leading to a time-varying ⟨μ (t) ⟩. In general, we are able to recognize the double power-law kernel as long as the average rate ⟨μ (t) ⟩ is much lower than the triggering rate at the transition point θ_1+θ_2/θ_1θ_2x_c. Around and above this value, the average rate makes the secondary power-law regime unobservable and renders a single power-law decay in triggering rates resembling the standard Omori Utsu relation, but with an exponent value that is unphysically low, as was found in Refs. <cit.>.In the case of ⟨μ (t) ⟩∼θ_1+θ_2/θ_1θ_2x_c and n_b∼ 1,MASR (blue lines in Fig. <ref>.d) render an effective power-law behavior with a low exponent extending up to 5 decades around x_c. This behavior is a consequence of the interplay between the transition point,the rate of independent events, and the contribution of higher generation triggering. Fig. <ref> shows this region in more detail for both a simulation with constant background rate (Fig. <ref>.a) and for σ=3 (Fig. <ref>.b). In the MASR for σ=1, we can fit the effective power-law with an exponent lower than 0.3, not directly related to θ_1 nor θ_2, nor observed in the compound rates. The scaling relations of the triggering rates with m_0 are also affected. Thus, a blind estimation of Omori (p) and the productivity (α) exponents limited within this interval by fitting and collapsing the MASR curves according to the scaling relations is unlikely to retrieve the right form of the triggering kernel. Finally, we evaluate the distribution of waiting times (DWT) for different magnitude thresholds, as shown in Fig. <ref>. We compare the results obtained with (a) the uniform (σ=1) and (b) time-dependent (σ=3) background rate, for different values of x_c and n_b=0.95. No significant differences are found for n_b=0.20 (not shown). The distributions are scaled with the mean waiting time for each threshold, following the scaling relation expected for a Poisson process and observed also for other processes such as seismicity <cit.> and rock fracture <cit.>, for example. In the standard ETAS model, the situation is more complicated <cit.>. This is also true for the modified ETAS model we consider here. Due to the additional scaling parameter (α_τ) the collapsing of the curves to a single scaling function can only be fulfilled over certain ranges. If the events were independent, from the background sampling one would expect an exponential distribution in the DWT for σ=1 and a power law decay with exponent: 2σ -1/σ -1=2.5 for σ=3. Instead, both behaviors are only found for waiting times longer than the typical waiting time of background events(δ∼⟨δ⟩). Below these times, the distribution is dominated by the triggering process, returning the power-law exponents predicted from the relation 2-1/p: 0.67 for the regime with p=0.75 and 1.41 for the regime p=1.7, see Fig. <ref>.§ DISCUSSION As expected, the branching ratio, the ratio between the power-law transition and the background rate are essential to understand the results of MASR in terms of the triggering kernel.The existence of a characteristic scale in the temporal triggering kernel offers a plausible explanation to the detection of effective Omori exponents lower than one in unlocalized catalogs of acoustic emission during mechanical processes <cit.> and calorimetryin structural phase transitions <cit.>.In the specific case of the failure of porous materials under compression <cit.> an effective Omori exponent p∼ 0.7 was observed using MASR, compatible with the short time power-law regime found in localized catalogs <cit.> and the MASR of the modified ETAS model (Fig. <ref>) for transition values x_c≳θ_1θ_2/θ_1+θ_2. The explanation derived from the modified ETAS models is that the second power law regime is hidden by the background rate. As a consequence, the estimation of the productivity term k(m_0) will neglect the existance of a scaling relation in the temporal axis. Thus, only a single exponent can be estimated by collapsing the curves ϕ(τ|m_0). In the modified ETAS model, if the second power-law is not observed in the MASR, one would find a scaling relation k(m_0) ∼ 10^α' m_0, but this measured exponent α' does not correspond to the scaling parameter α, nor the productivity exponent α=α+α_τ. Instead it corresponds to an intermediate value α'=α+α_τ(1-θ_1)= α - θ_1α_τ. This relation directly follows from Eqs. (<ref>) and (<ref>) if one only considers the regime x≤ x_c. The specific value of α'=0.925 for our simulations (and consistent with the experiments in <cit.>) is, however, different from the value α'≈ 0.5 observed during the failure of porous materials under compression <cit.> using MASR. Provided the validity of the modified ETAS model, this suggests that the exponents α and α_τ are not universal across rock fracture experiments. Assuming positive values of α, α_τ and θ_1=0.25, both exponents are however limited within the range 0≤α≤ 2 and 0≤α_τ≤ 2/3.The distribution of waiting times in experimental data usually exhibits a sharp transition between two power-law regimes <cit.>.The power-law regime observed for long waiting times is consistent with the temporal variations of the background rate. Our modified ETAS model can reproduce both observations if the maximum background rate is comparable to the rate at the transition point between the two power laws in the triggering kernel, i.e. x_c = 0.0167 in our simulations.The long time regime of the triggering kernel is only observable before the regime dominated by the background rate for x_c < 0.0167, as shown in Fig. <ref>. The absence of the secondarytriggeringregime for x_c ≳ 0.0167is also consistent with the above mentioned absence of the secondary regime above the background rate in the experimentally measured MASR. Finally, the exact mathematical relation between the p-value of the direct triggering rates and the power-law exponent in the waiting time distribution (2-1/p) is not consistent with the exponents measured using MASR in some experiments <cit.>. While one might expect that this is related to the presence of a high branching ratio <cit.>, this is not supported by our findings for the modified ETAS model considered here and remains an open question. § CONCLUSIONSThe measurement of triggering rates is a non-trivial problem, even in simplified branching processes. Due to limitations in the acquisition systems, we often must rely on the indirect measurement of mean aftershock sequence rates (MASR) to infer the original triggering kernel. The performance of this technique will depend specifically on the background rate and the branching ratio. We can retrieve the triggering kernel with a good precision whenever the individual triggering trees can be separated at low background rates and if secondary triggering can be neglected due to low branching ratio. But, in general, one should expect a strong superposition of independent and secondary activity.When the data is sampled from a triggering kernel with characteristic time scales — such as proposed in <cit.> for rock fracture and implemented in this work in the form of a modified ETAS model — the interplay between the characteristic scale of the triggering and the background rate can render non-scaling regimes in both the measurement of triggering rates using MASR and the distribution of waiting times.As a specific case, if the characteristic time scale and the time scale of the background activity are comparable we find a crossover regime similar to another power law. Yet, its exponent cannot be trivially associated with the underlying parameters of the modified ETAS model.In more general terms, our study here shares light on the problem of separating overlapping triggering cascades or branching trees from limited information to establish the underlying causal relationships, which is not specific to slip and fracture events. Indeed, another prime example is neuronal activity for which such investigations are still at the very beginning <cit.>.unsrt | http://arxiv.org/abs/1707.08956v1 | {
"authors": [
"Jordi Baró",
"Jörn Davidsen"
],
"categories": [
"physics.geo-ph",
"cond-mat.stat-mech",
"physics.data-an"
],
"primary_category": "physics.geo-ph",
"published": "20170726212439",
"title": "Are Triggering Rates of Labquakes Universal? Inferring Triggering Rates From Incomplete Information"
} |
18pt Finite semisimple group algebra of a normally monomial group Shalini Gupta [Corresponding author] Department of Mathematics, Punjabi University, Patiala,India. email: [email protected] Sugandha Maheshwary [Research supported by SERB, India (PDF/2016/000731).] Department of Mathematical Sciences, Indian Institute of Science Education and Research, Mohali, Sector 81, Mohali (Punjab)-140306, India. email: [email protected] =========================================================================================================================================================================================================================================================================================================================================================================================================== In this paper, the complete algebraic structure of finite semisimple group algebra of a normally monomial group is described. The main result is illustrated by computing the explicit Wedderburn decomposition of finite semisimple group algebras of various normally monomial groups. The automorphism groups of these group algebras are also determined. Keywords : semisimple group algebra, normally monomial groups, primitive central idempotents,Wedderburn decomposition. MSC2000 : 16S34; 20C05; 16K20 §INTRODUCTION Let 𝔽_q denote the field containing q elements and let G be a finite group of order relatively prime to q, so thatthe group algebra 𝔽_qG is semisimple. The knowledge of the algebraic structure of 𝔽_qG has applications in coding theory and is useful in describing the automorphism group as well as the unit group of𝔽_qG. This has attracted the attention of several authors <cit.>. Broche et al. <cit.> gavedescription of semisimple group algbra 𝔽_qG, when G is an abelian-by-supersolvable group, by computing its primitive central idempotents and the corresponding simple components, in terms of subgroups of G. Basing on the work in <cit.> and <cit.>, a more precise description of 𝔽_qG, where G is a finite metabelian group, has been given by Bakshi et al. in a series of papers<cit.>.The present paper is a contribution to the work in same series. Recall that a group G is called normally monomial, if every irreducible character of G isnormally monomial, i.e., induced from a linear character of a normal subgroup of G.We provide a complete set of primitive central idempotents and the Wedderburn decomposition of 𝔽_qG, when G is a finite normally monomial group. Since metabelian groups are normally monomial <cit.>, this generalises the main result of <cit.>. It may be remarked that normally monomial groups form a substantial class of monomial groups <cit.> and the rational group algebra of this class of groups has been studied <cit.>.The main result is given in Section 2, after setting up necessary notation and preliminariesand is illustrated on a family of metabelian groups. In Section 3, we give applications of the main result on certain p-groups, which include a family of non-metabelian but normally monomial groups of order p^7, p prime, p≥ 5. It may be pointed out that for most of these group algebras, thepackage<cit.> practically fails to compute the Wedderburn decomposition. Further, we provide the explicit structure of 𝔽_qG, for any group G of order p^n, p prime, n<5. For these groups, thegroup Aut(𝔽_qG) of 𝔽_q-automorphisms of 𝔽_qGhas also been computed. § NOTATION Throughout the paper, G denotes a finite group, and 𝔽_qG denotes the group algebra of G over the field 𝔽_q containing q elements, q relatively prime to the order |G| of G. The notation used are mostly standard and are listed for the ease of reader. [H ≤ ( ) G H is a subgroup (normal subgroup) of G; N_G(H)the normalizer of H in G, H ≤ G;[G:H]the index of the subgroup H in G, H ≤ G;H^g g^-1Hg, g ∈ G, H ≤ G;core_G(H) ⋂_g∈ GH^g , the largest normal subgroup of G contained in H, H ≤ G;[h,g]h^-1g^-1hg, g, h ∈ G; G' {[h,g] | g, h ∈ G};] [|X| the cardinality of the set X; ⟨ X⟩the subgroup generated by the subset X of G;ψ^Gthe character ψ of a subgroup of G, induced to G;𝔽_q an algebraic closure of 𝔽_q; Irr(G) the set of all the distinct irreducible characters of G over 𝔽_q; ker(χ){ g ∈ G| χ(g) = χ(1)}, χ∈Irr(G); 𝔽_q(χ) the field obtained by adjoining χ (g), g ∈ G, to 𝔽_q , χ∈Irr(G);Gal(𝔽_q(χ)/𝔽_q)the Galois group of 𝔽_q(χ) over 𝔽_q; e(χ)χ(1)/|G|∑ _g ∈ Gχ(g) g^-1, χ∈Irr(G); e_𝔽_q(χ)∑_σ ∈ Gal(𝔽_q(χ)/ 𝔽_q) e( σ∘χ), the primitive central idempotent of 𝔽_qG; associated to the character χ , χ∈Irr(G); a| b (a∤ b) a divides (does not divide) b, a,b ≥ 1;φ Euler's phi function; gcd(k,n)the greatest common divisor of the integers k and n;S_n the symmetric group on n symbols, n≥ 1;ℤ_nthe cyclic group of order n, n ≥ 1;SL_n(𝔽) the group of n× n matrices of determinant 1, over the field 𝔽, n ≥ 1;H_1⋊H_2the split extension of the group H_1 by the group H_2; M_n(𝔽) the ring of n × n matrices over the field 𝔽, n ≥ 1; M_n(𝔽)^(s) M_n(𝔽)⊕ M_n(𝔽)⊕...⊕ M_n(𝔽), the direct sum of s copies, s≥ 1;] § NORMALLY MONOMIAL GROUPS §.§ Strong Shoda pairs and idempotents For H K ≤ G, defineε (K, H) :=K̂, K = H ∏ (Ĥ-L̂), otherwise,where K̂ :=1/|K|∑_k ∈ K k and L runs over the normal subgroups of K which are minimal over the normal subgroups of K containing H properly. Set e(G,K,H) to be the sum of distinct G-conjugates of ε(K,H).A strong Shoda pair <cit.> of G is a pair (K, H) of subgroups of G with the property that (i) HK N_G(H); (ii) K/H is cyclic and a maximal abelian subgroup of N_G(H)/H;(iii) the distinct G-conjugates of ε(K, H) are mutually orthogonal.Two strong Shoda pairs (K_1, H_1) and (K_2, H_2) are said to be equivalent, ife(G, K_ 1, H_1) = e(G, K_2, H_2) and a complete set of representatives from distinct equivalence classes of strongShoda pairs of G is called a completeirredundant set of strong Shodapairs of G. We recall the algorithm to compute a complete irredundant set of strong Shoda pairs of a finite normally monomial group G, as described in <cit.>. Let 𝒩 be the set of all normal subgroups of G and for N∈𝒩,let A_N be a normal subgroup of G containing N such that A_N/N is an abelian normal subgroup of maximal order in G/N. Note that the choice of A_N is not unique. However, we need to fix one such A_N. For a fixed A_N, set[ 𝒟_N: the set of all subgroups D of A_N containing N such that core_G(D)=N,;A_N/D is cyclic and is a maximal abelian subgroup of N_G(D)/D.; 𝒯_N: a set of representatives of 𝒟_N under the equivalence relation defined by; conjugacy of subgroups in G.; 𝒮_N:{( A_N,D) | D ∈𝒯_N}. ] It thus follows that if N ∈𝒩 is such that G/N is abelian, then𝒮_N = {(G, N)}, if G/N is cyclic, ∅, otherwise. Observe that every pair (A,D)∈𝒮(G), where𝒮(G):=⋃_N ∈𝒩𝒮_N,is a strong Shoda pair of G. It has been proved (<cit.>, Corollary 1) that 𝒮(G) is a complete irredundant set of strong Shoda pairs of G, if G is a finite normally monomial group. Remark 1. A crucial observation in the above algorithm to compute 𝒮(G), for a given finite group G, is that the choice of A_N is irrelevant. For N ∈𝒩, let A_N^(1) be another normal subgroup of G containing N such that A_N^(1)/N is an abelian normal subgroup of maximal order in G/N and let 𝒟_N^(1), 𝒯_N^(1) and 𝒮_N^(1) be defined corresponding to A_N^(1). Then, any pair in 𝒮_N^(1) is equivalent to a pair in 𝒮_N and vice-versa. This is because, if (A_N^(1),D^(1))∈𝒮_N^(1) and ψ is a complex linear character of A_N^(1) with kernel D^(1), then ψ^G is irreducible and hence by (<cit.>, Lemma 1), there exists (A_N,D)∈𝒮_N such that e_ℚ(ψ^G), the primitive central idempotent of the rational group algebra ℚG, associated to ψ^G, is given by e(G,A_N,D). However, in view of (<cit.>,Theorem 2.1), e_ℚ(ψ^G)=e(G,A_N^(1),D^(1)). This gives that (A_N,D) is equivalent to (A_N^(1) ,D^(1)). The reverse conclusion holds similarly. For a strong Shoda pair (K,H) of G, let 𝒞(K/H) denote the set of q-cyclotomic cosets of (K/H) containing its generators, i.e., if χ is a generator of (K/H), then an element C of 𝒞(K/H) containing χ is the set {χ, χ^q,…, χ^q^o-1}, where n=|K/H| and o=o_n(q), the order of q modulo n. Suppose that N_G(H) acts on 𝒞(K/H) by conjugation, i.e., for g∈ N_G(H), C ∈𝒞(K/H) and χ∈ C, we define C^g={χ^g, χ^g^q,…, χ^g^q^o-1}, where χ^g(k)=χ(gkg^-1). Let ℛ(K/H) denote the set of distinct orbits of 𝒞(K/H)and let E_G(K/H) be the stabilizer of any C∈𝒞(K/H) under the above action. It is easy to see that |ℛ(K/H)|=ϕ(n)|E_G(K/H)|/|N_G(H)|o_n(q). For C∈𝒞(K/H) and χ∈ C, set ε_C(K, H) =1/|K|∑_g ∈ K (tr(χ(g)))g^-1, where g denotes the image of g in K/H, ζ_n a primitive n^th root of unity in 𝔽_q and tr:=tr_𝔽_q(ζ_n)/𝔽_q denotes the trace of the field extension 𝔽_q(ζ_n)/𝔽_q. Let e_C(G, K, H) denote the sum of distinct G-conjugates of ε_C(K, H). Broche and Rio <cit.> proved that e_C(G,K,H) is a primitive central idempotent of 𝔽_qG and 𝔽_qGe_C(G,K,H)≅ M_[G:K](𝔽_q^o/[E :K]), where E=E_G(K/H) and o=o_[K:H](q). §.§ Main theorem The following theorem gives a complete set of primitive central idempotents of finite semisimple group algebra 𝔽_qG, when G is a normally monomial group. Consequently, the complete algebraic structure of 𝔽_qG is obtained. Let𝔽_q be a finite field with q elements and let G be a finitegroup. Suppose that gcd(q, |G|) = 1. Then, E:={e_C(G, A, D) | (A, D)∈𝒮(G),C∈ℛ(A/D)}is a complete set of primitive central idempotents of 𝔽_qG if, and only if, G is normally monomial. In order to prove the above theorem, we need the following lemmas:(<cit.>, Lemma 1) Let 𝔽_q be a finite field with q elements and let G be a finitegroup. Suppose that gcd(q, |G|) = 1. Let ψ be a linear character of a normal subgroup A of G with kernel D. If ψ^G∈(G), then e_𝔽_q (ψ^G)=e_C(G, A, D), for someC∈𝒞(A/D). (<cit.>, Lemma 2.2) Let A be an abelian normal subgroup of maximal order in a group G. If χ is a faithful irreducible normally monomial character of G, then χ is induced from A.Proof. By assumption, χ = β^G for some linear character β of a normal subgroup B of G. As B'⊆core_G(ker (β))=ker (χ)= ⟨ 1⟩, we get that B is abelian and hence χ(1)=[G:B] ≥ [G:A]. Let α be an irreducible constituent of χ_A, where χ_A denotes the restriction of χ to A. By Frobenius reciprocity (<cit.>, Corollary 4.2.2), χ is a constituent of α^G, so that χ(1) ≤α^G(1) = [G : A]. Thus, χ(1) = [G : A] = α^G(1) and χ = α^G. Proof of Theorem <ref>. Assume first that E is a complete set of primitive central idempotents of 𝔽_qG. Let χ∈ (G), so thate_𝔽_q (χ)=e_C(G, A, D), where (A, D)∈𝒮(G) and C∈ℛ(A/D). But then, it follows from Lemma <ref> that e_C(G, A, D)= e_𝔽_q (ψ^G), for some linear character ψ of A with kernel D. Hence e_𝔽_q (χ)=e_𝔽_q (ψ^G), which implies that χ = σ∘ψ^G, where σ∈Gal(𝔽_q(ψ^G)/𝔽_q), i.e., χ is induced from a linear character σ∘ψ of a normal subgroup A of G. Thus, G is normally monomial. Conversely, if G is a normally monomial group, then for χ∈Irr(G), χ = ψ^G, where ψ is a linear character of some normal subgroup K of G with kernel H. Now, ker(χ)= core_G(H)=N(say). Let χ be the corresponding character of G/N. It follows from Lemma <ref> that χ = η^G/N for some linear character η of A_N/N, where A_N/N is an abelian normal subgroup of maximal order in G/N. Let (η)= L/N and let η : A_N→𝔽_q be givenby η(g)= η(gN), g∈ A_N. We have, by Eq. (<ref>),χ = η^G, and therefore, byLemma <ref>, it follows that e_𝔽_q(χ)= e_𝔽_q(η^G)=e_C(G, A_N, L), C∈𝒞(A_N/L). Now, if D∈𝒯_N is a representative of conjugacy class of L, then D=L^x, for some x∈ G and e_C(G, A_N, L)= e_C^x(G, A_N, L^x), with (A_N, D)∈𝒮(G).Consequently, e_𝔽_q(χ)∈ E. Moreover, it is easy to check that all elements in E are distinct and hence, E is a complete set of primitive central idempotents of 𝔽_qG. It thus follows from Theorem <ref> and Eq. (<ref>) that if G is normally monomial, then 𝔽_qG≅⊕_(A,D)∈𝒮(G)M_[G:A](𝔽_q^o/[E :A])^(|ℛ(A/D)|), where E=E_G(A/D) and o=o_[A:D](q). §.§ An illustration We first illustrate the results of Theorem 1 on a family of metabelian groups of order 2sp^2, p an odd prime and s≥ 1. The structure of the rational group algebra of this family has been described in <cit.>. We undertake the case of finite semisimple group algebra for these groups. Let G be a group generated by a, b, x, y satisfying a^p=b^p=x^s=y^2=1, [a,b]=[x,y]=1, x^-1ax=a^r, x^-1bx=b^r, y^-1ay=b, y^-1by=a, where p is an odd prime, r is a positive integer such that gcd(r, p)=1 ands(>1) is the order o_p(r) of r modulo p. The Wedderburn decomposition of𝔽_qG is given as follows: (i) If s is odd, then 𝔽_qG ≅ {[ ⊕_d|s𝔽_q^o_d^(2ϕ(d)/o_d)⊕ M_s(𝔽_q^fg/s)^(2(p-1)/fg)⊕ M_2s(𝔽_q^fg/s)^(p^2-p/2fg), 2∤ f; ⊕_d|s𝔽_q^o_d^(2ϕ(d)/o_d)⊕ M_s(𝔽_q^fg/s)^(2(p-1)/fg)⊕ M_2s(𝔽_q^fg/2s)^(p-1/fg)⊕ M_2s(𝔽_q^fg/s)^((p-1)^2/2fg),2|f ]; . (ii) If s is even, then 𝔽_qG ≅ {[ ⊕_d|s𝔽_q^o_d^(2ϕ(d)/o_d)⊕ M_s(𝔽_q^fg/s)^(2(p-1)/fg)⊕ M_2s(𝔽_q^fg/s)^((p-1)^2/2fg)⊕ M_s(𝔽_q^2fg'/s)^((p-1)/fg'),2∤ f;⊕_d|s𝔽_q^o_d^(2ϕ(d)/o_d)⊕ M_s(𝔽_q^fg/s)^(4(p-1)/fg)⊕ M_2s(𝔽_q^fg/s)^ ((p-1)^2/2fg), 2|f ], . where o_d=o_d(q), f=o_p(q)=o_2p(q), g=gcd{d|s : s/d|f} and g'=gcd{d | s/2 : s/2d , s/2d | f}. Proof. The distinct normal subgroups of G (<cit.>, Lemma 2) areN_0:=⟨ 1 ⟩, N_1:=⟨ ab⟩, N_2:=⟨ ab^-1⟩, N_3:=⟨ ab^-1, y⟩, G_d:= ⟨ a, b, x^d⟩,H_d:= ⟨ a, b, x^d, y⟩, d|s, if s is odd; and, in addition, N_4:= ⟨ ab, x^s/2y⟩,K_d:= ⟨ a, b, x^d/2y ⟩, with d even and d|s, if s is even. Further, for each normal subgroup N of G, the corresponding set 𝒮_N of strong Shoda pairs of G (<cit.>, Lemma 3) is as follows: (i) 𝒮_N_0 ={(⟨ a,b ⟩, ⟨b ⟩)}∪{( ⟨ a, b ⟩, ⟨ ba^λ ^i⟩) |1≤ i ≤p-3/2}, where λ is generator of the multiplicative groupof reduced residue classes modulo p. (ii) 𝒮_N_1 = {(G_s,N_1)}, s odd{( K_s,N_1)},s even. (iii) 𝒮_N_i ={(H_s,N_i)}, i =2, 3. (iv) 𝒮_N_4 ={( K_s,N_4)}. (v) 𝒮_G_d = {( G,G_d)}, d odd ∅,d even. (vi)𝒮_H_d= { ( G,H_d)}, d|s. (vii) 𝒮_K_d= { (G,K_d)}, d|s, d even. We tabulate the computations of parameters involved in applying Theorem <ref>. s odd =.5pt1.4(A_N, D)∈𝒮(G) E_G(A_N,D) o(A_N,D) |ℛ(A_N/D)| (G,G_d), d|s G o_2d(q)=o_d(q) ϕ(d)/o_d(q) (G,H_d), d|s G o_d(q) ϕ(d)/o_d(q) (G_s,N_1) {[ <a,b,x^g,y>,2|f; <a,b,x^g>, 2∤ f ]. {[ fg/2s,2|f;fg/s, 2∤ f ]. {[p-1/fg,2|f; p-1/2fg, 2∤ f ]. (H_s,N_2) <a,b,x^g,y> fg/s p-1/fg (H_s,N_3) <a,b,x^g,y> fg/s p-1/fg (<a,b>,<b>) <a,b,x^g> fg/s p-1/fg (<a,b>,<ba^λ^i>) 2*<a,b,x^g> 2*fg/s 2*p-1/fg 1≤ i ≤p-3/2 s even =.5pt1.4(A_N,D) E_G(A_N,D) o(A_N,D) |ℛ(A_N/D)| (G,G_d), d|s d G o_2d(q)=o_d(q) ϕ(d)/o_d(q) (G,K_d), d|s, d G o_d(q) ϕ(d)/o_d(q) (K_s,N_1) {[ <a,b,x^g,y>,2|f; <a,b,x^g'y>, 2∤ f ]. {[ fg/s, 2|f; 2fg'/s,2∤ f ]. {[ p-1/fg, 2|f; p-1/2fg',2∤ f ]. (K_s,N_4) {[ <a,b,x^g,y>,2|f; <a,b,x^g'y>, 2∤ f ]. {[ fg/s, 2|f; 2fg'/s,2∤ f ]. {[ p-1/fg, 2|f; p-1/2fg',2∤ f ]. (H_s,N_2,) <a,b,x^g,y> fg/s p-1/fg (H_s,N_3) <a,b,x^g,y> fg/s p-1/fg (<a,b>,<b>) <a,b,x^g> fg/s p-1/fg (<a,b>,<ba^λ^i>) 2*<a,b,x^g> 2*fg/s 2*p-1/fg 1≤ i ≤p-3/2 Consequently, Theorem <ref> follows using Eq. (<ref>).§ APPLICATIONS ON P-GROUPS §.§ A family of normally monomial groups of order p^7 We now illustrate the main result of this paper on a normally monomial group of order p^7,p a prime, p≥ 5, which is not metabelian. Let p≥ 5 be a prime and let G be the group generated by a, b, c, d, s, r with the following defining relations: a^p=b^p=c^p=d^p=s^p=r^p=1, [a,c]=[b,c]=[a,d]=[b,d]=1, [a,b]=[c,d], s^-1as=a, s^-1bs=bc, s^-1cs=acd, s^-1ds= ad, [s,r]=[a,b], [r,a]=[r,b]=[r,c]=[r,d]=1. Then, 𝔽_qG ≅ 𝔽_q⊕𝔽_q^f^(p^3-1/f)⊕ M_p( 𝔽_q^f)^(p^4-p/f)⊕ M_p^3( 𝔽_q^f)^(p-1/f), where f=o_p(q). Proof. Let G be as in statement of the theorem.How <cit.> proved that G is a normally monomial group, which is not metabelian. We find 𝒮(G) to write the Wedderburn decomposition of 𝔽_qG. First of all, we compute the set 𝒩 of normal subgroups of G. For this, we begin by observing that if ⟨ 1⟩≠ N∈𝒩, then[a,b]∈ N. Let 1≠ g=a^αb^βc^γd^δr^ρs^ζ[a,b]^η∈ N, where 0≤α, β, γ, δ, ρ, ζ, η < p. Since r^-1gr=g[a,b]^ζ∈ N, we obtain that ζ=0, if [a,b]∉N. Therefore, g=a^αb^βc^γd^δr^ρ[a,b]^η and b^-1gb=g[a,b]^α now yields α=0, if [a,b]∉N. Proceeding similarly, we obtain that if [a,b]∉N, then g=1, which contradicts the assumption and proves (<ref>). We next check that if a∉N, thenN=⟨[a,b]⟩ or ⟨[a,b], ra^i⟩ , where 0≤ i<p. In view of (<ref>), we already have that either N=⟨[a,b]⟩ or there exists an element g=a^αb^βc^γd^δr^ρs^ζ∈ N∖⟨ [a,b]⟩, where 0≤α, β, γ, δ, ρ, ζ < p. Observe that d^-1gd=g[a,b]^γa^-ζ∈ N implies ζ=0, if a∉N. Continuing in this manner, repeated conjugacy by s implies δ=γ=β=0, if a∉N and g=a^αr^ρ, ρ≠ 0.Thus, (<ref>) follows. Next, assumea∈ N, so that ⟨ a, b^-1ab⟩ (∈𝒩)≤ N, i.e., we look for normal subgroups N of G containing ⟨ a, b^-1ab⟩, which are in one-one correspondence with the normal subgroups N of G:=G/⟨ a, b^-1ab⟩. Proceeding as previously, we see that if 1≠g=b^βc^γd^δr^ρs^ζ∈N, for some 0≤β, γ, δ, ρ, ζ < p, then either d∈N or g=d^δr^ρ, ρ≠ 0, and in the later case,N=⟨ a, b^-1ab⟩ or ⟨ a, b^-1ab, rd^i⟩ , where 0≤ i<p. Now, if d∈N, then N is a normal subgroup of G, which contains the normal subgroup ⟨d⟩ of G. Hence, N corresponds to a normal subgroup N of G:=G/⟨d⟩. Continuing this process, the set of normal subgroups of G, is obtained as 𝒩={G,N_0,N_1,N_2,N_3,N_4,N_5,N_6,G^(1)_i,G^(2)_i,G^(3)_i,G^(4)_i,G^(5)_i,H_ij,K_ij | 0≤ i,j < p}, whereN_0:=⟨ 1 ⟩; N_1:=⟨ a,b,c,d,r ⟩; N_2:=⟨ a,b,c,d ⟩; N_3:=⟨ a,c,d ⟩; N_4:=⟨ a,b^-1ab, d ⟩; N_5:=⟨ a,b^-1ab ⟩; N_6:=⟨ [a,b] ⟩; G^(1)_i:=⟨ a,b,c,d,sr^i⟩; G^(2)_i:=⟨ a,c,d,rb^i⟩; G^(3)_i:=⟨ a,b^-1ab,d,rc^i⟩; G^(4)_i:=⟨ a,b^-1ab,rd^i⟩; G^(5)_i:=⟨ [a,b],ra^i⟩; H_ij:=⟨ a,c,d,rb^i,sb^j⟩; and K_ij:=⟨ a,c,d,sr^ib^j⟩, for 0≤ i,j < p. Note that if N=G,N_1,H_ij,G^(1)_i,N_2,K_ij,G^(2)_i or N_3, then G/N is abelian and hence, by Eq. (<ref>), we have 𝒮_N = {(G, N)}, if N∈𝒩_1:={G,N_1,H_ij,G^(1)_i}, ∅, if N∈𝒩_2:={N_2,K_ij,G^(2)_i,N_3}. Further, if N≠ N_0 is such that G/N is not abelian, then N_1/N is abelian.Consequently, A_N=N_1, if N∈𝒩_3:={G^(3)_i,N_4,G^(4)_i,N_5,G^(5)_i,N_6}. Also, it is easy to observe thatA_N=G^(3)_0, if N∈𝒩_4:={N_0}serves the purpose. Since 𝒩=𝒩_1 ∪ 𝒩_2 ∪ 𝒩_3 ∪ 𝒩_4, in view of (<ref>)-(<ref>), we need to find 𝒯_N,when N∈𝒩_3 ∪ 𝒩_4. We first take up the case when N∈𝒩_4, i.e., N=N_0=⟨ 1⟩. In this case, since A_N_0=G^(3)_0 is an abelian p-group, the set 𝒞_0 of subgroups ofA_N_0, which give cyclic quotient can be computed using the algorithm given in Section 5 of <cit.> and is as follows: 𝒞_0={H^(0),H^(1),H_β,H_αβ,H_αβδ | 0≤α,β,δ<p}, where H^(0):=⟨ a, d, r , b^-1ab⟩, H^(1):=⟨ a, d, r⟩, H_β:=⟨ d, r , b^-1aba^β⟩, H_αβ:=⟨ ad^α, r , b^-1abd^β⟩, H_αβδ:=⟨ ar^α, b^-1abr^β, dr^δ⟩, for 0≤α, β, δ<p.In view of (<ref>), we note that for H∈𝒞_0, core_G(H)=N_0=⟨ 1 ⟩, if, and only if, [a,b]∉H. This yields, 𝒟_N_0={H^(1)} ∪ {H_β | β≠ -1}∪{H_αβ, H_αβδ | α≠β, 0≤α,β,δ<p}. It turns out that all subgroups of 𝒟_N_0 are G-conjugates and hence 𝒯_N_0={H^(1)}. For ease of the reader, we list the conjugating element g∈ G such that H^g=H^(1), for every H∈𝒟_N_0. * H_β^x=H^(1), with x=b^ν, where ν(β+1)≡ -1 (mod p), β≠ -1. * H_αβ^y=H^(1), with y=(b^αc)^λ, where λ(β-α)≡ 1 (mod p), α≠β. * H_αβδ^z=H^(1), with z=(b^αc^-δ)^λs^λ, where λ(β-α)≡ 1 (mod p), α≠β. Finally, we proceed to find 𝒮_N, when N∈𝒩_3. For this, we find the set 𝒞_1 of subgroups of A:=N_1=⟨ a, b , c, d , r⟩ ( see (<ref>)) which give cyclic quotient, so that 𝒟_N:={D∈𝒞_1 | core_G(D)=N}. Let D∈𝒞_1. Then, A/D being cyclic ensures [a,b]∈ D. Moreover, any element g=a^αb^βc^γd^δr^ρ∈A:=A/D, where 0≤α, β, γ, δ, ρ< p, is such that g^p=1, so that A=⟨1⟩ or a cyclic group of order p. If a∉D, then a∈A yields A=⟨a⟩. Also, b∈A implies ba^β∈ D, for some 0≤β<p. Similarly, ca^γ, ra^ρ, da^δ∈ D, for some 0≤γ, ρ, δ<p. This, along with (<ref>) and order considerations yield D=D_ρδγβ:=⟨ [a,b],ra^ρ,da^δ,ca^γ,ba^β⟩,for some 0 ≤ρ,δ,γ,β<p. Assuming next, a∈ D and b∉D gives D=D_ρδγ:=⟨ a,[a,b],rb^ρ,db^δ ,cb^γ⟩, 0 ≤ρ,δ,γ<p. Iterating in the above manner, we obtain 𝒞_1:={D^(0),D^(1),D_ρ,D_ρδ,D_ρδγ,D_ρδγβ| 0≤ρ,δ,γ,β<p}, where D^(0):=A, D^(1):=⟨ a,b,c,d ⟩,D_ρ:=⟨ a,b,c,rd^ρ⟩, D_ρδ:=⟨ a,b,dc^δ,rc^ρ⟩, andD_ρδγ, D_ρδγβ are as defined above, 0 ≤ρ,δ,γ,β<p. Since core_G(D) is the largest element of 𝒩,contained in D, in view of (<ref>), (<ref>) & (<ref>),we obtain: * 𝒟_N=∅, if N=N_4,N_5 or N_6. * 𝒟_N={ D_i0}∪{ D_ρ0γ | ρ≡-γ i (mod p), 0< γ<p},if N=G^(3)_i, 0≤ i < p. * 𝒟_N={ D_i}∪{D_ρδ, D_ρδγ | ρ≡-δ i (mod p), 0<δ<p, 0≤γ<p},if N=G^(4)_i, 0≤ i < p. * 𝒟_N={D_iδγβ | 0≤δ,γ,β<p}, if N=G^(5)_i,0≤ i < p. We now omit the details for the remaining part and enlist the set𝒯_N, whenN=G^(3)_i, G^(4)_i,G^(5)_i, 0≤ i < p. * 𝒯_N={ D_i0},if N=G^(3)_i, 0≤ i < p. * 𝒯_N={ D_i}∪{ D_iδ0 | 0<δ<p},if N=G^(4)_i, 0≤ i < p. * 𝒯_N={D_i0γβ, 0≤γ, β<p}, if N=G^(5)_i, 0≤ i < p. In view of (<ref>) and Eq. (<ref>), the information gathered above yields the complete set of strong Shoda pairs of G, and their respective simple components, as tabulated below: =6.5pt1.41Normal Subgroup, N A_N 𝒮_N |ℛ(A_N/D)| ⊕_(A,D)∈𝒮_N𝔽_qGe_𝒞(G,A_N,D) G G {(G, G)} 1𝔽_q N_1=⟨ a, b, c, d, r⟩ G {(G, N_1)} p-1/f 𝔽_q^f H_i,j=⟨ a, c, d, rb^i, sb^j⟩ 2*G 2*{(G, H_ij)} 2*p-1/f 𝔽_q^f 0≤ i, j < p0≤ i, j < p G_i^(1)=⟨ a,b,c,d,sr^i⟩ 2*G 2*{(G, G_i^(1))} 2*p-1/f 𝔽_q^f 0≤ i < p0≤ i < p N_2= ⟨ a,c,d,b⟩ G ∅-- K_i,j=⟨ a, c, d, sr^ib^j⟩,2*G 2*∅ 2*- 2*- 0≤ i, j < p G_i^(2)= ⟨ a, c, d, rb^i⟩ ,2*G 2*∅ 2*- 2*- 0≤ i < p N_3=⟨ a,c,d⟩ G ∅ --G_i^(3)= ⟨ a, d, b^-1ab, rc^i⟩, 2*N_1 2*{(N_1, ⟨ a, b, d,rc^i⟩)} 2*p-1/f M_p(𝔽_q^f) 0≤ i < p0≤ i < p N_4=⟨ a,b^-1ab, d⟩ N_1 ∅ --G_i^(4)= ⟨ a,b^-1ab, rd^i⟩, 2*N_1 {(N_1, ⟨ a,b,c,rd^i⟩)} ∪ 2*p-1/f M_p(𝔽_q^f)^(p) 0≤ i < p{(N_1, ⟨ a,b^-1ab,c,db^δ,rd^i⟩) | 1 ≤δ < p}0≤ i < p N_5=⟨ a, b^-1ab⟩ N_1 ∅- -G_i^(5)= ⟨ [a,b], ra^i⟩, 2*N_1 2*{(N_1, ⟨ [a,b],d,ca^γ,ba^β,ra^i⟩) | 0 ≤β,γ< p} 2*p-1/f M_p(𝔽_q^f)^(p^2) 0≤ i < p0≤ i < p N_6=⟨ [a,b]⟩ N_1 ∅ - - N_0=⟨ 1 ⟩ G_0^(3) {(⟨ a, d, r, b^-1ab⟩, ⟨ a, d, r⟩)} p-1/f M_p^3(𝔽_q^f) Hence, the desired result is obtained. Remarks. Let G be as in statement of Theorem <ref>. Then, by the irredundant set of strong Shoda pairs of G, obtained in Theorem <ref> and (<cit.>, Corollary 1), it follows that ℚG≅ℚ⊕ℚ(ζ_p)^(1+p+p^2)⊕ M_p(ℚ(ζ_p))^(p(1+p+p^2))⊕ M_p^3(ℚ(ζ_p)). A complete irredundant set of strong Shoda pairs of G for the case when p=5 has been computed in <cit.>, using<cit.>, to find the Wedderburn Decomposition of the rational group algebra ℚG. However, it may be pointed out thatpackage<cit.> fails to compute the Wedderburn decomposition of ℚG or 𝔽_qG, for any p≥ 5. In fact, though theoretically,<cit.> could handle the calculation of the Wedderburn decomposition of group algebras of groups of arbitrary size but in practice, if the order of the group is greater than 5000 then the program may crash.featuresthe functionthat determines a complete irredundant set of strong Shoda pairs of G and the functionthat computes the set of primitive central idempotents of semisimple group algebra 𝔽G. For the case when 𝔽=ℚ, these functions are based on the search algorithms provided by Olivieri and del Río <cit.>. Based on the work in <cit.>, alternative and more efficient algorithms have been given in <cit.>.Analogously, in view of Theorem <ref>,improved algorithms can be written and implemented in , for the case when 𝔽 is a finite field. §.§ Groups of order p^n,p prime, n<5 The groups of order p^n, p prime, n<5 are metabelian and hence normallymonomial. Therefore, by applying Theorem <ref>, we now give the explicit structure of 𝔽_qG, and its automorphism group when G is a group of order p^n, p prime, n<5. The Wedderburn decomposition of 𝔽_qG, when G is abelian is well known <cit.> and the automorphism group in this case can be computed as in (<cit.>, Theorem 4). We therefore restrict to the case when G is non-abelian. §.§.§ Groups of order p^3 p=2 If p=2, then G is either Q_8, the quaternion group of order 8 or D_8, the dihedral group of order 8. In this case, the set of primitive central idempotents and the Wedderburn decomposition of 𝔽_qG can be read from (<cit.>, Examples 4.3 & 4.4). In fact, 𝔽_qQ_8≅𝔽_qD_8≅𝔽_q^(4)⊕ M_2(𝔽_q), and it follows from (<cit.>, Theorem 5) that Aut(𝔽_qQ_8)≅Aut(𝔽_qD_8)≅ S_4⊕ SL_2(𝔽_q). p≠ 2 If p ≠ 2, then G is isomorphic to either ⟨ a, b|a^p^2=b^p=1, b^-1ab= a^1+p⟩ or ⟨ a,b,c|a^p=b^p=c^p=1, ab=bac, ac=ca, bc=cb ⟩ (<cit.>, 4.4). The strong Shoda pairs of each of these groups, found in (<cit.>, Theorems 3 & 4) along with Theorem <ref>, Eq. (<ref>) and(<cit.>, Theorem 5), yield the following: Let G be a non-abelian group of order p^3, where p is an odd prime. Then, 𝔽_qG≅𝔽_q⊕𝔽_q^f^((1+p)e)⊕ M_p(𝔽_q^f)^(e), and Aut(𝔽_qG)≅ (ℤ_f^((1+p)e)⋊ S_(1+p)e)⊕((SL_p(𝔽_q^f)⋊ℤ_f)^(e)⋊ S_e), f≠ 1 S_(2+p)e⊕( SL_p(𝔽_q)^(e)⋊ S_e), f=1, where f=o_p(q) and e=p-1/f. §.§.§ Groups of order p^4 p=2 If p=2, then up to isomorphism, there are 9 non-isomorphic groups of order 2^4 (<cit.>, 118) as listed below: * ℋ_1:=⟨ a, b : a^8 =b^2 =1, ba=a^5b⟩; * ℋ_2:=⟨ a, b, c : a^4 =b^2=c^2 =1, cb=a^2bc, ab=ba,ac=ca⟩; * ℋ_3:=⟨ a, b : a^4 =b^4 =1, ba=a^3b⟩; * ℋ_4:=⟨ a, b, c : a^4 =b^2 =c^2=1, ca=a^3c, ba=ab, cb=bc⟩; * ℋ_5:=⟨ a, b, c : a^4 =b^2 =c^2=1, ca=abc, ba=ab, cb=bc ⟩; * ℋ_6:=⟨ a, b, c : a^4 =b^4 =c^2=1, ba=a^3b, ca=ac, cb=bc,a^2=b^2⟩; * ℋ_7:=⟨ a, b : a^8 =b^2 =1, ba=a^7b⟩; * ℋ_8:=⟨ a, b : a^8 =b^2 =1, ba=a^3b⟩; * ℋ_9:=⟨ a, b : a^8 =b^4 =1, ba=a^7b, a^4=b^2⟩. The Wedderburn decomposition of 𝔽_qℋ_i, 1≤ i ≤ 9, is as follows: (i) 𝔽_qℋ_1≅𝔽_q^(8)⊕M_2(𝔽_q)^(2)q≡ 1,5 (mod 8) 𝔽_q^(4)⊕𝔽_q^2^(2)⊕ M_2(𝔽_q^2) q≡ 3,7 (mod 8); (ii) 𝔽_qℋ_2≅𝔽_q^(8)⊕M_2(𝔽_q)^(2)q≡ 1 (mod 4) 𝔽_q^(8)⊕ M_2(𝔽_q^2) q≡ 3 (mod 4); (iii) 𝔽_qℋ_3≅𝔽_q^(8)⊕M_2(𝔽_q)^(2)q≡ 1 (mod 4) 𝔽_q^(4)⊕𝔽_q^2^(2)⊕M_2(𝔽_q)^(2)q≡ 3 (mod 4); (iv) 𝔽_qℋ_4≅𝔽_q^(8)⊕M_2(𝔽_q)^(2); (v) 𝔽_qℋ_5≅𝔽_q^(8)⊕M_2(𝔽_q)^(2)q≡ 1 (mod 4) 𝔽_q^(4)⊕𝔽_q^2^(2)⊕M_2(𝔽_q)^(2)q≡ 3 (mod 4); (vi) 𝔽_qℋ_6≅𝔽_q^(8)⊕M_2(𝔽_q)^(2); (vii) 𝔽_qℋ_7≅𝔽_q^(4)⊕M_2(𝔽_q)^(3)q≡ 1,7 (mod 8) 𝔽_q^(4)⊕ M_2(𝔽_q^2) ⊕ M_2(𝔽_q) q≡ 3,5 (mod 8); (viii) 𝔽_qℋ_8≅𝔽_q^(4)⊕M_2(𝔽_q)^(3)q≡ 1,3 (mod 8) 𝔽_q^(4)⊕ M_2(𝔽_q^2) ⊕ M_2(𝔽_q) q≡ 5,7 (mod 8); (ix) 𝔽_qℋ_9≅𝔽_q^(4)⊕M_2(𝔽_q)^(3)q≡ 1,7 (mod 8) 𝔽_q^(4)⊕ M_2(𝔽_q^2) ⊕ M_2(𝔽_q) q≡ 3,5 (mod 8). Proof. (i) Define N_0:= ⟨ 1 ⟩, N_1:= ⟨ a^4⟩, N_2:= ⟨ a^2⟩, N_3:= ⟨ a ⟩, H_i:= ⟨ a^4, a^2ib ⟩, K_j:= ⟨ a^2, a^jb ⟩ where 0 ≤ i,j ≤ 1. Observe that these subgroups are normal in ℋ_1. Using Eq. (<ref>), we have 𝒮_N_1= 𝒮_N_2= ϕ, 𝒮_N_3={(ℋ_1,N_3 )}, 𝒮_H_i={(ℋ_1,H_i)}, 𝒮_K_j={(ℋ_1,K_j)}, 0 ≤ i,j ≤ 1. In order to find 𝒮_N_0, we see that ⟨ a ⟩ is a maximal abelian subgroup of ℋ_1. Further, the only subgroup D of ⟨ a ⟩ satisfying core_G(D)=⟨ 1⟩ is D= ⟨1 ⟩. This gives 𝒮_N_0={(⟨ a ⟩, ⟨ 1 ⟩)}. Define𝒩_1= {⟨ 1 ⟩, ⟨ a^4⟩, ⟨ a^2⟩, ⟨ a ⟩, ⟨ a,b⟩} ∪ {⟨ a^4, a^2ib ⟩, ⟨ a^2, a^jb ⟩ | 0 ≤ i,j ≤ 1}. It follows from Eqs. (<ref>)-(<ref>) that ⊕_N∈𝒩_1⊕_(A_N,D)∈𝒮_N⊕_C∈ℛ(A_N/D)𝔽_qGe_C(G,A_N,D), is a direct summand of 𝔽_qG and has same 𝔽_q-dimension as 𝔽_qG. This yields that if 𝒩 is the set of all normal subgroups of ℋ_1, then 𝒮_N= ϕ, if N∉𝒩_1, i.e., 𝒮(ℋ_1)=⋃_N∈𝒩_1𝒮_N and 𝔽_qℋ_1≅𝔽_q^(8)⊕M_2(𝔽_q)^(2)q≡ 1,5 (mod 8) 𝔽_q^(4)⊕𝔽_q^2^(2)⊕ M_2(𝔽_q^2) q≡ 3,7 (mod 8) . (ii)-(ix) For 2 ≤ i ≤ 9, consider the following set 𝒩_i of normal subgroups of ℋ_i: [𝒩_2 ={⟨ 1 ⟩, ⟨ a^2⟩, ⟨ a^2,b ⟩, ⟨ a,b ⟩, ⟨ a,b,c ⟩} ∪; {⟨ a^2, b^ic ⟩, ⟨ a, b^ic ⟩, ⟨ ab^ic^j⟩, ⟨ a^2, a^ib, a^jc⟩ | 0≤ i,j ≤ 1 };;𝒩_3 ={⟨ 1 ⟩, ⟨ a^2⟩, ⟨ a ⟩, ⟨ a,b^2⟩, ⟨ a,b ⟩} ∪;{ ⟨ a^2ib^2⟩, ⟨ a^2, a^ib ⟩, ⟨ a^2,a^ib^2⟩ | 0≤ i ≤ 1 };;𝒩_4 = {⟨ 1 ⟩, ⟨ a^2⟩, ⟨ a^2,b ⟩, ⟨ a,b ⟩, ⟨ a,b,c ⟩} ∪; {⟨ a^2ib⟩, ⟨ a^2, b^ic ⟩, ⟨ a, b^ic⟩, ⟨ a^2, ab^ic^j⟩, ⟨a^2,a^ib, a^jc ⟩ | 0≤ i,j ≤ 1 ⟩};;𝒩_5 ={⟨ 1 ⟩, ⟨ b ⟩, ⟨ a^2⟩, ⟨ a^2b ⟩, ⟨ a^2,b ⟩, ⟨ a, b ⟩, ⟨ b,ac⟩, ⟨ a^2,b,c ⟩, ⟨ a,b,c ⟩} ∪; {⟨ b,a^2ic ⟩ | 0≤ i ≤ 1 ⟩};;𝒩_6 = {⟨ 1 ⟩, ⟨ a^2⟩, ⟨ a^2,c ⟩, ⟨ a,b ⟩, ⟨ a,b,c ⟩} ∪;{⟨ a^ic⟩, ⟨ bc^i⟩, ⟨ ab^ic^j⟩, ⟨ a, b^ic ⟩, ⟨a^2,a^ib, a^jc ⟩ | 0≤ i,j ≤ 1 ⟩};;𝒩_7 = {⟨ 1 ⟩, ⟨ a^4⟩, ⟨ a^2⟩, ⟨ a ⟩, ⟨ a,b ⟩} ∪ {⟨ a^2,a^ib ⟩ | 0≤ i ≤ 1 };;𝒩_8 = {⟨ 1 ⟩, ⟨ a^4⟩, ⟨ a^2⟩, ⟨ a ⟩, ⟨ a,b ⟩} ∪ {⟨ a^2,a^ib ⟩ | 0≤ i ≤ 1 };;𝒩_9 = {⟨ 1 ⟩, ⟨ a^4⟩, ⟨ a^2⟩, ⟨ a ⟩, ⟨ a,b ⟩} ∪ {⟨ a^2,a^ib ⟩ | 0≤ i ≤ 1 }.;] Proceeding as in (i), we get the following complete and irredundant set of strong Shoda pairs of ℋ_i, 2 ≤ i ≤ 9, which yield the desired result. [ (ii) 𝒮(ℋ_2) ={(ℋ_2, ℋ_2), (⟨ a, b⟩, ⟨ b ⟩), (ℋ_2, ⟨ a, b ⟩)} ∪; {(ℋ_2, ⟨ a,b^ic⟩), (ℋ_2, ⟨ a^2,a^ib, a^jc⟩) | 0 ≤ i,j ≤ 1 } ;;(iii) 𝒮(ℋ_3) = {(ℋ_3, ℋ_3), (ℋ_3, ⟨ a,b^2⟩), (ℋ_3, ⟨ a⟩), (ℋ_3, ⟨ a^2, ab^2⟩)} ∪;{(⟨ a,b^2⟩, ⟨ a^2ib^2⟩), (ℋ_3, ⟨ a^2, a^ib⟩) | 0≤ i≤ 1} ;; (iv) 𝒮(ℋ_4) = {(ℋ_4, ℋ_4), (ℋ_4, ⟨ a, b ⟩)} ∪; {(⟨ a, b ⟩, ⟨ a^2ib⟩), (ℋ_4, ⟨ a,b^ic⟩), (ℋ_4, ⟨ a^2,a^ib, a^jc⟩) | 0≤ i,j≤ 1} ;;(v) 𝒮(ℋ_5) = {(ℋ_5, ℋ_5), (ℋ_5, ⟨ a,b⟩), (ℋ_5, ⟨ a^2, b,c ⟩), (ℋ_5,⟨ b,ac⟩) } ∪; {(⟨ a, b ⟩, ⟨ a ⟩), (⟨ a, b ⟩, ⟨ a^2b⟩)} ∪{(ℋ_5, ⟨ b,a^2ic⟩) | 0 ≤ i ≤ 1};; (vi) 𝒮(ℋ_6) ={(ℋ_6, ℋ_6), (ℋ_6, ⟨ a, b ⟩)} ∪; {(⟨ a, c ⟩, ⟨ a^ic⟩), (ℋ_6, ⟨ a,b^ic⟩ ), (ℋ_6, ⟨ a^2,a^ib,a^jc⟩ ) | 0 ≤ i,j ≤ 1};;(vii) 𝒮(ℋ_7) = {(ℋ_7, ℋ_7), (ℋ_7, ⟨ a⟩)} ∪; {(⟨ a⟩, ⟨a^4i⟩), (ℋ_7, ⟨a^2,a^ib ⟩) | 0 ≤ i ≤ 1} ;; (viii) 𝒮(ℋ_8) = {(ℋ_8, ℋ_8), (ℋ_8, ⟨ a⟩)} ∪; {(⟨ a⟩, ⟨a^4i⟩), (ℋ_8, ⟨a^2,a^ib ⟩) | 0 ≤ i ≤ 1} ;; ] [ (ix) 𝒮(ℋ_9) = {(ℋ_9, ℋ_9), (ℋ_9, ⟨ a⟩)} ∪; {(⟨ a⟩, ⟨a^4i⟩), (ℋ_9, ⟨a^2,a^ib ⟩) | 0 ≤ i ≤ 1}.; ] The automorphism group of 𝔽_qℋ_i, 1≤ i ≤ 9, is as follows: * Aut(𝔽_qℋ_1 )≅ S_8⊕ (SL_2(𝔽_q)^(2)⋊ S_2), q≡ 1,5 (mod 8) S_4⊕ (ℤ_2^(2)⋊ S_2) ⊕ (SL_2(𝔽_q^2) ⋊ℤ_2), q≡ 3,7 (mod 8) ; * Aut(𝔽_qℋ_2 )≅ S_8⊕ (SL_2(𝔽_q)^(2)⋊ S_2), q≡ 1 (mod 4) S_8⊕ (SL_2(𝔽_q^2) ⋊ℤ_2), q≡3 (mod 4) ; * Aut(𝔽_qℋ_3 )≅ S_8⊕ (SL_2(𝔽_q)^(2)⋊ S_2), q≡ 1 (mod 4) S_4⊕ (ℤ_2^(2)⋊ S_2) ⊕ (SL_2(𝔽_q)^(2)⋊ S_2),q≡ 3 (mod 4) ; * Aut(𝔽_qℋ_4 )≅ S_8⊕ (SL_2(𝔽_q)^(2)⋊ S_2) ; * Aut(𝔽_qℋ_5 )≅ S_8⊕ (SL_2(𝔽_q)^(2)⋊ S_2), q≡ 1 (mod 4) S_4⊕ (ℤ_2^(2)⋊ S_2) ⊕ (SL_2(𝔽_q)^(2)⋊ S_2),q≡ 3 (mod 4) ; * Aut(𝔽_qℋ_6 )≅ S_8⊕ (SL_2(𝔽_q)^(2)⋊ S_2); * Aut(𝔽_qℋ_7 )≅ S_4⊕ (SL_2(𝔽_q)^(3)⋊ S_3), q≡ 1,7 (mod 8) S_4⊕ (SL_2(𝔽_q^2) ⋊ℤ_2) ⊕ SL_2(𝔽_q),q≡ 3,5 (mod 8) ; * Aut(𝔽_qℋ_8 )≅ S_4⊕ (SL_2(𝔽_q)^(3)⋊ S_3), q≡ 1,3 (mod 8) S_4⊕ (SL_2(𝔽_q^2) ⋊ℤ_2) ⊕ SL_2(𝔽_q),q≡ 5,7 (mod 8) ; * Aut(𝔽_qℋ_9 )≅ S_4⊕ (SL_2(𝔽_q)^(3)⋊ S_3), q≡ 1,7 (mod 8) S_4⊕ (SL_2(𝔽_q^2) ⋊ℤ_2) ⊕ SL_2(𝔽_q),q≡ 3,5 (mod 8) . p≠ 2 If p is an odd prime, then, up to isomorphism, the following are non-abelian groups of order p^4 (<cit.>, 117): * 𝒢_1:= ⟨ a, b : a^p^3= b^p=1, ba=a^1+p^2b ⟩; * 𝒢_2:=⟨ a, b, c: a^p^2 =b^p =c^p=1, cb=a^pbc, ab=ba, ac=ca ⟩; * 𝒢_3:=⟨ a, b : a^p^2 =b^p^2 =1, ba=a^1+pb⟩; * 𝒢_4:=⟨ a, b, c: a^p^2 =b^p =c^p=1, ca=a^1+pc, ba=ab, cb=bc ⟩; * 𝒢_5:=⟨ a, b, c: a^p^2 =b^p =c^p=1, ca=abc, ab=ba, bc=cb ⟩; * 𝒢_6:=⟨ a, b, c: a^p^2 =b^p =c^p=1, ba=a^1+pb, ca=abc, cb=bc ⟩; * 𝒢_7:=⟨ a, b, c: a^p^2 =b^p=1, c^p=a^p, ab=ba^1+p, ac=cab^-1, cb=bc ⟩, if p=3,⟨ a, b, c: a^p^2 =b^p =c^p=1, ba=a^1+pb, ca=a^1+pbc, cb=a^pbc ⟩, if p>3 ; * 𝒢_8:=⟨ a, b, c: a^p^2 =b^p=1, c^p=a^-p, ab=ba^1+p, ac=cab^-1, cb=bc ⟩, if p=3,⟨ a, b, c: a^p^2 =b^p =c^p=1, ba=a^1+pb, ca=a^1+dpbc, cb=a^dpbc ⟩, if p>3 d≢0,1 ( mod p); * 𝒢_9:=⟨ a, b, c, d: a^p =b^p =c^p=d^p=1, dc=acd, bd=db, ad=da, bc=cb, ac=ca,ab=ba ⟩; * 𝒢_10:=⟨ a, b, c: a^p^2 =b^p=c^p=1, ab=ba, ac=cab, bc=ca^-pb ⟩, if p=3, ⟨ a, b, c,d: a^p =b^p =c^p=d^p=1, dc=bcd, db=abd,ad=da, bc=cb,ac=ca,ab=ba ⟩, if p >3. For 1 ≤ i ≤ 10,a complete irredundant set𝒮(𝒢_i) of strong Shoda pairs of 𝒢_i has been computed in (<cit.>, Theorem 3) which in view of Theorem <ref>, and Eq. (<ref>) yields the following: The Wedderburn decomposition of 𝔽_q𝒢_i, 1≤ i ≤ 10, is as follows: * 𝔽_q𝒢_1≅𝔽_q⊕𝔽_q^f^((1+p)e)⊕𝔽_q^fp^(pe)⊕ M_p(𝔽_q^fp)^(e); * 𝔽_q𝒢_2≅𝔽_q⊕𝔽_q^f^((1+p+p^2)e)⊕ M_p(𝔽_q^fp)^(e); * 𝔽_q𝒢_3≅𝔽_q⊕𝔽_q^f^((1+p)e)⊕𝔽_q^fp^(ep)⊕ M_p(𝔽_q^f)^(pe); * 𝔽_q𝒢_4≅𝔽_q⊕𝔽_q^f^((1+p+p^2)e)⊕ M_p(𝔽_q^f)^(pe); * 𝔽_q𝒢_5≅𝔽_q⊕𝔽_q^f^((1+p)e)⊕𝔽_q^fp^(pe)⊕ M_p(𝔽_q^f)^(pe); * 𝔽_q𝒢_6≅𝔽_q⊕𝔽_q^f^((1+p)e)⊕ M_p(𝔽_q^f)^((1+p)e); * 𝔽_q𝒢_7≅𝔽_q⊕𝔽_q^f^((1+p)e)⊕ M_p(𝔽_q^f)^(e)⊕ M_p(𝔽_q^fp)^(e); * 𝔽_q𝒢_8≅𝔽_q⊕𝔽_q^f^((1+p)e)⊕ M_p(𝔽_q^f)^(e)⊕ M_p(𝔽_q^fp)^(e); * 𝔽_q𝒢_9≅𝔽_q⊕𝔽_q^f^((1+p+p^2)e)⊕ M_p(𝔽_q^f)^(pe); * 𝔽_q𝒢_10≅{[ 𝔽_q⊕𝔽_q^f^((1+p)e)⊕ M_p(𝔽_q^f)^(e)⊕ M_p(𝔽_q^fp)^(e),p=3; 𝔽_q⊕𝔽_q^f^((1+p)e)⊕ M_p(𝔽_q^f)^((1+p)e),p>3 ], . where o_p(q)=f and e=p-1/f. The automorphism group of finite semisimple group algebra of groups of order p^4, p an odd prime, can now be computed similarly. Acknowledgement The authors are grateful to I. B. S. Passi and G. K. Bakshi for their valuable comments and suggestions. amsplain | http://arxiv.org/abs/1707.08311v1 | {
"authors": [
"Shalini Gupta",
"Sugandha Maheshwary"
],
"categories": [
"math.RA",
"16S34, 20C05, 16K20"
],
"primary_category": "math.RA",
"published": "20170726074708",
"title": "Finite semisimple group algebra of a normally monomial group"
} |
Hydrogen Bonding in Protic Ionic Liquids]Hydrogen Bonding in Protic Ionic Liquids: Structural Correlations, Vibrational Spectroscopy, andRotational Dynamics of Liquid Ethylammonium Nitrate^1Institut für Physik, Universität Rostock, Albert-Einstein-Str. 23-24, D-18059 Rostock, Germany ^2Abteilung für Physikalische Chemie, Institut für Chemie, Universität Rostock, Dr.-Lorenz-Weg 1, D-18059 Rostock, [email protected] July 26, 2017 The properties of the hydrogen bonds in ethylammonium nitrate are analyzed by using molecular dynamics simulations and infrared as well as nuclear magnetic resonance experiments.Ethylammonium nitrate features a flexible three-dimensional network of hydrogen bonds with moderate strengths, which makes it distinct from related triethylammonium-based ionic liquids. First, the network's flexibility is manifested in a not very pronounced correlation of the hydrogen bond geometries, which is caused by rapid interchanges of bonding partners. The large flexibility of the network leads to a substantial broadening of the mid-IR absorption band, with the contributions due to N-H stretching motions ranging from 2800 to 3250 . Finally, the different dynamics are also seen in the rotational correlation of the N-H bond vector, where a correlation time as short as 16.1 ps is observed.§ INTRODUCTION Ethylammonium nitrate (EAN) has been synthesized more than hundred years ago by Paul Walden <cit.>. EAN is formed by the neutralization of ethylamine with concentrated nitric acid. It consists solely of ions, namely ethylammonium and nitrate, and melts significantly below 100^∘C, i.e. at 13-14^∘C. Thus, EAN can be regarded as the first room temperature ionic liquid (IL) <cit.>. EAN belongs to the subset of protic ionic liquids (PILs) <cit.>, which are of practical interest due to their ongoing development forhigh-performance Li-ion cells in electric vehicles, instead of the flammable carbonate electrolytes <cit.>, as well as electrolytes for advanced fuel cells <cit.>. Recent studies mainly addressed the structure and thermodynamics of EAN. This archetypical PIL forms a three-dimensional hydrogen bonded (H-bonded) network, similar to that of water. In principle, these structural features were supported by molecular dynamics (MD) simulations <cit.>. In addition, symmetric and asymmetric stretching modes of the H-bonds (HBs) were observed in far infrared (FIR) spectra <cit.>.EAN evaporates as contact ion pairs leading to low vapor pressures and an enthalpy of vaporization of about 105.3 kJmol^-1 <cit.>. This stability makes EAN attractive as an electrolyte for new generation fuel cells up to 419 K. In contrast, EAN evaporates as molecules near the boiling point at 513 K. Although structure and thermodynamics of EAN are widely studied, a detailed analysis of the HB network and the related dynamical information is still lacking. This is somewhat surprising because the “flickering HB network”, similar to that of water, suggests strongly enhanced dynamics of molecular vectors that are involved in H-bonding compared to those in the one-dimensional H-bonded PIL analogues <cit.>. For EAN, wherein the proton exchange is fast compared to the time scale of nuclear magnetic resonance (NMR) spectroscopy, the reorientational correlation times for the N-H molecular vector, τ_ NH, canonly be obtained from ^15N-enhanced proton relaxation time experiments. To avoid the demanding and costly synthesis of isotopic substituted compounds, one can instead measure the deuteron relaxation times of the N-D deuterons. This relaxation mechanism is strong and purely intramolecular in nature. However, proper correlation times, τ_ NH, can be obtained from the relaxation times, T_1, only, when reliable deuteron quadrupole coupling constants, χ_ D, are known. Unfortunately, this is not the case for ILs, neither for the solid, the liquid nor the gas phase. For this reason we recently developed a method for deriving χ_ D from a relation between density functional theory (DFT) calculated χ_ D (for N-D) and proton chemical shifts δ^1H (for N-H) within clusters of ILs <cit.>. Thus, with simple measurements of the proton chemical shifts,one can then determine accurate χ_ D values for the liquid phase.MD simulationsare an established tool to study ILs <cit.>. To get a reliable description of the complex atomic motions intrinsic to HBs, methods beyond simple force field-based MDare needed. In principle, DFT-based ab initio MD would be the method of choice, but it is hampered by the high computational costs <cit.>. An alternative is provided by the self-consistent charge DFT-based tight-binding approach (DFTB) <cit.>. Although applications of DFTB to the dynamics of ILs are scarce, available simulations provided evidence, e.g., for reliable prediction ofX-ray diffraction data <cit.> and vibrational spectra as well as geometric correlations related to the HB motion of PILs <cit.>. In the present work, we continue our investigation of the molecular dynamics of HBs in alkylammonium nitrate PILs <cit.> by focussing on liquid EAN. Using a combined experimental and theoretical approach, the N-H⋯O HBs will be analyzed in terms of their geometric correlations, FIR and mid infrared (MIR) signatures as well as the rotational correlation of the N-H bondvectors. It is the purpose of this work to provide detailed information about the structure and dynamics of the very first IL EAN, with particular focus on the HB network, which resembles that of water. Section <ref> starts with an outline of the DFTB MD simulation protocol. Next, the method for obtaining the reorientational correlation time, τ_ NH, for N-H/D bonds in EAN is sketched, which is based on the δ^1Hversus χ_ D relation calculated for EAN clusters. Further, experimental details for NMR and FIR/MIR measurements are provided. In Section <ref> results for structure, infrared (IR) spectra, androtational correlation time are discussed. A summary is provided in Section <ref>.§ METHODS§.§ Molecular Dynamics A simulation box consisting of 60 EAN ion pairs and 900 atoms with periodic boundary conditions was built from 10 randomly positioned hexamer clusters.To obtain a bulk structure, the system was equilibratedusing the force field parameters from <cit.>. To converge the box size a1.5 ns simulation in the NPT ensemble at 300 K was performed using aParrinello-Rahman barostat. The resulting density of 1.244 g/cm^3 is very close to the literature value of 1.216 g/cm^3 <cit.>. To control the temperature in the subsequent NVT simulation (at 300 K), a Nosé-Hover thermostat with chain length 10 is used. The length of the NVT trajectory has been5 ps. The force field-based MD simulations were performed with the Gromacs 4.5.5 software <cit.>.The final equilibrated force field structurewas used as input for subsequent DFTB simulations with the DFTB+ software package <cit.>. Here, 3rd order as well as dispersioncorrections have been used together with the3-obSlater-Koster parameter set <cit.>.A 25 ps NVT trajectory has been simulated (time step 0.5 fs), which was used to sample starting pointsfor NVE trajectories every 5 ps. The NVE trajectories had a length of 30 ps with a time step of 0.5 ps.The IR spectrum, I(ω), is obtained by Fourier transformation ofthe dipole moment trajectory, μ⃗(t), using a standard Kaiser window function, κ(t), with Kaiser parameter 10,i.e. <cit.>I(ω) = ω^2 ⟨ | ∫_0^Texp(-i ω t) μ⃗(t) κ(t) dt|^2 ⟩.Here, the canonical ensemble average denoted by < ⋯> is obtained by sampling initial configurations forNVE trajectories as described above. The window function ensures that the dipole is damped to zero at the edges. The Kaiser parameter controls the time scale of the decay and is chosen large enough to avoid artificial broadening ofthe spectra. The Mulliken charges from DFTB simulations are used to calculate the dipoles of the individual ions andof the NH_3 and CH_3 groups. The total box dipole is obtained as the sum of the dipoles of the individual ions. Note that thedipole of an ionic system is not uniquely defined, as discussed in Ref. <cit.>. Here, we employed the center of mass of the considered subsystem as the reference for the dipole calculations. The spectra presented below are smoothed by convolution with a Gaussian function with width parameter σ= 25cm^-1 in the MIR and σ= 12.5cm^-1 in the FIR region.§.§ Reorientational Correlation Time For the determination of the reorientational correlation time of the N-H molecular vectors, τ_ NH,we measured the deuteron relaxation time (T_1)_ D of the deuterated species [EtND_3][NO_3] at 303 K. We have chosen quadrupolar relaxation because it is the strongest relaxation mechanism and presents a purely intramolecular process. Deuteron nuclear magnetic relaxation is driven by the interaction of the electrostatic quadrupole moment, eQ, of the deuteron nucleus with the main component of the electric field gradient (EFG) tensor at the nucleus, q_zz, generated by the electron distribution surrounding the nucleus along the N-H bonds. The relaxation rate (1/T_1)_ D is given by <cit.>(1/T_1)_ D=3/10π^2 ( 1+ η_ D^2/3)χ_ D^2( τ_ NH/1+ω_0^2τ_ NH^2 +4τ_ NH/1+4ω_0^2τ_ NH^2) ,where χ_ D is the deuteron nuclear quadrupole coupling constant and η_ D=(q_xx-q_yy)/q_zz is the related asymmetry parameter. If the relaxation process does not depend on frequency and the extreme narrowing condition, ω_0τ_ NH≪ 1, holds, this equation simplifies to (1/T_1)_ D=3/2π^2 ( 1+ η_ D^2/3)χ_ D^2 τ_ NH.Note that we tacitly assumed that the reorientational correlation time of the principal axis of the deuterium EFG, actually entering these equations, is identical to τ_ NH <cit.>. Additionally, experimental data indicate that there is a negligible isotope effect for the rotational diffusion <cit.>.Before the reorientational correlation timescan be determined from the quadrupolar relaxation rates, the deuteron quadrupole coupling constantmust be known. Despite what the term `coupling constant' implies, χ_ D has been shown to be temperature and solvent dependent<cit.>. Thus χ_ D is a sensitive probe for H-bonding. Unfortunately, for EAN the deuteron quadrupole coupling constant is not known at all. Thus, we used an approach for the determination of the proper deuteron quadrupole coupling constant in the liquid phase, which has been successfully applied for N-D coupling constants in other PILs and is briefly described here <cit.>.The relation between the principal component of the EFG tensor, eq_zz, and the deuteron nuclear quadrupole coupling constant, χ_ D, is given by (h is Planck's constant) χ_ D=(eQeq_zz/h) . In principle, the deuteron quadrupole coupling constantcan be now obtained by multiplying calculated EFGby a calibrated nuclear quadrupole moment, eQ. The latteris obtained by plotting the measured gas phase quadrupole coupling constants from microwave spectroscopy versus the calculated EFGs for small molecules such as CD_4, CD_3OH, ND_3 etc. as described by Huber et al. <cit.>.The slope gives a reasonable value of eQ =295.5 fm^2. This approach holds regardless of whether gas phase molecules, H-bonded clusters or IL complexes are investigated. Recently, we could show for a large set of DFT-calculated clusters of trialkylammonium-based PILs that there exists a linear correlation between the calculated proton chemical shiftsand the calculateddeuteron quadrupole coupling constants <cit.>. The advantage now is that δ^1H can be easily measured in the liquid phase, and thus, by virtue of the computationally established linear correlation, provides access totheχ_ D values. Thus, we applied this approach to the present EAN. First, we obtained the proton chemical shifts, δ^1H, and the quadrupole coupling constants, χ_ D, from DFT calculated clusters including n=2, 4, 6, 8 ion-pairs.Throughout, the geometries of the clusters as well as the proton chemical shifts and the EFGsof the deuterons were calculated using Gaussian 09 <cit.> at the B3LYP-D3/6-31+G* level of theory, including D3 dispersion correction as introduced by Grimme et al. <cit.>.The proton chemical shifts were referenced againstTMS (tetramethylsilane), as it was done in the experiment (see below). The χ_ D values were derived by multiplying the calculated main components of the EFG tensor q_zz with the calibrated nuclear quadrupole moment eQ =295.5 fm^2.The asymmetry parameter of the EFGfor N-D, η_ D, was found to be negligible and did not have to be considered in (<ref>). This has been shown for molecular systems such as ammonia, formamide, N-methyl formamide and N-methyl acetamide<cit.>.The reorientational correlation times, τ_ NH, thus obtained from (<ref>) will be compared with simulation results. To this end, the rotational time correlation function, C(t), is computed according toC(t) = < P_2( u⃗(0) u⃗(t)) >. Here, P_2 is the second Legendre polynomial and u⃗(t) the unit vector along the N-H bond <cit.>. From C(t) the rotational correlation timefollows asτ_ NH = ∫_0^∞< P_2( u⃗(0) u⃗(t) > dt . §.§ Experimental Methods TheEAN samples were of commercial origin (IOLITEC) with purity degrees of>97%. Prior to experiments, the sample was subjected to vacuum evaporation at 333 K for more than 72 hours to remove possible traces of solvents and moisture. Once purified, the sample was transferred under inert argon atmosphere and stored in a hermetically sealed bottle. The purity of the EAN sample was confirmed with ^1H and ^13C NMR (see Supplement). The residual water concentration in the sample for combustion calorimetry was determined by Karl Fischer titration and was lower than 100 ppm. The ^1H NMR spectra of the protonated EAN were recorded on a Bruker Avance 500 MHz spectrometer using 5 mm probes. The δ^1H chemical shifts were measured versus TMS.The EAN sample was deuterated by H/D exchange in D_2O and dried again. Longitudinal magnetic relaxation times T_1 were measured with the same spectrometer at a resonance frequency of ν_0=ω_0/2π= 76.7 MHz employing the inversion recovery (180^∘ - τ - 90^∘) pulse sequence <cit.> (see Supplement). T_1 is estimated to be accurate to within ±2%. Temperature calibrations were carried out using methanol andethylene glycol NMR thermometers. TheFIR measurements were performed with a Bruker Vertex 70 FTIR spectrometer. The instrument was equipped with an extension for measurements in this spectral region. This equipment consists of a multilayer mylar beam splitter, a room temperature DLATGS detector with preamplifier and polyethylene windows for the internal optical path. The accessible spectral region for this configuration lies between 30 and 680 . The spectrometer was purged continuously with dry air during the experiments in order to minimize contributions from atmospheric water vapor. Further reduction of these signals was achieved by utilization of telescope tubes with polyethylene windows reducing the optical path in the sample chamber to a minimum. In this way, fluctuations in the atmospheric water content can be prevented. The L.O.T.-Oriel cell was equipped with polyethylene windows. The 0.012 mm path length was realized by tin spacers. For each spectrum 100 scans at a spectral resolution of 1 were recorded. The experimental spectrum shown in Figure <ref> has been published elsewhere <cit.>.The MIR measurements were performed with a Bruker Vector 22 FTIR spectrometer. An L.O.T.-Oriel variable-temperature cell equipped with CaF_2 windows was used. For each spectrum 128 scans were recorded at a spectral resolution of 1 .§ RESULTS AND DISCUSSION§.§ H-Bond GeometriesEAN facilitates multiple HB opportunities and thus is capable of forming a complex network structure, which has been discussed to be comparable to that of water <cit.>. There exist three acceptor oxygen atoms per anion and three donor positions per cation, so that in principle a fully saturated HB network is possible. Note that the trajectory simulations discussed below did not exhibit any proton transfer events leading to a stable product of two neutral molecules. This might be related to the relatively short propagation time.Nitrate is a strongly interacting anion and we assume all HB donors to be involved in H-bonding. In the following a HB is counted always between an ammonium hydrogen and its closest oxygen.The average HB distances are r_NO=2.91 Å, r_NH=1.04 Å, andr_OH=2.02 Å. These values are in the rangeexpectedfor moderate strong HBs <cit.>. Further, EAN exhibits a bent HB geometry with an average angle, α, between the NH and OH vector of 34^∘ (see also figure <ref>). Only 16% of the HBs are linear (α <15^∘). Hayes et al. investigated the structure of various PILs including EAN with Empirical Potential Structure Refinement (EPSR) and Monte Carlo methods <cit.>. They found 12% of the HBs being linear (α <15^∘) and 1.6oxygen atoms per ammonium hydrogen. In total this classifies theHBs in EAN as being substantially bifurcated.The continuous lifetime of a H-bonded ion pair is (average lifetime of non-interrupted HBs) and the intermittent lifetime is(total average lifetime along trajectories). It is not unusual that the two lifetimes do not match <cit.>. However, it should be noted that in the current analysis only one HB per ammonium hydrogen is possible. Thus, thermal motion of a bifurcated HB might, according to the distance criterion, be counted as bond breaking and reforming. In this case, the HB lifetimes would beunderestimated. The HBs of one ammonium group are almost exclusively formed with three different anions (less than 1% of the HBs of one ammonium group are to the same counter anion). In 27% of the cases an oxygen atom is acceptor of two HBs, leaving some of the oxygen atoms unpaired. Furthermore, there exist a number of ring-like structures, where two cations areH-bonded to the same two anions, an exemplary structure is shown in figure <ref>. This can be either via different oxygen atoms of the same anion or via a single oxygen involved in two HBs (as shown). In the present bulk MD simulations containing 60 ion pairs, on average 18.4 rings exist, consisting of 73.6 ions. Since there are three HB possibilities per ionit is possible for ions to be involved in more than one ring. Especially interesting are 'double rings' that form a HB network cluster consisting of two cations that share all three anions. On average 2.6 cluster structures are found.We can conclude that thering-like geometries appear to bean abundant structural motif in EAN and can be taken as evidence for the generally assumed three-dimensional network character of the HBs in EAN. In order to quantify geometric correlations of the HBs in EAN, the valence bond model of Pauling is used in the following <cit.>. Here, starting from the HB distances, r_NH and r_OH, bond orders are defined as p_i = exp(-(r_i - r_i^eq )/b_i) with i = { NH, OH} and r_i^eq being the gas phase equilibrium bond distances. Under the constraint that the sum of the two bond orders must equal one, the two coordinates depend on each other and the HB geometry change along a proton transfer reaction can be described by a single coordinate <cit.>. This path is commonly plotted into a graph showing the correlation between the coordinates r_1=0.5(r_ NH-r_ OH) and r_2=r_ NH+r_ OH.For the case of EAN the gas phase equilibrium distances r_i^eq are obtained from single molecule gas phase DFTB optimizations. The NH bond length is set to the average of the NH bond lengths of the optimized structure. The bond order decay parameters were changed simultaneously starting from the values reported for HBs in crystal structures <cit.> until a good visual fit with the DFTB data sampled along the trajectory was observed. This results in b_OH=0.3 Å and b_NH=0.33 Å.All parameters of the Pauling model are collected in table <ref>.In passing we note that the same values were found for EAN gas phase cluster simulations <cit.>. The resultingreaction path is drawn in figure <ref> (blue line) together with the distribution of geometries according to the trajectory. If the distances r_ NH and r_OH were fully uncorrelated, the geometries would lie on a straight line withslope 2, plotted for reference as a gray line. From the apparent deviation from linearity one can conclude on a weak correlation between the bond distances across the HB.This analysis was previously performed for triethylammonium nitrate (TEAN) using a similar setup <cit.>. The resulting parameters of the valence bond order model are also given in table <ref>. In particular thedecay parameters were found to be b_NH =0.35 Å and b_OH=0.32 Å. Thus, judging from the geometric correlations the HBs in EAN are slightly weaker than in TEAN. This can be attributed to the HB angle. Compared with TEAN, in EAN not only the average angle is larger (26^∘ versus 34^∘), but also the distribution is broader (cf. figure <ref>b). §.§ IR SpectraIn figure <ref> the FIR spectrum calculated via (<ref>) is presented together with an experimental spectrum taken from<cit.>. The calculated spectrum (blue) shows three peaks in the FIR region, i.e. at 92 cm^-1, 201 cm^-1 (with a shoulder at 260 cm^-1), and 439 cm^-1. To understand the origin of these peaks, the spectrum has been additionally computed from the dipoles of the anions, μ⃗_anion (red), and cations, μ⃗_cation (green), only. In general the intensities of the cation and anion spectrado not add up to the full spectrum, because cross-terms μ⃗_cation·μ⃗_anion originating from the square of the total dipole,μ⃗_cation + μ⃗_anion, in (<ref>) are not included in the separate spectra.The peak at the lowest frequencystems almost exclusively from cationic motion and is attributed to be essentially due to dispersion interactions, which usually are dominated by alkyl tails, with smaller contributions due to interionic HB motion.Note that this peak was not present in previous hexamer gas phase cluster simulations, since in this case the alkyl chains are pointing outwards what reduces their interaction <cit.>. The peak at 201 cm^-1 has a low intensity in cationic as well as anionic spectra. Hence, it can be attributed to cross-terms, which suggests that the band is due to a correlated cation-anion motion likely originating from interionic HBs. The peak around 440 cm^-1 was previously assigned by normal mode calculations to a N-C-C bending motion of the cation <cit.>. These assignments are in line with previous FIR experimental investigations of EAN <cit.>. Overall the simulation results reproduce the experimental spectra very well. The largest difference is the absence of a clearlow-frequency dispersion band around 92 cm^-1 in the experiment. This could be due to a larger broadening of this band in the experimental data. In fact related bands around 100 are discernible in less flexible TEA-like systems <cit.>. The MIR spectral region is shown with spectra from experiment (black) and simulation (blue) in figure <ref>. The experimental signal covers a wide range from 2700 to 3300 cm^-1 and shows a left-skewed distribution. The maximum intensity is located at 3092 cm^-1 with a side peak at 3159 cm^-1 and a shoulder at 3250 cm^-1. Further, there are several local maxima on the low-frequency side. In general, the MIR spectrum results from the extended three-dimensional HB network, including possibleproton transfers between anions and cations.Due to H-bonding the symmetric and anti-symmetric NH stretching vibrations are strongly red-shifted and overlap with the corresponding C-H vibrational bands of the ethyl group. To assign theobserved structure, the separate contributions to the calculated spectrum of the NH_3 (green) and CH_3 (red) dipoles are also shown in figure <ref>. For better visual comparison the maximum of the CH_3-only simulation is used as reference to scale all theoretical spectra by aligning it to the position found for the respective modes in experimental Raman spectra <cit.>. This results in a scaling of the frequencies by a factor of 0.978.Overall, the agreement between simulation and experiment is very good, i.e. position and general shape of the spectra match very well. As expected the contribution ofthe polar NH_3 groups has a much higher intensity than those of the non-polar CH_3groups.The IR signal above 3050 cm^-1 stems exclusively from NH_3 motion and the maximum coincides with that of the full spectrum. Note the perfect match of the global maximum at 3092 (unscaled 3162) cm^-1, mainly originating from the NH_3 dipole, which is achieved even though only the CH_3 peak was used to obtain the scaling factor. Additionally, the contribution from the NH_3 dipoles shows the same left-skewed behavior as the full spectrumand it is very broad, i.e. spanning the rangefrom 2800 to 3250 cm^-1. The latter indicates a substantial distribution of HB strengths. The alkyl CH stretching vibrations have a much weaker intensity and span a narrower range in between two maxima at 2923 (3060) cm^-1 and 2863 (2927) cm^-1.We note that the spectrum according to hexamer gas phase simulations using the DFTB method <cit.> deviates from that of the present bulk case. Especially, the cluster shows a larger HB strength, as indicated by a larger blue shift in FIR and a largerred shift in MIR of related spectral features. The results of the present simulations can be further compared with bulk phase calculation for TEAN reported in <cit.>. One finds that for EAN the red shift of the NH-stretching vibrations in the MIR region is smaller, but the blue shift in the FIR region is more pronounced. At first sight, this is a surprising result since the magnitudes of both shifts are associated with the HB strength. According to the above analysis of HB geometries, EAN should feature a weaker HB than TEAN. Thus, the comparison of shifts in the FIR region is misleading insofar as the interionic motion in EAN is influenced by three possible HBs per ion, whereas there is only one for TEAN. On average this may cause a tighter potential and hence a larger blue-shift. §.§ Rotational Correlation In figure <ref> we show thecalculated deuteron quadrupole coupling constants versus calculated proton chemical shifts for HBs taken from different sized EAN clusters. Apparently, there isan almost linear dependence. From linear regression we obtain the relation χ_ D=274.86 kHz - 12.342 δ^1H kHz/ppm.We observe that the proton chemical shifts vary between 2 and 11 ppm depending on the strength of the NH⋯O cation-anion interaction at the differentpositions within the clusters. This range for the chemical shifts corresponds to deuteron quadrupole coupling constants varying from 250 down to 140 kHz. The weaker HBs are characterized by smaller δ^1Hshifts, and larger χ_ D values, and vice versa. It does not matter, whether these pairs of properties are calculated for N-H/N-D in varying sized clusters or for different configurations within these clusters, they all show a linear dependence. This is not at all surprising, because both properties, δ^1Has well as χ_ D, are sensitive to local and directional interactions, such as H-bonding. The ultimate test for the reliability of the linear relationship between both properties in PILs is carried out by interpolating the chemical shifts to 0 ppm, indicating the absence of intermolecular interactions.For this case we expect the calculated χ_ D values to be similar to those measured for the gas phase of similar molecules. And indeed, the estimated value of 275 kHz for χ_ D in EAN is only slightly lower than the measured gas phase and calculated monomer χ_ D values for ammonia (290.6 kHz), formamide (292 kHz) and N-methylformamide (294 kHz), respectively<cit.>.We now use this relation for deriving the deuteron quadrupole coupling constant of EAN in the liquid phase. From the measured proton chemical shift δ^1H=7.375 ppm at 303 K we obtain a deuteron quadrupole constant of χ_ D=184 kHz (see table <ref>). In figure <ref> we compare the χ_ D value for EAN with those of triethylammonium-based PILs, including bis(trifluoromethylsulfonyl)-imide [NTf_2], trifluoromethyl sulfonate [CF_3SO_3], and methylsulfonate [CH_3SO_3] as anions (PILs I-III). At a first glance, it seems to be surprising that EAN, which includes the strongest interacting nitrate anion, does not give the lowest χ_ D value.However, in the TEA-based PILs we have single, one-dimensional HBs between cation and anion, whereas in EAN a three-dimensional HB network is formed. Thus the charge transfer from the anions to N-H bonds of the cations is distributed on three bonds rather than one as it is the case for the other PILs.Clearly, the quadrupole coupling constant is a sensitive probe for H-bonding in ILs, as it has been previously shown for molecular liquids <cit.>.The obtained χ_ D value of 184 kHz can now be used for the determination of the reorientational correlation timeusing (<ref>), which gives τ_ NH= 16.1 ps at 303 K. Compared to the other PILs, which are described by single N-H bonds and weaker interacting anions, the correlation time of EAN are surprisingly low. The τ_ NH values for PILs I-III are 154.2 ps, 190.2 ps and 316.9 ps, respectively and thus one order of magnitude larger than those of EAN (see table <ref>). In PILs I-III the molecular motion of the cations was slowed down in the order [NTf_2] < [CF_3SO_3] < [CH_3SO_3]. For the strongly interacting [NO_3] anion even larger correlation times should be expected. The strongly enhanced dynamics in EAN can ony be explained by the high flexibility of the HB network and fast proton exchange. That becomes obvious if we compare the correlation times for all four PILs with the corresponding viscosities. We recently discussed the relation between reorientational correlation times and viscosities, η, for PILs I-III. The observed linear behavior is expected from the Stokes-Einstein-Debye (SED) relation <cit.>, τ_ NH= η V_ eff/k_ BT if the volume/size of the solute is similar for all PILs. Here, V_ eff is the effective volume obtained by multiplying the volume V with the so-called Gierer-Wirtz factor <cit.>, which for neat liquids has a value of about 0.16. From the slope of the plot in figure <ref>, we can estimate the effective volume, V_ eff, to be about 0.1217 nm^3. If we assume a spherical shape of the solute particles, we obtain an effective radius, R_ eff, of about 3.07 Å. This radius is in reasonable agreement with the one of the TEA cation, which has been calculated to be 3.22 Å <cit.>. Since we use reorientational correlation times of the cation only, which are identical in all three PILs, a linear behavior between τ_ NHandη follows. That the straight line goes through zero further supports the idea that the microscopic correlation times and macroscopic viscosities describe the dynamical behavior of similar molecular species.If we add τ_ NH and η for EAN to figure <ref>, we observe that this relation does not hold for the smaller ethylammonium cation.Even if we take into account, that the volume of ethylammonium is half as large as that of the TEA cation, the measured correlation time is too small to be described by the SED relation.Obviously, the correlation time reflects the shorter life times of the HBs in the three-dimensional network of EAN.In contrast, in PILs I-III τ_ NH reflects the rotational dynamics of the strong HB within the ion pair and is thus more suitable to describe the simple hydrodynamic basis of the SED relation. Further insight into this conclusion comes from the MD simulations.The rotational correlation function C(t), calculated according to (<ref>), is presented in figure <ref>. C(t) shows an initially fast decay, followed by a much slower decay, which doesn't reach zero in the presented time interval. For the following analysis C(t) has been modeled by a tri-exponential function, i.e. C(t) = ∑_i=1^3 a_i exp(- t/τ_i ). The resulting three time scales (amplitudes) are τ_1=0.167 ps (0.24), τ_2=1.69 ps (0.34) and τ_3=13.4 ps (0.33). The rotational correlation time according to (<ref>) for this tri-exponential fit follows as 4.4 ps, i.e. it is about four times smaller than the experimental one of 16.1 ps. However, the simulations clearly confirm the order of magnitude for EAN as compared to the TEA-based systems. We note in passing, that force field MD simulationsof PILs I-IIIby Strauch et al. yielded correlation times, which for II and III substantially exceeded the experimental values <cit.>.The three different decay times contributing to C(t) can be discussed in the context of the findings by Hunger et al., who reported on dielectric relaxation and femtosecond IR experiments <cit.>.Focussing on the transient dynamics of the initially excited ND-stretching vibration, they observed three time scales at 298 K, i.e. 0.2 ps, 2.5 ps, and 15.7 ps. The former two have been related to vibrational energy relaxation via intermediate vibrational states.The slowest one was attributed to reorientational dynamics of the HBs.By comparison with the time constant obtained from dielectric relaxation, they proposed a cooperative reorientation through large angle jumps around the CN bond axis <cit.>. Comparison with the present correlation function analysis yields a good agreement of the slowest component of the tri-exponential fit (13.4 ps) with the above time scale (15.7 ps). However, we must note that the large angle jumps, which were considered to be at the origin for this time scale in <cit.>, have not been clearly observed in the present simulations. Relating the two shorter time scales of <cit.> to the present findings is not that straightforward, although the agreement is striking. Of course, one can argue that any change in HB geometry leading to the loss of orientational correlation is connected to anharmonic couplings, which could be responsible for the observed vibrational relaxation. This is supported by the fact that the τ_1=0.167 ps decay time is equal to the continuous lifetime τ_cont of a HB in EAN. Hence, this time scale reflects the simultaneous effect of HB widening/shortening and NH-bending, which is driven by anharmonic couplings. In passing we note that the disruption of the HBs can also be viewed as being due to librational motion.§ SUMMARYWe provide valuable information about the structure and dynamics of the HB network in EAN, which can be regarded as archetype of protic ionic liquids.The properties of the HBs have been analyzed using simulations and experiments within a multi-dimensional approach, which focussed on geometric correlations, IR, and NMR spectroscopy. DFTB-based molecular dynamics provided the frame for unraveling molecular details behind the macroscopic observables. EAN features a flexible three-dimensional network of HBs with moderate strength. This makes EANdistinct from related TEA-based PILs, which have a one-dimensional network only. Geometric correlations between HBand NH bond lengths are not very prononced, which can be seen as a consequence of the rapid interchange of HB partners, expressed by the rather short continuous lifetime. Signatures of H-bonding are also observed in the MIR and FIR spectra as red and blue-shifted bands, respectively. DFTB has the advantage of providing a means to calculate IR spectra due to molecular fragments in a very efficient way, which allowed to unravel to origin of broad spectral bands. This way we could show that the width of NH vibrational contributions spans the range from 2800 to 3250 . Rotational correlation of the NH bond vectors within the HB provides an additional means for scrutinizing the HB network. Whereas the deuteron quadrupole coupling constants for EAN and TEA-based PILs suggest comparable cation-anion interaction strengths, the dynamics is strikingly different as is manifest in the order of magnitude difference of the respective rotational correlation times. § ACKNOWLEDGMENTSThe authors thank the Deutsche Forschungsgemeinschaft (DFG) for financial support through the Sfb 652. § REFERENCES10 url<#>1#1urlprefixURL walden14_405 Walden P 1914 Bull. Acad. Imper. Sci. 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"authors": [
"Tobias Zentel",
"Viviane Overbeck",
"Dirk Michalik",
"Oliver Kühn",
"Ralf Ludwig"
],
"categories": [
"physics.chem-ph"
],
"primary_category": "physics.chem-ph",
"published": "20170726134140",
"title": "Hydrogen Bonding in Protic Ionic Liquids: Structural Correlations, Vibrational Spectroscopy, and Rotational Dynamics of Liquid Ethylammonium Nitrate"
} |
Wigner tomography of multispin quantum states Steffen J. Glaser January 15, 2018 ============================================= The star chromatic indexof a multigraph G, denoted χ'_s(G),is the minimum number of colors needed to properly color the edges of G such that no path or cycle of length four is bi-colored.A multigraph G is star k-edge-colorable if χ'_s(G)≤ k. Dvořák, Mohar and Šámal [Star chromatic index,J. Graph Theory72 (2013), 313–326] provedthat everysubcubic multigraph is star 7-edge-colorable, andconjecturedthat everysubcubic multigraph should be star 6-edge-colorable. Kerdjoudj, Kostochkaand Raspaudconsidered the list version of this problem for simple graphs and proved that every subcubic graph with maximum average degreeless than 7/3 isstar list-5-edge-colorable. It is known thata graph with maximum average degree 14/5is not necessarilystar 5-edge-colorable. In this paper, weprove that every subcubic multigraphwith maximum average degreeless than 12/5 is star 5-edge-colorable. Keywords: star edge-coloring; subcubic multigraphs; maximum average degreeAMS subject classification 2010: 05C15 § INTRODUCTION 17pt All multigraphsin this paper are finite and loopless; and all graphsare finite and without loops or multiple edges. Given a multigraph G, let c: E(G)→ [k] be a proper edge-coloring of G, where k≥1 is an integer and [k]:={1,2, …, k}. We say that c is astar k-edge-coloring of G if no path or cycle of length four in G is bi-colored under the coloring c; andG is star k-edge-colorable if G admits a star k-edge-coloring. The star chromatic indexof G, denoted χ'_s(G),is the smallest integer k such that G is star k-edge-colorable. As pointed out in<cit.>, the definition of star edge-coloring of a graph G is equivalent to the star vertex-coloring of its line graph L(G).Star edge-coloring of a graph was initiatedby Liu and Deng <cit.>, motivated by the vertex version (see <cit.>). Given a multigraph G, we use|G| to denote the number of vertices,e(G) the number of edges, δ(G) the minimum degree, and Δ(G) the maximum degree of G, respectively. We use K_n and P_n to denote the complete graph and the path on n vertices, respectively.A multigraph G is subcubic if all its vertices have degree less than or equal to three. The maximum average degree ofa multigraph G, denotedmad(G), is defined as the maximumof2 e(H)/|H| taken over all the subgraphs H of G.The following upper bound is a result ofLiu and Deng <cit.>.For any graphG withΔ(G)≥7, χ'_s(G)≤⌈16(Δ(G)-1)^3/2⌉. Theorem <ref>below is a result of Dvořák, Mohar and Šámal <cit.>, which gives an upper and a lower bounds for complete graphs. The star chromatic index of the complete graph K_n satisfies 2n(1+o(1))≤χ'_s(K_n)≤ n 2^2√(2)(1+o(1))√(log n)/(log n)^1/4.In particular, for every ϵ>0, there exists a constant c such thatχ'_s(K_n)≤ cn^1+ϵ for every integer n≥1. The true order of magnitude ofχ'_s(K_n) is still unknown. From Theorem <ref>,an upper bound in terms of the maximum degree for generalgraphs is also derived in <cit.>, i.e., χ'_s(G)≤Δ· 2^O(1)√(logΔ) foranygraph G with maximum degree Δ.In the same paper, Dvořák, Mohar and Šámal <cit.> also considered the star chromatic index of subcubic multigraphs. To state their result, we need to introduce one notation. A graph G covers a graph H if there is a mapping f: V(G)→ V(H) such that for any uv∈ E(G), f(u)f(v)∈ E(H), and for any u∈ V(G), f is a bijection between N_G(u) and N_H(f(u)).They proved the following. Let G be a multigraph. * If G issubcubic,then χ'_s(G)≤7.* If G iscubic and has no multiple edges, then χ'_s(G)≥4 and the equality holds if and only if G covers the graph of 3-cube.As observed in <cit.>,K_3,3 isnot star 5-edge-colorable but star 6-edge-colorable. No subcubic multigraphs with star chromatic index sevenare known. Dvořák, Mohar and Šámal <cit.> proposed the following conjecture.Let G be a subcubic multigraph. Then χ'_s(G)≤ 6. It wasshown in <cit.> that every subcubic outerplanar graph is star 5-edge-colorable.Lei, Shi and Song <cit.> recently proved that every subcubic multigraph G with mad(G)<24/11 is star 5-edge-colorable, andevery subcubic multigraph G with mad(G)<5/2 is star 6-edge-colorable. Kerdjoudj,Kostochka and Raspaud <cit.> considered the list version of star edge-colorings of simple graphs. They proved that every subcubic graph is star list-8-edge-colorable, and further proved the following stronger results.Let G be a subcubic graph. * If mad(G)<7/3, then G isstar list-5-edge-colorable. * If mad(G)<5/2, then G isstar list-6-edge-colorable.As mentioned above, K_3,3 has star chromatic index 6, and isbipartite and non-planar.The graph, depicted in Figure <ref>,has star chromatic index 6, and is planar andnon-bipartite.We see that not every bipartite, subcubic graph is star 5-edge-colorable; and not every planar, subcubic graph is star 5-edge-colorable. It remains unknown whether every bipartite, planar subcubic multigraph is star 5-edge-colorable. In this paper, we improve Theorem <ref>(a) by showing the following main result.Let G be a subcubic multigraph with mad(G)<12/5. Then χ'_s(G)≤ 5. We don't know if the bound 12/5in Theorem <ref> is best possible.The graph depictedin Figure <ref>hasmaximum average degree 14/5but is not star 5-edge-colorable.Thegirth of a graph G is the length of a shortest cycle in G.It was observed in <cit.> thatevery planar graph with girth g satisfies mad(G)< 2g/g-2. This, together with Theorem <ref>,implies the following. Let G be a planar subcubic graph with girth g. If g≥12, then χ'_s(G)≤ 5. We need to introduce more notation. Given a multigraph G,a vertex of degree kin G is a k-vertex, and a k-neighbor of a vertex vin G is a k-vertex adjacent to v in G.A 3_k-vertex in G is a 3-vertexincident to exactly k edges e in G such that the other end-vertexof e is a 2-vertex. For any proper edge-coloringc of a multigraph G and for any u∈ V(G), let c(u) denote the set of all colors such that each is used to color an edge incident with u under the coloring c. For any two setsA, B, letA B := A-B. IfB={b}, we simply writeA b instead of A B.§ PROPERTIES OF STAR 5-CRITICAL SUBCUBIC MULTIGRAPHSAmultigraph G isstar 5-critical if χ'_s(G)>5 andχ'_s(G-v)≤ 5 for any v∈ V(G). In this section,we establish some structure results on star 5-critical subcubic multigraphs. Clearly, every star 5-critical multigraphmust be connected. Throughout the remainderof this section, let G be a star 5-critical subcubic multigraph,and let N(v) and d(v) denote the neighborhood anddegree of a vertex v in G, respectively.Since every multigraph with maximum degree at mosttwo ornumber of vertices at most fouris star 5-edge-colorable, we see thatΔ(G)=3and |G|≥5.As observed in <cit.>, any 2-vertex in G must havetwo distinct neighbors.The following Lemma <ref> and Lemma <ref> are proved in <cit.> and will be used in this paper. For any 1-vertex x in G,letN(x)={y}. The following are true. * |N(y)|=3.* N(y) is an independent set in G,d(y_1)=3 and d(y_2)≥ 2, where N(y)={x, y_1,y_2} with d(y_1)≥ d(y_2). * If d(y_2)=2, then for any i∈{1,2} and any v∈ N_G(y_i) y, |N(v)|≥2, |N(y_1)|=3, |N(y_2)|=2,and N[y_1]∩ N[y_2]={y}. * If d(y_2)=2, thend(w_1)=3, where w_1 is the other neighbor of y_2 in G. * Ifd(y_2)=3, then either d(v)≥2 for any v∈ N(y_1) or d(v)≥2 for any v ∈ N(y_2).For any 2-vertex x in G,let N(x)={z, w} with |N(z)|≤ |N(w)|. The following are true. * If zw∈ E(G), then |N(z)|=|N(w)|=3 and d(v)≥2 for any v∈ N(z)∪ N(w). * Ifzw∉ E(G), then|N(w)|=3 or|N(w)|=|N(z)|=2,and d(w)=d(z)=3. * If d(z)=2 andz^*w∈ E(G),then |N(z^*)|=|N(w)|=3, and d(u)=3 for any u∈ (N[w] ∪ N[z^*]){x,z}, where z^* is the other neighbor of z in G.* Ifd(z)=2, then|N(z^*)|=|N(w)|=3, and |N(v)|≥2 for any v∈ N(w)∪ N(z^*), where N(z)={x, z^*}. Let H be the graph obtained from G by deleting all 1-vertices. By Lemma <ref>(a,b), H is connected andδ(H)≥2.Throughout the remaining of the proof,a 2-vertex in H is bad ifit has a2-neighborin H, and a 2-vertex in H is good ifit is not bad.For any 2-vertex r in H, we use r' to denote the unique 1-neighbor of r in G if d_G(r)=3. By Lemma <ref>(a) and the fact thatany 2-vertex in G has two distinct neighbors in G, we obtain the following two lemmas.For any 2-vertex x in H,|N_H(x)|=2.For any 3_k-vertex x in H with k≥2,|N_H(x)|=3. Proofs of Lemma <ref> and Lemma <ref> below can be obtainedfrom the proofsof Claim 11and Lemma 12 in <cit.>, respectively. Since a star 5-critical multigraph is not necessarily the edge minimal counterexample in the proof ofTheorem 4.1 in <cit.>, we include new proofs ofLemma <ref> and Lemma <ref> here for completeness. H has no 3-cycle such thattwo of itsverticesare bad. Suppose that H does contain a 3-cycle with vertices x, y, z such that both y and z are bad.Then x must be a 3-vertexin G because G is 5-critical. Let w be the third neighbor of x in G. Since G is 5-critical,let c: E(G{y,z})→ [5] be any star 5-edge-coloring of G{y,z}.Let α and β be two distinct numbers in [5] c(w) and γ∈ [5]{α, β, c(xw)}. Now coloring the edges xy, xz, yzby colorsα, β, γin order,and furthercoloring all the edges yy', zz' by color c(xw) if y' orz' exists,we obtain a star 5-edge-coloring of G, a contradiction. height3pt width6pt depth2pt H has no 4-cyclewith verticesx,u,v,w in ordersuch that all of u,v,w arebad. Furthermore, if H contains a path with vertices x,u,v,w,y in order such that all ofu, v,w arebad, then both x and y are 3_1-vertices in H.Let P be a path in H withvertices x,u,v,w,y in order such that all ofu, v,w arebad, where x and y may be the same.Since all ofu, v,w are bad, by the definition of H, uw∉ E(G). By Lemma <ref>(b,c,e) applied to the vertex v,d_G(v)=2.By Lemma <ref>(b) applied to v, d_G(u)=d_G(w)=3. Thus both w' andu' exist.Now by Lemma <ref>(c) applied to u' and w',d_H(x)=d_H(y)=3, and x y. This provesthat H has no 4-cyclewith verticesx,u,v,w in ordersuch that all of u,v,w arebad. We next show that both x and y are 3_1-vertices in H. Suppose that one of x and y, say y, is not a 3_1-vertex in H.Theny is either a 3_2- vertex or 3_3-vertex in H.By Lemma <ref>,|N_H(y)|=3.Let N_H(y)={w,y_1,y_2} with d_H(y_1)=2. Theny_1≠ u, otherwise H would have a 4-cycle with verticesy,u,v,w in ordersuch that all of u,v,w arebad.Note that y_2 and x are not necessarily distinct. By Lemma <ref>, letr be the other neighbor of y_1 in H. Since G is 5-critical,let c: E(G{v,u',w'})→ [5] be any star 5-edge-coloring of G{v,u',w'}. We may assume that c(wy)=3, c(yy_1)=1 and c(yy_2)=2. We first coloruv by a color α in [5] (c(x)∪{3}) and uu' by a color β in [5] (c(x)∪{α}). Then 3∈ c(y_1)∩ c(y_2), otherwise, we may assume that 3∉ c(y_i) for some i∈{1,2}, now coloring vw by a color γ in {i,4,5}α andww' by a color in {i,4,5}{α,γ} yields a star 5-edge-coloring of G, a contradiction. It follows that4, 5∈ c(y_1)∪ c(y_2), otherwise, say θ∈{4,5} is not in c(y_1)∪ c(y_2), now recoloring wy by color θ, uv by a color α' in {α,β}θ, uu' by {α,β}α', and then coloring ww' by a color in {1,2}α' and vw by a color in {3,9-θ}α', we obtain a star 5-edge-coloring of G, a contradiction.Thus c(y_1)={1,3,θ} and c(y_2)={2,3,9-θ}, where θ∈{4,5}.If c(y_1y_1')≠3 or c(y_1r)=θ and 1∉ c(r), then we obtain a star 5-edge-coloring of G by recoloring wy by color θ, uv by a color α' in {α,β}θ,uu' by {α,β}α', and then coloring ww' by a color γ in {2,3,9-θ}α',and vw by a color in {2,3,9-θ}{α',γ}. Therefore,c(y_1y_1')=3 and 1∈ c(r). Now recoloring y_1y_1' by a color in {2,9-θ} c(r),we obtain a star 5-edge-coloring c of G{v,u',w'} satisfying c(wy)=3, c(yy_1)=1 and c(yy_2)=2 but 3∉ c(y_1)∩ c(y_2), a contradiction.Consequently,each ofx and y must be a3_1-vertex in H. This completes the proof of Lemma <ref>.height3pt width6pt depth2ptFor any 3_3-vertex u in H,no vertex in N_H(u) isbad. Let N_H(u)={ x, y, z} withd_H(x)=d_H(y)=d_H(z)=2. By Lemma <ref>, u,x,y,z are all distinct.By Lemma <ref>, let x_1,y_1 and z_1 bethe other neighbors of x, y, z in H, respectively. Suppose thatsome vertex, say x, in N_H(u)is bad. Then d_H(x_1)=2.By Lemma <ref>, let w be the other neighborof x_1 in H.By Lemma <ref> and Lemma <ref>, N_H(u) is an independent set and x_1∉{y, z,y_1, z_1}.Notice thaty_1,z_1and ware not necessarily distinct. Let A:={x} when d_G(x_1)=2 and A:= {x,x_1'} whend_G(x_1)=3.Let c: E(G A)→ [5] be any star 5-edge-coloring of G A.We may assume thatc(uy)=1 andc(uz)=2. We next provethat(*) 1 ∈ c(y_1) and 2∈ c(z_1).Suppose that1 ∉ c(y_1) or 2∉ c(z_1), say the former.If c(w) ∪{1,2}[5],then we obtain a star 5-edge-coloring of Gfrom c by coloring the remaining edges of G as follows (we only consider the worst scenario whenboth x' and x_1'exist):color the edge xx_1 by a color α in [5](c(w) ∪{1,2}),x_1x_1' by a color β in [5](c(w) ∪{α}), ux by a color γ in [5] {1,2, α, c(zz_1)} and xx' by a color in [5] {1,2, α, γ}, a contradiction. Thusc(w) ∪{1,2}= [5]. Thenc(w)={3,4,5}. We may assume thatc(x_1w)=3. Ifc(z) ∪{1,3} [5], then {4,5} c(z)∅ and we obtain a star 5-edge-coloring of Gfrom cby coloringthe edge xx_1 bycolor2, x_1x_1' by color 1, ux by a color α in {4,5} c(z) and xx' by a color in{4,5}α, a contradiction.Thus c(z) ∪{1,3} = [5] and so c(z) ={2,4,5}. In particular,z' must exist. We again obtain a star 5-edge-coloring of Gfrom cby coloringux,xx', xx_1, x_1x_1' by colors 3, c(zz_1), 2,1 in order and then recoloring uz,zz' by colors c(zz'),2 in order,a contradiction.Thus 1 ∈ c(y_1) and 2∈ c(z_1). This proves (*).By (*), 1 ∈ c(y_1) and 2∈ c(z_1).Then y_1 z_1,and c(yy_1), c(zz_1)∉{1,2}. We may further assume thatc(zz_1)=3.Letα, β∉c(z_1) and let γ, λ∉c(y_1), where α, β,γ, λ∈ [5].Since α, β∉c(z_1), we may assume that c(yy_1)α. We may further assume that γα. If λα or γ∉{3,β}, then we obtain a star 5-edge-coloring, say c', ofG A from c by recoloring the edges uz, zz', uy, yy' by colors α, β, γ, λ, respectively. Thenc' is a star 5-edge-coloring ofG A with c'(uz)∉ c'(z_1),contrary to (*).Thusλ=α andγ∈{3,β}.By (*), 1∈ c(y_1) and soα=λ1 andγ1.Let c' be obtained from c by recoloring the edges uz, zz',yy' by colors α, β, γ, respectively. Thenc' is a star 5-edge-coloring ofG A with c'(uz)∉ c'(z_1), which again contradicts (*). This completes the proof of Lemma <ref>. height3pt width6pt depth2pt For any3-vertex u in H with N_H(u) = {x, y, z}, ifboth x and y are bad, then zx_1, zy_1∉ E(H), and z must be a 3_0-vertex in H, where x_1 and y_1 are the other neighbors of x and y in H, respectively. Let u, x, y, z, x_1, y_1 be given as in the statement.Sinced_H(x)=d_H(y)=2, by Lemma <ref>, u, x, y, zare all distinct. By Lemma <ref>,d_H(z)=3.Clearly,both x_1 and y_1 are bad and so z x_1, y_1. By Lemma <ref>, xy∉ E(G)and soN_H(u) is an independent set in H.By Lemma <ref>, x_1 y_1.It follows that u, x, y, z, x_1, y_1 are all distinct. We first show that zx_1, zy_1∉ E(H). Suppose that zx_1∈ E(H) orzy_1∈ E(H), say the latter.Then zy_1 is not a multiple edge because d_H(y_1)=2.Let z_1 be the third neighbor of z in H.By Lemma <ref>, letv be the other neighbor of x_1 in H. Then v y_1.Notice thatx_1 and z_1are not necessarily distinct. Let A={u, x, y, y_1, x_1'}.Since G is 5-critical,let c: E(G A)→ [5] be any star 5-edge-coloring of G A.We may assume that 1,2∉ c(z_1) and c(zz_1)=3. Let α∈ [5]( c(v)∪{1}) and β∈ [5](c(v)∪{α}). Then we obtain a star 5-edge-coloring of G from cbyfirst coloring theedges uz, zy_1, xx_1, x_1x_1' by colors 1, 2,α, β in order, and then coloring ux by a color γ in [5]{1,α,β,c(x_1v)}, xx' by a color in [5]{1,α,γ,c(x_1v)}, uy by a color θ in [5]{1,2,3, γ}, yy_1 by a color μ in [5]{1,2,γ,θ}, yy' by a color in [5]{2,γ,θ,μ}, y_1y_1' by a color in [5]{1,2, μ}, a contradiction. This proves thatzx_1, zy_1∉ E(H). It remains to show that z must be a 3_0-vertex in H.Suppose that z is not a 3_0-vertex in H. Since d_H(u)=3, we see that z is either a 3_1-vertex or a 3_2-vertex in H. Let N_H(z)={u,s,t} with d_H(s)=2.By Lemma <ref> applied to the vertex s,s t. Since zx_1, zy_1∉ E(H), we see that x_1, y_1, s,t are all distinct. By Lemma <ref>, let v, w, r be the other neighbor of x_1, y_1, s in H, respectively. Notethat r, t, v, w are not necessarily distinct. By Lemma <ref>, both v and w must be3-vertices in H. We next prove that(a)if x' or y' exists, then forany star 5-edge-coloring c^*of G{x', y'},c^*(xx_1)∈ c^*(v) or c^*(yy_1)∈ c^*(w).To see why(a) is true, suppose that there exists a star5-edge-coloring c^*: E(G{x', y'})→ [5] such that c^*(xx_1)∉ c^*(v) andc^*(yy_1)∉ c^*(w).Then we obtain a star 5-edge-coloring of G from c^*bycoloringxx' by a color in [5]({c^*(xx_1)}∪ c^*(u)) and yy' by a color in [5]({c^*(yy_1)}∪ c^*(u)), a contradiction. This proves (a).Let A be the set containing x, yand the1-neighbor ofeach of x_1, y_1 in G if it exists.Since G is 5-critical,let c_1: E(G A)→ [5] be any star 5-edge-coloring of G A.Let c be a star 5-edge-coloring of G{x,x', y', x_1'} obtained from c_1 bycoloringyy_1 by a color α in [5] (c_1(w)∪{c_1(uz)}),uy by a colorin [5](c_1(z)∪{α}), andy_1y_1'by a color β in [5] (c_1(w)∪{α}).We may assume that c(uz)=1, c(zs)=2 and c(zt)=3. By the choice of c(uy), we may further assume that c(uy)=4. We next obtain a contradictionby extending c to be a star5-edge-coloring of G(when neither ofx' and y'exists) or a star 5-edge-coloring of G{x', y'} (when x' or y' exists)which violates (a).We consider the worst scenario when x' andy' exist.We first prove two claims. Claim 1:β=4 orc(y_1w)=4. Suppose that β 4 and c(y_1w)4.We next showthat c(v)∪{1,4} [5]. Suppose that c(v)∪{1,4}= [5]. Then c(v)={2,3,5}. Clearly, c(x_1v)=5, otherwise, coloring ux, xx_1, x_1x'_1 by colors5,1,4 in order,we obtaina star 5-edge-coloring of G{x', y'}which violates (a), a contradiction.We see that 1∈ c(s)∩ c(t), otherwise,we may assumethat 1∉ c(s), we obtain a star 5-edge-coloring of G{x', y'}which violates (a) as follows:whenα2, color ux, xx_1, x_1x_1'by colors 2, 4,1 in order; when α=2,first color ux, xx_1, x_1x_1' bycolors 2, 4,1 in order and thenrecoloryy_1, y_1y_1' by colors β,2 in order.It follows that4, 5∈ c(s)∪ c(t), otherwise, say θ∈{4,5} is not in c(s)∪ c(t),let α'∈{2,3}α, now eithercoloring ux, xx_1, x_1x_1' by colors α', 4,1 in order andthen recoloringuz by color 5 when θ=5; orcoloring ux, xx_1, x_1x_1' by colors α', 1, 4 in order and then recoloringuz, uy by colors 4, 1 in order when θ=4,we obtaina star 5-edge-coloring of G{x', y'}which violates (a).Thus c(s)={1,2,θ} and c(t)={1,3,9-θ}, where θ∈{4,5}. Ifc(ss')=θor c(sr)= θand 2∉ c(r), then we obtaina star 5-edge-coloring of G{x', y'} (which violates (a)) as follows: when θ=5,color ux, xx_1, x_1x_1' by colors 3,1,4 in orderand then recoloruz by color 5; when θ=4 and α∈{2,5}, firstcolorux, xx_1, x_1x'_1 by colors 3, 1, 4 in order, and then recoloruz, uy by colors 4, 1 in order; when θ=4 and α=3 and β 5,colorux, xx_1, x_1x'_1 by colors 5, 1, 4 in order and then recoloruz, uy, yy_1, y_1y_1' by colors 4, 3, β, 3 in order;when θ=4 and α=3 and β=5,colorux, xx_1, x_1x'_1 by colors 3, 1, 4 in order and then recoloruz, uy, yy_1, y_1y_1' by colors 4, 1, 5, 3 in order.Thus c(ss')=1, c(sr)=θ and 2∈ c(r).Now recoloring the edge ss' by a color in {3,9-θ} c(r)yields a star 5-edge-coloring c of G{x,x', y', x_1'} satisfying β 4, c(y_1w)4,c(v)∪{1,4}= [5] and c(x_1v)=5 but 1∉ c(s)∩ c(t), a contradiction. This proves thatc(v)∪{1,4} [5].Since c(v)∪{1,4} [5], we see that [5] (c(v)∪{1,4})= {5}, otherwise, coloring ux by color 5,xx_1 by a color γ in [5](c(v)∪{1,4,5}), andx_1x'_1 bya color in [5] (c(v)∪γ), we obtain a star 5-edge-coloring of G{x', y'} which violates (a).Clearly, 2,3∈ c(v) and {1,4} c(v)∅. Let γ∈{1,4} c(v) and α'∈{2,3}α. Then 1∈ c(s)∩ c(t), otherwise,we may assumethat 1∉ c(s), nowcoloring ux, xx_1, x_1x_1' by colors 2, 5,γ in orderyields a star 5-edge-coloring of G{x', y'}which violates (a). It follows that4, 5∈ c(s)∪ c(t), otherwise, say θ∈{4,5} is not in c(s)∪ c(t),first recoloring uz by color θ and theneithercoloring ux, xx_1, x_1x_1' by colors α', 5,γ in order and then recoloring uy by color 1 when θ=4; or coloring ux, xx_1, x_1x_1' by colors α', 1, 5 in order when θ=5 and γ=1; or coloring ux, xx_1, x_1x_1' by colors 1,4,5 in orderwhenθ=5, γ=4 and c(x_1v)1; or coloring ux, xx_1, x_1x_1' by colors α',4,5 in orderwhenθ=5, γ=4 and c(x_1v)=1, we obtain a star 5-edge-coloring of G{x', y'}which violates (a). Thus c(s)={1,2,θ} and c(t)={1,3,9-θ}, where θ∈{4,5}. Ifc(ss')=θor c(sr)= θand 2∉ c(r), then we obtaina star 5-edge-coloring of G{x', y'} (which violates (a)) as follows: when θ=5 and γ=1,color ux, xx_1, x_1x_1' by colors 3, 1, 5 in order and then recoloruz by colors 5; when θ=5,γ=4 and c(x_1v)1,color ux, xx_1, x_1x_1' by color 1, 4, 5 in order and then recoloruz by colors 5; when θ=5,γ=4 and c(x_1v)=1,color ux, xx_1, x_1x_1' by color 3, 4, 5 in order and then recoloruz by colors 5 (andfurther recolor yy_1 by β and y_1y_1' by α when α=3); when θ=4 and β1,color ux, xx_1, x_1x_1' by color 3, 5, γ in order and then recoloruz,uy by colors4, 1 in order, and finally recolor yy_1 by a colorβ'∈{α,β} 3 andy_1y'_1 by a color in {α,β}β'; when θ=4,β=1 andγ=1,color ux, xx_1, x_1x_1' by color 5,1,5 in order and then recoloruz,uy, yy_1, y_1y_1' by colors4, 3, 1, α in order; when θ=4,β=1, γ=4 and α3,color ux, xx_1, x_1x_1' by color 3, 5, 4 in order and then recoloruz,uy by colors4, 1 in order; when θ=4,β=1, γ=4 and α=3, let γ'∈{1,3} c(x_1v),color ux, xx_1, x_1x_1' by color γ', 5, 4 in order and then recoloruz by color 4,uy by color5,yy_1 by a color β' in {1,3}γ' andy_1y_1' by a color in {1,3}β'. Thus c(ss')=1, c(sr)=θ and 2∈ c(r).Now recoloring the edge ss' by a color in {3,9-θ} c(r)yields a star 5-edge-coloring c of G{x,x', y', x_1'}satisfying β 4,c(y_1w)4and [5] (c(v)∪{1,4})= {5} but 1∉ c(s)∩ c(t), a contradiction.This completes the proof of Claim 1.height3pt width6pt depth2pt Claim 2:β=4. Suppose thatβ 4. By Claim 1,c(y_1w)=4.We first consider the case whenc(w)={2,3,4}. Thenα=5 and β=1. We claim that c(v)∪{1,4} [5]. Suppose that c(v)∪{1,4}= [5]. Then c(v)={2,3,5}. Clearly,1∈ c(s)∩ c(t), otherwise,we may assume that 1∉ c(s), now coloring ux, x x_1, x_1x'_1 by colors 5,4,1 in order and thenrecoloring uy by 2,weobtain a star 5-edge-coloring of G{x', y'} which violates (a).It follows that4, 5∈ c(s)∪ c(t), otherwise, say θ∈{4,5} is not in c(s)∪ c(t),now coloring ux, x x_1, x_1x'_1 by colors 3,1,4 in order and thenrecoloring uz,uy, yy_1, y_1y_1' by colors θ, 2, 1,5 in order we obtain a star 5-edge-coloring of G{x', y'}which violates (a). Thus c(s)={1,2,θ} and c(t)={1,3,9-θ}, where θ∈{4,5}. Ifc(ss')=θor c(sr)= θand 2∉ c(r), then coloring ux, x x_1, x_1x'_1 by colors 3,1,4 in order and thenrecoloring uz,uy, yy_1, y_1y_1' by colors θ, 9-θ, 1,5 in order yileds a star 5-edge-coloring of G{x', y'} which violates (a). Thus c(ss')=1,c(sr)= θand 2∈ c(r). Now recoloring the edge ss' by a color in {3,9-θ} c(r)yields a star 5-edge-coloring c of G{x,x', y', x_1'}satisfying α=5, β=1, c(y_1w)=4 andc(v)∪{1,4}= [5] but 1∉ c(s)∩ c(t), a contradiction. This proves thatc(v)∪{1,4} [5].Let η=5 when 5∉ c(v) or η∈{2,3} c(v) when 5∈ c(v).Let μ∈ [5]( c(v)∪{η}).By Claim 1 and the symmetry between x and y, either 4∉ c(v) or 5∉ c(v). We see that μ=4 when η 5. Then1∈ c(s)∩ c(t), otherwise, we may assume 1∉ c(s), we obtain a star 5-edge-coloring of G{x', y'}(which violates (a)) as follows: whenη≠ 2,color ux, xx_1, x_1x_1' by colors 2, η, μ in order; when η=2, then μ=4,first recolor uy by color 2 andthen color ux, xx_1, x_1x'_1 by colors5,4,2 in order. It follows that4, 5∈ c(s)∪ c(t), otherwise, say θ∈{4,5} is not in c(s)∪ c(t), now first recoloring uz, yy_1, y_1y_1' by colors θ, 1,5in order, and then coloringxx_1, x_1x'_1 by colors η,μ in order,ux by a color γ in [5]{μ,η,θ, c(x_1v)},and finallycoloring uy either by a color in {2,3}η whenγ=1 or by a color in{2,3}γ when γ1, we obtain a star 5-edge-coloring of G{x', y'}which violates (a). Thus c(s)={1,2,θ} and c(t)={1,3,9-θ}, where θ∈{4,5}. If c(ss')=θor c(sr)= θand 2∉ c(r), we obtain a star 5-edge-coloring of G{x', y'}(which violates (a)) as follows: when θ=4 and η=5, color ux, xx_1, x_1x_1' by colors 3, 5, μ in order and then recolor uz, uy by colors 4,1 in order; when θ=4 and η∈{2,3}, then μ=4,first recolor uz, uy by colors 4,3 in order and then color xx_1, x_1x_1' by colors η, 4 in order and finally color ux by a color γ in {1,5} c(x_1v), yy_1 by a color λ in {1,5}γ, and y_1y_1' by a color in {1,5}λ; when θ=5 and η∈{2,3}, then μ=4, color ux, xx_1, x_1x_1' by colors 1, 4, η in order and then recolor uz, uy, yy_1, y_1y_1' by colors 5,3, 1,5 in order; when θ=5,η=5 and μ3, color ux, xx_1, x_1x_1' by colors 1, μ, 5 in order and then recolor uz, uy, yy_1, y_1y_1' by colors 5,3, 1,5 in order; when θ=5,η=5 and μ=3, first recolor uz, uy, yy_1, y_1y_1' by colors 5,3, 1,5 in order, then color xx_1, x_1x_1' by colors 5, 3 in order andfinally color ux by a color in {1,4} c(x_1v).Thus c(ss')=1, c(sr)= θand 2∈ c(r).Now recoloring the edge ss' by a color in {3,9-θ} c(r)yields a star 5-edge-coloring c of G{x,x', y', x_1'}satisfying α=5, β=1, c(z)={1,2,3}, c(uy)=c(y_1w)=4 andc(v)∪{1,4}[5] but 1∉ c(s)∩ c(t), a contradiction. We next consider the case whenc(w)≠{2,3,4}. If α, β5, then recoloring uy by color 5 yieldsastar 5-edge-coloring c of G{x,x', y', x_1'} with c(uy) c(y_1y_1'), c(y_1w),contrary toClaim 1. Thus either α=5 orβ=5.Then 1∈ c(w) because c(w)≠{2,3,4} and |c(w)|=3.It follows that α, β∈{2,3,5} and 5∈{α, β}. We may assume that α∈{2,3} and β=5 by permuting the colors on yy_1 and y_1y_1' if needed.Then 4,5∈ c(s)∪ c(t), otherwise, say θ∈{4,5} is not in c(s)∪ c(t), we obtain a a star 5-edge-coloring c of G{x,x', y', x_1'}which contradicts Claim 1 by recoloring uz, uy by colors θ, 1 in order.Let α'∈{2,3}α.We next show that c(ss')=1,c(sr)= θand 2∈ c(r).Suppose firstthat c(v)∪{1,4}= [5]. Then c(v)={2,3,5}. We see thatc(x_1v)=5, otherwise, coloring ux, xx_1, x_1x'_1 by colors5,1,4 in order,we obtaina star 5-edge-coloring of G{x', y'}which violates (a).Clearly,1∈ c(s)∩ c(t), otherwise,we may assume that 1∉ c(s), now coloring ux, x x_1, x_1x'_1 by colors 2,4,1 in order and thenrecoloring yy_1,y_1y_1' by colors 5,α, weobtain a star 5-edge-coloring of G{x', y'} which violates (a).Since 4,5∈ c(s)∪ c(t), we see thatc(s)={1,2,θ} and c(t)={1,3,9-θ}, where θ∈{4,5}. Ifc(ss')=θor c(sr)= θand 2∉ c(r), thenrecoloring uz, uy by colors θ, 1 in orderyieldsastar 5-edge-coloring c of G{x,x', y', x_1'} with c(uy) c(y_1y_1'), c(y_1w),contrary toClaim 1. Thus c(ss')=1,c(sr)= θand 2∈ c(r).Next suppose that c(v)∪{1,4}[5]. Let η=5 when 5∉ c(v) or η∈{2,3} c(v) when 5∈ c(v).Let μ∈ [5]( c(v)∪{η}). By Claim 1 and the symmetry between x and y, either 4∉ c(v) or 5∉ c(v). We see that μ=4 when η 5. Then1∈ c(s)∩ c(t), otherwise, we may assume 1∉ c(s), we obtain a star 5-edge-coloring of G{x', y'}(which violates (a)) as follows: whenη=5,color ux, xx_1, x_1x_1' by colors 4, 5, μ in order and then recolor uy, yy_1, y_1y_1' by colors 2,5,α in order; when η∈{2,3}, then μ=4,color ux, xx_1, x_1x'_1 by colors5,η,4 in order.Since 4,5∈ c(s)∪ c(t), we see thatc(s)={1,2,θ} and c(t)={1,3,9-θ}, where θ∈{4,5}. Ifc(ss')=θor c(sr)= θand 2∉ c(r), thenrecoloring uz, uy by colors θ, 1 in orderyieldsastar 5-edge-coloring c of G{x,x', y', x_1'} with c(uy) c(y_1y_1'), c(y_1w),contrary toClaim 1. Thus c(ss')=1,c(sr)= θand 2∈ c(r). Now recoloring the edge ss' by a color in {3,9-θ} c(r)yields a star 5-edge-coloring c of G{x,x', y', x_1'}satisfying α∈{2,3}, β=5, c(y_1w)=4 and c(w)≠{2,3,4} but 1∉ c(s)∩ c(t), a contradiction. This completes the proof of Claim 2. height3pt width6pt depth2ptBy Claim 2, β=4. Suppose that α5.Thenα∈{2,3}.Note that α∉ c(w)∪{1}. Nowrecoloring uy by color 5, we obtain a star 5-edge-coloring c of G{x,x', y', x_1'} satisfying c(uz)=1, c(zs)=2 and c(zt)=3 but β c(uy), contrary to Claim 2. Thus α=5 and so c(w)={1,2,3}.By the symmetry of x and y, c(v)={1,2,3}.Then 1∈ c(s)∩ c(t), otherwise, we may assumethat 1∉ c(s), nowcoloring ux, xx_1, x_1x_1' by colors 2,5,4 in orderyields a star 5-edge-coloring of G{x', y'}which violates (a).It follows that4, 5∈ c(s)∪ c(t), otherwise, say θ∈{4,5} is not in c(s)∪ c(t), now first coloring ux, xx_1, x_1x_1' by colors 2,9-θ,θ in order and then recoloring uz, uy, yy_1, y_1y_1' by colors θ, 3, 9-θ, θ in order, we obtaina star 5-edge-coloring of G{x', y'}which violates (a).Thus c(s)={1,2,θ} and c(t)={1,3,9-θ}, where θ∈{4,5}.If c(ss')=θ or c(sr)=θ and 2∉ c(r), then we obtaina star 5-edge-coloring of G{x', y'} (which violates (a)) by coloring ux, xx_1, x_1x_1' by colors 1, 9-θ, θ in order, and thenrecoloring uz, uy, yy_1, y_1y_1' by colors θ, 3, 9-θ, θ in order. Thus c(ss')=1 and 2∈ c(r). Now recoloring ss' by a color in {3,9-θ} c(r),we obtain a star 5-edge-coloring c of G{x,x', y', x_1'} satisfying c(uz)=1, c(zs)=2,c(zt)=3, β= 4 andα=5but 1∉ c(s)∩ c(t).This completes the proof of Lemma <ref>.height3pt width6pt depth2pt§ PROOF OF THEOREM <REF>We are now ready to prove Theorem <ref>.Suppose theassertion is false. Let G be a subcubic multigraph with mad(G)<12/5 andχ'_s(G)>5. Among all counterexamples we choose G so that |G| is minimum. By the choice of G, G is connected,star 5-critical,andmad(G)<12/5.For all i∈[3], let A_i={v∈ V(G): d_G(v)=i} and letn_i=|A_i| for all i∈[3]. Sincemad(G)<12/5, we see that3n_3<2n_2+7n_1 and soA_1∪ A_2∅. By Lemma <ref>(a), A_1is anindependent set in G and N_G(A_1)⊆ A_3.Let H=G A_1.Then H is connected and mad(H)<12/5. By Lemma <ref>(b), δ(H)≥2.By Lemma <ref>, every3_2-vertex in H has three distinct neighbors in H. We say that a 3_2-vertex in H is bad ifboth of its2-neighborsare bad.A vertex u is a good (resp. bad) 2-neighbor of a vertex v in Hif uv∈ E(H) and u is a good (resp. bad) 2-vertex. By Lemma <ref>,every bad 3_2-vertex in H has a unique 3_0-neighbor. We nowapply the discharging method to obtain a contradiction.For each vertex v∈ V(H), let ω(v):= d_H(v)-12/5 be the initial charge of v. Then ∑_v∈ V(H)ω(v) =2e(H)-12/5|H|=|H|(2e(H)/|H|-12/5)<0. Notice that for each v∈ V(H),ω(v)=2-12/5=-2/5 if d_H(v)=2,and ω(v)=3-12/5=3/5 if d_H(v)=3. We will redistribute thecharges of vertices inH as follows. (R1):every bad 3_2-vertex in H takes1/5 from its unique 3_0-neighbor.(R2): every 3_1-vertex in H gives 3/5 to itsunique 2-neighbor.(R3):every3_2-vertexin H gives1/5 to each of its good 2-neighbors (possibly none) and 2/5 to each of its bad 2-neighbors (possibly none).(R4): every 3_3-vertex in H gives 1/5 to each of its 2-neighbors. Let ω^* be the new charge of H after applying the above discharging rules in order. It suffices to show that ∑_v∈ V(H)ω^*(v)≥0. For any v∈ V(H) with d_H(v)=2, by Lemma <ref>, v has two distinct neighbors in H. If v is a good 2-vertex, then v takesat least 1/5 from each of its 3-neighbors under(R2), (R3) and (R4), and soω^*(v)≥0.Next, ifv is a bad 2-vertex, let x, y be thetwo neighbors of v inH. We may assume thaty is a bad 2-vertex. By Lemma <ref>, let z be the otherneighborof y in H.By Lemma <ref>, we may assume that d_H(x)=3.By Lemma <ref>, x is either a 3_1-vertex or a 3_2-vertex in H.Under (R2) and (R3), v takes at least 2/5 from x.If d_H(z)=3, thenby a similar argument,y must take at least 2/5 from z.In this case, ω^*(v)+ω^*(y)≥0.If d_H(z)=2, then z is bad. By Lemma <ref>, let w be the other neighbor of z.By Lemma <ref>, each of x and w must be a 3_1-vertex in H. Under (R2), v takes 3/5 from x and z takes 3/5 from w. Hence, ω^*(v)+ω^*(y)+ω^*(z)≥0. For any v∈ V(H) with d_H(v)=3, if v is a bad 3_2-vertex,then v has a unique 3_0-neighbor by Lemma <ref>. Under (R1) and (R3), v first takes 1/5 from its unique 3_0-neighbor and then gives 2/5 to each of its bad 2-neighbors, we see thatω^*(v)≥0.Ifv is not a bad 3_2-vertex, thenv gives either nothingor one of 1/5,2/5,and 3/5 in total to its neighbors under(R1), (R2), (R3) and (R4). In either case, ω^*(v)≥0. Consequently,∑_v∈ V(H)ω^*(v)≥0, contrary to the fact that∑_v∈ V(H)ω^*(v)=∑_v∈ V(H)ω(v)<0. This completes the proof of Theorem <ref>. height3pt width6pt depth2pt Acknowledgments.Zi-Xia Song would like to thank Yongtang Shi andthe Chern Institute of Mathematics at Nankai University for hospitality and support during her visitin May 2017. HuiLei and YongtangShi are partially supported by the National Natural Science Foundation of China and the Natural Science Foundation of Tianjin (No.17JCQNJC00300). TaoWang is partially supported by the National Natural Science Foundation of China (11101125) and the Fundamental Research Funds for Universities in Henan (YQPY20140051).10ACKKR2004 M. O. Albertson, G. G. Chappell, H. A. Kierstead,A. Kündgen andR. Ramamurthi, Coloring with no 2-colored P_4's, Electron. J. Combin.11 (2004), #R26. BLM2016 Ľ. Bezegová, B. Lužar, M. Mockovčiaková, R. Soták and R. Škrekovski, Star edge coloring of some classes of graphs, J. Graph Theory 81 (1) (2016) 73–82. girth O. V.Borodin, A. V. Kostochka, J. Nešetřil,A.Raspaud andE. Sopena, On the maximum average degree and the oriented chromatic number of a graph, Discrete Math.206 (1999)77–89. BCMRW2009 Y. Bu, D. W. Cranston, M. Montassier, A. Raspaud andW. Wang, Star coloring of sparse graphs,J Graph Theory62(2009) 201–219. CRW2013 M. Chen, A. Raspaud andW. Wang, 6-star-coloring of subcubic graphs, J Graph Theory72(2013) 128–145. DMS2013 Z. Dvořák, B. Mohar and R. Šámal, Star chromatic index, J Graph Theory72(2013) 313–326. KKP2017 S. Kerdjoudj, A. V. Kostochka and A. Raspaud, List star edge coloring of subcubic graphs, to appear in Discuss. Math. Graph Theory. KKT2009 H. A.Kierstead andA.Kündgen,C. Timmons, Star coloring bipartite planar graphs, J Graph Theory60(2009) 1–10. LSS2017 H. Lei, Y. Shi and Z-X. Song, Star chromatic index of subcubic multigraphs, to appear in J. Graph Theory. DL2008 X.-S.Liu and K. Deng, An upper bound on the star chromatic index of graphs with Δ≥7, J Lanzhou Univ (Nat Sci)44(2008) 94–95. NM2003 J. Nešetřil andP. Ossona de Mendez, Colorings and homomorphisms of minor closed classes,Algorithms and Combinatorics, Vol. 25, Springer, Berlin, 2003, pp.651–664. | http://arxiv.org/abs/1707.08892v3 | {
"authors": [
"Hui Lei",
"Yongtang Shi",
"Zi-Xia Song",
"Tao Wang"
],
"categories": [
"math.CO"
],
"primary_category": "math.CO",
"published": "20170726134913",
"title": "Star 5-edge-colorings of subcubic multigraphs"
} |
University of Bari University of Naples Federico IIProviding Self-Aware Systems with Reflexivity Alessandro Valitutti1 Giuseppe Trautteur2 December 30, 2023 =============================================We propose a new type of self-aware systems inspired by ideas from higher-order theories of consciousness. First, we discussed the crucial distinction between introspection and reflexion. Then, we focus on computational reflexion as a mechanism by which a computer program can inspect its own code at every stage of the computation. Finally, we provide a formal definition and a proof-of-concept implementation of computational reflexion, viewed as an enriched form of program interpretation and a way to dynamically “augment" a computational process. § INTRODUCTIONSelf-aware computing is a recent area of computer science concerning autonomic computing systems capable of capturing knowledge about themselves, maintaining it, and using it to perform self-adaptive behaviors at runtime<cit.><cit.><cit.>.Almost all self-aware systems share one or more of three properties dealt with extensively in the AI literature: self-representation, self-modification, and persistence.Examples of self-aware behaviors are the introspection and reflection features implemented in some programming languages such as Java. Type introspection is the ability of a program to examine the type or properties of an object at runtime, while reflection[The term reflection should not be confused with the term reflexion, which will be discussed in Sections <ref> and <ref>.]additionally allows a program to manipulate objects and functions at runtime.However, neither of them have all of the above three properties. In fact, introspection implies self-representation but not self-modification. Moreover, reflection is temporally-bound, since it occurs in a small portion of the program execution. Even self-monitoring, considered as a periodic sequence of introspective events, implies persistence but not self-modification.We may wonder if we could have a type of computational self-awareness in which persistent self-representation and self-modification would occur simultaneously and yet being functionally distinct. In this paper, we address this issue and present a computational architecture provided with this property, which we call computational reflexivity.Specifically, we propose to introduce introspection and reflection at every step of the execution, enriching the interpretation loop with additional instructions aimed to represent the program at a meta level, combine local and global information, and perform a second-order execution.The enriched interpreter is thus capable of running a program and, concurrently, generating and executing a corresponding modified (or “augmented”) version.This separation between “observed” (or target) and “observing” (or augmented) process allows the system to perform self-modification at a virtual level (i.e., on the augmented process). As a consequence, the system can choose whether and when the modification should be applied to the target process.In addition to the formal definition of computational reflexivity, we provideaproof-of-concept prototype, implemented through the modification of a meta-circular interpreter. It allows us todemonstrate that the proposed mechanism is computationally feasible and even achievable with a small set of instructions.In our definition of computational reflexivity, we have been inspired by several concepts discussed in the literature on consciousness studies. Some of them will be reported in the following sections. Our main source of inspiration is, however, the notion of self-conscious reflexivity, as discussed in higher-order theories of consciousness, and the attempts to describe it in neuroscientific <cit.> and computational <cit.> terms.The rest of the paper is organized as follows. In the next section, we present an overview of self-awareness, introspection, and reflexion in the context of both computer science and consciousness studies. Section <ref> introduces the formal definitions of computational reflexion, and Section <ref> introduces the prototype.Finally, we present a short discussion in Section <ref> and draftpossible applications and next research steps in Section <ref>. § BACKGROUND§.§ Procedural IntrospectionIn the context of the present work, we use the term computational introspection to indicate a program capable of accessing itself, create a self-representation, and manipulate it.A crucial distinction should be made between the meaning of “knowledge” underlying the notion of “representation” and “manipulation”.For this reason, we distinguish between procedural knowledge and declarative knowledge, the former based on computable functions, and the latter on logical statements.Depending on which meaning of “knowledge” is adopted, there are two different ways to define computational introspection, called here procedural introspection and declarative introspection, respectively.Batali <cit.> claims that “introspection is the process of thinking about one's own thoughts and feelings. [...] To the degree that thoughts and feelings are computational entities, computational introspection would require the ability of a process to access and manipulate its own program and its current context” (See Valdemir and Neto <cit.> on self-modifying code). In other words, computational introspection corresponds to the ability of a program to process its own code as data and modify it[In this definition, we put together self-representation and self-modification and, thus, the introspection and reflection features mentioned in Section <ref>.]. By contrast, in declarative introspection, the access corresponds to the generation of a set of logical statements, while their manipulation is performed by logical inference <cit.><cit.>. Batali <cit.> says that “The general idea is that a computational system (an agent preferably) embodies a theory of reasoning (or acting, or whatever). This is what traditional Al systems are – each system embodies a theory of reasoning in virtue of being the implementation of a program written to encode the theory.”As discussed by Cox <cit.>, “From the very early days of AI, researchers have been concerned with the issues of machine self-knowledge and introspective capabilities. Two pioneering researchers, Marvin Minsky and John McCarthy, considered these issues and put them to paper in the mid-to-late 1950’s.[...] Minsky's <cit.>contention was that for a machine to adequately answer questions about the world, including questions about itself in the world, it would have to have an executable model of itself. McCarthy <cit.> asserted that for a machine to adequately behave intelligently it must declaratively represent its knowledge. [...] Roughly Minsky's proposal was procedural in nature while McCarthy's was declarative.” On the basis of these ideas, Stein and Barnden performed a more recent work to enable a machine to procedurally simulate itself <cit.>.Interestingly, Johnson-Laird <cit.>, inspired by Minsky, proposes a definition of procedural introspection closer to the concept of computable function. He claims that “Minsky's formulation is equivalent to a Turing machine with an interpreter that consults a complete description of itself (presumably without being able to understand itself), whereas humans consult an imperfect and incomplete mental model that is somehow qualitatively different.” According to Smith <cit.>, “the program must have access, not only to its program, but to fully articulated descriptions of its state available for inspection and modification.” [...] Moreover, “the program must be able to resume its operation with the modified state information, and the continued computation must be appropriately affected by the changes.”Unlike the use of `procedural' discussed above, actually consisting of a “declarative” representation of the “procedural knowledge”, we employ the term in a more restrictive way. Procedural introspection is here limited to program code access and modification, without any logical modeling and inference. In this way, we want to avoid the possible dependence of a particular declarative modeling from the choices of the human designer, instead focusing on aspects connected to program access and modification. §.§ Introspection in Consciousness StudiesHistorically, all the uses of the term `introspection' in computer science have been influenced by the meaning of the same term in philosophy of mind and, later on, neurosciences and cognitive science.In consciousness studies, introspection is often discussed in the context of the so-called higher-order (HO) theories, based on the assumption that there are different “levels” or “orders” of mental states.Perceptions, emotions, and thoughts are instances of first-order mental states. Higher-order mental states are mental states about other mental states.For example, a thought about thinking something. Introspection is considered as “an examination of the content of the first-order states” <cit.>. It is not clear, however,if introspection itself is a high-order state or it is involved in the occurrence of first-order states. §.§ Self-Conscious ReflexivityIntrospection is not generally consideredthe main characteristic of conscious states.In contrast, as claimed by Peters <cit.>, “consciousness is reflexivity”, where reflexion isthe “awareness that one is perceiving”. Unlike other defining characteristics, such as intentionality, reflexivity is the only one that is considered unique to consciousness. Trautteur remarked that Damasio was the firstscientistto describe reflexion in the context of neuroscience <cit.>.Damasio's definition of reflexion (referred to by the term core self) is reported in the following statement: * It is the process of an organism caught in the act of “representing its own changing state as it goes about representing something else” (<cit.>).This definitionis meant to be based on biological (and, thus, physicalist, objective) terms since the term `representation' here denotes specific neural patterns.The next statementexpresses the attempt by Trautteur to translate the above “metaphorical” definition in computational terms: * [It] is the process of an agent “processing its own processing while processing an input[This statement is extracted from unpublished notes by Trautteur.].”In this version,the organism is reformulated as a computational agent and representationas a computationalprocess.Both the above statementspresent a logical issue. We refer to it as the identity paradox. It consists of the fact that the object and the subject of the experience are perceived as the same entity. It is a violation of the identity principle, also detectable in other expressions used by the same and other authors such as “presence o the self to itself” or “the identity of the owner (of experience) and the owned” <cit.>.§.§ Elements of Inspiration and Informal Definition of Computational Reflexivity To overcome this logical contradiction, in the present research we moved the focus from identity to simultaneity. This frame shifting was inspired by Van Gulick <cit.>, which emphasizes the simultaneity of observed and observer: “what makes a mental state M a conscious mental state is the fact that it is accompanied by a simultaneous and non-inferential higher-order (i.e., meta-mental) state whose content is that one is now in M”. The above statement triggered the insight that reflexion can be seen as the simultaneous occurrence of two distinct and synchronized processes.It implies three underlying assumptions in it: temporal extension (i.e., `state' means that we are dealing with processes), distinction (i.e., we have two separate processes), and synchronicity (i.e., the two processes are simultaneous). Because of the temporal extension, the term `simultaneity' is employed here in the sense of interval simultaneity, which refers to sequences of events <cit.>.Interval simultaneity does not necessarily imply, here,the simultaneity of the single events. Our assumption of synchronicity requires that each step in one of the two processes must occur only after a corresponding step in the other one. As shown in the next section, each pair of steps are part of the same interpretation loop.Using the second statement as a reference, we informally define computational reflexion as the concurrent (i.e. at every step of the interpretation loop) and synchronized execution of a computer program and manipulation of its code. Correspondingly, an interpreter capable of performing computational reflexion is said to be provided with computational reflexivity. This definition impliesthat computational reflexivity is a characteristic of a particular class of universal machines.§ FORMAL DEFINITION OF COMPUTATIONAL REFLEXIONIn this section, we provide, a step a time, all building blocks for the formal definition of computational reflexion. We assume reflexivity as a property applicable to the execution of any computer program, instead of a property of a single program. For this reason, it must rely on a particular type of program interpretation. From the point of view of an interpreter, the execution of a program can be reduced to a number of iterations of the same interpretation loop. We use the term step to denote a single occurrence of the interpretation loop, despite its internal complexity. We unravel below the definition of computational reflexivity as a sequence of incremental enrichments of the interpretation loop. Each enrichment, referred by both a textual symbol and a graphic mark, is meant to induce a corresponding modification at the process level. 1. Lower Step and Standard ExecutionThe original computational step (i.e., the unmodified interpretation loop) is called here lower step, indicated by the symbol (S_L) and the graphic mark< g r a p h i c s > . At the process level, we call target process the overall program execution. 2. Single Introspection and Tracing In this modified step, the interpreter executes a local procedural introspection on the current step, returning the code of the current instruction. It is called single introspection, indicated by the symbol (S_L, I_S) and the graphic mark of the interpretation loop is< g r a p h i c s > . At the process level, the system generates a trace of execution, similar to the one produced by a debugger. 3. Single Upper Step and Mirroring The interpreter executes the instruction just extracted by introspection. We call it upper step, denoted by (S_L, I_S, S_SU). The overall loop is graphically represented as< g r a p h i c s > . At the process level, we have two identical programs simultaneously executed. We use the term mirroring to indicate this real-time duplication of the target process. 4. Double Upper Step and Augmentation Here the interpretation loop is enriched with two further operations: the modification of the current step of the “mirrored program” by introduction of an additional instructions, and the next step execution[Although a more general class of code modification is conceivable, we limit the focus on the modification by instruction insertion. As explained in the next point, the aim is to enrich the second process with information about the target process.]. The term double upper step, with the symbol (S_L, I_S, S_DU), indicates the execution of the “mirrored” instruction and the additional one. The overall loop is graphically represented as< g r a p h i c s > . We call computational augmentation the modification of the interpretation loop performed so far. Correspondingly, we have two simultaneous processes: the target process and the augmented process. The latter one is based on the former but modifiedat the step level. 5. Double Introspection and Reflexion Now, we consider a particular type of computational augmentation, in which the additional instruction of the double upper step is a further operation of global procedural introspection. While the local introspection returns the code of the current instruction of the target program (i.e., the lower step defined above), the global introspection returns the code of the entire target program or a subset of it. In this case, the upper step consists of an execution of the mirrored instructions of the target program plus additional global instructions about it. We call double introspection this type of double upper step, and denote it by the symbol (S_L, I_D, S_DU). The overall loop is represented by the graphical mark< g r a p h i c s > . Finally, we define computational reflexion as the process generated by the loop composed by lower step, double introspection and double upper step. Table <ref> summarizes the schema of all components. Each row reports the symbolic representation, the graphical mark, and the corresponding terminology at both step and process level. In summary, the addition of specific groups of instructions to the interpretation loop underlies the generation of different processes, each built on the previous one: standard execution, tracing, mirroring, augmentation, and reflexion. Given a target process, the enriched interpreter executed the related program and a concurrent version executed, at every step, with its own code.Our definition of computational reflexion is thus a formal specificationof the informal one reported in Section <ref>. § PROTOTYPICAL IMPLEMENTATIONAs a proof of concept of the feasibility to implement computational introspection, as defined in the previous section, we developed a prototypical version. Specifically, we employed and modified the code of a Lisp meta-circular interpreter <cit.><cit.> (i.e., an interpreter of the Lisp programming language, implemented in the same language), called here Lisp in Lisp.The main reason for using Lisp in Lisp is that it is one of the simplest ways to implement a general-purpose interpreter. Indeed, it is a specific model of computation based on Church's Lambda Calculus <cit.>. As reported by McCarthy <cit.>, “Another way to show that Lisp was neater than Turing machines was to write a universal Lisp function and show that it is briefer and more comprehensible than the description of a universal Turing machine. This was the Lisp function eval [...]” The program is just a few lines of code and the definition of its main function, , is based on the composition of a few primitive operators. Thefunction is what is performing the interpretation (or evaluation) process.In this case, we call computational step (and, equivalently, interpretation loop) the Lisp in Lisp execution between two next calls of thefunction. Therefore, using the sequence of steps described in the previous section, we modified the definition ofadding additional function calls. For example, the single introspection event correspond to a call of the function , which returns the code of the argument (i.e. the instruction under execution). The complete code of the program and applied examples of executions are free available for research purpose § DISCUSSIONThe intuitions formalized in this paper are aimed to envision a new type of self-aware systems. While almost all state-of-the-art systems are based on introspection, we propose to consider reflexion as the main aspect of self-awareness. We could intuitively define computational reflexion as “a mechanism for making a computational process continuously self-informed”. The expression “mechanism of making” expresses the fact that reflexion is defined as a particular type of interpreter. Indeed, we focused on the interpretation loop and modified it. Reflexion is not the property of a specific class of computer programs but, instead, something that can be provided to any executable programs through this form of interpretation. Through reflexion, the standard program execution (i.e., the target process) is dynamically “reflected” into the execution of its augmented counterpart (i.e., the reflexive process).As explained in Section <ref>, each instruction of the target program is executed twice: the first time (as sequence of lower steps) to achieve the standard execution (and generate the target process), and the second time (as sequence of upper steps) as part of the reflexive process. In the above definition, the term “self” is not referring to a single entity but to a couple of mutually interactive entities. This duality between the two processes is the way we theoretically addressthe identity paradox mentioned in Section <ref>. § POSSIBLE APPLICATIONS AND FUTURE WORKThe properties identified in the previous section allow us to conceive some interesting uses of the reflexive augmentation of program execution. For example, we could see the execution of the target program and the corresponding reflexive augmentation as performed by two separate but synchronized devices. Specifically, we could have an autonomous agent (e.g. a robot in a physical environment) and an interfaced web service implementing reflexion. Therefore, computational reflexion could be used as a way to provide a system with a temporary “streaming of self-awareness”.The aimed next steps of our research are focused on the following aspects. Firstly, we intend to further develop the proposed formalization and achieve possible interesting implications as formal theorems. Secondly, we aim to study the degree to which the reflexive process should give feedback to the target process and modify the related program. In other words, we would like to investigate aspects of run-time “virtual” self-modification, not yet taken into account, at this stage of the research, in our prototype.A crucial issue is about efficiency. We need to investigate to what degree the combination of step-level local and global introspection and corresponding execution can be feasible performed. If the target program is sufficiently complex, there is a limitation in the number of instructions capable of being executed along the duration of the interpretation loop. 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Peters, Theories of consciousness as reflexivity, The Philosophical Forum 44 (2013) 341–372.VanGulick2014 R. Van Gulick, Consciousness, in: E. N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy, spring 2014 Edition, The Metaphysics Research Lab, 2014.Jammer2006 M. Jammer, Concept of simultaneity: from antiquity to Einstein and beyond, The Johns Hopkins University Press, 2006.Landauer_Bellman2001 C. Landauer, K. L. Bellman, Self-modeling systems, in: Proceedings of the 2nd international conference on Selfadaptive software: applications (IWSAS'01), Balatonfüred, Hungary, 2001, pp. 238–256.Graham2002 P. Graham. http://lib.store.yahoo.net/lib/paulgraham/jmc.psThe roots of lisp [online] (18 January 2002) [cited 5 September 2016].Church1941 A. Church, The Calculi of Lambda-Conversion, Princeton University Press, Princeton, N.J., 1941.McCarthy1960 J. McCarthy, Recursive functions of symbolic expressions and their computation by machine, part i, Communications of the ACM 3 (1960) 184–195.McCarthy1978 J. McCarthy, History of lisp, ACM SIGPLAN Notices - Special issue: History of programming languages conference 13 (8) (1978) 217–223.Steele1990 G. L. Steele, Common Lisp the Language, 2nd Edition, Digital Press, 1990.§ APPENDIX A: CODE DESCRIPTION The following integrative material is provisionally provided as appendix of the paper. In the final version, this material will be provided in a website.[0.5ex]1ptWe employed the code of an interpreter of the Lisp programming language, implemented in the same language, and called here Lisp in Lisp. Specifically, since the term Lisp is currently used to denote an entire family of programming languages sharing common characteristics, we tested and run the code in Common Lisp <cit.>. Rather than the original formulation ofby McCarthy <cit.>, we adopted the simpler version by Paul Graham <cit.>, which also found a bug in the original version and removed it[We have found a small bug in Graham's code as well. In the definition body of the <> function, there is a call to the <> function, which is a system function. Since the set of primitive operators should not include <>, we defined the function <> andused it to replace all the occurrences of<>in the definition body of <> In this note, we rounded the function names with angular parenthesis to separate them more clearly from the rest of the text.].In this context, we define computational step (and, equivalently, interpretation loop) as the Lisp in Lisp execution between two next calls of thefunction. Appendix <ref> contains the code of the Lisp in Lisp (i.e., thefunction) and its modified version (called ) implementing reflexion.§ APPENDIX B: COMPONENTS OF COMPUTATIONAL REFLEXION In the same way proceeded in the previous section, we focused on the interpretation loop and gradually enriched to obtain the version implementing computational reflexion. * Lower Step.It is equivalent to the interpretation loop defined above.* Single (Local) Introspection.The currentcall returns the code of the current instruction, consisting of a function call. * Single Upper Step.The code of the current function call, produced by the local introspection, is in turn executed (i.e.is called on it). * Double Upper Step.The code generated by local introspection is enriched with additional instructions. As an example, we added acall to the output of the current call. In this way, the interpreter will display on the terminal the trace of execution. * Double Introspection.Finally, the interpretation loop is enriched with the instruction for global introspection. In other words, it returns the code of the entire program. In summary, at any stage of the computation, the interpreter accesses and executes the code both locally and globally. In particular, the program code could be modified at each step and, thus, influence the next execution.As a specific example, Appendix <ref> shows the Lisp definition of the function , which gets a list as input and returns its last element as output.§ APPENDIX C: LISP CODEThe code of the functioncorresponds to the version of the Lisp in Lisp by Paul Graham <cit.>. We modified it and defined , as a proof-of-concept version of the reflexive interpreter, with the following instruction: . The functionapplies the predicateto the input and output of the current step.The specific implementation ofin this example is , which extract the code of the current instruction and execute it again, thus performing the “mirroring” discussed in Section 3 of the paper. [language=lisp, basicstyle=](defun eval. (e a) (cond ((atom e) (assoc. e a)) ((atom (car e))(cond((eq (car e) 'quote) (cadr e))((eq (car e) 'atom)(atom (eval. (cadr e) a)))((eq (car e) 'eq)(eq (eval. (cadr e) a) (eval. (caddr e) a)))((eq (car e) 'car) (car(eval. (cadr e) a)))((eq (car e) 'cdr) (cdr(eval. (cadr e) a)))((eq (car e) 'cons)(cons (eval. (cadr e) a) (eval. (caddr e) a)))((eq (car e) 'cond)(evcon. (cdr e) a))('t (eval. (cons (assoc. (car e) a) (cdr e)) a)))) ((eq (caar e) 'label)(eval. (cons (caddar e) (cdr e)) (cons (list. (cadar e) (car e)) a))) ((eq (caar e) 'lambda)(eval. (caddar e) (append. (pair. (cadar e) (evlis. (cdr e) a))a)))))(defun null. (x) (eq x '()))(defun and. (x y) (cond (x (cond (y 't) ('t '()))) ('t '())))(defun not. (x) (cond (x '()) ('t 't)))(defun append. (x y) (cond ((null. x) y) ('t (cons (car x) (append. (cdr x) y)))))(defun list. (x y) (cons x (cons y '())))(defun pair. (x y) (cond ((and. (null. x) (null. y)) '()) ((and. (not. (atom x)) (not. (atom y)))(cons (list. (car x) (car y))(pair. (cdr x) (cdr y))))))(defun assoc. (x y) (cond ((null. y) '())((eq (caar y) x) (cadar y))('t (assoc. x (cdr y)))))(defun evcon. (c a) (cond ((eval. (caar c) a)(eval. (cadar c) a)) ('t (evcon. (cdr c) a))))(defun evlis. (m a) (cond ((null. m) '()) ('t (cons (eval.(car m) a) (evlis. (cdr m) a)))))(defun eval-augment (e a pred) (let* ((input (list e a))(output (cond ((atom e) (assoc. e a)) ((atom (car e)) (cond ((eq (car e) 'quote) (cadr e)) ((eq (car e) 'atom)(atom (eval-augment (cadr e) a pred))) ((eq (car e) 'eq)(eq (eval-augment (cadr e) a pred) (eval-augment (caddr e) a pred))) ((eq (car e) 'car) (car(eval-augment (cadr e) a pred))) ((eq (car e) 'cdr) (cdr(eval-augment (cadr e) a pred))) ((eq (car e) 'cons)(cons (eval-augment (cadr e) a pred) (eval-augment (caddr e) a pred))) ((eq (car e) 'cond)(evcon. (cdr e) a)) ('t (eval-augment (cons (assoc. (car e) a) (cdr e)) a pred)))) ((eq (caar e) 'label) (eval-augment (cons (caddar e) (cdr e)) (cons (list. (cadar e) (car e)) a) pred)) ((eq (caar e) 'lambda) (eval-augment (caddar e) (append. (pair. (cadar e) (evlis. (cdr e) a)) a) pred))))) (augment input output pred) output)) (defun augment (input output pred) (setq *done* (append *done* (list (list input output)))) (funcall pred *done*)) (setq *pred* #'(lambda (done) (let* ((next (car (last done))) (input (car next)) (e (car input)) (a (cadr input)) (output1 (eval. e a))) (format t (concatenate 'string (write-to-string input) " "-> " (write-to-string output1) " t))) § APPENDIX E: EXAMPLES OF EXECUTIONAs a simple example, the functionand the predicateare applied to simple data (the atomwith value , and the functionreturning the first element of the list . In particular, it is applied to the simple recursive function , returning the last element of a list. [language=lisp, basicstyle=]CL-USER(356): (eval-augment 'a '((a 1)) *pred2*) 1 CL-USER(357): (eval-augment 'a '((a 1)) *pred1*) (A ((A 1))) -> 1 1 CL-USER(360): (eval. '(car '(a b)) nil) A CL-USER(361): (eval-augment '(car '(a b)) nil *pred2*) A CL-USER(362): (eval-augment '(car '(a b)) nil *pred1*) ('(A B) NIL) -> (A B) ((CAR '(A B)) NIL) -> A A CL-USER(370): (setq e '(my-last '(a b c))) (MY-LAST '(A B C)) CL-USER(371): (setq a '((my-last (label my-last (lambda (x) (cond ((null. x) 'nil) ((null. (cdr x)) (car x)) ('t (my-last (cdr x))) )))) (null. (label null. (lambda (x) (eq x nil)))))) ((MY-LAST (LABEL MY-LAST (LAMBDA (X) (COND # # #))))(NULL. (LABEL NULL. (LAMBDA (X) (EQ X NIL))))) CL-USER(372): (eval. e a) C CL-USER(373): (eval-augment e a *pred2*) C CL-USER(374): (eval-augment e a *pred1*) ((COND ((NULL. X) 'NIL)((NULL. (CDR X)) (CAR X))('T (MY-LAST (CDR X))))((X (A B C)) (MY-LAST(LABEL MY-LAST (LAMBDA (X) (COND ((NULL. X) 'NIL) ((NULL. (CDR X)) (CAR X)) ('T (MY-LAST (CDR X))))))) (MY-LAST(LABEL MY-LAST (LAMBDA (X) (COND ((NULL. X) 'NIL) ((NULL. (CDR X)) (CAR X)) ('T (MY-LAST (CDR X))))))) (NULL. (LABEL NULL. (LAMBDA (X) (EQ X NIL)))))) -> C (((LAMBDA (X) (COND ((NULL. X) 'NIL) ((NULL. (CDR X)) (CAR X)) ('T (MY-LAST (CDR X))))) '(A B C))((MY-LAST(LABEL MY-LAST (LAMBDA (X) (COND ((NULL. X) 'NIL) ((NULL. (CDR X)) (CAR X)) ('T (MY-LAST (CDR X))))))) (MY-LAST(LABEL MY-LAST (LAMBDA (X) (COND ((NULL. X) 'NIL) ((NULL. (CDR X)) (CAR X)) ('T (MY-LAST (CDR X))))))) (NULL. (LABEL NULL. (LAMBDA (X) (EQ X NIL)))))) -> C (((LABEL MY-LAST(LAMBDA (X)(COND ((NULL. X) 'NIL)((NULL. (CDR X)) (CAR X))('T (MY-LAST (CDR X)))))) '(A B C))((MY-LAST(LABEL MY-LAST (LAMBDA (X) (COND ((NULL. X) 'NIL) ((NULL. (CDR X)) (CAR X)) ('T (MY-LAST (CDR X))))))) (NULL. (LABEL NULL. (LAMBDA (X) (EQ X NIL)))))) -> C ((MY-LAST '(A B C))((MY-LAST(LABEL MY-LAST (LAMBDA (X) (COND ((NULL. X) 'NIL) ((NULL. (CDR X)) (CAR X)) ('T (MY-LAST (CDR X))))))) (NULL. (LABEL NULL. (LAMBDA (X) (EQ X NIL)))))) -> C C | http://arxiv.org/abs/1707.08901v1 | {
"authors": [
"Alessandro Valitutti",
"Giuseppe Trautteur"
],
"categories": [
"cs.AI",
"F.1.1; F.3.3; I.2.0; I.2.2"
],
"primary_category": "cs.AI",
"published": "20170727150527",
"title": "Providing Self-Aware Systems with Reflexivity"
} |
OMLcmmbitϵ λ | http://arxiv.org/abs/1707.08680v2 | {
"authors": [
"Jacob Lambert",
"Lee Clement",
"Matthew Giamou",
"Jonathan Kelly"
],
"categories": [
"cs.RO"
],
"primary_category": "cs.RO",
"published": "20170727013704",
"title": "Entropy-Based $Sim(3)$ Calibration of 2D Lidars to Egomotion Sensors"
} |
Example 1Observations collectedwith the Cassini Telescope at Loiano station of the INAF-Bologna Astronomical Observatory, with the Copernico Telescope of the INAF-Padova Astronomical Observatory and with the TACOR Telescope of the Università “La Sapienza”, Roma.2INAF/IAPS, Roma Italy. 3INAF/ Osservatorio di Monte Porzio, Roma, Italy. 4Università La Sapienza Roma, Italy. 5 Ambartsumian Byurakan Astrophysical Observatory (BAO). Gaudenzi, Nesci & al. BIS sources : optical results.* S.Gaudenzi and R.Nesci: INAF/IAPS, via Fosso del Cavaliere 100, 00133 Roma, Italy([email protected]); ([email protected]).* C. Rossi: INAF-Osservatorio Astronomico di Roma, Via Frascati 33, 00040, Monte Porzio Catone (RM), Italy ([email protected]).* S. Sclavi and C. Rossi: Dipartimento di Fisica, Università La Sapienza, Piazzale Aldo Moro 3, 00185 Roma, Italy.* K.S.Gigoyan and A.M. Mickaelian: V. A. Ambartsumian Byurakan Astrophysical Observatory (BAO) and Isaac Newton Institute of Chile, Armenian Branch, Byurakan 0213, Aragatzotn province, Armenia. S.Gaudenzi, R.Nesci& al. Gaudenzi, S. Nesci, R. Rossi, C. Sclavi, S. K. S. Gigoyan. Mickaelian, A.M.Through the analysis and interpretation of newly obtained and of literature data we have clarified the nature of poorly investigated IRAS point sources classified as late type stars, belonging to the Byurakan IRAS Stars catalog.From medium resolution spectroscopy of95 stars we have strongly revised 47 spectral types andnewly classified 31 sources. Ninestars are of G or K types, fourare N carbon stars in the Asymptotic Giant Branch, the othersbeing M-type stars.From literatureand new photometric observations we have studied their variability behaviour.For the regular variables we determined distances, absolute magnitudes and mass loss rates. For the other stars we estimated the distances, ranging between 1.3 and 10 kpc with a median of 2.8 kpcfrom the galactic plane, indicating that BIS stars mostly belong to the halo population.Un gran nmero de entradas del catlogo de fuentes puntuales IRAS tienen todava una clasificacin aproximada. A travs del anlisis y la interpretacin de nuevos datos obtenidos y datos de la literatura tuvimos como objetivo clarificar la naturaleza de las pobremente investigadas fuentes puntuales de IRAS clasificadas como estrellas de tipo tardo, pertenecientes al catlogo de estrellas Byurakan IRAS. En 95 estrellas realizamos espectroscopa de mediana resolucin para mejorar la clasificacin espectral que previamente se haba basado solo en ndices de color en el ptico y/o espectros de baja resolucin. Recolectamos datos fotomtricos de la literatura y obtuvimos nuevas observaciones fotomtricas para estudiar su variabilidad; asignamos o revisamos la clasificacin de variabilidad de nuestros objetos. Revisamos los tipos espectrales y encontramos diferencias sustanciales con respecto al catlogo BIS original en 45% de las estrellas; adems el 30% no tenan clasificacin previa. Para las estrellas variables regulares determinamos sus distancias, sus magnitudes absolutas y sus tasas de prdida de masa. De las otras estrellas, estimamos sus distancias a partir de sus tipos espectrales y sus magnitudes aparentes: Se encontraron distancias desde 1.3 a 10 kpc, con una distancia media desde el plano Galctico de 2.8 kpc, lo cual indica que las estrellas del catlogo BIS en su mayora, pertenecen a la poblacin del halo de la Galaxia. Un gran nmero de estas estrellas se encontraran al alcance de una medicin directa de GAIA. Los espectros muestran que nueve estrellas son de tipo espectral G-K,cuatro estrellas de Carbn N se encuentran en la rama gigante asinttica, las otrasson del tipo espectral M.Stars: late type Stars: variables Stars: Mass lossSpectral Properties and Variabilityof BIS objects.1 S. Gaudenzi,2R. Nesci,2C. Rossi,3,4 S. Sclavi,4K. S. Gigoyan,5andA. M. Mickaelian,5 December 30, 2023 ====================================================================================================== § INTRODUCTION: BYURAKAN – IRAS STARS CATALOG Asymptotic Giant Branch (AGB) stars are very luminous objects, in a fast evolutionary phase which mix up the atomic nuclei produced by nuclear reaction in the inner part of the star into the external envelope and then into the interstellar space through the stellar wind and the final formation of a Planetary Nebula. A better census of the AGB stars population and good knowledge of their physical characteristics and variability is a basic requirement to perform satisfactory checks of our models of stellar evolution and to study the chemical evolution of our Galaxy <cit.>. The forthcoming direct distance measures by the GAIA mission of the stars within several kpc from the Sun will allow a substantial improvement in this field. With this aim in mind, we performed a study of a sample of AGB candidates collected in the Byurakan Infrared Stars (BIS) catalog <cit.>, based on the spectra of galactic sources visible in the objective prism plates of the First Byurakan Survey (FBS)<cit.> and on the IRAS Point Source Catalog <cit.>.In 1983, the Infra Red Astronomical Satellite (IRAS) surveyed about 96% of thesky in bands centredat 12, 25, 60, and 100μm.More than 245000 point sources were detected and their fluxes and positionswere listed in the IRAS PSC. A recent work by <cit.> made a cross-checkof the IRAS Point Source Catalog (PSC) and Faint Sources Catalog (FSC) <cit.>improving their positions and correlation with infrared sources detected by more recent missions (WISE <cit.> , AKARI <cit.>, 2MASS <cit.>). The IRAS archive alsoincludes several spectra in the Low-Resolution Spectra catalog <cit.> in the range 7.7-22.6 μmproviding useful indications on the chemical composition of the dust shells around the stars (see , and the ). The BIS catalogcontains data for a final census of276 IR sourcesas being potentially stars of late spectral types. The stars were selected on the basis of their low resolution spectra of FBS and of the Dearbon Astronomical Observatory <cit.>. Imagesof the Palomar Observatory Sky Survey (DSS, ) were also used to check the identification.The most recent citations of literature spectral types are to be found in <cit.>. The identifications on the FBS plates were carried out in the region with δ > +61^o and galactic latitude b > +15,coveringa surface of 1504 deg^2.For each object several information are given, such as accurate optical positions for two epochs (B1950 and J2000), photometric data after cross-correlation with MAPS <cit.>, USNO-B1.0 <cit.> and 2MASS catalogs, proper motions (PM) and aclassification based on the Digitized First Byurakan Survey spectra (DFBS, ) accessible online from the webpage http://ia2.oats.inaf.it/ . The stars have intermediate values of galactic coordinates, ranging in longitude between 90 and 151 degrees, and in latitude between 14 and 45 degrees, with average value of 30 degrees: they are therefore outside the solar circle andare likely members of the halo / thick-disk population.Aiming atclarifying the nature of the stars included in the BIS catalog we made a systematic collection of all the information available in literature and performed the acquisition of new photometric and spectroscopic data.This paper is devoted to the optical properties of a subsample of 95 stars randomly selected, while a companionpaper will be dedicated to the infrared characteristicsof all the stars of theBIS catalogusing data collected from public archives. <ref>, describes our targets; <ref>, and <ref>, are devoted to the observations and data analysis; <ref>, is devoted to estimate and discussion ofthe parameters of the IR sources; our conclusions are summarised in <ref>. § OUR TARGETS TheBIS catalogcontains several types of sources, includingM and CH stars, Mira-type and Semi-Regular (SR) variables, OH and SiO sources, N-type carbon stars and unknown sources surrounded by thick circumstellar shells.Most of our targets are poorly studied both from the photometric and spectroscopic point of view and only a few of them are classified as variables in the General Catalog of Variable stars (GCVS, ) or in the Variable Star Index (VSX, ) catalogs. A preliminary clue ofvariabilitywas given in<cit.> using the differences B1-B2 and R1-R2 of the two epochs of the DSS as reported in the USNO-B1.0 catalog.Among the 276 stars of the BIS catalog, 13 have color index 1.5≤ B-R≤2.5,the remaining have B-R ≥ 2.5 mag. We obtained newoptical CCD multi-band photometry and medium resolution spectra of asampleof95 stars: we have randomly selected 9 with color index ≤2.5 in order to check thereliability of the original classification and to classify two starshavingno previous classification. We also obtained the spectrum of BIS028, already known to be a planetary nebula. § OBSERVATIONSSpectrain the range 3940-8500 Å, 3.9 Å/pixel dispersion, were obtained with the Cassini telescope of the Bologna Astronomical Observatory (Italy) at Loiano, equipped with the Bologna Faint Objects Spectrometer and Camera (BFOSC) and a EEV P129915 CCD. Photometric observations were also obtained with BFOSC, possibly on the same dates as for spectra. Some objects were observed also at the Copernico Cima Ekar telescope of the Padova Astronomical Observatory (Italy) equipped with the Asiago Faint Objects Spectrometer and Camera (AFOSC) mounting aTK1024ABCCD. All the stars were always observed in the Red Johnson filter.Photometric observations were also performed in the R filter in the period July-November 2011 with the TACOR [TACOR = Telescopio A COntrollo Remoto] telescope of the Department of Physics of the University “La Sapienza” in Rome equipped with an Apogee U2CCD. The data were reduced by means of standard IRAF procedures [IRAF is distributed by the NOAO, which is operated by AURA, under contract with NSF.]. Table <ref> presents the journal of observations; the columns have thefollowing meaning: 1 - BIS number in the catalog;2 - IRAS FSC designation of the objects;3 - date of observations;4 - spectral type according to the classification from our CCD spectra;5 - Red magnitude at the date of observation and observatory: ^1 Loiano; ^2 Cima Ekar;all the other data were obtained with the TACOR telescope.6 - magnitude range from the archive of the Northern Sky Variability Survey (NSVS, ; see Section <ref>); 7 - our variability classification defined in Section <ref>.A large number of spectroscopic standards (from K7 to M9 giants and dwarfs) were observed with the same instrumental configuration as our target stars and used as the basis for our classification. We have selected the templates of M standards from several catalogs and Spectral Libraries<cit.>.Spectra of M standards were also downloaded from the sitehttp://kellecruz.com/M_standards/.For the carbon stars we used <cit.> and <cit.>. § DATA ANALYSIS In the following sections we describe the general characteristics of the spectra and the optical light curves for individual stars, when available.§.§ General characteristics for spectral classification Spectral types were derived by overlappingthe spectral tracingof the targets with those of the reference stars, overplotting them with IRAF/splot. Classification was made independently by three of us and the typical uncertainty is one subtype. We have checked the self-consistency of the reference stars in the same way. Most ofthe reference stars arenearby. Thespectral resolution used allows to discriminate luminosity classesbut not the metal content.The spectra of almost all our targets are typical for M-type stars. In these stars the most prominent absorptions belong to the TiO bands at 4761, 4954, 5167, 5448, 5862, 6159, 6700, 7055 and 7600 Å.In some cases the VO bands of the red system are also present, with several band heads in the range 7334-7472 Å, and 7851-7973 Å, seen only in very late type stars. We paid special attention in looking for features used as typical dwarf/giant discriminators, like the Mg b triplet 5167, 5173, 5184 Å; the NaD doublet 5890, 5896 Å; the CaOHdiffuse bands centered at 5550 and 6230 Å, the MgH bands at 4780, 5211 Å; the CaH at 6382, 6908, 6946 Å <cit.>. Few atomic lines belonging to Fe and Ti are also present in some spectra. From their spectral characteristicsfour targets appears to be N-type AGB carbon stars. Three of them are embedded in a dense envelope with the blue region stronglyunderexposed.Two stars, BIS 104and BIS 106, were erroneously classified as carbon stars on the basis of the IRAS spectrum <cit.> while our spectra clearly show the typical features of intermediate M-type stars.The nine stars in our samplewithB-R≤2.5showed spectra typical for G and early K type stars, as expected.BIS 094, 105, 109, 229,251, 259, and 286 arein fair agreement with the old classification;BIS 060 and BIS 131 had no previous classification. BIS 131, has an infrared excess and has been the subject of a previous publication <cit.>. All these starswill not be discussed further.None of our stars appears to be of luminosity class V (Main Sequence), one isof class I and two of class II.Our spectra are by far of too low resolution to measure the radial velocities of stars, so we cannot give a kinematic indication of the kind of stellar population (halo, thick disk) to which our stars belong. None of the stars have an appreciable (≥20mas) proper motion in the USNO-B1 catalog.Figures<ref> and <ref> show a selection ofrepresentativeM type spectra, the other spectra of the same class being similar to thosepresented. The atmospheric absorption bands of O_2 at 6867 and 7594 Å, and ofH_2O at 7186 Å, are not removed. In these figuresthe ordinates are relative intensities corrected for atmospheric extinction, normalised with the maximum set to 100. §.§ Photometric variabilityMost of our targets are present in the NSVS database, collected between 1997 and 2001 from theROTSE-I (Robotic Optical Transient Search Experiment I) experiment <cit.>.Theobservationsspanning up to one year gave us reliable indications about the photometric variability of our sample. We have downloaded all the available light curves from the NSVS web site to check the photometric behaviour and compare the ROTSE magnitudes (R_r) with our new data.Some objects are not present in the NSVS archive,being fainter than ROTSE-I detection limit (15.5 mag). Fewstars have been observed by theCatalina Real-time Transient Survey (CRTS [http://www.lpl.arizona.edu/css/]). Magnitudes from other catalogshave been only considered asindicative, being obtained from theDSS plates where manystars are saturated.It is worth to remember that the NSVS data were obtained with an unfiltered CCD, so that the quantum efficiency of the sensor makes the effective band most comparable to the Johnson R band <cit.>, or better a mix of V and R colors, which is a function of the spectral type of the star. To inter-calibrate the NSVS and our magnitudes we used the stars in our sample with a very stable NSVS light curve. Our photometry is tied to the R magnitude scale of the GSC2.3.2 catalog <cit.>, and a good calibration would require a bigger set of non variable stars in the M0-M8 spectral types range to define a reliable color correction.In fact, we have verified that for M8 stars, which emit most the photons in the IR tail of the unfiltered detector, the ROTSE instrumental magnitude are generally brighter than for M1 stars of similar R magnitude. In any case, even with this caveat, our data have been useful to confirm thevariability/stability of the stars of our sample. We assignthree main variability indices on the basis of the light curves: (1) regular, large amplitude variables, larger than 1.1 mag;(2) irregular, large amplitude variables, between 0.5 and 1.1 mag; (3) small amplitudevariables stars ( up to 0.5 mag) or non-variable ( up to 0.3 mag).A few starsshow small amplitude either irregular or quasi regularvariability. To these stars we assigned intermediatevariability classes (2/3, 2/1), reported in Table <ref>only. Here below we grouped stars with similar characteristics to avoid useless repetitions. For a number of stars we add spectroscopic and/or photometric details.Variability class (1) BIS 007;BIS 116; BIS 133 (IY Dra); BIS 196;BIS 267 :BIS 007: For this star <cit.> report an estimated period of 341 days. The light curve is compatible with that of a Mira-variable star, but the photometric behaviour deserves a long term observational program to improve the accuracy of the period. The star is erroneously classifiedin BIS catalogas an N-type carbon star. No other spectroscopic information was found in literature.We observed BIS 007 in July 2007 and July 2008. The energy distribution is similar in the two epochs as well as thespectral features; in both epochs Hδ and Hγ are in emission, other hydrogen lines being hidden by the strong molecular absorptions. We classify this object as an S-type star, similar to HD56567 (S5/6 subtype). In Fig. <ref> the 2008 spectrum is presented. BIS 116: During the ROTSE monitoring the star showed a continuous modulation with period of 160 days, in agreement with <cit.>; our photometric dataarein agreement with the expected values.<cit.> classified this star as a Semi-Regular variable but the light curve pushes toward a classification as a Short-Period Mira.We have obtainedthree spectra, in 2007, 2008 and 2015. Thecontinuum andtheintensity of molecular absorptions are variable. In the three spectrathe hydrogen Balmer lines are in emission, though with different intensities. In Fig.<ref> the 2008 spectrum is presented. BIS 133 ( IY Dra ): This is a Mira-type variable star whose photographic magnitudes at minimum and maximum luminosities are presented in <cit.>. A lower limit of 351 days for the period is given by <cit.>.We carefully verified our R magnitudes which are all at the faint limits of the ROTSE magnitudes even taking into account the color correction. The star is strongly saturated in the POSS red plates. The Sloan Digital Sky Survey (SDSS [http://www.sdss.org/]) reports r = 14.83 mag, corresponding to an R magnitude in the range 13.80-14.30, depending on the adopted transformation equation; we remember anyhow that all the transformations are based on main sequence stars. From the spectroscopic point of view IY Dra is a very late type star very similar to the 2008 spectrum of BIS 007.BIS 267: The ROTSE light curve shows strong variations with continuous modulation between R_r 10.2 and 13.2 magnitudes.The bibliography based on the POSS plates gives fainter R magnitudes, while our R magnitude ( 9.75 ) corresponds to a brighter object. This impliesthat the period should be longer thanthe 255 days reported by the automatic calculation of NSVS.From the spectroscopic point of view BIS 267 is an interesting object with the higher lines of the hydrogen Balmer series in emission ( extremely strong Hγ and Hδ, no Hα, no Hβ ). The absorption spectrum is remarkable for the strength ofthe TiO molecular bands BIS 196: The ROTSE light curve, one year long, shows strong variations with continuous modulation between R_r 11.0 and 12.6 magnitudes,The automatic calculation of NSVS reports a period of 313 days.<cit.> from objective prism spectroscopy already classified this object as an M8 type star. Our data confirm his classification. There were no emission lines in the optical spectrumat the epoch of our observation while the shape is very similar to that of BIS 007. <cit.>, using the data from <cit.> and IR photometry, include this star in a group of possible Miras with period longer than 350 days. At the epoch of our observations the star was definitely fainter than the ROTSE minimum, even taking into account the IRAS color correction. The photometric behavior of this giant deserves a long term observational program to compute the exact period.Variability class (2e)BIS 002; BIS 122; BIS 207; BIS 219;BIS 264:Emission lines of the hydrogen Balmer series characterise the spectra of these stars (see Figures <ref> and <ref>); the NSVS archive classify BIS 122, BIS 207 and BIS 264 as Semi-Regular variable with a period of 147, 181 and 154 days, respectively, but modulations with shorter periodicity and small magnitude oscillations are also present in the light curves. Our observations show variability inside the ROTSE range. Spectral features are typical for middle to late-type giants. Variability class (2) Semi RegularBIS 043 (KP Cam); BIS 198;BIS 276 ;BIS 001; BIS 006;BIS 014;BIS 032; BIS 037;BIS 038;BIS 088 ;BIS 103 ;BIS 104; BIS 106;BIS 120 ; BIS 123; BIS 132; BIS 138;BIS 168 ;BIS 173;BIS 200;BIS 211; BIS 213; BIS 214; BIS 271:These stars have been classified by<cit.> as large amplitude,Semi-Regular variables. The light curves showa semi-regular pattern with continuous modulation and large amplitude (>0.4 mag) variability. Forthe first three stars quasiperiodicities have also been evaluated. The spectra aretypical of M5-M7 giant stars. BIS 043 is also present as a Semi-Regular variable in 76th list of variable stars <cit.>.The visual inspection of its light curve does not support convincing evidence for the quasi periodicity of about 150 days automatically calculated by NSVS.BIS 138 is the only other star (besides the carbonBIS 184 quoted below), to have an IRAS LRS infrared spectrum classified as 24 (star with "not too thick oxigen-rich envelope").BIS 276 shows two minima at the same magnitude at the beginning and the end of the ROTSE monitoring, with a period automatically calculated of 436 days. At variance with expectation, in all our 6 observations we found the source always nearly at the same luminosity, inside the ROTSE range. Variability class (2) Irregular. BIS 003;BIS 004;BIS 015;BIS 039;BIS 126;BIS 136;BIS 142;BIS 145;BIS 154;BIS 156;BIS 167;BIS 170;BIS 209;BIS 212;BIS 216;BIS 226:During the ROTSE monitoringthe light curves showed irregular variabilitywith maximum amplitude of 0.5 mag. Our measures are generally slightly fainter than the ROTSE values, in agreement with the expected difference due to the color correction for the spectral type of the stars, and in better agreement with the R values from literature. All these stars have spectral types between M5 and M8. The light curve of BIS 226 shows irregular variability with two minima at R_r about 11.7 whileour R magnitude 12.85 is much fainterin agreement with other VIZIER catalogs. Variability class (2/3) BIS 010;BIS 107; BIS 113;BIS 172; BIS 174;BIS 201; BIS 224;BIS 255;BIS 275;BIS 285:During the ROTSE monitoring these stars showed random variability with maximum excursion of 0.3 magnitudes. The R magnitudes from our observations are also in agreement with the ROTSE range. Most ofthese stars have an early M spectrum.We included the M5 starBIS 010 in this variability class in spite of the fact that this star was neither monitored by ROTSE because of its faintness nor by the Catalina survey which does not cover its declination zone. The brightness of this star is probably overestimated in every optical catalog being the northern component of an apparent close binary (3 arcsec ). The southern component almost disappears in the POSS2 IR plate and we found that its spectrum is early G. From the comparison of the images ofDSS1 with DSS2 no magnitude variations of BIS010 are evident. Ourobservationsin2013, 2016, 2017 did not reveal spectroscopic or photometric variation. Its extremely red infrared magnitudes (see our companion paper)cannot be justified by the immediate explanation of a long period variable. Avery long monitoring should be anyhow recommendable. Variability class (3).BIS 034;BIS 044;BIS 067;BIS 087;BIS 099;BIS 102;BIS 110;BIS 137; BIS 143;BIS 155; BIS 197;BIS 199;BIS 203; BIS 210;BIS 215 BIS 228;BIS 247*; BIS 248;BIS 256;BIS 258; BIS 260 : All these stars are very stable during the ROTSE monitoringwith a maximum variability of 0.1 magnitude. The photometric data available in literature and our values indicate small fluctuations compatible with ROTSE or Catalina survey data.No spectroscopic information was found in the literature except those by <cit.>. Our revised spectral classification turned out to be early M-type for all stars. Fig. <ref> shows the spectrum of BIS 034 whichcan be better classified as an S type star of subtype S2/3, very similar tothe prototypes HD 49368 and HD22649. May be interesting is the very stable BIS 247showingemission lines in the spectrum indicating a circumstellar envelope, whose origin would deserve further studies.Carbon stars BIS 036; BIS 184 (HP Cam); BIS 222; BIS 194: BIS 194 is stable while the other stars areIrregular variables. BIS 036 and BIS 222 have been identified asR Coronae Borealis candidates by different authors. These stars could not be observed by ROTSE being too faint. We made repeated observations of these stars; the first spectra obtained were presented in <cit.>. Here we will only report the spectral evolution of BIS 036 and BIS 184 shown in Fig. <ref>. BIS 194did not change and BIS 222 progressively faded making impossiblethe acquisition of new spectra with our instruments.For BIS 036 we getthe light curve from the Catalina Survey where the V magnitude ranges from 13.5 to 16.5.Actually we verified that on the POSS1 plate taken on February 5, 1954 the star was quite luminous,brighter than the nearby star having RA(2000) 05:28:56.5 and DEC(2000) +69:20:31, while on the POSS2 plate taken on October 31, 1994 it appears much fainter,likely after an episode of mass ejection. On the FBS plate 0138a taken on November 21, 1969 the star appears as a very short faint segment, consistent with R≤16.On the basis of the Catalina light curve and ofthe energy distribution,recently <cit.> included this starin a list of R Coronae Borealiscandidates, although the strong photometric variations, typical for R CrB stars have never been reported.We have observed this star in 2008, 2016 and 2017 detectingchanges in thephotometry and spectrum. The spectrum of BIS 036 is typical for a very late carbon star : in addition to the CN bands, the (0,1) transition at λ 5636 Å (Swan System) of the C_2 molecule is barely visible.A deep NaD absorption is present at λ 5895 Å possibly produced in the circumstellar envelope. The 2016 and 2017 spectra are almost identical, so we present only the last one to avoidconfusion. An important differencebetween the 2008 and 2016 spectra is the appearance in 2016 of a strongHα in emission: this feature, added to the photometric behavior, while compatible with a long period variable star rules out the R CrB hypothesis. BIS 184:This N type carbon staris a known Semi-Regular variable star (HP Cam),listed in the General Catalog of Variable Stars <cit.>.A photometric variability with a period of 296 days is reported by <cit.>.The Low Resolution Spectrum from IRAS is the only spectroscopicreference <cit.> where this object is classified as a carbon star based on the presence of the SiC emission feature at 11.2 μm (LRS classification is 44, according to ). This classification is confirmed by our optical spectra which also showed variations, as expectedfrom its classification.A substantialstrengtheningof Hα occurred between January and March 2016. InFig. <ref> we do not show the February 2017 spectrum, which is practically the same as January 2016.The variability ofBIS 222, wasascertained from the comparison of the historical plates ofthe POSS and the FBS: it was as strong as the two nearby stars in the Red POSS1 plate taken in 1955, when its red magnitude was about 17.3. It is barely visible in the FBS plate No. 1332 (1975): taken into account that 17.5 mag is the limit in the photographic band for the FBS plates, this could be a clue for the magnitude at that epochs. The star is invisible inthe POSS2 red plate (1996) and very strong in the POSS2 photographic IR (1997) plate; on January 2010 we have measured an R magnitude of 18.5. For all these reasons we assigned toBIS 222 the variability class (2).Our spectrum is consistent with that of a dust-enshrouded carbon star of a very late (N8-N9) subtype, similar to the well known IRC +10216 (CW Leo), an archetype of post AGB star and Pre-Planetary Nebula object. The only spectral features visible in the optical spectrum are the red and Near IR bands of CN molecule at λ 6952, 7088 and 7945 Å, 8150 Å.Hα is present inemission ( see Fig.1 of).Note that<cit.> included this star among the R CrB candidates on the basis of the infrared colors and the energy distribution ( star No. 1542 ), butthe presence of Hα could make questionable this classification. Being the star fainter during our more recent observations,we couldobtain photometric dataonly.BIS 194: the short monitoring (120 days only) by ROTSEindicates avery small modulation between 9.4 and 9.6 mag. Our R magnitude, 9.3 ± 0.2, is in good agreement with the value indicated by ROTSE. We tentatively assign the variability class (3) taking into account the ROTSE monitoring and our photometry only. No spectroscopic information were found in the literature. Our spectrum is that of an N type carbon star ( see Fig.1 of) with well expressed absorption of C_2molecule (Swan system) with band heads at λλ 4737, 5165, 5636, 6122 and 6192 Å. The CN molecule is also present with the bands at λλ5264, 5730, 5746, 5878, 6206, 6360, 6478, 6631, 6952, 7088, 7259, 7876-7945 and 8150 Å. In the threeFBS platescontaining thestar, the blue molecular bands arewell visiblebut the red part of the spectrum is saturated.§ PHYSICAL PARAMETERSHaving defined the above mentioned data, we are now in a better position to discussthe results from the optical photometric behavior and the spectral types of the stars. Where possible, wederive absolute magnitudes, distances and mass loss rates by applying different methods, according to the different types and characteristics of our objects.These parameters are commonly derived from empirical relations based on stars of well known distances: the mass loss rate determination is the most uncertain. In fact the various models generating dust, wind and circumstellar shells, arebased on similar hydrodynamical equations but ondifferent hypothesis of gas/dust ratios and different time dependence.Several stars in our sample are variable; in these cases calculations to obtain physical parameters should imply time-averaged IR magnitudes. To this purpose we have used2MASS and IRAS catalogs. Being aware that results from sparse observations must be considered with care, but, having a limited phase coverage of the measurements, obtained at random phases, we usedthe data as if they were the mean values of the magnitudes and, as uncertainties, we adopted the typical amplitude.Carbon starsTo estimate the absolute luminosity and distance of the “naked” carbon star BIS 194, we used the relation between the color index J-K and the absolute K magnitude M_K obtained by <cit.>, valid for J-K in the range between 1.4 and 2.3 magnitudes. We obtainedM_K = -7.88 mag and d=3.8 kpc.Concerningthe N-type dusty carbon starsBIS 036, BIS 184 and BIS 222, we applied various empirical relations and modelsavailable in literature. The results are in general agreement except for the mass loss rates, where the model dependence is strong and the differences are significant from one method to another.Our procedures are described below and the results are summarized in Table <ref> where the columns have the following meaning: 1 - BIS number; 2 - absolute K magnitude M_K; 3 - range of distances d; 4 derived distance to the galactic plane Z 5 - apparent bolometric magnitude m_bol; 6 - absolute bolometric magnitude M_bol; 7 - mass loss from <cit.>; 8 - mass loss from <cit.>.We have derived the range of absolute K magnitudes using the relations between M(K_s) and (J-K_s)_o from the calibration by <cit.>. From these values we then computed the range of distances to the stars.A cross check of the estimated absolute K magnitude can be made in the case of BIS 184 using the cited period of 296 days.From the relationby <cit.>: M_K= -1.34 LogP(d)-4.5 ⟶ M_K= -7.8 ± 0.4 mag.From thecalibration by <cit.>:M_K= -7.18-3.69 (LogP-2.38)⟶ M_K = -7.5 ± 0.3 mag.Both values are in good agreement with those obtained using the <cit.> calibration reported in Table <ref>.We havecomputed the apparent bolometric magnitudem_bol = K + BC_K usingthe calibrations to the bolometric correctionsby<cit.>.Weobtained a range of values inside the limits reported in column 4 of Table <ref>. The results obtained using the calibrations by <cit.> are in good agreement with those listed in the table.From thederived distances and from m_bol we could infer the range of absolute bolometric magnitudes. For BIS 184 M_bol can also be obtained in an independent way, using the period-luminosity relation by <cit.> (see their Fig. 2 and eq. 1 with c=2.06). The result, M_bol = -4.22 ± 0.24 mag. is in perfect agreement with thatreported in Table <ref>. We have finally estimated a range of mass loss rates being aware that the uncertainties are quite large. The combinations between several IR colors and mass loss rate was studied by <cit.> starting from stationary and from time-dependent models, with fixed assumptions on the parameters of the circumstellar shells; for the stars in common with a paper by <cit.> they obtained a satisfactory agreement, though with lower values of Ṁ. A good correlation between K-[12] color and the total mass loss rate was derived by <cit.> refining the previous work.Our results confirm the systematic difference of a factor of three between the two methods. In the last two columns of Table <ref> we report the results (in M⊙/year), the uncertainties are about ± 0.2 in the logarithm.O-rich Miras For the M-type Mira variables a number of relations involving period and IR luminosities (discussed in our companion paper)can be applied. The results are summarisedin Table <ref> where the meaning of the columns is the following:1 - BIS number;2 - variability period P reported by NSVS;3 - absolute K magnitude M_K;4 - apparent bolometric magnitude m_bol;5 - absolute bolometric magnitude M_bol;6 - distances and corresponding errors; 7 derived distance to the galactic plane;8 - mass loss.The results for BIS 133 reported in the table are lower limits, being the period longer than 351 days. Similarly we haveassigned class 1 to BIS 267 although this star is not known to be a Mira type but shows a regular pattern in the light curve of ROTSE experiment, as described above.At the opposite side, BIS 116 has the shortest period, the smallest amplitude and the earliest spectral type: its position in almost all color-color diagrams is aligned with the long-period Miras.<cit.> divided the Mira with period below 225 days in two groups, “Short Period-Blue” and “Short Period-Red”depending on their infrared colors and average spectral types. All spectroscopic and photometric characteristics of BIS 116 lead us to place this star in thegroup of the “Short Period-Blue” Miras.We have computed the absolute K and bolometric magnitudes using the relation between period and magnitude given in <cit.> and references therein:M_K = -3.69* (Log(P) - 2.38) - 7.33M_bol= -3.00* Log(P) + 2.8 For the uncertainty on the period, after accurate inspection of the NSVS light curves, we assumed five days for BIS 116 and ten days for the other stars. By propagating the errors we obtained ΔM∼0.1 mag for BIS 116 andBIS 267 and ΔM∼0.07 mag for the other stars. The uncertainties are practically the same for M_bol and M_K.Wehave also computed the apparent bolometric magnitude with BC_K = 3.15 magfor BIS 007, BIS 133, BIS 196,BC_K = 2.9 mag for BIS 267, and BC_K = 2.8 mag for BIS 116, derived from the calibrations by <cit.>. As uncertainties we assumed the typical amplitude of the K magnitude thatis 0.4 mag for the late-type Miras and 0.2 mag for BIS 116and BIS 267.We could then infer a crude estimate of the distances and of the mass loss rate. The distances derived from the bolometric and the K magnitudes agree very well within the errors. To compute the mass loss we used calibration between Ṁ and K-[12]color index obtained by <cit.>. Here the contribution to the uncertainties is also due to thenon-simultaneous observations of the different sets of IR data.Semi-Regular variables Our sample includes several Semi-Regular variables, but only five, namelyBIS 122, BIS 198, BIS 207, BIS 209 and BIS 276 have well sampled light curve in the NSVS. We could derive a range of absolute magnitudes and distances by applying the relations found by <cit.> and by <cit.>. These last authors found different relations P-M_K for different kinematics characteristics and obtained significant results by dividing their data into four groups, representative of four different populations.Group 1 and 2have kinematics characteristics corresponding to old disk stars; group 3 has kinematics indicating a younger population, group 4 contains only high velocity stars.Only group 1 contains long periods. We report the resultsin Table <ref> where the columns have the following meaning: 1 - BIS number; 2 - variability period P; 3 - K magnitude from 2MASS;4 - absolute K magnitude M_K from <cit.>; 5 - range of M_K from <cit.>, obtained applying the relation giving the best agreement with <cit.>; 6 - corresponding group number following<cit.>;7 - range of distances from the minimum and maximum of columns 4 and 5. 8 derived distance to the galactic plane. Non variable stars All the non-variable or small amplitude variable stars aregiant ofM0-M4 sub-classes. For these stars we derived a range of distances between 1.0 and 3.3 kpc, by adoptingabsolute visual magnitudes-0.7≤ M_V ≤ -1.6(<cit.>; <cit.>; <cit.>) and averaging the apparent visual magnitudes retrieved from the catalogs UCAC4 <cit.> and GSC2.3. Consideringthe galactic latitude yields to distances to the galactic planeranging from thick disk to halo, ( 0.45-2.1 kpc, Z=1.03 , σ=0.45 kpc). Regarding the supergiant BIS 137 (M0 I), by adopting an absolute visual magnitudeM_V∼ 5.0,we estimate a distance ∼ 6 kpc, and Z∼ 2.6 kpc. Many stars are likely within the reach of a direct parallax measure by GAIA. § CONCLUDING REMARKSIn this work we have studied spectroscopic and photometric optical characteristics of a sample (95) oflate type stars from the BIS catalog. The original situationof the entire catalog (276 stars) included 30% unclassified objects and 55% with very uncertain classification.From our new spectroscopic data we have revised the spectral classification of the observed targets: in 45% of the cases we have deeply modified or improved the previous classification. Four objects came out to be carbon N-type stars, nine are earlier than M, the others are M-type giants.We have divided our stars into three main variability classes: regular variable,irregular variable, photometrically stable. About 60% of the stars show large amplitude, irregular or semi-regular light curves; in this category we have included the three dust enshrouded carbon stars.All the early M-type stars and the naked carbon star BIS 194 were found to be stable. Only 5 stars are Mira variables.Our study pointed out some peculiar stars which deserve more detailed studies. Our spectral classification together with the datacollected from literature allowed us toestimate absolute magnitudes, mass loss and distances fora number of targets, finding a good agreement between results obtained from different methods. Knowing the distances to the Sun and the galactic latitude, we can infer that the median distance of the variable starsfrom the galactic plane is 3.0 kpc, (σ=1.5) none being farther than 5.4 kpc.Our spectral resolution does not allow to investigatefor chemical differences between thick diskand halo stars.Anyhow, given the distance from the galactic plane, most of these stars are likely to be halo members.For the non-variable stars we derived an average distance of 1.03 kpc from the galactic plane, suggestive of a mixed population of thick-disk and halo. Regarding the evolutionary status of our sample, most of the stars have an absolute magnitude appropriate for being in the AGB phase.In a companion study(Gaudenzi et al.,submitted) we have analysed the Infrared properties of all the stars of the BIS catalog. To this purpose we have made use ofnear-IR (2MASS), mid-IR (WISE), and far- IR (IRAS and AKARI) photometric data to investigate their behaviour on various color-magnitude and color-color diagrams and graphically distinguish various types of sources.Acknowledgements. This research has made use of the SIMBAD database, operated at CDS, Strasbourg,France. This publication has made use of data products from:the Two Micron All-Sky Survey database, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology; the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/CaliforniaInstitute of Technology, funded by the National Aeronautics and SpaceAdministration;the Northern Sky Variability Survey (NSVS) created jointly by the Los Alamos National Laboratory and University of Michigan;the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory (JPL), California Institute of Technology, under contract with the National Aeronautics and SpaceAdministration; the International Variable Star Index (VSX) database, operated at AAVSO, Cambridge, Massachusetts, USA. 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(2006)]wfm06 Whitelock P.A., Feast M.W, Marang F., Groenewegen M.A.T. 2006, MNRAS, 369, 751 [Whitelock (2012)] white12 Whitelock P.A. 2012, Ap&SS, 341, 123 [Wozniak et al. (2004a)]woz04a Wozniak P.R., Vestrand W.T., Akerlof C.W., et al. 2004a, AJ, 127, 2436 http://skydot.lanl.gov/nsvs/nsvs.pht/ [Wozniak et al. (2004b)]woz04b Wozniak P.R., Williams S.J., Vestrand W.T., Gupta V.2004b, AJ, 128, 2965, VizieR On-line Data Catalog: J/AJ/128/2965 [Zacharias et al. (2012)]zac12 Zacharias,N., Finch, C.T., Girard,T.M., et al. 2012,The fourth US Naval Observatory CCD Astrograph Catalog (UCAC4).VizieR On-line Data Catalog: I/322a0.92, 0.45,1.34,2.17,0.80,0.54,1.43,0.81,1.82,0.91,1.15,1.42,0.33,0.60,1.12, 0.90,0.90,1.0,1.22 | http://arxiv.org/abs/1707.08628v1 | {
"authors": [
"S. Gaudenzi",
"R. Nesci",
"C. Rossi",
"S. Sclavi",
"K. S. Gigoyan",
"A. M. Mickaelian"
],
"categories": [
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"primary_category": "astro-ph.SR",
"published": "20170726200842",
"title": "Spectral Properties and Variability of BIS objects"
} |
from universal laws to realistic modellingComplex delay dynamics on railway networksB. Monechi Institute for Scientific Interchange Foundation,Via Alassio 11/c, 10126, Turin, Italy and SONY Computer Science Lab, 6 Rue Amyot, Paris, France, 75005, France [email protected]. Gravino "Sapienza" University of Rome,Piazzale Aldo Moro 5, 00185, Romaand SONY Computer Science Lab, 6 Rue Amyot, Paris, 75005, France [email protected] R. Di Clemente University College London,The Bartlett Centre for Advanced Spatial Analysis,London, WC1E 6BT, United Kingdom Massachusetts Institute of Technology,77 Massachusetts Ave, MA 02139 - Cambridge, [email protected] V. D. P. Servedio Complexity Science Hub Vienna,Josefstädter Str. 39, 1080 Vienna, [email protected] delay dynamics on railway networks Bernardo MonechiPietro Gravino Riccardo Di Clemente Vito D. P. ServedioReceived: date / Accepted: date ==================================================================================Railways are a key infrastructure for any modern country.The reliability and resilience of this peculiar transportation system may be challenged by different shocks such as disruptions, strikes and adverse weather conditions. These events compromise the correct functioning of the system and trigger the spreading of delays into the railway network on a daily basis.Despite their importance, a general theoretical understanding of the underlying causes of these disruptions is still lacking.In this work, we analyse the Italian and German railway networks by leveraging on the train schedules and actual delay data retrieved during the year 2015.We use these data to infer simple statistical laws ruling the emergence of localized delays in different areas of the network and we model the spreading of these delays throughout the network by exploiting a framework inspired by epidemic spreading models.Our model offers a fast and easy tool for the preliminary assessment of the effectiveness of traffic handling policies, and of the railway network criticalities.§ INTRODUCTIONTransportation networks are a critical infrastructure with an enormous impact on local, national, and international economies <cit.>. At the micro-level, the commuting time impacts the economy and the shape of cities <cit.>, directlyinfluencing our life style and choices <cit.>.At the macro-level the travel time regulates trade and stimulates economic activities <cit.>. Railroads, when correctly developed, foster interregional trades within the country, and increase income levels within the city boundaries <cit.>.Whether we are traveling in crowded coaches of the commuters rail during rush hours or in the long-medium high speed train trips, the reliability of the transportation system is a key factor in determining travel behavior <cit.>.The travel time unreliability in rail service can have substantial consequences for its users <cit.> and the growth of cities <cit.>. Delays areone of the causes that corruptthe travel time reliability of the transportation systems. To address delay propagation, traditional approaches apply either stochastic models<cit.> or use propagation algorithms on a timed event graph representation of a scheduled railway system<cit.>. Complex equations describing delay were obtained and solved by means of iterative refinement algorithms to predict positive delays in urban trains <cit.>, while traffic control models were proposed to manage the safety of timetables after perturbations occur <cit.>.The development of models addressing the dynamics of trains over railway networks are usually focused either on delimited aspects, e.g., single lane dynamics <cit.> or the issuing of schedule<cit.>, or are comprehensively including a wide variety of different microscopic ingredients that can be hardly validated by real data <cit.>. In the last years a new perspective bloomed to provide a different understanding of delay dynamics through the lens of Complex Systems. Railway networks have largely been studied and exhibit small-world scaling properties<cit.>, their topology have geo-spatial restriction<cit.>andstructural redundancy<cit.>. On the other end, a railway delay dynamics framework within a complex systems approach is still lacking.Our aim is to develop a model capable of reproducingthe actual delay structure and the emergence of congested areas, by exploiting some essential ingredients without relying on the detailed knowledge of the microscopic mechanisms underlying delay generation. A similar approach has already been successfully applied in the context of the Air Transport System of USA and Europe <cit.>, where interaction models were used to characterize and forecast the spreading of delays among flights.These modelling schemes are driven by the necessity of simple and novel frameworks to study transportation systems, allowingfast scenario simulations for schedule testing, which could be easily applied to resilience studies that are nowadays performed independently of the dynamics taking place over the network <cit.>.Following a similar approach, we develop a new data driven model of delay propagation among trains diffusing over a railway network.We start by inferring universal laws governing the emergence of delays over the railway network as a consequence of the occurrence of adverse conditions that are not depending on the (possible) interactions between trains.Hence, inspired by models for epidemic spreading <cit.>, we introduce the Delay PropagationModel as a novel framework to asses andtest how such emerging delays spawn and spread over the network.We name the first kind of delays as “exogenous delays”, while the second ones stemming from the interaction between trains as “endogenous delays”. While the occurrence of exogenous delays can be detected and analysed from our data, the endogenous delays, on the contrary, since the (supposed) direct interaction between trains cannot be inferred from the dataset we collected, will be modeled with a one parameter interaction. The effects and consistency of such endogenous delay modelling scheme will be checked a posteriori by means of numerical simulations. Our model is close to an SIS model of epidemic spreading <cit.>, since trains can either get infected by delay, recover from it and then get infected again.We show that this model is capable of reproducing the empirical distribution of delays measured in the data as well asthe emergence of large congested areas.Moreover, we show that the removal of the delay propagation mechanism prevents the modeled system from generating large disruptions, hence strongly suggesting the existence of this kind of interaction in the real system. Finally, we propose an application of our model in studying a scenario where the propagation of a localized delay leads to theemergence of a vast non-functioning area in the Italian Railway Network.The paper is structured as follows: in the first section we describe the dataset and some empirical findings that inspired the assumptions used in the development of the model;in the second section we discuss the model in detail and will show its capability of reproducing empirical findings such as the distribution of delays and the size of congested areas; we will also demonstrate the usefulnessof the model as a scenario simulation tool; in the last section we will summarize our findings and discuss possible future developments.§ METHODS§.§ Data In order to extract relevant patterns related with the emergence of train delays, we collected and analysed data about the daily operations of the Italian and German Railway Systems during the year 2015.Such data were collected in the first case through the ViaggiaTreno website <cit.> and for the second one through the OpenDataCity website <cit.>.The information we acquired allowed for a complete reconstruction of the schedules of the trains, the structure of the Railway Network and the delays that affected the trains during their movements. For more information about the data collection please refer to Supplementary Information (SI).The choice of the Italian and German Systems as subjects for our studies lies in their similarities in structure and management.In fact, their networks have remarkable and comparable sizes (41,315 and 16,723 km, respectively) and densities (8.22 and 12.46 km^2 per km of tracks, respectively <cit.>).Moreover, these two countries share also a crucial characteristic: in both railway systems traffic is handled mainly by a single national company. In other countries with similar networks, like in the United Kingdom, the railway network is managed by different companies resulting in a more complex system where trains are additionally subject to the commercial policies of different operators.Fig. <ref> shows a set of topological analyses of the tworailway networks, together with a preliminary analysis of the traffic load and delays in the systems.Following existing literature, we refer to a railway network as the network whose nodes represent stations that are connected by a link whenever there is a train connecting them with two consecutive stops in its schedule.The network is directed since it is possible that a connection between two stations is travelled just in one direction.In this paper we refer to the nodes in a railway network either as “nodes” or as “stations”.Moreover, the action performed by a train travelling from station A to station B will be referred as “travelling over the link” connecting A and B.In both networks the distribution of the degree k is peaked on k=4 and for larger degrees has exponential-like decrease <cit.>proportional to e^-k/k_0 with k_0 ≃ 4.5 ± 0.1 .This finding is in agreement with other geographical networks <cit.>. The network assortativity (Fig. <ref>A)<cit.> is for Italy and Germany, 0.18 and 0.24 respectively.This indicates that while there is a slight preference for stations with the same degree to be connected each other, yet the various degrees are mostly mixing, i.e., smaller and larger stations are typically connected. The local clustering coefficient (Fig. <ref>D),defined as the fraction of pairs of neighbours of a given station that are connected over all pairs of neighbours of that station <cit.>, can be used to infer the redundancy of the network. When a disruption occurs, a station with an high clustering coefficient can be bypassed easily.Complementarily, betweenness centrality (Fig. <ref>E) is a measure of the centrality of a node in a network based on the number of shortest paths that pass through it <cit.>. This measure highlights how strategic a given station is at a global level.The clustering and the betweenness outline the typical small word topology of transportation networks <cit.>. German stationsshow a distribution between 10 and 100 trains per day, narrower that the train distribution of the Italian stations, which instead display a broader distribution, suggesting a more heterogeneous handling of train traffic load (Fig. <ref>F).Finally, Fig. <ref>G shows the histogram of the average delays of trains aggregated by stations for the two nations. Both distributions show a peak andheavy tail, whereas compared to the German distribution, the Italian distribution is clearly shifted towards higher delays. The topological similarities between the two networks suggest that the differences in delays and traffic load might be the result of differences in the trains dynamics.§.§ Train interaction on railways networks To focus on the analysis of delays, their outbreak and evolution, we choosetrains instead of stations as our reference systems. The intermediate delays for a train i travelling from station A to station B on the link e is defined as Δ_i t(e),The departure delays at the initial station have been subtracted to analyse only the delays that have been generated during the travel.Hence Δ_i t(e) can be negativewhenever the train is in advance, resulting in the train waiting at the station for the correct time of departure.Fig. <ref>A shows the delay distributions for both national systems, considering the delays at intermediate stations along the path of a train, or just at the final station.We observe similar shapes of these distributions, with the Italian one exhibiting broader tails than the German.More than 10% of the train stops are on-time.The distributions of both countries exhibit an asymmetric pattern: the right tail (labeled Delay) shows a power-law like behaviour compatible with a q-exponential distribution <cit.>, while the left tail (labelled Advance) has an exponential steeper slope. We expect this distribution to be the result of the interplay between the occurrence of adverse conditions and the interaction between trains, influencing their dynamics. Despite the fact that the microscopic details in our data do not allow for a precise investigation of possible interactions between trains, we can highlight how the possibility of interaction might affect the delays in railway networks. Hence, we study the relation betweenthe first order co-activity of a link, i.e., the probability that at a certain time t (with time steps of 30 minutes) a link which is active has at least one active neighbouring link, to the average delay of the link itself.In Fig. <ref>B we show this quantity as a function of the average delay for both the Italian and German cases.We notice a slight increase in average delay as the co-activity increases, confirmed also by the Spearman's coefficients, 0.43 for Italy and 0.56 for Germany.Hence, the possibility of interaction between trainsin a certain part of the railway network seems to increase the delay localized in that area.Note that we have defined the first order co-activity between neighbouring links, i.e. links that have at least one node in common.We can define a k-order co-activity considering links connecting at least two nodes that are less than (k-1) links apart following the shortest-path connecting them.We report the same measurements for k=2 and k=3 showing that in general, the same relation with the average delay is confirmed even though the curves are shifted towards lower values.This indicates that considering the interaction between non-neighbouring links is relevant but might include less important contributions to the delay.Hence, in the following we will limit our analysis to the first order neighbour links and assume that interactions between trains are possible only when they are in nearby links.Fig. <ref>B supports the thesis of a propagation effect but an important feature has yet to be determined.In fact the direction of the propagation still has to be determined.§.§ Defining possible interactions Let us consider a train i travelling between two stations A and B (i.e., on the link e=AB, see Fig. <ref>) with some delay.We can argue that the propagation of this delay to other trains can occur only if they travel on the neighbour links of e.Due to the fact that the railway networks are directed, there are four different configurations of the links with respect to e: (i)links entering A, i.e. trains moving towards the last station crossed by i; (ii) links entering B, i.e. trains travelling towards the same station i is currently travelling to; (iii) links exiting from A, i.e. trains departed from the last station crossed by i; (iv) links exiting from B, i.e. trains leaving the station i is currently travelling to.We can exclude the last two case: (iii) given that all the trains in such configuration will have no interaction with i; (iv) is less important because it describes scheduled connections. In the latter case, schedules foresee extra-time between the two trains exactly to avoid delay propagation. We checked whether the propagation occurs in the case (i) of backward propagation, in the case (ii) of forward propagation, whose definitions are depicted in the graphic Fig. <ref>.To discriminate which mechanism is at play, we measured the average delay time sequence Δ t (e) of each link e of the network, defined as the average delay of all the trains that are currently travelling on e. Successively we measured the cross-correlation functions of the average delay time series of all the pairs of links, i.e.,CC_e,e'(dt)=∑_t Δ t_e(t) Δ t_e'(t+dt)/σ_eσ_e'being e and e' generic neighbours links of the network, σ_e and σ_e' the standard deviation of the whole time series Δ t_e(t) and Δ t_e'(t). Then we averaged, aggregating the pairs of neighbours links according to their configuration (forward, backward, etc). In this way, for each of the four configuration, we obtained an average cross-correlation function.In the backward propagation configuration can be defined :C_B(dt)= ⟨CC_e,e' (dt)⟩_(e,e') ∈Bwith B as the ensemble of links pairs. For both networksthe Backward mechanism is dominating while the Forward can be neglected (Fig. <ref>D ).Thehigh-speed layer of the railway network shows the similar backward mechanism, while there is no cross-correlations between the delays of high-speed vs. low-speed (see Supplementary Material Fig. S1) acting as two independent layers. §.§ Exogenous generation of delay We define two kinds of delays: endogenous and exogenous. By “endogenous” we mean that its origin is inside the railway system dynamics, i.e. it has been caused by another train.Conversely, by “exogenous” we mean that its cause is of a different nature: strikes, malfunctioning, bad weather or anything else which is not the result of the interaction with another train.We measure directly this types of delay in our datasets.Let us consider a train i travelling from a station A to a station B on the link e and further to a station C on the link e'.It will travel first on the link e and then on the link e'. The delay are indicate respectively as Δ t_i(e) and Δ t_i(e') .If there is a increase in the delay Δ t_i(e') > Δ t_i(e) it might be “endogenous” or “exogenous”.The exogenous delay is defined as δ t =Δ t_i(e') - Δ t_i(e), the variation of the train delay traveling on links whose neighbouring links were empty or hosted trains perfectly on time.It is worth noticing that δ t might also be negative, for example, if the train managed to make up for lateness.Results are reported in Fig. <ref>, showing the distribution of positive exogenous delays as well as negative ones for the Italian and German cases.In order to model these distributions, we adopted the same approach used in <cit.> for departure delays. We fitted both the positive and negative parts of the distributions with q-exponential functions, where the parameter q modulates from an exponential distribution q→ 1 to a fat-tailed distribution for q∈ (1,2]<cit.>:e_q,b(δ t) ∝ (1 + b(q-1) δ t)^1/(1-q) q∈[1,2], b>0.It has been shown that such distribution can be obtained starting from a poissonian process p(δ t| α)= α e^-α t, where α is a random variable extracted from of n independent gaussian random variables X_i with ⟨ X_i ⟩=0 and ⟨ X_i^2 ⟩≠0, so that α=∑_i=1^n X_i^2 <cit.>.In this way it can be proven that n = 2/(q-1) - 2, i.e. the parameter q, is related to the number of random variables composing α. The parameter b is proportional to the average value of α, so that large values of b at fixed q result in a distribution biased toward shorter delays. The parameter q quantifies how much equation (<ref>) deviates from being exponential, which is the case q=1. This model has already been applied to the departure delays in the UK railway system, showing that the value of q where so that 4≤ n ≤ 11 and thus estimating the number of independent occurrences contributing to the delay.For thepositive exogenous delays in the Italian and German case respectively, we found q=1.23 and q=1.32, corresponding to n≃ 7 and n≃ 4.The negative part of the distribution is exponential-like for the Italian railway network and broader for the German, this outlines the delay recovery strategies in the second case. To characterize the effect of the spatial distance on the delay distribution, we subdivided the links e of the railway networks in classes according to the geodesic distance d(e).Fig. <ref> shows the behavior of the q and b parameters of the q-exponential fit as functions of d(e). The parameter q remains constant in every case, while on the other hand the parameter b decreases as b ∼ d^-a. Fig. F-I of Appendix show the different best fit for equation (<ref>) as d(e) varies.This result suggest that while the causes of the delay remain the same, the distribution of disturbances gets closer to a power-law as the length of the links increases, this outlines a relation between link length and delay. Finally, we can assume that the occurrence of positive or negative exogenous delays on links, P(δ t>0|d(e)) andP(δ t<0|d(e)), are not roughly constant with d(e) and hence are not depending on the length of the links (Fig. J of Appendix).§.§ Generation of delay at departure Departure delays, i.e. the delay a train acquires right before leaving the first station on its route, cannot be considered in principle completely exogenous.In other words, due to the fact that different trains in our datasets can actually be the same physical train (e.g., the same convoy travelling back and forth along the same path on the railway network – this is denoted as “rotation” –), the delay at departure might suffer from the influence of the traffic.However, railway administrators envisage suitable time buffers at the endpoints of the paths of each train so that it is reasonable to assume, at least as crude approximation, that departure delay is exogenous in character.It has already been shown that this kind of delay can be described by a q-exponential distribution in <cit.>.However, the dependence on the parameters of the obtained distributions with respect to the topological properties of the network has not been investigated yet.Following the same spirit of the previous paragraph for the exogenous delay on links, we divide the nodes in the network (the train stations), with respect to their out-degree.The out-degree k_out represents roughly the number of different railway lines starting from a certain stations and hence can be considered as a proxy for the complexity of the station itself. Once the nodes of the networks have been divided according to k_out, we fitted these distributions using a q-exponential following the procedure defined in <cit.> (see Fig. L, Fig. M and Fig. N of Appendix).Fig. <ref> shows the behaviour of the parameters q and b of the q-exponential distribution as functions of k_out for the positive and negative departure delays in the two considered railway networks. Negative departure delays were never reported in the German dataset and hence we assume they are not present.Despite the fact that better proxies for station complexity than k_out might exist (weighting each link with the actual number of railway lines on it is a valuable alternative example), it is possible to see that we have again a constant parameter q indicating that the sources of delays can be assumed to be the same independently from the station, while on the other hand the parameter b decreases exponentially with k_out.The small value of R^2 in the case of Italy might reflect the above mentioned possibility of having a non negligible endogenous contribution to the departure delays because of train rotations. In Germany the departure delay can be considered constant and independent on the size of the station. In Italy theP(δ t>0 | k_out) rows linearly with k_out meaning that stations with high degree are generating larger disruptions in the network.§ MODELLING A REALISTIC RAIL TRANSPORT SYSTEMThe analyses proposed so far suggest the existence of two sources of delays: exogenous delays spontaneously occurring due to external adverse conditions at departure and during travel, depending on the topological properties of the Railway Network; endogenous delays resulting from the interaction between trains. While in the first case we were able to characterize the statistical laws governing the emergence of delays, we cannot directly investigate the mechanisms of interaction between trains. Following the ideas exploited to model real world epidemics <cit.> which have already been proven effective on air traffic delay modeling <cit.>, we will define a propagation process for delays i.e. we will suppose that trains can spread their delays from one another according to some fixed probability and in certain conditions. Such probability will be derived by comparing the results from the simulations with the model and the empirical data. §.§ The Delay Propagation Model We define the model scheme by leveragingthe train-to-train propagation mechanism, with the aim of reproducing the real dynamics of delay spreading across the railroad network.The model starts reproducing the normal schedule of trains on a certain day. Train schedules are organized so that each train has its own “time window” to travel over a certain link along its path. Each train departure is at a fixed time and at a certain station and each intermediate station is going to be visited at a given time. However, our interest is in the deviance from the expected schedule.If some disruptions occurs a train might go out of the window of use for a certain link and overlap the window of another train for the same link.In this case, the second train will be forced to wait for its path to be cleared.Hence, the model adopts three different sources of delay as depicted in Fig. <ref>: Departure delay This delay is assigned at the beginning of the path (originating station) of each train and is considered exogenous and unrelated with the current traffic conditions at the departing stations or in the nearby links. We assign to a train either a positive or negative delay according to the empirically found law of the corresponding probabilities p_dep^+=P(δ t_dep > 0| k_out) andp_dep^-=P(δ t_dep < 0| k_out) respectively (and no delay with complementary probability 1 - p_dep^+-p_dep^-) (see Fig. O of Appendix). Once the sign of the delay has been decided, we assign a positive or negative delay value so that |δ t_dep| ∼ e_q(k_out), b(k_out)(δ t), i.e. distributed according to a q-exponential distribution with the parameters q and b depending on k_out and on the sign of the delay itself as in Fig. <ref>. Exogenous Link Delay This delay is assigned whenever a train starts travelling on a link for the first time (Fig. <ref> reports an exemplification of the model). Considering a train i passing from the link AB to the link BC, we adopt the same modelling scheme as in the departure delay case assigning to the train a positive delay with probability p_exo^+=P(δ t_exo > 0| d(BC)), a negative one with probability p_exo^-=P(δ t_exo < 0| d(BC)) and no delay with 1 - p_exo^+-p_exo^-, where d(BC) is the geodesic length of the links BC (see Fig. J of Appendix for the corresponding law of probability). Having decided the sign of the delay, we fix the parameter of a q-exponential distribution q and b according to d(BC) (Fig. <ref>) and we extract the magnitude of the delay so that |δ t_exo| ∼ e_q(d(BC)), b(d(BC))(δ t). Delay Propagation We model the interaction between trains with a mechanism of propagation of delay from other trains to train i. The investigations performed on the datasets suggest that such interaction can occur only when nearby trains are in the Backward propagation configuration of Fig. <ref>. Following the picture of the model in Fig. <ref>, when the train i starts travelling on the link BC leading to the C station is susceptible of propagation from the train j, which is travelling from C towards D. Inspired by the SIS models of epidemic spreading <cit.>, in which agents can get infected and recover from the illness with a fixed probability, we introduce the delay propagation parameter β∈ [0,1] describing the probability that the delay of the train j propagates to i. We assume that the propagation occurs at the time i starts travelling onto BC. When train i travels on BC from time t_1 to time t_2, we check whether delayed trains are travelling on links in the Backward propagation configuration with respect to BC in [t_1,t_2]. Thus, we randomly pick one of those trains, say j, and with probability β its delay δ t_j is added to the delay of i. The model simulates the theoretic schedule applying all these delay-generating mechanisms.More in detail, we start the dynamics of each train i by adding a departure delay δ t_dep according to the Departure delay mechanism. After the departure, each time the train starts traveling over a link in its route, we update its delay according to the following rule:δ t_i →δ t_i + δ t_exo + δ t_jwhereδ t_exo is sample following Exogenous Link Delay mechanism, so that the delay will be positive with probability p_exo^+ and negative with p_exo^- and the parameters of the sampling distributions are depending on the length of the link as in Fig. <ref>. The term δ t_j the contribution of the Delay Propagation which will different from 0 with probability β and just if there is at least one delayed train in the right configuration.This mechanism for intermediate stations reproduces the general noise associated with external causes, but does not account for long correlations such those present in case of large scale adverse conditions, e.g., bad weather, or national strikes. Finally, when the train reaches its final destination it is simply removed from the simulation. We used the described model to reproduce the delay spreading dynamics starting from a theoretic timetable and local exogenous delays distribution. Results of the simulation for both the Italian and German national railways systems are reported in the next section. § RESULTSThe model is based on the exogenous sources of delay that represent the spontaneous emergence of disruptions due to the aggregation of a finite number of external causes (trains malfunctions, accidents, bad weather, etc.). These sources are modelled according to a universal probabilistic law, whose parameters are inferred from the data at our disposal. The propagation parameter β modulates the mutual interaction between trains. For simplicity, we chose this parameter to be uniform all over the network, which is a strong assumption since the propagation might depend on the considered part of the railway system.§.§ Reproducing the emergence of delays and congestion In the Appendix we show that it is possible to estimate the best value for β by trying to reproduce the average delay of each station. These values have been estimated as β=0.15 for Italy and β=0.1 for Germany. In the top pabels of Fig.<ref>A, we show that with this choices for β the system is capable of reproducing the empirical distributions of arrival delays. However, in order to disentangle the effect of the β parameter and the two other sources of exogenous delay, we also report the same result for the case in which β=0 and β is larger than the optimal value. In the latter, we clearly see that there is an increase in the number of extremely large delays. On the other hand, turning off the interaction by setting β=0 completely removes delays larger than 2 hours. Hence according to our model, the presence of exogenous delays by themselves would not be able to predict the presence of large disruptions.Since β represents the probability of delay propagation between two trains, these estimations give a quantitative account of the difference between the two countries.These results suggest that the Italian trains propagate each other delay more often than German trains.The microscopic reasons of this difference could be connected to different properties of the railway networks, to the peculiar geographical structure of the territory, to different delay handling policies, etc., and it is out of the scope of the present paper.As it happens in other transportation systems, we expect that disruptions occur clustered in certain areas of the network <cit.>.For this purpose we provide a definition of “cluster of congested stations”and how to discriminate whether a station is “congested”, i.e. when its functioning is inefficient because of the hoarding of delays. For each station we can define a threshold between the “functioning” and the “congested” status based on the value of incoming train delays.We calculated such threshold for each station as the average value of the delays of all the trains arriving at the considered station in the dataset. Considering a period of two months (March and April 2015), we examined each station every 5 minutes during the day, checking whether the average delay of all the trains moving towards that station during that lapse of time was above the average.We consider the station as “congested”, while if the average delay is below that threshold we consider it as “functioning”.We identify, at each time step, the “clusters of congested stations” as the connected components of the railway network leftonce we removed all the functioning stations from the network.(i.e. the stations whose congestion might be correlated with one another because of the propagation of delay). In order to check whether our model is able to reproduce the emergence of these areas, we focused on two measures:(i) the size of the clusters in terms of number of stations and(ii) the relation between the size and the diameter of the clusters. This latter measure is capable of giving insights about the topology of the congested clusters. Bottom panels of Fig. <ref>A show real and simulated cumulative distributions of cluster sizes for both countries.We find a strong accordance between model and reality when β is set to the optimal value, pointing out how the model is also able to reproduce this aspect of delays dynamics.Similarly to the arrival delay distribution, here β affects the tail of the distribution so that β=0 (i.e. no propagation of endogenous delays) implies the absence of large congested areas.We study the diameter of the cluster defined as the distance between the farthest couple of nodes in the cluster.We compute the diameter both in the case where just the topological distance between nodes is considered and the case where each link is weighted with the geodesic distance between the stations it connects.To panels of Fig. <ref>B shows the dependence of the cluster diameter computed with both distances on the cluster size. In the cases where the diameter is computed by using the geodesic distance, the dotted line represents the diameter of a cluster of size n, assuming that all the links of the clusters are as long as the average link length of the network.Such pattern corresponds to the case where all the clusters are randomly sampled from the whole network without any constraint.From the lower panels of Fig. <ref>B we can see that while clusters do have a path-like structure, they cannot be considered randomly distributed over the networks. Instead, they seem to be deployed in areas where the links are larger than the average links length of the corresponding network, resulting in the same dependence of the dotted lines but shifted upwards.The fact that the geographical diameter of large clusters grows up to hundreds of kilometers indicates that disturbances can propagate between far away parts of the network.The topological measure in the top panels of Fig. <ref>B exhibits strong deviations from the path-like behaviour especially when large clusters are involved.The mismatch between the two ways of characterizing clusters could be due to the fact that the geographical distance may hide, by stretching them, non path-like structures that are actually present in the delay propagation patterns. The appearance of these clusters, and their features, are typical outcomes of a non-trivial complex dynamics arising from train interactions. For this reason it is a remarkable result that, as can be neatly be seen in Fig. <ref>, the model proposed, despite its simplicity, captures this behaviour correctly. The relations between cluster size and cluster diameter is show very good agreements with the empirical measures for both nations and, perhaps more importantly, for both clusters diameter definition: the topological and the spatial one. This clear adherence with reality of the results from a model with only one parameter trained on a so small training set (only one week) is a strong proof of the fact that the model hypothesis are very likely to be correct.Our model is capable of reproducing some global patterns of the delay dynamics in railway systems. However, some discrepancies at a more microscopic level can be observed.In particular our model fails to correctly describe the behaviour of stations with low traffic. For these stations, the hypothesis of a constant in time and homogeneous coupling parameter β may not be well justified. We refer to Section 5 of Appendix for a thorough discussion of this point. §.§ Scenario Simulation: Prediction of the Effects of a Strong Localized Disturbance According to our model, the emergence of large clusters of congested stations is due to the propagation of delay between trains. The source of this delay is however exogenous, in the sense that comes from some adverse conditions which are external with respect to the interaction of the trains.In order to investigate how the emergence of a real large cluster is linked to exogenous effects, we study the case of the cluster emerged in the eastern part of the Italian Railway Network on the 28th of February 2015.As can be seen in the on-line visualization[The visualization can be found at <www.riccardodiclemente.com/trainsimulation.html>], a large congestion starts to emerge in the early afternoon around 12am and propagates to a large part of the network until night.We empirically identified the interested part of the network as the shortest-path connecting the station of “NAPOLI CENTRALE” to the station of “ROMA TERMINI”, indicated in Fig. <ref>A.The shortest-path has been computed weighting each link with the geographical distance between the stations it connects. Fig. <ref>B shows the fraction of stations in this path that are congested in different times of the day, clearly indicating that such fraction starts growing until a peak is reached in the afternoon.We have been able to identify the beginning of this adverse occurrence as a disruption on the “NAPOLI CENTRALE-AVERSA” link (highlighted in Fig. <ref>A) in the first part of the afternoon from 11 am to 14 pm, resulting in a large delay acquired by the trains travelling on that links.We argued that this disruption was the spark that lightened the emergence of the congested cluster. In order to check this hypothesis, we ran a simulation in which a large delay of 100 minutes is assigned with probability 1 to each of the trains crossing the “NAPOLI CENTRALE-AVERSA”link in that period of time.Due to the non-deterministic nature of the model, we performed the simulation of 200 different realizations in order to have a set of scenarios to be compared with the real data.This comparison is shown in Fig. <ref>B.We can see that with this simple modification and not considering other possible correlations between the occurrences of external adverse conditions in nearby links, we are able to reproduce a pattern which is qualitatively similar to the one observed in the data, clearly pointing out that the “NAPOLI CENTRALE-AVERSA” link played a major role in the congestion of the network. Moreover, our scenarios indicates the probability of having a minor congestion in the line also in the early morning, probably due to usual minor disruption occurring on other links.The proposed scenario simulation could be easily extended to less localized adverse occurrences, which might comprehend spatially correlated disruptions due to natural events or strikes. Moreover, the same approach could be used to study more dramatic effect of node, link removal or the positive effects due to the introduction of more resilient and optimized schedules. In this cases, it is sufficient to use the same structure of the model and modify the input schedule and/or the Railway Network itself. § CONCLUSIONSRailways and the railway transport systems have been historically of utmost importance for the development of modern countries. Nowadays their importance is on the rise again due to their relevance in the reduction of CO_2 emissions and their competitiveness in short and middle range movements, so that huge investment have been made in the European Union to improve their efficiency.However, the emergence of large disruptions and the intrinsic inefficiencies seem to be endemic in this kind of transportation.The understanding of the universal features governing the dynamics of the system from a theoretical point of view could give an important contribution to the solution of such problems. The development of a universal model explaining the emergence of delays in Railway Network would allow for a quantification of the causes behind the occurrence of large disruptions and for a more direct link between the effects that localized interventions might have on the global system. In this work, we developed a novel model of delay emergence and propagation between trains moving on Railway Networks, partly based on empirical laws inferred from the data governing the accidental emergence of delays from the influence of adverse occurrences.Remarkably, the statistical models used for the description of this “exogenous delays” showed how these adverse occurrences are the result of the same finite number of causes(e.g., bad weather conditions and malfunctions) independently from the topology, as opposed to the scale of these disruptions, which are largely influenced by the part of the network the trains are travelling on. We find, in fact, that both the complexity of a station as measured by the number of different routes originating from it and the length of a route, are connected to the magnitude of the microscopic delays composing the overall delay by simple statistical laws with two parameters, whose value is the same all over the network and can be inferred and fixed by data.This kind of universality is somehow surprisingand opens the possibility of easily simulating the behaviour of any railway system once the complete schedules are known.These emergent delays can then spread from one train to another by means of a delay propagation probability β, which is in fact the only free parameter of our model.Despite the high level of abstraction used in our approach, our model is capable of reproducing the empirical patterns found indata when the delay-spreading probability β is fine tuned to an optimal value.The model reproduces correctly the final delay distribution and the distribution of the sizes of congested areas. Moreover, the model grasps the topological properties of such congested areas, which appears to be spreading in some cases for many kilometres through the network. According to our model, the emergence of large disruptions is the result of the interplay between the occurrence of localized exogenous delays and the propagation of such delays between trains. In other words, turning off the delay propagation mechanism prevents the system from generating extremely large delays and congested areas, pointing out that interactions are the driving force behind the emergence of major spontaneous adverse occurrences. The modelling scheme seems quite promising so far, but it still suffers from some major assumptions that might limit its predictive power. In fact, the interactions between trains could occur in longer ranges and not just between neighboring links and the propagation probability could not be uniform all over the Railway Network but instead depending on specific operational conditions. Moreover, the modelization of exogenous delays could be refined by taking into account more static topological feature of the Railway Networks or by introducing dynamical variations due to different traffic conditions. Another interesting possibility would be to increase the number of Railway Systems under study (e.g., the French SNFC sharing similar characteristics with respect to the Italian and German systems whose data is publicly available <cit.>), in order to understand the reasons behind the quantitative differences observed, e.g. larger delays in the Italian case.In the final part of the paper, we show how the model can be applied to study the capability of functioning of the system after a localized large disruption occurring in a single node of the network. This approach can be easily extended to more complex case studies of distributed disruptions, the occurrence of strikes, the removal of trains or parts of the network, also including the possibility to introduce delay management strategies to increase the resilience of the system. The model is also useful in order to have a fast test of changes in the overall schedule of trains so to have a more precise assessment of their global effects.Finally, despite its simplicity the model is open an increase of complexity that might lead to a better adherence to the empirical findings such as long-range interactions between trains and a more detailed model of exogenous delays including correlations between nearby link and the dependence on traffic conditions.§ FUNDINGThis work has been supported by the KREYON Project, funded by the John Templeton Foundation under contract n. 51663; VDPS acknowledges financial support from the Austrian Research Promotion Agency FFG under grant #857136. RDC as Newton International Fellow of the Royal Society acknowledges support from The Royal Society, The British Academy and the Academy of Medical Sciences (Newton International Fellowship, NF170505).§ ACKNOWLEDGEMENTSThe authors acknowledge Fabio Lamanna for the initial discussion about the datasets to be used for the work. § APPENDIX § DATASETS INFORMATION Novel information technologies enabled real-time monitoring and sharing of any kind of traffic data.Impressive instances are the visualised datasets about marine traffic that can be easilyfound on the Internet[E.g.: <https://www.shipmap.org>, <https://www.marinetraffic.com>, <https://www.vesselfinder.com>]. Also, several websites display live air traffic, by gathering and visualising official data from various sources[E.g.: <https://www.flightradar24.com>, <https://flightaware.com>, <https://planefinder.net>].These sources of information were found to be crucial in order to improve the understanding of the related transportation systems <cit.>. However, it is still a hard task to aggregate and analyse global or continental datasets about railway systems.In fact, due tohistorical reasons and to the typical usage scale, each nation has a network with few international connections and the available datasets are not homogeneous in coverage and format.Thus, tailoring the analysis on national systems seems a natural choice, whereas the most interesting characteristic behaviours appear in all systems, suggesting some kind of universality in the dynamics.We focused our analysis on the European continent both for the historical importance of railways and for the recent institutional efforts to raise their adoption.The actual railway system is composed by three distinct layers: high-speed passenger trains, normal-speed trains (mostly of regional type) and freight/military trains. While the freight trains use different stations and traffic handling rules (e.g., they operate mainly during night time) and can be discarded from our analysis, the other two layers can possibly interact each other.In the high-speed layer, correlations are identical to those in the regular layer, while no appreciable correlations can be spotted across the two layers so that considering them as independent is a fairly good approximation. Since no additional information can be gained by studying the whole system, we focus on the regular-speed layer only, both for Italy and Germany §.§ The Italian Dataset The dataset regarding the Italian Railways has been collected by means of the “ViaggiaTreno" website[<http://www.viaggiatreno.it>].The purpose of this website is to provide real-time information to travellers regarding the position of a certain train on the network, its delay and possible adverse occurrences like cancellations or strikes. Despite the fact that the information is in real-time, i.e., the instantaneous delay of a train can be checked at any time during the day, whenever the train arrives at its final destination its record is not deleted from the site.Instead, it is possible to check its route and its delay at each intermediate stop from the departing station to the arrival one until the end of the day, at 23:59. Hence, we downloaded all the information displayed on the website each day at 23:30 in order to be sure that each train would have arrived at destination. Starting from the 1st of January 2015 and for the whole 2015, we collected 12 months of historical data about the dynamics of regular and high-speed trains in Italy.For each train we get an identifier, the ordered list of stations the train has to cross, the scheduled arrival time at each station and its delay.The resulting dataset comprehends the traffic running on 2253 stations, with a daily average schedule pertaining 8112 trains on 7062 links.Note that “ViaggiaTreno” does not collect information about the geographical position of the stations.Such information has been integrated by means of Wikipedia and Google Maps, allowing us to represent the geo-localized network of Italian railways. Each dot corresponds to a station and each link corresponds to a route between two stations. In other words, a line between Rome and Naples means that there is a direct train route linking them without intermediate stops. Lacking real point-wise tracks data, the route has been simply represented with a straight line. §.§ The German Dataset The data about German Railways have been collected through the OpenDataCity[<https://www.opendatacity.de>] website. This site gathers different datasets collected by a variety of on-going or terminated projects dealing with open data.In particular, the data we analysed come from the “Zugmonitor” project, which aimed at providing a web-app and an API to German travellers in order to have real time information on the position of the trains on the German Railway Network and their delay.The project is no-longer running and the API is not accessible anymore.However, some dumps of historical data collected during the project are still available. In particular, we downloaded all the data regarding year 2015, coveringthe same period of the Italian dataset.This dumps collected not only the delay at each station like in the Italian case, but also the delay at intermediate points between two stations.All the points are also geo-localized so that it is possible to reconstruct a quite accurate trajectory of the trains.In order to be consistent with the Italian dataset we used the geo-localization only to identify the position of the stations in the map. Scheduled arrival times at each station were also stored in the dump, so that in the end we managed to reconstruct a dataset with a structure identical to the Italian one.The resulting dataset includes data for 5979 stations with a daily average schedule containing 11,975 trains on 16,277. § HIGH-SPEED LAYER The structure Italian and the German Railway Networks is the overlap of two distinct layers, the normal-speed and the high-speed one. These two layers are different from the structural point of view. The high-speed layer in fact has to allow for fast travelling trains and have a different kind of rails connecting stations and, in general, it is reasonable to assume that high-speed trains and normal-speed ones do not interact when travelling from a station to another. However, the nodes of the network (i.e. stations) are shared between the layers, making the network a “multiplex” <cit.> and allowing for interactions of the two different kind of trains. Our datasets contain information about high-speed and normal-speed trains, for both the Italian and German case and in principle it could be possible to study the dynamics of the high-speed layer and its interaction with the normal one. In the main text we decided though to focus on the normal layer cutting-out the high-speed part. This choice was made for sake of simplicity, since the rules of interaction between the layers might have been hard to understand or derive with data analysis. Here, we will show that this approximation is reasonable due to the smaller numbers of the high-speed trains travelling and their poor effects on the dynamics of the normal-speed one. In our datasets it is possible to identify high-speed trains thanks a specific identifier (“ES*” for the Italian Network and “EC”, “IC”, “ICE” for the German Network)and use them to build the High-speed layer of the Railway Network in a similar way that has been done for the normal layer in the main text. The number of travelling high-speed trains per day is considerably smaller with respect to the normal-speed trains, being of ∼ 210 and ∼ 1055 for the Italian and German case respectively. As a consequence also the two networks are smaller compared to the normal-speed ones as shown in table <ref>. The smaller number of nodes indicates the fact that high-speed trains usually connects fewer, more important and distant stations, since it is used mainly for mid-long range movements. This is also reflect in the distribution of the length of the links in the network (Fig. <ref>), showing a tail which is considerably longer with respect to the normal-speed layer. Other topological properties are similar in the two case, like the degree distribution (Fig. <ref>) and the associativity coefficient (table <ref>). As an example of the dynamics taking place over the high-speed layer, we show in Fig. <ref> the distribution of the final (positive) delays of high-speed and normal-speed trains. We can see that both for the Italian and German case, the distributions are fat tailed, so that also trains on the high-speed layer can experience large delays and major disruptions. As a final remark, we validated the approximation of neglecting this layer by looking at the cross-correlations between the time-series of average delays on the links in the networks, similarly to what we have done for the normal-speed layer in the main text. In this case though we checked for correlations not just between the links of the same layer, but also between couples of links from different layers in order to see whether we can spot a signal of a possible inter-layer interaction. Fig. <ref> shows the cross-correlations in the “Forward”, “Backward” configurations, between the couples of links of the high-speed layer and the couples of links made by a link in the high-speed layer and one in the normal-speed layer. As for the normal-speed layer, decaying correlations exist for non inter-layer couples of links in the Backward configuration, while in all the other cases the signal of correlation is very close to 0. Hence, it is possible to considered the high-speed and normal-speed layer as independent and non-interacting. It is worth noticing that this measure of correlation might hide possible local interaction effects due to the fact that it is an aggregation of all the couples of links in the network. Such approximation will be then valid when considering global or aggregated metrics (e.g. the delay distributions), but it is not unlikely that more “fine-grained” observations (e.g. the distribution of delays on a single link or station) might be influenced by our choice. § EXOGENOUS DELAY DISTRIBUTIONS The most trivial way to group the links of the railway networks is according to the geodesic distance between the stations they connect, behind this a rough estimate of the length of the railway between them. Fig. <ref>show the distribution of these distances d(e) for all the edges e in the Italian and German Railway Networks. From these distributions we can see that the distances are distributed around a typical value of ∼ 5km, but then span with a long tail until ∼ 100km.In order to characterize correctly the exogenous delay on the links, we measured the positive and negative exogenous delay distributions aggregating the links according to d(e) as can be seen from Figures <ref>, <ref>, <ref>, and <ref>. In all these cases, we modelled the distribution using a q-exponential functional form<cit.>:e_q,b(δ t) ∝ (1 + b(q-1) δ t)^1/(1-q), q∈[1,2], b>0,so that in these cases the parameter q and b are depending on d(e).The behavior of the parameters with respect to d(e) are shown in the main text. We find that in general:q(d) = const, b(d) = A d^-a. The parameters for equation(<ref>) can be found in table <ref>: Since these distributions are all conditioned on the fact that the acquired exogenous delay is either positive or negative, we can check whether the probability of these conditions are influenced or not by the length of the link the train is travelling on. Fig. <ref> shows these dependencies for both the considered Railway Networks. Despite the fact that a small dependence can be observed in the probability of having positive delays (i.e. it is slightly increasing with d), assuming that such probabilities are constant is a good zero-order approximation that we have used in the main text.§ DEPARTURE DELAY DISTRIBUTIONS An approach similar to the one adopted for the exogenous delays on the links of the networks can also be adopted for the departure delays at the stations. In this case we categorized the departing stations (i.e. a subset of the nodes in the network) according to their out-degree k_out whose distributions are shown in Fig. <ref>.Having divided the nodes of the network according to k_out, we can fit the aggregated departure delay distributions as k_out varies as shown in Fig. <ref>, <ref> and <ref>. Note that negative departure delays are not present in the German dataset.These distribution have been fitted using a q-exponential functional form as in equation <ref>. The behavior of the q and b parameters for these distributions according to k_out can be summarized by the equations:q(k_out) =conts,b(k_out) = Ae^-a k_out.The parameters for equations (<ref>) are obtained by fitting the empirical data as shown in the main text. Table <ref> shows the values obtained with the fit:To conclude the investigation about departure delays, it is necessary to study the occurrences of positive and negative ones as k_out varies. Fig. <ref> shows these dependencies for the Italian and German Railway Networks. Similarly to what we have found for the dependency of the exogenous delay with the length of the links, in the German Railway Network no dependence can be observed and the probabilities of having a positive or negative departure delay can be considered constant in every station. However, this is not true for the Italian Railway Network where just the probability of having a negative delay can be considered constant. On the contrary, the probability of having a positive departure delay increases linearly with k_out. § OPTIMAL CHOICE OF Β In order to determine the optimal value of the β, we tune our model to reproduce with the highest probability the delay that each train gets whenever it crosses a station during its path. Considering a train i arriving at a station n on a given day, we call δ t_i,n^emp its measured arrival delay at that station as recorded in the dataset.Hence, we perform 200 simulations of the schedule of the considered day in order to compute the distribution P(δ t_i,n) of the corresponding δ t_i,n.Hence, considering the null hypothesis that δ t_i,n^emp is produced by our model (i.e. it is extracted by P(δ t_i,n)), we calculate the double tailed p-value for such hypothesis for every pair (i,n), i.e. for every train and for every station. For each day we average the p-values of all the train-station pairs to obtain the performance metrics for β. Fig. <ref> shows the dependence of the average p-value as a function of β. The curves have been computed from the simulation of a week of daily schedules.The values of β=0.15 for Italy andβ=0.10 for Germany allows the model to maximize the probability of reproducing the correct arrival delay for each train at a particular station. These values will be used in all the numerical simulations in the following, if not otherwise specified.§ PREDICTIVE LIMITS OF THE MODEL By means the definition of the p-value of the previous paragraph, we can explore a bit which are the predictive limits of the model, i.e. in which part of the Railway Network it is more likely to not reproduce correctly the delays. In other words, we computed for each station n the average p-values, by averaging all the p-values assigned to the couple (i,n). Fig. <ref> shows the distribution of these average p-values for the stations in the networks, obtained with the optimal value of β.The largest parts of the stations have p-values centred around a typical value of ∼0.6 for Italy and ∼ 0.7 for Germany, yet there is a large percentage (∼ 11% for the Italian and ∼ 20% for the German case) with a p-value smaller than 0.05. In these stations the predictive power of the model is particularly unsatisfactory and it is interesting to understand something about their features. Fig. <ref> shows the distribution of the average weight of the links (in terms of the number of trains that have travelled in the links in our whole datasets) and the distribution of the length of the links connected to a station, in the case of stations with p-value larger and smaller than 0.05.We can see that in the latter, the distribution of the weight is considerably narrower, indicating that in these stations the traffic is usually low.Hence, the disagreement might be the result of a poor sampling of the exogenous disturbances around these stations leading to poor predictions or to a dependence of the transmission parameter β on the traffic conditions that have been ruled out when we assumed them to be constant in time and uniform all over the network. 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"authors": [
"Bernardo Monechi",
"Pietro Gravino",
"Riccardo di Clemente",
"Vito D. P. Servedio"
],
"categories": [
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"primary_category": "physics.soc-ph",
"published": "20170726203256",
"title": "Complex delay dynamics on railway networks: from universal laws to realistic modelling"
} |
Department of Mechanical Engineering, National Taiwan University, Taipei 106, TaiwanDepartment of Mechanical Engineering, National Taiwan University, Taipei 106, TaiwanDepartment of Mechanical Engineering, National Taiwan University, Taipei 106, TaiwanDepartment of Mechanical Engineering, National Taiwan University, Taipei 106, [email protected], [email protected] of Mechanical Engineering, National Taiwan University, Taipei 106, Taiwan Department of Mathematical and Systems Engineering, Shizuoka University, Hamamatsu, Shizuoka 432-8561, Japan We design a vehicle with a steering system made of two independently rotatable wheels on the front. We quantify the effectiveness of the steering system in the mobility and maneuverability of the vehicle running in a box containing a layer ping-pong balls with a packing density 0.8, below the random close packing value 0.84 in 2D. The steering system can reduce the resistance exerted by the jammed balls formed ahead of the fast-moving vehicle. Moreover, if only one of the two steering wheels rotates, the vehicle can turn into the direction opposite to the rotating wheel. The steering system performs more efficiently if the wheels engage the ping-pong balls better by increasing the contact area between the wheels and the balls. We advocate applying our design to machines moving in granular materials with a moderate packing density. A vehicle with a two-wheel steering system mobile in shallow dense granular media Guo-Jie Jason Gao December 30, 2023 =================================================================================§ INTRODUCTIONThere has been a long history of designing mobile machines, for example airplanes and ships, in continuously deformable thermal media including air and water. However, similar attempts in deformable athermal media such as granular materials are only very recent. Partly it is because of the complexity of designing a mobile machine within a medium with equal stiffness. More importantly, the nonequilibrium and discrete features of athermal media make formulating a universal governing equation describing their rheology a very difficult task <cit.>, except under specific conditions approximating the media as a continuum <cit.>. In the literature, most designs for a vehicle in granular materials were initiated by mimicking creatures in nature, including sandfish and lizard <cit.>, insects <cit.> and clam <cit.>. Similar efforts using artificial designs are only lately <cit.>. Compared with designs inspired by living creatures such as a clam that uses an intricate way to move forward by swallowing and discharging sands, artificial ones have to meet the challenges of being simpler and more maneuverable. They should perform more precise movements from moving along a straight line to turning at a sharp angle. On the contrary, a sandfish can only move in a more or less sinuous way. These challenges make designing artificial vehicles a demanding job, but at the same time they are better candidates to satisfy more practical applications, for example, a robotic rover used on a low gravity extraterrestrial planet covered by sand that traps it easily. Recent studies on horizontally pulling an object in dense granular materials revealed that there is a highly jammed region formed ahead of the fast-moving object, where force chains between jammed grains frequently build and destruct <cit.>. Unlike a vehicle in thermal fluids, which usually has its steering system located at the end, in this study we design a vehicle in athermal granular materials, equipped with a steering system placed at its front by utilizing this jammed region. The steering system is composed of two independently rotatable wheels which can interact with the jammed region to reduce the resistance experienced by the vehicle and allow the vehicle to change its moving direction freely in a dense granular medium.We test our vehicle in a pool containing a layer of moderately packed ping-pong balls and quantitatively measure its performance in several vehicle setups. Our experimental results show clear evidence of the existence of the reported jammed region, formed in front of the fast-moving vehicle. The steering system can mobilize the jammed region to allow the vehicle to move faster or escape from being trapped by the jammed balls. Moreover, if only one of the two steering wheels rotates, the vehicle can turn to the direction away from the rotating wheel. The turning mechanism becomes more effective if the rotating wheel engages the ping-pong balls better by increasing the contact area with the balls.Below we introduce the design of the vehicle and the two-wheel steering system in section <ref>, and show our experimental results of operating the vehicle in a shallow ping-pong ball pool in section <ref>. We conclude our study in section <ref>.§ EXPERIMENTAL SETUPWe use ready-made parts manufactured by Tamiya, a Japanese model company. The vehicle is composed of a moving unit (Tamiya item No. 70104) and a two-wheel steering system, located in front of the moving unit. The moving unit, which drives the vehicle forward or backward, contains a twin-motor gearbox (Tamiya item No. 70097), equipped with two FA-130 regular motors powered together by two 1.2 volts, 2,450 mAh Ni-MH batteries. The twin-motor gearbox drives two independent but otherwise identical caterpillar tracks and allows certain differential during turning. The steering system contains two separate gearboxes (Tamiya item No. 70103), each equipped with a FA-130 regular motor powered by two 1.2 volts, 2,450 mAh Ni-MH batteries and driving a rotatable plastic steering wheel in the steering system. Each steering wheel basically has a cross-shape, or can be replaced by a round-shaped one to test the shape effect. The above design guarantees that no battery supplies electricity to more than one action of the vehicle, which maintains the consistency and controllability of the experiments. Besides, we make sure that all batteries are drained by no more than 3% of the fully-charged voltage in all experiments. Therefore, comparing results from experiments with different motor performance due to batteries drained variously is not a concern. All gearboxes are set at their highest gear ratios to output the maximum available torque. A schematic of the vehicle is shown in Fig. <ref>.The vehicle, having approximate dimensions of 11 cm in width (W) and 23 cm in length (L), can move freely in a wood box of 63 cm (W_b) by 108 cm (L_b), which contains a layer of N loosely packed plastic ping-pong balls with an average diameter of 3.8 cm (d). The area packing density ϕ of the system is defined as ϕ= [Nπ(d/2)^2 + WL]/W_bL_b. The two steering wheels measure about 2.8 cm high (H) and 6.0 cm in diameter (D). Left and right wheels viewed from above the vehicle are labeled by L and R, respectively. Their vertical geometric centers are located at d/2 above the bottom of the box. Each of them can be either still (status: 0), or rotate counterclockwise (status: +) or clockwise (status: -). The status of the steering system can be expressed as (L: status / R: status). The time-lapse positions of the vehicle, (x,y), is captured by a digital camera (Logitech C-920r) with a top-down view of the box. The optical distortion of the recorded area introduced by the camera is negligible. We calculate the average velocity v_x of the vehicle using v_x=Δ x/Δ t. In the following analysis, Δ x=x_2- x_1=10 cm and Δ t = t_2 - t_1 sec, calculated by obtaining the time difference between two still images, 1 and 2, taken at t=t_1 and t=t_2, respectively.To take the averages and their error bars of the position and velocity of the vehicle, we repeat one experiment with a given set of parameter three times, and the results are independently verified by all authors, who separately build their own vehicles using the same design. We observe the same results qualitatively in all experiments executed by different authors. We prepare a layer of ping-pong balls in the box at a required packing fraction ϕ by randomly pouring balls into the box and waiting until all balls stop moving. Although there are some regions where balls are packed orderly, the whole packing is overall disordered as long as the area packing fraction ϕ remains reasonably smaller than the random close packing density, 0.84, in two dimensional (2D). There is enough space above balls so that they can pile on top of one another if needed and therefore the system is quasi-2D. A snapshot of an initial condition is shown in Fig. <ref>. In the following section we present and discuss the findings extracted from experiments conducted by one representative person.§ RESULTS AND DISCUSSIONHere we describe the results of our experiments on the vehicle with a two-wheel steering system, introduced in section <ref>. The vehicle moves in a shallow granular medium of plastic ping-pong balls with a given area packing density, ϕ. Initially, we examine the mobility and maneuverability of the vehicle moving at three values of ϕ, 0.6, 0.7 and 0.8 (results not shown). No substantial jammed region forms in front of the vehicle at the two lower densities of 0.6 and 0.7, and therefore the two-wheel steering system performs poorly in these two situations. As a result, we decide to use ϕ=0.8, which best demonstrates the effectiveness of the steering system, throughout this study. The following experiments are all performed with the same initial condition, N=460 ping-pong balls and ϕ = 0.8.To demonstrate that ping-pong balls can jam ahead of the moving vehicle with still steering wheels (L: 0 / R: 0), which slows down or even blocks the motion of the vehicle, we measure its normalized velocity in the x-direction, v_x/v_x^0, where v_x^0 is the velocity of the vehicle moving in the box containing zero ping-pong balls. Then we turn on the steering-system (L: + / R: -), and the two wheels rotate in opposite directions and symmetrically push jammed balls ahead of the vehicle to its sides. Another setting, (L: - / R: +), where the two wheels swipe jammed balls toward the center of the vehicle does not work, because eventually balls get into the space between the two steering wheels and block their rotation. The results are shown in Fig. <ref>. We can clearly see that, in the presence of balls and still steering wheels, the speed of the vehicle in the beginning decreases to about 60% of the speed value in an empty box. The vehicle slows down further as it moves forward and can occasionally reaches a full stop by jammed ping-pong balls in the way. On the other hand, with the steering system rotating, the vehicle moves faster by about 10% and never gets trapped by jammed balls until it reaches the other side of the box.Then we test if the vehicle can turn if only one steering wheel rotates and the other stays still. We find that the vehicle turns to the direction away from the rotating wheel. The scenario stays the same while both wheels rotate in the same direction, but there is no significant change in the turning efficiency. The results are presented in Fig. <ref>. As expected, a symmetric rotation setting of the steering wheels, (L: + / R: -), allows the vehicle to move alone an almost straight path, while an asymmetric setting, (L: + / R: 0) and (L: 0 / R: -), deviates the vehicle from the straight course. Settings (L: + / R: +) and (L: + / R: 0) produce almost the same effect, so do settings (L: 0 / R: -) and (L: - / R: -). In all experiments, the vehicle never fails to perform a turn with the attempted setting. We can observe that the curved courses in Fig. <ref> do not mirror each other perfectly across the horizontal axis. This is because it is difficult to build a vehicle with a perfect symmetric steering system. This asymmetry can be minimized if we average the trajectories using more trials.Finally, to understand the influence of the shape of the steering wheels, we replace one of the cross-shaped wheels with a round-shaped one with a spiky rubber surface, while keeping all other characteristics of the vehicle untouched. The cross-shaped steering wheel and the round-shaped one have similar dimensions and weights. The length of an arm of the cross-shaped wheel and the depth of a spike on the round-shaped wheel are 0.66d and 0.11d, respectively. The results are shown in Fig. <ref>. It can be obviously seen that the cross-shaped steering wheel works almost twice as effective as the round one in terms of turning the vehicle, because its shape endows larger contact area with the jammed ping-pong balls and exchange momentum with them more efficiently.We can explain the above results by considering the momentum exchange between the moving vehicle with a rotating steering wheel and the jammed ping-pong balls in front of it, as depicted schematically in Fig. <ref>. The rotating wheel installed on the vehicle liquidizes its neighboring jammed balls and transfers momentum into them. In exchange, the vehicle gains opposite momentum according to Newton's third law of motion and turns away from the rotating wheel. It is worth noting that this mechanism works only when the vehicle moves fast enough in a fairly packed sea of balls, so that a jammed region can form ahead of the moving vehicle, as described before. The two-wheel steering system becomes ineffective if the vehicle travels too slow or in a loosely packed granular medium.In this study, we do not adopt a single steering wheel design, because it renders the vehicle a forward-triangle shape, and therefore the vehicle plunges easily into ping-pong balls of smaller size than the steering wheel. Superior to the single-wheel design, the two-wheel steering system offers a larger area to engage balls and makes the vehicle more controllable. Besides, if only one of the steering wheels rotates, the vehicle can turn into the direction opposite to the rotating steering wheel, while the other static one maintains a jammed region of balls ahead of the vehicle, which makes the turning mechanism more stable.Although we achieve consistent results of controlling the vehicle, there are still several issues should be taken into account. First, the system size is still very small and therefore the finite size effect and the influence from boundaries cannot be ignored. For example, the ping-pong balls in the box ahead of the vehicle can be fully jammed when the vehicle moves beyond x/L > 2. Second, we need a well-defined way to prepare an initial condition of randomly placed balls with a given ϕ. Using a map of computer-generated positions may be a solution to this issue. Finally, it is important to use much smaller balls than the sizes of the steering wheels and the vehicle so that the shape of the jammed region in front of the vehicle can be more flexible and avoid artificial crystallization. We expect the jammed region becomes smaller with decreasing the particle size of the granular media. We will address these issues in our next study.§ CONCLUSIONSIn summary, a region formed by jammed ping-pong balls ahead of a fast-moving vehicle is now believed to play a crucial role in affecting the motion of the vehicle. We take advantage of this jammed region and design a vehicle with a front two-wheel steering system, a feature different from most mobile machines used in thermal fluids, where rear steering systems are more common. The proposed steering system not only reduces the resistance experienced by the moving vehicle, but also helps it change directions freely. The turning capability of the vehicle can be improved if the steering wheels interact with the jammed balls more effectively by increasing the contact area between the steering wheels and the balls.The results of our study have many practical applications. For example, a space rover on another planet, offering very low gravity and covered by sand. Compared with the regular gravity on Earth, the lower gravity makes the sand loosely packed and maybe flow more easily, which cause the rover to sink into the sand more frequently. In order to move smoothly, the rover has to overcome the resistance from the sand jammed and piled up ahead of it, a scenario closely resembles what we have investigated in this study. We believe a similar maneuvering mechanism can be applied to designing a wide range of mobile machines used in dense athermal media, a field still in its infancy. § ACKNOWLEDGMENTSGJG gratefully acknowledges financial support from National Taiwan University funding 104R7417, MOST Grant No. 104-2218-E-002-019 (Taiwan), and startup funding from Shizuoka University (Japan). | http://arxiv.org/abs/1707.08716v1 | {
"authors": [
"Po-Yi Lee",
"Meng-Chi Tsai",
"I-Ta Hsieh",
"Pin-Ju Tseng",
"Guo-Jie Jason Gao"
],
"categories": [
"cond-mat.soft",
"cs.RO"
],
"primary_category": "cond-mat.soft",
"published": "20170727061717",
"title": "A vehicle with a two-wheel steering system mobile in shallow dense granular media"
} |
University of Regensburg Università della Svizzera italiana, LuganoWe present new results of full QCD at nonzero chemical potential. In PRD 92, 094516 (2015) the complex Langevin method was shown to break down when the inverse coupling decreases and enters the transition region from the deconfined to the confined phase. We found that the stochastic technique used to estimate the drift term can be very unstable for indefinite matrices. This may be avoided by using the full inverse of the Dirac operator, which is, however, too costly for four-dimensional lattices. The major breakthrough in this work was achieved by realizing that the inverse elements necessary for the drift term can be computed efficiently using the selected inversion technique provided by the parallel sparse direct solver package PARDISO. In our new study we show that no breakdown of the complex Langevin method is encountered and that simulations can be performed across the phase boundary.Selected inversion as key to a stable Langevin evolution across the QCD phase boundary Jacques Bloch1Speaker, [email protected] Olaf Schenk2 December 30, 2023 ======================================================================================== § INTRODUCTION The lattice simulations of QCD at nonzero quark chemical potential are strongly hampered by the sign problem, caused by the complex fermion determinant. The complex Langevin (CL) method has drawn a lot of attention in recent years as a potential solution to this problem <cit.>. Nevertheless, careful studies have shown that the method can break down or, even worse, can converge to the wrong solution, if the trajectories make excursions too far into the SL(3,ℂ) plane or come too close to a singularity of the drift <cit.>. Although conditions were derived that have to be satisfied for the CL solutions to be valid, the matching of theseconditions can only be verified a posteriori. Therefore, it cannot be excluded that the violation of the validity condition is due to numerical inaccuracies rather than to a theoretical deficiency of the method for the model being considered.The first successful application of the CL method to QCD was made in the heavy dense approximation <cit.>. For full QCD, the method was shown to work correctly in the deconfined phase, when the inverse coupling β is large enough; however, it breaks down when β gets smaller and the system crosses the phase boundary, such that no solutions are found in the confined phase <cit.>. Other studies at larger β and larger volumes also seem to converge to incorrect solutions <cit.>. Recently there were suggestions to modify the CL evolutions through dynamical stabilization <cit.> or deformation <cit.>, but the extrapolations needed to recover the original theory are not yet well controlled.Figure <ref> gives a sketch of the QCD phase diagram as a function of temperature and chemical potential. The simulations reported by Fodor et al. <cit.> follow the gray arrow through the roof of the phase transition; however, the CL simulations break down when crossing the phase boundary, and no results were found inside the confined phase. In these simulations, the temperature was lowered by decreasing β on an 8^3× 4 lattice with μ/T=1 and m=0.05. For these parameter values the critical temperature corresponds to β_c ≈ 5.04 at μ=0. The results for the Polyakov loop and its inverse, the temporal and spatial plaquettes, the chiral condensate, and the quark number density published in <cit.> are reproduced in Fig. <ref>. In these plots the CL results are compared with data reweighted from the μ=0 ensemble, however, CL results are only available for β≥ 5.1 as the CL simulations became unstable below this value. In this presentation we will show that the breakdown observed in <cit.> can be cured and stable CL solutions can be found for smaller β when the drift is computed exactly, rather than being estimated with stochastic techniques.§ COMPLEX LANGEVIN FOR QCD The lattice QCD partition function is given byZ=[∏_x=1^V∏_ν=1^d∫ d U_xν] exp[-S_g]D(m;μ)withWilson gauge action S_g, staggered Dirac operator D, and linksU_xν = exp[i∑_a=1^8 z_axνλ_a ]with Gell-Mann matrices λ_a and link parameters z_axν. After discretization of the Langevin time, the CL evolution of the links in 3c is described byU_xν(t+1) = R_xν (t) U_xν(t) ,where, in the stochastic Euler discretization, R_xν = exp[i∑_a λ_a (ϵ K_axν + √(ϵ) η_axν)] ∈3c,with Langevin step ϵ and Gaussian noise η_axν. The evolution is driven by the driftK_axν= - ∂_axν S = K_axν^g + K_axν^fwith complex action S=S_g - log D. In the following we will focus on the fermionic drift,K_axν^f = [D^-1∂_axν D],where ∂_axν D is the partial derivative of D wrt the variables z_axν. § FERMIONIC DRIFT TERM AND SELECTION INVERSION When looking at the breakdown of the CL method in <cit.> it is not immediately clear if the violation of the CL validity conditions is genuine or rather due to numerical approximations. The traces in the fermionic drift (<ref>) involve the inverse Dirac operator. Computing the full inverse, for example, with Lapack, is too expensive for four-dimensional QCD, both in terms of CPU time and storage space. Until now, the explicit computation of D^-1 was avoided by using stochastic estimators for the traces, [D^-1∂_axν D] ≈η^† D^-1∂_axν D η .Although the merit of this technique is undisputed for positive-definite matrices, it can be problematic when applied to indefinite matrices, as the number of noise vectors needed to get a good and stable estimate may be extremely large. As current CL algorithms typically estimate traces using a single noise vector, they rely on choosing ϵ tiny, such that consecutive CL steps are highly correlated and effectively provide an improved estimator to the trace after many Langevin steps. A potential danger is that this strategy may destabilize the discrete time evolution beyond repair.To clarify this situation we decided to investigate how using exact traces affects the stability of the CL evolution. As already mentioned, the exact traces require the inverse of the Dirac matrix, which is too costly to compute in full. Below we will analyze the drift term further and show that a relatively new numerical method, called selected inversion, can be used to compute the drift exactly, while saving CPU time and storage space. This method also allows us to use a larger ϵ, as the very small ϵ values used before were only needed to stabilize the drift when using the stochastic technique. ncbar angle/.initial=90, ncbar/.style= to path=() – (()!#1!/tikz/ncbar angle:()) – (()!(()!#1!/tikz/ncbar angle:())!/tikz/ncbar angle:()) – () , ncbar/.default=0.5cm,register/.style=rectangle,rounded corners,draw,single_entry/.style=rectangle, rounded corners, draw, thick,single_entry_not_broadcasted/.style=rectangle, rounded corners, draw,direction_line/.style=->, line width=0.25mm,reduction_line/.style=->, thick, dashed,connection_line/.style=thick,my_node/.style=midway, draw, circle, thick, inner sep=3pt, fill=white, square left brace/.style=ncbar=0.5cm, square right brace/.style=ncbar=-0.5cm, Consider the fermionic drift K^f_axν of Eq. (<ref>). The derivative matrix ∂_axν D is zero except for two 3×3 blocks at the positions of the link U_xν, so that the drift looks like[matrix of math nodes,left delimiter=(,right delimiter=)] (lhs) 1 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 1;[above=1.6cm] at (lhs.south) dense;[below=1mm] at (lhs.south) D^-1;(times) [right=.4cm] at (lhs.east) ×;[matrix of math nodes,left delimiter=(,right delimiter=), right=.4cm of times.east] (op)1 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 1; [single_entry,color=red](op-3-6.north west) rectangle (op-4-7.north west) node[midway] F; [single_entry,color=blue](op-6-3.north west) rectangle (op-7-4.north west) node[midway] B; [below=1mm] at (op.south) ∂_axν D; (i) [above right=5mm and 9.3mm] at (op.north west) x; (i1) [below=1mm] at (i) ; (i2) [below=5mm] at (i1) ;[->] (i1)– (i2); (id)[above right=5mm and 19.0mm] at (op.north west) x+ν̂; (id1) [below=1mm] at (id) ; (id2) [below=5mm] at (id1) ;[->] (id1)– (id2); (ih2) [below right=9mm and 5mm] at (op.north east) ; (ih1) [right=5mm] at (ih2) ; (ih) [right=-1mm] at (ih1) x;[->] (ih1)– (ih2); (idh2) [below right=20mm and 5mm] at (op.north east) ; (idh1) [right=5mm] at (idh2) ; (idh) [above right=-2mm and -1mm] at (idh1) x+ν̂;[->] (idh1)– (idh2); [left=24mm] at (lhs) K^f_axν=; [very thick] ((lhs)+(-17mm,-19mm)) to [square left brace] ((lhs)+(-17mm,19mm)); [very thick] ((op)+(17mm,-19mm)) to [square right brace] ((op)+(17mm,19mm));In the matrix product the sparse derivative matrix effectively selects out the two corresponding columns of D^-1 such that this can be rewritten as [matrix of math nodes,left delimiter=(,right delimiter=)] (lhs)1 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 1;[single_entry,color=red]((lhs.north west)+(9mm,-2mm)) rectangle((lhs.north west)+(13mm,-30mm));[single_entry,color=blue] ((lhs.north west)+(22mm,-2mm)) rectangle ((lhs.north west)+(26mm,-30mm)); [below=1mm] at (lhs.south) D^-1; (i) [above right=5mm and 8.1mm] at (lhs.north west) C_x; (i1) [below=1mm] at (i) ; (i2) [below=5mm] at (i1) ;[->] (i1)– (i2); (id)[above right=5mm and 19.7mm] at (lhs.north west) C_x+ν̂; (id1) [below=1mm] at (id) ; (id2) [below=5mm] at (id1) ;[->] (id1)– (id2); (times) [right=.4cm] at (lhs.east) ×;[matrix of math nodes,left delimiter=(,right delimiter=), right=.4cm of times.east] (op)1 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 1;[single_entry,color=red](op-3-6.north west) rectangle (op-4-7.north west) node[midway] F; [single_entry,color=blue](op-6-3.north west) rectangle (op-7-4.north west) node[midway] B; [below=1mm] at (op.south) ∂_axν D; (i) [above right=5mm and 9.3mm] at (op.north west) x; (i1) [below=1mm] at (i) ; (i2) [below=5mm] at (i1) ;[->] (i1)– (i2); (id)[above right=5mm and 19.0mm] at (op.north west) x+ν̂; (id1) [below=1mm] at (id) ; (id2) [below=5mm] at (id1) ;[->] (id1)– (id2); (ih2) [below right=9mm and 5mm] at (op.north east) ; (ih1) [right=5mm] at (ih2) ; (ih) [right=-1mm] at (ih1) x;[->] (ih1)– (ih2); (idh2) [below right=20mm and 5mm] at (op.north east) ; (idh1) [right=5mm] at (idh2) ; (idh) [above right=-2mm and -1mm] at (idh1) x+ν̂;[->] (idh1)– (idh2); [left=24mm] at (lhs) K^f_axν=; [very thick] ((lhs)+(-17mm,-19mm)) to [square left brace] ((lhs)+(-17mm,19mm)); [very thick] ((op)+(17mm,-19mm)) to [square right brace] ((op)+(17mm,19mm));where C_x+ν̂× B and C_x × F, respectively, give the columns x and x+ν̂ of D^-1∂_axν D. Only two 3×3 blocks of D^-1 contribute to the trace of this matrix product, so this simplifies further to[matrix of math nodes,left delimiter=(,right delimiter=)] (lhs)1 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 1;[single_entry,color=blue](lhs-3-6.north west) rectangle (lhs-4-7.north west) node[midway] P; [single_entry,color=red](lhs-6-3.north west) rectangle (lhs-7-4.north west) node[midway] Q; [below=1mm] at (lhs.south) D^-1; (i) [above right=5mm and 9.2mm] at (lhs.north west) x; (i1) [below=1mm] at (i) ; (i2) [below=5mm] at (i1) ;[->] (i1)– (i2); (id)[above right=5mm and 19.0mm] at (lhs.north west) x+ν̂; (id1) [below=1mm] at (id) ; (id2) [below=5mm] at (id1) ;[->] (id1)– (id2);(times) [right=.4cm] at (lhs.east) ×;[matrix of math nodes,left delimiter=(,right delimiter=), right=.4cm of times.east] (op)1 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 1;[single_entry,color=red](op-3-6.north west) rectangle (op-4-7.north west) node[midway] F; [single_entry,color=blue](op-6-3.north west) rectangle (op-7-4.north west) node[midway] B; [below=1mm] at (op.south) ∂_axν D; (i) [above right=5mm and 9.2mm] at (op.north west) x; (i1) [below=1mm] at (i) ; (i2) [below=5mm] at (i1) ;[->] (i1)– (i2); (id)[above right=5mm and 19.0mm] at (op.north west) x+ν̂; (id1) [below=1mm] at (id) ; (id2) [below=5mm] at (id1) ;[->] (id1)– (id2); (ih2) [below right=9mm and 5mm] at (op.north east) ; (ih1) [right=5mm] at (ih2) ; (ih) [right=-1mm] at (ih1) x;[->] (ih1)– (ih2); (idh2) [below right=20mm and 5mm] at (op.north east) ; (idh1) [right=5mm] at (idh2) ; (idh) [above right=-2mm and -1mm] at (idh1) x+ν̂;[->] (idh1)– (idh2); [left=24mm] at (lhs) K^f_axν=; [very thick] ((lhs)+(-17mm,-19mm)) to [square left brace] ((lhs)+(-17mm,19mm)); [very thick] ((op)+(17mm,-19mm)) to [square right brace] ((op)+(17mm,19mm)); Each drift term can thus be written asK^f_axν =(P · B) +(Q· F)and, hence, the computation of the drift only requires elements of D^-1 where D itself is nonzero. Note that, the same statement holds for the computation of the fermionic observables.This observation is what brought us to consider the selected inversion technique to compute the drift term. The method consists of a sparse LU-factorization followed by a selected inversion, which exactly computes selected elements of the inverse D^-1 of a general matrix D. The subset of selected elements is defined by the set of nonzero entries in D. The method is based on the fact that this specific subset of D^-1 can be evaluated without computing any inverse entry from outside of the subset. This is what substantially speeds up the computation compared to routines computing the full inverse. Moreover, as only the inverse elements of this subset are computed, they can be stored in sparse format. The parallel implementation of the selected inversion technique, as described in <cit.>, can be found in the latest version of the parallel sparse direct solver PARDISO <cit.>.It is worthwhile to note that the selected inversion method is optimally used in the CL evolution, as the selected inversion method precisely yields all the elements of D^-1 needed for the drift terms and for the fermionic observables.In contrast to the stochastic technique, the selected inversion allows us to compute the drift exactly, and this in a way that is much more efficient than Lapack, both in terms of CPU times and storage space.In Fig. <ref> we compare the scaling of the wall-clock time between the selected inversion from PARDISO, the full Lapack inversion, and the stochastic technique (where we used 100 intermediate steps due to the smaller choice of ϵ as explained after (<ref>)) as a function of the lattice volume. There is a clear performance gain when using the selected inversion, which is a factor of 100 compared to the Lapack dense inverse for a lattice size of 8^4. The comparison with the stochastic technique is more delicate as the stability of the CL evolution can be affected by the latter and merely comparing timings does not tell the whole story.The volume scaling of the selected inversion seems to be somewhat better than N^3. From other applications, the selected inverse is known to scale like N^2 for three-dimensional problems, however, as this is the first application to a four-dimensional problem the scaling has to be investigated further.§ RESULTS§.§ CL evolution for varying βIn this section we present the first results of the CL method using the selected inversion.The simulations were performed on the Xeon cluster at the ICS, Lugano. We performed the same study as in <cit.> and investigate QCD across the phase boundary as sketched in Fig. <ref>, decreasing the temperature from the deconfined to the confined phase by varying β for constant μ/T=1 on an 8^3×4 lattice with m=0.05 (β_c ≈ 5.04 at μ=0). Whereas the CL method broke down below β=5.1 in the original study, we are now able to generate stable CL trajectories for all investigated β values without any further tuning. Moreover, this was done using ϵ=0.001, which is much larger than before.The length of the trajectories is 30 Langevin time (Lt) of which 5 Lt are discarded as thermalization. From the results, shown in Fig. <ref>, we see that the simulations perform well across the phase boundary, and the complete range from β=5.45 to β=4.6 can be simulated without any problem. The numerical implementation includes gauge cooling, to avoid excessive excursions in SL(3,ℂ), and an adaptive step size such that the continuum trajectory is properly followed, even when the drift is large. The fact that a stable solution is found does not necessarily mean that it is correct, as the CL method can converge to the wrong solution in some instances. In order to validate the CL results we show the histograms for the chiral condensate and the Polyakov loop for β=5.4 and β=4.8, in Fig. <ref>. The CL results are only to be trusted if the tails of the histograms decay exponentially. For the chiral condensate it seems that the validity might be problematic for β=4.8, as the tails of the histogram are somewhat broad, but even for β=5.4 the exponential decay in the tails is not so clear, even though the results agree with the reweighting results. The histograms for the Polyakov loop look fine so far. From these data we conclude that the results may be incorrect below the phase transition, even though the histograms do not give a clear cut way to validate or invalidate CL measurements. §.§ CL evolution for varying μ As we found stable CL evolutions when decreasing β through the phase boundary, we also performed a partial study of the phase diagram and investigated QCD for varying μ at two values of the temperature.We chose β=5.0 and m=0.05 for which the pion mass is a m_π=0.5588 ± 0.0002 and a=[(0.3045 ± 0.0001)]fm, such that m_π≈[362]MeV. We work on an 8^3× N_t lattice and consider temporal extents N_t=4 and N_t=8, corresponding to T=[161.74]MeV and T=[80.87]MeV, respectively.In Fig. <ref> we show the results for the various observables as a function of μ for both temperatures. Although the results do not show any conspicuous behavior, there are no other results to compare with when the sign problem becomes large at nonzero μ. One value that can easily be checked is the value at μ=0 as it can be computed using standard importance sampling method. From the figure it is clear that the chiral condensate and even the plaquette measured from the CL simulations is incorrect for μ=0, even though they result from a convergent CL evolution. For μ=0 the Langevin evolution should be real, and the wrong result is merely due to numerical inaccuracies. This is in fact easily remedied by reunitarizing the links after each Langevin step; however, we decided against this to be consistent with the μ≠0 simulations. It is interesting to note that the measurements at nonzero chemical potential smoothly connect to the wrong μ=0 value, and so we expect all of them to be incorrect. This argument is further supported through the lack of Silver Blaze phenomenon, as one would have expected a very slow μ-evolution of the observables up to the phase transition. The validity of the CL results can again be verified using the histograms of the measured observables. The histograms are very similar to those of the bottom row of Fig. <ref>: that of the chiral condensate does not seem to have the required exponential falloff, while that of the Polyakov loop does not show a problem. Still, the histograms are not what one would brand as extremely broad, and so the decision of the validity is a difficult call, even though the chiral condensate at μ=0 is wrong by a factor three.§ SUMMARY AND OUTLOOK In this presentation we have argued that the breakdown of the CL method at the QCD phase boundary observed in <cit.> is in fact a numerical artifact due to the stochastic estimation of the drift. To compute the drift exactly, one needs the exact inverse of the Dirac operator, which cannot be computed in full with standard direct methods because it is too expensive in terms of both computer time and storage. However, we showed that the drift and the fermionic observables only require those elements of the inverse Dirac operator at the positions where the Dirac operator itself is filled, and, therefore, the selected inversion method, implemented in the sparse direct solver library PARDISO, can be applied. This allows for the exact computation of the drift term in a much faster way, using little storage space. We observed that the Langevin evolution became stable and convergent when the exact drift term was used. This allowed us to study the QCD phase transition across the roof of the phase diagram, i.e., when decreasing temperature from the deconfined to the confined phase at constant μ/T=1. Moreover, we were able to measure QCD observables as a function of μ at two values of the temperature below the deconfinement temperature.Although these are preliminary results, there are clear indications that the CL results obtained at small mass in the confined region are incorrect, even though they are stable. There is therefore a need for further investigation of these CL results to understand if the validity problems are of a fundamental or numerical nature. The fact that we now get stable CL trajectories should allow us to dig deeper into this problem and to search for improved methods. §.§ AcknowledgmentsThis work was supported by the DFG collaborative research center SFB/TRR-55. We would like to thank Falk Bruckmann and Piotr Korcyl for useful discussions, Denes Sexty for providing the zero μ data, Radim Janalík for helping with the computer resources, and the ICS, Lugano for providing computing time for the simulations. | http://arxiv.org/abs/1707.08874v1 | {
"authors": [
"Jacques Bloch",
"Olaf Schenk"
],
"categories": [
"hep-lat"
],
"primary_category": "hep-lat",
"published": "20170727141825",
"title": "Selected inversion as key to a stable Langevin evolution across the QCD phase boundary"
} |
University of Bordeaux, CNRS, CRPP-UPR8941,115 Avenue Schweitzer, F-33600 Pessac, France University of Bordeaux, CNRS, CRPP-UPR8941,115 Avenue Schweitzer, F-33600 Pessac, France [email protected] University of Bordeaux, CNRS, CRPP-UPR8941,115 Avenue Schweitzer, F-33600 Pessac, France Huygens sources are elements that scatter light in the forward direction as used in the Huygens-Fresnel principle. They have remained fictitious until recently when experimental systems have been fabricated. In this letter, we propose isotropic meta-atoms that act as Huygens sources. Using clusters of plasmonic or dielectric colloidal particles, Huygens dipoles that resonate at visible frequencies can be achieved with scattering cross-sections as high as 5 times the geometric cross-section of the particle surpassing anything achievable with a hypothetical simple spherical particle. Examples are given that predict extremely broadband scattering in the forward direction over a 1000 nm wavelength range at optical frequencies. These systems are important to the fields of nanoantennas, metamaterials and wave physics in general as well as any application that requires local control over the radiation properties of a system as in solar cells or bio-sensing. Valid PACS appear here Isotropic Huygens dipoles and multipoles with colloidal particles Alexandre Baron December 30, 2023 ================================================================= In 1690, Christiaan Huygens enunciated the famous principle that carries his name. It states that every point on a wave-front may be considered as a source of secondary sphericalwavelets which spread in the forward direction at the speed of light. The new wave front is the tangential surface to all of these secondary wavelets <cit.>. Huygens sources can be obtained by overlapping the radiation from electric and magnetic dipoles that have equal polarizabilities<cit.>. The forward scattering arises from interferences between their emission patterns. In the years 1980, this requirement of forward scattering was analyzed by Kerker et al. <cit.> and the condition for forward scattering was given in the case of spherical particles. The ability to produce such sources with large dipole moments has important consequences because the stored potential energy inside the source can be large while maintaining a free-space matched forward radiation. As a result a net phase-delay can be imparted onto a wavefront with high efficiency, making it possible to make metasurfaces that act as optical surface components with 100% transmission <cit.>, polarization beam controllers, splitters, converters and analysers <cit.> and perfect absorbers <cit.>. The ability to produce such sources also has important implications for the fields of nanoantennas, where large scattering is desired with a well controlled radiation direction <cit.>. Generally speaking, controlling the radiation of nanoparticles is essential to the design of both resonant and broadband anti-reflection coatings<cit.>. We identify two classes of Huygens sources. The first kind are composed of anisotropicparticles - disks or cylinders - in which the magnetic ∼λ/(2n) and electric ∼λ/(n) modes resonate at the same wavelength <cit.>. Both dielectric<cit.> and plasmonic systems have been proposed <cit.>. The second class of proposed Huygens sources are isotropic and composed of core shell nano-hybrids that overlap the electric dipole resonance of a plasmonic core with the magnetic Mie resonance of a high-index dielectric shell <cit.>. These two kind of Huygens dipoles both present severe limitations. The anisotropic systems make the Huygens featurehighly dependent on the incident angle and an angle as small as 4 reduces the transmission of a metasurface composed of such systems by almost 80% for both s an p-polarizations <cit.>. In the end, this means that the metasurface optical component has a close-to-zero field of view. Alternatively the core-shell systems preserve their functionality for oblique incidences but at the expense of properties that critically depend on the geometrical parameters of the designs <cit.>. There is a dire need for experimentally accessible designs that act as efficient and isotropic Huygens sources either resonant or broadband. The recent progress of colloidal nanochemistry has made it possible to realize optical scatterers of both simple geometries containing a single material or more complex geometrical shapes composed of several materials that exhibit unusual properties <cit.>. These bottom-up fabrication schemes are good candidates for the large-scale synthesis of Huygens sources at optical frequencies. The purpose of this article is to propose designs of efficient and brodband Huygens dipoles and multipoles based on colloidal systems. Let us first consider the best performances that may be reached by a lossless dielectric sphere of radius a with a refractive index n_p. Let m = n_p/n_h be the refractive index contrast, where n_h is the host medium refractive index in which the sphere is immersed. We note a_n(m,μ,x) and b_n(m,μ,x) the n^th order Mie scattering coefficients of the electric and magnetic mutlipole that describe the behavior of the particle. They are both functions of the dimensionless parameter x = 2π n_ha/λ and the magnetic permeability of the particle μ.Whenever a_n=b_n in the complex plane, the n^th multipole will scatter forward. In the following, we will refer to this condition as Kerker I. We may proceed to list three situations under which Kerker I will be satisfied. When μ=m (i), we have a_n = b_n. In the case where ϵ_h=1, Kerker I simplifies to ϵ =μ, which means that the particle impedance is matched to that of the host medium, ensuring that any impinging wave perfectly couples to the particle <cit.>. Since μ = 1 at optical frequencies for all natural materials, this condition cannot be met with simple spherical particles. When mx = ψ'_n,1 (ii), which is the first zero of the first derivative of the Ricatti-Bessel function ψ_n, we have a_n=b_n.When mx = j_n,1 (iii), which is the first zero of the spherical Bessel function J_n, we have a_n=b_n. Next we consider Q^(ii)_n and Q^(iii)_n, the scattering efficiencies of the n^th multipole under conditions (ii) and (iii) respectively, defined as the scattering cross-section of the n^th multipole normalized to the geometrical cross-section of the sphere (π a^2).Condition (iii) occurs at higher frequencies than condition (ii) in a regime where several multipoles may interfere, so we shall focus on condition (ii) and noteQ^(ii)_n,max the maximum achievable efficiency that satisfies Kerker I. Is is solely determined by m (see supplemental materials). In the dipolar case, as already noticed by Luk'yanchuk et al. <cit.>, the maximum efficiency reached is Q_1,max^(ii)≈ 3.72, when m = m^* ≈ 2.455, which means that x = x* = ψ'_n,1/m^* ∼ 1.118.Figure <ref>(a) shows how the total efficiency evolves as a function of x for a sphere when m = m^*.When x =x^*, the optimal efficiency is reached and the scattering efficiency of the electric and magnetic dipoles (a_1 and b_1) are equal, but this scattering does not occur at the resonance frequency of either dipole and the fraction of total scattered energy emitted in the forward direction ρ peaks at x^* and rapidly decreases as x increases (see Fig. <ref>(d)).We may then proceed to find a situation in which both the electric and magnetic dipoles resonate at the same frequency. This situation is given when m=m^**≈ 1.87. The scattering is shown on Fig. <ref>(b). We see that a maximum efficiency of 3.5 is reached for this dipole. Not all the scattering occurs in the forward direction because of a mismatch in amplitude between the electric and magnetic resonances, however a large fraction (>95%) is still scattered forward above the first crossing. This situation can be considerably bettered by considering a lossy system. Indeed, since the magnetic dipole mode has a stronger interaction with the particle medium than does the electric dipole mode, losses decrease the scattering amplitude of the magnetic dipole and an optimum is reached when m=m^***≈ 2.1+0.095i (see supplemental material), for which both amplitudes are equal and resonate at the same frequency (see Fig. <ref>(c)). In this case the maximum scattering efficiency reached is ∼ 3.2, which is still almost 86% of the value reached when m=m^*. We may refer to such dipoles as ideal Huygens dipoles.Unfortunately it is hard to find materials that will reach m^***. A solution can be found by resorting to an effective medium sphere of radius a made of spherical inclusions. Practically, this can be achieved by making a spherical cluster of nanospheres. In what follows, we shall consider clusters composed of identical inclusions of radius a_i in repulsive interaction arranged by distributing them quasi-homogenously into the effective medium sphere of radius a. Fixing the values of a_i and a determines the range of the number N of inclusions required to form the cluster. We will refer to it as an N-cluster and the volume fraction of inclusions in the cluster is f=Na_i^3/a^3.Targeting a peak resonance wavelength of λ≈ 500 nm using extended Maxwell-Garnett theory<cit.> (see supplemental material), we explored parameters f and a_i in the cases of silver and silicon inclusions with the aim of getting optical scatterers with overlapping electric and magnetic resonances at their peak. The optical properties of silver were taken from Palik's compendium <cit.>, those of silicon were taken from Aspnes and Studna <cit.> and the scattering from the clusters were calculated using the T-matrix solver developped by Mackowski <cit.>, which uses Generalized Mie Theory and is rigorous in calculating the scattering of ensembles of spheres. Following this procedure we arrived at two spherical clusters that act as ideal Huygens sources. The first is a 60-cluster made of silver inclusions in water of radius a_i = 15 nm, with f = 20%. The cluster radius is a ≈ 100 nm and m ≈ 1.9 +0.18i. The second is a 13-cluster made of silicon inclusions of radius a_i = 41 nm, with f = 47% in air. The cluster radius is a ≈ 123 nm and m ≈ 2.0+0.13i, as predicted by extended Maxwell-Garnett theory. For both designs, the effective index contrast m is close to m^***. Figures <ref> (a,b) are the main result of this paper and show the total scattering efficiency of both systems. We see that the ideal Huygens dipole is reached in both cases near the designed wavelength. At the peak total scattering, both the electric and magnetic resonance overlap at the resonance wavelength of 500 nm (525 nm) for the silver cluster (silicon cluster) with equal amplitudes. The peak efficiencies reached are ∼ 2.5 for the silver cluster and an impressive value of ∼ 5 for the silicon cluster. Such a strong value for the the 13-cluster is a major result as the scattering efficiency reached by this Huygens dipole beats the theoretical maximum scattering efficiency reachable by an ordinary sphere with a complex refractive index following Mie theory as evidenced by Fig. <ref>(c). Furthermore this large response is unaccounted for by extended Maxwell-Garnett theory because μ_eff is underestimated, while the real system is actually closer to condition (i). For each example, the amount of forward-scattered far-field energy relative to the total scattered energy is shown on Fig. <ref>(c,d) and shows that the scattered energy is radiated in the forward direction over a broad range of frequencies around the dipole resonance. The radiation diagrams in both the electric (blue) and magnetic planes (red) almost perfectly overlap as shown on Fig. <ref> (e,f). The results of the 13-cluster case are identical to those obtained using the commercial finite-element method solver COMSOL Multiphysics. Up to now we have considered ideal Huygens dipole scattering. However, this conditionis resonant and only occurs over a narrow band around the resonance frequency. For numerous applications, it may be interesting to maximize the forward scattering over a broad range of wavelengths. Given a specific geometry and value of m, the Kerker I condition cannot be fulfilled exactly for all multipoles at once. But by relaxing the requirement that the electric multipole be exactly equal to the magnetic multipole and just satisfying the condition lossely, i.e. for all n, to obtain a_n ∼ b_n, a broadband isotropic system may be designed. Indeed, by examining the trajectories taken by the scattering resonances of the electric (a_n) and magnetic (b_n) multipoles in the (m,x) plane (see Fig. 3),we see that, while they remain well separated for large values of m, resonances all bundle together towards larger values of x as m becomes smaller than 2. This is apparent from the L shapes taken by the |a_n|^2+|b_n|^2 shown on Fig. <ref>.The yellow area in Fig. <ref>, indicates multipoles satisfying the required condition that |a_n|^2∼|b_n|^2 ∼ 1, which means that multipoles have nearly equal amplitudes near their resonances. We see that the rigorous Kerker I condition under situations (ii), which are represented by the white dashed curves, even cross this merging area, when m<2. Moreover for decreasing values of m, an increasing number of multipoles tend to overlap. In particular, for values close to 1.6,the yellow area appears simultaneously for the first four multipoles. This means that the Kerker I condition can be approached multiple times and simultaneously for each couple of multipoles and forward scattering, though not perfect, will occur over a broad range of wavelengths. Such conditions can be met for colloidally synthesized nanoparticles such as TiO_2 or Al_2O_3, which are commercially available. Using empirical dispersions of the refractive index for both these compounds, we calculate the scattering efficiencies of a TiO_2 nanosphere of a = 150 nm immersed in water and of a Al_2O_3 nanosphere of a = 200 nm immersed in air in Fig. <ref>. The scattering by the electric and magnetic multipoles coincide over a broad range of wavelengths spanning over the interval [550 nm - 850 nm] with a scattering efficiency larger than 2 for TiO_2 and over the interval [400 nm - 1000 nm] with a scattering efficiency larger than 1 for Al_2O_3. The peak efficiency reached for TiO_2 (Al_2O_3) is ∼ 5 (∼ 4). With a silicon cluster, the range of wavelengths can be considerably increased. Figure <ref>(c) shows that a broad Huygens multipole spanning over an impressive 1000 nm range can be achieved starting at 500 nm and for which ρ is close to 1.It should be noted that the condition we use that the electric and magnetic multipoles should overlap is loose and not rigorous. This condition is only used to get a broadband forward scattering system and not so much to explain the exact features of the forward scattering. Overall, this is satisfied as we see that the rough broad overlap between the scattering efficiencies of electric and magnetic multipoles coincides with a broad forward scattering ratio. However, it is not precise in determining the extrema of the forward scattering ratio nor the exact position of the cutoff. The spectral overlap of the efficiencies for a given wavelength is dominated by the overlap of a certain couple of electric and magnetic multipoles of order n, but is also slightly perturbed by the presence of non-overlapping multipoles of order p ≠ n. For instance, this explains why the forward-scattering ratio in Fig. <ref>(e) reaches maxima at wavelengths slightly detuned compared to those where the electric and magnetic multipole efficiencies cross in Fig. <ref>(b).The interesting feature of these systems is that they are realistically achievable through colloidal self-assembly. Several recent examples show that it is possible to self-assemble plasmonic nanoparticles in clusters of submicrometric sizes<cit.>. The cluster system is rich as it is scalable and its optical properties can be tuned by varying the nature, amount, size and volume fraction of inclusions. Aside from these spherical clusters being experimentally feasible systems that produce isotropic Huygens sources, we believe they may serve as excellent building blocks for future nanoantennas and metasurfaces and inspire Huygens sources in other areas of wave physics such as acoustics or mechanics. Preliminary simulations suggest that the Huygens sources presented here can be used to produce phase-control metasurfaces. Several approaches exist to produce two-dimensional periodic surfaces with colloids as building blocks with impressive results <cit.>. These approaches typically consist in coating suspensions of nanoparticles on patterned templates. Combining emulsion-based fabricationof clusters and template-assisted self-assembly is a promising route to the fabrication of Huygens metasurfaces.The authors acknowledge support from the LabEx AMADEus (ANR-10-LABX-42) in the framework of IdEx Bordeaux (ANR-10-IDEX-03-02), France. | http://arxiv.org/abs/1707.08902v3 | {
"authors": [
"Romain Dezert",
"Philippe Richetti",
"Alexandre Baron"
],
"categories": [
"physics.optics"
],
"primary_category": "physics.optics",
"published": "20170727150531",
"title": "Isotropic Huygens dipoles and multipoles with colloidal particles"
} |
Spectral sequences of Type Ia supernovae. I. Connecting normal and sub-luminous SN Ia and the presence of unburned carbon [=========================================================================================================================Sports channel video portals offer an exciting domain for research on multimodal, multilingual analysis. We present methods addressing the problem of automatic video highlight prediction based on joint visual features and textual analysis of the real-world audience discourse with complex slang, in both English and traditional Chinese. We present a novel dataset based on League of Legends championships recorded from North American and Taiwanese Twitch.tv channels (will be released for further research), and demonstrate strong results on these using multimodal, character-level CNN-RNN model architectures.§ INTRODUCTIONOn-line eSports events provide a new setting for observing large-scale social interaction focused on a visual story that evolves over time—a video game. While watching sporting competitions has been a major source of entertainment for millennia, and is a significant part of today's culture, eSports brings this to a new level on several fronts.One is the global reach, the same games are played around the world and across cultures by speakers of several languages.Another is the scale of on-line text-based discourse during matches that is public and amendable to analysis. One of the most popular games, League of Legends, drew 43 million views for the 2016 world series final matches(broadcast in 18 languages) and a peak concurrent viewership of 14.7 million[<http://www.lolesports.com/en_US/articles/2016-league-legends-world-championship-numbers>]. Finally, players interact through what they see on screen while fans (and researchers) can see exactly the same views.This paper builds on the wealth of interaction around eSports to develop predictive models for match video highlights based on the audience's online chat discourse as well as the visual recordings of matches themselves.ESports journalists and fans create highlight videos of important moments in matches.Using these as ground truth, we explore automatic prediction of highlights via multimodal CNN+RNN models for multiple languages.Appealingly this task is natural, as the community already produces the groundtruth and is global, allowing multilingual multimodal grounding.Highlight prediction is about capturing the exciting moments in a specific video (a game match in this case), and depends on the context, the state of play, and the players.This task of predicting the exciting moments is hence different from summarizing the entire match into a story summary.Hence, highlight prediction can benefit from the available real-time text commentary from fans, which is valuable in exposing more abstract background context, that may not be accessible with computer vision techniques that can easily identify some aspects of the state of play. As an example, computer vision may not understand why Michael Jordan's dunk is a highlight over that of another player, but concurrent fan commentary might reveal this. We collect our dataset from Twitch.tv, one of the live-streaming platforms that integrates comments (see Fig. <ref>), and the largest live-streaming platform for video games. We record matches of the game League of Legends (LOL), one of the largest eSports game in two subsets, 1) the spring season of the North American League of Legends Championship Series (NALCS), and 2) the League of Legends Master Series (LMS) hosted in Taiwan/Macau/HongKong, with chat comments in English and traditional Chinese respectively.We use the community created highlights to label each frame of a match as highlight or not.In addition to our new dataset, we present several experiments with multilingual character-based models, deep-learning based vision models either per-frame or tied together with a video-sequence LSTM-RNN, and combinations of language and vision models.Our results indicate that while surprisingly the visual models generally outperform language-based models, we can still build reasonably useful language models that help disambiguate difficult cases for vision models, and that combining the two sources is the most effective model (across multiple languages). § RELATED WORK We briefly discuss a small sample of the related work on language and vision datasets, summarization, and highlight prediction.There has been a surge of vision and language datasets focusing on captions over the last few years, <cit.>, followed by efforts to focus on more specific parts of images <cit.>, or referring expressions <cit.>, or on the broader context <cit.>.For video, similar efforts have collected descriptions <cit.>, while others use existing descriptive video service (DVS) sources <cit.>. Beyond descriptions, other datasets use questions to relate images and language <cit.>.This approach is extended to movies in MovieQA.The related problem of visually summarizing videos (as opposed to finding the highlights) has produced datasets of holiday and sports events with multiple users making summary videos <cit.> and multiple users selecting summary key-frames <cit.> from short videos.For language-based summarization, Extractive models <cit.> generate summaries by selecting important sentences and then assembling these, while Abstractive models <cit.> generate/rewrite the summaries from scratch. Closer to our setting, there has been work on highlight prediction in football (soccer) and basketball based on audio of broadcasts <cit.> <cit.> where commentators may have an outsized impact or visual features <cit.>.In the spirit of our study, there has been work looking at tweets during sporting events <cit.>, but the tweets are not as immediate or as well aligned with the games as the eSports comments.More closely related to our work, yahoo_esports collects videos for Heroes of the Storm, League of Legends, and Dota2 on online broadcasting websites of around 327 hours total. They also provide highlight labeling annotated by four annotators. Our method, on the other hand, has a similar scale of data, but we use existing highlights, and we also employ textual audience chat commentary, thus providing a new resource and task for Language and Vision research.In summary, we present the first language-vision dataset for video highlighting that contains audience reactions in chat format, in multiple languages. The community produced ground truth provides labels for each frame and can be used for supervised learning. The language side of this new dataset presents interesting challenges related to real-world Internet-style slang.§ DATA COLLECTIONOur dataset covers 218 videos from NALCS and 103 from LMS for a total of 321 videos from week 1 to week 9 in 2017 spring series from each tournament. Each week there are 10 matches for NALCS and 6 matches for LMS. Matches are best of 3, so consist of two games or three games. The first and third games are used for training. The second games in the first 4 weeks are used as validation and the remainder of second games are used as test. Table <ref> lists the numbers of videos in train, validation, and test subsets. Each game's video ranges from 30 to 50 minutes in length which contains image and chat data linked to the specific timestamp of the game. The average number of chats per video is 7490 with a standard deviation of 4922. The high value of standard deviation is mostly due to the fact that NALCS simultaneously broadcasts matches in two different channels (nalcs1[<https://www.twitch.tv/nalcs1>] and nalcs2[<https://www.twitch.tv/nalcs2>]) which often leads to the majority of users watching the channel with a relatively more popular team causing an imbalance in the number of chats. If we only consider LMS which broadcasts with a single channel, the average number of chats are 7210 with standard deviation of 2719. The number of viewers for each game averages about 21526, and the number of unique users who type in chat is on average 2185, i.e., roughly 10% of the viewers. Highlight Labeling For each game, we collected community generated highlights ranging from 5 minutes to 7 minutes in length. For the purpose of consistency within our data, we collected the highlights from a single Youtube channel, Onivia,[<https://www.youtube.com/channel/UCPhab209KEicqPJFAk9IZEA>] which provided highlights for both championship tournaments in a consistent arrangement. We expect such consistency will aid our model to better pick up characteristics for determining highlights. We next need to align the position of the frames from the highlight video to frames in the full game video. For this, we adopted a template matching approach. For each frame in the video and the highlight, we divide it into 16 regions of 4 by 4 and use the average value of each color channel in each region as the feature. The feature representation of each frame ends up as a 48-dim vector as shown in Figure <ref>. For each frame in the highlight, we can find the most similar frame in the video by calculating distance between these two vectors. However, matching a single frame to another suffers from noise. Therefore, we alternatively concatenate the following frames to form a window and use template matching to find the best matching location in the video. We found out that when the window size is 60 frames, it gives consistent and high quality results. For each frame, the result contains not only the best matching score but also the location of that match in the video.[When the window contains a moment of clip transition in highlights, the best matching score appears low. This is used to separate all clips in the highlight. Then we can use the starting and end locations of each clip to label the video.] Figure <ref> illustrates this matching process.§ MODELIn this section, we explain the proposed models and components. We first describe the notation and definition of the problem, plus the evaluation metric used. Next, we explain our vision model V-CNN-LSTM and language model L-Char-LSTM. Finally, we describe the joint multimodal model lv-LSTM. Problem Definition Our basic task is to determine if a frame of the full input video should be labeled as being part of the output highlight or not.To simplify our notation, we use X={x_1, x_2, ..., x_t} to denote a sequence of features for frames.Chats are expressed as C={(c_1, ts_1), ... ,(c_n, ts_n) }. where each chat c comes with a timestamp ts.Methods take the image features and/or chats and predict labels for the frames,Y={y_1, y_2, ..., y_t}.Evaluation Metric: We refer to the set of frames with positive ground truth label as S_gt and the set of predicted frames with a positive label as S_pred. Following <cit.>, we use the harmonic mean F-score in Eq.<ref> widely used in video summarization task for evaluation:P=S_gt∩ S_pred/|S_pred| , R = S_gt∩ S_pred/ |S_gt|F= 2PR/P+R× 100 % V-CNN We use the ResNet-34 model <cit.> to represent frames, motivated by its strong results on the ImageNet Challenge <cit.>.Our naive V-CNN model (Figure <ref>) uses features from the pre-trained version of this network [<https://github.com/pytorch/pytorch>] directly to make prediction at each frame (which are resized to 224x224). V-CNN-LSTMIn order to exploit visual video information sequentially over time, we use a memory-based LSTM-RNN on top of the image features, so as to model long-term dependencies. All of our videos are 30FPS. As the difference between consecutive frames is usually minor, we run prediction every 10th frame during evaluationand interpolate predictions between these frames. During training, due to the GPU memory constraints, we unfold the LSTM cell 16 times. Therefore the image window size is around 5-seconds (16 samples every 10th frame from 30fps video). The hidden state from the last cell is used as the V-CNN-LSTM feature. This process is shown in Figure <ref>. L-Word-LSTM and L-Char-LSTM Next, we discuss our language-based models using the audience chat text. Word-level LSTM-RNN models <cit.> are a common approach to embedding sentences. Unfortunately, this does not fit our Internet-slang style language with irregularities, “mispelled" words (hapy, happppppy), emojis (94_94), abbreviations (LOL), marks (?!?!?!?!), or onomatopoeic cases (e.g., “4” which sounds like “yes” in traditional Chinese). People may type variant length of “4”, e.g.,, “4444444” to express their remarks.Therefore, alternatively, we model the audience chat with a character-level LSTM-RNN model <cit.>. Characters of the language, Chinese, English, or Emojis, are expanded to multiple ASCII characters according to the two-character Unicode or other representations used on the chat servers.We encode a 1-hot vector for each ASCII input character. For each frame we use all chats that occur in the next W_t seconds which are called text window size to form the input for L-Char-LSTM. We concatenate all the chats in a window, separating them by a special stop character, and then fed to a 3-layer L-Char-LSTM model.[The number of these stop characters is then an encoding of the number of chats in the window. Therefore, the L-Char-LSTM could learn to use this #chats information, if it is a useful feature. Also, some content has been deleted by Twitch.tv or the channel itself due to the usage of improper words. We use symbol "\n" to replace such cases.] This model is shown in Figure <ref>.Following the setting in Sec. <ref>, we evaluate the text window size from 5 seconds to 9 seconds, and got the following accuracies:32.1%, 29.6%, 41.5%, 28.2%, 34.4%. We achieved best results with text window size as 7 seconds, and used this in rest of the experiments. Joint Lv-LSTM Model Our final lv-LSTM model combines the best vision and language models: V-CNN-LSTM and L-Char-LSTM.For the vision and language models, we can extract features F_v and F_l from V-CNN-LSTN and L-Char-LSTM, respectively. Then we concatenate F_v and F_l, and feed it into a 2-layer MLP. The completed model is shown in Figure <ref>. We expect there is room to improve this approach, by using more involved representations, e.g., Bilinear Pooling <cit.>, Memory Networks <cit.>, and Attention Models <cit.>; this is future work. § EXPERIMENTS AND RESULTS Training DetailsIn development and ablation studies, we use train and val splits of the data from NALCS to evaluate models in Section <ref>. For the final results, models are retrained on the combination of train and val data (following major vision benchmarks e.g. PASCAL-VOC and COCO), and performance is measured on the test set. We separate the highlight prediction to three different tasks based on using different input data: videos, chats, and videos+chats. The details of dataset split are in Section <ref>. Our code is implemented in PyTorch. To deal with the large number of frames total, we sample only 5k positive and 5k negative examples in each epoch.We use batch size of 32 and run 60 epochs in all experiments. Weight decay is 10^-4 and learning rate is set as 10^-2 in the first 20 epochs and 10^-3 after that. Cross entropy loss is used.Highlights are generated by fans and consist of clips. We match each clip to when it happened in the full match and call this the “highlight clip” (non-overlapping).The action of interest (“kill”, “objective control”, etc.) often happens in the later part of a highlight clip, while the clip contains some additional context before that action that may help set the stage.For some of our experimental settings (Table <ref>), we used a heuristic of only including the last 25% frames in every highlight clip as positive training examples. During evaluation, we used all frames in the highlight clip.Ablation StudyTable <ref> shows the performance of each module separately on the dev set.For the basic L-Char-LSTM and V-CNN models, using only the last 25% of frames in highlight clips in training works best. In order to evaluate the performance of L-Char-LSTM model, we also train a Word-LSTM model by tokenizing all the chats and only considering the words that appeared more than 10 times, which results in 10019 words. We use this vocabulary to encode the words to 1-hot vectors. The L-Char-LSTM outperforms L-Word-LSTM by 22.3%.Test Results Test results are shown in Table <ref>. Somewhat surprisingly, the vision only model is more accurate than the language only model, despite the real-time nature of the comment stream. This is perhaps due to the visual form of the game,where highlight events may have similar animations.However, including language with vision in the lv-LSTM model significantly improves over vision alone, as the comments may exhibit additional contextual information.Comparing results between ablation and the final test, it seems more data contributes to higher accuracy. This effect is more apparent in the vision models, perhaps due to complexity. Moreover, L-Char-LSTM performs better in English compared to traditional Chinese. From the numbers given in Section <ref>, variation in the number of chats in NALCS was much higher than LMS, which one may expect to have a critical effect in the language model. However, our results seem to suggest that the L-Char-LSTM model can pickup other factors of the chat data (e.g. content) instead of just counting the number of chats. We expect a different language model more suitable for the traditional Chinese language should be able to improve the results for the LMS data. § CONCLUSIONWe presented a new dataset and multimodal methods for highlight prediction, based on visual cues and textual audience chat reactions in multiple languages. We hope our new dataset can encourage further multilingual, multimodal research. § ACKNOWLEDGMENTS We thank Tamara Berg, Phil Ammirato, and the reviewers for their helpful suggestions, and we acknowledge support from NSF 1533771.emnlp_natbib | http://arxiv.org/abs/1707.08559v1 | {
"authors": [
"Cheng-Yang Fu",
"Joon Lee",
"Mohit Bansal",
"Alexander C. Berg"
],
"categories": [
"cs.CL",
"cs.AI",
"cs.CV",
"cs.LG",
"cs.MM"
],
"primary_category": "cs.CL",
"published": "20170726174438",
"title": "Video Highlight Prediction Using Audience Chat Reactions"
} |
STN-OCR: A single Neural Network for Text Detection and Text Recognition C. Conti^4,5 December 30, 2023 ========================================================================§.§.§ Abstract In this paper, we describe an algorithm that, for a smooth connected curve X over a field k with normal completion having arithmetic genus p_a(X), a finite locally constant sheaf A on X_ of abelian groups of torsion invertible in k, represented by a smooth curve with normal completion having arithmetic genus p_a( A) and degree n over X, computes the first étale cohomology ^1(X_k^,, A) and the first étale cohomology with proper support ^1_c(X_k^,, A) as sets of torsors, in arithmetic complexity exponential in n^log n, p_a(X), and p_a( A). This is done via the computation of a groupoid scheme classifying the relevant torsors (with extra rigidifying data).§.§.§ Acknowledgements This paper is in part based on Chapter 3 of the author's dissertation, which was funded by the Netherlands Organisation for Scientific Research (project no. 613.001.110), and the author thanks his supervisors Bas Edixhoven and Lenny Taelman for their guidance during the author's PhD candidacy. The author also thanks the Max-Planck-Institut für Mathematik in Bonn for their support during the production of this paper.§ INTRODUCTION The motivating question for this paper is the following;this question is posed e.g. by Poonen, Testa, and van Luijk in <cit.>.Is there an algorithm that takes an algebraic variety X over a field k, and a positive integer n invertible in k, and computes (k^/k)-modules isomorphic to the étale cohomology groups ^q(X_k^,, Z/nZ) for q=0,1,…,2 X? We assume affine schemes of finite presentation over some base ring R to be given by generators and relations over R, and we assume X to be given by a gluing datum of affine varieties over k. The output will be given as a pair (l,X) of a Galois extension l/k and a finite (l/k)-set X. <cit.> guarantees that in the situation of this question, the groups ^q(X_k^,, Z/nZ) are indeed finite.The existence of an algorithm as in the question computing the étale cohomology groups in time polynomial in n for a fixed variety over Q implies, via the Lefschetz trace formula <cit.> and by the argument of <cit.>, the computation of the number of F_q-points of some fixed finite type scheme X (over Z) in time polynomial in log q. Here, we note that the problem of computing #X( F_q) has many efficient solutions in practice, see e.g. <cit.>, <cit.>, and <cit.>;however, none of them run in time polynomial in the characteristic of the finite field. One other application of a positive answer to the question above is the computation of Néron-Severi groups by Poonen, Testa, and van Luijk in <cit.> using the computation of the étale cohomology groups.<ref> is already known to have a positive answer. In 2015, Poonen, Testa, and van Luijk, in the aforementioned article <cit.>, showed that the étale cohomology groups are computable if X is a smooth, projective, geometrically irreducible variety over a field of characteristic 0. Later that year, Madore and Orgogozo <cit.> showed that they are computable for any variety over any field, and, assuming computations with constructible sheaves can be performed, with coefficient sheaf any constructible sheaf of abelian groups (of torsion invertible in the base field). However, both of these results are fundamentally merely computability results, without any bounds on the complexity, even for a fixed instance.So a natural extension of <ref> is (in addition to allowing more general coefficients) to also ask for explicit upper bounds for the complexity;beyond the classical case of smooth curves with constant coefficients, the author doesn't know of any such result. In this paper, we will describe an algorithm computing, for smooth connected curves, the first étale cohomology group (proper support or not) with coefficients in a finite locally constant sheaf of abelian groups (of torsion invertible in k), together with theoretical upper bounds for the complexity.We will assume the field k is given together with black box field operations (see <ref> for more details) and we measure the complexity only in the number of field operations performed. While this is a good approximation for the time complexity in case k is finite, for infinite k this is usually not the case because of coefficient size growth. Algorithms will be deterministic (except for the use of the black boxes);for an actual implementation of the algorithm to be presented, it may be more efficient in practice to use randomised algorithms. Moreover, the choice of algorithms is motivated by their theoretical worst-case complexities;for an actual implementation, it may be significantly more efficient in practice to use different algorithms than the ones used in this paper.With this in mind, let us state this paper's main theorem, deferring the description of the in- and output mainly to <ref> and <ref>.There exist an algorithm that takes as input a smooth connected curve X over k, and a (curve representing a) finite locally constant sheaf A of abelian groups of degree n over X with n invertible in k, and return as output ^1(X_k^, A|_X_k^) (resp. ^1_c(X_k^, A|_X_k^)) as (k^/k)-modules in a number of field operations exponential in n^log n, p_a(X), and p_a( A), where p_a denotes the arithmetic genus of the normal completion. More precise (and slightly more general) versions of this theorem will be given in <ref> and <ref>.§ THE GEOMETRIC IDEA BEHIND THE ALGORITHM Let X be a smooth connected curve over a field k, and let G be a finite locally constant sheaf of groups. Then the set ^1(X_k^, G|_X_k^), resp. ^1_c(X_k^, G|_X_k^), is the set of isomorphism classes of G-torsors on X_k^, resp. the set of isomorphism classes of j_! G-torsors on X_k^. Here, jX →X is the open immersion of X into its normal completion X. If G is a sheaf of abelian groups, a priori we have two possible definitions of j_! G;one arising from viewing j_! as the left adjoint of j_* on the category of sheaves of groups, and one arising from viewing j_! as the left adjoint of j_* on the category of sheaves of abelian groups. Let us call these j^G_!G and j^A_!G for now. There is a natural map j^G_!G → j^A_!G, which induces an isomorphism on stalks since j is an open immersion (and direct sums of zero, resp. one object in the category of groups and that of abelian groups have the same underlying sets). Hence j^G_!G = j^A_!G, so there is no confusion possible if we just write j_!G in this case, like we did above. Now, in short, the idea behind the algorithm is to compute a moduli scheme of torsors with some additional structure (which is to be specified) and use this moduli scheme to compute the first cohomology. The moduli space in question is constructed in <ref>.We describe this idea in some more detail below, using the language of stacks.Let T be the stack on (/X)_ of G-torsors, resp. the stack on (/X)_ of j_! G-torsors;note that this stack is presented by the group(oid) scheme G → X, resp. j_! G →X. Let f be the structure morphism X → k, resp. the structure morphism X→ k, and let p denote the morphism from the big étale topos to the small étale topos for which p_* is the restriction to the small site. Note that for this p, the functor p^-1 is the espace étalé functor. Let f_,* and f_,* denote the big and small pushforward, respectively.We then have a stack f_,*p_*T = p_*f_,* T on ( k)_, to which we can attach the sheaf π_0(f_,*p_*T) on ( k)_, and a morphism f_,*p_*T →π_0(f_,*p_*T) of stacks on ( k)_. The Galois set to be computed now corresponds to the sheaf π_0(f_,*p_*T) on ( k)_, or in other words, to the étale k-scheme p^-1π_0(f_,*p_*T).We show in <ref> that the diagonal of f_,* T is representable and finite étale, which simply means that for any k-scheme S and any two objects X,Y of f_,* T(S), the corresponding sheaf _f_,* T(S)(X,Y) on (/k)_ is representable by a finite étale S-scheme.In <ref> we compute a groupoid scheme R ⇉ U with R and U affine schemes of finite type over k, together with an obvious (non-explicit) morphism [ U/ R] → f_,* T of stacks on (/k)_. There, we also show that p^-1p_*[ U/ R] → p^-1f_,*p_*T is an equivalence (after some purely inseparable base change, but we ignore this technical point for now), and that the morphisms R ⇉ U are smooth and have geometrically irreducible fibres.Hence we are (after some purely inseparable base change) in the situation of the following proposition.Let T be a stack on (/k)_ of which the diagonal is representable and finite étale. Let R ⇉ U be a groupoid scheme such that both morphisms R → U are smooth and have geometrically connected fibres, and such that R and U are of finite type over k. Let [ U/ R] → T be a morphism of stacks on (/k)_ such that p^-1p_*[ U/ R] → p^-1p_*T is an equivalence, or in other words, such that for each separable extension l/k, the functor [ U/ R](l) → T(l) is an equivalence. Then the map U(k^) →π_0( T)(k^) is a (k^/k)-equivariant surjection, and factors through an isomorphism π_0( U) →π_0( T).If in addition the morphisms R → U have geometrically irreducible fibres, then the connected components of U_k^ are irreducible.First note that the equivalence p^-1p_*[ U/ R] → p^-1p^*T induces a surjective (k^/k)-equivariant map U(k^) →π_0( T)(k^).Let x ∈ U(k^) and let jU → U_k^ be the open immersion of the connected component U containing x into U_k^. Moreover, let fU → k^ denote the structure morphism, and let pU_k^→ U be the projection morphism. Let Y ∈ U( U) denote the “universal object”; i.e. the object corresponding to the identity map on U.Define Y_1 = j^-1p^-1 Y, Y_2 = f^-1 x^-1 Y ∈ U(U), and consider their images in T(U). Then _ T(U)(Y_1,Y_2) is representable by a finite étale U-scheme by assumption.Moreover, it is surjective as by construction _ T(U)(Y_1,Y_2)( x) is non-empty and U is connected. Hence for any point x'∈ U(k^), the set _ T(U)(Y_1,Y_2)(x') is non-empty as well. Therefore the morphism U(k^) →π_0( T)(k^) factors through a surjective (k^/k)-equivariant map π_0( U)(k^) →π_0( T)(k^). In other words, the morphism π_0( U) →π_0( T) of sheaves on ( k)_ is surjective.Denote the morphisms R → U by α and ω. As α and ω have geometrically connected fibres, it follows that the morphism π_0( U) →π_0( T) is an isomorphism; if x,x' ∈ U(k^) are isomorphic, then x' ∈α*[big]ω^-1( x), hence x,x' lie in the same geometric connected component of U.Finally, if α and ω have geometrically irreducible fibres, then the same argument implies that every geometric connected component of U is irreducible. As a corollary, we see that in <ref> any set X of points l_i → U (with l_i/k finite algebraic) such that every connected component of U contains a point X induces a finite cover of π_0( T). We describe in <ref> how to use such a set X to compute π_0( T).§ PRELIMINARIESIn this paper we will mainly consider generic field algorithms; i.e. algorithms that take a finite number of bits and a finite number of a field k, which are only allowed to operate on the field elements through a number of black box operations, and, aside from the black box operations, are deterministic. The assumptions that follow here are essentially the assumptions as mentioned in <cit.>.First, we assume the constants 0 and 1 (in k) and the characteristic exponent p (in Z) are given. Moreover, we assume the imperfectness degree p^e of k to be finite, and that k^1/p/k is given explicitly as a finite k-algebra (i.e. as a k-vector space with given unit and multiplication table).In this paper these black box operations are: * =, which takes x,y ∈ k, and returns 1 if x=y, and 0 if x ≠ y;* -, which takes x ∈ k, and returns -x;* ·^-1, which takes x ∈ k, and returns nothing if x=0, and x^-1 if x ≠ 0;* +, which takes x,y ∈ k, and returns x+y;* ×, which takes x,y ∈ k, and returns xy;* ·^1/p, which takes x ∈ k, and returns x^1/p∈ k^1/p;* F, which takes a polynomial f ∈ k[x], and returns its factorisation into irreducibles in k[x].For any field finitely generated over a finite field or Q, there are algorithms for each of the above black box operations, however, the most efficient implementations of the factorisation algorithm for finite fields are randomised. To such a generic field algorithm we attach a number of functions (from the set of inputs to N). * The bit-complexity N_; for an input I, the number N_(I) is the number of bit-operations the algorithm performs when given I.* The arithmetic complexity N_; for an input I, the number N_(I) is the number of black box operations the algorithm performs when given I.We will usually not mention the bit-complexity of the algorithms in this paper;in all cases, the bit-complexity will be small compared to the arithmetic complexity. As is customary, as a measure of size for inputs, we take the pair (b,f), where b is the number of bits in the input, and f is the number of field elements in the input;so for Φ a function from the set of inputs to N, we will denote by Φ(b,f) the maximum of the Φ(I) with I ranging over all the inputs with at most b bits and f field elements.We note that a lot of linear algebraic operations, like matrix addition, matrix multiplication, computation of characteristic polynomial, and by extension, reduced row echelon form, rank, kernels, images, quotients, etc. can all be performed in arithmetic complexity polynomial in the size of the input.By <cit.>, the primary decomposition of a finite k-algebra A can also be computed in arithmetic complexity polynomial in [A:k], and if k is perfect, the same holds for the computation of nilradicals. In fact, in our case <cit.> computes an l-basis (and therefore a k-basis) for the nilradical of A ⊗_k l (where l = k^1/p^*log_p [A:k]), and therefore also a k-basis for the nilradical of A, in arithmetic complexity polynomial in [A:k]^e+1.Moreover, using the criteria that a reduced finite k-algebra A is separable if [A:k] < p, and if and only if A is spanned over k by t_i^p for t_i any k-basis for A, one can compute separable closures of k in finite field extensions l in arithmetic complexity polynomial in [l:k], using the obvious recursive algorithm.By <cit.> we have algorithms which compute for a finite field extension l/k the extension l^1/p/l and the operations listed above; aside from the computation of l^1/p/l, that of characteristic roots, which have arithmetic complexity polynomial in [l:k]^e+1, and that of factorisation, which has arithmetic complexity polynomial in [l:k]^e+1 and the degree of the polynomial to be factored, every operation has arithmetic complexity polynomial in [l:k]. Moreover, l has the same characteristic exponent and imperfectness degree as k.Now consider the purely transcendental extension k(x)/k. We present its elements by pairs of polynomials;for f,g ∈ k[x] we set the height of f/g to be h(f/g) = max( f, g). Then note that for k(x)/k, we have k(x)^1/p = k^1/p(x^1/p) and therefore an obvious k(x)-basis for k(x)^1/p, and we can compute the listed operations for elements of k(x) of height at most h in arithmetic complexity polynomial in h(x). (Again, with the exceptions of characteristic roots, which has arithmetic complexity polynomial in h(x)^e+2 and polynomial factorisation, which has arithmetic complexity polynomial in h(x)^e+2 and the degree of the polynomial to be factored, see e.g. <cit.>.)As is customary, we will use the standard big-oh notation when bounding complexities; moreover, we will use O(x,y) as a shorthand for O*[big]max(x,y).§ PARAMETRISING MORPHISMS OF MODULESWe will use the following characterisation of isomorphism classes of vector bundles over P^1_k with k a field. Let k be a field, and let E be a vector bundle on P^1_k. Then there exists an up to permutation unique finite sequence (a_i)_i=1^s of integers such thatE ⊕_i=1^sO_ P^1_k(a_i).This motivates the following definition. Let S be a scheme, and let a be a finite sequence of integers of length s. The standard module of type a on S is the O_ P^1_S-moduleO_ P^1_S(a) = ⊕_i=1^sO_ P^1_S(a_i).So every vector bundle E over P^1_k is isomorphic to a standard module over k, say of type a; in this case, we simply say that E has type a.Let, for finite sequences a,b of integers, of lengths s,t, respectively, H_a,b define the functor ^→ sending S to _ O_ P^1_S*[big] O_ P^1_S(b), O_ P^1_S(a). Moreover, let N(a,b) = ∑_i=1,j=1^s,tmax(a_i-b_j+1,0).Then the functor H_a,b is representable by A^N(a,b):in fact, as_ O_ P^1_S*[big] O_ P^1_S(b_j), O_ P^1_S(a_i) =O(S)[x,y]_a_i-b_jfunctorial in S, we get an identification_ O_ P^1_S*[big] O_ P^1_S(b), O_ P^1_S(a) = *[Big]M ∈_s × t*[big] O(S)[x,y] : M_ij∈ O(S)[x,y]_a_i - b_j,and under this identification, all the relevant operations on morphisms of standard modules (i.e. identity map, composition, direct sum, tensor product, dual, exterior powers) correspond to their usual counterparts on matrices. In particular, if these operations are viewed as operations on the representing scheme A^N(a,b), then the degrees of the polynomials defining them are as expected.To an element of H_a,b(S), one way to give its fibre at 0 ∈ P^1_S is by substituting (0,1) for (x,y), and one way to give its fibre at ∞∈ P^1_S is by substituting (1,0) for (x,y). Moreover, a way to give the first infinitesimal neighbourhood at ∞∈ P^1_S is by substituting x = 1 and setting y^2 = 0.§ TORSORSA curve over a field k in this paper is a separated k-scheme of finite type, of pure dimension 1 over k.Consider the following situation.Let k be a field, let fX → k be a smooth connected curve, let iZ → X be a closed immersion, let jU → X be its open complement, and let G be a finite locally constant sheaf of groups on U_. We wish to compute R^1f_*j_!G (or equivalently, ^1(X_k^,,j_! G) as a Galois set) under some minor conditions, and as stated before, we will do this by reduction to a computation with standard modules. §.§ j_! G-torsors and recollement We first recall recollement. Let X be a scheme, let iZ → X be a closed immersion, and let jU → X be its open complement.Define the category _Z,U(X_) as follows. The set of objects of _Z,U(X_) is the set of triples ( F_Z, F_U,ϕ) of a sheaf F_Z on Z_, a sheaf F_U on U_, and a morphism ϕ F_Z → i^-1j_*F_U. For objects ( F_Z, F_U,ϕ) and ( F'_Z, F'_U,ϕ') of _Z,U(X_), the set of morphisms from ( F_Z, F_U,ϕ) to ( F'_Z, F'_U,ϕ') is the set of pairs (f_Z,f_U) of a morphism f_ZF_Z → F'_Z and a morphism f_UF_U → F'_U such that the following diagram commutes.F_Z [d,"ϕ"'] [r,"f_Z"] F'_Z [d,"ϕ'"] i^-1j_*F_U [r,"i^-1j_*(f_U)"'] i^-1j_*F'_U Let X be a scheme, let iZ → X be a closed immersion, and let jU → X be its open complement.Then the functor (X_) →_Z,U(X_) sending F to *[big]i^-1 F,j^-1 F,i^-1(υ), where υ F → j_*j^-1 F is the unit map of the adjoint pair (j^-1,j_*) of functors, is an equivalence of categories, and a quasi-inverse _Z,U(X_) →(X_) is given by sending ( F_Z, F_U,ϕ) to i_* F_Z ×_i_*(ϕ),i_*i^-1j_*F_U,υ j_*F_U, where υ j_*F_U → i_* i^-1 j_*F_U is the unit map of the adjoint pair (i^-1,i_*) of functors. Note that the functor i^-1j_* is left exact, hence commutes with finite limits. Let T denote the category of j_! G-torsors on X_, and let T_Z,U denote the category of which the objects are pairs ( F,s) of a G-torsor F on U_, and a section s ∈ i^-1j_* F(Z), and in which the morphisms ( F,s) → ( F',s') are the morphisms fF → F' such that i^-1j_*(f) sends s to s'. Let X be a scheme, let iZ → X be a closed immersion, and let jU → X be its open complement. Let G be a sheaf of groups on U_. The rule attaching to a j_! G-torsor F on X_ the pair *[big]j^-1 F,i^-1(υ), where υ F → j_*j^-1 F denotes the unit map of the adjoint pair (j^-1,j_*) of functors, defines an equivalence T → T_Z,U of categories.First, note that the sheaf j_!G is under recollement equivalent to the triple (1, G,1).Now giving a j_! G-action ρ on a sheaf F on X_ is equivalent to giving (i^-1 F,j^-1 F,i^-1(υ)) together with an action of G on j^-1 F; the commutativity of1 × i^-1 F [d,"1 ×i^-1(υ)"'] [r,"ρ_Z"] i^-1 F [d,"i^-1(υ)"] i^-1j_*G × i^-1j_*j^-1 F [r,"i^-1j_*(ρ_U)"'] i^-1j_*j^-1 Fis automatic since both ρ_Z and i^-1j_*(ρ_U) are group actions. (Of course, one can also deduce this equivalence by noting that a morphism j_! G →( F) is equivalent to a morphism G → j^-1( F) = ( F_U).)Now F is a j_! G-torsor if and only if the map j_! G × F → F × F given on local sections by (g,s) ↦ (s,gs) is an isomorphism, and F locally has a section. This is equivalent to the following. * i^-1 F is the terminal sheaf on Z_; therefore i^-1(υ) is an element of i^-1j_*F(Z), and it follows that the given rule indeed defines a functor;* j^-1 F is a G-torsor on U_,so the given rule defines an equivalence, as desired.§.§ Pushforward and normalisation Next, we consider a description of the pushforward of a finite locally constant sheaf along certain open immersions. This is mostly well-known, but the author doesn't know of a reference, so proofs are included here for completeness.Recall that, for a scheme X, the category of sheaves on X_ is equivalent to that of algebraic spaces étale over X. Quasi-inverses are given by the functor sending an algebraic space étale over X to its functor of points, and the functor sending a sheaf on X_ to its espace étalé. By descent, finite locally constant sheaves on X_ are precisely those of which the espace étalé is a finite étale X-scheme.Let X be a scheme, and let jU → X be a quasi-compact open immersion such that the normalisation of X in U is X. Let F be a finite locally constant sheaf on U_, or equivalently, a finite étale U-scheme. Let F be the normalisation of X in F. Then for all étale X-schemes T, we have j_* F(T) =F(T) functorial in T.First of all, note that we may restrict ourself to étale X-schemes T that are affine, and therefore to quasi-compact separated étale X-schemes T. So let T be an étale quasi-compact separated X-scheme. Let T be the normalisation of X in T, and let U ×_X T be the normalisation of X in U ×_X T. Then j_* F(T) =F(U ×_X T), and we have a map F(U ×_X T) → F(U ×_X T). Since for every Y-morphism U ×_X T→ F, the composition with U ×_X T →U ×_X T factors through F (as F is a finite étale X-scheme), it follows that F(U ×_X T) =F(U ×_X T).Now note that since normalisation commutes with smooth base change (see e.g. <cit.>), it follows that the normalisation of T in U ×_X T is simply T. Therefore U ×_X T→ T is an isomorphism, and we have F(U ×_X T) =F(T) =F(T), as desired.Let X be a scheme, and let jU → X be a quasi-compact open immersion such that the normalisation of X in U is X. Then for all finite sets F, we have j_* F = F.Let k be a field, let X be a k-scheme of finite type, and let jU → X be an open immersion such that the normalisation of X in U is X. Let F be a finite locally constant sheaf on U_, or equivalently, a finite étale U-scheme. Then j_*F is representable by an étale, quasi-compact, separated X-scheme.First note that by <cit.> j_*F is constructible, i.e. of finite presentation as an X-space.Note that F is finite locally constant, so F × F is the disjoint union of the diagonal and its complement, inducing a morphism F × F → Z/2Z such that the equaliser with the constant map with value 0 is the diagonal. Applying the left exact functor j_* to this gives a morphism j_*F × j_*F → j_* ( Z/2Z) =Z/2Z such that the equaliser with the constant map with value 0 is the diagonal. Therefore j_*F is separated as an X-space.It follows by <cit.> that j_*F is representable by an étale, quasi-compact, separated X-scheme.Let k be a field, let X be a k-scheme of finite type, and let jU → X be an open immersion such that the normalisation of X in U is X. Let F be a finite locally constant sheaf on U_, or equivalently, a finite étale U-scheme. Let F be the normalisation of X in F. Then F is the normalisation of X in j_*F.First note that we have a canonical morphism j_* F → F corresponding to the identity section of j_* F. Let j_* F → Y → X be a factorisation with Y → X integral. As F is the normalisation of X in F, it follows that there exists a unique morphism F→ Y such that the diagramF [d] [r] F[d] [dl] Y [r] Xcommutes. We show that this morphism also makes the diagramj_*F [r] [d] F[dl] Ycommute. Let T be an étale quasi-compact separated X-scheme, and consider the following diagram.j_*F(T) [r] [d] F(T) [dl] Y(T)As the normalisations of X in T and U ×_X T are equal, as in the proof of <ref>, the commutativity of this diagram is equivalent to the commutativity of the following one.F(U ×_X T) [r] [d] F(U ×_X T) [dl] Y(U ×_X T)It follows that the commutativity of (<ref>) holds when restricted to X_.Therefore, applying this to the identity section on j_*F, it follows that (<ref>) itself commutes. By Zariski's Main Theorem, we have the following. The canonical morphism j_* F → F is an open immersion identifying j_*F with the étale locus of F over X.The étale locus V of F over X is open in F and étale over X, therefore factors through j_*F. By maximality of V we get j_*F = V.§.§ Galois actions on finite locally constant sheaves Let X be a scheme, and let Γ be a group acting on X. Then recall that a Γ-sheaf on X_ is a sheaf F on X_ of which the espace étalé is a Γ-equivariant X-space.Let X be a connected scheme, let Γ be a finite group, and let fY → X be a finite étale connected Galois cover with Galois group Γ. Note that pullback of sheaves defines an equivalence from the category of sheaves on X_ to that of Γ-sheaves on Y_. A quasi-inverse is given in terms of sheaves by sending F to the sheaf of Γ-invariants of f_*F; in terms of espaces étalés, it sends an algebraic space Z étale over Y to the quotient Γ\ Z.If G is a finite locally constant sheaf of groups on X_ such that f^-1 G is constant, let G be the group of connected components of f^-1 G, and note that Γ acts on G by automorphisms. Therefore we see that a finite locally constant sheaf on X_ with G-action corresponds to a Γ- and G-equivariant finite étale Y-scheme.Let us now apply this to the following situation.Let k be a field. Suppose we have a finite group Γ, and a diagram of schemes of finite type over kV [d,"g"'] [r,"j'"] Y [d,"f"] W [d,"h"] [l,"i'"'] U [r,"j"'] X Z [l,"i"]where U and X are connected, g is finite étale Galois with Galois group Γ, Y is the normalisation of X in V, W = Y ×_X Z, and j is the open complement of i. Let G be a finite locally constant sheaf of groups on U such that g^-1 G is constant, say with group of connected components G. Let T_Z,U be as in the previous section, and let T_W,Y^Γ be the category of which the objects are pairs ( F,s) of a Γ-equivariant G-torsor F on Y_, and a Γ-equivariant section s ∈ (i')^-1 F(W), and in which the morphisms ( F,s) → ( F',s') are the Γ-equivariant morphisms fF → F' such that (i')^-1(f) sends s to s'.In <ref>, the rule attaching to a pair ( F,s) of a G-torsor F and a section s ∈ i^-1j_* F(Z) the pair (j'_*g^-1 F,s) defines an equivalence T_Z,U→ T_W,Y^Γ of categories.First note that giving a G-torsor F on U_ is equivalent to giving the Γ-equivariant G-torsor g^-1 F on V_. Moreover, giving the section sZ → i^-1j_*F is the same as giving a Γ-invariant section Z → i^-1j_* g_* g^-1 F = i^-1f_*j'_*g^-1 F = h_* (i')^-1 j'_* g^-1 F, where the last step uses proper base change. This is the same as giving a Γ-equivariant section W → (i')^-1 j'_* g^-1 F.Now j'_*g^-1 F is a Γ-equivariant G-pseudotorsor which étale locally has a section, i.e. a Γ-equivariant G-torsor. Therefore giving the pair (g^-1 F,s) is equivalent to giving (j'_*g^-1 F,s), as desired. §.§ Stacks of torsors Let S be a scheme. A topologically finite étale S-scheme T is a morphism T → S that factors as a composition T → T' → S with T' → S finite étale and T → T' a universal homeomorphism.Let fX → S be a proper smooth curve, let iY → X be a closed immersion, topologically finite étale over S, and let jU → X denote its open complement. Write h = fi and g = fj. Let G be a finite group; if Y is non-empty, we also assume that the order of G is invertible on S.Let T denote the fppf stack of G-torsors on U_; i.e. its objects are pairs (T, F) of an S-scheme T and a G-torsor F on (U ×_S T)_, and the morphisms (T, F) → (T',F') are the pairs of a morphism ϕ T → T' and an isomorphism ϕ^-1 F' → F. We show that T has a representable and finite étale diagonal, or equivalently, the relevant Isom-sheaves are representable by finite étale schemes.Without loss of generality, and to ease notation a bit, we will only consider the Isom-sheaves on S. More precisely, let F and F' be G-torsors on U_, and let I denote the sheaf on U_ sending ϕ T → U to the set _T(ϕ^-1 F,ϕ^-1F') of isomorphisms of G-torsors. We denote by g_,* the big pushforward functor U_→ S_. The sheaf g_,* I on S_ is representable by a finite étale S-scheme.By the theory of Hilbert schemes, the fppf sheaf g_big,* I is representable by an algebraic space, locally of finite presentation over S, since both F and F' are representable by finite étale algebraic spaces over U. Moreover, we easily see that it is formally étale over S, therefore étale over S. Hence it is representable by the espace étalé of (g_big,* I)|_S_ = g_*( I|_U_).Now we note that I|_U_ is a G-torsor on U_, so its pushforward under g is finite locally constant by <cit.>; this uses the additional assumption on the order of G if Y is non-empty. Hence g_big,* I is representable by a finite étale S-scheme. In addition, let Γ be a finite group acting on G, and on X over S, such that Y (and therefore U) is stable under Γ, and let kZ → U be a closed immersion stable under Γ, and let lV → U be its open complement.Let T' denote the fppf stack of Γ-equivariant l_!G-torsors on U_; i.e. its objects are pairs (T, F) of an S-scheme T and a Γ-equivariant l_!A-torsor F on (U ×_S T)_, and the morphisms (T, F) → (T',F') are the pairs of a morphism ϕ T → T' and a Γ-equivariant isomorphism ϕ^-1 F' → F. We show that T' too has a representable and finite étale diagonal.Again, without loss of generality, we will only consider the relevant Isom-sheaves on S. Let F and F' be Γ-equivariant l_! G-torsors on U_, and let I' denote the sheaf on U_ sending ϕ T → U to the set of Γ-invariant isomorphisms ϕ^-1 F →ϕ^-1 F' of l_!G-torsors. The sheaf g_,* I' on S_ is representable by a finite étale S-scheme.By the last step in the proof of <ref>, we see that we can write g_,* I' as a finite limit of finite étale S-schemes, which therefore is finite étale over S as well. Finally, let k be a field, let fX → k be a smooth connected curve, let iZ → X be a closed immersion, and let jU → X denote its open complement. Let G be a finite locally constant sheaf of groups on U_, let gV → U be a finite étale Galois cover with finite Galois group Γ such that g^-1 G is constant, say with Γ-module of connected components G, and let Y be the normal completion of V; by replacing k by a finite purely inseparable extension if necessary, we may assume Y is smooth over k. Let i'W → Y be the base change of i to Y, let j'V → Y be the canonical open immersion, and note that Z and W are automatically topologically finite étale over k.Now in <ref>, under the additional assumption that X and Y are curves that have smooth normal completions, we see that the stack of j_! G-torsors is equivalent to that of Γ-equivariant j'_! G-torsors by <ref>, which satisfies the condition of <ref> by the above. We will take this as the starting point of our computation in the next section.§ COMPUTATION OF A GROUPOID SCHEMEWe will consider the following situation.Let k be a field. Suppose we have a finite group Γ, and a diagram of connected curves over kV [r] [d,"g"'] Y [r] [d,"f"]Y[d] U [r] [d] X [r] [d]X[d] P^1_k - S_0 - S_∞[r] P^1_k - S_∞[r] P^1_kin which: * S_0 ∈*0,∅; S_∞∈*∞,∅;* all squares are cartesian;* X→ P^1_k is finite locally free;* V → U is finite étale Galois with Galois group Γ;* Y is the normal completion of V.Assume that X and Y are smooth over k.Let G be a finite locally constant sheaf of groups on U_ such that g^-1 G is constant, say with group of connected components G. We assume the situation above is given by the smooth curves X and Y given using the description in <ref>, the finite group Γ and its action on Y/X, the group G with Γ-action, and the choice of sets S_0 and S_1;note that we haven't explained yet how to decide whether such an input is valid, but the characterisations in this section and the introduction of the next section will allow us to do so. We wish to compute the Galois set ^1(X,j_! G), keeping <ref> in mind (after base change to k^). We therefore would like to give a description of, for every perfect field extension l of k, the category T(l) of j_! G-torsors on (X_l)_, purely in terms of (commutativity relations between) morphisms of vector bundles over P^1_l and free modules over l, and we can use <ref> to construct a groupoid with the desired properties.By <ref>, T(l) is equivalent to that of Γ-equivariant G-torsors on (Y_l)_, together with a Γ-equivariant section from Y_l×_ P^1_l S_0. (We note that in case S_0 = ∅, the empty morphism is Γ-equivariant.) By taking normal completions, we obtain the following. Let l be a perfect field extension of k. Then the category T(l) is equivalent to that of finite locally free P^1_l-schemes T, smooth over l, together with a Γ-equivariant G-action, a Γ-equivariant morphism T →Y_l and a Γ-equivariant section Y_l×_ P^1_l S_0 → T ×_ P^1_l S_0, such that T ×_ P^1_l ( P^1_l - S_∞) is a G-torsor on (Y_l)_; here, the morphisms are the Γ-equivariant, G-equivariant morphisms of Y_l-schemes. So aside from the conditions “smooth over l” and “T ×_ P^1_l ( P^1_l - S_∞) is a G-torsor on (Y_l)_”, the data in the description above can easily be expressed in terms of morphisms of vector bundles on P^1_l, and the relations in the description above can be easily expressed in terms of commutativity relations between these morphisms. §.§ TorsorsLet us first consider the condition “T ×_ P^1_l ( P^1_l - S_∞) is a G-torsor on (Y_l)_”. To this end, we first express the condition “T is finite locally free over P^1_l of constant rank”, using the fibre of Y_l above 0 ∈ P^1_l; this is not automatic as we didn't assume Y to be geometrically connected. Let S be a scheme, let X be a finite locally free P^1_S-scheme that is smooth over S. Let Y be an X-scheme that is finite locally free over P^1_S, such that Y ×_ P^1_S 0 is finite locally free over X ×_ P^1_S 0 of constant rank n. Then Y is a finite locally free X-scheme of constant rank n.We can check fibrewise on S that X ×_ P^1_S 0 intersects all components of X, from which our claim follows. Since finite locally free modules over an Artinian ring are free, we have the following. Let l be a perfect field extension of k. Then the category of finite locally free Y_l-schemes of constant rank is equivalent to that of finite locally free P^1_l-schemes T together with a morphism T →Y_l and an O(Y_l×_ P^1_l 0)-basis for O(T ×_ P^1_l 0) (morphisms in this category are simply morphisms of Y_l-schemes). Next, we want to express the condition “T ×_ P^1_l ( P^1_l - S_∞) is étale over Y_l” in terms of vector bundles on P^1_l. To this end, we will use the transitivity of the discriminant.First, we recall the definitions of the discriminant and the norm of a finite locally free morphism Y → X. Recall that, for a finite locally free morphism Y → X of schemes, we view O_Y as a (finite locally free) O_X-algebra. Let fY → X be a finite locally free morphism of schemes of constant rank, and let μ be the multiplication map O_Y ⊗_ O_X O_Y → O_Y. The trace form τ_f of f is the morphism O_Y →_ O_X( O_Y,O_X) corresponding to the composition _f μ O_Y ⊗_ O_X O_Y → O_X. The discriminant Δ_f of f is the determinant (over O_X) of the trace form τ_f.Let fY → X be a finite locally free morphism of schemes of constant rank, and let L be a line bundle on Y. The norm _fL of L is the line bundle_ O_X(_ O_X f_*O_Y, _ O_X f_*L).Let fY → X be a finite locally free morphism of schemes of constant rank, and let E and F be finite locally free O_Y-modules of the same constant rank. By <cit.> and the fact that norms (of line bundles) commute with tensor products and duals (see <cit.> and <cit.>), we see that there is a unique isomorphism_ O_X(_ O_X E, _ O_X F) = _ O_X(_f _ O_Y E, _f _ O_Y F)satisfying the following properties. * It is compatible with base change by open immersions.* For any isomorphism α F → E, we have induced isomorphisms_ O_X(_ O_X E, _ O_X F) →_ O_X(_ O_X E)and_ O_X(_f _ O_Y E, _f _ O_Y F) →_ O_X(_f _ O_Y E).Therefore they induce isomorphisms_ O_X(_ O_X E, _ O_X F) →_ O_X(_ O_X E) =G_m,Xand_ O_X(_f _ O_Y E, _f _ O_Y F) →_ O_X(_f _ O_Y E) =G_m,X.These isomorphisms are equal under the given identification. Therefore, we have the following.Let fY → X be a finite locally free morphism of schemes of constant rank, and let E be a finite locally free O_Y-module of constant rank r. Then_ O_X E= _f _ O_Y E ⊗_ O_X (_ O_X O_Y)^⊗ r _ O_X(_ O_X E, O_X)= _f _ O_Y_ O_Y( E,O_Y) ⊗_ O_X*[big]_ O_X(_ O_X O_Y,O_X)^⊗ rUsing the two identifications above, we may now state the transitivity of the discriminant. A proof can be found in e.g. <cit.>. Let fY → X and gZ → Y be finite locally free morphisms of schemes of constant rank, and suppose that g has rank r. ThenΔ_fg = _f Δ_g ⊗Δ_f^⊗ r.Let fY → X and gZ → Y be finite locally free morphisms of schemes of constant rank, and suppose that g has rank r. Then g is étale if and only if we have _ O_X O_Z(_ O_X O_Y)^⊗ r and Δ_fg and Δ_f^⊗ r differ by a unit. Therefore we have the following. Let l be a perfect extension of k. Then the category of finite étale Y_l-schemes is equivalent to the full subcategory of that of finite locally free Y_l-schemes T of constant rank (say r) such that _ O_ P^1_l O_T(_ O_ P^1_l O_Y_l)^⊗ r and Δ_T/ P^1_l and Δ_Y_l/ P^1_l^⊗ r differ by a unit. Note that the condition on the determinants is simply a condition on the types of the standard modules over l isomorphic to O_T and O_Y_l, so if S_∞ = ∅, this gives an expression of the desired form. If S_∞ = ∞, then we use the following instead. Let l be a perfect extension of k, and assume that S_∞ = ∞. Then the category of finite locally free Y_l-schemes étale over Y_l is equivalent to the full subcategory of that of finite locally free Y_l-schemes T of constant rank (say r) such that Δ_T/ P^1_l and Δ_Y_l/ P^1_l^⊗ r differ by a unit times a power of y.It suffices to show that for integers a,b, a map ϕ O_ P^1_l(b) → O_ P^1_k(a) is an isomorphism when restricted to A^1_l if and only if it is given by multiplication by s y^a-b with s ∈ l^×. Since y becomes invertible after restricting to A^1_l, it follows that if ϕ is multiplication by s y^a-b, then ϕ|_ A^1_l is an isomorphism. Conversely, ϕ is multiplication by some f ∈ l[x,y]_a-b, which after restriction becomes the multiplication by f(x,1) map l[x] → l[x]. Since this map is an isomorphism, f(x,1) must be an invertible constant in l[x], i.e. f = s y^a-b for some s ∈ l^×. We are almost ready to express the condition “T ×_ P^1_l ( P^1_l - S_∞) is a G-torsor on (Y_l)_” in terms of vector bundles on P^1_l.Let fY → X be a morphism of schemes, and let G be a finite group acting on Y/X. Then Y is a G-torsor on X if and only if f is flat, surjective, locally of finite presentation, and G acts freely and transitively on geometric fibres.The necessity of the condition is clear. Hence suppose that f is flat, surjective, locally of finite presentation, and G acts freely and transitively on geometric fibres. Then for any geometric point x of S, Y_ x is the trivial G-torsor, hence étale. As the property of being étale is fpqc local on the base, it follows that all fibres of f are étale, and since f is flat and locally of finite presentation, it follows that f is finite étale.Now consider the morphism ϕ G × Y → Y ×_X Y of finite étale Y-schemes given on the functor of points by (g,y) ↦ (gy,y), where the occurring schemes are viewed as Y-schemes via the projection on the second coordinate. Then ϕ is itself finite étale surjective, and as G × Y and Y ×_X Y have the same rank over Y, it follows that ϕ is an isomorphism. After base change with itself, it admits a section, so as f is finite étale, it also follows that Y is a G-torsor, as desired. Let fY → X be a finite étale morphism of schemes of constant rank n, and let G be a finite group of order n acting on Y/X. Then the locus in X where f is a G-torsor is open and closed in X.Consider the locus U in Y ×_X Y on which the morphism G × Y → Y ×_X Y given on the functor of points by (g,y) ↦ (gy,y) is an isomorphism (i.e. where the rank is equal to 1). It is an open and closed subset of Y ×_X Y as this morphism is finite étale. As the rank of f is equal to n, the X-locus where the same morphism is an isomorphism is the image of U in X, and hence is open and closed as well. This locus equals the X-locus where f is a G-torsor, as desired. Therefore we have the following. Let l be a perfect extension of k. Then the category of finite locally free G-equivariant Y_l-schemes T such that T ×_ P^1_l ( P^1_l - S_∞) is a G-torsor on (Y_l)_ is equivalent to the category of finite locally free G-equivariant Y_l-scheme T such that T ×_ P^1_l ( P^1_l - S_∞) is étale, and such that T ×_ P^1_l 0 is a G-torsor on (Y_l×_ P^1_l 0)_. Since in the description of finite locally free Y_l-schemes T, an O(Y_l×_ P^1_l 0)-basis for O(T ×_ P^1_l 0) occurred, in terms of which we can express the condition that T ×_ P^1_l 0 is a G-torsor on (Y_l×_ P^1_l 0)_. §.§ Smoothness at ∞Finally, we consider the condition “T is smooth over l”. If S_∞ = ∅, then this follows automatically from T having to be étale over Y_l, so assume that S_∞ = ∞. As T ×_ P^1_l A^1_l has to be étale over Y_l, it suffices to consider the condition “T is smooth over l at T ×_ P^1_l∞”.To this end, assume that we have a scheme S, a positive integer r, and the structure of an algebra A on O_S^r, given by, for the standard basis e_1,…,e_r on O_S^r, e_ie_i' = ∑_j μ_jii' e_j and 1 = ∑_j ϵ_j e_j. Then the relative differentials Ω_ A/ O_S over S are generated by the e_j, with relations e_i'e_i + e_ie_i' - ∑_j μ_jii'e_j = 0 for all i,i' and ∑_j ϵ_j e_j = 0. Therefore we get a canonical presentation ω_ A/ O_S A^r^2+1→ A^r of the A-module Ω_ A/ O_S, which is compatible with base change. Let l be any extension of k. The category of finite locally free P^1_l-schemes T smooth over l at T ×_ P^1_l∞ is equivalent to that of finite locally free P^1_l-schemes T, together with morphismsiO_T ×_ P^1_l∞^(2)→ O_T ×_ P^1_l∞^(2)^2r,jO_T ×_ P^1_l∞^(2)^2r→ O_T ×_ P^1_l∞^(2)^(2r)^2+2of O_T ×_ P^1_l∞^(2)-modules such that (ω_O_T ×_ P^1_l∞^(2)/l⊕ i)j is the identity on O_T ×_ P^1_l∞^(2)^2r; the morphisms in the latter category are simply the morphisms of P^1_l-schemes.We will first show that T is smooth over l at T ×_ P^1_l∞ if and only there exist i and j as in the proposition.Write B for the ring of global sections of T ×_ P^1_l ( P^1_l - 0), and note that it is a finite locally free l[y]-algebra. Then T ×_ P^1_l∞^(2) =B/y^2B. First suppose that there exist morphisms i(B/y^2B) → (B/y^2B)^2r,j(B/y^2B)^2r→ (B/y^2B)^(2r)^2+2such that for the canonical presentation ω_(B/y^2B)/l (B/y^2B)^(2r)^2+1→ (B/y^2B)^2r of Ω_(B/y^2B)/l as a (B/y^2B)-module, we have (ω_(B/y^2B)/l⊕ i) j = 𝕀. It immediately follows that Ω_(B/y^2B)/l is generated by one element as B/y^2B-module.Conversely, if Ω_(B/y^2B)/l is generated by one element, we let i be a morphism from (B/y^2B) to (B/y^2B)^2r sending 1 to (a lift of) a generator of Ω_(B/y^2B)/l. Hence (ω_(B/y^2B)/l⊕ i) is a surjective morphism to a free B/y^2B-module, so it has a section j, as desired.It remains to show that Ω_(B/y^2B)/l is generated as a B/y^2B-module by one element if and only if T is smooth over l at all points lying over ∞∈ P^1_l. Note thatwe have an isomorphismΩ_B/l⊗_B (B/yB) →Ω_(B/y^2B)/l⊗_B/y^2B (B/yB),and that by Nakayama's lemma, the right hand side (and therefore the left hand side) is generated as a B/yB-module by one element if and only if Ω_(B/y^2B)/l is generated as a B/y^2B-module by one element. Therefore, again by Nakayama's lemma, there exists some f ∈ 1+yB such that Ω_B/l⊗_B B_f is generated as a B_f-module by one element. So the left hand side is a B/yB-module generated by one element if and only if there exists a neighbourhood of T ×_ P^1_l∞ that is smooth over l, which holds if and only if T is smooth over l at all points lying over ∞∈ P^1_l.So now we have a forgetful functor from the category of finite locally free P^1_l-schemes T together with morphisms i and j as in the proposition, to that of finite locally free P^1_l-schemes T smooth over l at T ×_ P^1_l∞, which is essentially surjective by the above, and fully faithful by construction.§.§ Bounds on types In order to construct a groupoid scheme with the desired properties using the above, we first need to bound the number of possible types.Let S be a scheme, let a be a finite sequence of integers, let X be a finite locally free P^1_S-scheme of which the underlying O_ P^1_S-modules is standard of type a, and suppose that X has geometrically reduced fibres over S. Then a is non-positive (i.e. all of its elements are non-positive). By taking a geometric fibre if necessary, we assume without loss of generality that S is the spectrum of an algebraically closed field k. Let X_1, …, X_t be the connected components of X. Then there exist finite sequences a_1,…,a_t such that for all i, the algebra O_X_i is of type a_i. These have the property that their concatenation is equal to a up to a permutation. Hence we assume without loss of generality that X is connected. In this case X is a reduced curve over S, so O_X( P^1_S) =O_X(X) =O_S(S) = k, where π is the structure morphism of X, so we deduce that a is non-positive.Of course, the converse is not true; a counterexample is theO_ P^1_k⊕ O_ P^1_k(-1) ϵ with multiplication given by ϵ^2 = 0. In <ref>, let l be a perfect extension of k, and let T be the normal completion of a j'_!G-torsor on (Y_l)_. Let a be the type of O_Y, let b be the type of O_T, and let s,t be their respective lengths. Then by the above, both a and b are non-negative. As the degree of the finite locally free morphism T → Y_l is equal to #G, we see that t = s ·#G. Moreover, if S_∞ = ∅, then by <ref>, we have ∑_j b_j = #G ·∑_i a_i; so up to permutation, we only have finitely many possibilities for b. So suppose that S_∞ = ∞.Let S be a scheme, let a be a finite sequence of integers, and let X be finite locally free P^1_S-scheme such that O_X is a standard module over S of type a, where a has length s, and such that X is smooth over S. Then X is a family of curves over S of Euler characteristic s + ∑_i a_i.It suffices to check this on geometric fibres, so we may assume that S is the spectrum of an algebraically closed field k. Then_k ^0(X, O_X) - _k ^1(X, O_X)= _k ^0*[big] P^1_k, O_ P^1_k(a) - _k ^1*[big] P^1_k, O_ P^1_k(a)= ∑_i (1+a_i) = s + ∑_i a_i.Let S be a scheme, and let Y → X be a morphism of finite locally free P^1_S-schemes, with O_X and O_Y standard modules over S of respective types a and b, which have respective lengths s,t. Let G be a finite group of order invertible in S acting on Y over X, such that Y ×_ P^1_S A^1_S is a G-torsor over X ×_ P^1_S A^1_S. Then∑_j b_j ≥#G ∑_i a_i - 12 t. It suffices to check this on geometric fibres, so we may assume that S is the spectrum of an algebraically closed field k. As G acts transitively on Y over X, and the order of G is invertible in k, it follows that Y is tamely ramified over X. Therefore the ramification degree of Y over X is at most t, as Y ×_ P^1_k A^1_k is étale over X ×_ P^1_k A^1_k, and Y has degree t over P^1_k. So by the Riemann-Hurwitz formula, we have-2t - 2∑_j b_j ≤ -2 ts s - 2 ts ∑_i a_i + t,as desired (note that t = s ·#G). So therefore, also in the case that S_∞ = ∞, we see that there are only finitely many possibilities for the type b of T. §.§ Computation of the groupoid scheme Now we see that, in <ref>, the description of the category of j_! A-torsors on X_ in terms of vector bundles on P^1_k gives, for each of the (finitely many) possibilities for the type b, a groupoid scheme R_b ⇉ U_b of which R_b and U_b are (explicitly given) closed subschemes of some A^N_k. Let R = ∐_bR_b and U = ∐_bU_b. The groupoid scheme R ⇉ U satisfies the conditions of <ref>. Moreover, for fixed b, every isomorphism class in U_b,k^ has the same dimension.By construction, it remains to check that the isomorphism classes in U_b are irreducible for all finite sequences b of integers. So let a be the type of the underlying P^1_k-vector bundle of Y, and s its length, and let b be a finite sequence of integers, and t its length. Denote the morphisms R_b ⇉ U_b by α_b, ω_b, with α_b sending a morphism to its source, and with ω_b sending a morphism to its target. For this, it suffices to show that for all x ∈ U_b(k^), the geometric fibre H of α_b above x is irreducible, since the image ω_b(H) in U_b,k^ is by definition the isomorphism class of x.Note that, for any k^-scheme S, giving an isomorphism with fixed source (say with underlying Y_S-scheme T) in the groupoid R_b(S) ⇉ U_b(S) is, by transport of structure, the same as giving: * an O_ P^1_S-linear automorphism of O_ P^1_S(b);* an O_Y_S ×_ P^1_S 0-linear automorphism of O_Y_S ×_ P^1_S 0^#A;* if S_∞ = ∞, morphisms iO_T ×_ P^1_S∞^(2)→ O_T ×_ P^1_S∞^(2)^2#A,jO_T ×_ P^1_S∞^(2)^2#A→ O_T ×_ P^1_S∞^(2)^(2#A)^2+2such that (ω_T ×_ P^1_S∞^(2)/S⊕ i)j = 𝕀. In case S_∞ = ∅, we obtain an obvious isomorphism H → H_1 × H_2, and in case S_∞ = ∞, we obtain an obvious map H → H_1 × H_2 × H_3, where * H_1 is the functor sending a k^-scheme S to _ O_ P^1_S*[big] O_ P^1_S(b);* H_2 is the functor sending a k^-scheme S to _ O_Y_S ×_ P^1_S 0*[big] O_Y_S ×_ P^1_S 0^#G;* H_3 is the functor sending a k^-scheme S to the subset of_ O_T ×_ P^1_S∞^(2)( O_T ×_ P^1_S∞^(2), O_T ×_ P^1_S∞^(2)^2#G)of i such that ω_T ×_ P^1_S∞^(2)⊕ i is surjective. First note that using the description of standard modules, we easily see that H_1 is representable by a finite product of factors of the form G_m,k and A^1_k, so therefore by a smooth, irreducible k^-scheme. Moreover, note that H_2 is isomorphic to the functor sending a k^-scheme S to _S(Y_S ×_ P^1_S 0, _#G,S), which, as O_Y_S ×_ P^1_S 0 is finite free over O_S with a given basis functorial in S, is representable by a non-empty open subscheme of A^s(#G)^2_k^. Hence H_2 is a smooth, irreducible k^-scheme as well. Similarly, we see that H_3 is representable by a non-empty open subscheme of A^4t#G_k^, and therefore by a smooth, irreducible k^-scheme.Finally, we show that H is a smooth, irreducible k^-scheme. We do this by showing that H is Zariski locally on H_1 × H_2 × H_3 isomorphic to H_1 × H_2 × H_3 × A^N_k^ for some fixed N.First note that we have a morphism H_3 → A^M_k^ of k^-schemes, which is given on the functor of points by sending i ∈ H_3(S) to the corresponding 4t#G ×*[big]2t(2#G)^2+4t-matrix with coefficients in O(S), with respect to the basis subordinate to both the standard bases and the given k-basis of T ×_ P^1_k∞^(2), so that M = 4t^2*[big](2#G)^3+4#G. So let i ∈ H_3(k^), and view it as a 4t#G ×*[big]2t(2#G)^2+4t-matrix over k^. As this matrix corresponds to a surjective map of k^-vector spaces, there is a 4t#G × 4t#G-minor which is invertible. Let UA^M_k^ be the locus on which this minor is invertible, and let V be the inverse image of U in H_3; V is an open neighbourhood of i.Now let j ∈ H_3(V) be the open inclusion.By construction, the kernel of ω_T ×_ P^1_V∞^(2)⊕ j is free over O_V. Since an O_T ×_ P^1_V∞^(2)-linear section of this map is well-defined up to a unique tuple of elements from this kernel, it follows that the inverse image of H_1 × H_2 × V in H is isomorphic to H_1 × H_2 × V × A^N_k^ for some fixed N that is independent of the choices made. Hence H is a smooth, irreducible k^-scheme, as desired.Finally note that the dimension of H only depends on the type b, and that the induced morphism H → U_b,k^ has finite fibres, so every isomorphism class in U_b,k^ has the same dimension.There is a canonical bijection from π_0( U_k^) to the set of isomorphism classes of G-torsors on X_k^. Moreover, all U_b are equidimensional. Next, note that we now have an obvious algorithm which, given a diagram as in <ref>, computes R and U; the remainder of this section will be devoted to bounding the complexity of this algorithm. We will in the following restrict ourselves to the case in which S_0 = 0 and S_∞ = ∞; the bounds we obtain in this case will also hold in the other cases.Let a be the type of Y, say of length s, and write γ = ∑_i -a_i. Note that by <ref>, γ = s - 1 + p_a(Y), where p_a denote the arithmetic genus. Also note that #Γ≤ s, so by <ref> below, the number of field elements needed to give <ref> is polynomial in s, γ, and #G.First, let us bound the number of possible types b that can occur as the type of an object of U. The logarithm of the number of b that can occur as the type of an object of U is O*[big]s#G log(sγ#G).For convenience, write N = *#G*12s + γ.By <ref>, a possible type b must be non-positive. By <ref>, a possible type b must satisfy ∑_j -b_j ≤#G*[big]1/2s + γ. Such a type corresponds to a unique tuple (c_0,…,c_N) of non-negative integers with ∑_k=0^N c_k = t and ∑_k=0^N kc_k ≤ N by setting c_k to be the number of -b_j equal to k. The number of tuples satisfying the first of these conditions is N+tt≤ (N+t)^t. Next, we will bound the size of R_b, i.e. for the given closed immersion R_b → A^N_k, the number N, the number of polynomials generating the defining ideal, and the degree of these polynomials. Note that bounds for R_b will also hold for U_b. To this end, note that we have the following trivial bound. Let a and b be finite sequences of non-positive integers, of lengths s and t, respectively. Then _k _ O_ P^1_k( O_ P^1_k(a), O_ P^1_k(b)) ≤ st + t ∑_i -a_i. Let a be a finite sequence of non-positive integers, of length s. Then_k _ O_ P^1_k*[big] O_ P^1_k, O_ P^1_k(a) ≤ s + ∑_i -a_i_k _ O_ P^1_k*[big] O_ P^1_k(a), O_ P^1_k(a) ≤ s^2 + s∑_i -a_i_k _ O_ P^1_k*[big] O_ P^1_k(a)^⊗ 2, O_ P^1_k(a) ≤ s^3 + 2s^2∑_i -a_i_k _ O_ P^1_k*[big] O_ P^1_k(a)^⊗ 3, O_ P^1_k(a) ≤ s^4 + 3s^3∑_i -a_i.Therefore, working out everything, which is straightforward but tedious, gives the following.For the given closed immersion R_b → A^N_k, we have N = O*[big]s^4(#G)^4γ, its defining ideal is given by O*[big]s^4(#G)^4γ polynomials, which have degree at most s#G. Note that a polynomial ring in N variables has N+dd monomials of degree at most d; so by the proposition above, we see that the size of the output is*[Big]O*[big]s#G log(sγ#G). Now that we have bounds for the sizes of R and U, we now turn to the degrees of the defining polynomials of the morphisms defining the structure of a groupoid scheme.Recall for this that points of R_b are given by two objects of U_b, together with an O_ P^1-linear map connecting the two objects. So the source and target maps R_b → U_b are induced by projections between their ambient affine spaces. Therefore the affine k-scheme R_b ×_ U_b R_b of finite type is given by O*[big]s^4(#G)^4γ variables, O*[big]s^4(#G)^4γ relations of degree at most s#G. Moreover, the composition map R_b ×_ U_b R_b → R_b forgets the middle object and composes the two O_ P^1-linear maps, so it is given by polynomials of degree at most 2. The obvious algorithm computes the groupoid scheme R ⇉ U given <ref> as input, and has arithmetic complexity *[Big]O*[big]s#Glog(sγ#G). Simply note that every individual coefficient can be computed in arithmetic complexity bounded by a fixed polynomial in s,γ,#G. § GEOMETRIC POINTS AND FIRST COHOMOLOGYNext, we will use the groupoid scheme R ⇉ U to compute ^1(X_,j_! G). We will do this in a slightly more general situation, namely the following (compare with the conditions of <ref>).Let k be a field, let R ⇉ U be a groupoid scheme in which the morphisms R → U are smooth with geometrically irreducible fibres, and with R,U affine and equidimensional, given by at most r polynomials – which are of degree at most d – in at most N variables. Let T be a stack on (/k)_ with representable and finite étale diagonal, and let [ U/ R] → T be a morphism such that p^-1p_* [ U_k^/ R_k^] → p^-1p_*T_k^ is an equivalence. Here, for any field k, p(/k)_→ ( k)_ denotes the change-of-site morphism for which p_* is the restriction. Let us, for a finite reduced k-algebra A, denote by A^† the separable closure of k in A. Moreover, if A is a finite product ∏_i k_i of fields, denote by A^ the product ∏_i k_i^. Suppose we are in <ref>. Then to any morphism xl → U with l/k finite, we can attach an induced morphism l^→ U. This in turn induces a morphism l^→ p^-1p_* T_k^, which is étale as both l^ and p^-1p_* T_k^ are étale over k^. We hence get a morphism l^†→ p^-1p_* T.We prove a couple of lemmas regarding this construction.In <ref>, let *x_il_i → U be a family of points on U. Then the image of ∐_il_i → U intersects every geometric connected component of U if and only if ∐_il_i^†→ p^-1p_* T is surjective.We note that the image of ∐_il_i → U intersects every geometric connected component if and only if the image of ∐_il_i^→ U does so. This is equivalent to ∐_il_i^→ p^-1p_* T_k^ being surjective, i.e. to ∐_il_i^†→ p^-1p_*T being surjective. In <ref>, let xl → U and ym → U be two points on U. Let A be the coordinate ring of α^-1x ×_ Rω^-1y. Then A^† is the coordinate ring of l ×_p^-1p_* T m.Let x'l^→ U and y'm^→ U. Then A^ is the coordinate ring of α^-1x' ×_ Rω^-1y' =l^×_p^-1p_* T_k^ m^, so A^† is the coordinate ring of l^†×_p^-1p_* T m^†, being the unique finite k-subalgebra of A^ of which the base change to k^ is A^. So in <ref>, by finding enough points on U, one can construct a presentation of the stack p^-1p_*T, which then can be used to compute π_0 of this. Let us do so explicitly below.<ref> takes as input <ref> and computes a finite set X of morphisms x_il_i → U with l_i/k finite, such that the induced map ∐_il_i →π_0(p_*T) is surjective. Moreover, it does so in arithmetic complexity*[Big]O*[big] N^2 log(d), e N log(d), log(r)Compute a Noether normalisation ν U → A^ U using e.g. <cit.>. Note that this also works for finite fields, but only after a base change to a finite field extension;so for finite fields, one needs to keep track of the Galois action as well.Then set R =O*[big]ν^-1(0). Compute a k-basis for R using a Gröbner basis computation for the ideal defining R. Compute the primary decomposition of R and for each local factor S of R, compute the composition of O( U) → R, R → S, and S → S^.First note that, as U is equidimensional, every geometric connected component maps surjectively to A^ U. Hence R is the ring of global sections of a closed subscheme of U that intersects every geometric connected component, so this procedure indeed computes a set X as desired. It remains to prove the claims on the arithmetic complexity.By <cit.> (for Noether normalisation and the zero-dimensional Gröbner basis computation, respectively), R can be computed as finite k-algebras in arithmetic complexity*[Big]O*[big] N^2 log(d), log(r) . Let us bound the k-vector space dimension of the R. First note that U ≤ N. Therefore R is given by at most N generators, and by relations that are of degree at most d. Hence _k R = *[Big]O*[big] N log(d) .So by the methods of <cit.>, we see that the primary decomposition of R can be computed in arithmetic complexity*[Big]O*[big] N log(d) . Moreover, for each of the local factors, the degree of the purely inseparable extension l/k to be taken doesn't exceed (_k R)^e, as the degree of the polynomials to be factored doesn't exceed _k R. It then follows by that the maps R → S^ can be computed in arithmetic complexity*[Big]O*[big] (e+1)Nlog(d) .as _k (S ⊗_k l) ≤ (_k R)^e+1. <ref> takes as input <ref>, xl → U, and ym → U and computes the finite k-scheme α^-1 x ×_ Rω^-1 y in arithmetic complexity*[Big]O*[big] N_x,y^2 log(d_x,y), log(r) ,where N_x,y = max(N,log [l:k],log [m:k]) and d_x,y = max(d,[l:k],[m:k]). We first compute for l and m a “small” set of generators. Start by setting X = F = ∅. For s in a k-basis for l, compute k[X][s]l and the minimal polynomial f of s over k[X]l, and then, if f is linear, do nothing, otherwise add s to X and f to F. Write l = k[X]/(F) afterwards, and repeat this for m.Now compute x, y in terms of the “small” descriptions of l and m obtained above, and compute α^-1 x ×_ Rω^-1 y. Finally, compute a k-basis for its coordinate ring via Gröbner bases, and the unit and multiplication table with respect to this basis.Note that in the first step, we write l (resp. m) using O(log [l:k]) (resp. O(log [m:k])) generators and relations, of degree at most [l:k] (resp. [m:k]). Moreover, U and R are given by at most N generators and r relations, of degree at most d. Therefore α^-1 x ×_ Rω^-1 y is given by O*[big]log [l:k], log [m:k], N generators, O*[big]log [l:k], log [m:k], N, r relations, of degree at most max*[big][l:k],[m:k],d. Hence the arithmetic complexity follows from <cit.> in the same way as before. As a corollary, using <ref> and <ref>, we have the following. There exists an algorithm that takes <ref> as input and computes a diagramY^†[r,shift left] [r,shift right] X^† X [u] [d]Uwith X^†, Y^† finite étale over k, Y^†⇉ X^† a presentation for p^-1p_* T, X → X^† a finite purely inseparable morphism between finite k-schemes, and X → U having image intersecting every geometric connected component of U, in arithmetic complexity*[Big]O*[big] (e+1)N^3log(d)^3, log(r) .In the corollary above, we can assume that if U, V are two distinct connected components of X^†, then α^-1U ×_Y^†ω^-1V is empty, since whenever we encounter distinct connected components U, V for which α^-1U ×_Y^†ω^-1V is not empty, then we may simply omit one of U, V. It follows that the groupoid scheme Y^†⇉ X^† is a finite disjoint union of groupoid schemes Y_i^†⇉ X_i^† with finite étale arrows in which each X_i is the spectrum of a finite separable field extension of k. Therefore the problem of computing π_0(p^-1p_* T) reduces to computing π_0 of each of these groupoid schemes.The following lemma suggests how to compute π_0 in this case. Let S be a connected scheme, and let Y ⇉ X be groupoid scheme over S with X and Y finite étale S-schemes. Then the image R of Y → X ×_S X is an étale equivalence relation on X, and π_0([X/Y]) = X/R.This is trivial once we view X and Y as finite π_1(S)-sets. <ref> takes a groupoid scheme Y ⇉ X over k with X and Y finite étale k-schemes, and outputs π_0([X/Y]) in arithmetic complexity polynomial in the degrees of X and Y over k. Let l, B be the respective coordinate rings of X, Y, and let A = l ⊗_k l. Let A = ∏_i A_i be the primary decomposition of A; since A is separable over k, all A_i are fields.Compute the morphism A → B, and compute the set I of indices i for which the induced map A_i → B is non-zero. Set A_I = ∏_i ∈ I A_i. Compute the morphisms l → A_I sending s ∈ l to s ⊗ 1 and 1 ⊗ s, respectively, and return their equaliser k'.Note that, since A_I is the image of Y in X ×_ k X by construction, k' is the coequaliser of the two morphisms A_I → X constructed, in other words, it is the quotient of X by the étale equivalence relation A_i on X, as desired. Applying the above to the groupoid scheme obtained in <ref>, we get the following. There exists an algorithm that takes as input <ref> and computes a finite étale k-scheme representing ^1(X_k^,,j_! G) in arithmetic complexity*[Big]O*[big] (e+1)s^12(#G)^12γ^3 log(s#G)^3 .In order to be able to compute additional structures on ^1(X_k^,,j_! G), it will turn out to be useful to compute this set as a finite (k^/k)-set, together with some additional structure. The first step in this is to compute a finite Galois extension l/k such that the Galois action on ^1(X_k^,,j_! G) factors through (l/k). This is done in the standard recursive way.<ref> takes as input a finite separable k-algebra A, and computes the minimal Galois extension l/k such that A ⊗_k l is a product of copies of l, in arithmetic complexity polynomial in (_k A)!. If A is the underlying scheme of a group scheme over k, then the arithmetic complexity is*[Big]O*[big]log (_k A)^2 .Set A' = A, and compute a primary decomposition A' = ∏_i A'_i. Set l = k. While the number of factors is not _k A, choose A'_i of maximal dimension, and set A' = A' ⊗_l A'_i, l = A'_i, and compute a primary decomposition A' = ∏_i A'_i. Return l.There exists an algorithm that takes as input a finite separable k-algebra A, and computes the corresponding finite (k^/k)-set in arithmetic complexity polynomial in (_k A)! (or*[Big]O*[big]log (_k A)^2if A is the underlying scheme of a group scheme over k). Now by base change to l, we get the following.There exists an algorithm that takes as input <ref> and computes: * a finite Galois extension l/k such that the (k^/k)-action on the finite set ^1(X_k^,,j_! G) factors through (l/k),* the finite (l/k)-set ^1(X_k^,,j_! G),* for each h ∈^1(X_k^,, j_! G), a finite extension l_h/l and a morphism l_h → U representing h,in arithmetic complexity*[Big]O*[big] (e+1)s^12(#G)^12γ^3 log(s#G)^3, (e+1) log [l:k] .§ REDUCTIONS AND APPLICATIONSIn the previous sections, we assumed normal proper curves to be presented as finite locally free P^1_k-schemes as described in <ref>. Alternatively, normal proper connected curves can be presented using their function fields, as a finite extension of k(x), and in this case, we will present morphisms between normal proper connected curves by morphisms between their function fields.Passing from the presentation as finite locally free P^1_k-scheme to that as a finite extension of k(x) is simple:given a finite locally free P^1_k-scheme X with type a of length s, one can compute the finite k(x)-algebra k(X) corresponding to it in arithmetic complexity polynomial in s and ∑_i -a_i, simply by, in the conventions of <ref>, substituting y=1 in the multiplication table and unit defining O_X;this computation is functorial in X.Conversely, given a finite field extension A of k(x) of degree d defined by elements of height at most h, one can compute α_1,…,α_n ∈ A such that A = k(x,α_1,…,α_n), and minimal polynomials for α_i+1 over k(x,α_1,…,α_i) in arithmetic complexity polynomial in d, h, using the methods of <cit.>;note that n ≤log_2 d, and the minimal polynomials have degree at most d, and their coefficients have height at most d^3h. By multiplying by suitable polynomials in k[x], one can make each α_i+1 have a minimal polynomial of which the coefficients lie in k[x,α_1,…,α_i], in arithmetic complexity polynomial in d, h;the minimal polynomials in this case will have x-degree at most d^5h. Therefore we obtain a k[x]-order in A consisting of products of the α_i. Similarly, we can compute a k[x^-1]-order in A, in arithmetic complexity polynomial in d and h.Then, by <cit.> (which we can apply since we are able to compute nilradicals of finite k-algebras) one can compute the corresponding maximal orders over k[x] and k[x^-1] in arithmetic complexity polynomial in d^e+1 and h. Moreover, they define the same k[x,x^-1]-submodule of A, so it follows from <cit.> that one can compute a sequence (a_i) of integers, and bases (b_i) and (c_i) of the respective maximal orders such that b_i = x^a_i c_i for all i, and therefore a presentation of the normal proper connected curve as a finite locally free P^1_k-scheme, in arithmetic complexity polynomial in d^e+1 and h as well.In fact, as the computation of nilradicals of finite k-algebras as described in <ref> proceeds by first computing the nilradical of the base change to k^, it follows that one can compute a purely inseparable extension l of k and a smooth proper connected curve with function field k(x)l in arithmetic complexity polynomial in d^e+1 and h;we will refer to this as the construction of a smooth completion.As for functoriality, given a morphism K → L of function fields, with K given as a finite k(x)-algebra, and L as a finite k(y)-algebra, one can compute a k(x)-basis of L (and therefore a K-basis of L) by successively computing a k(x)-basis of k(x,y) and a k(x,y)-basis of L;with respect to this k(x)-basis of L, the computation given above is functorial. The above gives us an algorithm for the computation of the normalisation (over k and over k^) of a finite locally free P^1_k-scheme of type a of length s, in arithmetic complexity polynomial in s^e+1 and ∑_i -a_i;for the type a' (of length s') of the resulting normal proper curve, we have s = s' and ∑_i -a'_i ≤∑_i -a_i.To present divisors on proper normal curves, we will mainly use the so-called free ideal presentation. Roughly speaking, in this presentation, divisors on a proper normal curve X are given as formal sums of closed points of X, which in turn are given by maximal ideals of O_X. For more details on this and other related presentations, see e.g. <cit.>, or <cit.> for a more detailed exposition. For the purposes of this paper, we simply note that we can compute images and pre-images of closed points of a morphism of proper normal curves in arithmetic complexity polynomial in the size of the input.Now an arbitrary normal curve X will be presented by the product of the function fields of its connected components, and the finite complement of X in its normal completion X;as a measure for the size of an affine curve, we take the k(x)-degree of the corresponding k(x)-algebra, an upper bound h for the height of the elements of k(x) defining this algebra, the number of closed points in X - X, and the maximum degree of these closed points over k. Morphisms Y → X between normal curves will be presented by morphisms between their normal completions, such that for every closed point in the complement of X in X there is a closed point of Y in Y lying over it. §.§ Topological invariance of the small étale site In our reduction to <ref>, we will make use of finite locally free, purely inseparable morphisms between normal proper curves and the topological invariance of the small étale site, which states that for a universal homeomorphism fY → X, the functors f_* and f^-1 are quasi-inverse functors between (X_) and (Y_). Given a finite locally free, purely inseparable morphism fY → X between normal proper connected curves, we will make this explicit for étale sheaves representable by étale separated X-schemes (resp. Y-schemes), i.e. by normal curves.For F an étale sheaf representable by an étale and separated X-scheme, the pullback is simply Y ×_XF, which clearly can be computed in arithmetic complexity polynomial in the size of the input.<ref> takes a finite locally free, purely inseparable morphism fY → X between normal proper connected curves, and a normal curve F, étale over Y, and computes f_*F, in arithmetic complexity polynomial in the size of the input. Write K, L, for the function fields of X, Y, respectively, let F be an étale and separated Y-scheme (with normal completion F), and let B denote its corresponding L-algebra. Let A be the Weil restriction of B from L to K. Compute the L-algebra isomorphism A ⊗_K L → B, and therefore a morphism A → B. Let F' denote the corresponding normal proper curve, and F→ F' be the corresponding morphism. Output F' together with the image of the complement of F in F.The output is correct since the output F' needs to satisfy F' ×_X Y =F, and since taking Weil restrictions sends finite separable L-algebras to finite separable K-algebras, and is left adjoint to base changing from K to L.§.§ Presentation of torsors Now note that in <ref>, we have a second presentation of a j_! G-torsor on X_k^, (the first being as a geometric point on the groupoid scheme R ⇉ U constructed in the previous section). First, any j_! G-torsor is representable by an étale separated X-scheme and therefore by a normal curve, so we can present a j_! G-torsor T by a normal curve together with the group action j_! G ×_X T → T. In fact, j_! G is representable by the disjoint union of X (acting as the zero section) and G - 1 (which is finite étale over U). We indicate how to pass between these presentations.Starting with x ∈ U(l), we let k' be the separable closure of k in l. Note that x defines a Γ-equivariant A-torsor T on Y_l, together with a Γ-equivariant section Y×_ P^1_k S_0 →T. Using the function field presentations, one can then compute quotients under Γ using linear algebra, which gives us a j_! G-torsor T on X_l,. Therefore we can compute the corresponding j_! G-torsor on X_k', in arithmetic complexity polynomial in s^e+1, (#G)^e+1, γ^e+1, [l:k]^e+1.Conversely, let T be a j_! G-torsor on X_k', with k'/k separable. Pull T back to a Γ-equivariant G-torsor on Y_k',, together with Γ-equivariant section Y_k'×_ P^1_k S_0 → Y_k', and compute a smooth completion. Now some linear algebra suffices to compute the additional data (see <ref> and <ref>) required to obtain a point of U(k^) in arithmetic complexity polynomial in s^e+1, (#G)^e+1, γ^e+1, [k':k]^e+1.Therefore, using this, <ref> and <ref>, we have the following.<ref> takes as input <ref> (but with A =G a sheaf of abelian groups, so A = G is abelian) and computes the (k^/k)-module ^1(X_k^,,j_! A) in arithmetic complexity*[Big]O*[big](e+1)^3s^16(#A)^16γ^4 log(s#A)^3.It remains to compute addition of classes of j_! A-torsors on X_k^,. For this, it suffices to note that if T_1, T_2 are j_! A-torsors on X_k', with k'/k separable, then the sum of the classes of T_1 and T_2 in ^1(X_k^,,j_! A) is given by the quotient of T_1 ×_X_k' T_2 by the j_! A-action given by a(t_1,t_2) = (at_1,a^-1t_2);compute this using linear algebra over k'(x), and find the element in ^1(X_k^,,j_! A) isomorphic to it using <ref>.It remains to find an upper bound for log[l:k]. Note that the cardinality of the set ^1(X_k^,,j_! A) is*[Big]O*[big]s^8(#A)^8γ^2 log(s#A),es^4(#A)^4γlog(s#A)by <ref>. As this set is an abelian group, it follows that log[l:k] = O*[big](e+1)^2s^16(#A)^16γ^4 log(s#A)^2,from which the arithmetic complexity follows.§.§ Reduction to <ref> We will now indicate how to reduce to <ref> for “most” smooth connected curves X over k, non-empty open subschemes U of X, and finite locally constant sheaves A on U_. We would like to use explicit computation of Riemann-Roch spaces as in <cit.> to compute a suitable cover of X over P^1_k, however, this requires the curve to be given as a generically étale finite locally free P^1_k-scheme (or equivalently, a separating element for the function field of X must be given).<ref> takes as input a normal proper connected curve X over k of type a, and computes a finite purely inseparable extension l/k, a finite locally free purely inseparable morphism X' → X of degree at most [X: P^1_k]^e+2, and a generically étale finite locally free morphism X' → P^1_l of degree at most [X: P^1_k], in arithmetic complexity polynomial in ∑_i -a_i and [X: P^1_k]^e+2. By computing the separable closure K' of k(x) in K, compute the minimal p-power q such that f^q ∈ K' for all f ∈ K. Let L = K · k^1/q(x^1/q), which is the reduction of K ⊗_k(x) k^1/q(x^1/q), and output l = k^1/q and x^1/q, together with a K-basis of L.We note that q is bounded from above by [K:k(x)], and that therefore L can be computed in arithmetic complexity polynomial in ∑_i -a_i and [K:k(x)]^e+2. <ref> takes as input a smooth connected curve X generically étale over P^1_k, given by its normal completion X, a finite set *Q_1,…,Q_t of closed points of X, and an open immersion jU → X, given by a finite set *P_1,…,P_s of closed points of X, and computes a finite locally free morphism π X → P^1_k with π^-1(∞) = *Q_1,…,Q_t and π^-1(0) *P_1,…,P_s in arithmetic complexity polynomial in the size of the input. Write Z_0 = ∑_i=1^s P_i, Z_∞ = ∑_j=1^t Q_j, and let g be the genus of X. Let m be the smallest integer such that m(t-1) - s > 2g - 2 and 2^m > t. Using <cit.>, compute a k-basis B for O_X(-Z_0 + mZ_∞), and compute the subspaces O_X(-Z_0 + mZ_∞ - mQ_j) for j=1,…,t. Find a linear combination f = ∑_b ∈ Bϵ_b b with ϵ_b ∈*0,1 for all b ∈ B such that f is not in any of the O_X(-Z_0 + mZ_∞ - mQ_j) for j=1,…,t.Compute a minimal polynomial for f over k(x), and write it as a minimal polynomial for x over k(f), and compute successively a k(f)-basis for k(x,f) and a k(f)-basis for k(X). Output the corresponding map X → P^1_k.Note that by choice of m, we see that O_X(-Z_0 + mZ_∞ - mQ_j) has dimension #B - m, so f ranges over a set of 2^#B elements, of which at most t2^#B-m lie in one of the given subspaces. By choice of m, we have t2^#B-m < 2^#B, so there exists such f not lying in any of the given subspaces. In particular, if either U = X or X = X, then we can get a finite locally free morphism X→ P^1_k with U and X inverse images of P^1_k, P^1_k - 0, or P^1_k - ∞. Therefore, using smooth completions, we now have an obvious reduction to <ref>, and therefore the following corollary, which in turn implies <ref>. Let S_0 ∈*[big]∅,0 and S_∞∈*[big]∅,∞. There is an algorithm that takes a finite locally free P^1_k - S_∞-scheme X, smooth over k, and a finite étale commutative group scheme A over U = X ×_ P^1_k ( P^1_k - S_0 - S_∞), and computes the (k^/k)-module ^1(X_k^,,j_! A) in arithmetic complexity exponential in e, [X: P^1_k - S_∞], [ A:U]^log [ A:U], γ_X, and γ_ A. Here, γ_X (resp. γ_ A) is ∑_i -a_i, where a is the type of the normal completion of X (resp. A).We need to prove that the size of the cover Y constructed is polynomial in [X: P^1_k - S_∞], [ A:U]^log [ A:U], γ_X, and γ_ A. Recall that Y is constructed by setting X' = X, A' =A and then repeatedly base changing A'/X' to a non-trivial connected component of A'.For each such base change, let a be the type of A', let a' be the type of the chosen connected component of A', and let a” be the type of their fibre product A” over X. Note that as then O_ P^1_k(a') is a direct summand of O_ P^1_k(a), we have max_j -a'_j ≤max_i -a_i. Moreover, as O_ P^1_k(a) ⊗_ O_ P^1_k O_ P^1_k(a') surjects onto O_ P^1_k(a”), it follows that max_k -a”_k ≤max_i,j -a_i-a'_j ≤ 2 max_i -a_i. Since log_2 [ A:U] such base changes suffice for the construction of a finite locally free X-scheme of which the normalisation is Y, it follows that for the type b of Y, its length t is at most [X: P^1_k - S_∞][ A:U]^log_2 [ A:U], and ∑_j -b_j ≤ t max_j b_j ≤ t [ A:U] γ_ A.The result now follows from <ref>.§.§ Application to computation for constructible sheaves In this section, we will indicate how to compute ^1(X_k^,, A) for X a smooth connected curve and A an arbitrary constructible sheaf of abelian groups, of torsion invertible in k, under the following assumptions. We will assume a presentation of constructible sheaves (and morphisms between them) to be given, with respect to which one can perform certain operations. These operations are: * one can compute finite direct sums of constructible sheaves;* one can compute kernels and cokernels of morphisms;* for a constructible sheaf A, one can compute a non-empty open subscheme U of X such that A|_U is finite locally constant;* for a closed immersion iZ → X and its open complement jU → X, one can compute the functors i^-1, i_*, i^!, j_!, j^-1, j_* and the corresponding units and counits of adjunction; given A_Z on Z_, A_U on U_, and ϕ A_Z → i_* j^-1 A_U, one can compute the corresponding constructible sheaf on X_. In theory, one should be able to give such a presentation using recollement (as done in <ref> for j_! A-torsors), but we will not work this out in this paper.So suppose X is a smooth connected curve, and A is a constructible sheaf of abelian groups on X_, of torsion invertible in k. Let U be a non-empty open subscheme for which A|_U is finite locally constant. Use <ref> and <ref> to find l/k finite purely inseparable, VU_l open and a finite locally free morphism X_l → P^1_l such that V and X_l are finite locally free over their images in P^1_l. Write jV → X_l for the inclusion, and write iZ → X_l for its closed complement.Compute the canonical short exact sequence 0 → j_!j^-1 A → A → i_*i^-1 A → 0, and the morphism δ(i,j) ^0(X_k^,,i_*i^-1 A) →^1(X_k^,,j_!j^-1 A) which sends a section of i^-1 A to the constructible sheaf defined by j^-1 A on V_, 0 on Z_, and the section of i^-1j_*j^-1 A obtained from the given section of i^-1 A by composition with i^-1 A → i^-1j_*j^-1 A. Then, as ^1(X_k^,,i_*i^-1 A) = 0, we see that ^1(X_k^,, A) = δ(i,j).This is independent of the choice of (i,j) in the following sense. If j'V' → X_l and i'Z' → X_l are given, with V'V and a given morphism Z → Z' over X_l, then we can compute a commutative diagram0 [r] j'_!(j')^-1 A [r] [d] A [r] [d,equals] i'_*(i')^-1 A [r] [d] 0 0 [r] j_!j^-1 A [r] A [r] i_*i^-1 A [r] 0a commutative diagram^0(X_k^,,i'_*(i')^-1 A) [r] [d]^1(X_k^,,j'_!(j')^-1 A) [d]^0(X_k^,,i_*i^-1 A) [r]^1(X_k^,,j_!j^-1 A)and therefore the morphism δ(i',j') →δ(i,j) corresponding to the identity map on ^1(X_k^,, A).In the same way, we see that for constructible sheaves A, B on X_, and a morphism A → B, we can compute the induced morphism ^1(X_k^,, A) →^1(X_k^,, B), and that for a morphism fY → X of smooth connected curves, we can compute the pullback ^1(X_k^,, A) →^1(Y_k^,,f^-1 A).abbrvnat | http://arxiv.org/abs/1707.08825v2 | {
"authors": [
"Jinbi Jin"
],
"categories": [
"math.AG",
"14F20, 14Q05"
],
"primary_category": "math.AG",
"published": "20170727121337",
"title": "Explicit computation of the first étale cohomology on curves"
} |
Importance sampling for metastable and multiscale dynamical systems K. Spiliopoulos[Department ofMathematics and Statistics, Boston University, Boston, MA, 02215, [email protected]. This work was partially supported by the National Science Foundation CAREER award DMS 1550918 ] December 30, 2023 ========================================================================================================================================================================================================================== In this article, we address the issues that come up in the design of importance sampling schemes for rare events associated to stochastic dynamical systems. We focus on the issue of metastability and on the effect of multiple scales.We discuss why seemingly reasonable schemes that follow large deviations optimal paths may perform poorly in practice, even though they are asymptotically optimal. Pre-asymptotic optimality is important when one deals with metastable dynamicsand we discuss possible ways as to how to address this issue. Moreover, we discuss how the effect of the multiple scales (either in periodic or random environments) on the efficient design of importance sampling should be addressed. We discuss the mathematical and practical issues that come up, how to overcome some of the issues and discuss future challenges.§ INTRODUCTION In this paper, we discuss recent developments on importance sampling methods for metastable dynamics that may also have multiple scales. Development of accelerated Monte Carlo methods for metastable, multiple-scale processes is of great interest.Importance sampling is a variance reduction technique in Monte-Carlo simulation, which is especially relevant when dealing with rare events. Since its introduction, importance sampling has been one of the most popular techniques for rare event simulation. There is a vast literature of papers investigating its applications from a broad set of sciences including engineering, chemistry, physics, biology, finance, insurance, e.g., <cit.>.Consider a sequence {X^ϵ}_ϵ>0 of random elements and assume that we want to estimate the probability 0<p^ϵ=ℙ[X^ϵ∉𝒟∪∂𝒟]≪ 1 for a given set 𝒟, such that the event {X^ϵ∉𝒟∪∂𝒟} is unlikely for small ϵ. If closed form formulas are not available, or numerical approximations are either too crude or unavailable, then one has to resort in simulation. It is well known that standard Monte-Carlo simulation techniques (i.e., using the unbiased estimator p̂^ϵ=1/N∑_j=1^N1_X^ϵ,j∉𝒟∪∂𝒟) perform rather poorly in the rare-event regime. As it is known, see for example <cit.>,in order toachieve relative error smaller than one using standard Monte Carlo, one needs an effective sample size N≈ 1/p^ϵ. In other words, for a fixed computational cost, relative errors grow rapidly as the event becomes more rare. Thusstandard Monte-Carlo is infeasible for rare-event simulation. The goal of importance sampling is to simulate the system under an alternative probability distribution ℙ̅ instead of the original probability ℙ. Let's say for example that we are interested in the estimation of 𝔼_y[e^-1/ϵh(X^ϵ_T)] or ℙ_y[τ^ϵ_𝒟∪∂𝒟≤ T]where h:ℝ^d↦ℝ is a positive function, T>0, ϵ>0, y∈ D is the initial point, τ^ϵ_𝒟∪∂𝒟 is exit time from the set 𝒟∪∂𝒟, X^ϵ is a stochastic process modeling the dynamics.Also, notice that the probabilityabove can be considered (modulo the important technical point of lack of continuity) as a special case of 𝔼_y[e^-1/ϵh(X^ϵ_T)], when h is for example chosen such that h(x)=0 for x∉𝒟∪∂𝒟 and h(x)=+∞ for x∈𝒟∪∂𝒟.When rare events dominate, then standard Monte-Carlo methods perform poorly in the small noise limit. Then, to estimate𝔼_y[e^-1/ϵh(X^ϵ_T)], one generates iid samples X^ϵ_(k) from ℙ̅ and uses the importance sampling estimator 1/N∑_k=1^Ne^-1/ϵh(X^ϵ_(k))dℙ/dℙ̅(X^ϵ_(k)).The key question is the design of ℙ̅ such that the second moment 𝔼̅_y[e^-1/ϵh(X^ϵ_T)(dℙ/dℙ̅)(X^ϵ_·)]^2 (and hence the variance) is minimized. 𝔼̅ is the expectation operator under ℙ̅. The choice of the appropriate alternative measure ℙ̅is closely related to certain Hamilton-Jacobi-Bellman (HJB) equations. The first issue that we address is the effect of rest points (and metastability in general) on importance sampling. It turns out that when dealing with metastability, even seemingly reasonable schemes that are also asymptotically optimal, may perform poorly in practice. This includes also changes of measure that try to enforce the simulated trajectories to follow large deviations most likely paths. The reason for the degradation in performance is the role of prefactors. Prefactors can become very important when rest points are included in the domain of interest for the simulation. Large deviations based change of measures may not account for the prefactors, as they rely on logarithmic asymptotics. We elaborate on these issues and discuss potential ways on how the issue can be addressed.The second issue that we address is the effect of multiple scales on the design of provably-efficient importance sampling methods. It turns out that when the dynamical system has widely separated multiple scales, then one can use averaging and homogenization techniques. However, as we shall see, it is not sufficient to base the design of importance sampling on the effective homogenized dynamics. The local information needs to be taken into account. Mathematically this is done using the so called cell problem, or macroscopic problem, in the theory of periodic and random homogenization.The rest of the article is summarized as follows. In Section <ref> we review the classical large deviations theory and the setup of importance sampling for small noise diffusions. In Section <ref> we discuss the effects of rest points, i.e. of stable and unstable equilibrium points, in the design of importance sampling. We argue why asymptotic optimality may actually not mean good practical performance and we also argue that following large deviations most likely optimal paths may lead to poor performance. In addition, we present constructions that lead to guaranteed good performance. We supplement the theoretical arguments by simulation studies. We refer the interested reader to <cit.> for more details. In Sections <ref> and <ref>, we address the design of importance sampling schemes in the presence of multiple scales. We construct asymptotically optimal schemes in the presence of multiple scales. To be more precise, in Section <ref> we consider overdamped Langevin dynamics in periodic multiscale environments and we review the related large deviations theory and importance sampling theory, presenting simulation studies.The interested reader can also consult <cit.>. In Section <ref> we review recent developments in large deviations and importance sampling for multiscale dynamics in random environments, see also <cit.>. In Section <ref> we describe how one can combine the results of Section <ref> with those of Sections <ref> and <ref> and also review future directions.For the sake of concreteness and for exposition purposes we restrict the presentation of this article in the case of diffusions with gradient drift and constant diffusivity, which also implies reversible diffusion dynamics. However, we mention that almost all of the arguments can and have been generalized to the case with general state dependent drift and diffusion coefficient, especially those about the effect of multiple scales on importance sampling, see <cit.>. For results in the infinitely dimensional case we refer the interested reader to <cit.>. § REVIEW OF LARGE DEVIATIONS AND IMPORTANCE SAMPLING THEORY FOR DIFFUSIONSLet us briefly review the setup for small noise diffusions in ℝ^d (e.g. <cit.>) without the effect of multiple scales. Let W_t be a standard d-dimensional Wiener process and considerdX_t^ϵ=-∇ V( X_t^ϵ)dt+√(ϵ)Γ dW_t,X_t_0^ϵ=y.Large deviations principle for the process X^ϵ_t is well known (e.g, <cit.>). In particular, the action functionalfor the process X^ϵ_t, t_0≤ t≤ T, in 𝒞([t_0,T]) as ϵ↓ 0 has the form 1/ϵS_t_0T(ϕ), whereS_t_0T(ϕ)=1/2∫_t_0^T(ϕ̇_s+∇ V(ϕ_s))^T[ΓΓ^T]^-1(ϕ̇_s+∇ V(ϕ_s))ds,if ϕ∈𝒜𝒞([t_0,T]) +∞,otherwise.Here 𝒞([t_0,T]), 𝒜𝒞([t_0,T]) are the sets of continuous and absolutely continuous functions on [t_0,T] respectively.Then, under fairly general conditions,𝔼_y[e^-1/ϵh(X^ϵ_T)]≈ e^-1/ϵinf{S_t_0T(ϕ)+h(ϕ_T):ϕ,ϕ_t_0=y},as ϵ↓ 0. A simple application of Jensen's inequality together with Varadhan's integral lemma (e.g., <cit.>) shows that an asymptotically optimal ℙ̅ should satisfylim_ϵ→ 0ϵln𝔼̅[e^-1/ϵh(X^ϵ_T)dℙ/ dℙ̅]^2=-2G(t_0,y),withG(t,x)=inf_ϕ∈𝒜𝒞([t,T]), ϕ_t=x{S_tT(ϕ)+h(ϕ_T)}Turning to importance sampling, for ℙ̅ that are absolutely continuous with respect to ℙ, Girsanov's formula impliesdℙ̅/dℙ=e^-1/2ϵ∫_0^T|v_s|^2 ds+1/√(ϵ)∫_0^Tv_sdW_swhere v_t is a progressively measurable process (control) such that the right hand sideis a martingale (with respect to an appropriate filtration). Under ℙ̅, X^ϵ satisfies dX_t^ϵ=[-∇ V( X_t^ϵ)+Γ v_t]dt+√(ϵ)Γ dW̅_t,withW̅_t=W_t-1/√(ϵ)∫_t_0^tv_ρdρ So, the problem is restricted to choosing the control v_toptimally (i.e., such that the second moment is minimized)and then using the estimator based on iid samples generated from ℙ̅ under (<ref>). Under appropriate conditions, the zero-variance (i.e. the best) change of measure is based on the control v_t given by the formula v_t=u̅(t,X^ϵ_t) wherev̅(t,x)=-Γ^T∇ G^ϵ(t,x) where G^ϵ(t,x), with terminal condition G^ϵ(T,x)=h(x),is thesolution to the PDE, of HJB type:∂_tG^ϵ(t,x)-∇ V(x)·∇ G^ϵ(t,x)-1/2|Γ^T∇ G^ϵ(t,x)|^2+ϵ/2tr[ΓΓ^T∇^2G^ϵ(t,x)]=0. Since (<ref>) is not tractable, it is standard approach to go to the viscosity limit ϵ↓ 0. Then G(t,x)=lim_ϵ↓ 0G^ϵ(t,x)is the viscosity solution to theHJB equation with HamiltonianH(x,p)=<-∇ V(x), p>-1/2‖Γ^T p‖^2 i.e., to the equation ∂_tG(t,x)-∇ V(x)· DG(t,x)-1/2|Γ^T DG(t,x)|^2=0,G(T,x)=h(x). Notice that by control arguments, e.g., see <cit.>, we can also writeG(t,x)=lim_ϵ↓ 0G^ϵ(t,x)=inf_ϕ∈𝒜𝒞([t,T]), ϕ_t=x{S_tT(ϕ)+h(ϕ_T)}. In fact, more is true. A smooth function U̅(t,x):[0,T]×ℝ^d↦ℝ is called a subsolution to the HJB equation (<ref>) with ϵ=0 if∂_tU̅(t,x)-∇ V(x)·∇U̅(t,x)-1/2|Γ^T∇U̅(t,x)|^2≥ 0, U̅(T,x)≤ h(x). It turns out (Theorem 4.1 in <cit.>), that appropriate, smooth subsolutions are enough. If U̅(t,x)∈𝒞^1,1([t_0,T]×ℝ^d) satisfies (<ref>) and the feedback control to use in (<ref>) is v_t=-Γ^T∇U̅(t,X^ϵ_t), then G(t_0,y)+U̅(t_0,y)≤lim inf_ϵ→0-ϵln𝔼̅[e^-1/ϵh(X^ϵ_T)dℙ/dℙ̅]^2≤ 2G(t_0,y).Therefore,asymptotic optimality is attained if U̅ satisfies U̅(t_0,y)=G(t_0,y)=lim_ϵ↓ 0G^ϵ(t_0,y) since then lower and upper bound agree. The design and analysis of importance sampling schemes based on the systematic connection with subsolutions to the appropriate HJB and Isaacsequations goes back to <cit.>. See also<cit.> for the closely related concept of Lyapunov inequalities.The importance sampling simulation scheme in order to estimate θ^ϵ(t_0,y)≐𝔼_t_0,y[e^-1/ϵh(X_T^ϵ )] goes as follows. Let X^ϵ,v be the solution to the SDEdX_t^ϵ,v =(-∇ V( X_t^ϵ,v)+Γ v_t) dt+√(ϵ)Γ dW_t,X_t_0^ϵ,u=y. * Consider v_t=u̅(t,X_t^ϵ,v)=-Γ^T∇_xU̅(t,X_t^ϵ,v)with U̅ an appropriate subsolution, i.e., it satisfies (<ref>)* Consider the estimatorθ̂^ϵ(y)≐1/N∑_j=1^N[e^-1/ϵh(X_T^ϵ,v(j))Z_j^v]whereZ_j^v≐ e^-1/2ϵ∫_0^Tu̅( t,X_t^ϵ,v(j)) ^2dt-1/√(ϵ) ∫_0^T<u̅(t,X_t^ϵ,v(j)),dW_t (j)>and (W(j),X^ϵ,v(j)) is an independent sample generated from (<ref>) with control v_t=u̅(t,X_t^ϵ ,v(j)).We conclude this section, with the remark that a choice of the control v_t based on a subsolution as defined by (<ref>)only guarantees logarithmic asymptotic optimality and does not say something about the important effect of pre-factors. As we will see in Section <ref>, this can imply degradation in the performance of the algorithm in problems with metastability.When dealing with metastability issues, things may be even more problematic if one is using the exact solution to the association HJB equation, G(t,x). While this may be not be a problem for problems that do not involve rest points (i.e. does not involve stable or unstable equilibrium points) in the domain of interest, it does become problematic when dealing with metastability issues. Obtainingaccurately the solution G(t,x) to the HJB equation (<ref>), analytical or numerical,is challenging in high dimensions. However, even if this were possible, the solution by itself is not alwayssuitable for importance sampling when one is interested in computing escape or transition probabilities. The issue is that in these cases, the solution is a viscosity solution with a discontinuous derivative at the rest point (stable or unstable equilibrium points) and with negative definite generalized second derivative there. Physically, the exact solution to the HJB equation attempts at each point in time and space to force the simulated trajectories to follow a most likely large deviations optimal path. However, by standard control arguments, see <cit.>, the discontinuity of the spatial derivative at the rest point, implies that multiple most likely optimal paths exist. As a consequence, the noise can cause trajectories to return to a neighborhood of the origin, thereby producing large likelihood ratios.In Section <ref>, we will see that this is a serious issue, leading to poor performance, even in dimension one where one can solve the HJB equation analytically. Importance sampling, when dealing with state dependent metastable dynamical systems, needs to be addressed from a global point of view and not local.§ THE EFFECT OF REST POINTSON IMPORTANCE SAMPLINGAs it is shown, mathematically and numerically, in <cit.>, in dynamical systems that exhibit metastable behavior standard simulation methods do not readily apply. Asymptotic optimality is necessary but not sufficient for good performance due to the non-trivial effect of the pre-factors.The pre-factor computations in <cit.> prove that there is non-trivial interaction of parameters such as the strength of the noise ϵ and the terminal time T. We remark here that this is in contrast to escape probabilities for other well studied problems, such as stochastic networks, e.g., <cit.>, because there the proximity of the rest point has little impact on either the asymptotic rate of decay or the pre-exponential term.These interactive effects vanish in the logarithmic limit as the noise goes to zero, but they have a significant effect on the performance of the algorithms. The following question immediately presents itself: * Is it sufficient to have schemes that are only asymptotically logarithmical optimal, in the sense that the second moment of the estimator satisfies (<ref>)? What about pre-factors? Are they truly negligible in practice in the rare event regime?* Can we construct a subsolution U̅(t,x) that not only satisfies (<ref>) but it also takes care of the prefactor effects?§.§ Effects in the prelimitLet us demonstrate the effect of prefactors on the behavior of estimators in the following classical simple setting. Let us assume that the diffusion coefficient Γ=I, and that x=O is the global minimum for V(x). In particular, let us assume that DV(O)=0 and that DV(x)≠ 0 for every x≠ O. Define𝒟={ x∈ℝ^d: 0≤ V(x)<L }and let A_c={ x∈ℝ^d: V(x)=c}. Then for an initial point y such that 0≤ V(y)<L, let us assume that we want to estimateθ^ϵ(t,y)=ℙ_t,y{ X^ϵ hitsA_L before time T}. A classical quantity if interest in metastability theory is the quasipotential, see <cit.>. The quasipotential with respect to the equilibrium point O is defined as followsW(O,x)={ S_0T(ϕ): ϕ∈𝒞([0,T]), ϕ(O)=0, ϕ(T)=x, T∈(0,∞)} Under the assumptions of this section, thequasi-potential is computable in closed form <cit.>:W (O,x) = 2V(x)forx∈{y∈𝒟∩∂𝒟: V(y)≤inf_z∈∂𝒟V(z)}. Now, if we define τ^ϵ=inf{t>0: X_t^ϵ∉𝒟}, then, as it is shown in <cit.> we have that lim_ϵ↓ 0ϵln𝔼τ^ϵ=inf_z∈∂𝒟W(O,z).Thus, the quasi-potential allows to approximate exit times in the logarithmic large deviations regime,<cit.>.Many quantities in the theory of metastability are defined via the quasi-potential. The quasi-potential characterizes the leading asymptotics of exit times and exit probabilities, approximates transition rates for reversible and irreversible systems and allows to qualitatively describe transitions between stable attractors if the system has many of them; see also <cit.> for more details. These conclusions hold for both gradient and non-gradient cases, but in the gradient case the quasi-potential is computable in closed form.Turning now to importance sampling,it is easy to verify that the quasi-potential is a stationary subsolution to the associated HJB equation (<ref>) with ϵ=0, by adding an appropriate constant C in order to justify the necessary boundary and terminal conditions. In particular, U̅_QP(x)=2L-W(O,x) defines a subsolution for (<ref>). It turns out, see <cit.>, that the quasipotential yields areasonable change of measure if rest points are not part of the domain of interest. However, this is no longer true if rest points are included in the domain of interest.Let us denote Q^ϵ(0,y;u̅)=𝔼̅[e^-1/ϵh(X^ϵ_T)dℙ/ dℙ̅]^2 to be the second moment of the estimator constructed using the control u̅. Based now on the arguments of <cit.> one can prove the following representation for the second moment of the estimator estimator based on the change of measure induced by the control u̅(t,x)=-∇U̅_QP(x)-ϵlog Q^ϵ(0,y;u̅)=inf_v∈𝒜 𝔼[1/2∫_0^τ̂^ϵ‖ v(s)‖ ^2ds-∫_0^τ̂^ϵ‖u̅(X̂^ϵ_s)‖ ^2ds+∞1_{τ̂^ϵ>T}]. where X̂^ϵ_s is the unique solution to the SDE dX̂^ϵ_s=-DV(X̂^ϵ_s)ds+[ √(ϵ)dW_s-[u̅(X̂^ϵ_s)-v(s)]ds]with initial condition X̂^ϵ_0=y and τ̂^ϵ is the first time that X̂^ϵ exits from 𝒟.It is important to note that (<ref>) provides a non-asymptotic representation for the second moment of the estimator. By the arguments of <cit.>, we can choose a particular admissible control v(s) in (<ref>) so that the following takes place. Let T be large and let 0<K<T so that the time interval [0,T] is split into [0,T-K) and [T-K,T]. Set v(s)=0 for s∈[0,T-K). The resulting dynamics for X̂^ϵ is stable for s∈[T-K,T] and with high probability the process will stay around the point y for s∈[0,T-K). In the time interval [T-K,T], we set v(s) so that escape happens prior to T. Then, it can be shown that there are positive constants C_1,C_2<∞, so thatQ^ϵ(0,y;u̅)≥ e^-1/ϵC_1+C_2(T-K). This bound indicates that if T is large, one may need to go to considerably small values of ϵ in order to achieve the theoretical optimal asymptotic performance. We also remark that if T is large (see Chapter 4 of <cit.>), G(0,y) and U̅(y) get closer in value. Thus, by (<ref>) and for large enough T, the particular importance sampling scheme is asymptotically optimal.Hence, we have just seen an example where an importance sampling estimator is almost asymptotically optimal, but it does not perform that well pre-asymptotically due to the effect of the possibly long time horizon T and its interplay with ϵ. §.§ The problems arising when following large deviations asymptotically most likely paths and a remedy to the problemThe connection of change of measures with HJB equations via large deviations is well situated for a systematic treatment of dynamic importance sampling schemesfor state dependent processes like diffusions (<ref>). For small noise diffusions the theoretical framework of subsolutions to HJB equations and their use for Monte Carlo methods can be found in <cit.>. It was a common belief for sometime that if the underlying stochastic process has a large deviations principle and if the change of measure is consistent withthe large deviations asymptotically most likely path leading to the rare event (an open-loop control), then the resulting importance sampling scheme would be optimal. However, such heuristics have been shown to be unreliable in general and simple examples have been constructed showing the failure of the corresponding importance schemes even in very simple settings <cit.>. This is due to the presence of “rogue-trajectories", i.e., unlikely trajectories, that are likely enough to increase likelihood ratios to the point thatthe performance is comparable to standard Monte Carlo. This is especially true for metastability problems (i.e., when transitions between fixed points occur at suitable (large) timescales)where multiple nearly optimal paths may exist. Use of dynamic changes of measure, i.e. based on feedback controls (time and location dependent) becomes important, see <cit.>. However, even changes of measures that are based on feedback controls, that are consistent with large deviations and lead to asymptotically optimal change of measures can also be problematic in practice. We demonstrate this below in Table <ref>. Namely, as it turns out, in the presence of rest points and metastability, the prefactors may affect negatively the behavior of estimators even if one is using asymptotically optimal changes of measure in the spirit of (<ref>). Hence, it becomes important to use dynamic change of measures that are based on subsolutions but lead to good performance even pre-asymptotically. To that end, novel explicit simulation schemes are then constructed in <cit.> that perform provably-well both asymptotically and non-asymptotically, even when thesimulation time is long. These constructions are based on large deviations asymptotics <cit.>, stochastic control arguments and asymptotic expansions <cit.>and detailed asymptotic analysis of the subsolution to the associated HJB in the neighborhood of the rest point where the potential can be thought of as being approximately quadratic. Essentially, due to the fact that near the rest point, the potential can be thought of as being approximately quadratic, one can hope to solve or to approximate the solution to the associated variational problem there. Then one needs to patch this solution together with the quasipotential based subsolution (which is a good subsolution away from the rest point) in the right way. Then, the combined subsolution, see U̅^δ(t,x) in (<ref>), turns out to be a good approximation to the zero variance change of measure. Such schemes lead to importance sampling algorithms with provably-good performance for all small ϵ>0 and without suffering from bad prefactor effects.In order to illustrate the point, let us briefly demonstrate such a construction in the case of dimension one, see <cit.>. So, let us assume that V(x)=λ/2x^2 with λ>0 and let us assume that we study the problem of crossing a level set, say L, of the potential function V(x). Here, we can compute G(t,x) in closed form and we getG(t,x)=inf_ϕ_t=x,V(ϕ_T)=L{1/2∫_t ^T‖ϕ̇_s+λϕ_s‖ ^2ds} =inf_x̂∈ V^-1(L)λ(x̂-xe^λ (t-T))^2/1-e^2λ(t-T). Notice, that G(t,x) is also a viscosity solution to the ϵ=0 HJB equation(<ref>) when supplemented with the appropriate boundary conditions. Hence, based on (<ref>) a change of measure based on G(t,x), i.e., using the control u(t,x)=-∂_xG(t,x), is expected to yield an asymptotically efficient estimator. While this is true, we will see below that this is not sufficient to yield good performance. The fact that the function G(t,x) is not continuously differentiable in the domain of interest, implies that multiple optimal paths exist, which is an intuitive reason for the degradation in performance that will be demonstrated below.However, by appropriately mollifyingG(t,x) and combining it with the quasipotential subsolution (as constructed in Section <ref>), one can construct a global subsolution which performs provably well even pre-asymptotically. The pointis that G(t,x) provides a good change of measure while near the rest point, whereas the quasipotential induced subsolution U̅_QP(x)=2L-W(O,x) provides a good change of measure away from the rest point. There are a few more issues to deal with though. The first one is that G(t,x) is discontinuous near t=T. The second one is that we need to put them together in a smooth way that will define a global subsolution.Since G(t,x) is discontinuous at t=T,we introduce two mollification parameters t^∗ and M that will be appropriately chosen as functions of ϵ. Motivated by the fact that G(t,x) is a good subsolution near the equilibrium point, we fix another parameter L̂∈(0,L]. In the one-dimensional case, it is easy to solve the equation V(x^*)=L̂ and in particular we get that x^*=±x̂ where x̂ =√(2L̂/λ). As a matter of fact, instead of using G(t,x) directly, we setF^M(t,x;x̂) = λ(x̂-xe^λ (t-T))^2/1/M+1-e^2λ(t-T)In order now to pass smoothly between the U̅_QP(x) and F^M(t,x;x̂) or F^M(t,x;-x̂) without violating the subsolution property, we use the exponential mollification, see <cit.>U^δ(t,x)=-δlog(e^-1/δU̅_QP(x)+e^-1/δ[F^M(t,x;x̂)+U̅_QP(x̂)]+e^-1/δ[F^M(t,x;-x̂)+U̅_QP(-x̂)])It is easy to see that as δ↓ 0lim_δ↓ 0U^δ(t,x)=min{U̅_QP(x), F^M(t,x;x̂), F^M(t,x;-x̂)} Clearly, if we choose L̂=L, then we get U̅_QP(x̂)=0. Based on these constructions, a provably efficient importance sampling scheme is constructed in <cit.>, based on the subsolutionU̅^δ(t,x)={[ U̅_QP(x), t>T-t^∗; U^δ(t,x),t≤ T-t^∗ ]. It turns out that U̅^δ(t,x) is a global smooth subsolution which has provably good performance both pre-asymptotically and asymptotically.The role of the exponential mollification is to allow a smooth transition between the region that is near the equilibrium point and the region that is far away from it.The precise optimality bound and its proof guide the choice of the parameters δ,t^∗,M and L̂. For the convenience of the reader, we present in Table <ref> the suggested values for (δ,L̂,M,t^*), given the value of the strength of the noise ϵ>0. We refer the interested reader to <cit.> for further details on the theoretical performance of the algorithm and on the choice of parameters.In order to illustrate in a simple setting the effect of prefactors in the presence of metastable effects, we record in Table <ref> Monte Carlo estimates based on K=10^7 trajectories for the exit time distribution ℙ_y[τ^ϵ_𝒟∪∂𝒟≤ T]from the basin of attraction of the left attractor of the potential of Figure 1 for the process X^ϵ given by (<ref>) with Γ=I. We used the importance sampling (IS) methods of <cit.>, i.e., the change of measure based on the subsolution (<ref>) and record estimates for different pairs (ϵ,T). In the figures next to Table <ref>, wecompare the relative errors per sample of (a): the algorithm, which is optimal for all ϵ>0, i.e the one based on the subsolution U̅^δ(t,x), with (b): the IS algorithm that is consistent with the large deviations asymptotically most likely path leading to the rare event, i.e the one based on the actual solution G(t,x) of the associated HJB equation. Noticehowever that the IS algorithm based on G(t,x) is only asymptotically optimal in the large deviations logarithmic sense as ϵ↓ 0 (i.e., it satisfies (<ref>)). Using relative error per sample as comparison criterium, we compare the two algorithms for two values of ϵ, one for which the events are not so rare (ϵ=0.13) and one for which the events are very rare (ϵ=0.05). Exact values are in the table, and we remark for completeness that intermediate behavior is qualitatively the same. Both algorithms perform well when T is small, but the algorithm that is based on the solution of the associated HJB equation, which is only logarithmic asymptotically optimal, starts deteriorating considerably as T gets large. The latteris an effect of the pre-factors becoming important. On the other hand, the change of measure constructed in <cit.> that takes into account the pre-factor information and is pre-asymptotically optimal, yields optimal performance independently of the values ϵ and T with relative errors around one, meaning that the values recorded at the table are reliable. It is important to note that due to large deviations, exit happens in long time scales, which implies that reliable estimates, especially when T is large, are essential. § IMPORTANCE SAMPLING FOR ROUGH ENERGY LANDSCAPESIn Section <ref>, we reviewed some of the practical issues that come up whenone is trying to apply importance sampling techniques to metastable dynamics. While in Section <ref> we ignored the effect of multiple scales, the goal of this section is to address the role of multiple scales in the design of asymptotically optimal importance sampling schemes. A particular model of interest in chemical physics is the first order Langevin equation (<ref>).Let us considerdX_t^ϵ,δ=[ -ϵ/δ∇ Q( X_t^ϵ,δ/δ) -∇ V( X_t^ϵ,δ)] dt+√(ϵ)√(2D)dW_t,X_0^ϵ,δ=y, , 0<ϵ,δ≪ 1,wherethe two-scale potential is composed by a large-scale part, V(x), and a fluctuating part, ϵ Q(x/δ). If Q is periodic then we have a periodic environment, whereas if Q is random then we have a random environment. Models like (<ref>) can be used to model rough energy landscapes <cit.>.As it has been suggested (e.g., <cit.>), the associated energy landscapes of certain biomolecules can be rugged (i.e., consist of many local “small" minima within local deep minima separated by barriers of varying heights). When one is interested in rare events, large deviations and Monte Carlo methods are relevant. If Q(y) is periodic,large deviations for multiscale diffusions in periodic environments are obtained in <cit.> for all possible interactions between ϵ and δ, setting the ground for the mathematical formulation of the related importance sampling theory, <cit.>. The novel feature is that the optimal change of measure for importance sampling is not based only on the gradient of the homogenized HJB equation (as in Subsection <ref>). The effect of fluctuations, which is quantified via the solution to the “cell problem" inhomogenization <cit.>, is equally important. The cell problem is the solution to a Poisson type PDE. It is used to define the so called “corrector", which characterizes the first order correction in the approximation of the multiscale HJBby its homogenized limit.Therefore, when compared to the case without multiple scales, one needs more detailed information in order to guarantee, at least, asymptotic optimality.For example, consider model (<ref>) in the case ϵ/δ↑∞. Define the Gibbs measureμ(dy)=1/L e^-Q(y)/Ddy, L=∫_𝕋^d e^-Q(y)/Ddy. Then denote by χ(y)the smooth solution to the “cell problem”-∇ Q(y)·∇χ(y)+ Dtr[∇^2χ(y)] = ∇ Q(y),∫χ(y)μ(dy) = 0.The following large deviations result holds which is a special case of the results of <cit.>. In particular, <cit.> covers the case of general state dependent drift (not necessarily of gradient form) and state dependent diffusion coefficient. Assume that the functions ∇ Q( y ) and ∇ V(x) are continuous and globally bounded, as are their partial derivatives up to order 1 in y and order 2 in x respectively. Let {X^ϵ,δ,ϵ,δ>0} be the unique strong solution to (<ref>). Letr(x) =-∫_𝕋^d(I+∂χ(y)/∂ y)μ(dy) ∇ V(x),q =2D ∫_𝕋^d(I+∂χ(y)/∂ y)(I+∂χ(y) /∂ y)^Tμ(dy),where I denotes the identity matrix. If ϵ/δ→∞, then {X^ϵ,δ,ϵ,δ>0} converges in probability as ϵ,δ→ 0 to the solution of the ODEdX̅_t=r(X̅_t)dtand satisfies a large deviations principle with rate functionS_tT(ϕ)= 1/2∫_t^T(ϕ̇_s-r(ϕ_s)) q^-1(ϕ̇_s-r(ϕ_s))^T dsif ϕ∈𝒜𝒞 ([t,T]),ϕ_t=x+∞ otherwise. In addition, it turns out that an asymptotically efficient change of simulation measure can be constructed analogously to Section <ref>, butbased on thefeedback control (see Theorem 4.1 in <cit.>)v_t=u̅(t,X^ϵ_t,X^ϵ_t/δ),withu̅(t,x,y)=-√(2D)( I+ ∂χ(y)/∂ y) ^T∇_xU̅(t,x). U̅(t,x) satisfies the inequalities in (<ref>) with the homogenized (averaged) coefficients r(x) and q in place of the original ones -∇ V(x) and Γ= √(2D) I (compare with (<ref>)). In particular, the second moment of an estimator with change of measure based on the control v_t by (<ref>) will satisfy (<ref>); this is Theorem 4.1 in <cit.>.Thus, compared to the case without multiscale features, one needs to consider the extra factor ( I+ ∂χ(y)/∂ y), that can be thought as the appropriate weight function, to achieve asymptotic optimality. In the absence of multiple scales, i.e., when Q=0, we have χ=0 and we recover the case studied in Section <ref>. The numerical simulation studies of <cit.> verify the need for accounting for the local environment via the weights ( I+ ∂χ(y)/∂ y) in the change of simulation measure.Before illustrating the performance of this importance sampling scheme in a simulation study, let us demonstrate theoretically the necessity to include the cell problem information in the design of the change of measure. For simplicity purposes, let us restrict attention to dimension one. As we have seen before, the effective diffusion coefficient is given byq=2D ∫_𝕋(1+∂χ/∂ y) ^2μ(dy) In this case, the optimal change of measure is based on the controlu̅(t,x,y)=-√(2D)( 1+ ∂χ(y)/∂ y) ∂_xU̅(t,x). So, let us assume that one is using instead the change of measure, based on the control dictated by the averaged dynamics. Namely, let us assume that the control in question is û(t,x)=-√(q)∂_xU̅(t,x).A verification theorem, see <cit.> for details,shows that one would need a statement of the form"𝔼∫_t^T[√(2D)(1+∂χ/∂ y( X^ϵ,δ_s/δ))-√(q)] ds→ 0" By averaging principle, this is true if√(q)=∫_𝕋√(2D)(1+∂χ(y)/∂ y) μ(dy). However, this is impossible, since(∫(1+∂χ(y)/∂ y)μ(dy))^2≠∫(1+∂χ(y)/∂ y)^2μ(dy). This last property explains mathematically why, the local information, as quantified via the cell problem, needs to be taken into account in the design of importance sampling. In Section <ref>, we will also see numerical evidence of this issue.§.§ A simulation study Let us demonstrate the performance of the importance sampling scheme in a simple simulation study. Consider the one well potential function with diffusion coefficient D=1,V(x)=1/2x^2,Q(y)=cos(y)+sin(y)Assume that we want to estimate θ(ϵ,δ)=𝔼[e^-1/ϵh(X^ϵ,δ_1)], where h(x)=(|x|-1)^2. It is easy to see that we are dealing with a rare event here, as the function h(x) is minimized at |x|=1. Let us compare the following three different estimatorsθ̂_0(ϵ,δ) = 1/K∑_j=1^K[e^-1/ϵh(X^ϵ,δ_1(j))]- standard Monte Carloθ̂_1(ϵ,δ) = 1/K∑_j=1^K[e^-1/ϵh(X̅^ϵ,δ, u̅_1(j))Z^u̅_j]- optimal θ̂_2(ϵ,δ) = 1/K∑_j=1^K[e^-1/ϵh(X̅^ϵ,δ, û_1(j))Z^û_j]- ignores local information where we have defined the controls * u̅(t,x,y)=-√(2)( 1+ ∂χ(y)/∂ y)G_x(t,x)–asymptotically optimal.* û(t,x)=-√(q)G_x(t,x)–based only on the homogenized system.and the likelihood ratio is Z^u_j=dP/dP̅(X̅^ϵ,δ, u_1(j)). Notice that in this case, we can compute1+∂χ(y)/∂ y=e^Q(y)/∫_𝕋 e^Q(y)dy,which justifies the interpretation of the term 1+∂χ(y)/∂ y as the proper weight term needed that takes into account the local information.In Table <ref>, we see simulation studies based on N=10^7 simulation trajectories each, for the estimation of θ(ϵ,δ) using the three different estimators. The measure of comparison is chosen to be the relative error per sample, defined to beρ̂_i(ϵ,δ)≐√(N)√(Var(θ̂_i(ϵ,δ)))/θ̂_1(ϵ,δ).It is clear, that the importance sampling scheme based on the asymptotically optimal change of measure u̅(t,x,y) outperforms the standard Monte Carlo estimator in which no change of measure is being done. It also outperforms,the estimator based solely on the homogenized system, which ignores the local information characterized by solution to the cellproblem χ(y). § IMPORTANCE SAMPLING FOR MULTISCALE DIFFUSIONS IN RANDOM ENVIRONMENTS Let 0<ϵ,δ≪ 1 and consider the process (X^ϵ, Y^ϵ)={(X^ϵ_t, Y^ϵ_t), t∈[0,T]} taking values in the space ℝ^m×ℝ^d-m that satisfies the system of SDEs dX^ϵ_t = [ϵ/δb(Y^ϵ_t,γ)+c(X^ϵ_t ,Y^ϵ_t,γ)] dt+√(ϵ) σ(X^ϵ_t,Y^ϵ_t,γ) dW_t,dY^ϵ_t = 1/δ[ϵ/δf(Y^ϵ_t,γ)+g(X^ϵ_t ,Y^ϵ_t,γ)] dt+√(ϵ)/δ[ τ_1(Y^ϵ_t,γ) dW_t+τ_2(Y^ϵ_t,γ)dB_t], X^ϵ_0 = x_0,Y^ϵ_0=y_0 We assume non-degeneracy of the diffusion coefficients as well 𝒞^1 smoothness and boundedness of the drift and diffusion coefficients. Moreover, we assume that δ=δ(ϵ)↓0 such that ϵ/δ↑∞ as ϵ↓0.(W_t, B_t) is a 2κ-dimensional standard Wiener process. We assume that for each fixed x∈ℝ^m,b(·,γ), c(x,·,γ),σ(x,·,γ),f(·,γ), g(x,·,γ), τ_1(·,γ) and τ_2(·,γ) are stationary and ergodic random fields in an appropriate probability space (Γ,𝒢,ν) with γ∈Γ. Notice that if we choose b(y,γ)=f(y,γ)=-∇_yQ(y,γ)for a periodic function Q(·), c(x,y,γ)=-∇_xV(x),σ(x,y,γ)=τ_1(y,γ)=√(2D) and τ_2(y,γ)=0, and set y_0=x_0/δ, we then get the Langevin equation (<ref>). In particular, if we make these choices, then we simply have Y^ϵ_t=X^ϵ_t/δ and the model can be interpreted asdiffusion in the rough potential ϵ Q(x/δ,γ)+ V(x), where the roughness is dictated byQ. In general, Q may not be modelled as a periodic function. One may model Q as a random field; see the simulation study in Subsection <ref>.§.§ Description of the random environmentThe large deviations and importance sampling results for (<ref>), see <cit.>, are true under certain assumptions on the random medium that we recall here for convenience.We assume that there is a group of measure preserving transformations {τ_y, y∈ℝ^d-m} acting ergodically on Γ that is defined as follows. * τ_y preserves the measure, namely ∀ y∈ℝ^d-m and ∀ A∈𝒢 we have ν(τ_yA)=ν(A). * The action of {τ_y: y∈ℝ^d-m} is ergodic, that is if A=τ_yA for every y∈ℝ^d then ν(A)=0 or 1. *For every measurable function f on (Γ, 𝒢, ν), the function (y,γ)↦ f(τ_yγ) is measurable on (ℝ^d-m×Γ, 𝔹(ℝ ^d-m)⊗𝒢).Let ϕ̃ be a square integrable function in Γ and define the operator T_yϕ̃(γ)=ϕ̃(τ_yγ). The operator T_y·is a strongly continuous group of unitary maps in L^2(Γ), see <cit.>. Denote by D_i the infinitesimal generatorof T_y in the direction i, which is a closed and densely defined generator, see <cit.>.In order to guarantee that the involved functions are ergodic and stationary random fields on ℝ^d-m, for ϕ̃∈ L^2(Γ), let us define the operator ϕ(y,γ)=ϕ̃(τ_yγ).Similarly, for a measurable function ϕ̃:ℝ^m×Γ↦ℝ^m we consider the (locally) stationary random field (x,y) ↦ϕ̃(x,τ_yγ)=ϕ(x,y,γ). Then, it is guaranteed that ϕ(y,γ) (respectively ϕ(x,y,γ)) is a stationary (respectively locally stationary) ergodic random field. The coefficients, b,c,σ,f,g,τ_1,τ_2 of (<ref>) are defined through this procedure and therefore are guaranteed to be ergodic and stationary random fields. For example in the case of the c drift term,we start with an L^2(Γ) function c̃(x,γ) and we define the corresponding coefficientsvia the relation c(x,y,γ)=c̃(x,τ_yγ).For every γ∈Γ, let us the operatorℒ^γ=f(y,γ)∇_y·+tr[( τ_1(y,γ)τ^T_1(y,γ)+τ_2(y,γ)τ^T_2(y,γ))∇_y∇_y·]which is the infinitesimal generator of a Markov process, say Y_t,γ. Using the Markov process Y_t,γ, we can define the so-called environment process, see <cit.>, denoted by γ_t. The environment process γ_t has continuous transition probability densities with respect to the d-dimensional Lebesgue measure, see <cit.>, and is defined by the equationsγ_t =τ_Y_t,γγ γ_0 =τ_y_0γThe infinitesimal generator of the Markov process γ_t is given byL̃=f̃(γ)D·+tr[ ( τ̃_1(γ)τ̃_1^T(γ)+τ̃_2(γ)τ̃_2^T(γ))D^2·]. In order to simplify the presentation, let us assume that the operatorL̃ is in divergence form. In particular, let us set f̃(γ)=-D Q̃(γ) and τ̃_1(γ)=√(2D)θ=constant and τ̃_2(γ)=√(2D)√(1-θ^2)=constant.Then, we can write the unique ergodic invariant measure for the environment process {γ_t}_t≥ 0 in closed form; see <cit.> for more general case which is not necessarily restricted to the gradient case. Denote by 𝔼^νthe expectation operator with respect to the measure ν. Then , themeasure π(dγ) definedon (Γ,𝒢) byπ(dγ)≐m̃(γ)/𝔼^νm̃(· )ν(dγ),with m̃(γ)=exp[-Q̃(γ)/D].is the unique ergodic invariant measure for the environment process {γ_t}_t≥0.Next, we need to define the equivalent to the cell problem in the case of periodic coefficients, also known as the macroscopic problem in the homogenization theory. To do so, we first define ℋ^1=ℋ^1(ν) to be the Hilbert spaceequipped with the inner product(f̃,g̃)_1=∑_i=1^d(D_if̃,D_ig̃).Let us consider ρ>0 and consider the following problem on Γρχ̃_ρ-L̃χ̃_ρ=b̃. Under the condition b̃∈ L^2(ν) with ‖b̃‖_ℋ^-1<∞,Lax-Milgram lemma, see <cit.>, guarantees that equation (<ref>) has a unique weak solution in the abstract Sobolev space ℋ^1 or equivalently in ℋ^1(π). At this point, we note that in the periodic case one also considers (<ref>), but one can then take ρ=0 given that b averages to zero when is integrated against the invariant measure π. However, in the random case, (<ref>) with ρ=0 does not necessarily have a well defined solution (even if b averages to zero when is integrated against the invariant measure π), see for example <cit.>.In the general random case, we consider the equation with ρ>0 and in the end, the homogenization theorem is proven by taking appropriate sequences ρ=ρ(ϵ) such that ρ(ϵ)↓ 0 as ϵ↓ 0. Taking ρ↓ 0 is allowed by the following well known properties of the solution to (<ref>), (see <cit.>), * There is a constant K that is independent of ρ such thatρ𝔼^π[χ̃_ρ(·)]^2 +𝔼^π[Dχ̃_ρ(·)]^2≤ K * χ̃_ρ has an ℋ^1 strong limit, i.e., there exists a χ̃_0∈ℋ^1(π) such thatlim_ρ↓0‖χ̃_ρ(·)-χ̃ _0(·)‖ _1=0 and lim_ρ↓0ρ𝔼^π[χ̃_ρ (·)]^2=0.§.§ Large deviations and importance sampling theory for diffusion in random environments.Now that we have defined the random environment and explained its properties, let us review the related large deviations and importance sampling theory from <cit.>. Set for notational convenience ξ̃=Dχ̃_0. Let {(X^ϵ,γ,Y^ϵ,γ),ϵ>0} be, for fixed γ∈Γ, the unique strong solution to (<ref>). Assume non-degeneracy of the diffusion coefficients as well as 𝒞^1 smoothness and boundedness of the drift and diffusion coefficients. Consider the regime where ϵ,δ↓ 0 such that ϵ/δ↑∞. Then, {X^ϵ,γ,ϵ>0} converges in probability, almost surely with respect to the random environmentγ∈Γ, as ϵ,δ↓ 0 to the solution of the ODEdX̅_t=r(X̅_t)dtand satisfies, almost surely with respect to γ∈Γ, the large deviations principle with rate functionS_t_0T(ϕ)= 1/2∫_t_0^T(ϕ̇_s-r(ϕ_s))^Tq^-1(ϕ_s )(ϕ̇_s-r(ϕ_s))dsif ϕ∈𝒜𝒞 ([t_0,T]) and ϕ_t_0=x_0+∞ otherwise.wherer(x) =lim_ρ↓0𝔼^π[c̃(x,·) +Dχ̃_ρ(·) g̃(x,·)]=𝔼^π[c̃(x,·)+ξ̃ (·)g̃(x,·)]q(x) =lim_ρ↓0𝔼^π[(σ̃(x,·)+Dχ̃_ρ (·)τ̃_1(·))(σ̃(x,·)+Dχ̃_ρ (·)τ̃_1(·))^T+(Dχ̃_ρ (·)τ̃_2(·))(Dχ̃_ρ (·)τ̃_2(·))^T]=𝔼^π[(σ̃(x,·)+ξ̃(·)τ̃_1(·))(σ̃(x,·)+ξ̃ (·)τ̃_1(·))^T+(ξ̃ (·)τ̃_2(·))(ξ̃ (·)τ̃_2(·))^T]and ρ=ρ(ϵ)=δ^2/ϵ.Notice that the coefficients r(x) and q(x) are obtained by homogenizing (<ref>) by taking δ↓ 0 with ϵ fixed. Theform of the action functional can be recognized as the one that would come up when considering large deviations forthe homogenized system. This is also implied by the fact that δ goes to zero faster than ϵ, sinceϵ/δ↑∞.We also remark here that if b=0, then χ_ρ=0. In this caser(x),q(x) take the simplified forms r(x)=𝔼^π[c̃(x,·)] and q(x)=𝔼^π[σ̃(x,·) σ̃(x,·)^T].Turning now to importance sampling, given controls u_1 and u_2 one considers the controlled dynamics under the importance sampling measure ℙ̅dX̅^ϵ_s = [ϵ/δb(Y̅^ϵ_s,γ)+c(X̅^ϵ_s ,Y̅^ϵ_s,γ)+σ(X̅_s^ϵ,Y̅_s^ϵ,γ)u_1(s)] dt+√(ϵ) σ(X̅^ϵ_s,Y̅^ϵ_s,γ) dW̅_s, dY̅^ϵ_s = 1/δ[ϵ/δf(Y̅^ϵ_s,γ)+g(X̅^ϵ_s ,Y̅^ϵ_s,γ)+τ_1(Y̅^ϵ_s,γ)u_1(s)+ τ_2(Y̅^ϵ_s,γ)u_2(s)] dt +√(ϵ)/δ[ τ_1(Y̅^ϵ_s,γ) dW̅_s+τ_2(Y̅^ϵ_s,γ)dB̅_s], X̅^ϵ_t_0 = x_0,Y̅^ϵ_t_0=y_0 where (v_1(s), v_2(s)) denote the first and second component of the controlu(s,X̅^ϵ_s,Y̅^ϵ_s)=(u_1(s,X̅^ϵ_s,Y̅^ϵ_s), u_2(s,X̅^ϵ_s,Y̅^ϵ_s)). Then, for a given cost function h(x), under ℙ̅Δ^ϵ,γ(t_0,x_0,y_0)=exp{-1/ϵh(X̅^ϵ_T)}dℙ/dℙ̅(X̅^ϵ, Y̅^ϵ),is an unbiased estimator for 𝔼[exp{-1/ϵh(X^ϵ_T)}].Consider next the HamiltonianH(x,p)=<r(x), p>-1/2‖ q^1/2(x)p‖^2 with r(x),q(x) the coefficients defined in Theorem <ref> and consider the HJB equation associated to this Hamiltonian, letting U̅(t,x) be a smooth subsolution to it (analogously to Section <ref> with r(x) and q(x) in place of -∇ V(x) and Γ respectively). Then, the following theorem guarantees at least logarithmic asymptotically good performance.Let {(X^ϵ_s, Y^ϵ_s),ϵ>0} be the solution to (<ref>) for s∈[t_0,T] with initial point (x_0,y_0) at time t_0.Consider a non-negative, bounded and continuous function h:ℝ^m↦ℝ. Let U̅(s,x) be a subsolution to the associated HJB equation that has continuous derivatives up to order 1 in t and order 2 in x, and the first and second derivatives in x are uniformly bounded. Assume non-degeneracy of the diffusion coefficients as well 𝒞^1 smoothness and boundedness of the drift and diffusion coefficients. In the general case where b≠ 0, consider ρ>0 and define the (random) feedback control u_ρ(s,x,y,γ)=(u_1,ρ(s,x,y,γ), u_2,ρ(s,x,y,γ)) byu_ρ(s,x,y,γ)=(-(σ+Dχ_ρτ_1)^T(x,y,γ)∇_xU̅(s,x), -(Dχ_ρτ_2)^T(y,γ)∇_xU̅(s,x)) Then for ρ=ρ(ϵ)=δ^2/ϵ↓ 0 we have that almost surely in γ∈Γlim inf_ϵ→0-ϵln Q^ϵ,γ(t_0,x_0,y_0;u_ρ(·))≥ G(t_0,x_0)+U̅(t_0,x_0). If b=0, then set u(s,x,y,γ)=(-σ^T(x,y,γ)∇_xU̅(s,x), 0) and (<ref>) holds with u_ρ(·)=u(·).§.§ A simulation studyConsider for instance the case of Example <ref>dX^ϵ,δ_t=-∇ V^ϵ(X^ϵ,δ_t,X^ϵ,δ_t/δ)dt+√(2ϵ)dW_t,where the potential function V^ϵ(x,x/δ)=ϵ Q(x/δ)+V(x). Q(y) is a stationary ergodic random field on a probability space (𝒳,𝒢,ν). We may consider for instance V(x)=1/2x^2 andQ(y) mean zero Gaussian with E^ν[Q(x)Q(y)]=exp[-|x-y|^2] Making the connection with (<ref>), the fast Y motion essentially is Y=X/δ. Referring to Theorems <ref> and <ref> we have r(x)=-V^'(x)/(KK̂) and q=2/(KK̂) where K=E^ν[e^-Q(z)], K̂=E^ν[e^Q(z)]. Given a classical subsolution U̅, one expects that the corresponding change of simulation measure that guarantees at least asymptotic optimality, is based on the control u̅(s,x,y,γ)=(-√(2)(1+∂χ(y,γ)/∂ y)U̅_x(s,x),0) where one can compute that the weight function is 1+∂χ(y,γ)/∂ y=e^Q(y,γ)/K̂. Note that in contrast to the periodic case, the control u is random in that it implicitly depends on γ∈Γ, via the random field Q(y,γ). Assume that we want to estimateθ^ϵ,δ=P[X^ϵ,δ hits1before0|X^ϵ,δ_0=0.1] As in Subsection <ref>, we compare the asymptotical optimal change of measure with standard Monte Carlo, which corresponds to no change of measure, and with the importance sampling that corresponds to the change of measure based only on the homogenized problem, which ignores the macroscopic problem. Based on 10^7 trajectories, we have the following simulation data It is clear, that the importance sampling scheme based on the asymptotically optimal change of measure u̅(t,x,y,γ) outperforms the standard Monte Carlo estimator in which no change of measure is being done. It also outperforms,the estimator based solely on the homogenized system, which ignores the local information characterized by solution to the macroscopic problem. Of course, this behavior is parallel to the behavior observed in the periodic case of Subsection <ref>. Additional simulation studies can be found in <cit.>.In <cit.>, the interested reader can find further simulation studies in the case of the general model (<ref>) where one does not necessarily have the Y motion to be X/δ. However, we do point out that the theoretical results of <cit.> are valid for the system (<ref>) where the process (X^ϵ,Y^ϵ) has initial point (x_0,y_0) and both x_0 and y_0 are of order one as δ↓ 0. This is not exactly the same to the case where Y=X/δ, as then y_0=x_0/δ, which is no longer of order one as δ↓ 0. But, simulation studies, as the one presented in Table <ref>, indicate that the theoretical results should be also valid for the Y=X/δ case.§ IMPORTANCE SAMPLING FOR METASTABLE MULTISCALE PROCESSES AND FURTHER CHALLENGESIn Section <ref> we elaborated on the effects of rest points and metastable dynamics on importance sampling schemes. The end conclusion was that extra care is needed when stable or unstable equilibrium points are in the domain of interest. In this case, asymptotic optimality is not enough in that asymptotically optimal schemes may not perform well in practice unless one goes to really small values of ϵ, in which case the events may be too rare to be of any practical interest. Then, in Section <ref> and <ref> we summarized the issues that come up in the design of asymptotically efficient importance sampling schemes when the dynamics have multiple scales.In <cit.> we have systematically addressed the effects of rest points onto the design of importance sampling schemes and have identified what the main issues are. In <cit.>, we have suggested a potential provably appropriate remedy to the issue, by constructions as the ones mentioned in Section <ref>. The subsolution constructed there effectively yields a very good approximation to the zero variance change of measure.Even though, the constructions in <cit.> work provably well pre-asymptotically and asymptotically and do not degrade as parameters such as the time horizon T getting large, the performance in higher dimensions can be worse than the correspondingperformance in the lower-dimensional cases. While this is expected to be the case as the dimension gets larger, due to further approximations and simplifications that need to be made, there is a clear room for improvement here. This is part of ongoing work of the author and we refer the interested reader to <cit.> for some results in the infinitely dimensional small noise SPDE case.Moreover, it is clear that the constructions of Sections <ref> and <ref> guarantee only asymptotic optimality. If in addition to multiscale dynamics one has to also face metastability, then, as it was seen in Section <ref>, theoretical asymptotic optimality is not sufficient for good numerical performance. One can of course combine the results of Section <ref> with those of Sections <ref> and <ref>. To be more precise, one can combine the results of <cit.> with those of <cit.>. In practice, one can just use the changes of measure as indicated in <cit.> that guarantee asymptotic optimality, but construct the subsolution U̅(t,x) as indicated in <cit.>. 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Acad. Sci. USA, Vol. 85, (1988), pp. 2029-2030. | http://arxiv.org/abs/1707.08868v1 | {
"authors": [
"Konstantinos Spiliopoulos"
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"categories": [
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"primary_category": "math.PR",
"published": "20170727140130",
"title": "Importance sampling for metastable and multiscale dynamical systems"
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apsrev | http://arxiv.org/abs/1707.08433v3 | {
"authors": [
"Jean David Wurtz",
"Chiu Fan Lee"
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"published": "20170726133551",
"title": "Chemical reaction-controlled phase separated drops: Formation, size selection, and coarsening"
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A New Framework for Synthetic Aperture Sonar Micronavigation Salvatore Caporale and Yvan Petillot S. Caporaleand Y. Petillot are with the Institute of Sensors, Signals and Systems, Heriot-Watt University, Edinburgh, Scotland, UK (e-mail: [email protected]; [email protected]). December 30, 2023 =================================================================================================================================================================================================================================== In this paper, we consider smooth convex optimization problems with simple constraints and inexactness in the oracle information such as value, partial or directional derivatives of the objective function. We introduce a unifying framework, which allows to construct different types of accelerated randomized methods for such problems and to prove convergence rate theorems for them. We focus on accelerated random block-coordinate descent, accelerated random directional search, accelerated random derivative-free method and, using our framework, provide their versions for problems with inexact oracle information. Our contribution also includes accelerated random block-coordinate descent with inexact oracle and entropy proximal setup as well as derivative-free version of this method. Moreover, we present an extension of our framework for strongly convex optimization problems. § INTRODUCTION In this paper, we consider smooth convex optimization problems with simple constraints and inexactness in the oracle information such as objective value, partial or directional derivatives of the objective function. Different types of randomized optimization algorithms, such as random coordinate descent or stochastic gradient descent for empirical risk minimization problem, have been extensively studied in the past decade with the main application being large-scale convex optimization problems. Our main focus in this paper is on accelerated randomized methods: random block-coordinate descent, random directional search, random derivative-free method. As opposed to non-accelerated methods, these methods have complexity proportional to 1/√() iterations to achieve objective function residual , as opposed to 1/ for non-accelerated methods. Accelerated random block-coordinate descent method was first proposed in <cit.>, which was the starting point for active research in this direction. The idea of the method is, on each iteration, to randomly choose a block of coordinates in the decision variable and make a step using the derivative of the objective function with respect to the chosen coordinates. Accelerated random directional search and accelerated random derivative-free method were first proposed in 2011 and published recently in <cit.>, but there was no extensive research in this direction. The idea of random directional search is to use a projection of the objective's gradient onto a randomly chosen direction to make a step on each iteration. Random derivative-free method uses the same idea, but random projection of the gradient is approximated by finite-difference, i.e. the difference of values of the objective function in two close points. This also means that it is a zero-order method which uses only function values to make a step.Existing accelerated randomized methods have different convergence analysis. This motivated us to pose the main question, we address in this paper, as follows. Is it possible to find a crucial part of the convergence rate analysis and use it to systematically construct new accelerated randomized methods? To some extent, our answer is "yes". We determine three main assumptions and use them to prove convergence rate theorem for our generic accelerated randomized method. Our framework allows both to reproduce known and to construct new accelerated randomized methods. The latter include new accelerated random block-coordinate descent with inexact block derivatives and entropy proximal setup.§.§ Related WorkIn the seminal paper <cit.>, the author proposed random block-coordinate descent for convex optimization problems with simple convex separable constraints and accelerated random block-coordinate descent for unconstrained convex optimization problems. In <cit.>, the authors proposed accelerated random block-coordinate descent with non-uniform probability of choosing a particular block of coordinates. They also developed an efficient implementation without full-dimensional operations on each iteration. The authors of <cit.> introduced accelerated block-coordinate descent for composite optimization problems, which include problems with separable constraints. Later, the paper <cit.> extended this method for strongly convex problems. The papers <cit.> (this work first appeared in May, 2015), <cit.> and <cit.> proposed an accelerated block-coordinate descent with complexity, which does not explicitly depend on the problem dimension. We also mention special type of accelerated block-coordinate descent developed in <cit.> for empirical risk minimization problems. All these accelerated block-coordinate descent methods work in Euclidean setup, when the norm in each block is Euclidean and is defined using some positive semidefinite matrix. Non-accelerated block-coordinate methods, but with non-euclidean setup, were considered in <cit.>. All the mentioned methods rely on exact block derivatives and exact projection on each step. Inexact projection in the context of non-accelerated random coordinate descent was considered in <cit.>. Research on accelerated random directional search and accelerated random derivative/free methods for smooth problems started in <cit.>, where also non-smooth problems were considered. Derivative-free methods for non-smooth problems were further developed in <cit.> for the case of exact computations and in <cit.> for the case of inexact function values. Non-accelerated gradient-free methods for smooth problems were developed in <cit.> for the case of exact objective value evaluation and in <cit.> for inexact values, see also extensive reviews <cit.>. Accelerated gradient-free methods with inexact oracle were introduced in <cit.> and later extended for the case of random directional search in <cit.>. The main difference with this work is that here we develop a unifying framework for all the three types of accelerated randomized methods including coordinate descent method.We should also mention that there are other accelerated randomized methods in <cit.>. Most of these methods were developed deliberately for empirical risk minimization problems and do not fall in the scope of this paper. We also list the following works, which consider accelerated full-gradient methods with inexact oracle <cit.>.§.§ Our Approach and ContributionsOur framework has two main components, namely, Randomized Inexact Oracle and Unified Accelerated Randomized Method. The starting point for the definition of our oracle is a unified view on random directional search and random block-coordinate descent. In both these methods, on each iteration, a randomized approximation for the objective function's gradient is calculated and used, instead of the true gradient, to make a step. This approximation for the gradient is constructed by a projection on a randomly chosen subspace. For random directional search, this subspace is the line along a randomly generated direction. As a result a directional derivative in this direction is calculated. For random block-coordinate descent, this subspace is given by randomly chosen block of coordinates and block derivative is calculated. One of the key features of these approximations is that they are unbiased, i.e. their expectation is equal to the true gradient. We generalize two mentioned approaches by allowing other types of random transformations of the gradient for constructing its randomized approximation. The inexactness of our oracle is inspired by the relation between derivative-free method and directional search. In the framework of derivative-free methods, only the value of the objective function is available for use in an algorithm. At the same time, if the objective function is smooth, the directional derivative can be well approximated by the difference of function values at two points which are close to each other. Thus, in the context of zero-order optimization, one can calculate only an inexact directional derivative. Hence, one can construct only a biased randomized approximation for the gradient when a random direction is used. We combine previously mentioned random transformations of the gradient with possible inexactness of this transformations to construct our Randomized Inexact Oracle, which we use in our generic algorithm to make a step on each iteration. The basis of our generic algorithm is Similar Triangles Method of <cit.> (see also <cit.>), which is an accelerated gradient method with only one proximal mapping on each iteration, this proximal mapping being essentially the Mirror Descent step. The notable point is that, we only need to substitute the true gradient with our Randomized Inexact Oracle and slightly change one step in the Similar Triangles Method, to obtain our generic accelerated randomized algorithm, which we call Unified Accelerated Randomized Method (UARM), see Algorithm <ref>. We prove convergence rate theorem for UARM in two cases: the inexactness of Randomized Inexact Oracle can be controlled and adjusted on each iteration of the algorithm, the inexactness can not be controlled.We apply our framework to several particular settings: random directional search, random coordinate descent, random block-coordinate descent and their combinations with derivative-free approach. As a corollary of our main theorem, we obtain both known and new results on the convergence of different accelerated randomized methods with inexact oracle. To sum up, our contributions in this paper are as follows. * We introduce a general framework for constructing and analyzing different types of accelerated randomized methods, such as accelerated random directional search, accelerated block-coordinate descent, accelerated derivative-free methods. Our framework allows to obtain both known and new methods and their convergence rate guarantees as a corollary of our main Theorem <ref>. * Using our framework, we introduce new accelerated methods with inexact oracle, namely, accelerated random directional search, accelerated random block-coordinate descent, accelerated derivative-free method. To the best of our knowledge, such methods with inexact oracle were not known before. See section <ref>. * Based on our framework, we introduce new accelerated random block/coordinate descent with inexact oracle and non-euclidean setup, which was not done before in the literature. The main application of this method is minimization of functions on a direct product of large number of low-dimensional simplexes. See subsection <ref>. * We introduce new accelerated random derivative-free block-coordinate descent with inexact oracle and non-euclidean setup. Such method was not known before in the literature. Our method is similar to the method in the previous item, but uses only finite-difference approximations for block derivatives. See subsection <ref>.The rest of the paper is organized as follows. In section <ref>, we provide the problem statement, motivate and make three our main assumptions, illustrate them by random directional search and random block-coordinate descent. In section <ref>, we introduce our main algorithm, called Unified Accelerated Randomized Method, and, based on stated general assumptions, prove convergence rate in Theorem <ref>. Section <ref> is devoted to applications of our general framework for different particular settings, namely * Accelerated Random Directional Search (<ref>), * Accelerated Random Coordinate Descent (<ref>), * Accelerated Random Block-Coordinate Descent (<ref>), * Accelerated Random Derivative-Free Directional Search (<ref>), * Accelerated Random Derivative-Free Coordinate Descent (<ref>), * Accelerated Random Derivative-Free Block-Coordinate Descent (<ref>). * Accelerated Random Derivative-Free Block-Coordinate Descent with Random Approximations for Block Derivatives (<ref>).§ PRELIMINARIES §.§ Notation Let finite-dimensional real vector space E be a direct product of n finite-dimensional real vector spaces E_i, i=1,...,n, i.e. E=⊗_i=1^n E_i and dim E_i = p_i, i=1,...,n. Denote also p=∑_i=1^n p_i. Let, for i=1,...,n, E_i^* denote the dual space for E_i. Then, the space dual to E is E^*=⊗_i=1^n E_i^*. Given a vector x^(i)∈ E_i for some i ∈ 1,...,n, we denote as [x^(i)]_j its j-th coordinate, where j ∈ 1,...,p_i. To formalize the relationship between vectors in E_i, i=1,...,n and vectors in E, we define primal partition operators U_i : E_i → E, i=1,...,n, by identityx = (x^(1),...,x^(n)) = ∑_i=1^n U_i x^(i),x^(i)∈ E_i,i=1,...,n,x ∈ E.For any fixed i ∈ 1,...,n, U_i maps a vector x^(i)∈ E_i, to the vector (0,....,x^(i),...,0) ∈ E. The adjoint operator U_i^T: E^* → E_i^*, then, is an operator, which, maps a vector g = (g^(1),...,g^(i),..., g^(n)) ∈ E^*, to the vector g^(i)∈ E_i^*.Similarly, we define dual partition operators U_i : E_i^* → E^*, i=1,...,n, by identityg = (g^(1),...,g^(n)) = ∑_i=1^n U_i g^(i),g^(i)∈ E_i^*,i=1,...,n,g ∈ E^*.For all i=1,...,n, we denote the value of a linear function g^(i)∈ E_i^* at a point x^(i)∈ E_i by g^(i), x^(i)_i. We define g, x= ∑_i=1^ng^(i), x^(i)_i,x ∈ E,g ∈ E^*.For all i=1,...,n, let ·_i be some norm on E_i and ·_i,* be the norm on E_i^* which is dual to ·_ig^(i)_i,* = max_x^(i)_i ≤ 1 g^(i), x^(i)_i.Given parameters β_i ∈^n_++, i=1,...,n, we define the norm of a vector x=(x^(1),...,x^(n)) ∈ E as x_E^2 = ∑_i=1^n β_i x^(i)_i^2. Then, clearly, the dual norm of a vector g=(g^(1),...,g^(n)) ∈ E^* is g_E,*^2 = ∑_i=1^n β_i^-1g^(i)_i^2. Throughout the paper, we consider optimization problem with feasible set Q, which is assumed to be given as Q=⊗_i=1^n Q_i ⊆ E, where Q_i ⊆ E_i, i=1,...,n are closed convex sets. To have more flexibility and be able to adapt algorithm to the structure of sets Q_i, i=1,...,n, we introduce proximal setup, see e.g. <cit.>. For all i=1,...,n, we choose a prox-function d_i(x^(i)) which is continuous, convex on Q_i and * admits a continuous in x^(i)∈ Q_i^0 selection of subgradients ∇ d_i(x^(i)), where x^(i)∈ Q_i^0 ⊆ Q_i, and Q_i^0is the set of all x^(i), where ∇ d_i(x^(i)) exists; * is 1-strongly convex on Q_i with respect to ·_i, i.e., for any x^(i)∈ Q_i^0, y^(i)∈ Q_i, it holds that d_i(y^(i))-d_i(x^(i)) -∇ d_i(x^(i)) ,y^(i)-x^(i)_i ≥1/2y^(i)-x^(i)_i^2. We define also the corresponding Bregman divergence V_i[z^(i)] (x^(i)) := d_i(x^(i)) - d_i(z^(i)) - ∇ d_i(z^(i)), x^(i) - z^(i)_i, x^(i)∈ Q_i, z^(i)∈ Q_i^0, i=1,...,n.It is easy to see that V_i[z^(i)] (x^(i)) ≥1/2x^(i) - z^(i)_i^2,x^(i)∈ Q_i, z^(i)∈ Q_i^0, i=1,...,n. Standard proximal setups, e.g. Euclidean, entropy, ℓ_1/ℓ_2, simplex can be found in <cit.>. It is easy to check that, for given parameters β_i ∈^n_++, i=1,...,n, the functions d(x) = ∑_i=1^n β_i d_i(x^(i)) and V[z] (x) = ∑_i=1^n β_i V_i[z^(i)] (x^(i)) are respectively a prox-function and a Bregman divergence corresponding to Q.Also, clearly,V[z] (x) ≥1/2x - z_E^2,x ∈ Q,z ∈ Q^0 := ⊗_i=1^n Q_i^0.For a differentiable function f(x), we denote by ∇ f(x) ∈ E^* its gradient. §.§ Problem Statement and AssumptionsThe main problem, we consider, is as followsmin_x∈ Q ⊆ Ef(x),where f(x) is a smooth convex function, Q=⊗_i=1^n Q_i ⊆ E, with Q_i ⊆ E_i, i=1,...,n being closed convex sets. We now list our main assumptions and illustrate them by two simple examples. More detailed examples are given in section <ref>. As the first example here, we consider random directional search, in which the gradient of the function f is approximated by a vector ∇ f(x) , ee, where ∇ f(x) , e is the directional derivative in direction e and random vector e is uniformly distributed over the Euclidean sphere of radius 1. Our second example is random block-coordinate descent, in which the gradient of the function f is approximated by a vector U_iU_i^T ∇ f(x), where U_i^T ∇ f(x) is i-th block derivative and the block number i is uniformly randomly sampled from 1,...,n. The common part in both these randomized gradient approximations is that, first, one randomly chooses a subspace which is either the line, parallel to e, or i-th block of coordinates. Then, one projects the gradient on this subspace by calculating either ∇ f(x) , e or U_i^T ∇ f(x). Finally, one lifts the obtained random projection back to the whole space E either by multiplying directional derivative by vector e, or applying dual partition operator U_i. At the same time, in both cases, if one scales the obtained randomized approximation for the gradient by multiplying it by n, one obtains an unbiased randomized approximation of the gradient_e n ∇ f(x) , ee = ∇ f(x), _i n U_iU_i^T ∇ f(x) = ∇ f(x),x ∈ Q.We also want our approach to allow construction of derivative-free methods. For a function f with L-Lipschitz-continuous gradient, the directional derivative can be well approximated by the difference of function values in two close points. Namely, it holds that ∇ f(x) , e = f(x+τ e) - f(x)/τ + o(τ),where τ > 0 is a small parameter. Thus, if only the value of the function is available, one can calculate only inexact directional derivative, which leads to biased randomized approximation for the gradient if the direction is chosen randomly. These three features, namely, random projection and lifting up, unbiased part of the randomized approximation for the gradient, bias in the randomized approximation for the gradient, lead us to the following assumption about the structure of our general Randomized Inexact Oracle.[Randomized Inexact Oracle] We access the function f only through Randomized Inexact Oracle (x), x ∈ Q, which is given by (x) = ρ(^T∇ f(x) + ξ(x)) ∈ E^*, where ρ> 0 is a known constant;is a random "`projection"' operator from some auxiliary space H to E, and, hence, ^T, acting from E^* to H^*, is the adjoint to ; : H^* → E^* is also some random "`reconstruction"' operator; ξ(x) ∈ H^* is a, possibly random, vector characterizing the error of the oracle. The oracle is also assumed to satisfy the following properties ρ^T∇ f(x) = ∇ f(x), ∀ x ∈ Q, ξ(x)_E,*≤δ, ∀ x ∈ Q, where δ≥ 0 is oracle error level. Let us make some comments on this assumption. The nature of the operator ^T is generalization of random projection.For the case of random directional search, H =, ^T: E^* → is given by ^T g =g, e, g ∈ E^*. For the case of random block-coordinate descent, H = E_i, ^T: E^* → E_i^* is given by ^T g = U_i^T g, g ∈ E^*. We assume that there is some additive error ξ(x) in the generalized random projection ^T ∇ f(x). This error can be introduced, for example, when finite-difference approximation of the directional derivative is used. Finally, we lift the inexact random projection ^T ∇ f(x)+ξ(x) back to E by applying operator . For the case of random directional search, : → E^* is given by t = te, t ∈. For the case of random block-coordinate descent, : E_i^* → E^* is given by g^(i) = U_i g^(i), g^(i)∈ E_i^*. The number ρ is the normalizing coefficient, which allows the part ρ^T∇ f(x) to be unbiased randomized approximation for the gradient. This is expressed by equality (<ref>). Finally, we assume that the error in our oracle is bounded, which is expressed by property (<ref>). In our analysis, we consider two cases: the error ξ can be controlled and δ can be appropriately chosen on each iteration of the algorithm; the error ξ can not be controlled and we only know oracle error level δ.Let us move to the next assumption. As said, our generic algorithm is based on Similar Triangles Method of <cit.> (see also <cit.>), which is an accelerated gradient method with only one proximal mapping on each iteration. This proximal mapping is essentially the Mirror Descent step. For simplicity, let us consider here an unconstrained minimization problem in the Euclidean setting. This means that Q_i = E_i = ^p_i, x^(i)_i = x^(i)_2, i=1,...,n. Then, given a point u ∈ E, a number α, and the gradient ∇ f(y) at some point y ∈ E, the Mirror Descent step isu_+ = min_x ∈ E{1/2x-u_2^2 + α∇ f(y),x } = u - α∇ f(y).Now we want to substitute the gradient ∇ f(y) with our Randomized Inexact Oracle (y). Then, we see that the step u_+ = u - α(y) makes progress only in the subspace onto which the gradient is projected, while constructing the Randomized Inexact Oracle. In other words, u - u_+ lies in the same subspace as (y). In our analysis, this is a desirable property and we formalize it as follows. [Regularity of Prox-Mapping] The set Q, norm ·_E, prox-function d(x), and Randomized Inexact Oracle (x) are chosen in such a way that, for any u,y ∈ Q, α >0, the point u_+ = min_x ∈ Q{V[u](x) + α(y),x } satisfies ^T∇ f(y), u - u_+= ∇ f(y), u - u_+ . The interpretation is that, in terms of linear pairing with u - u_+, the unbiased part ^T∇ f(y) of the Randomized Inexact Oracle makes the same progress as the true gradient ∇ f(y).Finally, we want to formalize the smoothness assumption for the function f. In our analysis, we use only the smoothness of f in the direction of u_+ - u, where u ∈ Q and u_+ is defined in (<ref>). Thus, we consider two points x,y ∈ Q, which satisfy equality x = y + a (u_+ - u), where a ∈. For the random directional search, it is natural to assume that f has L-Lipschitz-continuous gradient with respect to the Euclidean norm, i.e. f(x) ≤ f(y) + ∇ f(y), x-y+ L/2x-y_2^2,x,y ∈ Q.Then, if we define x_E^2 = L x_2^2, we obtain that, for our choice x = y + a (u_+ - u),f(x) ≤ f(y) + ∇ f(y), x - y+ 1/2x-y_E^2.Usual assumption for random block-coordinate descent is that the gradient of f is block-wise Lipschitz continuous. This means that, for all i=1,...,n, block derivative f'_i(x) = U_i^T ∇ f(x) is L_i-Lipschitz continuous with respect to chosen norm ·_i, i.e.f'_i(x+U_ih^(i)) - f'_i(x)_i,*≤ L_i h^(i)_i,h^(i)∈ E_i,i=1,...,n,x ∈ Q.By the standard reasoning, using (<ref>), one can prove that, for all i=1,...,n,f(x+U_ih^(i)) ≤ f(x) +U_i^T∇ f(x), h^(i) + L_i/2h^(i)_i^2,h^(i)∈ E_i,x ∈ Q.In block-coordinate setting, (x) has non-zero elements only in one, say i-th, block and it follows from (<ref>) that u_+ - u also has non-zero components only in the i-th block. Hence, there exists h^(i)∈ E_i, such that u_+ - u = U_i h_i and x = y + a U_i h^(i). Then, if we define x_E^2 = ∑_i=1^n L_i x^(i)_i^2, we obtainf(x)= f(y + a U_i h^(i)) (<ref>)≤f(y) +U_i^T∇ f(y), a h^(i) + L_i/2a h^(i)_i^2= f(y) + ∇ f(y), a U_i h^(i) + 1/2a U_i h^(i)_E^2 = f(y) + ∇ f(y), x - y+ 1/2x-y_E^2.We generalize these two examples and assume smoothness of f in the following sense. [Smoothness] The norm ·_E is chosen in such a way that, for any u,y ∈ Q, a ∈, if x = y + a (u_+ - u) ∈ Q, then f(x) ≤ f(y) + ∇ f(y), x - y+ 1/2x-y_E^2. § UNIFIED ACCELERATED RANDOMIZED METHOD In this section, we introduce our generic Unified Accelerated Randomized Method, which is listed as algorithm <ref> below, and prove <ref>, which gives its convergence rate. The method is constructed by a modification of Similar Triangles Method (see <cit.>) and, thus, inherits part of its name. Algorithm <ref> is correctly defined in the sense that, for all k≥ 0, x_k,y_k ∈ Q. The proof is a direct generalization of Lemma 2 in <cit.>. By definition (<ref>), for all k≥ 0, u_k ∈ Q. If we prove that, for all k≥ 0, x_k ∈ Q, then, from (<ref>), it follows that, for all k≥ 0, y_k ∈ Q. Let us prove that, for all k≥ 0, x_k is a convex combination of u_0 … u_k, namely x_k = ∑_l=0^kγ_k^l u_l, where γ_0^0 = 1, γ_1^0 = 0, γ_1^1 = 1, and for k ≥ 1, γ_k+1^l =(1 - α_k+1/A_k+1)γ_k^l, l = 0,…,k - 1α_k+1/A_k+1(1 - ρα_k/A_k) + ρ(α_k/A_k - α_k+1/A_k+1),l = kρα_k+1/A_k+1,l = k+1. Since, x_0=u_0, we have that γ_0^0 = 1. Next, by (<ref>), we have x_1 = y_1 + ρα_1/A_1 (u_1 - u_0) = u_0 + ρα_1/A_1 (u_1 - u_0) = (1 - ρα_1/A_1) u_0 + ρα_1/A_1 u_1. Solving the equation (<ref>) for k=0, and using the choice α_0 = 1 - 1/ρ, we obtain that α_1 = 1/ρ and α_1/A_1 (<ref>)=α_1/ρ^2 α_1^2 = 1/ρ. Hence, x_1=u_1 and γ_1^0 = 0, γ_1^1 = 1. Let us now assume that x_k = ∑_l=0^kγ_k^l u_l and prove that x_k+1 is also a convex combination with coefficients, given by (<ref>). From (<ref>), (<ref>), we have x_k+1 = y_k+1 + ρα_k+1/A_k+1 (u_k+1 - u_k) = α_k+1u_k + A_k x_k/A_k+1 + ρα_k+1/A_k+1 (u_k+1 - u_k) = A_k/A_k+1x_k + (α_k+1/A_k+1 - ρα_k+1/A_k+1)u_k + ρα_k+1/A_k+1u_k+1= (1 - α_k+1/A_k+1) ∑_l = 0^kγ_k^l u_l + (α_k+1/A_k+1 - ρα_k+1/A_k+1)u_k + ρα_k+1/A_k+1u_k+1. Note that all the coefficients sum to 1. Next, we have x_k+1 = (1 - α_k+1/A_k+1) ∑_l = 0^k-1γ_k^l u_l+ (γ_k^k (1 - α_k+1/A_k+1) + (α_k+1/A_k+1 - ρα_k+1/A_k+1))u_k + ρα_k+1/A_k+1u_k+1= (1 - α_k+1/A_k+1) ∑_l = 0^k-1γ_k^l u_l+ (ρα_k/A_k(1 - α_k+1/A_k+1) + (α_k+1/A_k+1 - ρα_k+1/A_k+1))u_k + ρα_k+1/A_k+1u_k+1=(1 - α_k+1/A_k+1) ∑_l = 0^k-1γ_k^l u_l+ (α_k+1/A_k+1(1 - ρα_k/A_k) + ρ(α_k/A_k -α_k+1/A_k+1))u_k + ρα_k+1/A_k+1u_k+1. So, we see that (<ref>) holds for k+1. It remains to show that γ_k+1^l ≥ 0, l = 0,…,k+1. For γ_k+1^l, l = 0,…,k-1 and γ_k+1^k+1 it is obvious. From (<ref>), we have α_k+1 = 1+√(1+4 ρ^2 A_k)/2 ρ^2. Thus, since {A_k}, k≥ 0 is non-decreasing sequence, {α_k+1}, k≥ 0 is also non-decreasing. From (<ref>), we obtain α_k+1/A_k+1 = α_k+1/ρ^2α_k+1^2, which means that this sequence is non-increasing. Thus, α_k/A_k≥α_k+1/A_k+1 and α_k/A_k≤α_1/A_1≤1/ρ for k ≥ 1. These inequalities prove that γ_k+1^k≥ 0. Let the sequences {x_k, y_k, u_k, α_k, A_k }, k≥ 0 be generated by algorithm <ref>. Then,for all u ∈ Q, it holds that α_k+1(y_k+1), u_k - u≤ A_k+1(f(y_k+1) - f(x_k+1)) + V[u_k](u) - V[u_k+1](u)+ α_k+1ρξ(y_k+1), u_k - u_k+1. Using assumptions <ref> and <ref> with α = α_k+1, y=y_k+1, u=u_k, u_+ = u_k+1, we obtain α_k+1(y_k+1), u_k - u_k+1 (<ref>)=α_k+1ρ(^T∇ f(y_k+1) + ξ(y_k+1)), u_k - u_k+1 (<ref>)=α_k+1ρ∇ f(y_k+1), u_k - u_k+1+ α_k+1ρξ(y_k+1), u_k - u_k+1 (<ref>)= A_k+1∇ f(y_k+1), y_k+1 - x_k+1+ α_k+1ρξ(y_k+1), u_k - u_k+1. Note that, from the optimality condition in (<ref>), for any u ∈ Q, we have ∇ V[u_k](u_k+1) + α_k+1(y_k+1), u - u_k+1≥ 0. By the definition of V[u](x), we obtain, for any u ∈ Q, V[u_k](u) - V[u_k+1](u) - V[u_k](u_k+1)=d(u) - d(u_k) - ∇ d(u_k), u-u_k - ( d(u) - d(u_k+1) - ∇ d(u_k+1), u-u_k+1) - ( d(u_k+1) - d(u_k) - ∇ d(u_k), u_k+1-u_k )= ∇ d(u_k) - ∇ d(u_k+1) , u_k+1 - u=- ∇ V[u_k](u_k+1) , u_k+1 - u . Further, for any u ∈ Q, by assumption <ref>, α_k+1(y_k+1), u_k - u= α_k+1(y_k+1), u_k - u_k+1 + α_k+1(y_k+1), u_k+1 - u (<ref>)≤α_k+1(y_k+1), u_k - u_k+1 +-∇ V[u_k](u_k+1) , u_k+1 - u (<ref>)=α_k+1(y_k+1), u_k - u_k+1 + V[u_k](u) - V[u_k+1](u) - V[u_k](u_k+1) (<ref>)≤α_k+1(y_k+1), u_k - u_k+1 + V[u_k](u) - V[u_k+1](u) - 1/2u_k-u_k+1_E^2 (<ref>), (<ref>)= A_k+1∇ f(y_k+1), y_k+1 - x_k+1+ α_k+1ρξ(y_k+1), u_k - u_k+1 ++ V[u_k](u) - V[u_k+1](u) - A_k+1^2/2ρ^2 α_k+1^2y_k+1 - x_k+1_E^2 (<ref>)= A_k+1(∇ f(y_k+1), y_k+1 - x_k+1 - 1/2y_k+1 - x_k+1_E^2) ++ α_k+1ρξ(y_k+1), u_k - u_k+1 + V[u_k](u) - V[u_k+1](u) (<ref>), (<ref>)≤ A_k+1(f(y_k+1) - f(x_k+1))+ V[u_k](u) - V[u_k+1](u) + +α_k+1ρξ(y_k+1), u_k - u_k+1. In the last inequality, we used assumption <ref> with a=ρα_k+1/A_k+1, x=x_k+1, y=y_k+1, u=u_k, u_+=u_k+1. Let the sequences {x_k, y_k, u_k, α_k, A_k }, k≥ 0 be generated by Algorithm <ref>. Then,for all u ∈ Q, it holds that α_k+1∇ f(y_k+1), u_k - u≤A_k+1(f(y_k+1) - _k+1f(x_k+1)) + V[u_k](u)- _k+1V[u_k+1](u) + _k+1α_k+1ρξ(y_k+1), u - u_k+1, where _k+1 denotes the expectation conditioned on all the randomness up to step k. First, for any u∈ Q, by assumption <ref>, _k+1α_k+1(y_k+1), u_k - u (<ref>)=_k+1α_k+1ρ(^T∇ f(y_k+1) + ξ(y_k+1)), u_k - u (<ref>)=α_k+1∇ f(y_k+1), u_k - u + _k+1α_k+1ρξ(y_k+1), u_k - u . Taking conditional expectation _k+1 in (<ref>) of Lemma <ref> and using (<ref>), we obtain the statement of the Lemma. Let the sequences {x_k, y_k, u_k, α_k, A_k }, k≥ 0 be generated by Algorithm <ref>. Then,for all u ∈ Q, it holds that A_k+1_k+1f(x_k+1) - A_kf(x_k) ≤ α_k+1( f(y_k+1) + ∇ f(y_k+1), u - y_k+1) + V[u_k](u) - _k+1V[u_k+1](u) + _k+1α_k+1ρξ(y_k+1), u - u_k+1. For any u ∈ Q, α_k+1∇ f(y_k+1), y_k+1 - u = α_k+1∇ f(y_k+1), y_k+1 - u_k+ α_k+1∇ f(y_k+1), u_k - u (<ref>), (<ref>)= A_k∇ f(y_k+1), x_k - y_k+1 + α_k+1∇ f(y_k+1), u_k - uconv-ty≤ A_k(f(x_k) - f(y_k+1) ) + α_k+1∇ f(y_k+1), u_k - u (<ref>)≤ A_k(f(x_k) - f(y_k+1) ) + A_k+1(f(y_k+1) - _k+1f(x_k+1)) ++ V[u_k](u) - _k+1V[u_k+1](u) + _k+1α_k+1ρξ(y_k+1), u - u_k+1 (<ref>)=α_k+1 f(y_k+1) + A_kf(x_k) - A_k+1_k+1f(x_k+1) + V[u_k](u) - _k+1V[u_k+1](u)+ _k+1α_k+1ρξ(y_k+1), u - u_k+1. Rearranging terms, we obtain the statement of the Lemma. Let the assumptions <ref>, <ref>, <ref> hold. Let the sequences {x_k, y_k, u_k, α_k, A_k }, k≥ 0 be generated by Algorithm <ref>. Let f_* be the optimal objective value and x_* be an optimal point in problem (<ref>). Denote P_0^2 = A_0(f(x_0)-f_*) + V[u_0](x_*). * If the oracle error ξ(x) in (<ref>) can be controlled and, on each iteration, the error level δ in (<ref>) satisfies δ≤P_0/4ρ A_k, then, for all k ≥ 1, f(x_k) - f_*≤3P_0^2/2A_k, wheredenotes the expectation with respect to all the randomness (up to step k in this case). * If the oracle error ξ(x) in (<ref>) can not be controlled, then, for all k ≥ 1, f(x_k) - f_*≤2P_0^2/A_k + 4 A_k ρ^2 δ^2. Let us change the counter in Lemma <ref> from k to i, fix u=x_*, take the full expectation in each inequality for i=0,...,k-1 and sum all the inequalities for i=0,...,k-1. Then, A_k f(x_k) - A_0f(x_0) ≤ ∑_i=0^k-1α_i+1( f(y_i+1) + ∇ f(y_i+1), x_* - y_i+1) + V[u_0](x_*) -V[u_k](x_*) + ∑_i=0^k-1α_i+1ρξ(y_i+1), x_* - u_i+1conv-ty, (<ref>), (<ref>)≤ (A_k-A_0)f(x_*) + V[u_0](x_*)-V[u_k](x_*) + ∑_i=0^k-1α_i+1ρδx_* - u_i+1_E. Rearranging terms and using (<ref>), we obtain, for all k ≥ 1, 0 ≤ A_k( f(x_k) - f_* )≤ P_0^2-V[u_k](x_*) + ρδ∑_i=0^k-1α_i+1 R_i+1, where we denoted R_i = u_i-x_*_E, i≥ 0. 1. We first prove the first statement of the Theorem. We have 1/2 R_0^2 = 1/2x_*-u_0_E^2 (<ref>)≤ V[u_0](x_*) (<ref>)≤ P_0^2. Hence, R_0 = R_0 ≤ P_0 √(2)≤ 2 P_0. Let R_i ≤ 2 P_0, for all i=0,...,k-1. Let us prove that R_k≤ 2 P_0. By convexity of square function, we obtain 1/2( R_k)^2 ≤1/2 R_k^2 (<ref>)≤ V[u_k](x_*) (<ref>)≤ P_0^2 + ρδ∑_i=0^k-2α_i+1 2P_0 + α_kρδ R_k (<ref>)= P_0^2 + 2 ρδ P_0 (A_k-1-A_0) + α_kρδ R_k≤ P_0^2 + 2 ρδ P_0 A_k + α_kρδ R_k . Since α_k ≤ A_k, k ≥ 0, by the choice of δ (<ref>), we have 2 ρδ P_0 A_k≤P_0^2/2 and α_kρδ≤ A_kρδ≤P_0/4. So, we obtain an inequality for R_k 1/2( R_k)^2 ≤3P_0^2/2 +P_0/4 R_k. Solving this quadratic inequality in R_k, we obtain R_k≤P_0/4 + √(P_0^2/16+3P_0^2) = 2P_0. Thus, by induction, we have that, for all k ≥ 0, R_k ≤ 2 P_0. Using the bounds R_i ≤ 2 P_0, for all i=0,...,k, we obtain A_k( f(x_k) - f_* ) (<ref>)≤ P_0^2 + ρδ∑_i=0^k-1α_i+1 R_i (<ref>), (<ref>)≤P_0^2 + ρP_0/4ρ A_k· (A_k-A_0) ·2 P_0 ≤3P_0^2/2. This finishes the proof of the first statement of the Theorem. 2. Now we prove the second statement of the Theorem. First, from (<ref>) for k=1, we have 1/2( R_1)^2 ≤1/2 R_1^2 (<ref>)≤ V[u_1](x_*)(<ref>)≤ P_0^2 + ρδα_1 R_1. Solving this inequality in R_1, we obtain R_1≤ρδα_1+ √((ρδα_1 )^2+2P_0^2)≤ 2ρδα_1 +P_0√(2), where we used that, for any a,b ≥ 0, √(a^2+b^2)≤ a +b. Then, P_0^2 + ρδα_1 R_1 ≤P_0^2+ 2 (ρδα_1)^2 +ρδα_1 P_0√(2)≤(P_0 + ρδ√(2) (A_1-A_0) )^2. Thus, we have proved that the inequality P_0^2 + ρδ∑_i=0^k-2α_i+1 R_i+1≤(P_0 + ρδ√(2) (A_k-1-A_0))^2 holds for k=2. Let us assume that it holds for some k and prove that it holds for k+1. We have 1/2( R_k)^2 ≤1/2 R_k^2 (<ref>)≤ V[u_k](x_*)(<ref>)≤ P_0^2 + ρδ∑_i=0^k-2α_i+1 R_i+1 + α_kρδ R_k(<ref>)≤(P_0 + ρδ√(2) (A_k-1-A_0))^2 + α_kρδ R_k. Solving this quadratic inequality in R_k, we obtain R_k ≤α_kρδ + √((α_kρδ)^2+2(P_0 + ρδ√(2) (A_k-1-A_0))^2)≤ 2 α_kρδ +(P_0 + ρδ√(2) (A_k-1-A_0)) √(2), where we used that, for any a,b ≥ 0, √(a^2+b^2)≤ a +b. Further, P_0^2 + ρδ∑_i=0^k-1α_i+1 R_i+1 (<ref>)≤(P_0 + ρδ√(2) (A_k-1-A_0))^2 + ρδα_kR_k(<ref>)≤(P_0 + ρδ√(2) (A_k-1-A_0))^2+ 2 (ρδα_k)^2 +ρδα_k (P_0 + ρδ√(2) (A_k-1-A_0)) √(2)≤(P_0 + ρδ√(2) (A_k-1-A_0) + ρδα_k √(2))^2 = (P_0 + ρδ√(2) (A_k-A_0) )^2, which is (<ref>) for k+1. Using this inequality, we obtain A_k( f(x_k) - f_* )(<ref>)≤ P_0^2 + ρδ∑_i=0^k-1α_i+1 R_i+1≤(P_0 + ρδ√(2) (A_k-A_0) )^2 ≤ 2P_0^2 + 4 ρ^2 δ^2 A_k^2, which finishes the proof of the Theorem. Let us now estimate the growth rate of the sequence A_k, k ≥ 0, which will give the rate of convergence for <ref>. Let the sequence {A_k }, k≥ 0 be generated by Algorithm <ref>. Then,for all k ≥ 1 it holds that (k-1+2ρ)^2/4ρ^2≤ A_k ≤(k-1+2ρ)^2/ρ^2. As we showed in <ref>, α_1=1/ρ and, hence, A_1=α_0+α_1=1. Thus, (<ref>) holds for k=1. Let us assume that (<ref>) holds for some k≥ 1 and prove that it holds also for k+1. From (<ref>),we have a quadratic equation for α_k+1 ρ^2α_k+1^2 - α_k+1 - A_k = 0. Since we need to take the largest root, we obtain, α_k+1= 1 + √( 1 + 4ρ^2A_k)/2ρ^2 = 1/2ρ^2 + √(1/4ρ^4 + A_k/ρ^2)≥1/2ρ^2 + √(A_k/ρ^2)≥1/2ρ^2 + k-1+2ρ/2ρ^2 = k+2ρ/2ρ^2, where we used the induction assumption that (<ref>) holds for k. On the other hand, α_k+1= 1/2ρ^2 + √(1/4ρ^4 + A_k/ρ^2)≤1/ρ^2 + √(A_k/ρ^2)≤1/ρ^2 + k-1+2ρ/ρ^2 = k+2ρ/ρ^2, where we used inequality √(a+b)≤√(a)+√(b), a,b ≥ 0. Using the obtained inequalities for α_k+1, from (<ref>) and (<ref>) for k, we get A_k+1 = A_k + α_k+1≥(k-1+2ρ)^2/4ρ^2 + k+2ρ/2ρ^2≥(k+2ρ)^2/4ρ^2 and A_k+1 = A_k + α_k+1≤(k-1+2ρ)^2/ρ^2 + k+2ρ/ρ^2≤(k+2ρ)^2/ρ^2. In the last inequality we used that k ≥ 1, ρ≥ 0. According to Theorem <ref>, if the desired accuracy of the solution is , i.e. the goal is to find such x̂∈ Q that f(x̂) - f_* ≤, then the Algorithm <ref> should be stopped when 3P_0^2/2A_k≤. Then 1/A_k≤2/3 P_0^2 and the oracle error level δ should satisfy δ≤P_0/4 ρ A_k≤/6 ρ P_0. From <ref>, we obtain that 3P_0^2/2A_k≤ holds when k is the smallest integer satisfying (k-1+2ρ)^2/4ρ^2≥3P_0^2/2 . This means that, to obtain an -solution, it is enough to choose k = max{⌈ρ√(6P_0^2/) + 1 -2 ρ⌉,0}. Note that this dependence onmeans that the proposed method is accelerated. § EXTENSION FOR STRONGLY CONVEX FUNCTIONS In this section, we consider strongly-convex objective functions and show how the restart technique can be used to obtain faster rates of convergence under this additional assumption.[Strong convexity] Assume that the function f(x) is strongly convex: f(x) ≥ f(y) + ⟨∇ f(y), x-y⟩ + μ/2x - y_E^2, ∀ x, y ∈ Q, where the constant μ > 0. We also introduce additional assumption on the Bregman divergence V[y](x): [Prox-function bound] We assume that the function d(x) satisfies conditions 0 = _x∈ Q d(x) and d(0) = 0. Also for a fixed point x_* and random point y ∈ Q^0, if y - x_*^2_E ≤ R^2 then d(y - x_*/R) ≤V^2/2. Some examples of prox-functions satisfying this assumption can be found in <cit.>, the simplest being the squared Euclidean norm. The following algorithm is obtained by restarting Algorithm <ref>. As in Algorithm <ref> our algorithm has two variations. For simplicity we introduce a constant Ĉ which is equal to 0 if the oracle error ξ(x) in (<ref>) can be controlled and Ĉ = 4(k+2ρ)^2 δ^2 if the oracle error ξ(x) in (<ref>) can not be controlled. Variable k is a constant which is predefined in (<ref>). Let the assumptions <ref>, <ref>, <ref>, <ref>,<ref> hold. Let the sequences {y_j, _j}, j≥ 0 be generated by Algorithm <ref>. Let f_* be the optimal objective value and x_* be an optimal point in problem (<ref>). * For the second statement of Theorem <ref> when the oracle error ξ(x) in (<ref>) can not be controlled for all j ≥ 0, f(y_j) - f_* ≤_j + 8(1 - 2^-j) (k+2ρ)^2 δ^2 = _0 2^-j + 8(1 - 2^-j) (k+2ρ)^2 δ^2, where k is defined in (<ref>). * For the first statement of Theorem <ref> when the oracle error ξ(x) in (<ref>) can be controlled for all j ≥ 0, f(y_j) - f_* ≤_j = _0 2^-j, wheredenotes the expectation with respect to all the randomness up to step j. Also Algorithm <ref> finds an -solution of the function f(x) in the number of <ref> steps not greater than ⌈log_2(_0/) ⌉max{⌈ρ√(16(1-1/ρ + V^2/μ)) + 1 -2 ρ⌉,0}. Let us denote Ĉ = 4(k+2ρ)^2 δ^2. We start our proof with a remark that if d(x) is a prox-function then a^2 d(1/ax) is also a prox-function, it is enough to show that a ^2d(1/ax) is 1-strong convex function. From definition d(x) - d(y) - ⟨∇ d(y), x - y⟩≥1/2x - y_E^2,x ∈ Q,y ∈ Q^0 := ⊗_i=1^n Q_i^0, obviously, we can get a^2d(1/ax) - a^2d(1/ay) - ⟨∇( a^2d(1/ay)), x - y⟩≥1/2x - y_E^2, x ∈ Q, y ∈ Q^0 := ⊗_i=1^n Q_i^0. Let us denote P̂_0^j =√(( A_0 + V^2/μ)_j) for all j=0,...,N, we can show following equality for k: k= max{⌈ρ√(16(1-1/ρ + V^2/μ)) + 1 -2 ρ⌉,0}= max{⌈ρ√(8(A_0 + V^2/μ)_j-1/_j) + 1 -2 ρ⌉,0}= max{⌈ρ√(8(P̂_0^j-1)^2/_j) + 1 -2 ρ⌉,0}. Thus we have equivalent definition of k: k = max{⌈ρ√(8(P̂_0^j-1)^2/_j) + 1 -2 ρ⌉,0} . 1. We first prove the first statement of Theorem <ref>. For j = 0 inequality (<ref>) follows from Algorithm <ref> input. Let us assume that this is true for j and prove that it holds for j+1. Let us define P_0^j-1 equal to P_0 from the j's call of Algorithm <ref>. From assumption <ref>, induction, and definition of Ĉ, we obtain μ/2y_j - x_*_E^2 ≤(f(y_j) - f_*) ≤_j + 2(1 - 2^-j)Ĉ. We can write following upper bound for P_0^j: (P_0^j)^2= A_0(f(y_j)-f_*) + V̂[y_j](x_*) =A_0(f(y_j)-f_*) + (d̂(x_*) - d̂(y_j) - ⟨∇d̂(y_j), x_* - y_j⟩) From assumption <ref> and (<ref>), we have d̂(y_j) = 0 and ∇d̂(y_j) = 0. Therefore, (P_0^j)^2 = A_0(f(y_j)-f_*)+ 2(_j + 2(1 - 2^-j)Ĉ)/μ d(√(μ/2(_j+ 2(1 - 2^-j)Ĉ))(y_j - x_*)). In assumption <ref> we can take R = √(2(_j + 2(1 - 2^-j)Ĉ)/μ) and get (P_0^j)^2 ≤ A_0(f(y_j)-f_*) + (_j+ 2(1 - 2^-j)Ĉ)V^2/μ≤(A_0 + V^2/μ)(_j+ 2(1 - 2^-j)Ĉ) and hence (P_0^j)^2 ≤ (P̂_0^j)^2 + 2(A_0 + V^2/μ)(1 - 2^-j)Ĉ. Using Theorem <ref> we obtain: f(y_j+1) - f_* ≤2 (P_0^j)^2/A_k + 4 A_kρ^2 δ^2(<ref>)≤2 (P_0^j)^2/A_k + Ĉ≤2(P̂_0^j)^2/A_k + 2(A_0 + V^2/μ)(1 - 2^-j)Ĉ/A_k + Ĉ. Using the second definition (<ref>) for k as well as in Remark <ref> with P_0 = P̂_0^j we have: 2(P̂_0^j)^2/A_k≤_j+1. Finally, we can conclude: f(y_j+1) - f_*≤_j+1+2(A_0 + V^2/μ)(1 - 2^-j)Ĉ/(P̂_0^j)^2_j+1+ Ĉ=_j+1+2(A_0 + V^2/μ)(1 - 2^-j)Ĉ/( A_0 + V^2/μ)_j_j+1+ Ĉ=_j+1+2(1 - 2^-j-1)Ĉ. 2. Obviously, the proof for the second statement of Theorem <ref> is the same. The only difference is that Ĉ = 0. It is enough to take N = ⌈log_2(_0/) ⌉ in order to find an-solution. Due to (<ref>) we can estimate total number of Algorithm <ref> steps: K_T = Nk= Nmax{⌈ρ√(16(1-1/ρ + V^2/μ)) + 1 -2 ρ⌉,0}=⌈log_2(_0/) ⌉max{⌈ρ√(16(1-1/ρ + V^2/μ)) + 1 -2 ρ⌉,0}. Note that this dependence onand μ means that the proposed method is accelerated. § EXAMPLES OF APPLICATIONS In this section, we apply our general framework, which consists of assumptions <ref>, <ref>, <ref>, UARM as listed in Algorithm <ref> and convergence ratein theorem <ref>, to obtain several particular algorithms and their convergence rate. We consider problem (<ref>) and, for each particular case, introduce a particular setup, which includes properties of the objective function f, available information about this function, properties of the feasible set Q. Based on each setup, we show how the Randomized Inexact Oracle is constructed and check that the assumptions <ref>, <ref>, <ref> hold.Then, we obtain convergence rate guarantee for each particular algorithm as a corollary of theorem <ref>. Our examples include accelerated random directional search with inexact directional derivative, accelerated random block-coordinate descent with inexact block derivatives, accelerated random derivative-free directional search with inexact function values, accelerated random derivative-free block-coordinate descent with inexact function values. Accelerated random directional search and accelerated random derivative-free directional search were developed in <cit.>, but for the case of exact directional derivatives and exact function values. Also, in the existing methods, a Gaussian random vector is used for randomization. Accelerated random block-coordinate descent was introduced in <cit.> and further developed in by several authors (see Introduction for the extended review). Existing methods of this type use exact information on the block derivatives and also only Euclidean proximal setup. In the contrast, our algorithm works with inexact derivatives and is able to work with entropy proximal setup. To the best of our knowledge, our accelerated random derivative-free block-coordinate descent with inexact function values is new. This method also can work with entropy proximal setup. §.§ Accelerated Random Directional Search In this subsection, we introduce accelerated random directional search with inexact directional derivative for unconstrained problems with Euclidean proximal setup. We assume that, for all i=1,...,n, Q_i=E_i=,x^(i)_i^2=(x^(i))^2, x^(i)∈ E_i, d_i(x^(i))=1/2(x^(i))^2, x^(i)∈ E_i and, hence, V_i[z^(i)](x^(i)) = 1/2(x^(i)-z^(i))^2, x^(i), z^(i)∈ E_i. Thus, Q=E=^n. Further, we assume that f in (<ref>) has L-Lipschitz-continuous gradient with respect to Euclidean norm, i.e. f(x) ≤ f(y) + ∇ f(y), x-y+ L/2x-y_2^2,x,y ∈ E.We set β_i=L, i=1,...,n. Then, by definitions in subsection <ref>, we have x_E^2 = L x_2^2, x ∈ E, d(x) = L/2x_2^2=1/2x_E^2, x ∈ E, V[z](x)=L/2x-z_2^2=1/2x-z_E^2, x, z ∈ E. Also, we have g_E,*^2 = L^-1g_2^2, g ∈ E^*.We assume that, at any point x ∈ E, one can calculate an inexact derivative of f in a direction e ∈ Ef̃'(x,e) = ∇ f(x), e+ ξ(x), where e is a random vector uniformly distributed on the Euclidean sphere of radius 1, i.e. _2(1):={s ∈^n: s_2=1}, and the directional derivative error ξ(x) ∈ is uniformly bounded in absolute value by error level Δ, i.e. |ξ(x)| ≤Δ, x ∈ E. Since we are in the Euclidean setting, we consider e also as an element of E^*. We use n (∇ f(x), e+ ξ(x))e as Randomized Inexact Oracle.Let us check the assumptions stated in subsection <ref>.Randomized Inexact Oracle. In this setting, we have ρ = n, H =, ^T: E^* → is given by ^T g =g, e, g ∈ E^*, : → E^* is given by t = te, t ∈. Thus,(x) = n (∇ f(x), e+ ξ(x))e.One can prove that _e n ∇ f(x), ee =n_eee^T∇ f(x) = ∇ f(x), x ∈ E, and, thus, (<ref>) holds. Also, for all x ∈ E, we have ξ(x)_E,* = 1/√(L)ξ(x)e_2 ≤Δ/√(L), which proves (<ref>) if we take δ = Δ/√(L).Regularity of Prox-Mapping. Substituting particular choice of Q, V[u](x), (x) in (<ref>), we obtainu_+= min_x ∈^n{L/2x-u_2^2+ α n (∇ f(y), e+ ξ(y))e,x } = u - α n/L (∇ f(y), e+ ξ(y))e.Hence, since e, e= 1, we have^T∇ f(y), u - u_+ = ∇ f(y), ee, α n/L (∇ f(y), e+ ξ(y))e =∇ f(y), ee , eα n/L (∇ f(y), e+ ξ(y))= ∇ f(y),α n/L (∇ f(y), e+ ξ(y))e = ∇ f(y), u - u_+ ,which proves (<ref>).Smoothness. By definition of ·_E and (<ref>), we have f(x) ≤ f(y) + ∇ f(y), x - y+ L/2x-y_2^2 = f(y) + ∇ f(y), x - y+ 1/2x-y_E^2,x, y ∈ Eand (<ref>) holds.We have checked that all the assumptions listed in subsection <ref> hold. Thus, we can obtain the following convergence rate result for random directional search as a corollary of Theorem <ref> and Lemma <ref>. Let Algorithm <ref> with (x) = n (∇ f(x), e+ ξ(x))e, where e is random and uniformly distributed over the Euclidean sphere of radius 1, be applied to Problem (<ref>) in the setting of this subsection. Let f_* be the optimal objective value and x_* be an optimal point in Problem (<ref>). Assume that directional derivative error ξ(x) satisfies |ξ(x)| ≤Δ, x ∈ E. Denote P_0^2 = (1-1/n)(f(x_0)-f_*) + L/2u_0-x_*_2^2. * If the directional derivative error ξ(x) can be controlled and, on each iteration, the error level Δ satisfies Δ≤P_0√(L)/4nA_k, then, for all k ≥ 1, f(x_k) - f_*≤6n^2P_0^2/(k-1+2n)^2, wheredenotes the expectation with respect to all the randomness up to step k. * If the directional derivative error ξ(x) can not be controlled, then, for all k ≥ 1, f(x_k) - f_*≤8n^2P_0^2/(k-1+2n)^2 + 4/L (k-1+2n)^2 Δ^2. According to Remark <ref> and due to the relation δ = Δ/√(L), we obtain that the error level Δ in the directional derivative should satisfy Δ≤√(L)/6nP_0. At the same time, to obtain an -solution for Problem (<ref>), it is enough to choose k = max{⌈ n√(6P_0^2/) + 1 -2 n ⌉,0}.§.§ Accelerated Random Coordinate Descent In this subsection, we introduce accelerated random coordinate descent with inexact coordinate derivatives for problems with separable constraints and Euclidean proximal setup. We assume that, for all i=1,...,n, E_i=, Q_i ⊆ E_i are closed and convex, x^(i)_i^2=(x^(i))^2, x^(i)∈ E_i, d_i(x^(i))=1/2(x^(i))^2, x^(i)∈ Q_i, and, hence, V_i[z^(i)](x^(i)) = 1/2(x^(i)-z^(i))^2, x^(i), z^(i)∈ Q_i. Thus, Q=⊗_i=1^n Q_i has separable structure. Let us denote e_i ∈ E the i-th coordinate vector. Then, for i=1,...,n, the i-th coordinate derivative of f is f_i'(x) = ∇ f(x), e_i. We assume that the gradient of f in (<ref>) is coordinate-wise Lipschitz continuous with constants L_i, i=1,...,n, i.e.|f'_i(x+he_i) - f'_i(x)| ≤ L_i |h|,h ∈,i=1,...,n,x ∈ Q.We set β_i=L_i, i=1,...,n. Then, by definitions in subsection <ref>, we have x_E^2 = ∑_i=1^n L_i (x^(i))^2, x ∈ E, d(x) = 1/2∑_i=1^n L_i(x^(i))^2, x ∈ Q, V[z](x)=1/2∑_i=1^n L_i (x^(i)-z^(i))^2, x, z ∈ Q. Also, we have g_E,*^2 = ∑_i=1^n L_i^-1 (g^(i))^2, g ∈ E^*. We assume that, at any point x ∈ Q, one can calculate an inexact coordinate derivative of ff̃_i'(x) = ∇ f(x), e_i+ ξ(x), where the coordinate i is chosen from i=1,...,n at random with uniform probability 1/n, the coordinate derivative error ξ(x) ∈ is uniformly bounded in absolute value by Δ, i.e. |ξ(x)| ≤Δ, x ∈ Q. Since we are in the Euclidean setting, we consider e_i also as an element of E^*. We use n (∇ f(x), e_i+ ξ(x))e_i as Randomized Inexact Oracle.Let us check the assumptions stated in subsection <ref>.Randomized Inexact Oracle. In this setting, we have ρ = n, H = E_i =, ^T: E^* → is given by ^T g =g, e_i, g ∈ E^*, : → E^* is given by t = te_i, t ∈. Thus,(x) = n (∇ f(x), e_i+ ξ(x))e_i,x ∈ Q.One can prove that _i n ∇ f(x), e_ie_i =n_ie_ie_i^T∇ f(x) = ∇ f(x), x ∈ Q, and, thus, (<ref>) holds. Also, for all x ∈ Q, we have ξ(x)_E,* = 1/√(L_i)|ξ(x)| ≤Δ/√(L_0), where L_0 = min_i=1,...,nL_i. This proves (<ref>) with δ = Δ/√(L_0).Regularity of Prox-Mapping. Separable structure of Q and V[u](x) means that the problem (<ref>) boils down to n independent problems of the form u_+^(j) = min_x^(j)∈ Q_j{L_j/2(u^(j)-x^(j))^2+ α(y), e_jx^(j)},j = 1,...,n.Since (y) has only one, i-th, non-zero component, (y), e_j is zero for all ji. Thus, u-u_+ has one, i-th, non-zero component and e_i, u - u_+e_i = u - u_+.Hence, ^T∇ f(y), u - u_+ = ∇ f(y), e_ie_i, u - u_+= ∇ f(y), e_i e_i, u - u_+= ∇ f(y), e_i, u - u_+e_i= ∇ f(y), u - u_+ ,which proves (<ref>).Smoothness. By the standard reasoning, using (<ref>), one can prove that, for all i=1,...,n,f(x+h e_i) ≤ f(x) + h ∇ f(x),e_i + L_i h^2/2,h ∈,x ∈ Q.Let u,y ∈ Q, a ∈, and x = y + a (u_+-u) ∈ Q. As we have shown above, u_+-u has only one, i-th, non-zero component. Hence, there exists h ∈, such that u_+-u = he_i and x = y + a he_i. Thus, by definition of ·_E and (<ref>), we have f(x)= f(y + a he_i )≤ f(y) + a h ∇ f(y), e_i+ L_i/2(a h)^2= f(y) + ∇ f(y), a he_i+ 1/2a h e_i_E^2 = f(y) + ∇ f(y), x - y+ 1/2x-y_E^2.This proves (<ref>).We have checked that all the assumptions listed in subsection <ref> hold. Thus, we can obtain the following convergence rate result for random coordinate descent as a corollary of <ref> and <ref>. Let Algorithm <ref> with (x) = n (∇ f(x), e_i+ ξ(x))e_i, where i is uniformly at random chosen from 1,...,n, be applied to Problem (<ref>) in the setting of this subsection. Let f_* be the optimal objective value and x_* be an optimal point in Problem (<ref>). Assume that coordinate derivative error ξ(x) satisfies |ξ(x)| ≤Δ, x ∈ Q. Denote P_0^2 = (1-1/n)(f(x_0)-f_*) + ∑_i=1^nL_i/2(u_0^(i)-x_*^(i))^2. * If the coordinate derivative error ξ(x) can be controlled and, on each iteration, the error level Δ satisfies Δ≤P_0√(L_0)/4nA_k, then, for all k ≥ 1, f(x_k) - f_*≤6n^2P_0^2/(k-1+2n)^2, wheredenotes the expectation with respect to all the randomness up to step k. * If the coordinate derivative error ξ(x) can not be controlled, then, for all k ≥ 1, f(x_k) - f_*≤8n^2P_0^2/(k-1+2n)^2 + 4/L_0 (k-1+2n)^2 Δ^2. According to Remark <ref> and due to the relation δ = Δ/√(L_0), we obtain that the error level Δ in the coordinate derivative should satisfy Δ≤√(L_0)/6nP_0 . At the same time, to obtain an -solution for Problem (<ref>), it is enough to choose k = max{⌈ n√(6P_0^2/) + 1 -2 n ⌉,0}. §.§ Accelerated Random Block-Coordinate Descent In this subsection, we consider two block-coordinate settings. The first one is the Euclidean, which is usually used in the literature for accelerated block-coordinate descent. The second one is the entropy, which, to the best of our knowledge, is analyzed in this context for the first time. We develop accelerated random block-coordinate descent with inexact block derivatives for problems with simple constraints in these two settings and their combination.Euclidean setup. We assume that, for all i=1,...,n, E_i=^p_i; Q_i is a simple closed convex set; x^(i)_i^2= B_i x^(i), x^(i), x^(i)∈ E_i, where B_i is symmetric positive semidefinite matrix; d_i(x^(i))=1/2x^(i)_i^2, x^(i)∈ Q_i, and, hence, V_i[z^(i)](x^(i)) = 1/2x^(i)-z^(i)_i^2, x^(i), z^(i)∈ Q_i.Entropy setup. We assume that, for all i=1,...,n, E_i=^p_i; Q_i is standard simplex in ^p_i, i.e., Q_i = {x^(i)∈^p_i_+: ∑_j=1^p_i[x^(i)]_j = 1}; x^(i)_i=x^(i)_1=∑_j=1^p_i|[x^(i)]_j|, x^(i)∈ E_i; d_i(x^(i))=∑_j=1^p_i[x^(i)]_j ln [x^(i)]_j, x^(i)∈ Q_i, and, hence, V_i[z^(i)](x^(i)) = ∑_j=1^p_i[x^(i)]_j ln[x^(i)]_j/[z^(i)]_j, x^(i), z^(i)∈ Q_i. Note that, in each block, one also can choose other proximal setups from <cit.>. Combination of different setups in different blocks is also possible, i.e., in one block it is possible to choose the Euclidean setup and in an another block one can choose the entropy setup.Using operators U_i, i=1,...,n defined in (<ref>), for each i=1,...,n, the i-th block derivative of f can be written as f_i'(x) = U_i^T ∇ f(x). We assume that the gradient of f in (<ref>) is block-wise Lipschitz continuous with constants L_i, i=1,...,n with respect to chosen norms ·_i, i.e.f'_i(x+U_ih^(i)) - f'_i(x)_i,*≤ L_i h^(i)_i,h^(i)∈ E_i,i=1,...,n,x ∈ Q.We set β_i=L_i, i=1,...,n. Then, by definitions in subsection <ref>, we havex_E^2 = ∑_i=1^n L_i x^(i)_i^2,x ∈ E, d(x) = ∑_i=1^n L_id_i(x^(i)),x ∈ Q, V[z](x)=∑_i=1^n L_i V_i[z^(i)](x^(i)),x, z ∈ Q. Also, we have g_E,*^2 = ∑_i=1^n L_i^-1g^(i)_i,*^2, g ∈ E^*. We assume that, at any point x ∈ Q, one can calculate an inexact block derivative of ff̃_i'(x) = U_i^T ∇ f(x) + ξ(x), where a block number i is chosen from 1,...,n randomly uniformly, the block derivative error ξ(x) ∈ E_i^* is uniformly bounded in norm by Δ, i.e. ξ(x)_i,*≤Δ, x ∈ Q, i=1,...,n. As Randomized Inexact Oracle, we use n U_i(U_i^T ∇ f(x) + ξ(x)), where U_i is defined in (<ref>).Let us check the assumptions stated in subsection <ref>.Randomized Inexact Oracle. In this setting, we have ρ = n, H = E_i, ^T: E^* → E_i^* is given by ^T g = U_i^T g, g ∈ E^*, : E_i^* → E^* is given by g^(i) = U_i g^(i), g^(i)∈ E_i^*. Thus,(x) = n U_i(U_i^T ∇ f(x) + ξ(x)),x ∈ Q.Since i ∈ R[1,n], one can prove that _i n U_iU_i^T ∇ f(x) =∇ f(x), x ∈ Q, and, thus, (<ref>) holds. Also, for all x ∈ Q, we have ξ(x)_E,* = U_i ξ(x)_E,* = 1/√(L_i)ξ(x)_i,*≤Δ/√(L_0), where L_0 = min_i=1,...,nL_i. This proves (<ref>) with δ = Δ/√(L_0). Regularity of Prox-Mapping. Separable structure of Q and V[u](x) means that the problem (<ref>) boils down to n independent problems of the form u_+^(j) = min_x^(j)∈ Q_j{L_jV[u^(j)](x^(j))+ α U_j^T(y),x^(j)},j = 1,...,n.Since (y) has non-zero components only in the block i, U_j^T(y) is zero for all ji. Thus, u-u_+ has non-zero components only in the block i and U_iU_i^T (u - u_+) = u - u_+.Hence, ^T∇ f(y), u - u_+ = U_iU_i^T∇ f(y), u - u_+= ∇ f(y), U_iU_i^T (u - u_+)= ∇ f(y), u - u_+ ,which proves (<ref>).Smoothness. By the standard reasoning, using (<ref>), one can prove that, for all i=1,...,n,f(x+U_ih^(i)) ≤ f(x) +U_i^T∇ f(x), h^(i) + L_i/2h^(i)_i^2,h^(i)∈ E_i,x ∈ Q.Let u,y ∈ Q, a ∈, and x = y + a (u_+-u) ∈ Q. As we have shown above, u_+-u has non-zero components only in the block i. Hence, there exists h^(i)∈ E_i, such that u_+-u = U_i h^(i) and x = y + a U_i h^(i). Thus, by definition of ·_E and (<ref>), we have f(x)= f(y + a U_i h^(i))≤ f(y) +U_i^T∇ f(y), a h^(i) + L_i/2a h^(i)_i^2= f(y) + ∇ f(y), a U_i h^(i) + 1/2a U_i h^(i)_E^2 = f(y) + ∇ f(y), x - y+ 1/2x-y_E^2.This proves (<ref>).We have checked that all the assumptions listed in subsection <ref> hold. Thus, we can obtain the following convergence rate result for random block-coordinate descent as a corollary of <ref> and <ref>. Let <ref> with (x) = n U_i(U_i^T ∇ f(x) + ξ(x)), where i is uniformly at random chosen from 1,...,n, be applied to Problem (<ref>) in the setting of this subsection. Let f_* be the optimal objective value and x_* be an optimal point in Problem (<ref>). Assume that block derivative error ξ(x) satisfies |ξ(x)| ≤Δ, x ∈ Q. Denote P_0^2 = (1-1/n)(f(x_0)-f_*) + V[u_0](x_*). * If the block derivative error ξ(x) can be controlled and, on each iteration, the error level Δ satisfies Δ≤P_0√(L_0)/4nA_k, then,for all k ≥ 1, f(x_k) - f_*≤6n^2P_0^2/(k-1+2n)^2, wheredenotes the expectation with respect to all the randomness up to step k. * If the block derivative error ξ(x) can not be controlled, then, for all k ≥ 1, f(x_k) - f_*≤8n^2P_0^2/(k-1+2n)^2 + 4/L_0 (k-1+2n)^2 Δ^2. According to Remark <ref> and due to the relation δ = Δ/√(L_0), we obtain that the block derivative error Δ should satisfy Δ≤√(L_0)/6nP_0. At the same time, to obtain an -solution for Problem (<ref>), it is enough to choose k = max{⌈ n√(6P_0^2/) + 1 -2 n ⌉,0}. §.§ Accelerated Random Derivative-Free Directional Search In this subsection, we consider the same setting as in subsection <ref>, except for Randomized Inexact Oracle. Instead of directional derivative, we use here its finite-difference approximation. We assume that, for all i=1,...,n, Q_i=E_i=,x^(i)_i=(x^(i))^2, x^(i)∈ E_i, d_i(x^(i))=1/2(x^(i))^2, x^(i)∈ E_i, and, hence, V_i[z^(i)](x^(i)) = 1/2(x^(i)-z^(i))^2, x^(i), z^(i)∈ E_i. Thus, Q=E=^n. Further, we assume that f in (<ref>) has L-Lipschitz-continuous gradient with respect to Euclidean norm, i.e. f(x) ≤ f(y) + ∇ f(y), x-y+ L/2x-y_2^2,x,y ∈ E.We set β_i=L, i=1,...,n. Then, by definitions in subsection <ref>, we have x_E^2 = L x_2^2, x ∈ E, d(x) = L/2x_2^2=1/2x_E^2, x ∈ E, V[z](x)=L/2x-z_2^2=1/2x-z_E^2, x, z ∈ E. Also, we have g_E,*^2 = L^-1g_2^2, g ∈ E^*.We assume that, at any point x ∈ E, one can calculate an inexact value (x) of the function f, s.t. |(x)-f(x)|≤Δ, x ∈ E. To approximate the gradient of f, we use (x) = n(x+τ e)-(x)/τe, where τ > 0 is small parameter, which will be chosen later, e ∈ E is a random vector uniformly distributed on the Euclidean sphere of radius 1, i.e. on _2(1):={s ∈^n: s_2=1}. Since, we are in the Euclidean setting, we consider e also as an element of E^*. Let us check the assumptions stated in subsection <ref>.Randomized Inexact Oracle. First, let us show that the finite-difference approximation for the gradient of f can be expressed in the form of (<ref>). We have(x)= n(x+τ e)-(x)/τe= n (⟨∇ f(x), e⟩+ 1/τ( (x+τ e)-(x) - τ⟨∇ f(x), e⟩))e.Taking ρ = n, H =, ^T: E^* → be given by ^T g =g, e, g ∈ E^*, : → E^* be given by t = te, t ∈, we obtain(x) = n (∇ f(x), e+ ξ(x))e,where ξ(x) = 1/τ( (x+τ e)-(x) - τ⟨∇ f(x), e⟩). One can prove that _e n ∇ f(x), ee =n_eee^T∇ f(x) = ∇ f(x), x ∈ E, and, thus, (<ref>) holds. It remains to prove (<ref>), i.e., find δ s.t. for all x ∈ E, we have ξ(x)_E,*≤δ.ξ(x)_E,*=1/√(L)ξ(x)e_2= 1/√(L)1/τ( (x+τ e)-(x) - τ⟨∇ f(x), e⟩) e _2 = 1/√(L)1/τ( (x+τ e) - f(x+τ e) - ((x) - f(x)) _2+ 1/√(L)(f(x+τ e) - f(x) - τ⟨∇ f(x), e⟩)) e _2 ≤2Δ/τ√(L)+ τ√(L)/2.Here we used that |(x)-f(x)|≤Δ, x ∈ E and (<ref>). So, we have that (<ref>) holds with δ = 2Δ/τ√(L)+ τ√(L)/2. To balance both terms, we choose τ = 2 √(Δ/L), which leads to equality δ = 2 √(Δ).Regularity of Prox-Mapping. This assumption can be checked in the same way as in subsection <ref>.Smoothness. This assumption can be checked in the same way as in subsection <ref>.We have checked that all the assumptions listed in subsection <ref> hold. Thus, we can obtain the following convergence rate result for random derivative-free directional search as a corollary of Theorem <ref> and Lemma <ref>. Let Algorithm <ref> with (x) = n(x+τ e)-(x)/τe, where e is random and uniformly distributed over the Euclidean sphere of radius 1, be applied to Problem (<ref>) in the setting of this subsection. Let f_* be the optimal objective value and x_* be an optimal point in Problem (<ref>). Assume that function value error (x)-f(x) satisfies |(x)-f(x)| ≤Δ, x ∈ E. Denote P_0^2 = (1-1/n)(f(x_0)-f_*) + L/2u_0-x_*_2^2. * If the error in the value of the objective f can be controlled and, on each iteration, the error level Δ satisfies Δ≤P_0^2/64n^2A_k^2, and τ = 2 √(Δ/L) then, for all k ≥ 1, f(x_k) - f_*≤6n^2P_0^2/(k-1+2n)^2, wheredenotes the expectation with respect to all the randomness up to step k. * If the error in the value of the objective f can not be controlled and τ = 2 √(Δ/L), then, for all k ≥ 1, f(x_k) - f_*≤8n^2P_0^2/(k-1+2n)^2 + 16 (k-1+2n)^2 LΔ. According to Remark <ref> and due to the relation δ = 2 √(Δ), we obtain that the error level in the function value should satisfy Δ≤^2 /144 n^2P_0^2. The parameter τ should satisfy τ≤/6 n P_0 √(L) . At the same time, to obtain an -solution for Problem (<ref>), it is enough to choose k = max{⌈ n√(6P_0^2/) + 1 -2 n ⌉,0}.§.§ Accelerated Random Derivative-Free Coordinate Descent In this subsection, we consider the same setting as in subsection <ref>, except for Randomized Inexact Oracle. Instead of coordinate derivative, we use here its finite-difference approximation. We assume that, for all i=1,...,n, E_i=, Q_i ⊆ E_i are closed and convex, x^(i)_i=(x^(i))^2, x^(i)∈ E_i, d_i(x^(i))=1/2(x^(i))^2, x^(i)∈ Q_i, and, hence, V_i[z^(i)](x^(i)) = 1/2(x^(i)-z^(i))^2, x^(i), z^(i)∈ Q_i. Thus, Q=⊗_i=1^n Q_i has separable structure. Let us denote e_i ∈ E the i-th coordinate vector. Then, for i=1,...,n, the i-th coordinate derivative of f is f_i'(x) = ∇ f(x), e_i. We assume that the gradient of f in (<ref>) is coordinate-wise Lipschitz continuous with constants L_i, i=1,...,n, i.e.|f'_i(x+he_i) - f'_i(x)| ≤ L_i |h|,h ∈,i=1,...,n,x ∈ Q.We set β_i=L_i, i=1,...,n. Then, by definitions in subsection <ref>, we have x_E^2 = ∑_i=1^n L_i (x^(i))^2, x ∈ E, d(x) = 1/2∑_i=1^n L_i(x^(i))^2, x ∈ Q, V[z](x)=1/2∑_i=1^n L_i (x^(i)-z^(i))^2, x, z ∈ Q. Also, we have g_E,*^2 = ∑_i=1^n L_i^-1 (g^(i))^2, g ∈ E^*.We assume that, at any point x in a small vicinity Q̅ of the set Q, one can calculate an inexact value (x) of the function f, s.t. |(x)-f(x)|≤Δ, x ∈Q̅. To approximate the gradient of f, we use (x) = n(x+τ e_i)-(x)/τe_i, where τ > 0 is small parameter, which will be chosen later, and the coordinate i is chosen from i=1,...,n randomly with uniform probability 1/n. Since, we are in the Euclidean setting, we consider e_i also as an element of E^*. Let us check the assumptions stated in subsection <ref>.Randomized Inexact Oracle. First, let us show that the finite-difference approximation for the gradient of f can be expressed in the form of (<ref>). We have(x)= n(x+τ e_i)-(x)/τe_i = n (⟨∇ f(x), e_i⟩+ 1/τ( (x+τ e_i)-(x) - τ⟨∇ f(x), e_i⟩))e_i.Taking ρ = n, H =, ^T: E^* → is given by ^T g =g, e_i, g ∈ E^*, : → E^* is given by t = te_i, t ∈, we obtain(x) = n (∇ f(x), e_i+ ξ(x))e_i,where ξ(x) = 1/τ( (x+τ e_i)-(x) - τ⟨∇ f(x), e_i⟩). One can prove that _i n ∇ f(x), e_ie_i =n_ie_ie_i^T∇ f(x) = ∇ f(x),x ∈ Q, and, thus, (<ref>) holds. It remains to prove (<ref>), i.e., find δ s.t. for all x ∈ Q, we have ξ(x)_E,*≤δ.ξ(x)_E,*=1/√(L_i)|ξ(x)|= 1/√(L_i)|1/τ( (x+τ e_i)-(x) - τ⟨∇ f(x), e_i⟩)| = 1/τ√(L_i)| (x+τ e_i) - f(x+τ e_i) - ((x) - f(x))|+ 1/τ√(L_i)|f(x+τ e_i) - f(x) - τ⟨∇ f(x), e_i⟩| ≤2Δ/√(L_i)τ+ τ√(L_i)/2.Here we used that |(x)-f(x)|≤Δ, x ∈Q̅ and (<ref>), which follows from (<ref>). So, we obtain that (<ref>) holds with δ = 2Δ/√(L_i)τ+ √(L_i)τ/2. To balance both terms, we choose τ = 2 √(Δ/L_i)≤ 2 √(Δ/L_0), where L_0 = min_i=1,...,nL_i.This leads to equality δ = 2 √(Δ).Regularity of Prox-Mapping. This assumption can be checked in the same way as in subsection <ref>.Smoothness. This assumption can be checked in the same way as in subsection <ref>.We have checked that all the assumptions listed in subsection <ref> hold. Thus, we can obtain the following convergence rate result for random derivative-free coordinate descent as a corollary of Theorem <ref> and Lemma <ref>. Let Algorithm <ref> with (x) = n(x+τ e_i)-(x)/τe_i, where i is random and uniformly distributed in 1,...,n, be applied to Problem (<ref>) in the setting of this subsection. Let f_* be the optimal objective value and x_* be an optimal point in Problem (<ref>). Assume that function value error (x)-f(x) satisfies |(x)-f(x)| ≤Δ, x ∈Q̅. Denote P_0^2 = (1-1/n)(f(x_0)-f_*) + ∑_i=1^nL_i/2(u_0^(i)-x_*^(i))^2. * If the error in the value of the objective f can be controlled and, on each iteration, the error level Δ satisfies Δ≤P_0^2/64n^2A_k^2, and τ = 2 √(Δ/L_0) then, for all k ≥ 1, f(x_k) - f_*≤6n^2P_0^2/(k-1+2n)^2, wheredenotes the expectation with respect to all the randomness up to step k. * If the error in the value of the objective f can not be controlled and τ = 2 √(Δ/L_0), then, for all k ≥ 1, f(x_k) - f_*≤8n^2P_0^2/(k-1+2n)^2 + 16 (k-1+2n)^2 Δ. According to Remark <ref> and due to the relation δ = 2 √(Δ), we obtain that the error level in the function value should satisfy Δ≤^2 /144 n^2P_0^2. The parameter τ should satisfy τ≤/6 n P_0 √(L_0). At the same time, to obtain an -solution for Problem (<ref>), it is enough to choose k = max{⌈ n√(6P_0^2/) + 1 -2 n ⌉,0}. §.§ Accelerated Random Derivative-Free Block-Coordinate Descent In this subsection, we consider the same setting as in subsection <ref>, except for Randomized Inexact Oracle. Instead of block derivative, we use here its finite-difference approximation.As in subsection <ref>, we consider Euclidean setup and entropy setup. Euclidean setup. We assume that, for all i=1,...,n, E_i=^p_i; Q_i is a simple closed convex set; x^(i)_i^2= B_i x^(i), x^(i), x^(i)∈ E_i, where B_i is symmetric positive semidefinite matrix; d_i(x^(i))=1/2x^(i)_i^2, x^(i)∈ Q_i, and, hence, V_i[z^(i)](x^(i)) = 1/2x^(i)-z^(i)_i^2, x^(i), z^(i)∈ Q_i.Entropy setup. We assume that, for all i=1,...,n, E_i=^p_i; Q_i is standard simplex in ^p_i, i.e., Q_i = {x^(i)∈^p_i_+: ∑_j=1^p_i[x^(i)]_j = 1}; x^(i)_i=x^(i)_1=∑_j=1^p_i|[x^(i)]_j|, x^(i)∈ E_i; d_i(x^(i))=∑_j=1^p_i[x^(i)]_j ln [x^(i)]_j, x^(i)∈ Q_i, and, hence, V_i[z^(i)](x^(i)) = ∑_j=1^p_i[x^(i)]_j ln[x^(i)]_j/[z^(i)]_j, x^(i), z^(i)∈ Q_i. Note that, in each block, one also can choose other proximal setups from <cit.>. Combination of different setups in different blocks is also possible, i.e., in one block it is possible to choose the Euclidean setup and in an another block one can choose the entropy setup.Using operators U_i, i=1,...,n defined in (<ref>), for each i=1,...,n, the i-th block derivative of f can be written as f_i'(x) = U_i^T ∇ f(x). We assume that the gradient of f in (<ref>) is block-wise Lipschitz continuous with constants L_i, i=1,...,n with respect to chosen norms ·_i, i.e.,f'_i(x+U_ih^(i)) - f'_i(x)_i,*≤ L_i h^(i)_i,h^(i)∈ E_i,i=1,...,nx ∈ Q.We set β_i=L_i, i=1,...,n. Then, by definitions in subsection <ref>, we have x_E^2 = ∑_i=1^n L_i x^(i)_2^2, x ∈ E, d(x) = 1/2∑_i=1^n L_ix^(i)_2^2, x ∈ Q, V[z](x)=1/2∑_i=1^n L_i x^(i)-z^(i)_2^2, x, z ∈ Q. Also, we have g_E,*^2 = ∑_i=1^n L_i^-1g^(i)_2^2, g ∈ E^*.We assume that, at any point x in a small vicinity Q̅ of the set Q, one can calculate an inexact value (x) of the function f, s.t. |(x)-f(x)|≤Δ, x ∈Q̅. To approximate the gradient of f, we use (x) = nU_i ( (x+τ U_ie_1)-(x)/τ,...,(x+τ U_ie_p_i)-(x)/τ)^T,where τ > 0 is small parameter, which will be chosen later, a block number i is chosen from i=1,...,n randomly with uniform probability 1/n, e_1,...,e_p_i are coordinate vectors in E_i, U_i is defined in (<ref>), U_i is defined in (<ref>).Let us check the assumptions stated in subsection <ref>.Randomized Inexact Oracle. First, let us show that the random derivative-free block-coordinate approximation for the gradient of f can be expressed in the form of (<ref>). Denote g̃_i = 1/τ( (x+τ U_i e_1)-(x),...,(x+τ U_i e_p_i)-(x))^T ∈ E_i, i=1,...,n. We have(x) = nU_ig̃_i= nU_i ( U_i^T∇ f(x) + (g̃_i - U_i^T∇ f(x))).Taking ρ = n, H = E_i, ^T: E^* → E_i^* be given by ^T g = U_i^T g, g ∈ E^* and : E_i^* → E^* be given by g^(i) = U_i g^(i), g^(i)∈ E_i^*, we obtain(x) = n U_i(U_i^T∇ f(x) + ξ(x)),where ξ(x) = g̃_i - U_i^T∇ f(x). Since i ∈ R[1,n], one can prove that _i n U_iU_i^T ∇ f(x) =∇ f(x), x ∈ Q, and, thus, (<ref>) holds.It remains to prove (<ref>), i.e., find δ s.t. for all x ∈ Q, we have ξ(x)_E,*≤δ.Let us fix any i from 1,...,n and j from 1,...,p_i. Then, for any x ∈Q̅, the j-th coordinate of ξ(x) = g̃_i - U_i^T∇ f(x) can be estimated as follows|[ξ(x)]_j|= |1/τ( (x+τ U_i e_j)-(x) - τ U_i^T ∇ f(x), e_j|= 1/τ| (x+τ U_i e_j) - f(x+τ U_i e_j) - ((x) - f(x))|+ 1/τ| (f(x+τ U_i e_j) - f(x) - τ U_i^T ∇ f(x), e_j ) | ≤2Δ/τ+ τ L_i/2.Here we used that |(x)-f(x)|≤Δ, x ∈Q̅, (<ref>), which follows from (<ref>).In our setting, for any i = 1,...,n, ·_i,* is either max-norm (for the entropy case) or Euclidean norm (for the Euclidean case). Thus, in the worst case of Euclidean normξ(x)_E,* =U_i ξ(x)_E,*= 1/√(L_i)ξ(x) _i,*(<ref>)≤√(p_i)/√(L_i)(2Δ/τ+ τ L_i/2)≤√(p_max)(2Δ/τ√(L_i)+ τ√(L_i)/2),where p_max = max_i=1,...,n p_i. So, we obtain that (<ref>) holds with δ = √(p_max)(2Δ/τ√(L_i)+ τ√(L_i)/2). To balance both terms we choose τ =2 √(Δ/L_i)≤ 2 √(Δ/L_0), where L_0 = min_i=1,...,nL_i. This leads to equality δ = 2√(p_maxΔ).Regularity of Prox-Mapping. This assumption can be checked in the same way as in subsection <ref>.Smoothness. This assumption can be checked in the same way as in subsection <ref>.We have checked that all the assumptions listed in subsection <ref> hold. Thus, we can obtain the following convergence rate result for random derivative-free block-coordinate descent as a corollary of Theorem <ref> and Lemma <ref>. Let Algorithm <ref> with (x) defined in (<ref>), be applied to Problem (<ref>) in the setting of this subsection. Let f_* be the optimal objective value and x_* be an optimal point in Problem (<ref>). Assume that function value error (x)-f(x) satisfies |(x)-f(x)| ≤Δ, x ∈Q̅. Denote P_0^2 = (1-1/n)(f(x_0)-f_*) + V[u_0](x_*). * If the error in the value of the objective f can be controlled and, on each iteration, the error level Δ satisfies Δ≤P_0^2/64n^2p_maxA_k^2 , and τ = 2 √(Δ/L_0) then, for all k ≥ 1, f(x_k) - f_*≤6n^2P_0^2/(k-1+2n)^2 , wheredenotes the expectation with respect to all the randomness up to step k. * If the error in the value of the objective f can not be controlled and τ = 2 √(Δ/L_0), then, for all k ≥ 1, f(x_k) - f_*≤8n^2P_0^2/(k-1+2n)^2 + 16 (k-1+2n)^2 p_maxΔ. According to Remark <ref> and due to the relation δ = 2 √(p_maxΔ), we obtain that the error level in the function value should satisfy Δ≤^2 /144 n^2p_maxP_0^2. The parameter τ should satisfy τ≤/6 n P_0 √(L_0). At the same time, to obtain an -solution for Problem (<ref>), it is enough to choose k = max{⌈ n√(6P_0^2/) + 1 -2 n ⌉,0}. §.§ Accelerated Random Derivative-Free Block-Coordinate Descent with Random Approximations for Block Derivatives In this subsection, we combine random block-coordinate descent of subsection <ref> with random derivative-free directional search described in subsection <ref> and random derivative-free coordinate descent described in <ref>.We construct randomized approximations for block derivatives based on finite-difference approximation of directional derivatives. Unlike subsection <ref>, we consider only Euclidean setup. We assume that, for all i=1,...,n, E_i=^p_i; x^(i)_i^2=x^(i)_2^2, x^(i)∈ E_i;Q_i is either E_i, or ⊗_j=1^p_iQ_ij, where Q_ij⊆ are closed convex sets; d_i(x^(i))=1/2x^(i)_i^2, x^(i)∈ Q_i and, hence, V_i[z^(i)](x^(i)) = 1/2x^(i)-z^(i)_i^2, , x^(i), z^(i)∈ Q_i. For the case, Q_i = E_i, we consider randomization on the Euclidean sphere of radius 1, as in subsection <ref>. For the case, Q_i = ⊗_j=1^p_iQ_ij, we consider coordinate-wise randomization, as in subsection <ref>. Using operators U_i, i=1,...,n defined in (<ref>), for each i=1,...,n, the i-th block derivative of f can be written as f_i'(x) = U_i^T ∇ f(x). We assume that the gradient of f in (<ref>) is block-wise Lipschitz continuous with constants L_i, i=1,...,n with respect to chosen norms ·_i, i.e.,f'_i(x+U_ih^(i)) - f'_i(x)_i,*≤ L_i h^(i)_i,h^(i)∈ E_i,i=1,...,nx ∈ Q.We set β_i=L_i, i=1,...,n. Then, by definitions in subsection <ref>, we have x_E^2 = ∑_i=1^n L_i x^(i)_2^2, x ∈ E, d(x) = 1/2∑_i=1^n L_ix^(i)_2^2, x ∈ Q, V[z](x)=1/2∑_i=1^n L_i x^(i)-z^(i)_2^2, x, z ∈ Q. Also, we have g_E,*^2 = ∑_i=1^n L_i^-1g^(i)_2^2, g ∈ E^*.We assume that, at any point x in a small vicinity Q̅ of the set Q, one can calculate an inexact value (x) of the function f, s.t. |(x)-f(x)|≤Δ, x ∈Q̅. To approximate the gradient of f, we first randomly choose a block i ∈ 1,...,n with probability p_i/p, where p=∑_i=1^np_i. Then we use one the following types of random directions e ∈ E_i to approximate the block derivative f'_i(x) by a finite-difference. * If Q_i = E_i, we take e ∈ E_i to be random vector uniformly distributed on the Euclidean sphere of radius 1, i.e. _2(1):={s ∈^p_i: s_2=1}. We call this unconstrained case. * If Q_i = ⊗_j=1^p_iQ_ij, we take e to be random uniformly chosen from 1,...,p_i coordinate vector, i.e. e = e_j ∈ E_i with probability 1/p_i. We call this separable case. Based on these randomizations and inexact function values, our randomized approximation for the gradient of f is (x) = p U_i (x+τ U_i e)-(x)/τe, where τ > 0 is small parameter, which will be chosen later, U_i is defined in (<ref>) and U_i is defined in (<ref>).Let us check the assumptions stated in subsection <ref>.Randomized Inexact Oracle. First, let us show that the random derivative-free block-coordinate approximation for the gradient of f can be expressed in the form of (<ref>). We have(x) = pU_i(x+τ U_i e)-(x)/τe= pU_i ( ⟨ U_i^T∇ f(x), e⟩ e + 1/τ( (x+τ U_i e)-(x) - τ⟨ U_i^T ∇ f(x), e⟩) e ).Taking ρ = p, H = E_i, ^T: E^* → E_i^* be given by ^T g =U_i^T g,ee, g ∈ E^* and : E_i^* → E^* be given by g^(i) = U_i g^(i), g^(i)∈ E_i^*, we obtain(x) = p U_i(⟨ U_i^T∇ f(x), e⟩ e + ξ(x)),where ξ(x) = 1/τ( (x+τ U_i e)-(x) - τ⟨ U_i^T ∇ f(x), e⟩) e. By the choice of probability distributions for i and e and their independence, we have, for all x ∈ Q, _i,e p U_i⟨ U_i^T∇ f(x), e⟩ e =p_i,eU_ie e^T U_i^T∇ f(x) = p_iU_i (E_ee e^T) U_i^T∇ f(x) = p_i1/p_iU_i U_i^T ∇ f(x) = ∇ f(x)and, thus, (<ref>) holds.It remains to prove (<ref>), i.e., find δ s.t. for all x ∈ Q, we have ξ(x)_E,*≤δ. We haveξ(x)_E,* =U_i ξ(x)_E,*= 1/√(L_i)1/τ( (x+τ U_i e)-(x) - τ⟨ U_i^T ∇ f(x), e⟩) e _i,* =e _i,*/τ√(L_i)| (x+τ U_i e) - f(x+τ U_i e) - ((x) - f(x)) + (f(x+τ U_i e) - f(x) - τ⟨ U_i^T ∇ f(x), e⟩ ) | ≤2Δe _i,*/τ√(L_i)+ τe _i,*e _i^2 √(L_i)/2= 2Δ/τ√(L_i)+ τ√(L_i)/2.Here we used that |(x)-f(x)|≤Δ, x ∈Q̅, (<ref>), which follows from (<ref>), and that the norms ·_i, ·_i,* are standard Euclidean. So, we obtain that (<ref>) holds with δ = 2Δ/τ√(L_i)+ τ√(L_i)/2. To balance both terms we choose τ =2 √(Δ/L_i)≤ 2 √(Δ/L_0), where L_0 = min_i=1,...,nL_i. This leads to equality δ = 2√(Δ). Regularity of Prox-Mapping. Separable structure of Q and V[u](x) means that the problem (<ref>) boils down to n independent problems of the form u_+^(l) = min_x^(l)∈ Q_l{L_l/2u^(l)-x^(l)_2^2+ α U_l^T(y),x^(l)},l = 1,...,n.Since (y) has non-zero components only in the block i, U_l^T(y) is zero for all li. Thus, u-u_+ has non-zero components only in the block i and U_i(u^(i)-u_+^(i)) = u - u_+. In the unconstrained case, similarly to subsection <ref>, we obtain that u^(i)-u_+^(i)=γ e, where γ is some constant. Using these two facts, we obtain^T∇ f(y), u - u_+ = U_i U_i^T∇ f(y), ee , u - u_+=U_i^T∇ f(y), ee ,U_i^T(u - u_+)=U_i^T∇ f(y), ee ,u^(i)-u_+^(i) =U_i^T∇ f(y), ee ,γ e=U_i^T∇ f(y), γ ee , e=U_i^T∇ f(y), u^(i)-u_+^(i) = ∇ f(y), U_i(u^(i)-u_+^(i))= ∇ f(y), u - u_+ ,which proves (<ref>) for the unconstrained case.In the separable case, similarly to subsection <ref>, we obtain that u^(i)-u_+^(i) has only one j-th non-zero coordinate, where j ∈ 1,...,p_i. Hence, e_j, u^(i)-u_+^(i) e_j = u^(i)-u_+^(i). So, we get, ^T∇ f(y), u - u_+ = U_i U_i^T∇ f(y), e_je_j , u - u_+=U_i^T∇ f(y), e_je_j ,U_i^T(u - u_+)=U_i^T∇ f(y), e_je_j ,u^(i)-u_+^(i) =U_i^T∇ f(y), e_je_j ,u^(i)-u_+^(i) =U_i^T∇ f(y),e_j ,u^(i)-u_+^(i) e_j=U_i^T∇ f(y), u^(i)-u_+^(i) = ∇ f(y), U_i(u^(i)-u_+^(i))= ∇ f(y), u - u_+ ,which proves (<ref>) for the separable case.Smoothness. This assumption can be checked in the same way as in subsection <ref>.We have checked that all the assumptions listed in subsection <ref> hold. Thus, we can obtain the following convergence rate result for random derivative-free block-coordinate descent with random approximations for block derivatives as a corollary of Theorem <ref> and Lemma <ref>. Let <ref> with (x) defined in (<ref>), be applied to Problem (<ref>) in the setting of this subsection. Let f_* be the optimal objective value and x_* be an optimal point in Problem (<ref>). Assume that function value error (x)-f(x) satisfies |(x)-f(x)| ≤Δ, x ∈Q̅. Denote P_0^2 = (1-1/p)(f(x_0)-f_*) + ∑_i=1^nL_i/2u_0^(i)-x_*^(i)_2^2. * If the error in the value of the objective f can be controlled and, on each iteration, the error level Δ satisfies Δ≤P_0^2/64p^2A_k^2 , and τ = 2 √(Δ/L_0) then, for all k ≥ 1, f(x_k) - f_*≤6p^2P_0^2/(k-1+2p)^2 , wheredenotes the expectation with respect to all the randomness up to step k. * If the error in the value of the objective f can not be controlled and τ = 2 √(Δ/L_0), then, for all k ≥ 1, f(x_k) - f_*≤8p^2P_0^2/(k-1+2p)^2 + 16 (k-1+2p)^2 Δ. According to Remark <ref> and due to the relation δ = 2 √(Δ), we obtain that the error level in the function value should satisfy Δ≤^2 /144 p^2P_0^2. The parameter τ should satisfy τ≤/6 p P_0 √(L_0). At the same time, to obtain an -solution for Problem (<ref>), it is enough to choose k = max{⌈ p√(6P_0^2/) + 1 -2 p ⌉,0}. § MODEL GENERALITY IN A NON-ACCELERATED RANDOM BLOCK-COORDINATE DESCENTAbove we propose a unified view on different randomized gradient-free and (block) coordinate-wise schemes. In the last decade an interest to structural optimization has grown <cit.>. The most general results (see<cit.>) allow to unify different structures of the problem (composite problems, max-type problems e.t.c.) in one envelop (model generality <cit.>). But this envelop deals with the first-order methods. Motivated by Random Block Coordinate Descent (RBCD)<cit.> we try to develop RBCD in model generality. We expect that further unification of model generality conception <cit.> with the results of this paper is also possible.Typical randomized algorithm has such kind of auxiliary problemx_k+1=min_x ∈ Q{V[x_k](x) + α(x_k), x - x_k },see for example (<ref>) and (<ref>) in Algorithm <ref>. Here the term (x_k), x - x_k is a (random) linear <<model>> of target function in considered point x_k. The idea is to replace this model (x_k), x - x_k on something more general.For example, if f(x):= f(x) + h(x) we may expect, that composite model (x_k), x - x_k+ h(x) - h(x_k) is also well suited and may lead to a better results than standard model (x_k), x - x_k. Assume that V[z](x)=1/2∑_i=1^n L_i x^(i)-z^(i)_2^2,Q = ⊗_i=1^nQ_i. Function ψ_i(y,x) determines i-th part of the model of target function, that corresponds to block Q^(i). For example, in composite case if h(x) = ∑_i=1^n h_i(x^(i)) we can considerψ_i(y,x) = U_i U_i^T ∇ f(x), y - x+ h_i(y^(i)) - h_i(x^(i)). [convexity]Assume that for all x ∈ Q model ψ_i(y, x) is convex function of y∈ Q and there exists such γ > 0 that for any x, y ∈ Q_i ψ_i(y, x) ≤1/γ(f(y) - f(x)). Parameter γ = n for composite case.[smoothness] For anyx_k,x_k+1 generated by Algorithm <ref> f(x_k+1) ≤ f(x_k) + ψ_i(x_k+1, x_k) + V[x_k](x_k+1). Assumption <ref> typically holds true with additional condition that x_k,x_k+1 generated by Algorithm <ref> satisfy x_k^(j) = x_k+1^(j) for j ≠ i. Let the assumptions <ref> and <ref> hold. Let x_k be generated by Algorithm <ref>. Let f_* be the optimal objective value and x^* be an optimal point in Problem (<ref>). Then, for all N ≥ 1,f(x_N) - f(x^*) ≤γ/N(f(x_0) - f(x^*)) + γ/NV[x_0](x^*),where x_N = 1/N∑_k=0^N-1 x_k.From (<ref>), (<ref>) follows formula (3.12)of <cit.> -ψ_i(x, x_k) ≤f(x_k) - f(x_k+1) + V[x_k](x) - V[x_k+1](x)for all x∈ Q.From this formula and (<ref>)1/γ(f(x_k) - f(x)) ≤ -_iψ(x, x_k)≤f(x_k) -_if(x_k+1) + V[x_k](x) - _iV[x_k+1](x).Let start the proof using (<ref>)_if(x_k+1) ≤_if(x_k) + _i ψ_i(x_k+1, x_k) + _iV(x_k+1, x_k)Using optimality condition for (<ref>)∃ g ∈∂ψ(x_k+1, x_k), ⟨ g + V[x_k](x_k+1), x - x_k+1⟩≥ 0, ∀ x ∈ Q.From strongly convex ψ(x, x_k) + V[x_k](x) we obtain thatψ(x, x_k) + V[x_k](x) ≥ψ(x_k+1, x_k) + V[x_k](x_k+1) + ⟨ g + V[x_k](x_k+1), x - x_k+1⟩ + V[x_k+1](x)Using inequalities for proximal operator (<ref>) and (<ref>) we obtain that ψ(x_k+1, x_k) + V[x_k](x_k+1) ≤ψ(x, x_k) + V[x_k](x) - V[x_k+1](x),From (<ref>) and (<ref>) we can get1/n(f(x_k) - f(x)) ≤ -_i(ψ(x, x_k)) ≤ -_if(x_k+1) + _if(x_k) + _iV[x_k](x) - _iV[x_k+1](x)Then take the full mathematical expectation from each inequality, where k = 0, …, N-1, put x = x_* and sum all the inequalities for k = 0, …, N-1, we obtain that 1/γ∑_k=0^N-1 f(x_k) - N/γf(x_*) ≤(f(x_0) - f(x_N)) + V[x_0](x_*) -V[x_N](x_*)Since V[x_N](x_*) ≥ 0, then1/γ∑_k=0^N-1 f(x_k) - N/γf(x_*) ≤(f(x_0) - f(x_N)) + V[x_0](x_*)Using Yensen inequality and the fact that f(x_N) ≥ f(x_*) we obtain final resultf(x_N) - f(x_*) ≤γ/N(f(x_0) - f(x_*)) + γ/NV[x_0](x_*).§ CONCLUSIONIn this paper, we introduce a unifying framework, which allows to construct different types of accelerated randomized methods for smooth convex optimization problems and to prove convergence rate theorems for these methods. As we show, our framework is rather flexible and allows to reproduce known results as well as obtain new methods with convergence rate analysis. At the moment randomized methods for empirical risk minimization problems are not directly covered by our framework. It seems to be a n interesting direction for further research. Another directions, in which we actually work, include generalization of our framework for strongly convex problems based on well-known restart technique. Another direction of our work is connected to non-uniform probabilities for sampling of coordinate blocks and composite optimization problems. Acknowledgments. The authors are very grateful to Yu. Nesterov and V. Spokoiny for fruitful discussions. Our interest to this field was initiated by the paper <cit.>. siamplain | http://arxiv.org/abs/1707.08486v2 | {
"authors": [
"Pavel Dvurechensky",
"Alexander Gasnikov",
"Alexander Tiurin",
"Vladimir Zholobov"
],
"categories": [
"math.OC",
"cs.NA",
"math.NA",
"90C25, 90C30, 90C06, 90C56, 68Q25, 65K05, 49M27, 68W20, 65Y20, 68W40",
"G.1.6"
],
"primary_category": "math.OC",
"published": "20170726151602",
"title": "Unifying Framework for Accelerated Randomized Methods in Convex Optimization"
} |
UTF8gbsnFRIB/NSCL Laboratory, Michigan State University, East Lansing, Michigan 48824, USAFRIB Laboratory, Michigan State University, East Lansing, Michigan 48824, USADepartment of Physics and Astronomy and FRIB Laboratory, Michigan State University, East Lansing, Michigan 48824, USASchool of Physics, Peking University, Beijing 100871, ChinaBackground Weakly bound and unbound nuclear states appearing around particle thresholds are prototypical open quantum systems. Theories of such states must take into accountconfiguration mixing effects in the presence of strong coupling to the particle continuum space.Purpose To describe structure and decays of three-body systems, we developed a Gamow coupled-channel (GCC) approach in Jacobi coordinates by employing the complex-momentum formalism. We benchmarked the new frameworkagainst the complex-energy Gamow Shell Model (GSM).Methods The GCC formalism is expressed in Jacobi coordinates, so that the center-of-mass motion is automatically eliminated. To solvethe coupled-channel equations, we use hyperspherical harmonics to describe the angular wave functions while the radial wave functions are expandedin the Berggren ensemble, which includes bound, scattering and Gamow states. Results We show that the GCC method is both accurate and robust.Its results for energies, decay widths, and nucleon-nucleon angular correlations are in good agreement with the GSM results. Conclusions We have demonstrated that a three-body GSM formalism explicitly constructed in cluster-orbital shell model coordinates provides similar results to a GCC framework expressed inJacobi coordinates, provided that a large configuration space is employed.Our calculations for A=6 systems and^26Oshow that nucleon-nucleon angular correlations aresensitive to the valence-neutron interaction.The new GCC technique has many attractive features when applied to bound and unbound states of three-body systems: it is precise, efficient, and canbe extendedby introducing a microscopic model of the core.Structure and decays ofnuclear three-body systems: the Gamow coupled-channel method in Jacobi coordinates F.R. Xu (许甫荣) December 30, 2023 ===========================================================================================================§ INTRODUCTION Properties of rare isotopes that inhabit remote regions of the nuclear landscape at and beyond the particle driplines are in the forefront of nuclear structure and reaction research <cit.>. The next-generation of rare isotope beamfacilities will provide unique data on dripline systems thatwill testtheory, highlight shortcomings,andidentify areas for improvement.The challenge for nucleartheory is to develop methodologies to reliably calculate and understand the properties and dynamics of new physical systemswith different properties due to large neutron-to-proton asymmetriesand low-lying reaction thresholds.Here, dripline systems are of particular interest as they can exhibit exotic radioactivedecay modes such as two-nucleon emission <cit.>. Theories of such nuclei must take into account their open quantum nature. Theoretically, a powerful suite of A-body approaches based on inter-nucleon interactions provides a quantitative description of light and medium-mass nuclei andtheir reactions <cit.>. To unify nuclear bound states with resonances and scattering continuum within one consistent framework, advanced continuum shell-model approaches have been introduced <cit.>. Microscopic models of exotic nuclear states have been supplemented by a suite of powerful, albeit more phenomenologicalmodels,based on effective degrees of freedom such as cluster structures. While such models provide a“lower resolution" picture of the nucleus, they can be extremely useful when interpreting experimental data, providing guidance for future measurements, and provide guidance for more microscopic approaches.The objective ofthis work is to develop a new three-body method to describe both reaction and structure aspects of two-particle emission. A prototype system of interestis the two-neutron-unbound ground state of ^26O <cit.>. According to theory,^26O exhibits the dineutron-typecorrelations <cit.>. To describe such a system, nuclear model should be based on a fine-tuned interaction capable of describing particle-emission thresholds, a sound many-body method, and a capability to treat simultaneously bound and unbound states.If one considers bound three-body systems, few-body models are very useful <cit.>, especially models basedon the Lagrange-mesh technique <cit.> or cluster-orbital shell model (COSM) <cit.>. However, for the description ofresonances, the outgoing wave function in the asymptotic region need to be treated very carefully. For example, one can divide the coordinate space into internal and asymptotic regions, where the R-matrix theory <cit.>, microscopic cluster model <cit.>, and the diagonalization of the Coulomb interaction <cit.> can be used. Other useful techniques include the Green function method <cit.> and the complex scaling<cit.>. Our strategy is to construct a precise three-body framework toweakly bound and unbound systems similar to that of the GSM <cit.>. The attractive feature of the GSM is that – by employing the Berggren ensemble <cit.> – it treats bound, scattering, and outgoing Gamow states on the same footing. Consequently,energies and decay widths are obtained simultaneously as the real and imaginary parts of the complex eigenenergies of the shell model Hamiltonian <cit.>. In this study, we develop a three-body Gamow coupled-channel (GCC) approach in Jacobi coordinates with the Berggren basis. Since the Jacobi coordinates allow for the exact treatment ofnuclear wave functions in both nuclear and asymptotic regions, and as the Berggren basis explicitly takes into account continuum effects, a comprehensive description of weakly-bound three-body systems can be achieved. As the GSM is based on theCOSM coordinates,a recoil term appears due to the center-of-mass motion.Hence, it is of interest to compare Jacobi- and COSM-based frameworks for the description of weakly bound and resonant nuclear states. This article is organized as follows.Section <ref>contains the description ofmodels andapproximations. In particular, it lays out the new GCC approach and GSM model used for benchmarking, and defines the configuration spaces used.The results for A=6 systems and^26O are contained in Sec. <ref>. Finally, the summary and outlook are given in Sec. <ref>.§ THE MODEL §.§ Gamow Coupled Channel approach In the three-body GCC model, the nucleus is described in terms of a core and two valence nucleons (or clusters). The GCC Hamiltoniancan be written as:Ĥ = ∑^3_i=1p̂⃗̂^2_i/2 m_i +∑^3_i>j=1 V_ij(r⃗_ij)-T̂_ c.m.,where V_ij is the interaction between clusters i and j, including central, spin-orbit and Coulomb terms, andT̂_ c.m. stands forthe kinetic energy of the center-of-mass.The unwanted featureof three-body models is the appearance of Pauli forbidden states arising from the lack of antisymmetrization between core and valence particles. In order to eliminate the Pauli forbidden states, we implemented the orthogonal projection method <cit.> by adding to the GCC Hamiltionan the Pauli operatorQ̂= Λ∑_c |φ ^j_c m_c⟩⟨φ ^j_c m_c|,where Λ is a constant and | φ^j_c m_c⟩ is a 2-body state involving forbidden s.p. states of core nucleons. At large values of Λ, Pauli-forbidden states appear at highenergies, so that they are effectively suppressed. In order to describe three-body asymptotics and to eliminate the spurious center-of-mass motion exactly, we expressthe GCC model in the relative(Jacobi) coordinates <cit.>:x⃗ = √(μ _ij) (r⃗_i - r⃗_j), y⃗ = √(μ _(ij)k)(r⃗_k - A_ir⃗_i + A_jr⃗_j/A_i + A_j),where r⃗_i is the position vector of the i-th cluster, A_i isthe i-th cluster mass number, and μ _ij and μ _(ij)k are the reduced masses associated with x⃗ and y⃗, respectively:μ _ij = A_iA_j/A_i+A_j, μ _(ij)k = (A_i+A_j)A_k/A_i+A_j+A_k.As one can see in Fig. <ref>, Jacobi coordinates can be expressed asT- and Y-types, each associated witha complete basis set. In practice, it is convenient to calculate the matrix elements of the two-body interaction individually in T- and Y-type coordinates, and then transform them to one single Jacobi set. To describe the transformation between different types of Jacobi coordinates, it is convenient to introduce the basis of hyperspherical harmonics (HH) <cit.>. The hyperspherical coordinatesare constructed from a five-dimensional hyperangular coordinates Ω_5 and a hyperradial coordinate ρ=√(x^2 + y^2). The transformation between different sets of Jacobi coordinates is given by the Raynal-Revai coefficients <cit.>.Expressed in HH, the total wave-function can be written as <cit.>:Ψ ^JMπ (ρ, Ω_5) = ρ ^-5/2∑_γ Kψ ^Jπ_γ K(ρ) 𝒴 ^JM_γ K (Ω_5),where K is the hyperspherical quantum number and γ = {s_1,s_2,s_3,S_12,S,ℓ_x,ℓ_y,L} is a set of quantum numbers other than K. The quantum numbers s and ℓ stand for spin and orbital angular momentum, respectively, ψ ^Jπ_γ K(ρ) is the hyperradial wave function, and 𝒴 ^JM_γ K (Ω_5) is the hyperspherical harmonic. The resultingSchrödinger equation for the hyperradial wave functions can be written as a set of coupled-channel equations:[ -ħ^2/2m(d^2/dρ^2 - (K+3/2)(K+5/2)/ρ^2)-Ẽ] ψ ^Jπ_γ K(ρ)+ ∑_K'γ' V^Jπ_K'γ', Kγ(ρ) ψ ^Jπ_γ'K'(ρ) +∑_K'γ'∫_0^∞ W_K'γ', Kγ(ρ,ρ')ψ ^Lπ_γ'K'(ρ')dρ'=0,whereV^Lπ_K'γ', Kγ(ρ) = ⟨𝒴 ^JM_γ' K'|∑^3_i>j=1 V_ij(r⃗_ij)| 𝒴 ^JM_γ K⟩and W_K'γ', Kγ(ρ,ρ') = ⟨𝒴 ^JM_γ' K' | Q̂|𝒴 ^JM_γ K⟩is thenon-local potential generated by the Pauli projection operator (<ref>).In order to treat the positive-energy continuum space precisely, we use the Berggrenexpansion technique for the hyperradial wave function:ψ ^Jπ_γ K(ρ) = ∑_ n C^Jπ M_γ n Kℬ ^Jπ_γ n(ρ),where ℬ ^Jπ_γ n(ρ) represents a s.p. state belonging to to the Berggrenensemble <cit.>. The Berggren ensemble defines a basis in the complex momentum plane, which includes bound, decaying, and scattering states. The completeness relation for the Berggren ensemble can be written as:∑_ n∈ b,dℬ_ n(k_ n,ρ)ℬ_ n(k_ n,ρ^')+∫_L^+ℬ(k,ρ)ℬ(k,ρ^')dk = δ(ρ-ρ^'),where b are bound states andd aredecaying resonant (or Gamow)states lying between the real-k momentum axis in the fourth quadrant of thecomplex-k plane, and the L^+ contour representing the complex-k scattering continuum.For numerical purposes,L^+has to be discretized, e.g., by adopting the Gauss-Legendre quadrature <cit.>.In principle, the contour L^+can be chosen arbitrarily as long as it encompasses the resonances of interest. If the contour L^+ is chosen to lie along the real k-axis, the Berggren completeness relation reduces to the Newton completeness relation <cit.> involving bound and real-momentum scattering states.To calculate radial matrix elementswith the Berggren basis, we employthe exterior complex scaling <cit.>, where integrals are calculated along a complex radial path: ⟨ℬ_ n| V(ρ)|ℬ_ m⟩ =∫_0^Rℬ_ n(ρ)V(ρ)ℬ_ m(ρ)dρ+ ∫_0^+∞ℬ_ n(R+ρ e^iθ)V(R+ρ e^iθ)ℬ_ m(R+ρ e^iθ)dρ.For potentials that decrease as O(1/ρ^2) (centrifugal potential) or faster (nuclear potential),R should be sufficiently large to bypass all singularities and the scaling angle θ ischosen so that the integral converges, seeRef. <cit.> for details. As the Coulomb potential is not square-integrable, its matrix elements diverge whenk_n = k_m. A practical solution is provided by the so-called “off-diagonal method" proposed in Ref. <cit.>. Basically, a small offset ±δ k is added to the linear momenta k_n and k_m of involved scattering wave-functions, so that the resulting diagonal Coulomb matrix element converges. §.§ Gamow Shell Model In the GSM, expressed inCOSM coordinates, one deals with the center-of-mass motion by adding a recoil term (p̂⃗̂_1·p̂⃗̂_2/m_nA_ core) <cit.>. The GSM Hamiltonian is diagonalized in a basis of Slater determinants built from the one-body Berggren ensemble. In this case, it is convenientto deal with the Pauli principle by eliminating spurious excitationsat a level of the s.p. basis. In practice, one just needs to construct a valence s.p.space that does not contain the orbits occupied in the core. It is equivalent to the projection technique used in GCC wherein the Pauli operator (<ref>) expressed in Jacobi coordinates has a two-body character. The treatment of the interactions is the same inGSM and GCC. In both cases,we use thecomplex scaling method to calculatematrix elements <cit.> and the “off-diagonal method" to deal with the Coulomb potential <cit.>. Thetwo-bodyrecoil term istreated in GSMby expanding itin a truncated basis of harmonic oscillator (HO). The HO basis depends on the oscillator length b and the number of states used in the expansion. As it was demonstrated in Refs. <cit.>, GSM eigenvalues and eigenfunctions converge for a sufficient number of HO states, and the dependence of the results on b is very weak.Let us note in passing thatone has to be careful when usingarguments based on the variational principle when comparingthe performance of GSM with GCC. Indeed,the treatment of the Pauli-forbidden states is slightly different in the two approaches. Moreover, the recoil effect in the GSM is not removed exactly. (There is no recoil term in GCC as the center-of-mass motion is eliminated through the use of Jacobi coordinates.) §.§ Two-nucleon correlations In order to study the correlations between the two valence nucleons, weutilize thetwo-nucleon density <cit.> ρ_nn'(r,r',θ) = ⟨Ψ|δ(r_1-r)δ(r_2-r^')δ(θ_12 - θ)|Ψ⟩, where r_1, r_2, and θ_12 are defined in Fig. <ref>(a). In the following, we apply the normalization convention of Ref. <cit.> in which the Jacobian 8π^2 r^2 r'^2 sinθ is incorporated into the definition of ρ_nn', i.e., it does not appear explicitly. The angular density of the two valence nucleons is obtainedby integrating ρ_nn'(r,r',θ) over radial coordinates: ρ(θ) = ∫ρ_nn'(r,r^',θ) dr dr^'.The angular density is normalized to one:∫ρ(θ) dθ = 1.While it is straightforward to calculate ρ_nn' with COSM coordinates, the angular density cannot be calculated directly with the Jacobi T-type coordinates used to diagonalize the GCC Hamiltonian. Consequently, one can either calculate the density distribution ρ_ T(x,y,φ) in T-type coordinates and then transform it toρ(r_1,r_2,θ_12) in COSM coordinates by using the geometric relations of Fig. <ref>(a), or – as we do in this study – one can apply theT-type-to-COSMcoordinate transformation. This transformation <cit.>, provides an analytical relation between hyperspherical harmonics in COSM coordinates 𝒴 ^JM_γ^' K^' (r⃗_1^', r⃗_2^' ) and the T-type Jacobi coordinates 𝒴 ^JM_γ K (x⃗^', y⃗^' ), where r⃗_1^', r⃗_2^', x⃗^' and y⃗^' are:r⃗_1^' = √(A_i)r⃗_1, r⃗_2^' = √(A_j)r⃗_2, x⃗^' = x⃗= √(μ_ij)(r⃗_1-r⃗_2), y⃗^' = √(A_i+A_j/μ_(ij)k)y⃗ = A_ir⃗_1+A_jr⃗_2/√(A_i+A_j).§.§ Model space and parametersIn order to compare approachesformulated in Jacobi and COSM coordinates, we consider model spaces defined by the cutoff value ℓ_ max, which is the maximum orbital angular momentum associated with (r⃗_1, r⃗_2) in GSM and(x⃗, y⃗) in GCC.The remaining truncations come from the Berggren basis itself.The nuclear two-body interaction between valence nucleons has been approximated by thefinite-range Minnesota force with the original parameters of Ref. <cit.>. For the core-valence Hamiltonian, we took a Woods-Saxon (WS) potentialwithparametersfitted to the resonances of the core+n system. The one- and two-body Coulomb interaction has been considered when valence protons are present.In the case of GSM, we use the Berggren basis for the spd partial waves and a HO basis for the channels withhigher orbital angular momenta.For ^6He, ^6Li and ^6Be we assume the ^4He core. For ^6He and ^6Be, GSM we took a complex-momentum contour defined by the segmentsk= 0 → 0.17-0.17i → 0.34 → 3 (all infm^-1)for thep_3/2 partial wave,and0 → 0.5 → 1 → 3 fm^-1 for the remainingspd partial waves. For ^6Li, we took the contours 0 → 0.18-0.17i → 0.5 → 3 fm^-1 for p_1/2; 0 → 0.15-0.14i → 0.5 → 3 fm^-1 for p_3/2;and 0 → 0.25 → 0.5 → 3 fm^-1 for the sd partial waves. Each segment was discretized with10 points. This is sufficient for the energies and most of other physical quantities, but one may need more points to describe wave functions precisely, especially for the unbound resonant states that areaffected by Coulomb interaction. Hence, we choose 15 points for each segment to calculate the two-proton angular correlation of the unbound^6Be.TheHO basiswas defined through the oscillator length b = 2 fm and the maximum radial quantum number n_ max=10. The WS parametersfor the A = 6 nuclei are: the depth of the central term V_0= 47 MeV; spin-orbit strength V_ s.o. = 30 MeV; diffuseness a=0.65 fm; and the WS (and charge) radiusR=2 fm.With theseparameters we predictthe 3/2^- ground state (g.s.) of ^5He at E=0.732 MeV (Γ=0.622 MeV), and its first excited 1/2^- state at E=2.126 MeV (Γ=5.838 MeV). For ^26O, we consider the^24Ocore <cit.>. In the GSM variant, we used the contour0 → 0.2-0.15i → 0.4 → 3 fm^-1 for d_3/2,and 0 → 0.5 → 1 → 3 fm^-1 for the remaining spd partial waves. For the HO basis we took b = 1.75 fm and n_ max=10. TheWS potential for ^26Ohas fitted in Ref. <cit.> to the resonances of ^25O. Its parameters are: V_0= 44.1 MeV, V_ s.o.= 45.87 MeV, a= 0.73 fm, and R = 3.6 fm. The GCC calculations have been carried out with the maximal hyperspherical quantum number K_ max = 40, which is sufficient for all the physical quantities westudy. We checked that the calculated energiesdiffer by as little as 2 keV when varyingK_ max from 30 to 40. Similar as inGSM, in GCC we used the Berggren basis for the K ⩽ 6 channels and the HO basis for the higher angular momentum channels. The complex-momentum contour of the Berggren basis is defined as:k= 0 → 0.3-0.2i → 0.5 → 0.8 → 1.2 → 4 (all in fm^-1), with each segment discretized with 10 points. We took theHO basis withb = 2 fm andn_ max = 20. As k_ρ^2 = k_x^2 + k_y^2,the energy range covered by the GCC basis is roughly doubled as compared to that of GSM.For the one-body Coulomb potential, we usethe dilatation-analytic form <cit.>: U^(Z)_ c(r) = e^2Z_ c erf(r/ν_ c)/r,where ν_c=4R_0/(3√(π)) fm,R_0 is the radius of the WS potential, and Z_ c is the number of core protons.We emphasize that the large continuum space, containing states of both parities, is essential for the formation of the dineutron structure in nuclei such as ^6He or ^26O<cit.>. In the following, we shall studythe effect of including positive and negative parity continuum shells on the stability of threshold configurations. § RESULTS§.§ Structure of A=6 systems We begin withthe GCC-GSM benchmarking for the A=6 systems. Figure <ref> shows the convergence rate for the g.s. energies of^6He, ^6Li, and ^6Be with respect to ℓ_ max.(See Ref. <cit.> for a similar comparison between GSM and complex scaling results.) Whilethe g.s. energies of ^6He and ^6Be are in a reasonableagreement with experiment, ^6Li is overbound. This is because theMinnesota interaction does not explicitly separate the T = 0 and T = 1 channels.The structure of ^6He and ^6Be is given by the T=1 force, whilethe T=0channel that is crucial for ^6Li has not been optimized. This is of minor importance for this study, as our goal is to benchmark GCC and GSMnot to provide quantitative predictions.As we use different coordinates in GCC and GSM, theirmodel spaces are manifestly different. Still for ℓ_ max=10 both approaches provide very similar results, which is most encouraging. Onecan see in Fig. <ref> that the calculations done with Jacobi coordinates convergefaster than thosewith COSM coordinates. This comes from the attractive character of the nucleon-nucleon interaction,whichresults in the presence of a di-nucleonstructure (see discussion below). Consequently, as T-type Jacobi coordinateswell describethedi-nucleon cluster,they are able to capturecorrelations in a more efficient way than COSM coordinates. This is in agreement with the findings of Ref. <cit.> based onthe complex scalingmethod with COSM coordinates, who obtainedthe g.s. energy ^6Hethat was slightly less bound as compared to results of Ref. <cit.> usingJacobi coordinates. In any case, our calculations havedemonstrated that one obtains very similar results in GCC and GSM whensufficiently large model spaces are considered. As shown in Table <ref>, theenergy difference betweenGCC and GSM predictions for A=6 systems is very small, around 20 keV for majority of states. The maximum deviation of ∼70 keV is obtained for the 3^+ state of ^6Li. However, because of the attractive character of the T=0 interaction, the GSM calculation for thisstate has not fully converged at ℓ_ max = 10.Motivated by the discussion in Ref. <cit.>, we have also studied the effect of the ℓ-dependent core-nucleus potential. To this end, wechanged the WS strength V_0 from 47 MeV to 49 MeV for the ℓ=1 partial waves while keeping the standard strength for the remaining ℓ values. As seen in Fig. <ref>, the convergence behavior obtained withJacobi and COSM coordinates is fairlysimilar to that shown in Fig. <ref>, where the WSstrength V_0 is the same for all partial waves.For ℓ_ max=12, the difference between GSM and GCC energies of ^6Hebecomes very small.This result is consistentwith the findings ofRef. <cit.> that therecoileffect can indeedbesuccessfully eliminated using COSM coordinatesat the expense of reduced convergence.In order to see whether the difference between the model spaces of GCC and GSM can be compensated by renormalizing the effectiveHamiltonian, we slightly readjusted the depthof the WS potential in GCC calculationsto reproduce the g.s. GSM energy of ^6He at model space ℓ_ max=7. As a result, the strength V_0 changed from 47 MeV to 46.9 MeV. Except for the 2^+ state of ^6He, the GSM andGCC energies for A=6 systems got significantly closer as a result of such a renormalization. This indicates that the differences between Jacobi coordinates and COSM coordinates can be partly accounted for by refitting interaction parameters, even though model spaces and asymptotic behavior are different.GCC is also in rough agreement with GSM when comparing decay widths, considering that they are very sensitive to the asymptotic behavior of the wave function, which is treated differently with Jacobi and COSM coordinates. Also, the presence of the recoil term in GSM, which is dealt with by means of the HO expansion, is expected to impact the GSM results fordecay widths. In order to checkthe precision of decay widthscalculated with GCC, we adopted the current expression <cit.>:Γ=i∫( Ψ^† Ĥ Ψ - Ψ Ĥ Ψ^† ) dx⃗dy⃗/∫|Ψ|^2 dx⃗dy⃗,which can be expressed inhyperspherical coordinates as <cit.>:Γ=i ħ^2/m. ∫ dΩ_5Im[ψ∂/∂ρψ^†]|_ρ=ρ_ max/∫^ρ_ max_0 |ψ|^2 dρ dΩ_5,where ρ_ maxis larger than the nuclear radius (in general, the decay width should not depend on the choice of ρ_ max).By using the current expression, we obtain Γ=42 keV for 2^+ state of ^6He and Γ=54 keV for 0^+ state of ^6Be, which are practically the same as the GCC values of Table <ref> obtainedfrom the direct diagonalization. We now discuss the angular correlationof the two valence neutrons in the g.s. of^6He. Figure <ref> shows GSM and GCC results formodelspaces defined bydifferent values of ℓ_ max.The distribution ρ(θ) shows two maxima <cit.>. The higher peak, at a small opening angle, can be associated with a dineutron configuration.The second maximum, found in the region of large angles, represents the cigarlike configuration. The GCC results for ℓ_ max=2 and 10 are already very close. This is not the case for the GSM, which shows sensitivity to the cutoff value of ℓ. This is because the large continuum space, including states of positive and negative parity is needed in the COSM picture to describe dineutron correlations <cit.>. Indeed, as ℓ_ max increases, the angular correlations obtained in GSM and GCC are very similar. This indicates that Jacobi and COSM descriptions of ρ(θ) areessentially equivalentprovided that the model space is sufficiently large.In order to benchmark GCC and GSM calculations for the valence-proton case, in Fig. <ref> we compare two-nucleon angular correlations for A = 6 nuclei ^6He, ^6Li, and ^6Be.Similar to Refs. <cit.>, we find that theT=1 configurations have a dominantS = 0 component, in which the two neutrons in ^6He or two protons in ^6Be are in the spin singlet state. The amplitude of the S = 1 density componentissmall. For all nuclei, GCC and GSM angular correlations are close.Similar to ^6He, the two peaks in ^6Be indicate diproton and cigarlike configurations <cit.> (see also Refs. <cit.>). It is to be notedthat the dineutron peak in ^6He is slightly higher than the diproton maximum in ^6Be. This isdue to the repulsive character of the Coulomb interaction between valence protons.The large maximum at small opening angles seen in ^6Li corresponds to the deuteron-like structure. As discussed in Ref. <cit.>, this peak is larger that the the dineutron correlation in ^6He.Indeed, the valence proton-neutron pair in ^6Li is very strongly correlated because the T=0 interaction is much stronger than the T=1 interaction. The different features in the two-nucleon angular correlationsin the threeA=6 systems shown in Fig. <ref> demonstrate that the angular correlationscontain useful information on the effective interaction between valence nucleons. §.§ Structure of unbound ^26O After benchmarking GSM and GCC for A=6 systems, we apply both models to ^26O, which is believed to be a threshold dineutron structure <cit.>.It is a theoretical challenge to reproduce the resonances in ^26O as both continuum and high partial waves must be considered.As ^24O can be associated with thesubshell closure in which the 0d_5/2 and 1s_1/2 neutron shells are occupied <cit.>, it can be used as core in our three-body model. Figure <ref> illustrates the convergenceof the g.s. of ^26O with respect toℓ_ max in GSM and GCC calculations. It is seen that in the GCC approach the energy convergesnearly exponentiallyand that the stable result is practically reached at ℓ_ max=7.While slightly higher in energy, the GSM results are quite satisfactory, as they differ onlyby about30 keV from the GCC benchmark. Still, it is clear that ℓ_ max=12 is not sufficient to reach the full convergence in GSM. Thecalculated energies and widthsof g.s. and 2^+ state of ^26O are displayedin Table <ref>; they are bothconsistent with the most recentexperimental values <cit.>. The amplitudes of dominant configurations listed inTable <ref>illustrate the importance of considering partial waves of different parity in the GSM description of a dineutron g.s. configuration in ^26O <cit.>.The g.s. wave function of ^26Ocomputed in GCC is shown inFig. <ref> in the Jacobi coordinates. The corresponding angular distribution is displayed in Fig. <ref>.Three pronounced peaks associated with the dineutron, triangular, and cigarlike configurations <cit.> can be identified.In GCC, the (ℓ_x, ℓ_y) = (s, s), (p, p)components dominate the g.s. wave function of^26O; this is consistent with a sizable clusterization of the two neutrons. In COSM coordinates,it is the (ℓ_1, ℓ_2) = (d,d) configuration that dominates, but the the negative-parity (f, f) and (p, p) channels contribute with ∼20%. Again, it is encouraging to see that with ℓ_ max=10 both approaches predict very similar two-nucleon densities.In Table <ref>we also display the predicted structure of the excited 2^+ state of ^26O . The predicted energy is close to experiment <cit.> and other theoretical studies, see, e.g., <cit.>.We obtain a small width for this state, which is consistent with the GSM+DMRG calculations of Ref. <cit.>. The GCC occupations of Table <ref> indicate that the wave function of the 2^+ state is spreadout in space, as the main three configurations, of cluster type, only contribute to the wave function with only 65%. When considering the GSM wave function, the (d, d) configurationdominates.Thecorresponding two-neutron angular correlation shown in Fig. <ref>(b)exhibits a broad distribution with a maximumaround90^∘.This situation is fairly similar to what has beenpredicted for the 2^+ state of ^6He <cit.>. Finally, it is interesting to studyhow the neutron-neutron interaction impacts the angular correlation. To this end, Fig. <ref>(a) shows ρ(θ) obtained with the Minnesota neutron-neutron interaction whose strength has been reduced by 50%.While there are still three peaks present, the distribution becomes more uniform andthe dineutron component no longer dominates. We can this conclude that the nn angular correlation can be used as an indicator of theinteraction between valence nucleons. § CONCLUSIONS We developed a Gamow coupled-channel approach in Jacobi coordinates with the Berggren basis to describe structure and decays of three-body systems. We benchmarkedthe performance of the new approach against the Gamow Shell Model. Both methods are capable of considering large continuum spaces but differ in their treatment ofthree-body asymptotics, center-of-mass motion, and Pauli operator. In spite of these differences, we demonstrated that the Jacobi-coordinate-based framework (GCC) and COSM-based framework (GSM) can produce fairly similar results, provided that the continuum space is sufficiently large.For benchmarking and illustrative examples we choose^6He, ^6Li, and ^6Be, and ^26O – all viewed as a core-plus-two-nucleon systems. We discussed the spectra, decay widths, and nucleon-nucleon angular correlations in these nuclei. The Jacobi coordinates capturecluster correlations (such as dineutron and deuteron-type) more efficiently; hence, the convergence rate of GCC is faster than that of GSM. For^26O, we demonstrated the sensitivity of nn angular correlation to the valence-neutron interaction. It will be interesting to investigate this aspect further to provideguidance for future experimental investigations of di-nucleon correlations in bound and unbound states of dripline nuclei.In summary, we developed an efficient approach to structure and decays of three-cluster systems. The GCC method is based on a Hamiltonian involving a two-body interaction between valence nucleons and a one-body field representing the core-nucleon potential. The advantage of the model is its ability to correctly describe the three-body asymptotic behavior and the efficient treatment of the continuum space, which is of particular importance for the treatment of threshold states and narrow resonances. The model can be easily extended along the lines of the resonating group method by introducing a microscopic picture of the core <cit.>. 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Płoszajczak,10.1103/PhysRevC.89.034624 journal journal Phys. Rev. C volume 89, pages 034624 (year 2014)NoStop | http://arxiv.org/abs/1707.08954v1 | {
"authors": [
"S. M. Wang",
"N. Michel",
"W. Nazarewicz",
"F. R. Xu"
],
"categories": [
"nucl-th"
],
"primary_category": "nucl-th",
"published": "20170727175750",
"title": "Structure and decays of nuclear three-body systems: the Gamow coupled-channel method in Jacobi coordinates"
} |
Juan J. Valencia-Cardona 1,Quentin Williams 2, Gaurav Shukla 3,Renata M. Wentzcovitch 4,5 1Scientific Computing Program, University of Minnesota, Minneapolis, Minnesota, USA 2Department of Earth and Planetary Sciences, University of California Santa Cruz, Santa Cruz, California, USA 3Department of Earth, Ocean, and Atmospheric Science, Florida State University, Tallahassee, Florida, USA 4Department of Applied Physics and Applied Mathematics, Columbia University, New York City, New York, USA 5Lamont-Doherty Earth Observatory, Columbia University, Palisades, New York, USA Elastic anomalies produced by the spin crossover in ferropericlase have been documented by both first principles calculations and high pressure-temperature experiments. The predicted signature of this spin crossover in the lower mantle is, however, subtle and difficult to geophysically observe within the mantle. Indeed, global seismic anomalies associated with spin transitions have not yet been recognized in seismologic studies of the deep mantle. A sensitive seismic parameter is needed to determine the presence and amplitude of such a spin crossover signature. The effects of spin crossovers on Bullen's parameter, η, are assessed here for a range of compositions, thermal profiles, and lateral variations in temperature within the lower mantle. Velocity anomalies associated with the spin crossover in ferropericlase span a depth range near 1,000 km for typical mantle temperatures. Positive excursions of Bullen's parameter with a maximum amplitude of ∼ 0.03 are calculated to be present over a broad depth range within the mid-to-deep lower mantle: these are largest for peridotitic and harzburgitic compositions. These excursions are highest in amplitude for model lower mantles with large lateral thermal variations, and with cold downwellings having longer lateral length-scales relative to hot upwellings. We conclude that predicted deviations in Bullen's parameter due to the spin crossover in ferropericlase for geophysically relevant compositions may be sufficiently large to resolve in accurate seismic inversions of this parameter, and could shed light on both the lateral variations in temperature at depth within the lower mantle, and the amount of ferropericlase at depth. § INTRODUCTION The adiabatic nature of the convecting mantle is a frequently used concept in the geophysical sciences. For instance, equation of state parameters, which are used to calculate the elastic and thermodynamic properties of minerals at mantle conditions, are commonly assumed to be adiabatic within the convecting mantle, e.g., the adiabatic bulk modulus and its derivative. However, various geodynamic simulations and seismological models <cit.> suggest that the mantle is regionally nonadiabatic, particularly in the shallow and deep mantle regions, and in some cases, at mid lower mantle pressures. The latter is important because deviations from adiabaticity within the mantle provide insights into temperature gradients, heat flux, thermal history, thermal boundary layers, phase transitions, chemical stratification, and compositional heterogeneities. Therefore, knowledge about the degree of adiabaticity of the mantle helps us to constrain its composition and thermal structures related to mantle convection <cit.>. A common observable that quantifies the adiabaticity level of the mantle is Bullen's parameter, η. Introduced and developed by <cit.>, η is a measure of the ratio between the actual density increase with pressure within the Earth (as constrained by a combination of seismology, the Earth's moment of inertia, and mass) with respect to the profile derived from adiabatic self-compression. As such, it is expected to be unity where the mantle is homogeneous, adiabatic, and free of phase transitions. Thus, deviations of η from unity (generally ∼± 0.1 or less), indicate super(sub)adiabatic regions, and consequently, the presence of thermal boundary layers, compositional variations or phase transitions. Moreover, there is also the possibility that due to internal heating within the mantle, the mantle may be systematically subadiabatic <cit.>.Evaluations of η in geodynamic simulations are generally done by probing the parameter space associated with plausible convection models. This includes examining the effects of possible variations of the thermal conductivity, thermal expansion coefficient, viscosity, internal heating, and heat flux from the core, each of which directly impact the inferred geotherms <cit.>. For instance, if internal heating is relatively significant, subadiabaticity is expected. Additionally, differences in elastic properties between individual phases within an aggregate can also produce variations in Bullen's parameter, and hence apparent deviations from adiabaticity. This is a bulk attenuation effect. Specifically, bulk attenuation phenomena are attributed to internal shear stresses generated from the local mismatch of the elastic moduli of neighboring grains in a given aggregate. One formulation of bulk attenuation by <cit.> characterizes it through the ratio of the adiabatic bulk modulus K_S and an effective modulus (Reuss bound) K_E, since the mantle can be assumed to be under hydrostatic pressure. Attenuation is a complicated problem to tackle, because it involves calculating complex moduli with an associated time dependency <cit.>. Such bulk attenuation effects are beyond the scope of this study, since the calculations we conduct are not time-dependent, but certainly needs to be addressed to understand systematic deviations of Bullen's parameter from 1. Here, we study how anomalies in bulk modulusinduced by spin crossovers affect the Bullen's parameter, and hence inferred adiabaticity of the lower mantle.Elastic anomalies produced by the spin crossover in ferropericlase (fp) and bridgmanite (bdg), have been documented by both first principles calculations and high pressure-temperature experiments <cit.>. The predicted signatures of this spin crossover in the lower mantle are subtle. Despite the fact that thermally induced velocity heterogeneities associated with this spin crossover appear to correlate statistically with seismic tomographic patterns observed in deeply rooted plumes <cit.>, spherically averaged anomalies have not yet been recognized in seismologic studies of the deep mantle. This may be due to difficulties associated with resolving gradual changes in the slopes of seismic velocities as a function of depth, and the trade-offs involved in seismic inversions of depth-dependent velocity and density structures. In particular, velocity anomalies associated with the spin crossover in fp are anticipated to span a depth range greater than 1000 km at mantle temperatures. Thus, a sensitive seismic parameter is needed to determine the presence or absence of this spin crossover signature, which would in turn shed light on the amount of ferropericlase in the lower mantle. Bullen's parameter η is an ideal candidate as it relates seismic wave speeds with density variations, and sensitively records deviations from adiabaticity. Moreover, deviations from Bullen's parameter can be readily identified because it has a clear reference value (unity) in an adiabatic mantle that is heated from below. We calculated one dimensional perturbations of η due to changes in composition, temperature, and spin crossover.We achieved this by computing η of different relevant mantle aggregates along their own adiabats. We also approximate lateral variations in temperature by modeling differing areas and temperature differences between upwellings and downwellings. The mantle phases of the aggregates considered are bridgmanite (bdg: Al- Fe- bearing MgSiO_3 perovskite), CaSiO_3 perovskite (CaPv), and ferropericlase (fp: (Mg,Fe)O). The aggregates have Mg/Si ratios that range from 0.82 to 1.56 and are harzburgite (Mg/Si ∼ 1.56) <cit.>, chondrite (Mg/Si ∼ 1.07) <cit.>, pyrolite (Mg/Si ∼ 1.24) <cit.>, peridotite (Mg/Si ∼ 1.30) <cit.>, and perovskite only (Mg/Si ∼ 0.82) <cit.>. The predicted deviations in η due to the spin crossover are comparable to previously reported variations <cit.>, and may be sufficiently large to turn up in accurate seismic inversions of this parameter. § METHOD AND CALCULATION DETAILSWe used bdg Mg_1-xFe^2+_xSiO_3, (Mg_1-xAl_x)(Si_1-xAl_x)O_3, (Mg_1-xFe^3+_x)(Si_1-xAl_x)O_3, (Mg_1-xFe^3+_x)(Si_1-xFe^3+_x)O_3(x=0 and 0.125) and fp Mg_1-yFe_yO (y = 0 and 0.1875) thermoelastic properties from <cit.> and <cit.>. Results for other x and y values were obtained by linear interpolation. All compositions account for the spin crossover in fp unless otherwise noted, i.e., bdg's iron (ferrous and/or ferric) is in the high spin (HS) state and fp is in a mixed spin (MS) state of HS and low spin (LS) states. For CaPv, we used thermoelastic properties from <cit.>, which were reproduced within the Mie-Debye-Grüneisen<cit.> formalism. The mantle aggregates in this study, namely, harzburgite<cit.>, chondrite <cit.>, pyrolite<cit.>, peridotite <cit.>, and perovskititic only <cit.>, are mixtures within the SiO_2 - MgO - CaO - FeO - Al_2O_3 system (ignoring alkalis and TiO_2 is not anticipated to resolvably affect the results). In addition, the Fe-Mg partition coefficient K_D = x/(1-x-z)/y/(1-y) between bdg and fp, which is known to be affected by the spin crossover <cit.>, was assumed to beuniform throughout the mantle with a value of 0.5. Further details about these compositions can be found in <cit.>. The adiabats of the different minerals and aggregates were integrated from their adiabatic gradient,( ∂ T/∂ P)_S =α V T/ C_pWe denote the molar fraction, molar volume, molar mass, thermal expansion coefficient, and isobaric specific heat of the i^th mineral in the mixture as μ_i, V_i, M_i, α_i, and Cp_i respectively. The aggregate properties such as volume, thermal expansion coefficient, and isobaric specific heat are then V=∑_i μ_iV_i, α= ∑_i α_i μ_i V_i / V, and C_p = ∑_i μ_i C_p_i. The adiabatic aggregate bulk moduli K_S were obtained from the Voigt-Reuss-Hill (VRH) average. Moreover, the aggregate density ρ = ∑_i μ_i M_i/V and seismic parameter ϕ = K_S/ρ were calculated along the aggregate adiabat, in order to compute adiabatic changes of density with respect to pressure as, η =ϕdρ/dPwhere η is the Bullen's parameter. If η =1 the mantle is homogeneous and adiabatic, whereas values of η > 1 can indicate a phase change as ρ varies more rapidly with depth than predicted by the adiabat. Furthermore, values of η < 1 may signify the presence of a thermal boundary layer or substantial internal heat production. Details about equation (<ref>) can be found in the supplementary information.§ RESULTS AND DISCUSSION§.§ Observations of η in the lower mantle Figure <ref> shows different η calculations from previous geodynamic <cit.>, seismic <cit.>, and seismic plus mineral physics models with a priori starting conditions <cit.>.Overall, η oscillates between values of ∼ 1.04 to 0.96 for most of these models, except for the AK135F model <cit.>, which displays the largest fluctuations. AK135F exhibited an average value of η ∼ 0.92 from 1000 km to 2700 km in depth,and variations in η above and below those depths were at least of the order of ∼ 0.1, which is substantially larger than the other inversions and calculations. For other seismic models such as PEM <cit.> and PREM <cit.> such large fluctuations are not observed, but they could be suppressed by the continuity requirements of the polynomial formulations of these models. However, the Bullen's parameters of these seismic models do suggest the presence of a thermal boundary layer at the bottom of the lower mantle, as shown by the negative slope of all models in the bottommost hundred to few hundred km of the mantle. Notably, for the mineral physics plus inverse model calculation by <cit.>, η values less than one from 800 km to 1300 km were attributed to iron depletion from their initially pyrolitic compositional model. Two and three dimensional geodynamic calculations of η were first done by <cit.>, where the effect of varying parameter space properties, such as thermal conductivity, thermal expansion coefficient, and viscosity, lead to different perturbations in η, but with an average value of ∼ 1.01. This average value is in general agreement with other geodynamic calculations by <cit.>, which also showed that the presence of internal heat sources lead to subadiabatic regions. Other thermal contributions, like core heating, cause superadiabatic temperature gradients at the bottom of the mantle and thus the presence of a thermal boundary layer, as manifested by the negative slopes of η near the base of the mantle.§.§ Spin-crossover effect on the adiabaticity of the lower mantle We studied the effect of spin crossovers on lower mantle adiabaticity by examining η excursions for different lower mantle aggregates along their self consistent adiabats. All of the adiabats of the different aggregates are listed in <cit.>. Figure <ref> shows the variations of η only due to the spin crossover in fp: only a portion of trivalent iron in bdg is anticipated to undergo a spin transition within the mantle (e.g., <cit.> and <cit.>). For compositions with fp, fluctuations inη were ∼ 0.02 max, which are well within the variations in seismological observations and geodynamical calulations shown in Figure <ref>. Furthermore, larger deviations from adiabaticity occur as the aggregate’s Mg/Si ratio, i.e., fp content, is increased. The sensitivity to Mg/Si content of the Bullen's parameter maximum near 1900 km depth, induced by the spin crossover, is relatively large: peridotitic and harzburgitic compositions have anη anomaly which is nearly twice that of the chondritic composition. The η excursions for the perovskitic composition, Pv only, depict the profile of a composition without fp in the lower mantle. §.§ Lateral temperature variationsWe have characterized what Bullen parameter anomalies, due to spin crossovers, might generate for one-dimensional seismic models of an isochemical adiabatic mantle. However, the lack of maxima in most Bullen parameter observations (Figure <ref>) that are at the appropriate depth and have the right breadth to correspond to the spin crossover of fp, led us to probe the effect of lateral temperature variations on deviations ofη. Since lateral temperature variations and their areal distribution at a given depth of hot/upwelling and cold/downwelling material are not well-constrained in the deep mantle (e.g., <cit.>), we conducted a sensitivity analysis for the effect of thermal variations on η in a pyrolitic mantle. Here, material at each depth is distributed along adiabats with potential temperatures above (hot) and below (cold) a reference adiabat pinned at 1873 K at 23 GPa as in <cit.> (B&S) (See also <cit.>). The lateral temperature variations between hot and cold regions that we probed were± 250 K, ± 500 K, and ± 750 K in a sequence of 25%:75%, 50%:50% and 75%:25% ratio of the mantle at a given depth being hot:cold (See Figure <ref>).For all the temperature-average distributions (Figures <ref>a, <ref>b, and <ref>c), we observed that the spin crossover anomalies, i.e. deviations from adiabaticity, became more prominent at lower temperatures: this is a natural consequence of the broadening of the spin transition that occurs at high temperatures. Conversely, greater amounts of hot material tend to make spin crossovers more difficult to resolve. Furthermore, we also observed that for large temperature variations, ± 750 K, two peaks in η can also be generated at different depths in an isochemical thermally heterogeneous mantle (Figure <ref>c). This phenomenon is attributed to the volume increase with temperature, which increases the pressures that are required for the spin crossover to occur. Since the amplitude of the perturbations in η increases also with higher fp content, it is expected that regions with larger cold harzburgitic chemistry present within the lower mantle, such as subducting slabs, should have substantially greater local fluctuations in the Bullen's parameter if a local vertical sampling of η over such regions is performed. Beyond lateral temperature variations, we examined the case of coupled compositional and thermal lateral heterogeneities. The rationale here is that cold, downwelling subducted material is likely to have a larger concentration of harzburgite than ambient mantle. We utilized a similar temperature averaging scheme, but with cold η values being harzburgitic. Figure <ref> shows different η profiles with the mantle being 75% hot(pyrolite) and 25% cold(harzburgite). For this scenario, perturbations in η due to the spin crossover vary their magnitude and reach a maxima at different depths, depending on the temperature difference between the cold downwellings of harzburgitic chemistry and ambient pyrolitic mantle. If the temperature difference is sufficiently large, e.g. ± 750 K, multiple peaks can be observed. Thus, the relative amplitudes and locations of multiple peaks could, if observed/observable, provide strong constraints on lateral variations in the geotherm and/or composition of the deep mantle. In particular, the depth at which the spin transition-induced peak occurs in Bullen's parameter is highly sensitive to temperature (Figure <ref>), while the amplitude of its variation is sensitive to composition (Figure <ref>). § GEOPHYSICAL SIGNIFICANCEWe have utilized η as an observable for spin crossovers in the lower mantle for the first time, in an attempt to reconcile mineral physics with seismic observations and to understand how such spin crossovers may affect observations of deviations from adiabaticity within the mantle. Our results suggest that the spin crossover signatures in η should be sufficiently large to turn up in accurate (ca. 1%) seismic inversions for this parameter. Whether such accuracies are achievable is unclear: several decades ago, <cit.> concluded that η variations from seismic observations could be resolved with a precision no better than 2%. Recent results from an inverse Bayesian method, deployed via a neural network technique by <cit.>, showed that ρ, Vp, and Vs may each be resolvable to somewhat better than 1% in the ∼ 2000 km depth range, based on their observed probability density functions. A linear combination of these uncertainties will certainly lead to values of order 1-2.5% for the net uncertainty in 1-D inversions for Bullen's parameter. Nevertheless, given markedly improved and more accurate seismic inversions coupled with substantially larger data sets, it is possible that better constraints on η might be developed.We also highlight the importance of the chosen temperature profile, as it has a direct impact on η. Elastic moduli, seismic velocities, and aggregate densities stronglydepend ontemperature. Hence, super(sub)adiabatic geotherms will lead to different interpretations of η. As recently showed by <cit.>,the spin crossover in fp and bdg induces an increment in the adiabat's temperature of a given aggregate and such a temperature increment will impactη's sensitivity. Because of the potentially complex coupling of lateral temperature differences with compositional variations, further work on the effect of spin crossovers on η would likely benefit from an assessment within a three dimensional convective scheme, such as the formulation proposed by <cit.>.§ CONCLUSIONSApparent deviations from adiabaticity due to spin crossover, as recorded by the Bullen's parameter, increased in proportion to the aggregate's ferropericlase content. The magnitude of these perturbations is generally consistent with the magnitude of variations in η present in previous seismological and geodynamic inversions of η in the lower mantle. Our results provide a sense of how much of a perturbation in η, given the spin crossover and lateral temperature variations, might be expected in one dimensional seismic models, with the net result being of order 1-2%. Accurate characterization of η either globally or locally could provide constraints on both the lateral temperature distribution and the fp content at depth, although such determinations hinge critically on achieving sufficient seismic resolution to resolve spin transitions. Also, the perturbations found in η for different mantle temperature averages highlight the importance of doing vertical seismic velocity profiles with sufficient precision to allow η to be characterized on a regional basis. Our results provide a guide for possible a priori models of η in regionalized inversions of velocity as a function of depth: inversions without spin crossover induced perturbations in η implicitly assume that spin transitions are absent at depth, and hence that no ferropericlase is present in the deep mantle. We thank two reviewers for helpful comments. This work was supported primarily by grants NSF/EAR 1319368 and 1348066. Q. Williams was supported by NSF/EAR 1620423. Results produced in this study are available in the supporting information. The 2016 CIDER-II program (supported by NSF/EAR 1135452) is thanked for providing a portion of the original impetus of this study.§ SUPPLEMENTARY MATERIAL The supporting information consists of Text S1, Figure S1, and Table S1.Figure S1 shows η for prystine lower mantle minerals, namely, MgSiO_3, MgO, and CaSiO_3.Text S1 shows the Bullen's parameter η derivation.Table S1 the values of η for the different aggregates.§.§ Text S1The bulk modulus of a mineral under adiabatic self compression is given by,K_S = ρ( ∂ P/∂ρ)_SHence,K_S/ρ = ( ∂ P/∂ρ)_S = ϕ where ϕ is the seismic parameter ϕ = V_P^2 - ( 4/3)V_S^2.Furthermore, assuming a homogeneous media under hydrostatic changes in pressures with respect to depth,dP/dr = -ρ g where g and r are the acceleration due gravity and depth, respectively. Thus,dP/dρdρ/dr = -ρ g and using equation (<ref>),1 = - ϕρ^-1 g^-1dρ/dr Equation (<ref>) is known as the Adams-Williamson equation. 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Justo, Z. Wu, C. R. S da Silva, D. A. Yuen, and D. Kohlstedt (2009), Anomalous compressibility of ferropericlase throughoutthe iron spin cross-over, Proc. Natl. Acad. Sci., 106, 8447-8452.[Williams and Knittle(2005)Williams and Knittle]WilliamsWilliamsQ. and E. Knittle (2005), The uncertain major element bulk composition of Earth's mantle,Earth's Deep Mantle: Structure, Composition, and Evolution. Geophysical Monograph Series 160. AGU, 10.1029/160GM12.[Wu et al.(2013)Wu, Justo and Wentzcovitch ]Wu1 Wu Z., J. F. Justo, and R. M. Wentzcovitch (2013), Elastic anomalies in a spin-crossover system: Ferropericlase at lower mantle conditions, Phys. Rev. Lett., 110, 228501. [Wu et al.(2009)Wu, Justo, da Silva, de Gironcoli, and Wentzcovitch ]Wu4 Wu Z., J. F. Justo,C. R. S da Silva, S de Gironcoli, and R. M. Wentzcovitch (2009), Anomalous thermodynamicproperties in ferropericlase throughout its spin crossover transition, Phys. Rev. B., 80, 014409. 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"authors": [
"Juan J. Valencia-Cardona",
"Quentin Williams",
"Gaurav Shukla",
"Renata M. Wentzcovitch"
],
"categories": [
"physics.geo-ph"
],
"primary_category": "physics.geo-ph",
"published": "20170726204531",
"title": "Bullen's parameter as a seismic observable for spin crossovers in the lower mantle"
} |
Bandit Convex Optimization forScalable and Dynamic IoT ManagementTianyi Chen and Georgios B. Giannakis Work in this paper was supported by NSF 1509040, 1508993, and 1711471.T. Chen and G. B. Giannakis are with the Department of Electrical and Computer Engineering and the Digital Technology Center, University of Minnesota, Minneapolis, MN 55455 USA. Emails: {chen3827, georgios}@umn.edu December 30, 2023 =============================================================================================================================================================================================================================================================================================================================================The present paper deals with online convex optimization involving both time-varying loss functions,and time-varying constraints.The loss functions are not fully accessible to the learner, and instead only the function values (a.k.a. bandit feedback) are revealed at queried points. The constraints are revealed after making decisions, and can be instantaneously violated, yet they must be satisfied in the long term. This setting fits nicely the emerging online network tasks such as fog computing in the Internet-of-Things (IoT), where online decisions must flexibly adapt to the changing user preferences (loss functions), and the temporally unpredictable availability of resources (constraints).Tailored for such human-in-the-loop systems where the loss functions are hard to model,a family of bandit online saddle-point (BanSaP) schemes are developed, which adaptively adjust the online operations based on (possibly multiple) bandit feedback of the loss functions, and the changing environment.Performance here is assessed by: i) dynamic regret that generalizes the widely used static regret; and, ii) fit that captures the accumulated amount of constraint violations.Specifically, BanSaP is proved to simultaneously yield sub-linear dynamic regret and fit, provided that the best dynamic solutions vary slowly over time. Numerical tests in fog computation offloading tasks corroborate that our proposed BanSaP approach offers competitive performance relative to existing approaches that are based on gradient feedback. Online learning, bandit convex optimization, saddle-point method, Internet of Things, mobile edge computing.§ INTRODUCTIONInternet-of-Things (IoT) envisions an intelligent infrastructure of networked smart devices offering task-specific monitoring and control services <cit.>.Leveraging advances in embedded systems, contemporary IoT devices are featured with small-size and low-power designs, but their computation and communication capabilities are limited.A prevalent solution during the past decade was to move computing, control, and storage resources to the remote cloud (a.k.a. data centers).Yet, the cloud-based IoT architecture is challenged by high latency due to directly communications with the cloud, which certainly prevents real-time applications <cit.>. Along with other features of IoT, such as extreme heterogeneity and unpredictable dynamics, the need arises for innovations in network design and management to allow for adaptive online service provisioning, subject to stringent delay constraints <cit.>. From the network design vantage point, fog is viewed as a promising architecture for IoT that distributes computation, communication, and storage closer to the end IoT users, along the cloud-to-things continuum <cit.>.In the fog computing paradigm, service provisioning starts at the network edge, e.g., smartphones, and high-tech routers, and only a portion of tasks will be offloaded to the powerful cloud for further processing (a.k.a. computation offloading) <cit.>. Existing approaches for computation offloading either focus on time-invariant static settings, or, rely on stochastic optimization approaches such as Lyapunov optimization to deal with time-varying cases; see <cit.> and references therein.Nevertheless, static settings cannot capture the changing IoT environment, and the stationarity commonly assumed in stochastic optimization literature may not hold in practice, especially when the stochastic process involves human participation as in IoT. From the management perspective, online network control, which is robust to non-stationary dynamics and amenable to light-weight implementations, remains a largely uncharted territory <cit.>.Indeed, the primary goal of this paper is an algorithmic pursuit of online network optimization suitable for emerging tasks in IoT.Focusing on such algorithmic challenges, online convex optimization (OCO) is a promising methodology for sequential tasks with well-documented merits, especially when the sequence of convex costs varies in an unknown and possibly adversarial manner <cit.>. Aiming to empower traditional fog management policies with OCO, most available OCO works benchmark algorithms with a static regret, which measures the difference of costs (a.k.a. losses) between the online solution and the best static solution in hindsight <cit.>. However, static regret is not a comprehensive performance metric in dynamic settings such as those encountered with IoT <cit.>.Recent works extend the analysis of static to that of dynamic regret <cit.>, but they deal with time-invariant constraints that cannot be violated instantaneously. Tailored for fog computing setups that need flexible adaptation of online decisions to dynamic resource availability, OCO with time-varying constraints was first studied in <cit.>,along with its adaptive variant in <cit.>, and the optimal regret bound in this setting was first established in <cit.>. Yet, the approaches in <cit.> remain operational under the premise that the loss functions are explicitly known, or, their gradients are readily available.Clearly, none of these two assumptions can be easily satisfied in IoT settings, because i) the loss function capturing user dissatisfaction, e.g., service latency or reliability, is hard to model in dynamic environments; and, ii) even if modeling is possible in theory, the low-power IoT devices may not afford the complexity of running statistical learning tools such as deep neural networks “on-the-fly.”In this context, targeting a gradient-free light-weight solution, alternative online schemes have been advocated leveraging point-wise values of loss functions (partial-information feedback) rather than their gradients (full-information feedback). They are termed bandit convex optimization (BCO) in machine learning <cit.>, or referred as zeroth-order schemes in optimization circles <cit.>. While <cit.> and <cit.> employed on BCO with time-invariant constraints that cannot be violated instantaneously, the long-term effect of such instantaneous violations was studied in <cit.>, where the focus is still on static regret and time-invariant constraints. Building on full-information precursors <cit.>, the present paper broadens the scope of BCO to the regime with time-varying constraints, and proposes a class of online algorithms termed online bandit saddle-point (BanSaP) approaches.With an eye on managing IoT with limited information, our contribution is the incorporation of long-term and time-varying constraints to expand the scope of BCO, as well as an improved regret-fit tradeoff relative to that in <cit.>; see a summary in Table <ref>. In a nutshell, relative to existing works, the main contributions of the present paper are summarized as follows.c1) We generalize the standard BCO framework with only time-varying costs <cit.>, to account for both time-varying costs and constraints. Performance here is established relative to the best dynamic benchmark, via metrics that we term dynamic regret and fit (Section III).c2) We develop a class of BanSaP algorithms to tackle this novel BCO problem, and analytically establish that BanSaP solvers yield simultaneously optimal sub-linear dynamic regret and fit, given that the accumulated variations of per-slot minimizers are known to grow sub-linearly with time (Section IV).c3) Our BanSaP algorithms are applied to computation offloading tasks emerging in IoT management, and simulations demonstrate that the BanSaP solvers have comparable performance relative to full-information alternatives (Section V). Notation. (·)^⊤ stands for vector and matrix transposition, and 𝐱 denotes the ℓ_2-norm of a vector 𝐱. Inequalities for vectors 𝐱 > 0, and the projection [𝐚]^+:=max{𝐚,0} are entry-wise. § BANDIT ONLINE LEARNING WITH CONSTRAINTS In this section, a generic BCO formulation with long-term and time-varying constraints will be introduced, along with its real-world application in IoT management.§.§ Online learning with constraints under partial feedback Before introducing BCO with long-term constraints, we begin with the classical BCO setting, where constraints are time-invariant, and must be strictly satisfied <cit.>.Akin to its full-information counterpart <cit.>, BCO can be viewed as a repeated game between a learner and nature.Consider that time is discrete and indexed by t. Per slot t, a learner selects an action 𝐱_t from a convex set X⊆ℝ^d, and subsequently nature chooses a loss function f_t(·): ℝ^d→ℝ through which the learner incurs a loss f_t(𝐱_t).The convex feasible set X is a-priori known and fixed over the entire time horizon.Different from the OCO setup, at the end of each slot, only the value of f_t(𝐱_t) rather than the form of f_t(𝐱) is revealed to the learner in BCO. Although this standard BCO setting is appealing to various applications such as online end-to-end routing <cit.> and task assignment<cit.>, it does not account for potential variations of (possibly unknown) constraints, and does not deal with constraints that can possibly be satisfied in the long term rather than a slot-by-slot basis <cit.>.Online optimization with time-varying and long-term constraints is well motivated for applications from power control in wireless communication <cit.>, geographical load balancing in cloud networks <cit.>, to computation offloading in fog computing <cit.>.Motivated by these dynamic network management tasks, our recent works <cit.> studied OCO with time-varying constraints in full information setting, where the gradient feedback is available.Complementing <cit.> and <cit.>, the present paper broadens the applicability of BCO to the regime with time-varying long-term constraints.Specifically, we consider that per slot t, a learner selects an action 𝐱_t from a known and fixed convex set X⊆ℝ^d, and then nature chooses not only a loss function f_t(·): ℝ^d→ℝ, but also a time-varying penalty function 𝐠_t(·): ℝ^d→ℝ^N. The later gives rise to the time-varying constraint 𝐠_t(𝐱)≤0, which is driven by the unknown application-specific dynamics.Similar to the standard BCO setting, only the value of f_t(𝐱_t) at the queried point 𝐱_t is revealed to the learner here; but different from the standard BCO setting, besidesX, the constraint 𝐠_t(𝐱)≤0 needs to be carefully taken care of.And the fact that 𝐠_t is unknown to the learner when performing her/his decision, makes it impossible to satisfy in every time slot.Hence, a more realistic goal here is to find a sequence of solutions {𝐱_t} that minimizes the aggregate loss, and ensures that the constraints {𝐠_t(𝐱_t)≤0} are satisfied in the long term on average.Specifically, extending the BCO framework <cit.> to accommodate such time-varying constraints, we consider the following online optimization problem[box=]align min_{𝐱_t∈X,∀t} ∑_t=1^T f_t(𝐱_t) s. to ∑_t=1^T 𝐠_t(𝐱_t) ≤0where T is the entire time horizon, 𝐱_t∈ℝ^d is the decision variable, f_t represents the cost function, 𝐠_t:=[g_t^1,…,g_t^N]^⊤ denotes the constraint function with nth entry g_t^n(·):ℝ^d→ℝ, and X∈ℝ^d is a convex set.In the current setting, we assume that only the values of loss function are available at queried points since e.g., its complete form related to user experience is hard to approximate, but the constraint function is revealed to the learner as it represents measurable physical requirements e.g., power budget, and data flow conservation constraints.Before the algorithm development in Section <ref> and performance analysis in Section <ref>, we will introduce a motivating example of fog computing in IoT. §.§ Motivating setup: mobile fog computing in IoTThe online computational offloading task of fog computing in IoT <cit.> takes the form of BCO with long-term constraints (<ref>).Consider a mobile network with a sensor layer, a fog layer, and a cloud layer <cit.>. The sensor layer contains heterogeneous low-power IoT devices (e.g., wearable watches and smart cameras), which do not have enough computational capability, and usually offload their collected data to the local fog nodes (e.g., smartphones and high-tech routers) in the fog layer for further processing <cit.>.The fog layer consists of N nodes in the set N:={1,…,N} with moderate processing capability; thus, part of workloads will be collaboratively processed by the local fog servers to meet the stringent latency requirement, and the rest will be offloaded to the remote data center in the cloud layer <cit.>; also see Fig. <ref>. Per time t, each fog node n collects data requests b_t^n from all its nearby sensors.Once receiving these requests, node n has three options: i) offloading the amount z_t^n to the remote data center; ii) offloading the amount y_t^nk to each of its nearby node k for collaborative computing; and, iii) locally processing the amount y_t^nn according to its resource availability.The optimization variable 𝐱_t in this case consists of the cloud offloading, local offloading, and local processing amounts; i.e., 𝐱_t:=[z_t^1,…,z_t^N,y_t^11,…,y_t^1N,…,y_t^N1,…,y_t^NN]^⊤. Assuming that each fog node has a data queue to buffer unserved workloads, the instantaneously served workloads (offloading plus processing) is not necessarily equal to the data arrival rate. Instead, a long-term constraint is common to ensure that the cumulative amount of served workloads is no less than the arrived amount at each node n over time <cit.> ∑_t=1^T g_t^n(𝐱_t):=∑_t=1^T(b_t^n+∑_k∈ N_n^ in y_t^kn-∑_k∈ N_n^ out y_t^nk-z_t^n-y_t^nn) ≤ 0where N_n^ in and N_n^ out represent the sets of fog nodes with in-coming links to node n and those with out-going links from node n, respectively. The bandwidth limit of communication link (e.g., wireline) from fog node n to the remote cloud is z̅^n; the limit of the transmission link (e.g., wireless) from node n to its neighbor k is y̅^nk, and the computation capability of node n is y̅^nn.With 𝐱̅ collecting all the aforementioned limits, the feasible region can be expressed by 𝐱_t ∈ X:={0≤𝐱_t≤𝐱̅}. Performance is assessed by the user dissatisfaction of the online processing and offloading decisions, e.g., aggregate delay <cit.>. Specifically, as the computation delay is usually negligible for data centers with thousands of high-performance servers, the latency for cloud offloading amount z_t^n is mainly due to the communication delay, which is denoted as a time-varying cost c_t^n(z_t^n) depending on the unpredictable network congestion during slot t.Likewise, the communication delay of the local offloading decision y_t^nk from node n to a nearby node k is denoted as c_t^nk(y_t^nk), but its magnitude is much lower than that of cloud offloading. Regarding the processing amount y_t^nn, its latency comes from the computation delay due to its limited computational capability, which is presented as a time-varying function h_t^n(y_t^nn) capturing the dynamic CPU capability during the computing processes. Per slot t, the network delayf_t(𝐱_t) aggregates the computation delay at all nodes plus the communication delay at all links, namelyf_t(𝐱_t):=∑_n∈ N(c_t^n(z_t^n)+∑_k∈ N_n^ outc_t^nk(y_t^nk)_ communication+h_t^n (y_t^nn)_ computation).Clearly, the explicit form of functions c_t^n(·), c_t^nk(·), and h_t^n(·) is unknown to the network operator due to the unpredictable traffic patterns <cit.>; but they are convex (thus f_t(𝐱_t) is convex) with respect to their arguments, which implies that the marginal computation/communication latency is increasing as the offloading/processing amount grows.Aiming to minimize the accumulated network delay while serving all the IoT workloads in the long term, the optimal offloading strategy in this mobile network is the solution of the following online optimization problem (cf. (<ref>))min_{𝐱_t∈ X,∀ t}∑_t=1^T f_t(𝐱_t), s. to (<ref>) for n=1,…,N.Comparing to the generic form (<ref>), we consider an online fog computing problem in (<ref>), where the loss (network latency) function f_t(·) and the data requests {b_t^n} within slot t are not known when making the offloading and local processing decision 𝐱_t; after performing 𝐱_t, only the value of f_t(𝐱_t) (a.k.a. loss) as well as the measurements {b_t^n} are revealed to the network operator.In this example, measuring {b_t^n} is tantamount to knowing the constraint function g_t^n(·) in (<ref>).Therefore, (<ref>) is in the form of (<ref>).§ ONLINE BANDIT SADDLE-POINT METHODS To solve the problem in Section <ref>, an online saddle-point method is revisited first, before developing its bandit variants for network optimization with only partial feedback. §.§ Online saddle-point approach with gradient feedbackSeveral works have studied the OCO setup with time-varying long-term constraints (cf. (<ref>)), including <cit.>, and the recent variant <cit.> incorporating with adaptive stepsizes.Consider now the per-slot problem (<ref>), which contains the current objective f_t(𝐱), the current constraint 𝐠_t(𝐱)≤0, and a time-invariant feasible set X. With λ∈ℝ^N_+ denoting the Lagrange multiplier associated with the time-varying constraint, the online Lagrangian of (<ref>) can be expressed asL_t(𝐱,λ):=f_t(𝐱)+λ^⊤𝐠_t(𝐱). Serving as a basis for developing the bandit approaches, we next revisit the online saddle-point scheme with full-information <cit.>, that is also equivalent to <cit.> when 𝐠_t(𝐱) is linear. Specifically, given the primal iterate 𝐱_t and the dual iterate λ_t at each slot t, the next decision 𝐱_t+1 is generated by𝐱_t+1∈min_𝐱∈ X∇_𝐱^⊤ L_t(𝐱_t,λ_t)(𝐱-𝐱_t)+1/2α𝐱-𝐱_t^2where α is a pre-defined constant, and ∇_𝐱 L_t(𝐱_t,λ_t)=∇ f_t(𝐱_t)+∇^⊤𝐠_t(𝐱_t)λ_t is the gradient of L_t(𝐱,λ_t) with respect to (w.r.t.) the primal variable 𝐱 at 𝐱=𝐱_t.The minimization (<ref>) admits the closed-form solution, given by𝐱_t+1= P_ X(𝐱_t-α∇_𝐱 L_t(𝐱_t,λ_t))where P_ X(𝐲):=_𝐱∈ X𝐱-𝐲^2 denotes the projection operator. In addition, the dual update takes the modified online gradient ascent formλ_t+1=[λ_t+μ (𝐠_t(𝐱_t)+∇^⊤𝐠_t(𝐱_t)(𝐱_t+1-𝐱_t))]^+where μ is a positive stepsize, and ∇_λ L_t(𝐱_t,λ_t)=𝐠_t(𝐱_t) is the gradient of L_t(𝐱_t,λ) w.r.t. λ at λ=λ_t.Note that (<ref>) is a modified gradient update since the dual variable is updated along the first-order approximation of 𝐠_t(𝐱_t+1) at the previous iterate 𝐱_t rather than 𝐠_t(𝐱_t) used in <cit.>, which will be critical in our subsequent analytical derivations. To perform the online saddle-point recursion (<ref>)-(<ref>) however, the gradient ∇ f_t(𝐱) and the constraint 𝐠_t(𝐱) should be known to the learner at each slot t.When the gradient of f_t(𝐱) (or its explicit form) is unknown as it is in our setup, additional effort is needed.In this context, the systematic design of the online bandit saddle-point (BanSaP) methods will be leveraged to extend the online saddle-point method to the regime where gradient information is unavailable or computationally costly.§.§ BanSaP with one-point partial feedbackThe key idea behind BCO is to construct (possibly stochastic) gradient estimates using the limited function value information <cit.>.Depending on system variability, the online learner can afford one or multiple loss function evaluations (partial-information feedback) per time slot<cit.>.Intuitively, the performance of a bandit algorithm will improve if multiple evaluations are available per time slot; see Fig. <ref> for a comparison of full- versus partial-information feedback settings. To begin with, we consider the case where the learner can only observe the function value of f_t(𝐱) at a single point per slot t. The crux here is to construct a (possibly unbiased) estimate of the gradient using this single piece of feedback. Interestingly though, a stochastic gradient estimate of f_t(𝐱) can be obtained by one point random function evaluation <cit.>.The intuition can be readily revealed from the one-dimensional case (d=1): For a binary random variable u taking values {-1,1} equiprobable, and a small constant δ>0, the idea of forward differentiation implies that the derivative f_t' at x can be approximated by f_t'(x)≈f_t(x+δ )-f_t(x-δ)/2δ=𝔼_u[u/δf_t(x+δ u)]where the approximation is due to δ>0, and the equality follows from the definition of expectation. Hence, f_t(x+δ u)u/δ can serve as a stochastic estimator of f_t'(x) based only single function evaluation f_t(x+δ u). Generalizing this approximation to high dimensions, with a random vector 𝐮 drawn from the unit sphere (a.k.a. the surface of a unit ball), the scaled function evaluation at a perturbed point 𝐱+δ𝐮 yields an estimate of the gradient ∇ f_t(𝐱), given by <cit.>∇ f_t(𝐱)≈𝔼_𝐮[d/δf_t(𝐱+δ𝐮)𝐮]:=𝔼_𝐮[∇̂^1 f_t(𝐱)]where we define one-point gradient ∇̂^1 f_t(𝐱):=d/δf_t(𝐱+δ𝐮)𝐮.Building upon this intuition, consider a bandit version of the online saddle-point iteration, for which the primal update becomes (cf. (<ref>))𝐱̂_t+1= P_(1-γ) X(𝐱̂_t-α∇̂_𝐱^1 L_t(𝐱̂_t,λ_t))where (1-γ) X:={(1-γ)𝐱:𝐱∈ X} is a subset of X,γ∈[0,1) is a pre-selected constant depending on δ, and the one-point Langragian gradient is given by (cf. (<ref>))∇̂_𝐱^1 L_t(𝐱̂_t,λ_t):=∇̂^1 f_t(𝐱̂_t)+∇^⊤𝐠_t(𝐱̂_t)λ_t. In the full-information case, 𝐱_t in (<ref>) is the learner's action, but in the bandit case the learner's action is 𝐱_1,t:=𝐱̂_t+δ𝐮_t, which is the point for function evaluation but not 𝐱̂_t in (<ref>).Furthermore, the projection is performed on a smaller convex set (1-γ) X in (<ref>), which ensures feasibility of the perturbed 𝐱_1,t∈ X.Similar to the full-information case (<ref>), the dual update of BanSaP is given byλ_t+1=[λ_t+μ (𝐠_t(𝐱̂_t)+∇^⊤𝐠_t(𝐱̂_t)(𝐱̂_t+1-𝐱̂_t))]^+where μ is again the stepsize, and the learning iterate 𝐱̂_t rather than the actual decision 𝐱_t is used in this update.Compared with the gradient-based recursions (<ref>)-(<ref>), the updates (<ref>)-(<ref>) with one-point bandit feedback do not increase computation or memory requirements, and thus provide a light-weight surrogate for gradient-free online bandit network optimization.§.§ BanSaP with multipoint partial feedbackFeaturing a simple update given minimal information, the BanSaP with one-point bandit feedback is suitable for fast-varying environments, where multiple function evaluations are impossible. As shown later in Sections <ref> and <ref>, the theoretical and empirical performance of BanSaP with single-point evaluation is degraded relative to the full-information case. To improve the performance of BanSaP with one-point feedback, we will first rely on two-point function evaluation at each slot <cit.>, and then generalize to multipoint evaluation.Intuitively, this approach is justified when the underlying dynamics are slow, e.g., when the load and price profiles in power grids are piece-wise stationary.In this case, each slot can be further divided into multiple mini-slots, and one query is performed per mini-slot, over which the loss function and the constraints do not change.Compared to (<ref>)-(<ref>), the key difference is that the one-point estimate in (<ref>) is replaced by∇̂^2 f_t(𝐱̂_t):=d/2δ(f_t(𝐱̂_t+δ𝐮_t)-f_t(𝐱̂_t-δ𝐮_t))𝐮_twhere the function values are evaluated on two points around the learning iterate 𝐱̂_t, namely, 𝐱_1,t:=𝐱̂_t+δ𝐮_t and 𝐱_2,t:=𝐱̂_t-δ𝐮_t with 𝐮_t again drawn uniformly from the unit sphere 𝕊:={𝐮∈ℝ^d:𝐮=1}. The primal update becomes 𝐱̂_t+1= P_(1-γ) X(𝐱̂_t-α∇̂_𝐱^2 L_t(𝐱̂_t,λ_t)), with Lagrangian gradient ∇̂_𝐱^2 L_t(𝐱̂_t,λ_t):=∇̂^2 f_t(𝐱̂_t) +∇𝐠_t(𝐱̂_t)^⊤λ_t.Similar to the one-point case, it is instructive to consider the two-point gradient estimate in the one-dimensional case (d=1), where the expectation of the differentiation term in (<ref>) approximates well the derivative of f_t at x̂_t; that is, 𝔼_u[u_t/2δ(f_t(x̂_t+δ u_t)-f_t(x̂_t-δ u_t))] = 1/2δ(f_t(x̂_t+δ )-f_t(x̂_t-δ))≈ f_t'(x̂_t)where the equality follows because the random variable u_t takes values {-1,1} equiprobable. Relative to the one-point feedback case, the advantage of the two-point feedback is variance reduction in the gradient estimator.Specifically, the second moment of the stochastic gradient can be uniformly bounded, 𝔼[d/2δ(f_t(𝐱̂_t+δ𝐮_t)-f_t(𝐱̂_t-δ𝐮_t))𝐮_t^2]≤ d^2G^2, where G is the Lipschitz constant of f_t(𝐱). This is in contrast to the one-point feedback where the second moment is inversely proportional to δ, since 𝔼[d/δf_t(𝐱̂_t+δ𝐮_t)𝐮_t^2]≤ d^2 F^2/δ^2, with F denoting an upper-bound of f_t(𝐱). The proof of this argument can be found in the Appendix (Lemma <ref>). In fact, a bias-variance tradeoff emerges in the one-point case, but not in the two-point case.This subtle yet critical difference will be responsible for an improved performance of BanSaP with two-point feedback, and its stable empirical performance, as will be seen later.With the insights gained so far, the next step is to endow the BanSaP with more than two function evaluations <cit.>.With M>2 points, the gradient estimator is obtained by querying the function values over M points in the neighborhood of 𝐱̂_t. These points include 𝐱_m,t:=𝐱̂_t+δ𝐮_m,t, 1≤ m ≤ M-1, and the learning iterate 𝐱_m,t:=𝐱̂_t, where 𝐮_m,t is independently drawn from 𝕊. Specifically, the gradient becomes (cf. (<ref>))∇̂_𝐱^M L_t(𝐱̂_t,λ_t):=d/δ (M-1)∑_m=1^M-1(f_t(𝐱̂_t+δ𝐮_m,t)-f_t(𝐱̂_t))𝐮_m,t+∇𝐠_t(𝐱̂_t)^⊤λ_twhere we define the M-point stochastic gradient as ∇̂^M f_t(𝐱̂_t):=d/δ (M-1)∑_m=1^M-1(f_t(𝐱̂_t+δ𝐮_m,t)-f_t(𝐱̂_t))𝐮_m,t.At the price of extra computations, simulations will validate that the BanSaP with multipoint feedback enjoys improved performance.The family of the BanSaP approaches with one- or multiple-point feedback is summarized in Algorithm <ref>.The BanSaP solvers here adopt uniform sampling for gradient estimation, meaning 𝐮 is drawn uniformly from the unit sphere. However, other sampling rules can be incorporated without affecting the order of regret bounds derived later. For example, one can sample 𝐮 from the canonical basis of a d-dimensional space uniformly at random <cit.>, or, sample 𝐮 from a normal distribution <cit.>. The effectiveness of these schemes will be tested using simulations. § PERFORMANCE ANALYSISIn this section, we will introduce pertinent metrics to evaluate BanSaP algorithms in the online bandit learning with long-term constraints, and rigorously analyze the performance of the proposed algorithms. §.§ Optimality and feasibility metricsWith regard to performance of BCO schemes, static regret is a common metric, under time-invariant and strictly satisfied constraints, which measures the difference between the aggregate loss and that of the best fixed solution in hindsight <cit.>. Extending the definition of static regret to accommodate M-point function evaluations and time-varying constraints, let us first considerReg^ s_T:=1/M∑_t=1^T ∑_m=1^M 𝔼[f_t(𝐱_m,t)]-∑_t=1^T f_t(𝐱^*)where the actual loss per slot is averaged over the losses of M actions (queried points), 𝔼 is taken over the sequence of random actions (due to δ𝐮 perturbations), and the best static solution is 𝐱^*∈min_𝐱∈ X ∑_t=1^T f_t(𝐱); s. to 𝐠_t(𝐱) ≤0, ∀ t. A BCO algorithm yielding a sub-linear regret implies that the algorithm is “on average” no-regret <cit.>; or, in other words, asymptotically not worse than the best fixed solution 𝐱^*.Though widely used, the static regret relies on a rather coarse benchmark, which is not as useful in dynamic IoT settings. Specifically, the gap between the loss of the best static and that of the best dynamic benchmark is as large as O(T) <cit.>.In response to the quest for improved benchmarks in this dynamic setup with constraints, two metrics are considered here: dynamic regret and dynamic fit. The notion of dynamic regret has been recently adopted in <cit.> to assess performance of online algorithms under time-invariant constraints. For our BCO setting of (<ref>), we adoptReg^ d_T:=1/M∑_t=1^T ∑_m=1^M 𝔼[f_t(𝐱_m,t)]-∑_t=1^T f_t(𝐱_t^*)where 𝔼 is again taken over the sequence of random actions, and the benchmark is now formed via a sequence of best dynamic solutions {𝐱_t^*} for the instantaneous cost minimization problem subject to the instantaneous constraint, namely𝐱_t^*∈min_𝐱∈ Xf_t(𝐱) s. to 𝐠_t(𝐱) ≤0.Quantitatively, the dynamic regret is always larger than the static regret, i.e., Reg^ s_T ≤ Reg^ d_T, since∑_t=1^Tf_t(𝐱^*) is always no smaller than ∑_t=1^Tf_t(𝐱^*_t) according to the definitions of 𝐱^* and 𝐱^*_t. Hence, a sub-linear dynamic regret implies a sub-linear static regret, but not vice versa.Regarding feasibility of decisions generated by a BCO algorithm, the notion of dynamic fit will be used to measure the accumulated violation of constraints <cit.>, that isFit^ d_T:=[1/M∑_t=1^T ∑_m=1^M 𝐠_t(𝐱_m,t)]^+.Note that the dynamic fit is zero if the accumulated violation 1/M∑_t=1^T ∑_m=1^M 𝐠_t(𝐱_m,t) is entry-wise less than zero. Hence, enforcing 1/M∑_t=1^T ∑_m=1^M 𝐠_t(𝐱_m,t) ≤0 is different from restricting 𝐱_t to meet 1/M∑_m=1^M 𝐠_t(𝐱_m,t)≤0 in every slot. While the latter readily implies the former, the long-term constraint implicitly assumes that the instantaneous constraint violations can be compensated by the later strictly feasible decisions, and thus allows adaptation of online decisions to the unknown dynamics.Under this broader BCO setup, an ideal online algorithm is the one that achieves both sub-linear dynamic regret and sub-linear dynamic fit. A sub-linear dynamic regret implies “no-regret” relative to the clairvoyant dynamic solution on the long-term average; i.e., lim_T→∞ Reg_T^ d/T=0; and a sub-linear dynamic fit indicates that the online strategy is also feasible on average; i.e., lim_T→∞ Fit_T^ d/T=0. Unfortunately, the sub-linear dynamic regret is not achievable in general, even when the time-varying constraint in (<ref>) is absent <cit.>. Therefore, we aim at designing an online strategy that generates a sequence {𝐱_m,t} ensuring sub-linear dynamic regret and fit, under the suitable conditions on the underlying dynamics.§.§ Main resultsBefore formally analyzing the dynamic regret and fit for BanSaP, we assume that the following conditions are satisfied. (as1) For every t, the functionsf_t(𝐱) and𝐠_t(𝐱) are convex.(as2) Function f_t(𝐱) is bounded over the set X, meaning |f_t(𝐱)|≤ F, ∀𝐱∈ X; while f_t(𝐱) and g_t^n(𝐱) have bounded gradients; that is, ∇ f_t(𝐱)≤ G, and max_n∇ g_t^n(𝐱)≤ G.(as3) For a small constant γ, there exists a constant η>0, and an interior point 𝐱̃∈ (1-γ) X such that 𝐠_t(𝐱̃)≤ -η1, ∀ t.(as4) With 𝔹:={𝐱∈ℝ^d:𝐱≤ 1} denoting the unit ball, there exist constants 0<r≤ R such that r𝔹⊆ X⊆ R𝔹. Assumptions (as1)-(as2) are typical in OCO with both full- and partial-information feedback <cit.>; (as3) is Slater's condition modified for our BCO setting, which guarantees the existence of a bounded Lagrange multiplier <cit.> in the constrained optimization; and, (as4) requires the action set to be bounded within a ball that contains the origin.When (as4) appears to be restrictive, it is tantamount to assuming X is compact and has a nonempty interior, because one can always apply an affine transformation (a.k.a. reshaping) on X to satisfy (as4); see <cit.>. Under these assumptions, we are on track to first provide upper bounds for the dynamic regret, and the dynamic fit of the BanSaP solver with one-point feedback. Suppose that (as1)-(as4) are satisfied, and consider the parameters α, μ, δ, γ defined in (<ref>)-(<ref>), and constants F, G, r, R defined in (as2)-(as4).If the dual variable is initialized by λ_1=0, then the BanSaP with one-point feedback in (<ref>)-(<ref>) has dynamic regret bounded byReg^ d_T≤R/αV(𝐱_1:T^*) +R^2/2α+d^2G^2R^2α T/δ^2+2Gδ T+γ GRT(1+λ̅)+2μ G^2R^2Twhere λ̅:=max_t λ_t, and the accumulated variation of the per-slot minimizers 𝐱^*_t in (<ref>) is given byV(𝐱_1:T^*):=∑_t=1^T 𝐱^*_t-𝐱^*_t-1.In addition, the dynamic fit defined in (<ref>) is bounded byFit^ d_T≤λ̅/μ+G^2√(N)T/2β +δ G√(N)T+β√(N)T(α^2 d^2F^2 /δ^2+α^2 G^2λ̅^2)where β>0 is a pre-selected constant.Furthermore, if we choose the stepsizes as α=μ= O(T^-3/4), and the parameters δ= O(T^-1/4), β=T^1/4 and γ=δ/r, then the online decisions generated by BanSaP are feasible, i.e., 𝐱_1,t∈ X; and also yield the following dynamic regret and fit [box=]align Reg^d_T= O(V(𝐱_1:T^*)T^3/4) and Fit^d_T =O(T^3/4). See Appendix <ref>. For BanSaP with one-point feedback, Theorem <ref> asserts that its dynamic regret and fit are upper-bounded by some constants depending on the those parameters, the time horizon, and the accumulated variation of per-slot minimizers. Interestingly, the crucial constant δ controlling the perturbation of random actions appears in both the denominator and numerator of (<ref>) and (<ref>), which correspond to the variance and bias of the gradient estimator.Therefore, simply setting a small δ will not only reduce the bias, but it will also boost the variance - a clear manifestation of the that is known as bias-variance tradeoff in BCO <cit.>. Optimally choosing parameters implies that the dynamic fit is sub-linearly growing, and the dynamic regret is sub-linear given that the variation of the per-slot minimizer is slow enough; i.e., V(𝐱_1:T^*)=𝐨(T^1/4).Regarding BanSaP with two-point feedback, we can prove the following result that parallels Theorem <ref>.Consider the assumptions and the definitions of constants in Theorem <ref>.If the dual variable is initialized by λ_1=0, then BanSaP with two-point feedback has dynamic regret bounded byReg^ d_T≤R/αV(𝐱_1:T^*)+R^2/2α +2μ G^2R^2T+α d^2G^2 T+γ GRT(1+λ̅)+2δ GTand has dynamic fit in (<ref>) bounded byFit^ d_T≤λ̅/μ+G^2√(N)T/2β +δ G√(N)T+β√(N)T(α^2 d^2G^2+α^2 G^2λ̅^2).In this case, if we choose the stepsizes as α=μ= O(T^-1/2), and set the parameters as β=T^1/2,δ= O(T^-1), and γ=δ/r, then the online decisions generated by BanSaP are feasible, and its dynamic regret and fit are bounded by [box=]align Reg^d_T= O(V(𝐱_1:T^*)T^1/2) and Fit^d_T =O(T^1/2)where V(𝐱_1:T^*) is the accumulated variation of the per-slot minimizers 𝐱^*_t in (<ref>). See Appendix <ref>. Comparing with the bounds in (<ref>) and (<ref>), the perturbation constant δ only appears in the numerator of (<ref>) and (<ref>) because our gradient estimator here replies on two points.In this case, the additional function evaluation allows BanSaP to choose an arbitrarily small δ to minimize the bias of stochastic gradient, without increasing its variance. This observation is aligned with those in BCO without long-term constraints <cit.>.Furthermore, Theorem <ref> establishes that the dynamic regret and fit are sub-linear if V(𝐱_1:T^*)=𝐨(T^1/2), which markedly improves those in Theorem <ref> under one-point feedback.For the case of BanSaP with M>2 points, slightly improved bounds can be proved without changing the order of regret and fit, but they are omitted here for brevity.In addition, the bounds in Theorems <ref> and <ref> can be achieved without any knowledge of V(𝐱_1:T^*). When the order of V(𝐱_1:T^*) is known, or, can be estimated a-priori, tighter regret and fit bounds can be obtained by adjusting stepsizes accordingly. Formally, we can arrive at the following corollary. Under the conditions of Theorems <ref> and <ref>, suppose that there exists a constant ρ∈[0,1) such that the variation satisfies V(𝐱_1:T^*)=𝐨(T^ρ). If the stepsizes of BanSaP with one-point feedback are chosen as α=μ= O(T^3/4(ρ-1)), and the parameters are δ= O(T^1/4(ρ-1)), β=T^3/4(1-ρ),and γ=δ/r, then the dynamic regret and fit in (<ref>) become Reg^ d_T= O(T^1/4(ρ+3)) and Fit^ d_T= O(T^1/4(ρ+3)).Likewise, if the stepsizes of BanSaP with two-point feedback are chosen such that α=μ= O(T^1/2(ρ-1)), and the parameters are δ= O(T^1/2(ρ-1)), β=T^1/2(1-ρ),and γ=δ/r, then the dynamic regret and fit in (<ref>) become Reg^ d_T= O(T^1/2(ρ+1)) and Fit^ d_T= O(T^1/2(ρ+1)).Apparently, Corollary <ref> implies that sub-linear dynamic regret and fit are both possible, provided that the accumulated variation of the minimizers is growing sub-linearly (ρ<1), and it is available to the learner in advance. It provides valuable insights for choosing optimal stepsizes in dynamic environments. Specifically, adjusting stepsizes to match the variability of the environment is the key to achieving the optimal dynamic regret and fit. Intuitively, when the variation is fast (large ρ), slowly decaying stepsizes (thus larger stepsizes) can better track the potential changes; and vice versa. As a special case of Theorems <ref> and <ref>, by confining 𝐱_1^*=⋯=𝐱_T^* so that V(𝐱_1:T^*)=0, the dynamic regret bounds (<ref>) and (<ref>) reduce to the static ones, which correspond to O(T^3/4) in the one-point feedback case, and to O(√(T)) in the two-point case.This pair of bounds markedly improves the regret versus fit tradeoff in <cit.>, and matches the order of regret in <cit.>, and <cit.>, which are the best possible ones that can be achieved by efficient algorithms even in the BCO setup without the long-term constraints. Theorems <ref>, <ref> and Corollary <ref> extend the dynamic regret analysis in <cit.> to the regime of bandit online learning with long-term time-varying constraints. Interestingly though, in the BCO setting of our interest, sub-linear dynamic regret and fit are possible to achieve when the per-slot minimizer does not vary on average, that is, V(𝐱_1:T^*) is sub-linearly growing with T.§ NUMERICAL TESTS In this section, we demonstrate how the fog computation offloading task can benefit from our novel BanSaP solvers. §.§ BanSaP for fog computation offloadingRecall that the computation offloading problem (<ref>) is in the form of (<ref>).Therefore, the BanSaP solver of Section <ref> can be customized to solve (<ref>) in an online fashion, with provable performance and feasibility guarantees.Specifically, with 𝐠_t(𝐱_t) as in (<ref>) and f_t(𝐱_t) as in (<ref>), the primal update (<ref>) boils down to a simple closed-form gradient update amenable to decentralized implementation; the cloud offloading amount at node n is ẑ_t+1^n=[ẑ_t^n-α(∇̂ c^n_t(ẑ_t^n)-λ_t^n)]_0^z̅^nand the offloading amount from node n to node k is given byŷ_t+1^nk=[ŷ_t^nk-α(∇̂ c^nk_t(ŷ_t^nk)-λ_t^n+λ_t^k)]_0^y̅^nkwhile the local processing decision at node n is generated byŷ_t+1^nn=[ŷ_t^nn-α(∇̂ h^n_t(ŷ_t^nn)-λ_t^n)]_0^y̅^nn where α is chosen according to Theorems <ref> and <ref>.Using two-point feedback (M=2) as an example, the gradients involved in (<ref>) can be estimated as ∇̂^2 c^n_t(ẑ_t^n):=d/2δ(f_t(𝐱̂_t+δ𝐮_t_𝐱_1,t)-f_t(𝐱̂_t-δ𝐮_t_𝐱_2,t))u_t(ẑ^n)and with respect to the offloading variable, as ∇̂^2 c^nk_t(ŷ_t^nk):=d/2δ(f_t(𝐱̂_t+δ𝐮_t)-f_t(𝐱̂_t-δ𝐮_t))u_t(ŷ^nk)and with respect to the local processing variable, as∇̂^2 h^n_t(ŷ_t^nn):=d/2δ(f_t(𝐱̂_t+δ𝐮_t)-f_t(𝐱̂_t-δ𝐮_t))u_t(ŷ^nn) where u_t(ẑ^n), u_t(ŷ^nk), and u_t(ŷ^nn) represent the corresponding entries of the random vector 𝐮_t∈ℝ^| E| at slot t.The dual update (<ref>) at each node n reduces to λ_t+1^n=[λ_t^n+μ(b_t^n+∑_k∈ N_n^ inŷ_t+1^kn-∑_k∈ N_n^ outŷ_t+1^nk-ẑ_t+1^n-ŷ_t+1^nn)]^+where μ is chosen according to Theorems <ref> and <ref>.Intuitively, to guarantee completion of the service requests, the dual variable increases (increasing penalty) when there is instantaneous service residual, and decreases when over-serving incurs inthe mobile-edge computing systems. Following its generic form in Algorithm <ref>, BanSaP for online fog computation offloading tasks, is summarized in Algorithm <ref>.§.§ Numerical experimentsConsider the fog computing task in Section <ref> with N=10 nodes and a cloud center.Each fog node has an outgoing link to the cloud, and two outgoing links to two nearby fog nodes for local collaborative computing.For a communication link offloading loads from node n to k, the offloading limit is y̅^nk=10, the local computation limit at node n is y̅^nn=50, and the fog-cloud offloading limits {z̅^n} are all set to 100. The online cost (a.k.a. service latency) in (<ref>) is specified byf_t(𝐱_t):=∑_n∈ N(e^p_t^n z_t^n+∑_k∈ N_n^ outl^nky_t^nk+l^nn(y_t^nn)^2)where p_t^n=0.015sin(π t/96)+0.05, n∈ N\{4,5}, p_t^n=0.045sin(π t/96)+0.15, n∈{4,5}, and the local coefficients are set to l^nk=8/y̅^nk and l^nn=8/y̅^nn.Regarding the data arrival rate b_t^n, it is generated according to b_t^n=q^nsin(π t/96)+ν_t^n, with q^n and ν_t^n uniformly distributed over [40,50] and [45,55] for n∈ N\{1,2,3}⋃{4,5}, and q^n∈[32,40],ν_t^n∈[36,44], n∈{1,2,3}, and q^n∈[20,25], ν_t^n∈[22.5,27.5], n∈{4,5}. Notice that the scales of p_t^n and b_t^n vary between nodes, mimicking heterogeneity of real IoT systems; and the periods of p_t^n and b_t^n correspond to a 24-hour interval with slot duration 7.5 minutes.When the parameters of BanSaP need to be slightly adjusted in each test, they are set to γ=0.05, and δ=4 for with M=1, and δ=0.05 for M≥ 2.Finally, BanSaP is benchmarked by: i) the full-information MOSP method in <cit.> that takes gradient-based update for primal-dual variables; ii) the heuristic cloud-only approach that offloads all data requests to the remote cloud; and, iii) the heuristic fog-only approach that processes all data requests locally without collaboration.For both cloud-only and fog-only approaches, unoffloaded and unprocessed requests are buffered at the fog nodes for later processing; thus, these amounts are measured by their fit.As different stepsizes of BanSaP and MOSP lead to different behaviors, we manually optimized stepsizes in each test so that they have similar fit, and focus on their cost comparison. All simulated tests were averaged over 500 Monte Carlo realizations. Effect of complexity and sampling schemes. In a simplified setting with N=5 nodes, the fit and average cost are compared among the BanSaP variants with M-point feedback under different sampling schemes in Figs. <ref> and <ref>. Clearly, for both sampling schemes, the cost and fit of BanSaP solvers decrease as the amount of bandit feedback increases. However, such performance gain varnishes when feedback increases; e.g., M≥ 4.Regarding the sampling schemes,Fig. <ref> demonstrates that when all the BanSaP variants have low dynamic fit, the uniform sampling-based BanSaP with one-point feedback has large initial fit; and Fig. <ref> confirms thatfor M=1, the coordinate sampling-based BanSaP outperforms that with uniform sampling; and, for M≥ 2, the BanSaP solvers with uniform sampling incur lower cost. Therefore, to optimize empirical performance in the subsequent tests, coordinate sampling is adopted by BanSaP with M=1, while uniform sampling is used in BanSaP with M≥2. Optimality and feasibility. With optimized sampling schemes for BanSaP solvers, the dynamic fit and average cost are then compared among three BanSaP variants, MOSP, and two heuristic schemes in Figs. <ref> and <ref>.Without queueing at the fog side, the cloud-only scheme has much lower dynamic fit since all user demands are offloaded to the remote cloud.However, it incurs a much higher average cost (service latency) as the network latency between fog and cloud becomes high due to the large offloading amount.By increasing the amount of feedback, the BanSaP solver tends to have a lower fit and a lower average cost, both of which are comparable to those of MOSP when M≥ 2. On the other hand, the BanSaP with only one-point bandit feedback still has a similar fit relative to the fog-only scheme, but enjoys much lower cost.Interestingly enough, when the variance (cf. the shaded area in Fig. <ref>) of the one-point BanSaP's cost is high, it markedly varnishes when multiple function values become available, which corroborates our claims in Theorems <ref>-<ref>. Effect of network size. The third test evaluates the performance of all schemes under different number of fog nodes (i.e., network size). For each algorithm, the fit averaged over all fog nodes and time is presented in Fig. <ref>, and the cost averaged over the time is shown in Fig. <ref>.Clearly, the one-point BanSaP has lower average fit than the fog-only approach in most scenarios, and also incurs less average cost in all tested settings. Similar to those in Figs. <ref> and <ref>, the average fit of BanSaP with multiple function evaluations is still comparable to that of the full-information MOSP as the network size grows.An interesting observation here is that as the number of fog nodes increases, the performance gain of the BanSaP solver with a large M becomes more evident; see e.g., Fig. <ref>.This implies that for a larger network, BanSaP benefits from more bandit information to learn and track the network dynamics. § CONCLUSIONS AND THE ROAD AHEAD Bandit convex optimization (BCO) in dynamic environments was studied in this paper.Different from existing works in bandit settings, the focus was on a broader setting where part of the constraints are revealed after taking actions, and are also tolerable to instantaneous violations but have to be satisfied on average. The novel BCO setting fits well the emerging fog computing tasks in IoT. A class of online bandit saddle-point (BanSaP) approaches were proposed, and their online performance was rigorously analyzed.It was shown that the resultant regret bounds match those attained in BCO setups without long-term constraints.Furthermore, the BanSaP solvers can simultaneously yield sub-linear dynamic regret and fit, if the dynamic solutions vary slowly over time. Our algorithmic and theoretical results serve as an exciting first step to innovate online bandit learning tailored for dynamic network management tasks, emerging from contemporary IoT applications.Interesting future directions include designing asynchronous variants of BanSaP, and incorporating predictable dynamic models in online network optimization. § ACKNOWLEDGEMENTThe authors would like to thank Yanning Shen and Qing Ling for helpful feedback on the early version of this manuscript. 10url@samestyle samie2016b F. Samie, V. Tsoutsouras, S. Xydis, L. Bauer, D. Soudris, and J. 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Zeevi, “Non-stationary stochastic optimization,” Operations Research, vol. 63, no. 5, pp. 1227–1244, Sep. 2015.bertsekas1999 D. P. Bertsekas, Nonlinear Programming.1em plus 0.5em minus 0.4emBelmont, MA: Athena scientific, 1999.The proof generalizes the result in <cit.> from static regret with full-information gradient feedback to the dynamic regret with partial-information bandit feedback.§.§ Supporting lemmasBefore proving the main theory, we first establish several key lemmas and propositions. The following lemma establishes the unbiasedness of one- and two-point estimations <cit.>. With 𝐮 drawn uniformly from the surface of the unit ball 𝕊:={𝐮:𝐮=1}⊆ℝ^d, we have for given a constant δ>0 that 𝔼_𝐮[d/δf_t(𝐱+δ𝐮)𝐮]=∇f̌_t(𝐱)where ∇f̌_t(𝐱) is the gradient of the smoothed function f̌_t(𝐱):=𝔼_𝐯[f_t(𝐱+δ𝐯)] with 𝐯 drawn from a unit ball 𝔹, and d is the dimension of the variable 𝐱. Likewise, for the two-point case, we have that 𝔼_𝐮[d/2δ(f_t(𝐱+δ𝐮)-f_t(𝐱-δ𝐮))𝐮]=∇f̌_t(𝐱). Lemma <ref> provides valuable insights for performing gradient-based algorithms in bandit setting. Namely, ∇̂^1 f_t(𝐱̂_t) and ∇̂^2 f_t(𝐱̂_t) are the unbiased gradient estimators of the smoothed function f̌_t(𝐱), which is an approximation of f_t(𝐱). Note that (as1)-(as2) also imply that the smoothed function f̌_t(𝐱) is convex and G-Lipschitz continuous <cit.>, which will be used frequently in the subsequent analysis.The following lemma establishes the norm (or variance) of one- and two-point gradient estimations <cit.>. For the gradient ∇̂^1 f_t(𝐱̂_t) in (<ref>), we have that∇̂^1 f_t(𝐱̂_t)≤d/δ Fwhere F is an upper-bound of the function. For the gradient estimator ∇̂^2 f_t(𝐱̂_t) in (<ref>), we have that∇̂^2 f_t(𝐱̂_t)≤ d Gwhere G is the Lipschitz constant of the loss function. For the gradient ∇̂^1 f_t(𝐱̂_t) in (<ref>), it holds that∇̂^1 f_t(𝐱̂_t)=d/δ|f_t(𝐱̂_t+δ𝐮_t)|𝐮_t≤d/δ Fwhere F is an upper-bound of the function. Likewise for ∇̂^2 f_t(𝐱̂_t) in (<ref>), we have that∇̂^2 f_t(𝐱̂_t) =d/2δ(f_t(𝐱̂_t+δ𝐮_t)-f_t(𝐱̂_t-δ𝐮_t))𝐮_t=d/2δ|f_t(𝐱̂_t+δ𝐮_t)-f_t(𝐱̂_t-δ𝐮_t)|𝐮_t≤dG/2δ2δ𝐮_t≤ d Gwhere G is the Lipschitz constant of the loss function and 𝐮_t=1 by design. Thus, the proof is complete.Having bounded the norm of stochastic gradients, the next lemma is useful to ensure feasibility of actual online actions.Consider a constant r>0 so that r𝔹⊆ X, where 𝔹:={𝐯:𝐯≤ 1}⊆ℝ^d is the unit ball. If we choose γ=δ/r, and the iterate satisfies 𝐱̂_t∈(1-γ) X, then 𝐱̂_t+δ𝐮_t∈ X, where 𝐮_t is drawn uniformly from the unit sphere 𝕊:={𝐮:𝐮=1}⊆ℝ^d. The next lemma is crucial to establish the dynamic fit <cit.>. Considering the BanSaP recursion (<ref>), we have the following bound for the cumulative constraint violation∑_t=1^T 𝐠_t(𝐱̂_t)≤λ_T+1/μ+G^2T1/2β+β/2∑_t=1^T𝐱̂_t+1-𝐱̂_t^21where μ>0 is the stepsize of the dual iteration (<ref>), and β>0 is a pre-defined constant. From the n-th entry of λ in (<ref>), we haveλ_t+1^n ≥λ_t^n+μ (g_t^n(𝐱̂_t)+∇^⊤ g_t^n(𝐱̂_t)(𝐱̂_t+1-𝐱̂_t))(a)≥λ_t^n+μ g_t^n(𝐱̂_t)-μ/2β∇ g_t^n(𝐱̂_t)^2-μβ/2𝐱̂_t+1-𝐱̂_t^2(b)≥λ_t^n+μ g_t^n(𝐱̂_t)-μ G^2/2β-μβ/2𝐱̂_t+1-𝐱̂_t^2where (a) uses the Cauchy-Schwarz inequality, and (b) follows from the bound on the gradients in (as2). The proof is then complete after summing up (<ref>) over t=1,…,T. Consider the BanSaP recursions (<ref>) and (<ref>) with a generic gradient ∇̂ f_t(𝐱̂_t), which is estimated from one- or multi-point feedback. The following holds ∀𝐱∈ (1-γ) X1/μ𝔼[Δ(λ_t)]≤f̌_t(𝐱)-𝔼[f̌_t(𝐱̂_t)]+𝔼[λ_t^⊤𝐠_t(𝐱)]+2μ G^2R^2+1/2α𝔼[𝐱-𝐱̂_t^2]-1/2α𝔼[𝐱-𝐱̂_t+1^2]+α∇̂ f_t(𝐱̂_t)^2where the constants G, R and F are as in (as2) and (as3).Taking the norm square in (<ref>), we have λ_t+1^2 ≤λ_t^2+2μλ_t^⊤(𝐠_t(𝐱̂_t)+∇^⊤𝐠_t(𝐱̂_t)(𝐱̂_t+1-𝐱̂_t))+μ^2𝐠_t(𝐱̂_t)+∇^⊤𝐠_t(𝐱̂_t)(𝐱̂_t+1-𝐱̂_t)^2≤λ_t^2+2μλ_t^⊤(𝐠_t(𝐱̂_t)+∇^⊤𝐠_t(𝐱̂_t)(𝐱̂_t+1-𝐱̂_t))+2μ^2𝐠_t(𝐱̂_t)^2+2μ^2∇^⊤𝐠_t(𝐱̂_t)(𝐱̂_t+1-𝐱̂_t)^2. With Δ(λ_t):=1/2(λ_t+1^2-λ_t^2),(<ref>) implies that1/μΔ(λ_t)≤λ_t^⊤(𝐠_t(𝐱̂_t)+∇^⊤𝐠_t(𝐱̂_t)(𝐱̂_t+1-𝐱̂_t))+2μ G^2R^2where G and R are the bounds on the gradient and the radius of the feasible set. On the other hand, recall that the primal iterate 𝐱̂_t+1 is the optimal solution to the following optimization problem𝐱̂_t+1=min_𝐱∈(1-γ) X∇̂_𝐱^⊤ L_t(𝐱̂_t,λ_t)(𝐱-𝐱̂_t)+1/2α𝐱-𝐱̂_t^2.Recalling the definition of ∇̂_𝐱 L_t(𝐱̂_t,λ_t), we thus have that𝐱̂_t+1 =min_𝐱∈(1-γ) X∇̂^⊤ f_t(𝐱̂_t)(𝐱-𝐱̂_t) + λ_t^⊤(𝐠_t(𝐱̂_t)+∇^⊤𝐠_t(𝐱̂_t)(𝐱-𝐱̂_t))+1/2α𝐱-𝐱̂_t^2where we add λ_t^⊤𝐠_t(𝐱_t) to the RHS of (<ref>).Note that it will not change the minimizer of (<ref>), since the added term is constant, and not coupled with the variable 𝐱.To connect (<ref>) with the bound obtained in (<ref>), adding ∇̂^⊤ f_t(𝐱̂_t)(𝐱̂_t+1-𝐱̂_t)+1/2α𝐱̂_t+1-𝐱̂_t^2 to the RHS of (<ref>), we have that1/μΔ(λ_t)+∇̂^⊤ f_t(𝐱̂_t)(𝐱̂_t+1-𝐱̂_t)+1/2α𝐱̂_t+1-𝐱̂_t^2 ≤λ_t^⊤(𝐠_t(𝐱̂_t)+∇^⊤𝐠_t(𝐱̂_t)(𝐱̂_t+1-𝐱̂_t))+1/2α𝐱̂_t+1-𝐱̂_t^2 + ∇̂^⊤ f_t(𝐱̂_t)(𝐱̂_t+1-𝐱̂_t)+2μ G^2R^2. Note that 𝐱̂_t+1 is the minimizer of (<ref>), where the objective on the RHS of (<ref>) is strongly-convex, thus we have that1/μΔ(λ_t)+∇̂^⊤ f_t(𝐱̂_t)(𝐱̂_t+1-𝐱̂_t)+1/2α𝐱̂_t+1-𝐱̂_t^2 ≤λ_t^⊤(𝐠_t(𝐱̂_t)+∇^⊤𝐠_t(𝐱̂_t)(𝐱-𝐱̂_t))+1/2α𝐱-𝐱̂_t^2+2μ G^2R^2+∇̂^⊤ f_t(𝐱̂_t)(𝐱-𝐱̂_t)-1/2α𝐱-𝐱̂_t+1^2 (a)≤λ_t^⊤𝐠_t(𝐱)+∇̂^⊤ f_t(𝐱̂_t)(𝐱-𝐱̂_t)+2μ G^2R^2+1/2α𝐱-𝐱̂_t^2-1/2α𝐱-𝐱̂_t+1^2, ∀𝐱∈ (1-γ) Xwhere (a) uses the non-negativity that λ_t≥0, and the convexity such that 𝐠_t(𝐱̂_t)+∇^⊤𝐠_t(𝐱̂_t)(𝐱-𝐱̂_t)≤𝐠_t(𝐱).Using the Cauchy-Schwarz inequality, we have that-∇̂^⊤ f_t(𝐱̂_t)(𝐱̂_t+1-𝐱̂_t) ≤α∇̂ f_t(𝐱̂_t)^2+𝐱̂_t+1-𝐱̂_t^2/4α. Plugging (<ref>) into (<ref>) and rearranging terms, for ∀𝐱∈ (1-γ) X, we have that1/μΔ(λ_t)+1/4α𝐱̂_t+1-𝐱̂_t^2 ≤λ_t^⊤𝐠_t(𝐱)+∇̂^⊤ f_t(𝐱̂_t)(𝐱-𝐱̂_t)+2μ G^2R^2+1/2α𝐱-𝐱̂_t^2-1/2α𝐱-𝐱̂_t+1^2+α∇̂ f_t(𝐱̂_t)^2. Taking expectation over 𝐮_t on both side of (<ref>) conditioning on 𝐱̂_t, it follows that 1/μ𝔼[Δ(λ_t)]+1/4α𝔼[𝐱̂_t+1-𝐱̂_t^2] ≤λ_t^⊤𝐠_t(𝐱)+𝔼[∇̂^⊤ f_t(𝐱̂_t)(𝐱-𝐱̂_t)]+2μ G^2R^2+1/2α𝐱-𝐱̂_t^2-1/2α𝔼[𝐱-𝐱̂_t+1^2]+α∇̂ f_t(𝐱̂_t)^2 (c)=λ_t^⊤𝐠_t(𝐱)+∇^⊤f̌_t(𝐱̂_t)(𝐱-𝐱̂_t)+2μ G^2R^2+1/2α𝐱-𝐱̂_t^2-1/2α𝔼[𝐱-𝐱̂_t+1^2]+α∇̂ f_t(𝐱̂_t)^2where (c) holds since the randomness 𝐮_t in ∇̂ f_t(𝐱̂_t) is independent of 𝐱̂_t, and ∇̂ f_t(𝐱̂_t) is an unbiased estimator of ∇f̌_t(𝐱̂_t).The convexity of f_t(𝐱) implies that f̌_t(𝐱) is also convex, and thus ∇^⊤f̌_t(𝐱̂_t)(𝐱-𝐱̂_t)≤f̌_t(𝐱)-f̌_t(𝐱̂_t). Plugging into (<ref>) and taking expectation over all possible 𝐱̂_t, it follows that 1/μ𝔼[Δ(λ_t)]+1/4α𝔼[𝐱̂_t+1-𝐱̂_t^2] ≤f̌_t(𝐱)-𝔼[f̌_t(𝐱̂_t)]+𝔼[λ_t^⊤𝐠_t(𝐱)]+2μ G^2R^2 + 1/2α𝔼[𝐱-𝐱̂_t^2]-1/2α𝔼[𝐱-𝐱̂_t+1^2]+α𝔼[∇̂ f_t(𝐱̂_t)^2] which completes the proof by dropping the nonnegative term 𝔼[𝐱̂_t+1-𝐱̂_t^2] in the LHS.§.§ Proof of Theorem <ref>With γ=δ/r, the feasibility of actions {𝐱_1,t} readily follows from Lemma <ref>, i.e., 𝐱_1,t∈ X, ∀ t.To prove the dynamic regret and fit bounds, the following result is needed. For the BanSaP recursions (<ref>)-(<ref>), if we choose α=μ= O(T^-3/4) and δ= O(T^-1/4), the dual iterates are uniformly bounded by λ_t≤ C= O(1),with the constant C given byC:=max{2GR,(1/η+1)GR+2G^2R^2μ/η+d^2F^2α/ηδ^2+μ R^2/2αη}where the constants G, R, and η are as in (as2)-(as4).Plugging the bounded norm of the one-point gradient estimator (<ref>) into (<ref>), it holds that1/μ𝔼[Δ(λ_t)]≤GR+𝔼[λ_t^⊤𝐠_t(𝐱)]+2μ G^2R^2+d^2 F^2α/δ^2+1/2α𝔼[𝐱-𝐱̂_t^2]-1/2α𝔼[𝐱-𝐱̂_t+1^2] where we used the Lipschitz condition on (<ref>); i.e.,𝔼[f̌_t(𝐱)-f̌_t(𝐱̂_t)]≤ GR. Selecting the interior point 𝐱=𝐱̃∈ (1-γ) X so that 𝐠_t(𝐱̃)≤ -η1, it follows from (<ref>) that1/μ𝔼[Δ(λ_t)]≤GR-η𝔼[λ_t^⊤1]+2μ G^2R^2+d^2F^2α/δ^2+1/2α𝔼[𝐱̃-𝐱̂_t^2]-1/2α𝔼[𝐱̃-𝐱̂_t+1^2]. Using -ηλ_t^⊤1=-ηλ_t_1≤ -ηλ_t, we arrive at1/μ𝔼[Δ(λ_t)]≤GR-η𝔼[λ_t]+2μ G^2R^2+d^2F^2α/δ^2+1/2α𝔼[𝐱̃-𝐱̂_t^2]-1/2α𝔼[𝐱̃-𝐱̂_t+1^2]. Now we are ready to show that the norm of the dual variable is uniformly bounded by a constant C that is independent of time; that is, λ_t≤ C, ∀ t.For 1≤ t≤1/μ, it follows readily thatλ_t ≤λ_t-1+μ𝐠_t(𝐱̂_t)+∇^⊤𝐠_t(𝐱̂_t)(𝐱̂_t+1-𝐱̂_t)≤λ_t-1+2μ GR≤λ_1+2μ tGR≤ Cwhere the last inequality follows from λ_1=0, t≤1/μ, and the definition of C in (<ref>). For 1/μ≤ t≤ T, we will prove the claim by contradiction. Assume T_0 is the first slot for which λ_T_0>C. Therefore, we have λ_T_0>C≥λ_T_0-1/μ, which after recalling (<ref>) and the definition of Δ(λ_t), yields1/μ∑_t=T_0-1/μ^T_0-1𝔼[Δ(λ_t)]=1/2μ(𝔼[λ_T_0^2-λ_T_0-1/μ^2])>0.On the other hand however, summing up (<ref>), we obtain 1/μ∑_t=T_0-1/μ^T_0-1𝔼[Δ(λ_t)]≤GR/μ-η∑_t=T_0-1/μ^T_0-1𝔼[λ_t]+2G^2R^2 +d^2F^2α/μδ^2+1/2α𝔼[𝐱̃-𝐱̂_T_0-1/μ^2]-1/2α𝔼[𝐱̃-𝐱̂_T_0^2] (a)≤ GR/μ-η∑_t=T_0-1/μ^T_0-1𝔼[λ_t]+2G^2R^2+d^2F^2α/μδ^2+R^2/2αwhere (a) uses again the bound 𝐱̃-𝐱̂_T_0-1/μ≤ R.Note that since λ_T_0>C and λ_T_0-λ_T_0-1≤ 2μ GR, we have thatλ_T_0-τ>C-2τμ GR. Combining (<ref>) with (<ref>), we deduce1/μ∑_t=T_0-1/μ^T_0-1𝔼[Δ(λ_t)]≤GR/μ-Cη/μ+η GR/μ+2G^2R^2+d^2F^2α/μδ^2+R^2/2α. Together with (<ref>), recursion (<ref>) implies thatC<GR/η+GR+2G^2R^2μ/η+d^2F^2α/ηδ^2+μ R^2/2αηwhich contradicts the definition of C in (<ref>).Hence, there is no T_0 satisfying λ_t≤ C, which implies that λ_t≤ C, ∀ t.By choosing the stepsizes α=μ= O(T^-3/4), and the parameter δ= O(T^-1/4), it follows thatC= O(GR/η+GR+2G^2R^2/η T^3/4+d^2F^2/η T^1/4+R^2/2η)= O(1) which completes the proof of the lemma. Dynamic regret in Theorem 1:Recall that 𝐱_t^* is the minimizer of the following time-varying problem (<ref>), and note that (1-γ)𝐱_t^*∈ (1-γ) X. Hence, plugging (1-γ)𝐱_t^* into (<ref>), we have 1/μ𝔼[Δ(λ_t)]≤f̌_t((1-γ)𝐱_t^*)-𝔼[f̌_t(𝐱̂_t)]+ 1/2α𝔼[(1-γ)𝐱_t^*-𝐱̂_t^2]-1/2α𝔼[(1-γ)𝐱_t^*-𝐱̂_t+1^2]+ 𝔼[λ_t^⊤𝐠_t((1-γ)𝐱_t^*)]+α/δ^2d^2F^2+2μ G^2R^2.From the Lipschitz condition, we can bound the inner product in (<ref>) by 𝔼[λ_t^⊤𝐠_t((1-γ)𝐱_t^*)] ≤ 𝔼[λ_t^⊤(𝐠_t(𝐱_t^*)+γ GR·1)](a)≤γ GR 𝔼[λ_t](b)≤γ GRλ̅where (a) follows from λ_t^⊤𝐠_t(𝐱_t^*)≤ 0 since 𝐠_t(𝐱_t^*)≤0, and λ_t≥0; and (b) uses the upper bound of λ̅:=max_tλ_t. The two distance terms in (<ref>) can be bounded by (1-γ)𝐱^*_t -𝐱̂_t^2-(1-γ)𝐱^*_t-𝐱̂_t+1^2 = (1-γ)𝐱^*_t -𝐱̂_t^2-(1-γ)𝐱^*_t-1-𝐱̂_t^2+(1-γ)𝐱^*_t-1-𝐱̂_t^2-(1-γ)𝐱^*_t-𝐱̂_t+1^2= (1-γ)𝐱^*_t -𝐱^*_t-1(1-γ)(𝐱^*_t+𝐱^*_t-1)-2𝐱̂_t+(1 -γ)𝐱^*_t-1-𝐱̂_t^2-(1-γ)𝐱^*_t-𝐱̂_t+1^2.Using the triangle inequality, it follows that (1-γ)(𝐱^*_t+𝐱^*_t-1)-2𝐱̂_t ≤ (1-γ)𝐱^*_t-𝐱̂_t+(1-γ)𝐱^*_t-1-𝐱̂_t≤ 2Rwhich together with (<ref>), implies that (1-γ)𝐱^*_t -𝐱̂_t^2-(1-γ)𝐱^*_t-𝐱̂_t+1^2 ≤2(1 -γ)R 𝐱^*_t-𝐱^*_t-1+(1-γ)𝐱^*_t-1-𝐱̂_t^2 -(1-γ)𝐱^*_t-𝐱̂_t+1^2. Plugging (<ref>) and (<ref>) into (<ref>), and summing up over t=1,…,T, we find1/2μ(𝔼[λ_T+1^2-λ_1^2])+∑_t=1^T(𝔼[f̌_t(𝐱̂_t)]-f̌_t((1-γ)𝐱^*_t)) ≤ γ GRλ̅T+∑_t=1^T(1-γ)R/α𝐱^*_t-𝐱^*_t-1+2μ G^2R^2T+α d^2F^2 T/δ^2 + 1/2α(𝔼[(1-γ)𝐱^*_0-𝐱̂_1^2]-𝔼[(1-γ)𝐱^*_T-𝐱̂_T+1^2]) (c)≤γ GRλ̅T+R/αV(𝐱_1:T^*)+2μ G^2R^2T+R^2/2α+α d^2F^2 T/δ^2 where (c) uses (1-γ)𝐱^*_0-𝐱̂_1≤𝐱^*_0-𝐱̂_1≤ R, and (1-γ)𝐱^*_T-𝐱̂_T+1^2≥ 0, and the accumulated variation of the per-slot minimizers defined as V(𝐱_1:T^*):=∑_t=1^T 𝐱^*_t-𝐱^*_t-1.Since 𝔼[λ_T+1^2]≥ 0, initializing the dual variable with λ_1=0, and rearranging (<ref>), we have that ∑_t=1^T(𝔼[f̌_t(𝐱̂_t)]-f̌_t((1-γ)𝐱^*_t)) ≤ γ GRλ̅T+R/αV(𝐱_1:T^*)+2μ G^2R^2T+R^2/2α+α d^2F^2 T/δ^2.The iterates {𝐱̂_t} in this bound are not the actual decisions taken by the learner.To obtain the regret bound, our next step is to decompose the regret as∑_t=1^T( 𝔼[f_t(𝐱_1,t)]-f_t(𝐱_t^*)) =∑_t=1^T(𝔼[f_t(𝐱_1,t)]-𝔼[f̌_t(𝐱_1,t)]_U_1+𝔼[f̌_t(𝐱_1,t)]-𝔼[f̌_t(𝐱̂_t)]_U_2+𝔼[f̌_t(𝐱̂_t)]-f̌_t((1-γ)𝐱_t^*))_U_3+f̌_t((1-γ)𝐱_t^*))-f̌_t(𝐱_t^*)_U_4+f̌_t(𝐱_t^*)-f_t(𝐱_t^*)_U_5).We next bound each under-braced, starting withU_1= 𝔼[f_t(𝐱_1,t)-𝔼_𝐯[f_t(𝐱_1,t+δ𝐯_t)]] (d)≤ 𝔼[f_t(𝐱_1,t)-f_t(𝔼_𝐯[𝐱_1,t+δ𝐯_t])](e)=0 where (d) uses Jensen's inequality, and (e) follows from 𝔼_𝐯[δ𝐯_t]=0 since 𝐯_t is drawn from 𝔹:={𝐯:𝐯≤ 1}.Regarding the second term, it follows that U_2=𝔼[f̌_t(𝐱̂_t+δ𝐮_t)-f̌_t(𝐱̂_t)] (f)≤𝔼[Gδ𝐮_t]=δ Gwhere (f) uses the Lipschitz condition of f̌_t(𝐱). The third term U_3 has been already bounded as in (<ref>). Using the Lipschitz condition of f̌_t(𝐱), we can further bound the fourth term U_4=f̌_t((1-γ)𝐱_t^*))-f̌_t(𝐱_t^*)≤γ GR and likewise for the last term for which U_5=𝔼_𝐯[f_t(𝐱_t^*+δ𝐯_t)]-f_t(𝐱_t^*) ≤𝔼_𝐯[Gδ𝐯_t]≤δ G.Plugging (<ref>) and (<ref>)-(<ref>) into (<ref>), we arrive that ∑_t=1^T(𝔼[f_t(𝐱_1,t)]-f_t(𝐱_t^*)) ≤R/αV(𝐱_1:T^*) +R^2/2α+d^2G^2R^2α T/δ^2+γ GRT(1+λ̅)+2μ G^2R^2T +2Gδ T.Upon choosing α=μ= O(T^-3/4), and δ= O(T^-1/4) along with γ=δ/r, it follows that (cf. Lemma <ref>) Reg^ d_T= O(RV(𝐱_1:T^*)T^3/4+GRCT^3/4+2G^2R^2T^1/4+d^2G^2R^2T^3/4)from which the proof is complete. Dynamic fit in Theorem 1: To bound the dynamic fit, recall that the constraint violations in (<ref>) depend on the magnitude of the dual variable and the difference of two consecutive primal iterates.The distance between iterates 𝐱̂_t and 𝐱̂_t+1 can be bounded as𝐱̂_t+1-𝐱̂_t(a)≤α∇̂_𝐱^1 L_t(𝐱̂_t,λ_t) (b)≤ d/δ|f_t(𝐱̂_t+δ𝐮_t)|+∇𝐠_t(𝐱̂_t)λ_t(c)≤α dF/δ+α Gλ_twhere (a) uses the non-expansive property of the projection operator, (b) relies on (<ref>) and the Cauchy-Schwarz's inequality; and (c) uses the bounds in (as2).On the other hand, using the Lipschitz continuity of 𝐠_t(𝐱) and (<ref>), it follows that∑_t=1^T 𝐠_t(𝐱_1,t)≤∑_t=1^T 𝐠_t(𝐱̂_t)+δ GT1 ≤ λ_T+1/μ+G^2T1/2β+β/2∑_t=1^T𝐱̂_t+1-𝐱̂_t^21+δ GT1 (d)≤ λ_T+1/μ+G^2T1/2β+β T(α^2 d^2F^2/δ^2+α^2 G^2λ̅^2)1+δ GT1where (d) uses (<ref>), and the fact that (a+b)^2≤ 2(a^2+b^2).Taking [·]^+ and · on both sides of (<ref>), we have (cf. (<ref>))Fit^ d_T≤λ̅/μ +G^2√(N)T/2β+δ G√(N)T+β√(N) T(α^2 d^2F^2 /δ^2+α^2 G^2λ̅^2)which establishes (<ref>). Upon selecting α= O(T^-3/4), and δ= O(T^-1/4), we find from Lemma <ref> that λ̅≤ C= O(1). Together with μ= O(T^-3/4) and β= O(T^1/4), it holds from (<ref>) thatFit^ d_T≤CT^3/4+G^2√(N)T^3/4/2+G√(N)T^3/4+√(N) T^5/4(d^2F^2 T^-1+T^-3/2 G^2C^2)= O(T^3/4)which completes the proof of (<ref>). §.§ Proof of Theorem <ref>Similar to the proof of Theorem <ref>, feasibility of actions {𝐱_1,t,𝐱_2,t} readily follows from Lemma <ref>; hence, 𝐱_1,t,𝐱_2,t∈ X, ∀ t.To prove the dynamic regret and fit bounds in this setup, the following result is needed. For the BanSaP recursion (<ref>), (<ref>), and (<ref>), selecting α=μ= O(T^-1/2) ensures that the dual iterates are uniformly bounded by λ_t≤ C= O(1), with the constant C given byC:=max{2GR,(1/η+1)GR+2G^2R^2μ/η+d^2G^2α/η+μ R^2/2αη}where the constants G, R, and η are as in (as2)-(as4).It follows steps similar to those used to prove Lemma <ref>. Similar to Lemma <ref>, Lemma <ref> asserts that the dual variable in BanSaP with two-point bandit feedback is also uniformly bounded from above.Now, we are ready to prove the regret bound in Theorem <ref>.Dynamic regret in Theorem 2:To obtain the regret bound in the case of two-point feedback, our first step is to connect the regret with the optimality loss induced by the sequence of iterates {𝐱̂_t}, given by1/2∑_t=1^T(𝔼[f_t(𝐱_1,t)]+𝔼[f_t(𝐱_2,t)])-∑_t=1^Tf_t(𝐱_t^*)(a)≤ 1/2∑_t=1^T(𝔼[f_t(𝐱̂_t)]+δ G+𝔼[f_t(𝐱̂_t)]+δ G)-∑_t=1^Tf_t(𝐱_t^*)= ∑_t=1^T(𝔼[f_t(𝐱̂_t)]-f_t(𝐱_t^*))+δ G where (a) follows from the Lipschitz condition in (as2).The LHS of (<ref>) can be further decomposed as∑_t=1^T(𝔼[f_t(𝐱̂_t)]-f_t(𝐱_t^*))= ∑_t=1^T(𝔼[f_t(𝐱̂_t)]-𝔼[f̌_t(𝐱̂_t)]_U_1+𝔼[f̌_t(𝐱̂_t)]-f̌_t((1-γ)𝐱_t^*))_U_2 +f̌_t((1-γ)𝐱_t^*))-f̌_t(𝐱_t^*)_U_3+f̌_t(𝐱_t^*)-f_t(𝐱_t^*)_U_4). For the first term, following the steps in (<ref>), we have thatU_1≤ 𝔼[f_t(𝐱̂_t)-𝔼_𝐯[f_t(𝐱̂_t+δ𝐯_t)]] ≤ 𝔼[f_t(𝐱̂_t)-f_t(𝔼_𝐯[𝐱̂+δ𝐯_t])]≤0.Similar to (<ref>), we have for the case of two-point feedback ∑_t=1^TU_2 =∑_t=1^T(𝔼[f̌_t(𝐱̂_t)]-f̌_t((1-γ)𝐱^*_t))≤γ GRλ̅T+R/αV(𝐱_1:T^*)+2μ G^2R^2T+R^2/2α+α d^2G^2 T. Using the Lipschitz condition of f̌_t(𝐱), we can bound the third term U_3=f̌_t((1-γ)𝐱_t^*))-f̌_t(𝐱_t^*)≤γ GR and likewise for the last term, it follows from the Lipschitz condition of f_t(𝐱) that U_4=𝔼_𝐯[f_t(𝐱_t^*+δ𝐯_t)]-f_t(𝐱_t^*)≤δ G. Plugging (<ref>)-(<ref>) into (<ref>), we arrive at 1/2∑_t=1^T(𝔼[f_t(𝐱_1,t)]+𝔼[f_t(𝐱_2,t)])-∑_t=1^Tf_t(𝐱_t^*) ≤R/αV(𝐱_1:T^*)+R^2/2α+2μ G^2R^2T+α d^2G^2 T+γ GRT(1+λ̅)+2δ GT.Upon choosing α=μ= O(T^-1/2), and δ= O(T^-1) along with γ=δ/r, it follows that (ignoring constant terms) Reg^ d_T= O(RV(𝐱_1:T^*)T^1/2+1/2R^2T^1/2+2G^2R^2T^1/2+d^2G^2T^1/2)where we used the upper bound of dual variables in Lemma <ref>. This completes the proof of (<ref>).Dynamic fit in Theorem 2: To derive the bound on dynamic fit, recall that the constraint violations in (<ref>) depend on the magnitude of the dual variable as well as on the difference of two consecutive primal iterates.The distance between iterates 𝐱_t and 𝐱̂_t+1 can be bounded by𝐱̂_t+1-𝐱̂_t(a)≤α∇̂_𝐱^2 L_t(𝐱̂_t,λ_t) (b)≤ ∇̂^2 f_t(𝐱̂_t)+∇𝐠_t(𝐱̂_t)λ_t(c)≤α dG+α G λ̅where (a) uses the non-expansive property of the projection operator, (b) applies the Cauchy-Schwarz inequality, and (c) relies on the bounds in (as2).On the other hand, using the Lipschitz continuity of 𝐠_t(𝐱) and (<ref>), we have1/2∑_t=1^T (𝐠_t(𝐱_1,t)+𝐠_t(𝐱_2,t))≤∑_t=1^T 𝐠_t(𝐱̂_t)+δ GT1 ≤ λ_T+1/μ+G^2T1/2β+β/2∑_t=1^T𝐱̂_t+1-𝐱̂_t^21+δ GT1 (c)≤ λ_T+1/μ+G^2T1/2β+β T(α^2 d^2G^2+α^2 G^2λ̅^2)1+δ GT1where (c) uses (<ref>), and the fact that (a+b)^2≤ 2(a^2+b^2). In this case, if we take [·]^+ and then · on both sides of (<ref>), and further choose α=μ= O(T^-1/2), δ=T^-1, and β= O(T^1/2), we arrive atFit^ d_T≤ λ_T+1/μ+G^2N^1/2T/2β+β N^1/2T(α^2 d^2G^2+α^2 G^2λ̅^2)= CT^1/2+N^1/2T^1/2G^2(1/2+d^2+C^2)= O(T^1/2)where we used the bound on dual variables in Lemma <ref>. This completes also the proof of (<ref>), and also that of Theorem <ref>. | http://arxiv.org/abs/1707.09060v1 | {
"authors": [
"Tianyi Chen",
"Georgios B. Giannakis"
],
"categories": [
"cs.LG",
"cs.NI"
],
"primary_category": "cs.LG",
"published": "20170727221733",
"title": "Bandit Convex Optimization for Scalable and Dynamic IoT Management"
} |
firstpage–lastpage 2017Modelling the Scene Dependent Imaging in Cameraswith a Deep Neural Network Seonghyeon NamYonsei [email protected] Joo KimYonsei [email protected], Received; in original form========================================================================================================================= We study the chemical abundances of a wide sample of 142 Galactic planetary nebulae (PNe) with good quality observations, for which the abundances have been derived more or lesshomogeneously, thus allowing a reasonable comparison with stellar models.The goal is the determination of mass, chemical composition and formation epoch of their progenitors, throughcomparison of the data with results from AGB evolution. The dust properties of PNe, whenavailable, were also used to further support our interpretation. We find that the majority (∼60%) of the Galactic PNestudied has nearly solar chemical composition, while ∼40% of the sourcesinvestigated have sub-solar metallicities.About half of the PNe have carbon star progenitors, in the 1.5 M_⊙ < M < 3 M_⊙ mass range, which have formed between 300 Myr and 2 Gyr ago. The remaining PNe are almost equally distributed among PNe enriched in nitrogen, which we interpret as the progenyof M > 3.5 M_⊙ stars, younger than 250 Myr, and a group of oxygen-rich PNe,descending from old (> 2 Gyr) low-mass (M < 1.5 M_⊙) stars that never becameC-stars.This analysis confirms the existence of an upper limit to the amount of carbon which can be accumulated at the surface of carbon stars, probably due tothe acceleration of mass loss in the late AGB phases. The chemical composition of the present sample suggests that in massive AGB stars of solar (or slightly sub-solar) metallicity, the effects of third dredge up combine with hot bottom burning, resulting in nitrogen-rich - but not severely carbon depleted - gaseous material to be ejected.140.252.118.146Planetary Nebulae: individual – Stars: abundances – Stars: AGB and post-AGB. Stars: carbon§ INTRODUCTIONThe stars of mass below ∼ 8, after the end of core helium burning, enter the AGB phase, during which a CNO burning shell provides the thermonuclear energy supplyalmost entirely. Periodically, a helium-rich layer above the degenerate core is ignited in conditions of thermal instability, hence the name "thermal pulse" (hereinafter TP),used to define these episodes.Despite relatively short in comparison with the core hydrogen and helium burning phases, the AGB evolution proves extremely important to understand the feedback of these stars on their host system, because it is during this phase that most of the massloss occurs, with the consequent pollution of gas and dust of the interstellar medium.A full comprehension of the AGB evolution proves crucial for a number of astrophysical contexts, such as the determination of the masses of galaxies at high redshifts <cit.>,the formation and chemical evolution of galaxies <cit.>, the dustcontent of high-redshift quasars <cit.>, and the formation of multiplepopulations of stars in globular clusters <cit.>.The modelling of the AGB phase has made significant progresses in the recent years, withthe description of the dust formation process in the winds, coupled to theevolution of the central star <cit.>.The results are still rather uncertain though, because of the poor knowledge of convection and mass loss, two physical mechanisms having a strong impact on the physical and chemical features of the AGB evolution <cit.>. Therefore, the comparisonwith the observations is extremely important to allow a qualitative step towards an exhaustive knowledge of the main properties of these stars.The Magellanic Clouds (MC) have been so far the most investigated environments to this aim,owing to their relatively short distances (51 kpc and 61 kpc respectively, for the Largeand Small Magellanic Cloud, Cioni et al. 2000; Keller & Wood 2006) and the low reddening(E_B-V=0.15 mag and 0.04 mag, respectively, for the LMC and SMC, Westerlund 1997). The near- and mid-infrared observations of stars in the MC, particularly suitable to study AGB stars, given the cool (below ∼ 4000K) surface temperatures of these objects, have been extensively used to constrain the AGB models <cit.>. These studieshave been completed by more recent investigations, focused on dust production expectedfrom this class of stars, and the relative effects on the infrared colours, <cit.>.A complementary approach to infer valuable information on the evolution of AGB stars is offered by the study of PNe. The chemical composition of these objects reflects the final surface chemistry, at the end of the AGB phase, and is determined by the combination of the two mechanisms potentially able to change their surface chemistry, namely third dredge up (TDU) and hot bottom burning (HBB). The determination of the abundances of the individual species in the PN winds provides a unique tool to understand the efficiency of the two mechanisms. Motivated by these arguments, we have recently started a research project aimed at interpreting the observed sample of PNe based on recent AGB models, accounting for the formation of dust in the circumstellar envelope. In the first two papers of this series we focused on the PNe population of the LMC (Ventura et al. 2015b, hereinafter paper I) and of theSMC (Ventura et al. 2016b, paper II). These works allowed the characterization of the observed PNe in terms of the mass and metallicity of the progenitors.Here we extend this study to the population of Galactic PNe, whose α-elementabundances indicate a wider range of progenitor metallicities. Our aim is twofold: a) we attempt to characterize the individual PNe observed and to identify the progenitors; b) we test AGB evolution and the dust formation process against a differentand more complex environment than that of Magellanic Cloud PNe.The paper is organised as follows: section 2 provides a description of the most important physical and chemical input used to model the AGB phase; in section 3 we discuss the modification of the surface chemical composition of AGB stars, and the expected, final abundances of the various chemical species; the interpretation of the two samples of Galactic PNe studied in the present work is addressed in section 4; in section 5 we discuss the results obtained on the basis of the dust features detected in the spectra of some of the PNe observed;the conclusions are given in section 6.§ AGB MODELLINGOur aim is to interpret the abundances of specific Galactic PN samples on the basis of AGBmodels of different mass and chemical composition. Our goal is to deduce the mass and the metallicity of the progenitor of the individual PNe observed, by comparing the abundances of the various chemical species at the end of the AGB phase with the values derived from the observations. Before entering this detailed comparison, we provide a brief description of the AGB models adopted.The evolutionary sequences have been calculated by the ATON code for stellar evolution. The details of the numerical structure of the code are extensively discussed in <cit.>, whereas the most recent updates are presented in <cit.>. Here we provide a short description of the physical and chemical input most relevant to this work. §.§ ConvectionThe temperature gradient within regions unstable to convective motions is found by the full spectrum of turbulence (FST) model for convection <cit.>. Mixing of chemicals and nuclear burning are treated simultaneously, by means of a diffusive-like approach. During the two major core burning phases we assume that convective velocities decay exponentially from the border of the core within radiatively stable regions, with an e-folding distance of 0.02H_p; this choice is motivated by a calibration of core overshooting based on the observed extension of the main sequences of open clusters, given in <cit.>. During the AGB phase we assume overshoot from the base of the envelope and from the borders of the convective shell which forms at the ignition of each TP; in this case the e-folding distance of convective velocities is 0.002H_p, according to a calibration based on the luminosity function of carbon stars in the LMC, given in <cit.>. §.§ Mass lossWe adopt the formalism by <cit.> to describe mass loss of oxygen-rich AGB stars. Blocker's formula consists in the canonical Reimers' mass loss ratemultiplied by a power of the luminosity, L^2.7. The free parameter entering the Reimers' rate was set to η_R=0.02, following the study on the luminosity function of lithium-rich stars in the MC, by <cit.>. For what regards carbon stars, we adopt the results from hydrodynamical models of carbon stars, published by <cit.>. §.§ OpacitiesRadiative opacities are calculated according to the OPAL release, in the versiondocumented by <cit.>. The molecular opacities in the low-temperatureregime (T < 10^4 K) are calculated by means of the AESOPUS tool <cit.>.The opacities are constructed to follow the changes of the envelope chemical composition,in particular carbon, nitrogen and oxygen individual abundances. §.§ Chemical compositionThe AGB models used here have metallicities Z=10^-3, 2× 10^-3, 4× 10^-3,8× 10^-3, Z=0.014, Z=0.018, Z=0.04. In the Z=1,2 × 10^-3 models we assume the mixture by <cit.>, with an alpha- enhancement [α/Fe]=+0.4; the Z=4,8 × 10^-3 models were calculated with the <cit.> mixture and an α- enhancement [α/Fe]=+0.2; the Z=0.014 models are based on thesolar-scaled mixture by <cit.>; finally, the Z=0.018 and Z=0.04 models have asolar-scaled mixture, with the distribution by <cit.>. The initial abundances of the chemical species mostly used in the present work for the various metallicitiesis reported in Table 1.§.§ Dust formationDust formation in the winds of AGB stars is described according to the schematization introduced by <cit.>. The wind is assumed to expandisotropically under the effects of radiation pressure, acting on dust grains, partly counterbalanced by gravity. The dynamics of the wind is described by means of the momentum conservation equation and by mass conservation, giving the radial stratification of density as a function of the gas velocity and of the rate of mass loss.The effects of the radiation pressure is calculated by means of the opacity coefficient, which, in turn, depends on the number and the size of the dust particles formed. The growth rate of the dust particles of a given species are found via thedifference between the growth and the vaporisation terms.All the relevant equations, with an exhaustive discussion on the role played by various physical factors, are given in <cit.>.The dust species considered depend on whether the surface of the star is oxygen-rich or carbon rich: in the former case we consider the formation of silicates and of alumina dust, whereas for carbon stars we model the formation and growth of silicon carbide and of solid carbon grains <cit.>.§ CHANGES IN THE SURFACE CHEMISTRY OF AGB STARS The AGB models used in the present analysis were introduced and discussed in previous papers by our group. We address the interested reader to <cit.>(Z=4× 10^-3), <cit.> (Z=1,8× 10^-3, initial mass above3 M_⊙), <cit.> (low–mass models of metallicity Z=1,8× 10^-3and initial mass below 3 M_⊙), paper II (Z=2× 10^-3) and<cit.> (Z=0.018). To complete the array of comparison models we alsointroduce here a series of updated, unpublished models with solar metallicityZ=0.014 and with Z=0.04.§.§ Low mass domain: the formation of carbon starsThe surface chemistry of stars of mass below ∼ 3 M_⊙is altered only by the first dredge up (FDU) and by a series of TDU events, which may eventually turn the star into a carbon star. The number of TDU experienced is higher the larger is the initial mass, as more massive objects start the AGB phase with a more massive envelope: this is the reason why only stars with initial mass above a threshold value will eventually become carbon stars. The minimum mass required to become carbon star, shown in Table 1 (col. 9) depends on the metallicity: thehigher is Z, the more difficult is to achieve C/O ratios above unity, owing to the larger quantity of oxygen in the star. For sub-solar metallicities, the lowest mass becoming carbon star is ∼ 1.25 M_⊙;in the solar case this lower limit is ∼ 1.5 M_⊙, whereas no carbon stars are expected to form for Z=0.04. The upper limit in mass for carbon stars coincides withthe minimum mass required to ignite HBB; the latter process prevents the achievement of the C-star stage, via destruction of the surface carbon. The chemical composition of carbon stars will be enriched in nitrogen, as a consequence of the FDU. An increase in the surface oxygen, significantly smaller in comparison tocarbon, is also expected, particularly in low-metallicity stars.§.§ Hot bottom burning and helium enrichment in massive AGB starsStars with with initial mass above a given threshold experience HBB during the AGB phase. The minimum mass required for the ignition of HBB depends on the metallicity. Table 1 (col. 8) reports the valuescorresponding to the different metallicities.The ignition of HBB strongly affects the AGB evolution, because the proton-capturenucleosynthesis activated at the base of the convective envelope significantly changes thesurface chemical abundances of these stars. Among all, it provokes the destruction of thesurface carbon and the production of great quantities of nitrogen. While this is a commonproperty of all M ≥ 3.5 M_⊙ models, the destruction of the surface oxygen, which requires higher HBB temperatures (∼ 80 MK), is sensitive to the metallicity, andis higher the lower is Z <cit.>. The destruction of oxygen is extremelysensitive to the modelling of convection: in the present analysis, based on the FSTdescription, we find significant depletion of oxygen in metal poor AGB stars; conversely,when a less efficient convective model is used, the HBB experienced is weaker, thuslimiting the efficiency of oxygen burning <cit.>. In this context, the detection of oxygen-poor PNe, enriched in nitrogen, would be an important evidencein favour of a very efficient convective transport of energy in the internal regions of the envelope of AGB stars.The stars experiencing HBB are also exposed to the second dregde-up (SDU), after theconsumption of the helium in the core. The main effect of the SDU is the increase in the surface helium (Δ Y), which is sensitive to the initial mass of the star: typically, Δ Y is negligible in stars with mass close to the minimum threshold required to start HBB, and increases with the initial mass, up to Δ Y ∼ 0.1 for M = 8 M_⊙. This result is much more robustthan the predictions concerning the depletion of oxygen, because the SDU takes placebefore the TP phase, thus the results are unaffected by most of the uncertainties affecting AGB modelling. §.§ Surface chemistry at the end of AGB evolution The final chemical composition of the models used here, which will be used to interpret the PN abundances, are shown in Fig. <ref>, in the CN (left) and ON (right) planes.The distribution of the mass fractions of the individual species at the end of the AGBevolution models in these planes allows us to understand the role played by mass andmetallicity on the evolution of the chemistry of the surface layers in AGB stars.For all the metallicities investigated, the lines connecting models of different massdefine a typical counterclockwise shape, moving from the lowest (1 M_⊙) to thehighest 8 M_⊙ mass stars considered. As shown in the left panel of Fig. <ref>, the range of carbon abundances spanned by the models extends over two order of magnitudes,independently of the metallicity. Conversely (see right panel of Fig. <ref>),the distribution of the oxygen abundances is much more sensitive to metallicity: Z=0.04 modelsexhibit a negligible variation in oxygen, whereas in the Z=2× 10^-3 case we find an overall variation of a factor ∼ 30. This behaviour is due to the larger sensitivity of C to HBB and TDU, compared to O. TDU favours a significant increase in the surface C, whereas the effects on O are much smaller. The activation of HBB provokes thedestruction of the surface C, independently of Z, whereas the destruction of the surfaceoxygen via HBB is limited to the stars with the lowest metallicity. The lowest masses considered never become carbon stars, because they loose theexternal mantle before the surface carbon exceeds the oxygen content. Compared to the initial chemical composition, with which they formed, their chemistry is enriched in nitrogen, as a consequence of the FDU. For carbonthe situation is more tricky. If no TDU occurred, the final carbon is smaller than the initial quantity, because during the FDU the surface convection reached regions where carbon was consumed by CN nucleosynthesis; however, if some TDU events take place, the situation is reversed, and the final carbon is above the starting abundance. We remark here that for these stars, owing to the null effects of HBB and the small effects of TDU, the final chemical composition is extremely dependent on the assumptions regarding the chemical mixture of the gas from which the stars formed. As more massive objects, up to ∼ 3 M_⊙, are considered, thetheoretical sequences move to the right on the CN plane, owing to the increase in thesurface carbon, as a consequence of repeated TDU events. On the contrary, nitrogen keepsapproximately constant in this range of mass. The final carbon, could be potentially usedas a mass indicator for M < 3 M_⊙ stars, because stars of higher mass reach highercarbon abundances in the final AGB phases.A word of caution on the predictions regarding the evolution of the surface nitrogen of M ≤ 3 M_⊙ stars is needed here. In the present models, when modelling the red giant branch (RGB) phase, neither thermohaline nor any sort of extra-mixing was considered; this is going to underestimate the N increase, which occurs during the ascending of the RGB <cit.>. Therefore, the N abundances of the modelsof initial mass below ∼ 3 M_⊙ are to be considered as lower limits, with anoverall uncertainty of ∼ 0.2 dex. Regarding the possibility of using the abundances of some chemical elements to infer the metallicity of the progenitors of the PNe, it is evident from the models lociiof Fig. <ref> (left panel), that carbon should not be used to infer metallicity ofthe PN progenitors, since the abundance of this element is determined by the number ofTDU events experienced by the star, which is scarcely related to metallicity.PN oxygen abundances, on the other hand, can be used as probes of the progenitor'smetallicity, at least in the higher metallicity domain. In fact, in the right panelof Fig. <ref>, we observe a tight correlation between metallicity and oxygenabundances, for Z>4× 10^-3. For lower metallicities, we should also consider:a) for massive progenitors, HBB in low-Z AGB stars favoursa significant decrease in the surface O, which breaks out any O-Z relationship; b) in low-mass progenitors the surface oxygen increases under the effects of TDU (see the Z = 4× 10^-3 line in the figure). <cit.> interpreted oxygen enhancement observed in low-metallicity GalacticPNe based on these same modelling feature.The nitrogen abundance is also correlated to metallicity in the low-mass domain;however, the afore mentioned uncertainties affecting the extent of the nitrogen increaseduring the RGB phase prevents the use of the measured N as a robust metallicity indicator.Stars with mass above the minimum threshold required to activate HBB,M > 3-3.5 M_⊙, show the imprinting of CN or, in some cases, CNO nucleosynthesis, in their surface chemical composition. A robust prediction in this case is the significant increase in the nitrogen content, which at the end of the AGB phase is a factor of ∼ 10 higher than the N initially present in the star.The final carbon of these stars is more uncertain, as it is sensitive to the relative importance of HBB and TDU. In an HBB-dominated environment the final Cwill be almost a factor ∼ 10 smaller than the initial value; however, a few TDUevents, in the very final AGB phases, after HBB was turned off, might partlycounterbalance the effects of the proton-capture nucleosynthesis; thisargument is still debated. According to the models used here, shown in the left panel of Fig. <ref>, we find that stars of initial mass ∼ 4-5 M_⊙ evolve to final phases characterised by a great increase in N and a carbon content similar to the matter from which they formed; this is because in these stars a few final TDU episodes make the surface carbon, previously destroyed by HBB, to rise again;conversely, 6-8 M_⊙ stars reach the final evolutionary stages with a surfacecarbon reduced by almost an order of magnitude in comparison with the initial chemistry. The effect of metallicity in this region of the CN plane is that higher Z stars evolve to higher N: this is because the equilibrium abundance of N in a region where CNO nucleosynthesis is active is proportional to the overall C+N+O content.Concerning oxygen, the results are metallicity-dependent: as shown in the right panel of Fig. <ref>, the extent of the depletion of oxygen in massive AGB stars is negligible in solar chemistry models, whereas it amounts to almost a factor 10 in the Z=2× 10^-3 case.§ COMPARISON OF THE OBSERVED PN ABUNDANCES WITH THE AGB FINAL YIELDSOur aim is to study a large sample of Galactic PNe, for which the chemical abundances were derived homogeneously, in the framework of AGB evolution.To trace AGB evolution and especially the HBB and TDU phenomena, the most importantabundances are those of carbon, oxygen, and nitrogen. Dust properties of the PNe addanother handle to determine carbon enrichment (see Stanghellini et al. 2012) even in thecases where carbon abundances are not available.Stanghellini et al. (2007) found clear correlations between gas and dust composition for a sample of PNe in the LMC. PNe with carbon-rich dust (CRD) features were found to have typically carbon-rich gas as well (i.e. C/O > 1), while PNe with oxygen-rich dust (ORD) features had C/O < 1. The same analysis could not be done for Galactic PNe (Stanghellini et al. 2012) for theunavailability of dust and gas chemistry in the same sample of Galactic PNe.It is worth recalling here that oxygen and nitrogen abundances in Galactic PNe are withineasy reach from optical spectra. On the other hand, carbon abundances, and information onthe nature of the nebular dust, are observationally harder to determine. Carbon in PNe can be measured via collissionally-excited emission lines (CELs), themajority of which are emitted in the UV regime, with C II] at 2626-28 Å, and C III] at1907-09 Å. These two intensities are usually sufficient for a complete carbondetermination for low and intermediate excitation PNe. For high excitation PNe, the UVrecombination line C IV 1548-50 Å is also used. An additional possibility is to estimate the C/O ratio from optical recombinationlines (ORLs), which are much easier to acquire. There are several C/O estimates in theliterature from OLRs analysis, both from the assumption that C^2+/O^2+∼ C/O,and from ICF evaluation (see for a discussionDelgado-Inglada & Rodríguez 2014).However, we prefer to avoid carbon estimates from ORLs in this study, to avoid mixingdeterminations from CELs and RLs.The dust content of PNe has been studied in recent years with the advent of the SpitzerSpace Telescope. IRS/Spitzer spectra have been exploited to determine whether PNe havecarbon or oxygen rich dust, or a mixture of both (see Stanghellini et al. 2012;García-Hernández & Górny 2014). The PNe whose dust content has beenclassified intocarbon-rich and oxygen-rich classes to date do not overlap with those with a CELs carbondetermination. For this reason, we decided to select two PN samples for the AGB modelcomparison: the first (Sample 1) is driven by the availability of carbon abundancesdetermined from CELs in the literature; the second (Sample 2) is driven by beingclassified based on their dust contents based on IRS/Spitzer data. §.§ Sample 1 PNeSample 1 PNe are Galactic PNe whose carbon, oxygen, and nitrogen abundances areavailable in the literature to date, with carbon abundances determined from UV emissionlines. Before the HST had became available, UV spectra of Galactic PNe have been acquiredwith the IUE satellite. Data have been accumulated in the decades, and two main groupshave revisited the IUE spectra and derived carbon abundances of Galactic PNe. Kingsburgand Barlow (1994), and Henry et al. (2000) publishedcarbon abundances, with one targetin common. We also searched the literature for Galactic PNe whose spectra has been acquiredwith the HST. Henry et al. (2015) and Dufour et al. (2015) examined 7 carbon determinationsin Galactic PNe (4 in common with the IUE samples described above). Furthermore, Henry etal. (2008) observed the halo PN DdDm 1 (PN G061.9+41.3), and Bianchi et al. (2001) observeda globular cluster PN (K648 in M15). In summary, reliable carbon abundances are available for 40 Galactic PNe from UV data, 7of which are from HST spectra. For all these PNe, the original papers also gave abundancesof He, N, O, Ne. We list Sample 1 PNe in Table 2, where we give their PN G numbers(column 1), their usual names (column 2), their dust and morphological types (columns 3and 4, see table note for description of the keys), and their C, N, and O abundances inthe usual scale of log(X/H)+12 (columns 5 through 7). The asymmetric log uncertaintieshave been calculated from the uncertainties in the original references, when given. Thereferences for the abundances are given in column 8.It is worth noting that all abundance references use the same ICF scheme(Kingsburgh & Barlow 1994), and are thus homogeneous, with the exception of the 7 PNewhose abundances are from Dufour et al. (2015). We comment on possible use of the modelabundances from Henry et al. (2015), based on Dufour et al.' s (2015) data, in specificcases in the following sections. It is important to note that we have checked all PNe in this sample for possibly beinglocated in the bulge or halo of the Galaxy, according to the definition which is commonlyadopted (see Stanghellini & Haywood 2010). We found none of the Sample 1 PNe tobelong to the bulge, while H 4-1 (PN G049.3+88.1), BoBn 1 (PN G108.4-76.1), DdDm 1(PN G061.9+41.3), and Me 2-1 (PN G342.1+27.5) may belong to the Galactic halo. §.§.§ The origin of the Sample 1 PNeWe can take full advantage of the availability of gaseous carbon abundances in sample 1PNe to interpret them by comparison with the AGB yields.Combined with the measurements of nitrogen and oxygen, this allows the knowledge of the overall CNO chemistry, which can be used to infer the progenitors of the individual sources in the sample. We therefore followed an approach similar to paper I and paper IIFig. <ref> shows the Sample 1 PN chemical abundances in the CN (left panel) and ON (right panel) planes. Superimposed to the data we show the final yieldsof AGB models of various mass and metallicity, discussed in the previous section. The symbols used to indicate the PNe reflect our understanding of the mass and chemical composition of the progenitors. We reiterate here that for those cases when the N and O abundances observed provided different directions for interpretation, werelied on the O determinations, because of the on-the-average smaller errorsassociated to the measurements of oxygen compared to nitrogen, and for the uncertainty affecting the predictions of the variation of the surface nitrogen in low-mass AGB stars,owing to the still debated effects of extra-mixing during the RGB ascending.Approximately 65% of Sample 1 PNe descend from solar metallicitystars, whereas ∼ 35% have a slightly sub-solar chemistry, with metallicity Z∼ 4-8× 10^-3.About half of Sample 1 PNe descend from carbon stars. Their surface chemical composition was modified mainly by TDU, with no effects of HBB. These sources areindicated in Fig. <ref> with open pentagons (solar metallicity) and circles(sub-solar chemistry). The progenitors of this group of PNe, characterised by masses inthe range 1.5 M_⊙ < M < 3 M_⊙, formed between 2 Gyr and 500 Myr ago.According to our interpretation, the PNe belonging to this groupwith the largest carbon abundance are younger anddescend from higher mass progenitors. In this sub-sample we find NGC 3242 (PN G261.0+32.0),NGC 6826 (PN G083.5+12.7), and IC 418 (PN G215.2-24.2), whose IR spectra, analysed by<cit.>, exhibit traces of carbon dust, consistently with the interpretationgiven in <cit.> and confirmed in the present study.30% of the PNe in this sample descend from low-mass progenitors, with mass in the ∼ 1-1.5 M_⊙ range, that never reached the C-star stage. These stars, indicatedwith open squares (solar chemistry) and diamonds (sub-solar metallicity) in Fig. <ref>,are the oldest PNe, formed between 10 Gyr and 2 Gyr ago. This group of PNe includes NGC6210 (PN G043.1+37.7), also present in the sample studied by <cit.>, andinterpreted by <cit.> as the progeny of a low-mass progenitor.This sample also include a group of objects that experienced HBB. These PNe areindicated as asterisks and open triangles in Fig. <ref>, according to whether their metallicity is, respectively, solar, or sub-solar. We used once more thecombination of the O and N abundances to deduce the metallicity. These PNe descend from stars of mass above 3.5 M_⊙, formed in more recent epochs, younger than 250 Myr. Their surface chemistry, largely contaminated by HBB, exhibits extremely large N abundances; the carbon of these PNe, in all cases above 8, suggeststhe additional effects of TDU and seems to rule out 7-8 M_⊙ progenitors. Thehelium abundances of these sources, in all but one case log(He/H)+12 > 11.05,further supports this interpretation. Inthis group we include Hu2-1 (PN G051.4+09.6), surrounded by carbon dust, suggestingthe combined effects of HBB and TDU.§.§.§ Similarities and differences of Sample 1 Galactic PNe with the PNein the Magellanic Clouds The PNe of Sample 1 are the only Galactic PNe with measured carbon from CELs.There are two other notabe samples of PNe with measured carbon abundances from CELs,namely, those in the SMC and the LMC, discussed, respectively, in paper I and paper II.We can now compare results from the three different galaxies, based on carbon abundances.According to the results found in this paper, we confirm one of the main findings ofpapers I, and II: the present models of carbon stars nicely reproduce the largestabundances of gaseous carbon observed. The observations indicate log(C/H)+12<9.2 acrossthe galaxies studied, in agreement with the models. This confirms the existence of an upperlimit to the amount of carbon which can be accumulated in the external regions of AGB stars.This limit is likely due to the formation of large quantities of carbon dust in the windsof carbon stars, which favours a fast loss of the external mantle, owing to the effects ofradiation pressure, acting on dust grains. In Sample 1 PNe we did not find pure HBB contamination, at odds with what found forthe Magellanic Cloud PNe in papers I and II: while a few N-rich PNe in the LMC (see leftpanel of Fig 4 of paper I) and the SMC (see left panel of Fig 2 of paper II) disclosedextremely low carbon abundances (log(C/H)+12<8), here we find log(C/H)+12>8.4 forall N-rich PNe. The small number of N-rich PNe in all studied galaxies does not allow anyrobust statistics. However, part of the explanation of this result could reside in theaveragely higher metallicities of Sample 1 PNe compared with those of paper I andpaper II, because the HBB experienced by massive AGB stars is stronger the lower is Z. §.§.§ A few outliers In the analysis of the PNe in the present sample, as stated previously, we attempted to deduce the main properties of the progenitors based on the combination of the CNO abundances observed. While the agreement between the observations and the theoretical expectations was generally extremely satisfactory, in a few cases we could not fit simultaneously the abundances of all the elements. We analyse these PNe individually, in the following.IC 4593 (PN G025.3+40.8). The abundances of N, O and Ne suggest a low-mass (∼ 1 M_⊙) progenitor, with solar metallicity. As evident in the left panel of Fig. <ref>, the only problem with this interpretation is the carbon content, with is ∼ 0.5 lower than expected. A possible explanation could be that the chemistry of IC 4593 reflects the sole effects of mixing during the RGB ascending, and that some additional carbon depletion occurred, owing to unusually large extra-mixing during the RGB phase. It is worth adding that both carbon and nitrogenerror bars are very large for this PN and the inconsistency with the models could beascribed to the low-quality data available.DdDm 1 (PN G061.9+41.3). The O and N abundances indicate a low-mass,metal-poor chemistry, with Z=2× 10^-3. The presence of traces of silicate dust in the spectra is compatible with this hypothesis. While the carbon abundance is substantially compatible with this interpretation, the measured N is a factor ∼ 4 higher than expected (see the left panel of Fig. <ref>). Possible explanations are an overestimation of the surface N and/or a difference in the original N contentin comparison to the typical pop II chemistry, although the large errorbar for nitrogenmay indicate a poor S/N spectrum. It is worth recalling that this is a halo PNe(Henry et al. 2008), thus the initial chemical mixture ratios used for the AGB models maynot be the ideal choice to model it. NGC 7662 (PN G106.5-17.6). The N, O and Ne abundances indicate a low-mass progenitor,of sub-solar metallicity, Z=8× 10^-3. As shown in the left panel of Fig. <ref>, the surface carbon (7.37) is too small for this interpretation (but note the huge errorbar).Note that the carbon abundance whose derivation is based on photoionisation models C/H=8.13 (Henry et al. 2015) is in much better agreement with our interpretation.NGC 7662 is a inhomogeneous PN, with a lot ofstratification (see Dufour et al. 2015). This may help to explain the disagreement betweenthe carbon and other abundance indicators. IC 418 (PN G215.2-24.2). The O and Ne abundances point in favour of a ∼ 2 M_⊙low metallicity progenitor, Z∼ 4× 10^-3. This interpretation would also be in agreement with the presence of carbon dust in the surroundings of this object.If this is the case, we deduce from the right panel of Fig. <ref> that the Ncontent is overestimated by a factor of ∼ 2. The carbon content is also an issue in this case, as according to the models a higher amount of carbon is expected (see left panel of Fig. <ref>). IC 418 was discussed in <cit.>: the interpretation was different in terms of metallicity, as the oxygencontent derived by <cit.> issignificantly lower compared to the values upon which thepresent analysis is based.§.§ Sample 2 PNe Sample 2 PNe are those Galactic PNe whose dust spectrum has been observed byIRS/Spitzer, and whose dust properties have been uniformly analyzed in the recent past byStanghellini et al. (2012), <cit.> and Gutenkunst et al. (2008).Elemental abundances for these PNe are available from <cit.> (hereinafter GG14),who targeted explicitly these dust-analyzed PNe and recalculated the principal elementalabundances from published and newly observed optical emission lines.As in turned out,unfortunately none of the Sample 2 PNe have carbon abundance measured from the CELsUV lines, thus, while interesting, they lack one of the major observing constraints forthis type of work. There is still purpose of using this large sample to confront with theAGB yields.In Table 3 we give the Sample 2 PN names, dust and morphological properties(note that the dust content codes and the morphological type codes are the same as inTable 2), and the elemental abundances (nitrogen, oxygen, and argon, see Fig. <ref>)with their uncertainties. It is worth noting that the Sample 2 PNe, consisting of 101targets, is more restricted than the whole sample discussed by GG14. In fact, we haveeliminated from the GG14 sample those PNe whose ionization correction factor were deemedby the authors to be uncertain (S. K. Górny, private communication), i.e.,those with O^2+/O<0.4. In the Tables we mark with ^1 and ^2 the PNe that arelikely to belong to the Galactic bulge or halo respectively, based on the prescription inStanghellini & Haywood (2010). We base our comparison between abundances and yields on the O-N plane. The N and Oabundances observed are shown in the left panel of Fig. <ref>, overimposed to theresults from AGB modelling. The observations have been indicated with different symbols,according to the dust properties. The yellow-shaded region indicates the zone of the O-Nplane where we expect to find the progeny of carbon stars; in the interpretation of thePNe located close to the lower and upper borders of this region, we will consider theuncertainties related to the final nitrogen of low-mass stars, discussed in section<ref>. The right panel of Fig. <ref> shows the observed oxygen and argon abundances. The PNe shown in green in both panels are those with helium abundances 12+log(He/H)>11.1, which we will take as the typical threshold above which we see the signature of the second dredge-up, operating in stars of initial mass above ∼ 4 M_⊙.To derive the mass, age, and metallicity of the progenitors of the Sample 2 PNewe rely on their position on the O-N plane to understand the relative importance of HBB and TDU in modifying the surface chemical composition, which provides an indication on the initial mass (see discussion in section <ref>). The metallicity of the progenitors is deduced on the basis of the position on the O-Ar plane, shown in the right panel of Fig. <ref>. Among the various species unaffected by AGB evolution, we prefer to use Argon as metallicity indicator, because: a) the chlorine abundance is available only for part of the PNe in the sample; b) the sulphur detected might not reflectthe original content, because part of this element is absorbed in dust particles, particularly in carbon-rich environments <cit.>.§.§ Mass and metallicity distributionThe comparison between the Sample 2 PN abundances and the models indicate thatapproximately half of the PNe in this sample have solar/supersolar metallicity, the remaining∼ 50% exhibit a sub-solar chemistry, with Z_⊙/3 < Z < Z_⊙/2. We findalso a few metal-poor objects, namely M 2-39 (PN G008.1-04.7), Pe 2-7 (PN G285.4+02.2),M 4-6 (PN G358.6+01.8): based on the N, O and Ar abundances, we interpret these PNe as theprogeny of low-mass stars, with mass below ∼ 1.5 M_⊙ and metallicityZ=1-2 × 10^-3. The relative distribution of PNe of different metallicityexhibits a slight change according to the position in the Galaxy: in the bulge, the "solar" component exceeds by ∼ 50% the "sub-solar"group, whereas in the disk the solar metallicity PNe account for ∼ 45 % of the total, the more numerous component beingthe sub-solar one. The diagonal line in the left panel of Fig. <ref> represents an approximateseparation between the stars that experienced some HBB, with the consequent nitrogen enrichment of the surface layers, and those that experienced only dredge-up effects. The PNe above this line are identified as the progeny of stars of mass M≥ 4 M_⊙, which underwent SDU and HBB. Thefact that most of the PNe in this zone of the O-N plane are also helium-rich adds more robustness to this interpretation. These PNe are the objects formed more recently in the sample examined, and are younger than ∼ 250 Myr. The fraction of PNe which havebeen exposed to HBB during the AGB phase is approximately 30%. The PNe within the shaded region in the O-N plane are generally interpreted as the progeny of stars with mass in the range 1.5 M_⊙ < M < 3 M_⊙, which reachedthe carbon star stage during the AGB evolution. This is the dominant component in the sample, including ∼ 50% of the PNe observed. Similarly to the analysis ofSample 1 PNe, we may conclude that Sample 2 PNe have ages in the range300 Myr - 2 Gyr, with no straightforward trend between age and the position on the O-Nplane. We reiterate here that the vertical extension of this region is somewhat uncertain, as it is sensitive to the extent of the N enrichment occurring during the RGB, which is still debated, and may be different between stars of the same mass. The remaining 20% of Sample 2 PNe are located below the shaded region in the left panel of Fig. <ref>; these are the descendants of low-mass stars, which did not reach the C-star stage during the AGB phase. These are the oldest PNe, formed between 2 Gyr and 10 Gyr ago. The spectra of the majority of the PNe in thissub-sample exhibit the feature of silicate dust, in agreement with our interpretation. We do not expect any carbon enhancement in the surface chemical composition of these objects. §.§ Galactic distribution of Sample 2 PNeThe mass and metallicity distribution of Sample 2 PNe can be used to outline some important points, regarding how PNe of different mass, chemistry and dust type aredistributed across the Galaxy. According to our interpretation, PNe with solar/supersolar metallicity progenitors arecomposed by ∼ 50% of objects that experienced HBB during their AGB life, anindication of massive and relatively young progenitors. The remaining half of solarmetallicity PNe are divided among the progeny of C-stars and of low-mass stars, inapproximately equivalent percentages. These relative numbers hold both for the bulgeand the disk.§.§ A few outliers The PNe Mac1-2 (PN G309.5-02.9) and H1-33 (PN G355.7-03.0, in the bulge), represented bythe two open squares in the HBB region (i.e., above the straight line) in the left panelof Fig. <ref>, are enriched in nitrogen, thus indicating the signature of HBB.Their surface chemical composition suggests a ∼ 4-5 M_⊙ progenitor, of solarmetallicity. The problem with this interpretation is that their spectra exhibit thetypical features of carbon dust; this is at odds with our understanding, as the starsthat experience HBB during the AGB phase destroy the surface carbon, thus leaving no roomfor the formation of carbon dust in the circumstellar envelope. The only possibility toreconcile these results with our theoretical description is that the nitrogen of thesePNe are overestimated (the errors associated to the N determination are of the order of0.2 dex), so that in the O-N plane they fall into the carbon star zone, yellow-shaded inFig. <ref>. Alternatively, we are left with three possibilities: a) carbon dust canbe formed around PNe, despite the surface oxygen is in excess of carbon; b) in the veryfinal AGB phases, after HBB is turned off by the loss of the external mantle, a sequenceof TDU events may favour the formation of a carbon star; (c) the PN is of the MCD type,but the oxygen dust features are too weak to be seen in the spectra. It goes without sayingthat the knowledge of the carbon content of Mac1-2 and H1-33 would be crucial to answerthese questions.The bulge PNe Th3-4 (PN G354.5+03.3), M3-44 (PN G359.3-01.8), and H1-61(PN G006.5-03.1)exhibit MCD type. In the O-N plane they are represented with the three asterisks atlog(O/H)+12 ∼ 8.1, log(N/H)+12 ∼ 8. The O and N abundances indicate gasprocessed by HBB, typical of massive (M∼ 6-8 M_⊙) AGB stars; the low oxygensuggests a metallicity Z∼ 4× 10^-3. The surface helium is not significantlyenriched, at odds with the expectations regarding massive AGB stars. For whatregards H1-61 and Th3-4, the observed helium is close to ∼ 11.1; taking into accountthe errors (∼ 0.04), these results can be reconciled with the theoretical expectationsfrom SDU computations, thus confirming low-metallicity, massive ∼ 6-7 M_⊙ progenitors for these two PNe. The interpretation for M3-44 is more tricky, because the observed helium is log(He/H)+12 = 9.92 with an error of ∼ 0.05; this value is definitively too low to be compatible with SDU effects, thus ruling out a massive progenitor. Given the very small argon abundances, our best interpretation is that M3-44 descends from a metal-poor star (Z∼ 2× 10^-3), with initial mass∼ 3 M_⊙, which experienced some HBB, triggering the increase in thesurface N.Mac 1-11 (PN G008.6-02.6), H1-46 (PN G358.5-04.2), M2-50 (PN G097.6-02.4), He2-62(PN G295.3-09.3) and H1-1 (PN G343.4+11.9, in the halo) exhibit evidences of thepresence of silicates in their spectra, despite being in the region of the O-N plane where we expect to find carbon stars. They are represented by open triangles within the shaded region in the left panel of Fig. <ref>.Mac 1-11 is in the lower region of the shaded region in the left panel ofFig. <ref>. Given the uncertainties associated with the measurement of nitrogen ∼ 0.1 dex and the poor understanding of the nitrogen enrichment during the RGB ascending of low-mass stars, we suggest that this PNe descends from a low mass ∼ 1 M_⊙ progenitor, and is indeed oxygen-rich.H1-46 is located in the middle of the C-star region in Fig. <ref>. Either the N is largely overestimated, or there is no way of explaining this PNe within our modelling. The N content of M-50 and He2-62 is too large to be compatiblewith a low-mass progenitor that failed to reach the C-star stage. The discrepancy between the models and the observation in this case amounts to ∼ 05. dex,far in excess of the uncertainties associated to the measurements (very small in these cases) or to the modelling of extra-mixing during the RGB phase. Our favourite possibility here is that the N is slightly underestimated, thus rendering the chemical composition ofM-50 and He2-62 compatible with a ∼ 3 M_⊙ progenitor, that experienced some HBB, thus inhibiting the formation of a carbon star.For what concerns H1-1, there is the possibility that it descends from a low-mass(M<1.5 M_⊙) progenitor, which never reached the C-star stage. The reason for itsanomalous position, within the C-star region, might be related to a different nitrogencontent, possibly enhanced, in the halo region where it formed. In all these outlier cases, it is almost impossible to proceed without a measure of theatomic carbon content of the PNe. With these measurements on hand, we are confidentthat the evolutionary paths to the observed chemistry, given initial mass and metallicity,would be more obvious.§ DISCUSSIONDust properties of PNe have been linked to their progenitors in a variety of environments(e.g., Stanghellini et al. 2007, 2012). With Sample 1 we find similar conclusions that were addressed by Stanghellini et al. (2007)for LMC PNe. 4 out of the 6 PNe for which dust identification is available confirm that ORD corresponds to C/O<1, and CRD to C/O>1. Statistics is very limitedfor sample 1 though, and we do find 2 exceptions of CRD PNe with C/O<1 (NGC 6826 and NGC 3242).We can not use Sample 2 PNe for this type of analysis, given the lack of carbon abundancesfor these PNe. In this work, using Sample 2 PNe,we found a similar fraction of subsolar and solarmetallicity PNe with ORD dust chemistry. GG14 analyzed several of the Sample 2 PNe,together with other PNe with known dust properties,andfound that solar ORD PNepredominate. It is worth noting that we did not include in Sample 2 PNe with uncertainabundances, while GG14 had, so this selection may explain the different population fraction.CRD PNe are rare in the bulge, as noted by GG14. On the other hand, MCD PNe are frequentin the bulge. These PNe lie in the carbon star progeny locus of the ON diagram (theyellow region of Fig. <ref>). Naturally, the conclusion about these progenitors islinked to the reliability of the ON diagnostics to predict that these AGB stars do notexperience HBB. It is also worth adding that, as noted by <cit.>,we can not disregard other formation channels for MCD PNe, such as extra-mixing, rotation,binary interaction, or pre-He enrichment. These channels will only be constrained whencarbon gas abundances will be available for these PNe.We found that the majority (∼ 80 %) of disk PNe with sub-solar metallicity are theprogeny of carbon stars. This results, in agreement with the analysis by GG14, could bepartly due to the intrinsic difficulty in detecting PNe with very massive progenitors,owing to the short time they evolve at high luminosity. This paucity of HBB PNe withvery massive progenitors is seen in both the disc and the bulge.By comparing the observed chemistry of each object with the model predictions we foundthat both sub-solar and solar metallicity MCD PNe could be the progeny of massive(M >5 M_⊙) AGB evolution. Once again, gas phase carbon abundances will clarifythis point, especially since there are chemical outliers that could have a very differentorigin (see a relevant discussion in GG14). We also found that the fraction of PNedescending from AGB stars that experienced HBB in the bulge is higher (40%) than inthe disc, suggesting a higher percentage of young PN progenitors in the bulge than inthe disc. The latter result is based on 12 PNe, thus it must be taken with some caution, and follow up when carbon abundances of these PNe will be available. § CONCLUSIONSWe study the PNe population of the Milky Way, based on the properties of two samples ofGalactic PNe, selected according to the availability of their physical parameters in theliterature. Sample 1 includes 40 PNe, for which the CNO abundances are available, withcarbon measurements are from ultraviolet emission lines. Sample 2 includes 102 PNe whosedust properties and abundances of several elements are available, but atomic carbon hasnot been measured from CELs so far.By comparing the PNe chemical composition data samples at hand with the yields from AGBevolution of an array of models, we discussed the possible progenitors of the PNe observed.Particular importance have the abundances of elements related to CNO cycling, which arethe most sensitive to the efficiency of the physical processes able to alter the surfacechemical composition of stars during the AGB evolution. The mass fraction of neon and argon are also used in the comparison, mainly to infer themetallicity of the progenitor populations, because these elements are not expected toundergo significant changes during the AGB phase. Furthermore, the enrichment in the helium abundance helps the interpretation, being a signof SDU, operating only in massive AGB stars.According to our interpretation the majority of PN progenitors, about 60%,have a solar chemical composition, the remaining 40% having metallicities in the rangeZ_⊙/3 < Z < Z_⊙/2. A few metal-poor objects are also present in the samples.Half of the sources in both samples disclose a carbon-rich chemistry, thus suggesting aC-star origin. These PNe descend from 1.5-3 M_⊙ progenitors, formed between 500Myr and 2 Gr ago. A small fraction (20%) ofSample PNe 1 are nitrogen enriched, indicatingthat they have been exposed to HBB during the AGB evolution. The progenitors of these PNeare the youngest stars to have ejected a PN, formed not earlier than 250 Myr ago, and withmass above 3.5 M_⊙. The fraction of nitrogen rich PNe is slightly higher (∼ 30%) in Sample 2 than in Sample 1. The remaining PNe of both samples are the progeny of low-mass (M < 1.5 M_⊙)stars, which are older than 2 Gyr. These old stars failed to reach the carbon star stagebecause they lost the external envelope before achieving the C/O>1 condition at thesurface of the star.We conclude that measuring gas phase carbon abundance in PNe is crucial to allow arobust classification of PNe andtheir progenitors. Carbon is the most sensitive elementto the two physical mechanisms potentially able to alter the surface chemical compositionof AGB stars, namely TDU and HBB. In this context, the analysis of Sample 1 PNe providesa more robust analysis of the AGB progenitors than that of Sample 2 PNe. The observedcarbon abundances are nicely reproduced by the yields of AGB evolution used in thepresent analysis, in particular for the upper limits of the carbon amounts. This findingsupports the outcome of AGB modeling, indicating an upper limit to the quantity ofcarbon which can be accumulated to the surface regions of the stars, and that can beobserved directly in PNe. This is due to the low effective temperature reached during thecarbon star stage, which favours the formation of great quantities of carbon dust, leadingto a rapid loss of the external envelope, once the surface carbon is largely in excess ofoxygen.A few Sample 2 PNe present a HBB contaminated chemistry, yet they are unexpectedly aresurrounded by carbon dust. Measuring atomic carbon from UV CELs in these PNe would beextremely important to shed new light on the very final AGB phases of massive AGB stars,particularly on the possibility, still highly debated, that late TDU events could favourthe C-star stage, after HBB is extinguished. § ACKNOWLEDGMENTSL.S. is indebted to the Observatory of Rome for the warm hospitality during her sabbaticleave and to S. K. Górny for useful discussions. 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"authors": [
"P. Ventura",
"L. Stanghellini",
"F. Dell'Agli",
"D. A. Garcia-Hernandez"
],
"categories": [
"astro-ph.SR",
"astro-ph.GA"
],
"primary_category": "astro-ph.SR",
"published": "20170726095541",
"title": "The evolution of Galactic planetary nebula progenitors through the comparison of their nebular abundances with AGB yields"
} |
[ Jorge Drumond Silva December 30, 2023 =======================fancyRecent studies show that the state-of-the-art deep neural networks (DNNs) are vulnerable to adversarial examples, resulting from small-magnitude perturbations added to the input. Given that that emerging physical systems are using DNNs in safety-critical situations, adversarial examples could mislead these systems and cause dangerous situations. Therefore, understanding adversarial examples in the physical world is an important step towards developing resilient learning algorithms. We propose a general attack algorithm, (), to generate robust visual adversarial perturbations under different physical conditions. Using the real-world case of road sign classification, we show that adversarial examples generated using achieve high targeted misclassification rates against standard-architecture road sign classifiers in the physical world under various environmental conditions, including viewpoints. Due to the current lack of a standardized testing method, we propose a two-stage evaluation methodology for robust physical adversarial examples consisting of lab and field tests. Using this methodology, we evaluate the efficacy of physical adversarial manipulations on real objects. With a perturbation in the form of only black and white stickers, we attack a real stop sign, causing targeted misclassification in of the images obtained in lab settings, and in of the captured video frames obtained on a moving vehicle (field test) for the target classifier. § INTRODUCTIONDeep Neural Networks (DNNs) have achieved state-of-the-art, and sometimes human-competitive, performance on many computer vision tasks <cit.>. Based on these successes, they are increasingly being used as part of control pipelines in physical systems such as cars <cit.>, UAVs <cit.>, and robots <cit.>.Recent work, however, has demonstrated that DNNs are vulnerable to adversarial perturbations <cit.>. These carefully crafted modifications to the (visual) input of DNNs can cause the systems they control to misbehave in unexpected and potentially dangerous ways.This threat has gained recent attention, and work in computer vision has made great progress in understanding the space of adversarial examples, beginning in the digital domain (by modifying images corresponding to a scene) <cit.>, and more recently in the physical domain <cit.>. Along similar lines, our work contributes to the understanding of adversarial examples when perturbations are physically added to the objects themselves. We choose road sign classification as our target domain for several reasons: (1) The relative visual simplicity of road signs make it challenging to hide perturbations. (2) Road signs exist in a noisy unconstrained environment with changing physical conditions such as the distance and angle of the viewing camera, implying that physical adversarial perturbations should be robust against considerable environmental instability. (3) Road signs play an important role in transportation safety. (4) A reasonable threat model for transportation is that an attacker might not have control over a vehicle's systems, but is able to modify the objects in the physical world that a vehicle might depend on to make crucial safety decisions. The main challenge with generating robust physical perturbations is environmental variability. Cyber-physical systems operate in noisy physical environments that can destroy perturbations created using current digital-only algorithms <cit.>. For our chosen application area, the most dynamic environmental change is the distance and angle of the viewing camera. Additionally, other practicality challenges exist: (1) Perturbations in the digital world can be so small in magnitude that it is likely that a camera will not be able to perceive them due to sensor imperfections. (2) Current algorithms produce perturbations that occupy the background imagery of an object. It is extremely difficult to create a robust attack with background modifications because a real object can have varying backgrounds depending on the viewpoint. (3) The fabrication process (e.g., printing of perturbations) is imperfect. Informed by the challenges above, we design (), which can generate perturbations robust to widely changing distances and angles of the viewing camera. creates a visible, but inconspicuous perturbation that only perturbs the object (a road sign) and not the object's environment. To create robust perturbations, the algorithm draws samples from a distribution that models physical dynamics (varying distances and angles) using experimental data and synthetic transformations (Figure <ref>). Using the proposed algorithm, we evaluate the effectiveness of perturbations on physical objects, and show that adversaries can physically modify objects using low-cost techniques to reliably cause classification errors in DNN-based classifiers under widely varying distances and angles. For example, our attacks cause a classifier to interpret a subtly-modified physical Stop sign as a Speed Limit 45 sign. Specifically, our final form of perturbation is a set of black and white stickers that an adversary can attach to a physical road sign (Stop sign). We designed our perturbations to resemble graffiti, a relatively common form of vandalism. It is common to see road signs with random graffiti or color alterations in the real world as shown in Figure <ref> (the left image is of a real sign in a city). If these random patterns were adversarial perturbations (right side of Figure <ref> shows our example perturbation), they could lead to severe consequences for autonomous driving systems, without arousing suspicion in human operators. Given the lack of a standardized method for evaluating physical attacks, we draw on standard techniques from the physical sciences and propose a two-stage experiment design: (1) A lab test where the viewing camera is kept at various distance/angle configurations; and (2) A field test where we drive a car towards an intersection in uncontrolled conditions to simulate an autonomous vehicle. We test our attack algorithm using this evaluation pipeline and find that the perturbations are robust to a variety of distances and angles.Our Contributions. Figure <ref> shows an overview of our pipeline to generate and evaluate robust physical adversarial perturbations. * We introduce () to generate physical perturbations for physical-world objects that can consistently cause misclassification in a DNN-based classifier under a range of dynamic physical conditions, including different viewpoint angles and distances (Section <ref>). * Given the lack of a standardized methodology in evaluating physical adversarial perturbations, we propose an evaluation methodology to study the effectiveness of physical perturbations in real world scenarios(Section <ref>).* We evaluate our attacks against two standard-architecture classifiers that we built: with 91% accuracy on the LISA test set and with 95.7% accuracy on the GTSRB test set. Using two types of attacks (object-constrained poster and sticker attacks) that we introduce, we show that produces robust perturbations for real road signs. For example, poster attacks are successful in 100% of stationary and drive-by tests against , and sticker attacks are successful inof stationary testing conditions and in of the extracted video frames against . * To show the generality of our approach, we generate the robust physical adversarial example by manipulating general physical objects, such as a microwave. We show that the pre-trained Inception-v3 classifier misclassifies the microwave as “phone" by adding a single sticker. * We design (), the first algorithm that generates robust physical adversarial examples for real objects, which can fool DNN based classifiers in different viewpoints.* We apply to generate physical adversarial examples for road sign recognition systems. We achieve high attack success rate against varying real-world control variables, including distances, angles, and resolutions, in both lab and drive-by settings.* Focusing on the DNN application domain of road sign recognition, we introduce two attack classes on different physical road signs: Poster-Printing, where an attacker prints an actual-sized road sign with adversarial perturbations and then overlays it over an existing sign. Sticker Perturbation, where an attacker prints perturbations on paper, and then sticks them to an existing sign.For these attacks, we physically realize two types of perturbations: (1) perturbations that occupy the entire region of the sign, and (2) perturbations that take the form of graffiti and abstract art. These attacks do not require special resources—only access to a color printer. * Given the lack of a standardized methodology in evaluating physical adversarial perturbation, we propose an evaluation methodology to study the effectiveness of physical perturbations in real world scenarios. For the road sign recognition system, the methodology consists of two stages: a stationary test, and a drive-by test using a vehicle. The tests aim to capture dynamic real-world conditions which an autonomous vehicle might experience (Section <ref>).* We provide a thorough evaluation of our physical adversarial examples against and using the proposed methodology. We find that: how about we remove this part with the details below to make the contribution more clear? In a stationary test: a poster attack causes a Stop sign to be misclassified as a Speed Limit 45 sign in of the testing conditions (15 out of 15 images); a graffiti attack and an abstract art attack cause a Stop sign to be misclassified as a Speed Limit 45 sign in (10 out of 15) and (15 out of 15) of the test cases, respectively. In a drive-by test: a poster attack causes a Stop sign to be misclassified as a Speed Limit 45 sign in of the extracted video frames (37 out of 37 frames); a abstract art attack has the same misclassification effect in of the extracted video frames (28 out of 33 frames; we sampled every 10 frames in both cases).In a stationary test: a abstract art attack causes a Stop sign to be misclassified as a Speed Limit 80 sign in of all test cases (12 out of 15 images).In a drive-by test: a abstract art attack causes a Stop sign to be misclassified as a Speed Limit 80 sign in of the extracted video frames (28 out of 32 frames; we sampled every 10 frames).* add the new exp against robust models trained on ImageNet hereOur work, thus, contributes to understanding the susceptibility of image classifiers to robust adversarial modifications of physical objects. These results provide a case for the potential consequences of adversarial examples on deep learning models that interact with the physical world through vision. Our overarching goal with this work is to inform research in building robust vision models and to raise awareness on the risks that future physical learning systems might face.We include more examples and videos of the drive-by tests on our webpage § RELATED WORK We survey the related work in generating adversarial examples. Specifically,given a classifier f_θ(·) with parameters θ and an input x with ground truth label y for x, an adversarial example x' is generated so that it is close to x in terms of certain distance, such as L_p norm distance. x' will also cause the classifier to make an incorrect prediction as f_θ(x')y (untargeted attacks), or f_θ(x') = y^* (targeted attacks) for a specific y^* ≠ y. We also discuss recent efforts at understanding the space of physical adversarial examples.Digital Adversarial Examples.Different methods have been proposed to generate adversarial examples in the white-box setting, where the adversary has full access to the classifier <cit.>. We focus on the white-box setting as well for two reasons: (1) In our chosen autonomous vehicle domain, an attacker can obtain a close approximation of the model by reverse engineering the vehicle's systems using model extraction attacks <cit.>. (2) To develop a foundation for future defenses, we must assess the abilities of powerful adversaries, and this can be done in a white-box setting. Given that recent work has examined the black-box transferability of digital adversarial examples <cit.>, physical black-box attacks may also be possible. Goodfellow proposed the fast gradient method that applies a first-order approximation of the loss function to construct adversarial samples <cit.>.Optimization based methods have also been proposed to create adversarial perturbations for targeted attacks <cit.>. These methods contribute to understanding digital adversarial examples. By contrast, our work examines physical perturbations on real objects under varying environmental conditions.Physical Adversarial Examples. Kurakin showed that printed adversarial examples can be misclassified when viewed through a smartphone camera <cit.>. Athalye and Sutskever improved upon the work of Kurakin and presented an attack algorithm that produces adversarial examples robust to a set of two-dimensional synthetic transformations <cit.>. These works do not modify physical objects—an adversary prints out a digitally-perturbed image on paper. However, there is value in studying the effectiveness of such attacks when subject to environmental variability.Our object-constrained poster printing attack is a reproduced version of this type of attack, with the additional physical-world constraint of confining perturbations to the surface area of the sign. Additionally, our work goes further and examines how to effectively create adversarial examples where the object itself is physically perturbed by placing stickers on it. Concurrent to our work,[This work appeared at arXiv on 30 Oct 2017.] Athalye improved upon their original attack, and created 3D-printed replicas of perturbed objects <cit.>. The main intellectual differences include: (1) Athalye only use a set of synthetic transformations during optimization, which can miss subtle physical effects, while our work samples from a distribution modeling both physical and synthetic transformations. (2) Our work modifies existing true-sized objects. Athalye 3D-print small-scale replicas. (3) Our work simulates realistic testing conditions appropriate to the use-case at hand.Sharif attacked face recognition systems by printing adversarial perturbations on the frames of eyeglasses <cit.>. Their work demonstrated successful physical attacks in relatively stable physical conditions with little variation in pose, distance/angle from the camera, and lighting. This contributes an interesting understanding of physical examples in stable environments. However, environmental conditions can vary widely in general and can contribute to reducing the effectiveness of perturbations. Therefore, we choose the inherently unconstrained environment of road-sign classification. In our work, we explicitly design our perturbations to be effective in the presence of diverse physical-world conditions (specifically, large distances/angles and resolution changes). Finally, Lu performed experiments with physical adversarial examples of road sign images against detectors and show current detectors cannot be attacked <cit.>. In this work, we focus on classifiers to demonstrate the physical attack effectiveness and to highlight their security vulnerability in the real world. Attacking detectors are out of the scope of this paper, though recent work has generated digital adversarial examples against detection/segmentation algorithms <cit.>, and our recent work has extended to attack the YOLO detector <cit.>. § PROBLEM STATEMENT Our goal is to examine whether it is possible to create robust physical perturbations of real-world objects that trick deep learning classifiers into producing incorrect class labels even when images are taken under different physical conditions, and even at extreme angles and distances. In this work, we focus on deep neural networks applied to road sign recognition because of the critical role of these objects in road safety and security. §.§ U.S. Road Sign Classification To the best of our knowledge, there is current currently no publicly available road-sign classifier for U.S. road signs.Therefore, we use the LISA dataset <cit.> of U.S. traffic signs containing 47 different road signs to train a DNN-based classifier. This dataset does not contain equal numbers of images for each sign. In order to balance our training data, we chose 17 common signs with the most number of training examples. Furthermore, since some of the signs dominate the dataset due to their prevalence in the real world (Stop and Speed Limit 35), we limit the maximum number of examples used for training to 500 per sign. Our final dataset includes commonly used signs such as Stop, Speed Limits, Yield, and Turn Warnings. Finally, the original LISA dataset contains image resolutions ranging from 6 × 6 to 167 × 168 pixels. We resized all images to 32 × 32 pixels, a common input size for other well-known image datasets such as CIFAR10 <cit.>. Table <ref> summarizes our final training and testing datasets.[speedLimitUrdbl stands for unreadable speed limit. This means the sign had an additional sign attached and the annotator could not read due to low image quality] We set up and trained our road sign classifier in TensorFlow using this refined dataset. The network we used was originally defined in the Cleverhans library <cit.> and consists of three convolutional layers followed by a fully connected layer. Our final classifier accuracy is 91% on the test dataset. For the rest of the paper, we refer to this classifier as for simplicity. §.§ Improving the ClassifierIn order to test the versitility of , we also run our attack against a classifier trained with a larger road sign dataset, the German Traffic Sign Recognition Benchmark (GTSRB) <cit.>. For this purpose, we use a publicly available implementation <cit.> of a multi-scale CNN architecture that has been known to perform well on road sign recognition <cit.>.Our goal is to guarantee that our attack is effective across different training datasets and network architectures. Because we did not have access to German Stop signs for our experiments, we replace the German Stop signs in GTSRB with the entire set of U.S. Stop sign images in LISA. After training, our classifier achieves95.7% accuracy on the test set, which also had the German Stop signs replaced with U.S. Stop signs. This test set consists of all GTSRB test images except the German Stop sign images. In addition, we include our own images of 181 US Stop signs. None of the images in the test set are present in either the training set or the validation set of the network. We will release the set of 181 real-world US Stop sign images we took after publication. or the rest of the paper, we refer to this classifier as .§.§ Threat Model In contrast to prior work, we seek to physically modify an existing road sign in a way that causes a road sign classifier to output a misclassification while keeping those modifications inconspicuous to human observers. Here, we focus on evasion attacks where attackers can only modify the testing data and do not have access to the training data (as they would in poisoning attacks).We assume that attackers do not have digital access to the computer systems running the classifiers. If attackers have this superior level of access, then there is no need for adversarial perturbations—they can simply feed malicious input data directly into the model to mislead the system as they want, or they can compromise other control software, completely bypassing all classifiers. Following Kerckhoffs' principle <cit.>, it is often desirable to construct defenses that are robust in the presence of white-box attackers. As one of the broader goals of our work is to inform future defense research, we assume a strong attacker with white-box access, an attacker gains access to the classifier after it has been trained <cit.>. Therefore, although the attacker can only change existing physical road signs, they have full knowledge of the classifier and its architecture. Finally, due to the recent discovery of transferability <cit.>, black-box attacks can be carried out using perturbations computed with white-box access on a different model. As our goal is to inform future defenses, we will focus on white-box attacks in this paper.Specific to the domain of autonomous vehicles, future vehicles might not face this threat as they might not need to depend on road signs. There could be databases containing the location of each sign. Moreover, with complete autonomy, vehicles might be able to manage navigating complex traffic flow regions by means of vehicle-to-vehicle communication alone. However, these are not perfect solutions and some have yet to be developed. An autonomous vehicle might not be able to rely solely on a database of road sign locations. Such records might not always be kept up-to-date and there might be unexpected traffic events such as construction work and detours due to accidents that necessitate traffic regulation with temporary road signs. We demonstrate that physically modifying real-world objects to fool classifiers is a real threat, but we consider the rest of the control pipeline of autonomous vehicles to be outside the scope of our work.Attack Generation Pipeline. Based on our threat model, the attack pipeline proceeds as follows:* Obtain several clean image of the target road sign without any adversarial perturbation under different conditions, including various distances and angles.* Use those images after appropriate pre-processing as input to and generate adversarial examples.* Reproduce the resulting perturbation physically by printing out either the entire modified image in the case of poster-printing attacks or just the relevant modified regions in the case of sticker attacks.* Apply the physically reproduced perturbation to the targeted physical road signs. § ADVERSARIAL EXAMPLES FOR PHYSICAL OBJECTS Our goal is to examine whether it is possible to create robust physical perturbations for real-world objects that mislead classifiers to make incorrect predictions even when images are taken in a range of varying physical conditions. We first present an analysis of environmental conditions that physical learning systems might encounter, and then present our algorithm to generate physical adversarial perturbations taking these challenges into account. §.§ Physical World Challenges Physical attacks on an object must be able to survive changing conditions and remain effective at fooling the classifier. We structure our discussion of these conditions around the chosen example of road sign classification, which could be potentially applied in autonomous vehicles and other safety sensitive domains. A subset of these conditions can also be applied to other types of physical learning systems such as drones, and robots.Environmental Conditions. The distance and angle of a camera in an autonomous vehicle with respect to a road sign varies continuously. The resulting images that are fed into a classifier are taken at different distances and angles. Therefore, any perturbation that an attacker physically adds to a road sign must be able to survive these transformations of the image. Other environmental factors include changes in lighting/weather conditions, and the presence of debris on the camera or on the road sign. Spatial Constraints. Current algorithms focusing on digital images add adversarial perturbations to all parts of the image, including background imagery. However, for a physical road sign, the attacker cannot manipulate background imagery. Furthermore, the attacker cannot count on there being a fixed background imagery as it will change depending on the distance and angle of the viewing camera.Physical Limits on Imperceptibility. An attractive feature of current adversarial deep learning algorithms is that their perturbations to a digital image are often so small in magnitude that they are almost imperceptible to the casual observer. However, when transferring such minute perturbations to the real world, we must ensure that a camera is able to perceive the perturbations. Therefore, there are physical limits on how imperceptible perturbations can be, and is dependent on the sensing hardware.Fabrication Error. To fabricate the computed perturbation, all perturbation values must be valid colors that can be reproduced in the real world. Furthermore, even if a fabrication device, such as a printer, can produce certain colors, there will be some reproduction error <cit.>. In order to successfully physically attack deep learning classifiers, an attacker should account for the above categories of physical world variations that can reduce the effectiveness of perturbations. §.§ Robust Physical Perturbation We derive our algorithm starting with the optimization method that generates a perturbation for a single image x, without considering other physical conditions; then, we describe how to update the algorithm taking the physical challenges above into account. This single-image optimization problem searches for perturbation δ to be added to the input x, such that the perturbed instance x' = x + δ is misclassified by the target classifier f_θ (·):min H(x+δ, x), s.t. f_θ(x + δ) = y^*where H is a chosen distance function, and y^* is the target class.[For untargeted attacks, we can modify the objective function to maximize the distance between the model prediction and the true class. We focus on targeted attacks in the rest of the paper.] To solve the above constrained optimization problem efficiently, we reformulate it in the Lagrangian-relaxed form similar to prior work <cit.>.δargmin λ ||δ||_p + J(f_θ(x + δ), y^*) Here J(· , ·) is the loss function, which measures the difference between the model's prediction and the target label y^*. λ is a hyper-parameter that controls the regularization of the distortion. We specify the distance function H as ||δ||_p, denoting the ℓ_p norm of δ. Next, we will discuss how the objective function can be modified to account for the environmental conditions. We model the distribution of images containing object o under both physical and digital transformations X^V. We sample different instances x_i drawn from X^V. A physical perturbation can only be added to a specific object o within x_i.In the example of road sign classification, o is the stop sign that we target to manipulate. Given images taken in the physical world, we need to make sure that a single perturbation δ, which is added to o, can fool the classifier under different physical conditions. Concurrent work <cit.> only applies a set of transformation functions to synthetically sample such a distribution. However, modeling physical phenomena is complex and such synthetic transformations may miss physical effects. Therefore, to better capture the effects of changing physical conditions, we sample instance x_i from X^V by both generating experimental data that contains actual physical condition variability as well as synthetic transformations. For road sign physical conditions, this involves taking images of road signs under various conditions, such as changing distances, angles, and lightning. This approach aims to approximate physical world dynamics more closely. For synthetic variations, we randomly crop the object within the image, change the brightness, and add spatial transformations to simulate other possible conditions. To ensure that the perturbations are only applied to the surface area of the target object o (considering the spatial constraints and physical limits on imperceptibility), we introduce a mask. This mask serves to project the computed perturbations to a physical region on the surface of the object (road sign). In addition to providing spatial locality, the mask also helps generate perturbations that are visible but inconspicuous to human observers. To do this, an attacker can shape the mask to look like graffiti—commonplace vandalism on the street that most humans expect and ignore, therefore hiding the perturbations “in the human psyche.” Formally, the perturbation mask is a matrix M_x whose dimensions are the same as the size of input to the road sign classifier. M_x contains zeroes in regions where no perturbation is added, and ones in regions where the perturbation is added during optimization.In the course of our experiments, we empirically observed that the position of the mask has an impact on the effectiveness of an attack. We therefore hypothesize that objects have strong and weak physical features from a classification perspective, and we position masks to attack the weak areas. Specifically, we use the following pipeline to discover mask positions: (1) Compute perturbations using the L_1 regularization and with a mask that occupies the entire surface area of the sign. L_1 makes the optimizer favor a sparse perturbation vector, therefore concentrating the perturbations on regions that are most vulnerable. Visualizing the resulting perturbation provides guidance on mask placement. (2) Recompute perturbations using L_2 with a mask positioned on the vulnerable regions identified from the earlier step.To account for fabrication error, we add an additional term to our objective function that models printer color reproduction errors. This term is based upon the Non-Printability Score (NPS) by Sharif <cit.>. Given a set of printable colors (RGB triples) P and a set R(δ) of (unique) RGB triples used in the perturbation that need to be printed out in physical world, the non-printability score is given by:𝑁𝑃𝑆 = ∑_p̂∈ R(δ)∏_p' ∈ P |p̂ - p'| Based on the above discussion, our final robust spatially-constrained perturbation is thus optimized as: δargmin λ ||M_x·δ||_p + 𝑁𝑃𝑆+𝔼_x_i ∼ X^V J(f_θ(x_i + T_i( M_x·δ)), y^*)Here we use function T_i(·) to denote the alignment function that maps transformations on the object to transformations on the perturbation (if the object is rotated, the perturbation is rotated as well).Finally, an attacker will print out the optimization result on paper, cut out the perturbation (M_x), and put it onto the target object o. As our experiments demonstrate in the next section, this kind of perturbation fools the classifier in a variety of viewpoints.[For our attacks, we use the ADAM optimizer with the following parameters: β_1=0.9, β_2=0.999, ϵ=10^-8, η∈ [10^-4, 10^0]]§ EXPERIMENTS In this section, we will empirically evaluate the proposed . We first evaluate a safety sensitive example, Stop sign recognition, to demonstrate the robustness of the proposed physical perturbation. To demonstrate the generality of our approach, we then attack Inception-v3 to misclassify a microwave as a phone. §.§ Dataset and Classifiers We built two classifiers based on a standard crop-resize-then-classify pipeline for road sign classification as described in <cit.>.Our uses LISA, a U.S. traffic sign dataset containing 47 different road signs <cit.>. However, the dataset is not well-balanced, resulting is large disparities in representation for different signs. To alleviate this problem, we chose the 17 most common signs based on the number of training examples. 's architecture is defined in the Cleverhans library <cit.> and consists of three convolutional layers and an FC layer. It has an accuracy of 91% on the test set.Our second classifier is , that is trained on the German Traffic Sign Recognition Benchmark (GTSRB) <cit.>. We use a publicly available implementation <cit.> of a multi-scale CNN architecture that has been known to perform well on road sign recognition <cit.>. Because we did not have access to German Stop signs for our physical experiments, we replaced the German Stop signs in the training, validation, and test sets of GTSRB with the U.S. Stop sign images in LISA. achieves95.7% accuracy on the test set. When evaluating on our own 181 stop sign images, it achieves 99.4% accuracy.§.§ Experimental Design To the best of our knowledge, there is currently no standardized methodology of evaluating physical adversarial perturbations. Based on our discussion from Section <ref>, we focus on angles and distances because they are the most rapidly changing elements for our use case. A camera in a vehicle approaching a sign will take a series of images at regular intervals. These images will be taken at different angles and distances, therefore changing the amount of detail present in any given image. Any successful physical perturbation must cause targeted misclassification in a range of distances and angles because a vehicle will likely perform voting on a set of frames (images) from a video before issuing a controller action. Our current experiments do not explicitly control ambient light, and as is evident from experimental data (Section <ref>), lighting varied from indoor lighting to outdoor lighting.Drawing on standard practice in the physical sciences, our experimental design encapsulates the above physical factors into a two-stage evaluation consisting of controlled lab tests and field tests.Stationary (Lab) Tests. This involves classifying images of objects from stationary, fixed positions. * Obtain a set of clean images C and a set of adversarially perturbed images ({c}, ∀ c∈ C)at varying distances d ∈ D, and varying angles g ∈ G. We use c^d,g here to denote the image taken from distance d and angle g. The camera's vertical elevation should be kept approximately constant.Changes in the camera angle relative the the sign will normally occur when the car is turning, changing lanes, or following a curved road. * Compute the attack success rate of the physical perturbation using the following formula:1.5∑_c ∈ C1_{f_θ(c^d,g) = y^*∧ f_θ(c^d,g) = y}/∑_c ∈ C1_{f_θ(c^d,g) = y} where d and g denote the camera distance and angle for the image, y is the ground truth, and y^* is the targeted attacking class.[For untargeted adversarial perturbations, change f_θ(e^d,g) = y^* to f_θ(e^d,g)y.]Note that an image c that causes misclassification is considered as a successful attack only if the original image c with the same camera distance and angle is correctly classified, which ensures that the misclassification is caused by the added perturbation instead of other factors.Drive-By (Field) Tests. We place a camera on a moving platform, and obtain data at realistic driving speeds. For our experiments, we use a smartphone camera mounted on a car.* Begin recording video at approximately 250 ft away from the sign. Our driving track was straight without curves. Drive toward the sign at normal driving speeds and stop recording once the vehicle passes the sign. In our experiments, our speed varied between 0 mph and 20 mph. This simulates a human driver approaching a sign in a large city. * Perform video recording as above for a “clean” sign and for a sign with perturbations applied, and then apply similar formula as Eq. <ref> to calculate the attack success rate, where C here represents the sampled frames. An autonomous vehicle will likely not run classification on every frame due to performance constraints, but rather, would classify every j-th frame, and then perform simple majority voting. Hence, an open question is to determine whether the choice of frame (j) affects attack accuracy. In our experiments, we use j = 10. We also tried j = 15 and did not observe any significant change in the attack success rates. If both types of tests produce high success rates, the attack is likely to be successful in commonly experienced physical conditions for cars.§.§ Results forWe evaluate the effectiveness of our algorithm by generating three types of adversarial examples on (91% accuracy on test-set). For all types, we observe high attack success rates with high confidence. Table <ref> summarizes a sampling of stationary attack images. In all testing conditions, our baseline of unperturbed road signs achieves a 100% classification rate into the true class.Object-Constrained Poster-Printing Attacks. This involves reproducing the attack of Kurakin <cit.>. The crucial difference is that in our attack, the perturbations are confined to the surface area of the sign excluding the background, and are robust against large angle and distance variations. The Stop sign is misclassified into the attack's target class of Speed Limit 45 in of the images taken according to our evaluation methodology. The average confidence of predicting the manipulated sign as the target class is 80.51% (second column of Table <ref>).§.§ Poster-Printing Attacks We first show that an attacker can overlay a true-sized poster-printed perturbed road sign over a real-world sign and achieve misclassification into a target class of her choosing. The attack has the following steps: * The attacker obtains a series of high resolution images of the sign under varying angles, distances, and lighting conditions. We use 34 such images in our experiments. None of these images were present in the datasets used to train and evaluate the baseline classifier.* The attacker then crops, rescales, and feeds the images into and uses equation (<ref>) as the objective function. She takes the generated perturbation, scales it up to the dimensions of the sign being attacked, and digitally applies it to an image of the sign.* The attacker then prints the sign (with the applied perturbation) on poster paper such that the resulting print's physical dimensions match that of a physical sign. In our attacks, we printed 30”× 30” Stop signs and 18”× 18” Right Turn signs.* The attacker cuts the printed sign to the shape of the physical sign (octagon or diamond), and overlays it on top of the original physical sign.We use our methodology from Section <ref> to evaluate the effectiveness of such an attack. In order to control for the performance of the classifier on clean input, we also take images of a real-size printout of a non-perturbed image of the sign for each experiment. We observe that all such baseline images lead to correct classification in all experiments.For the Stop sign, we choose a mask that exactly covers the area of the original sign in order to avoid background distraction. This choice results in a perturbation that is similar to that in existing work <cit.> and we hypothesize that it is imperceptible to the casual observer (see the second column of Table <ref> for an example). In contrast to some findings in prior work, this attack is very effective in the physical world. The Stop sign is misclassified into the attack's target class of Speed Limit 45 in of the images taken according to our evaluation methodology. The average confidence of the target class is 80.51% with a standard deviation of 10.67%. For the Right Turn warning sign, we choose a mask that covers only the arrow since we intend to generate perturbations. In order to achieve this goal, we increase the regularization parameter λ in equation (<ref>) to demonstrate small magnitude perturbations. Table <ref> summarizes our attack results—we achieve a targeted-attack success rate (Table <ref>). Out of 15 distance/angle configurations, four instances were not classified into the target. However, they were still misclassified into other classes that were not the true label (Yield, Added Lane). Three of these four instances were an Added Lane sign—a different type of warning. We hypothesize that given the similar appearance of warning signs, small perturbations are sufficient to confuse the classifier. Sticker Attacks. Next, we demonstrate how effective it is to generate physical perturbations in the form of stickers, by constraining the modifications to a region resembling graffiti or art. The fourth and fifth columns of Table <ref> show a sample of images, and Table <ref> (columns 4 and 6) shows detailed success rates with confidences. In the stationary setting, we achieve a targeted-attack success rate for the graffiti sticker attack and a targeted-attack success rate for the sticker art attack. Some region mismatches may lead to the lower performance of the LOVE-HATE graffiti.The steps for this type of attack are: * The attacker generates the perturbations digitally by using just as in Section <ref>.* The attacker prints out the Stop sign in its original size on a poster printer and cuts out the regions that the perturbations occupy. * The attacker applies the cutouts to the sign by using the remainder of the printed sign as a stencil.Drive-By Testing. Per our evaluation methodology, we conduct drive-by testing for the perturbation of a Stop sign. In our baseline test we record two consecutive videos of a clean Stop sign from a moving vehicle, perform frame grabs at k = 10, and crop the sign. We observe that the Stop sign is correctly classified in all frames. We similarly test subtle and abstract art perturbations for using k = 10. Our attack achieves a targeted-attack success rate of for the poster attack, and a targeted-attack success rate of for the abstract art attack. See Table <ref> for sample frames from the drive-by video. §.§ Results for To show the versatility of our attack algorithms, we create and test attacks for (95.7% accuracy on test-set). Based on our high success rates with the -art attacks, we create similar abstract art sticker perturbations. The last column of Table <ref> shows a subset of experimental images. Table <ref> summarizes our attack results—our attack fools the classifier into believing that a Stop sign is a Speed Limit 80 sign in of the stationary testing conditions. Per our evaluation methodology, we also conduct a drive-by test (k = 10, two consecutive video recordings). The attack fools the classifier of the time.§.§ Results for Inception-v3To demonstrate generality of , we computed physical perturbations for the standard Inception-v3 classifier <cit.> using two different objects, a microwave and a coffee mug. We chose a sticker attack since poster printing an entirely new surface for the objects may raise suspicions. Note that for both attacks, we have reduced the range of distances used due to the smaller size of the cup and microwave compared to a road sign (Coffee Mug height- 11.2cm, Microwave height- 24cm, Right Turn sign height- 45cm, Stop Sign- 76cm). Table <ref> summarizes our attack results on the microwave and Table <ref> summarizes our attack results on the coffee mug. For the microwave, the targeted attack success rate is 90%. For the coffee mug, the targeted attack success rate is 71.4% and the untargeted success rate is 100%. Example images of the adversarials stickers for the microwave and cup can be seen in Tables <ref> and <ref>. § DISCUSSIONBlack-Box Attacks. Given access to the target classifier's network architecture and model weights, can generate a variety of robust physical perturbations that fool the classifier. Through studying a white-box attack like , we can analyze the requirements for a successful attack using the strongest attacker model and better inform future defenses. Evaluating in a black-box setting is an open question. Image Cropping and Attacking Detectors. When evaluating , we manually controlled the cropping of each image every time before classification. This was done so the adversarial images would match the clean sign images provided to . Later, we evaluated the camouflage art attack using a pseudo-random crop with the guarantee that at least most of the sign was in the image. Against LISA-CNN, we observed an average targeted attack rate of 70% and untargetedattack rate of 90%. Against GTSRB-CNN, we observed an average targeted attack rate of 60% and untargeted attack rate of 100%. We include the untargeted attack success rates because causing the classifier to not output the correct traffic sign label is still a safety risk. Although image cropping has some effect on the targeted attack success rate, our recent work shows that an improved version of can successfully attack object detectors, where cropping is not needed <cit.>.§ CONCLUSIONWe introduced an algorithm () that generates robust, physically realizable adversarial perturbations. Using , and a two-stage experimental design consisting of lab and drive-by tests, we contribute to understanding the space of physical adversarial examples when the objects themselves are physically perturbed. We target road-sign classification because of its importance in safety, and the naturally noisy environment of road signs. Our work shows that it is possible to generate physical adversarial examples robust to widely varying distances/angles. This implies that future defenses should not rely on physical sources of noise as protection against physical adversarial examples.Acknowledgements. We thank the reviewers for their insightful feedback. This work was supported in part by NSF grants 1422211, 1616575, 1646392, 1740897, 1565252, Berkeley Deep Drive, the Center for Long-Term Cybersecurity, FORCES (which receives support from the NSF), the Hewlett Foundation, the MacArthur Foundation, a UM-SJTU grant, and the UW Tech Policy Lab. ieee | http://arxiv.org/abs/1707.08945v5 | {
"authors": [
"Kevin Eykholt",
"Ivan Evtimov",
"Earlence Fernandes",
"Bo Li",
"Amir Rahmati",
"Chaowei Xiao",
"Atul Prakash",
"Tadayoshi Kohno",
"Dawn Song"
],
"categories": [
"cs.CR",
"cs.LG"
],
"primary_category": "cs.CR",
"published": "20170727173722",
"title": "Robust Physical-World Attacks on Deep Learning Models"
} |
Degue et al.: Differentially Private Kalman Filtering and LQG ControlA Two-Stage Architecture for Differentially Private Kalman Filtering and LQG Control Kwassi H. Degue and Jerome Le Ny, Senior Member, IEEE A preliminary version of this paper appeared in <cit.>.This work was supported by NSERC under Grant RGPIN-5287-2018 and RGPAS-2018-522686,by the Pierre Arbour Foundation doctoral scholarship and by an FRQNT doctoral scholarship.The authors are with the Department of Electrical Engineering, Polytechnique Montreal and GERAD, Montreal, QC H3T-1J4, Canada{kwassi-holali.degue, jerome.le-ny}@polymtl.ca. December 30, 2023 ========================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== Large-scale monitoring and control systems enabling a more intelligent infrastructureincreasingly rely on sensitive data obtained from private agents, e.g., locationtraces collected from the users of an intelligent transportation system.In order to encourage the participation of these agents, it becomes then criticalto design algorithms that process information in a privacy-preserving way.This article revisits the Kalman filtering and Linear Quadratic Gaussian (LQG)control problems, subject to privacy constraints. We aim to enforce differential privacy, a formal, state-of-the-artdefinition of privacy ensuring that the output of an algorithm is not toosensitive to the data collected from any single participating agent. A two-stage architecture is proposed that first aggregates and combines the individualagent signals before adding privacy-preserving noise and post-filtering the result tobe published. We show a significant performance improvement offered by this architectureover input perturbation schemes as the number of input signals increases and thatan optimal static aggregation stage can be computed by solving a semidefinite program. The two-stage architecture, which we develop first for Kalman filtering, is thenadapted to the LQG control problem by leveraging the separation principle. Numerical simulations illustrate the performance improvements over differentiallyprivate algorithms without first-stage signal aggregation.Differential privacy; Kalman filtering; Estimation; Filtering;LQG control; Optimal control§ INTRODUCTION To monitor and control intelligent infrastructure systems such as smart grids, smart buildings or smart cities, data needs to be continuously collectedfrom the people interacting with these systems, either through sensors installedin the environment such as cameras and smart meters, or through personal devicessuch as smartphones.Hence, in contrast to more traditional control systems, the measured signals for such systems often contain highly privacy-sensitive information, e.g., relatedto the real-time location or health of a person. For example, the accuracy of crowd-sourced traffic maps and congestion-aware routingapplications is increased by using data provided by smartphones and connected vehicles <cit.>. However, individual location data turns out to be very difficult to properlyanonymize because individuals have highly unique mobility patterns<cit.>,and in fact individual trajectories can be reconstructed even fromjust aggregate location data <cit.>. Similarly, fine-grained measurements of a house's electric power consumptioncollected by a smart meter can enable demand-response schemes, but canalso be used to infer the activities of the occupants,by identifying the usage of individual appliances<cit.>. Therefore, it is necessary to implement privacy-preserving mechanismswhen sensitive data must be shared to improve a system's performance. Various definitions of privacy have been proposed that are amenable to formal analysis. While a survey of such definitions is out of the scope of this paper, we can mentionsome recent work focusing on signal processing and control problems. Privacy is measured by a lower bound on the mutual information between published and private signals in <cit.>,on the Fisher information in <cit.>, or on the error covarianceof the estimator of a sensitive signalin <cit.>. The concept of k-anonymity and its extensions has been applied to the publicationof location traces in <cit.>. But much of the recent research on privacy-preserving data analysis relies on the notion of differential privacy <cit.>.In the standard set-up, which is also the situation considered in this article, a data holder aims to release the results of computations based on private data. Differential privacy is enforced by adding an appropriate amount of noise to the published results,in such a way that the probability distribution over the outputs does not depend toomuch on the data of any single individual. As a result, the ability of a third party observing the outputs to makenew inferences about a given person is roughly the same, whether or not that person chooses to contribute its data. This guarantee can then be used toweigh the risks of information disclosure against the benefits of publishingmore accurate analyses. A large number of techniques have been developed to compute differentially private versions of various statistics, see <cit.> for an overview.Nevertheless, the differentially private analysis of streaming data remains relatively less explored <cit.>, despite itsimportance for signal processing and control applications.Some previous work has focused on the design of differentially private dynamic estimators <cit.>,controllers <cit.>,consensus algorithms <cit.>, or anomaly detectors<cit.>. In particular, <cit.> discusses the Kalman filtering problem undera differential privacy constraint and compares schemes introducing noise eitherdirectly on the measured signals (input perturbation mechanisms) or on thepublished estimate (output perturbation mechanisms). Output perturbation provides better performance as the number of input signals increases, but has the drawback of leaving unfiltered noise on the output, which motivates thetwo-stage architecture that we consider here. In <cit.>, the authors consider a multi-agent linear quadratic trackingproblem where the trajectory tracked by each agent should remain private, while <cit.> considers an LQG control problem where each agent wishesto keep its individual state private. In both cases, noise is added directly on the individual measurements, a form of input perturbation. In this paper we study the design of Kalman filters and LQG controllers subjectto a differential privacy constraint on the measured signals.These problems, stated formally in Section <ref>, arise when a data collector measures private signals originating from a population of agents, whosedynamics can be modeled as linear Gaussian systems, in order to publish in real-timeeither an estimate of an aggregate state of the agent population, or a control signalshared by the agents and aimed at regulating such an aggregate state. As a motivating example, one can consider the problem of controlling the distribution of vehicles on a road network by means of traffic messages broadcasted to all cars, with the current density estimated from location data obtained from the smartphones ofindividual drivers.Section <ref> presents the main contribution of this paper, namely, a two-stage architecture for differentially private Kalman filtering,where the privacy-preserving noise is added only after an input stage appropriatelycombining the measured signals of the individual agents, while an output stage filtersout this noise. Such two-stage architectures were discussed in <cit.> but have not yet been applied to the Kalman filtering problem, and we argue first inSection <ref> via a simple example that significant performanceimprovements can be expected compared to input perturbation mechanisms.We show that the optimal input stage can be computed by solving a semidefinite program (SDP), hence, a tractable convex optimization problem.The fact that the input stage design problem admits an SDP formulationis reminiscent of other Kalman filtering problems subject to resourceor communication constraints, see, e.g.,<cit.>, but the SDP capturing specifically the differential privacy constraint is new. The design procedure is then adapted in Section <ref>to the LQG control problem. By exploiting the classical properties of the optimal LQG controller (linearity and separation principle), we can view the control problemas the problem of estimating a certain linear combination of the agent states asin Section <ref>, but for a specific cost on the estimation error. A conference version of this paper appeared in <cit.>, but contained no proof and did not discuss the LQG control problem.The numerical examples discussed in Sections <ref> and <ref> are also new. Finally, we introduce some notation used throughout this paper.We fix a generic probability triple (Ω,ℱ,ℙ), where ℱis a σ-algebra on Ω and ℙ a probability measure defined on ℱ.The notation X ∼𝒩(μ,Σ) means that X is a Gaussianrandom vector with mean vector μ and covariance matrix Σ. “Independent and identically distributed” is abbreviated iid. We denote the p-norm of a vector x∈ℝ^k by|x| _p:=(∑_i=1^k|x_i|^p)^1/p, for p ∈ [1,∞).For a matrix A, the induced 2-norm (maximum singular value of A) is denotedA_2 and the Frobenius norm A_F := √(Tr(A^TA)). If A and B are symmetric matrices, A ≽ B (resp. A ≻ B) means thatA-B is positive semi-definite (resp. positive definite). We use the notation diag(A_1,…,A_n) to represent a block-diagonal matrix with the matrices A_i on the diagonal. The column vector of size n with all components equal to 1 is denoted 1_n. Finally, for a discrete-time signal x we denote x_0:t := {x_0,…,x_t}.§ PROBLEM STATEMENT §.§ Privacy-Preserving State Estimation and LQG Control for a Population of Dynamic AgentsConsider a set of n privacy-sensitive signals {y_i,t}_0 ≤ t ≤ T,i=1,…,n, with y_i,t∈ℝ^p_i, collected by a data aggregator,and which could originate from n distinct agents. Let p = ∑_i=1^n p_i. We assume that a mathematical model capturing known dynamic and statistical properties of these signals is publicly available, consisting of a linear system with n independent(vector-valued) states associated to the n measured signalsx_i,t+1 = A_i,tx_i,t +B_i,tu_t+ w_i,t,0 ≤ t ≤ T-1, y_i,t = C_i,tx_i,t + v_i,t,0 ≤ t ≤ T,for i=1,…,n, where x_i,t, w_i,t∈ℝ^m_i. Here w_i,t∼𝒩(0,W_i,t) and v_i,t∼𝒩(0,V_i) areindependent sequences of iid zero-mean Gaussian random vectors with covariance matricesW_i,t≻ 0, V_i≻ 0, for i=1, …, n.In particular, assuming that the matrices W_i,t are invertible is necessary in thefollowing to be able to use the “information filter” form of the Kalman filterequations <cit.>. The sequence u with u_t∈ℝ^h represents a control input that is sharedby the n individuals. This is motivated by scenarios in which a common signal is broadcast to drive the aggregate state of a population, while individual signals can still be subject to privacy constraints.The initial conditions x_i,0 are independent Gaussian random vectors that are alsoindependent of the noise processes w and v, with mean x_i,0and covariance matrices Σ_i,0^-≻ 0.Let x_t := [x_1,t^T,…,x_n,t^T]^T, y_t := [y_1,t^T,…,y_n,t^T]^T,w_t = [w_1,t^T,…,w_n,t^T]^T andv_t = [v_1,t^T,…,v_n,t^T]^T denotethe global state, measurement and noise signals of (<ref>).Define A_t := (A_1,t,…,A_n,t), B_t = [B_1,t^T,…,B_n,t^T]^T, C_t = (C_1,t,…,C_n,t), W_t = (W_1,t,…,W_n,t), V = (V_1,…,V_n). Then the system (<ref>) can be rewritten more compactly asx_t+1= A_t x_t+ B_t u_t + w_t, 0 ≤ t ≤ T-1, y_t= C_t x_t+v_t,0 ≤ t ≤ T,with w_t ∼𝒩(0,W_t) and v_t ∼𝒩(0,V). Throughout the paper, the model parameters x_i,0, Σ^-_i,0, A_i,t, B_i,t, C_i,t,W_i,t, V_i, are assumed to be publicly known information.In Section <ref>, we first consider a filtering problem (with u a known signal)where the data aggregator aims to publish at each period t a causal estimate ẑ_t ofa linear combination z_t= L_t x_t = ∑_i=1^nL_i,tx_i,t of the individual states,computed from the signals y_i, with L_t := [ L_i,t, …, L_n,t ]some given (publicly known) matrices.This estimator should minimize the Mean Square Error (MSE) performance measureE_T := 1/T+1∑_t=0^T 𝔼[z_t - ẑ_t_2^2].For privacy reasons, the signals y_i are not released by the data aggregator and moreoverthe publicly released estimate ẑ should also guarantee the differential privacy of the inputsignals y_i, as defined precisely in Section <ref>. For example, the signals y_i could represent position measurements of n individuals,each state x_i could consist of the position and velocity of individual i,and the goal might be to publish only a real-time estimate of the average velocityof all individuals. Note that in the absence of privacy constraint,the optimal estimator is ẑ_t=∑_i=1^nL_i,tx̂_i,t, with x̂_i,tprovided by the (time-varying) Kalman filter estimating the state x_i of subsystem i from thesignal y_i <cit.>, and in particular the estimation problem then decouples for then subsystems.Next, in Section <ref> we build on the results obtained for the filtering problem to study the following privacy-constrained Linear Quadratic Gaussian (LQG)regulation problem. The data aggregator uses the measured signals y_i, 1 ≤ i ≤ n, to compute and broadcast a (causal) control signal u that minimizesthe following quadratic cost for the n agentsJ_T= 1/T+1𝔼[ ∑_t=0^T-1(x_t^T Q_t x_t+u_t^T R_t u_t) + x_T^T Q_T x_T],where Q_t ≽ 0 for 0 ≤ t ≤ T and R_t ≻ 0 for 0 ≤ t ≤ T-1 are publicly known weight matrices. Again, only the signal u is published (in particular, it is available to the n agents),and releasing u must guarantee the differential privacy of the measured signals y_i. It is worth noting that the cost function (<ref>) can be used to drive an aggregate value of the global population state toward 0 rather than the individual agent states, since trying to do the latter might be in direct conflict with the privacy requirement (which, essentially, aims to hide the value of the individual signals y_i, and hence indirectly of the individual states x_i). For example, we can havex_t Q_t x_t =( 1/n∑_i=1^n x_i,t)^2 if Q_t = 1/n^21_n 1_n^T, in order to regulate the average population state. Figure <ref> represents the estimation and control problem setups, including a basic scheme to enforce differential privacy by injecting noise directly in the signals {y_i}_1 ≤ i ≤ n, as described in Section <ref>.Note that the steady-state versions of both the filtering and LQG control problems are also considered, with themodel (<ref>)-(<ref>) in this case assumed time-invariant and the performancemeasures defined asE_∞ := lim_T →∞ E_T,J_∞ := lim_T →∞ J_T.To finish stating the above problems formally, we define in the next section the differential privacyconstraint imposed on the signal ẑ for the filtering problem and u for the LQG problem. §.§ Differential Privacy ConstraintDifferential privacy <cit.> is a property satisfied by certain randomized algorithms (also calledmechanisms), which in an abstract setting compute outputs in a space 𝖱 based on sensitivedata in a space 𝖣. To be differentially private, the algorithm must ensure that the probability distribution of its randomized output is not very sensitive to certain variations in the input data, which are specified as part of the privacy requirement.More concretely, we equip the input space 𝖣 with a symmetric binary relation called adjacency and denoted Adj, which captures the variations in the input datasets that we want to make hard to detect by observing the outputs. In our case, the input space 𝖣 is the vector space ℝ^p(T+1) of global measurementsignals y, and a mechanism is a causal stochastic system producing an output signal (ẑ or uin the previous section) based on its input y.We define two measured signals to be adjacent if the following condition holdsAdj(y,y') iff for some1 ≤ i ≤ n,y_i-y_i' _2≤ρ_i,and y_j=y_j' for all j≠ i,with {ρ_i}_i=1^n∈ℝ_+^n a given set of positive numbers and with the definition of the ℓ_2-norm v_2 := (∑_t=0^T |v_t|_2^2)^1/2 for a vector-valued signal v. In other words, two adjacent measurement signals can differ by the values ofa single participant, with only ℓ_2-bounded signal deviations allowed for each individual. For any two inputs that satisfy the adjacency relation, the following definition characterizesthe deviation that is allowed for a differentially private mechanism's output distribution. Let 𝖣 be a space equipped with a symmetric binary relation denoted Adjand let (𝖱,) be a measurable space, where is a given σ-algebra over 𝖱.Let ϵ≥ 0, 1 ≥δ≥ 0. A randomized mechanism M from𝖣 to 𝖱 is (ϵ,δ)-differentially private (for Adj) iffor all d,d'∈𝖣 such that Adj(d,d'), ℙ(M(d)∈ S) ≤ e^ϵ ℙ(M(d')∈ S) + δ, ∀ S ∈ℛ.In Definition <ref>, smaller values of ϵ and δ correspondto stronger privacy guarantees, i.e., distributions for M(d) and M(d') that are closer in (<ref>). Next, we need tools that can be used to enforce the property ofDefinition <ref>. The following definition is useful for mechanismssuch as ours that produce outputs in vector spaces. Let 𝖣 be a space equipped with an adjacency relation Adj. Let 𝖱 be a vector space equipped with a norm ·_𝖱.The sensitivity of a mapping q:𝖣↦𝖱 is defined asq := sup_{d,d':Adj(d,d')} q(d)-q(d') _𝖱.For 𝖱=ℝ^h(T+1) or (ℝ^h)^ℕ equipped withthe ℓ_2-norm, this defines the ℓ_2-sensitivity of q, denoted _2q.The Gaussian mechanism <cit.> consists in adding Gaussian noiseproportional to the ℓ_2-sensitivity of a mapping to enforce (ϵ,δ)-differential privacy.A fairly tight upper bound on the proportionality constant is provided in <cit.>. Recall first the definition of the 𝒬-function 𝒬(x):=1/√(2π)∫_x^∞exp(-u^2/2)du, which is monotonicallydecreasing from (-∞,∞) to (0,1). Then, for 1 > δ > 0, define κ_δ,ϵ := 1/2ϵ( Q^-1(δ)+√((Q^-1(δ))^2+2ϵ)). The following theorem can be found in <cit.>. Let ϵ > 0, 1 > δ > 0. Let G be a dynamic system with p inputs and q outputs.Then the mechanism M(y) = Gy + ν, where ν is a white Gaussian noise (sequence of iid zero-meanGaussian vectors) with ν_t ∼𝒩(0,κ_δ,ϵ^2 (Δ_2 G)^2 I_q), is (ϵ,δ)-differentially private. In other words, Theorem <ref> says that we can produce a differentially private signal by adding white Gaussian noise at the output of a system G processing the sensitive signal y, with covariance matrix σ^2 I_q, and σ proportional to the ℓ_2-sensitivity of G.§.§ A First Solution: Input Perturbation MechanismAn important property of differential privacy is its resilience topost-processing <cit.>, i.e., applying further computations to an output that isdifferentially private does not degrade the differential privacy guarantee, as long as the original dataset is not reaccessed for these computations.This leads immediately to a first solution for the estimation and control problems stated in the previoussection, called the input perturbation mechanism, which consists in perturbing each measured signal y_i directly to release differentially private versions of these signals. First, note that the memoryless system defined by (G y)_t = M y_t, where M is the diagonal matrix M = diag (1/ρ_1, …, 1/ρ_n), has the sensibility boundΔ G := sup_Adj(y,y')My - My'_2 ≤ 1for the adjacency relation (<ref>). Hence, by Theorem <ref>,releasing the signals {y_i/ρ_i + ν_i}_1 ≤ i ≤ n, where each signal ν_i isa white Gaussian noise with covariance matrix κ_δ,ϵ^2 I_p_i,is (ϵ,δ)-differentially private. Equivalently, using the resilience topost-processing to multiply this output by M^-1, we see that releasing the signals{ỹ_i := y_i + ρ_i ν_i}_1 ≤ i ≤ n is (ϵ,δ)-differentially private.Once these signals are released, applying further processing on them does not impact thedifferential privacy guarantee. Moreover, these signals are of the same form as the outputs of system (<ref>), except for a higher level of (still Gaussian) noise due to the additionof the artificial privacy-preserving noise. One can therefore produce an (ϵ,δ)-differentiallyprivate estimate ẑ or control signal u discussed in Section <ref>by applying standard Kalman filtering and LQG design techniques to the signals ỹ_i, see Figure <ref>. An advantage of input perturbation mechanisms is that each agent can release directly thedifferentially private signal ỹ_i, and hence does not need to trust the data aggregator to enforce the differential privacy property. Moreover, this mechanism has the potentially useful feature of publishing the individual signals ỹ_i, which could be used for other purposes than the original estimation or control problem. Nonetheless, as we discuss in the following sections, input perturbation typically leadsto a high level of noise and hence performance degradation, which motivates the searchfor better mechanisms.§ DIFFERENTIALLY PRIVATE KALMAN FILTERING Input perturbation for the differentially private Kalman filtering problem, as discussedin Section <ref> and represented on Figure <ref>,was considered in <cit.>.Here, we show first in Section <ref> via a simple example that the performance of this mechanism can be significantly improved by combiningthe individual input signals before adding the privacy preserving noise. This leads to the two-stage mechanism of Figure <ref>, whose systematic design is discussed in Section <ref>. §.§ A Scalar Example Consider a scalar homogeneous version of model (<ref>) withA_i,t=a ∈ℝ, B_i,t = b, C_i,t=c, W_i,t=σ_w^2, V_i=σ_v^2and L_i,t=1 (so z_t = ∑_i=1^n x_i,t),and assume ρ_i = ρfor all i in (<ref>).In other words, A_t = a I_n, B_t = b 1_n, C_t = c I_n,W_t = σ_w^2 I_n, V = σ_v^2 I_n in (<ref>).We also let T →∞ in the problem statement and consider the steady-state MSE E_∞, see (<ref>), as performance measure for a given estimator ẑ of z. Moreover in this sectionwe consider for simplicity the minimum mean square error (MMSE) estimateẑ_t of x_t given the measurements up to time t-1 only, since the correspondingMSE is directly obtained by solving an algebraic Riccati equation (ARE). Let α := κ_δ,ϵ ρ. Sincesup_y,y':Adj(y,y')‖ y-y'‖ _2 = ρ,by Theorem <ref> and as explained in Section <ref>, releasing the signal r_t=y_t+ν_t, with white noise ν such that ν_t∼𝒩(0,α^2I_n),is (ϵ,δ)-differentially private for the adjacency relation (<ref>).The steady-state MSE of a Kalman filter estimating z for the system with dynamics as in (<ref>) and measurements r is obtained by solving ascalar ARE, which leads to the following expression E_∞^1 = n/2c^2(-β+√(β^2+4(α^2 +σ_v^2)σ_w^2c^2)),where β = (1-a^2) (σ_v^2+α^2)-c^2 σ_w^2. Instead of input perturbation, we can use the architecture shown onFig. <ref> with D=1_n^T,a 1 × n row vector of ones. Consider the same adjacency relation (<ref>) and denote η_t = D y_t = ∑_i=1^ny_i,t, θ_t=∑_i=1^nw_i,tand λ_t=∑_i=1^nv_i,t. We havez_t+1= a z_t+θ_t, η_t= c z_t+λ_t.Since again sup_y,y':Adj(y,y')‖ Dy-Dy'‖ _2 = ρ, releasing the scalar signal s_t= η_t + ζ_t, with ζ_t∼𝒩(0,α^2),is (ϵ,δ)-differentially private for the adjacency relation (<ref>).The MSE of a Kalman filter estimating z from this signal s, with the dynamics of the model (<ref>), can again be obtained by solving an ARE, which leads to the following expressionE_∞^2 = n/2c^2(-β_(n)+√(β_(n)^2+4(α^2/n +σ_v^2) σ_w^2 c^2)),where β_(n) = (1-a^2) (σ_v^2+α^2/n)-c^2 σ_w^2.Comparing (<ref>) and (<ref>), we see that the only difference is the vanishing influence of the privacy preserving noise on E_∞^2 as n increases, with the term α^2/n replacing α^2 in E_∞^1. For example, if n=100, a=1, c=1, ρ=50, ϵ=ln(3), δ=0.05, σ_w^2=0.5,σ_v^2=0.9, we obtain E_∞^1 ≈ 6235 and E_∞^2 ≈ 650. It is indeed desirable that as the number of agents n increases, differential privacy becomes easier to enforce and the impact of the privacy requirement on achievable performance decreases, a feature that the architectureof Figure <ref> has the potential to achieve. The design of this architecture is discussedin the next section for the general filtering problem of Section <ref>. §.§ Design of the Two-Stage Mechanism Following Figure <ref>, we construct a differentially privateestimate ẑ_t of z_t by first multiplying the global signal y witha constant matrix D=[ D_1 D_2 … D_n ],with the matrices D_i ∈ℝ^q× p_i, 1 ≤ i ≤ n, to be designed and q to be determined. Then, we add white Gaussian noise ζ according to the Gaussian mechanism, in order to make the signal s differentially private, withs_t = D y_t + ζ_t =D C_t x_t + D v_t + ζ_t, 0 ≤ t ≤ T.Therefore, the role of the matrix D is to combine the individual signals appropriately before adding the privacy-preserving noise, in order to decrease the overall sensitivity (see Definition <ref>), while preserving enough information for z to be estimated with sufficient accuracy. Finally, we construct a causal MMSE estimator ẑof z from s, a task for which it is optimal to use a Kalman filter, since the systemmodel producing s with the state dynamics of (<ref>) is still linear and Gaussian.This Kalman filter produces a state estimate x̂ of x and then ẑ_t = L_t x̂_tfor all t.Given D, for measurement signals y and y' adjacent according to (<ref>)and differing at index i, we haveDy-Dy' _2 =D_iy_i-D_iy_i'_2≤ρ_i D_i_2,where D_i_2 denotes the maximum singular value of the matrix D_i, and there are adjacent signals y_i, y_i' achieving the bound. Hence, we can bound the sensitivity of the memoryless system y ↦ Dy as follows _2D :=sup_y,y':Adj(y,y')‖ Dy-Dy'‖ _2 =1 ≤ i ≤ nmax{ρ_i D_i_2 }. Therefore, from Theorem <ref>, for any matrix D, releasings_t=D y_t+ζ_t, with ζ_t∼𝒩(0,(κ_δ,ϵ _2D)^2I_q), is (ϵ,δ)-differentially private for the adjacency relation (<ref>).The estimate ẑ is then also (ϵ,δ)-differentially private, since it is obtained by post-processing s, without re-accessing the sensitive signal y.§.§.§Input Transformation Optimization We can now consider the problem of optimizing the choice of matrix D.Let x̂_t^- = 𝔼[x_t | s_0:t-1] and x̂_t = 𝔼[x_t | s_0:t] be thestate estimates produced by the Kalman filter of Figure <ref> after theprediction step and the measurement update step respectively <cit.>. Let Σ̅_t = 𝔼[(x_t - x̂_t^-) (x_t - x̂_t^-)^T | s_0:t-1] andΣ_t= 𝔼[(x_t - x̂_t) (x_t - x̂_t)^T | s_0:t] be the corresponding error covariance matrices.We also denote by Σ̅_0 = diag(Σ^-_1,0, …Σ^-_n,0)the covariance matrix for the initial state x_0.For completeness, we recall here the equations of the Kalman filter. Given the dynamics (<ref>) and the measurementequation (<ref>), with ζ a Gaussian noise with covariance matrix(κ_δ,ϵ _2D)^2 I_q, we have for t ≥ 0 and starting from x̂_0^- := x̅_0x̂_t= x̂_t^- + K^f_t (s_t - DC_t x̂_t^-), x̂_t+1^-= A_t x̂_t + B_t u_t, with K^f_t= Σ̅_t C_t^T D^T (D (C_t Σ̅_t C_t^T + V) D^T+ κ_δ,ϵ^2 Δ_2 D^2 I_q)^-1.The error covariance matrices evolve for t ≥ 0 asΣ_t^-1 = Σ̅_t^-1+C_t^T Π C_t, Σ̅_t+1 = A_tΣ_tA_t^T+W_t,whereΠ = D^T(D V D^T+(κ_δ,ϵ _2D)^2I_q)^-1D. With z = L_t x_t and its estimator ẑ_t = L_t x̂_t, we can rewrite the MSE E_T in (<ref>) asE_T = 1/T+1∑_t=0^TTr (L_tΣ_t L_t^T).As a result, a matrix D minimizing the MSE can be found by solving the following optimization problemmin_q ∈ℕ, D ∈ℝ^q × p 1/T+1∑_t=0^TTr( L_tΣ_t L_t^T ) s.t. Σ_0^-1 =Σ̅_0^-1 + C_0^T Π C_0, Σ_t+1^-1 = (A_tΣ_t A_t^T+ W_t)^-1 + C_t+1^T Π C_t+1, 0 ≤ t ≤ T-1,Π = D^T (D V D^T + κ_δ,ϵ^2 (Δ_2D)^2 I_q)^-1 D. In the minimization (<ref>), we have emphasized that finding the firstdimension q of the matrix D is part of the optimization problem. Note also that we can write the optimization problem above equivalently as a minimizationover the variables D, Π, and {Σ_t}_0 ≤ t ≤ T, but the variablesother than D can be immediately eliminated using the equalityconstraints (<ref>)-(<ref>).With our assumption V ≻ 0, we obtain an equivalent form for (<ref>) by using the matrix inversion lemma Π =V^-1 - V^-1( V^-1 + D^T D/(κ_δ,ϵΔ_2 D)^2)^-1 V^-1,or, alternatively,κ_δ,ϵ^2 [ ( V - V Π V )^-1 - V^-1] =D^T D/Δ_2 D^2.§.§.§ Semidefinite Programming-based SynthesisIn this section, we show that the optimization problem (<ref>)-(<ref>) can be recast as a semidefinite program (SDP) and hence solved efficiently <cit.>, if we impose the following additional constraints on DΔ_2 D = 1 = ρ_1 D_1_2 = … = ρ_n D_n_2.First, the following Lemma shows that in fact no loss of performance occursby adding the constraint (<ref>) to(<ref>)-(<ref>), i.e., that this constraintis satisfied automatically by some matrix D^* that is optimalfor (<ref>)-(<ref>). For any feasible solutionD of (<ref>)-(<ref>) that does not satisfy(<ref>), there exists a feasible solution that does satisfy this constraintand gives a lower or equal cost in (<ref>).In particular, adding the constraint (<ref>) to the problem(<ref>)-(<ref>) does not change the value of the minimum nor the existence of a minimizer.Consider a matrix D and a corresponding sequence {Σ_t}_0 ≤ t ≤ T defined by theiterations (<ref>)-(<ref>). First, rescaling D to λ D for any λ≠ 0 does not impact the constraint(<ref>) (note that Δ_2 (λ D) = λΔ_2 D), and so we can add the constraint Δ_2 D = 1 without changing the solution of the optimization problem(<ref>)-(<ref>).Next, if the other constraints of (<ref>) are not satisfied by D, constructthe p × p matrix M = D^T D + diag({η_i I_p_i}_1 ≤ i ≤ n), with η_i = (Δ_2 D / ρ_i)^2 - D_i^2_2. Since η_i ≥ 0 by (<ref>), M is positive semi-definite. The i^th diagonal block of M is M_ii = D_i^T D_i + [(Δ_2 D / ρ_i)^2 - D_i^2_2] I_p_i, which has maximum eigenvalue (Δ_2 D / ρ_i)^2. Define some matrix D̃ such that D̃^T D̃ = M and group the columns of D̃ asD̃ = [D̃_1 …D̃_n ] as for D, so that D̃_i consists of p_i columns. In particular M_ii = D̃_i^T D̃_i, so D̃_i^T D̃_i has maximum eigenvalue (Δ_2 D / ρ_i)^2 and hence D̃_i has maximum singular value (Δ_2 D / ρ_i). In other words, D̃ satisfies (<ref>) with a sensibility Δ_2 D = Δ_2 D̃ that is unchanged, and moreover D̃^T D̃ = M ≽ D^T D.Therefore, when we replace D by D̃, Δ_2 D in the denominator of (<ref>) remains unchanged, and moreover ( V^-1 + D̃^TD̃/(κ_δ,ϵΔ_2 D̃)^2)^-1 ≼( V^-1 + D^T D/(κ_δ,ϵΔ_2 D)^2)^-1hence Π̃≽Π, where Π̃ and Π are defined according to (<ref>) or equivalently (<ref>)for D̃ and D respectively. Let K := Π̃- Π≽ 0.Replacing Π by Π̃ in (<ref>), we obtain a matrix Σ̃_0 satisfying Σ̃_0^-1 = Σ_0^-1 + C_0^T K C_0≽Σ_0^-1, so Σ̃_0≼Σ_0. Now if we have two matrices Σ̃_t≼Σ_t, and we use these two matricestogether with Π̃ and Π to define Σ̃_t+1, Σ_t+1 according to (<ref>),then immediately Σ̃_t+1^-1 = (A_ tΣ̃_ t A_ t^T+ W_ t)^-1 + C_t+1^T Π̃C_t+1≽ (A_ tΣ_ t A_ t^T + W_ t)^-1 +C_t+1^T Π̃C_t+1= Σ_t+1^-1 + C_t+1^T K C_t+1≽Σ^-1_t+1.Therefore, Σ̃_t+1≼Σ_t+1.Hence, by induction, starting from D̃ we obtain a sequence {Σ̃_t}_t suchthat Σ̃_t≼Σ_t for all t ≥ 0. This gives a smaller or equal cost1/T+1∑_t=0^T+1Tr (L_tΣ̃_t L_t^T) ≤1/T+1∑_t=0^T+1Tr (L_tΣ_t L_t^T),and so the lemma is proved. By Lemma <ref>, we can add without loss of optimality the constraints(<ref>) to (<ref>)-(<ref>), which allows us in the following to recast the problem as an SDP.Let α_i = κ_δ,ϵρ_i, for all 1 ≤ i ≤ n. Denote E_i=[ 0 … I_p_i … 0 ]^Tthe p × p_i matrix whose elements are zero except for an identity matrix inits i^th block. The next lemma converts the constraints(<ref>)-(<ref>) to linearmatrix inequalities. If Π, D satisfy the constraints (<ref>)-(<ref>), then Π satisfies V - V Π V ≽ 0 together with the following constraints,for all 1 ≤ i ≤ n,[ I_p_i/α_i^2 + V_i^-1E_i^T;E_iV-V Π V ]≽0, [ I_p_i/α_i^2 + V_i^-1E_i^T;E_iV-V Π V ]≻0. Conversely, if Π satisfies these constraints, then there exists a matrix Dsuch that Π, D satisfy (<ref>)-(<ref>).One such D can be obtained by the factorization ofκ_δ,ϵ^2 [ ( V - V Π V )^-1 - V^-1] = D^T D(e.g., via singular value decomposition (SVD)) and will then satisfy Δ_2 D = 1. V - V Π V ≽ 0 is immediate from (<ref>), since it is equal to( V^-1 + D^T D/(κ_δ,ϵΔ_2 D)^2)^-1. Together with (<ref>), the right-hand sideof (<ref>) then represents any positivesemidefinite matrix M = D^T D such that its diagonal blocks M_ii = D^T_i D_i have maximum eigenvalue equal to 1/ρ_i^2, since D_i_2 = 1/ρ_i by (<ref>). These constraints are equivalent to saying that for all 1 ≤ i ≤ n,E_i^T[ ( V - V Π V )^-1 - V^-1] E_i≼ I_p_i/α_i^2, and not E_i^T[ ( V - V Π V )^-1 - V^-1] E_i≺ I_p_i/α_i^2. Indeed, this comes from the standard fact that the maximum value λ_i, maxof M_ii is the smallest λ satisfying M_ii≼λ I_p_i.The constraints given in the Lemma are obtained bynoting that E_i^T V^-1E_i = V_i^-1 and taking Schur complements in (<ref>) and (<ref>).Note that the fact that the left-hand side of (<ref>) ispositive semidefinite is a simple consequence of V ≽ V - V Π V, hence adding the constraint ( V - V Π V )^-1 - V^-1≽ 0 is unnecessary. Next, define the information matrices Ω_t = Σ_t^-1, for 0 ≤ t ≤ T.If the matrices W_t are invertible, denoting Ξ_t = W_t^-1 and using the matrix inversion lemma in (<ref>), one getsC_t+1^TΠ C_t+1 - Ω_t+1 + Ξ_t - Ξ_t A_t (Ω_t + A_t^TΞ_t A_t)^-1 A_t^TΞ_t = 0. Replacing the equality in (<ref>) by ≽ 0 and taking a Schur complement, together withthe inequalities of Lemma <ref>, leads to the following SDP with variablesΠ≽ 0, {X_t≽ 0,Ω_t≻ 0}_0 ≤ t ≤ Tmin_Π≽ 0, {X_t, Ω_t}_0 ≤ t ≤ T1/T+1∑_t=0^TTr (X_t) s.t.[ X_t L_t; L_t^T Ω_t ]≽ 0,0 ≤ t≤ T, Ω_0 = Σ̅_0^-1 + C_0^TΠ C_0, [ C_t+1^TΠ C_t+1 - Ω_t+1 + Ξ_tΞ_t A_t; A_t^TΞ_t Ω_t + A_t^TΞ_t A_t ]≽ 0,0 ≤ t ≤ T-1, [ I_p_i/α_i^2 + V_i^-1E_i^T;E_iV-V Π V ]≽0, 1 ≤ i≤ n. Here the minimization of the cost (<ref>) has been replacedby the minimization of (<ref>), after introducing theslack variable X_t satisfying (<ref>), or equivalentlyX_t ≽ L_t Ω_t^-1 L_t^T by taking a Schur complement. Since we replaced the equality in (<ref>) by an inequality, the SDP above is a relaxation of the original problem (<ref>)-(<ref>). The purpose of the next theorem is to show that this relaxation is tight. Once an optimal solution for this SDP is obtained, we recover an optimal matrix D from Π by the factorization (<ref>). Let Π^* ≽ 0,{ X^*_t≽ 0,Ω^*_t≻ 0}_0 ≤ t ≤ Tbe an optimal solution for (<ref>)-(<ref>). Suppose that for some 0 ≤ t ≤ T, we have L_t(Ω^*_t)^-1C_t^T ≠ 0. Let D^* be a matrix obtained from Π^* by the factorization (<ref>). Then D^* is an optimal solution for (<ref>)-(<ref>), which moreover satisfies D_i^*_2 = 1/ρ_i for 1 ≤ i ≤ n, with the decomposition (<ref>). The corresponding optimal covariance matrices {Σ_t^*}_0 ≤ t ≤ T for the Kalman filter can be computed using the equations (<ref>)-(<ref>). Finally, the optimal costs of (<ref>)-(<ref>) and(<ref>)-(<ref>) are equal, i.e., the SDP relaxation is tight. Even though the condition L_t(Ω^*_t)^-1C_t^T ≠ 0 introduced to guaranteethe possibility of constructing the matrix D in the proof is not an explicit condition expresseddirectly in term of the problem parameters, it appears to be a weak requirement in practice. Consider Π^*, {X_t^*,Ω_t^*}_0 ≤ t ≤ T an optimal solutionof the SDP (<ref>)-(<ref>).As explained in the proof of Lemma (<ref>), the constraint (<ref>) is equivalent toα_i^2 E_i^T[ ( V - V Π^* V )^-1 - V^-1] E_i≼ I_p_i.We show that we cannot have α_i^2 E_i^T[ ( V - V Π^* V )^-1 - V^-1] E_i ≺ I_p_i. Indeed, otherwise there exists η > 0 such that the matrix Π̃= Π^* + η I_p still satisfies (<ref>). Using this matrix Π̃ in (<ref>), we obtain a matrix Ω̃_0 = Ω_0^* + η C_0^T C_0 feasible for (<ref>). Now define Ω̃_1 = Ω_1^* + η C_1^T C_1. One can immediately check thatΠ̃, Ω̃_0 and Ω̃_1 satisfy (<ref>) for t = 0, using the fact that Ω_0^*, Ω_1^*, Π^* are feasible and that C_0^T C_0 ≽ 0. Similarly the matrices Ω̃_t = Ω_t^* + η C_t^T C_t are feasible in (<ref>) for all 0 ≤ t ≤ T. Now in (<ref>), taking a Schur complement, we obtain that the matrices X̃_t = L_t Ω̃_t^-1 L_t^T are feasible. By the matrix inversion lemma we can writeX̃_t = X^*_t - L_t (Ω^*_t)^-1 C_t^T K_t C_t (Ω^*_t)^-1 L_t^T.for some matrices K_t ≻ 0. These matrices X̃_t give a cost1/T+1∑_t=0^T Tr(X^*_t) - L_t (Ω^*_t)^-1 C_t^T K_t^1/2_F, which is a strict improvement over the assumed optimal solution as soon as one matrixL_t (Ω^*_t)^-1 C_t^T is not zero (since the K_t's are invertible). Hence, we have a contradiction and so we cannot haveα_i^2 E_i^T[ ( V - V Π^* V )^-1 - V^-1] E_i ≺ I_p_i.We can then apply Lemma <ref> and construct a matrix D^* from Π^*as in (<ref>), so that the pair Π^*, D^* satisfies(<ref>)-(<ref>).Let 𝒱^* be the optimum value of (<ref>)-(<ref>), and V^* that of (<ref>)-(<ref>). First, 𝒱^* ≤ V^* since the constraints of the original problem have been relaxed to obtain the SDP. We now show how to construct a sequence {Σ_t^* }_0 ≤ t ≤ T, which together with Π^* satisfy the constraints of (<ref>)-(<ref>) and achieve the cost 𝒱^*, thereby proving the remaining claims of the theorem. Note that since Ω^*_t + A_t^TΞ_t A_t≻ 0, (<ref>)is equivalent to ℛ_t(Ω^*_t, Ω^*_t+1) ≽ 0, whereℛ_t(Ω_t, Ω_t+1) := C_t+1^TΠ^* C_t+1 - Ω_t+1 + Ξ_t- Ξ_t A_t (Ω_t + A_t^TΞ_t A_t)^-1 A_t^TΞ_t.First, we take Σ_0^* = (Ω_0^*)^-1.If ℛ_t(Ω^*_t, Ω^*_t+1) = 0 for all 0 ≤ t ≤ T-1, then the matrices Ω^*_t satisfy (<ref>) and we can take Σ_t^* = (Ω_t^*)^-1 for all t, since these matrices satisfy the equivalent condition (<ref>). Otherwise, let t̃ be the first time index such thatℛ_t̃(Ω^*_t̃, Ω^*_t̃+1) is not zero. For t ≤t̃, we take Σ_t^* = (Ω_t^*)^-1 and so in particular we have ℛ_t̃((Σ_t̃^*)^-1, Ω^*_t̃+1) ≽ 0 and not zero. Consider the matrixΩ̃_t̃+1 = Ω^*_t̃+1 + ℛ_t̃(Ω^*_t̃, Ω^*_t̃+1),which then satisfies ℛ_t̃(Ω^*_t̃, Ω̃_t̃+1) = 0by definition. We set Σ_t̃+1^* = Ω̃_t̃+1^-1.Now note that we again haveℛ_t̃+1((Σ_t̃+1^*)^-1, Ω̃_t̃+2) ≽ 0, by verifying that (<ref>) is satisfied at t+1, using the fact that (Σ_t̃+1^*)^-1 = Ω̃_t̃+1≽Ω^*_t̃+1. From here, we can proceed by induction, assuming that Σ_0^*, …, Σ_t^*are set and takingΣ_t+1^* = (Ω̃_t+1)^-1 := (Ω_t+1^*+ℛ_t((Σ_t^*)^-1, Ω^*_t+1))^-1,which reduces to (Ω_t+1^*)^-1 if ℛ_t((Σ_t^*)^-1, Ω^*_t+1) = 0. The procedure above provides matrices Π^*, {Σ_t^* }_0 ≤ t ≤ Tsatisfying the constraints of the original program (<ref>)-(<ref>). By construction, we have (Σ_t^*)^-1≽Ω_t^* and the matrices (Σ_t^*)^-1also satisfy (<ref>). Therefore, replacing, for each 0 ≤ t ≤ T, Ω_t^* by(Σ^*_t)^-1 and X_t^* by L_t Σ_t L_t^T in the solutionof (<ref>)-(<ref>) that we started with gives a cost 𝒱≤𝒱^* for the SDP, hence 𝒱 = 𝒱^* by optimality of 𝒱^*. But this cost 𝒱 is also equal to the cost 1/T+1∑_t=0^TTr (L_tΣ_t^* L_t^T)of (<ref>). Hence, we have shown that (<ref>)-(<ref>) and (<ref>)-(<ref>) have the same minimum value, and constructed an optimal solution D^*, Π^*, {Σ_t^* }_0 ≤ t ≤ T to (<ref>)-(<ref>) achieving this value.§.§ Stationary problemIn the stationary case with T →∞ and the model(<ref>)-(<ref>) now assumed time-invariantand detectable, we wish to find a signal aggregation matrix D followedby a time-invariant Kalman filter to minimize the steady-state MSE E_∞. This can be done by solving the following SDP with variables Π≽ 0,X ≽ 0, Ω≻ 0 min_Π≽ 0, X, ΩTr (X) s.t.[ X L; L^T Ω ]≽ 0, [ C^TΠ C - Ω + ΞΞ A; A^TΞ Ω + A^TΞ A ]≽ 0, [ I_p_i/α_i^2+ V_i^-1 E_i^T; E_i V-V Π V ]≽0, 1 ≤ i≤ n.Compared to (<ref>)-(<ref>), this SDP is of much smaller size, due to the fact that the transient behavior is neglected in the performance measure. The proof of the following theorem is similar to that of Theorem <ref>. Let Π^* ≽ 0, X^* ≽ 0, Ω^* ≻ 0 be an optimal solutionfor (<ref>)-(<ref>). Suppose that we have L (Ω^*)^-1 C^T ≠ 0. Let D^* be a matrix obtained from Π^* by the factorization (<ref>). Then D^* minimizes the steady-state MSE E_∞ among all possible matrices Dintroduced as in Figure <ref>, and the corresponding value of E_∞ is equal to the optimal value of the SDP.Note that given the optimum matrix D^* and corresponding Π^*, an alternative way ofcomputing E_∞ is by solving an ARE to obtain the steady-state prediction errorcovariance matrix Σ̅_∞ for x̂^-, then compute the steady-state errorcovariance matrix Σ_∞ = (Σ̅_∞^-1 + C^T Π^* C)^-1 forthe estimator x̂, and finally E_∞ = (L Σ_∞ L^T).§.§ Syndromic Surveillance ExampleTo illustrate the differentially private filtering methodology, including issues related to the choice of model and adjacency relation (<ref>), we discuss in this section an example motivated by the analysis of epidemiological data.Consider a scenario in which Public Health Services (PHS) must publish for a population infected by a disease the number I_t of infectious people, i.e., those who have the diseaseand are able to infect others.PHS use privacy-sensitive data collected from n=12 hospitals,with each hospital i recording the number I_i,t of infectious people in its area, as well as the number R_i,t of recovered people, i.e, those who were infected by the disease and are now immune. For each area i, these numbers are assumed to followa discrete-time SEIR epidemiological model for the specific disease <cit.>, written here first without process noise, obtained by discretizing a classical continuous-time model <cit.> usinga forward Euler discretization <cit.>S_i,t+1=S_i,t-β_i S_i,tI_i,t/N_i, E_i,t+1=(1-τ_i)E_i,t+β_i S_i,tI_i,t/N_i,I_i,t+1=(1-ϑ_i)I_i,t+τ_i E_i,t, R_i,t+1=R_i,t+ϑ_i I_i,t,where S_i represents the number of susceptible people, i.e., those who are not infectedbut could become infected, the number of exposed people, i.e., those infected but not yetable to infect others, and N_i the total number of people. The parameters τ_i, β_i and ϑ_i represent the transition rates fromone disease stage to the next. For each hospital i's area, N_i in (<ref>) is assumed constant for the time interval of interest. Moreover, let us make an approximation that this period is short enough or the disease at an early-enough stage so that S_i,t can also be assumed approximately constant, equal to S_i,0. Then, the remaining statesη_i,t=[E_i,t, I_i,t, R_i,t]^T ∈ℝ^3 evolve as a linear system of the formη_i,t+1 = 𝒜_i η_i,t+ φ_i,t,0 ≤ t ≤ T-1,where 𝒜_i=[ 1-τ_i β_i S_i,0/N_i 0; τ_i 1-ϑ_i 0; 0 ϑ_i 1 ],and we introduced the white Gaussian process noise φ_i,t∼𝒩(0,Φ_i) in the model, with covariance matrices Φ_i≻ 0, for 1 ≤ i ≤ 12.Now, consider the problem of choosing the level ρ_i in the adjacency relation (<ref>) to provide a meaningful privacy guarantee to the patients. If we assume that the measurement for hospital i in our model isy_i,t = [I_i,t, R_i,t]^T, then once a person becomes sick, he or she iscounted at each period either in the signal I_i or, after recovery, in thesignal R_i. As a result, the impact of a single individual on the measurement y_icould be quite large, proportional to the time horizon T, requiring in turn a largevalue of ρ_i to provide strong privacy guarantees, and hence a high level of noise. A practical solution to this issue is to not continuously record the same individuals,but simply count at each time period t the number of newly infectious individuals, i.e., y_i,t^(1) = I_i,t-I_i,t-1, as well as the number of newly recovered individuals, y_i,t^(2) = R_i,t-R_i,t-1. Such a measurement model is much more beneficial from a privacy point of view. Indeed, assuming that a given individual can only become sick once, he or she can affect y_i,t^(1) by ± 1 for at most 2 periods t (when the person becomes sick and recovers), and y_i,t^(2) for at most one period by 1. As a result, one can take ρ_i = √(3) in (<ref>) to provide a strong privacy guarantee, i.e., insensitivity of the published output to the complete record ofa single individual. The measurement noise v_i,t in the model represents counting errors, e.g., due to people not being diagnosed by the hospital. To obtain a dynamic model compatible with the measurements y_i,t = [y_i,t^(1), y_i,t^(2)]^T, define for each hospital i the 4-dimensional statex_i,t = [I_i,t-1, R_i,t - R_i,t-1, E_i,t, I_i,t]^T. These states x_i,t evolve as (<ref>) withA_i=[ 0 0 0 1; 0 0 0 ϑ_i; 0 0 1-τ_i β_i S_i,0/N_i; 0 0 τ_i 1-ϑ_i ],B_i= [ 0; 0; 0; 0 ]and the noise w_i,t = [ϖ_i,t, φ_1,i,t, φ_2,i,t, φ_3,i,t]^T includes the components of φ_i introduced in (<ref>) as well as a small independent Gaussian noise ϖ_i with small variance σ^2,added so that the covariance matrices of W_i = diag(σ^2,Φ_i) are invertibleas required by our algorithms (ideally ϖ_i,t would be 0 since the first line of A_i corresponds to the delay in the model). The measurement matrices are immediatelyC_i=[ -1001;0100 ], 1 ≤ i ≤ 12,and one can then verify that the model is observable. Let us assume the following values for the parameters in (<ref>)τ_i=0.2,β_i S_i,0/N_i = 0.5,ϑ_i=0.1,for 1 ≤ i ≤ 3 τ_i=0.3,β_i S_i,0/N_i = 0.3,ϑ_i=0.5,for 4 ≤ i ≤ 6 τ_i=0.5,β_i S_i,0/N_i = 0.7,ϑ_i=0.15, for 7 ≤ i ≤ 9 τ_i=0.7,β_i S_i,0/N_i = 0.6,ϑ_i=0.3,for 10 ≤ i ≤ 12.Moreover, assume for all 1 ≤ i ≤ 12V_i=0.4 I_2, Φ_i = [ 0.3 -0.15 0; -0.15 0.3 -0.15; 0 -0.15 0.3 ].The goal ofthe surveillance system is to continuously release, at each period t, an estimate of the quantity z_t=∑_i=1^n( [ 0 0 0 1 ]× x_i,t).Let us set the privacy parameters to δ=0.02 and ϵ=ln(3) for example (and ρ_i to √(3) as discussed above). We design the two-stage architecture of Figure <ref> by firstsolving the stationary optimizationproblem (<ref>)-(<ref>), which provides an optimal matrix Π^*. Recall that the matrix D can be then obtained from the factorization (<ref>). The number of rows q of D is then equal to the rank of the matrix M^* := κ_δ,ϵ^2 [ ( V - V Π^* V )^-1 - V^-1]. Due to the numerical procedures, this rank will typically be maximal(here equal to 24). However, we plot on Fig. <ref> theratios σ_i/σ_max of the singular values of M^*, with σ_max the maximum singular value. We see that if we select for example only the singular values σ_i ≥ 10^-4σ_max in the SVD of M^* and set the smaller ones manually to 0, we obtain a matrix of rank 14, hence a matrix D with 14 rows instead of 24. We then verify (by solving an algebraic Riccati equation) that theperformance of the steady-state Kalman filter is left virtually unchanged by thistruncation, with a steady-state MSE of about 160, i.e., a root mean square error (RMSE)of 12.65 for the estimate of the number I_t of infectious people.In contrast, the input perturbation mechanism (i.e., taking D = I) gives a steady-stateMSE of 777 or RMSE on I_t of 27.87.Reducing the number of rows of D is also beneficial for example in termsof processing complexity of the Kalman filter, which now has fewer inputs. For δ=0.01, we compare on Fig. <ref> the steady-state RMSEof the two-stage mechanism and the input perturbation architecture fordifferent values of the privacy parameter ϵ. One can seethat by aggregating the input signals, we obtain a much better performance,especially in the high-privacy regime (when ϵ is small). Other measures of performance could also be of interest for the final filtering architecture, such as the convergence time of the estimates.For illustration purposes, Fig. <ref> shows sample pathsof differentially private estimates both for the two-stage mechanism andfor the input perturbation mechanism.§ DIFFERENTIALLY PRIVATE LQG CONTROLWe now turn to the LQG control problem introduced at the end ofSection <ref>. Forconcreteness, we assume here that the control input u_t at time tcan depend on the measurements y_0:t up to time t.It is straightforward to adapt the discussion to the case whereonly y_0:t-1 are available to compute u_t. By the separationprinciple <cit.> for the standard LQG control problem (i.e., with no privacy constraint),the optimal control law for the system (<ref>)-(<ref>)and quadratic cost (<ref>) is of the formu_t = K^c_t x̂_t, where: i) x̂_t = 𝔼[x_t | y_0:t] isthe MMSE estimator, computed by the Kalman filter (<ref>) independentlyof the design of the optimal control law; and ii) K^c_t is the optimalgain for the deterministic linear quadratic regulator (LQR) problem, i.e.,assuming that w = 0 in (<ref>) and C = I_n, v = 0in (<ref>).In particular, since the sequence of control gains K^c_t can be precomputed,the LQG control problem is similar to the filtering problem considered in theprevious section, with the desired published output ẑ_t = L_t x̂_tsimply replaced by u_t = K^c_t x̂_t. This motivates the architecture proposedon Figure <ref> for differentially privateLQG control, which, compared to Figure <ref>, aggregates themeasured signals y_i before adding the privacy-preserving noise.Essentially, the only difference with the Kalman filtering problem is that theperformance is measured by (<ref>) instead of theMSE (<ref>), so that the cost function in the optimizationproblem for the matrix D needs to be changed.The following theorem summarizes the discussion above and the classical results (see for example <cit.>) that allow us to formulate in the followingan efficiently solvable optimization problem for the choice of aggregation matrix D onFigure <ref>.Given a choice of matrix D for the differentially private LQG control architecture ofFigure <ref>, the control law u_t(y_0:t), t ≥ 0,minimizing the cost function (<ref>) takes the formu_t = K^c_t x̂_t,where x̂_t is computed by the Kalman filter (<ref>) and the gainsK^c_tare precomputed independently of the filtering problem as K^c_t := - (R_t + B_t^T P_t+1 B_t)^-1 B_t^T P_t+1 A_twith the matrices P_t ≽ 0 given by P_T = Q_T and the backward Riccatidifference equationP_t =A_t^T P_t+1 A_t + Q_t - A_t P_t+1 B_t (R_t + B_t^T P_t+1 B_t)^-1 B_t^T P_t+1 A_t.Moreover, the optimal objective (<ref>) corresponding to thiscontrol law can be writtenJ_T = J^c_T + J^f_T(D)where J^c_T = 1/T+1( x̅_0^T P_0 x̅_0 + (P_0 Σ̅_0)+ ∑_t=1^T (P_t W_t) )is a term independent of D andJ_T^f(D)= 1/T+1∑_t=0^T-1(N_t Σ_t),with N_t= Q_t + A_t^T P_t+1 A_t - P_t,0 ≤ t ≤ T-1,and the matrices Σ_t for 0 ≤ t ≤ T-1 defined by (<ref>)-(<ref>). Note that the dependence on D in (<ref>) is due to the factthat the error covariance matricesΣ_t depend on D via (<ref>)-(<ref>). Moreover, from (<ref>) we see that N_t defined in (<ref>) is positive semidefinite.Hence, we can define for all 0 ≤ t ≤ T-1 matrices L_t such that N_t = L_t^T L_t,and L_T = 0, to rewrite the cost (<ref>) as 1/T+1∑_t=0^T(L_t Σ_t L_t^T). Minimizing this cost over the matrices D with the relations(<ref>)-(<ref>) leads to an optimalaggregation matrix D for the architecture of Figure<ref>.The reformulation of this optimization problem as an SDP then follows exactly from thesame argument as in Section <ref>, which led to Theorem <ref>. In other words, we havethe following result. Let L_t, 0 ≤ t ≤ T-1, be any matrices obtained from the factorizationL_t^T L_t = N_t, 0 ≤ t ≤ T-1,with N_t defined by (<ref>), and let L_T = 0. Let Π^* ≽ 0,{ X^*_t≽ 0,Ω^*_t≻ 0}_0 ≤ t ≤ Tbe an optimal solution for (<ref>)-(<ref>) with this choice of matrices L_t. Suppose that for some 0 ≤ t ≤ T, we have L_t(Ω^*_t)^-1C_t^T ≠ 0. Let D^* be a matrix obtained from Π^* by the factorization (<ref>). Then D^* minimizes the LQG cost (<ref>) among all the aggregation matrices D of Figure <ref>. This cost is equal to J_T^c + 1/T+1∑_t=0^T (X_t^*), with J_T^c defined in (<ref>). §.§ Stationary ProblemAs in Section <ref> for the filtering problem, we can considerthe steady-state LQG problem by letting T →∞ and assuming themodel (<ref>)-(<ref>) and the weight matrices Q and Rin the cost (<ref>) to be time-invariant. We assume the model to be detectable and stabilizable and the pair (A,Q^1/2) to be detectable, in order to be able to implement a stabilizing LQG controller. We can take the optimal gains K^c and K^f of the controller andthe Kalman filter respectively to be also independent of time.Following Theorem <ref>and Proposition <ref>, we then immediately have thefollowing result for the design of the optimal D matrix. Let P be the positive semidefinite solution of the following algebraic Riccati equationP = A^T P A + Q - A^T P B(R+B^T P B)^-1 B^T PA.Let L be any matrix obtained from the factorizationL^T L = A^T P A + Q- P.Let Π^* ≽ 0, X^* ≽ 0, Ω^* ≻ 0 be an optimal solutionfor (<ref>)-(<ref>), for this choice of matrix L. Suppose that we have L (Ω^*)^-1 C^T ≠ 0. Let D^* be a matrix obtained from Π^* by the factorization (<ref>). Then D^* minimizes the steady-state LQG cost J_∞among all possible matrices D introduced as in Figure <ref>, the corresponding value of the cost is J_∞ = (PW) + (X^*). §.§ Numerical SimulationsWe illustrate the above results numerically for n = 10 independent scalar systems,with states x_i,t evolving as first order systems with time-invariant dynamics (<ref>), whereA_1= 1.1, A_2 = 0.85, A_3 = 0.84, A_4 = 0.7, A_5 = 0.75, A_6= 0.9, A_7 = 0.8, A_8 = 1.05, A_9 = 0.99, A_10 = 1,C_i = 1, W_i = 0.02 and V_i = 0.1 for all 1 ≤ i ≤ 10, and B is a 10 × 3 matrix with B_ij = 0 except forB_3,1 = B_6,1 = B_9,1 = 1 B_1,2 = B_4,2 = B_7,2 = B_10,2 = 1 B_2,3 = B_5,3 = B_8,3 = 1.In other words, the published control signal is 3-dimensional, with control input u_1 simultaneously actuating systems 3, 6, 9, u_2 actuating systems 1, 4, 7, 10 and u_3 actuating systems 2, 5 and 8. We wish to regulate the sum of the states ∑_i=1^10 x_i to 0,hence we takeQ to be the 10 × 10 all-ones matrixand R = I_3 in (<ref>).We set the privacy parameters to δ=0.05 and ϵ=ln(3), and ρ_i=1 for 1 ≤ i ≤ 10. To design the differentially private LQG controller with signal aggregation for the stationary problem, we compute the matrix L ofProposition <ref> and solve the optimizationproblem (<ref>)-(<ref>). Following the methodology discussed at the end of Section <ref>, we find that one can take the matrix D to be a 4 × 10 matrix at the matrix factorization stage (<ref>). The corresponding steady-state cost J_∞ is found to be 1.37, whereas it is 2.17 for the input perturbation mechanism (i.e., with D = I_10). Hence, signal aggregation results in a significant improvement. Figure <ref> shows a comparison of the cost J_∞ for this problem, with the two architectures,for different values of ϵ.Finally, Figure <ref> illustrates the sample paths obtained under closed-loop control with the differentially private controllers.We see in particular on Figure <ref> that the two-stage architecture provides a much better transient behavior for the regulated average trajectory (or sum of trajectories) compared to the input perturbation architecture, in addition to a better steady-state performance. § CONCLUSIONThis paper considers the Kalman filtering and LQG optimal control problems under a differential privacy constraint. We proposean architecture combiningan input stage aggregating the individual signals appropriately, the Gaussianmechanism to enforce differential privacy and a Kalman filter to reconstruct the desired estimate. Optimizing the parameters of this architecture can be recast as an SDP. Examples illustrate the performance improvementscompared to the input perturbation mechanism, which adds noise directly on the individual signals. The methodology is then adapted to propose a similar two-stage architecture for an LQG control problem, where the goal is to compute a shared control broadcasted to the agent population. Future research could consider the extension of these ideas to nonlinear systems,improving on the input and output perturbation mechanisms of <cit.>. In addition, since the size of the SDP increases rapidly with the number of agents (and the time horizon in the non-stationary case), it would be useful to developnumerical methods and a problem-specific solver that take advantage of the sparsityof the matrices involved in the constraints, as in <cit.> for example. IEEEtran | http://arxiv.org/abs/1707.08919v2 | {
"authors": [
"Kwassi H. Degue",
"Jerome Le Ny"
],
"categories": [
"cs.SY",
"cs.CR"
],
"primary_category": "cs.SY",
"published": "20170727160037",
"title": "A Two-Stage Architecture for Differentially Private Kalman Filtering and LQG Control"
} |
[email protected] ^1)Institut de Ciencies de lEspai (IEEC-CSIC),Carrer de Can Magrans, s/n, 08193 Barcelona, Spain ^2)ICREA, Passeig LluAs Companys, 23, 08010 Barcelona, [email protected] ^3)Laboratory for Theoretical Cosmology, Tomsk State University of Control Systems and Radioelectronics (TUSUR), 634050 Tomsk, Russia^4)Tomsk State Pedagogical University, 634061 Tomsk, Russia [email protected] ^5)Joint Institute for Nuclear Research, Joliot-Curie 6, 141980 Dubna, Moscow region, Russia^6)Institute of Physics, Kazan Federal University, Kremlevskaya street 18, 420008 Kazan, Russia We study in detail the phase space of a Friedmann-Robertson-Walker Universe filled with various cosmological fluids which may or may not interact. We use various expressions for the equation of state, and we analyze the physical significance of the resulting fixed points. In addition we discuss the effects of the stability or an instability of some fixed points. Moreover we study an interesting phenomenological scenario for which there is an oscillating interaction between the dark energy and dark matter fluid. As we demonstrate, in the context of the model we use, at early times the interaction is negligible and it starts to grow as the cosmic time approaches the late-time era. Also the cosmological dynamical system is split into two distinct dynamical systems which have two distinct de Sitter fixed points, with the early-time de Sitter point being unstable. This framework gives an explicit example of the unification of the early-time with late-time acceleration. Finally, we discuss in some detail the physical interpretation of the various models we present in this work. 04.50.Kd, 95.36.+x, 98.80.-k, 98.80.Cq, 11.25.-w Phase Space Analysis of the Accelerating Multi-fluid UniversePetr V. Tretyakov^5,6 December 30, 2023 ============================================================== § INTRODUCTIONIn modern theoretical cosmology, the most striking event was the observation of the late-time acceleration <cit.> that our Universe undergoes at present time. Admittedly this observation has utterly changed the way of thinking of modern cosmologists, since this late-time acceleration is a feature of our Universe that was never thought it would actually occur. Consequently, the focus for the last nearly 20 years is to model in a successful way this late-time acceleration and also to harbor the late and early-time acceleration era in a unified theoretical framework. Towards this unified description, many proposals, especially those suggesting to modify the gravitational sector, have been introduced ever since, see the reviews <cit.> for details. From the first moment that the late-time acceleration has been observed, it was realized that no perfect matter fluid known at that time was able to realize the late-time acceleration era, and therefore the need for alternative generalized cosmological fluids was compelling. By using generalized cosmological fluids, both the late and early-time acceleration era can be realized, and up to date there are many theoretical proposals that use generalized fluids, for example in Refs. <cit.> imperfect fluids are used in order to describe the cosmological evolution of our Universe, and in some cases certain particular examples are used, called viscous fluids are (see <cit.> for reviews). It is notable that the imperfect fluids may describe even phantom evolution of our Universe, without using phantom scalar fields, which violate the energy conditions, see Refs. <cit.> for details. Furthermore, other cosmological evolutions like bouncing cosmology, in the context of both classical and loop quantum cosmology imperfect fluids were studied in <cit.>, and also singular cosmology can be realized by imperfect fluids <cit.>. Furthermore, several models which take into account bulk viscosity were discussed in Refs. <cit.>, and also an important class of models which assume an interaction between dark matter and dark energy fluids, can be found in Refs. <cit.>.In this paper we shall perform a detailed phase-space analysis of a Friedmann-Robertson-Walker Universe, filled with different fluid components, which may or may not interact between them. The dynamical evolution of such kind of model is described by the Friedmann equations,3/κ^2H^2=∑ρ_i, -1/κ^2(2Ḣ + 3H^2)=∑ p_i,or equivalently,-2/κ^2Ḣ = ∑(ρ_i+ p_i),and also the energy conservation equations hold true,ρ̇_i=-3H(ρ_i+p_i).In the above, κ^2=8π G and also the equation of state (EoS) p_i=p_i(ρ_i) may be highly non-trivial for some fluid components. We shall appropriately choose the variables in order to capture the phase space dynamics in the most optimal way, and we shall analyze the structure of the phase space by providing an analytic treatment of the cosmological dynamical equations. Seeing the cosmological equations as a dynamical system is a particularly appealing way to investigate the phase space in many cosmological contexts, see for example <cit.>, but also in modified gravity too, see for example <cit.>. In most cases, the choice of the dynamical system variables plays a crucial choice, and in some cases the resulting cosmological dynamical system may be rendered autonomous <cit.>. Also it is possible to choose dimensionless variables, see for example Refs. <cit.> for an F(R) gravity cosmological dynamical system, and also see Ref. <cit.>, for a cosmological theory with higher derivatives of the scalar curvature. The dynamical systems approach for cosmological systems has many attributes, with the most important being the fact that the fixed points of the dynamical system actually provide new insights with regards to the behavior of the attractor solutions and also reveals the stability structure of the dynamical system near the attractors. It is conceivable that the choice of the variables plays an important role, as we also demonstrate by this work. This paper is organized as follows: In section <ref> we present some well known features of dynamical systems approach in cosmological context with generalized fluids, in section <ref> we discuss in brief how interactions between dark matter and dark energy may be introduced and by using appropriately chosen variables we present how the dynamical systems analysis can be performed in this case. In section <ref> we generalize the formalism we developed in the previous sections and by using dimensionless variables, we investigate the physical consequences of having various equations of state for the fluid components of the cosmological system. In section <ref> by using appropriately chosen dimensionless variables, we investigate in detail how the interaction of dark matter and dark energy fluids may affects the phase space structure. We study the stability and behavior of the fixed points of the dynamical system and we also discuss how the early and late-time acceleration eras are affected by the various functional forms of the interaction coupling between dark matter and dark energy. Finally the conclusions follow in the end of the paper.§ STANDARD APPROACH ON DYNAMICAL SYSTEMS AND COSMOLOGICAL DYNAMICS In this section we present the simplest case of the dynamical systems approach in cosmological dynamics. We consider the simplest case in which the Universe is filled with radiation ρ_r and a perfect fluid with a non-trivial EoS of the form p=-ρ+f(ρ)+G(H) <cit.>. Such non-trivial EoS can be considered as some sort of viscous fluid or generalized EoS fluid, see <cit.> for a review on this topic. In addition, such an EoS can be considered as an effective fluid presentation of some modified gravity theory <cit.>. In the case at hand, the full dynamical system takes the following form,3/κ^2H^2=ρ_r+ρ=ρ_tot, ρ̇_r=-4Hρ_r, ρ̇=-3H[f(ρ)+G(H)].The appearance of the term G(H) might seem unconventional, from a thermodynamic point of view, and we need to briefly describe the motivation for using such a term. This term encompasses the viscous part of the cosmological fluid, so it mainly quantifies the viscosity of the fluid. In the Universe, nd especially in the very early stages of it's evolution, the effects of a viscous cosmological component are most likely expected to occur during the neutrino decoupling process, which occurs at the end of the lepton era <cit.>. Hence, viscosity is encompassed in the very own fabric of the Universe. In addition, a strong motivation for using viscous fluid components comes from the fact that the perfect fluid approach among cosmologists-hydrodynamicists is just an ideal approach, and does not describe the real world. Finally, due to the fact that early and late-time acceleration may be described by an unknown form of a cosmological fluid, it is natural to assume that the fluid has the most general form, which means that a viscous component is needed[Note that terms of the form G(H,Ḣ) in the cosmological relation between the effective pressure and the energy density, naturally occur in modified gravity thermodynamics <cit.>].Having described the motivation for the use of viscous fluid components, we can rewrite the above cosmological equations in terms of dimensionless variables. In the case at hand, there is only one independent variable due to the constraint equation (<ref>). By using the e-foldings number[In this case time derivatives for some variable X transform as X'=dX/dN=Ẋ/H] N=ln a and by introducing a new dimensionless variable defined as,x=κ^2/3H^2ρ=ρ/ρ_tot,we obtain the next dynamical equation,x'=3/ρ_tot [ f(ρ) + G(H) ](x-1) -4x^2 +4x,where the variables ρ, H and ρ_tot must be expressed in terms of x depending on the choices of the functions f and G. Let us here discuss the simplest choice of EoS, which is p=w_0ρ +w_1H^2, which implies that f(ρ)=ρ(1+w_0) and G(H)=w_1H^2. It is easy to see that in this case, Eq. (<ref>) takes the following form,x'= [ 3(1+w_0)x +κ^2w_1 ](x-1) -4x^2 +4x≡ m(x),where the “prime” denotes differentiation with respect to the e-foldings number. The equation that determines the stationary points for the above dynamical equation, is quadratic with discriminant 𝔇=(κ^2w_1 -4 + 3(1+w_0) )^2, so the existing solutions are always real. For the case 𝔇=0 there is the only one solution, which is, x=1/2-κ^2w_1/2(3(1+w_0)-4). For the case 𝔇>0 there are two solutions, which are:x_1=1,x_2=-κ^2w_1/3(1+w_0)-4,and this case is the most interesting, since it provides us plenty dynamical solutions. Now note that the physical values that the variable x can take, are 0⩽ x⩽ 1, but for x_2 we have 0<x_2<1, which give us the following restrictions for the free parameters:w_1<0,w_0>-1/4,-κ^2w_1<-4+3(1+w_0), w_1>0,w_0<-1/4,-κ^2w_1>-4+3(1+w_0).Let us study the stability conditions of the existing stationary points. It is clear that for every point x_i there is only one eigenvalue μ_i=m'(x=x_i), and for the fixed points at hand, we have,μ_1(x_1)=-4+3(1+w_0)+κ^2w_1, μ_2(x_2)=4-3(1+w_0)-κ^2w_1=-μ_1 .Therefore, we find that for any values of parameters w_0, w_1 one of the fixed points is stable and the other one is unstable. Note that for the case 𝔇=0 we have μ=0 and therefore it is compelling to investigate the corresponding center manifold, but fortunately this is not interesting case from a physical point of view. The physical significance of the stationary point x_1 is that it corresponds to a Universe with ρ_tot=ρ and ρ_r=0. With regard to the fixed point x_2, it corresponds to a Universe with some fixed relation between ρ and ρ_r.§ MODELS WITH DARK MATTER INTERACTING TO DARK ENERGY Let us discuss at this point some models which describe interactions of dark matter with dark energy. The dynamical system in this case can be written in the following form,3/κ^2H^2=ρ_r+ρ+ρ_dm=ρ_tot, ρ_r'=-4ρ_r, ρ'=-3[f(ρ)+G(H)]-Q/H, ρ'_dm=-3ρ_dm+Q/H-3(-3Hζ),where the prime denotes as previously differentiation with respect to the e-foldings number N, and Q quantifies the interaction between dark energy and dark matter, and the term -3Hζ corresponds to the bulk viscous pressure of the dark matter fluid.We will assume that the bulk viscous coefficient ζ has the following form <cit.>,ζ=ζ_0/√(3κ^2)ρ_tot^1/2=1/κ^2Hζ_0.The motivation for using this kind of ansatz, comes from astronomical estimates on the relation between ζ and ρ_tot, which was studied in Ref. <cit.>, where a general form of the ζ-ρ_tot relation was assumed, and it was of the form ζ∼ρ^λ_tot. The choice ζ∼ρ^1/2_tot we used in Eq. (<ref>) yields a good fit between astronomical constraints and the fluid approach.In this case we introduce the following set of dimensionless variables,x=κ^2/3H^2ρ=ρ/ρ_tot,y=κ^2/3H^2ρ_dm=ρ_dm/ρ_tot,q=κ^2/3H^3Q=Q/Hρ_tot,where only x and y are independent dynamical variables. By using the set of dimensionless variables, the dynamical equations (<ref>)-(<ref>) take the form[Note that the constraint (<ref>) and the equation for ρ_r are already took into account in this system.]:x'=-4x^2+4x-yx-3xζ_0-q+3(f+G)/ρ_tot(x-1),y'=-y^2-4xy+y(1-3ζ_0)+3ζ_0+q+3(f+G)/ρ_toty,and by using the EoS we used in the previous section, namely, f(ρ)=ρ(1+w_0), G(H)=w_1H^2 we obtain,x'=x^2(3w_0-1)+x(w_1κ^2-3ζ_0-3w_0+1)-xy-(q+w_1κ^2),y'=-y^2+xy(3w_0-1)+y(w_1κ^2+1-3ζ_0)+3ζ_0+q.We can easily find the stationary points for the above dynamical system, by multiplying (<ref>) by y, (<ref>) by (1-x) and by combining the resulting equations we find,(1-x_0-y_0)(y+3ζ_0+q)=0. The above equation indicates that there exist at most three stationary points, and it is clear that even in the most general case, the stationary points may be found analytically, and these are equal to,x_0=q+w_1κ^2/1-3w_0, y_0=-3ζ_0-q, x_0=1-y_0, y_0=1/6w_0 [ 3w_0+w_1κ^2-3ζ_0 ±√((3w_0+w_1κ^2-3ζ_0)^2+12w_0q+36w_0ζ_0) ].Since there are a many free parameters, we investigate some particular cases, which are interesting from a physical point of view. We start off with the case w=0, in which case the first fixed point takes the form,y_0=-3ζ_0-q, x_0=q+w_1κ^2,and the corresponding eigenvalues are,μ_1=1, μ_2=1-w_1κ^2+3ζ_0,Accordingly, the second fixed point is,y_0=3ζ_0+q/3ζ_0-w_1κ^2, x_0=1-y_0,and the corresponding eigenvalues are,μ_1=w_1κ^2-3ζ_0, μ_2=w_1κ^2-3ζ_0-1,so the first point always unstable, whereas the second one may be stable, depending on the choice of the free parameters. However, if the first point lies in the physically allowed region, the second fixed point is rendered always unstable. Typical phase portrait with the two fixed points in the physical region, are presented in Fig.<ref>.Now let us consider another physically interesting case, for which q=-3ζ_0. In this case, there exist three fixed points which are,y_0=0,x_0=3ζ_0-w_1κ^2/3w_0,μ_1=w_1κ^2-3ζ_0+1+x_0(3w_0-1),μ_2=2μ_1-w_1κ^2+3ζ_0-3w_0-1,y_0=0,x_0=1,μ_1=w_1κ^2-3ζ_0+3w_0,μ_2=w_1κ^2-3ζ_0+3w_0-1,y_0=3w_0+w_1κ^2-3ζ_0/3w_0,x_0=1-y_0,μ_1=-1,μ_2=-3w_0-w_1κ^2+3ζ_0.A particularly interesting subcase of the above is if we further choose, 3ζ_0=w_1κ^2, then the fixed points become,x_0=0,y_0=1,μ_1,2=-1,-3w_0,x_0=1,y_0=0,μ_1,2=3w_0,3w_0-1,x_0=0,y_0=0,μ_1,2=1,1-3w_0.We can see that two fixed points are always unstable, whereas one of the three, the first or the second one depending on the sign of the parameter w_0, is stable.In Fig. <ref> we plotted the phase portrait for the case w_0=0.5, while in Fig. <ref> we plotted the phase portrait for w_0=-0.5, and finally in Fig. <ref> we can see the phase portrait for w_0=0. As it can be seen in all figures, two of the three fixed points are unstable and one of the three is stable. Furthermore, it can be seen that by using Eq. (<ref>), it is possible to find the case (by appropriately choosing the parameters) for which there will be some stationary point with fixed relation x/y≠0,1. This task may be solved numerically but we refrain from going into details on this. § GENERALIZED FORM OF COSMOLOGICAL FLUIDSIn this section we extend the cases we presented in section <ref> to include generalized form of the EoS. We consider the simplest scenario for which the Universe is filled with ρ_r and a perfect fluid with non-trivial EoS p=-ρ+f(ρ)+G(H) <cit.>. In this case, the dynamical system takes the following form,3/κ^2H^2=ρ_r+ρ=ρ_tot, -2/κ^2Ḣ= 4/3ρ_r + [f(ρ)+G(H)], ρ̇_r=-4Hρ_r, ρ̇=-3H[f(ρ)+G(H)].By using the e-foldings number as independent variable and also by introducing the dimensionless variables,x=κ^2/3H^2ρ=ρ/ρ_tot, z=1/κ^2H^2=3/ρ_totκ^4the dynamical system can be cast in the following form,x'=κ^4z [ f(ρ) + G(H) ](x-1) -4x^2 +4x,z'=4z(1-x)+z^2κ^4 [ f(ρ) + G(H) ],where according to the definition, ρ=3x/κ^4z and H^2=1/κ^2z. Moreover we can see from the system (<ref>)-(<ref>) that the fixed point x_0=1,z_0=0 always exists, except for some very special choices of functions f and G. It is interesting to note that actually due to the constraint equation (<ref>), in the case at hand, there is only one independent variable, but by introducing the variable z and by taking into account the equation (<ref>), allows us to obtain more information about the evolution of the dynamical system. For instance, one of the most hard tasks in such kind of dynamical systems analysis, is to interpret correctly the physical meaning of the stationary points. This approach was firstly proposed in Ref. <cit.>. Let us introduce an additional parameter, which will be very helpful for this interpretation, which is the effective equation of state, which we denote as w_eff, and it is defined as follows:w_eff≡ -1-2Ḣ/3 H^2.The parameter w_eff can be expressed in terms of the dimensionless variables (<ref>)-(<ref>) as follows, w_eff=1/3-4/3x+1/ρ_tot(f+G).By specifying the EoS, it is possible to obtain various physically interesting evolution scenarios, so in the rest of this section we shall specify the EoS and we study in detail the dynamical evolution stemming from the choice of EoS. §.§ A Simplified Form of EoS Let us discuss the simplest case of EoS, which is p=w_0ρ +w_1H^2, which in turn implies f(ρ)=ρ(1+w_0), G(H)=w_1H^2. It is easy to verify that in this case, the equations (<ref>)-(<ref>) take the following form,x'= x^2(3w_0-1) +x(1-3w_0+w_1κ^2)-w_1κ^2,z'= z[ x(3w_0-1) +4+w_1κ^2].Thus we have the following stationary points for the dynamical system above, * First fixed pointx_0=w_1κ^2/1-3w_0, z_0=0.* Second fixed point x_0=1, z_0=0. For the first fixed point we need to note that, this point lies in the physical region only if 0<w_1κ^2/1-3w_0<1. The corresponding eigenvalues are μ_1=1-3w_0-w_1κ^2, and μ_2=4. Using Eq. (<ref>) we find that at this point we have w_eff=1/3.With regard to the second fixed point, the eigenvalues are μ_1=3w_0+w_1κ^2-1, and μ_2=μ_1+4. Correspondingly we find that at this point w_eff=w_0+1/3w_1κ^2. Thus we can see that the first fixed point always unstable and also that the second fixed point may be stable only if one (or both) of parameters w_0, w_1 are strictly negative. For instance if we require that the first point lies in the physical region, we find that the second point is stable for w_0<-1. In Figs.(<ref>-<ref>) we plot the typical behavior of the phase trajectories.Particularly, Fig. <ref> corresponds to w_0=-2, and w_1κ^2=0.5. The left fixed point corresponds to the effective EoS parameter w_eff=1/3 and for the right fixed point, we have w_eff=-11/6. Clearly the left fixed point represents radiation, while the right one corresponds to some phantom evolution. In Fig. <ref>, the phase portrait corresponds to the following choices for the parameters, w_0=0.1, w_1κ^2=0.3. The left point corresponds to an effective EoS parameter w_eff=1/3, while the right point corresponds to w_eff=0.2, which describes a form of collisional matter <cit.>.§.§ More Complicated Forms of the EoS Now let us study more complicated forms of the EoS, and we choose it to be of the form f(ρ)+G(H)=Aρ^α+BH^2β. This EoS is known to lead the cosmological system to finite-time singularities, as this was demonstrated in Ref. <cit.>. For this EoS, the equations (<ref>)-(<ref>) take the form,x'= -4x^2+4x+κ^4z[ A(3x/κ^4z)^α+B(1/κ^2z)^β](x-1),z'=4z(1-x)+κ^4z^2[ A(3x/κ^4z)^α+B(1/κ^2z)^β].Correspondingly, the effective EoS in this case reads, w_eff=1/3 [1-4x + Aκ^4z(3x/κ^4 z)^α+Bκ^4 z1/(κ^2 z)^β ]. It is clear that in the most general case, this system may be solved only numerically, so let study some appropriately chosen cases which admit analytical solutions. Consider first the case for which α=1, β=2, in which case, the dynamical system (<ref>)-(<ref>) takes the following form,x'= ( 3Ax-4x+B/z)(x-1),z'=4z(1-x)+3Axz+B.In this case there are two stationary points: the first is x_0=1, z_0=0[Note here that in all these three cases existence of the point x_0=1, z_0=0 is not an obvious solution, but it can be confirmed by numerical investigations.] and the second is x_0=1, z_0=-B/3A. For the first point, we have w_eff=sign(B)∞[Infinite values of w_eff looks like the Ruzmaikin solution at t→ 0.] and for the second one we have w_eff=-1.We can see that the second point corresponds to some new non-trivial de Sitter state with ρ_tot=ρ and H^2=H_0^2≠ 0. The eigenvalues of the second fixed point are μ_1=-4, and μ_2=3A. The typical behavior of the phase trajectories corresponding to this case, can be found in Fig.<ref>, for A=-1, and B=1. As it can be seen, there exist trajectories which start from the first fixed pointx_0=1, z_0=0 (note that zero values of z correspond to infinite values of H, which indicates a singularity) and end up to the second fixed point, with non-singular and non-zero values of H. In effect, the second fixed point may be interpreted as a late-time acceleration de Sitter point of the cosmological dynamical system. Now let us consider the case for which α=2, and β=1, in which case, the dynamical system (<ref>)-(<ref>) takes the form,x'= ( 9A/κ^4x^2/z-4x+κ^2 B)(x-1),z'=4z(1-x)+9A/κ^4x^2+κ^2 B z.In this case there are two stationary points, namely, x_0=1, z_0=0 and also x_0=1, z_0=-9A/κ^6B and the situation is very similar to the previous case. For the first fixed point we have w_eff=sign(A)∞ and for the second one w_eff=-1. The eigenvalues of the second point are μ_1=-4, and μ_2=κ^2B. Note also that for both these cases, A and B must have opposite sign, in order for the second fixed point to lie in the physical region. We also need to note that for this case there are additional fixed points, which are difficult to find analytically, but the most physically interesting cases of fixed points are the ones we just presented. The phase space behavior corresponding to this case can be found in Fig. <ref>, for A=1/2, B=-9, κ^2=1, and as it can be seen, the behavior of the trajectories is similar to the previous case.Concluding this section, let us briefly discuss another choice of parameters for which it is possible to find analytically the fixed points, and this occurs for the choice α=2, β=2, in which case, the dynamical system (<ref>)-(<ref>) takes the form,x'= ( 9A/κ^4x^2/z-4x+B/z)(x-1),z'=4z(1-x)+9A/κ^4x^2+B.In this case, the stationary points are the following, x_0=1, z=0 and x_0^2=-Bκ^4/9A, z=0. For the first fixed point we have w_eff=sign(Bκ^4+9A)∞ and for the second one w_eff=1/3-4/9κ^2√(-B/A). The behavior of the trajectories corresponding to this case can be found in Fig. <ref>, for the choice A=-1, B=1, and in Fig. <ref>, for the choice A=1, B=-1. § DARK MATTER INTERACTING WITH DARK ENERGY: SOME NON-TRIVIAL MODELS Let us now discuss some non-trivial models which describe interactions between the dark matter and dark energy fluids. The dynamical system in this case may be written in the following form,3/κ^2H^2=ρ_r+ρ+ρ_dm=ρ_tot, -2/κ^2Ḣ= 4/3ρ_r + [f(ρ)+G(H)]+ρ_dm+(-3Hζ), ρ̇_r=-4Hρ_r, ρ̇=-3H[f(ρ)+G(H)+Q/3HH^2k], ρ̇_dm=-3H[ρ_dm-Q/3HH^2k+(-3Hζ)],where we modified the interaction between dark energy and dark matter by using the multiplier H^2k[Note that case k=0 corresponds to the standard interaction we presented in a previous section.], and also the term -3Hζ, which corresponds to the bulk viscous pressure of the dark matter fluid. We will assume that the bulk viscous coefficient ζ has the following form,ζ=ζ_0/√(3κ^2)ρ_tot^1/2=1/κ^2Hζ_0.We define the set of dimensionless variables as follows,x=κ^2/3H^2ρ=ρ/ρ_tot,y=κ^2/3H^2ρ_dm=ρ_dm/ρ_tot,z=1/κ^2H^2=3/ρ_totκ^4,q=κ^2/3H^3Q=Q/Hρ_tot,where only x and y are dynamical independent variables. In the new variables system, Eqs. (<ref>)-(<ref>) take the form[Note that the constraint (<ref>) and the equation for ρ_r have already been taken into account for this system.]:x'=-4x^2+4x-yx-3xζ_0-q/(κ^2z)^k+κ^4z(f+G)(x-1),y'=-y^2-4xy+y(1-3ζ_0)+3ζ_0+q/(κ^2z)^k+κ^4zy(f+G),z'=4z(1-x-y)+z^2κ^4(f+G)+3yz-3zζ_0,and using that the EoS has the form we used in the previous section, namely, f(ρ)=ρ(1+w_0), G(H)=w_1H^2, we obtain,x'=(x-1)(w_1κ^2+3xw_0-x)-x(y+3ζ_0)-q/(κ^2z)^k,y'=-y^2+y(1-3ζ_0+3xw_0-x+w_1κ^2)+3ζ_0+q/(κ^2z)^k,z'=z(4-x-y+3xw_0-3ζ_0+w_1κ^2).The effective EoS in this case (for arbitrary functions f and G) reads, w_eff=1/3 [1-4x -y + κ^4z(f+G) -3ζ_0],which in the case that f(ρ)=ρ(1+w_0), G(H)=w_1H^2, becomes, w_eff=1/3 [1-x -y + 3xw_0 +w_1κ^2 -3ζ_0].Now we consider some special cases, and we start with the case k=0. This case corresponds to the usual interaction term. Since the first twoequations are identical to (<ref>)-(<ref>), we have the solutions (<ref>), (<ref>), where we need to add z_0=0.Also, since the first two equations do not depend on z, the first and second eigenvalues will be totally identical to the ones we obtained in section <ref>, and one additional eigenvalue appears for every stationary point, which is,μ_3=4-x_0-y_0+3x_0w_0-3ζ_0+w_1κ^2.Also note that the stability or instability of the point z_0=0 with respect to this additional coordinate z, implies stability in the past or in the future correspondingly. It mean that a stable with respect to coordinates x, y point which have z_0=0 will be stable in the past if it is stable with respect to coordinate z, or equivalently will be stable for infinite values of H. In addition, it will stable in the future, if it is unstable with respect to coordinate z, or equivalently it will be stable for small (zero) values of H.The effective EoS (<ref>) for the fixed point (<ref>) is equal to w_eff=1/3 for any values of parameters, which clearly describes a radiation dominated era. However, the effective EoS for the fixed points (<ref>), has a more complicate structure, which is given below,w_eff=1/2w_0+1/6w_1κ^2-1/2ζ_0 ±1/6√((3w_0+w_1κ^2-3ζ_0)^2+12w_0q+36w_0ζ_0).Let us now further analyze the case at hand, by specifying the values of the free parameters, so we start with the parameter w_0, and assume for the moment that w_0=0. In this case the fixed point (<ref>) has an additional eigenvalue μ_3=4 and the corresponding effective EoS becomes w_eff=1/3, which describes radiation. Moreover, the fixed point (<ref>) has the additional eigenvalue μ_3=3-3ζ_0+w_1κ^2 and the corresponding effective EoS for this point is w_eff=1/3w_1κ^2-ζ_0. Thus in Fig. <ref> the left fixed point corresponds to radiation with w_eff=1/3, and the right point has w_eff=11/3, and for both the fixed points, H has infinite values. In Table <ref> we have gathered all the fixed points which correspond to the case k=0, w_0=0.As it can be seen in Table <ref>, the fixed point P_1b may describe late-time acceleration. Indeed, if we set w_eff=-1+α, with 0<α≪ 1, we obtain μ_1=-3+3α, μ_2=-4+3α and μ_3=α and as we already noted, this means that this point is stable in the future (for small values of H). Moreover, by changing the parameter q, we can provide any interesting relation between ρ_dm and ρ. For instance if put q=3/4-3ζ_0 we obtain for this point y_0≡ρ_dm/ρ_tot=1/4.Let us discuss some alternative choices for the parameters, so consider the case q=-3ζ_0, in which case the fixed point (<ref>) has an additional eigenvalue, which we denote μ_3, and it is equal to, μ_3=4-3ζ_0-w_1κ^2/3w_0 and the corresponding effective EoS is w_eff=1/3. In addition, the fixed point (<ref>) has the additional eigenvalue μ_3=3+3w_0-3ζ_0+w_1κ^2 with w_eff=0, which describes a matter dominated state. Finally, the fixed point (<ref>) has μ_3=3 with w_eff=w_0+1/3w_1κ^2-ζ_0. In Table <ref> we have gathered all the fixed points for the case q=-3ζ_0.As it can be seen in Table <ref>, the fixed point P_2a is always unstable, if it lies in the physical region, that is, when x_0⩽ 1. The fixed point P_2b may be stable in the past and in the future, depending on the values of the parameters. Finally, the fixed point P_2c may be stable only if w_eff>0. So in this case, no fixed point describes late-time acceleration, however, the fixed points P_2a and P_2b may describe a radiation dominated era and matter dominated era respectively. Consider now the case q=-3ζ_0, q=-w_1κ^2, in which case, the fixed point (<ref>) has the additional eigenvalue μ_3=3 and the corresponding effective EoS is w_eff=w_0. Accordingly, the fixed point (<ref>) has μ_3=3+3w_0 with w_eff=0 and finally the fixed point (<ref>) has μ_3=4 with w_eff=1/3. Thus in Fig.<ref> we have w_eff=1/3 for the fixed point (x_0=0,y_0=0), w_eff=0 for the fixed point (x_0=1,y_0=0) and w_eff=1/2 for the fixed point (x_0=0,y_0=1). Correspondingly, in Fig. <ref> we have w_eff=1/3 for the fixed point (x_0=0,y_0=0), w_eff=0 for the fixed point (x_0=1,y_0=0) and w_eff=-1/2 for the fixed point (x_0=0,y_0=1). Note that all the aforementioned fixed points correspond to states which have infinite values of H.In Table <ref> we have gathered all the fixed points for the case q=-3ζ_0, q=-w_1κ^2.Now let us consider some alternative choices for the parameter k, and we start with the case k=1, in which case the only stationary point is the following,x_0=3ζ_0-3-w_1κ^2/3w_0, y_0=1-x_0, z_0=qw_0/κ^2(3+3w_0+w_1κ^2-3ζ_0-3w_0ζ_0),and the expression (<ref>) yields for this fixed point,w_eff=-1.The effective EoS parameter above describes a de Sitter evolution, with a finite value for the Hubble parameter, namely H=H_0, which may be arbitrarily small, depending on the values of the other parameters. The eigenvalues for this fixed point in the most general case, have a quite complicate form, but there is one special case for which we can compute these analytically, and this occurs when w_0=1/3. In this case, the eigenvalues are,μ_1=-4, μ_2,3=1/2[ 3ζ_0-7-w_1κ^2±√((3ζ_0-7-w_1κ^2)^2+12(4-4ζ_0+w_1κ^2)) ],where the first eigenvalue corresponds to variable x. We can see that the stability with respect to all dimensions, requires the following conditions to hold true,ζ_0<7/3+1/3w_1κ^2, ζ_0>1+1/4w_1κ^2,which is quite compatible with 0⩽ x_0⩽ 1, but incompatible with z_0>0, which reads ζ_0<1+1/4w_1κ^2. This means that by varying the parameters, we can make this point stable with respect to the coordinates x and y, and unstable with respect z, so this point may be used for the construction of the late-time acceleration phase. Note also that by changing value of w_0, it is quite possible to make this point stable with respect to all the coordinates. Consider now the case k=-1, in which case there are four stationary points, which can be found in Table <ref>. Note that the parameter a appearing in Table <ref> is equal to a=(3w_0-w_1κ^2+3ζ_0)^2+12κ^2w_0w_1. The corresponding eigenvalues can be found in Table <ref>.By looking the Table <ref> it can be seen that the most phenomenologically interesting fixed point is P_4b. First of all it is easy to make this fixed point a stable attractor. Moreover w_eff is equal to -1 exactly, so this point corresponds to some de Sitter solution. Finally, for this point we have z_0≠ 0 which means that this point corresponds to some state with non-zero (and non-infinite) value of the Hubble parameter H=H_0. So by changing the values of the parameters it is easy to make z_0 sufficiently large, which implies sufficiently small values of H_0. For instance, by choosing w_0>0, w_1>0 and ζ_0<0 we can make this fixed point stable. By assigning sufficiently large values of w_1 for sufficiently small (positive) values of q, we get large values for z_0. And finally we can see that by changing the values of the parameters, it is easy to construct some fixed relation y_0/x_0≡ρ_dm/ρ. In conclusion, this de Sitter attractor may be viewed as a late-time attractor.It is interesting to note that there are five different regions in the parameter space, for which all four stationary points lie in the physical region, and these regions are the following,[i.e. 0⩽ x_0⩽ 1, 0⩽ y_0⩽ 1, z⩾0 for all points simultaneously.]w_1κ^2⩽-4,q>0, -1/3⩽ζ_0<a,w_0⩾ b,w_1κ^2⩽-4,q>0, a⩽ζ_0⩽0,w_0⩾ c,-4<w_1κ^2<-2/3√(13)-2/3,q>0, -c+4/3⩽ζ_0<a,w_0⩾ b,-4<w_1κ^2<-2/3√(13)-2/3,q>0, a⩽ζ_0<0,w_0⩾ c,-2/3√(13)-2/3⩽ w_1κ^2⩽-3,q>0, -c+4/3⩽ζ_0⩽ 0,w_0⩾ c,wherea=1/3(2w_1κ^2-1)+2/3√(w_1^2κ^4-w_1κ^2),b=-1/3w_1κ^2-ζ_0+2/3√(3w_1κ^2ζ_0),c=1/3(1-w_1κ^2).However, the above cases are quite hard to be tackled analytically, so a numerical study is needed, which exceeds the purposes of this article.§.§ Oscillating Dark Energy-Dark Matter InteractionIn the previous sections we considered cases of dark energy and dark matter interactions, by specifying the free parameters, and in this section we follow a different approach: we will directly modify the functional form of the dark energy-dark matter interaction by making it oscillating. So assume that in Eq. (<ref>) we make the following replacement, -QH^2k→ -Q cos(h_0H^2). In effect, instead of Eqs. (<ref>)-(<ref>) we havex'=(x-1)(w_1κ^2+3xw_0-x)-x(y+3ζ_0)-qcos(h_0H^2),y'=-y^2+y(1-3ζ_0+3xw_0-x+w_1κ^2)+3ζ_0+qcos(h_0H^2),z'=z(4-x-y+3xw_0-3ζ_0+w_1κ^2),where cos(h_0H^2)=cos (h_0/κ^2 z ). Let us suppose that the argument of the cosine function changes monotonically, from π/2 during the inflationary era, when H^2 M_Pl, to 0 which corresponds at the present time. In practice we may realize this kind of behavior by assuming sufficiently small values of the parameter h_0. Then, the physical interpretation of the situation at hand is the following, at early times (inflationary era), there is no interaction between the various matter fluid components, and as the time grows, this interaction starts to develop, and it grows until it reaches its maximumvalue at H=0. From a mathematical point of view, this would mean that there are two distinct asymptotic states at t→ 0 and t→∞, which are described by two distinct dynamical systems. In the following we discuss these two asymptotic dynamical systems separately.§.§.§ Asymptotic State I: The Inflationary Epoch Let us now focus on the inflationary asymptotic state, in which case instead of (<ref>)-(<ref>), we have the following dynamical system,x'=(x-1)(w_1κ^2+3xw_0-x)-x(y+3ζ_0),y'=-y^2+y(1-3ζ_0+3xw_0-x+w_1κ^2)+3ζ_0,z'=z(4-x-y+3xw_0-3ζ_0+w_1κ^2),where for the stationary points in the last equation above, we must put z_0=0. We can see that the last equation is decoupled from the system, and mainly governs the stability with respect to the variable z only. The stationary points of the system (<ref>)-(<ref>) are already found in section <ref> and these are presented in Eqs. (<ref>)-(<ref>), where we need to put q=0. The first fixed point should be excluded from the future investigation, since it has w_eff=1/3. Thus we have one possible fixed point candidate that may describe inflation (let us denote it P_5a):x_0=1-y_0, y_0=1/6w_0 [ 3w_0+w_1κ^2-3ζ_0 ±√((3w_0+w_1κ^2-3ζ_0)^2+36w_0ζ_0) ],with, w_eff=1/3 [ 3x_0w_0 +w_1κ^2 -3ζ_0].Now in order to describe inflation, we need to construct a de Sitter solution, which should be stable with respect to the coordinates x, y and unstable with respect to z, in order to provide an exit from inflation. Thus let us put for (<ref>) w_eff=-1+γ, where 0<γ≪ 1. In this case, the third eigenvalue associated with the coordinate z will exactly be μ_3=γ, so this provides an instability of de Sitter point. Moreover deriving x_0 from (<ref>) and equating it to (<ref>), we find an additional relation between the free parameters, ζ_0=(1-γ)(3-3γ+3w_0+w_1κ^2)/3(1+w_0-γ),or if we require that all the values of parameters are not very small in comparison with γ, we have, ζ_0=3+3w_0+w_1κ^2/3(1+w_0)-6+3w_0+w_1κ^2/3(1+w_0)γ.§.§.§ Asymptotic State II: Present Time and Late-time Acceleration Now let us consider the present-time asymptotic case, in which case instead of the dynamical system of Eqs. (<ref>)-(<ref>) we have the following system,x'=(x-1)(w_1κ^2+3xw_0-x)-x(y+3ζ_0)-q,y'=-y^2+y(1-3ζ_0+3xw_0-x+w_1κ^2)+3ζ_0+q,z'=z(4-x-y+3xw_0-3ζ_0+w_1κ^2),where for stationary points in the last equation we must require z_0≠ 0. Thus, from the last equation we have y_0=4-3ζ_0+w_1κ^2+x_0(3w_0-1).Substituting the above in Eq. (<ref>) we find,x_0=-q-w_1κ^2/3(w_0+1),and by substituting in Eq. (<ref>) we find,x_0=12ζ_0+q-3w_1κ^2-12/3(3w_0-1).In effect, the system will compatible only if the following condition holds true,-q-w_1κ^2/3(w_0+1)=12ζ_0+q-3w_1κ^2-12/3(3w_0-1),which gives us, q=3w_0+3-3ζ_0+w_1κ^2-3ζ_0w_0/w_0,and the corresponding stationary point is (let us denote it P_5b) x_0=3ζ_0-w_1κ^2-3/3w_0,y_0=3w_0+3-3ζ_0+w_1κ^2/3w_0,x_0+y_0=1.Combining (<ref>) with (<ref>) we find, q=(6+3w_0+w_1κ^2)γ,so we can see that q is very small but does not vanish.[Putting w_eff=-1 exactly for the first point, we get q=0.] Note also that the effective EoS for this point, calculated for (<ref>) yields exactly w_eff=-1 for any values of the parameters. §.§.§ combining and stability analysis.By looking the differential equations in Eqs. (<ref>)-(<ref>) and (<ref>)-(<ref>), it can be seen that these are identical apart for some additive constants, and in effect the eigenvalues will be identical for the two systems. Let us denote x'=f and y'=g for notational simplicity. In effect, the equation that determines the eigenvalues, takes the following form,| [ (f_x)_0-μ (f_y)_0;; (g_x)_0 (g_y)_0-μ ] | =0,where (f_x)_0, (f_y)_0, (g_x)_0 and (g_y)_0, are equal to,(f_x)_0=w_1κ^2+6w_0x_0-x_0-3ζ_0-3w_0,(f_y)_0=-x_0,(g_x)_0=(3w_0-1)(1-x_0),(g_y)_0=-1+x_0-3ζ_0+3w_0x_0+w_1κ^2,and in the above relations we took into account that for both points we have x_0+y_0=1.The solution of Eq. (<ref>) for the fixed point P_5a with positive sign in (<ref>) reads, μ_1=-√((3w_0+w_1κ^2-3ζ_0)^2+36w_0ζ_0),μ_2=1/2[ -2+ 3w_0+w_1κ^2-3ζ_0 +μ_1 ],and for the point P_5b μ_1=-4,μ_2= -6 - 3w_0 - w_1κ^2+3ζ_0,and by substituting the expression (<ref>) for ζ_0, for the fixed point P_5a, we have, μ_1=-3 + 6w_0 + 3w_0^2 + w_0w_1κ^2/w_0+1,μ_2=-4,and for the point P_5b we get, μ_1=-4,μ_2= -3 + 6w_0 + 3w_0^2 + w_0w_1κ^2/w_0+1.Note that we have set γ=0, since all the eigenvalues, even in this case, do not vanish. Thus we can see that both points P_5a and P_5b are stable or unstable with respect to coordinates x and y simultaneously, and stability condition reads, (3 + 6w_0 + 3w_0^2 + w_0w_1κ^2)(w_0+1)>0.Moreover, we need to require that both fixed points lie in the physical region, namely at 0⩽ x_0⩽ 1 and 0⩽ y_0⩽ 1, so the following conditions must be satisfied,w_1κ^2<-3,w_0⩾ c,w_1κ^2=-3,w_0> 0,-3<w_1κ^2<0,c⩽ w_0 <0w_0>0,w_1κ^2>0,w_0⩽ c,where,c=-1/3(w_1κ^2+3).In Table <ref> we have gathered all the fixed points and the corresponding eigenvalues for the case of an oscillating form of the interaction term Q.It is worth discussing some interesting scenarios, so assume that w_0=1, w_1κ^2=-1, then the fixed points and the eigenvalues become,P_5a: x_0=1/6,y_0=5/6, μ_1=-11/2,μ_2=-4,P_5b: x_0=5/6,y_0=1/6, μ_1=-4,μ_2=-11/2.So by changing the values of the parameters w_0 and w_1, it is possible to appropriately fix the fixed point P_5b, which recall that it corresponds to late-time acceleration, in order some fixed relation between ρ_dm and ρ_de holds true. Note that in this section we studied only case for which the choices of the functions f(ρ) and G(H^2) were the simplest choices, but in principle a more involved functional form for these functions may lead to more interesting phenomenology. In such a case though, the analytical treatment will be possibly insufficient, so a concrete numerical analysis will be needed. In conclusion, we constructed a cosmological model that describes the inflationary and the late-time acceleration era, due to an oscillating interaction term between the dark energy and dark matter fluids. We found two fixed points, with the first being P_5a, which describes the initial de Sitter solution, namely the inflationary era, and this fixed point was unstable, a feature which indicates that the graceful exit is triggered. The second fixed point was also found to be a de Sitter fixed point, which describes the present day acceleration. As we demonstrated, in the oscillating model we discussed, during the early Universe there was no interaction between dark energy and dark matter, however during the late-time the interaction was present. Finally, as we showed, by appropriately choosing the parameters, it is possible to produce some fixed relation for ρ_dm/ρ_de.Before closing this section, let us briefly discuss an interesting issue with regards the early-time era. It would be interesting to calculate the slow-roll indices and the corresponding observational indices for the inflationary era we presented in section <ref>. For example by assuming the perfect fluid approach <cit.>, we can express the spectral index of primordial curvature perturbations n_s and the scalar-to-tensor ratio r in the usual way these are given in the case of a canonical scalar field,n_s=1-6ϵ+2η,r=16ϵ.with the slow-roll indices being defined in terms of the Hubble rate as follows,ϵ= -Ḣ/H^2,η= ϵ -Ḧ/2HḢ.By combining Eqs. (<ref>) and (<ref>) it is easy to calculate the slow-roll parameter ϵ, which reads,ϵ=1/2 [ 4- y +x(3w_0-1) +w_1κ^2 -3ζ_0].Differentiating (<ref>) and by combining with equations (<ref>)-(<ref>) we can find a similar expression for the slow-roll parameter η. Note here that the resulting expression for η is much more complicated so we omit it. There is a major obstacle in calculating the slow-roll indices however, since in the expressions for ϵ and η, we must use values for x and y not on the stationary point, but from some point near the stationary point. The resulting slow-roll indices must be expressed in terms of the e-foldings number N, but doing this analytically is a rather formidable task, that exceeds the purposes of this work. A numerical approach though might be less difficult to perform, so we hope to address this task in a future work. § CONCLUSIONSIn this work we analyzed in detail the phase space of a cosmological system that contains cosmological fluids that have various forms of equation of state. We firstly discussed the simplest forms of EoS, and we found the fixed points of the cosmological dynamical system and we discussed the physical significance of these fixed points. In addition, we discussed theories that admit interactions between the dark energy and dark matter fluids. In addition we introduced a new class of interaction between dark energy and dark matter, in which theories the interaction term is oscillating, allowing different form of interactions for various eras during the cosmological evolution. As we demonstrated, it is possible to have almost negligible interactions at early times, that is, during the inflationary era, and for the same model the interaction is turned on at late times. The cosmological dynamical system of the oscillating interaction term is in turn decomposed into two distinct dynamical systems at early and late times. Interestingly enough the two dynamical systems predict two de Sitter fixed points corresponding to early and late times, with the early-time de Sitter point being unstable in one of the coordinates, a feature which indicates the possible exit from the inflationary era. This framework gives us the interesting fluid-filled Universe evolution, unifying the early-time acceleration with late-time acceleration.The fluid description offers an alternative viewpoint in modern cosmology, which may describe successfully many eras of our Universe's evolution. The dynamical system approach offers many new insights since the fixed points of the dynamical system reveal the attractors of the whole theory and their stability indicates whether these attractors are final attractors of the system. A compelling extension of this work is to include Loop Quantum Cosmology effects in the EoS, as was performed in Ref. <cit.>, so we defer this task to a future work. Another strong motivation to adopt the fluid approach in cosmology, is the late-time acceleration era, since this can be modeled by fluid cosmology <cit.>. Furthermore, as modified gravity maybe easily presented in the effective fluid representation, the dynamical systems approach turns out to be useful also for the study of the cosmology in modified gravity. More interestingly, the dynamical system approach can be applied to the study of bouncing cosmology, described by a multi-component fluid. We aim to address this latter issue in a future work. 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"authors": [
"S. D. Odintsov",
"V. K. Oikonomou",
"Petr V. Tretyakov"
],
"categories": [
"gr-qc",
"astro-ph.CO",
"hep-th"
],
"primary_category": "gr-qc",
"published": "20170726230537",
"title": "Phase Space Analysis of the Accelerating Multi-fluid Universe"
} |
[classification=text] TA]Theresa Anderson BC]Brian Cook KH]Kevin Hughes AK]Angel KumchevWe improve the range of ℓ^p(ℤ^d)-boundedness of the integral k-spherical maximal functions introduced by Magyar. The previously best known bounds for the full k-spherical maximal function require the dimension d to grow at least cubically with the degree k. Combining ideas from our prior work with recent advances in the theory of Weyl sums by Bourgain, Demeter, and Guth and by Wooley, we reduce this cubic bound to a quadratic one. As an application, we improve upon bounds in the authors' previous work <cit.> on the ergodic Waring–Goldbach problem, which is the analogous problem of ℓ^p(ℤ^d)-boundedness of the k-spherical maximal functions whose coordinates are restricted to prime values rather than integer values.§ INTRODUCTION Our interest lies in proving ℓ^p(ℤ^d)-bounds for the integral k-spherical maximal functions when k ≥ 3. These maximal functions are defined in terms of their associated averages, which we now describe. Define a positive definite function 𝔣 on ℝ^d by 𝔣(𝐱) = 𝔣_d,k(𝐱) := |x_1|^k + … + |x_d|^k,and note that when 𝐱∈ℝ_+^d, 𝔣(𝐱) is the diagonal form x_1^k + … + x_d^k. For λ∈ℕ, let R(λ) denote the number of integral solutions to the equation𝔣_d,k(𝐱) = λ.When R(λ)>0, define the normalized arithmetic surface measure σ_λ(𝐱) := 1/R(λ)1_{𝐲∈ℤ^d: 𝔣(𝐲)=λ}(𝐱). We are interested in averages given by convolution with these measures:σ_λ * f(𝐱) = 1/R(λ)∑_𝔣 (𝐲) = λ f(𝐱 - 𝐲)for functions f: ℤ^d →ℂ.We know from the literature on Waring's problem that as λ→∞, one has the asymptotic R(λ) ∼𝔖_d,k(λ) λ^d/k-1, where 𝔖_d,k(λ) is a convergent product of local densities:𝔖_d,k(λ) = ∏_p ≤∞μ_p(λ).Here μ_p(λ) with p < ∞ is related to the solubility of (<ref>) over the p-adic field ℚ_p, and μ_∞(λ) to solubility over the reals. It is known that when d is sufficiently large in terms of k, one has 1 ≲𝔖_d,k(λ) ≲ 1. In particular, these bounds hold for d ≥ 4k when k ≥ 4 is a power of 2, and for d ≥3/2k otherwise (see Theorems 4.3 and 4.6 in Vaughan <cit.>). Throughout the paper we use the notation f(x) ≲ g(x) or g(x) ≳ f(x) to mean that there exists a constant C > 0 so that |f(x)| ≤ C|g(x)| for all sufficiently large x ≥ 0.The implicit constant C may depend on `inessential' or fixed parameters, but will be independent of `x';below the implicit constants will often depend on the parameters k,d and p.For instance, (<ref>) means that there exists positive constants C_1 and C_2 depending on d and k so that C_1 ≤𝔖_d,k(λ) ≤ C_2. In view of (<ref>) and (<ref>), we may replace the convolution σ_λ * f above by the averageA_λ f(𝐱) := λ^1-d/k∑_𝔣 (𝐲) = λ f(𝐱 - 𝐲).Our k-spherical maximal function is then defined, for 𝐱∈ℤ^d, as the pointwise supremum of all averagesA_* f(𝐱) := sup_λ∈ℕ |A_λ f(𝐱)|.Variants of this maximal function were introduced by Magyar <cit.> and studied later in <cit.>. In particular,Magyar, Stein and Wainger <cit.> considered the above maximal function when k=2 and d ≥ 5 and proved that it is bounded on ℓ^p(ℤ^d) when p > d/d-2. This result is sharp except at the endpoint, for which the restricted weak-type bound was proved later by Ionescu <cit.>. To the best of our knowledge, the sharpest results on the boundedness of A_* for degrees k ≥ 3 are those obtained by the third author in <cit.>. In the present paper, we give a further improvement.For k ≥ 3, define d_0(k) := k^2 - max_2 ≤ j ≤ k-1{k j - min(2^j+2,j^2+j)/k-j+1}. Further, set τ_k = max{ 2^1-k,(k^2 - k)^-1} and define the function δ_0(d,k) by kδ_0(d,k) :=(d-d_0)/(k^2+k - d_0)ifd_0(k) ≤ d ≤ k^2+k, 1 + (d-k^2-k)τ_kifd > k^2+k.Finally, definep_0(d,k):=max{d/d-k, 1+1/1+2δ_0(d,k)}.We remark that when d > d_0(k), p_0(d,k) always lies in the range (1,2). Our main result is the following. If k ≥ 3, d > d_0(k) and p>p_0(d,k), then the maximal operator A_*, defined by (<ref>), is bounded on ℓ^p(ℤ^d): that is,A_*f _ℓ^p(ℤ^d)≲ f _ℓ^p(ℤ^d) .Let d_0^*(k) := 1 + ⌊ d_0(k) ⌋ denote the least dimension in which Theorem <ref> establishes that A_* is bounded on ℓ^2(ℤ^d). We emphasize that for large k, we have d_0^*(k) = k^2 - k + O(k^1/2), whereas in previous results, such as the work of the third author <cit.>, one required d > k^3-k^2.While our results improve on these previous results by a factor of the degree k, the conjecture is that the maximal function is bounded on ℓ^2(ℤ^d) for d ≳ k (see <cit.> for details), but such a result appears to be way beyond the reach of present methods. It is also instructive to compare d_0^*(k) to the known bounds for the function G̃(k) in the theory of Waring's problem (defined as the least value of d for which the asymptotic formula (<ref>) holds). It transpires that the values of d_0^*(k) match the best known upper bounds on G̃(k) for all but a handful of small values of k, and even in those cases, we miss the best known bound on G̃(k) only by a dimension or two. For an easier comparison, we list the numerical values of d_0^*(k), k ≤ 10, their respective analogues in earlier work, and the bounds on G̃(k) in Table <ref>.A key ingredient in the proof of Theorem <ref> and its predecessors is an approximation formula generalizing (<ref>). First introduced in <cit.> when k=2, such approximations are obtained for the average's corresponding Fourier multiplier:A_λ() = λ^1-d/k∑_𝔣(𝐱)=λe(𝐱·ξ),where ξ∈𝕋^d and e(z) = e^2π iz. We need to introduce some notation in order to state our approximation formula. Given an integer q ≥ 1, we write ℤ_q = ℤ/qℤ and ℤ_q^* for the group of units; we also write e_q(x) = e(x/q). The d-dimensional Gauss sum of degree k is defined as G(q; a, 𝐛) = q^-d∑_𝐱∈ℤ_q^d e_q ( a𝔣(𝐱) + 𝐛·𝐱)for a ∈ℤ_q and 𝐛∈ℤ_q^d. If Σ_λ denotes the surface in ℝ_+^d defined by (<ref>) and dS_λ(𝐱) denotes the induced Lebesgue measure on Σ_λ, we define a continuous surface measure dσ_λ(𝐱) on Σ_λ bydσ_λ(𝐱):= λ^1-d/kdS_λ(𝐱)/|∇𝔣(𝐱)|.We note that dσ_λ is essentially a probability measure on Σ_λ for all λ. We also fix a smooth bump function ψ, which is 1 on [-1/8,1/8]^d and supported in [-1/4, 1/4]^d. Finally, we write μ for the ℝ^d-Fourier transform of a measure μ on ℝ^d and f for the ℤ^d-Fourier transform (which has domain 𝕋^d) of a function f on ℤ^d. If k ≥ 3, d > d_0(k) and λ∈ℕ is sufficiently large, then one hasA_λ(ξ) = ∑_q=1^∞∑_a ∈ℤ_q^* e_q( -λ a) ∑_ w ∈{± 1 }^d∑_𝐛∈ℤ^d G(q; a,w𝐛) ψ(qξ-𝐛) dσ_λ( w(ξ-q^-1𝐛)) + E_λ(ξ),where the error terms E_λ are the multipliers of convolution operators satisfying the dyadic maximal inequalitysup_λ∈ [Λ/2, Λ) |E_λ| _ℓ^2(ℤ^d) →ℓ^2(ℤ^d)≲Λ^-δfor each Λ≥ 1 and all sufficiently small δ > 0. Our Approximation Formula takes the same shape as those in <cit.> and <cit.>, but with an improved error term that relies on two recent developments:* the underlying analytic methods were improved in the authors' previous work <cit.>,* and the recent resolution of the main conjecture about Vinogradov's mean value integral <cit.> and related refinements of classical mean value estimates <cit.>.The most dramatic improvement follows from our improved analytic methods originating in <cit.> where we improve the range of ℓ^2(ℤ^d) by a factor of the degree k.In <cit.>, this sort of improvement - which also used the recent resolution of the Vinogradov mean values theorems <cit.> - was limited to maximal functions over sufficiently sparse sequences.Here, our bounds supersede those for integral k-spherical maximal functions over sparse sequences in <cit.> because our treatment of the minor arcs in the error term is more efficient.The reader may compare Lemmas <ref> and <ref> below to Lemmas 2.1 and 2.2 of <cit.> to determine the efficacy of our method here.Consequently, <cit.> and <cit.> allow us to further improve slightly upon a more direct application of the Vinogradov mean value theorems from <cit.> and <cit.>.One minor drawback is that in our method the ϵ-losses in <cit.> do not allow us to deduce endpoint bounds. As an application, we deduce that the maximal function of the "ergodic Waring–Goldbach problem" introduced in our recent work <cit.> is bounded on the same range of spaces as above. That maximal function is associated with averages where, instead of sampling over integer points, we sample over points where all coordinates are prime. To be precise, let R^*(λ) denote the number of prime solutions to the equation (<ref>) weighted by logarithmic factors: that is, R^*(λ) := ∑_𝔣(𝐱) = λ1_ℙ^d(𝐱)(log x_1) ⋯ (log x_d),where 1_ℙ^d is the indicator function of vectors 𝐱∈ℤ^d with all coordinates prime. When R^*(λ)>0, define the normalized arithmetic surface measure ω_λ(𝐱) := 1/R^*(λ) 1_{𝐲∈ℙ^d: 𝔣(𝐲)=λ}(𝐱) (log x_1) ⋯ (log x_d) and the respective convolution operatorsW_λ f := ω_λ * f.Similarly to (<ref>), we know that as λ→∞, one has the asymptotic R^*(λ) = (𝔖_d,k^*(λ) + o(1)) λ^d/k-1,where 𝔖_d,k^*(λ) is a product of local densities similar to 𝔖_d,k(λ) above. Moreover, when d > 3k and λ is restricted to a particular arithmetic progression Γ_d,k, we have 1 ≲𝔖_d,k^*(λ) ≲ 1, and the above estimate turns into a true asymptotic formula for R^*(λ) (see the introduction and references in <cit.>). By Theorem 6 of <cit.> and Theorem <ref> above, we immediately obtain the following result. Here, as in <cit.>, d_1(3) = 13 and d_1(k) = k^2+k+3. If k ≥ 3, d ≥ d_1(k) and p > p_0(d,k), then the maximal function defined byW_* f(𝐱) := sup_λ∈Γ_d,k |W_λ f(𝐱)|,is bounded on ℓ^p(ℤ^d).As another application one may give analogous improvements of the ergodic theorems obtained in <cit.>, but we do not consider this here.To establish our theorems, we follow the paradigm in <cit.> and strengthen the connection to Waring's problem as initiated in <cit.> by using a lemma from <cit.>; we then use recent work on Waring's problem to obtain improved bounds.We remark that <cit.> and <cit.> previously connected mean values (Hypothesis K^* and Vinogradov's mean value theorems respectively) to discrete fractional integration.In Section <ref>, we outline the proofs of Theorems <ref> and <ref>; we recall some results from <cit.> and state the key propositions required in the proofs. The remaining sections establish the propositions.In Section <ref> we deal with the minor arcs; particularly, in Section <ref>, we use the recent work of Bourgain, Demeter and Guth <cit.> on Vinogradov's mean value theorem and a method of Wooley <cit.> for estimation of mean values over minor arcs.In Section <ref>, we establish the relevant major arc approximations. Finally, in Section <ref>, we establish the boundedness of the maximal function associated with the main term in the Approximation Formula.§ OUTLINE OF THE PROOF Since A_* is trivially bounded on ℓ^∞(ℤ^d), we may assume through the rest of the paper that p ≤ 2.Fix Λ∈ℕ and consider the dyadic maximal operatorA_Λ f := sup_λ∈ [Λ/2,Λ) |A_λf|.When λ≤Λ, we haveA_λ f = λ^1-d/k∫_𝕋 (h_Λ(θ) * f)e(-λθ) dθ,where h_Λ(θ) = h_Λ(θ; 𝐱) := e(θ𝔣(𝐱))1_[-N,N]^d(𝐱) with N = Λ^1/k. This representation allows us to decompose A_λ into operators of the formA_λ^𝔅 f = λ^1-d/k∫_𝔅 (h_Λ(θ) * f)e(-λθ) dθ,for various measurable sets 𝔅⊆𝕋.Our decomposition of A_λ, is inspired by the Hardy–Littlewood circle method. When q ∈ℕ and 0 ≤ a ≤ q, we define the major arc 𝔐(a/q) by𝔐(a/q) = [ a/q - 1/4kqN^k-1,a/q + 1/4kqN^k-1].We then decompose 𝕋 into sets of major and minor arcs, given by𝔐 = 𝔐(Λ) = ⋃_q ≤ N⋃_a ∈ℤ_q^*𝔐(a/q)and 𝔪(Λ) = 𝕋∖𝔐(Λ).Since the major arcs 𝔐(a/q) are disjoint, this yields a respective decomposition of A_λ as A_λ = ∑_q ≤ N∑_a ∈ℤ_q^* A_λ^a/q + A_λ^𝔪,where A_λ^a/q := A_λ^𝔐(a/q). We will use the notations A_*^𝔅 and A_Λ^𝔅 to denote the respective maximal functions obtained from the operators A_λ^𝔅. For example, from (<ref>) and the trivial bound for the trigonometric polynomial h_Λ(θ) * f, we obtain the trivial ℓ^1-boundA_Λ^𝔅_ℓ^1(ℤ^d) →ℓ^1(ℤ^d)≲Λ |𝔅|.In Section <ref>, we analyze the minor arc term and prove the following result. If k ≥ 3, d > d_0(k), and Λ≳ 1, thenA_Λ^𝔪_ℓ^2(ℤ^d) →ℓ^2(ℤ^d)≲Λ^-δfor all δ∈ (0,δ_0(d,k)). For 1 < p ≤ 2, interpolation between (<ref>) and (<ref>) yields A_Λ^𝔪_ℓ^p(ℤ^d) →ℓ^p(ℤ^d)≲Λ^-α_pwith α_p = 2(1+δ)(1-1/p) - 1. When p_0(d,k) < p ≤ 2, we have α_p > 0, and hence,A_*^𝔪_ℓ^p(ℤ^d) →ℓ^p(ℤ^d)≤∑_Λ = 2^j ≳ 1 A_Λ^𝔪_ℓ^p(ℤ^d) →ℓ^p(ℤ^d)≲∑_Λ = 2^j ≳ 1Λ^-α_p≲ 1. The estimation of the major arc terms is more challenging, because an analogue of (<ref>) does not hold for A_Λ^𝔐. Still, it is possible to establish a slightly weaker version of Proposition <ref>. The following result was first established by Magyar <cit.>, for d ≥ 2^k, and then extended by the third author <cit.> in the present form. If k ≥ 3, d > 2k, d/d-k < p ≤ 2, and Λ≳ 1, one has A_Λ^𝔐_ℓ^p(ℤ^d) →ℓ^p(ℤ^d)≲ 1.This proposition suffices to establish the ℓ^p-boundedness of the dyadic maximal functions A_Λ (this is the main result of Magyar <cit.>), but falls just short of what is needed for an equally quick proof of Theorem <ref>. Instead, we use Theorem <ref> to approximate A_*^𝔐 by a bounded operator. Let M_λ^a/q denote the convolution operator on ℓ^p(ℤ^d) with Fourier multiplierM_λ^a/q(ξ) :=e_q( -λ a) ∑_ w ∈{± 1 }^d∑_𝐛∈ℤ^d G(q; a,w𝐛) ψ(qξ-𝐛) dσ_λ( w(ξ-q^-1𝐛)),and defineM_λ := ∑_q ∈ℕ∑_a ∈ℤ_q^* M_λ^a/q andM_* := sup_λ∈ℕ |M_λ|. In Section <ref>, we will handle the major arc approximations and prove the following proposition.If k ≥ 3, d > 2k, and d/d-k < p ≤ 2, then there exists an exponent β_p = β_p(d,k) > 0 such that sup_λ∈ [Λ/2, Λ)| A_λ^𝔐-M_λ| _ℓ^p(ℤ^d) →ℓ^p(ℤ^d)≲Λ^-β_p. Theorem <ref> is an immediate consequence of Propositions <ref> and <ref>. Moreover, since when p = 2 and d ≥5/2k, inequality (<ref>) holds for any β_2 < 1/(2k) (see inequality (<ref>) and the comments after it), the error bound (<ref>) holds for all δ with 0 < δ < min{δ_0(d,k), 1/(2k) }.When we sum (<ref>) over dyadic Λ = 2^j, we deduce thatsup_λ≳ 1| A_λ^𝔐-M_λ| _ℓ^p(ℤ^d) →ℓ^p(ℤ^d)≲ 1.Combining this bound and (<ref>), we conclude that the ℓ^p-boundedness of the maximal operator A_* follows from the ℓ^p-boundedness of M_*. The following proposition, which we establish in Section <ref>, then completes the proof of Theorem <ref>. If k ≥ 3, d> 2k, and d/d-k < p ≤ 2, then M_* is bounded on ℓ^p(ℤ^d).§ MINOR ARC ANALYSISOur minor arc analysis splits naturally in two steps. The first step is a reduction to mean value estimates related to Waring's problem; for this we use a technique introduced in <cit.>. We then apply recent work on Waring's problem to estimate the relevant mean values and to prove Proposition <ref>. §.§ Reduction to mean value theoremsThe reduction step is based on the following lemma, a special case of Lemma 7 in <cit.>. In the present form, the result is a slight variation of Lemma 4.2 in <cit.> and is implicit also in <cit.>.For λ∈ℕ, let T_λ be a convolution operator on ℓ^2(ℤ^d) with Fourier multiplier given byT_λ(ξ) := ∫_𝔅 K(θ; ξ) e(-λθ) dθ,where 𝔅⊆𝕋 is a measurable set and K( ·; ξ) ∈ L^1(𝕋) is a kernel independent of λ. Further, for Λ≥ 2, define the dyadic maximal functionsT_*f(𝐱) = T_*(𝐱; Λ) := sup_λ∈ [Λ/2,Λ) |T_λ f(𝐱)|.Then T_* _ℓ^2(ℤ^d) →ℓ^2(ℤ^d)≤∫_𝔅sup_ξ∈𝕋^d| K(θ; ξ) | dθ.For a measurable set 𝔅⊂𝕋, we haveA_λ^𝔅(ξ)= λ^1-d/k∫_𝔅ℱ_N(θ; ξ) e(-λθ) dθ,whereℱ_N(θ; ξ) := ∏_j=1^d S_N(θ, ξ_j)with S_N(θ, ξ) := ∑_|n| ≤ N e(θ |n|^k + ξ n).Thus, in the proof of Proposition <ref>, we apply (<ref>) with K = ℱ_N and 𝔅 = 𝔪. The supremum over ξ on the right side of (<ref>) then stands in the way of a direct application of known results from analytic number theory. Our next lemma overcomes this obstacle; its proof is a variant of the argument leading to (12) in Wooley <cit.>. If k ≥ 2, l ≤ k-1 and s are natural numbers and 𝔅⊆𝕋 a measurable set, then∫_𝔅sup_ξ∈𝕋 |S_N(θ, ξ) |^2sdθ≲ N^l(l+1)/2∫_𝔅∫_𝕋^l| ∑_n=1^N e(θ n^k + ξ_l n^l + … + ξ_1 n) |^2s dξ dθ + 1. DefineS_N(θ, ξ) = ∑_1 ≤ n ≤ N e(θ n^k + ξ n).Sincesup_ξ∈𝕋 |S_N(θ, ξ)| ≤ 2sup_ξ∈𝕋 |S_N(θ, ξ)| + 1,it suffices to establish (<ref>) with S_N in place of S_N. Set H_j = sN^j. For 𝐡 = (h_1, …, h_l) ∈ℤ^l, we define a_𝐡(θ) = ∑_1 ≤ n_1, …, n_s ≤ Nδ(𝐧; 𝐡) e(θ𝔣_s,k(𝐧)),where δ(𝐧; 𝐡) =1if 𝔣_s,j(𝐧) = h_jfor allj = 1, …, l, 0otherwise.We haveS_N(θ, ξ)^s = ∑_h_1 ≤ H_1⋯∑_h_l ≤ H_l a_𝐡(θ)e(ξ h_1),so by applying the Cauchy–Schwarz inequality we deduce thatsup_ξ |S_N(θ, ξ)|^2s≤ H_1 ⋯ H_l ∑_h_1 ≤ H_1⋯∑_h_l ≤ H_l |a_𝐡(θ)|^2.Hence, ∫_𝔅sup_ξ |S_N(θ, ξ) |^2sdθ≲ N^l(l+1)/2∫_𝔅∑_h_1 ≤ H_1⋯∑_h_l ≤ H_l a_𝐡(θ)a_𝐡(θ)dθ.We have ∑_h_1 ≤ H_1⋯∑_h_l ≤ H_l a_𝐡(θ)a_𝐡(θ) = ∑_1 ≤𝐧, 𝐦≤ N e(θ( 𝔣_s,k(𝐧) - 𝔣_s,k(𝐦))) ∑_h_1 ≤ H_1⋯∑_h_l ≤ H_lδ(𝐧; 𝐡)δ(𝐦; 𝐡).By orthogonality, ∑_h_1 ≤ H_1⋯∑_h_l ≤ H_lδ(𝐧; 𝐡)δ(𝐦; 𝐡)= ∑_h_1 ≤ H_1⋯∑_h_l ≤ H_lδ(𝐦; 𝐡) ∫_𝕋^l e( ∑_j=1^l ξ_j( 𝔣_s,j(𝐧) - h_j ) ) dξ= ∫_𝕋^l e( ∑_j=1^l ξ_j( 𝔣_s,j(𝐧) - 𝔣_s,j(𝐦)) ) dξ,since for a fixed 𝐦, the sum over 𝐡 has exactly one term (in which h_j = 𝔣_s,j(𝐦)). Hence,∑_h_1 ≤ H_1⋯∑_h_l ≤ H_la_𝐡(θ)a_𝐡(θ) = ∫_𝕋^l| ∑_n=1^N e( θ n^k + ξ_ln^l + … + ξ_1n) |^2s dξ. The lemma follows from (<ref>)–(<ref>).With slight modifications, the argument of Lemma <ref> yields also the following estimate∫_𝔅∫_𝕋 |S_N(θ, ξ) |^2sdξ dθ≲ N^l(l+1)/2-1∫_𝔅∫_𝕋^l| ∑_n=1^N e(θ n^k + ξ_l n^l + … + ξ_1 n) |^2s dξ dθ + 1. §.§ Mean value theoremsWe now recall several mean value estimates from the literature on Waring's problem. The first is implicit in the proof of Theorem 10 of Bourgain <cit.>, which is a variant of a well-known lemma of Hua (see Lemma 2.5 in <cit.>). The present result follows from eqn. (6.6) in <cit.>. We note that when l = k, the left side of (<ref>) turns into Vinogradov's integral J_s,k(N) and the lemma turns into the main result of Bourgain, Demeter and Guth <cit.>. If k ≥ 3, 2 ≤ l ≤ k and s ≥1/2l(l+1) are natural numbers, then ∫_𝕋∫_𝕋^l-1| ∑_n=1^N e(θ n^k + ξ_l-1 n^l-1 + … + ξ_1 n) |^2s dξ dθ≲N^2s - l(l+1)/2 + .For small k, we will use another variant of Hua's lemma due to Brüdern and Robert <cit.>. The following is a weak form of Lemma 5 in <cit.>.If k ≥ 3 and 2 ≤ l ≤ k are natural numbers, then∫_𝕋^2| ∑_n=1^N e(θ n^k + ξ n) |^2^l+2 dξ dθ≲ N^2^l-l+1+. Note that by Remark <ref> (with l-1 in place of l) and Lemma <ref> we obtain a version of (<ref>) with l(l+1) in place of 2^l + 2. Together with Lemma <ref>, this observation yields the following bound. If k ≥ 3 and 2 ≤ l ≤ k are natural numbers and r ≥min{ l(l+1),2^l+2 }, then∫_𝕋^2| ∑_n=1^N e(θ n^k + ξ n) |^r dξ dθ≲ N^r-l-1+.We also use a variant of Lemma <ref> that provides extra savings when the integration over θ is restricted to a set of minor arcs. Lemma <ref>, a slight modification of Theorem 1.3 in Wooley <cit.>, improves on (<ref>) in the case l = k-1.Here, 𝔪 is the set of minor arcs defined at the beginning of <ref>. If k ≥ 3 and s ≥1/2k(k+1) are natural numbers, then∫_𝔪∫_𝕋^k-2| ∑_n=1^N e(θ n^k + ξ_k-2 n^k-2 + … + ξ_1 n) |^2s dξ dθ≲ N^2s - k(k - 1)/2-2+. The main point in the proof of Theorem 2.1 in Wooley <cit.> is the inequality (see p. 1495 in <cit.>)∫_𝔪∫_𝕋^k-2| ∑_n=1^N e(θ n^k + ξ_k-2 n^k-2 + … + ξ_1 n) |^2s dξ dθ≲ N^k-2(log N)^2s+1 J_s,k(2N).The lemma follows from this inequality and the Bourgain–Demeter–Guth bound for Vinogradov's integral J_s,k(2N) (the case l = k of (<ref>)).Now we interpolate between the above bounds.If k ≥ 3 and 2 ≤ l ≤ k-1 are natural numbers and r is real, with r_1(k,l) := k^2-k l - min(l^2+l, 2^l+2)/k-l+1≤ r ≤ k^2+k,thenI_r,k(N) := ∫_𝔪sup_ξ∈𝕋| S_N(θ, ξ) |^r dθ≲ N^r-k-δ(r) +,where δ(r) is the linear function of r with values δ(r_1) = 0 and δ(k^2+k) = 1.Let r_0 = min{ l^2+l, 2^l+2 }. The hypothesis on r implies that r_0 < r ≤ k^2+k, so we can find t ∈ [0,1) such that r = t r_0 + (1-t) (k^2 + k). By Lemma <ref> with l = 1 and Corollary <ref>, I_r_0,k(N) ≲ N ∫_𝕋^2| ∑_n=1^N e(θ n^k + ξ n) |^r_0 dξ dθ≲ N^r_0 - l + .On the other hand, Lemma <ref> with l = k-2 and Lemma <ref> yieldI_k(k+1),k(N) ≲ N^(k-1)(k-2)/2 N^k(k+1)-k(k-1)/2-2+ = N^k^2-1+.Using Hölder's inequality and the above bounds, we getI_r,k(N)≲ I_k(k+1),k(N)^1-t I_r_0,k(N)^t ≲ N^(1-t)(k^2-1+) N^t(r_0-l+)≲ N^(1-t)(k^2+k) + t r_0 N^-(1-t)(k+1)-tl += N^r-k-1+t(k-l+1) + .This inequality takes the form (<ref>) with δ(r) = 1 - t(k-l+1). Since t depends linearly on r and t = 0 when r = k^2+k, δ(r) is a linear function of r with δ(k^2+k) = 1. The value of r_1(k,l) in (<ref>) is the unique solution of the linear equation δ(r) = 0.§.§ Proof of Proposition <ref> By Lemma <ref> and the arithmetic-geometric mean inequality,sup_λ∈ [Λ/2,Λ) |A_λ^𝔪| _ℓ^2(ℤ^d) →ℓ^2(ℤ^d)≤ N^k-d∫_𝔪sup_ξ∈𝕋^d| ℱ_N(θ; ξ) | dθ≤ N^k - dI_d,k(N),where I_d,k(N) is the integral defined in (<ref>). Thus, the proposition will follow, if we prove the inequalityI_d,k(N) ≲ N^d - k - δ +with δ = kδ_0(d,k).Let l_0(k) denote the value of l for which the maximum in the definition of d_0(k) is attained (recall (<ref>)). When d_0(k) < d ≤ k^2+k, we may apply (<ref>) with r=d and l = l_0 to deduce (<ref>) with δ = δ(d). We now observe that when d ≤ k^2+k, we have δ(d) = kδ_0(d,k) and that the hypothesis d > d_0(k) ensures that δ(d) > 0. When d > k^2+k, we enhance our estimates with the help of the L^∞-bound for S_N(θ, ξ) on the minor arcs: by combining a classical result of Weyl (see Lemma 2.4 in Vaughan <cit.>) and Theorem 5 in Bourgain <cit.>, we havesup_(θ, ξ) ∈𝔪×𝕋| S_N (θ, ξ) | ≲ N^1-τ_k+,τ_k being the quantity that appears in the definition of δ_0(d,k). Thus, when d > k^2+k, we haveI_d,k(N) ≲ N^(d-k^2-k)(1-τ_k) + I_k(k+1),k(N) ≲ N^d-k-1 - (d-k^2-k) τ_k + .We conclude that (<ref>) holds with δ = 1 + (d - k^2-k)τ_k.§ MAJOR ARC ANALYSISWe will proceed through a series of successive approximations to A_λ^a/q, which we will define by their Fourier multipliers. Our approximations are based on the following major arc approximation for exponential sums that appears as part of Theorem 3 of Brüdern and Robert <cit.>. In this result and throughout the section, we writeG(q; a, b) := q^-1∑_x ∈ℤ_q e_q( ax^k + bx ) and v_N(θ, ξ) := ∫_0^N e( θ t^k + ξ t) dt.Let θ, ξ∈𝕋, q ∈ℕ, a ∈ℤ_q^*, and b ∈ℤ, and suppose that| θ - a/q| ≤1/4kqN^k-1, | ξ - b/q| ≤1/2q.Then∑_n ≤ N e(θ n^k + ξ n) = G(q; a,b)v_N(θ - a/q, ξ - b/q) + O( q^1-1/k+).Recall the definition of S_N(θ,ξ) from Section <ref>. When a/q + θ lies on a major arc 𝔐(a/q) and ξ = b/q + η, with |η| ≤ 1/(2q), the above lemma yieldsS_N(a/q+θ, ξ) = G(q; a,b)v_N(θ, η) + G(q; a,-b)v_N(θ, -η) + O( q^1-1/k+).We will use this approximation in conjunction with the following well-known bounds (see Theorems 7.1 and 7.3 in Vaughan <cit.>):G(q; a, b) ≲ q^-1/k+,v_N(θ, η) ≲ N(1 + N|η| + N^k|θ|)^-1/k.We will make use also of the inequalityv_N(θ, η) ≲ N(1 + N|η|)^-1/2,which follows from the representationv_N(θ, η) = 1/k∫_0^N^k u^1/k-1 e(θ u + η u^1/k) duand the second-derivative bound for oscillatory integrals (see p. 334 in <cit.>). §.§ Proof of Proposition <ref> The bulk of the work concerns the case p=2 of the proposition.SinceA_λ^a/q(ξ) = λ^1-d/k e_q(-λ a) ∫_𝔐(0/q)ℱ_N(a/q + θ; ξ) e(-λθ) dθ,the asymptotic (<ref>) suggests that the following multiplier should provide a good approximation to A_λ^a/q(ξ):B_λ^a/q(ξ) := λ^1-d/k e_q(-λ a) ∫_𝔐(0/q)𝒢_N(θ; ξ)e(-λθ) dθ,where 𝒢_N(θ; ξ) := ∏_j=1^d { G(q; a, b_j)v_N(θ, η_j) + G(q; a, -b_j)v_N(θ, -η_j)},with b_j the unique integer such that -1/2≤ b_j - qξ_j < 1/2 and η_j = ξ_j - b_j/q. Let B_λ^a/q denote the operator on ℓ^2(ℤ^d) with the above Fourier multiplier. To estimate the ℓ^2-error of approximation of A_λ^a/q by B_λ^a/q, we use that when θ∈𝔐(0/q), (<ref>) and (<ref>) yield| ℱ_N(a/q+θ; ξ) - 𝒢_N(θ; ξ) | ≲ q^1-d/k+N^d-1(1 + N^k|θ|)^(1-d)/k.Thus, if d > k+1, we have∫_𝔐(0/q)sup_ξ∈𝕋^d | ℱ_N(a/q+θ; ξ) - 𝒢_N(θ; ξ) | dθ ≲∫_ℝq^1-d/k+N^d-1dθ/(1 + N^k|θ|)^(d-1)/k≲ q^1-d/k+N^d-k-1.Lemma <ref> then yieldssup_λ∈ [Λ/2,Λ)| A_λ^a/q - B_λ^a/q| _ℓ^2(ℤ^d) →ℓ^2(ℤ^d)≲ q^1-d/k+N^-1. Next, we approximate B_λ^a/q by the operator C_λ^a/q with multiplierC_λ^a/q(ξ) := λ^1-d/k e_q(-λ a) ∑_𝐛∈ℤ^dψ(qξ - 𝐛)∫_𝔐(0/q)𝒢_N(θ; ξ)e(-λθ) dθ.By the localization of ψ, the above sum has at most one term in which 𝐛 matches the integer vector that appears in the definition of 𝒢_N(θ; ξ). Hence, 𝒢_N(θ; ξ)( 1 - ∑_𝐛∈ℤ^dψ(qξ - 𝐛) )is supported on a set where 1/8≤ |qξ_j - b_j| ≤1/2 for some j. For such j, by (<ref>),v_N(θ, ξ_j - b_j/q) ≲ (qN)^1/2,and we conclude that|𝒢_N(θ; ξ)|( 1 - ∑_𝐛∈ℤ^dψ(qξ - 𝐛) ) ≲ q^1/2-d/k+N^d-1/2(1 + N^k|θ|)^(1-d)/k.So, when d > k+1, Lemma <ref> givessup_λ∈ [Λ/2,Λ)| B_λ^a/q - C_λ^a/q| _ℓ^2(ℤ^d) →ℓ^2(ℤ^d) ≲∫_ℝq^1/2-d/k+N^k-1/2dθ/(1 + N^k|θ|)^(d-1)/k≲ q^1/2-d/k+N^-1/2. In our final approximation, we replace C_λ^a/q by the operator D_λ^a/q with multiplierD_λ^a/q(ξ) := λ^1-d/k e_q(-λ a) ∑_ w ∈{± 1 }^d∑_𝐛∈ℤ^dψ(qξ - 𝐛) G(q; a,w𝐛) J_λ( w(ξ - q^-1𝐛)),where w𝐛 = (w_1b_1, …, w_db_d) andJ_λ(η) := ∫_ℝ{∏_j=1^d v_N(θ; η_j) } e(-λθ) dθ.We remark that C_λ^a/q(ξ) can be expressed in a matching form, with J_λ(ξ - q^-1 w𝐛) replaced by the analogous integral over 𝔐(0/q).Thus, when d > k, we deduce from Lemma <ref> and (<ref>) thatsup_λ∈ [Λ/2,Λ)| C_λ^a/q - D_λ^a/q| _ℓ^2(ℤ^d) →ℓ^2(ℤ^d) ≲∫_𝔐(0/q)^cq^-d/k+N^kdθ/(1 + N^k|θ|)^d/k≲ q^-1+N^1-d/k.Here, 𝔐(0/q)^c denotes the complement of the interval 𝔐(0/q) in ℝ. Finally, we note that D_λ^a/q is really M_λ^a/q.Indeed, by the discussion on p. 498 in Stein <cit.> (see also Lemma 5 inMagyar <cit.>), we have J_λ(η)= ∫_ℝ∫_ℝ^d1_[0,N]^d(𝐭)e(η·𝐭) e(θ(𝔣(𝐭) - λ)) d𝐭 dθ= λ^d/k-1∫_ℝ^d1_[0,N]^d(𝐭)e(η·𝐭) dσ_λ(𝐭) = λ^d/k-1dσ_λ(η),since the surface measure dσ_λ is supported in the cube [0,N]^d. Combining this observation and (<ref>)–(<ref>), and summing over a,q, we conclude that when d > 2k, ∑_q ≤ N∑_a ∈ℤ_q^*sup_λ∈ [Λ/2,Λ)| A_λ^a/q - M_λ^a/q| _ℓ^2(ℤ^d) →ℓ^2(ℤ^d)≲ N^-γ+, with γ = min(d/k-2, 1/2) > 0. In particular, when d ≥5/2 k, we have γ = 1/2. We can now finish the proof of Proposition <ref>. For brevity, we write p_1 = d/d-k. From (<ref>), we obtain thatsup_λ∈ [Λ/2,Λ)| A_λ^𝔐 - M_λ| _ℓ^2(ℤ^d) →ℓ^2(ℤ^d)≲ N^-γ+.On the other hand, we know from Propositions <ref> and <ref> that both A_Λ^𝔐 and M_* are bounded on ℓ^p(ℤ^d) when p_1 < p ≤ 2; hence,sup_λ∈ [Λ/2, Λ)| A_λ^𝔐-M_λ| _ℓ^p(ℤ^d) →ℓ^p(ℤ^d)≲ 1.When p ∈ (p_0(d,k), 2], we interpolate between (<ref>) and (<ref>) with p = p_1 + for a sufficiently small > 0. The interpolation yields (<ref>) with β_p > 0 that can be chosen arbitrarily close to 2γ(p-p_1)/ (kp(2-p_1)).§ MAIN TERM CONTRIBUTIONIn this section, we prove Proposition <ref>. First, we obtain L^p(ℝ^d)-bounds for the maximal function of the continuous surface measures dσ_λ. If k ≥ 2, d>3/2k and p > 2d-k/2d - 2k, then for all f ∈ L^p(ℝ^d), sup_λ > 0 |f * dσ_λ| _L^p(ℝ^d)≲f_L^p(ℝ^d). We will deduce the lemma from a result of Rubio de Francia – Theorem A in <cit.> – which reduces (<ref>) to bounds for the Fourier transform of the measure dσ_λ. First, we majorize the measure dσ_λ by a smooth one. By the choice of normalization of dσ_λ, we havef * dσ_λ(𝐱) = ∫_ℝ^d f(𝐱 - 𝐲) dσ_λ(𝐲) = ∫_ℝ^d f(𝐱 - t𝐲) dσ_1(𝐲),where t = λ^1/k. Let ϕ be a smooth function supported in [-1/2, 3/2] and such that 1_[0,1](x) ≤ϕ(x), and write ϕ(𝐱) = ϕ(x_1) ⋯ϕ(x_d). Since dσ_1 is supported inside the unit cube [0,1]^d, we have|∫_ℝ^d f(𝐱 - t𝐲) dσ_1(𝐲) | ≤∫_ℝ^d |f(𝐱 - t𝐲)| ϕ(𝐲) dσ(𝐲) =: ∫_ℝ^d |f(𝐱 - t𝐲)| dμ(𝐲),where dσ is the surface measure on the smooth manifold x_1^k + … + x_d^k = 1. By Rubio de Francia's theorem, the maximal function𝒜_tf(𝐱) := sup_t > 0∫_ℝ^d |f(𝐱 - t𝐲)| dμ(𝐲)is bounded on L^p(ℝ^d), provided thatμ(ξ) ≲ (|ξ| + 1)^-afor somea > min{1/2p-2,1/2}. Thus, the lemma will follow, if we establish (<ref>) with a = d/k - 1 and d > k+1 because a>1/2 once d>3/2k.We now turn to (<ref>). Similarly to (<ref>), we haveμ(ξ) = ∫_ℝ∫_ℝ^dϕ(𝐱)e(ξ·𝐱) e(θ(𝔣(𝐱)-1)) d𝐱 dθ = ∫_ℝ{∏_j=1^d v_ϕ(θ, ξ_j) }e(-θ) dθ,wherev_ϕ(θ, ξ) := ∫_ℝϕ(x)e(θ x^k + ξ x) dx.By the corollary on p. 334 of Stein <cit.>, we havev_ϕ(θ, ξ) ≲ (1 + |θ|)^-1/k, uniformly in ξ. On the other hand, if k2^k|θ| ≤ |ξ|, we have| d/dx( ξ^-1θ x^k + x ) | ≥1/2, | d^j/dx^j( ξ^-1θ x^k + x ) | ≲_j 1on the support of ϕ. Hence, Proposition VIII.1 on p. 331 of Stein <cit.> yieldsv_ϕ(θ, ξ) ≲_M (1 + |ξ|)^-M for any fixed M ≥ 1. We now choose an index j, 1 ≤ i ≤ d, such that |ξ| ≤ d|ξ_i| and set θ_0 = |ξ_i|/(k2^k). We apply (<ref>) to the trigonometric integrals v_ϕ(θ,ξ_j), ji, and to v_ϕ(θ,ξ_i) when |θ| > θ_0; we apply (<ref>) to v_ϕ(θ,ξ_i) when |θ| ≤θ_0. From these bounds and the integral representation for μ(ξ), we obtainμ(ξ)≲∫_|θ| ≤θ_0(1 + |ξ_i|)^-M/(1 + |θ|)^(d-1)/kdθ + ∫_|θ| > θ_0dθ/(1 + |θ|)^d/k≲ (1 + |ξ_i|)^-M + (1 + |ξ_i|)^1-d/k≲ (1+|ξ|)^1-d/k,provided that M ≥ d/k - 1 and d > k+1.This completes the proof.§.§ Proof of Proposition <ref> To prove Proposition <ref>, it suffices to prove that (uniformly in a and q)sup_λ∈ℕ| M_λ^a/q| _ℓ^p(ℤ^d) →ℓ^p(ℤ^d)≲q^-d/k(2-2/p)+,for all d/d-k < p ≤ 2 and d>k. The proposition then follows by summing over a and q (the hypothesis on p ensures that the resulting series over q is convergent). Fix q ∈ℕ and a ∈ℤ_q^* and write ψ_1(ξ) = ψ(ξ/2) (so that ψ = ψψ_1). We borrow a trick from Magyar, Stein and Wainger <cit.> to express M_λ^a/q (ξ) as a linear combination of Fourier multipliers that separate the dependence on λ and from the dependence on a/q: M_λ^a/q(ξ)= e_q(-λ a) ∑_ w ∈{± 1 }^d∑_𝐛∈ℤ^dψ(qξ - 𝐛) G(q; a,w𝐛) dσ_λ( w(ξ - q^-1𝐛)) = e_q(-λ a) ∑_ w ∈{± 1 }^d( ∑_𝐛∈ℤ^dψ_1(qξ - 𝐛) G(q; a,w𝐛) ) ( ∑_𝐛∈ℤ^dψ(qξ - 𝐛) dσ_λ( w(ξ - q^-1𝐛)) )=: e_q(-λ a) ∑_ w ∈{± 1 }^dS_ w^a/q(ξ) T_λ, w^q(ξ).Since w takes on precisely 2^d values for each a/q, it suffices to prove that sup_λ∈ℕ|T_λ, w^q∘ S_ w^a/q| _ℓ^p(ℤ^d) →ℓ^p(ℤ^d)≲ q^-d/k(2-2/p)+,uniformly for w ∈{-1,1}^d. To prove (<ref>), we will first bound the maximal function over the `Archimedean' multipliers T_λ, w^q, and then we will bound the non-Archimedean multipliers S_ w^a/q.This is possible because S_ w^a/q is independent of λ∈ℕ. For d > 3/2k and p>2d-k/2d-2k, Lemma <ref> and Corollary 2.1 of <cit.> (the `Magyar–Stein–Wainger transference principle') give the boundsup_λ∈ℕ| T_λ, w^qg | _ℓ^p(ℤ^d)≲g_ℓ^p(ℤ^d),with an implicit constant independent of q and w. We now observe that S_ w^a/q does not depend on λ and apply (<ref>) with g = S_ w^a/qf to find that sup_λ∈ℕ|T_λ, w^q∘ S_ w^a/q| _ℓ^p(ℤ^d) →ℓ^p(ℤ^d)≲ S_ w^a/q_ℓ^p(ℤ^d) →ℓ^p(ℤ^d),under the same assumptions on d and p which we note are weaker than the hypotheses of the proposition. Finally, observe that G(q; a, b) is a q-periodic function with ℤ_q-Fourier transform equal to ∑_b ∈ℤ_q e_q(-mb) G(q; a, b) =q^-1∑_x ∈ℤ_q e_q( ax^k ) ∑_b ∈ℤ_q e_q( b(x-m) ) = e_q( am^k ),for each m ∈ℤ_q. Hence, we may apply Proposition 2.2 in <cit.> and the bound (<ref>) for G(q; a, b) to deduce that S_ w^a/q_ℓ^p(ℤ^d) →ℓ^p(ℤ^d)≲ q^-d/k(2-2/p)+.The desired inequality (<ref>) follows immediately from (<ref>) and (<ref>). § ACKNOWLEDGMENTSThe first author was supported by NSF grant DMS-1502464. The second author was supported by NSF grant DMS-1147523 and by the Fields Institute. Last but not least, the authors thank Alex Nagel and Trevor Wooley for several helpful discussions.amsplain 99ACHK:WaringGoldbach T. C. Anderson, B. Cook, K. Hughes, and A. Kumchev, On the ergodic Waring–Goldbach problem, preprint arXiv:1703.02713.AS M. Avdispahić and L. Smajlović,On maximal operators on k-spheres in ℤ^n, Proc. Amer. Math. Soc. 134 (2006), no. 7, 2125–2130. Bourgain_Vinogradov J. Bourgain,On the Vinogradov mean value, Tr. Mat. Inst. Steklova 296 (2017), 36–46. BourgainDemeterGuth J. Bourgain, C. Demeter, and L. Guth, Proof of the main conjecture in Vinogradov's mean value theorem for degrees higher than three, Ann. Math. (2) 184 (2016), no. 2, 633–682. BrudernRobert J. Brüdern and O. Robert, Rational points on linear slices of diagonal hypersurfaces, Nagoya Math. J. 218 (2015), 51–100. Hughes_Vinogradov K. Hughes, Maximal functions and ergodic averages related to Waring's problem, Israel J. Math. 217 (2017), no. 1, 17–55.Hughes_restricted K. Hughes, Restricted weak-type endpoint estimates for k-spherical maximal functions, Math. Z. (2017), in press.Ionescu A. D. Ionescu, An endpoint estimate for the discrete spherical maximal function, Proc. Amer. Math. Soc. 132 (2004), no. 5, 1411–1417.Magyar_dyadic A. Magyar, L^p-bounds for spherical maximal operators on ℤ^n, Rev. Mat. Iberoam. 13 (1997), no. 2, 307–317.Magyar:ergodic A. Magyar, Diophantine equations and ergodic theorems, Amer. J. Math. 124 (2002), no. 5, 921–953. MSW A. Magyar, E. M. Stein, and S. Wainger, Discrete analogues in harmonic analysis: Spherical averages, Ann. Math. (2) 155 (2002), no. 1, 189–208.Pierce_mean_values L. B. Pierce, On discrete fractional integral operators and mean values of Weyl sums, Bull. Lond. Math. Soc. 43 (2011), no. 3, 597–612.Rubio J. L. Rubio de Francia, Maximal functions and Fourier transforms, Duke Math. J. 53 (1986), no. 2, 395–404. SteinHA E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993; with the assistance of T. S. Murphy. Stein_Wainger_discrete_fractional_integration_revisited E. M. Stein and S. Wainger, Two discrete fractional integral operators revisited, J. Anal. Math. 87 (2002), 451–479.Vaughan_cubes R. C. Vaughan, On Waring's problem for cubes, J. Reine Angew. Math. 365 (1986), 122–170.Vaughan R. C. Vaughan, The Hardy–Littlewood Method, Second ed., Cambridge University Press, Cambridge, 1997.Wooley_asymptotic T. D. Wooley, The asymptotic formula in Waring's problem, Int. Math.Res. Not. IMRN (2012), no. 7, 1485–1504. Wooley_cubic T. D. Wooley, The cubic case of the main conjecture in Vinogradov's mean value theorem, Adv. Math. 294 (2016), 532–561.[TA] Theresa C. AndersonDepartment of Mathematics University of Wisconsin–Madison Madison, WI 53705, U.S.A.tcandersonmathwiscedu[BC] Brian CookDepartment of Mathematicl Sciences Kent State University Kent, OH 44242U.S.A.bcook25kentedu [KH] Kevin HughesSchool of Mathematics The University of Bristol Bristol, BS8 1TW, UKKevin.Hughesbristolacuk [AK] Angel KumchevDepartment of Mathematics Towson University Towson, MD 21252, U.S.A.akumchevtowsonedu | http://arxiv.org/abs/1707.08667v2 | {
"authors": [
"Theresa C. Anderson",
"Brian Cook",
"Kevin Hughes",
"Angel Kumchev"
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"published": "20170726235141",
"title": "Improved $\\ell^p$-Boundedness for Integral $k$-Spherical Maximal Functions"
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p_c, p_u and graph limitsItai Benjamini2.7.17 ========================== A question relating the critical probability for percolation, the critical probability for a unique infinite cluster and graph limits is presented, together with some partial results. MSC subversive math § A QUESTION Let G be an infinite Cayley graph rooted at o. Denote by B(o, n) the ball of radius n centered at o, and let H be the limit, in the sense of <cit.>, of the sequence of balls B(o, 1),B(o,2)..., possibly passing to a subsequence to get convergence. H is a unimodular random graph, <cit.>.Does p_c(H) = p_u(G) a.s. ? In p-Bernoulli site percolation on a graph, each vertex is open with probability p independently. p_c denotes the critical probability for the existence of an infinite open cluster (connected component), and p_u isthe critical probability for uniqueness of the infinite open cluster,see <cit.>. A limit, a la <cit.>, of finite graphs G_nis a random rooted infinite graph with the property that neighborhoods of G_n around a random uniform vertex converge in distribution to neighborhoods of the infinite graph around the root. When G is a Cayley graph it might be the case that the limit along balls exists, and there is no need to pass to a subsequence? Neither one of the directions in question <ref> seems immediate. For a very simple example take G to be the 3-regular tree, T_3, then H is the canopy tree, which can be thought of as an “infinite binary tree viewed from a leaf". p_c(H) = p_u(G) =1. While for amenable Cayley graphs p_c = p_u <cit.>, it is conjectured that p_c < p_u for non-amenable Cayley graphs <cit.>. It was shown in <cit.> that every non-amenable group admits a generating set for which p_c < p_u. Showing that p_c(H) > p_c(G) fornon-amenable G, will bean interesting progress towards question <ref>. In <cit.>it is proved that p > p_u is equivalent to non-decay of the connection probability. For an amenable Cayley graph G, the limit along a Følner sequence is a.s. G.Recall the critical parameters for weak survival of the contact process λ_c and of strong survival λ_s, see <cit.> <cit.>.Does λ_c(H) = λ_s(G)a.s. ?Question <ref>is already non trivial for T_d, and it will be great to see a proof. § PARTIAL RESULTS AND FURTHER QUESTIONS Given a Cayley graph G, it is often hard to understand the structure of H. Still we now present several families of graphs for which the answer to question <ref> is yes. Planar triangular latticesWhen G is a cocompact triangular lattice in the hyperbolic plane, H is a.s. a triangulated quasi-canopy tree. That is, a tree quasi isometric to the canopy tree. For simplicity we will assume it's the canopy tree. View the canopy tree as an infinite ray with growing trees rooted along the ray. For every p_c(G) < p < p_u(G), and any fixed vertex von the ray, there is a positive probabilityof an open path from v to the leaves. This gives a cutset separating o from infinity. When p > p_u(G),the probabilities of (1-p)- open paths from v to the leaves decays exponentially with the distance from v to o. As the open path has to exit a ball of radius dist(v, o) centered at v, which is isometric to a ball in the transitive lattice G. Therefore restricted to the ball thisevent has identical probability to the event of an open path of a subcritical percolationon G, <cit.> exiting a ball of radius dist(v, o), and by <cit.> this probability decays exponentially. Verify question <ref>forplanar stochastic hyperbolic triangulations, <cit.>?Graph productsIt is interestingto consider next the conjecture for T_d×, having a non planar instance with p_u < 1 analyzed. Tom Hutchcroft challenged us with the following comment. Grimmett and Newman's paper <cit.> implies the following statement: Take the product of the d-regular tree with , and replace every -edge with a path of length n. Then no matter how large n is, p_u of the resulting graph is always bounded from above by (d-1)^-1/2. If question <ref>has a positive answer, that would imply that if we take the product of the canopy tree withand stretch the lengths of the -edgesas much as we want, then p_c is always bounded above by (d-1)^-1/2 also. Tom (private communication) proved that this is indeed the case. This supports a positive answer for T_d ×. A first step towards a positive answer to question <ref> suggested by Tom, is to show that p_u(T_d ×) = p_u(T_d ×). Is there non-uniqueness at p_u on T_d ×?Percolation on T_d × at p_u admits infinitely many infinite clusters a.s., while on planar hyperbolic lattices there is a unique infinite cluster at p_u, <cit.>, <cit.>. What is the behaviour at p_c for the corresponding H's?Does p_c(×) = p_c(×)? Show no percolation at criticality on ×?When a Cayley graph G is transient for the simple random walk, one expects that the infinite clusters of Bernoulli percolation are a.s. transient as well. This is known for many families of Cayley graphs but not in general. Show that the infinite cluster of supercritical Bernoullipercolation on × is a.s. transient? Free productsIn <cit.> it was suggested that for any infinite Cayley graph G, the graph limit along a converging subsequence of balls, H, is invariantly amenable, and therefore p_c(H) = p_u(H) and λ_c(H) = λ_s(H). In <cit.> it is proved that if G has Yu's property A, then H is hyperfinite or invariantly amenable. These include hyperbolic groups, groups with finiteasymptotic dimension and amenable groups. (<cit.> contains many further questions regarding graph limits along balls).Other Cayley graphs Tom Hutchcroftsuggested to look at, as a possible counterexamples, are free product of ^2 and an edge, anda free product of a hyperbolic Cayley graph and an edge.The free product of ^2 and an edge has finite asymptotic dimension, <cit.>. Hence the limit along balls is invariantly amenable (or hyperfinite) unimodular random graphand thus the limiting graph has one end a.s., see <cit.>. H is not contained in a finite number of copies of ^2as H has a uniform exponential lower bound on it's volume growth, as G contained a regular tree and therefore H contained a canopy.H intersects any copy of ^2 in a finite set, as otherwise we will get more than one end. Thus p_c(H) =1 a.s., since the infinitecluster has to cross infinitely many cut edges.The free product of an hyperbolic group G and an edge is hyperbolic. Limits along balls of hyperbolic groups is invariantly amenable, So the limit has one end a.s. and hencep_c(H) = 1 a.s., by the same argument as above.Assume a Cayley graph G has infinitely many ends. Show that p_c(H) =1.To answer question <ref>, one needs to show that the limit along balls of any Cayley graph which isnot quasi isometric to , has one end a.s. Final commentsIt is still open if there is a Cayley graph in which the sequencesof balls B(o, 1),B(o,2)... is an expander family, <cit.>? I conjecture that there is no such Cayley graph. In this case H will be non-amenable.Recall a sequence of finite subgraphsin a graphis exhausting, ifeach graph in the sequences contains all the previous subgraphs and the union is the whole graph.If G is hyperbolic and we consider the limit along an arbitrarily sequence ofexhausting finite sets, we expect to get p_c(H) ≥ p_u(G). This might hold for anyexhausting sequence of finite sets on any Cayley graph. Gromov's monster, <cit.>, might serve as a counter example. Let G be the Diestel-Leader graph DL(3,2), <cit.>,which is a non-amenable, non-unimodularvertex transitive graph. The limit along balls ofDL(3,2)is a horocyclic product of two canopy trees, see figure. Show that p_c(H) < 1. Does p_c(H) = p_u(G)? Does p_c(G) < p_u(G)?Maybe a version of question <ref> for the critical temperature formagnetization of the ferromagnetic Ising model, will be more tractable? We believe that the answer to question <ref>will still be positive if rather than taking limit along balls, weconsider "round" sets such as limit of isoperimetric minimizers, or minimizers along Green balls, or taking a sequence sets {K_n}, where K_n is a set of vertices of size n minimizing the expected escape probability of simple random walk starting from a random uniform element on the set.Assume G and G' are quasi isometric. Let H be the limit along some exhausting sequence of G. Is there an exhausting sequences of G', so that H' is quasi isometric to H? That is, there is a coupling of the two random rooted graphs H and H' which is a quasi isometry. We would like to end with a random walk question of similar spirit. In <cit.> it was proved that the limit of bounded degree finite planar graphs, is a.s. recurrent for the simple random walk.Which Cayley graphs has an exhausting sequence of subgraphs with a limit that is a.s. transient? One feature of transient bounded degreeplanar graphs, is that they admit non constant Dirichlet harmonic functions, that is, harmonic functions with a gradient in l^2, see <cit.> and <cit.>. Let G be an infinite transient Cayley graph. Assume G has no non constant Dirichlet harmonic functions. Is there an exhaustingsequence of finite subgraphs, such that H is a.s. transient? If there is such a sequence,probably the limit along growing balls is a.s. transient. There are no non constant Dirichlet harmonic functions on T_3 ×. The limit along boxes of T_3 × is a.s. a ×, which is transient. All examples we know, including limits along balls inproducts of trees, are Liouville,Are there examples in which limits along balls are a.s. non-Liouville?Acknowledgements: Thanks to Lewis Bowen,Tom Hutchcroft forvery useful discussions.BKCAL D. Aldous and R. Lyons, Processes on unimodular random networks. Electron. J. Probab. 12 (2007), no. 54, 1454–-1508.BDK G. Bell, A. Dranishnikov and J. Keesling, On a formula for the asymptotic dimension of free products. Fund. Math. 183 (2004), no. 1, 39–-45.BS96 I. Benjamini and O. Schramm, Harmonic functions on planar and almost planar graphs and manifolds, via circle packings. Invent. Math. 126 (1996), no. 3, 565-–587. B I. Benjamini, Expanders are not hyperbolic. Israel J. Math. 108 (1998), 33-–36.BE I. Benjamini and G. Elek, Sparse graph limits along balls (2017).BS96 I. Benjamini and O. Schramm, Percolation beyond ^d, many questions and a few answers. Electron. Comm. Probab. 1 (1996), no. 8, 71-–82.BS I. Benjamini and O. Schramm, Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6 (2001), no. 23, 13 pp.BS01 I. Benjamini and O. Schramm, Percolation in the hyperbolic plane. J. Amer. Math. Soc. 14 (2001), no. 2, 487-–507. BK R. Burton and M. Keane,Density and uniqueness in percolation. Comm. Math. Phys. 121 (1989), no. 3, 501–-505.C N. Curien, Planar stochastic hyperbolic triangulations. Probab. Theory Related Fields 165 (2016), no. 3-4, 509–-540.DL R. Diestel and I.Leader, A conjecture concerning a limit of non-Cayley graphs. J. Algebraic Combin. 14 (2001), no. 1, 17-–25. DT H. Duminil-Copin and V. Tassion, A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. Comm. Math. Phys. 343 (2016), no. 2, 725-–745.GN G. Grimmett and C. Newman, Percolation in ∞ +1 dimensions. Disorder in physical systems, 167–-190, Oxford Sci. Publ., Oxford Univ. Press, New York, 1990.G M. Gromov, Random walk in random groups. Geom. Funct. Anal. 13 (2003), no. 1, 73–-146. H T. Hutchcroft, Harmonic Dirichlet Functions on Planar Graphs. https://arxiv.org/abs/1707.07751L R. Lyons, Phase transitions on nonamenable graphs. Probabilistic techniques in equilibrium and nonequilibrium statistical physics. J. Math. Phys. 41 (2000), no. 3, 1099–-1126. LS R. Lyons and O. Schramm, Indistinguishability of percolation clusters. Ann. Probab. 27 (1999), no. 4, 1809–-1836.PS I. Pak and T. Smirnova-Nagnibeda, On non-uniqueness of percolation on nonamenable Cayley graphs. C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 6, 495-–500.P R. Pemantle, The contact process on trees. Ann. Probab. 20 (1992), no. 4, 2089-–2116.S H. Sako, Property A and the operator norm localization property for discrete metric spaces. J. Reine Angew. Math. 690 (2014) 207–216.Sh R. Schonmann,Percolation in ∞+1 dimensions at the uniqueness threshold. Perplexing problems in probability, 53–-67, Progr. Probab., 44, Birkhäuser Boston, Boston, MA, 1999. | http://arxiv.org/abs/1707.08544v3 | {
"authors": [
"Itai Benjamini"
],
"categories": [
"math.PR"
],
"primary_category": "math.PR",
"published": "20170726171048",
"title": "$p_c$, $p_u$ and graph limits"
} |
[pages=1-last]final_wacv_release.pdf | http://arxiv.org/abs/1707.08682v2 | {
"authors": [
"Wei Xiang",
"Dong-Qing Zhang",
"Heather Yu",
"Vassilis Athitsos"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170727015017",
"title": "Context-Aware Single-Shot Detector"
} |
Structure and decays ofnuclear three-body systems: the Gamow coupled-channel method in Jacobi coordinates F.R. Xu (许甫荣) December 30, 2023 ===========================================================================================================We present the results from a Gemini snapshot radial-velocity survey of 44 low-mass white dwarf candidates selected from the Sloan Digital Sky Survey spectroscopy. To find sub-hour orbital period binary systems, our time-series spectroscopyhad cadences of 2 to 8 min over a period of 20-30 min. Through follow-up observations at Gemini and the MMT, we identify four double degenerate binary systems with periods ranging from 53 min to 7 h. The shortest period system, SDSS J123549.88+154319.3, was recently identified as a subhour period detached binary by Breedt and collaborators. Here we refine the orbital and physical parameters of this system. High-speed and time domain survey photometry observations do not reveal eclipses or other photometric effects in any of our targets. We compare the period distribution of these four systems with the orbital period distribution of known double white dwarfs; the median period decreases from 0.64 to 0.24 d for M=0.3-0.5 M_⊙ to M<0.3 M_⊙ white dwarfs. However, we do not find a statistically significant correlation between the orbital period and white dwarf mass. binaries: close — white dwarfs — stars: individual (SDSS J083446.91+304959.2, J123549.88+154319.3, J123728.64+491302.6, J234248.86+081137.2) — supernovae: general — gravitational waves § INTRODUCTION There are now more than 90 short period binary white dwarfs known <cit.>, including more than three dozen systems that will merge within a Hubble time. The majority of the merger systems were found in the last 7 years, thanks to the Extremely Low Mass Survey <cit.>, which targets white dwarfs with logg<7 and M<0.3 M_⊙. Given the finite age of the universe, the only way to form ELM white dwarfs is through binary evolution, and we do in fact find almost 100% of ELM white dwarfs in short period systems. This is significantly higher than the binary fraction of 10% for the overall population of white dwarfs that were observed as part of the Supernova-Ia Progenitor surveY <cit.>. <cit.> estimate an ELM white dwarf merger rate of 3 × 10^-3 yr^-1 over the entire disk of the Milky Way. This is significantly larger than the AM CVn formation rate, indicating that most ELM white dwarf systems will merge. These merger systems, depending on the total mass of the binary, will likely form single subdwarfs, extreme helium stars, or single massive white dwarfs. The most likely outcome is an R Cor Bor star, since the ELM white dwarf merger rate is statistically identical to the R Cor Bor formation rate. These merger rates are dominated by the quickest merger systems, the ones with the shortest periods. There are currently five sub-hour orbital period detached double white dwarfs known; J0106-1000, J0651+2844, J1630+4233, WD 0931+444 <cit.>, and J1235+1543 <cit.>. The two shortest period systems with P<20 min, J0651+2844 and WD 0931+444, are verification sources for the Laser Interferometer Space Antenna <cit.>.The discovery of additional sub-hour orbital period systems is important for both precise white dwarf merger rate estimates and future space-based gravitational wave missions <cit.>.Here we present the results from a targeted search for sub-hour period binary white dwarfs from Gemini Observatory, with additional follow-up observations from the MMT. We discuss our target selection in Section 2, describe our follow-up spectroscopy and photometry in Section 3, and present the orbital solutions for four binaries in Section 4, including J1235+1543. <cit.> independently identified the latter as a sub-hour orbital period system based on the SDSS subspectra. Here we refine the orbital parameters of this system based on extended follow-up observations. We discuss the parameters of the four confirmed binary systems in our sample, as well as the implications of the results from this search in Section 5.§ TARGET SELECTION Figure <ref> shows the temperature versus period distribution of the binary white dwarfs in the ELM Survey <cit.>. This figure demonstrates that the shortest period systems also happen to be the hotter white dwarfs with T_ eff> 12,000 K. This is a direct consequence of gravitational wave emission: white dwarfs in the shortest period systems merge before they have a chance to cool down. Hence, we only see them when they are relatively young and hot. This provides an excellent, but currently under-utilized, selection mechanism for the shortest period binary systems. For example, 39% of the previously observed ELM white dwarfs hotter than 12,000 K are in binaries with P<0.1 d, with a median period of 65 min.We take advantage of this selection mechanism to search for short period binary white dwarfs in the SDSS Data Release 10 spectroscopy sample. One of the authors (CAP) fitted all of the DR10 optical spectra with stellar templates for main-sequence stars <cit.> and white dwarfs using the FERRE code <cit.>. Among these objects, we identify 49 relatively hot low-mass white dwarfs with T_ eff> 12,000 K, M<0.4 M_⊙, S/N>10 SDSS spectroscopy, and with no previous radial velocity observations.Note that our target selection would have included the eclipsing double white dwarf systems J0651 <cit.> and CSS 41177 <cit.>. § OBSERVATIONS We obtained follow-up optical spectroscopy of 34(10) targets using the 8m Gemini North (South) telescope equipped with the Gemini Multi-Object Spectrograph (GMOS) as part of the programs GN-2016A-Q-54, GN-2016B-Q-45, GS-2015A-Q-10, GS-2016A-Q-58, and GS-2016B-Q-48. Since we are only interested in finding sub-hour orbital period systems, and not constraining the binary periods for all targets, we limited our observations to ≈30 min per target. Depending on the target brightness, we obtained a sequence of 4-11 × 2-8 min long exposures with the B600 grating and a0.5 slit, providing wavelength coverage from 3570 Å to 6430 Åand a resolving power of 1850 for GMOS-North, and coverage from 3620 Å to 6780 Å and a resolving power of 1940 for GMOS-South. Each spectrum has a comparison lamp exposure taken within 10 min of the observation time. Based on the initial velocity measurements from GMOS, we obtainedadditional follow-up data for six targets (J1113+2712, J1237+4913, J1323+3254, J1407+1241, J1633+3030, and J1716+2838) using the same setup on Gemini North as part of the Fast Turnaround program GN-2016A-FT-34. Most of these targets were observed with back-to-back exposures over ≈1.8 h, but some of the observations were split into multiple nights due to weather conditions and the constraints imposed by queue scheduling.We used the 6.5m MMT with the Blue Channel spectrograph to obtain follow-up data on five targets (J0834+3049, J1032+2147, J1235+1543, J1237+4913, and J2342+0811) between 2016 Jan and 2017 Mar. We operated the spectrograph with the 832 line mm^-1 grating in second order, providing wavelength coverage from 3600 Å to 4500 Å and a spectral resolution of 1.0 Å.We obtained all observations at the parallactic angle, with a comparison lamp exposure paired with every observation. We flux-calibrated using blue spectrophotometric standards <cit.>.We also used the Kitt Peak National Observatory 4m telescope + KOSMOS <cit.> in 2016 Dec and the Apache Point Observatory 3.5m telescope with the Dual Imaging Spectrograph (DIS) in 2017 Mar to obtain additional follow-up spectroscopy of J1237+4913. We operated the KOSMOS (as part of the program 2016B-0160) and DIS spectrographs with the b2k and B1200 gratings, providing wavelength coverages of 3500-6200 Å and 3750-5000 Å, and spectral resolutions of 2.0 Å and 1.8 Å, respectively.We obtained follow-up time-series photometry of one of our targets, J1235+1543, using the McDonald Observatory 2.1m Otto Struve telescope with the ProEM camera and the BG40 filter. We used an exposure time of 10 s with a total integration time of 3230 s, which covers the entire orbital period for this short period system. We binned the CCD by 4×4, which resulted in aplate scale of 0.38 pixel^-1. We adopted the external IRAF package ccd_hsp <cit.> for aperture photometry. There was only one bright comparison star available in the field of view, and we corrected for transparency variations by dividing the sky-subtracted light curve by the light curve of this comparison star. § RESULTS§.§ Stellar Atmosphere Fits We employed a pure-hydrogen model atmosphere grid covering 4000-35,000 K and logg= 4.5-9.5 to fit the normalized Balmer line profiles of our targets in the summed, rest-frame Gemini spectra. The models and our fitting procedures are described in <cit.>. We used the evoluationary sequences from <cit.> forlow-mass He-core white dwarfs and <cit.> for C/O core white dwarfs to estimate masses and absolute magnitudes for each object.Figure <ref> shows our model fits to a dozen targets inour sample. Given our initial target selection (T_ eff≥ 12,000 K)based on the SDSS data, this figure uses the hot solution in the model fits. Balmer lines are strongest at T_ eff∼ 14,000 K for average mass C/O white dwarfs. This usually leads to a degeneracy in the best-fitsolution for model atmosphere analysis where a hot and a cool solution can both fit the normalized Balmer line profiles reasonably well, but optical photometry can help identify the correct solution in the majority of the cases. Five of our targets have ugriz photometry that implies an effective temperature below 10,000 K. Using the cool solutions in our spectroscopic model fits, these five objects (J1113+2712, J1321+1758, J1323+3254, J1011+0242, and J1132+0751) are best-fit with T_ eff≤9,000K and logg<7 models, i.e. sdA stars <cit.>. <cit.> demonstrate that ∼99% of the sdA starsare metal-poor main-sequence stars in the halo. Hence, we do not consider these five stars as white dwarfs. Table 1 presents the physical parameters for all 44 stars in our sample. Our model atmosphere analysis shows that nine of these stars have masses above 0.5 M_⊙. Excluding these nine stars and the sdAs, there are 30 low-mass white dwarfs in our sample. §.§ Radial Velocities and Errors We measure radial velocities by cross-correlating the spectra against high signal-to-noise templates of known velocity.We use the RVSAO package documented in <cit.> and based on the <cit.> algorithm.The cross-correlation is the normal product of the Fourier transform of an object spectrum with the conjugate of the Fourier transform of a template spectrum. The software package includes extra steps such as Fourier bandpass filtering, to dampen the high frequency (pixel-to-pixel) and low frequency (slow continuum roll) noise in spectra.Velocity errors are measured from the full-width-at-half-maximum of the cross-correlation peak using the r-statistic <cit.>.Empirical validation using repeat low- and high-spectral resolution observations of galaxies and stars confirm the precision of the cross-correlation errors <cit.>. However, systematic errors can arise from poorly-matched templates, which skew the shape of the cross-correlation peak.We address this systematic issue by shifting-to-rest-frame and summing together all observations of a given target, and then cross-correlating the individual spectra against the summed spectrum of itself.This approach minimizes systematic error, but hides statistical error.The location of a star on the spectrograph slit changes how it illuminates the spectrograph and disperses its light onto the detector.We use our higher resolution and higher signal-to-noise ratio MMT data to investigate this issue. Back-to-back exposures of constant velocity targets at the MMT demonstrate a 10 to 15dispersion slit illumination effect, most apparent in targets observed in sub-arcsec seeing with short exposure times (like J1235+1543).Wavelength calibration errors also contribute.The MMT arc line fits have 3residuals, however the blue end <3900 Å is anchored by weak lines that have larger 5 to 10 residuals.The cross-correlation does not discriminate between slit illumination and wavelength calibration errors and real velocity change.The upshot is that we must add statistical error in quadrature to the cross-correlation error.Our approach is to add 20in quadrature to the velocity errors of objects observed at Gemini and MMT and 30in quadrature to the velocity errors of objects observed at KPNO and APO. When fitting orbital parameters to the confirmed binaries, this choice of errors yields reduced χ^2 values of 1 (see the discussion in Section 4.4). We also test for zero point offsets between telescopes when fitting binary orbital parameters.We see no evidence for zero point offsets greater than the 1-σ error in γ, the systemic velocity. §.§ Constraints on Radial Velocity Variability <cit.> presented a robust method for identifying radial velocity variable objects. They used the weighted mean radial velocity for each star in their sample to calculate the χ^2 statistic for a constant-velocity model. They then calculatedthe probability, p, of obtaining the observed value of χ^2 or higher from random fluctuations of a constant velocity, taking into account the appropriate number of degrees of freedom. They identify objects with log(p)<-4 as binary systems. This selection leads to a false detection rate of <0.5% in a sample of 44 objects. We adopt the same method to identify radial velocity variable objects in our sample. Table 1 lists the number of spectra (N), weighted mean radial velocity, χ^2 for a constant-velocity fit, and the probability of obtaining this χ^2 value given the number of degrees of freedom (N-1) for each star. The majority of the stars in our sample do not show significant velocity variations. Given the brevity of our Gemini observations (except for the stars with extensive follow-up observations), this is not surprising.Figure <ref> shows our initial set of Gemini observations for six stars, three of which are excellent examples of non-velocity-variable objects. Our Gemini data for J1011+0242, J1237-0039, and J1542+2936 (top panels) are consistent with a constant velocity fit with log(p)=-0.01 to -0.13. On the other hand, there are several targets with log(p)<-4, indicating that they are likely in short period binary systems (bottom panels). For example, our 2 min cadence data on J1235+1543 sample a significant portion of the binary orbit, and our 8 min cadence data on J1237+4913 reveal a positive velocity trend in that system. Similarly, the initial set of Gemini observations on J2342+0811 reveal a ≈250velocity change over two consecutive nights. We discuss these three objects further in the next section.The five sdA stars in our sample (J1113+2712, J1321+1758, J1323+3254, J1011+0242, and J1132+0751) do not show significant velocity variations in our data. Metal-poor main sequence stars in detached binaries must have orbital periods above about 9 hr <cit.>. Hence, the lack of significant velocity variations in these stars, as well as the majority of the stars in our sample of low-mass white dwarfs is consistent with the expectation that they are likely in longer period binary systems. §.§ Four Binary Systems There are eight objects in Table 1 with log(p)<-4; a constant velocity model is a poor representation of the data for these stars. These are likely binary systems. We have limited follow-up data on four of them, J0738+3241, J1407+1241, J2214+0550, and J2247+2951, and we are unable to constrain the orbital parameters for these four systems. However, the remaining four stars with log(p)<-4 have extensive follow-up observations, and they do show significant radial velocity variations with periods ranging from 53 min to 7 h.We determine orbital parameters by minimizing χ^2 for a circular orbit. Figures <ref> and <ref> show the radial velocity observations, phased velocity curves, and periodograms for these four white dwarfs.Each panel also includes a blow-up of the frequency range where the minimum χ^2 is found. <cit.> discussed the problems with identifying the correct orbital period from radial velocity data given problems with aliasing. They found that the reduced χ^2 values from circular orbit fits were significantly larger than 1 for some of their targets. They attributed this to an unaccounted source of error in their velocity measurements, perhaps the true variability of the star or slit illumination effects. They estimated the level of this uncertainty in their data such that when systematic and statistical errors are added in quadrature they give reduced χ^2 values of 1. We estimate statistical uncertainties of 20 and 30for the Gemini/MMT and KPNO/APO data, respectively (Section 4.2). Adding the cross-correlation errors from RVSAO and the statistical uncertainties in quadrature, we find the best-fit circular orbits with reduced χ^2 ranging from 0.97 to 1.16 for our four binary systems.Out of these four binaries, three have unique orbital period solutions, whileJ2342+0811 has a significant period alias (at P = 0.14369 ± 0.0029 d and K = 126.0 ±10.1 ). Its second period alias at P=5 h differs by 20 in χ^2 and is unlikely to be significant. The χ^2 minima have substructure due to the sampling (see the insets in Figures <ref> and <ref>), however we do not fit the substructure.We measure the orbital period from the envelope of χ^2, which is well-defined and symmetric in all four binaries.We estimate errors by re-sampling the radial velocities with their errors and re-fitting orbital parameters 10,000 times. This Monte Carlo approach samples χ^2 space in a self-consistent way. We report the median period, semi-amplitude, and systemic velocity along with the average 15.9% and 84.1% percentiles of the distributions in Table <ref>.The distributions are symmetric, and so we average the percentiles for simplicity. We also fit J2342+0811's second and third minima at P=3.5 and 5 hr; the semi-amplitudes differ by 2and are thus statistically identical to the best fit.Table 2 presents the orbital elements for these four binary systems with well constrained orbits. Note that <cit.> also identified J1235+1543 as a subhour orbital period binary with P=49.5 min and K=176 ± 21 km s^-1. However, they only used 5 radial velocity measurements from 800-1000 s long exposures. Based on 39 exposures, with exposure times as short as 2 min, we refine the period and velocity semi-amplitude for J1235+1543 to P= 52.9 minand K=166.5 ± 6.2 .The observed velocity semi-amplitudes are relatively modest (K<200 ) for these stars, even for the 53 min period system J1235+1543. The median semi-amplitude of the ELM white dwarf binaries is 220 <cit.>. However, our targets are about twice as massive as the typical ELM white dwarfs, hence the observed smaller velocity amplitudes are not surprising.Table 2 also presents the mass functions, constraints on the companion masses, and the merger times due to gravitational wave radiation. Note that we define the visible low-mass white dwarf in each system as the primary star. The minimum mass companions to our targets range from 0.17 to 0.47 M_⊙, with gravitational wave merger times of roughly 100 Myr for J1235+1543 to ≤13 Gyr for J0834+3049.All but one of these objects, J0834+3049, have minimum mass companions that are smaller in mass than the visible white dwarfs. Since lower mass white dwarfs should form last, and hence appear brighter, these three single-lined spectroscopic binary systems are likely low inclination systems where the companions are more massive than the visible white dwarfs. §.§ Photometric Constraints Spectral types and temperatures of the companions to single-lined spectroscopic binaries can be inferred through photometric effects like Doppler boosting, ellipsoidal variations, and eclipses, or through excess flux in the red or infrared bands.Based on our model atmosphere analysis, the absolute magnitudes of our four binary white dwarfs range from 9.9 to 10.4 in the i-band. If the companions are M dwarfs, the minimum mass companions would be comparable in brightness <cit.> or even brighter than our white dwarf targets in the i-band. We do not see that. Hence, these four binary systems are double degenerates.The probability of eclipses increases with decreasing orbital period. To search for eclipses and other photometric effects, we obtained high-speed photometry of the shortest period system in our sample, J1235+1543, with a cadence of 10 s. Figure <ref> shows these observations over a binary orbit. There is no significant variability in this system, ruling out eclipses and ellipsoidal variations. The amplitude of the ellipsoidal effect is proportional to (M_2/M_1)(R_1/a)^3, where a is the orbital semi-major axis and R_1 is the radius of the primary <cit.>. Compared to the ELM white dwarfs that show ellipsoidal variations <cit.>, (M_2/M_1) and R_1 are relatively small for J1235+1543. Therefore, the lack of ellipsoidal variations is not surprising.All four of the binary white dwarfs in our sample were observed by the Catalina Sky Survey <cit.>. Figure <ref> shows these light curves phased with the best-fit period from the radial velocity data. The Catalina data are sparse for J2342+0811 and part of the orbit is not covered. In addition, the data are noisy for these relatively faint stars. There is a 4σ dip in the J0834+3050 light curve that might be an eclipse, however there are several other >4σ outliers in the same light curve. We suspect that the photometric errors are underestimated. We conclude that there is no significant evidence for eclipses or other photometric effects in any of these systems given the Catalina observations.§ DISCUSSION Our snapshot radial velocity survey of relatively hot and young low-mass white dwarfs has revelaed four double degenerates with periods ranging from 53 min to about 7 h. Figure <ref> compares the mass and period distribution for these systems against the period distribution of ELM <cit.> and low-mass white dwarfs <cit.>. The dashed line shows the predicted mass (of the brighter white dwarf) versus period relation from the rapid binary-star evolution (BSE) algorithm of <cit.> for an initial binary of main-sequence stars with masses 2 M_⊙ + 1 M_⊙.The BSE calculations depend on two important parameters, α_ CE and α_ int (or α_ rec). The former parameter is the efficiency in converting orbital energy into kinetic energy to eject the envelope, and the latter describes the fraction of the internal energy (thermal, radiation and recombination energy) used to eject the envelope. Note that the latest version of the BSE code treats the binding energy parameter λ as a variable.<cit.>, <cit.>, and <cit.> demonstrate that both of these efficiency parameters are small. We adopt α_ CE = α_ rec = 0.25 as in <cit.> for the evolutionary sequence shown in Figure <ref>. The BSE calculations demonstrate that the closest stellar pairs that survive the common-envelope evolution should form lower mass white dwarfs.This is also consistent with the binary population synthesis calculations of <cit.>.Studying the orbital period distribution of post-common-envelope binaries containing C/O and He-core white dwarfs separately, <cit.> found median periods of 0.57 d and 0.28 d for the two samples respectively. This difference is consistent with our understanding of the common-envelope evolution. If the mass transfer starts when the primary star is on the red giant branch, this leads to a He-core white dwarf, whereas if the mass transfer starts while the primary is on the asymptotic giant branch, this leads to a C/O core white dwarf. Hence, stellar evolution theory predicts the C/O core white dwarfs in post-common-envelope binaries to be in longer period systems.The period distribution of the double white dwarfs presented in Figure <ref> shows a trend with mass, at least in the observed lower limit in period. The shortest period binaries with ∼0.4 M_⊙ white dwarfs are in 0.1 d systems, whereas the shortest period 0.2-0.3 M_⊙ white dwarfs are in 0.01 d systems. The median period decreases from 0.64 d to 0.24 d for M=0.3-0.5 M_⊙ to M<0.3 M_⊙ white dwarfs in Figure <ref>. With masses ranging from 0.29 to 0.43 M_⊙, the period distribution for the four binaries presented in this paper is consistent with the period distribution of the post-common-envelope binaries presented here. The rest of the low-mass white dwarfs in our sample are also likely in binary systems with ∼day long periods. However, our Gemini snapshot survey is not sensitive to such long periods.<cit.> looked for a significant correlation between the white dwarf mass and orbital period for He-core and C/O-core white dwarfs separately, but they could not reject the null hypothesis based on an F-test. Adding the low-mass white dwarfs from this paper, the ELM Survey, and the literature (the sample shown in Figure <ref>) does not change the results; we still cannot reject the null hypothesis (no correlation with mass) based on an F-test of the current binary white dwarf sample. This is almost certainly due to the fact that the sample of long period systems with high masses is incomplete, as it is relatively hard to identify these systems and constrain their parameters. The population synthesis calculations <cit.> predict many low-mass (∼0.4 M_⊙) white dwarfs at P∼10 d, yet they are missing from the observational samples. Hence, larger samples of binary white dwarfs that include longer period systems are needed to definitively find a trend between orbital period and primary white dwarf mass.All four binary systems presented in this paper will merge within a Hubble time, with total masses of ≥ 0.76, 0.52, 0.68, and 0.68 M_⊙, respectively. The quickest merger system is J1235+1543, which contains a 0.35 M_⊙ white dwarf with a M≥0.17 M_⊙ companion. Note that a 0.17 M_⊙ companion is expected to form after the 0.35 M_⊙ white dwarf and it should be brighter. Hence, this is likely a low inclination system (i≤36^∘) with a companion that is comparable to or more massive than 0.35 M_⊙.Based on its distance of 386 pc, J1235+1543 has a gravitational wave strain of logh≥ -22.2 at logν = -3.2. This strain is comparable to that of the AM CVn binary GP Com. 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"authors": [
"Mukremin Kilic",
"Warren R. Brown",
"A. Gianninas",
"Brandon Curd",
"Keaton J. Bell",
"Carlos Allende Prieto"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170727173935",
"title": "A Gemini Snapshot Survey for Double Degenerates"
} |
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