polarimetry has become a fundamental part of solar physics and is rapidly gaining importance in many other fields of astrophysics .one of the most challenging problems that one faces when doing astronomical polarimetry is the determination of the instrumental polarization introduced by the telescope in the observed beam .there are two approaches that have been used to deal with this problem .the first one is to place calibration optics at the telescope entrance , which allows one to feed light in a known state of polarization into the system .the measured stokes parameters can then be used to determine the telescope response with a very high degree of accuracy .two solar telescopes currently use this strategy : the dunn solar telescope ( dst ) , at the sacramento peak observatory ( nm , usa ) operated by the national solar observatories , and the german vacuum tower telescope ( vtt ) at the observatorio del teide of the instituto de astrof ' isica de canarias ( spain ) . the swedish solar tower at the observatorio del roque de los muchachos ( also of the instituto de astrof ' isica de canarias )will soon have this capability , as well .  the second approach to determine the telescope polarimetric response is to observe relatively well - known astronomical sources of polarization .this is sometimes done in night - time astronomy by observing `` standard stars '' . in solar physics , this may be accomplished with observations of a sunspot .the procedure considers milne - eddington fits ( mef ) to sunspot umbra spectra , from which the elements of the telescope mueller matrix are adjusted to minimize the residuals of the mef to the observed data .it is generally desirable to have a purely instrumental telescope calibration , so that no _ a priori _ assumptions on the observed targets are needed .unfortunately , the construction of calibration optics for a large aperture ( larger than  m ) is impractical .this had not been an issue in the past because existing solar telescopes have apertures that do not exceed 1 m in diameter , whereas larger night - time telescopes did not require absolute polarimetric accuracies higher than what can be routinely achieved by means of standard stars .this situation , however , is rapidly changing due to the ever increasing accuracy demands of astronomical polarimetry and the development of new large - aperture solar telescopes , like the german telescope gregor or the advanced technology solar telescope ( atst ) .a previous paper introduced the `` beam - expansion '' method ( bem ) for the instrumental calibration of a large telescope .this work presents a different procedure , the sub - aperture method , which has some interesting advantages over the bem ( at least for the particular case of the atst design ) .we shall hereafter consider the problem of calibrating the atst , but the proposed procedure may be easily applicable to other telescopes as well .the sub - aperture method is based on the following basic idea .one starts by effectively turning the atst into a small - aperture telescope .this is done by inserting a mask into the beam at any pupil image .the telescope sub - aperture is calibrated as usual by means of calibration polarization optics above the primary mirror . an astronomical polarization source ( e.g. , a sunspot )is observed through the sub - aperture and then again with the full aperture .the two observations are only a few seconds apart in time .the sub - aperture calibration is used to remove instrumental polarization in the first observation , which allows one to determine the actual stokes vectors emerging from the astronomical source .once these input vectors are known , the full aperture observation can be used to calibrate the telescope .the sub - aperture method proposed in this work requires the following elements .note that , for the case of the atst , requirement 1 is probably the most demanding one .  1 .  a mechanical mount is needed to slide calibration optics in and out of the beam before the primary mirror .the calibration optics may have a small diameter ( see next point ) and consist of an achromatic linear polarizer and a retarder .the polarizer and retarder can rotate and be inserted and removed independently of one another .it is important to realize that the calibration optics does not need to be placed over the center of the aperture .it could be at any location over the primary mirror , e.g. near the outer edge of the telescope aperture .this consideration probably simplifies the design of the mechanical mount , since it does not need to extend all the way into the center of the aperture .moreover , there is an additional advantage for off - axis telescopes like the atst . by having the sub - aperture closer to the axis of the system , rather than the aperture center ,the instrumental polarization of the sub - aperture is minimized .this is of relevance because , as discussed in [ sec : simul ] , the level of accuracy achieved is limited by the accuracy of the sub - aperture calibration .2 .  the diameter of the calibration optics ( ) may be as small as desired , provided only that it fills an aperture large enough to observe the source of polarization .typically this would be a sunspot. however , there may be times of low solar activity in which no sunspots are available .in such cases plage regions may be employed as sources .a suitable value for is probably between 20 cm and 40 cm ( see [ sec : simul ] ) .3 .  a blocking mask that can be inserted in the beam independently of the calibration optics is required .the location of the mask can be any position in the system where an image of the telescope pupil is formed .it may be at the entrance ( e.g. , the dome shutter ) or , perhaps more convenient , at any location downstream in a collimated beam ( which would make it a small element ) .the purpose of the mask is to block the rays in the light beam that did not pass through the calibration optics . in other words ,when the mask is in the beam the telescope aperture is reduced to .the alignment between the mask and the calibration optics does not need to be very accurate .we simply require that only light that passed through the calibration optics can get through the mask .the procedure proposed to calibrate the optical train is as follows :  1 .  slide calibration optics into the beam 2 .slide mask into the beam . only polarizedlight passes through .calibration stokes vectors are measured for various configurations of the calibration optics , in the same way as it is presently done with other solar telescopes .if represents the mueller matrix of the small - aperture telescope+instrument system , we have that : 4 .  from ( known ) inputs and ( measured ) outputs ,determine . in general the measured mueller matrix , that we denote by ,is affected of small errors . 5 .remove calibration optics , _ but leave mask in_. 6 .observe solar source of polarization ( e.g. , a sunspot ) through mini - aperture of diameter : 7 .remove mask and repeat same observation using full aperture .only a few seconds pass between the observations with and without the mask .if is the mueller matrix of the full - aperture telescope+instrument system , we have : 8 .calibrate small - aperture solar observations to obtain : 9 .use known and measured to determine .the following assumption is implicit :   there are two important assumptions in the procedure as outlined above . the first one is given by eq ( [ appr1 ] ) and requires that the calibration of the sub - aperture must be done with sufficient accuracy .possible errors in this step will affect the full aperture calibration through eq ( [ appr2 ] ) .the second assumption is that of eq ( [ approx12 ] ) . in order for it to be accurate ,it is important that the two observations and be taken as close in time as possible .it is also important that the field of view remain the same , especially when doing slit spectroscopy .possible discrepancies in the telescope pointing between the two observations need to be much smaller than the spatial structure of the polarization source . this would probably not be an issue when an adaptive optics system is employed to compensate for seeing - induced wave - front distortion .however , it is not yet clear that the adaptive optics system of the atst will be able to operate while in calibration mode .the validity of these two assumptions , and the degree to which they limit the accuracy of the telescope calibration , is explored by means of numerical simulations in [ sec : simul ] .this section describes some numerical experiments aimed at estimating the accuracy of the sub - aperture method and its limitations .the simulations consist of several `` hound and hares '' type of tests , in which one seeks to determine a telescope matrix which is known beforehand .i have used calibrated data from the asp instrument to simulate a set of solar stokes vectors ( and ) .the simulated solar data is observed through a small sub - aperture system ( characterized by its mueller matrix ) and through the full system ( with mueller matrix ) .obviously , the actual mueller matrix of the atst is still unknown . using that of an existing telescope is probably not very helpful , since their mueller matrices are likely to be very different .the approach taken in this work is to perform montecarlo simulations with a large number of randomly chosen mueller matrices that represent the telescope . according to reference , off - diagonal elements of up to% are to be expected due to the off - axis geometry of the atst .therefore , it seems appropriate to consider matrices of the form : where is the 4 identity matrix and is a particular realization of a 4 matrix whose elements are random real numbers obeying a normal distribution of width 1 .two different asp maps are used in the simulations .the first one ( map a) contains a sunspot and some plage regions .the second ( map b) is a quiet sun map and is used here to explore the feasibility of the sub - aperture method with quiet sun signals .fig [ fig : maps ] shows the observations and the three slit positions that have been used as solar polarization sources ( sunspot , plage and network ) .let us discuss the numerical experiments in some detail .we start by taking some spectra from one of the slit position marked in fig [ fig : maps ]. these will be our solar input stokes vectors from the first observation [ , where denotes spatial position along the slit and represents the spectral variable ] . an observation through the telescope sub - aperture is simulated as : the variation of with wavelength is slow and can be neglected over the spectral range recorded by the detector ( typically a few ) .a second set of stokes spectra is obtained from the same map , but at a slightly different slit position .let us denote the offset between the two observations by .these vectors are observed using the full aperture :   random noise is added to both and .the noise has a normal distribution and has a larger amplitude for the case of as a result of the smaller observing aperture ( ) .i have simulated an exposure time of  seconds in both cases , which results in a noise amplitude of ( with given in meters ) for and for .obviously , it would be possible to use longer exposure times , thus reducing the noise in the observations .  at this pointwe have the simulated solar observations and the telescope matrices used to generate them .next we need to apply the procedure described in [ sec : proc ] to the simulated data and compare the retrieved matrix to the actual generated earlier .  in a real situation ,one starts by using the calibration optics to obtain the mueller matrix ( or at least an approximation ) of the sub - aperture system .the matrix resulting from the measurements should be a good approximation to the actual .for instance , the asp calibration yields an accuracy better than .the tests presented in this section consider that : where is a parameter that accounts for the accuracy of the sub - aperture calibration ( e.g. , for asp , as stated above ) and is again a matrix of random elements .several experiments with different values of have been carried out in order to explore the propagation of errors from the calibration of the sub - aperture to the full aperture system ( see below ) .finally , one needs to perform steps 8 and 9 of the procedure detailed in [ sec : proc ] .a least - squares algorithm is employed to determine the matrix by solving the following system : instead of using all the possible stokes vectors observed at every point along the slit , it is numerically convenient to select a suitable less redundant subset .the strength of the polarization signal of the is used as a selection criterion .a total of 280 vectors , consisting of the 40 points with the largest positive values of q , -q , u , -u , v , -v plus the 40 points with lowest total polarization , are selected .although the number of vectors has been chosen rather arbitrarily , the procedure is not very sensitive to it .this value was adopted after verifying that , in selected simulations , doubling it resulted in no accuracy improvement .the least - squares fitting results in a matrix that can be straightforwardly compared to . in this manner it is possible to study the propagation of errors and the overall accuracy of the sub - aperture method .the entire process has been repeated 100 times for each one of the simulations , with different realizations of the random matrices that appear in eqs ( [ random ] ) .the results presented below are the mean - square errors from the montecarlo simulations .let us start the discussion in this section with a simple reference case .consider for instance a diameter of  m for the calibration optics and a calibration accuracy .the solar source is a sunspot ( slit position 1 in fig [ fig : maps ] ) .let us assume for the moment that the telescope pointing has not changed between the two observations ( ) .the propagation of errors and noise through the calibration procedure ( as explained above ) leads to the following error matrix :   notice that the second and third columns , which correspond to cross - talk from q and u , are larger than the other two .this is due to the fact that the linear polarization signals are considerably weaker than i and v , even in the penumbra where the linear polarization is strongest .fortunately , the atst polarimetric accuracy requirement ( the amount of cross - talk must be smaller than ) does not necessarily imply that the matrix elements of have to be all smaller than .typically , v is ( where is the continuum intensity ) and q , u are .this means that , in order to meet this requirement , the elements in the first column of must be smaller than , those in the second and third smaller than , and those in the fourth smaller than ( see reference for a more detailed discussion ) .these values are used as derived requirements for the atst polarimetric accuracy in the discussion below .fig [ fig : tests1 ] shows how the calibration error varies with the sub - aperture diameter for and .the horizontal dotted line marks the atst accuracy requirements for each column of .cross - talk terms from i are almost at the required level for  cm ( upper left panel ) . a slightly lower would be needed to fully meet the atst requirement ( see below ) . the situation is better for the other stokes parameters , but only when using a sunspot as the polarization source .plage and quiet sun fields can be used to determine the first and fourth columns of at atst levels .however , the second and third columns ( cross - talk from q and u ) exhibit very large inaccuracies .the reason for this is that the observations used here have been taken at disk center and show very little linear polarization . in order to fully determine the q and u terms of the telescope matrix from plage or quiet sun regions, one would also need near - limb observations which exhibit linear polarization signals .it is possible to achieve the required accuracy by calibrating the sub - aperture to the level of , as shown in fig [ fig : tests2 ] .notice in that figure that the telescope calibration accuracy is strongly related to that of the sub - aperture .also in fig [ fig : tests2 ] we can see the uncertainties introduced by pointing errors between the observation of and .the method is not compromised by errors as large as 0.75 arc - seconds and is therefore robust enough to be employed even in the absence of adaptive optics .  ) .solid : no pointing error ( ) . dashed : =0.75 arc - seconds .dash - dots : =1.5 arc - seconds .the horizontal dotted line represents the atst polarimetric accuracy requirement .[ fig : tests2 ] , scaledwidth=100.0% ]the sub - aperture method has the following advantages over the bem and mef :  * unlike mef , no _ a priori _ assumptions are made on the observed targets ( physical model , profile shape or symmetry , etc ) . * simpler design modifications than bem .* unlike mef , can be employed at any observable wavelength .* unlike bem , uses the exact same optical setup as the actual observations .* unlike bem , the full optical train is calibrated , from the primary mirror to the detectors . * unlike mef , can be employed in the absence of sunspots ( e.g. , looking at plage regions or any other suitable source of polarization vectors ) .* unlike mef , can be applied to non - solar telescopes .there are also some drawbacks that should be mentioned :  * unlike mef , requires design modifications .* unlike bem , requires an astronomical source of polarization .the sub - aperture method proposed in this work provides a suitable means of calibrating a large telescope .this paper explores its possibilities and limitations by means of hound - and - hares tests .potential advantages and disadvantages with respect to two other alternatives ( the bem and mef ) are discussed .the simulations indicate that the sub - aperture method is robust enough to meet the stringent atst polarimetric requirements in practical situations .the amount of time required for the calibration procedure is probably inferior to  minutes .most of the time is spent observing the various configurations of the calibration optics , which is the same procedure employed at the dst ( sacramento peak ) or the vtt ( tenerife ) for the polarimeter calibration .this is done currently on a routine basis about once per day during the observing runs .this work utilizes data from the advanced technology solar telescope ( atst ) project , managed by the national solar observatory , which is operated by aura , inc . under a cooperative agreement with the national science foundation .     ,r. , von der l " uhe , o. , kneer , f. , staude , j. , berkfeld , t. , schmidt , w. , soltau , d. , nicklas , h. , wiehr , e. , wittman , a. , hofmann , a. , sobotka , m. , & klvana , m. , `` current status of the 1.5 m solar telescope gregor '' , astronomische nachrichten supplement , * 324 * , 112 - 115 ( 2003 ) .

Abstract: a new method for absolute polarimetric calibration of large telescopes is presented . the proposed method is highly accurate and is based on the calibration of a small sub - aperture , which is then extended to the full system by means of actual observations of an astronomical source . the calibration procedure is described in detail along with numerical simulations that explore its robustness and accuracy . the advantages and disadvantages of this technique with respect to other possible alternatives are discussed .


distributed storage systems ( dss ) are widely used today for storing data reliably over long periods of time using a distributed collection of storage nodes , which may be individually unreliable .application scenarios include large data centers such as total recall , oceanstore and peer - to - peer storage systems such as wuala , that use nodes across the internet for distributed file storage .one of the challenges for dss is the _ repair problem _ : if a node storing a coded piece fails or leaves the system , in order to maintain the same level of reliability , we need to create a new encoded piece and store it at a new node with the minimum repair bandwidth . to solve this problem , dimakis _ et al ._ introduced a generic framework based on _ regenerating codes _ ( rc ) in .rc use ideas from network coding to define a new family of erasure codes that can achieve different trade - offs in the optimization of storage capacity and communication costs .the optimal tradeoff curve for achievable codes has two extremal points which are of particular interest : the minimum storage regenerating ( msr ) codes with minimum possible storage size for a given repair capability , and the minimum bandwidth regenerating ( mbr ) codes with minimum possible repair bandwidth .consider a minimum storage system , where a source file of size units is split into parts , defined over a finite field and stored across nodes in the dss . while the economy in storage is highly desirable, issues may arise when the system tries to repair failure at the optimal repair bandwidth . specifically ,if or grows to be arbitrarily large , then the system may become inefficient and impractical due to the high computational complexity or the fast growing storage consumption .another challenge for dss is data availability , which is of critical importance to a peer - to - peer ( p2p ) storage / backup system that relies on a swarm of distributed and independent nodes for file storage . as the nodes not only differ in their storage capacity and traffic bandwidth , but they may not be online or available at all times . hence , there is a pressing need to increase the data availability , such that infomation is available with a probability approaching 1 .clearly , p2p enrironments are heterogeneous by nature , and code design for such systems must explicitly account for this heterogeneity .the primary interest of this paper is to study a _non - homogeneous dss _ , where one _ super node _ has a larger storage size and higher reliability and availability than the other storage nodes .we study a class of high - rate mds storage codes , and show that with mds code such non - homogeneous dss can achieve the optimal bound in when repairing single or double - node failures , but require smaller and than the traditional homogeneous model in . another proposed scheme based on non - mds codesis shown to repair 1-node failure below the optimal repair bandwidth bound in .moreover , we show that our proposed non - homogeneous dss schemes can achieve a higher data availability than the traditional homogeneous dss scheme .this paper is organized as follows .section ii shows the definition of non - homogeneous dss .section iii shows three proposed schemes of exact repair with storing codes in non - homogeneous dss .section iv shows the numerical results of our schemes and the comparison with previous methods .finally , the paper is concluded in section v.  ]in this section , we present a brief review of the traditional homogeneous dss proposed in .then , a new model of non - homogeneous dss is proposed to realize the practical dss .we follow the definition of traditional homogeneous dss using regenerating codes over finite field .this network has storage nodes and every nodes suffice to reconstruct all the data .the size of the file to be stored is units and partitioned into equal parts where . after encoding them into coded parts using an maximum distance separable ( mds ) code , we store them at nodes .we define here the mds property of a storage code using the notion of data collectors as presented in .a storage code where each node contains worth of storage , has the mds property if a data collector can reconstruct the all the units by connecting to any out of storage nodes .when a node fails , the data stored therein is recovered by downloading packets each from any of the remaining nodes ; the total repair bandwidth is then as shown in fig .[ fig : traditionalrepair ] .it has been shown in that there exists an optimal tradeoff between the storage per node , , and the bandwidth to repair one node , . in this paper , we focus on the extreme point where the smallest corresponds to a _ minimum - storage regenerating _ ( _ msr _ ) code . to minimize , let and we get . in the case of high - rate codes ,a lower bound for repair bandwidth of 1-node failure was shown as :    a non - homogeneous dss with the parameter is a distributed storage systems with nodes based on storing codes and the amount of data stored and downloaded from any nodes are variable .node in the network stores units .when node fails then it is repaired by downloading packets from node , .  it is clear that we must have for all since a node can not transmit more information than it is storing . when for all , we obtain the traditional homogeneous dss . when , there are more redundant blocks than the storage nodes .the storage process has to decide which node(s ) to store more blocks .  in this paper, we present the idea of non - homogeneous dss using the following setting : there is one big node , called the _ super node _ , which has a larger storage capacity and higher reliability and availability than the other nodes .such scenario is possible in practical system , e.g. in a peer - to - peer backup system , the super node could be the service provider that has higher availability and provides higher storage capacity than other peers .consider a system with one super node and three other storage nodes non - homogeneous dss based on a mds code , which can be denoted as .assume a file of size , then this file is divided into parts , each part containing packets . after encoding them into encoded parts or packets , we store the first packets in the super node , and each of the remaining three nodes stores packets as shown in fig . [fig : examplenon - homogeneousdss ] .mds codes and storage nodes.(node is the super node.)[fig : examplenon - homogeneousdss ] ]in this paper , we limit our study to high - rate exact - repair storing codes . this homogeneous problem has been considered in .we propose three efficient dss schemes using mds and non - mds storage codes in such non - homogeneous dss , which are denoted as scheme a , b , and c in table .[ table:(k+2,k)model ] . scheme a and c use mds codes while scheme b uses non - mds codes .the new system consists of nodes which include nodes of storage size and one super node of size .  [ cols="^,^,^,^",options="header " , ]   in this subsection , we employ the framework proposed in to measure and compare the data availability between our proposed non - homogenous dss schemes and traditional homogenous dss schemes to show the efficiency of our proposed schemes .let  ] .it can be seen that if , then )\leq1\leq\chi$ ] .therefore , for all .we run the simulations for the case of and and obtain the result in fig .[ fig : dataavailable ] .it can be seen that for , data availability of non - homogeneous dss scheme outperforms the homogeneous dss scheme . for , the non - homogeneous schemes also have a big improvement when has a high online availability .therefore , it can be seen that our proposed non - homogeneous dss schemes achieve a higher data availability than the traditional homogeneous dss .the gap between the two becomes larger when the online availability of the super node increases , e.g. when is greater than 25% of , the data availability of the proposed non - homogenous over homogenous dss is increased by 10% .  ]we proposed three distributed storage schemes for _ non - homogeneous dss _ with high rate codes .two of the schemes make use of mds code , and can achieve optimal repair bandwidth of at smaller finite field and 75% smaller fragment than .small and are desirable , because they reduce the update bandwidth and complexity .another scheme based on non - mds code can achieve a smaller repair bandwidth than the optimal bandwidth based on mds code by for 1-node failure .we further demonstrate that in such non - homogeneous dss , if we can ensure one super node with a higher online probability than the other nodes , we can achieve a higher data availability than the homogeneous dss .this research is partly supported by the international design center ( grant no .idg31100102 & idd11100101 ) .li s work is supported in part by the national science foundation under grants no .ccf-0829888 , cmmi-0928092 , and eager-1133027 .papailiopoulos , a.g .dimakis , v.r .cadambe , `` repair optimal erasure codes through hadamard designs , '' _ the 49th annual allerton conference on communication , control and computation _ , pp.1382 - 1389 , sep .2011 .r. bhagwan , k. tati , y .- c .cheng , s. savage , and g. m. voelker , total recall : system support for automated availability management , _ the 1st symp .networked systems design and implementation ( nsdi ) _ , mar .cadambe , c. huang , j. li , `` permutation code : optimal exact - repair of a single failed node in mds code based distributed storage systems , '' _ ieee international symposium on information theory , _pp.1225 - 1229 , aug .cadambe , s.a .jafar , h. maleki , `` asymptotic interference alignment for exact repair in distributed storage systems , '' _ the 44th signals , systems and computers ( asilomar ) _ , pp.1617 - 1621 , nov . 2010 .f. dabek , j. li , e. sit , j. robertson , m. kaashoek , and r. morris , designing a dht for low latency and high throughput , _ the 1st _ _ symp .networked systems design and implementation ( nsdi ) _ , mar .2004 .l. pamies - juarez , p. garcia - lopez , m. sanchez - artigas , b. herrera , `` towards the design of optimal data redundancy schemes for heterogeneous cloud storage infrastructures '' , computer network , pp .1100 - 1113 , nov . 2011

Abstract: this paper describes a _ non - homogeneous _ distributed storage systems ( dss ) , where there is one super node which has a larger storage size and higher reliability and availability than the other storage nodes . we propose three distributed storage schemes based on ( ) maximum distance separable ( mds ) codes and non - mds codes to show the efficiency of such non - homogeneous dss in terms of repair efficiency and data availability . our schemes achieve optimal bandwidth when repairing 1-node failure , but require only one fourth of the minimum required file size and can operate with a smaller field size leading to significant complexity reduction than traditional homogeneous dss . moreover , with non - mds codes , our scheme can achieve an even smaller repair bandwidth of . finally , we show that our schemes can increase the data availability by 10% than the traditional homogeneous dss scheme .   exact - repair mds codes , non - homogeneous dss , minimum storage regenerating ( msr ) codes .


admission control is a queue management tool that can increase the efficiency of resource utilization in many service systems . depending on the particular application, it can be used to preserve system capacity for future customers who bring higher profit , to limit the number of admitted customers in order to provide a better quality of service to those already in , etc .admission control is often employed indirectly via dynamic pricing , or by price discrimination ( direct or indirect ) among different customer types .however in several situations frequent price changes may not be feasible , and in order to contain congestion denying service to certain arriving customers may be necessary .  on the other hand , in several queueing systems it is possible to alleviate the congestion effects by adjusting the service capacity .for example in banks or call centers the number of servers may change during the day to follow variations of the arrival rate .the service rate may also be varied dynamically when the queue length becomes too long . increasing ( and in some cases decreasing )the service rate comes at a cost , but it has the advantage over admission control policies that fewer or no customers are turned away .it is thus of interest to explore to what extent a flexibility in service capacity can interact with admission control and when it can alleviate its effects .  in this paperwe investigate the interaction of service rate flexibility and admission control in an m / m/1 queueing system .the flexibility is modeled as the option to switch dynamically between a low and a high service rate .we analyze the problem of joint admission and service rate control by formulating a markovian decision process model to maximize the infinite horizon expected discounted profit .we also explore the effect of service rate flexibility on the optimal profit and the admission thresholds .specifically , the benefit of the service rate switch option is compared against a baseline case where only the low service rate is used and admission control is employed .thus , in principle , the benefit is due both to the existence of a higher service rate as well as the flexibility to use it . on the other hand ,the higher service rate is not free but comes at a higher cost , thus the benefit of using it reflects the tradeoff between serving at a faster rate and paying higher operational costs .when we use the term benefit of flexibility we mean exactly how this tradeoff manifests itself in the presence of admission control .the paper develops a model of dynamic optimization of queueing systems , a large area with very extensive literature .both admission and service control models have been studied thoroughly . and survey several dynamic optimization models developed for queueing control .  for single class customers problems , as the one analyzed in this paper, admission control makes sense when there is an exogenous holding cost rate function . a simple model in this direction was first presented in , where arriving customers are admitted or not based on the observed queue length with the objective to maximize customer s overall ( social ) benefit from receiving a reward after service completion minus a linear increasing holding cost per unit time of delay .it is shown that a socially optimal policy admits fewer customers than those who would decide to enter based on an individual optimality criterion . considers a queue under infinite horizon discounted cost , assuming a convex and nondecreasing holding cost rate function .it is shown that the optimal policy has a threshold structure if and only if the optimal benefit is concave in the number of customers in the system which in turn depends on convexity of the holding cost rate function .as in , we also consider a convex , nondecreasing holding cost rate which implies a threshold property of the optimal admission policy .  on the other hand in multi - class systems with finite capacity admission controlmay be useful even in the absence of holding costs , because in this case admitting a customer implies the possibility of a loss of profit from a future higher class customer . considers a system with parallel and identical servers , no waiting room and customer classes , which contribute to the system , different fixed rewards .this model results in a threshold type optimal policy with a preferred class . the optimal admission rule for a system with one server and no waiting room which receives offers from customers according to a joint service time and reward probability distribution . and also investigate properties of optimal admission policies for certain loss systems . develop an optimal static admission policy in an queueing system with customer classes with generally different service requirements and service rewards . examine the problem of dynamic admission control in a two class loss markovian queueing system with different service rates and different fixed rewards for the two customer classes .the admission control problem has been also analyzed in queueing systems under heavy traffic .an often used approach in this framework the dynamic optimization is usually approximated by a diffusion control problem , following the approach of .recent works in this area include and , both analyzing admission control under customer abandonments . analyze a queue in the balanced heavy traffic regime , where the optimal control depends on the sample path of the diffusion and the resulting asymptotically admission control policy of threshold type depends on second moment data of the interarrival and service times . consider the same problem in amulti - class environment under discounted expected cost minimization .dynamic service control in queueing systems is an equally large field .several problems can be viewed as service control models , including controlled server vacations , server allocation policies in polling systems , etc . in an early work examines dynamic service control under infinite horizon expected average expected cost in a maintenance system with finite available service rates , a linear holding cost rate and a reward collected in service completions .it is shown that the optimal service rate is increasing in the number of customers waiting in line .the monotonicity of the optimal service rate is also shown in in the framework of an queue , with service rates varying in a closed set and the holding cost rate increasing and convex . the service control problem in an queue where service rates are dynamically selected from a close subset of  ] let be the randomized policy under which the service rate is always set to and arriving customers are admitted with probability . if is such that , then under policy the system is equivalent to a stable queue , with state denoting the number of customers in the system , arrival rate , service rate admission reward and cost rate while in state .the discrete - time equivalent of this process corresponds to a positive recurrent markov chain , with transition probabilities  and one - step rewards .let denote the first passage time to state 0 .it is well known that the expected first passage times to state 0 in the continuous - time m / m/1 queue are equal to , thus in the discrete time model   now consider the first passage expected profit   thus , to show that , it suffices to show that the expected first passage holding cost is finite , i.e. ,   from corollary c.2.4 of it follows that in order to show , it is sufficient to establish that there exists a nonnegative finite function such that   we will prove that there exist a sufficiently small and a sufficiently large such that these inequalities are satisfied by function , where is the constant appearing in assumption 1 .first , is immediate , since . for , on the other hand , from assumption 1 ( i ), it follows that .therefore , if we take , so that and , then it is true that   summarizing , we have shown that for sufficiently small there exists a function satisfying , , thus policy is 0 standard and the proof of the theorem is complete .  from corollary 7.5.4 of , we also obtain that the relative value function is nonnegative and nonincreasing in for any .the results on the structure of the discounted optimal policy are extended to the average reward case , since the average optimal policy is obtained as a limit of a sequence of discounted optimal policies .specifically , there exist admission and service thresholds as in theorem [ threspolicy2 ] and if the thresholds are ordered as in proposition [ optpol ] , and in this case the service rate flexibility is of no value .in the previous sections it was assumed that the reward is collected upon admitting a customer . while this is plausible in many situations such as ticket - based operations or call centers with upfront charge , it is also often the case that the service reward is collected at the time of departure of the customer , for example in jobshops with payment upon delivery . in this sectionwe formulate the corresponding mdp model for the second case , point out the similarities and differences and show that essentially all the conclusions obtained so far still hold .when is collected at departure epochs , the markov decision process in finite horizon corresponding to - now takes the following form : note that in - the term for is added at transitions from to , whereas at admission epochs there is no reward collected .  as in the original model ,let .one and , in essense , the only difference between the two models is that lemma [ burdenprp1 ] is not true anymore , i.e. , the value function is not nonincreasing in .this is intuitively expected , since an additional customer in the queue brings with him the prospect of a future reward as well as a burden due to the higher holding costs .mathematically , the induction proof of lemma [ burdenprp1 ] breaks down at state .indeed , it is now true that which is not necessarily nonnegative due to terms and .therefore , it is not generally true anymore that , thus it can not be interpreted as a burden , but rather as the net effect of an additional customer , which can be either a burden or a benefit .  on the other hand , by following the remaining proofs in the original model, it can be verified that all the results on the monotonicity of in , the threshold structure of the optimal policy and the properties of the value of flexibility function still hold .the admission and service rate thresholds now take the form where corresponds to the new net effect function and is the same generic threshold function defined in .in this section we present the results of some computational experiments , which explore the value of service rate flexibility and the influence of the service rate switch option on the optimal admission policy . in the previous sectionsit was established analytically that when the service rate flexibility is essentially of no value .this is so because from proposition [ cntcombadm ] , , thus either the system stops accepting customers before the higher service rate is used , or the higher service rate is employed at the last state before the rejection sign is posted .theorem [ vof_novalue ] further implies that in this case the optimal admission thresholds are the same with and without the service rate switch option .however , even if , it may still be true that and thus .our first numerical experiment demonstrates that , although the relationship between and by itself does not uniquely characterize the value of service rate flexibility , it provides a good approximation for such a characterization . to do so, we consider a system with and varying value of .figure [ serflexdg ] shows the relative value of the service rate flexibility as a fraction of the total profit of the combined problem at state ( panel ( a ) ) .we observe that the relative value of the service flexibility is increasing in , when .panel ( b ) shows the service and admission thresholds for the combined and the admission problem .the service threshold is decreasing in , while both admission thresholds and are increasing .   for ,scaledwidth=80.0% ]figure [ serflexdg ] also verifies that when , it is true that , and .however these are still true for values of slightly greater than .specifically , there exists a critical value such that , when and otherwise , and this value is greater than .  to examine this issue further, we perform a more general numerical experiment , where are the same as before and we vary parameters and .figure [ threscombdg ] presents the value of for varying values of .the curve of is indeed above the diagonal , however only slightly .this shows that the condition is a good approximation to characterize the value of service flexibility . in all numerical cases we analyzedwe observed similar behavior , i.e. , there exists a critical value of above which it is true that , and this critical value is higher but close to .  in the third numerical experiment we explore the sensitivity of the service flexibility with respect to the relative increase in service capacity , offered by introducing the high service rate option .this effect is presented for two values of the traffic intensity , , one low , , and one high , .to do so , we consider a system with and varying value of the high service rate .  in table[ sensmh ] we present numerical results for the relative value of the service flexibility as a fraction of the total profit of the combined problem at state and for the values of service and admission thresholds of the combined and the admission problem , respectively , for the two values of the traffic intensity .we observe that the value of flexibility and the admission thresholds are increasing in , where on the other hand , the service threshold is decreasing .we also note that , as expected , when the flexibility is essentially of no value and . finally , the effect of the higher traffic rate is more pronounced in the value of flexibility than in the threshold values .  note that in all numerical experiments the parameters do not generally satisfy the normalization assumption , .however , the required rescaling has been performed in the computations ..relative value of service flexibility and thresholds of the combined problem and the admission control subproblem as a function of , for and .[ cols="^,^,^,^,^,^,^,^,^ " , ]in this paper we analyzed the problem of joint dynamic admission and service control in an queue under expected discounted profit maximization .we established a threshold structure for the optimal service rate - admission control policy .we defined the value of the service rate flexibility as the benefit that the higher service rate option brings to a system with pure admission control , and showed that the value of flexibility is increasing with system congestion .we finally identified a simple condition between the admission reward and the relative cost of high service rate , under which the admission policy is not affected and the value of service flexibility is zero in low congestion states .the analysis in this paper was conducted in the context of an queue , where the high service rate means that the server operates with higher speed .however in many real applications such as banks or call centers the service capacity is affected by dynamically varying the number of operating servers .this corresponds to an queue with being determined by a dynamic policy .we conjecture that the admission policy will still be threshold - based . on the other hand, it would be interesting to study how the service rate policy is structured under different assumptions on the service rate switch option , e.g. assuming that all servers are required to use the same service rate , or that each server is allowed to select its own rate .furthermore , the assumption of identical customers may be relaxed by assuming multiple customer classes differentiated by admission reward and/or service rate .if the service rates are identical among classes , it is expected that the optimal admission policy is determined by class - dependent admission thresholds which are increased when the high service rate option becomes available .it would also be interesting to consider the value of service rate flexibility when there is a fixed service rate switch cost , which brings hysteretic policies into play . finally , in the present model we considered maximization of net profit from the point of view of a single decision maker , who may be the system owner or a collective representative maximizing the total customer benefit. it would be interesting to consider equilibrium strategies in a game theoretic model where the server determines the service rate and arriving customers respond by deciding individually whether to join the queue or balk .such a model could also include pricing as an additional policy component available to the server .the extension to equilibrium models with dynamic service control is the object of our current research work .a. p. ghosh and a. p. weerasinghe .optimal buffer size and dynamic service rate control for a queueing system with impatient customers in heavy traffic ._ stochastic processes and their applications _ , 120:0 21032141 , 2010.  j. m. harrison .brownian models of queueing networks with heterogeneous customer populations . in c.w. fleming , p. l. lions , editor , _ stochastic differential systems , stochastic control theory and appplications .i m a volumes in mathematics and its applications _ , volume 10 , pages 147186 .springer - verlag , new york , 1988 .

Abstract: we consider a single server queueing system with admission control and the possibility to switch dynamically between a low and a high service rate , and examine the benefit of this service rate flexibility . we formulate a discounted markov decision process model for the problem of joint admission and service control , and show that the optimal policy has a threshold structure for both controls . regarding the benefit due to flexibility , we show that it is increasing in system congestion , and that its effect on the admission policy is to increase the admission threshold . we also derive a simple approximate condition between the admission reward and the relative cost of service rate increase , so that the service rate flexibility is beneficial . we finally show that the results extend to the expected average reward case .


the mathematical model of quantum computation has appeared in the 1980-th with the works of david deutsch [ 1 ] and others [ 2 - 4 ] . in 1994peter shor [ 5 ] proposed an effective quantum algorithm for solving the npi - candidate task factorisation of big composite numbers .this breakthrough allowed to expect , that any task from np could have an effective solution on a quantum computer .this hope was weakened after the lov grover s work [ 6 ] in 1996 , which proposed the general quantum algorithm for solving any task from np .the algorithm was designed to search the particular point in the unsorted database of size n with only oracle queries .grover has shown that under the considered assumptions the algorithm is optimal , so only quadratic speedup can be achived by the quantum computer compared to the classical one , if we do nt have additional information about the oracle functioning .this paper describes the improvement of the quantum database search algorithm , exhibiting the exponential speedup over the classical search , by changing some preliminary assumptions about the model of computation .the reasoning goes as follows . in section 2we recall the grover s database search algorithm .section 3 describes the new quantum computation model with von neumann architecture applied , and the new algorithm for the unsorted database search is proposed along with its complexity estimation .section 4 glues everything together thus serving as a conclusion .first , let us recall the algorithm proposed by lov grover . let s consider the function f :     which is some decision function over the set of n - bit strings . is the number of all possible n - bit strings , thus the number of all possible inputs to f. function f is implemented as a black box .our purpose is to find the n - bit string on which f returns 1 .grover defines the quantum oracle :    where x is n - bit string , and y is one bit .on the first n qubits acts as follows :     acts identically on any vector orthogonal to , while changes the sign of itself , thus it can be considered as the reflection of any vector over the hyperplane orthogonal to .  then the vector and operator are introduced as follows :     where h is n - qubit hadamard transform .operator reflects any vector over the vector .the grover iteration is    the operator rotates the initial vector towards the desired vector by the angle , where .the action of the first two grover iterations is shown on the figures [ grov_1 ] and [ grov_2 ] .[ grov_1 ]   [ grov_2 ]the von neumann architecture was introduced by john von neumann [ 7 ] in 1945 .one of its most famous concepts is the idea of representing algorithms as a special type of data , which can be stored in computer memory along with the information they process .the computer hardware may then become universal and not depending on the solved tasks .this idea is considered to be very common and natural for the classical computing model .however , the quantum computing model still distinguish data ( quantum system state ) from a program ( sequence of unitary transforms in the system state space ) .some of these unitary tranforms have the natural mapping on the system states .for example , the state defines the transform in the previous section .if we assume , that the state of one quantum system can be used as a reflection tranform in another quantum system , then we can introduce a much simplier and effective algorithm for the unsorted database search .  following grover, we will denote :     we still consider the case of searching of the special input among all n - bit strings .the number of grover s iterations needed to get the result with high probability after measurement is estimated by .for example , if n equals to 8 ( each input is one byte ) , then the number of grover s iterations must be close to 12 .let s suppose that we already have performed a half of these 12 iterations in one system , while in another system we ve prepared the initial state ( figure [ new_alg_1 ] ) .[ new_alg_1 ]  with 6 grover s iterations.,title="fig:",width=336 ]  if the vector can be used as the operator in the left system with the initial state , then we need only one iteration to get close to instead of the remained 6 grover s iterations ( figure [ new_alg_2 ] ) .[ new_alg_2 ]  applied.,title="fig:",width=336 ]  again , the vector itself can be constructed from the vector with the same process , thus omitting 3 grover s iterations ( figure [ new_alg_3 ] ) . continuously applying this trick we can reduce the number of all grover s iterations to .each such reflection divides the number of nesessary grover s iterations by 2 .[ new_alg_3 ]  applied.,title="fig:",width=336 ]  this means that for solving the unsorted database search problem we need to :  1 .perform one grover s iteration to get vector .2 .  prepare the state on another system .apply the vector from the first system as a reflection to the second system .4 .  repeat times the steps 2 - 3 .the implementation of operator , which depends on some quantum state and acts on another quantum system is an open question in the scope of this work .the proposed quantum algorithm allows us to solve the unsorted database search problem exponentialy faster than that described by grover .however , the assumption of possibility of applying states as operators to other systems does nt correspond to the classical quantum computation model , developed to these days .note that with the new algorithm we need only one call to the oracle to get the direction .after that all the remained iterations do not call the oracle directly . to implement the new algorithmwe need only two quantum systems , which will change their roles on each iteration . the `` algorithm '' system ( which is used as an operator ) on the next iteration must prepare the initial state and become a `` data '' system and vice versa .w. shor , _ algorithms for quantum computation : discrete logarithms and factoring_.1em plus 0.5em minus 0.4emproceedings of the 35th ieeesymposium on foundations of computer science .pp . 124 134 .

Abstract: the new model of quantum computation is proposed , for which an effective algorithm for solving any task in np is described . the work is based on and inspired by the grover s algorithm for solving np - tasks with quadratic speedup compared to the classical computation model . the provided model and algorithm exhibit the exponential speedup over that described by grover . the main idea of the model improvement is employing the von neumann architecture for a quantum computer , allowing program parts to be stored as quantum states along with the processed data .   quantum computing , class np , quantum algorithm , grover s algorithm , von neumann architecture


the scientific goal of applying the doppler imaging technique to ap stars , is to derive observational constraints on the diffusion mechanism in the presence of a stellar magnetic field .theoretical studies ( michaud et al , 1981 ; mgessier , 1984 ) which predict that certain elements raise , sink or move horizontally in magnetic stellar atmospheres need to be verified . this can only be achieved if an unambiguous correlation between the elemental abundance structures and the magnetic field geometry is obtained . however , two major problems are responsible for difficulties in reaching that goal .firstly , doppler imaging can ( presently ) only be applied to stars which have a fairly weak magnetic field and hence only the variation of the integrated longitudinal field strength ( effective field , ) is usually determined .this sets only weak limits to the ` real ' magnetic geometry . for objects with stronger fieldsthe spectral line profiles are severely influenced by zeeman broadening and are not mainly caused by abundance variations .this limitation can be avoided by obtaining all four stokes parameters as a function of the stellar rotational phase .then zeeman doppler imaging codes ( donati , 1995 ) can be applied to calculate the magnetic field strength and surface orientation in combination with the abundance distribution .but these measurements of the stokes parameters are not yet available , though strong efforts are made to get them from polarimetry .secondly , in the past , only a few elements were mapped , mostly silicon , rarely iron and chromium , not even all of them for each object .furthermore , these maps were based on inversions from single ` unblended ' spectral lines , which are difficult to find , especially in cooler ap stars .this introduced uncertainties leading to a questionable reliability of the derived maps and weakening their relevance to theoretical modelling .considering this situation , we focused our programme on the production of abundance maps of up to five and more elements on the surface of each of our target stars . a brief report on the stars andthe progress of our project is given hereafter , followed by the presentation of results obtained for the magnetic ap star cu virginis .as pointed out before , one crucial limitation of doppler imaging in the past was that only a few elements were mapped , mostly silicon and occasionally iron and chromium .furthermore , the number of stars studied was not sufficient to draw general conclusions by relating the abundance maps to the approximate magnetic field structure , thereby providing new constraints on theory ( hatzes 1995 ) .in particular , the abundance maps of elements which are pushed upwards by radiatively driven diffusion should look severely different from those of elements that have a tendency to sink .    the helium map ( fig .1 ) is characterized by a dominant spot which appears to be at the approximate position of the positive magnetic pole ( phase 0.5 ) . only in the central part of this spotdoes helium reach the solar abundance ; on all other parts of the stellar surface , it is depleted by about 1.5 dex .this result confirms theoretical predictions by vauclair et al .( 1991 ) which have been obtained by modelling helium abundance in main sequence magnetic stars by introducing a weak wind of ionized metals at the magnetic poles , where the field lines are vertical .  in total contrast to that ,silicon ( fig .2 ) is strongly depleted in the helium spot but overabundant on the remaining visible surface . in these regionsthe magnetic field lines are mainly horizontal and as predicted by e.g. mgessier ( 1994 ) , silicon accumulates at the magnetic equator band . chromium and ironhave surface structures very similar to that of silicon , but both elements are less enhanced compared to their solar abundances.  however , the magnesium ( fig .3 ) distribution differs much from that of the other metals .the main feature is a ring centred at the helium spot ( magnetic pole ) , but with an the overall abundance deficiency of about 1 dex and much less contrast than all other elements .the helium spot distribution of cu vir was already found by hiesberger et al .( 1995 ) , on the basis of data obtained in 1980 , which is an evidence for the high reliability of this result .also , the silicon abundance structure confirms earlier results published by hatzes ( 1995 ) .furthermore , there are strong indications that cu virginis has slowed down its rotation rate abruptly in the year 1985 ( pyper et al , 1998 ) .this effect has not been found in any other ap star and no theoretical explanation can be given at the moment .  nevertheless , the doppler imaging results for cu virgins are a good demonstration of how important it is to obtain the surface distribution of different elements for the same star .cu virginis is one of the very few objects for which a correlation of the abundance structures and the magnetic field geometry can be given with high significance .furthermore , it is planned to obtain the abundance distributions for elements like c , n , o and some rare earth species .this should complete the picture and may lead to a detailed theoretical modelling of diffusion processes interacting with the magnetic field of this star .donati , j .- f . : 1995 , in _ stellar surface structure , iau symp .176 _ , eds : k.g .strassmeier and j.l .linsky , kluwer , dordrecht , 53 hatzes , a.p . : 1995 , in _ stellar surface structure , iau symp .176 _ , eds : k.g .strassmeier and j.l .linsky , kluwer , dordrecht , 305

Abstract: doppler imaging , a technique which inverts spectral line profile variations of an ap star into a two - dimensional abundance maps , provides new observational constraints on diffusion mechanism in the presence of a global magnetic field . a programme is presented here with the aim to obtain abundance distributions of at least five elements on each star , in order to study how different diffusion processes act under influence of a stellar magnetic field . the importance of this multi - element approach is demonstrated , by presenting the abundance maps of helium , magnesium , silicon , chromium and iron for the magnetic b9psi star cu vriginis .   # 1#2#3=#3 cm 2 2


analyses of spiking activity recorded from sensory neurons have revealed three main features : first , neuronal activity is stochastic and exhibits significant variability across trials ; second , the spiking statistics often undergo rapid changes referred to as neuronal plasticity , in order to adapt to changing stimulus salience and behavioral context ; and third , the tuning characteristics of sensory neurons to the stimuli exhibit a degree of sparsity .examples include place cells in the hippocampus and spectrotemporally tuned cells in the primary auditory cortex .hence , in order to gain insight into the functional mechanism of the underlying neural system , it is crucial to have a mathematical theory to simultaneously capture the stochasticity , dynamicity and sparsity of neuronal activity .on one hand , the theory of point processes has been recently adopted as a mathematical framework to model the stochasticity of neuronal data .traditionally , these models have been used to predict the likelihood of self - exciting processes such as earthquake occurrences , but have recently found significant applications in the analysis of neuronal data .  on the other hand , classic results in signal processing such as the least mean squares ( lms ) and recursive least squares ( rls )algorithms have created a framework to efficiently capture the dynamics of the parameters in linear observation models .existing solutions in computational neuroscience have adopted this framework to estimate the dynamics of neuronal activity .for instance , in an lms - type point process filter was introduced to study plasticity in hippocampal neurons . in ,more general adaptive filtering solutions based on approximations to the chapman - kolmogorov equation were introduced .although quite powerful in analyzing neuronal data , these solutions do not account for the sparsity of the underlying parameters .finally , the theory of compressed sensing ( cs ) has provided a novel methodology for measuring and estimating statistical models governed by sparse underlying parameters . in particular , for _ static _ linear and generalized linear models ( glm ) with random covariates and sparsity of the parameters , the cs theory characterizes sharp trade - offs between the number of measurement , sparsity , and estimation accuracy .the sparse solutions of cs are typically achieved using batch - mode convex programs and greedy techniques . in online settings ,sparse adaptive filters have only been introduced in the context of linear systems governed by sparse parameters such as communication channels .  despite significant progress in all these research fronts , a unified framework to simultaneously capturethe stochasticity , dynamicity and sparsity of neuronal data is lacking . in this paper, we close this gap by integrating techniques from point process theory , adaptive filtering , and compressed sensing . to this end , we consider the problem of estimating time - varying stimulus modulation coefficients ( e.g. , receptive fields ) from a sequence of binary observations in an online fashion . we model the spiking activity by a conditional bernoulli point process , where the conditional intensity is a logistic function of the stimulus and its time lags .we will then design a novel objective function by incorporating the forgetting factor mechanism of rls - type algorithms into the -regularized maximum likelihood estimation of the point process parameters .we will present theoretical guarantees that extend those of cs theory and characterize fundamental trade - offs between the number of measurements , forgetting factor , model compressibility , and estimation error of the underlying point processes in the non - asymptotic regime .we will next develop two adaptive filters for recursive estimation of the objective function based on proximal gradient techniques , as well as a filter for recursive computation of statistical confidence regions .  in order to validate our algorithms ,we provide simulation studies which reveal that the proposed adaptive filtering algorithms significantly outperform existing point process filters in terms of goodness - of - fit , mean square error and trackability .we finally apply our proposed filters to multi - unit spiking data from ferret primary auditory cortex ( a1 ) during passive stimulus presentation and during performance of a click rate discrimination task in order to characterize the spectrotemporal receptive field ( strf ) plasticity of a1 neurons .application of our algorithm to these data provides new insights into the time course of attention - driven strf plasticity , with over 3 orders of magnitude increase in temporal resolution from minutes to centiseconds , while capturing the underlying sparsity in a robust fashion . aside from their theoretical significance ,our results are particularly important in light of the recent technological advances in neural prostheses , which require real - time robust neuronal system identification from limited data .the rest of the paper is organized as follows : in section [ prelim ] , we present our notational conventions , preliminaries and problem formulation . in section[ sec : main ] , we introduce the main theoretical results of this paper , including the construction and stability analysis of the objective function , recursive filter development , and computation of confidence regions .section [ sec : applications ] provides numerical simulations as well as application to real data , followed by our concluding remarks in section [ sec : conclusion ] .technical details of section [ sec : main ] are presented in appendices [ proof][app : conf ] .we first give a brief introduction to point process models ( see for a detailed treatment ) .we will use the following notation throughout the paper .parameter vectors are denoted by bold - face greek letters .for example , ' ] denoting the transpose operator .consider a stochastic process defined by a sequence of discrete events at random points in time , noted by ' ] by bins of length , that is where . throughout this paper , will be considered as the observed spiking sequence , which will be used for estimation purposes . also , by approximating eq .( [ cif_def ] ) for small , and defining , we have : in discrete time , the orderliness of the process is equivalent to the requirement that with high probability not more than one event fall into any given bin . in practice , this can always be achieved by choosing small enough. an immediate consequence of eq .( [ bernoulli ] ) is that can be approximated by a sequence of bernoulli random variables with success probabilities .  a popular class of models for the cif is given by generalized linear models ( glm ) . in its general form ,a glm consists of two main components : an observation model ( which is given by ( [ bernoulli ] ) in this paper ) and an equation expressing some ( possibly nonlinear ) function of the observation mean as a _ linear _ combination of the covariates . in neuronal systems ,the covariates consist of extrinsic covariates ( e.g. , neural stimuli ) as well as intrinsic covariates ( e.g. , the history of the process ) . in this paper , we only consider glm models with purely extrinsic covariates , although most of our results can be generalized to incorporate intrinsic covariates as well .let denote the stimulus at time bin , ' ] and  ] represents the time - varying parameter vector at time . throughout the rest of the paper, we refer to and as the covariate vector and the modulation parameter vector at time , respectively .  in our applications of interest ,the modulation parameter vector exhibits a degree of sparsity .that is , only certain components in the stimulus modulation have significant contribution in determining the statistics of the process .these components can be thought of as the preferred or intrinsic tuning features of the underlying neuron . to be more precise , for a sparsity level ,we denote by the support of the highest elements of in absolute value , and by the best -term approximation to .we also define to capture the compressibility of the parameter vector .recall that for , the -norm is defined as .when , the parameter is called -sparse , and when is small compared to , the parameter is called -compressible .  finally , the main estimation problem of this paper can be stated as follows : _ given binary observations and covariates from a point process with a cif given by eq .( [ glmdynamic ] ) , the goal is to estimate the -dimensional -compressible parameter vectors in an online and stable fashion . _in this section , we will first describe the construction of an appropriate objective function for addressing our main estimation problem .we will then present a rigorous analysis of the maximizers of the objective function , which extends the results of cs to our setting .next , we will introduce two adaptive filters to recursively maximize the objective function based on proximal gradient techniques .finally , we will outline how statistical confidence bounds can also be constructed in a recursive fashion for our estimates .   before proceeding with the construction of the objective function, we need to introduce more notational conventions . in order to have a framework allowing multi - timescale dynamics, we consider piece - wise constant dynamics for the parameter .that is , we assume that remains constant over windows of arbitrary length samples , for some integer . by segmenting the corresponding spiking data into windows of length samples each , we assume that the cif for each time point is governed by , for .note that number of spiking samples is assumed to be an integer multiple of window size , without loss of generality .  invoking the bernoulli approximation to the poisson statistics for , the log - likelihood of the observation at time can be expressed as : assuming conditional independence of the spiking events , the joint log - likelihood of the observations within window is given by :   in order to explicitly enforce adaptivity in the log - likelihood function , we adopt the forgetting factor mechanism of the rls algorithm , where the log - likelihood of each window is exponentially weighted regressively in time , with a forgetting factor .that is , the effective data log - likelihood up to and including window is taken to be : for some .note what for , coincides with the natural data log - likelihood .moreover , if we replace the bernoulli log - likelihood with the gaussian log - likelihood , then coincides with the conventional rls objective function .next , in order to explicitly enforce sparsity , we adopt the -regularization mechanism of cs .that is , at window , we seek an estimate of the form :    where is a regularization parameter controlling the trade off between the log - likelihood fit and the sparsity of estimated parameters .our theoretical analysis in the next subsection reveals appropriate choices for , and the trade - offs therein .   in order to quantify the trade - offs involving our choice of the objective function in eq .( [ mainprob ] ) , we proceed in the tradition of performance analysis result of the rls algorithm by characterizing the geometric properties of the estimates in a stationary environment where for all .our analysis , however , is quite general and avoids ad hoc assumptions such as direct averaging or covariate independence which are usually invoked in the analysis of least squares problems .we need to make the following technical assumptions for our analysis :  \1 ) the stimulus sequence consists of independent ( but not necessarily identically distributed ) random variables with a variance of which are uniformly bounded by a constant in absolute value .note that with this assumption , two successive covariate vectors , say at times and , given respectively by  ] are highly _ dependent _ , as they have random variables in common .hence , the independent assumption used in studying least squares problem is violated .\2 ) we further assume that for all times , , for some constants and , i.e. , the probability of spiking does not reach its extremal values of and , but can get arbitrarily close .this assumption can be realized due to the boundedness of the covariates and appropriate normalization of the stimulus modulation coefficients , and does not result in any practical loss of generality .we have the following theoretical result regarding the stability of the maximizers of the objective function :  [ thm : main ] _ suppose that binary observations from a point process with a cif given by eq .( [ glmdynamic ] ) are given over windows of length each .consider a stationary environment with for all and suppose that is -compressible . then, under assumptions ( 1 ) and ( 2 ) , for a fixed positive constant , there exist constants , , and such that for , and a choice of , any solution to ( [ mainprob ] ) satisfies the bound {\!(1\!-\!\beta ) l \log m},\end{aligned}\ ] ] with probability at least .the constants , , and are only functions of , , , , , and , and are explicitly given in appendix [ proof ] . _  the proof is given in appendix [ proof ] .* remarks . *the result of theorem [ thm : main ] has four major implications .first , the error bound scales with , the sparsity level , as opposed to , the ambient dimension of the parameter vector , which is consistent with results from cs , and results in the robustness of the estimate when the underlying parameter is sparse . note that the bounds holds for general non - sparse , but is sharpest when is negligible , i.e. , the parameter vector is nearly -sparse .second , the theorem prescribes a lower bound on the forgetting factor akin to the bounds obtained in cs theory for the total number of observations .for instance , the result of for cs under toeplitz sensing measurements for the linear model requires number of measurements to achieve a similar scaling of the error bound . in our case , the role of the number of measurements is transferred to forgetting factor by taking as the _ effective _ length of the measurements . in the absence of the forgetting factor ( ) , by a careful limiting process ,our results require measurements .the latter case can be compared to the result of for point process models with independent and identically distributed covariate vectors , which requires for stability .the loss of is incurred due to the shift structure and hence high dependence of the covariate vectors in our case , as exemplified in assumption ( 1 ) .third , the theorem reveals the scaling of the regularization parameter in terms of and .in particular , this scaling is significant as it reveals another role for the forgetting factor mechanism : not only the forgetting factor mechanism allows for adaptivity of the estimates , it also controls the scaling of the -regularization term with respect to the log - likelihood term .fourth , unlike conventional results in the analysis of adaptive filters which concern the expectation of the error in the asymptotic regime , our result holds for a single realization with probability polynomially approaching 1 , in the non - asymptotic regime .note that the objective function is clearly concave , and assuming that the matrix of the covariate vectors is full - rank , will be strictly concave with a unique maximizer .however , the result of theorem [ thm : main ] does not require the uniqueness of the maximizer and holds for any maximizer of the objective function .in the next section , we will proceed with the development of recursive filters to track the maximizer of the objective function in the more general time - varying setting .several standard optimization techniques , such as interior point methods , can be used to find the maximizer of ( [ mainprob ] ) . however , most of these techniques operate in batch mode and do not meet the real - time requirements of the adaptive filtering setting where the observations arrive in a streaming fashion . in order to avoid the increasing runtime complexity and memory requirements of the batch - mode computation , we seek a recursive approach which can perform low - complexity updates in an online fashion upon the arrival of new data in order to form the estimates . to this end , we adopt the proximal gradient approach . a version of the proximal gradient algorithm is given in appendix [ proximal ] .each iteration of the algorithm moves the previous iterate along the gradient of the log - likelihood function , which will then pass through a shrinkage operator .  before describing further details, we introduce a more compact notation for convenience .let ' ] denote the vector of cifs within window . by extending the domain of the to vectors in a component - wise fashion, we can express as :   where ' ] .the shrinkage step will be then given by :   we refer to the resulting filter as the -regularized point process filter of the zeroth order ( -ppf ) .a pseudo - code is given in algorithm [ alg0 ] ._ * first order expansion : * _ if instead , we retain the first two terms in the taylor expansion , eq .( [ eq:0 ] ) will be replaced by : where is a diagonal matrix with the diagonal element given by . using the first order approximation above, we can improve the resulting approximation to the gradient , denoted by , as : by defining :   we can express as :   .  ,    ]  in what follows , we give a brief description of how the methods of apply to our setting , and leave the details to appendix [ app : conf ] . using the result of , the estimate as the maximizer of ( [ mainprob ] )can be decomposed as : where is an approximate inverse to the hessian of evaluated at , is the gradient of previously defined in eq .( [ gradserie ] ) , and is an unbiased and asymptotically gaussian random vector with a covariance matrix of , with the first term in eq .( [ eq : decomp ] ) is a bias term which can be directly computed given .given , statistical confidence bounds for the second term at desired levels can be constructed in a standard way .the main technical issue in the aforementioned procedure in our setting is the computation of in a recursive fashion .since the rows of are computed using -regularized least squares , we use the sparls algorithm as an efficient method to carry out the computation in a recursive fashion .algorithm [ alg3 ] summarized the recursive computation of confidence intervals for the component of .in this section , we will apply the proposed algorithms to simulated data as well as real spiking data from the ferret primary auditory cortex . in our simulation studies ,we compare the performance of our proposed filters with two of the state - of - the - art point process filters , namely the steepest descent point process filter ( sdppf ) and the stochastic state point process filter ( ssppf ) .these adaptive filters are based on approximate solutions to the chapman - kolmogorov forward equation obtained by a steepest descent and a gaussian approximation procedure , respectively .first , we consider a stationary environment where is constant over time .we use a bin size of and window size of sample , for a total observation window of ( ) .the length of the parameter vector  ] . the stimulus vector at time , is given by the vectorized version of the spectrogram of the acoustic stimulus with frequency bands and lags .in order to capture the sparsity of the strf in the time - frequency plane , we further represent over a gabor time - frequency dictionary consisting of gaussian windows centered around a regular subset of the time - frequency plane .that is , for , where is the dictionary matrix and is the sparse representation of the strf. the estimation procedures of this paper can be applied to , by absorbing the dictionary matrix into the data matrix at window .we apply our proposed adaptive filter -ppf to multi - unit spike recordings from the ferrets a1 during a series of passive listening conditions and active auditory task conditions . during each active task, ferrets attended to the temporal dynamics of the sounds , and discriminated the rate of acoustic clicks .the strfs were estimated from the passive condition , where the quiescent animal listened to a series of broadband noise - like acoustic stimuli known as temporally orthogonal ripple combinations ( torc ) .the experiment consisted of 2 active and 11 passive blocks . within each passive block, 30 torcs were randomly repeated a total of 4 - 5 times each . in our analysis , we pool the spiking data corresponding to the same repeated torc within each block. therefore , the time axis corresponds to the experiment time modulo repetitions within each block .we discretize the resulting duration of to time bins of size , and segment data to windows of size samples ( ) .the strf dimensions are , regularly spanning lags of 1 to and frequency bands of to ( in logarithmic scale ) .the dictionary consists of gabor atoms , evenly spaced within the strf domain .each atom is a two - dimensional gaussian kernel with a variance of per dimension , where denotes the spacing between the atoms .we selected a forgetting factor of , a step size of , where is the average variance of the spectrogram components , iterations per sample , and a regularization parameter of via two - fold even - odd cross validation .figure [ fig : f5 ] , top row , depicts five snapshots taken at corresponding to the end - points of the passive tasks .the bottom row shows the time - course of five selected points ( marked as a through d in the leftmost panel of the top row ) of the strf during the experiment .the strf snapshots at times and correspond to after the two active tasks , respectively , and verify the sharpening effect of the excitatory region ( ) due to the animal s attentive behavior following the active task reported in . moreover, the strf snapshots at times and reveal the weakening of the excitatory region long after the active task and returning to the pre - active state , highlighting the plasticity of a1 neurons .previous studies have revealed the strf dynamics with a resolution of the order of minutes .our result in figure [ fig : f5 ] provides a temporal resolution of the order of centiseconds ( 3 orders of magnitude increase ) , while capturing the strf sparsity in a robust fashion .in this paper , we considered recursive estimation of the time - varying parameter vectors in a logistic regression model for binary time series driven by continuous input . to this end , we integrated several techniques from compressed sensing , adaptive filtering , optimization and statistics .we constructed an objective function which enjoys from the trackability features of the rls - type algorithms , sparsifying features of -minimization , and unlike the rate - based linear models commonly used to analyze spiking data , takes into account the binary statistics of the observations .we analyzed the maximizers of the objective function in a rigorous fashion , revealing novel trade - offs between various model parameters .we constructed two adaptive filters , with respective linear and quadratic complexity requirements , for recursive maximization of the objective function in an online setting .moreover , we characterized the statistical confidence regions for our estimates , and devised a recursive procedure to compute them efficiently .although we specialized our treatment to logistic statistics and -regularization , our approach to algorithm development has a plug - and - play feature : other glm link functions ( e.g. , log - link ) with possibly history dependent covariates and other regularization schemes ( e.g. , re - weighted , or group - sparse regularization ) can be used instead and result in a large class of adaptive filters for sparse point process regression .we tested the performance of our algorithms on simulated as well as experimentally recorded spiking data .our simulation studies revealed that the proposed filters outperform several existing point process filters .application of our filters to real data from the ferret primary auditory cortex provided a high - resolution characterization of the time - course of spectrotemporal receptive field plasticity , with 3 orders of magnitudes increase in temporal resolution .although we focused on auditory neurons , we expect a similar superior performance of our filters when applied to other sensory or motor neurons ( e.g. , cells in primary or supplementary motor cortex ) .the proof is mainly based on the beautiful treatment of negahban et al . . the major difficulty in our case lies in the high inter - dependence of the covariates , which form a toeplitz structure due to the setup of adaptive filtering .we address the latter issue by adopting techniques from another remarkable paper by haupt et al . to deal with the underlying interdependence . in the process, we also employ concentration inequalities for dependent random variables due to van de geer .  in order to proceed ,we adopt the notion of strong restricted convexity ( rsc ) introduced in . for a twice differentiable log - likelihood with respect to , the rsc property or order implies the existence of a lower quadratic bound on the negative log - likelihood : for a positive constant and all satisfying : for any index set of cardinality .the following key lemma establishes the rsc for :  _ let denote a sequence of covariates and let denote the corresponding logistic parameters .then , for a fixed positive constant , there exist constants and such that for and the negative log - likelihood satisfies the rsc of order with constant with probability greater than .the constants and are only functions of , , , , , , and are explicitly given in the proof . _  the proof is inspired by the elegant treatment of negahban et al . .the major difficulty in our setting is the high interdependence of successive covariates due to the shift structure induced by the adaptive setting , whereas in , the matrix of covariates is composed of i.i.d .rows . using the taylor s theorem , can be written as : with for some . since by hypothesis , we have : we can therefore further lower bound by : where , and note that the matrix has highly inter - dependent elements due to the toeplitz structure in the adaptive design . in order to establish the rsc condition, we show the stronger restricted eigenvalue ( re ) property , which in turn implies rsc .let be fixed .to do so , we need to bound the eigenvalues of , the restriction of to a subset of columns and rows corresponding to indices in with , for some integer .  without loss of generality , we assume that the first entry of the covariate vectors is replaced by instead of , for presentational simplicity of the following treatment . for , we have : for , and for , we also have . using hoeffding s inequality we get : since , for  ] with probability at least : hence , by choosing , the probability above is greater than .  next , by invoking lemma 4.1 ( ii ) of , we have that satisfies the rsc condition over the set given by eq .( [ eq : cone ] ) with a constant given by : hence , the negative log - likelihood satisfies the rsc with a constant given by . finally , by taking , we have that , which makes independent of and , given by : and with .next , the result of theorem 1 of implies : for .we have , for , now , let be the -field generated by , i.e. , .we have that & = \mathbb{e } \left [ \mathbb{e } \left [ \left \ { n_{t } - \lambda_{t } \delta\right\ } { s}_{t } | \mathcal{f}_{t } \right ] \right ] \\\nonumber & = \mathbb{e } \left [ s_t \mathbb{e } \left [ \left \ { n_{t } - \lambda_{t } \delta\right\ } | \mathcal{f}_{t } \right ] \right ] \\\nonumber & = \mathbb{e } \big [ s_t \mathbb{e } \big [ \underbrace{\left \ { \lambda_{t } \delta - \lambda_{t } \delta\right\}}_{=0 } | \mathcal{f}_{t } \big ] \big ] = 0.\end{aligned}\ ] ] hence for all , .we next invoke the following result for concentration of dependent random variables :  [ hoeff_dep ] consider a sequence of -fields .suppose that is -measurable with for some constant , and .then for all ,   this result is a special case of theorem 2.5 of for _ bounded _ and possibly dependent random variables , which generalizes hoeffding s inequality .  in our case, we can similarly show that = \mathbb{e } \left [ s_t \mathbb{e } \left [ \left \ { n_{t } - \lambda_{t } \delta\right\ } | \mathcal{f}_{t-1 } , \mathcal{f}_t \right ] \right ] = 0 ] , we get : using the union bound , we have : by choosing , we have that with probability at least .  hence , for a fixed , , and , by taking with and with , any maximizer satisfies : {\!(1\!-\!\beta ) l \log m},\end{aligned}\]]with probability at least , where is given by this appendix , we give an overview of the proximal gradient algorithm for minimization of convex function . the corresponding algorithm for maximization of concave functionscan be obtained by negating the objective functions .consider the general optimization problem where functions and are assumed to be closed proper convex functions .suppose that is differentiable with a lipschitz continuous gradient with constant .the function can be possibly non - smooth .a wide range of practical optimization problems can be cast in this form , particularly in the context of machine learning , where the objective function can be decomposed into a loss function and a regularization term .the proximal gradient algorithm provides an iterative procedure for solving ( [ genprob ] ) in the following form : ,  \label{proxgrad}\end{aligned}\ ] ] where the parameter is an appropriately chosen step size at iteration so that , and the _ proximal operator _ of function with parameter is defined as among the several interpretations available for the proximal gradient method , we have adopt a quadratic approximation - based model to derive the main iterative scheme in ( [ proxgrad ] ) .this interpretation , is based on the majorization - minimization algorithm ( see for a detailed discussion ) . in the approximation - based derivation , the -th iteration for solving the general problem ( [ genprob ] )can be written in the following form : where the original objective function is replaced with a quadratically - regularized linear approximation around the previous iterate , given by where the quadratic term is referred to as the _ trust region penalty_. modulo constants , the objective function in ( [ approxprob ] ) can be rearranged to get the proximal gradient form . \end{aligned}\ ] ]  the proximal operator often admits closed form expressions . as for -regularization , the proximal operator takes the simple form of the _ soft thresholding shrinkage operator _ whose component is given by with denoting the standard signum function , and . in this case, the proximal algorithm leads to a family of algorithms referred to as iterative shrinkage algorithms , where each iteration involves a simple gradient descent step followed by a shrinkage operation .finally , in our setting , the function is taken to be the exponentially weighted log - likelihood . due to the smoothness of the logistic function, the lipschitz constant for can be upper bounded by the trace of the hessian given in eq .( [ eq : hess ] ) . noting that the elements of are at most equal to , we get . using assumption ( 1 ) of section [ maintheorem ] and an application of hoeffding s inequality , shows that the sum is concentrated around its mean given by , for large enough .therefore , we choose the step size , for some constant .the -regularized ml estimate of eq .( [ mainprob ] ) can be written in the following form where denotes the log - likelihood function over a generic window with spiking vector , data matrix and parameter vector , and the operator is defined for a function as the empirical expectation exponentially weighted with a forgetting factor : where we have suppressed the dependence of on and on the left hand side for notational simplicity . following the treatment of theorem 3.1 of , the corresponding empirical gradient vector and hessian are respectively given by : the kkt conditions for the estimator can be then written as : where is a subgradient vector from the subdifferential of the norm , with components for and otherwise , for .substituting by its quadratic approximation around the true parameter vector , and inverting the corresponding kkt conditions , the de - sparsified estimator can be obtained as : where the matrix is the approximate inverse of hessian matrix , and can be computed using the following node wise regression procedure . to compute the row of , first the solution to the following lasso problemis obtained : where the dependence of on is suppressed for notational convenience , and the subscript notations are the same as those described in the footnote of algorithm [ alg3 ] .then , we define the vector as : and the scaling constant as finally , the row of is given by .the variance and the confidence interval at a level of for the component of can then be computed as given in lines 9 and the output of algorithm [ alg3 ] , where using taylor expansion similar to that in the development of -ppf , the matrix can be recursively updated as given in line 2 of algorithm [ alg3 ] .finally , the node wise regression can be recursively computed using the sparls algorithm , which is given in lines 35 of algorithm [ alg3 ] .the parameter can be chosen to be in the same order of in ( [ mainprob ] ) .

Abstract: we consider the problem of estimating the sparse time - varying parameter vectors of a point process model in an online fashion , where the observations and inputs respectively consist of binary and continuous time series . we construct a novel objective function by incorporating a forgetting factor mechanism into the point process log - likelihood to enforce adaptivity and employ -regularization to capture the sparsity . we provide a rigorous analysis of the maximizers of the objective function , which extends the guarantees of compressed sensing to our setting . we construct two recursive filters for online estimation of the parameter vectors based on proximal optimization techniques , as well as a novel filter for recursive computation of statistical confidence regions . simulation studies reveal that our algorithms outperform several existing point process filters in terms of trackability , goodness - of - fit and mean square error . we finally apply our filtering algorithms to experimentally recorded spiking data from the ferret primary auditory cortex during attentive behavior in a click rate discrimination task . our analysis provides new insights into the time - course of the spectrotemporal receptive field plasticity of the auditory neurons .   adaptive filtering ; point process models ; compressed sensing ; neural signal processing ; receptive field plasticity .