in 1988 , aharonov , albert , and vaidman introduced the idea of weak measurement  . for a very weak interaction between a system and measuring apparatus, they showed that the measurement results obtained can be much larger than the eigenvalues of the observable by selecting appropriate initial and final states .the counterintuitive results have been implemented in experiments  .weak measurement has provided a new perspective to the famous hardy s paradox  , and the illustrations have been realized in experiments  .using the idea of weak measurement , hosten and kwiat have observed the tiny spin hall effect of light  .this idea has also been used to amplify small transverse deflections and frequency changes of optical beams  .because of the importance of weak measurement in applications , there has been much research work on this issue  .  in the seminal paper  , aharonov , albert , and vaidman ( aav ) introduced the weak value of an observable , where and are the preselected and postselected states ( pps ) , respectively . for the interaction hamiltonian , and the weak value ( where ) , jozsa  has derived that the final mean pointer momentum and position satisfy where and are the original mean pointer position and momentum , is the variance of position in the initial pointer state , and is the mass of the pointer .it seems that for , the shifts of the pointer s position and momentum might be arbitrarily large . in factarbitrarily large shifts can not be obtained , as there is one approximation step in the derivation : where and are the original and final states of the measuring device . as pointed out in , the validity of the approximation in the above equation requires that for all . when , the above requirement is not satisfied , so the measurement results given by eq .( ) are not valid .thus , we need some more precise descriptions of weak measurements .there has been some research on weak measurements beyond aav s formalism  . by retaining the second order terms of ,the formalism of weak measurements is generalized to the general preselected and postselected states in  . in that reference, it was shown that the weak measurement outcomes can not increase arbitrarily by decreasing the overlap between the pps .  in this paper, we calculate the shifts of the pointer position and momentum without any approximation .the interaction between the system and measuring device is not assumed to be very weak .so , the expressions we obtained are not only valid for the weak interaction cases , but also legitimate for the strong interaction cases .we also give the maximum shifts of the measuring device for the two - dimensional systems by choosing appropriate pps . using weak measurement , the improvements of the signal - to - noise ratio ( snr ) and the enhancements of the measure sensitivity ( ms )are discussed in the paper .when we consider the probabilities reduced by postselection , we find the snr and ms can not be effectively improved in weak measurement .we also show that the snr and ms can be effectively improved without considering the probabilities .this paper is organized as follows . in sec. ii we will give the general precise expressions of the pointer shifts .the maximum pointer shifts with a two - dimensional system and the explicit results for the stern - gerlach experiment are derived in sec .section iv is devoted to studying the improvement of the snr brought by a weak measurement .the discussion of the enhancements of the ms is given in sec .v. a short conclusion is presented in sec . vi .in this section , we derive the precise expressions of the measurement results with preselection and postselection , without any restriction on the strength of the interaction between the system and the measuring device .the preselected and postselected states of the quantum system are denoted by and respectively , and the initial state of the measuring device is denoted by .the interaction hamiltonian between the system and the device is described as where is the coupling strength with , is an observable of the quantum system , and is a variable of the measuring device . throughout this article , for the purpose of convenience , by , and ,we redefine and such that becomes a dimensionless operator with a unit norm .it is assumed that the complete orthonormal eigenstates of the observable are , the corresponding eigenvalues are .so , the preselected state can be written as and the overall state of the system and measuring device is after the interaction described by eq .( [ x1 ] ) the overall state evolves into with throughout this paper .the postselected state can also be written as .after postselection , the final device state is which is not a normalized state .the probability of obtaining the final state is the expectation value of a pointer observable is given by where .thus , we have given the general and precise expression of the device s readings .without loss of generality , the initial state of the pointer device is assumed to be a gaussian wave function centered on and , where and are the position and momentum variables , and the standard deviations of them are and . in this paper , we consider , and the interaction . from eq .( [ eq : c3 ] ) , the final state of the pointer device after postselection is and the probability of obtaining the final state is from eq .( ) , we can directly calculate the expectation values of the pointer momentum and position and as the expectation values of the pointer momentum and position in the initial state are both zero , the values given in eqs .( ) and ( ) are also the average shifts of the pointer momentum and position , which are denoted by and , respectively .  in the rest of this section ,we discuss two extreme cases in terms of the interaction strength : the weak interaction case and the strong interaction case .first , we discuss the weak interaction case ( i.e. , the coupling strength is weak ) .we have , and and if the real part of the weak value is big , the shift can be much larger than the shift of the pointer without postselection .this amplification effect is implemented in experiments  .the results in eqs .( [ eq : c10 ] ) and ( [ eq : c11 ] ) are the same as those given in eq .( [ eq : jozsa ] ) .however , it should be pointed out that the approximation step may not be legitimate when .therefore , in a general discussion without restriction on the overlap between preselection and postselection states , we use eqs .( ) and ( ) instead .next , we discuss the strong interaction case ( i.e. , ) .we have for , and then . if the eigenvalues of are nondegenerate , the shift of the pointer momentum can be approximately expressed as where is the state of the system after projection onto the eigenstates .when the eigenvalues are degenerate , let the eigenvalue be -fold degenerate , we can re - label the eigenstates of as , where .then the pps can be rewritten as and . from eq .( [ eq : c8 ] ) and for , we get where .the shifts given by eqs .( [ eq : c12 ] ) and ( [ e1 ] ) are the results for a strong measurement , and they can be regarded as the ideal quantum measurement results which are independent of the device s states .so , we have got the ideal measurement results by increasing the strength of the interaction between the quantum system and measuring device . from the eq .( [ eq : c12 ] ) and eq .( [ e1 ] ) , we can get that where and are the minimum and maximum eigenvalues of the observable , respectively .the measurement results given in eq .( [ eq : c13 ] ) mean that the amplification effect disappears when the interaction strength is strong .in this section , we derive the maximum shifts of the pointer position and momentum for a measurement with preselection and postselection on a qubit system .the interaction between the system and the measuring device is described by the hamiltonian , which is not assumed to be weak unless specified . as the system to be measured is two - dimensional ,so the orthonormal eigenstates of the observable can be denoted as , and the corresponding eigenvalues are denoted as , respectively . without loss of generality , we assume . then the pps and can be written as where and . from eqs .( ) and ( ) , we get and where .when the two states and are orthogonal to each other , we obtain that while aav s formalism can not give any measurement result for two orthogonal pps .now , for a given pointer state and coupling strength , we search for the extreme values of and over all the pps .let , from , we get , and where . by , we get the extreme values of the shift of the pointer momentum as , it is obtained that let and . when and , reaches its minimum value ; when and , reaches its maximum value , where . for the weak interaction case , we have , the extreme shift is achieved when .it is obvious that and imply .so for the weak interaction cases , the shift of the pointer reaches its extreme value when the pps are approximatively orthogonal .  furthermore , for the weak interaction cases with , we get and which only depend on the original state of the measuring device . for the strong interaction cases with , we get , and , which depend on the eigenvalues of and the coupling strength .  by a similar derivation , we obtain that the minimum shift of the pointer position is achieved when and ; while the maximum shift is achieved when and . for , , so the extreme shift of the pointer position is achieved only when . for the weak interaction cases , the minimum and maximum shifts of the pointer position are given by , and ; while for the strong interaction cases , we get .  here , we have obtained the extreme values of the pointer sshifts for all pps . when , we find that the maximum shifts are independent of the coupling strength , and by choosing the appropriate pps , the maximum values of pointer shifts and can be obtained .so for the weak interaction case , due to the postselection , the signal is amplified by a factor of which could be very large . when , it is obtained that and , and these results mean that there is no amplification effect at all under the strong interaction circumstance .now we consider the stern - gerlach experiment which is widely discussed in weak measurements  . in this experiment , spin- particles with the spin pointed in the direction travel through an inhomogeneous ( in the direction ) magnetic field , andthen are postselected in the direction .so , we have and .the initial pointer state is , where is a normalized wave function .the interaction hamiltonian can be written as , where depends on the magnetic field .after postselection , the final pointer state is substituting into eq .( ) , and from eqs .( ) and ( ) , we get that   .the extremum values are obtained near and .,title="fig : " ] .the extremum values are obtained near and .,title="fig : " ]  in fig .1 , the shifts are depicted for .it can be seen that , the maximal pointer shifts are and which are the extreme values obtained in eqs .( ) and ( ) .the maximum amplification factor obtained here is , which can not be increased anymore by decreasing the overlap between the pps ( it can never reach as in  ) .  from and , we get the maximal shift of the pointer momentum the maximum is achieved when and . for , , we have .it means the optimum pps to achieve the maximum shift of the pointer momentum are approximatively orthogonal . andthe probability of obtaining this maximum value is given by .  from and , we get the maximal shift of the pointer position which can be achieved when and . for , , and we have .the probability of obtaining the maximum shift is .as , we relabel the probability .  in fig2 , the maximal pointer shifts and the probability are pictured with different coupling strength .it can be seen from figs .2(a ) and 2(b ) that when , the maximum shifts and are given by and respectively .however , the probability of obtaining the maximal shifts is very small , . when , we have and , which is expected from a strong measurement ., we have and ; when , we have and , which are the strong measurement results . when , the probability to obtain the maximum shifts is very small.,title="fig : " ] , we have and ; when , we have and , which are the strong measurement results . when , the probability to obtain the maximum shifts is very small.,title="fig : " ] , we have and ; when , we have and , which are the strong measurement results . when , the probability to obtain the maximum shifts is very small.,title="fig : " ]in this section , we study the improvements of the signal - to - noise ratio ( snr ) by weak measurements .snr has been studied in many reference  ; it was claimed that the snr can be largely improved by the weak measurement processes . here, we show that the snr can be improved for some practical cases when the probability decrease due to postselection is not considered .however , we also show that the snr can not be effectively improved when the probability decrease due to postselection is considered .the signal considered here is the average shift of the pointer variable , and the noise is represented by the standard deviation .if the number of the measurement repetition is , as a result of the central limit theorem , the mean standard deviation is .then we get the snr this definition is similar to the one given in  .  without postselection , the state of the measuring device after the interaction is where is the composite state of the system and measuring device given by eq .( [ eq : c2 ] ) .the shift of the pointer momentum without postselection is where is the original mean pointer momentum , and .we get the snr without postselection as , where is the eigenvalue of with the maximum absolute value , by choosing an appropriate input state we can get the optimal snr without postselection is the expectation values of and with postselection are given in eqs .( ) and ( ) , respectively .the standard deviations of them are changed by postselection , and it can be obtained that and where , the final device state after postselection , is given by eq .( [ eq : c6 ] ) .so , we can get the standard deviations after postselection where and are given by eqs .( [ eq : c8 ] ) and ( [ eq : c9 ] ) .after postselection , the number of the measurement repetition is changed into , where is the probability of obtaining the postselected results .then , we get the snr with the postselection procedure now we define the improvements of snr by weak measurement as they are referred to as type - i improvements of snr , and in the definitions the probabilities reduced by postselection are taken into account. next we will search the values of the type - i improvements in the stern - gerlach experiment . from eqs .( ),(),( ) , ( ) , and ( ) , we can obtain the standard deviations after postselection and then we can get the type - i improvements of snr and where . for , the values of and pictured in fig .3 . from fig .3 , we can see that the maximum values of and are both approximately . although we can get very largely amplified shifts , the probabilities of obtaining the large shifts are very low .the low probabilities cancel the advantage of the amplification effect  . by numerical calculation , when , we get that the maximum type - i improvements . in the stern - gerlach experiment ,the post - selected state is fixed .we have also searched for the values of the improvements for all pps by numerical calculation , and obtained that the upper bound of them is also 1.038 .those results mean that the snr can not be effectively improved by the weak measurements .however , this does not mean that the weak measurement methods are not helpful for actual experiments .as we know , measurement errors are divided into random errors and system errors  .the standard deviations considered here are the random errors which can be reduced by repeated measurements . for a given system error which can not be reduced by repeated measurements ,the relative system error is  , where is the shift of the measuring device .by the idea of weak measurement , the shift could be largely amplified .so , the relative system errors could be effectively reduced by weak measurement in actual experiments .  .from the four pictures , it can be seen that the values of and are both less than 1 ; meanwhile the maximum value of and are all approximatively equal to .,title="fig : " ] . from the four pictures, it can be seen that the values of and are both less than 1 ; meanwhile the maximum value of and are all approximatively equal to .,title="fig : " ] . from the four pictures, it can be seen that the values of and are both less than 1 ; meanwhile the maximum value of and are all approximatively equal to .,title="fig : " ] . from the four pictures, it can be seen that the values of and are both less than 1 ; meanwhile the maximum value of and are all approximatively equal to .,title="fig : " ]   in the practical case , most experiments are limited by technical issues . here we discuss the improvements of snr under the detectors saturation intensity technical limit  . in this circumstance ,the input quantum systems are sufficient , and the numbers of the measurement repetition are determined by the detectors saturation intensity .so , the numbers of repetition with and without postselection are almost the same .so the probability reduced by postselection needs not be considered in defining the improvements of snr they are referred to as type - ii improvements of snr , and in the definitions the probabilities reduced by postselection are not taken into account .one can see that and .  in the stern - gerlach experiment , the type ii improvements of snr are and   in fig .3 , for , it is shown that the maximum values of and can approximately reach about 53.7 which means that the snr can be efficiently improved .when , we can neglect the orders of higher than in eq .( [ eq : e03 ] ) : where . by , we get . as we want to search the maximum of , by ensuring , we get , then denoting , let , from , we get . substituting this solution into , we obtain that , so . for , we have which is presented in fig .3 . through a similar procedure, we can also get .so , under the measurement repetition limited by the detectors saturation intensity , the snr can be greatly improved by weak measurement .using the idea of weak measurement , a series of tiny physical effects has been effectively detected in experiments  . in those experiments ,what has been actually measured could be regarded as the interaction strength between the systems and measuring devices .a natural question arises , whether weak measurement with preselection and postselection enhances the measurement precision in estimating the strength of a tiny interaction ? in this section we will study the enhancements of the measurement sensitivity ( ms ) brought by weak measurement .we consider the enhancements with and without considering the probabilities reduced by postselection .   from the error propagation theory  ,the error on the estimated coupling strength is where is the shift of the measuring device , and is the average standard deviation of after the measurement is repeated for times . if the interaction considered is , our purpose now is to estimate the value of the coupling strength . without postselection ,the error is where is the shift of the pointer given in eq .( ) and is the standard deviation of momentum . by choosing an appropriate input state, we obtain the optimal error without postselection where is the eigenvalue of observable with the maximum absolute value .the errors and on the estimation of with postselection are where and are given in eqs .( [ eq : c8 ] ) and ( [ eq : c9 ] ) . with considering the probabilities reduced by postselection ,the enhancements of ms are defined as they are referred to as type - i enhancements of the ms , and in the definitions the probabilities reduced by postselection are taken into account . if the value of the enhancement is bigger than 1 , weak measurement could indeed improve the precision in estimating the interaction strength .now , we consider the enhancements of ms in the stern - gerlach experiment . in this experiment , and the type - i enhancements are and where . for ,the enhancements are pictured in fig .we can see that the values of and are both less than 1 .  comparing and with and , we get that , and . for , we have denoting , by and , we get the maximum values of , and . as , we get and .so , for , we obtain that , .we also search the values of the enhancements for all pps by numerical calculation , and we find that the upper bound of the type - i enhancements is 1 .therefore , the ms can not be efficiently enhanced by weak measurement if the probability decrease due to postselection is taken into account .this was also pointed out in  .  .for all pps , and are both less than 1 ; meanwhile the maximum values of and are both approximatively .,title="fig : " ] . for all pps , and both less than 1 ; meanwhile the maximum values of and are both approximatively .,title="fig : " ] . for all pps , and both less than 1 ; meanwhile the maximum values of and are both approximatively .,title="fig : " ] . for all pps , and are both less than 1 ; meanwhile the maximum values of and are both approximatively .,title="fig : " ]   in this section , we consider the same technical limit discussed in sec .iv b. under this technical limit , the number of the outcomes is determined by the saturation of the detectors without considering the probability decrease due to postselection , and we have an alternative definition of the enhancements of ms they are referred to as type - ii enhancements of ms , and in the definitions the probabilities reduced by postselection are not taken into account .similarly , we have that and .  in the stern - gerlach experiment, we can have and where .we can see that , and . as for , we have , so we can get , . for , the values of enhancements and pictured in fig .4 . it can be seen that the maximum value of the enhancements is about 28 , which means the ms can be largely improved by the weak measurements when the probability decrease due to postselection is not considered . for , by numerical calculation , we get , which implies that weak measurements can effectively enhance ms when the probability decrease due to postselection is not considered , such as for practical cases when the measurement repetition frequency is determined by the intensity saturation of the detectors .significantly enhanced ms has been obtained in observing the spin hall effect of light  .we have studied quantum measurement with preselection and postselection , and have given the precise expressions of the measurement results without any restriction on the system - device interaction strength . for the very weak interaction ,a significant amplification effect can be obtained . for a qubit system ,the maximal pointer shifts of momentum and position are obtained .when the interaction between the system and device is strong , we have also obtained the ideal quantum measurement results .we also have studied the improvements of the snr and the enhancements of the ms by weak measurements .we have shown that weak measurements can not effectively improve the snr and the ms when the probability decrease due to postselection need to be considered ; while for practical cases when the probability reduced by postselection need not be considered , weak measurements can significantly improve both the snr and the ms . in addition , the large shifts obtained in weak measurements can also effectively reduce the relative system error .the idea of weak measurement is very useful in practical experiments .this work is financially supported by the national natural science foundation of china(grants no . 11075148 , and no .11175063 ) , the fundamental research funds for the central universities , cas , and the national fundamental research program , china .j. r. taylor , _ an introduction to error analysis : the study of uncertainties in physical measurements ( university science books , 1999)_. i. g. hughes and t. p. a. hase , _ measurements and their uncertainties : a practical guide to modern error analysis ( oxford university press , 2010)_. v. giovannetti , s. lloyd , and l. maccone science 306 , 1330 ( 2004 ) .

Abstract: we study quantum measurement with preselection and postselection , and derive the precise expressions of the measurement results without any restriction on the coupling strength between the system and the measuring device . for a qubit system , we derive the maximum pointer shifts by choosing appropriate initial and finial states . a significant amplification effect is obtained when the interaction between the system and the measuring device is very weak , and typical ideal quantum measurement results are obtained when the interaction is strong . the improvement of the signal - to - noise ratio ( snr ) and the enhancement of the measurement sensitivity ( ms ) by weak measurements are studied . without considering the probability decrease due to postselection , the snr and the ms can be both significantly improved by weak measurements ; however , neither snr nor ms can be effectively improved when the probability decrease is considered .   # 1|#1 # 1#1| # 1


understanding the distinct roles of noise and determinism is important for all experimental chaotic systems .we examine a scale - dependent lyapunov exponent ( sdle ) of a boolean chaotic system that iterates the dynamics a one - dimensional ( 1d ) map .a sdle was studied previously as a way to distinguish the entropy of microscopic noise from the entropy of macroscopic chaos .this distinction and the transition from a microscale entropy to a macroscale entropy is of increasing importance as physical random number generators that use chaotic systems as entropy sources continue to be developed .our experiment iterates a macroscopic tent map with microscopic noise and an intermediate scaling caused by mesoscopic discontinuities in the map . here, we demonstrate experimentally that the scaling of these discontinuities manifests as a structured scale - dependence in the lyapunov exponent of our boolean chaotic system .boolean chaos is a term used to describe the phenomenon of deterministic dynamics in unclocked boolean networks with an exponential divergence of neighboring trajectories .originally , theories of continuous , ideal boolean networks predicted non - repeating switching in certain networks , but without chaos .contrary to this prediction , recent experiments showed that physical logic gates can introduce non - ideal effects that give rise to such chaos .boolean chaos has been reported in both autonomous and driven networks , both of which yield complex , multi - dimensional dynamics .one - dimensional ( 1d ) boolean chaotic maps were theorized for specific non - ideal effects .here , we examine the first experimental 1d boolean chaotic map and report its unique , multi - scaled features .our experimental setup is influenced by recent studies of non - chaotic autonomous boolean systems .in particular , asynchronous networks of logic gates were studied as excitable systems with synchronization patterns and phase oscillators with chimera states .one appealing feature of these boolean systems is that they can be implemented entirely on a field - programmable gate array ( fpga ) , a common component in modern electronics .this platform allows for large dynamical networks of asynchronous logic gates to easily be built .the implementation of our experiment is also facilitated by an fpga , where we note that our system is not a simulation on a finite - state - machine ( see and references therein ) ; it is an unclocked system with non - zero entropy and a potential continuum of dynamical states that are subject to analog effects and experimental noise .our experimental system is shown in fig .[ fig : dynamics]a , which is initialized by an input voltage pulse of initial pulse - width .this pulse drives a map operator which consists of two separate functions : a pulse - width folding - function and a pulse - width gain - function that approximately doubles a pulse s width ( details provided later ) , where this combination of folding and stretching are sufficient conditions to see chaos .the output voltage of is labeled as , and a delay line routes back to the input of , where the delay is long enough to ensure only one pulse is in this feedback loop at a time .we note that this is a time - delay system with an infinite dimensional phase space , but neighboring pulses do not interact , allowing for 1d dynamics .this system remains in a stable steady state v until we inject a pulse , and thus serves as its initial condition .  after an initial pulseis injected , the system produces a self - sustaining pulse - train . in figs .[ fig : dynamics]b - c , we plot the temporal evolution of , which contains pulses that occur with non - repeating pulse - widths and non - repeating spacings .thus , the transition times in are the state variables of the chaos .this is different from analog chaotic circuits that iterate 1d maps in discrete or continuous time and use voltage or current as the state variables .the power spectral density of ( not shown ) is broadband with prominent frequency components at integer multiples of , where ns is the average pulse - repetition period and can be adjusted with the feedback delay .  to analyze the dynamics , we study , which is plotted in fig . [fig : dynamics]d ; as we will show later , is a function of and contains no new information about the dynamics .we construct the return map using in fig .[ fig : dynamics]e , which shows a 1d structure similar to a tent map .figure [ fig : dynamics]e also shows that the density of the return map is non - uniform , which differs from an ideal tent map .we fit fig .[ fig : dynamics]e using a piecewise - linear function where is the average slope and is the folding point .the fit yields and ns , which shows that the map is not a tent - map of full height .we note that a tent map of full height in the experimental boolean implementation can only show transient chaos before collapsing to the steady state v due to short - pulses rejection ( spr ) by the physical logic gates . the slope limits the grammar of the map and prevents pulse widths ns , allowing for non - transient chaos .interestingly , we might approximate the lyapunov exponent using such that . however , using this fit of the return map to estimate assumes a continuous map . as we will discuss later, the discrete logic gates in the system s design create discontinuities in the output of the operator .to avoid assumptions about the experimental system , we instead compute a sdle from .we define a sdle as the divergence of neighboring trajectories along the return map , where neighbors and satisfy where determines the scale of and is a fixed width that determines the window size to search for neighboring points . for a given , we find all and that satisfy eq . [ [ eq : epsil ] ] and define the initial trajectory separation as , where is the index of total neighboring pairs .next , we follow the trajectories of each neighboring pair and calculate their normalized divergences for to .an example of two neighboring trajectories and is shown in fig .[ fig : lyapunov]a .we then average over all .an example of for a given shown in fig .[ fig : lyapunov]b on a log scale .lastly , we use a linear fit of ( up to ) , where the slope of the linear fit gives the initial exponential separation of the neighboring trajectories for a given .we define this initial exponential separation as .the example in fig .[ fig : lyapunov]b yields , where we divide by to provide a unitless quantity . following this procedure , we calculate for a scan of in 10 ps steps and the result is shown in fig .[ fig : lyapunov]c . in the figure ,we use a scaling reference of , which is the approximate delay time through a single logic element .we note that is an indicator of trajectory separation at an exponential rate .figure [ fig : lyapunov]c demonstrates that the experimental system exhibits exponential separation from both noise and chaos .neighboring points on the return map with have an initial separation that is dominated by microscopic noise , while for the sdle shows oscillations in the divergence rate at frequency near ( the value of the lyapunov exponent for a continuous tent - map of the full height ) .thus , the rate of divergence shows a structured scale - dependence about a global rate that is described by the macroscopic features of the return map . studying this phenomenon is important for understanding the transition from microscopic noise to macroscopic chaos . in the remainder of this paper, we briefly outline the design of fig .[ fig : dynamics]a and examine the map operator s functions to motivate a simple model that exhibits a similarly structured sdle .our experimental system exploits the propagation delays of pulses as they transmit through logic gates . in the simplest example , the feedback loop in fig .[ fig : dynamics]a acts as a continuous delay line that routes pulses from the output of the map operator back to its input .this delay line is constructed using cascaded not gates , where even numbers of not gates are used to reduce asymmetries between rise and fall times of pulse edges that propagate and preserve pulse widths .the number of not gates sets the propagation delay .  in the map operator ,a folding function is implemented with the circuit in fig .[ fig : tentmapfolding]a . in the figure , and are signals for input pulses of width and output pulses of width , respectively , such that for and for . to illustrate this circuit s folding , in fig .[ fig : tentmapfolding]b we plot experimental examples of of the folding circuit , and in fig .[ fig : tentmapfolding]c , we scan and plot the respective . we model from eq .[ [ eq : tentfold ] ] with , and we note does not address , but based on the folding circuit s operation , we derive for and otherwise .  the pulse - width gain function of the map operator is shown fig .[ fig : stretching]a . in the figure ,an input pulse is launched into a delay line of not gates , where and gates compare the outputs of gates , where is the index number of total not gates such that .the and - gate outputs drive a multi - input or - gate , which outputs a pulse of width for , where is the delay through gates . to illustrate the pulse - width gain from this circuit , we plot examples of and a scan of in figs .[ fig : stretching]b - c , respectively .the resulting waveforms show approximate pulse - width doubling and the characterization of has average slope .however , the discrete nature of the and - gate comparisons of the delay line in fig .[ fig : stretching]a creates regularly - spaced discontinuities that are not resolved in fig .[ fig : stretching]c . based on these discrete comparisons ,we model the discontinuous pulse - width gain as where is a measure of in single gate - delays , and is a function that describes the width of an output pulse for a single gate . as increases in eq .[ [ eq : discont ] ] , is discontinuous and increases by steps of .we define such that , when is an integer multiple of ( , the pulse - width gain is exactly 2 .when is not an integer multiple of , provides a corrective term that describes the continuous growth of pulse widths , where the input to resets at each multiple of .thus , the function is has an average slope with discontinuities spaced regularly by and local slope(s ) in between each discontinuity .  for simplicity, we let such that the map is an example of a piecewise - linear system that exhibits noise - induced chaos .noise - induced chaos occurs in chaotic systems that only show periodic or steady state dynamics without the presence of noise .similar models with discontinuities have been previously studied with the use of a sdle , where with enough noise , these simulated chaotic maps exhibit characteristics of their macroscopic map structures .a different choice for , such as a nonlinear function , can yield chaos without noise , but we choose the simplest chaotic model with mesoscopic discontinuities and microscopic noise .noise in the fpga causes jitter in , where we measure the jitter to be approximately gaussian with standard deviation ( std ) ps .we simulate the map using eqs . [[ eq : tentfold]-[eq : discont ] ] with , ns , ns , and additive - white - gaussian - noise at every iteration ( std ) .the simulated return map is shown in fig .[ fig : simulations]a with a 1d structure similar to a tent map with slopes .the probability density of the map is also non - uniform , where clustering occurs at evenly - spaced intervals .this differs from the experimental density because , in the model , we can guarantee that , for integer . even though there are an integer number of logic gates in the experimental , heterogeneities in gate delays due to physical effects and fpga routing cause , where is a cumulative timing difference .we implement a timing difference in the model ( not shown ) and note that clustering in the return map changes with .next , we calculate the simulated sdle for the return map in fig . [fig : simulations]a .we acknowledge that there are different methods for the treatment of noise when computing lyapunov exponents in simulated chaotic systems with noise . in our simulations, we compute iterations of the model with additive noise at each iteration and search for neighboring points .this method is equivalent to specifying nearby initial conditions and using different realizations of noise along each trajectory .furthermore , this method allows us to emulate the process of collecting experimental data and use the same algorithm for finding the sdle that is applied in the experiment .the resulting simulated sdle is plotted in fig .[ fig : simulations]b . in the figure ,the sdle increases rapidly as goes to zero , which is similar to the observed experimental sdle . for , the divergence of neighboring pointsis dominated by a stochastic , uncorrelated process provided by the additive gaussian noise .in addition , fig .[ fig : simulations]b demonstrates that contains mesoscopic features ) that oscillate about the average divergence rate of the macroscopic chaos .similar to the experiment , the oscillatory structure in the simulated sdle has a frequency of and not .based on the model s construction , we attribute the frequency to grammar restrictions in the dynamics that are caused by symmetric gaps on either side of the map due to the discontinuities in . though the symmetry of these gaps is not visible in the experimental map , it is inherent in its design and we hypothesize that , on average , similar grammar restrictions occur in the experimental dynamics .thus , the results from the simple model are quantitatively similar to those from the experimental boolean system , where more agreement can likely be achieved by introducing individual gate - delay heterogeneities , using a macroscopic slope , and exploring nonlinear functions for .furthermore , we note that the structures in the simulated slde become more ( less ) pronounced for lower ( higher ) levels of noise , and for sufficiently high noise levels , the structures are no longer detectable . as examples , in figs . [ fig : simulations]c - d , we include the results from simulations of the model with noise levels ps and ps . for ps , the amplitude of the oscillatory structure grows while its frequency remains fixed . for , the noise is on a comparable scaling to the mesoscopic , which removes the structure in the sdle .this suggests that some physical chaotic systems may have underlying fine - scaled features that do not play a role in the divergence of trajectories because they are on the same scale or smaller than the microscopic noise levels .in addition , as shown in figs . [fig : simulations]b - d , the amplitude of the noise influences the size of the microscopic region in the sdle ; larger noise amplitudes extend the microscopic region to larger epsilon .lastly , introducing irregularly - spaced discontinuities in the model ( not just at but at , , etc . ) also blurs these structures ( not shown ) and suggests that heterogeneities may be a mechanism that removes structures in the sdle . in our experiment ,the mesoscopic spacings between discontinuities can be tuned by moving the and - gate inputs in fig .[ fig : stretching]a , and thus our 1d boolean chaotic system is a good candidate to begin exploring these concepts .  in summary , we present an experimental chaotic system with a macroscopic 1d return map , microscopic noise , and mesoscopic discontinuities that are regularly spaced .we demonstrate these three scalings in a sdle and observe that the mesoscopic features create an oscillatory structure in the sdle . using a physically - motivated , simple model, we reproduce a similarly structured scale - dependent lyapunov exponent and note its effects at different noise levels .s.d.c acknowledges the financial support of the us army aviation and missile research development and engineering center and is thankful for discussions with dr .ned corron and dr .jonathan blakely .also thanks dr .david rosin his procedures in ref . .16ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1'''' [ 0]secondoftwosanitize [ 0 ] + 12$12 & 12#1212_12%12[1][0] link:\doibase 10.1103/physrevlett.82.1132 [ * * ,  ( ) ] link:\doibase  10.1103/physreve.62.427 [ * * ,  ( ) ] link:\doibase  10.1103/physreve.74.066204 [ * * ,  ( ) ] * * ,  ( ) link:\doibase 10.1103/physrevlett.103.024102 [ * * ,  ( ) ] * * ,  ( ) * * ,  ( ) link:\doibase 10.1103/physreve.87.040902 [ * * ,  ( ) ] * * ( ) link:\doibase 10.1103/physreve.80.045202 [ * * ,  ( ) ] _ _ , edited by , , and ( , ) chap .  * * ( ) link:\doibase 10.1209/0295 - 5075/100/30003 [ * * ( ) , 10.1209/0295 - 5075/100/30003 ] link:\doibase 10.1103/physrevlett.110.104102 [ * * ,  ( ) ] link:\doibase 10.1103/physreve.89.042907 [ * * ,  ( ) ] link:\doibase  10.1103/physreve.90.030902 [ * * ,  ( ) ] _ _ , edited by and ( , ) chap . _ _ ( , ) * * ,  ( ) * * ,  ( ) * * ,  ( ) _ _ , springer theses ( , , )

Abstract: we report on structures in a scale - dependent lyapunov exponent of an experimental chaotic map that arise due to discontinuities in the map . the chaos is realized in an autonomous boolean network which is constructed using asynchronous logic gates to form a map operator that outputs an unclocked pulse - train of varying widths . the map operator executes pulse - width stretching and folding and the operator s output is fed back to its input to continuously iterate the map . using a simple model , we show that the structured scale - dependence in the system s lyapunov exponent is the result of the discrete logic elements in the map operator s stretching function .


the issue of the pairwise packing between helices in proteins was addressed soon after helical structures had been suggested .a number of models were developed , mostly devoted to surface complementarities upon packing .the `` knobs into holes '' model , first introduced by crick and elaborated by richmond and richards , aimed to find the best steric fit between regular helices .chothia , levitt and richardson recognized the importance of `` ridges '' and `` grooves '' formed by residues with different sequential distances ( `` ridges into grooves '' model ) .efimov tried to relate the packing angle between the two helices with the preferred rotational states of the side - chains along them .a comprehensive analysis was eventually carried out by walther et . , by modeling helix packing as the superposition of the two regular lattices that result from unrolling the helix cylinders onto a plane and contain points representing each residue .the six `` preferred '' angles predicted by this last model are consistent with earlier results and with the histograms of the experimentally observed packing angles ( see fig. 7 of ref .the agreement between theoretical modeling and experimental data was remarkable , although not perfect ( see ref . for a more detailed discussion ) .the success of steric models in providing an explanation for the most prevalent packing angles was however put under discussion after the observation made by bowie that statistical corrections must be applied to the values collected from experimentally determined structures before true interaxial angle preferences can be revealed .indeed , the helix - helix packing is defined to occur only when :  * [ c1 ] the segment of closest approach ( sca ) , of length , between the two finite helix axes is shorter than a prefixed threshold ; * [ c2 ] this segment intersects both helix axes at a perpendicular angle .  in ref . , two helices were considered to be in contact if they were satisfying both conditions and .note that walther _ selected interacting helical pairs from their database of native protein conformations , by considering distances between all possible inter - helical heavy atom pairs .we used a different selection procedure involving distances between inter - helical atom pairs ( see methods for details ) .none of these particular choices really affects our geometrical analysis , and condition c1 , as defined above , is just the simplest way of defining an equivalent distance constraint .condition c2 is needed to ensure face - to - face packing of the two helices , justifying thus the use of theoretical modeling based on the steric interdigitation of the helices .  if the sca is coincident with the global segment of closest approach ( gsca ) , of length , between the two straight lines which are obtained by indefinitely prolonging the helix axes , condition c2 is then automatically satisfied . in such a situation , which corresponds to effectively dealing with helices of infinite length ( see fig .1a ) , the reference probability distribution of interaxial angles is simply the spherical - polar distribution between any two random vectors , namely  .walther and co - workers realized that the finiteness of helix axes is crucial in modifying the angular dependence of , because requiring that the sca intersects both axes at a perpendicular angle introduces new restrictions depending on the packing angle .this can be understood by looking at fig .2a , where it is shown that , fixing one helix position , and assuming the gsca to be orthogonal to the page , the second finite helix axis may then be placed only within a plane parallel to the paper plane and such that its starting point lies inside the dark shaded parallelogram .the probability of placing the starting point of the second helix in a is proportional to its area and therefore to .consequently the probability for selecting a particular packing angle is proportional to the product of the spherical - polar contribution and of the effect due to the finite length condition :    with this new random reference distribution the actual angular propensities need to be reconsidered . as is clear from fig . 2 of ref . , the normalized frequencies reveal a prominent representation of packing angles near and ( not expected by theoretical models ) whereas the predicted optimal steric packing angles manifest themselves , at most as shoulders in the distribution of propensities .the predictivity of geometrical models is cast under serious doubts after this analysis .our first observation is that the random distribution is correct only and only if the condition of mutual perpendicularity between the sca and the two axes is strictly fulfilled . but is this the real situation when statistical histograms are derived ?  as a matter of fact , condition was relaxed by walther _, by admitting a small tolerance : a total deviation ( and are the complementary angles to the angles and formed between the sca and helix 1 and helix 2 , respectively ( see fig .1b ) ) was accepted up to a threshold ( in and in ) .  at first, one might wonder why such a threshold needs to be used since it does not effectively increase the number of data contributing to the histograms .however , we believe that this choice is indeed necessary because of the ambiguity in the definition of the axis direction in natural helices , which introduces an intrinsic uncertainty in the computation of the angles ( see methods for a detailed explanation of how the axis is reconstructed in the typical case of a non - ideal bent helix and for an estimation of such uncertainty ) . a relaxation of condition is thus crucial , if we want to analyse correctly data extracted from the database of real native protein structures.  we will show in this paper , by means of partly semi - analytical geometrical arguments and eventually numerical simulations , that such threshold effect does drastically change the random reference distribution .having measured helix - helix packing angles from a set of proteins representative of the pdb native structures , we will then reanalyse the packing angle distribution after its proper rescaling with the newly found reference distribution .we employed the same ensemble of 600 proteins considered by chang et .al , which consisted of sequences varying in length from 44 to 1017 , with low sequence homology and covering many different three - dimensional folds according to the structural classification of proteins ( scop ) scheme .the structures were monomeric and determined using x - ray crystallography .we collected helices each with at least four consecutive residues classified as helical in the pdb files .the average number of residues of these helices was .two helices were defined to be in close contact , if at least one interhelical contact between atoms was present , with a maximal threshold distance of ( analogous to condition in the text ) .only helix pairs separated in sequence by at least intervening residues were considered , to get rid of possible correlations induced by short loops .the resulting data set consisted of closely packed helix pairs .   the reconstruction of the helix axis from the coordinates of the atoms of the corresponding residues is a critical step in the determination of packing angle preferences . since real helices can be bent , we adopted the procedure described by walther _ , in which a local axis is associated to every consecutive residue pair along the helix .the overall axis is thus a broken line consisting of short segments .  a good starting approximation for the local axis , based on the the four atom positions , , , , was introduced by chothia _we employed a sligthly modified definition , where we first define the normalized bond vectors . in this way ,the set of three orthonormal vectors , , , the natural reference system associated with the atom trace , can be defined without the distortions due to the fluctuations of the bond length between consecutive atoms in the following way : , , .if the atom trace followed a perfect ideal helix , the vector would be parallel to the helix axis .we then initially determined the local axis between residues , , as the vector parallel to , of length , which has the geometric centre of the closest four consecutive atom positions , , , , , as its midpoint .the two local axes at the helix termini were obtained by simply prolonging the neighbouring ones .we then applied a smoothing procedure similarly to walther _the ` new ' local axes were obtained by averaging the direction of the closest three ` old ' ones , , , , while conserving its midpoints . for the two local axes at the helix termini we averaged over the direction of the closest two available ones . to ensure the continuity of the overall axis ,the inner hinges of the broken axial line were computed as the midpoints between extremities of consecutive local vectors obtained from the running average . after repeating the whole procedure twice ,the standard deviation of the distances of each atom from the reconstructed helix axis was .the interaxial packing angle between two helices was computed bewteen the local axis pair for which the minimum distance of closest approach was achieved . in casethe segment of closest approach ( sca ) intersected an inner hinge of the broken global axis , the local axis direction was defined as the average of the two corresponding local vectors .the whole discussion concerning the perpendicularity of both local axis direction with the sca applies only to cases involving a terminal local axis , since face - to - face packing is anyway ensured in case of contact between inner local axes .packing angles are positive if the background helix is rotated clockwise with respect to the frontal helix when facing them .the angle ( ) corresponds to parallel ( antiparallel ) helices with respect to their sequence direction .  imposing the further requirement that the sca intersected both local axis directions at a perpendicular angle within a threshold plays a critical role , as discussed in the text .for the three different values considered in this work , , we collected datasets of , , closely packed helix pairs , respectively .  as a final remark we report the mean value of the angle which is formed between consecutive local directions averaged over all inner hinges of all helical axes which were reconstructed using the procedure discussed above .we found , strongly supporting the necessity of allowing a similar threshold when imposing angular constraints as in condition .we will now sketch the geometrical consequences of relaxing condition within a given threshold , in order to understand by means of a simple argument how this effects the random reference distribution .a comprehensive analytical treatment is in principle feasible but quite cumbersome .numerical simulation will ultimately be the preferential approach to extract the corrected random reference distribution for interaxial angles .  within the admitted threshold , cases similar to the one described in fig .1b may now occur .the sca between the two helices , which intersects helix 1 at one of its axis ends and helix 2 at some internal point , forms an angle which deviates from by less than , whereas the gsca does not intersect both helices ( in this situation ) . if , the two helices are considered to be in contact .the condition is equivalent to the condition that gsca intersects helix 1 within a distance from the end of its axis .this limit is reached when .there are also cases in which and , when the sca intersects helix 2 at one of its axis ends and helix 1 at some internal point , and cases in which both and are not zero , and the sca intersects both helices at one of their axes ends ( fig .in this last case , the geometrical condition implied by , and involving the distances , from the end of both helix axes to their intersection with the gsca , is more complicated ( and is discussed in the caption of fig .1 ) .  using the visual representation of fig .2b , we can say that allowing condition to be satisfied within the threshold , opens up the possibility for the starting point of helix 2 to lie in the portion of space ( ) formed by the four lightly shaded parallelograms ( two parallel to helix 1 with sides and , and two parallel to helix 2 with sides and ) and by the four remaining white corner regions ( whose boundaries are defined by hyperboles , see figure 2 caption for details ) , which surround parallelogram .  to compute the area of is not trivial , because of the hyperbole - shaped regions , but the result is obviously dependent .for example a simple trigonometric calculation shows that :    this implies that the area of diverges for , which already points towards the fact that relevant corrections are possible even for small values of .  to conclude our analysis we need to enforce also condition . by using the pythagorean theoremit is easy to see that for any fixed , is satisfied for any point ( thought as starting point of helix 2 ) whose distance from parallelogram is less than .these points belong to the region , shown in fig .2c , formed by the 4 parallelograms ( two of sides and and two of sides and ) and the four circular sectors ( of radius ) surrounding a. the area of is independent on ( see caption of fig . 2c ) .all the points in the portion of space resulting from the intersection of and satisfy both condition and .therefore , the probability of selecting a particular angle with a tolerance , at a given distance , is given by the product of the spherical polar term  and a term proportional to the area of parallelogram _ plus _ the area of .when , eq.2 shows that diverges and thus : where is the area of the corner regions in .  since does not depend on , the second term in the bracket will eventually dominate the small behaviour of . in practice , the actual relevance of this effect can be appreciated only after integrating over , since the relative weights of the different terms in the bracket vary with .such computation is quite cumbersome and we have rather chosen to get by simulating random helices which satisfy the contact conditions within a threshold , and to extract numerically the normalized histograms .   in order to compute the reference distribution for interaxial packing angles we generated _ random _ helix pairs by means of computer simulations , and then selected them with the same conditions , and , used in extracting histograms from real helices in native protein structures .note that when computing the reference distribution we did not take steric effects into account ; in other words random helices might overlap .  more specifically , we constructed ideal discretized helices with the same geometrical properties of -helices in real proteins , i.e. twist per residue , rise per residue , radius ( as is the case when considering atoms ) .we chose to generate helices consisting of residues , the average length in the dataset of real helices that we collected from the pdb . keeping fixed the position of the first helix, the second helix was placed by firstly choosing randomly the midpoint of its axis within a sphere of radius centred in the midpoint of the first helix axis , and then selecting , again randomly , both the direction of its axis and the twist of its first residue .boundary effects might be relevant , when the radius of the sphere in which the second helix is generated is too small with respect to the helix length and helix finiteness is effectively reduced .we made sure that our results did not change when increasing the radius of the sphere in which the second helix axis is placed .we generated random helix pairs , out of which satisfied condition of having at least one pair of residues distant less then .condition was then applied , in the same way as explained in methods for the real helices data set , with the whole axis being now a segment in the ideal case . in this way we generated , , helix pairs , respectively , for the three different values considered in this work , .  in fig .3 we plot the corresponding random reference distributions , comparing them with the ideal ( ) case : .the difference , although due to a very subtle effect is substantial , and it is clearly seen that a new regime is present for , as expected from the previous discussion .   to test the effect of this new reference distribution we have recomputed an experimental distribution of interaxial angles ( see methods for details ) by analysing a databank of 600 proteins and using a contact threshold and a tolerance , corresponding the intrinsic uncertainty estimated when determining local axis directions ( see methods ) .the histogram is reported in the upper panel of fig .4 : in order to obtain good statistics for each single bin , we gathered results every . the histogram is consistent with previous analysis . in the middle panel of fig .4 we plot and , for reference , normalized with the same binning as in the previous panel . finally , in the lower panel of fig .4 we present the histogram both rescaled with and with .the results are clear and striking : the correct distribution removes the spurious peaks at and , whereas out of the predicted packing angles arise as clean local maxima in the expected positions. only two of the predicted packing angles are clearly not favoured according to our analysis . remarkably , the two peaks theoretically predicted at and , that were seen as shoulders in the unrescaled histogram , are instead placed in the correct position ( within binning uncertainty ) in the rescaled histogram .we also note that a residual preference for parallel and antiparallel alignment of two contacting helices can not be ruled out even after proper rescaling .because of small statistics , it is difficult to ascertain whether this is a genuine effect or is just due to the tail superposition of different preferential angle peaks . in order to further test both the robustness of the results and the full procedure we have obtained statistical histograms from the experimental data using three different thresholds , . since we can compare them with the corresponding random distribution we should expect three similar rescaled histograms .this is nicely confirmed by fig .5 , where the three rescaled histograms show a very good overlap , within the statistical uncertainty .we also note that generally , the greater the threshold used to relax condition , the more blurred the peaks corresponding to the sterically preferred angles .this confirms that condition is needed to ensure face - to - face packing and the applicability of steric models .in this paper we have shown that the calculation of the probability distribution of interaxial angles between random finite helices which are in contact is not a trivial geometric problem , because of the approximations introduced to ensure face - to - face packing between contacting helices .such approximations are unavoidable , due to the imperfect shape of natural occurring helices which do not have well defined axes . although analytical results can be found to estimate the correct random distribution , the simplest way to obtain it consists in using numerical simulations .we have presented a re - analysis of the distribution of packing angles rescaled with our new reference distribution and we have found a remarkable agreement with the packing angles predicted by steric models .  99  crick , f.h.c .( 1953 ) _ acta crystallog . _ * 6 * , 689 - 697 .richmond , t.j . ,richards , f.m .biol . _ * 119 * , 537 - 555 ( 1978 ) .chothia , c. , levitt , m. , richardson , d. ( 1977 ) _ proc .usa _ * 74 * , 4130 - 4134 .chothia , c. , levitt , m. richardson , d. ( 1981 ) _ j. mol .biol . _ * 145 * , 215 - 250 .efimov , a.v .( 1979 ) _ j. mol .biol . _ * 134 * , 23 - 40 .walther , d. , eisenhaber , f. , argos , p. ( 1996 ) _ j. mol .biol . _ * 255 * , 536 - 553 .bowie , j.u .( 1997 ) _ nature struct .biol . _ * 4 * , 915 - 917 .walther , d. , springer , c. , cohen , f.e .( 1998 ) _ proteins : struct .genet . _ * 33 * , 457 - 459 .chang , i. , cieplak , m. , dima , r.i ., maritan , a. , banavar , j.r .( 2001 ) _ proc .usa _ * 98 * , 14350 - 14355 .murzin , a.g . , brenner , s.e ., hubbard , t. , chothia , c. ( 1995 ) _ j. mol. biol . _ * 247 * , 536 - 540 .rey , a. , skolnick , j. _ j. comput .* 13 * , 443 - 456 ( 1992 ) .we are indebted to amos maritan for stimulating discussions and advice , giuseppe zanotti for helpful suggestions and iksoo chang for his invaluable help in providing us with the protein databank used in ref .we thank harvey dobbs for a critical reading of the manuscript .this work was supported by infm , miur cofin-2001 and fisr 2002 .\a ) the global segment of closest approach ( gsca ) , of length , between two straight lines is , by definition , perpendicular to both of them .the segment of closest approach ( sca ) , of length , between two helices ( schematically represented by cylinders ) surely coincides with the global one if it intersects both helices within their axis length .this fact is always guaranteed if the two helices are assumed to have infinite length ( hypothesis of ref . ) .\b ) the two finite helices have a sca that is not coincident with the gcsa .the latter intersects helix 1 at a distance from its end and may ( as in the picture ) or may not intersect helix 2 .the former joins the end point of helix 1 to a point inside helix 2 .the angle formed with helix 1 ( ) is deviating from by an amount .instead and . the trigonometric relation expressing as an increasing function of be easily obtained : . when , condition , defined in the text , is still fulfilled .this is equivalent to require that .notice that diverges when approaches from below , and consequently , when , diverges . in such regimethe condition c2 ( but obviously not c1 because may also diverge ) is always satisfied within the allowed tolerance .a similar situation and a similar analysis can be done for and .\c ) gsca is neither intersecting helix 1 nor helix 2 and the sca is joining two endpoints . in the case depicted in the figure the two helicesare placed in opposite sides with respect to the gsca .the threshold condition may be translated into the following one involving and :  .situations in which the two helices are placed on the same side with respect to the gsca are also possible . in this casethe condition becomes  .\a ) in this figure , the position of helix 1 ( represented by a vector of length ) and a vector orthogonal to the paper plane , determining the direction of the gsca to a second helix , are fixed .given a particular packing angle , in order to satisfy condition c2 without any tolerance , a second finite helix axis ( of length ) may then be placed only within a plane parallel to the paper plane and such that its starting point is inside the dark shaded parallelogram , otherwise the sca between the two finite axes would no longer be perpendicular to both of them . the area of parallelogram is .\b ) the tolerance with which the condition of perpendicularity between the sca and the two helices is implemented enlarges the portion of space where the starting point of helix 2 can be placed .the region formed by the four lightly shaded parallelograms and the four white corner regions , which together surround parallelogram , can now be exploited .the two parallelograms parallel to helix 2 , whose sides are and , are due to the geometric arrangement described in fig . 1b with and ( the vector with the dashed starting point represent the possible location of helix 2 in such context ) .the two other parallelograms , parallel to helix 1 , with sides and , correspond to the similar case and .the four remaining white regions , corresponding to the case and , are all limited by hyperbola - like curves .the two of them subtending an obtuse angle are due to the geometric arrangement described in fig .1c , corresponding to the two helices being placed in opposite sides with respect to the gsca ( the vector with the white starting point represent the possible location of helix 2 in such context ) .the two of them subtending an acute angle correspond to the two helices being placed in the same side with respect to the gsca . in both cases it is possible to find the equation describing such boundary curves , which we omit here for the sake of simplicity .the area of is plus a non trivial contribution coming from the four corner regions .\c ) the distance constraint c1 limits by itself the portion of plane surrounding parallelogram where the starting point of the second helix can be placed .this region is simply the locus of points whose euclidean distance from in the paper plane is less than . for given values of and the distance threshold , this regionis thus formed by the four lightly shaded parallelograms and by the four white circular sectors .the area of is .ideal ` random ' distribution of interaxial packing angles for four different values of the angle threshold used in applying condition ; ( upper left panel ) , ( upper right panel ) , ( lower left panel ) , ( lower right panel ) .the four distributions were computed by means of numerical simulations described in methods . since ideal helicesdo not have a preferential order , the histograms show in this figure have been restricted to the region .the packing angle values collected in the histogram were then successively filtered out by the more and more restrictive , , threshold conditions to generate the , , data , respectively , collected in the corresponding histograms .the dashed line is the fit of the data , obtained enforcing condition in a strict way , to the expected distribution , which is then reported for comparison in all other histograms .all histograms were constructed with a bin width of .histograms are normalized in such a way that a flat distribution would correspond to a constant height .upper panel .experimental ` bare ' unrescaled distribution of interaxial packing angles between contacting helices .data were obtained with a distance threshold for interhelical atom pairs and an angular threshold for condition .the histogram is normalized in such a way that a flat distribution would correspond to a constant height .middle panel .ideal ` random ' distribution which have to be used to rescale the ` bare ' experimental distribution , in order to reveal true packing angle preferences .the two different distributions obtained with either ( thick line ) or ( filled columns ) are shown .histograms are normalized in such a way that a flat distribution would correspond to a constant height .  lower panel .rescaled distribution of interaxial packing angles . the experimental ` bare ' distribution in the upper panel is divided by the ideal ` random ' one in the middle panel , obtained with either ( blue line ) or ( filled columns ) .histogram heights greater than correspond to preferential packing angles , whereas heights lower than correspond to disfavoured packing angles .all histograms were constructed with a bin width of .arrows in the upper and lower panels mark the values of the six preferred packing angles predicted by walther _et al . _ ; ,each represented twice with a periodicity of .rescaled distribution of interaxial packing angles between contacting helices for three different values of the angle threshold used in enforcing condition ; ( light grey filled columns ) , ( dark grey line ) , ( black line ) .each histogram was obtained by dividing the experimental unrescaled distribution by the corresponding ideal reference one ( we used the three distributions represented in fig .3 with a different binning ) .all histograms were constructed with a bin width of .note that the histograms are not normalized , being computed as the ratio of two normalized histograms .the arrows mark the values of the the six optimal packing angles predicted by walther _

Abstract: for many years it had been believed that steric compatibility of helix interfaces could be the source of the observed preference for particular angles between neighbouring helices as emerging from statistical analysis of protein databanks . several elegant models describing how side chains on helices can interdigitate without steric clashes were able to account quite reasonably for the observed distributions . however , it was later recognized ( bowie , 1997 and walther , 1998 ) that the `` bare '' measured angle distribution should be corrected to avoid statistical bias . disappointingly , the rescaled distributions dramatically lost their similarity with theoretical predictions casting many doubts on the validity of the geometrical assumptions and models . in this report we elucidate a few points concerning the proper choice of the random reference distribution . in particular we show the existence of crucial corrections due to the correct implementation of the approach used to discriminate whether two helices are in contact or not and to measure their relative orientations . by using this new rescaling , the `` true '' packing angle preferences are well described , even more than with the original `` bare '' distribution , by regular packing models .  # 1#2  # # 1##2#2##1  0.0 cm 0.0 cm


identifying complex dependence structures among predictors and responses is an important problem in statistics and machine learning , since these structures reveal hidden domain knowledge about the data .for example , in bioinformatics , identifying gene regulatory networks is crucial for understanding gene regulatory paths and gene functions , which helps disease prediction and diagnosis .similarly , in social media analysis , inferring the influence networks from user activities ( that is , _ diffusion network inference _problem ) is an important problem , and it has found applications in social media marketing and crisis management . in these big data applications , inferring the dependence structures is challenging since the responses and predictors may be related through a few latent pathways and/or associated through only a subset of responses and predictors . moreover , the curse of dimensionality and massive amounts of data , that is , scalability issues make the dependence structure discovery problem even harder to solve .recently , regularization methods such as lasso and group lasso , and reduced - rank regression approaches have been proposed to recover sparse response - predictor associations and latent predictors , respectively . and have proposed sparse reduced - rank regression approaches by combining the regularization and reduced - rank regression techniques to find the complex dependence structures between responses and predictors .sparse reduced - rank regression works by modeling the associations between the predictor and response variables via a sparse and low - rank representation of the coefficient matrix .it not only enhances the interpretability of the estimated matrix by eliminating irrelevant features , but also reduces the number of free parameters of the model and thus the number of observations required for desired estimation consistency .sparse reduced - rank regression has found applications in micro - array biclustering , subspace clustering , social network community discovery , and motion segmentation . in these applications ,joint sparsity and low - rankness has been used to enforce a clustered dependence structure among data points . in particular ,the key idea is to estimate a similarity matrix among data points that is simultaneously sparse and low - rank and then permute the rows and columns of the matrix to yield approximately block - diagonal structures , which naturally lead to clustering of data points into several groups .note that , , and the references therein have considered estimating matrices with a low - rank plus sparse representation which is different from our work as we are interested in estimating a matrix that is jointly low - rank and sparse .a natural approach to solving the sparse reduced - rank regression problem is to simultaneously penalize the parameter matrix using the and nuclear norm regularizers , as they are convex relaxations to sparsity and low - rankness of a matrix , respectively .the resulting optimization problem is convex and can be solved using the alternating direction method of multipliers ( admm ) as proposed by and . in , an alternative approach ,called rank constrained group lasso ( rcgl ) , was proposed which directly penalizes the rank and the number of nonzero rows of the parameter matrix .they showed oracle rates for the estimated matrix and also provided a practical algorithm which iteratively and jointly solves a -regularization and low - rank estimation problem .to further improve the estimation accuracy , borrowed ideas from adaptive lasso and proposed the iterative exclusive extraction algorithm ( ieea ) which finds a locally optimal solution in the neighborhood of the initial value. they also showed model selection consistency and asymptotic normality results along with the improved empirical performance of ieea on microarray biclustering analysis data .all the above approaches for sparse reduced - rank regression achieve both desirable theoretical properties and strong empirical results . however , they can not scale to large matrix estimation problems in many big data applications .the admm algorithm of and uses iterative singular value thresholding for solving the joint and nuclear norm regularization .iterative singular value thresholding is known to be computationally expensive since it performs a full singular value decomposition of the parameter matrix in each iteration . on the other hand ,rcgl is computationally much faster than admm since it only performs top- singular value decomposition for estimating a rank- matrix in each iteration . despite a lower computational cost per iteration ,it is unclear how many iterations rcgl needs for convergence .ieea performs nested loops of alternating -penalized regression for each singular vector which can be expensive , especially on parallel computing devices .the iterative nature of these three approaches makes them not scalable and renders them inefficient for large matrix estimation even on high performance computing devices .  to overcome the scalability issues of the previous approaches , we propose a simple and scalable sparse reduced - rank regression procedure called sequential estimation with eigen - decomposition ( seed ) .seed is designed for high - performing computing platforms .it converts the sparse and low - rank regression problem to a sparse generalized eigenvalue problem , and then solves the problem using the recent algorithms for sparse eigenvalue decomposition . as a pure learning algorithm , seed is expected to perform only a single top- sparse eigenvalue decomposition for estimating a rank- matrix , which makes it truly scalable and efficient for large matrix estimation problems .the main contributions of our paper are threefold .first , for the sparse reduced - rank regression problem , our proposed procedure seed provides a scalable approach to uncovering the sparse predictor - response association network while simultaneously achieving dimension reduction and variable selection .second , for the high - dimensional settings , our theoretical analysis shows that seed can consistently estimate the singular vectors , latent factors as well as the regression coefficient matrix , identify the correct rank of the matrix , accurately predict the multivariate response vector , and recover the support of the singular vectors under mild conditions . note that , compared to , we do not make any assumption on the positive definiteness of the design matrix for proving our consistency results .third , we empirically demonstrate that seed can not only be efficiently implemented on both central processing units ( cpu ) and graphics processing units ( gpu ) for large - scale applications , but it also outperforms the state - of - the - art sparse reduced - rank regression approaches .the rest of this paper is organized as follows .section [ sec : method ] introduces our seed method .we discuss the implementation details of seed in section [ sec : imp ] and present its asymptotic properties in section [ sec : analysis ] .we demonstrate the advantages of seed on both synthetic and real data sets in section [ sec : exp ] , and in section [ sec : dis ] we discuss some extensions of our seed method . all technical proofs and details are provided in the appendix .denote by  observations for the fixed design setting where and represent the predictor and the corresponding response vectors , respectively . given a predictor vector ,the corresponding response vector is drawn from the following model where the noise vector is a -dimensional zero mean multivariate gaussian random vector with the covariance matrix , and is the regression coefficient matrix .we can rewrite the data model in the matrix form as follows where {^{t}} ] , and {^{t}} ] , and ^{1/4}\right) ] , and ^{1/4}\right) ] , and ^{1/4}\right) ] and is the evolution matrix at lag .the influence network can be built from the evolution matrices by establishing an edge from node to node if is significantly larger than zero .  in this experiment, we gather a twitter data set with tweets on the `` haiti earthquake '' and apply vector auto - regressive model to identify the potential top influencers on this topic ( that is , those twitter accounts with the largest impact on the others ) .we divide the 17 days after the haiti earthquake on jan . 12 , 2010 into 1000 uniformly spaced intervals and generate a multivariate time series data set by counting the number of tweets on this topic for the top 1000 users who tweeted most about it .for accurate modeling , we remove the users that were highly correlated with each other , most of which were operated by the same users and tweeted exactly the same content .we also remove robot - like user - accounts which tweeted on very regular intervals , which in total led to a subset of 270 users .we analyze this data with a var( ) model which requires estimation of a dimensional response vector using predictors while we have only observations . the number of lags is chosen based on the intuition about the maximum retweeting delay .since we do not have access to the true influence network , we use the retweet network as a surrogate of the ground truth following the evaluation convention in the social networks community .the retweet network is constructed by adding an edge from user to user if user has retweeted at least of the tweets of user .clearly , the retweet network is not the actual underlying temporal dependency graph , mainly because there are possible implicit influence patterns as well .however , it is the best possible metric that we could obtain for graph estimation accuracy evaluation in our data set .the retweet network for the 270 selected users is sparse ; it has only of possible edges .we apply seed , ln admm , and rcgl algorithms to uncover the influence network in our twitter data set .figure [ fig : accuracytwitter ] shows the accuracy of the procedures in uncovering the true influence network in terms of auc . for every value of the rank parameter, we tune the sparsity by 5-fold cross - validation .given the fact that exact rank constraint can not be enforced directly in the ln admm algorithm , we find the best value of the nuclear norm regularization parameter by 5-fold cross - validation .then , we compute the low - rank approximations of the parameter matrix and evaluate the accuracy at each rank .    the results in figure [ fig : accuracytwitter ] show that seed significantly outperforms the baseline procedures .they also indicate that , in all of the algorithms , as we increase the rank of the solution matrix , the accuracy is improved initially and then quickly saturates .seed achieves the highest accuracy when the rank is 4 .note that this result also confirms other studies that the social network connections may be strongly influenced by a few unobserved exogenous variables .the results in table [ tab : speedtwitter ] demonstrate the significant speedup achieved by seed compared to the baselines ..run time ( in seconds ) of the algorithms on the application data set .[ cols="^,^,^",options="header " , ]   [ tab : speedtwitter ]in this paper , we propose to convert the problem of sparse reduced - rank regression into a sparse generalized eigenvalue problem , which allows us to efficiently employ the recently developed sparse eigenvalue decomposition techniques .after this transformation , the left singular vectors can be estimated in two simple steps , and the estimation of both sparse and dense right singular vectors is unified in a single framework . as a pure learning algorithm, seed deviates from traditional regularization frameworks ( i.e. , a loss function plus certain penalties ) , leading to computational efficiency and scalability .furthermore , we prove that seed achieves nice estimation and prediction accuracy that coincides with the minimax error bound in the univariate regression setting .some interesting problems for future research include extending the current formulation of the regression coefficient matrix in ( [ eq : c * ] ) to the case where the singular values can be repeated such that the left singular vectors ( which correspond to latent factors ) are not identifiable. then we will need to estimate the eigenspaces spanned by important singular vectors and characterize the estimation accuracy by some new criterion , such as the one in and .another research direction is to explore the theory of random design matrices and this can be addressed by using an extended version of perturbation theory where the perturbation in is also included in the analysis .moreover , it is computationally straightforward to extend seed to the generalized linear models by adapting the sequential quadratic programming framework . for this extension, we first approximate the loss function by the quadratic loss function and find the optimal unit rank matrix . then we can add the unit rank matrix to the solution and re - approximate the loss function with another quadratic function around this new solution . by performing these three steps sequentially , we can efficiently estimate the low - rank coefficient matrix .statistical properties of such estimator can be analyzed by extending the results in for greedy sparse procedures to reduced - rank regression .chandrasekaran , v. , parrilo , p. , willsky , a. s. , et al .latent variable graphical model selection via convex optimization . in _ communication , control , and computing ( allerton ) , 2010 48th annual allerton conference on _ , pages 16101613 .ieee .embar , v. r. , pasumarthi , r. k. , and bhattacharya , i. ( 2014 ) .a bayesian framework for estimating properties of network diffusions . in _ proceedings of the 20th acm sigkdd international conference on knowledge discovery and data mining _ , pages 12161225 .acm .myers , s. a. , zhu , c. , and leskovec , j. ( 2012 ) .information diffusion and external influence in networks . in _ proceedings of the 18th acm sigkdd international conference on knowledge discovery and data mining _ , pages 3341 .acm .starbird , k. and palen , l. ( 2012 ) .( how ) will the revolution be retweeted ? : information diffusion and the 2011 egyptian uprising . in _ proceedings of the acm 2012 conference on computersupported cooperative work _ , pages 716 .

Abstract: sparse reduced - rank regression is an important tool to uncover meaningful dependence structure between large numbers of predictors and responses in many big data applications such as genome - wide association studies and social media analysis . despite the recent theoretical and algorithmic advances , scalable estimation of sparse reduced - rank regression remains largely unexplored . in this paper , we suggest a scalable procedure called sequential estimation with eigen - decomposition ( seed ) which needs only a single top- singular value decomposition to find the optimal low - rank and sparse matrix by solving a sparse generalized eigenvalue problem . our suggested method is not only scalable but also performs simultaneous dimensionality reduction and variable selection . under some mild regularity conditions , we show that seed enjoys nice sampling properties including consistency in estimation , rank selection , prediction , and model selection . numerical studies on synthetic and real data sets show that seed outperforms the state - of - the - art approaches for large - scale matrix estimation problem .   # 1  0  0  1  0  _ running title _ : seed _ key words _ : reduced - rank regression ; scalability ; sparsity ; high dimensionality ; greedy algorithm ; sparse eigenvector estimation