quantum computation promises exponential speed - up over classical computation for certain problems , such as period finding and quantum simulation .traditional classical simulation of a composite quantum system requires updating each of the amplitudes characterizing the state of qubits , according to a hamiltonian made up of elements .the exponential growth of the state space with imposes a severe burden on resources for this type of simulation .finding classes of quantum computations that can be simulated efficiently is an active field of research  .  in quantum mechanics ,the qubits that comprise a composite system remain accessible , and it is through interactions with individual and pairs of qubits that computation is implemented .for example , a single - qubit transformation affects all computational basis states of an -qubit system .it therefore seems that one of the features of quantum systems that enables more efficient computing is the ability to harness the degrees of freedom of a vector space by interacting with only qubits .here we present a classical model of discrete quantum systems that is based on representing individual qubits and transformations applied to them .this enables us to address specific qubits in a composite system , the state of which is represented as a complex function of a single variable , and thereby directly replicate the steps in an algorithm as they would be implemented in a quantum system . while the resources required to carry out a computation exactly are comparable to other methods , this model may be compatible with new approximations that would enable simulating more qubits than currently is feasible .furthermore , the approach of building a classical model based on imitating quantum systems could offer an opportunity to gain insight into the difference in computational power of classical and quantum architectures .a qubit can be realized with any two - state quantum system that can be prepared in a general superposition of basis states of a 2d , complex vector space . for multipleuncoupled qubits , the state of the composite system is given by the tensor product of the individual states , and the number of computational basis states grows exponentially with the number of qubits .for example , the state of an -qubit system , with each qubit in an equal superposition of computational basis states and , is given by the state vector , application of a unitary operation to a qubit that is part of a composite system , which constitutes a step in an algorithm , affects all states in the superposition simultaneously , illustrating the massive parallelism inherent in quantum computation .any quantum algorithm can be approximated arbitrarily closely using just single qubit operations and a generic two - qubit interaction , such as the controlled - not ( cnot ) gate  .the effects of these operations can be visualized as rotations and inversions of the quantum - computer state vector .the state in eq .( [ e.tensor_prod ] ) , constructed as a product of individual qubit states , is a special case .almost all of the states the system can occupy in its vector space will be non - separable , implying that entanglement is required for general computation .qubits and operations on them are subject to perturbations from the environment and experimental imperfections . in general , it is believed that errors due to decoherence grow exponentially with the number of qubits in a system  .realizable quantum computation relies on the ability to diminish the effects of these errors quantum error correction and fault - tolerant quantum computation exploit entanglement and the discrete nature of quantum systems to make this possible .this brief introduction to the fundamental elements of quantum computation emphasizes the role played by the mathematical structure of the single- and multiple - qubit vector spaces .quantum systems require this mathematical description , and our classical model is developed according to this description by building into it the same state - space structure .the result is a new method of simulation and an architecture that may offer insight into the fundamental advantages of quantum systems for computing .for the representation of a qubit we use a harmonic function with frequency .orthogonal functions and serve as convenient basis states and span a 2d vector space .we assign these basis functions the role of the computational basis states of a qubit , applying a general unitary transformation to a qubit , requires isolating the coefficients and ; each coefficient can then be multiplied by the corresponding transformed basis function , yielding the transformed state : =\alpha\hspace{.02 in } u\big[\sin(\omega t)\big]+\beta\hspace{.02in}u\big[\cos(\omega t)\big].\ ] ] orthogonality of the basis functions makes it straightforward to isolate the coefficients by taking the inner product of the corresponding function with : a general transformation is illustrated in fig . [ f.unitary](a ) .the modulus squared of the coefficient , or amplitude , of a basis function gives the corresponding `` measurement probability '' .the function representing a qubit can be replaced with one or the other basis function according to these probabilities in order to represent the measurement process . the ability to determine both measurement probabilities for a given stateeliminates the need to introduce and carry along normalization coefficients the relative probabilities can be determined at the time of measurement .this model can be extended to composite systems by using a different frequency for the basis functions for each two - state system , creating a new 2d vector space for each qubit .the mapping of quantum states to functions for composite systems becomes refers to the qubit and is the total number in the system .  if single - qubit functions in equal superpositions are multiplied , the result is a linear combination of different products , this is analogous to the tensor - product state for a composite quantum system in eq .( [ e.tensor_prod ] ) , with the mapping here , is the binary value representing the state of the qubit of a quantum system , is the basis function ( sine or cosine ) representing the qubit in our model , and is the of the combinations of products of .  the functions that naturally arise when representing composite systems look like -qubit computational basis states , and we would like to determine whether they too span a state space that grows exponentially with qubit number .this will be the case if all of the functions are orthogonal .it is easy to see by expanding the products of harmonic functions in terms of sum and difference frequencies that , for qubits , the are comprised of fourier frequencies , where is 1 or .the are all multiples of a fundamental frequency given by the greatest common divisor of the qubit frequencies ,  .the orthogonality of the individual fourier components can be shown to lead to orthogonality of the for few qubits ; the three - qubit case is demonstrated in appendix [ a.dimension ] .   to demonstrate orthogonality in the general case whenthe fourier frequencies are unique , we consider the inner product between two -qubit functions and , \big[h_{j,2}(\omega_2 t)h_{k,2}(\omega_2 t)\big]\ldots \nonumber \\ \big[h_{j , n}(\omega_n t)h_{k , n}(\omega_n t)\big ] dt , \label{e.ortho}\end{aligned}\ ] ] where factors for a given qubit have been grouped together  .each product can be written as a function of frequency , as either or .the integrand in eq .( [ e.ortho ] ) can then be seen to consist of three types of terms .first , there will always be a term that is a product of a function for each qubit , .this can be written as a sum of fourier components with frequencies that are twice those of the and are , therefore , also multiples of .because integration of a harmonic function over an integral multiple of its period yields zero , this term s contribution to the inner product vanishes .second , there can be terms that only include factors for some qubits , such as , which arises when .the fourier frequencies for these terms are also multiples of fundamental frequency is the greatest common divisor of the set of all qubit frequencies , and necessarily divides any subset of them .these terms therefore also vanish when integrated over an interval of .  finally , the integrand in eq .( [ e.ortho ] ) can have a term of unity ( times ) .this only occurs when all qubit functions are the same for and ,_ i.e. _ .this argument is only valid if all of the fourier frequencies are unique .if there are at least two combinations of qubit frequencies that give the same fourier frequency , we can write , for . terms that enter this equation with the same sign for each fourier frequency cancel , and the remaining terms give , where the frequencies have been arranged so that all signs are positive . when considering all inner products , all possible combinations of harmonic factors in the integrand in eq .( [ e.ortho ] ) will arise ; for some inner product , there will be a term in the integrand like , with fourier frequencies that include .for the case(s ) in which this frequency is the argument of a cosine , the constant term results in a nonzero integral , and the different in this case are not orthogonal .  therefore , for sets of qubit frequencies that result in unique fourier frequencies , the functions that naturally arise when representing composite systems are orthogonal and span a space .unique fourier frequencies can be ensured by using a qubit - frequency definition such as .   in quantum computation ,single - qubit operations along with a generic two - qubit interaction , such as a cnot gate , are universal .the general approach to implementing operations in our model of composite quantum systems is the same as for a single qubit we need to isolate the factor multiplying each basis function ( or ) for a particular qubit . in this case, these factors will be expressions involving other qubit basis functions .once isolated , they can be multiplied by the transformed basis functions and these recombined to generate the transformed composite function .consider a general , similar to eq .( [ e.multi-c-bit ] ) but with arbitrary coefficients for the .we can write as a sum of two parts , one with terms that include and one with terms that include : the are products of harmonic functions , where is cosine or sine , and , are coefficients for the , terms . and are the functions that we need to be able to isolate to apply a linear transformation to qubit ; determining these functions can be considered to be `` addressing qubit . ''  for small , a procedure similar to the one for a single qubit can be adapted .multiplication of by the relevant basis function for the qubit leads to a different spectrum for terms containing that basis function .these different frequencies could be selected from the terms containing the orthogonal basis function , enabling the qubit to be addressed .as the qubit number grows , however , the number and density of frequencies grow dramatically , making this process unfeasible .a more general procedure can be used which determines exactly using the orthogonality of the .an inner product can be imposed between and a _ projector _ that forces all of the terms with one basis function for qubit to vanish while preserving the others .the construction of the projector for a given system is straightforward .a linear combination of all of the for a system of qubits can be generated by putting each qubit into an equal superposition of basis functions as in eq .( [ e.multi-c-bit ] ) .we define a similar function , the _ generator _ for qubit , as the product of equal superpositions of computational basis states for all qubits in the system except qubit : if this is multiplied by , the resulting function s inner product with gives the sum of the amplitudes of the terms in , but the functional form is lost .we can salvage the functional dependence by transferring it to a second variable introduced to exactly replicate the dependence on : when this generator is multiplied by , we get the projector for : taking the inner product of and , integrated over , gives us : this is an exact technique for addressing a qubit that is part of a composite system .the rest of the procedure for applying a transformation follows as in the single - qubit case and is illustrated in fig .[ f.unitary](b ) .iteration of this technique of addressing qubits allows for multiple - qubit gates .for example , for a controlled - not gate with qubit as the control and as the target , the gate would begin with determination of , which would then take on the role of the function for determining and . would be reconstructed after inverting the basis functions for , giving .finally , the transformed state would be generated as .gates involving more than two qubits can be implemented by further iteration .the measurement probability for a basis function is determined by  .due to the orthogonality of the , all cross terms in the inner product vanish and the result is the sum of the moduli squared of the amplitudes of all of the terms containing the corresponding basis function .the state of a general composite system can be represented as where represents an individual , unentangled qubit , and characterizes entangled qubits . while unentangled qubits can be stored and processed individually and with little overhead , the resources required to exactly represent entangled qubits scale exponentially with  .the qubit frequencies and the maximum fourier frequency can be kept finite , but the fundamental frequency decreases at least exponentially with number .the interval over which needs to be defined is given by the integration interval required for addressing a qubit .( [ e.fs ] ) , the integration limit of yields exact values for the ; coupled with the necessary resolution imposed by the highest fourier frequency , on order of points are required to define .we can consider the impact on addressing a qubit both for a unitary transformation and for measurement of simply truncating all of the functions that arise in a calculation . for unitary transformations ,we determine the accuracy of and , the functions determined by integrating eq .( [ e.fs ] ) to rather than to , by comparing to .we define a parameter to represent the error in : \big)^2 dx,\end{aligned}\ ] ] where is normalized over the interval 0 to , and n is the normalization constant for over the same interval .  in fig .[ f.delta](a ) we plot for the state , with and through 9 .we evaluate for the case of addressing the first qubit .the curves show that determination of and , which is exact for an integration limit of , abruptly becomes less precise as the integration interval is reduced .any calculation involving many gates will likely require a value for on the steep part of the curve , imposing an integration interval that scales exponentially with .  to assess the effect of truncation on measurement probabilities , we truncate the integrals used to determine and as above , and then integrate the square of each .we look at the ratio of the truncated probability to measure sine versus cosine as a function of integration time : this ratio is plotted in fig .[ f.delta](b ) for the same state used above , for which the actual ratio is one .the graph shows that reaching the actual ratio of to also scales exponentially with . however , precise values for measurement probabilities are often not needed and useful qualitative information can be obtained for integration intervals that are shorter than .for instance , for this example the curves indicate that for integration limits beyond , the ratio is about 0.25 , of the same order as the actual value .the most celebrated quantum algorithm is shor s method of finding the prime factors of a number .the problem of factorization can be related to the problem of determining the period of the function , for an integer that is co - prime with ; if is even , either or will have a common factor with  .shor s algorithm relies on application of the quantum fourier transform ( qft ) to to efficiently determine the period .the qubits involved in implementing this algorithm are divided into two registers , the states of which are treated as integers according to the states of the associated qubits . the first step in the procedure is to prepare the first register in a superposition of all computational basis states , by applying a hadamard transform to each of the qubits in the register  .a gate applied to both registers produces the entangled state if the second register is measured , it collapses to for some .the first register ends up in a superposition of all states for which , leaving the system in the state application of the qft to the first register imposes interference that results in a superposition of states that are close to integral multiples of the inverse period , for integer .measurement yields an integer close to one of the , and after several iterations the period can be determined .the minimum number of qubits required for each register is and .this ensures that the second register is large enough to represent , which satisfies , and that the first register is large enough to give a unique value for from the qft ( see reference  ) .we apply our model to the factorization of using , the first integer co - prime with 21 .this requires 14 qubits , nine for the first register and five for the second ; the alogorithm for our example is illustrated in fig .[ f.shor ] .the qubits are labeled 1 through 14 , with the convention that qubit 1 ( 14 ) is the most ( least ) significant qubit for the first ( second ) register .qubit is represented using the frequency , and the function representing the state of the system is defined with a resolution of over an interval of  . in fig .[ f.functions ] we show the function representing the state of the system at various stages in the calculation , as denoted by the dashed lines in fig .[ f.shor](a ) .we implement by generating the output state , shown in fig .[ f.functions](b ) , by summing all functions representing over to  . from here , we measure qubits 10 through 14 by comparing to and applying the rule that the outcome of a measurement is the state with the higher measurement probability , or is chosen at random if the probabilities are equal .this leaves the second register in the state , and the first register in a linear combination of all between 0 and for which ; the function representing the first register at this stage is shown in fig [ f.functions](c ) .qubits 10 through 14 are removed from the calculation and the qft is applied to the remaining nine qubits .the results of applying the qft and subsequent measurement of qubits 19 are shown in fig .[ f.qft](a ) , which displays the measurement probability for the state .the peaks at multiples of indicate a period of for the function , which in turn yields either of the prime factors of 21 from and .figures [ f.qft](b)-(d ) zoom in on the probability distributions near different multiples of .when is not an integer , the probability is distributed among integers closest to the fractional value , as in ( b ) and ( c ) .there is a dramatic simplification to the factoring algorithm when using a classical model for simulation .the need for the qft stems from the fact that , in a quantum system , measurement of the state in eq .( [ e.period ] ) results in one of the states in the superposition , with no opportunity to learn the others .repetition of the algorithm to that point likely results in a different , preventing the period from being learned in successive iterations . in a classical system , the state in eq .( [ e.period ] ) can be measured as many times as necessary to determine the period . if we apply this simplified scheme to the state in fig .[ f.functions](c ) , we find equal probability for any satisfying ; the first eight peaks are shown in fig . [ f.qft](e ) .in addition to avoiding all of the gates required for the qft , no imaginary numbers are required , and most importantly , we save on qubits . the first register ,whose size for shor s algorithm is dictated by the qft stage , in this case only needs enough qubits to represent , and .this provides a savings on order of qubits compared to the quantum case . applying this simplified factoring algorithm to require only 10 qubits , five for each register .we have presented a classical , qubit - based model of discrete quantum systems that offers a new framework for simulating quantum computations by providing access to individual qubits and their interactions .the dimension of the state space for a composite system grows exponentially with qubit number , and individual qubits continue to be accessible .application to shor s algorithm highlights the features of the model in action .the resources required for implementation scale exponentially with the number of entangled qubits , yet it is possible to save on qubits in a classical model .the can be fourier decomposed by expanding the products of harmonics in terms of sum and difference frequencies . herewe explicitly show the decomposition for the case of 3 qubits .the 8 functions can be expanded as + \cos\big[(\omega_1+\omega_2-\omega_3)t\big ] \nonumber \\ + \cos\big[(\omega_1-\omega_2+\omega_3)t\big]+\cos\big[(\omega_1-\omega_2-\omega_3)t\big]\big ) \nonumber \\\cos(\omega_1 t)\cos(\omega_2 t)\sin(\omega_3 t)= \frac{1}{4}\big(\sin\big[(\omega_1+\omega_2+\omega_3)t\big ] -\sin\big[(\omega_1+\omega_2-\omega_3)t\big ]  \nonumber\\ + \sin\big[(\omega_1-\omega_2+\omega_3)t\big]- \sin\big[(\omega_1-\omega_2-\omega_3)t\big]\big ) \nonumber \\\cos(\omega_1 t)\sin(\omega_2t)\cos(\omega_3 t)= \frac{1}{4}\big(\sin\big[(\omega_1+\omega_2+\omega_3)t\big ] + \sin\big[(\omega_1+\omega_2-\omega_3)t\big ]  \nonumber \\-\sin\big[(\omega_1-\omega_2+\omega_3)t\big]- \sin\big[(\omega_1-\omega_2-\omega_3)t\big]\big ) \nonumber \\\cos(\omega_1 t)\sin(\omega_2 t)\sin(\omega_3 t)= \frac{1}{4}\big(-\cos\big[(\omega_1+\omega_2+\omega_3)t\big]+ \cos\big[(\omega_1+\omega_2-\omega_3)t\big ]  \nonumber \\ + \cos\big[(\omega_1-\omega_2+\omega_3)t\big]- \cos\big[(\omega_1-\omega_2-\omega_3)t\big]\big ) \nonumber \\\sin(\omega_1 t)\cos(\omega_2 t)\cos(\omega_3 t)= \frac{1}{4}\big(\sin\big[(\omega_1+\omega_2+\omega_3)t\big ] + \sin\big[(\omega_1+\omega_2-\omega_3)t\big ]  \nonumber \\ + \sin\big[(\omega_1-\omega_2+\omega_3)t\big]+\sin\big[(\omega_1-\omega_2-\omega_3)t\big]\big ) \nonumber \\\sin(\omega_1 t)\cos(\omega_2 t)\sin(\omega_3 t)= \frac{1}{4}\big(-\cos\big[(\omega_1+\omega_2+\omega_3)t\big ] + \cos\big[(\omega_1+\omega_2-\omega_3)t\big ]  \nonumber \\-\cos\big[(\omega_1-\omega_2+\omega_3)t\big]+ \cos\big[(\omega_1-\omega_2-\omega_3)t\big]\big ) \nonumber \\\sin(\omega_1 t)\sin(\omega_2 t)\cos(\omega_3 t)= \frac{1}{4}\big(-\cos\big[(\omega_1+\omega_2+\omega_3)t\big ] -\cos\big[(\omega_1+\omega_2-\omega_3)t\big ]  \nonumber \\ + \cos\big[(\omega_1-\omega_2+\omega_3)t\big]+\cos\big[(\omega_1-\omega_2-\omega_3)t\big]\big ) \nonumber \\\sin(\omega_1 t)\sin(\omega_2 t)\sin(\omega_3 t)= \frac{1}{4}\big(-\sin\big[(\omega_1+\omega_2+\omega_3)t\big ] + \sin\big[(\omega_1+\omega_2-\omega_3)t\big ]  \nonumber \\ + \sin\big[(\omega_1-\omega_2+\omega_3)t\big]- \sin\big[(\omega_1-\omega_2-\omega_3)t\big]\big).\end{aligned}\ ] ]  we introduce a more compact notation to represent the 8 fourier components : for all of these terms , , where corresponds to the fourier component represented by the row vector with a one in the place and zeros everywhere else .   from thisit can be seen that all of the are orthogonal .for example , in this case and for small numbers of qubits , all of the different inner products can be verified to be zero , and .we can also see that not all of the functions are orthogonal if there is redundancy in the fourier frequencies . if , for instance , , then the second and third vectors in eq .( [ e.fourier_vector ] ) are identical ( as are the sixth and seventh ) . in this case , the can only be represented by six orthogonal vectors , not eight .then ,          clearly , if there are several subsets of entangled qubits with no entanglement between subsets , then each collection would be described by its own , and resources would scale exponentially only with the number within a subset .the general method of imposing via controlled modular exponentiation requires a minimum of additional qubits  , which is too computationally burdensome at this stage of the calculation .

Abstract: we present an approach to simulating quantum computation based on a classical model that directly imitates discrete quantum systems . qubits are represented as harmonic functions in a 2d vector space . multiplication of qubit representations of different frequencies results in exponential growth of the state space similar to the tensor - product composition of qubit spaces in quantum mechanics . individual qubits remain accessible in a composite system , which is represented as a complex function of a single variable , though entanglement imposes a demand on resources that scales exponentially with the number of entangled qubits . we carry out a simulation of shor s algorithm and discuss a simpler implementation in this classical model .


the dynamics of complex stochastic systems are often governed by rare events - examples include protein folding , macromolecular association , or phase transitions . these rare events lead to sampling problems when trying to compute expectation values from computer simulations , such as molecular dynamics ( md ) or markov chain monte carlo ( mcmc ) .one approach to alleviate such sampling problems is to increase the rate at which the rare events occur by generating and combining simulations at different thermodynamic states .for example , proteins can easily be unfolded at high temperatures , and protein - ligand complexes can dissociate at artificial hamiltonians , where protein and ligand have reduced interaction energies .generalized ensemble methods , such as replica exchange molecular dynamics , parallel tempering and simulated tempering exploit this observation by coupling simulations at different temperatures or hamiltonians within an mcmc framework .yet another example is umbrella sampling which uses a set of biased hamiltonians to ensure approximately uniform sampling along a set of pre - defined slow coordinates .all of the aforementioned enhanced sampling methods are constructed such that for long simulation times , the equilibrium distribution of each thermodynamic state will be sampled from . with that in mind , reweighting estimators make use of all simulation data by reweighting each probability density from the thermodynamic state sampled at to the condition of interest _ via _ the boltzmann density . the most frequently used reweighting estimators are the weighted histogram analysis method ( wham ) , bin - less wham and the multi - state bennett acceptance ratio ( mbar ) method .the most common use of reweighting is to obtain equilibrium expectations or free energy differences .reweighting can also be applied to obtain dynamical information from the available contiguous trajectory pieces .when the probability density of trajectories can be evaluated , mbar can be applied to trajectories instead of sample configurations , obtaining estimates of dynamical expectations .both wham and mbar are statistically optimal under specific assumptions as they can be derived from maximum - likelihood or minimum variance principles .however , a key assumption of both estimators is that data is given as statistically independent samples of the respective equilibrium distributions . in reality ,md and mcmc simulations provide time - correlated data trajectories .consequently , estimators exploiting the time - correlation in the data can achieve significantly better results .in particular , wham and mbar can not obtain unbiased estimates from datasets for which the initial conditions of harvested trajectories do not come from a probability density that is known a priori .examples of such datasets are swarms of short uncoupled trajectories or metadynamics and conformational flooding during the fill - up phase .a complementary approach to address sampling problems is markov state modeling . a markov state model ( msm ) transition matrix contains conditional transition probabilities between discrete configuration states at some lagtime . given the transition matrix ,an equilibrium distribution can be computed that is unbiased if each transition event used for estimating the msm originates from a local equilibrium distribution restricted to respective discrete state . in this case , msms are able to reweigh trajectories that are not sampled from global equilibrium without any a priori knowledge of the transition probabilities . in order to use msms for computing kinetics, must be long enough for transition events to approximately decorrelate for a given configuration space discretization . since msm estimators are purely based on observed transition statistics they can not combine the information from different thermodynamic states .therefore , the orders - of - magnitude speedup that can sometimes be achieved with enhanced sampling methods has not been accessible to msms as yet .the transition - based reweighting analysis method ( tram ) aims at combining the advantages of reweighting estimators and msms . in ref . , we have defined tram as a class of estimators that ( 1 ) take the statistical weights of samples at different thermodynamic states into account , in order to reweigh these samples ; and ( 2 ) exploits transitions observed in the sampled trajectories , without assuming that these trajectories are sampled from equilibrium . ref . a statistically optimal estimator for non - equilibrium trajectories given that the statistical weight of each trajectory can be evaluated .conceptually , an optimal tram estimator could be formulated from this principle . in practice , however , data is typically not stored at every integration time step , such that trajectory probability densities are not available .  in ref . , we have introduced the first tram estimator that is applicable to practical molecular dynamics data , and could show that it can provide superior estimates of equilibrium probabilities and free energy compared with wham .however , the estimator in ref . is only approximately optimal and is very tedious to compute .another tram estimator is presented in ref . and called the dynamic histogram analysis method ( dham ) .dham was shown to avoid systematic errors that may occur when analyzing umbrella sampling with wham .dham uses a dynamical model ( diffusion along a reaction coordinate ) to relate transition matrices of simulations at different bias potentials .this assumption is helpful when a diffusion model is appropriate and there is one or a few slow reaction coordinates only ( e.g. as in the case of umbrella sampling ) . in this way it regularizes the solution andtherefore makes the estimates statistically more stable . in ref . we introduced xtram , which does not assume a specific dynamical model but only exploits the fact that samples can be reweighted in order to couple different thermodynamic states .moreover , xtram is as yet the only tram estimator that avoids the discretization of reweighting factors required in the other tram methods and as well as in wham .xtram may provide substantial improvements compared to mbar for time - correlated data , is shown to be asymptotically exact , and is shown to converge to mbar when the data is not time correlated .however , xtram was not derived from a maximum - likelihood or minimum - variance principle and is therefore probably not statistically optimal for finite data sets .  in the present paper , we provide for the first time a statistically optimal tram method by presenting a maximum - likelihood solution to the discrete tram problem formulated in ref .we derive a set of self - consistent equations whose solution yields the maximum likelihood dtram estimator .it is shown that the dtram solution is an asymptotically correct estimator , i.e. it converges to the exact equilibrium probabilities and transition probabilities with an increasing amount of simulation data .the dtram equations can be solved using a newton method , as done before for wham and mbar , or an easy - to - implement iterative algorithm provided here .we show that dtram becomes identical to wham in the limit of statistically independent samples , i.e. sampled from the global equilibrium at each thermodynamic state .moreover , dtram becomes identical to a reversible msm when we have only a single thermodynamic state .a number of applications are shown to demonstrate the usefulness and versatility of dtram .we assume that a set of md or mcmc simulations have been performed , each in one of thermodynamic states ( indexed by the superscript ) . for simulations in which the thermodynamic state is frequently changed , such as in replica - exchange simulations , each contiguous sequence is treated as a separate trajectory at one of the thermodynamic states .furthermore , we assume that the data has been discretized to a configuration space partition ( indexed by subscripts ) .we are primarily interested in the free energy , or equivalently , the equilibrium probability of discrete states in some unbiased or reference ensemble .in addition we might be interested in the equilibrium probability of states under all biased ensembles .if the simulation trajectories are long enough , we will also be able to compute kinetic properties , as discussed later .we will be dealing with simulations where the unbiased , or reference probability and the biased probability at simulation condition , are related by a known and constant bias factor :    where is a normalization constant .thus , the bias is multiplicative in probabilities or additive in the potential .this formalism is applicable whenever one has simulations conducted at different thermodynamic states , such as replica - exchange methods or umbrella sampling , but also direct simulations at different temperatures or hamiltonians . in the results section, we will show how the reweighting factor can be computed for a few selected examples .the most common analysis method used in the present scenario is wham .wham uses the histogram counts , i.e. the number of samples falling into bin at thermodynamic state .although wham was originally derived as a minimum - variance estimator , it can be derived as a maximum - likelihood estimator using the likelihood which simply assumes that every count is independently drawn from the biased distribution , which is linked to the unbiased distribution _ via _ eq .( [ eq : reweighting_1 ] ) .let us now turn to reversible markov state models .the maximum likelihood markov model is the transition matrix between discrete configuration states , that maximizes the likelihood of the observed transitions between these states .the likelihood of a markov model is well known , and simply a product of all transition probabilities corresponding to the observed trajectory . to obtain a reversible markov state model ,this likelihood is maximized using the constraints that the transition probabilities must fulfill detailed balance with respect to the equilibrium distribution : where is the number of times the trajectories were observed in state at time and in state at a later time , where is the lag time at which the markov model is estimated . for an msm ,all simulation data producing counts , has to be generated at the same thermodynamic state ( e.g. temperature , hamiltonian ) , and the estimated is then only valid for this thermodynamic state .the reversibility of the msm is ensured by the constraint equations ( [ eq : msm - dbconstraint ] ) .estimators that maximize eqs .( [ eq : msm - likelihood]-[eq : msm - dbconstraint ] ) usually provide both and the equilibrium distribution  .  in tram , we combine these two approaches as follows : we avoid the wham assumption that every count is sampled from global equilibrium , and instead treat every trajectory at thermodynamic condition as a markov chain with the configuration - state transition counts . however , in contrast to markov models we exploit the fact that equilibrium probabilities can be reweighted between different thermodynamic states via ( [ eq : reweighting_1]-[eq : reweighting_2 ] ) .the resulting likelihood of all and , based on simulations at all thermodynamic states can be formulated as : here , is the markov transition matrix at thermodynamic state , and are the number of transitions observed at that simulation condition . is the vector of equilibrium probabilities of discrete states at each thermodynamic state .note that all of these equilibrium distributions are coupled through eqs .( [ eq : reweighting_1]-[eq : reweighting_2 ] ) . because each markov model must have the distribution as a stationary distribution , all markov models are coupled too .this is what makes the maximization of the tram likelihood eqs .( [ eq : tram - likelihood]-[eq : tram - dbconstraint ] ) difficult , and it can neither be achieved by wham , nor by existing msm estimators .we call eqs .( [ eq : reweighting_1],[eq : reweighting_2],[eq : tram - likelihood],[eq : tram - dbconstraint ] ) the tram equations . in the present paperwe will obtain the reweighting factors in eqs .( [ eq : reweighting_1]-[eq : reweighting_2 ] ) by a configuration space discretization or binning , such as in wham . for this reason we call the present solution a discrete tram method , which should be distinguished from approaches where the reweighting is done for individual samples .we will seek the maximum likelihood of eq .( [ eq : tram - likelihood ] ) . as in common practice ,we work with the logarithm of the likelihood , because it has the same maximum point as the likelihood but can be treated more easily : moreover , we have the following constraints .using detailed balance eq .( [ eq : tram - dbconstraint ] ) with the reweighting equations ( [ eq : reweighting_1]-[eq : reweighting_2 ] ) results in note that the normalization factors , , have cancelled .in addition , should be a transition matrix and should be a probability vector , so we have the normalization conditions :    the normalization of is achieved by the normalization constants in eq .( [ eq : reweighting_1])-([eq : reweighting_2 ] ) .  in order to solve the discrete tram problem we have to maximize the log likelihood ( [ eq : tram - log - likelihood ] ) under the constraints ( [ eq : tram - constraint1]-[eq : tram - constraint3 ] ) .the variables are both the unbiased equilibrium probabilities ( providing variables due to the constraint ( [ eq : tram_iteration_pi ] ) ) , and the biased transition matrices ( each having remaining free variables that are not fixed by constraints ( [ eq : tram - constraint1])-([eq : tram - constraint2 ] ) ) .note that changing the simulation conditions , such as bias or temperature , will modify the transition probabilities in a non - trivial way that depends on the simulation condition , the integrator and thermostat used , and the state space discretization .therefore we can not relate the different without restricting the generality of our estimator .the only general connection between these markov models is the coupling of their equilibrium distributions via eqs .( [ eq : reweighting_1]-[eq : reweighting_2 ] ) .  in appendix a.1 - 3, we use lagrange duality theory to show that the optimal solution of the discrete tram problem above fulfills the following two conditions :   where are unknown lagrange multipliers . in the setting with detailed balancewe can unfortunately not give a closed expression for them , but we can optimize them along with the equilibrium distribution . note that the equations above do not require the transition probabilities to be computed explicitly . if these are desired , they can be subsequently computed from the solution of eqs .( [ eq : dtram_equation1]-[eq : dtram_equation2 ] ) ( see sec . [ sub : kinetics ] below ) .  in appendixa.4 we prove that the dtram equations above are asymptotically correct .this means that in the limit of a lot of simulation data - either realized by long trajectories or many short trajectories - the estimate will converge to the correct stationary distributions and .we can rewrite the self - consistent equations ( [ eq : dtram_equation1],[eq : dtram_equation2 ] ) to derive the following iteration ( fixed - point method ) , that can be used to numerically solve the discrete tram problem .first we initialize and by the simple guess : and then we iterate the following equations until is converged : instead of the simple initialization for in eq .( [ eq : initial_guess_pi ] ) , we could use the standard wham algorithm to obtain a much better guess .while a better starting point might be relevant for optimizing computational performance , we have not observed the estimation result to depend on this choice .  as an alternative to the fixed - point iteration ( [ eq : tram_iteration_v],[eq : tram_iteration_pi ] ), we can solve equations ( [ eq : dtram_equation1],[eq : dtram_equation2 ] ) by using the multidimensional newton method for root finding available in many numerics packages .given and at their optimal values , the transition probabilities can be computed for any thermodynamic state simulated at by : see appendix a.2 for the derivation . in eq .( [ eq : pij ] ) we have explicitly stated that transition counts , and hence the transition probabilities are estimated at a given lag time .as a consequence of the asymptotic correctness of dtram ( appendix a.4 ) , the estimates of are also asymptotically correct , that is for either long trajectories or many short trajectories we will get an unbiased estimate of the transition probabilities .  in order to compute kinetics , such as transition rates or timescales , the transition matrices do not only have to be valid for the lag time estimated at , but they have to be markov models that predict the kinetics at longer times correctly .how adequate is as a markov model should be tested by validating that the relaxation timescales computed from the eigenvalues of are approximately constant in  and by checking that the chapman - kolmogorow ^{n}12 & 12#1212_12%12[1][0] ( , ) pp . * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) link:\doibase 10.1002/jcc.21186 [ * * ,  ( ) ] * * ,  ( ) * * ,  ( ) ( ) * * ,  ( ) * * ,  ( ) link:\doibase 10.1073/pnas.1103547108 [ * * , ( ) ] * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) link:\doibase 10.1063/1.3216567 [ * * ,  ( ) ] _ _ , , vol . ( , , ) * * ,  ( ) _ _ ( , ) _ _ ( , ) _ _ ( , )a detailed description of lagrange duality theory we refer to textbooks .here we just summarize a few aspects of the theory relevant for solving the tram problem .  consider a constrained minimization problem of the following form : the lagrangian functioncan be defined by if ( [ eq : convex - problem ] ) is a convex problem and some technical assumption such as slater condition holds , it can be proven that ( [ eq : convex - problem ] ) is _ equivalent _ to the following maximization problem : where is the lagrange dual function defined by the equivalence here means the optimal values of the two problems are equal , i.e. , the solution to ( [ eq : convex - problem ] ) and the solution to ( [ eq : dual - problem ] ) satisfies .moreover ,    we first consider the dual lagrange approach to solving dtram at a single thermodynamic state , i.e. the situation that is equivalent with a reversible markov state model . herewe first ignore the normalization of the stationary distribution , i.e. , may not be equal to .the maximum likelihood estimation of the transition matrix with a fixed equilibrium distribution is given by the following optimization problem : where is the log - likelihood function of . by using the lagrange duality theory, we have the following lemma :  * lemma * : the minimization problem ( [ eq : single-1 ] ) is equivalent to the maximization problem and the optimal solution to ( [ eq : single-1 ] ) and the optimal solution to ( [ eq : single-2 ] ) satisfy * proof * : the lagrangian function of ( [ eq : single-1 ] ) is defined as * * note that then the dual function of ( [ eq : single-1 ] ) is ,\boldsymbol{\lambda},\mathbf{v}\right)\nonumber \\  & = & -\sum_{i , j}c_{ij}\ln\left(\frac{c_{ij}}{\pi_{i}\left(\lambda_{ij}-\lambda_{ji}\right)+v_{i}}\right)\nonumber \\  &  & + \sum_{i , j}c_{ij}-\sum_{i}v_{i}\nonumber \\  & = & \sum_{i , j}c_{ij}\ln\left(\pi_{i}\left(\lambda_{ij}-\lambda_{ji}\right)+v_{i}\right)-\sum_{i}v_{i}\nonumber \\  &  & -\sum_{i , j}c_{ij}\ln c_{ij}+\sum_{i , j}c_{ij}\label{eq : dual - function}\end{aligned}\ ] ] and ( [ eq : single-1 ] ) is equivalent to the maximization problem from ( [ eq : dual - function ] ) , we can get and therefore , and the optimal value of ( [ eq : single-1 ] ) is equal to and where we now consider the case that the stationary distribution is unknown . in this case , the maximum likelihood estimate of is    if we further consider multiple biased simulations with , we can also conclude that the maximum likelihood estimate of can be given by the following min - max problem : for the sake of convenience , here we define .then ( [ eq : multiple - minimax - problem ] ) can be rewritten as because is a concave function of if is fixed and a convex function of if is fixed , the solution to ( [ eq : multiple - minimax - problem - u ] ) is a saddle point and characterized by ( see section 10.3.4 in )   considering that and we can conclude that the optimal solution to ( [ eq : multiple - minimax - problem ] ) should satisfy this leads to the following iterative algorithm for unbiased estimation of multiple simulations :    here we show that dtram converges to the correct equilibrium distribution and transition probabilities in the limit of large statistics . in this limit , either achieved through long simulation trajectories or many short simulation trajectories , the count matrices  ] is the matrix of exact transition probabilities ( no statistical error ) , which satisfies where $ ] are the exact stationary probabilities of configuration states . inserting ( [ eq : asymptotic - limit - counts ] ) into the dtram equations ( [ eq : dtram_equation1]-[eq : dtram_equation2 ] ) , we obtain : and thus the first dtram equation is satisfied .furthermore , we obtain : and thus the second dtram equation is satisfied as well . from the above two equations , we can conclude that in the statistical limit ( either achieved by long trajectories or many short trajectories ) , the solution of the dtram equations converges to the correct equilibrium distribution and the correct transition probabilities .note that we have assumed that all trajectory data is in local equilibrium within each starting state - if this is not the case the counts and thus also the estimated and will be biased .thus , if the data is of such a nature that local equilibrium is a concern ( e.g. metadynamics or uncoupled short md trajectories ) , all estimates should be computed as a function of the lag time .the simulation trajectory is generated by a metropolis simulation model , which is a reversible markov chain with initial state , stationary distribution , and transition probability where denotes the proposal distribution which satisfies and .

Abstract: we propose a discrete transition - based reweighting analysis method ( dtram ) for analyzing configuration - space - discretized simulation trajectories produced at different thermodynamic states ( temperatures , hamiltonians , etc . ) dtram provides maximum - likelihood estimates of stationary quantities ( probabilities , free energies , expectation values ) at any thermodynamic state . in contrast to the weighted histogram analysis method ( wham ) , dtram does not require data to be sampled from global equilibrium , and can thus produce superior estimates for enhanced sampling data such as parallel / simulated tempering , replica exchange , umbrella sampling , or metadynamics . in addition , dtram provides optimal estimates of markov state models ( msms ) from the discretized state - space trajectories at all thermodynamic states . under suitable conditions , these msms can be used to calculate kinetic quantities ( e.g. rates , timescales ) . in the limit of a single thermodynamic state , dtram estimates a maximum likelihood reversible msm , while in the limit of uncorrelated sampling data , dtram is identical to wham . dtram is thus a generalization to both estimators .


the interaction of soft substrates with fluid interfaces is common to many biological , medical , and industrial processes . for this reason , the field of elastocapillarity ,in which one studies how surface tension forces couple to the deformations of elastic substrates , has been the subject of much recent attention .even the most basic characterization of the wetting forces at the three - phase contact - line remains in dispute , especially for soft viscoelastic materials that have both liquid and solid properties .for such materials , the young - dupr law for a liquid wetting a hard solid , neumann s law for a liquid wetting another liquid , or some hybrid thereof may apply .theoretical efforts have been made to bridge the gap between the hard solid and liquid regime , such as introducing the concept of solid surface tension .alternative methods employ computational approaches such as density functional theory ( dft ) and molecular dynamic ( md ) simulations to gain a more thorough understanding of the wetting forces acting at the contact - line .  in general , the elastic resistance must be comparable in magnitude to the surface tension forces applied to the elastic medium for many elastocapillary phenomenon .the elastocapillary number is a typical measure of the relative importance of capillarity to elasticity . here is the surface tension , the elastic modulus and a characteristic length scale . in experiments ,it is typically easiest to adjust , as seen in the wrinkling of elastic sheets , capillary origami and buckling of elastic fibers .however , it is also possible to use a gel as the solid phase , which permits to be tuned over several orders of magnitude .  [ cols="^,^ " , ]   we end this section with a comment on the failure criteria .the results presented in this section have assumed that the substrate is always weak enough to generate deformations that exceed the failure threshold for agar gel . as such, we have focused our study on identifying a critical wavenumber and not the failure threshold .additional experiments would be needed to extract such information .the model developed above captures the central feature required to generate a starburst with a particular number of arms : a critical wavenumber for which the tangential displacement is maximal . in experiments, we observe that for a given set of experimental conditions , a narrow range of wavenumbers are selected ; this is not surprising given that values of can have a maximum displacement similar to .however , the failure criteria does not provide a failure mechanism . in physical gels such as agar or gelatin ,the stiffness of the material is provided by a network of entangled polymers . under strain, these polymers are stretched further away from equilibrium , although not necessarily to the point of failure .because polymers are subject to thermal fluctuations , even a sub - critical strain can be triggered by a sufficiently large thermal fluctuation .this is the mechanism behind delayed fracture , whereby there is an exponential distribution of waiting times ( indicating a poisson process ) before which a strained material fractures . this has been directly observed in both stiffer gelatin rods (  kpa ) and in vibration - controlled experiments in this system .( color online ) definition sketch of a fluid - filled crack of length driven by a pressure head ., scaledwidth=37.0% ]  once a fracture is initiated , fluid begins to flow into the crack tip and the crack propagates outward ( away from the central droplet ) until the reservoir of fluid is exhausted .we develop a two - dimensional model for the asymptotic propagation of a mode i fracture driven by a pressure head . a schematicis shown in figure [ fig : fracture ] .to begin , we consider an incipient crack of length , filled with an inviscid fluid of density , propagating into an elastic solid.in this section , we model the solid as either i ) a linear elastic material or ii ) an incompressible neo - hookean material , as is typically used for polymeric gels , and contrast the computed propagation rates .we assume the capillary pressure head at the fluid reservoir is greater than the elastic stress in the solid in front of that reservoir , so as to drive fluid into the crack tip .the flow of this inviscid fluid is governed by bernoulli s law , and , hence , driven by the gradient in fluid pressure , which is related to the elastic stress induced by deformations of the crack surface .here we assume there is no ambient stress field in the elastic solid , or the substrate is not pre - stressed. the localized stress ( pressure ) field near the crack tip of a linear elastic solid has the well - known asymptotic form ,   for an incompressible neo - hookean solid , the asymptotic stress field is given by , here and are stress intensity factors for a linear elastic and incompressible neo - hookean solid , respectively .in general , is a function of the crack geometry and far - field loading conditions , but we may simplify our model by utilizing several experimental observations .recent experiments have demonstrated self - healing behavior of fluid - filled cracks in soft substrates .this observation is characteristic of stable , or transport - limited , crack growth .that is , the speed of crack growth is limited by the extent to which the fluid can be supplied ( transported ) to the crack tip . for reference , unstable crack growthwould be controlled by the speed of an elastic wave in the solid , a characteristic which is not observed experimentally .hereafter , we assume the crack propagates in the marginal state , where is the fracture toughness , consistent with stable crack growth .we apply bernoulli s law ( [ bernoulli ] ) using the crack tip ( state 1 ) and a point upstream ( state 2 ) to derive an equation relating the crack - tip velocity to the pressure gradient in the linear elastic solid ( [ fstress ] ) :    whose solution gives this predicts a characteristic exponent of for the growth of the arm .a similar relationship is derived for an incompressible neo - hookean solid by applying the respective stress field ( [ nhstress ] ) instead : for this system , the characteristic exponent is instead :   note that both constitutive models predict power law behavior , but the neo - hookean model is able to reproduce the universal exponent observed experimentally .we attribute this result to the observation that the gel substrate is incompressible and that the large scale deformation of the soft solid are more aptly described by the nonlinear neo - hookean constitutive law .( color online ) histogram for the crack length exponent with mean ., scaledwidth=40.0% ]  because the gels are spatially - heterogeneous , cracks nucleate with different widths and propagate at varying rates .we fit the length of each fracture arm to the form and examine the distribution of values observed .no systematic variation with material parameters was found for the values of , but the width of the crack determines the pre - factor  .for this same data set of individual arms , we find that ( standard error ) .the full probability distribution is show in figure [ fig : alpha ] , which allows for comparison to the ( neo - hookean ) and ( linear elastic ) constitutive models .we have analyzed the deformations of a soft elastic substrate induced by the liquid / solid interactions with a liquid droplet , as well as the propagation of a fluid - filled crack in a soft elastic medium .the deformations associated with a corrugated contact - line yield a critical disturbance with wavenumber that generates the largest elastic response within the substrate .we quantify the response with the tangential displacement , a failure criteria which we correlate with the initiation of a starburst fracture ( see figure [ fig : intro ] ) .computations reveal that the location of the unbalanced contact - line force is the most important parameter in wavenumber selection .our theoretical predictions compare favorably to i ) previously - reported experiments with fixed volume droplets and ii ) experiments where the contact - angle is fixed and the droplet volume is varied , thereby confirming our interpretation of the initiation process .our model also predicts that the substrate thickness is also an important parameter in wavenumber selection , an observation which we see in experiments . for the crack propagation problem, we develop a model by considering the transport of an inviscid fluid into the crack tip of either a linear elastic or incompressible neo - hookean solid .while both elastic models yield power law growth , the neo - hookean model predicts the crack length grows with universal exponent and does not scale with any material parameters , consistent with experimental observations .studies on fracture necessarily take place on materials which are strong enough to resist fracture until the point when the measurement is made . however , many biological materials are soft enough that conventional fracture measurements are not possible because the material is too weak to support its own weight .the present study suggests that by depositing a droplet of known size and wettability , it could be possible to use starburst fractures to measure the fracture toughness of very soft materials so long as they remain fully supported by the dish . furthermore , because the growth rate of fractures is sensitive to whether or not the material is linear - elastic , this could also provide a new method for materials characterization .the authors are grateful for support from the national science foundation under grant number dms-0968258 , as well as nc state s undergraduate research office , to michael shearer for valuable discussions , and to mark schillaci for preliminary experiments useful in formulating the model .tadashi kajiya , adrian daerr , tetsuharu narita , laurent royon , francois lequeux , and laurent limat .advancing liquid contact line on visco - elastic gel substrates : stick - slip vs. continuous motions . , 9:454461 , 2013 .

Abstract: a liquid droplet resting on a soft gel substrate can deform that substrate to the point of material failure , whereby fractures develop on the gel surface that propagate outwards from the contact - line in a starburst pattern . in this paper , we characterize i ) the initiation process in which the number of arms in the starburst is controlled by the ratio of surface tension contrast to the gel s elastic modulus and ii ) the propagation dynamics showing that once fractures are initiated they propagate with a universal power law . we develop a model for crack initiation by treating the gel as a linear elastic solid and computing the deformations within the substrate from the liquid / solid wetting forces . the elastic solution shows that both the location and magnitude of the wetting forces are critical in providing a quantitative prediction for the number of fractures and , hence , an interpretation of the initiation of _ capillary fractures_. this solution also reveals that the depth of the gel is an important factor in the fracture process , as it can help mitigate large surface tractions ; this finding is confirmed with experiments . we then develop a model for crack propagation by considering the transport of an inviscid fluid into the fracture tip of an incompressible material , and find that a simple energy - conservation argument can explain the observed material - independent power law . we compare predictions for both linear elastic and neo - hookean solids finding that the latter better explains the observed exponent .


the discovery of multi - body extrasolar planetary systems during the past few years has once again confronted astrodynamicists with the old question of the stability of such systems .the discovery of gj 876 where two planets are locked in a 2:1 mean - motion resonance ( marcy et al .2001 ) , the confirmation of three planets in orbit about upsilon andromedae ( butler et al.1999 ) , and the discovery of planetary systems around 47 uma and 55 cancri with planets in orbits more closely resembling those in the solar system ( fischer et al . 2002 ; marcy et al .2002 ) , have set the grounds for a deeper look at the problem of the stability of multi - body systems .here we present a preliminary investigation of the dynamical stability of the 47 uma and 55 cancri systems .a more detailed analysis , including similar work on gj 876 and and , will be addressed in future work .we performed dynamical fits to the radial velocity data for the stars 47 uma and 55 cancri using a levenberg - marquardt minimization algorithm ( press et al .1992 ) , as in laughlin & chambers ( 2001 ) and rivera & lissauer ( 2001 ) .we assumed that the planets in each system were coplanar and that the plane of the planets contained the line of sight .we then used the resulting fitted parameters as initial conditions for n - body simulations . in order to give the resulting systems the ability to explore extra degrees of freedom, we artificially added a mutual inclination of one degree .we also performed simulations of each system in which hundreds of ( massless ) test particles were added .all test particles were started on circular orbits with respect to the central star .no test particles were placed in regions such that a test particle would initially cross the orbit of a planet .the results may be used as an indication of the presence and locations of potential terrestrial planets in these systems .the simulations were performed with the second - order mixed variable symplectic ( mvs ) integrator in the mercury integration package ( chambers 1999 ) , which is based on the technique pioneered by wisdom & holman ( 1991 ) .this code was modified to include the principal effects of general relativity , as in lissauer & rivera ( 2001 ) .in agreement with fischer et al .( 2002 ) , we find that in dynamical fits to the radial velocity data for 47 uma , the value of does not change significantly for a fit with the eccentricity of the outer companion in the range 0 to 0.2 .when we fit for the eccentricity of the outer planet , the fitting routine converged on a value of 0.19 . note that this value is near the boundary of stability determined by fischer et al .a long - term simulation of the system based on this fit is stable over billions of years . in this simulation ,the two companions are in a 5:2 mean motion resonance .also , throughout the simulation the orbits are nearly anti - aligned .table 1 gives the parameters from the fit at epoch jd 2446959.737 , and figure 1a shows the nearest and farthest points in the companions orbits vs. time for the first 1 gyr of a long - term simulation based on the parameters given in table 1 .figure 2 shows the eccentricities , inclinations , and the period ratio of the companions over short times .figure 1b shows the stability of test particles in a 10 myr simulation of the same system .it shows the time that a test particle was lost vs. its initial semimajor axis .the dots and lines toward the bottom of the figure indicate the initial semimajor axes and radial excursions of the planets .note that islands of instability occur at the locations of several mean motion resonances with the planets .also , exterior to the outer companion , there are islands of stability between mean motion resonances . since 47uma is similar to the sun , the region around 1 au is of particular interest .unfortunately , the 3:1 mean motion resonance with the inner companion lies in this region , and there are signs that the region may not be stable . note that while only the one test particle at 1 au was lost in less than 10 myr , proximity to the 3:1 mean motion resonance could endanger the stability of other nearby orbits over longer timescales .lrrcrrr & & & + parameter & inner & outer & & inner & middle & outer + period ( days ) & 1082 & 2735 & & 14.65 & 44.58 & 5582 + & 0.034 & 0.190 & & 0.021 & 0.169 & 0.141 + ( deg ) & 133 & 284 & & 104 & 56 & 202 + mean anomaly ( deg ) & 351 & 329 & & 340 & 345 & 189 + ( ) & 2.65 & 0.90 & & 0.84 & 0.20 & 4.21 +this system proved to be difficult to fit with the levenberg - marquardt routine .we obtained two fits with almost the same value with very different parameters for the outer planet .this is an indication of the large uncertainties in this planet s orbital parameters .table 1 gives the parameters from one of our fits at epoch jd 2450250.0 , and figure 3a shows the periastron and apastron of each planet vs. time from a simulation based on this fit .the simulation shows that the inner and middle companions are near a 3:1 mean motion resonance .these results are in agreement with marcy at al .( 2002 ) .figure 3b shows the stability of test particles in a 5 myr simulation of the same system .it clearly shows a large stable region between the middle and outer companions .this is in rough agreement with marcy et al .( 2002 ) , in which they showed that a terrestrial planet on a circular orbit at 1 au would be stable .we have presented results on the stability of the 47 uma and 55 cancri systems with and without massless particles .dynamical fits were used to determine the initial conditions for the simulations .a large area of parameter space is consistent with the observations .the area around 1 au in the 47 uma system may not harbor a terrestrial planet , while the opposite is true in the 55 cancri system .these results are in rough agreement with previous studies ( fischer et al .2002 ; marcy et al .2002 ) .butler , r. p. , marcy , g. w. , fischer , d. a. , brown , t. m. , contos , a. r. , korzennik , s. g. , nisenson , p. , & noyes , r. w. 1999 , , 526 , 916 chambers , j. e. 1999 , , 304 , 793 fischer , d. a. , marcy , g. w. , butler , r. p. , laughlin , g. , & vogt , s. s. 2002 , , 564 , 1028 laughlin , g. , & chambers , j. e. 2001 , , 551 , l109 lissauer , j. j. & rivera , e. j. 2001 , , 554 , 1141 marcy , g. w. , butler , r. p. , fischer , d. a. , vogt , s. s. , lissauer , j. j. , & rivera , e. j. 2001 , , 556 , 296 marcy , g. w. , butler , r. p. , fischer , d. a. , laughlin , g. , vogt , s. s. , henry , g. w. , & pourbaix , d. 2002 , , submitted press , w. h. , teukolsky , s. a. , vetterling , w. t. , flannery , b. p. 1992, numerical recipes in fortran ( 2nd ed . ;cambridge : cambridge univ .press ) rivera , e. j. & lissauer , j. j. 2001 , , 558 , 392 wisdom , j. & holman , m. 1991 , , 102 , 1528

Abstract: in this paper , we present preliminary results on the stability of massless particles in two and three - planet systems . the results of our study may be used to address questions concerning the stability of terrestrial planets in these systems and also the trapping of particles in resonances with the planets . the possibility of the existence of islands of stability and/or instability at different regions in multi - body systems and their probable correspondence to certain resonances are also discussed .   # 1_#1 _ # 1_#1 _ =  # 1 1.25 in .125 in .25 in


gesture and human action recognition from visual information is an active research topic in computer vision and machine learning .it has many potential applications including human - computer interactions and robotics .since first work  on action recognition from depth data captured by commodity depth sensors ( e.g. , kinect ) in 2010 , many methods  for action recognition have been proposed based on specific hand - crafted feature descriptors extracted from depth or skeleton data . with the recent development of deep learning ,a few methods have also been developed based on convolutional neural networks ( convnets )  and recurrent neural networks ( rnns )  .however , most of the works on gesture / action recognition reported to date focus on the classification of individual gestures or actions by assuming that instances of individual gestures and actions have been isolated or segmented from a video or a stream of depth maps / skeletons before the classification . in the cases of continuous recognition , the input stream usually contains unknown numbers , unknown orders and unknown boundaries of gestures or actions and both segmentation and recognition have to be solved at the same time .there are three common approaches to continuous recognition .the first approach is to tackle the segmentation and classification of the gestures or actions separately and sequentially .the key advantages of this approach are that different features can be used for segmentation and classification and existing classification methods can be leveraged .the disadvantages are that both segmentation and classification could be the bottleneck of the systems and they can not be optimized together .the second approach is to apply classification to a sliding temporal window and aggregate the window based classification to achieve final segmentation and classification .one of the key challenges in this approach is the difficulty of setting the size of sliding window because durations of different gestures or actions can vary significantly .the third approach is to perform the segmentation and classification simultaneously .this paper adopts the first approach and focuses on robust classification using convnets that are insensitive to inaccurate segmentation of gestures .specifically , individual gestures are first segmented based on quantity of movement ( qom )  from a stream of depth maps . for each segmented gesture , an improved depth motion map ( idmm )is constructed from its sequence of depth maps .convnets are used to learn the dynamic features from the idmm for effective classification .[ fig : framework ] shows the framework of the proposed method .the rest of this paper is organized as follows .section ii briefly reviews the related works on video segmentation and gesture / action recognition based on depth and deep learning .details of the proposed method are presented in section iii .experimental results on the dataset provided by the challenge are reported in section iv , and section v concludes the paper .there are many methods proposed for segmenting individual gestures from a video .the popular and widely used method employs dynamic time warping ( dtw ) to decide the delimitation frames of individual gestures  .difference images are first obtained by subtracting two consecutive grey - scale images and each difference image is partitioned into a grid of cells . each cellis then represented by the average value of pixels within this cell .the matrix of the cells in a difference image is flattened as a vector called motion feature and calculated for each frame in the video excluding the final frame .this results in a matrix of motion features for a video with frames .the motion feature matrix is extracted from both test video and training video which consists of multiple gestures .the two matrices are treated as two temporal sequences with each motion feature as a feature vector at an instant of time .the distance between two feature vectors is defined as the negative euclidean distance and a matrix containing dtw distances ( measuring similarity between two temporal sequences ) between the two sequences is then calculated and analysed by viterbi algorithm  to segment the gestures .another category of gesture segmentation methods from a multi - gesture video is based on appearance . upon the general assumption that the _ start _ and_ end _ frames of adjacent gestures are similar , correlation coefficients  and k - nearest neighbour algorithm with histogram of oriented gradient ( hog )  are used to identify the _ start _ and _ end _ frames of gestures .jiang et al . proposed a method based on quantity of movement ( qom ) by assuming the same _ start _ pose among different gestures .candidate delimitation frames are chosen based on the global qom .after a refining stage which employs a sliding window to keep the frame with minimum qom in each windowing session , the _ start _ and _ end _ frames are assumed to be the remained frames .this paper adopts the qom based method and its details will be presented in section [ subsec : video - seg ] .   with microsoft kinect sensorsresearchers have developed methods for depth map - based action recognition . sampled points from a depth map to obtain a bag of 3d points to encode spatial information and employ an expandable graphical model to encode temporal information .yang et al . stacked differences between projected depth maps as a depth motion map ( dmm ) and then used hog to extract relevant features from the dmm .this method transforms the problem of action recognition from spatio - temporal space to spatial space . in ,a feature called histogram of oriented 4d normals ( hon4d ) was proposed ; surface normal is extended to 4d space and quantized by regular polychorons . following this method , yang and tian cluster hypersurface normals and form the polynormal which can be used to jointly capture the local motion and geometry information .super normal vector ( snv ) is generated by aggregating the low - level polynormals . in ,a fast binary range - sample feature was proposed based on a test statistic by carefully designing the sampling scheme to exclude most pixels that fall into the background and to incorporate spatio - temporal cues .existing deep learning approach can be generally divided into four categories based on how the video is represented and fed to a deep neural network .the first category views a video either as a set of still images  or as a short and smooth transition between similar frames  , each color channel of the images is fed to one channel of a convnet . although obviously suboptimal , considering the video as a bag of static frames performs reasonably well .the second category represents a video as a volume and extends convnets to a third , temporal dimension  replacing 2d filters with 3d equivalents .so far , this approach has produced little benefits , probably due to the lack of annotated training data .the third category treats a video as a sequence of images and feeds the sequence to an rnn  . an rnn is typically considered as memory cells , which are sensitive to both short as well as long term patterns .it parses the video frames sequentially and encodes the frame - level information in their memory .however , using rnns did not give an improvement over temporal pooling of convolutional features  or over hand - crafted features .the last category represents a video as one or multiple compact images and adopts available trained convnet architectures for fine - tuning  .this category has achieved state - of - the - art results of action recognition on many rgb and depth / skeleton datasets .the proposed gesture classification in this paper falls into the last category .the proposed method consists of two major components , as illustrated in fig .[ fig : framework ] : video segmentation and construction of improved depth motion map ( idmm ) from a sequence of depth maps as the input to convnets . given a sequence of depth maps consisting of multiple gestures , the _ start _ and _ end _ frames of each gesture are identified based on quantity of movement ( qom )  .then , one idmm is constructed by accumulating the absolute depth difference between current frame and the _ start _ frame for each gesture segment .the idmm goes through a pseudo - color coding process to become a pseudo - rgb image as an input to a convnet for classification .the main objective of pseudo - color coding is to enchance the motin pattern captured by the idmm . in the rest of this section , video segmentation , construction of idmm , pseudo - color coding of idmm , and training of the convnetsare explained in detail .  [ cols="^,^,^,^,^,^,^,^",options="header " , ]this paper presents an effective yet simple method for continuous gesture recognition using only depth map sequences .depth sequences are first segmented so that each segmentation contains only one gesture and a convnet is used for feature extraction and classification .the proposed construction of idmm enables the use of available pre - trained models for fine - tuning without learning afresh .experimental results on chalearn lap congd dataset verified the effectiveness of the proposed method .how to exactly extract the neutral pose and fuse different modalities to improve the accuracy will be our future work .the authors would like to thank nvidia corporation for the donation of a tesla k40 gpu card used in this challenge .10 w. li , z. zhang , and z. liu , `` action recognition based on a bag of 3d points , '' in _ proc .ieee computer society conference on computer vision and pattern recognition workshops ( cvprw ) _ , 2010 , pp .j. wang , z. liu , y. wu , and j. yuan , `` mining actionlet ensemble for action recognition with depth cameras , '' in _ proc .ieee conference on computer vision and pattern recognition ( cvpr ) _ , 2012 , pp .12901297 .x. yang , c. zhang , and y. tian , `` recognizing actions using depth motion maps - 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Abstract: this paper addresses the problem of continuous gesture recognition from sequences of depth maps using convolutional neural networks ( convnets ) . the proposed method first segments individual gestures from a depth sequence based on quantity of movement ( qom ) . for each segmented gesture , an improved depth motion map ( idmm ) , which converts the depth sequence into one image , is constructed and fed to a convnet for recognition . the idmm effectively encodes both spatial and temporal information and allows the fine - tuning with existing convnet models for classification without introducing millions of parameters to learn . the proposed method is evaluated on the large - scale continuous gesture recognition of the chalearn looking at people ( lap ) challenge 2016 . it achieved the performance of 0.2655 ( mean jaccard index ) and ranked place in this challenge .   gesture recognition ; depth map sequence ; convolutional neural network , depth motion map