Stochastic matrix

Stochastic matrices for which both row and column sums are equal to 1 are called double stochastic matrices. Stochastic matrices are defined in the framework ofthe MARCH-INSIDE descriptors, TOMOCOMD descriptors, and —> walk counts. 

The correction for non-uniform instrumental broadening in SEC is solved through a non-recursive matrix stochastic technique. To this effect, Tung s equation ( ) must be reformulated in matrix form, and the measurements assumed contaminated with zero-mean noise. 

Let u be a vector valued stochastic variable with dimension D x 1 and with covariance matrix Ru of size D x D. The key idea is to linearly transform all observation vectors, u , to new variables, z = W Uy, and then solve the optimization problem (1) where we replace u, by z . We choose the transformation so that the covariance matrix of z is diagonal and (more importantly) none if its eigenvalues are too close to zero. (Loosely speaking, the eigenvalues close to zero are those that are responsible for the large variance of the OLS-solution). In order to liiid the desired transformation, a singular value decomposition of /f is performed yielding... 

Fig. 4. Computation of the stochastic matrix Pa via mapping of discretization boxes.

After the assembling of the stochastic matrix Pd we have to solve the associated non-selfadjoint eigenvalue problem. Our present numerical results have been computed using the code speig by Radke AND S0RENSEN in Matlab,... 

Closely related to the transition matrix is the stochastic matrix, whose elements are labelle a . TTiis matrix gives the probability of choosing the two states m and n between whic the move is to be made. It is often known as the underlying matrix of the Markov chain, the probability of accepting a trial move from m to n is then the probability of makir a transition from m to n (7r, ) is given by multiplying the probability of choosing states... 

The simplest scheme that accounts for the destruction of phase coherence is the so-called stochastic interruption model [Nikitin and Korst 1965 Simonius 1978 Silbey and Harris 1989]. Suppose the process of free tunneling is interrupted by a sequence of collisions separated by time periods vo = to do After each collision the system forgets its initial phase, i.e., the off-diagonal matrix elements of the density matrix p go to zero, resulting in the density matrix p ... 

In the framework of this ultimate model [33] there are m2 constants of the rate of the chain propagation kap describing the addition of monomer to the radical Ra whose reactivity is controlled solely by the type a of its terminal unit. Elementary reactions of chain termination due to chemical interaction of radicals Ra and R is characterized by m2 kinetic parameters k f . The stochastic process describing macromolecules, formed at any moment in time t, is a Markov chain with transition matrix whose elements are expressed through the concentrations Ra and Ma of radicals and monomers at this particular moment in the following way [1,34] ... 

This is the simplest of the models where violation of the Flory principle is permitted. The assumption behind this model stipulates that the reactivity of a polymer radical is predetermined by the type of bothjts ultimate and penultimate units [23]. Here, the pairs of terminal units MaM act, along with monomers M, as kinetically independent elements, so that there are m3 constants of the rate of elementary reactions of chain propagation ka ]r The stochastic process of conventional movement along macromolecules formed at fixed x will be Markovian, provided that monomeric units are differentiated by the type of preceding unit. In this case the number of transient states Sa of the extended Markov chain is m2 in accordance with the number of pairs of monomeric units. No special problems presents writing down the elements of the matrix of the transitions Q of such a chain [ 1,10,34,39] and deriving by means of the mathematical apparatus of the Markov chains the expressions for the instantaneous statistical characteristics of copolymers. By way of illustration this matrix will be presented for the case of binary copolymerization ... 

In order to obtain the expression for the components of the vector of instantaneous copolymer composition it is necessary, according to general algorithm, to firstly determine the stationary vector ji of the extended Markov chain with the matrix of transitions (13) which describes the stochastic process of conventional movement along macromolecules with labeled units and then to erase the labels. In this particular case such a procedure reduces to the summation ... 

An exhaustive statistical description of living copolymers is provided in the literature [25]. There, proceeding from kinetic equations of the ideal model, the type of stochastic process which describes the probability measure on the set of macromolecules has been rigorously established. To the state Sa(x) of this process monomeric unit Ma corresponds formed at the instant r by addition of monomer Ma to the macroradical. To the statistical ensemble of macromolecules marked by the label x there corresponds a Markovian stochastic process with discrete time but with the set of transient states Sa(x) constituting continuum. Here the fundamental distinction from the Markov chain (where the number of states is discrete) is quite evident. The role of the probability transition matrix in characterizing this chain is now played by the integral operator kernel ... 

In Eq. (13), the vector q denotes a set of mass-weighted coordinates in a configuration space of arbitrary dimension N, U(q) is the potential of mean force governing the reaction, T is a symmetric positive-definite friction matrix, and , (/) is a stochastic force that is assumed to represent white noise that is Gaussian distributed with zero mean. The subscript a in Eq. (13) is used to label a particular noise sequence For any given a, there are infinitely many... 

The dynamical behaviors of p(At) v and p(At)av av, have to be determined by solving the stochastic Liouville equation for the reduced density matrix the initial conditions are determined by the pumping process. For the purpose of qualitative discussion, we assume that the 80-fs pulse can only pump two vibrational states, say v = 0 and v = 1 states. In this case we obtain... 

For many synthetic copolymers, it becomes possible to calculate all desired statistical characteristics of their primary structure, provided the sequence is described by a Markov chain. Although stochastic process 31 in the case of proteinlike copolymers is not a Markov chain, an exhaustive statistic description of their chemical structure can be performed by means of an auxiliary stochastic process 3iib whose states correspond to labeled monomeric units. As a label for unit M , it was suggested [23] to use its distance r from the center of the globule. The state of this stationary stochastic process 31 is a pair of numbers, (a, r), the first of which belongs to a discrete set while the second one corresponds to a continuous set. Stochastic process ib is remarkable for being stationary and Markovian. The probability of the transition from state a, r ) to state (/i, r") for the process of conventional movement along a heteropolymer macromolecule is described by the matrix-function of transition intensities... 

The classical dynamics corresponding to a quantum graph defined by a unitary propagator SG is given by a stochastic process with transition matrix T defined by... 

The matrix T is clearly stochastic, as YJj=i Tij = 1 due to the unitarity of SG the set of transition matrices related to a unitary matrix as defined in (6) is a subset of the set of all stochastic transition matrices, referred to as the set of unitary-stochastic matrices. The topology of the set in the space of all stochastic matrices is in fact quite complicated, see Pakonski et.al. (2001). In what follows, we will only use that T... 

The Markov processes associated with quantum star graphs correspond to systems of weakly coupled edges. Its dynamical properties are determined by the spectrum of the stochastic matrix associated with (14) which is highly degenerate and can be given explicitly (Kottos and Smilansky 1999), that is,... 

The operators Fk(t) defined in Eq.(49) are taken as fluctuations based on the idea that at t=0 the initial values of the bath operators are uncertain. Ensemble averages over initial conditions allow for a definite specification of statistical properties. The statistical average of the stochastic forces Fk(t) is calculated over the solvent effective ensemble by taking the trace of the operator product pmFk (this is equivalent to sum over the diagonal matrix elements of this product), so that = Trace(pmFk) is identically zero (Fjk(t)=Fk(t) in this particular case). The non-zero correlation functions of the fluctuations are solvent statistical averages over products of operator forces,... 

In terms of the equivalent stochastic differential equation (6.106), p. 279, this choice yields a diffusion matrix of the form B() = S -(i(), where the matrix G does not depend on the molecular-diffusion coefficients. 

Working with Markov chains, confusion is bound to arise if the indices of the Markov matrix are handled without care. As stated lucidly in an excellent elementary textbook devoted to finite mathematics,24 transition probability matrices must obey the constraints of a stochastic matrix. Namely that they have to be square, each element has to be non-negative, and the sum of each column must be unity. In this respect, and in order to conform with standard rules vector-matrix multiplication, it is preferable to interpret the probability / , as the probability of transition from state. v, to state s (this interpretation stipulates the standard Pp format instead of the pTP format, the latter convenient for the alternative 5 —> Sjinterpretation in defining p ), 5,6... 

Unidirectional kinetic processes cannot be immediately interpreted as Markov chains, since only the (1,1) element of the /- -matrix would differ from zero, violating the stochastic matrix constraints (Section II. 1). An artificial Markov matrix complying with this constraint can be visualized, however, with the understanding that no other element of this imbedded P-matrix, past the (1,1) element, will have a physical meaning. It follows that the initial state probability vector is non-zero only in its (1,1)... 

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