The evolution of the attainable structures of a continuous time homogeneous Markov system with fixed size

1996 ◽  
Vol 33 (1) ◽  
pp. 34-47 ◽  
Author(s):  
George M. Tsaklidis
Keyword(s):  
Euclidean Space ◽  
Discrete Time ◽  
Continuous Time ◽  
Fixed Size ◽  
Markov System

In order to describe the evolution of the attainable structures of a continuous time homogeneous Markov system (HMS) with fixed size, we evaluate the volume of the sets of the attainable structures in Euclidean space in the course of time, and we find the value of the volume asymptotically. Then, using the concept of the volume of the attainable structures, we provide a method to evaluate the ‘age' of the system in continuous and discrete time. We also estimate the evolution of the distance of two (attainable) structures of the system as it changes following the transformations of the structures.

The evolution of the attainable structures of a continuous time homogeneous Markov system with fixed size

1996 ◽  
Vol 33 (01) ◽  
pp. 34-47 ◽  
Author(s):  
George M. Tsaklidis
Keyword(s):  
Euclidean Space ◽  
Discrete Time ◽  
Continuous Time ◽  
Fixed Size ◽  
Markov System

In order to describe the evolution of the attainable structures of a continuous time homogeneous Markov system (HMS) with fixed size, we evaluate the volume of the sets of the attainable structures in Euclidean space in the course of time, and we find the value of the volume asymptotically. Then, using the concept of the volume of the attainable structures, we provide a method to evaluate the ‘age' of the system in continuous and discrete time. We also estimate the evolution of the distance of two (attainable) structures of the system as it changes following the transformations of the structures.

The stress tensor and the energy of a continuous time homogeneous Markov system with fixed size

1999 ◽  
Vol 36 (1) ◽  
pp. 21-29 ◽  
Author(s):  
George M. Tsaklidis
Keyword(s):  
Velocity Field ◽  
Euclidean Space ◽  
Stress Tensor ◽  
Continuous Time ◽  
Suitable Model ◽  
Fixed Size ◽  
Markov System

The set of the attainable structures of a continuous time homogeneous Markov system (HMS) with fixed size, is considered as a continuum and the evolution of the HMS in the Euclidean space corresponds to its motion. Taking account of the velocity field of the HMS, a suitable model of continuum–defined by its stress tensor–is proposed in order to explain the motion of the system. The adoption of this model (equivalently of its stress tensor) enables us to establish the concept of the energy of a structure of the HMS.

The stress tensor and the energy of a continuous time homogeneous Markov system with fixed size

1999 ◽  
Vol 36 (01) ◽  
pp. 21-29 ◽  
Author(s):  
George M. Tsaklidis
Keyword(s):  
Velocity Field ◽  
Euclidean Space ◽  
Stress Tensor ◽  
Continuous Time ◽  
Suitable Model ◽  
Fixed Size ◽  
Markov System

The set of the attainable structures of a continuous time homogeneous Markov system (HMS) with fixed size, is considered as a continuum and the evolution of the HMS in the Euclidean space corresponds to its motion. Taking account of the velocity field of the HMS, a suitable model of continuum–defined by its stress tensor–is proposed in order to explain the motion of the system. The adoption of this model (equivalently of its stress tensor) enables us to establish the concept of the energy of a structure of the HMS.

scholarly journals Parametric analysis of the Bass model

2016 ◽  
Vol 12 (1) ◽  
pp. 29-40
Author(s):  
Yair Orbach
Keyword(s):  
Euclidean Space ◽  
Discrete Time ◽  
Continuous Time ◽  
Parametric Analysis ◽  
Inflection Point ◽  
Bass Model ◽  
Long Run ◽  
The Common ◽  
Special Case ◽  
Q Values

In this research, the authors explore the influence of the Bass model p, q parameters values on diffusion patterns and map p, q Euclidean space regions accordingly. The boundaries of four different sub-regions are classified and defined, in the region where both p, q are positive, according to the number of inflection point and peak of the non-cumulative sales curve. The researchers extend the p, q range beyond the common positive value restriction to regions where either p or q is negative. The case of negative p, which represents barriers to initial adoption, leads us to redefine the motivation for seeding, where seeding is essential to start the market rather than just for accelerating the diffusion. The case of negative q, caused by a declining motivation to adopt as the number of adopters increases, leads us to cases where the saturation of the market is at partial coverage rather than the usual full coverage at the long run. The authors develop a solution to the special case of p + q = 0, where the Bass solution cannot be used. Some differences are highlighted between the discrete time and continuous time flavors of the Bass model and the implication on the mapping. The distortion is presented, caused by the transition between continuous and discrete time forms, as a function of p, q values in the various regions

The evolution of the attainable structures of a homogeneous Markov system by fixed size

1994 ◽  
Vol 31 (02) ◽  
pp. 348-361
Author(s):  
George M. Tsaklidis
Keyword(s):  
Euclidean Space ◽  
Fixed Size ◽  
Markov Systems ◽  
Markov System

In order to describe the evolution of the attainable structures of a homogeneous Markov system (HMS) with fixed size, we evaluate the volume of the sets of the attainable structures in Euclidean space as they are changing in time and we find the value of the volume asymptotically. We also estimate the evolution of the distance of two (attainable) structures of the system as it changes following the transformations of the structures; extensions are obtained concerning results from the Perron–Frobenius theory referring to Markov systems.

The evolution of the attainable structures of a homogeneous Markov system by fixed size

1994 ◽  
Vol 31 (2) ◽  
pp. 348-361 ◽  
Author(s):  
George M. Tsaklidis
Keyword(s):  
Euclidean Space ◽  
Fixed Size ◽  
Markov Systems ◽  
Markov System

In order to describe the evolution of the attainable structures of a homogeneous Markov system (HMS) with fixed size, we evaluate the volume of the sets of the attainable structures in Euclidean space as they are changing in time and we find the value of the volume asymptotically. We also estimate the evolution of the distance of two (attainable) structures of the system as it changes following the transformations of the structures; extensions are obtained concerning results from the Perron–Frobenius theory referring to Markov systems.

The size order of the state vector of a continuous-time homogeneous Markov system with fixed size

2001 ◽  
Vol 38 (3) ◽  
pp. 635-646
Author(s):  
I. Kipouridis ◽  
G. Tsaklidis
Keyword(s):  
Lower Bound ◽  
Convergence Rate ◽  
Continuous Time ◽  
State Vector ◽  
The State ◽  
Specific Time ◽  
Fixed Size ◽  
Geometric Characteristics ◽  
Markov System ◽  
Selection Of

The variation of the state vectors p(t) = (pi(t)) of a continuous-time homogeneous Markov system with fixed size is examined. A specific time t0 after which the size order of the elements pi(t) becomes stable provides a criterion of the system's convergence rate. A method is developed to find t0 and a quickly evaluated lower bound for t0. This method is based on the geometric characteristics and the volumes of the attainable structures. Moreover, a condition concerning the selection of starting vectors p(0) is given so that the vector functions p(t) retain the same size order for every time greater than a given time t.

The size order of the state vector of a continuous-time homogeneous Markov system with fixed size

2001 ◽  
Vol 38 (03) ◽  
pp. 635-646
Author(s):  
I. Kipouridis ◽  
G. Tsaklidis
Keyword(s):  
Lower Bound ◽  
Convergence Rate ◽  
Continuous Time ◽  
State Vector ◽  
The State ◽  
Specific Time ◽  
Fixed Size ◽  
Geometric Characteristics ◽  
Markov System ◽  
Selection Of

The variation of the state vectors p (t) = (p i (t)) of a continuous-time homogeneous Markov system with fixed size is examined. A specific time t 0 after which the size order of the elements p i (t) becomes stable provides a criterion of the system's convergence rate. A method is developed to find t 0 and a quickly evaluated lower bound for t 0. This method is based on the geometric characteristics and the volumes of the attainable structures. Moreover, a condition concerning the selection of starting vectors p (0) is given so that the vector functions p (t) retain the same size order for every time greater than a given time t.

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