Denumerable Markov processes with bounded generators: a routine for calculating pij(∞)

1971 ◽  
Vol 8 (2) ◽  
pp. 423-427 ◽  
Author(s):  
Arne Jensen ◽  
David Kendall
Keyword(s):  
Banach Space ◽  
Markov Process ◽  
Bounded Operator ◽  
Transition Rates ◽  
Dense Domain ◽  
Transition Functions ◽  
The Matrix ◽  
Exponential Power ◽  
Image Position

1. Let the (honest) Markov process with transition functions (pij(0)) have transition rates (qij) and suppose that, for some M, so that the matrix Q = (qij) determines a bounded operator on the Banach space l1 by right-multiplication. Then in the terminology of [8], (pp. 12 and 19) Q will be bounded and ΩF will be a closed restriction of Q with dense domain, so that ΩF = Q; that is, we shall have a process whose associated semigroup has a bounded generator. In these circumstances Theorem 10.3.2 of [2] applies and the matrix Pt = (pij(t)) is given by where exp{·} denotes the function defined by the exponential power-series. We shall be interested here (as in [5] and [9]) in the determination of the limit matrix P∞ = (limt→∞pij(t)).

Denumerable Markov processes with bounded generators: a routine for calculating pij (∞)

1971 ◽  
Vol 8 (02) ◽  
pp. 423-427
Author(s):  
Arne Jensen ◽  
David Kendall
Keyword(s):  
Banach Space ◽  
Markov Process ◽  
Bounded Operator ◽  
Transition Rates ◽  
Dense Domain ◽  
Transition Functions ◽  
The Matrix ◽  
Exponential Power ◽  
Image Position

1. Let the (honest) Markov process with transition functions (pij (0)) have transition rates (qij ) and suppose that, for some M, so that the matrix Q = (qij ) determines a bounded operator on the Banach space l 1 by right-multiplication. Then in the terminology of [8], (pp. 12 and 19) Q will be bounded and Ω F will be a closed restriction of Q with dense domain, so that Ω F = Q; that is, we shall have a process whose associated semigroup has a bounded generator. In these circumstances Theorem 10.3.2 of [2] applies and the matrix Pt = (pij (t)) is given by where exp{·} denotes the function defined by the exponential power-series. We shall be interested here (as in [5] and [9]) in the determination of the limit matrix P ∞ = (lim t→∞ pij (t)).

Proportional intensities and strong ergodicity for Markov processes

1983 ◽  
Vol 20 (01) ◽  
pp. 185-190 ◽  
Author(s):  
Mark Scott ◽  
Dean L. Isaacson
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
Discrete Time ◽  
Continuous Time ◽  
Homogeneous Markov Process ◽  
Strong Ergodicity ◽  
Transition Functions ◽  
The Matrix ◽  
Intensity Functions ◽  
Time Point

By assuming the proportionality of the intensity functions at each time point for a continuous-time non-homogeneous Markov process, strong ergodicity for the process is determined through strong ergodicity of a related discrete-time Markov process. For processes having proportional intensities, strong ergodicity implies having the limiting matrix L satisfy L · P(s, t) = L, where P(s, t) is the matrix of transition functions.

Growth conditions on subharmonic functions and resolvents of operators

1985 ◽  
Vol 97 (2) ◽  
pp. 321-324
Author(s):  
J. R. Partington
Keyword(s):  
Banach Space ◽  
Numerical Range ◽  
Bounded Operator ◽  
Complex Banach Space ◽  
Growth Conditions ◽  
Subharmonic Functions ◽  
Image Position

Let X be a complex Banach space and T a bounded operator on X. The numerical range of T is defined by

The calculation of limit probabilities for denumerable Markov processes from infinitesimal properties

1973 ◽  
Vol 10 (01) ◽  
pp. 84-99 ◽  
Author(s):  
Richard L. Tweedie
Keyword(s):  
Markov Process ◽  
Continuous Time ◽  
Transition Matrix ◽  
Invariant Measures ◽  
Equilibrium Distribution ◽  
Transition Functions ◽  
The Matrix ◽  
Limit Probability ◽  
Derivatives Of ◽  
Regular Chains

The problem considered is that of estimating the limit probability distribution (equilibrium distribution) πof a denumerable continuous time Markov process using only the matrix Q of derivatives of transition functions at the origin. We utilise relationships between the limit vector πand invariant measures for the jump-chain of the process (whose transition matrix we write P∗), and apply truncation theorems from Tweedie (1971) to P∗. When Q is regular, we derive algorithms for estimating πfrom truncations of Q; these extend results in Tweedie (1971), Section 4, from q-bounded processes to arbitrary regular processes. Finally, we show that this method can be extended even to non-regular chains of a certain type.

The calculation of limit probabilities for denumerable Markov processes from infinitesimal properties

1973 ◽  
Vol 10 (1) ◽  
pp. 84-99 ◽  
Author(s):  
Richard L. Tweedie
Keyword(s):  
Markov Process ◽  
Continuous Time ◽  
Transition Matrix ◽  
Invariant Measures ◽  
Equilibrium Distribution ◽  
Transition Functions ◽  
The Matrix ◽  
Limit Probability ◽  
Derivatives Of ◽  
Regular Chains

The problem considered is that of estimating the limit probability distribution (equilibrium distribution) πof a denumerable continuous time Markov process using only the matrix Q of derivatives of transition functions at the origin. We utilise relationships between the limit vector πand invariant measures for the jump-chain of the process (whose transition matrix we write P∗), and apply truncation theorems from Tweedie (1971) to P∗. When Q is regular, we derive algorithms for estimating πfrom truncations of Q; these extend results in Tweedie (1971), Section 4, from q-bounded processes to arbitrary regular processes. Finally, we show that this method can be extended even to non-regular chains of a certain type.

Proportional intensities and strong ergodicity for Markov processes

1983 ◽  
Vol 20 (1) ◽  
pp. 185-190 ◽  
Author(s):  
Mark Scott ◽  
Dean L. Isaacson
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
Discrete Time ◽  
Continuous Time ◽  
Homogeneous Markov Process ◽  
Strong Ergodicity ◽  
Transition Functions ◽  
The Matrix ◽  
Intensity Functions ◽  
Time Point

By assuming the proportionality of the intensity functions at each time point for a continuous-time non-homogeneous Markov process, strong ergodicity for the process is determined through strong ergodicity of a related discrete-time Markov process. For processes having proportional intensities, strong ergodicity implies having the limiting matrix L satisfy L · P(s, t) = L, where P(s, t) is the matrix of transition functions.

Lateral resolution of the electron microbeam analysis

1978 ◽  
Vol 36 (1) ◽  
pp. 542-543
Author(s):  
H.J. Dudek
Keyword(s):  
Quantitative Analysis ◽  
Depth Resolution ◽  
Lateral Resolution ◽  
Cylindrical Particle ◽  
Correction Procedure ◽  
X Ray ◽  
Element Concentrations ◽  
Microbeam Analysis ◽  
The Matrix

The chemical inhomogenities in modern materials such as fibers, phases and inclusions, often have diameters in the region of one micrometer. Using electron microbeam analysis for the determination of the element concentrations one has to know the smallest possible diameter of such regions for a given accuracy of the quantitative analysis.In th is paper the correction procedure for the quantitative electron microbeam analysis is extended to a spacial problem to determine the smallest possible measurements of a cylindrical particle P of high D (depth resolution) and diameter L (lateral resolution) embeded in a matrix M and which has to be analysed quantitative with the accuracy q. The mathematical accounts lead to the following form of the characteristic x-ray intens ity of the element i of a particle P embeded in the matrix M in relation to the intensity of a standard S

Determination of the phase structure of polymerblends by electron microscopy

1978 ◽  
Vol 36 (1) ◽  
pp. 492-493
Author(s):  
Dr. G. Kaemof
Keyword(s):  
Screw Extruder ◽  
Electron Microscopic ◽  
Thin Sections ◽  
Single Screw Extruder ◽  
Transmission Electron ◽  
The Matrix ◽  
Twin Screw ◽  
Special Case ◽  
Microscopic Investigations

A mixture of polycarbonate (PC) and styrene-acrylonitrile-copolymer (SAN) represents a very good example for the efficiency of electron microscopic investigations concerning the determination of optimum production procedures for high grade product properties.The following parameters have been varied:components of charge (PC : SAN 50 : 50, 60 : 40, 70 : 30), kind of compounding machine (single screw extruder, twin screw extruder, discontinuous kneader), mass-temperature (lowest and highest possible temperature).The transmission electron microscopic investigations (TEM) were carried out on ultra thin sections, the PC-phase of which was selectively etched by triethylamine.The phase transition (matrix to disperse phase) does not occur - as might be expected - at a PC to SAN ratio of 50 : 50, but at a ratio of 65 : 35. Our results show that the matrix is preferably formed by the components with the lower melting viscosity (in this special case SAN), even at concentrations of less than 50 %.

Polishing process effect on the image of dispersed precipitates

1984 ◽  
Vol 42 ◽  
pp. 534-535
Author(s):  
C.T. Hu ◽  
C.W. Allen
Keyword(s):  
Matrix Material ◽  
Specimen Preparation ◽  
Second Phase ◽  
Precipitate Particle ◽  
Polishing Process ◽  
Second Phase Particles ◽  
The Matrix ◽  
Surface Profiles ◽  
Thinning Process

One important problem in determination of precipitate particle size is the effect of preferential thinning during TEM specimen preparation. Figure 1a schematically represents the original polydispersed Ni3Al precipitates in the Ni rich matrix. The three possible type surface profiles of TEM specimens, which result after electrolytic thinning process are illustrated in Figure 1b. c. & d. These various surface profiles could be produced by using different polishing electrolytes and conditions (i.e. temperature and electric current). The matrix-preferential-etching process causes the matrix material to be attacked much more rapidly than the second phase particles. Figure 1b indicated the result. The nonpreferential and precipitate-preferential-etching results are shown in Figures 1c and 1d respectively.

scholarly journals Simplified and Rapid Determination of Primaquine and 5,6-Orthoquinone Primaquine by UHPLC-MS/MS: Its Application to a Pharmacokinetic Study

Molecules ◽  
2021 ◽  
Vol 26 (14) ◽  
pp. 4357
Author(s):  
Waritda Pookmanee ◽  
Siriwan Thongthip ◽  
Jeeranut Tankanitlert ◽  
Mathirut Mungthin ◽  
Chonlaphat Sukasem ◽  
...  
Keyword(s):  
Pharmacokinetic Study ◽  
High Performance ◽  
Rapid Determination ◽  
Active Metabolites ◽  
Chromatography Tandem Mass Spectrometry ◽  
Accuracy And Precision ◽  
The Matrix ◽  
Liquid Chromatography Tandem Mass ◽  
Isocratic Mode

The method for the determination of primaquine (PQ) and 5,6-orthoquinone primaquine (5,6-PQ), the representative marker for PQ active metabolites, via CYP2D6 in human plasma and urine has been validated. All samples were extracted using acetonitrile for protein precipitation and analyzed using the ultra-high-performance liquid chromatography–tandem mass spectrometry (UHPLC-MS/MS) system. Chromatography separation was carried out using a Hypersil GOLDTM aQ C18 column (100 × 2.1 mm, particle size 1.9 μm) with a C18 guard column (4 × 3 mm) flowed with an isocratic mode of methanol, water, and acetonitrile in an optimal ratio at 0.4 mL/min. The retention times of 5,6-PQ and PQ in plasma and urine were 0.8 and 1.6 min, respectively. The method was validated according to the guideline. The linearity of the analytes was in the range of 25–1500 ng/mL. The matrix effect of PQ and 5,6-PQ ranged from 100% to 116% and from 87% to 104% for plasma, and from 87% to 89% and from 86% to 87% for urine, respectively. The recovery of PQ and 5,6-PQ ranged from 78% to 95% and form 80% to 98% for plasma, and from 102% to from 112% to 97% to 109% for urine, respectively. The accuracy and precision of PQ and 5,6-PQ in plasma and urine were within the acceptance criteria. The samples should be kept in the freezer (−80 °C) and analyzed within 7 days due to the metabolite stability. This validated UHPLC-MS/MS method was beneficial for a pharmacokinetic study in subjects receiving PQ.

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