Applications of space-time harmonic functions to branching processes

1975 ◽  
Vol 7 (2) ◽  
pp. 251-252
Author(s):  
T. H. Savits
Keyword(s):  
Harmonic Functions ◽  
Branching Processes ◽  
Space Time ◽  
Time Harmonic

Space-time harmonic functions and age-dependent branching processes

1975 ◽  
Vol 7 (02) ◽  
pp. 283-298
Author(s):  
Thomas H. Savits
Keyword(s):  
Harmonic Functions ◽  
Positive Root ◽  
Branching Processes ◽  
Random Variable ◽  
Space Time ◽  
Malthusian Parameter ◽  
Time Harmonic ◽  
Age Dependent ◽  
Image Position ◽  
Number Of Particles

LetXbe an age-dependent branching process with lifetime distributionGand age-dependent generating function π(y,s) = σk= 0∞pk(y)sk. We assume thatGis right-continuous andG(0+) =G(0) = 0. The base state spaceSis [0,T) whereT= inf{t:G(t) = 1}. Setm(y) = σk= 0∞k pk(y) andThen extinction occurs with probability one iffm≤ 1. In the case wherem> 1, define the Malthusian parameter λ to be the unique (positive) root ofand setonS.is a-space-time harmonic function of the processXand the corresponding non-negative martingaleconverges w.p.l to a random variableW; furthermore, under a regularity assumption,Wis non-trivial iffwhereandIf 0 <a≤ Φ ≤ β < ∞, for some constantsa, β, thenw.p.l, whereZtis the number of particles at timet.

Space-time harmonic functions and age-dependent branching processes

1975 ◽  
Vol 7 (2) ◽  
pp. 283-298 ◽  
Author(s):  
Thomas H. Savits
Keyword(s):  
Harmonic Functions ◽  
Positive Root ◽  
Branching Processes ◽  
Random Variable ◽  
Space Time ◽  
Malthusian Parameter ◽  
Time Harmonic ◽  
Age Dependent ◽  
Image Position ◽  
Number Of Particles

Let X be an age-dependent branching process with lifetime distribution G and age-dependent generating function π(y,s) = σk = 0∞pk(y) sk. We assume that G is right-continuous and G(0+) = G(0) = 0. The base state space S is [0,T) where T = inf{t : G(t) = 1}. Set m(y) = σk = 0∞k pk(y) and Then extinction occurs with probability one iff m ≤ 1. In the case where m > 1, define the Malthusian parameter λ to be the unique (positive) root of and set on S. is a -space-time harmonic function of the process X and the corresponding non-negative martingale converges w.p.l to a random variable W; furthermore, under a regularity assumption, W is non-trivial iff where and If 0 < a ≤ Φ ≤ β < ∞, for some constants a, β, then w.p.l, where Zt is the number of particles at time t.

scholarly journals Harmonic functions for a class of Markov chains

1979 ◽  
Vol 28 (4) ◽  
pp. 413-422 ◽  
Author(s):  
H. Cohn
Keyword(s):  
Markov Chains ◽  
Harmonic Functions ◽  
Branching Process ◽  
Space Time ◽  
Time Harmonic ◽  
Bounded Space ◽  
Tail Events ◽  
Bounded Harmonic Functions

AbstractA class of Markov chains is considered for which a certain property of the tail events makes bounded harmonic functions obtainable from bounded space-time harmonic functions. Applications to almost surely convergent Markov chains are given and, in particular, a representation of Martin-Doob-Hunt type is derived for all bounded harmonic functions of a finite mean supercritical branching process.

ON THE RELATION BETWEEN BPS SOLUTIONS IN 4D AND 5D

2006 ◽  
Vol 15 (10) ◽  
pp. 1603-1618 ◽  
Author(s):  
KLAUS BEHRNDT ◽  
GABRIEL LOPES CARDOSO ◽  
SWAPNA MAHAPATRA
Keyword(s):  
Black Holes ◽  
Harmonic Functions ◽  
Space Time ◽  
Black Rings

We discuss the explicit dictionary between general stationary four- and five-dimensional supersymmetric solutions in N = 2 supergravity theories with cubic prepotentials. All these solutions are completely determined in terms of the same set of harmonic functions and attractor equations. We discuss various examples like black holes and black rings in Taub-NUT space–time. Then we consider corrections to the four-dimensional solutions associated with more general prepotentials and comment on analogous corrections on the 5D side.

Influence of the Seven-Phase Winding Phase Timing Sequence on Producing the Resulting Voltage Vector

Vestnik MEI ◽  
2021 ◽  
pp. 83-91
Author(s):  
Vladimir M. Tereshkin ◽  
◽  
Dmitriy A. Grishin ◽  
Sergey P. Balandin ◽  
Vyacheslav V. Tereshkin ◽  
...  
Keyword(s):  
Function Analysis ◽  
Harmonic Component ◽  
Rotation Frequency ◽  
Space Time ◽  
Phase Voltage ◽  
Voltage Vector ◽  
Analysis Methods ◽  
Time Harmonic ◽  
Traction Drives ◽  
Electric Traction

The subject of the study is a seven-phase symmetric winding, namely, its properties due to the possibility to alter the phase timing sequence. The seven-phase ABCDEFG winding can be connected to a symmetrical seven-phase voltage with different phase timing sequences: ABCDEFG, ACEGBDF, or ADGCFBE. In the study, the vector analysis methods, methods of expanding in a Fourier series, and grapho-analytical function analysis methods were used. The results of theoretical studies are confirmed by experimental data obtained on a prototype seven-phase engine sample. It has been found that with the phase timing sequence ACEGBDF of the seven-phase winding ABCDEFG, the operating space-time vector is the vector of the fifth harmonic component, which rotates in the opposite direction in comparison with the space-time vector that is produced with the phase timing sequence ABCDEFG. With this phase timing sequence, it is only the fifth phase voltage time harmonic component that produces the space-time vector. With the time sequence ADGCFBE, it is only the third time harmonic component that produces the resulting space-time voltage vector. By changing, using the converter control algorithm, the phase sequence, it is possible to obtain three values of the field rotation frequency with the same converter frequency value. These properties broaden the seven-phase machine control capabilities, a circumstance that can be used in implementing modern electric traction drives.

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