scholarly journals Asymptotic Analysis of the kth Subword Complexity

Entropy ◽  
2020 ◽  
Vol 22 (2) ◽  
pp. 207 ◽  
Author(s):  
Lida Ahmadi ◽  
Mark Daniel Ward
Keyword(s):  
Mellin Transform ◽  
Complex Analysis ◽  
Factorial Moment ◽  
Probability Generating Function ◽  
Exact Expression ◽  
A Value ◽  
Auto Correlation ◽  
Point Analysis ◽  
Subword Complexity ◽  
Binary Strings

Patterns within strings enable us to extract vital information regarding a string’s randomness. Understanding whether a string is random (Showing no to little repetition in patterns) or periodic (showing repetitions in patterns) are described by a value that is called the kth Subword Complexity of the character string. By definition, the kth Subword Complexity is the number of distinct substrings of length k that appear in a given string. In this paper, we evaluate the expected value and the second factorial moment (followed by a corollary on the second moment) of the kth Subword Complexity for the binary strings over memory-less sources. We first take a combinatorial approach to derive a probability generating function for the number of occurrences of patterns in strings of finite length. This enables us to have an exact expression for the two moments in terms of patterns’ auto-correlation and correlation polynomials. We then investigate the asymptotic behavior for values of k = Θ ( log n ) . In the proof, we compare the distribution of the kth Subword Complexity of binary strings to the distribution of distinct prefixes of independent strings stored in a trie. The methodology that we use involves complex analysis, analytical poissonization and depoissonization, the Mellin transform, and saddle point analysis.

LÉVY INDEX AND MULTIFRACTAL CHARACTERISTICS OF RING-LIKE AND JET-LIKE EVENTS IN HIGH ENERGY NUCLEAR COLLISIONS

Fractals ◽  
2010 ◽  
Vol 18 (01) ◽  
pp. 75-86
Author(s):  
DIPAK GHOSH ◽  
ARGHA DEB ◽  
PRABIR KUMAR HALDAR ◽  
SIMA GUPTAROY ◽  
APARNA DHAR (MITRA)
Keyword(s):  
Nuclear Emulsion ◽  
Azimuthal Angle ◽  
Factorial Moment ◽  
High Energy ◽  
Emission Angle ◽  
Two Dimensional ◽  
Nuclear Collisions ◽  
A Value ◽  
Index Analysis ◽  
Angle Space

We perform a Lévy index analysis of relativistic shower particles produced in the interactions of 32 S nuclei at 200A GeV with nuclear emulsion using the results of factorial moment Fq in two-dimensional anisotropic (η - ϕ) space. We carry out the same investigation for target fragments produced in 32 S - AgBr interactions at 200A GeV energy for both events in emission angle space and azimuthal angle space. The analysis reveals that for pions the value of Lévy index μ is 1.491 ± 0.025 for ring-like events and μ ~ 2.004 ± 0.054 for jet-like events which indicates different degree of multifractality. In case of target fragments jet-like events show a value of μ ~ 1.871 ± 0.010 whereas ring-like events yields an unphysical value (μ > 2) the implication of which need to be explored.

scholarly journals Simple derivations of properties of counting processes associated with Markov renewal processes

2005 ◽  
Vol 42 (04) ◽  
pp. 1031-1043 ◽  
Author(s):  
Frank Ball ◽  
Robin K. Milne
Keyword(s):  
Renewal Process ◽  
Factorial Moment ◽  
Probability Generating Function ◽  
Stochastic Modelling ◽  
Markov Renewal Process ◽  
Counting Processes ◽  
Continuous Time Markov Chain ◽  
Factorial Moments ◽  
Asymptotic Expressions ◽  
Markov Renewal Processes

A simple, widely applicable method is described for determining factorial moments of N̂ t , the number of occurrences in (0,t] of some event defined in terms of an underlying Markov renewal process, and asymptotic expressions for these moments as t → ∞. The factorial moment formulae combine to yield an expression for the probability generating function of N̂ t , and thereby further properties of such counts. The method is developed by considering counting processes associated with events that are determined by the states at two successive renewals of a Markov renewal process, for which it both simplifies and generalises existing results. More explicit results are given in the case of an underlying continuous-time Markov chain. The method is used to provide novel, probabilistically illuminating solutions to some problems arising in the stochastic modelling of ion channels.

scholarly journals Simple derivations of properties of counting processes associated with Markov renewal processes

2005 ◽  
Vol 42 (4) ◽  
pp. 1031-1043 ◽  
Author(s):  
Frank Ball ◽  
Robin K. Milne
Keyword(s):  
Renewal Process ◽  
Factorial Moment ◽  
Probability Generating Function ◽  
Stochastic Modelling ◽  
Markov Renewal Process ◽  
Counting Processes ◽  
Continuous Time Markov Chain ◽  
Factorial Moments ◽  
Asymptotic Expressions ◽  
Markov Renewal Processes

A simple, widely applicable method is described for determining factorial moments of N̂t, the number of occurrences in (0,t] of some event defined in terms of an underlying Markov renewal process, and asymptotic expressions for these moments as t → ∞. The factorial moment formulae combine to yield an expression for the probability generating function of N̂t, and thereby further properties of such counts. The method is developed by considering counting processes associated with events that are determined by the states at two successive renewals of a Markov renewal process, for which it both simplifies and generalises existing results. More explicit results are given in the case of an underlying continuous-time Markov chain. The method is used to provide novel, probabilistically illuminating solutions to some problems arising in the stochastic modelling of ion channels.

KIC 5359678: A detached eclipsing binary with starspots

2021 ◽  
Author(s):  
Jiaxin Wang ◽  
Jianning Fu ◽  
Hubiao Niu ◽  
Yang Pan ◽  
Chunqian Li ◽  
...  
Keyword(s):  
Main Sequence ◽  
Function Analysis ◽  
Light Curves ◽  
Main Sequence Stars ◽  
Period Range ◽  
Eclipsing Binary ◽  
A Value ◽  
Auto Correlation ◽  
Auto Correlation Function ◽  
The Masses

Abstract We study the detached eclipsing binary, KIC 5359678, with starspot modulation using the high-quality Kepler photometry and LAMOST spectroscopy. The PHOEBE model, optimal for this binary, reveals that this system is a circular detached binary, composed of two F-type main-sequence stars. The masses and radii of the primary and the secondary are M1 = 1.31 ± 0.05M⊙, R1 = 1.52 ± 0.04R⊙, M2 = 1.12 ± 0.04M⊙, and R2 = 1.05 ± 0.06R⊙, respectively. The age of this binary is estimated to be about 2Gyr, a value much longer than the synchronization timescale of 17.8 Myr. The residuals of light curves show quasi-sinusoidal signals, which could be induced by starspots. We apply auto-correlation function analysis on the out-of-eclipse residuals and find that the spot with rotational period close to the orbital period, while, the decay timescale of starspots is longer than that on the single stars with the same temperature, period range, and rms scatter. A two-starspot model is adopted to fit the signals with two-dip pattern, whose result shows that the longitude decreases with time.

Series expansions of probability generating functions and bounds for the extinction probability of a branching process

1980 ◽  
Vol 17 (04) ◽  
pp. 939-947 ◽  
Author(s):  
D. J. Daley ◽  
Prakash Narayan
Keyword(s):  
Remainder Term ◽  
Branching Process ◽  
Factorial Moment ◽  
Probability Generating Function ◽  
Taylor Series Expansion ◽  
Random Variable ◽  
Extinction Probability ◽  
Series Expansions ◽  
Power Series Expansions ◽  
New Inequalities

In the Taylor series expansion about s = 1 of the probability generating function f(s) of a non-negative integer-valued random variable with finite nth factorial moment the remainder term is proportional to another p.g.f. This leads to simple proofs of other power series expansions for p.g.f.'s, including an inversion formula giving the distribution in terms of the moments (when this can be done). Old and new inequalities for the extinction probability of a branching process are established.

scholarly journals Chemical Kinetics Roots and Methods to Obtain the Probability Distribution Function Evolution of Reactants and Products in Chemical Networks Governed by a Master Equation

Entropy ◽  
2019 ◽  
Vol 21 (2) ◽  
pp. 181 ◽  
Author(s):  
José-Luis Muñoz-Cobo ◽  
Cesar Berna
Keyword(s):  
Probability Distribution ◽  
Master Equation ◽  
Factorial Moment ◽  
Probability Generating Function ◽  
Maximum Entropy Principle ◽  
Point Of View ◽  
Chemical System ◽  
Entropy Principle ◽  
Random Part ◽  
Multinomial Coefficients

In this paper first, we review the physical root bases of chemical reaction networks as a Markov process in multidimensional vector space. Then we study the chemical reactions from a microscopic point of view, to obtain the expression for the propensities for the different reactions that can happen in the network. These chemical propensities, at a given time, depend on the system state at that time, and do not depend on the state at an earlier time indicating that we are dealing with Markov processes. Then the Chemical Master Equation (CME) is deduced for an arbitrary chemical network from a probability balance and it is expressed in terms of the reaction propensities. This CME governs the dynamics of the chemical system. Due to the difficulty to solve this equation two methods are studied, the first one is the probability generating function method or z-transform, which permits to obtain the evolution of the factorial moment of the system with time in an easiest way or after some manipulation the evolution of the polynomial moments. The second method studied is the expansion of the CME in terms of an order parameter (system volume). In this case we study first the expansion of the CME using the propensities obtained previously and splitting the molecular concentration into a deterministic part and a random part. An expression in terms of multinomial coefficients is obtained for the evolution of the probability of the random part. Then we study how to reconstruct the probability distribution from the moments using the maximum entropy principle. Finally, the previous methods are applied to simple chemical networks and the consistency of these methods is studied.

Uniqueness of meromorphic functions sharing a value or small function

2016 ◽  
Vol 66 (4) ◽  
Author(s):  
Nguyen Van Thin ◽  
Ha Tran Phuong
Keyword(s):  
Nevanlinna Theory ◽  
Complex Analysis ◽  
Meromorphic Functions ◽  
Small Function ◽  
Uniqueness Problem ◽  
Differential Polynomials ◽  
A Value

AbstractThe paper concerns interesting problems related to the field of Complex Analysis, in particular Nevanlinna theory of meromorphic functions. The author have studied certain uniqueness problem on differential polynomials of meromorphic functions sharing a value or small function. The results of this paper are generalizations of some problems studied in [BOUSSAF, K.—ESCASSUT, A.—OJEDA, J.:

Series expansions of probability generating functions and bounds for the extinction probability of a branching process

1980 ◽  
Vol 17 (4) ◽  
pp. 939-947 ◽  
Author(s):  
D. J. Daley ◽  
Prakash Narayan
Keyword(s):  
Remainder Term ◽  
Branching Process ◽  
Factorial Moment ◽  
Probability Generating Function ◽  
Taylor Series Expansion ◽  
Random Variable ◽  
Extinction Probability ◽  
Series Expansions ◽  
Power Series Expansions ◽  
New Inequalities

In the Taylor series expansion about s = 1 of the probability generating function f(s) of a non-negative integer-valued random variable with finite nth factorial moment the remainder term is proportional to another p.g.f. This leads to simple proofs of other power series expansions for p.g.f.'s, including an inversion formula giving the distribution in terms of the moments (when this can be done). Old and new inequalities for the extinction probability of a branching process are established.

Théorie Exacte de la Distribution de la Somme des Photoélectrons Emis en L Points

1972 ◽  
Vol 50 (12) ◽  
pp. 1307-1314 ◽  
Author(s):  
Jacques Bures ◽  
Claude Delisle ◽  
Andrzej Zardecki
Keyword(s):  
Physical Meaning ◽  
Factorial Moment ◽  
Moment Generating Function ◽  
Exact Formula ◽  
Coherence Time ◽  
Exact Expression ◽  
Detection Time ◽  
Factorial Moments ◽  
Statistical Behavior ◽  
The Difference

The statistical behavior of photoelectrons emitted at L points of a photocathode in partially coherent Gaussian light is considered. The exact formula for the cumulants and the factorial moment generating function is derived. An exact expression for the photocount distribution and for the factorial moments is obtained in the case when the detection time is much smaller than the coherence time. Two cases of interest which were treated earlier by the same authors are also considered. In particular, the case of two points of detection is of special interest because a physical meaning can be given to the difference of the normalized eigenvalues. Finally, both the exact and approximate theories are compared with the experimental photocount distribution for four different geometrical arrangements.

Observations of the Thermal Decomposition of Brucite

1968 ◽  
Vol 26 ◽  
pp. 446-447
Author(s):  
P. L. Burnett ◽  
W. R. Mitchell ◽  
C. L. Houck
Keyword(s):  
Thermal Decomposition ◽  
Magnesium Oxide ◽  
Basal Plane ◽  
Magnesium Ions ◽  
A Value ◽  
Reaction Proceeds

Natural Brucite (Mg(OH)2) decomposes on heating to form magnesium oxide (MgO) having its cubic ﹛110﹜ and ﹛111﹜ planes respectively parallel to the prism and basal planes of the hexagonal brucite lattice. Although the crystal-lographic relation between the parent brucite crystal and the resulting mag-nesium oxide crystallites is well known, the exact mechanism by which the reaction proceeds is still a matter of controversy. Goodman described the decomposition as an initial shrinkage in the brucite basal plane allowing magnesium ions to shift their original sites to the required magnesium oxide positions followed by a collapse of the planes along the original <0001> direction of the brucite crystal. He noted that the (110) diffraction spots of brucite immediately shifted to the positions required for the (220) reflections of magnesium oxide. Gordon observed separate diffraction spots for the (110) brucite and (220) magnesium oxide planes. The positions of the (110) and (100) brucite never changed but only diminished in intensity while the (220) planes of magnesium shifted from a value larger than the listed ASTM d spacing to the predicted value as the decomposition progressed.

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