Coincident probabilities and applications in combinatorics

1988 ◽  
Vol 25 (A) ◽  
pp. 185-200 ◽  
Author(s):  
Samuel Karlin
Keyword(s):  
Markov Process ◽  
Individual Component ◽  
Combinatorial Problems ◽  
Particle Systems ◽  
Determinant Formula ◽  
Component Processes ◽  
The Individual ◽  
Multiple Particle ◽  
Inhomogeneous Random Walks ◽  
Strong Markov

For a strong Markov process on the line with continuous paths the Karlin–McGregor determinant formula of coincidence probabilities for multiple particle systems is extended to allow the individual component processes to start at variable times and run for variable durations. The extended formula is applied to a variety of combinatorial problems including counts of non-crossing paths in the plane with variable start and end points, dominance orderings, numbers of dominated majorization orderings, and time-inhomogeneous random walks.

Coincident probabilities and applications in combinatorics

1988 ◽  
Vol 25 (A) ◽  
pp. 185-200 ◽  
Author(s):  
Samuel Karlin
Keyword(s):  
Markov Process ◽  
Individual Component ◽  
Combinatorial Problems ◽  
Particle Systems ◽  
Determinant Formula ◽  
Component Processes ◽  
The Individual ◽  
Multiple Particle ◽  
Inhomogeneous Random Walks ◽  
Strong Markov

For a strong Markov process on the line with continuous paths the Karlin–McGregor determinant formula of coincidence probabilities for multiple particle systems is extended to allow the individual component processes to start at variable times and run for variable durations. The extended formula is applied to a variety of combinatorial problems including counts of non-crossing paths in the plane with variable start and end points, dominance orderings, numbers of dominated majorization orderings, and time-inhomogeneous random walks.

scholarly journals Invariance in ecological pattern

F1000Research ◽  
2019 ◽  
Vol 8 ◽  
pp. 2093 ◽  
Author(s):  
Steven A. Frank ◽  
Jordi Bascompte
Keyword(s):  
Maximum Entropy ◽  
Individual Component ◽  
Local Community ◽  
Neutral Theory ◽  
Simple Derivation ◽  
Shape Pattern ◽  
Ecological Patterns ◽  
Component Processes ◽  
General Answer ◽  
The Individual

Background: The abundance of different species in a community often follows the log series distribution. Other ecological patterns also have simple forms. Why does the complexity and variability of ecological systems reduce to such simplicity? Common answers include maximum entropy, neutrality, and convergent outcome from different underlying biological processes.  Methods: This article proposes a more general answer based on the concept of invariance, the property by which a pattern remains the same after transformation. Invariance has a long tradition in physics. For example, general relativity emphasizes the need for the equations describing the laws of physics to have the same form in all frames of reference.  Results: By bringing this unifying invariance approach into ecology, we show that the log series pattern dominates when the consequences of processes acting on abundance are invariant to the addition or multiplication of abundance by a constant. The lognormal pattern dominates when the processes acting on net species growth rate obey rotational invariance (symmetry) with respect to the summing up of the individual component processes. Conclusions: Recognizing how these invariances connect pattern to process leads to a synthesis of previous approaches. First, invariance provides a simpler and more fundamental maximum entropy derivation of the log series distribution. Second, invariance provides a simple derivation of the key result from neutral theory: the log series at the metacommunity scale and a clearer form of the skewed lognormal at the local community scale. The invariance expressions are easy to understand because they uniquely describe the basic underlying components that shape pattern.

scholarly journals Invariance in ecological pattern

2019 ◽  
Author(s):  
Steven A. Frank ◽  
Jordi Bascompte
Keyword(s):  
Maximum Entropy ◽  
Individual Component ◽  
Local Community ◽  
Neutral Theory ◽  
Simple Derivation ◽  
Shape Pattern ◽  
Ecological Patterns ◽  
Component Processes ◽  
General Answer ◽  
The Individual

The abundance of different species in a community often follows the log series distribution. Other ecological patterns also have simple forms. Why does the complexity and variability of ecological systems reduce to such simplicity? Common answers include maximum entropy, neutrality, and convergent outcome from different underlying biological processes. This article proposes a more general answer based on the concept of invariance, the property by which a pattern remains the same after transformation. Invariance has a long tradition in physics. For example, general relativity emphasizes the need for the equations describing the laws of physics to have the same form in all frames of reference. By bringing this unifying invariance approach into ecology, we show that the log series pattern dominates when the consequences of processes acting on abundance are invariant to the addition or multiplication of abundance by a constant. The lognormal pattern dominates when the processes acting on net species growth rate obey rotational invariance (symmetry) with respect to the summing up of the individual component processes. Recognizing how these invariances connect pattern to process leads to a synthesis of previous approaches. First, invariance provides a simpler and more fundamental maximum entropy derivation of the log series distribution. Second, invariance provides a simple derivation of the key result from neutral theory: the log series at the metacommunity scale and a clearer form of the skewed lognormal at the local community scale. The invariance expressions are easy to understand because they uniquely describe the basic underlying components that shape pattern.

scholarly journals Excision of a strong Markov process

1972 ◽  
Vol 23 (2) ◽  
pp. 114-120 ◽  
Author(s):  
F. B. Knight ◽  
A. O. Pittenger
Keyword(s):  
Markov Process ◽  
Strong Markov Process ◽  
Strong Markov

scholarly journals CONSTRUCTION OF ELLIPTIC DIFFUSIONS WITH REFLECTING BOUNDARY CONDITION AND AN APPLICATION TO CONTINUOUS N-PARTICLE SYSTEMS WITH SINGULAR INTERACTIONS

2008 ◽  
Vol 51 (2) ◽  
pp. 337-362 ◽  
Author(s):  
Torben Fattler ◽  
Martin Grothaus
Keyword(s):  
Boundary Condition ◽  
Markov Process ◽  
Diffusion Process ◽  
Lebesgue Measure ◽  
Dirichlet Form ◽  
Measure Zero ◽  
Particle Systems ◽  
Lebesgue Measure Zero ◽  
Compact Set ◽  
Singular Interactions

AbstractWe give a Dirichlet form approach for the construction and analysis of elliptic diffusions in $\bar{\varOmega}\subset\mathbb{R}^n$ with reflecting boundary condition. The problem is formulated in an $L^2$-setting with respect to a reference measure $\mu$ on $\bar{\varOmega}$ having an integrable, $\mathrm{d} x$-almost everywhere (a.e.) positive density $\varrho$ with respect to the Lebesgue measure. The symmetric Dirichlet forms $(\mathcal{E}^{\varrho,a},D(\mathcal{E}^{\varrho,a}))$ we consider are the closure of the symmetric bilinear forms\begin{gather*} \mathcal{E}^{\varrho,a}(f,g)=\sum_{i,j=1}^n\int_{\varOmega}\partial_ifa_{ij} \partial_jg\,\mathrm{d}\mu,\quad f,g\in\mathcal{D}, \\ \mathcal{D}=\{f\in C(\bar{\varOmega})\mid f\in W^{1,1}_{\mathrm{loc}}(\varOmega),\ \mathcal{E}^{\varrho,a}(f,f)\lt\infty\}, \end{gather*}in $L^2(\bar{\varOmega},\mu)$, where $a$ is a symmetric, elliptic, $n\times n$-matrix-valued measurable function on $\bar{\varOmega}$. Assuming that $\varOmega$ is an open, relatively compact set with boundary $\partial\varOmega$ of Lebesgue measure zero and that $\varrho$ satisfies the Hamza condition, we can show that $(\mathcal{E}^{\varrho,a},D(\mathcal{E}^{\varrho,a}))$ is a local, quasi-regular Dirichlet form. Hence, it has an associated self-adjoint generator $(L^{\varrho,a},D(L^{\varrho,a}))$ and diffusion process $\bm{M}^{\varrho,a}$ (i.e. an associated strong Markov process with continuous sample paths). Furthermore, since $1\in D(\mathcal{E}^{\varrho,a})$ (due to the Neumann boundary condition) and $\mathcal{E}^{\varrho,a}(1,1)=0$, we obtain a conservative process $\bm{M}^{\varrho,a}$ (i.e. $\bm{M}^{\varrho,a}$ has infinite lifetime). Additionally, assuming that $\sqrt{\varrho}\in W^{1,2}(\varOmega)\cap C(\bar{\varOmega})$ or that $\varrho$ is bounded, $\varOmega$ is convex and $\{\varrho=0\}$ has codimension at least 2, we can show that the set $\{\varrho=0\}$ has $\mathcal{E}^{\varrho,a}$-capacity zero. Therefore, in this case we can even construct an associated conservative diffusion process in $\{\varrho>0\}$. This is essential for our application to continuous $N$-particle systems with singular interactions. Note that for the construction of the self-adjoint generator $(L^{\varrho,a},D(L^{\varrho,a}))$ and the Markov process $\bm{M}^{\varrho,a}$ we do not need to assume any differentiability condition on $\varrho$ and $a$. We obtain the following explicit representation of the generator for $\sqrt{\varrho}\in W^{1,2}(\varOmega)$ and $a\in W^{1,\infty}(\varOmega)$:$$ L^{\varrho,a}=\sum_{i,j=1}^n\partial_i(a_{ij}\partial_j)+\partial_i(\log\varrho)a_{ij}\partial_j. $$Note that the drift term can be singular, because we allow $\varrho$ to be zero on a set of Lebesgue measure zero. Our assumptions in this paper even allow a drift that is not integrable with respect to the Lebesgue measure.

scholarly journals Parametrix construction for certain Lévy-type processes

2015 ◽  
Vol 0 (0) ◽  
Author(s):  
Victoria Knopova ◽  
Alexei Kulik
Keyword(s):  
Markov Process ◽  
Probability Density ◽  
Lower Bounds ◽  
Transition Probability ◽  
Local Operator ◽  
Upper And Lower Bounds ◽  
Transition Probability Density ◽  
Strong Markov Process ◽  
Non Local ◽  
Strong Markov

AbstractIn this paper, we show that a non-local operator of certain type extends to the generator of a strong Markov process, admitting the transition probability density. For this transition probability density we construct the intrinsic upper and lower bounds, and prove some smoothness properties. Some examples are provided.

scholarly journals The Composition of Cigarette Smoke. An Historical Perspective of Several Polycyclic Aromatic Hydrocarbons

2009 ◽  
Vol 23 (6) ◽  
pp. 384-410 ◽  
Author(s):  
A Rodgman ◽  
LC Cook
Keyword(s):  
Polycyclic Aromatic Hydrocarbons ◽  
Melting Point ◽  
Cigarette Smoke ◽  
Aromatic Hydrocarbons ◽  
Tobacco Smoke ◽  
Individual Component ◽  
Complex Mixtures ◽  
Polycyclic Aromatic ◽  
The Individual ◽  
Cigarette Mainstream Smoke

AbstractBecause of the significant advancements in fractionation, analytical, and characterization technologies since the early 1960s, hundreds of components of complex mixtures have been accurately characterized without the necessity of actually isolating the individual component. This has been particularly true in the case of the complex mixtures tobacco and tobacco smoke. Herein, an historical account of a mid-1950 situation concerning polycyclic aromatic hydrocarbons (PAHs) in cigarette smoke is presented. While the number of PAHs identified in tobacco smoke has escalated from the initial PAH, azulene, identified in 1947 to almost 100 PAHs identified by late 1963 to more than 500 PAHs identified by the late 1970s, the number of PAHs isolated individually and characterized by several of the so-called classical chemical means (melting point, mixture melting point, derivative preparation and properties) in the mid-1950s and since is relatively few, 14 in all. They were among 44 PAHs identified in cigarette mainstream smoke and included the following PAHs ranging from bicyclic to pentacyclic: Acenaphthylene, 1,2-dihydroacenaphthylene, anthracene, benz[a]anthracene, benzo[a]pyrene, chrysene, dibenz[a, h]anthracene, fluoranthene, 9H-fluorene, naphthalene, 1-methylnaphthalene, 2-methylnaphthalene, phenanthrene, and pyrene. One of them, benzo[a]pyrene, was similarly characterized in another study in 1959 by Hoffmann.

Superimposed non-stationary renewal processes

1971 ◽  
Vol 8 (01) ◽  
pp. 184-192 ◽  
Author(s):  
S. Blumenthal ◽  
J. A. Greenwood ◽  
L. Herbach
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Classical Result ◽  
Scale Parameter ◽  
Renewal Processes ◽  
Stationary Case ◽  
Waiting Time Distribution ◽  
Component Processes ◽  
The Individual

For superposition of independent, stationary renewal processes, it is well known that the distribution of waiting time between events for the superimposed process is approximately exponential if the number of processes involved is sufficiently large, (see Khintchine (1960), Ososkov (1956)). We assume that all component processes have the same age t, and we generalize the classical result to show that even for t finite (non-stationary case), the limiting waiting time distribution (as the number of processes increases) is exponential with a scale parameter which depends on t through the average of the individual process renewal densities.

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