scholarly journals Renewal theory in two dimensions: Basic results

1974 ◽  
Vol 6 (2) ◽  
pp. 376-391 ◽  
Author(s):  
Jeffrey J. Hunter
Keyword(s):  
Renewal Theory ◽  
Two Dimensions ◽  
Standard Theory ◽  
Renewal Processes ◽  
Counting Processes ◽  
Unified Theory ◽  
Two Dimensional ◽  
Renewal Function ◽  
Bivariate Exponential Distribution ◽  
Bivariate Exponential

In this paper a unified theory for studying renewal processes in two dimensions is developed. Bivariate generating functions and bivariate Laplace transforms are the basic tools used in generalizing the standard theory of univariate renewal processes. An example involving a bivariate exponential distribution is presented. This is used to illustrate the general theory and explicit expressions for the two-dimensional renewal density, the two-dimensional renewal function, the correlation between the marginal univariate renewal counting processes, and other related quantities are derived.

scholarly journals Renewal theory in two dimensions: Basic results

1974 ◽  
Vol 6 (02) ◽  
pp. 376-391 ◽  
Author(s):  
Jeffrey J. Hunter
Keyword(s):  
Renewal Theory ◽  
Two Dimensions ◽  
Standard Theory ◽  
Renewal Processes ◽  
Counting Processes ◽  
Unified Theory ◽  
Two Dimensional ◽  
Renewal Function ◽  
Bivariate Exponential Distribution ◽  
Bivariate Exponential

In this paper a unified theory for studying renewal processes in two dimensions is developed. Bivariate generating functions and bivariate Laplace transforms are the basic tools used in generalizing the standard theory of univariate renewal processes. An example involving a bivariate exponential distribution is presented. This is used to illustrate the general theory and explicit expressions for the two-dimensional renewal density, the two-dimensional renewal function, the correlation between the marginal univariate renewal counting processes, and other related quantities are derived.

scholarly journals Renewal theory in two dimensions: asymptotic results

1974 ◽  
Vol 6 (3) ◽  
pp. 546-562 ◽  
Author(s):  
Jeffrey J. Hunter
Keyword(s):  
Joint Distribution ◽  
Counting Process ◽  
Renewal Theory ◽  
Two Dimensions ◽  
Renewal Processes ◽  
Unified Theory ◽  
Two Dimensional ◽  
Renewal Function ◽  
Asymptotic Results ◽  
Renewal Counting Process

In an earlier paper (Renewal theory in two dimensions: Basic results) the author developed a unified theory for the study of bivariate renewal processes. In contrast to this aforementioned work where explicit expressions were obtained, we develop some asymptotic results concerning the joint distribution of the bivariate renewal counting process (Nx(1), Ny(2)), the distribution of the two-dimensional renewal counting process Nx,y and the two-dimensional renewal function &Nx,y. A by-product of the investigation is the study of the distribution and moments of the minimum of two correlated normal random variables. A comprehensive bibliography on multi-dimensional renewal theory is also appended.

scholarly journals Renewal theory in two dimensions: asymptotic results

1974 ◽  
Vol 6 (03) ◽  
pp. 546-562 ◽  
Author(s):  
Jeffrey J. Hunter
Keyword(s):  
Joint Distribution ◽  
Counting Process ◽  
Renewal Theory ◽  
Two Dimensions ◽  
Renewal Processes ◽  
Unified Theory ◽  
Two Dimensional ◽  
Renewal Function ◽  
Asymptotic Results ◽  
Renewal Counting Process

In an earlier paper (Renewal theory in two dimensions: Basic results) the author developed a unified theory for the study of bivariate renewal processes. In contrast to this aforementioned work where explicit expressions were obtained, we develop some asymptotic results concerning the joint distribution of the bivariate renewal counting process (N x(1), N y(2)), the distribution of the two-dimensional renewal counting process N x,y and the two-dimensional renewal function &N x,y. A by-product of the investigation is the study of the distribution and moments of the minimum of two correlated normal random variables. A comprehensive bibliography on multi-dimensional renewal theory is also appended.

Renewal theory in two dimensions: bounds on the renewal function

1977 ◽  
Vol 9 (03) ◽  
pp. 527-541 ◽  
Author(s):  
Jeffrey J. Hunter
Keyword(s):  
Renewal Theory ◽  
Random Variable ◽  
Two Dimensions ◽  
Upper And Lower Bounds ◽  
Renewal Processes ◽  
Renewal Function ◽  
Poisson Random Variable ◽  
Joint Distributions ◽  
The Family ◽  
Fréchet Bounds

In two earlier papers [6], [7] the properties of bivariate renewal processes and their associated two-dimensional renewal functions, H(x, y) were examined. By utilising the Fréchet bounds for joint distributions and the properties of univariate renewal processes, a collection of upper and lower bounds for H(x, y) are constructed. The evaluation of these bounds is carried out for the case of the family of bivariate Poisson processes. An interesting by-product of this investigation leads to a new inequality for the median of a Poisson random variable.

Renewal theory in two dimensions: bounds on the renewal function

1977 ◽  
Vol 9 (3) ◽  
pp. 527-541 ◽  
Author(s):  
Jeffrey J. Hunter
Keyword(s):  
Renewal Theory ◽  
Random Variable ◽  
Two Dimensions ◽  
Upper And Lower Bounds ◽  
Renewal Processes ◽  
Renewal Function ◽  
Poisson Random Variable ◽  
Joint Distributions ◽  
The Family ◽  
Fréchet Bounds

In two earlier papers [6], [7] the properties of bivariate renewal processes and their associated two-dimensional renewal functions, H(x, y) were examined. By utilising the Fréchet bounds for joint distributions and the properties of univariate renewal processes, a collection of upper and lower bounds for H(x, y) are constructed. The evaluation of these bounds is carried out for the case of the family of bivariate Poisson processes. An interesting by-product of this investigation leads to a new inequality for the median of a Poisson random variable.

Urbanicity, City Delegations, and Two-Dimensional Liberalism

2018 ◽  
Author(s):  
Thomas K. Ogorzalek
Keyword(s):  
National Level ◽  
Two Dimensions ◽  
Political Order ◽  
Two Dimensional ◽  
Local Institutions ◽  
Horizontal Integration ◽  
National Politics ◽  
United Front ◽  
Strategies For Success

This theoretical chapter develops the argument that the conditions of cities—large, densely populated, heterogeneous communities—generate distinctive governance demands supporting (1) market interventions and (2) group pluralism. Together, these positions constitute the two dimensions of progressive liberalism. Because of the nature of federalism, such policies are often best pursued at higher levels of government, which means that cities must present a united front in support of city-friendly politics. Such unity is far from assured on the national level, however, because of deep divisions between and within cities that undermine cohesive representation. Strategies for success are enhanced by local institutions of horizontal integration developed to address the governance demands of urbanicity, the effects of which are felt both locally and nationally in the development of cohesive city delegations and a unified urban political order capable of contending with other interests and geographical constituencies in national politics.

scholarly journals Interface Roughening in Two Dimensions

2021 ◽  
Vol 182 (3) ◽  
Author(s):  
Gernot Münster ◽  
Manuel Cañizares Guerrero
Keyword(s):  
Field Theory ◽  
Monte Carlo ◽  
Capillary Wave ◽  
System Size ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Wave Approximation ◽  
Statistical Field ◽  
Statistical Field Theory ◽  
Interface Roughening

AbstractRoughening of interfaces implies the divergence of the interface width w with the system size L. For two-dimensional systems the divergence of $$w^2$$ w 2 is linear in L. In the framework of a detailed capillary wave approximation and of statistical field theory we derive an expression for the asymptotic behaviour of $$w^2$$ w 2 , which differs from results in the literature. It is confirmed by Monte Carlo simulations.

scholarly journals Is contact angle a cause or an effect? – A cautionary tale

2020 ◽  
Vol 146 ◽  
pp. 03004
Author(s):  
Douglas Ruth
Keyword(s):  
Contact Angle ◽  
Solid Surface ◽  
Contact Angles ◽  
Two Dimensions ◽  
Experimental Results ◽  
Flow In Porous Media ◽  
Two Dimensional ◽  
Wide Range ◽  
Fluid Drop ◽  
Surrounding Fluid

The most influential parameter on the behavior of two-component flow in porous media is “wettability”. When wettability is being characterized, the most frequently used parameter is the “contact angle”. When a fluid-drop is placed on a solid surface, in the presence of a second, surrounding fluid, the fluid-fluid surface contacts the solid-surface at an angle that is typically measured through the fluid-drop. If this angle is less than 90°, the fluid in the drop is said to “wet” the surface. If this angle is greater than 90°, the surrounding fluid is said to “wet” the surface. This definition is universally accepted and appears to be scientifically justifiable, at least for a static situation where the solid surface is horizontal. Recently, this concept has been extended to characterize wettability in non-static situations using high-resolution, two-dimensional digital images of multi-component systems. Using simple thought experiments and published experimental results, many of them decades old, it will be demonstrated that contact angles are not primary parameters – their values depend on many other parameters. Using these arguments, it will be demonstrated that contact angles are not the cause of wettability behavior but the effect of wettability behavior and other parameters. The result of this is that the contact angle cannot be used as a primary indicator of wettability except in very restricted situations. Furthermore, it will be demonstrated that even for the simple case of a capillary interface in a vertical tube, attempting to use simply a two-dimensional image to determine the contact angle can result in a wide range of measured values. This observation is consistent with some published experimental results. It follows that contact angles measured in two-dimensions cannot be trusted to provide accurate values and these values should not be used to characterize the wettability of the system.

scholarly journals Calderón problem for Maxwell's equations in two dimensions

2016 ◽  
Vol 24 (3) ◽  
Author(s):  
Oleg Y. Imanuvilov ◽  
Masahiro Yamamoto
Keyword(s):  
Maxwell’S Equations ◽  
Maxwell Equations ◽  
Maxwell's Equations ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Global Uniqueness ◽  
Calderón Problem ◽  
Dirichlet To Neumann Map

AbstractWe prove the global uniqueness in determination of the conductivity, the permeability and the permittivity of the two-dimensional Maxwell equations by the partial Dirichlet-to-Neumann map limited to an arbitrary subboundary.

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