Some aspects of the non-asymptotic behaviour of a two-dimensional invasion process

Normal and abnormal cells are positioned at the vertices of a regular two-dimensional lattice. Abnormal cells divide k times as fast as normal cells. Whenever a cell divides, the daughter is the same type as the parent and replaces an adjacent cell. The Kolmogorov forwards and backwards equations are derived, and then used to obtain bounds for the distribution function of the time when all the abnormal cells are forced from the plane. These bounds are used to comment on the non-asymptotic variance of the number of abnormal cells at a given time and on a method of estimating k.

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The asymptotic behaviour of an invasion process

Journal of Applied Probability ◽

10.2307/3213461 ◽

1977 ◽

Vol 14 (3) ◽

pp. 584-590 ◽

Cited By ~ 17

Author(s):

F. P. Kelly

Keyword(s):

Cell Division ◽

Asymptotic Behaviour ◽

The Other ◽

Adjacent Cell ◽

Rectangular Lattice ◽

Parent Cell ◽

Invasion Process ◽

Black And White ◽

Daughter Cells ◽

A Cell

Black and white cells are positioned at the vertices of a rectangular lattice. When a cell division occurs, the daughter cells are of the same colour as the parent cell; one of them replaces an adjacent cell and the other remains in the position of the parent cell. In one variant of the model it is assumed that whenever a white cell appears at the origin it is transformed into a black cell; apart from this the black and white cells are equally competitive and in particular they divide at the same rate. Initially, only the cell at the origin is black. The asymptotic behaviour of the black clone is investigated.

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The asymptotic behaviour of an invasion process

Journal of Applied Probability ◽

10.1017/s0021900200025821 ◽

1977 ◽

Vol 14 (03) ◽

pp. 584-590 ◽

Cited By ~ 6

Author(s):

F. P. Kelly

Keyword(s):

Cell Division ◽

Asymptotic Behaviour ◽

The Other ◽

Adjacent Cell ◽

Rectangular Lattice ◽

Parent Cell ◽

Invasion Process ◽

Black And White ◽

Daughter Cells ◽

A Cell

Black and white cells are positioned at the vertices of a rectangular lattice. When a cell division occurs, the daughter cells are of the same colour as the parent cell; one of them replaces an adjacent cell and the other remains in the position of the parent cell. In one variant of the model it is assumed that whenever a white cell appears at the origin it is transformed into a black cell; apart from this the black and white cells are equally competitive and in particular they divide at the same rate. Initially, only the cell at the origin is black. The asymptotic behaviour of the black clone is investigated.

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Hyperplasia: The spread of abnormal cells through a plane lattice

Advances in Applied Probability ◽

10.2307/1426160 ◽

1971 ◽

Vol 3 (2) ◽

pp. 210-211 ◽

Cited By ~ 10

Author(s):

Trevor Williams ◽

Rolf Bjerknes

Keyword(s):

Basement Membrane ◽

Basal Cell ◽

Basal Layer ◽

Normal Cells ◽

Daughter Cells ◽

Plane Lattice ◽

Abnormal Cells ◽

A Cell

When a basal cell divides, both daughter cells remain in the basal layer of the epithelium, with one of the neighbouring cells being pushed out to make room. This fact opens the possibility that a cell with a heritable advantage over the normal cells may gradually produce a clone covering more and more of the basal layer. The advantage in question may consist in a faster rate of division than normal, or a more tenacious hold on the basement membrane; we shall limit consideration to the former situation.

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Hyperplasia: The spread of abnormal cells through a plane lattice

Advances in Applied Probability ◽

10.1017/s0001867800037836 ◽

1971 ◽

Vol 3 (02) ◽

pp. 210-211 ◽

Cited By ~ 2

Author(s):

Trevor Williams ◽

Rolf Bjerknes

Keyword(s):

Basement Membrane ◽

Basal Cell ◽

Basal Layer ◽

Normal Cells ◽

Daughter Cells ◽

Plane Lattice ◽

Abnormal Cells ◽

A Cell

When a basal cell divides, both daughter cells remain in the basal layer of the epithelium, with one of the neighbouring cells being pushed out to make room. This fact opens the possibility that a cell with a heritable advantage over the normal cells may gradually produce a clone covering more and more of the basal layer. The advantage in question may consist in a faster rate of division than normal, or a more tenacious hold on the basement membrane; we shall limit consideration to the former situation.

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DISCRETE AND CONTINUUM APPROACHES TO THREE-DIMENSIONAL QUANTUM GRAVITY

Modern Physics Letters A ◽

10.1142/s0217732391004140 ◽

1991 ◽

Vol 06 (39) ◽

pp. 3591-3600 ◽

Cited By ~ 40

Author(s):

HIROSI OOGURI ◽

NAOKI SASAKURA

Keyword(s):

Gauge Theory ◽

Moduli Space ◽

Three Dimensional ◽

Two Dimensional ◽

Analogue Model ◽

Gauge Invariant ◽

Invariant Functions ◽

Dimensional Lattice ◽

Chern Simons ◽

Dimensional Surface

It is shown that, in the three-dimensional lattice gravity defined by Ponzano and Regge, the space of physical states is isomorphic to the space of gauge-invariant functions on the moduli space of flat SU(2) connections over a two-dimensional surface, which gives physical states in the ISO(3) Chern–Simons gauge theory. To prove this, we employ the q-analogue of this model defined by Turaev and Viro as a regularization to sum over states. A recent work by Turaev suggests that the q-analogue model itself may be related to an Euclidean gravity with a cosmological constant proportional to 1/k2, where q=e2πi/(k+2).

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