scholarly journals Lattice points in a random parallelogram

1990 ◽  
Vol 22 (2) ◽  
pp. 484-485 ◽  
Author(s):  
Philip Holgate
Keyword(s):  
Fourier Series ◽  
Lattice Points ◽  
Square Lattice ◽  
Fixed Angle

A Fourier series is obtained for the variance of the number of points lying in a parallelogram thrown randomly on to a square lattice at a fixed angle.

scholarly journals Lattice points in a random parallelogram

1990 ◽  
Vol 22 (02) ◽  
pp. 484-485 ◽  
Author(s):  
Philip Holgate
Keyword(s):  
Fourier Series ◽  
Lattice Points ◽  
Square Lattice ◽  
Fixed Angle

A Fourier series is obtained for the variance of the number of points lying in a parallelogram thrown randomly on to a square lattice at a fixed angle.

scholarly journals Extension of three theorems For Fourier series on the disc to the torus

1986 ◽  
Vol 33 (3) ◽  
pp. 335-350 ◽  
Author(s):  
A. G. Miamee
Keyword(s):  
Fourier Series ◽  
Lattice Points ◽  
The Third

We extend three well-known facts of Fourier series on the disc to Fourier series on the torus, a theorem of Riesz, a theorem of Szegö, and the fact that any function in H1 can be factored as the product of two functions in H2. Here the rôle of negative integers is played by the lattice points in the third quadrant. In earlier extensions of these theorems this rôle was played by half-planes.

Cell Growth Problems

1967 ◽  
Vol 19 ◽  
pp. 851-863 ◽  
Author(s):  
David A. Klarner
Keyword(s):  
Cell Growth ◽  
Equivalence Class ◽  
Equivalence Classes ◽  
Lattice Points ◽  
The Other ◽  
Square Lattice ◽  
Cut Set ◽  
Point Set ◽  
Infinite Set ◽  
Interior Points

The square lattice is the set of all points of the plane whose Cartesian coordinates are integers. A cell of the square lattice is a point-set consisting of the boundary and interior points of a unit square having its vertices at lattice points. An n-omino is a union of n cells which is connected and has no finite cut set.The set of all n-ominoes, Rn is an infinite set for each n; however, we are interested in the elements of two finite sets of equivalence classes, Sn and Tn, which are defined on the elements of Rn as follows: Two elements of Rn belong to the same equivalence class (i) in Sn, or (ii) in Tn, if one can be transformed into the other by (i) a translation or (ii) by a translation, rotation, and reflection of the plane.

scholarly journals Osculating Paths and Oscillating Tableaux

10.37236/731 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Roger E. Behrend
Keyword(s):  
Young Diagram ◽  
Fixed Number ◽  
Lattice Points ◽  
Lattice Paths ◽  
Square Lattice ◽  
Vertex Model ◽  
Young Diagrams ◽  
Primary Interest ◽  
Alternating Sign Matrices ◽  
Positive Integers

The combinatorics of certain tuples of osculating lattice paths is studied, and a relationship with oscillating tableaux is obtained. The paths being considered have fixed start and end points on respectively the lower and right boundaries of a rectangle in the square lattice, each path can take only unit steps rightwards or upwards, and two different paths within a tuple are permitted to share lattice points, but not to cross or share lattice edges. Such path tuples correspond to configurations of the six-vertex model of statistical mechanics with appropriate boundary conditions, and they include cases which correspond to alternating sign matrices. Of primary interest here are path tuples with a fixed number $l$ of vacancies and osculations, where vacancies or osculations are points of the rectangle through which respectively no or two paths pass. It is shown that there exist natural bijections which map each such path tuple $P$ to a pair $(t,\eta)$, where $\eta$ is an oscillating tableau of length $l$ (i.e., a sequence of $l+1$ partitions, starting with the empty partition, in which the Young diagrams of successive partitions differ by a single square), and $t$ is a certain, compatible sequence of $l$ weakly increasing positive integers. Furthermore, each vacancy or osculation of $P$ corresponds to a partition in $\eta$ whose Young diagram is obtained from that of its predecessor by respectively the addition or deletion of a square. These bijections lead to enumeration formulae for tuples of osculating paths involving sums over oscillating tableaux.

Lattice Partitions with a Straight Line

1986 ◽  
Vol 29 (3) ◽  
pp. 287-294 ◽  
Author(s):  
D. R. K. Brownrigg
Keyword(s):  
Explicit Expression ◽  
Digital Filter ◽  
Lattice Points ◽  
Square Lattice ◽  
Straight Line

AbstractIn [1], the solution of a problem of distinct digital filter enumeration was expressed in terms of enumerating partitions of a rectangular set of lattice points with a straight line, under certain restrictions. Here, firstly, an explicit expression is derived for the number of such partitions in that and a more general case. Secondly, the asymptotic ratio of partitions to square of lattice dimensions is derived for a square lattice.

scholarly journals Coverage of a square lattice by an inclined rectangle

1989 ◽  
Vol 21 (3) ◽  
pp. 705-707 ◽  
Author(s):  
Dennis Rosen
Keyword(s):  
Lattice Points ◽  
Square Lattice ◽  
Rational Fraction ◽  
Arbitrary Size

By use of a method of dissection, a formula is derived for the variance in the number of lattice points covered by a rectangle of arbitrary size, lying on a square lattice of unit spacing and inclined to the lattice at an angle of which the tangent is a rational fraction.

scholarly journals Coverage of a square lattice by an inclined rectangle

1989 ◽  
Vol 21 (03) ◽  
pp. 705-707 ◽  
Author(s):  
Dennis Rosen
Keyword(s):  
Lattice Points ◽  
Square Lattice ◽  
Rational Fraction ◽  
Arbitrary Size

By use of a method of dissection, a formula is derived for the variance in the number of lattice points covered by a rectangle of arbitrary size, lying on a square lattice of unit spacing and inclined to the lattice at an angle of which the tangent is a rational fraction.

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