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.

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.

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 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 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.

scholarly journals Weakly prudent self-avoiding bridges

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Axel Bacher ◽  
Nicholas Beaton
Keyword(s):  
Generating Function ◽  
Square Lattice ◽  
Growth Constant ◽  
Arbitrary Size ◽  
New Class ◽  
Advantages And Disadvantages ◽  
International Audience ◽  
Self Avoiding Walks

International audience We define and enumerate a new class of self-avoiding walks on the square lattice, which we call <i>weakly prudent bridges</i>. Their definition is inspired by two previously-considered classes of self-avoiding walks, and can be viewed as a combination of those two models. We consider several methods for recursively generating these objects, each with its own advantages and disadvantages, and use these methods to solve the generating function, obtain very long series, and randomly generate walks of arbitrary size. We find that the growth constant of these walks is approximately 2.58, which is larger than that of any previously-solved class of self-avoiding walks.

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.

Influence of Flexible Side-Chains on the Breathing Phase Transition of Pillared Layer MOFs: A Force Field Investigation

2020 ◽  
Author(s):  
Julian Keupp ◽  
Johannes P. Dürholt ◽  
Rochus Schmid
Keyword(s):  
First Principles ◽  
Good Accuracy ◽  
Field Investigation ◽  
Square Lattice ◽  
Side Chain ◽  
Second Step ◽  
Side Chains ◽  
Dynamics Simulations ◽  
Complex Phenomena ◽  
Closed Pore

The prototypical pillared layer MOFs, formed by a square lattice of paddle-<br>wheel units and connected by dinitrogen pillars, can undergo a breathing phase<br>transition by a “wine-rack” type motion of the square lattice. We studied this not<br>yet fully understood behavior using an accurate first principles parameterized force<br>field (MOF-FF) for larger nanocrystallites on the example of Zn 2 (bdc) 2 (dabco) [bdc:<br>benzenedicarboxylate, dabco: (1,4-diazabicyclo[2.2.2]octane)] and found clear indi-<br>cations for an interface between a closed and an open pore phase traveling through<br>the system during the phase transformation [Adv. Theory Simul. 2019, 2, 11]. In<br>conventional simulations in small supercells this mechanism is prevented by periodic<br>boundary conditions (PBC), enforcing a synchronous transformation of the entire<br>crystal. Here, we extend this investigation to pillared layer MOFs with flexible<br>side-chains, attached to the linker. Such functionalized (fu-)MOFs are experimen-<br>tally known to have different properties with the side-chains acting as fixed guest<br>molecules. First, in order to extend the parameterization for such flexible groups,<br>1a new parametrization strategy for MOF-FF had to be developed, using a multi-<br>structure force based fit method. The resulting parametrization for a library of<br>fu-MOFs is then validated with respect to a set of reference systems and shows very<br>good accuracy. In the second step, a series of fu-MOFs with increasing side-chain<br>length is studied with respect to the influence of the side-chains on the breathing<br>behavior. For small supercells in PBC a systematic trend of the closed pore volume<br>with the chain length is observed. However, for a nanocrystallite model a distinct<br>interface between a closed and an open pore phase is visible only for the short chain<br>length, whereas for longer chains the interface broadens and a nearly concerted trans-<br>formation is observed. Only by molecular dynamics simulations using accurate force<br>fields such complex phenomena can be studied on a molecular level.

Circles on lattices and Hadamard matrices

2019 ◽  
pp. 2-9 ◽  
Author(s):  
N. A. Balonin ◽  
M. B. Sergeev ◽  
J. Seberry ◽  
O. I. Sinitsyna
Keyword(s):  
Hadamard Matrices ◽  
Lattice Points ◽  
Practical Significance ◽  
Upper And Lower Bounds ◽  
Problem Size ◽  
Orthogonal Matrices ◽  
Video Information ◽  
Scientific Schools ◽  
Gauss Theorem ◽  
Triangular Numbers

Introduction: The Hadamard conjecture about the existence of Hadamard matrices in all orders multiple of 4, and the Gauss problem about the number of points in a circle are among the most important turning points in the development of mathematics. They both stimulated the development of scientific schools around the world with an immense amount of works. There are substantiations that these scientific problems are deeply connected. The number of Gaussian points (Z3 lattice points) on a spheroid, cone, paraboloid or parabola, along with their location, determines the number and types of Hadamard matrices.Purpose: Specification of the upper and lower bounds for the number of Gaussian points (with odd coordinates) on a spheroid depending on the problem size, in order to specify the Gauss theorem (about the solvability of quadratic problems in triangular numbers by projections onto the Liouville plane) with estimates for the case of Hadamard matrices. Methods: The authors, in addition to their previous ideas about proving the Hadamard conjecture on the base of a one-to-one correspondence between orthogonal matrices and Gaussian points, propose one more way, using the properties of generalized circles on Z3 .Results: It is proved that for a spheroid, the lower bound of all Gaussian points with odd coordinates is equal to the equator radius R, the upper limit of the points located above the equator is equal to the length of this equator L=2πR, and the total number of points is limited to 2L. Due to the spheroid symmetry in the sector with positive coordinates (octant), this gives the values of R/8 and L/4. Thus, the number of Gaussian points with odd coordinates does not exceed the border perimeter and is no less than the relative share of the sector in the total volume of the figure.Practical significance: Hadamard matrices associated with lattice points have a direct practical significance for noise-resistant coding, compression and masking of video information.

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