REFLEXIVE POLYTOPES IN DIMENSION 2 AND CERTAIN RELATIONS IN SL2(ℤ)

2002 ◽  
Vol 01 (02) ◽  
pp. 159-173 ◽  
Author(s):  
LUTZ HILLE ◽  
HARALD SKARKE
Keyword(s):  
Lattice Points ◽  
Lattice Isomorphism ◽  
Two Dimensional ◽  
Winding Number ◽  
Reflexive Polytopes ◽  
Dimension 2

It is well known that there are 16 two-dimensional reflexive polytopes up to lattice isomorphism. One can check directly from the list that the number of lattice points on the boundary of such a polytope plus the number of lattice points on the boundary of the dual polytope is always 12. It turns out that two-dimensional reflexive polytopes correspond to certain relations of two generators A and B of SL 2(ℤ) of length 12. We generalize this correspondence to reflexive configurations with winding number w and relations of length 12w.

scholarly journals Winding number selection on merons by Gaussian curvature’s sign

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Ricardo Gabriel Elías ◽  
Nicolás Vidal-Silva ◽  
Vagson L. Carvalho-Santos
Keyword(s):  
Gaussian Curvature ◽  
Domain Walls ◽  
Curved Surfaces ◽  
Two Dimensional ◽  
Winding Number ◽  
Direct Relation ◽  
Topological Properties ◽  
The Core ◽  
Magnetic Surfaces ◽  
The Relationship

Abstract We study the relationship between the winding number of magnetic merons and the Gaussian curvature of two-dimensional magnetic surfaces. We show that positive (negative) Gaussian curvatures privilege merons with positive (negative) winding number. As in the case of unidimensional domain walls, we found that chirality is connected to the polarity of the core. Both effects allow to predict the topological properties of metastable states knowing the geometry of the surface. These features are related with the recently predicted Dzyaloshinskii-Moriya emergent term of curved surfaces. The presented results are at our knowledge the first ones drawing attention about a direct relation between geometric properties of the surfaces and the topology of the hosted solitons.

On entrance times of a homogeneous N-dimensional random walk: an identity

1988 ◽  
Vol 25 (A) ◽  
pp. 321-333 ◽  
Author(s):  
J. W. Cohen
Keyword(s):  
Performance Evaluation ◽  
Random Walk ◽  
Boundary Value Problem ◽  
Random Walks ◽  
Mathematical Problem ◽  
Lattice Points ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Computer Performance ◽  
Hilbert Type

Present developments in computer performance evaluation require detailed analysis of N-dimensional random walks on the set of lattice points in the first 2N-ant of Recent research has shown that for the two-dimensional case the inherent mathematical problem can often be formulated as a boundary value problem of the Riemann–Hilbert type. The paper is concerned with a derivation and analysis of an identity for the first entrance times distributions into the boundary of such random walks. The identity formulates a relation between these distributions and the zero-tuples of the kernel of the random walk; the kernel contains all the information concerning the structure of the random walk in the interior of its stage space. For the two-dimensional case the identity is resolved and explicit expressions for the entrance times distributions are obtained.

Extended continued fractions, recurrence relations and two-dimensional Markov processes

1989 ◽  
Vol 21 (2) ◽  
pp. 357-375 ◽  
Author(s):  
C. E. M. Pearce
Keyword(s):  
Markov Processes ◽  
Continued Fractions ◽  
Recurrence Relations ◽  
Equilibrium Distribution ◽  
Linear Recurrence ◽  
Natural State ◽  
Two Dimensional ◽  
State Spaces ◽  
General Order ◽  
Dimension 2

Connections between Markov processes and continued fractions have long been known (see, for example, Good [8]). However the usefulness of extended continued fractions in such a context appears not to have been explored. In this paper a convergence theorem is established for a class of extended continued fractions and used to provide well-behaved solutions for some general order linear recurrence relations such as arise in connection with the equilibrium distribution of state for some Markov processes whose natural state spaces are of dimension 2. Specific application is made to a multiserver version of a queueing problem studied by Neuts and Ramalhoto [13] and to a model proposed by Cohen [5] for repeated call attempts in teletraffic.

scholarly journals GENERALIZED HILBERT–KUNZ FUNCTION IN GRADED DIMENSION 2

2016 ◽  
Vol 230 ◽  
pp. 1-17 ◽  
Author(s):  
HOLGER BRENNER ◽  
ALESSIO CAMINATA
Keyword(s):  
Rational Number ◽  
Bounded Function ◽  
Closed Field ◽  
Two Dimensional ◽  
Prime Characteristic ◽  
Algebraically Closed Field ◽  
Graded Module ◽  
Dimension 2

We prove that the generalized Hilbert–Kunz function of a graded module $M$ over a two-dimensional standard graded normal $K$-domain over an algebraically closed field $K$ of prime characteristic $p$ has the form $gHK(M,q)=e_{gHK}(M)q^{2}+\unicode[STIX]{x1D6FE}(q)$, with rational generalized Hilbert–Kunz multiplicity $e_{gHK}(M)$ and a bounded function $\unicode[STIX]{x1D6FE}(q)$. Moreover, we prove that if $R$ is a $\mathbb{Z}$-algebra, the limit for $p\rightarrow +\infty$ of the generalized Hilbert–Kunz multiplicity $e_{gHK}^{R_{p}}(M_{p})$ over the fibers $R_{p}$ exists, and it is a rational number.

scholarly journals Harmonic morphisms with one-dimensional fibres on conformally-flat Riemannian manifolds

2008 ◽  
Vol 145 (1) ◽  
pp. 141-151 ◽  
Author(s):  
RADU PANTILIE
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Harmonic Morphisms ◽  
Two Dimensional ◽  
Conformally Flat ◽  
Harmonic Morphism ◽  
One Dimensional ◽  
Dimension 2 ◽  
Real Analytic ◽  
Flat Riemannian Manifold

AbstractWe classify the harmonic morphisms with one-dimensional fibres (1) from real-analytic conformally-flat Riemannian manifolds of dimension at least four (Theorem 3.1), and (2) between conformally-flat Riemannian manifolds of dimensions at least three (Corollaries 3.4 and 3.6).Also, we prove (Proposition 2.5) an integrability result for any real-analytic submersion, from a constant curvature Riemannian manifold of dimensionn+2 to a Riemannian manifold of dimension 2, which can be factorised as ann-harmonic morphism with two-dimensional fibres, to a conformally-flat Riemannian manifold, followed by a horizontally conformal submersion, (n≥4).

Extended continued fractions, recurrence relations and two-dimensional Markov processes

1989 ◽  
Vol 21 (02) ◽  
pp. 357-375 ◽  
Author(s):  
C. E. M. Pearce
Keyword(s):  
Markov Processes ◽  
Continued Fractions ◽  
Recurrence Relations ◽  
Equilibrium Distribution ◽  
Linear Recurrence ◽  
Natural State ◽  
Two Dimensional ◽  
State Spaces ◽  
General Order ◽  
Dimension 2

Connections between Markov processes and continued fractions have long been known (see, for example, Good [8]). However the usefulness of extended continued fractions in such a context appears not to have been explored. In this paper a convergence theorem is established for a class of extended continued fractions and used to provide well-behaved solutions for some general order linear recurrence relations such as arise in connection with the equilibrium distribution of state for some Markov processes whose natural state spaces are of dimension 2. Specific application is made to a multiserver version of a queueing problem studied by Neuts and Ramalhoto [13] and to a model proposed by Cohen [5] for repeated call attempts in teletraffic.

Bloch waves and higher order Laue zone effects in high energy electron diffraction

1976 ◽  
Vol 350 (1662) ◽  
pp. 335-361 ◽  
Keyword(s):  
Reciprocal Lattice ◽  
Wave Dispersion ◽  
High Energy ◽  
Zone Axis ◽  
Lattice Points ◽  
Two Dimensional ◽  
Perturbation Expansions ◽  
Bloch Waves ◽  
Bloch Functions ◽  
Dispersion Surface

The theory of the scattering of fast electrons by a thin crystalline slab is formulated in terms of the Bloch waves of an infinite perfect crystal. In the symmetric Laue case, effects due to the variation of the crystal potential U(r) along the zone axis parallel to the surface normal, are investigated by expanding these Bloch waves in terms of the Bloch functions of a two dimensional potential obtained by averaging U(r) along the zone axis. A high energy and forward scattering approximation is introduced which allows the scattering to be treated as an initial value problem. Perturbation expansions are used to analyse the changes in the dispersion surface and the Bloch waves when the variation of the potential along the zone axis is included. It is found that the most important perturbations are due to interactions associated with reciprocal lattice points in the Laue zones. These lead to hybridization of the Bloch functions of the two dimensional projected potential. A [111] zone axis of silicon at 293 K is studied as an example. In this case, the first order Laue zone leads to the strongest effects which can appear as fine bright lines in reflections in this zone, and also as fine lines in the strong reflexions in the zero Laue zone.The latter are usually dark, but can sometimes be bright. It is shown how it is often possible to separate the effects into the geometry of the intersection of free wave dispersion spheres centred on points in the non-zero Laue zones with the dispersion surface of the projected potential, and the strength of the matrix elements of the deviation of the potential U(r) from the projected potential.

A NEW SYMMETRIC TWO-DIMENSIONAL ALGORITHM

2006 ◽  
Vol 02 (04) ◽  
pp. 489-498
Author(s):  
PEDRO FORTUNY AYUSO ◽  
FRITZ SCHWEIGER
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Dual Graph ◽  
Plane Curve ◽  
Singularity Theory ◽  
Main Topic ◽  
Two Dimensional ◽  
Dimension 2 ◽  
Multidimensional Continued Fraction ◽  
Fraction Algorithm

Continued fractions are deeply related to Singularity Theory, as the computation of the Puiseux exponents of a plane curve from its dual graph clearly shows. Another closely related topic is Euclid's Algorithm for computing the gcd of two integers (see [2] for a detailed overview). In the first section, we describe a subtractive algorithm for computing the gcd of n integers, related to singularities of curves in affine n-space. This gives rise to a multidimensional continued fraction algorithm whose version in dimension 2 is the main topic of the paper.

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