On the hard-hexagon model and the theory of modular functions

1988 ◽  
Vol 325 (1588) ◽  
pp. 643-702 ◽  
Keyword(s):  
Exact Solution ◽  
Order Parameter ◽  
Modular Group ◽  
Lattice Gas ◽  
Modular Equations ◽  
Modular Functions ◽  
Congruence Subgroups ◽  
Geometrical Properties ◽  
Mathematical Properties ◽  
Hard Hexagon Model

The mathematical properties of the exact solution of the hard-hexagon lattice gas model are investigated by using the Klein-Fricke theory of modular functions. In particular, it is shown that the order-parameter R and the reciprocal activity z' for the model can be expressed in terms of hauptmoduls that are associated with certain congruence subgroups of the full modular group Known modular equations are then used to prove that R (z') is an algebraic function of A connection is established between the singular points of this function and the geometrical properties of the icosahedron .

ARITHMETIC PROPERTIES OF TRACES OF SINGULAR MODULI ON CONGRUENCE SUBGROUPS

2010 ◽  
Vol 06 (08) ◽  
pp. 1755-1768 ◽  
Author(s):  
SOON-YI KANG ◽  
CHANG HEON KIM
Keyword(s):  
Modular Form ◽  
Modular Group ◽  
Fourier Coefficients ◽  
Singular Values ◽  
Modular Functions ◽  
Congruence Subgroups ◽  
Arithmetic Properties ◽  
Holomorphic Modular Form ◽  
Singular Moduli ◽  
Arbitrary Level

After Zagier proved that the traces of singular moduli are Fourier coefficients of a weakly holomorphic modular form, various arithmetic properties of the traces of singular values of modular functions mostly on the full modular group have been found. The purpose of this paper is to generalize the results for modular functions on congruence subgroups with arbitrary level.

scholarly journals Exact solutions for the total variation denoising problem of piecewise constant images in dimension one

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Riccardo Cristoferi
Keyword(s):  
Exact Solution ◽  
Exact Solutions ◽  
Total Variation ◽  
Piecewise Constant ◽  
Geometrical Properties ◽  
Total Variation Denoising ◽  
Fidelity Term ◽  
Dimension One

AbstractA method for obtaining the exact solution for the total variation denoising problem of piecewise constant images in dimension one is presented. The validity of the algorithm relies on some results concerning the behavior of the solution when the parameter λ in front of the fidelity term varies. Albeit some of them are well-known in the community, here they are proved with simple techniques based on qualitative geometrical properties of the solutions.

scholarly journals Water ow on vegetated hill. 1D shallow water equation type model

2015 ◽  
Vol 23 (3) ◽  
pp. 83-96 ◽  
Author(s):  
Stelian Ion ◽  
Dorin Marinescu ◽  
Anca Veronica Ion ◽  
Stefan Gicu Cruceanu ◽  
Virgil Iordache
Keyword(s):  
Mathematical Model ◽  
Shallow Water ◽  
Numerical Approximation ◽  
Approximation Scheme ◽  
Shallow Water Equation ◽  
Simplified Model ◽  
Type Model ◽  
Curvature Radius ◽  
Geometrical Properties ◽  
Mathematical Properties

Abstract A mathematical model for the water ow on a hill covered by variable distributed vegetation is proposed in this article. The model takes into account the variation of the geometrical properties of the terrain surface, but it assumes that the surface exhibits large curvature radius. After describing some theoretical properties for this model, we introduce a simplified model and a well-balanced numerical approximation scheme for it. Some mathematical properties with physical relevance are discussed and finally, some numerical results are presented.

scholarly journals Multiplier systems for Hilbert's and Siegel's modular groups

1985 ◽  
Vol 27 ◽  
pp. 57-80 ◽  
Author(s):  
Karl-Bernhard Gundlach
Keyword(s):  
Modular Forms ◽  
Functional Equations ◽  
Modular Group ◽  
Meromorphic Functions ◽  
Invariance Property ◽  
Modular Functions ◽  
Multiplier System ◽  
Modular Groups ◽  
Image Position ◽  
Totally Real

The classical generalizations (already investigated in the second half of last century) of the modular group SL(2, ℤ) are the groups ГK = SL(2, o)(o the principal order of a totally real number field K, [K:ℚ]=n), operating, originally, on a product of n upper half-planes or, for n=2, on the product 1×− of an upper and a lower half-plane by(where v(i), for v∈K, denotes the jth conjugate of v), and Гn = Sp(n, ℤ), operating on n={Z∣Z=X+iY∈ℂ(n,n),tZ=Z, Y>0} byNowadays ГK is called Hilbert's modular group of K and Гn Siegel's modular group of degree (or genus) n. For n=1 we have Гℚ=Г1= SL(2, ℤ). The functions corresponding to modular forms and modular functions for SL(2, ℤ) and its subgroups are holomorphic (or meromorphic) functions with an invariance property of the formJ(L, t) for fixed L (or J(M, Z) for fixed M) denoting a holomorphic function without zeros on ) (or on n). A function J;, defined on ℤK×or ℤn×n to be able to appear in (1.3) with f≢0, has to satisfy certain functional equations (see below, (2.3)–(2.5) for ГK, (5.7)–(5.9) for Гn) and is called an automorphic factor (AF) then. In close analogy to the case n=1, mainly AFs of the following kind have been used:with a complex number r, the weight of J, and complex numbers v(L), v(M). AFs of this kind are called classical automorphic factors (CAP) in the sequel. If r∉ℤ, the values of the function v on ГK (or Гn) depend on the branch of (…)r. For a fixed choice of the branch (for each L∈ГK or M∈Гn) the functional equations for J, by (1.4), (1.5), correspond to functional equations for v. A function v satisfying those equations is called a multiplier system (MS) of weight r for ГK (or Гn).

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