An exploration of the interior or the Mandelbrot Set and the appearance of functions appearing to be similar to modular forms. This provides yet another example of the close interconnection between the structure of the Modular Group SL(2,Z) and fractals. The relationship is demonstrated computationally and visually, and not from first principles; visually, the interior resembles the Weierstrass...

Let R be a commutative ring and S a multiplicatively closed subset of R having no zero divisors. The pair (.R, S) is said to be stable if the ring of fractions of R, S R, defined by S is a ring for which all finitely generated projective modules are free. For a stable pair (R, S) assume 2 is a unit in R and V is a free R-module of dimension > 3. This paper examines the action of a group automor...

Similarly, for a non local ring R (of prime characteristic p), and an ideal I ⊆ R for which l(R/I) is finite, the Hilbert-Kunz function and multiplicity makes sense. Henceforth for such a pair (R, I), we denote the Hilbert-Kunz multiplicity of R with respect to I by HKM(R, I), or by HKM(R) if I happens to be an obvious maximal ideal. Given a pair (X,L), where X is a projective variety over an a...

The precise measurement for irradiance is important the performance evaluations of solar systems, because both beam and diffuse radiation measurements must be known to calculate efficiency a system. One widely used instruments pyranometer with ring. Hence, investigations on shadow band correction factors are necessary. normally calculated by expressions Drummond Burek et al. under isotropic con...

Richard Dedekind's characterization of the real numbers as the system of cuts of rational numbers is by now the standard in almost every mathematical book on analysis or number theory. In the philosophy of mathematics Dedekind is given credit for this achievement, but his more general views are discussed very rarely and only superrcially. For example, Leo Corry, who dedicates a whole chapter of...

In the title compound, C(22)H(16)O, the six-membered ring within the anthrone moiety adopts a shallow boat conformation, with puckering parameters Q = 0.2860 (17) Å, Θ = 99.1 (3)° and Φ = 114.8 (3)°. The dihedral angle between the outer benzene rings is 26.53 (8)°. The mean plane through the anthrone ring system makes a dihedral angle of 38.73 (6)° with the pendant benzene ring. In the crystal,...

Let n = p ν 1 1 · · · p νr r > 1 be an integer. An integer a is called regular (mod n) if there is an integer x such that a 2 x ≡ a (mod n). Let ̺(n) denote the number of regular integers a (mod n) such that 1 ≤ a ≤ n. Here ̺(n) = (φ(p ν 1 1) + 1) · · · (φ(p νr r) + 1), where φ(n) is the Euler function. In this paper we first summarize some basic properties of regular integers (mod n). Then in or...

The designation of the customary restricted three-body disturbing potential Φ as the perturbation Hamiltonian is believed to be the cause of Neptune ring arcs' radial offset between theories and observations. To identify the appropriate perturbation Hamiltonian, the energy integral in the fixed frame of a restricted three-body system, consisting of the central, primary, and test bodies, is reco...

We state several conditions under which comultiplication and weak comultiplication modulesare cyclic and study strong comultiplication modules and comultiplication rings. In particular,we will show that every faithful weak comultiplication module having a maximal submoduleover a reduced ring with a finite indecomposable decomposition is cyclic. Also we show that if M is an strong comultiplicati...