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References [1] Aizicovici , S. , McKibben , M. , Reich , S. 2001 Anti-periodic solutions to nonmonotone evolution equations with discontinuous nonlinearities Nonlinear Anal
] Chen , P. , Fang , H. 2007 Existence of periodic and subharmonic solutions for second-order p -Laplacian difference equations Adv. Difference Equ. 2007 1 – 9 10.1155/2007/42530 . [10
. , Y u , J. S. and G uo , Z. M. , Existence of periodic solutions for fourthorder difference equations , Comput. Math. Appl. , 50 ( 1–2 ) ( 2005 ), 49 – 55 . [8] C
Abstract
It is demonstrated that all stable (non-radioactive) isotopes are formally interrelated as the products of systematically adding alpha particles to four elementary units. The region of stability against radioactive decay is shown to obey a general trend based on number theory and contains the periodic law of the elements as a special case. This general law restricts the number of what may be considered as natural elements to 100 and is based on a proton:neutron ratio that matches the golden ratio, characteristic of biological and crystal growth structures. Different forms of the periodic table inferred at other proton:neutron ratios indicate that the electronic configuration of atoms is variable and may be a function of environmental pressure. Cosmic consequences of this postulate are examined.
Abstract
Let f be a real continuous 2π-periodic function changing its sign in the fixed distinct points y
i
∈ Y:= {y
i
}
i∈ℤ such that for x ∈ [y
i
, y
i−1], f(x) ≧ 0 if i is odd and f(x) ≦ 0 if i is even. Then for each n ≧ N(Y) we construct a trigonometric polynomial P
n
of order ≦ n, changing its sign at the same points y
i
∈ Y as f, and
Abstract
If a finite abelian group is factored into a direct product of its cyclic subsets, then at least one of the factors is periodic. This is a famous result of G. Hajós. We propose to replace the cyclicity of the factors by an abstract property that still guarantees that one of the factors is periodic. Then we present applications of this approach.
Abstract
We study those functions that can be written as a finite sum of periodic integer valued functions. On ℤ we give three different characterizations of these functions. For this we prove that the existence of a real valued periodic decomposition of a ℤ → ℤ function implies the existence of an integer valued periodic decomposition with the same periods. This result depends on the representation of the greatest common divisor of certain polynomials with integer coefficients as a linear combination of the given polynomials where the coefficients also belong to ℤ[x]. We give an example of an ℤ → {0, 1} function that has a bounded real valued periodic decomposition but does not have a bounded integer valued periodic decomposition with the same periods. It follows that the class of bounded ℤ → ℤ functions has the decomposition property as opposed to the class of bounded ℝ → ℤ functions. If the periods are pairwise commensurable or not prescribed, then we get more general results.
In this paper, the joint approximation of a given collection of analytic functions by a collection of shifts of zeta-functions with periodic coefficients is obtained. This is applied to prove the functional independence for these zeta-functions.
A ring R is called periodic if, for every x in R, there exist distinct positive integers m and n such that x m=x n. An element x is called potent if x k=x for some integer k≯1. A ring R is called weakly periodic if every x in R can bewritten in the form x=a+b for some nilpotent element a and some potent element b. A ring R is called weakly periodic-like if every x in R which is not in the center of R can be written in the form x=a+b, where a is nilpotent and b is potent. Our objective is to study the structure of weakly periodic-like rings, with particular emphasis on conditions which yield such rings commutative, or conditions which render the nilpotents N as an ideal of R and R/N as commutative.
