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Let S denote the unit circle on the complex plane and ☆:S2&rarr, S be a continuous binary, associative and cancellative operation. From some already known results, it can be deduced that the semigroup (S,☆) is isomorphic to the group (S,&middot, ), thus, it is a group, where &middot, is the usual multiplication of complex numbers. However, an elementary construction of such isomorphism has not been published so far. We present an elementary construction of all such continuous isomorphisms F from (S,&middot, ) into (S,☆) and obtain, in this way, the following description of operation ☆: x☆y=F(F&minus, 1(x)&middot, F&minus, 1(y)) for x,y&isin, S. We also provide some applications of that result and underline some symmetry issues, which arise between the consequences of it and of the analogous outcome for the real interval and which concern functional equations. In particular, we show how to use the result in the descriptions of the continuous flows and minimal homeomorphisms on S.

A function f analytic in a domain D &isin, C is called p-valent in D, if for every complex number w, the equation f ( z ) = w has at most p roots in D, so that there exists a complex number w 0 such that the equation f ( z ) = w 0 has exactly p roots in D. The aim of this paper is to establish some sufficient conditions for a function analytic in the unit disc D to be p-valent starlike in D or to be at most p-valent in D . Our results are proved mainly by applying Nunokawa&rsquo, s lemmas.

Ebisu and Iwassaki proved that there are three-term relations for 3F2(1) with a group symmetry of order 72. In this paper, we apply some specific three-term relations for 3F2(1) to partially answer the open problem raised by Miller and Paris in 2012. Given a known value 3F2((a,b,x),(c,x+1),1), if f−x is an integer, then we construct an algorithm to obtain 3F2((a,b,f),(c,f+n),1) in an explicit closed form, where n is a positive integer and a,b,c and f are arbitrary complex numbers. We also extend our results to evaluate some specific forms of p+1Fp(1), for any positive integer p≥2.

Symmetry and elementary symmetric functions are main components of the proof of the celebrated Hermite&ndash, Lindemann theorem (about the transcendence of e &alpha, for algebraic values of &alpha, ) which settled the ancient Greek problem of squaring the circle. In this paper, we are interested in similar results, but for powers such as e &gamma, log n . This kind of problem can be posed in the context of arithmetic functions. More precisely, we study the arithmetic nature of the so-called &gamma, th arithmetic zeta function &zeta, &gamma, ( n ) : = n &gamma, ( = e &gamma, log n ), for a positive integer n and a complex number &gamma, Moreover, we raise a conjecture about the exceptional set of &zeta, in the case in which &gamma, is transcendental, and we connect it to the famous Schanuel&rsquo, s conjecture.

We consider a quantum theory based on a Galois field. In this approach infinities cannot exist, the cosmological constant problem does not arise, and one irreducible representation (IR) of the symmetry algebra splits into independent IRs describing a particle an its antiparticle only in the approximation when de Sitter energies are much less than the characteristic of the field. As a consequence, the very notions of particles and antiparticles are only approximate and such additive quantum numbers as the electric, baryon and lepton charges are conserved only in this approximation. There can be no neutral elementary particles and the spin-statistics theorem can be treated simply as a requirement that standard quantum theory should be based on complex numbers., 43 pages, Latex, 1 figure

This paper essentially presents the last and important steps in the study of (practical) solvability of two-dimensional product-type systems of difference equations of the following form z n = α z n - k a w n - l b , w n = β w n - m c z n - s d , n ∈ N 0 , where k , l , m , s ∈ N , a , b , c , d ∈ Z , and where α , β and the initial values are complex numbers. It is devoted to the most complex case which has not been considered so far (the case k = l = s = 1 and m = 3 ). Closed form formulas for solutions to the system are found in all possible cases. The structure of the solutions to the system is considered in detail. The following five cases: (1) b = 0 ; (2) c = 0 ; (3) d = 0 ; (4) a c ≠ 0 ; (5) a = 0 , b c d ≠ 0 , are considered separately. Some of the situations appear for the first time in the literature.

Dispensing with the common property of distributivity and replacing classical trigonometric functions with their l p -counterparts in Euler’s trigonometric representation of complex numbers, classes of l p -complex numbers are introduced and some of their basic properties are proved. The collection of all points that leave the l p -absolute value of each l p -complex number invariant under l p -complex numbers multiplication is shown to be a group of elements that have l p -absolute value one but not the symmetry group.

The Ohtsuka–Vălean sum is extended to evaluate an extensive number of trigonometric and hyperbolic sums and products. The sums are taken over finite and infinite domains defined in terms of the Hurwitz–Lerch zeta function, which can be simplified to composite functions in special cases of integer values of the parameters involved. The results obtained include generalizations of finite and infinite products and sums of tangent, cotangent, hyperbolic tangent and hyperbolic cotangent functions, in certain cases raised to a complex number power.