Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of first-order arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and ...

Recent results on the reflection of waves from ridged mirrors are discussed. Numerical calculations, analytical estimates and the direct measurements of coefficient of specular reflection of atomic waves from solid-state mirrors are combined. The reflectivity is approximated as an elementary function of period L of ridges, their width , wavenumber K, grazing angle θ and effective depth w of the...

Dirichlet considered a question very similar to the one which inspired Euler’s introduction of the ζfunction: namely, how the primes are distributed modulo m. The simplest question of this type is whether are there infinitely many primes congruent to a modulo m. Obviously there can only be infinitely many primes of this form if a and m are relatively prime. Unfortunately, this problem turned ou...

We prove that if M0 is a model of a simple theory, and p(x) is a complete type of Cantor-Bendixon rank 1 over M0, then p is stationary and regular. As a consequence we obtain another proof that any countable model M0 of a countable complete simple theory T has infinitely many countable elementary extensions up to M0-isomorphism. The latter extends earlier results of the author in the stable cas...

Information Keywords: Newton method; quaternions; radial derivative. Abstract In this paper we propose a version of Newton method for finding zeros of a quaternion function of a quaternion variable, based on the concept of quater-nion radial derivative. Several numerical examples involving elementary functions are presented.

Eggert’s Conjecture says that if R is a finite-dimensional nilpotent commutative algebra over a perfect field F of characteristic p, and R(p) is the image of the p-th power map on R, then dimF R ≥ p dimF R(p). Whether this very elementary statement is true is not known. We examine heuristic evidence for this conjecture, versions of the conjecture that are not limited to positive characteristic ...

Let G be a finite group and H a normal subgroup such that G/H is cyclic. Given a conjugacy class g of G we define its centralizing subgroup to be HCG(g). Let K be such that H ≤ K ≤ G. We show that the G-conjugacy classes contained in K whose centralizing subgroup is K, are equally distributed between the cosets of H in K. The proof of this result is entirely elementary. As an application we fin...

*Correspondence: [email protected] Department of Physics and Basic Sciences, Faculty of Engineering Technology, Al-Balqa Applied University, Amman, Jordan Abstract In this paper, we aim to investigate q-analogues of the natural transform on various elementary functions of special type. We obtain results associated with classes of q-convolution products, Heaviside functions, q-exponential...

Simplifying a matrix of differential operators by elementary row and column operations is quite a fundamental task. If one considers matrices whose entries are ordinary timevarying differential operators with rational function coefficients (or more generally meromorphic functions) then a classical theorem tells you that by elementary row and column operations an analogue to the Smith form can b...

Let p be a prime = 1 (mod 16). We obtain extensions of known congruences involving parameters of bioctic Jacobi sums (modp). These extensions are used to give an elementary proof of an important congruence of Ήasse relating parameters of quartic and octic Jacobi sums (mod p). This proof leads directly to an elementary resolution of sign ambiguities of parameters of certain quartic, octic, and b...