Journal of Prime Research in Mathematics

Abstract: Newton interpolating series are constructed by means of Newton interpolating polynomials with coefficients in an arbitrary field \(K\) (see Section 1). If \(K = \mathbb{C}\) is the field of complex numbers with the ordinary absolute value, particular convergent series of this form were used in number theory to prove the transcendence of some values of exponential series (see Theorem 1). Moreover, if \(K = \mathbb{R}\), by means of these series it can be obtained solutions of a multipoint boundary value problem for a linear ordinary differential equation (see Theorem 2). If \(K = \mathbb{C}_{P}\), some particular convergent series of this type (so-called Mahler series) are used to represent all continuous functions from \(\mathbb{Z}_{P}\) in \(\mathbb{C}_{P}\) (see [4]). For an arbitrary field K, with respect to suitable addition and multiplication of two elements the set of Newton interpolating series becomes a commutative K-algebra \(K_{S}[[X]]\) which generalizes the canonical \(K\)-algebra of formal power series. If we consider K a local field, we construct a subalgebra of \(K_{S}[[X]]\), even for more variables, which is a generalization of Tate algebra used in rigid analytic geometry (see Section 3).

Abstract: Let \(k\) be a field and \(\overline{k}\) an algebraic closure of \(k\). Suppose that \(k\)
contains more than five elements if char \(k \neq 2\). Let \(h\) be a one-dimensional subalgebra of the Lie \(k-\)algebra \(sl_{2}\overline{k}\) consisting of semi-simple matrices. In this paper, it is proved that if g is a subring of the Lie ring \(sl_{2}\overline{k}\) containing h, then g is either solvable or there exists a quaternion algebra A over a subfield \(F\) of \(\overline{k}\) such that \(F ⊇ k\) and g is isomorphic to the Lie \(F-\)algebra of all elements in A that are skew-symmetric with respect to a symplectic type involution defined on A.

Abstract: Let \(0=U_{0,n}\leq U_{1,n}\leq … \leq U_{n,n}=1\) be an ordered sample from uniform [0,1] distribution \(D_{in}=U_{i,n}-U_{i-1,n}\), \(i=1,2,..,n\), \(n=1,2,…\) , be their spacings,and let \(f_{1n},…,f_{nn}\) be a set of measurable functions. In this paper theorems on the probabilities of deviations in the moderate zones for \(R_{n}D=f_{1n}(nD_{1n}, … , f_{nn}(nD_{nn}\) are presented. Application of these results to study an intermediate efficiencies of the tests based on statistic \(R_{n}(D)\) are also considered.

Abstract: In this paper an effective quantum particle beam-splitter in the momentum space is realized in the frame of open-loop control scheme. We demonstrate for small interaction time that the splitting effect \(±40\hbar k\) with summarized relative intensity in both main components is about 50 per cent from initial intensity of the atomic beam.

Abstract: In this note we associate to a compact subset of a metric space a subset of natural numbers. We give some interesting properties of this last subset. We also introduce a configuration matrix for a given compact set and propose a conjecture related to this.