Algebraic properties of special rings of formal series

JPRM-Vol. 1 (2007), Issue 1, pp. 178 – 185 Open Access Full-Text PDF
Azeem Haider
Abstract: The \(K\)-algebra \(K_{S}[[X]]\) of Newton interpolating series is constructed by means of Newton interpolating polynomials with coefficients in an arbitrary field K (see Section 1) and a sequence S of elements \(K\). In this paper we prove that this algebra is an integral domain if and only if \(S\) is a constant sequence. If K is a non-archimedean valued field we obtain that a \(K\)-subalgebra of convergent series of \(K_{S}[[X]]\) is isomorphic to Tate algebra (see Theorem 3) in one variable and by using this representation we obtain a general proof of a theorem of Strassman (see Corollary 1). In the case of many variables other results can be found in [2].
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JPRM-Vol. 1 (2007), Issue 1, pp. 169 – 177 Open Access Full-Text PDF
S.N.Hosseinimotlagh, M.Roostaie, H.Kazemifard
Abstract: Brajinskii’s equations are the fundamental relations governing the behavior of the plasma produced during a fusion reaction, especially ICF plasma. These equations contains six partial differential coupled together. In this paper we have tried to give analytical solutions to these equations using a one dimensional method. Laplace transform technique is the main tool to do that with an arbitrary boundary and initial conditions for some special cases.
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Algebraic properties of integral functions

JPRM-Vol. 1 (2007), Issue 1, pp. 162 – 168 Open Access Full-Text PDF
S.M. Ali Khan
Abstract: For \(K\) a valued subfield of \(\mathbb{C}_{p}\) with respect to the restriction of the p-adic absolute value | | of \(\mathbb{C}_{p}\) we consider the \(K\)-algebra \(IK[[X]]\) of integral (entire) functions with coefficients in \(K\). If \(K\) is a closed subfield of \(\mathbb{C}_{p}\) we extend some results which are known for subfields of \(C\) (see [3] and [4]). We prove that \(IK[[X]]\) is a Bezout domain and we describe some properties of maximal ideals of \(IK[[X]]\).
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A degenerate hydrodynamic dispersion model

JPRM-Vol. 1 (2007), Issue 1, pp. 140 – 153 Open Access Full-Text PDF
Sergey Sazhenkov
Abstract: A Cauchy problem for a two-dimensional ultra-parabolic model of filtration through a porous ground of a viscous incompressible fluid containing a solute (tracer) is considered. The fluid is driven by the buoyancy force. The phenomenon of molecular diffusion of the tracer into the porous ground is taken into account. The porous ground consists of one dimensional filaments oriented along some smooth non-degenerate vector field. Two cases are distinguished depending on spatial orientation of the filaments, and existence of generalized entropy solutions is proved for the both. In the first case, all filaments are parallel to the buoyancy (gravitational) force and, except for this, the equations of the model have rather general forms. In the second case, the filaments can be nonparallel to the buoyancy force and to each other, in general, but their geometric structure must be genuinely nonlinear. The proofs rely on the method of kinetic equation and the theory of Young measures and H-measures.
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Quality Surface Construction

JPRM-Vol. 1 (2007), Issue 1, pp. 129 – 139 Open Access Full-Text PDF
Cripps R. J
Abstract:  Current surface construction methods in CADCAM use parametric polynomial equations in the form of a NURBS. This representation is ideal for computer-based implementations, allowing efficient interrogation. However, issues exist in constructing and manipulating such surfaces. When constructing a NURBS surface there are difficulties in determining constraints such as parameterisation, tangent magnitudes and twist vectors. Controlling the geometric features like curvature profiles of sectional/longitudinal curves on a NURBS surface is problematical as is joining several such surfaces together. A cause of these difficulties in control is that the control points do not lie on the surface itself. An alternative approach to surface construction is to specify the curvature and construct the surface so that it satisfies the curvature constraints. Since NURBS does not directly allow this, a fundamentally different approach is required. The key is to adopt a point-based approach where the surface is defined by a small number of points lying on the surface. Intermediate points are then constructed using a recursive approach which is defined to ensure that the curvature profile between adjacent points is of a very  high quality. A case study is presented that illustrates the point-based approach.
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On newton interpolating series and their applications

JPRM-Vol. 1 (2007), Issue 1, pp. 120 – 128 Open Access Full-Text PDF
Ghiocel Groza
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).
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Matrix lie rings that contains a one-dimentional lie algebra of semi-simple matrices

JPRM-Vol. 1 (2007), Issue 1, pp. 111 – 119 Open Access Full-Text PDF
Evgenii L. Bashkirov
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.
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