Mathematics

In this paper, we implemented the power series variation iteration method for solving fourth order boundary value integro-differential equations. The method is a mixture of power series approximation method (PSAM) and the variation iteration method (VIM). In this method, the approximate solution is given as a power series. The series solution is made to satisfy the given boundary conditions and hence, is the initial approximation. The accuracy of the method was illustrated using two examples with known analytic solution. All numerical evidences were obtained by using the computer application software Maple 18. Keywords: Power series, Variation iteration method, Approximate solution, Lagrange multiplier, Integro-differential equations, Boundary value problems

In this paper, we proposed a numerical method called the improved Elzaki transform method for solving singular initial value problems in second order ordinary differential equations. The method is used in handling a generalization of this kind of problem. For this purpose, an Elzaki linear operator is defined and evaluated using certain Elzaki transforms properties. The method was experimented for linear and nonlinear singular initial value problems, and the resulting numerical evidences show that the method converges to the exact solution usually in the first iteration. The method requires less computational rigor such that computational, round-off and truncation errors are eliminated. Similarly, results obtained by the improved Elzaki transform method converge favourably to same results obtained using the modified ADM. Also, the standard Elzaki transform method was treated for the same examples considered, and the results obtained were compared with the improved Elzaki transform m...

Delay differential equations are relevant in the modeling of many areas in science and technology. Solving these problems either analytically or numerically is of great significance. This paper used a numerical algorithm called the Variational iteration homotopy perturbation method (VIHPM) in the approximation of the analytic solution of delay differential equations (DDE). This method is an elegant mixture of the Variational iteration method (VIM) and the homotopy perturbation method (HPM). The method is very efficient as no linearization or discretization is required. The rate of convergence of the method was also examined explicitly. Two examples were considered to verify the accuracy and applicability of the method. All computations were implemented with maple 18 software.

In this paper, the physical properties of a coupled nonlinear differential equation of aninviscid flow at different values of Mach angle α, is presented. The governing partial differential equation is reduced using dimensionless variables into a system of coupled nonlinear differential equation, thereby, establishing the physical properties of existence and uniqueness of solutions and analyzing the stability using Liapunov function to obtain an asymptotic value.

We consider an optimal liquidation problem with instantaneous price impact and stochastic resilience for small instantaneous impact factors. Within our modelling framework, the optimal portfolio process converges to the solution of an optimal liquidation problem with general semimartingale controls when the instantaneous impact factor converges to zero. Our results provide a unified framework within which to embed the two most commonly used modelling frameworks in the liquidation literature and provide a microscopic foundation for the use of semimartingale liquidation strategies and the use of portfolio processes of unbounded variation. Our convergence results are based on novel convergence results for BSDEs with singular terminal conditions and novel representation results of BSDEs in terms of uniformly continuous functions of forward processes. AMS Subject Classification: 93E20, 91B70, 60H30.

The aim of this paper is to build a new family of lattices related to some combinatorial extremal sum problems, in particular to a conjecture of Manickam, Mikl\"os and Singhi. We study the fundamentals properties of such lattices and of a particular class of boolean functions defined on them.

We study the quasi-projective variety Bir_d of plane Cremona transformations defined by three polynomials of fixed degree d and its subvariety Bir_d^o where the three polynomials have no common factor. We compute their dimension and the decomposition in irreducible components. We prove that Bir_d is connected for each d and Bir_d^o is connected when d < 7.

In the space of quaternions, we investigate the natural, invariant geometry of the open, unit disc Δ_ and of the open half-space ^+. These two domains are diffeomorphic via a Cayley-type transformation. We first study the geometrical structure of the groups of Möbius transformations of Δ_ and ^+ and identify original ways of representing them in terms of two (isomorphic) groups of matrices with quaternionic entries. We then define the cross-ratio of four quaternions, prove that, when real, it is invariant under the action of the Möbius transformations, and use it to define the analogous of the Poincaré distances on Δ_ and ^+. We easily deduce that there exists no isometry between the quaternionic Poincaré distance of Δ_ and the Kobayashi distance inherited by Δ_ as a domain of C^2, in accordance with a direct consequence of the classification of the non compact, rank 1, symmetric spaces.

The celebrated Schwarz-Pick lemma for the complex unit disk is the basis for the study of hyperbolic geometry in one and in several complex variables. In the present paper, we turn our attention to the quaternionic unit ball B. We prove a version of the Schwarz-Pick lemma for self-maps of B that are slice regular, according to the definition of Gentili and Struppa. The lemma has interesting applications in the fixed-point case, and it generalizes to the case of vanishing higher order derivatives.

The regular fractional transformations of the extended quaternionic space have been recently introduced as variants of the classical linear fractional transformations. These variants have the advantage of being included in the class of slice regular functions, introduced by Gentili and Struppa in 2006, so that they can be studied with the useful tools available in this theory. We first consider their general properties, then focus on the regular M\"obius transformations of the quaternionic unit ball B, comparing the latter with their classical analogs. In particular we study the relation between the regular M\"obius transformations and the Poincar\'e metric of B, which is preserved by the classical M\"obius transformations. Furthermore, we announce a result that is a quaternionic analog of the Schwarz-Pick lemma.

In this paper we investigate the Brolin's theorem over H, the skew field of quaternions. Moreover, considering a quaternionic polynomial p with real coefficients, we focus on the properties of its equilibrium measure, among the others, the mixing property and the Lyapunov exponents of the measure. We prove a central limit theorem and we compute the topological entropy and measurable entropy with respect to the quaternionic equilibrium measure. We prove that they are equal considering both a quaternionic polynomial with real coefficients and a polynomial with coefficients in a slice but not all real. Brolin's theorems for the one slice preserving polynomials and for generic polynomials are also proved.

In this paper we introduce the notion of {\it core} for two specific classes of boolean maps on finite involution posets (which are a generalization of the boolean lattices) and we prove some extension results for such families of boolean maps. Through the properties of the core, we provide a complete characterization of such maps. The main purpose of such abstract results is their application to the study of the compatibility of a particular class of systems of linear inequalities related to a conjecture of Manickam, Mikl\"os and Singhi (\cite{ManSin88}, \cite{ManMik87}), still unsolved and that can be considered dual to the theorem of Erd\"os-Ko-Rado \cite{erd-ko-rad}.

In this paper we study the following type of functions f : QR3 → R3, where QR3 is the quadratic cone of the algebra R3. From the fact that it is possible to write the algebra R3 as a direct sum of quaternions, we get the observation that it is possible to find a clever representation for QR3 . By using this result a slice regular theory was introduced and a Cauchy formula is discussed. Moreover, a detailed study of the zeros is performed. Finally, we find a formula for the determinant of a matrix with entries in QR3 .