Papers by Ibtesam Bajunaid
Journal of Mathematical Analysis and Applications, 2011
Let T be a tree rooted at e endowed with a nearest-neighbor transition probability that yields a ... more Let T be a tree rooted at e endowed with a nearest-neighbor transition probability that yields a recurrent random walk. We show that there exists a function K biharmonic off e whose Laplacian has potential theoretic importance and, in addition, has the following property: Any function f on T which is biharmonic outside a finite set has a representation, unique up to addition of a harmonic function, of the form f = βK + B + L, where β a constant, B is a biharmonic function on T , and L is a function, subject to certain normalization conditions, whose Laplacian is constant on all sectors sufficiently far from the root. We obtain a characterization of the functions biharmonic outside a finite set whose Laplacian has 0 flux similar to one that holds for a function biharmonic outside a compact set in R n for n = 2, 3, and 4 proved by Bajunaid and Anandam. Moreover, we extend the definition of flux and, under certain restrictions on the tree, we characterize the functions biharmonic outside a finite set that have finite flux in this extended sense.
Far East J. Appl. Math, 2007
Page 1. Far East J. Appl. Math. 28(1) (2007), 81-94 This paper is available online at http://www.... more Page 1. Far East J. Appl. Math. 28(1) (2007), 81-94 This paper is available online at http://www.pphmj.com :tion Classifica ject Sub s Mathematic 2000 35D05, 35J70, 35R35, 49J40. Keywords and phrases: approximate solution ...
The American Mathematical Monthly, 2005
His early work in algebraic topology and low-dimensional complexes led to an interest in combinat... more His early work in algebraic topology and low-dimensional complexes led to an interest in combinatorial group theory. The study of free groups led naturally to trees. The trees then led him astray to functional analysis, harmonic analysis, integral geometry, and potential theory. In his spare time he is active in politics, and is national chair of the liberal political group Americans for Democratic Action.
Advances in Applied Mathematics, 2009
Graphs, viewed as one-dimensional simplicial complexes, can be given harmonic structures satisfyi... more Graphs, viewed as one-dimensional simplicial complexes, can be given harmonic structures satisfying the Brelot axioms. In this paper, we describe all possible harmonic structures on graphs. We determine those harmonic structures which induce discrete harmonic structures when restricted to the set of vertices. Conversely, given a discrete harmonic structure on the set of vertices and an arbitrarily prescribed harmonic structure on each edge, we determine when these structures yield a harmonic structure on the graph. In addition, we provide a variety of interesting examples.
Advances in Applied Mathematics, 2011
Hiroshima Mathematical Journal, 2008
In this paper, we give a new definition of the flux of a superharmonic function defined outside a... more In this paper, we give a new definition of the flux of a superharmonic function defined outside a compact set in a Brelot space without positive potentials. We also give a new notion of potential in a BS space (that is, a harmonic space without positive potentials containing the constants) which leads to a Riesz decomposition theorem for the class of superharmonic functions that have a harmonic minorant outside a compact set. Furthermore, we give a characterization of the local axiom of proportionality in terms of a global condition on the space.
Hiroshima Mathematical Journal, Jul 1, 2007
Memoirs of the Faculty of Science and Engineering Shimane University Ser B Mathematical Science, 2009
An approximate numerical solution to the Generalized Burgers and the Burgers-Huxley Systems with ... more An approximate numerical solution to the Generalized Burgers and the Burgers-Huxley Systems with two coupled equations using ADM-Padé technique is presented along with an analysis of the convergence of the ADM series solutions to the systems using the properties of the involved differential operators. ADM-Padé technique is a combination of Adomian decomposition method and Padé approximation. The Padé technique is used to enhance the accuracy and speed up the convergence rate of the truncated series solution produced by the ADM. Numerical results are given in graphical form for explicit examples of the two systems.
Advances in Applied Mathematics, 2003
A Brelot space is a connected, locally compact, noncompact Hausdorff space together with the choi... more A Brelot space is a connected, locally compact, noncompact Hausdorff space together with the choice of a sheaf of functions on this space which are called harmonic. We prove that by considering functions on a tree to be functions on the edges as well as on the vertices (instead of just on the vertices), a tree becomes a Brelot space. This leads to many results on the potential theory of trees. By restricting the functions just to the vertices, we obtain several new results on the potential theory of trees considered in the usual sense. We study trees whose nearest-neighbor transition probabilities are defined by both transient and recurrent random walks. Besides the usual case of harmonic functions on trees (the kernel of the Laplace operator), we also consider as "harmonic" the eigenfunctions of the Laplacian relative to a positive eigenvalue showing that these also yield a Brelot structure and creating new classes of functions for the study of potential theory on trees.
Hiroshima Mathematical Journal, 2007
We will give an explicit description of non-commutative extensions over an F p-algebra of the add... more We will give an explicit description of non-commutative extensions over an F p-algebra of the additive group scheme (resp. the additive formal group scheme) by the group scheme (resp. the formal group scheme) which gives a deformation of the additive group scheme to the multiplicative group scheme (resp. the additive formal group scheme to the multiplicative formal group scheme).
Hiroshima Mathematical Journal, 2000
We give a characterization of the hyperbolic Riemannian manifolds R in which for any biharmonic f... more We give a characterization of the hyperbolic Riemannian manifolds R in which for any biharmonic function b outside a compact set, there exists a biharmonic function B in R such that B-b is bounded outside a compact set. 2000 Mathematics subject classification. 31C 12 Key words and phrases. Tapered manifold; Biharmonic extension.
Mathematical Reports
Poisson integrals are used to study the behaviour of biharmonic functions defined outside a compa... more Poisson integrals are used to study the behaviour of biharmonic functions defined outside a compact set in ℝ n , n≥2, and to obtain a few known global properties of biharmonic functions in ℝ n in a more general form. Some of these results are extended to the case of a Riemannian manifold.
Hiroshima Mathematical Journal
Potential theory on a Cartier tree T is developed on the lines of the classical and the axiomatic... more Potential theory on a Cartier tree T is developed on the lines of the classical and the axiomatic theories on harmonic spaces. The harmonic classifications of such trees are considered; the notion of a subordinate structure on T is introduced to consider more generally the potential theory on T associated with the Schrö dinger equation DuðxÞ ¼ QðxÞuðxÞ, QðxÞ b 0 on T; polysuperharmonic functions and poly-potentials on T are defined and a Riesz-Martin representation for positive polysuper-harmonic functions is obtained.
Defining biharmonic spaces in a general way in the axiomatic poten-tial theory of Brelot, we obta... more Defining biharmonic spaces in a general way in the axiomatic poten-tial theory of Brelot, we obtain some extension properties of biharmonic functions defined outside a compact set, and use them to study the removable singularity of bounded biharmonic functions.
Uploads
Papers by Ibtesam Bajunaid