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 Communications in Advanced Mathematical SciencesNumber of Followers: 0     Open Access journal ISSN (Print) 2651-4001 Published by DergiPark  [100 journals]
• GBS Operators of Bivariate Durrmeyer Operators on Simplex

• Authors: Harun ÇİÇEK ; Aydın İZGİ , Mehmet AYHAN
Abstract: We define GBS operators of Durrmeyer operators for bivariate functions on simplex and we give their approximations and rate of their approximations for B-continuous and B-differentiable functions. We show that the GBS type the operators of new Durrmeyer have better approximation than the new operators.
PubDate: Wed, 30 Jun 2021 00:00:00 +030

• Reachability Results in Plane Trees

• Authors: Albert NYARİARO OLOO; Isaac Owino OKOTH
Abstract: In this paper, we obtain closed formulas for the number of reachable vertices in labelled plane trees by paths lengths, sinks, leaf sinks, first children, left most path, non-first children, and non-leaves. Our counting objects are plane trees having their edges oriented from a vertex of lower label towards a vertex of higher label. For each statistic, we obtain the average number of reachable vertices. Moreover, we obtain a counting formula for the number of plane trees on $n$ vertices such that exactly $k\leq n$ are reachable from the root.
PubDate: Wed, 30 Jun 2021 00:00:00 +030

• On Noncrossing and Plane Tree-Like Structures

• Authors: Isaac Owino OKOTH
Abstract: Mathematical trees are connected graphs without cycles, loops and multiple edges. Various trees such as Cayley trees, plane trees, binary trees, $d$-ary trees, noncrossing trees among others have been studied extensively. Tree-like structures such as Husimi graphs and cacti are graphs which posses the conditions for trees if, instead of vertices, we consider their blocks. In this paper, we use generating functions and bijections to find formulas for the number of noncrossing Husimi graphs, noncrossing cacti and noncrossing oriented cacti. We extend the work to obtain formulas for the number of bicoloured noncrossing Husimi graphs, bicoloured noncrossing cacti and bicoloured noncrossing oriented cacti. Finally, we enumerate plane Husimi graphs, plane cacti and plane oriented cacti according to number of blocks, block types and leaves.
PubDate: Wed, 30 Jun 2021 00:00:00 +030

• Eigenvalue Expansion of Nonsymmetric Linear Compact Operators in Hilbert
Space

• Authors: Ludwig KOHAUPT
Abstract: For a symmetric linear compact resp. symmetric densely defined linear operator with compact inverse, expansion theorems in series of eigenvectors areknown. The aim of the present paper is to generalize the known expansion theorems to the case of corresponding operators without the symmetry property. For this, we replace the set of orthonormal eigenvectors in the symmetric case by a set of biorthonormal eigenvectors resp. principal vectors in the case of simple eigenvalues resp. general eigenvalues. The results for the operators without the symmetry property are all new. Furthermore, if the operators are symmetric, the generalized results deliver the known expansions. As an application of the results for nonsymmetric operators with simple eigenvalues, we obtain a known expansion in a series of eigenfunctions for a non-selfadjoint Boundary EigenvalueProblem with ordinary differential operator discussed in a book of Coddington/Levinson. But, we obtain a new result if the eigenvalues are general, thatis, not necessarily simple. In addition, for a differential operator of 2nd order with constant coefficients, the eigenfunctions and Green’s function are explicitly determined. This result is also new, as far as the author is aware.
PubDate: Wed, 30 Jun 2021 00:00:00 +030

• Contact Hamiltonian Description of 1D Frictional Systems

• Authors: Furkan Semih DÜNDAR ; Gülhan AYAR
Abstract: In this paper, we consider contact Hamiltonian description of 1D frictional dynamics with no conserved force. Friction forces that are monomials of velocity, and sum of two monomials are considered. For that purpose, quite general forms of contact Hamiltonians are taken into account. We conjecture that it is impossible to give a contact Hamiltonian description dissipative systems where the friction force is not in the form considered in this paper.
PubDate: Wed, 30 Jun 2021 00:00:00 +030

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