Mathematics

In this article, for positive integers n?�m?? , the parameter spaces for the isomorphism classes of the generic point arrangements of cardinality n , and the antipodal point arrangements of cardinality 2n in the Eulidean space R m are described using the space of totally nonzero Grassmannian G r tnz mn (R) . A stratification S tnz mn (R) of the totally nonzero Grassmannian G r tnz mn (R) is mentioned and the parameter spaces are respectively expressed as quotients of the space S tnz mn (R) of strata under suitable actions of the symmetric group S n and the semidirect product group ( R ??) n ??S n . The cardinalities of the space S tnz mn (R) of strata and of the parameter spaces S n ??S tnz mn (R),(( R ??) n ??S n )??S tnz mn (R) are enumerated in dimension m=2 . Interestingly enough, the enumerated value of the isomorphism classes of the generic point arrangements in the Euclidean plane is expressed in terms of the number theoretic Euler-totient function. The analogous enumeration questions are still open in higher dimensions for m?? .

Let G=(V,E) be a simple undirected and connected graph on n vertices. The Graovac--Ghorbani index of a graph G is defined as AB C GG (G)= ∑ uv∈E(G) n u + n v −2 n u n v − − − − − − − − − − √ , where n u is the number of vertices closer to vertex u than vertex v of the edge uv∈E(G) and n v is defined analogously. It is well-known that all bicyclic graphs with no pendant vertices are composed by three families of graphs, which we denote by B n = B 1 (n)∪ B 2 (n)∪ B 3 (n). In this paper, we give an lower bound to the AB C GG index for all graphs in B 1 (n) and prove it is sharp by presenting its extremal graphs. Additionally, we conjecture a sharp lower bound to the AB C GG index for all graphs in B n .

We use the Ramanujan's master theorem to evaluate the integral ∫ ∞ 0 x l−1 (1+x ) m+1 log n (1+x)dx in terms of the digamma function, the gamma function, and the Hurwitz zeta function.

Using elementary linear algebra, this paper clarifies and proves some concepts about a recently introduced octonion-like associative division algebra over R. The octonion-like algebra is the even subalgebra of Clifford algebra Cl_4,0(R), which is isomorphic to Cl_0,3(R) and to the split-biquaternion algebra. For two seminorms described in the paper (which differ from the norm used in the original papers on the octonion-like algebra), it is shown that the octonion-like algebra is a seminormed algebra over R with no zero divisors when using one of the two seminorms. Moreover, additional results related to the computation of inverse numbers in the octonion-like algebra are introduced in the paper, confirming that the octonion-like algebra is a division algebra over R as long as the two seminorms are non-zero. Additional results on normalization of octonion-like numbers and some involutions are also presented. The elementary linear algebra descriptions used in the paper also allow straightforward software implementations of the octonion-like algebra.

In this article we combinatorially describe the triangles that are present in two types of line arrangements, those which have global cyclicity and those which are infinity type line arrangements. A combinatorial nomenclature has been described for both the types and some properties of the nomenclature have been proved. Later using the nomenclature we describe the triangles present in both types of line arrangements in Theorems A,B . We also prove that the set of triangles uniquely determine, in a certain precise sense, the line arrangements with global cyclicity and not the infinity type line arrangements where counter examples have been provided. In Theorem 9.1 , given a nomenclature, we characterize when a particular line symbol in the nomenclature is a line at infinity for the arrangement determined by the nomenclature.

The cluster structures that can be observed in the first few level sets of the Collatz tree are maintained through all its levels, provided that the orbit steadiness ∏ k∈R(n) k≡4 (mod 6) k−1 k of the elements n of the Collatz tree is suitably bounded from below, where R(n) denotes the Collatz orbit of n .

In the first part of this article, we obtain a linear system whose the solution solves the time-independent incompressible Navier-Stokes system for the special case in which the external forces vector is a gradient. In a second step we develop approximate solutions, also for the time independent incompressible Navier-Stokes system, through the generalized method of lines. We recall that for such a method, the domain of the partial differential equation in question is discretized in lines and the concerning solution is written on these lines as functions of the boundary conditions and boundary shape. Finally, we emphasize these last main results are established through applications of the Banach fixed point theorem.

We construct a two-parameter complex function η κν :C→C , κ∈(0,∞) , ν∈(0,∞) that we call a holomorphic nonlinear embedding and that is given by a double series which is absolutely and uniformly convergent on compact sets in the entire complex plane. The function η κν converges to the Dirichlet eta function η(s) as κ→∞ . We prove the crucial property that, for sufficiently large κ , the function η κν (s) can be expressed as a linear combination η κν (s)= ∑ ∞ n=0 a n (κ)η(s+2νn) of horizontal shifts of the eta function (where a n (κ)∈R and a 0 =1 ) and that, indeed, we have the inverse formula η(s)= ∑ ∞ n=0 b n (κ) η κν (s+2νn) as well (where the coefficients b n (κ)∈R are obtained from the a n 's recursively). By using these results and the functional relationship of the eta function, η(s)=λ(s)η(1−s) , we sketch a proof of the Riemann hypothesis which, in our setting, is equivalent to the fact that the nontrivial zeros s ∗ = σ ∗ +i t ∗ of η(s) (i.e. those points for which η( s ∗ )=η(1− s ∗ )=0) are all located on the critical line σ ∗ = 1 2 .

We examine what integers are representable as sums of three cubes. We also provide formulas for the number of representations of x 3 + y 3 + z 3 =n under the condition x+y+z=t . Also we show how the problem of three cubes is related to abc− conjecture.

A matrix is well separated if all its Gershgorin circles are away from the unit circle and they are separated from each other. In this article, the region of relative errors in the eigenvalues is obtained as a quadratic oval for non diagonal perturbation of well seperated matrices. Thus giving a computable relative error bound in terms of Gershgorin circle parameters. When the separation is O(n) and the matrix is positive definite, an interlacing theorem for the eigenvalues under perturbation is presented. Further when the separation is O( n 2 ) , condition number of the eigenvector matrix is upper bounded to obtain the region of perturbed eigenvalue. Numerical results show the relation between diagonal entries and the magnitude of the eigenvector entries even when the matrix is not so well separated. We exploit this trend in estimating the Perron vector using power method.