Mohan Tikoo

A note on Pythagorean triple preserving matrices

Several authors have found many Pythagorean triple preserving matrices in recent years. The purpose of this note is to show that all these matrices, and in particular the results published in Deshpandes 2001 paper are special cases of the earlier results obtained by Palmer, Ahuja and Tikoo.

Remainders of H-closed extensions☆

Abstract The remainders of compactifications of Tychonoff spaces have been studied extensively during the last five decades, but, the remainders of H -closed extensions of Hausdorff spaces have not been studied in a concerted manner so far. In this paper, we show that two Hausdorff spaces X and Y have the same remainders in their H -closed extensions if there exists a perfect, irreducible and θ-continuous mapping from X onto Y . Further, a Hausdorff space Z is shown to be the remainder of some H -closed extension of a given space X if and only if there exists a perfect (not necessarily continuous) mapping from σX ⧹ X onto Z . We also show that if X is locally H -closed and |σX⧹X|⩾ℵ 0 , then the set of remainders of H -closed extensions of X contains all separable H -closed spaces. We give examples to illustrate the differences between the sets of remainders of compactifications and remainders of H -closed extensions of a fixed Tychonoff space.

A note on Morley's triangle theorem

In this note, we offer a proof of a variant of Morleys triangle theorem, when the exterior angles of a triangle are trisected. We also offer a generalization of Morleys theorem when angles of an n-gon are n-sected.

Two approaches of studying singularity of projective conics

The singularity of a projective conic can be determined via the associated matrix to the implicit equation of the projective conic. In this expository article, we will first derive a known result for determining the singularity of a projective conic via the associated matrix. Then we will introduce the concepts of μ-basis of the parametric projective conics, and close with the same criteria via μ-basis method.

On strict and simple type extensions

Let (Y,τ) be an extension of a space (X,τ′)⋅p∈Y, let 𝒪yp={W∩X:W∈τ,p∈W}. For U∈τ′, let o(U)={P∈Y:U∈𝒪yp}. In 1964, Banaschweski introduced the strict extension Y#, and the simple extension Y

Integrating geometry in a meaningful way (a point of view)

Some known examples from the literature are used to offer a point of view that could integrate different geometries and blend several topics, which have important applications both in higher mathematics and other disciplines, appropriately with those that are often discussed in a standard High School Curriculum. The aim is to make these topics, more attractive and understandable to students in grades 9‐12 in a way that reinforces logical thinking with visual understanding obtained through computer usage. In this process, students not only learn the materials better, but also gain the ability of making conjectures while exploring the computer screen This could add some spice to teaching and make it more enjoyable for students.