arXiv subjects

Soumen Sarkar

Publications and source records attributed to Soumen Sarkar.

At least 37 records · Page 2Linked to original sources

arXiv2023

Blowdown, $k$-wedge and evenness of quasitoric orbifolds

In this paper, we introduce polytopal $k$-wedge construction and blowdown of a simple polytope and inspect the effect on the retraction sequence of a simple polytope due to $k$-wedge construction and blowdown. In relation to this construction, we introduce the $k$-wedge and blowdown of a quasitoric orbifold. We compare the torsions in the integral cohomologies of $k$-wedges and blowdowns of a quasitoric orbifold with the original one. These two constructions provide infinitely many integrally equivariantly formal quasitoric orbifolds from a given one.

math.AT↗

arXiv2023

Invariant circles and phase portraits of cubic vector fields on the sphere

In this paper, we characterize and study dynamical properties of cubic vector fields on the sphere $\mathbb{S}^2 = \{(x, y, z) \in \mathbb{R}^3 ~|~ x^2+y^2+z^2 = 1\}$. We start by classifying all degree three polynomial vector fields on $\mathbb{S}^2$ and determine which of them form Kolmogorov systems. Then, we show that there exist completely integrable cubic vector fields on $\mathbb{S}^2$ and also study the maximum number of various types of invariant circles for homogeneous cubic vector fields on $\mathbb{S}^2$. We find a tight bound in each case. Further, we also discuss phase portraits of certain cubic Kolmogorov vector fields on $\mathbb{S}^2$.

math.DS↗

arXiv2023

Estimation of the error matrix in a linear least square fit to the data from an experiment performed by smartphone photography

Determination of the Young modulus of a metal bar in the form of a cantilever is an old experimental concept. However, we have taken the advantage of modern advanced technology of smartphone camera to find the load depression graph of the cantilever by taking photographs with the smartphone camera. Smartphone photography allows us to find a precise transverse magnification of an object from the size of the real image formed on the sensor of the camera. Image size on the sensor can be obtained with micron level accuracy. From the load depression graph, we have determined the Young modulus of the bar. The sensitive measurements of the depression of the cantilever at its free end by its own weight, have allowed us to determine the density of aluminium. We have added an analysis of the chi squred minimisation technique for determining the parameters and their uncertainities in a linear fit. Starting from the curvature matrix we have made a comprehensive analysis of the error matrix relevant for a two parameter linear fit. Then we have shown how to form the error matrix for the fitted parameters which includes the covariance term between the two correlated parameters, in the context of our specific experiment. We have propagated the errors in the parameters to find the uncertainties in the Young modulus and the density of the bar. We have shown that a precise measurement is possible by smartphone photography.

physics.ed-ph↗

arXiv2023

Quadratic, Homogeneous and Kolmogorov vector fields on $S^1\times S^2$ and $S^2 \times S^1$

In this paper, we consider the following two algebraic hypersurfaces $$S^1\times S^2=\{(x_1,x_2,x_3,x_4)\in \mathbb{R}^4:(x_1^2+x_2^2-a^2)^2 + x_3^2 + x_4^2 -1=0;~ a>1\}$$ and $$S^2\times S^1=\{(x_1,x_2,x_3,x_4)\in \mathbb{R}^4:(x_1^2+x_2^2+x_3^2-b^2)^2+x_4^2-1=0;~ b>1\}$$ embedded in $\mathbb{R}^4$. We study polynomial vector fields in $\mathbb{R}^4$ separately, having $S^1\times S^2$ and $S^2\times S^1$ invariant by their flows. We characterize all linear, quadratic, cubic Kolmogorov and homogeneous vector fields on $S^1\times S^2$ and $S^2\times S^1$. We construct some first integrals of these vector fields and find which of the vector fields are Hamiltonian. We give upper bounds for the number of the invariant meridian and parallel hyperplanes of these vector fields. In addition, we have shown that the upper bounds are sharp in many cases.

math.DS↗

arXiv2023

Various Topological Complexities of Small Covers and Real Bott Manifolds

In this paper, we compute the LS-category and equivariant LS-category of a small cover and its real moment angle manifold. We calculate a tight lower bound for the topological complexity of many small covers over a product of simplices. Then we compute symmetric topological complexity of several small covers over a product of simplices. We calculate the LS one-category of real Bott manifolds and infinitely many small covers.

math.AT↗

arXiv2023

Integral equivariant $K$-theory and cobordism ring of simplicial GKM orbifold complexes

In this paper, we define `simplicial GKM orbifold complexes' and study some of their topological properties. We introduce the concept of filtration of regular graphs and `simplicial graph complexes', which have close relations with simplicial GKM orbifold complexes. We discuss the necessary conditions to confirm an invariant $q$-CW complex structure on a simplicial GKM orbifold complex. We introduce `buildable' and `divisive' simplicial GKM orbifold complexes. We show that a buildable simplicial GKM orbifold complex is equivariantly formal, and a divisive simplicial GKM orbifold complex is integrally equivariantly formal. We give a combinatorial description of the integral equivariant cohomology ring of certain simplicial GKM orbifold complexes. We prove the Thom isomorphism theorem for orbifold $G$-vector bundles for equivariant cohomology and equivariant $K$-theory with rational coefficients. We extend the main result of Harada-Henriques-Holm (2005) to the category of $G$-spaces equipped with `singular invariant stratification'. We compute the integral equivariant cohomology ring, equivariant $K$-theory ring and equivariant cobordism ring of divisive simplicial GKM orbifold complexes. We describe a basis of the integral generalized equivariant cohomology of a divisive simplicial GKM orbifold complex.

math.AT↗

arXiv2023

LS-category and topological complexity of several families of fibre bundles

In this paper, we study upper bounds for the topological complexity of the total spaces of some classes of fibre bundles. We calculate a tight upper bound for the topological complexity of an $n$-dimensional Klein bottle. We also compute the exact value of the topological complexity of $3$-dimensional Klein bottle. We describe the cohomology rings of several classes of generalized projective product spaces with $\mathbb{Z}_2$-coefficients. Then we study the LS-category and topological complexity of infinite families of generalized projective product spaces. We reckon the exact value of these invariants in many specific cases. We calculate the equivariant LS-category and equivariant topological complexity of several product spaces equipped with $\mathbb{Z}_2$-action.

math.AT↗

arXiv2022

On cohomology of quasitoric manifolds over a vertex cut of a finite product of simplices

In this paper, we classify the characteristic matrices associated to quasitoric manifolds over a vertex cut of a finite product of simplices. We discuss the integral cohomology rings of these quasitoric manifolds with possibly minimal generators and show several relations among the product of these generators. We classify integral cohomology rings (up to isomorphism as graded rings) of the quasitoric manifolds over the vertex cut of a finite product of simplices.

math.AT↗

arXiv2022

Use of a smartphone camera to determine the focal length of a thin lens by finding the transverse magnification of the virtual image of an object

In this work we have determined the focal length of a concave lens by photographing the virtual image of an object by a smartphone camera. We have similarly determined the focal length of a convex lens by forming a virtual image of an object keeping it within the focal distance from the lens. When a photograph is taken by a smartphone, the transverse width of the image on the sensor of the camera in pixels can be read off by software available freely from the internet. By taking a photograph of the virtual image from two positions of the camera separated by a distance along the line of sight of the camera, we have determined the transverse width of the virtual image. From this we find the focal lengths of the lenses knowing the transverse width and the distance of the object from the lenses.

cs.CV↗

arXiv2022

Resolution of singularities of toric orbifolds and equivariant cobordism of contact toric manifolds

Toric orbifolds are a generalization of simplicial projective toric varieties. In this paper, we show that there is a resolution of singularities of a toric orbifold. In a different category, the class of quasi-contact toric manifolds contains the class of good contact toric manifolds. We prove that a quasi-contact toric manifold is equivariantly a boundary. Moreover, we conclude that good contact toric manifolds and generalized lens spaces are equivariantly boundaries.

math.AT↗

arXiv2022

Determination of the transverse width and distance of an object with a smartphone camera

A smartphone is a powerful learning aid in the hands of a large section of students around the world. The camera of the phone can be used for several learning purposes apart from its obvious purpose of photographing. If the focal length of the lens of the camera can be determined, several experiments in optics can be performed with it. In some recent works, the method of determination of the focal length has been discussed. When a real image of an object is formed by a lens of known focal length, one can determine either the distance or the transverse magnification of the object if the other is known. In this work we have shown that we can determine both the transverse size and the distance of an object, by photographing it from two positions, separated by a distance along the line of sight of the camera.

physics.ed-ph↗

arXiv2022

Integral generalized equivariant cohomologies of weighted Grassmann orbifolds

We introduce a new definition of weighted Grassmann orbifolds. We study their several invariant $q$-cell structures and the orbifold singularities on these $q$-cells. We discuss when the integral cohomology of a weighted Grassmann orbifold has no $p$-torsion. We compute the equivariant $K$-theory ring of weighted Grassmann orbifolds with rational coefficients. We introduce divisive weighted Grassmann orbifolds and show that they have invariant cell structures. We calculate the equivariant cohomology ring, equivariant $K$-theory ring and equivariant cobordism ring of a divisive weighted Grassmann orbifold with integer coefficients. We discuss how to compute the weighted structure constants for the integral equivariant cohomology ring of a divisive weighted Grassmann orbifold.

math.AT↗

arXiv2021

Relative LS categories and higher topological complexities of maps

In this paper, we study three relative LS categories of a map and study some of their properties. Then we introduce the `higher topological complexity' and `weak higher topological complexity' of a map. Each of them are homotopy invariants. We discuss some lower and upper bounds of these in invariants and compare them with previously known `topological complexities' of a map.

math.AT↗

arXiv2021

Equivariant K-theory of toric orbifolds

Toric orbifolds are a topological generalization of projective toric varieties associated to simplicial fans. We introduce some sufficient conditions on the combinatorial data associated to a toric orbifold to ensure the existence of an invariant cell structure on it and call such a toric orbifold retractable. In this paper, our main goal is to study equivariant cohomology theories of retractable toric orbifolds. Our results extend the corresponding results on divisive weighted projective spaces.

math.AG↗

arXiv2021

On generalized projective product spaces and Dold manifolds

D. Davis introduced projective product spaces in 2010 as a generalization of real projective spaces and discussed some of their topological properties. On the other hand, Dold manifolds were introduced by A. Dold in 1956 to study the generators of the non-oriented cobordism ring. Recently, in 2019, A. Nath and P. Sankaran made a modest generalization of Dold manifolds. In this paper we simultaneously generalize both the notions of projective product spaces and Dold manifolds, leading to infinitely many different classes of new smooth manifolds. Our main goal will be to study the integral homology groups. cohomology rings, stable tangent bundles, and vector field problems, on certain generalized projective product spaces and Dold manifolds.

math.AT↗

arXiv2020

Equivariant cohomological rigidity of certain $T$-manifolds

We introduce the category of {\it locally $k$-standard $T$-manifolds} which includes well-known classes of manifolds such as toric and quasitoric manifolds, good contact toric manifolds and moment-angle manifolds. They are smooth manifolds with well-behaved actions of tori. We study their topological properties, such as fundamental groups and equivariant cohomology algebras. Then, we discuss when the torus equivariant cohomology algebra distinguishes them up to weakly equivariant homeomorphism.

math.AT↗

arXiv2020

Maps of degree one, relative LS category and higher topological complexities

In this paper, we introduce relative LS category of a map and study some of its properties. Then we introduce `higher topological complexity' of a map, a homotopy invariant. We give a cohomological lower bound and compare it with previously known `topological complexity' of a map. Moreover, we study the relation between Lusternik-Schnirelmann category and topological complexity of two closed oriented manifolds connected by a degree one map.

math.AT↗

arXiv2020

GKM theory for orbifold stratified spaces and application to singular toric varieties

We study the GKM theory for a equivariant stratified space having orbifold structures in tis successive quotients. Then, we introduce the notion of an \emph{almost simple polytope}, as well as a \emph{divisive toric variety} generalizing the concept of a divisive weighted projective space. We employ the GKM theory to compute the generalized equivariant cohomology theories of toric varieties associated to almost simple polytopes and divisive toric varieties.

math.AT↗