arXiv subjects

Soumen Sarkar

Publications and source records attributed to Soumen Sarkar.

At least 19 recordsLinked to original sources

arXiv2026

On generalized complex projective product spaces

Free circle action on manifolds has been explored in several articles. Gonzalez and Velasco considered some free circle actions on the finite product of spheres. In this paper, we introduce generalized complex projective product spaces, extending their definition and the concept of Dold manifolds. This produces infinitely many different classes of new smooth manifolds. First, we study the integral cohomology rings and stable tangent bundles on certain generalized complex-projective product spaces. Then, we discuss the product inequality for strongly equivaraint TC. We exhibit the closeness of the lower and upper bounds for the LS-category and topological complexity of several classes of generalized complex projective product spaces. In many cases, we compute the exact value of the LS-category.

math.AT↗

arXiv2026

Discrete version of topological complexity of maps

We introduce and study discrete analogs of Scott's and Murillo-Wu's topological complexity of maps. We prove that these discrete analogs are contiguity invariants and are, in fact, equivalent. Furthermore, we establish the fundamental theoretical properties and computational aspects of the discrete topological complexity of simplicial maps.

math.AT↗

arXiv2026

A Simple Method of Demonstration of Characteristics of Rainbows Using a Glass of Water and a Few Laser Sources

A rainbow is a captivating natural phenomenon resulting from the refraction, dispersion, and reflection of sunlight within water droplets. Traditional classroom demonstrations often focus on qualitative explanations of the formation of rainbows using prisms or water bowls. This study presents a simple experimental approach to analysing the process of rainbow formation through quantitative analysis using a cylindrical glass filled with water, graph paper, and three semiconductor laser sources emitting red, green, and blue light. By measuring the angles of minimum deviation for different wavelengths, we have found that the experimental values closely match the theoretical predictions. This method offers a hands-on, cost-effective approach to enhance students' understanding of the physics behind rainbows.

physics.ed-ph↗

arXiv2026

Use of smartphone as a density measuring device

In this paper, we have proposed a simple method of measuring the density of a solid material. We have utilized the pressure sensor of a smartphone as a pressure-measuring device. By measuring the values of pressure when a solid object is in air and also in the fully immersed condition in a non-reactive liquid, we have determined the density of the object.

physics.ed-ph↗

arXiv2026

Study of the effect of electromagnetic damping force on a magnet oscillating near a non-ferromagnetic conducting plate

We have designed an experiment that involves studying the effects of a conducting plate on the motion of an oscillating disc magnet. We have employed the video analysis method by Tracker software to investigate the variation of electromagnetic damping coefficient with distance between the plate and the magnet. This experiment can indeed serve as a valuable educational tool for undergraduate students, covering topics such as damped oscillation, electromagnetic damping, Lenz's law, and eddy currents.

physics.ed-ph↗

arXiv2026

A smartphone-based simple method for determination of the free space permeability

A simple and novel method is designed to determine the free space permeability. This value is computed from the expression of the terminal velocity of a magnet falling through a conducting pipe using the magnetic sensor of a smartphone and a video player. This method deserves its importance because of the accuracy and precision of the results.

physics.ed-ph↗

arXiv2026

Determination of the acceleration due to gravity by studying magnet s motion through a conducting pipe

We determine the acceleration due to gravity (g) in a novel way using a magnetic sensor and video analysis technique of a smartphone. The same applications are used to measure the terminal velocity of a magnet falling through a conducting pipe and the magnetic moment of the magnet from its torsional oscillations. This experiment would appear to be intriguing, as it combines elements of magnetism, terminal velocity, and electromagnetic damping to determine g.

physics.ed-ph↗

arXiv2026

Determination of the magnetic moment of a magnet by letting it fall through a conducting pipe

A novel method is proposed to determine the magnetic moment of a magnet by studying its free-falling motion inside a non-ferromagnetic and conducting pipe. The dynamics of a neodymium magnet falling inside a pipe is tracked by using sound waves of a fixed frequency generated by one smartphone and detecting acoustic resonance in the pipe simultaneously by the other. This tracking technique leads to the measurement of the terminal velocity of the falling magnet, as the interaction between the magnet and the conducting pipe creates viscosity artificially. The result obtained is verified by studying torsional oscillations of the suspended magnet and conforms to the reported value in such a low-cost setup. The experiment is designed with concepts integrating the domains of general physics, electromagnetic induction, and acoustics.

physics.ed-ph↗

arXiv2026

Smartphone-based measurement of magnetic force and demonstration of Newton third law of motion

A fascinating approach to teaching Newton's Third Law using readily available technology is presented in this article. Magnetic forces are measured by using a smartphone's pressure sensor, two ring magnets, and common household items. Students can measure the magnitudes of forces, gain a more tangible understanding of the law, and see how 'action' and 'reaction' are quantitatively equal and opposite.

physics.ed-ph↗

arXiv2026

LiDAR based determination of spring constant using smartphones

A novel use of the LiDAR sensor of a smartphone in introductory physics experiments is discussed in this article. We have determined the spring constant for various combinations of springs using the LiDAR sensor of a smartphone through the phyphox application. An electrical heater coil is used as a spring, and the period of oscillation of a vertical spring-mass system is measured using a LiDAR sensor. The experimental values of spring constants agree with the theoretical values. A high school student can perform this simple experiment in a smart way at home.

physics.ed-ph↗

arXiv2026

On $\ZZ_2^n$-equivariant triangulation of $\RR P^n$

We study several properties of $\ZZ_2^n$-equivariant triangulations of $\RR P^n$. We show that a $\ZZ_2^n$-equivariant triangulation of $\RR P^n$ induces a triangulated subdivision of the orbit space $\bigtriangleup^n$. We show that any vertex minimum $\ZZ_2^3$-equivariant triangulation of $\RR P^3$ contains $11$ vertices.

math.GT↗

arXiv2026

On the equivariant triangulation of some small covers

In this paper, we study certain properties of $\mathbb{Z}_2^n$-equivariant triangulations of small covers. We show that any $\mathbb{Z}_2^n$-equivariant triangulation of a small cover naturally induces a triangulation of the orbit space. Then, we explicitly construct the minimal $\mathbb{Z}_2^3$-equivariant triangulation of $\mathbb{RP}^3$, which contains $11$ vertices and prove that this is the unique $\mathbb{Z}_2^3$-equivariant triangulation of $\mathbb{RP}^3$ with $11$ vertices. For a finite group $G$, we give a method for constructing some $G$-equivariant triangulations of connected sums of manifolds from their respective $G$-equivariant triangulations. In particular, we construct a $\mathbb{Z}_2^3$-equivariant triangulation of $\mathbb{RP}^3 \# \mathbb{RP}^3$ with $17$ vertices, which is the best known yet. This triangulation of $\mathbb{RP}^3 \# \mathbb{RP}^3$ provides another minimal $g$-vector improving one of the result of Lutz in \cite{LS}. Moreover, we prove that a $\ZZ_2^4$-equivariant triangulation of $\mathbb{RP}^4$ requires at least $18$ vertices.

math.AT↗

arXiv2026

Darboux first integrals of Kolmogorov systems with invariant $n$-sphere

In this paper, we characterize all polynomial Kolmogorov vector fields for which the standard $n$-sphere is invariant. We exhibit completely integrable Kolmogorov vector fields of degree $m$ on $\mathbb{S}^n$ for any $m >2$. Then, we show that there is no cubic Hamiltonian Kolmogorov vector field that makes an odd-dimensional sphere invariant. We examine the conditions under which a cubic Kolmogorov vector field has a Darboux first integral. In many cases, we determine whether they constitute necessary and sufficient conditions. Moreover, we study the complete integrability of cubic Kolmogorov vector fields having an invariant $n$-sphere.

math.DS↗

arXiv2025

Some $\ZZ_3^n$-equivariant triangulations of $\CP^n$

In 1983, Banchoff and Kuhnel constructed a minimal triangulation of $\CP^2$ with 9 vertices. $\CP^3$ was first triangulated by Bagchi and Datta in 2012 with 18 vertices. Known lower bound on number of vertices of a triangulation of $\CP^n$ is $1 + \frac{(n + 1)^2}{2}$ for $n \geq 3$. We give explicit construction of some triangulations of complex projective space $\CP^n$ with $\frac{4^{n+1}-1}{3}$ vertices for all $n$. No explicit triangulation of $\CP^n$ is known for $n \geq 4$.

math.GT↗

arXiv2025

Sectional category with respect to group actions and sequential topological complexity of fibre bundles

Let $X$ be a $G$-space. In this paper, we introduce the notion of sectional category with respect to $G$. As a result, we obtain $G$-homotopy invariants: the LS category with respect to $G$, the sequential topological complexity with respect to $G$ (which is same as the weak sequential equivariant topological complexity $\mathrm{TC}_{k,G}^w(X)$ in the sense of Farber and Oprea), and the strong sequential topological complexity with respect to $G$, denoted by $\mathrm{cat}_G^{\#}(X)$, $\mathrm{TC}_{k,G}^{\#}(X)$, and $\mathrm{TC}_{k,G}^{\#,*}(X)$, respectively. We explore several relationships among these invariants and well-known ones, such as the LS category, the sequential (equivariant) topological complexity, and the sequential strong equivariant topological complexity. In one of our main results, we give an additive upper bound for $\mathrm{TC}_k(E)$ for a fibre bundle $F \hookrightarrow E \to B$ with structure group $G$ in terms of certain motion planning covers of the base $B$ and the invariant $\mathrm{TC}_{k,G}^{\#,*}(F)$ or $\mathrm{cat}_{G^k}^{\#}(F^k)$, where the fibre $F$ is viewed as a $G$-space. As applications of these results, we give bounds on the sequential topological complexity of generalized projective product spaces and mapping tori.

math.AT↗

arXiv2024

Dynamics and integrability of polynomial vector fields on the $n$-dimensional sphere

In this paper, we characterize arbitrary polynomial vector fields on $S^n$. We establish a necessary and sufficient condition for a degree one vector field on the odd-dimensional sphere $S^{2n-1}$ to be Hamiltonian. Additionally, we classify polynomial vector fields on $S^n$ up to degree two that possess an invariant great $(n-1)$-sphere. We present a class of completely integrable vector fields on $S^n$. We found a sharp bound for the number of invariant meridian hyperplanes for a polynomial vector field on $S^2$. Furthermore, we compute the sharp bound for the number of invariant parallel hyperplanes for any polynomial vector field on $S^n$. Finally, we study homogeneous polynomial vector fields on $S^n$, providing a characterization of their invariant $(n-1)$-spheres.

math.DS↗

arXiv2024

LS-category and topological complexity of real torus manifolds and Dold manifolds of real torus type

The real torus manifolds are a generalization of small covers, and the Dold manifolds of real torus type are a class of non-trivial fibre bundles over the projective product spaces with real torus manifolds as fibres. In this paper, first, we compute the LS-category of these two types of manifolds and obtain sharp bounds on their topological complexities. We show that under certain hypotheses, the topological complexities of real torus manifolds of dimension $n$ are either $2n$ or $2n+1$.We figure out tight bounds for the topological complexity of generalized real Bott manifolds, and in many cases, the difference between these upper and lower bounds is less than 5. We compute the $\mathbb{Z}_2$-equivariant LS-category of small covers when the $\mathbb{Z}_2$-fixed points are path connected. In the end, we study the symmetric topological complexity of the above-mentioned manifolds and obtain exact values for infinitely many cases.

math.AT↗

arXiv2024

Invariant Hyperplane Sections of Vector Fields on the Product of Spheres

Let $S_{p,q}$ be the hypersurface in $\mathbb{R}^{p+q+1}$ defined by the following: $$ S_{p,q} := \left\lbrace (x_1,\ldots,x_{p+1},x_{p+2},\ldots,x_{p+q+1}) \in \mathbb{R}^{p+q+1} \big| \left( \sum_{i=1}^{p+1} x_i^2 - a^2 \right)^2 + \sum_{j=p+2}^{p+q+1} x_j^2 = 1 \right\rbrace,$$ where $a > 1$. We show that $S_{p,q}$ is homeomorphic to the product $S^p \times S^q$. We classify all degree one and two polynomial vector fields on $S_{p,q}$. We consider the polynomial vector field $\mathcal{X} = (R_1,...,R_{p+1},R_{p+2},...,R_{p+q+1})$ in $\mathbb{R}^{p+q+1}$ which keeps $S_{p,q}$ invariant. Then we study the number of certain invariant algebraic subsets of $S_{p,q}$ for the vector field $\mathcal{X}$ if either $p>1$ or $q>1$.

math.DS↗