arXiv subjects

Mohammad Hassan Murad

Publications and source records attributed to Mohammad Hassan Murad.

11 recordsLinked to original sources

arXiv2026

An Affine Approach to Metric Invariants of Poncelet Polygons

We prove that the total area of the power circles associated with an odd $p$-gon inscribed in and circumscribed about a pair of homothetic ellipses remains invariant throughout the Poncelet family. Our proof is based on a simple affine averaging principle, which also yields several related quadratic invariants, including explicit formulas for the sums of the squared distances from the center to the vertices, to the side midpoints, as well as for the sum of the squared side lengths. As an application, we show that a convex Poncelet pentagon and the corresponding star Poncelet pentagon, both circumscribed about the same inellipse, have equal total power-circle area. These results unify several metric invariants of odd Poncelet polygons within a common affine-geometric framework.

math.MG↗

arXiv2026

Siebeck--Marden Theorems and Symmetric Parametrizations of Cyclic Polygons

The classical Blaschke-product approach provides an elegant description of triangles inscribed in a circle and circumscribed about an ellipse. Motivated by the Siebeck--Marden theorem, we derive a symmetric parametrization for cyclic $p$-gons circumscribed about a Siebeck--Marden curve of class $p-1$, recovering the Blaschke product parametrization as the triangular case. We then use this parametrization to establish several geometric invariance results, including a characterization of when the sum of the squares of all sides and diagonals is invariant. We revisit Cayley's criterion for 3- and 4-Poncelet pairs and obtain a complete classification of the associated central conics solely in terms of their foci. In particular, we show that the formulas for the lengths of the major axis obtained under the assumption on the foci of the conic lying inside the circumcircle remain valid when one focus or both foci lie outside the circumcircle, thereby extending the classical theory from ellipses to all admissible central conics. Several geometric properties of the associated cyclic quadrilaterals are also obtained.

math.GM↗

arXiv2026

A Symmetric Polynomial Approach to Poncelet Triangles Inscribed in a circle and circumscribed about Central Conics

We develop a symmetric-polynomial framework for the study of one-parameter families of Poncelet triangles. For triangles inscribed in the unit circle and circumscribed about a central conic, we show that the elementary symmetric polynomials of the vertices depend linearly on a single complex parameter. It follows that every symmetric rational function of the vertices is a rational function of this parameter, providing a unified algebraic approach to the investigation of geometric invariants. As applications, we recover several known invariance results and establish new ones involving angles, orthic triangles, tangential triangles, polar circles, and other classical constructions. We also obtain explicit formulas and loci for several associated geometric objects, leading to new invariance phenomena within Poncelet families.

math.GM↗

arXiv2026

A Vector Generalization of Euler's Quadrilateral Theorem

In this paper, we develop a unified algebraic framework for Euler-type identities in real and complex inner product spaces. Starting from the parallelogram identity, we derive Apollonius' identity and recover Euler's classical theorem. We then establish a general Euler-type identity valid for every finite collection of $n\geq 4$ vectors. The proof is based on a combinatorial analysis of pairwise distances. The resulting identity recovers Euler's theorem when $n=4$. Several previously known identities thus arise naturally within a single algebraic framework.

math.GM↗

arXiv2026

Generalized Chapple--Euler Relation

We provide a new proof of the necessary and sufficient condition for a triangle to be circumscribed about a central conic (ellipse or hyperbola), expressed in terms of the circumradius and the distances from the circumcenter to the foci. If the inscribed conic is an ellipse, in the limiting case, where the foci coincide, the condition reduces to the classical Chapple--Euler relation. We also prove that the sum of the squares of the sides of a triangle in a family inscribed in a circle and circumscribed about a central conic remains invariant throughout the family if and only if the center of the circle coincides either with the center of the conic or with one of its foci. Using Blaschke products of degree three and affine transformations, we characterize all central $3$-Poncelet pairs with constant triangle area. We prove that the associated family of triangles has constant area if and only if the two conics are homothetic ellipses. We also prove two observations of Reznik concerning the invariance of the total area of the power circles of Poncelet triangles by relating them to Apollonius's identity on medians and establish a more general result valid for both ellipses and hyperbolas. Finally, we propose two conjectures on area invariance of power-circles.

math.GM↗

arXiv2025

Parable of the Parabola

We study triangles and quadrilaterals which are inscribed in a circle and circumscribed about a parabola. Although these are particular cases of the celebrated Poncelet's Theorem, in this paper we {\it do not assume} the theorem but prove it along the way. Similarly, our arguments here \emph{are logically independent} from Cayley's condition, describing points of a finite order on an elliptic curve or any other use of the theory of elliptic curves. Instead, we use purely planimetric methods, including the Joachimsthal notation, to fully describe such polygons and associated circles and parabolas. We prove that a circle contains the focus of a parabola if and only if there is a triangle inscribed in the circle and circumscribed about the parabola. We prove that if the center of a circle coincides with the focus of a parabola, then there exists a quadrilateral inscribed in the circle and circumscribed about the parabola. We further prove that the quadrilaterals obtained in such a way are antiparallelograms. If the center of a circle does not coincide with the focus of a parabola, then a quadrilateral inscribed in the circle and circumscribed about the parabola exists if and only if the directrix of the parabola contains the point of intersection of the polar of the focus with respect to the circle with the line determined by the center and the focus. That point coincides with the intersection of the diagonals of any quadrilateral inscribed in the circle and circumscribed about the parabola. In particular, for a given circle and a confocal pencil of parabolas with the focus different than the center of the circle, there is a unique parabola for which there exists a quadrilateral circumscribed about it and inscribed in the circle.

math.HO↗

arXiv2025

Poncelet pairs of a circle and parabolas from a confocal family and Painlevé VI equations

We study pairs of conics $(\mathcal{D},\mathcal{P})$, called \textit{$n$-Poncelet pairs}, such that an $n$-gon, called an \textit{$n$-Poncelet polygon}, can be inscribed into $\mathcal{D}$ and circumscribed about $\mathcal{P}$. Here $\mathcal{D}$ is a circle and $\mathcal{P}$ is a parabola from a confocal pencil $\mathcal{F}$ with the focus $F$. We prove that the circle contains $F$ if and only if every parabola $\mathcal{P}\in\mathcal{F}$ forms a $3$-Poncelet pair with the circle. We prove that the center of $\mathcal{D}$ coincides with $F$ if and only if every parabola $\mathcal{P}\in \mathcal{F}$ forms a $4$-Poncelet pair with the circle. We refer to such property, observed for $n=3$ and $n=4$, as \textit{$n$-isoperiodicity}. We prove that $\mathcal{F}$ is not $n$-isoperiodic with any circle $\mathcal{D}$ for $n$ different from $3$ and $4$. Using isoperiodicity, we construct explicit algebraic solutions to Painlevé VI equations.

math.CA↗

arXiv2023

Causally-interpretable meta-analysis: clearly-defined causal effects and two case studies

Meta-analysis is commonly used to combine results from multiple clinical trials, but traditional meta-analysis methods do not refer explicitly to a population of individuals to whom the results apply and it is not clear how to use their results to assess a treatment's effect for a population of interest. We describe recently-introduced causally-interpretable meta-analysis methods and apply their treatment effect estimators to two individual-participant data sets. These estimators transport estimated treatment effects from studies in the meta-analysis to a specified target population using individuals' potentially effect-modifying covariates. We consider different regression and weighting methods within this approach and compare the results to traditional aggregated-data meta-analysis methods. In our applications, certain versions of the causally-interpretable methods performed somewhat better than the traditional methods, but the latter generally did well. The causally-interpretable methods offer the most promise when covariates modify treatment effects and our results suggest that traditional methods work well when there is little effect heterogeneity. The causally-interpretable approach gives meta-analysis an appealing theoretical framework by relating an estimator directly to a specific population and lays a solid foundation for future developments.

stat.ME↗

arXiv2014

Some new Wyman-Adler type static relativistic charged anisotropic fluid spheres compatible to self-bound stellar modeling

In this work some families of relativistic anisotropic charged fluid spheres have been obtained by solving Einstein-Maxwell field equations with preferred form of one of the metric potentials, a suitable forms of electric charge distribution and pressure anisotropy functions. The resulting equation of state (EOS) of the matter distribution has been obtained. Physical analysis shows that the relativistic stellar structure for matter distribution obtained in this work may reasonably model an electrically charged compact star whose energy density associated with the electric fields is on the same order of magnitude as the energy density of fluid matter itself (e.g. electrically charged bare strange stars). These models permit a simple method of systematically fixing bounds on the maximum possible mass of cold compact electrically charged self-bound stars. It has been demonstrated numerically that the maximum compactness and mass increase in the presence of electric field and anisotropic pressures. Based on the analytic model developed in this present work, the values of the relevant physical quantities have been calculated by assuming the estimated masses and radii of some well known potential strange star candidates like PSR J1614-2230, PSR J1903+327, Vela X-1, and 4U 1820-30.

gr-qc↗

arXiv2012

A family of well behaved charge analogues of Durgapal's perfect fluid exact solution in general relativity

This paper presents a new family of interior solutions of Einstein-Maxwell field equations in general relativity for a static spherically symmetric distribution of a charged perfect fluid with a particular form of charge distribution. This solution gives us wide range of parameter, K, for which the solution is well behaved hence, suitable for modeling of superdense star. For this solution the gravitational mass of a star is maximized with all degree of suitability by assuming the surface density equal to normal nuclear density, 2.5E17 kg/m3. By this model we obtain the mass of the Crab pulsar, MCrab 1.3679 solar mass and radius 13.21 km, constraining the moment of inertia > 1.61E38 kg m2 for the conservative estimate of Crab nebula of 2 solar mass . And MCrab = 1.9645 solar mass with radius 14.38 km constraining the moment of inertia > 3.04E38 kg m2 for the newest estimate of Crab nebula mass, 4.6 solar mass. These results are quite well in agreement with the possible values of mass and radius of Crab pulsar. Besides this, our model yields moments of inertia for PSR J0737-3039A and PSR J0737-3039B, I_A = 1.4285E38 kg m2 and I_B=1.3647E38 kg m2 respectively. It has been observed that under well behaved conditions this class of solutions gives us the overall maximum gravitational mass of super dense object, Mmax, 4.7487 solar mass with radius R(Mmax) = 15.24 km, surface redshift 0.9878, charge 7.91E20 C, and central density 4.31 times nuclear density.

physics.gen-ph↗

arXiv2012

A new well behaved class of charge analogue of Adler's relativistic exact solution

The paper presents a new class of parametric interior solutions of Einstein-Maxwell field equations in general relativity for a static spherically symmetric distribution of a charged perfect fluid with a particular form of electric field intensity. This solution gives us wide range of parameter, K, for which the solution is well behaved hence, suitable for modeling of superdense star. For this solution the gravitational mass of a superdense object is maximized with all degree of suitability by assuming the surface density of the star equal to the normal nuclear density 2.5E17 kg/m3. By this model we obtain the mass of the Crab pulsar 1.401 Solar mass and the radius 12.98 km constraining the moment of inertia parameter greater than 1.61 for the conservative estimate of Crab nebula mass 2 Solar mass and 2.0156 Solar mass with radius 14.07 km constraining the moment of inertia parameter greater than 3.04 for the newest estimate of Crab nebula mass 4.6 Solar mass which are quite well in agreement with the possible values of mass and radius of Crab pulsar.Besides this, our model yields the moments of inertia for PSR J0737-3039A and PSR J0737-3039B are 1.4624E38 kgm2 and 1.2689E38 kgm2 respectively. It has been observed that under well behaved conditions this class of parametric solution gives us the maximum gravitational mass of causal superdense object 2.8020 Solar mass with radius 14.49 km, surface redshift 0.4319, charge 4.67E20 C, and central density 2.68 times nuclear density.

gr-qc↗