Cassini Oval Research Articles

A new Gershgorin-type result for the localisation of the spectrum of matrices

We present a Gershgorin's type result on the localisation of the spectrum of a matrix. Our method is elementary and relies upon the method of Schur complements, furthermore it outperforms the one based on the Cassini ovals of Ostrovski and Brauer. Furthermore, it yields estimates that hold without major differences in the cases of both scalar and operator matrices. Several refinements of known results are obtained.

Optimal Placement for Barrier Coverage in Bistatic Radar Sensor Networks

By taking advantage of active sensing using radio waves, radar sensors can offer several advantages over passive sensors. Although much attention has been given to multistatic and multiple-input-multiple-output (MIMO) radar concepts, little has been paid to understanding radar networks (i.e., multiple individual radars working in concert). In this context, we study the coverage problem of a bistatic radar (BR) sensor network, which is very challenging due to the Cassini oval sensing region of a BR and the coupling of sensing regions across different BRs. In particular, we consider the problem of deploying a network of BRs in a region to maximize the worst-case intrusion detectability, which amounts to minimizing the vulnerability of a barrier. We show that it is optimal to place BRs on the shortest barrier if it is the shortest line segment that connects the left and right boundary of the region. Based on this, we study the optimal placement of BRs on a line segment to minimize its vulnerability, which is a nonconvex optimization problem. By exploiting certain specific structural properties pertaining to the problem (particularly an important structure of detectability), we characterize the optimal placement order and the optimal placement spacing of the BR nodes, both of which present elegant balanced structures. Our findings provide valuable insights into the placement of BRs for barrier coverage. To our best knowledge, this is the first work to explore the barrier coverage of a network of BRs.

Elastic buckling of Cassini ovaloidal shells under external pressure - theoretical study

The paper is devoted to shells of revolution with positive and negative Gaussian curvature. The meridian of shells is a plane curve in the Cassini oval form. Geometrical properties of the middle surface of the shell of revolution based on this curve are presented. The membrane state of stress of a family of shells with constant capacity and mass under uniform pressure is described analytically and numerically with the use of the FEM (the ANSYS system). The critical pressure, buckling modes and equilibrium paths of analysed shells are calculated numerically. The advantages of a pressure vessel made in the form of Cassini ovaloidal shell, such as the lack of edge effect and a stable post-buckling behaviour, are pointed out. The results of the analytical and numerical investigations are compared and presented in tables and figures.

Towards an alternative gravitational theory

In 1680, Cassini proposed oval curves as alternative trajectories for the visible planets around the sun. The Cassini ovals were of course overshadowed by the Kepler's first law (1609), namely the planets move around the sun describing conic orbits. Here, we describe the possibility that the Cassini's idea works at larger or smaller scales. Indeed, we consider the Spiric curves (which are a generalization of the Cassini oval) and present the first steps towards a Spiric gravitational theory. We show that from our formalism an ellipse associated with a planet can be obtained as a particular case.

Cassini Ovals in Dynamic Geometry of Polynomials

In this paper, we investigate the behavior of critical points of some polynomials whose roots are the vertices of some parallelogram, in case of rotation two of them on a given circle. In this case, the trajectory is the Cassini ovals.

Simulation of Bernoulli Lemniscate holonomically constrained by Cassini oval function for Brunnian braiding mechanism and dynamics

This paper presents a novel braiding mechanism simulated on Bernoulli Lemniscate function. Spatial dynamics of thread sequence have been computed to find a simple and innovative drive mechanism for braiding complex pattern. Dynamic degree-of-freedoms (DOFs) of strings has been holonomically constrained by Cassini oval function in simulation and spatially driven by centrifugal force to move the strings over a 2-manifold surface. The paper presents the essence of a new mechanism, and the result reflects the correctness of concept and allows to indicate technological possibilities of the working principle. Drive system is first modelled in mathematics, and later, more accurately in a multi-degree-of-freedom dynamic model. Correlated experiment and simulation response validate mathematical phase model, for use as a designer’s tool.

The research on the coverage area of multistatic sonar under various work modes

Absorption loss, which is an important characteristic in the field of underwater acoustic, caused by sea water is often neglected in the research on the coverage area of bistatic sonar. In this article, the coverage area of bistatic sonar considering absorption loss is studied. The change of coverage area caused by absorption loss and the approximation are discussed. When multistatic sonar works in different modes, detection criterion varies, and the coverage area cannot be simply considered as the union set or the intersection of coverage area of each bistatic sonar unit. The definition of multistatic coverage area is discussed, and the coverage area respect to the most relaxed criterion and the strictest criterion is studied.

Kepler's ellipse, Cassini's oval and the trajectory of planets

In this article, we discuss the similarities and differences between Kepler's ellipse and Cassini's oval with a small eccentricity. We show that these curves are barely distinguishable when the planetary orbits of our solar system are considered and that, from a numerical viewpoint, it is difficult to decide in favour of one of them.

Scission Configurations for StI and StII Fission Modes in the Reaction 235U(nth, f)

Two scission configurations that can explain the fission fragment properties of the principal fission modes, short (StI) and standard (StII), in the reaction 235U(nth, f) are presented. These configurations are very close to the optimal scission shapes defined with three constraints: on fragment elongation, mass asymmetry and neck radius. They are subsequently approximated by the modified Cassini ovals in order to calculate fission fragment properties such as mass yields, excitation energies of individual fragments and total kinetic energies. For each configuration a comparison is made with the same quantities obtained from experimental data after decomposition into contributions from three fission modes.

A Multi Foci Closed Curve: Cassini Oval, its Properties and Applications

Cassini ovali, duzlem uzerindeki sabit iki noktaya olan mesafelerinin carpimi yine bir sabit olan noktalar kumesinin olusturdugu kuadratik bir egri olarak tanimlanabilir. Benzersiz ozellikleri ve geometrik profili, bu ovalleri pek cok askeri ve ticari alanda kullanilmasina imkan taniyan ustun bir arac haline getirmis, ayrica analitik geometriye ve genel elips konseptinin otesindeki matematik teorisiyle iliskili diger konulara yeni bir boyut kazandirmistir. Bu calisma kapsaminda, Cassini ovallarinin cesitli ozellikleriyle ilgili analitik ifadeler gelistirilmis ve farkli alanlardan ornekler kullanilarak soz konusu ovallerin uygulama alanlarina iliskin ozet bilgi verilmistir.

New regions including eigenvalues of Toeplitz matrices

Two new eigenvalue inclusion regions for matrices with a constant main diagonal are given. We then apply these results to Toeplitz matrices, and obtain two regions including all eigenvalues of Toeplitz matrices. Furthermore, it is proved that the new regions are tighter than those in [Melman A. Ovals of Cassini for Toeplitz matrices, Linear and Multilinear Algebra. 2012;60:189–199].

Simulation of translational dielectrophoretic velocity spectra of erythrocytes in traveling electric field using various volume models

Proper models of cell geometry are needed for biophysical analysis of cellular electrical phenomena. This work compares various mathematical volume-models for normal human erythrocytes (discocyte) possessing biconcave-discoid form to simulate translational dielectrophoretic velocity spectra of erythrocyte suspensions induced in a traveling electric field over a frequency range from 1 kHz to 15 MHz. The non-spherical volumes of the oblate-spheroid, the prolate-spheroid, and the oval of Cassini and the “Bun-model” were numerically evaluated according to the normal range of cellular dimension values for mammalian erythrocytes. The latter model is the novel approach derived to provide a more realistic model for the shape of discocytes in the thin biconcave-disc form with a toroidal rim. The bun model is also more rugged than the Cassini equation. Using the actual cell dimensions for calculations, the numerical results among these calculated cell volumes revealed large and significant differences with respect to the Bun-model of +32.09%, +8.95%, and −8.45% for prolate-spheroid, Cassini's equation, and oblate-spheroid, respectively. These large volume deviations shift the magnitude of the sharp peak in dielectrophoretic velocity spectra to lower values with differences of +189.28%, +7.66%, and −9.49%, respectively. For traveling wave dielectrophoresis, similar results were found for the sharp peak of +145.76%, +7.71%, and −9.50%, respectively. The suitability of the Bun-model was verified by curve-fitting of the cell velocity spectra between experimental and theoretical curves, which gave the maximum discrepancies of less than ±10%.

Angular Distribution of the Scission Neutrons with Respect to the Fission Axis

Abstract We study the angular distribution of the scission neutrons in a time dependent approach. This implies the numerical solution of the bi-dimensional time-dependent Schrodinger equation (TDSE) with time-dependent potential. To describe the axially symmetric extremely deformed nuclear shapes involved, we have used modified Cassini ovals. The Hamiltonian in cylindrical coordinates is discretized on a bi-dimensional grid, using finite difference approximations of the derivatives. The initial wave-functions for TDSE are the eigensolutions of the stationary Schrodinger equation whose potential corresponds to the pre-scission point (when the neck connecting the primary fission fragments starts to break). The time evolution is calculated by a Crank-Nicolson scheme until the neck dissappears (the post-scission point). The resulting wave-functions are then propagated keeping the last configuration to further intervals of time. We investigate the nucleus 236 U at different mass asymmetries. The numerical solutions can be used to evaluate relevant physical quantities. Among them, the current density, a key quantity in the angular distribution calculation. The angular distribution of the scission neutrons is a priori a way to separate them from other neutron components. Moreover our preliminary results show a striking similarity with the angular distribution of the neutrons evaporated from fully accelerates fragments.

Axisymmetric creeping motion of a prolate particle in a cylindrical pore

A theoretical study of the fluid flow caused by an axially symmetric prolate particle translating along its axis of revolution in a coaxial circular cylindrical pore at low Reynolds numbers is presented. A method of distribution of a set of spherical singularities along the axis of revolution within the particle is used to find the general solution of the fluid velocity field that satisfies the boundary conditions at the pore wall and at infinity. The no-slip condition on the particle surface is then satisfied by applying a boundary collocation technique to this general solution to determine the unknown constants. The drag force exerted on the particle by the ambient fluid is calculated with good convergence behavior for the cases of a prolate spheroid and a prolate Cassini oval (whose surface can be partly concave) with broad ranges of their aspect ratio (shape parameter) and distance from the pore wall. For the axially symmetric translation of a spheroid in a cylindrical pore, our drag results agree with the limited approximate solutions available in the literature. For a fixed particle–wall separation parameter, the normalized drag force increases with an increase in the axial-to-radial aspect ratio of the particle, and the boundary effect on the motion of the particle can be very significant when it gets close to the pore wall.

Cassini Curves in Normed Planes

We extend the concept of Cassini curves from the Euclidean plane to normed (or Minkowski) planes and show that geometric properties of (Minkowskian) Cassini curves are closely related to geometric properties of the unit disc determining the underlying normed plane.

A single oval of Cassini for the zeros of a polynomial

We derive two ovals of Cassini, each containing all the zeros of a polynomial. The computational cost to obtain these ovals is similar to that of the Brauer set for the companion matrix of a polynomial, although they are frequently smaller. Their derivation is based on the Gershgorin set for an appropriate polynomial of the companion matrix.

The Schur complement of strictly doubly diagonally dominant matrices and its application

It is known that the Schur complements of doubly diagonally dominant matrices are doubly diagonally dominant. In this paper, we obtain an estimate for the doubly diagonally dominant degree on the Schur complement of strictly doubly diagonally dominant matrices. Then, as an application we obtain that the eigenvalues of the Schur complements are located in the Brauer Ovals of Cassini of the original matrices under certain conditions. As another application, we obtain an upper bound for the infinity norm on the inverse on the Schur complement of strictly doubly diagonally dominant matrices. Further, based on the derived results, we give a kind of iteration called the Schur-based iteration, which can solve large scale linear systems though reducing the order by the Schur complement and can compute out the results faster.

On the Calculation of Two-Dimensional Added Mass Coefficients by the Taylor Theorem and the Method of Fundamental Solutions

ABSTRACTThis work integrates the Taylor theorem and the method of fundamental solutions to develop a numerical tool for estimating the added mass coefficient tensor for a solid object of any convex shape moving in potential flow. In potential flow theory, the Taylor theorem calculates the added mass coefficient tensor for a Rankine body with algebraic manipulations of the properties of the internal singularities employed to generate the corresponding flow. To apply this theorem for objects in other shapes, the singularity strength and locations are required information which is facilitated numerically in this work by the method of fundamental solutions (MFS). The developed scheme is tested on a circle, an ellipse, a square, and a rhombus and the numerical results are in good agreement with the corresponding analytical values. A final example of a Cassini oval is also considered to show the potential applications on bio-engineering problems.

Ovals of Cassini for Toeplitz matrices

Both the Gershgorin and Brauer eigenvalue inclusion sets reduce to a single disc when applied to a matrix with a constant main diagonal. We show that a union of ovals of Cassini containing all the eigenvalues of such a matrix can nonetheless be derived and that this union is guaranteed to lie inside the Brauer set. We then apply our results to Toeplitz matrices.

Cassini-oval Description of the Energy Balance at Scission During 235U(nth, f)

For a detailed description of the excitation energy of the fission fragments, that is used to evaporate neutrons and emit γ-rays, realistic nuclear shapes at scission are necessary. It is shown that the nuclear shapes around the scission point, along the main fission mode, are well described by Cassini ovals with only two parameters: α (elongation) and α1 (mass asymmetry). This shape parametrization is used in an attempt to solve, in the case of the low-energy fission of 236 U, the puzzle of the energy balance at scission. The deformation energy liberated during the neck rupture (αi=0.985 → αf =1.001)and the extra def deformation energy of the fragments immediately after scission are calculated as a function of eachdef fragment mass A. This allows us to estimate the excitation energy for each fragment pair as well as its partition among the light and the heavy fragment and compare with data extracted from measured prompt neutron and γ-ray multiplicities. The Coulomb repulsion of the fission fragments immediately after scission is also calculated as a function of their mass ratio and compared with experimental total fragment kinetic energies.