Plane Polar Coordinates Research Articles

Mathematics as a basis for the generation of cryptographic tools

This article is the product of the research “Mathematics as a basis for the generation of cryptographic tools”, developed at the Universidad de Cundinamarca, sectional Ubaté in the year 2021. Problem: Improve the security of code with differential calculus, plane and spatial geometry, polar coordinates, and functions; the resulting encryption should improve on the standards of current encryption systems. Objective: Explain the algorithm’s operation with fundamental issues that use and generate a modification proposal to make the algorithm more robust when subjected to an attack by unethical hackers. Methodology: A bibliographic review, encryption codes, history of algorithms, and operation of SHA-256 in the Ethereum platform and Bitcoin explains how it works and gives an idea of how to improve security. Result: The algorithm’s operation is explained step by step using subject topics such as precalculus and calculus. Also, topics such as functions, truth tables, logical operators, programming functions, and a way to complicate the algorithm were proposed by applying issues of plane geometry, space, polar coordinates, ceiling function, and absolute values. Conclusion: It is possible to assimilate the operation of SHA-256 (256-bit secure hash algorithm) and the mathematical basis for generating codes in hexadecimal, depending on the message to be encrypted. Algebraic models allow taking different points of the figures generated in the Cartesian and polar planes to use them as a means of encryption. Originality: Explaining an algorithm widely used in encryption issues; however, according to a survey conducted, only a few understood the code that explains the mathematics, all the aspects behind its operation, and why it is so secure. In addition, the project proposes a way to improve its security. Limitations: The algorithm used was a hyperbolic and linear function combined. In this sense, common random points between the two functions are selected for encryption, and it is not aligned with the encryption systems used in blockchains; it is a restriction for the implementation. Currently, several updates are aligned with the SHA-256 system, involving limiting the inclusion of updated models.

Converting translation operators into plane polar and spherical coordinates and their use in determining quantum-mechanical wavefunctions in a representation-independent fashion

Quantum mechanics is often developed in the position representation, but this is not necessary, and one can perform calculations in a representation-independent fashion, even for wavefunctions. In this work, we illustrate how one can determine wavefunctions, aside from normalization, using only operators and how those operators act on state vectors. To do this in plane polar and spherical coordinates requires one to convert the translation operator into those coordinates. As examples of this approach, we illustrate the solution of the Coulomb problem in two and three dimensions without needing to express any operators in position space.

Nanofluid flow in a converging and diverging channel of rectangular and heated walls

We have investigated the nanofluid flow in a converging and diverging channel of heated and inclined plane walls. The wellknown Buongiorno model has been undertaken for the transport of nanoparticles in a base fluid. The upper (lower) wall of the channel has the geometry and it is presented by an equation of the straight line y=mx+a0 (y=-mx-a0), in which m is the slope of the upper wall, a0 is the half width of the channel inlet (exit) for diverging (converging) flow, x and y are representing the Cartesian Coordinates. Note that the walls of the channel are uniformly and equally heated and the nanoparticle concentration at the walls is also constant. On the contrary, the flow problem is strictly described in Cartesian Coordinates. The problem is formulated for the upper half of the channel and later on the results are generalized for the lower half. The new observations are shown in different graphs. The four governing PD E’ s are simplified with the help of proper and appropriate similarity transformations and they constituted a system of ODE’s, whereas, it is equipped with several dimensionless parameters. Note that the convective terms in energy and concentration equations are disappeared in view of these similarity variables. The effects of all parameters are investigated on flow, heat and mass transfer characteristics. Moreover, effects of all parameters are seen on skin friction, rates of heat and mass transfer. In addition, classical models of nanofluid flow inside a converging and diverging channel in plane polar coordinates will be the special cases of the current modeled problem. Furthermore, the perturbation solution of the modeled equations is also determined for small values of the parameter. Note that slope m and Pr number are used as a perturbation parameters for the momentum equation and energy, species concentration equations and these solutions are valid for small values of all other parameters involved in the problem. In addition, exact solutions of the final ODE are also found for special ranges of parameters value. Moreover, the comparison is shown between the different solutions of the problem and classical solutions in tables and graphs. The temperature (concentration) profiles are increased (decreased) with the increasing of Nb, and Nt, however, both of them are increased with m. Moreover, the linear (non-linear) profiles of skin friction (rate of heat transfer) are decreased(increased) with the increasing of both Re (Pr,m) and m (Nb). In addition, the rate of mass transfer is increased linearly (non-linearly) against the increasing value of Nt (Pr) and Nb (Nb=Nt). Note that the temperature (concentration) profiles are effectively improved (declined) with the increase of thermophoretic forces, moreover, it is also rapidly increased (decreased) with the increase of concentration of nanoparticles (and their rapid fluctuations) in a converging and diverging channel.

Scattering and diffraction of a uniform complex-source beam by a slit in a perfectly conducting plane screen

Abstract. A new multipole solution in plane-polar coordinates for the scattering of an arbitrary TE- or TM-polarized incident field by an infinitely extended slit in a plane screen is derived. To this end a classical three-domain problem with a complete multipole expansion in each of the domains is formulated. The unknown multipole amplitudes are found from the continuity conditions of the tangential electric and magnetic field components. Finally an infinite system of linear equations is derived which can be approximated by a numerically tractable finite one leading to a solution with an arbitrary accuracy. The results are numerically successfully compared to those ones obtained by classically solving the strip problem using Mathieu functions and by a subsequent application of the rigorous form of the Babinet principle. Numerical results include the comparison of the scattered fields obtained for an incident uniform complex-source beam and for an incident plane wave.

Elzaki transform method for finding solutions to two-dimensional elasticity problems in polar coordinates formulated using Airy stress functions

In this paper, the Elzaki transform method is used for solving two-dimensional (2D) elasticity problems in plane polar coordinates. Airy stress function was used to express the stress compatibility equation as a biharmonic equation. Elzaki transform was applied with respect to the radial coordinate to a modified form of the stress compatibility equation, and the biharmonic equation simplified to a fourth order ordinary differential equation (ODE). The general solution for the Airy stress potential function in the Elzaki transform space was obtained by solving the ODE. By inversion, the general solution for the Airy stress potential function was obtained in the physical domain space variables in terms of four unknown integration constants. Normal stresses and shear stress fields were also determined for the general case of 2D elasticity problems. The Flamant problem was solved as a particular illustration of 2D elasticity problems. The stress boundary conditions and the requirement of equilibrium of the internal stress resultants and the external forces were used simultaneously to determine the four constants of integration. The Airy stress potential function and the normal and shear stress fields were thus completely determined. The principle of superposition was used to obtain the elasticity solutions for the stress fields in the elastic half plane due to strip load of infinite extent, and solutions for horizontal stresses on smooth rigid retaining walls due to strip loads and parallel line load of infinite extent acting on the elastic half plane.

Exact Solutions to the Three-Dimensional Navier–Stokes Equations Using the Extended Beltrami Method

Abstract In a 1966 publication, Chi-Yi Wang used the streamfunction in concert with the vorticity equations to develop a methodology for obtaining exact solutions to the incompressible Navier–Stokes equations, now known as the extended Beltrami method. In Wang's approach, the vorticity is represented by the sum of a linear function of the streamfunction and an assumed auxiliary function, such that the vorticity equation can be reduced to a quasi-linear partial differential equation, and exact solutions are obtainable for many choices of the auxiliary function. In the present work, a natural extension of Wang's formulation to three-dimensional flows in arbitrary orthogonal curvilinear coordinates has been derived, wherein two auxiliary functions are formed at the outset, with the caveat that the pressure and velocity components may vary in two spatial dimensions. As is the case with two-dimensional extended Beltrami flows, exact solutions are only obtainable when the forms of the auxiliary functions are “simple enough” to render the governing equations solvable. To demonstrate the solutions which may be obtained using the extended formulation, the well-known Kovasznay flow is generalized to a three-dimensional flow. A unique solution in plane polar coordinates is found. An extension to the solution to Burgers vortex has been derived and discussed in the context of existing literature. Finally, a new 3D swirling flow solution which is the angular analogue to Kovasznay flow has been developed.

GENERAL SOLUTION OF ELASTICITY PROBLEMS IN TWO DIMENSIONAL POLAR COORDINATES USING MELLIN TRANSFORM

Abstract In this work, the Mellin transform method was used to obtain solutions for the stress field components in two dimensional (2D) elasticity problems in terms of plane polar coordinates. the Mellin transformation was applied to the biharmonic stress compatibility equation expressed in terms of the Airy stress potential function, and the boundary value problem transformed to an algebraic problem which was solved to obtain the Mellin transformed Airy stress potential function. The Mellin transform was similarly used to obtain the Mellin transformed stress field components. The use of Mellin transform inversion formula yielded the solutions to the 2D elasticity problem in the physical space domain variables. Specific illustration was considered of the solution by using the Mellin transform method for the Flamant problem and the Mellin transform solutions found to agree with solutions from the literature.

Numerical MHD codes for modeling astrophysical flows

We describe a Godunov-type magnetohydrodynamic (MHD) code based on the Miyoshi and Kusano (2005) solver which can be used to solve various astrophysical hydrodynamic and MHD problems. The energy equation is in the form of entropy conservation. The code has been implemented on several different coordinate systems: 2.5D axisymmetric cylindrical coordinates, 2D Cartesian coordinates, 2D plane polar coordinates, and fully 3D cylindrical coordinates. Viscosity and diffusivity are implemented in the code to control the accretion rate in the disk and the rate of penetration of the disk matter through the magnetic field lines. The code has been utilized for the numerical investigations of a number of different astrophysical problems, several examples of which are shown.

Explicit examples of Lipschitz, one-homogeneous solutions of [formula omitted]-singular planar elliptic systems

We give examples of systems of Partial Differential Equations that admit non-trivial, Lipschitz and one-homogeneous solutions in the form u(R,θ)=Rg(θ), where (R,θ) are plane polar coordinates and g takes values in Rm, m≥2. The systems are singular in the sense that they arise as the Euler–Lagrange equations of the functionals I(u)=∫BW(x,∇u(x))dx, where DFW(x,F) behaves like 1|x| as |x|→0 and W satisfies an ellipticity condition. Such solutions cannot exist when |x|DFW(x,F)→0 as |x|→0, so the condition is optimal. The associated analysis exploits the well-known Fefferman–Stein duality (Fefferman and Stein, 1972). We also discuss conditions for the uniqueness of these one-homogeneous solutions and demonstrate that they are minimizers of certain variational functionals.

Novel bound states treatment of the two dimensional Schrödinger equation with pseudocentral potentials plus multiparameter noncentral potential

By converting the rectangular basis potential V(x,y) into the form as V(r)+V(r, phi) described by the pseudo central plus noncentral potential, particular solutions of the two dimensional Schrodinger equation in plane-polar coordinates have been carried out through the analytic approaching technique of the Nikiforov and Uvarov (NUT). Both the exact bound state energy spectra and the corresponding bound state wavefunctions of the complete system are determined explicitly and in closed forms. Our presented results are identical to those of the previous works and they may also be useful for investigation and analysis of structural characteristics in a variety of quantum systems

Flatland Position-Dependent-Mass: Polar Coordinates, Separability and Exact Solvability

The kinetic energy operator with position-dependent-mass in plane polar coordinates is obtained. The separability of the corresponding Schr\"odinger equation is discussed. A hypothetical toy model is reported and two exactly solvable examples are studied.

A central conservative scheme for general rectangular grids

We present an extension of the genuinely multi-dimensional semi-discrete central scheme developed in [A. Kurganov, S. Noelle, G. Petrova, Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton–Jacobi equations, SIAM J. Sci. Comput. 23 (3) (2001) 707–740.] to arbitrary orthogonal grids. The presented algorithm is constructed to yield the geometric scaling factors in a self-consistent way. Additionally, the order of the scheme is not fixed during the derivation of the basic algorithm. Based on the resulting general scheme it is possible to construct methods of any desired order, just by considering the corresponding reconstruction polynomial. We demonstrate how a second order scheme in plane polar coordinates and cylindrical coordinates can be derived from our general formulation. Finally, we demonstrate the correctness of this second order scheme through application to several numerical experiments.

End effects for a generalized biharmonic equation with applications to functionally graded materials

In this paper we consider a generalized biharmonic equation modeling two-dimensional inhomogeneous elastic state in the curvilinear rectangle a < r < b , 0 < θ < α , where ( r , θ ) denote plane polar coordinates. Such an arch-like region is maintained in equilibrium under self-equilibrated traction applied on one of the edges, while the other three edges are traction free. Our aim is to derive some explicit spatial estimates describing how some appropriate measures concerning the specific Airy stress function evolve with respect to the distance to the loaded edge. Two types of smoothly varying inhomogeneity are considered: (i) the elastic moduli vary smoothly with the polar distance, (ii) they vary smoothly with the polar angle. Such types of smoothly varying inhomogeneous elastic materials provide a model for technological important functionally graded materials. The results of the present paper prove how the spatial decay rate varies with the constitutive profile.

Spatial Decay Estimates for the Biharmonic Equation in Plane Polars with Applications to Plane Elasticity

The present paper considers an isotropic and homogeneous elastic body occupying the arch-like region a ≤ r ≤ b, 0 ≤ θ ≤ α, where (r, θ) denote plane polar coordinates. The arch-like body is in equilibrium under an (in plane) self-equilibrated load on the edge r = a, while the other three edges r = b, θ = 0 and θ = α are traction-free and the body forces are absent. An appropriate measure is defined in terms of the Airy stress function ϕ, provided that the opening angle of the arch-like region is lower than 2 . Then the spatial behavior of the solution is studied and a clear relationship is established with Saint-Venant's principle on such regions. In fact, for a bounded arch-like region it is shown that the measure decays at least algebraically with respect to r, while for an unbounded region our result reveals a relationship with the classical Phragmèn—Lindelöf theorem.

On a new method for the study of the spatial behavior in a homogeneous elastic arch-like region

We consider a two-dimensional homogeneous elastic state in the arch-like region a ≤ r ≤ b, 0 ≤ θ ≤ α, where (r,θ) denotes plane polar coordinates. We assume that three of the edges are traction-free, while the fourth edge is subjected to a (in plane) self-equilibrated load. The Airy stress function ‘ϕ’ satisfies a fourth-order differential equation in the plane polar coordinates with appropriate boundary conditions. We develop a method which allows us to treat in a unitary way the two problems corresponding to the self-equilibrated loads distributed on the straight and curved edges of the region. In fact, we introduce an appropriate change for the variable r and for the Airy stress functions to reduce the corresponding boundary value problem to a simpler one which allows us to indicate an appropriate measure of the solution valuable for both the types of boundary value problems. In terms of such measures we are able to establish some spatial estimates describing the spatial behavior of the Airy stress function. In particular, our spatial decay estimates prove a clear relationship with the Saint-Venant's principle on such regions.

On semi-inverse solutions for the time-dependent flows of a second-grade fluid

This paper deals with analytical solutions for the time-dependent equations arising in a second-grade fluid. The solutions have been developed by assuming certain forms of the stream function. Expressions for velocity components are obtained for flows in plane polar, axisymmetric cylindrical, and axisymmetric spherical polar coordinates. The obtained solutions are compared with existing results.

On Saint-Venant's Principle for a Homogeneous Elastic Arch-Like Region

In this paper, we consider a two-dimensional homogeneous isotropic elastic material state in the arch-like region a ≤ r ≤ b, 0 ≤ θ ≤ α, where (r, θ) denote plane polar coordinates. We assume that three of the edges r = a, r = b, θ = α are traction-free, while the edge θ = 0 is subjected to an (in plane) self-equilibrated load. We define an appropriate measure for the Airy stress function φ and then we establish a clear relationship with the Saint-Venant's principle on such regions. We introduce a cross-sectional integral function I(θ) which is shown to be a convex function and satisfies a second-order differential inequality. Consequently, we establish a version of the Saint-Venant principle for such a curvilinear strip, without requiring of any condition upon the dimensions of the arch-like region.

On Saint-Venant's Principle for an Inhomogeneous Curvilinear Rectangle

A two-dimensional inhomogeneous isotropic elastic material is considered in the arch-like region a ≤ r ≤ b,0 ≤ θ ≤ α, where (r,θ) denotes plane polar coordinates. It is envisaged that three of the edges r = a, r = b, θ = α are traction-free, while the edge θ = 0 is subjected to an (in-plane) self-equilibrated load. An appropriate energy-like measure E(θ) of the Airy stress function φ in the region between arbitrary θ and θ = α is defined, and it is proved to be positive definite provided that some appropriate assumptions are satisfied by the material and geometric characteristics of the arch-like region. Then, a version of Saint-Venant's Principle for the curvilinear strip is established.

The structure of an unstable circular vortex in a background straining flow

The formation of compound vortices from a ‘shielded’ vortex monopole embedded in a two-dimensional background straining flow is studied numerically. The computations are performed in plane polar coordinates using an infinite radial domain Navier–Stokes solver. The straining flow excites an azimuthal instability, leading to vortex tripole formation in the case of small strain rates. For larger strain rates, tripole formation proceeds briefly before the structure is destroyed by the dominant background straining flow.

Duelling idiots and other probability puzzlers

This book comprises 21 problems in probability, some having several parts, that are fairly discursively set out over 65 pages. Their solutions occupy 95 pages, and there are a further 90 pages including many MATLAB programs and an essay on random number generation. The text includes a good deal of personal, technical and historical background to the topics under discussion, between solid chunks of mathematical working. Most of the problems use discrete probability and are of a classical nature, with urns, dice, cards and Markov chains in the background. However, these questions are, for the most part, given a fresh twist in the context of family sizes, baseball games, duelling idiots and so on. A minority of around half a dozen problems are essentially geometrical probability. Once again these are classical examples, but not the ones most commonly seen in textbooks. Overall, even experienced probabilists would need to be quite well read (and with the memory of an elephant) not to find something unfamiliar in this collection. One principal difference from the general run of publications of this type lies in the emphasis on simulation and computation, as evidenced by the 42 MATLAB programs that appear at the end of the book. This may be due to the author's background in electrical engineering, and I would guess is likely to increase its appeal in these quarters. A curious feature of the book is the occasional reliance on brute force, even where a briefer and more attractive solution is not far to seek. There are several examples, but one might single out the solution to problem 18, where three pages are spent hammering out a double integral; but it does in fact yield quickly and quietly to a simple standard polar transformation. Possibly the author intended to minimize the technical expertise required of the reader, but plane polar coordinates will surely be accessible to anyone capable of following the given derivation. At the design level, it is very irritating that the running heads are in fact running feet, and that they do not include the name or number of the problem or solution in progress on the page. This elementary editorial omission wears the reader's patience very thin, very quickly. In brief, there is a place for this book in the collection of any keen probabilist with an eye for some of the more unusual elementary problems in the field. And in addition, it should be of interest to any computing- or mathematics-enabled reader looking for puzzles in probability. D Stirzaker