Open Access. Powered by Scholars. Published by Universities.®

Physics Commons™

Open Access. Powered by Scholars. Published by Universities.®

Journal

2018

Discipline
Institution
Keyword
Publication
File Type

Articles 1 - 30 of 137

Full-Text Articles in Physics

Table Of Contents Dec 2018

Table Of Contents

Journal of the South Carolina Academy of Science

No abstract provided.

Go to article

The Circular Restricted Four-Body Problem With Triaxial Primaries And Variable Infinitesimal Mass, Abdullah A. Ansari Dec 2018

The Circular Restricted Four-Body Problem With Triaxial Primaries And Variable Infinitesimal Mass, Abdullah A. Ansari

Applications and Applied Mathematics: An International Journal (AAM)

This paper investigates the circular restricted four-body problem in which three primaries are taken as triaxial rigid body which are placed at the vertices of an equilateral triangle and the fourth infinitesimal body is varying its mass with time. We used the Jeans law to determine equations of motion and then evaluated the Jacobi integral. In the next section, we have performed the computational work to draw the graphs of the equilibrium points in different planes, zero velocity curves, surfaces and the Newton-Raphson basins of attraction with the variations of the triaxiality parameters. Finally, we have examined the linear stability …

Go to article

Soret Effect On Transient Mhd Convective Flow Past A Semi-Infinite Vertical Porous Plate With Heat Sink And Chemical Reaction, K. Choudhury, N. Ahmed Dec 2018

Soret Effect On Transient Mhd Convective Flow Past A Semi-Infinite Vertical Porous Plate With Heat Sink And Chemical Reaction, K. Choudhury, N. Ahmed

Applications and Applied Mathematics: An International Journal (AAM)

This problem concerns with an analytical study of heat and mass transfer on unsteady MHD convective flow past a semi-infinite porous vertical plate in presence of thermal diffusion, heat sink and chemical reaction. A uniform magnetic field is imposed transversely to the plate. The governing equations are solved by perturbation technique. The expressions for the velocity, temperature and concentration fields as well as transport properties are obtained in non-dimensional form. The influence of the physical parameters like magnetic field parameter, Schmidt number, Prandtl number, heat sink parameter, chemical reaction parameter and Soret number on these fields is discussed through graphs …

Go to article

Numerical Treatment For The Flow Of Casson Fluid And Heat Transfer Model Over An Unsteady Stretching Surface In The Presence Of Internal Heat Generation/Absorption And Thermal Radiation, Mohammed M. Babatin Dec 2018

Numerical Treatment For The Flow Of Casson Fluid And Heat Transfer Model Over An Unsteady Stretching Surface In The Presence Of Internal Heat Generation/Absorption And Thermal Radiation, Mohammed M. Babatin

Applications and Applied Mathematics: An International Journal (AAM)

Several important industrial and engineering problems are very difficult to solve analytically since they are high nonlinear. The Chebyshev spectral collocation method possesses an ability to predict the solution behavior for a system of high nonlinear ordinary differential equations. This method which is based on differentiated Chebyshev polynomials is introduced to obtain an approximate solution to the system of ordinary differential equations which physically describe the flow and heat transfer problem of an unsteady Casson fluid model taking into consideration both heat generation and radiation effects in the temperature equation. Based on the spectral collocation method, the obtained solution is …

Go to article

Numerical Studies For Mhd Flow And Gradient Heat Transport Past A Stretching Sheet With Radiation And Heat Production Via Dtm, Khadijah M. Abualnaja Dec 2018

Numerical Studies For Mhd Flow And Gradient Heat Transport Past A Stretching Sheet With Radiation And Heat Production Via Dtm, Khadijah M. Abualnaja

Applications and Applied Mathematics: An International Journal (AAM)

This paper presents a numerically study for the effect of the internal heat generation, magnetic field and thermal radiation effects on the flow and gradient heat transfer of a Newtonian fluid over a stretching sheet. By using a similarity transformation, the governing PDEs can be transformed into a coupled non-linear system of ODEs with variable coefficients. Numerical solutions for these equations subject to appropriate boundary conditions are obtained by using the differential transformation method (DTM). The effects of various physical parameters such as viscosity parameter, the suction parameter, the radiation parameter, internal heat generation or absorption parameter and the Prandtl …

Go to article

Conformable Derivative Operator In Modelling Neuronal Dynamics, Mehmet Yavuz, Burcu Yaşkıran Dec 2018

Conformable Derivative Operator In Modelling Neuronal Dynamics, Mehmet Yavuz, Burcu Yaşkıran

Applications and Applied Mathematics: An International Journal (AAM)

This study presents two new numerical techniques for solving time-fractional one-dimensional cable differential equation (FCE) modeling neuronal dynamics. We have introduced new formulations for the approximate-analytical solution of the FCE by using modified homotopy perturbation method defined with conformable operator (MHPMC) and reduced differential transform method defined with conformable operator (RDTMC), which are derived the solutions for linear-nonlinear fractional PDEs. In order to show the efficiencies of these methods, we have compared the numerical and exact solutions of fractional neuronal dynamics problem. Moreover, we have declared that the proposed models are very accurate and illustrative techniques in determining to approximate-analytical …

Go to article

Thermoelastic Stress Analysis Of A Functionally Graded Transversely Isotropic Hollow Cylinder In Elliptical Coordinates, Tara Dhakate, Vinod Varghese, Lalsingh Khalsa Dec 2018

Thermoelastic Stress Analysis Of A Functionally Graded Transversely Isotropic Hollow Cylinder In Elliptical Coordinates, Tara Dhakate, Vinod Varghese, Lalsingh Khalsa

Applications and Applied Mathematics: An International Journal (AAM)

This paper is concerned with the axisymmetric thermoelastic problem to investigate the influence of nonlinear heat conduction equation, displacement functions and thermal stresses of a functionally graded transversely isotropic hollow cylinder that is presented in the elliptical coordinate system. The method of integral transform technique is used to produce an exact solution of the heat conduction equation in which sources are generated according to a linear function of the temperature. An explicit exact solution of the governing thermoelastic equation is proposed when material properties are power-law functions with the exponential form of the radial coordinate. Numerical calculations are also carried …

Go to article

Quasi-Linearization Method With Rational Legendre Collocation Method For Solving Mhd Flow Over A Stretching Sheet With Variable Thickness And Slip Velocity Which Embedded In A Porous Medium, M. M. Khader Dec 2018

Quasi-Linearization Method With Rational Legendre Collocation Method For Solving Mhd Flow Over A Stretching Sheet With Variable Thickness And Slip Velocity Which Embedded In A Porous Medium, M. M. Khader

Applications and Applied Mathematics: An International Journal (AAM)

The quasi-linearization method (QLM) and the rational Legendre functions are introduced here to present the numerical solution for the Newtonian fluid flow past an impermeable stretching sheet which embedded in a porous medium with a power-law surface velocity, variable thickness and slip velocity. Firstly, due to the high nonlinearity which yielded from the ordinary differential equation which describes the proposed physical problem, we construct a sequence of linear ODEs by using the QLM, hence the resulted equations become a system of linear algebraic equations. The comparison with the available results in the literature review proves that the obtained results via …

Go to article

Similarity Analysis Of Three Dimensional Nanofluid Flow By Deductive Group Theoretic Method, Hemangini Shukla, Hema C. Surati, M. G. Timol Dec 2018

Similarity Analysis Of Three Dimensional Nanofluid Flow By Deductive Group Theoretic Method, Hemangini Shukla, Hema C. Surati, M. G. Timol

Applications and Applied Mathematics: An International Journal (AAM)

The objective of this paper is to obtain similarity solution of three-dimensional nanofluid flow over flat surface stretched continuously in two lateral directions. Two independent variables from governing equations are reduced by applying deductive two parameter group theoretical method. Partial differential equations with boundary conditions are converted into ordinary differential equations with appropriate boundary conditions. Obtained equations are solved for temperature and velocity. The effect of nanoparticles volume fraction on temperature and velocity profile is investigated.

Go to article

The Basins Of Convergence In The Planar Restricted Four-Body Problem With Variable Mass, Amit Mittal, Monika Arora, Md S. Suraj, Rajiv Aggarwal Dec 2018

The Basins Of Convergence In The Planar Restricted Four-Body Problem With Variable Mass, Amit Mittal, Monika Arora, Md S. Suraj, Rajiv Aggarwal

Applications and Applied Mathematics: An International Journal (AAM)

We have studied the existence, location and stability of the libration points in the model of restricted four-body problem (R4BP) with variable mass. It is assumed that three primaries, one dominant primary and the other two with equal masses, are always forming an equilateral triangle. We have determined the equations of motion of the above mentioned problem for the fourth body which is an infinitesimal mass. The libration points have been determined numerically for different values of the parameters considered. It is found that there are eight or ten libration points out of which six are non-collinear and two or …

Go to article

Fractional Order Thermoelastic Thick Circular Plate With Two Temperatures In Frequency Domain, Parveen Lata Dec 2018

Fractional Order Thermoelastic Thick Circular Plate With Two Temperatures In Frequency Domain, Parveen Lata

Applications and Applied Mathematics: An International Journal (AAM)

The present investigation is concerned with thermomechanical interactions in the fractional theory of thermoelasticity for a homogeneous isotropic thick circular plate in the light of two-temperature thermoelasticity theory in frequency domain. The upper and lower surfaces of the thick plate are traction free and subjected to an axisymmetric heat supply. The solution is found by using Hankel transform technique and a direct approach without the use of potential functions. The analytical expressions of displacement components, stresses, conductive temperature, temperature change and cubic dilatation are computed in transformed domain. Numerical inversion technique has been applied to obtain the results in the …

Go to article

Analysis Of A Recent Heat Conduction Model With A Delay For Thermoelastic Interactions In An Unbounded Medium With A Spherical Cavity, Bharti Kumari, Anil Kumar, Manushi Gupta, Santwana Mukhopadhyay Dec 2018

Analysis Of A Recent Heat Conduction Model With A Delay For Thermoelastic Interactions In An Unbounded Medium With A Spherical Cavity, Bharti Kumari, Anil Kumar, Manushi Gupta, Santwana Mukhopadhyay

Applications and Applied Mathematics: An International Journal (AAM)

In this work, we study the thermoelastic interactions in an unbounded medium with a spherical cavity in the context of a very recently proposed heat conduction model established by Quintanilla (2011). This model is a reformulation of three-phase-lag conduction model and is an alternative heat conduction theory with a single delay term. We make an attempt to study the thermoelastic interactions in an isotropic elastic medium with a spherical cavity subjected to three types of thermal and mechanical loads in the contexts of two versions of this new model. Analytical solutions for the distributions of the field variables are found …

Go to article

Linear Stability Analysis With Solution Patterns Due To Varying Thermal Diffusivity For A Convective Flow In A Porous Medium, Dambaru Bhatta Dec 2018

Linear Stability Analysis With Solution Patterns Due To Varying Thermal Diffusivity For A Convective Flow In A Porous Medium, Dambaru Bhatta

Applications and Applied Mathematics: An International Journal (AAM)

Here we investigate the effect of the vertical rate of change in thermal diffusivity due to a hydrothermal convective flow in a horizontal porous medium. The continuity equation, the heat equation and the momentum-Darcy equation constitute the governing system for the flow in a porous medium. Assuming a vertically varying basic state, we derive the linear system and from this linear system, we compute the critical Rayleigh and wave numbers. Using fourth-order Runge-Kutta and shooting methods, we obtain the marginal stability curves and linear solutions to analyze the solution pattern for different diffusivity parameters.

Go to article

Restricted Three-Body Problem Under The Effect Of Albedo When Smaller Primary Is A Finite Straight Segment, Shipra Chauhan, Dinesh Kumar, Bhavneet Kaur Dec 2018

Restricted Three-Body Problem Under The Effect Of Albedo When Smaller Primary Is A Finite Straight Segment, Shipra Chauhan, Dinesh Kumar, Bhavneet Kaur

Applications and Applied Mathematics: An International Journal (AAM)

This paper addresses the dynamics of the infinitesimal body in the restricted three-body problem under the effect of Albedo when the smaller primary is a finite straight segment and bigger one is a source of radiation. The measure of diffusive reflection of solar radiation out of the total solar radiation received by a body is Albedo which is measured on a scale from 0 to 1. The equations of motion of the infinitesimal body are derived and it is found that there exist five libration points, out of which three are collinear and the rest are non-collinear with the primaries. …

Go to article

A New American Electron Ion Collider On The Way: What It Means For Particle Accelerator Science And Duquesne Students, Courtney Boyd Nov 2018

A New American Electron Ion Collider On The Way: What It Means For Particle Accelerator Science And Duquesne Students, Courtney Boyd

D.U.Quark

No abstract provided.

Go to article

Spectra Of One – And Two Photon Excited Luminescence Of Medical Drugsin Photon Traps, A.K. Kubanov, D.I. Semenov Oct 2018

Spectra Of One – And Two Photon Excited Luminescence Of Medical Drugsin Photon Traps, A.K. Kubanov, D.I. Semenov

Scientific Journal of Samarkand University

A technique for studying the photoluminescence spectra of pharmaceuticals (analgin, paracetamol) was developed for single- and two-photon excitation by a pulse-periodic laser. The technique is based on the fiber-optic recording of photoluminescence spectra using a small-size spectrometer, quartz light guides, and mini cavity cuvettes, which makes it possible to compare the analyzed spectrum with the spectrum of the reference substance.

Go to article

Laser Spectroscopy Of Ultra-Small Concentration Of Atoms And Aerosols In Various Phase States Of Substance, А.Т. Xalmanov Oct 2018

Laser Spectroscopy Of Ultra-Small Concentration Of Atoms And Aerosols In Various Phase States Of Substance, А.Т. Xalmanov

Scientific Journal of Samarkand University

In this work it is given studying of dynamics and types of physical processes with excited atoms and molecules in atomizer of "graphite furnace". Physical mechanism of increasing atomic-ionization signal for two stepwise excitation schemes was shown. It is shown that the maximum signal is observed for In atoms at the transition 5p 2 Р3/2→6s 2 S3/2→8p 2 P1/2 and for Li, Na, Ag, Au, Pt atoms at the transition (n)s→(m)p→(k)d respectively (where n, k, m and f principal quantum numbers of states). In the result in the experiment photoionization collisional ionization and mechanisms were developed. The results are show …

Go to article

Probabilities Involving Standard Trirectangular Tetrahedral Dice Rolls, Rulon Olmstead, Doneliezer Baize Oct 2018

Probabilities Involving Standard Trirectangular Tetrahedral Dice Rolls, Rulon Olmstead, Doneliezer Baize

Rose-Hulman Undergraduate Mathematics Journal

The goal is to be able to calculate probabilities involving irregular shaped dice rolls. Here it is attempted to model the probabilities of rolling standard tri-rectangular tetrahedral dice on a hard surface, such as a table top. The vertices and edges of a tetrahedron were projected onto the surface of a sphere centered at the center of mass of the tetrahedron. By calculating the surface areas bounded by the resultant geodesics, baseline probabilities were achieved. Using a 3D printer, dice were constructed of uniform density and the results of rolling them were recorded. After calculating the corresponding confidence intervals, the …

Go to article

The Battle Between Impeccable Intonation And Maximized Modulation, Timothy M. True Oct 2018

The Battle Between Impeccable Intonation And Maximized Modulation, Timothy M. True

Musical Offerings

Equal temperament represents a way of completing the musical circle, and systematically compensating for the Pythagorean comma. Pythagoras discovered this acoustical problem around 550 B.C., and since that time music theorists have debated how to deal with it. The problem is that no perfect solution exists—something must be compromised. As musical styles developed, specific factors and harmonic tendencies led to the gradual adoption of equal temperament. Early in music history, theorists preferred systems which kept acoustical purity relatively intact. Pythagorean intonation and just intonation serve as two examples. However, the move from modality to tonality decentralized the melody as the …

Go to article

Electronic Specific-Heat Anomalies In High-TC Cuprates, S. Dzhumanov, E. Karimboev, B. Yavidov Sep 2018

Electronic Specific-Heat Anomalies In High-TC Cuprates, S. Dzhumanov, E. Karimboev, B. Yavidov

Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences

In this work, we study the electronic specific heat $C_e(T)$ of underdoped to overdoped high-$T_c$ cuprates, and identify the nature of anomalies in $C_e(T)$ at the superconducting transition temperature $T_c$ and at temperatures above $T_c$. The doped cuprate superconductor is considered as a multi-carrier model system which is composed of different types of charge carriers. The normal-state electronic specific heat $C_n(T)$ of high-$T_c$ cuprates below a characteristic pseudogap (PG) temperature $T^*$ is calculated taking into account three contributions coming from the excited components of Cooper pairs, the ideal Bose-gas of incoherent Cooper pairs and the unpaired carriers in the impurity …

Go to article

Analysis Of Clusterization Of C70 Molecules In Benzene Solutions Prepared By Various Methods, U. Makhmanov, O. Ismailova, A. Kokhkharov, S. Bakhramov Sep 2018

Analysis Of Clusterization Of C70 Molecules In Benzene Solutions Prepared By Various Methods, U. Makhmanov, O. Ismailova, A. Kokhkharov, S. Bakhramov

Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences

Self-assembling features of molecules of C70 in benzene solution prepared in two various ways – equilibrium and non-equilibrium – has been investigated experimentally by method of high-resolution transmission electron microscopy. It was demonstrated, that the formation of densely packed monomolecular fullerene aggregates with a diameter of not more than 30 nm in solutions prepared by the equilibrium method (without the use of external mechanical influences on solution). In the case of solutions C70, which were prepared by the non-equilibrium method (stirring of solution of C70 by a mechanical rotator), large quasispherical aggregates of nanoporous structure with …

Go to article

Electron Exchange Between Impurity Centers Of Tin In Pbs And Pbse, S. Zaynabiddinov, K. Bobokhujaev Sep 2018

Electron Exchange Between Impurity Centers Of Tin In Pbs And Pbse, S. Zaynabiddinov, K. Bobokhujaev

Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences

Electron exchange between neutral and doubly ionized U-minus centers of tin in par-tially compensated solid solutions Pb1-xSnxS and Pb1-xSnxSe has been investigated using 119mmSn(119mSn) emission Mössbauer spectroscopy. The activation energy of the process for Pb1-xSnxS solid solutions is 0.11 ± 0.02 eV and is comparable to the depth of energy levels of tin in the PbS band gap, while the activation energy of this process for Pb1-xSnxSe solid solutions is comparable to the activation energy of donor U-minus centers …

Go to article

A Nonlinear Approximate Solution To The Damped Pendulum Derived Using The Method Of Successive Approximations, Justin A. Hill, Javier E. Hasbun 8394092 Sep 2018

A Nonlinear Approximate Solution To The Damped Pendulum Derived Using The Method Of Successive Approximations, Justin A. Hill, Javier E. Hasbun 8394092

Georgia Journal of Science

An approximate analytic solution to the damped pendulum is derived using the method of successive approximations to obtain a nonlinear approximation for the system. We take the approximate solution to the undamped pendulum using the method of successive approximations and compare it to the damped pendulum solution when a linear approximation is used. By looking at these two solutions, we can make an educated guess about the form of the general, approximate solution to the nonlinear damped pendulum. By adjusting the initial guesses and the initial conditions, we derive approximate solutions in three ways. Using MATLAB, the approximate solutions are …

Go to article

New Mechanism For Accelerated Removal Of Excess Radiogenic Heat, Russell Humphreys Jul 2018

New Mechanism For Accelerated Removal Of Excess Radiogenic Heat, Russell Humphreys

Proceedings of the International Conference on Creationism

In a technical paper (Humphreys, 2014), I presented Biblical and scientific evidence that (a) space is a physical material that we do not perceive, (b) this fabric of space, and objects within it, are thin in a 4th spatial direction we do not ordinarily perceive, and (c) the fabric is surrounded by a hyperspace of four spatial dimensions. End note 27 of the paper explained that light emitted by objects within the fabric ordinarily would be constrained to travel entirely within the fabric. The end note also proposed that under certain extraordinary conditions the Bible calls the opening of the …

Go to article

Modeling Of Flood And Post-Flood Ocean Floor Cooling, William J. Worraker, Richard Ward Jul 2018

Modeling Of Flood And Post-Flood Ocean Floor Cooling, William J. Worraker, Richard Ward

Proceedings of the International Conference on Creationism

Given that the earth’s ocean basins are geologically young, few areas being older than early Jurassic, and that most creation scientists regard Jurassic rocks as Flood deposits, these basins must have formed during and since the Flood, i.e. within no more than 4500 years. This paper represents a first attempt at modeling ocean basin formation by the separation of the continents and cooling of mantle material emplaced at spreading centres well within that limited time. We use a spreadsheet-based finite difference solution of the heat diffusion equation applied to a simple widely-used plate model of ocean lithosphere formation. Having verified …

Go to article

Numerical Investigation Of Strength-Reducing Mechanisms Of Mantle Rock During The Genesis Flood, Noah Cho, John Baumgardner, Jesse Sherburn, Mark Horstemeyer Jul 2018

Numerical Investigation Of Strength-Reducing Mechanisms Of Mantle Rock During The Genesis Flood, Noah Cho, John Baumgardner, Jesse Sherburn, Mark Horstemeyer

Proceedings of the International Conference on Creationism

This paper reports our efforts to model the effects of grain size, recrystallization, creep, and texture on overall rock strength within the Earth’s mantle during the Genesis Flood. Our study uses experimental rheological data obtained from the mineralogical literature for olivine, which is an important mantle mineral. We apply an Internal State Variable (ISV) constitutive model within the framework of the TERRA finite element code to capture the subscale structures and their associated dynamics, strength, and viscosity effects during the Flood episode. Our numerical investigations, in both 2D and 3D, that include the improved deformation model reveal even more clearly …

Go to article

Effect Of Aerosol Distributions On Precipitation Patterns Needed For A Rapid Ice Age, Steven M. Gollmer Jul 2018

Effect Of Aerosol Distributions On Precipitation Patterns Needed For A Rapid Ice Age, Steven M. Gollmer

Proceedings of the International Conference on Creationism

Introduced in the Genesis Flood by Whitcomb and Morris (1961) and fleshed out by Oard (1979) a model for an ice age in the wake of the Genesis flood was used to explain the evidence of glaciation in Canada and the United States without resorting to eons of time. It was proposed that this rapid ice age was the consequence of post flood warm oceans, barren land and volcanic aerosols. The impact of warm oceans was simulated by Vardiman (1998) and Gollmer (2013) using climate models. Although warm oceans increase precipitation in the Arctic, global surface temperatures become unbearably hot …

Go to article

59th Annual Rocky Mountain Conference On Magnetic Resonance Jul 2018

59th Annual Rocky Mountain Conference On Magnetic Resonance

Rocky Mountain Conference on Magnetic Resonance

Final program, abstracts, and information about the 59th annual meeting of the Rocky Mountain Conference on Magnetic Resonance, co-endorsed by the Colorado Section of the American Chemical Society and the Society for Applied Spectroscopy. Held in Snowbird, Utah, July 22-27, 2018.

Go to article

Fuel Flow Reduction Impact Analysis Of Drag Reducing Film Applied To Aircraft Wings, Damon Resnick, Chris Donlan, Nimish Sakalle, Cody Pinkerman Jul 2018

Fuel Flow Reduction Impact Analysis Of Drag Reducing Film Applied To Aircraft Wings, Damon Resnick, Chris Donlan, Nimish Sakalle, Cody Pinkerman

SMU Data Science Review

In this paper, we present an analysis of flight data in order to determine whether the application of the Edge Aerodynamix Conformal Vortex Generator (CVG), applied to the wings of aircraft, reduces fuel flow during cruising conditions of flight. The CVG is a special treatment and film applied to the wings of an aircraft to protect the wings and reduce the non-laminar flow of air around the wings during flight. It is thought that by reducing the non-laminar flow or vortices around and directly behind the wings that an aircraft will move more smoothly through the air and provide a …

Go to article

Gamma-Ray Burst Classification: New Insights From Mining Pulse Data, Stanley Mcafee, Jon Hakkila Jul 2018

Gamma-Ray Burst Classification: New Insights From Mining Pulse Data, Stanley Mcafee, Jon Hakkila

Journal of the South Carolina Academy of Science

Despite being the most energetic electromagnetic explosions in the universe, gamma-ray bursts (GRBs) are still poorly understood. The literature recognizes two potentially different types of GRB progenitors, although statistical data suggest the existence of three GRB classes. Reliable inference of GRB physics depends on the identification of appropriate classification attributes, as well as on the statistical classification techniques used. It has recently been shown that pulses are the basic unit of GRB emission. We use new data describing GRB pulse characteristics, in conjunction with data mining tools, to provide a more reliable gamma-ray burst classification system and place additional constraints …

Go to article

Next Last