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CHEMISTRY AND MEDICINE AND LAGRANGIAN OPERATOR
ABSTRACT

To find out the dependence of pressure on equilibrium temperature when two phases coexist.

Along a phase transition line, the pressure and temperature are not independent of each other, since the system is univariant, that is, only one intensive parameter can be varied independently.When the system is in a state of equilibrium, i.e., thermal, mechanical and chemical equilibrium, the temperature of the two phases has to be identical, the pressure of the two phases has to be equal and the chemical potential also should be the same in both the phases.
INTRODUCTION

Consider a system consisting of a liquid phase at state 1 and a vapour phase at state 1’ in a state of equilibrium. Let the temperature of the system is changed from T1 to T2 along the vaporization curve.

For
the phase transition for 1 to 1’:

or
or

In reaching

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METHODS IN MEDICINE AND CHEMISTRY WITH LAGRANGIAN
ABSTRACT

The J0 Bessel function
The equation for J0 Bessel is the zeroth order Bessel equation
.
The “standard form” of differential equations is often specified as having the coefficient of the highest order derivative cancelled through. Thus in standard form the equation would be written
.

Our procedure for the series solution of this equation is to take the assumed series

and substitute it into the equation. This involves using the first and second derivatives

and

In the expression for the second derivative we have (as in the sine/cosine case above) shifted the dummy summation variable by 2 so that the sum expression contains xn explicitly.

So far we have left the sum for the first derivative unchanged. The point here is that what the differential equation contains is , and the expression for this must be written as a sum in xn. For this reason we shift the dummy variable in the series for the first derivative by 2:

so that
.
We now substitute the series expressions into the original differential equation:
.
The term in the second sum may be treated separately. In that case everything else falls within a sum over n from 0 to .
1 INTRODUCTION
So the solution to the ODE which we have discovered is a constant times the J0 Bessel function
.
Thus far this is quite good; we have discovered a new function which solves the above differential equation. But it is a second order differential equation and therefore, as with the previous SHO equation, there should be two independent solutions. Where is the other solution?

When we examined the solution of the wave equation for a drumhead we found the separated radial equation took the form of the zeroth order Bessel equation. And at that stage we simply noted that Mathematica gave, as independent solutions to that equation, the two zeroth order Bessel functions J0(x) and Y0(x). We plotted the functions and the behaviour of the functions in the vicinity of x = 0 gives us an important clue about the “other” solution.

J0(x) and Y0(x) Bessel functions

The J0(x) function goes to 1 as x goes to 0. This we see on the plot and we have discovered this in the series solution. The Y0(x) function, on the other hand, looks as if it is heading for minus infinity as x goes to 0. That is the problem.

Recall the point made when we introduced the power series method. A series

will only work when the function is “well behaved”. This is OK for J0(x), but going off to infinity is an example of “bad behaviour”; then a simple power series won’t work. We will see how to overcome this in a later section.

The important concepts of this section are:

• The simple power series method works only for “well-behaved” functions; it cannot cope with “badly-behaved” functions.

• The basic idea is to substitute the power series into the differential equation.

• With a slight juggling of the dummy summation variables the equation is cast into the form . Special care must be taken with the first derivative term.

• Each power of x must equate to zero.

• The term in x-1 must be treated separately; this tells us that there is no simple series in odd powers of x. The series method is only going to give us one solution to the ODE, which is even in x.

• Equating the general term in xn to zero gives a recurrence relation for the coefficients.

• We build up one solution to the ODE from the a0 coefficient.

• A 2nd order ODE has two independent solutions; where is the other? We recognise that the simple power series method can’t cope with it as it is “badly-behaved”.

• Properties of the J0 function are obtained from the series solutions and the original ODE.

Legendre’s equation
Legendre’s equation follows from separating the laplacian in spherical polar coordinates. This equation arises from the separated equation in the polar angle . Legendre’s equation is
.
In this equation n is often a positive integer; we will explore this a little later.

As before, we start with a power series expression for the function

Now, however, we will use Mathematica to obtain the recurrence relation for the coefficients. This is outlined in the Mathematica Notebook “Legendre”. The recurrence relation is
.

A series in even powers of x will be built up from a0 and a series in odd powers of x will be built up from odd powers of x. The general solution to the Legendre equation is thus

where

Often n is a positive integer. In that case: if n is even then the series for will terminate at the xn term, while if n is odd then the series for will terminate at the xn term. These solutions, normalised to , are called the Legendre polynomials, denoted by . The first few are given by

They are plotted in the following figure

First few Legendre polynomials

When n is an integer one series solution of the Legendre terminates and we thus have the Legendre polynomials The other series solution does not terminate. These are denoted by , and they can be expressed in terms of logarithms:

They are plotted in the following figure

First few Legendre Qn functions

The general solution of the Legendre equation will be
.
But note that has a logarithmic divergence at ; the only well-behaved solutions of Legendre’s equation for integer n are the Legendre polynomials

The important concepts of this section are:

• Mathematica can be used to derive the coefficient recurrence relation by substituting some general terms of the power series into the differential equation.

• The recurrence relation connects every other coefficient.

• Therefore there are two independent solutions, one in the even powers of x and one in the odd powers.

• For the Legendre equation with integer n the coefficients of one of the series solutions will terminate.

• If n is even then the series for will terminate at the xn term, while if n is odd then the series for will terminate at the xn term.

• With appropriate normalisation these are the Legendre polynomials.

so here we go with some details as prof dr mircea orasanu and prof horia orasanu and
ABSTRACT
Alexandru NICA

Born 26 July 1931, Bacau.
Education: Polytechnic Institute (now ‘Politehnica’ University) Bucharest (graduated 1955); Ph.D. in Romanian Academy (1968).
Current employment: head of technical staff, Aeronautical College, Bucharest-Baneasa. Office address: Colegiul de Aeronautica, bd. Ficusului 44, RO-71544 Bucharest. Phone: (021 ) 6333616.
Home address: str. Gura Motrului 1-3, bl. XIII/4, sc. B, ap. 37, RO-71546 Bucharcst. Phone (021) 6662333.
Fields of research: tribology (friction, wear, lubricalion), materials, mechanics, as aplied to machine building and particulary to aeronautical structures and powerplants, economics and management.
Professional record: researcher, senior researcher, head of Hydronamic Lubrication Laboratory (1955-78) Institute of Applied Mechanics of the Romanian Academy (later on the Institute of Fluid Mechanics and Aerospace Constructions); head of Tribology Laboratory, National Institute of Thermal Engines; Professor of mechanical engineering, Aeronautical College, Bucharest-Baneasa (1978-84); on editorial board: Applied Mechanics Reviews, San Antonio, Texas (1968). Membership: Studsvik Nuclear AB, Nyköping, Sweden (1990), Romanian Association of Consultants in Management, AMCOR (1991). Awards: Traian Vuia Prize of the Romanian Academy (1965).
Humboldt Research Fellowship : machine building, Max Planck Institut für Strömungsforschung, Prof. Dr. Georg Vogelpohl (1970-71); Technische Universität Clausthal-Zellerfeld, Abteilung für Reibungsforschung, Prof. G. Noack (1990).
ublications include:

A. Books:

1. Lagare cu alunecare (“Sliding Bearings”). 1st ed., Bucharest: Editura Academiei, 1962; russian ed., Moscow: Mashghiz, 1965; 3th ed., Bucharest: Editura Tehnica, 1980; English ed., New York: Allerton Press, 1985 (in collaboration).
2. Sisteme de lubrificatie (“Lubrication Systems”). Bucharest: Editura Academici, 1968.
3. Theory and Practice of Lubrication Systems. Broseley: Scientific Publication, 1969.
4. Sisteme de lubrificatie, alimentare cu carburant si fluide hidraulice pentru constructiile aerospatiale (“Aerospace Lubrication, Fuel and Hydraulic Systemes”). Bucharest: Editura Academiei, 1976 (in collaboration).
5. Mecarica materialelor pentru constructii aerospatiale (“Mechanics of Aerospace Materials”). Bucharest: Editura Academici, 1978 (in collaboration); English ed., Amsterdam: Elsevier,1981.
6. Alegerea si utilizarea lubrifiantilor si combustibililor pentru motoare termice (” Selection and Utilizalion of Oils and Fuels for Thermal Engines”). Bucharest: Editura Tehnica, 1978 (in collaboration).
7. Bazele fabricatiei navelor aerospatiale (“Basic Engineering of Aerospace Vehicles”). Bucharest: Editura Tehnica, 1986 (in collaboration).
8. Ceramica tehnica (“Technical Ceramics”). Bucharest: Editura Tehnica,1988.
9. Motoare si instalatii ale aeronavelor (“Aviation Powerplants and Appliances”). Bucharest: Editura Didactica si Pedagogica, 1980 Ist ed.; 1982-2nd ed.; 3rd ed. (in collaboration).

B. Papers:

1. “Contributions to the Determination of Real Clearnce in Sliding Bearings.” Trans. ASME, Journal of Basic Eng., 87, 196, 3.
2. “Beitrag zur Bestimmung der Zähigkeit im Schmierfilm.” Freiherger Forschungshefte,1966.
3. “Tribological Aspects of PCMI in Fuel Rods: Stick-Slip Phenomena.” Studsvik Nuclear AB, 1992-1993.
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are known some observations and therefore these appear in studies as observed prof dr mircea orasanu and prof horia orasanu as followed
CHEMISTRY IN CONNECTION WITH MATHEMATICS OPERATORS
ABSTRACT
Velocity Potentials and Stream Functions
As we have seen, a two-dimensional velocity field in which the flow is everywhere parallel to the – plane, and there is no variation along the -direction, takes the form
(5.16)

Moreover, if the flow is irrotational then is automatically satisfied by writing , where is termed the velocity potential. (See Section 4.15.) Hence,
(5.17)
(5.18)

On the other hand, if the flow is incompressible then is automatically satisfied by writing , where is termed the stream function. (See Section 5.2.) Hence,
(5.19)
(5.20)

Finally, if the flow is both irrotational and incompressible then Equations (5.17)-(5.18) and (5.19)-(5.20) hold simultaneously, which implies that
(5.21)
(5.22)

It immediately follows, from the previous two expressions, that

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FROBENIUS CONCEPTS IN INTEGRATION AND APPLICATION. FOR FACULTIES Chemistry and Pharmaceutic Bucharest
ABSTRACT
In fact in all the mathematical work left by Fermat there is only one proof. Fermat proves that the area of a right triangle cannot be a square. Clearly this means that a rational triangle cannot be a rational square. In symbols, there do not exist integers x, y, z with
x2 + y2 = z2 such that xy/2 is a square. From this it is easy to deduce the n = 4 case of Fermat’s theorem.

with many aspects we consider that in chemistry and medicine are valuable more results as in case of F. VIETE ,KRONECKER or SYLVESTER and thus prof dr mircea orasanu and prof drd horia orasanu appear with
LAGRANGIAN AND HURWITZ CONSIDERATIONS
ABSTRACT
The following problems involve the use of l’Hopital’s Rule. It is used to circumvent the common indeterminate forms “0”0 and “∞”∞
when computing limits. There are numerous forms of l”Hopital’s Rule, whose verifications require advanced techniques in calculus, but which can be found in many calculus books. This link will show you the plausibility of l’Hopital’s Rule. Following are two of the forms of l’Hopital’s Rule.
THEOREM 1 (l’Hopital’s Rule for zero over zero): Suppose that limx→af(x)=0
, limx→ag(x)=0, and that functions f and g are differentiable on an open interval I containing

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