We spoke with Dana Powell, Assistant Professor of Anthropology at Appalachian State University, about her new book *Landscapes of Power: Politics of Energy in the Navajo Nation*. Powell examines the rise and fall of the controversial Desert Rock Power Plant initiative in New Mexico, tracing the political conflicts surrounding native sovereignty and energy development on Navajo (Diné) Nation land and emphasizing the potential of Navajo resistance to articulate a vision of autonomy in the face of colonial conditions.

**How does your book approach and examine the Desert Rock Energy Project initiative, a failed late-2000s attempt to establish a coal-burning power plant on Navajo land in New Mexico? **

This is a story about the sociocultural dynamics of intensive extraction. The book takes two tacks: first, I approach the problem of Desert Rock historically, telling the longer story of the Navajo Nation’s decades of economic dependency on energy minerals, but ask readers to critically understand this dependency through the double entanglements of settler colonialism and environmental harm. At the same time, I examine the parallel legacies of resistance and energy activism, which emerged from grassroots leaders who not only recognized this nefarious entanglement but saw how the 1960s ascendence of coal production intensified the risk of exposure already in place from Cold War uranium extraction. Second, I approach the problem ethnographically, inviting readers into some of the on-the-ground complexities of tribal sovereignty, economic development, policy change, and various interpretations of place, by following the work of one social movement organization in particular. Ethnography allows me to examine the situated experiences of Diné people on both sides of the debate—those for the power plant, and those against the plant—as an embodied struggle around science, technology, and the future of infrastructure in indigenous territory. By interspersing ethnographic vignettes in between longer chapters that examine policy, discourse, expressive arts, and resistance strategies, I hope readers gain a feel for the everyday life impacts of large-scale industrial development and their unique dynamics in Diné landscapes.

**You were a political organizer and assistant manager with the Indigo Girls, an activist folk-rock duo that campaigns and holds benefits for native communities. How did your involvement affect the direction and nature of your research?**

The national political organizing work that I did through my affiliation with Indigo Girls offered me privileged access to conversations within the Indigenous Environmental Network, Honor the Earth, and the Intertribal Council on Utility Policy, as well as dozens of tribal NGOs, which deeply shaped my sense of the double entanglement of colonialism and environmental harm in Native America (from early 20th-century extractive legacies to later 20th-century impacts of climate change). This involvement not only established certain alliances and relationships for me, which became crucial as my solidarity work morphed from activist-ally to activist-researcher, but allowed me to develop research questions in tandem with indigenous organizers and policy-makers. I came to see my work as nurturing a conversation among conversations, linking discussions within activist networks with similar discussions in academic debates. Later on, in my academic work with the Social Movements Working Group and Modernity/Coloniality groups at UNC-Chapel Hill and Duke, I came to see how the knowledge work of the environmental-social movements I’d been engaged in for many years established the epistemic framework for my newfound anthropological inquiries into the problems. Aesthetically, having spent years working closely with feminist artists who approached social justice through songwriting, performance, and music, I was tuned in to the ways that expressive and sonic arts flowed through Diné and other indigenous environmental justice movements; this orientation provided me with a much-needed balance to my emphasis on the policy and political economy of energy. Last, years of assisting with the production of community-based and larger market-based benefit concerts confirmed for me the power of spectacle and affect in public education and outreach, and I tried to enact this sensibility and intention in the writing that coalesced into this book.

**How did your thoughts about indigenous environmental activism shift over the course of your time with the Indigo Girls?**

Over time, I came to see environmental activism in the U.S. as social justice work with questions of indigenous political difference and matters of territory front and center. Amy Ray and Emily Saliers offered strong models of how to enact solidarity as white allies to an indigenous movement; their feminist and queer analysis brought new angles to thinking about “justice” in matters of environmental harm. For example, although large-scale environmental organizations were brought to task by grassroots groups of color a few decades ago (see the “Letter to the Group of Ten” authored by the Southwest Organizing Project, and others), changing public discourse on the racialized and gendered dimensions of environmental risk is still pretty poorly understood among wider publics. We were focusing on solutions: supporting community-led and tribal government-led wind and solar development in Native Nations; but, over time, my thoughts shifted from these national efforts toward the complexities of “transition” work in specific locations. Over the years, I came to see that the national (really, international) activism we were engaged in didn’t always line up with what people desired in specific locales, so I became increasingly interested in understanding these frictions and how building power in particular demanded more specific, rather than general kinds of knowledge.

**You describe how environmental journalists packaged Diné activism against the Desert Rock power plant as a “David and Goliath” story. Why was this frame harmful or misleading?**

This Biblical metaphor offers no simple alignment: who in this struggle was the godless Goliath? The energy company, the federal government, surrounding jurisdictional states, or the Navajo Council? And who was the liberatory David? Grassroots EJ groups, the Navajo Council, residents of the impacted area who fought back through their endangered status? In the case of Desert Rock, the “perpetrators” and the “underdog” were not so clearly defined adversarial positions. The only appropriate analogy or likeness in this figure of speech is, perhaps, activists’ questioning of the “god” of capital.

**You spent time in native communities both as an activist and as an anthropologist-observer. How did these two roles feel distinct from each other? How did your multiple visits to the Navajo Nation affect your understanding of the community and the nature of your research?**

Perhaps like anyone who begins working in movements and then shifts in/to the academy, I experienced the unsettling feeling of betrayal: were my newly constructed academic questions—despite being inspired by the knowledge-work on the ground—a departure from more urgently needed, different modes of labor for non-native allies? Could the two positions ever be reconciled? Over time, I came to feel they were not so different, after all: the activist questions, theorizes, experiments, observes, analyzes and expresses, as does the anthropologist, following differing registers of expressive practice and media. Striving to maintain this critical edge within myself, recognizing and valuing both roles, deepened my understanding of the matters at stake and how the “local” struggle was, indeed, a “global” story and critically relevant to other extractive contexts. But at times, these roles made different ethical demands, challenging me to constantly interrogate what I was following, and why. Certainly, the multiple visits (that I discuss through the ethnographic trope of “arrivals” into the field) stretched out over years (1999-present) enabled me to slowly establish what have become long-term relationships of trust and collaboration with particular Diné people, and the project would not have been possible without these connections. And because I was examining the sociocultural life of the contemporary landscape, I had to learn to “see” infrastructures of power (from livestock wells to power lines, from ceremonial hogans to well-worn pathways in the forests) and it took many years of encounters to develop this perspective.

**How can activists reconcile care for the environment with an understanding of the complex issues facing Native communities? What resonance do the lessons of Desert Rock hold for today’s activists?**

Activists should not start with a consideration of the “environment”: it’s an abstract idea. As Anna Tsing, Bruno Latour, and many activists like those I work with in Navajoland argue, its unquestioned universality occludes the particularities of sites of struggle, in which the matters at stake are often not “the environment” as (we) imagined. Native Nations in the 21st century are facing new kinds of challenges to indigenous territorial sovereignty, often enacted through large-scale energy technologies: this was visible on a new scale, thanks to social media, during the Standing Rock/NoDAPL movement in 2016-2017. As I discuss in the book and elsewhere, activists who yearn for “environmental sustainability” in the U.S. cannot continue to follow the conventional “three E’s” approach to environment/economics/equity: the *political *difference of American Indians must be front and center in any project of harm reduction or transition. The notion of “equity” cannot contain this political/historical difference or the conditions of violence, ongoing, wrought by centuries of settler colonialism. An idea of “sustainability” that does not include sovereignty, in the case of Native Nations, is bankrupt. Likewise, as Myles Lennon shows in his study of Black Lives Matter activists’ pursuit of solar power, the question of energy justice in the U.S. brings with it long histories of the structural “demattering” of people of color. Activists can take these lessons of historical and political difference from the Desert Rock struggle. In this moment of public lands and sacred lands continually coming under threat (e.g., Bears Ears Monument, Standing Rock, Chaco Canyon, and more), especially with the expansion of energy infrastructure, activists who care for “the environment” would be wise to begin with an inquiry into the patterns of displacement, labor, settlement, and significance in a particular landscape.

*Pick up Dana Powell’s *Landscapes of Power* for 30% off using coupon code E17LAND at dukeupress.edu.*

these are very important is observed by prof dr mircea orasanu

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ANALYTIC FUNCTIONS

Author Mircea Orasanu

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GEOMETRICAL THEOREMS

Author Mircea Orasanu

ABSTRACT

The fractal nature of space generates the breaking of differential time reflection invariance. In such a context, the usual definitions of the derivative of a given function with respect to time [6,7],

dFdt=limΔt→0+F(t+Δt)−F(t)Δt=limΔt→0−F(t)−F(t−Δt)Δt

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as sure and indeed we see as say prof dr mircea orasanu that must consider importance of mathematics and medicine and specially

CAUCHY ‘s MEAN THEOREM

ABSTRACT

Let f and g be differentiable functions on the interval [a,b], with a < b.

Consider the following function:

h(x) = (f(x) – f(a)(g(b) – g(a)) – (g(x) – g(a))(f(b) – f(a).

Then if we evaluate h at x = a and at x = b, we get:

h(a) = h(b) = 0.

So by Rolle's theorem there exists a point $\xi$ in the interval (a, b), such that $h'(\xi) = 0$.

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sure as that

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LAGRANGIAN AND FORMULAS

ABSTRACT

LAGRANGIAN AND INTERRPOLATION

Interpolation and Polynomial Approximation

3.1 Interpolation and the Lagrange Polynomial

3.1.1 Lagrangian Form

Consider a polynomial of degree (n 1):

P(x) = a1x n-1 + a2x n-2 +… + an-1x + an

where the ai are constants. The polynomial can be written in Lagrangian form:

P(x) = c1(x 2) (x 3)… (x n) + c2(x 1) (x 3)… (x n) + …

ci(x 1) (x 2) … (x i-1) (x i+1) … (x n) + …

cn(x 1) (x 2)… (x n-1)

where i, i = 1, 2, …, n are arbitrary scalars, while the constants ci are related to the constants ai.

Example 3.1-1 _____________________________________________________

Write the polynomial P(x) = x 2 4x + 3 in the Lagrangian form.

Solution

The Lagrangian form for P(x) = x 2 4x + 3 is

P(x) = c1(x 2) (x 3) + c2(x 1) (x 3) + c3(x 1) (x 2)

where i, i = 1, 2, 3 are arbitrary scalars. Let 1 = 1, 2 = 2, 3 = 3, then

P(x) = c1(x 2) (x 3) + c2(x 1) (x 3) + c3(x 1) (x 2)

The constants c1 can be evaluated from the above relation by substituting x = 1 = 1

P(x = 1) = 1 4 + 3 = c1(1 2) (1 3) c1 = 0

For x = 2 = 2

P(x = 2) = 4 8 + 3 = c2(2 1) (2 3) c2 = 1

For x = 3 = 3

1 INTRODUCTION

A short form notation for P(x) is

P(x) =

where denotes product of all terms (x k), for k varying from 1 to n except i. Let x = i then

P(i) = ci(i 1) (i 2) … (i i-1) (i i+1) … (i n)

The constant ci can be expressed as

ci =

3.1.2 Polynomial Approximation

Consider a function f(x) that passes through the two distinct points (x0, f(x0)) and (x1, f(x1)) as shown in Figure 3.1-1. The first order polynomial that approximates the function between these two points can be expressed as

P(x) = a + bx

where a and b are constants. P(x) can also be written in Lagrangian form as

P(x) = c0(x x1) + c1(x x0)

Figure 3.1-1 First and second order polynomial approximation.

where

ci =

or

c0 = = , and c1 = =

The approximating polynomial is finally

P(x) = f(x0) + f(x1)

The first order polynomial basis function L0(x) is defined as

L0(x) = =

Similarly, the first order polynomial basis function L1(x) is defined as

L1(x) = =

In terms of the basis function, P(x) can be written as

P(x) = L0(x) f(x0) + L1(x) f(x1)

If a second order polynomial is used to approximate the function using three points (x0, f(x0)), (x1, f(x1)), and (x2, f(x2)) then

P(x) = f(x0) + f(x1) + f(x2)

P(x) can also be written in terms of the second order polynomial basis function L2,k(x)

P(x) = L2,0(x) f(x0) + L2,1(x)f(x1) + L2,2(x)f(x2)

where L2,0(x) = =

In general: L2,k(xk) = 1 at node k, L2,k(xi) = 0 at other nodes.

We now seek a polynomial P(x) of degree n that interpolates a given function f(x) between the node xi of the grid for which there are n+1 nodes x0, x1, , xn and

P(xk) = f(xk) for each k = 1, 2, , n

The error associated with interpolation is then

En(x) = () = (h)[( 1)h] [( n)h] ()

The only variable in the above expression is h the spacing of the nodes, therefore

En(x) = Chn+1, x0 < < xn

where C is a coefficient independent of h.

We can therefore write En(x) = O(hn+1) meaning that the ratio En(x)/ hn+1 is bounded by a constant as h 0. As the increment h decreases, so also will the interpolation error En.

Example 3.1-4 _____________________________________________________

For the function f(x) = ln(x + 1), construct interpolation polynomials of degree one and two to approximate f(0.45) from the given nodes. Find the error bound and the actual error.

xk 0 0.6 0.9

ln(x + 1) 1 0.47000 0.64185

Solution

First degree polynomial

P1(x) = (0) + (0.47) = 0.78334x

P1(0.45) = 0.3525

Error bound: En(x) = (x x0)(x x1) (x xn) ()

E1(x) = | (x x0)(x x1)|

f(x) = ln(x + 1) f’(x) = f”(x) = f””(x) =

sure these must appear in an article

2 FORMULATION

Exercise: concepts from chapter 3

Reading: Fundamentals of Structural Geology, Ch 3

1) The natural representation of a curve, c = c(s), satisfies the condition |dc/ds| = 1, where s is the natural parameter for the curve.

a) Describe in words and a sketch what this condition means.

b) Demonstrate that the following vector function (3.6) is the natural representation of the circular helix (Fig. 1) by showing that it satisfies the condition |dc/ds| = 1.

(1)

c) Use (1) and MATLAB to plot a 3D image of the circular helix (a = 1, b = 1/2). An example is shown in Figure 1. Describe and label a and b.

Figure 1. Circular helix with unit tangent (red), principal normal (green), and binormal (blue) vectors at selected locations.

2) An arbitrary representation of a curve, c = c(t), satisfies the condition |dc/dt| = ds/dt, where t is the arbitrary parameter and s is the natural parameter for the curve.

a) Demonstrate that the following vector function (3.2) is an arbitrary representation of the circular helix by showing that it satisfies this condition.

(2)

b) Show how this condition and the chain rule are used to derive the equation (3.8) for the unit tangent vector for an arbitrary representation of a curve and then use this equation to derive the unit tangent vector for the circular helix (3.11). In the process show how t and s are related.

c) Using your result from part b) for t(t) write the equation for the unit tangent vector, t(s), as a function of the natural parameter. Use this equation and MATLAB to plot a 3D image of a set of unit tangent vectors on the circular helix (a = 1, b = 1/2) as in Figure 1.

3) The curvature vector, scalar curvature, and radius of curvature are three closely related quantities (Fig. 3.10) that help to describe a curved line in three-dimensional space.

a) Derive equations for the curvature vector, k(s), the scalar curvature, (s), and the radius of curvature, (s), for the natural representation of the circular helix (1).

b) Show how these equations reduce to the special case of a circle.

c) Derive an equation for the unit principal normal vector, n(s), for the circular helix as given in (1).

d) Use MATLAB to plot a 3D image (Fig. 1) of a set of unit principal normal vectors on the circular helix (a = 1, b = 1/2). Describe the orientation of these vectors with respect to the circular helix itself, and the Cartesian coordinates.

e) Derive an equation for the unit binormal vector, b(s), for the circular helix (1). This is the third member of the moving trihedron.

f) Use MATLAB to plot a 3D image (Fig. 1) of a set of unit binormal vectors on the circular helix (a = 1, b = 1/2).

4) If c = c(t) is the arbitrary parametric representation of a curve, then a general definition of the scalar curvature is given in (3.26) as:

(3)

a) Show how this relationship may be specialized to plane curves lying in the (x, y)-plane where c(t) = cxex + cyey and the components are arbitrary functions of t.

b) Further specialize this relationship for the plane curve lying in the (x, y)-plane where the parameter is taken as x instead of t, so one may write cx = x and cy = f(x) such that c(x) = xex + f(x)ey and the normal curvature is:

(4)

c) Evaluate the error introduced in the often-used approximation (x) ~ |d2f/dx2| by plotting the following ratio as a function of the slope, df/dx, in MATLAB:

(5)

Develop a criterion to limit errors to less than 10% in practical applications.

5) If c = c(t) is the arbitrary parametric representation of a curve, then a general definition of the scalar torsion is given is (3.50) as:

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JOURNAL OF LAGRANGIAN

ABSTRACT We describe surfaces and geodesics without assuming prior knowledge of differential geometry. This involves selecting and presenting basic definitions and theorems. Included in this discussion are definitions of surface, coordinate patch, geodesic, etc. This summary closes with a proof of the length-minimizing properties of geodesics. Examples of surfaces are given and plotted in Mathematica. We also describe geodesics on these surfaces and plot select examples. The surfaces chosen include some with Clairaut patches, some without, and some surfaces in R3 and some not in R3.

A well-knbetween the two cities.

To generalize the adage–and along the way to explain why planes travel this way–we will introduce a special class of curves on surfaces, called geodesics. Geodesics have the useful property that the shortest curve segment connecting two points on a surface is a segment of a geodesic. As we shall see, great circles are geodesics on the sphere, and they therefore have the property that they are the “shortest” curves on the sphere. To examine geodesics, we will develop connections between differential geometry, differential equations, and vector calculus. In order to see geodesics, even when they cannot be found explicitly, the

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JOURNAL OF LAGRANGIAN AND OPERATORS

ABSTRACT

The Geodesic Equations

Important curves on surfaces are curves called geodesics. Geodesics are essentially the extensions into M of straight lines in the plane—that is, relative to the surface, there appears to be no acceleration. Formally,

Definition: For a surface M in Euclidean three-space, a geodesic is a curve α:[0, 1]→M where α’’ is always normal to M.

Since α’’ is always normal to M, that means that the dot product of α’’ and a vector in in TpM is always zero. In particular, α’’ • α’ = 0. Let s(t) be the speed of α at t; . Differentiating s with respect to t,

(1)

This shows that s(t) is a constant function, and so a geodesic must be a constant-speed curve. In fact, we can parameterize any geodesic so that it is unit-speed.

Given an orthogonal coordinate patch x in a geometric surface M, geodesics can be defined by differential equations called, appropriately, the geodesic equations. Consider a curve α in M. Express α(t) = x(u(t), v(t)). Then, , and so

(2)

If α is a geodesic, then it is normal to the surface, and hence

(3) α•xu = 0 and α•xv = 0

So, using (2) and (3) and the fact that xu•xv = 0 yields the differential system

(4) |xu|2 u + u2 xuu•xu + 2uvxuv•xu + v2xvv•xu = 0

(5) |xv|2 v + v2 xvv•xv + 2uvxuv•xv + u2xuu•xv = 0

which a curve must satisfy be a geodesic

to1 INTRODUCTION

These two equations are called the geodesic equations, because solving them gives geodesics. In fact, these equations allow us to generalize the definition of a geodesic to a surface that is not in R3. For such a surface, we merely define a geodesic to be a solution of the geodesic equations. An immediate result of this system of differential equations is the following theorem:

Theorem: Given a surface M in R3 with a unit normal, a point pM, and vector vTpM, there exists a unique geodesic such that (0)=p and (0)=v.

Proof: Let (t)=x(u(t), v(t)). Then (0)=p gives initial conditions u(0) and v(0). (0)=v gives initial conditions u(0) and v(0). Then, by the fundamental existence and uniqueness theorems of ordinary differential equations [1], exists and is unique.

If every geodesic can be extended infinitely without leaving the surface, then the surface is called a complete surface.

Usually, the geodesic equations cannot be solved by hand. For this reason, it is useful to be able to solve the geodesic equations numerically; we give a Mathematica procedure for this below. On a few surfaces, such as the A Clairaut patch is an orthogonal patch in which |xu| and |xv| are both independent of v. This implies and , or simply xuv•xu = xvv•xv = 0. Also, since xu•xv = 0, . Therefore, the geodesic equations reduce in this case to

FORMULATION

There are four Key Stages for learning achievement along the years. KS3 and KS4 are described in The Secondary National Strategy. KS3, which seems closest to the end of compulsory schooling in other European countries, aims to raise standards by strengthening teaching and learning across the curriculum for all 11–14 year olds.

Except for formulations such as mathematics provides opportunities for pupils to develop the key skills of: Communication, through learning to express ideas and methods precisely, unambiguously and concisely and Working with others, through group activity and discussions on mathematical ideas language and communication is not really a significant issue in the national curriculum for mathematics in England at KS3.

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HISTORY AND LAGRANGE PROBLEM

ABSTRACT

Firstly, what is the reason behind the appearance of a Schwarzian derivative {z,w} and why is it sitting with the central charge term or the anomaly term c12

?

One observation is under a fractional linear transformation

z→az+bcz+d

the Schwarzian vanishes, thereby making the transformation look “normal”, but then why does this transformation play a special role in a CFT?

conformal-field-theory stress-energy-momentum-ten

This seems like two or three questions in one. It is a policy of this site to have only one question per post. Kindly modify your question to reflect that. – Prahar Oct 2 ’15 at 6:08

However, they are pretty related. I’d rather rewrite instead of asking multiple questions. Hope the edited version is coherent enough to meet the policy requirements. – Jon Snow Oct 2 ’15 at 6:29

1

The question seems fine to me, though it’s faaaaaaaar beyond my ability to answer 🙂 – John Rennie Oct 2 ’15 at 6:46

add a

5

Polchinski Vol. 1 (Sec. 2.4): I’m trying to understand the Eq. 2.4.26 where he shows how the stress tensor transforms under a conformal transformation (z→w

):

(∂w)2T(w)=T(z)−c12{w,z}(2.4.26)

Firstly, what is the reason behind the appearance of a Schwarzian derivative {z,w}

and why is it sitting with the central charge term or the anomaly term c12

?

One observation is under a fractional linear transformation

z→az+bcz+d

the Schwarzian vanishes, thereby making the transformation loo

1 INTRODUCTION

While the infinitesimal conformal transformations form the infinite-dimensional Witt algebra spanned by the vector fields

Ln=−zn+1∂z

we must be mindful that those vector fields are not globally defined on the Riemann sphere S2=C∪{∞}. Obviously, they are singular at z=0 for n1

.

Therefore, the only globally defined conformal generators areTherefore, the only globally defined conformal generators are L−1,L0,L1. These three generate precisely the group of Möbius transformations z↦az+bcz+d

.

Thus, the symmetry group of a conformal field theory on the Riemann sphere is just PSL(2,C)

, and we have the requirement that the stress-energy tensor also should be invariant under this symmetry group. No such requirement can be said for the infinitesimal transformation of the Witt algebra. Nevertheless, classically, the stress-energy tensor transforms with its usual conformal weight also under those, since there is no central charge.Thus, the symmetry group of a conformal field theory on the Riemann sphere is just PSL(2,C)

, and we have the requirement that the stress-energy tensor also should be invariant under this symmetry group. No such requirement can be said for the infinitesimal transformation of the Witt algebra. Nevertheless, classically, the stress-energy tensor transforms with its usual conformal weight also under those, since there is no central charge.

In the course of quantization, we incur a central charge for the Witt algebra, turning it into the Virasoro algebra1. Since the energy-momentum tensor is T(z)=∑nLnzn−2

, the appearance of the central charge means the classical transformation law under the infinitesimal transformations generated by the Ln may change by a quantum correction – this is precisely the Schwarzian derivative term

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now sure we establish important properties for some To find out the dependence of pressure on equilibrium temperature when two phases coexist.as say prof dr mircea orasanu and prof horia orasanu

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.

Representing in terms of Gibbs free energy, the criterion of equilibrium is:

at constant T and P

or,

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’:

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ELECTROMAGNETIC CLAPEYRON OF EQUATION AND LAGRANGIAN

ABSTRACT

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

1 INTRODUCTION 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.

Representing in terms of Gibbs free energy, the criterion of equilibrium is:

at constant T and P

or,

For the phase transition for 1 to 1’:

or

or

In reaching state 2 from state 1, the change in the Gibbs free energy of the liquid phase is given by:

Similarly, the change in the Gibbs free energy of the vapour phase in reaching the state 2’ from state 1’ is given by:

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and also here we consider that some situations appear as say prof dr mircea orasanu and prof horia orasanu and followed with

SOME FOUNDATIONS OF KLEIN GORDON EQUATION

ABSTRACT

The Clapeyron equation becomes:

or

which is known as the Clausius-Clapeyron equation.

Assume that is constant over a small temperature range, the above equation can be integrated to get,

or +constant

Hence, a plot of lnP versus 1/T yields a straight line the slope of which is equal to –(hfg/R).

Kirchoff Equation

Kirchoff relation predicts the effect of temperature on the latent heat of phase transition.

Consider the vaporization of a liquid at constant temperature and pressure as shown in figure. The latent heat of vaporization associated with the phase change 1 to 1’ is ( – ) at temperature T. When the saturation temperature is raised to (T+dT), the latent heat of vaporization is ( – ). The change in latent heat,

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the above are true as say prof dr mircea orasanu and prof horia orasanu

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here we mention as is known that as say prof dr mircea orasanu and prof horia orasanu that as followed with

EXPLORING OF ELLIPTICAL PROPERTIES FOR MEDICINE

ABSTRACT

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:

1 INTRODUCTION

This may be tidied into

,

which gives a recurrence relation for the coefficients:

.

We may now build up the coefficients from the term. Starting from we find

.

Now putting gives

and so on. We see that the coefficients of all the even powers of x are given in terms of and we obtain the solution to the ODE as

.

The series specifies the J0 Bessel function:

.

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

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is very important sure as say prof dr mircea orasanu and prof horia orasanu as followed

MEDICINE AND MATHEMATICS

ABSTRACT

equilibrium temperature when two phases coexist.

Along a phase transition line, the pressure and To find out the dependence of pressure on 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.

Representing in terms of Gibbs free energy, the criterion of equilibrium is:

at constant T and P

or,

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 state 2 from state 1, the change in the Gibbs free energy of the liquid phase is given by:

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