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

LikeLike

ab other aspects that we see is as say prof dr mircea orasanu as foloowed

ANALYTIC FUNCTIONS

Author Mircea Orasanu

LikeLike

also here sure we mention as say prof dr mircea orasanu and prof horia orasanu some as

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

LikeLike

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 $h'(\xi) = 0$.

LikeLike

sure as that

LikeLike

here sure we consider some important aspects as say prof dr mircea orasanu and prof horia orasanu as followed for

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:

LikeLiked by 1 person

here indeed we see as the above as say prof dr mircea orasanu and prof horia orasanu as followed with

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

LikeLike

here we see and consider some aspects as say prof dr mircea orasanu and prof horia orasanu concerning some FOUNDATIONS and POWER PLANT and as followed

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.

LikeLike

here we consider as we see some as say prof dr mircea orasanu and prof horia orasanu as followed for

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

LikeLike

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’:

LikeLike

here we consider as say prof dr mircea orasanu and prof horia orasanu and as with

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:

LikeLike

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,

LikeLike

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

LikeLike

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

LikeLike

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:

LikeLike

of course there are more situations and we see some as say prof dr mircea orasanu and prof horia orasanu as followed

KLEIN GORDON EQUATION AND APPLICATIONS

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

Thank you for signing up for Subject Matters email updates.

1. INTRODUCTION Now, in order to find the shortest path between points and , we need to minimize the functional with respect to small variations in the function , subject to the constraint that the end points, and , remain fixed. In other words, we need to solve

(676)

The meaning of the above equation is that if , where is small, then the first-order variation in , denoted , vanishes. In other words, . The particular function for which obviously yields an extremum of (i.e., either a maximum or a minimum). Hopefully, in the case under consideration, it yields a minimum of .

Consider a general functional of the form

(677)

where the end points of the integration are fixed. Suppose that . The first-order variation in is written

(678)

where . Setting to zero, we obtain

(679)

This equation must be satisfied for all possible small perturbations .

Integrating the second term in the integrand of the above equation by parts, we get

(680)

Now, if the end points are fixed then at and . Hence, the last term on the left-hand side of the above equation is zero. Thus, we obtain

(681)

The above equation must be satisfied for all small perturbations . The only way in which this is possible is for the expression enclosed in square brackets in the integral to be zero. Hence, the functional attains an extremum value whenever

(682)

LikeLike

the problem can be extended we see and more some as say prof dr mircea orasanu and prof horia orasanu how as followed for

LEBESGUE INTEGRATION AND FATOU THEORY

ABSTRACT

In this definition, we did not require the to be disjoint. However, it is easy enough to arrange this, basically by exploiting Venn diagrams (or, to use fancier language, finite boolean algebras). Indeed, any subsets of partition into disjoint sets, each of which is an intersection of or the complement for (and in particular, is measurable). The (complex or unsigned) simple function is constant on each of these sets, and so can easily be decomposed as a linear combination of the indicator function of these sets

1 INTRODUCTION

It is clear from construction that the space of complex-valued simple functions forms a complex vector space; also, also closed under pointwise product and complex conjugation . In short, is a commutative -algebra. Meanwhile, the space of unsigned simple functions is a -module; it is closed under addition, and under scalar multiplication by elements in .

. One easy consequence of this is that if is a complex-valued simple function, then its absolute value is an unsigned simple function.

It is geometrically intuitive that we should define the integral of an indicator function of a measurable set to equal :

LikeLike

for more situations we have some solutions as consider prof dr mircea orasanu and prof horia orasanu concerning CHEMISTRY AND MEDICINE WITH APPLICATIONS IN BOOLEAN SETS AND LAGRANGIAN TO PUBLISH

ABSTRACT According to the number of equations and variables (2, 3, 4, … equations; 1, 2, 3, 4, … variables)

According to the type and degree of the equations (polynomic, linear, non-linear, rational, irrational…)

According to the number of solutions

Solving simultaneous equations means finding the answers to several equations at the same time.

The first type of simultaneous equations we are going to solve is a set of two first-degree equations (linear equations) in two variables. That is any set of simultaneous equations equivalent to one like this:

We must find values for “x” and “y” for which both equations are true.

Solving two linear simultaneous equations with two unknowns

Always before applying any method:

-Rearrange both equations into the above form.

Always after applying any method:

-Once you get the value of one of the unknowns, substitute it into any equation to get the other.

The substitution method:

-Leave one variable alone rearranging the easiest equation.

-Plug in the resulting expression for that variable into the other equation.

-Solve the equation obtained.

Example:

1 INTRODUCTION Frequently in physical problems there is a need to know how a given Bessel or modified Bessel functions for large values of argument, that is, the asymptotic behavior. Using the method of stepest descent studied in Chapter 2, we are able to derive the asymptotic behaviors of Hankel functions (see page 450 in the text book for details) and related functions:

1. (6.99)

2. The second kind Hankel function is just the complex conjugate of the first (for real argument),

(6.100)

3. Since is the real part of ,

(6.101)

4. The Neumann function is the imaginary part of , or

(6.102)

5. Finally, the regular hyperbolic or modified Bessel function is given by

(6.103)

or

(6.104)

LikeLike

is very important to present some situations as appear prof dr mircea orasanu and prof horia orasanu and

LEBESGUE THEOREM OF CONVERGENCE AND LAGRANGIAN . YOUNG AND FREDHOLM

ABSTRACT

. The Riemann integral involves partitioning the interval to be integrated over without regards to the function being integrated at all; that is, if you were doing or you wouldn’t partition any differently.

2. The elements of any partition of the Riemann integral are intervals of finite length.

The Lebesgue integral changes these two features;

1. We’ll use information about the function being integrated to help us select partitions and

2. The elements of our partition need not be intervals of finite length; they just need to be measurable sets.

For example, suppose we wish to compute by using a Lebesgue integral.

Partition the range of into 4 subintervals:

Now consider the inverse image of these subintervals and label these:

Then we form something similar to upper and lower sums. Recall the measure of an interval is just its length.

So we obtain something like an upper sum:

and a lower sum as well:

1 INTRODUCTION The ideal is a collection of all algebraic integer multiples of a given algebraic integer. For example, the notation (2) represents such a particular collection, as . . . -8, -6, -4, -2, 0, 2, 4, 6, 8 . . . . The sum of two ideals is an ideal that is composed of all the sums of all their individual members. The product of two ideals is similarly defined. Ideals, considered as Once a point of origin, a unit length, and a direction have been picked for the latter, the two systems can be correlated systematically: each rational number corresponds, in a unique and order-preserving way, with a point on the line. But a further question then arises: Does each point on the line correspond to a rational number? Crucially, that question can be reformulated in terms of Dedekind’s idea of “cuts” defined directly on the rational numbers, so that any geometric intuition concerning continuity can be put aside. Namely, if we divide the whole system of rational numbers into two disjoint parts while preserving their order, is each such division determined by a rational number? The answer is no, since some correspond to irrational numbers (e.g., the cut consisting of {x:x22} corresponds to 2–√). In this explicit, precise sense, the system of rational numbers is not continuous, i.e. not line-complete.integers, can then be added, multiplied, and hence factored. By means of this theory of ideals, he allowed the process of unique factorization—that is, expressing a number as the product of only one set of primes, or 1 and itself—to be applied to many algebraic structures that hitherto had eluded analysis.

LikeLike

more situations appear as are known in many cases as well concern by prof dr mircea orasanu and prof horia orasanu for example followed with ZAMM, ZAMP and unfortunately

FACULT MATHEMATICS bucharest that do not life and

POTENTIAL AND STREAM FUNCTIONS.LEBESGUE ,LAGRANGIAN AND BOOLEAN

ABSTRACT

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

1 INTRODUCTION

Fluid motion is said to be two-dimensional when the velocity at every point is parallel to a fixed plane, and is the same everywhere on a given normal to that plane. Thus, in Cartesian coordinates, if the fixed plane is the – plane then we can express a general two-dimensional flow pattern in the form

(5.1)

Figure 5.1: Two-dimensional flow.

Let be a fixed point in the – plane, and let and be two curves, also in the – plane, that join to an arbitrary point . (See Figure 5.1.) Suppose that fluid is neither created nor destroyed in the region, (say), bounded by these curves. Because the fluid is incompressible, which essentially means that its density is uniform and constant, fluid continuity requires that the rate at which the fluid flows into the region , from right to left (in Figure 5.1) across the curve , is equal to the rate at which it flows out the of the region, from right to left across the curve . The rate of fluid flow across a surface is generally termed the flux. Thus, the flux (per unit length parallel to the -axis) from right to left across is equal to the flux from right to left across . Because is arbitrary, it follows that the flux from right to left across any curve joining points and is equal to the flux from right to left across . In fact, once the base point has been chosen, this flux only depends on the position of point , and the time . In other words, if we denote the flux by then it is solely a function of the location of and the time. Thus, if point lies at the origin, and point has Cartesian coordinates ( , ), then we can write

(5.2)

Th

LikeLike

we consider important forms of medicine and some aspects of mathematics as clarified prof dr mircea orasanu and prof horia orasanu and is followed APPLICATIONS IN MEDICINE LEBESGUE AND FUBINI THEOREM. ZAMP AND THEOREMS

ABSTRACT

Therefore, from the definition of the generalized momenta,

and we can write this as a vector-matrix equation,

.

That is, is a linear function of the ‘s. Hence, the inverse matrix will give us as a linear function of the , and then putting this expression for the into the Lagrangian gives the Hamiltonian as a function only of the and the , that is, the phase space variables.

The matrix is always invertible because the kinetic energy is positive definite (as is obvious from its Cartesian representation) and a symmetric positive definite matrix has only positive eigenvalues, and therefore is invertible.

Hamilton’s Equations

Having finally established that we can write, for an incremental change along the dynamical path of the system in phase space,

we have immediately the so-called canonical form of Hamilton’s equations of motion:

Evidently going from state space to phase space has replaced the second order Euler-Lagrange equations with this equivalent set of pairs of first order equations.

Simple Example

For a particle moving in a potential in one dimension,

Hence

Therefore

(Of course, this is just the total energy, as we expect.)

The Hamiltonian equations of motion are

So, as we’ve said, the second order Lagrangian equation of motion is replaced by two first order Hamiltonian equations. Of course, they amount to the same thing (as they must!): differentiating the first equation and substituting in the second gives immediately that is, the original Newtonian equation (which we derived earlier from the Lagrange equations).

The Hamiltonian formulation of classical mechanics – NYU

Hamilton’s Equations – Galileo and Einstein

LikeLike

in more situations we can be consider convergence and volumes and thus as look rof dr mircea orasanu and prof horia orasanu must observed

CONVERGENCE AND DEDEKIND THEOREMS

LikeLike

also we continued the more aspects and as look prof dr mircea orasanu and prof horia orasanu with the following

INTEGRATION IN MULTIPLE INTEGRATION. LEBESGUE CRITERION. FAC. MATHEM.Bucharest

ABSTRACT

In some instances, this tabular method might be quicker than the de Morgan laws method. The truth is that neither method is very speedy when we have Boolean functions to consider in which the number n of events in their generating class is very large. In some instances we will get lucky and be able to observe by inspection of a minterm map which minterms appear in an expansion of some Boolean function. When this does not happen, however, we know that there are two formally equivalent ways of expressing a Boolean function in canonical form, as I have just illustrated. As I noted above, and will illustrate further below, the virtue of minterm expansions is that they provide us with an account of all specific and unique instances of conjunctive combinations of the events in some generating class of events that occurs in a Boolean function of interest. The first two applications of Boolean functions I will now mention make use of minterm expansions.

1 . INTRODUCTION

SOME APPLICATIONS OF BOOLEAN FUNCTIONS

Here is a collection of thoughts about how Boolean functions and their canonical forms may be usefully employed in three related areas of ongoing research. All three of these areas involve processes having great complexity in which efforts are being made to discover or generate new ideas or to invent new engineering designs. As I noted earlier, I believe there to be common elements of these activities as they involve discovery and invention.

3.1 Inventor 2000/2001

In 2001 I tried my best to generate some ideas I hoped were useful in work with Tom Arciszewski, Ken De Jong, and Tim Sauer on Inventor 2000/2001. In another document [Schum, 2001] I have mentioned some thoughts involving Boolean functions and their minterm expansions that may serve to stimulate the process of inquiry regarding the evolutionary mechanism according to which Inventor 2000/2001 generates new wind-bracing designs for tall buildings. The computational engine in this system makes use of the evolutionary processes of mutations and recombinations [e.g. crossovers], both of which can be construed as search mechanisms [Kauffman, 2000, pp 16-20]. This evolutionary process also involves selection since at each new step of the evolutionary process, only the fittest designs are selected and allowed to “mate” to produce new designs at the next iteration. In early studies using Inventor 2000, the fitness criterion was very simple and involved only one measure, namely the physical weight of the design.

LikeLike

here we introduced some aspects and as observed prof dr mircea orasanu and prof horia orasanu we conclude with the following

DEDEKIND,LEBESGUE CONVERGENCE IN INTEGRATION. FAC MATH.

ABSTRACT

In some instances, this tabular method might be quicker than the de Morgan laws method. The truth is that neither method is very speedy when we have Boolean functions to consider in which the number n of events in their generating class is very large. In some instances we will get lucky and be able to observe by inspection of a minterm map which minterms appear in an expansion of some Boolean function. When this does not happen, however, we know that there are two formally equivalent ways of expressing a Boolean function in canonical form, as I have just illustrated. As I noted above, and will illustrate further below, the virtue of minterm expansions is that they provide us with an account of all specific and unique instances of conjunctive combinations of the events in some generating class of events that occurs in a Boolean function of interest. The first two applications of Boolean functions I will now mention make use of minterm expansions.

3.0 SOME APPLICATIONS

LikeLike

we have here that thus are important aspects of sets and Dedekind forms as Lebesgue infinity as observed prof dr mircea orasanu and prof horia orasanu in case when the sets contain these points and as

BOOLEAN FORMS AND LEBESGUE SUMS

ABSTRACT

Laws and Theorems of Boolean Algebra

1a. X • 0 = 0 1b. X + 1 = 1 Annulment Law

2a. X • 1 = X 2b. X + 0 = X Identity Law

3a. X • X = X 3b. X + X = X Idempotent Law

4a. X • X = 0 4b. X + X = 1 Complement Law

5. X = X Double Negation Law

6a. X • Y = Y • X 6b. X + Y = Y + X Commutative Law

7a. X (Y Z) = (X Y) Z = (X Z) Y = X Y Z Associative Law

7b. X + (Y + Z) = (X + Y) + Z = (X + Z) + Y = X + Y + Z Associative Law

8a. X • (Y + Z) = X Y + X Z 8b. X + Y Z = (X + Y) • (X + Z) Distributive Law

9a. X • Y = X + Y 9b. X + Y = X • Y de Morgan’s Theorem

10a. X • (X + Y) = X 10b. X + X Y = X Absorption Law

11a. (X + Y) • (X + Y) = X 11b. X Y + X Y = X Redundancy Law

12a. (X + Y) • Y = XY 12b. X Y + Y = X + Y Redundancy Law

13a. (X + Y) • (X + Z) • (Y + Z) = (X + Y) • (X + Z) Consensus Law

13b. X Y + X Z + Y Z = X Y + X Z Consensus Law

14a. X ⊕ Y = (X + Y) • (X + Y) 14b. X ⊕ Y = X Y + X Y XOR Gate

15a. X ⊙ Y = (X + Y) • (X • Y) 15b. X ⊙ Y = X Y + X Y XNOR Gate

15c. X ⊙ Y = (X + Y) • (X + Y) XNOR Gate

Gates

Standard DeMorgan’s

NAND X = A • B X = A + B

AND X = A • B X = A + B

NOR X = A + B X = A • B

OR X = A + B X = A • B

LAWS AND THEOREMS OF BOOLEAN ALGEBRA

Identity Dual

Operations with 0 and 1:

1. X + 0 = X (identity)

3. X + 1 = 1 (null element) 2. X.1 = X

4. X.0 = 0

Idempotency theorem:

5. X + X = X

6. X.X = X

Complementarity:

7. X + X’ = 1

8. X.X’ = 0

Involution theorem:

9. (X’)’ = X

Identities for multiple variables

Cummutative law:

10. X + Y = Y + X

11. X.Y = Y X

Associative law:

12. (X + Y) + Z = X + (Y + Z)

= X + Y + Z

13. (XY)Z = X(YZ)

= XYZ

Distributive law:

14. X(Y + Z) = XY + XZ

15. X + (YZ) = (X + Y)(X + Z)

DeMorgan’s theorem:

16. (X + Y + Z + …)’ = X’Y’Z’…

or {f(X1,X2,…,Xn,0,1,+,.)}

= {f(X1’,X2’,…,Xn’,1,0,.,+)} 17. (XYZ…)’ = X’ + Y’ + Z’ + …

Simplification theorems:

18. XY + XY’ = X (uniting)

20. X + XY = X (absorption)

22. (X + Y’)Y = XY (adsorption)

19. (X + Y)(X + Y’) = X

21. X(X + Y) = X

23. XY’ + Y = X + Y

Consensus theorem:

24. XY + X’Z + YZ = XY + X’Z 25. (X + Y)(X’ + Z)(Y + Z)

= (X + Y)(X’ + Z)

Duality:

26. (X + Y + Z + …)D = XYZ…

or {f(X1,X2,…,Xn,0,1,+,.)}D

= f(X1,X2,…,Xn,1,0,.,+)

27. (XYZ …)D = X + Y + Z + …

1 INTRODUCTION

first at FAC MATHEM Bucharest these are unknown

and The two operations used are + (addition) and * (multiplication), where A + B is read as either A or B. A * B is read as A and B.

Boolean algebra theorems are those theorems which are very helpful in simplifying the various complex problems of Boolean algebra with ease.

The boolean algebra laws are listed below:

1. The Commutative Law, where A + B = B + A and A * B = B * A

2. The Associate Law, where (A + B) + C = A + (B + C) and (A * B) * C = A * (B * C)

3. The Distributive Law, where A * (B + C) = A * B + A * C and A + (B * C) = (A + B) * (A + C)

4. The Identity Law, where A + A = A and A * A = A

5. The additive and multiplicative inverse, where 0 + A = A and 0 * A = 0, 1 + A = 1 and 1 * A = A.

6. The sign of negation, denoted as A bar, Not A, where 1¯

= 0 and 0¯ = 1, i.e. A * A¯ = 0 and A + A¯

• = 1.

• The Redundancy Laws, where A + (A * B) = A and A * (A + B) = A

• The DeMorgan’s Law, where (A+B)¯ = A¯∗B¯ and (A∗B)¯ = A¯+B¯

8.

Boolean Algebra Examples:

Given below are some of the examples on boolean algebra.

Example 1:

Prove that (x + y) * (x + z) = x + yz

Solution:

Given function f (x) = (x + y) * (x + z)

________________________________________

Boolean Algebra Theorems

Boolean algebra is related to logics which has values as either true or false or not at all. Therefore, there are only 3 operators which are used in Boolean algebra, which are True, False and Not. The corresponding value of True is given by 1 and the value of False is given by 0.

LikeLike

further we deduce other situations as observed prof dr mircea orasanu and prof horia orasanu and as followed with

FATOU ,LAGRANGIAN,DIRICHLET

ABSTRACT

Ideally, we want the validity of this equation to be independent of ε , that is, given x, for it to hold for all microquantities ε. In that case the derivative f ′(x) may be defined as the unique quantity D such that the equation

f(x + ε) − f(x) = εD

holds for all microquantities ε.

Setting x = 0 in this equation, we get in particular

f(ε) = f(0) + εD, (4)

for all ε. It is equation (4) that is taken as axiomatic in smooth infinitesimal analysis. Let us write Δ for the set of microquantities, that is,

Δ = {x: x ∈ R ∧ x2 = 0}.

Then it is postulated that, for any f: Δ → R, there is a unique D ∈ R such that equation (4) holds for all ε. This says that the graph of f is a straight line passing through (0, f(0)) with slope Δ. Thus any function on Δ is what mathematicians term affine, and so this postulate is naturally termed the principle of microaffineness. It means that Δ cannot be bent or broken: it is subject only to translations and rotations—and yet is not (as it would have to be in ordinary analysis) identical with a point. Δ may be thought of as an entity possessing position and attitude, but lacking true extension.

Now consider the space ΔΔ of maps from Δ to itself. It follows from the microaffineness principle that the subspace (ΔΔ)0 of ΔΔ consisting of maps vanishing at 0 is isomorphic to R[51]. The space ΔΔ is a monoid[52] under composition which may be regarded as acting on Δ by evaluation: for f ∈ ΔΔ, f • ε = f (ε). Its subspace (ΔΔ)0 is a submonoid naturally identified as the space of ratios of microquantities. The isomorphism between (ΔΔ)0 and R noted above is easily seen to be an isomorphism of monoids (where R is considered a monoid under its usual multiplication). It follows that R itself may be regarded as the space of ratios of microquantities. This was essentially the view of Euler, who regarded (real) numbers as representing the possible results of calculating the ratio 0/0. For this reason Lawvere has suggested that R be called the space of Euler reals.

If we think of a function y = f(x) as defining a curve, then, for any a, the image under f of the “microinterval” Δ + a obtained by translating Δ to a is straight and coincides with the tangent to the curve at x = a. In this sense each curve is “infinitesimally straight”.

From the principle of microaffineness we deduce the important principle of microcancellation, viz.

If εa = εb for all ε, then a = b.

For the premise asserts that the graph of the function g: Δ → R defined by g(ε) = aε has both slope a and slope b: the uniqueness condition in the principle of microaffineness then gives a = b. The principle of microcancellation supplies the exact sense in which there are “enough” infinitesimals in smooth infinitesimal analysis.

From the principle of microaffineness it also follows that all functions on R are continuous, that is, send neighbouring points to neighbouring points. Here two points x, y on R are said to be neighbours if x − y is in Δ, that is, if x and y differ by a microquantity. To see this, given f : R → R and neighbouring points x, y, note that y = x + ε with ε in Δ, so that

f(y) − f(x) = f(x + ε) − f(x) = εf ′(x).

But clearly any multiple of a microquantity is also a microquantity, so εf ′(x) is a microquantity, and the result follows.

In fact, since equation (3) holds for any f, it also holds for its derivative f ′; it follows that functions in smooth infinitesimal analysis are differentiable arbitrarily many times, thereby justifying the use of the term “smooth”.

LikeLike

in continues we pointing as observed prof dr mircea orasanu and prof horia orasanu that as followed appear more mistakes in English,German GRAMMAR as in example of many works and must consider

LAGRANGIAN FORMS. FATOU ,DIRICHLET Problems

ABSTRACT

We need to restrict to subsets of the real line for which translation invariance and countable additivity holds; when we use Lebesgue outer measure on these sets we call it Lebesgue measure.

So, what we are going to show is this: if we restrict our sets to those sets for which:

(where is the set complement of ) for ALL subsets , then Lebesgue outer measure IS countably additive with respect to those subsets. Lebesgue outer measure applied to these sets is called Lebesgue measure.

Note: If is true we say that “ splits and sometimes refer to as a “test set”.

So, why is this condition the one that we want? Well, well prove the following results assuming that the given sets splits all test sets .

1. If and are measurable sets (e. g., splits all sets ) then is also measurable (splits all sets) and is also measurable.

Proof: first recall that by countable subadditivity of Lebesgue outer measure. We need to show the other inequality.

LikeLike

thus that here we go with some important situations as look prof dr mircea orasanu and prof horia orasanu as followed

SOME OBSERVATION OF LAGRANGIAN OPERATORS

ABSTRACT

Example

Evaluate the integral

Solution

Try as you may, you will not find an antiderivative of and we don’t want to get into power series expansions. We have another choice. The picture below shows the region.

We can switch the order of integration. The region is bounded above and below by y = 1/3 x and y = 0. The double integral with respect to y first and then with respect to x is

The integrand is just a constant with respect to y so we get

This integral can be performed with simple u-substitution.

u = x2 du = 2x dx

and the integral becomes

________________________________________

LikeLike

in thus of cases we present some as that of more aspects so called found and as look prof dr mircea orasanu and prof horia orasanu concerning that so

INFINITY POINT IN JORDAN AND LEBESGUE THEORY

ABSTRACT

In 1873 Cantor demonstrated that the rational numbers, though infinite, are countable (or denumerable) because they may be placed in a one-to-one correspondence with the natural numbers (i.e., the integers, as 1, 2, 3,…). He showed that the set (or aggregate) of real numbers (composed of irrational and rational numbers) was infinite and uncountable. Even more paradoxically, he proved that the set of all algebraic numbers contains as many components as the set of all integers and that transcendental numbers (those that are not algebraic, as π), which are a subset of the irrationals, are uncountable and are therefore more numerous than integers, which must be conceived as infinite

.1 INTRODUCTION

ability.

Cantor’s theory became a whole new subject of research concerning the mathematics of the infinite (e.g., an endless series, as 1, 2, 3,…, and even more complicated sets), and his theory was heavily dependent on the device of the one-to-one correspondence. In thus developing new ways of asking questions concerning continuity and infinity, Cantor quickly became controversial. When he argued that infinite numbers had an actual existence, he drew on ancient and medieval philosophy concerning the “actual” and “potential” infinite and also on the early religious training given him by his parents. In his book on sets, Grundlagen einer allgemeinen Mannigfaltigkeitslehre (“Foundations of a General Theory of Aggregates”), Cantor in 1883 allied his theory with Platonic metaphysics. By contrast, Kronecker, who held that only the integers “exist” (“God made the integers, and all the rest is the work of man”), for many years heatedly rejected his reasoning and blocked his appointment to the faculty at the University of Berlin.

This work contains his conception of transfinite numbers, to which he was led by his demonstration that an infinite set may be placed in a one-to-one correspondence with one of its subsets. By the smallest transfinite cardinal number he meant the cardinal number of any set that can be placed in one-to-one correspondence with the positive integers

LikeLike

we go here and step as points for other aspects and as more in prof dr mircea orasanu and prof horia orasanu with a

CAUCHY’ s PROBLEM FOR LAGRANGIAN AND CLAIRAUT EQUATION

ABSTRACT

DEDEKIND’s mappings were also subject to FREGE’s objections, which are similar to his objections to sets. Mappings are not purely logical tools; instead one should speak in the intensional tongue, namely about relations. To end his discussion of DEDEKIND’s ideas, FREGE wrote: “Concept and relation are the basic stones on which I erect my building”, i.e., the Grundgesetze der Arithmetik (FREGE 1893, 3). FREGE’s definition of natural numbers, for instance, cannot be stated without making use of concepts, extensions of concepts, and relations.

In spite of FREGE’s criticism, the parallelism between his pair of basic notions, concept and relation, and DEDEKIND’s basic ideas of system and mapping, is remarkable. If we ignore the choices made by FREGE on the basis of his preference for the intensional, this is just a confirmation that DEDEKIND’s was indeed a logical theory. Most important, it shows that DEDEKIND’s and FREGE’s logicisms rested essentially on the same basis.

1 .INTRODUCTION

The parallelism between FREGE’s and DEDEKIND’s theories of arithmetic shows how the logicist program needed set theory, or an equivalent device, in order to subsume arithmetic─or classical mathematics generally─under pure logic. Logicism depended the notions of set and relation, conceived extensionally or otherwise. Thus set theory was an indispensable ingredient of the logicist’s logic, which is why this first logicist program was shaken by the antinomies. For as we will see, the antinomies undermined the traditional justification of the logical character of sets: the connection between concept and set, the principle of comprehension.

6.2. Effect of the antinomies.

This is not the place to detail the complex ‘discovery’ of the antinomies, from CANTOR’s clear perception of the issue during the 1890s, through the changing and sometimes obscure reflections of Cesare BURALI-FORTI (1861-1931) and RUSSELL, to FREGE’s despair, which finally led to a general recognition of the problem. The important point for us is that the antinomies showed the unrestricted transition from concept to set, or alternatively the notion of a universal set, to entail contradictions. Thus they forced changes that affected core elements of traditional logic, and that altered the whole relation of set theory and logic.

The traditional transition from a concept to its corresponding extension, class or set had led to a naive acceptance that any (apparently) well-defined concept determined an acceptable─i.e., non-antinomical─set. In terms of ZERMELO’s axiom of subsets or ‘separation’, it was as if a universal set existed, to which any property could be applied in order to get a subset. (In fact, most 19th century logicians and mathematicians seem to have accepted the notion of a universal set─with the notable exceptions of CANTOR and SCHRÖDER.)

In 1897, as an editor of Mathematische Annalen, HILBERT corresponded with CANTOR on the latter’s project of adding a third part to the ‘Beiträge zur Begründung der transfiniten Mengenlehre’. In this third part, the well-ordering theorem would be proven on the basis of the antinomical character of the set of all alephs. CANTOR communicated to HILBERT this antinomy, but at first HILBERT could not accept the contradictory character of the set of alephs. His reply, textually quoted by CANTOR, is a perfect example of the traditional mentality:

LikeLike

in more situations appear other considerations and as look prof dr mircea orasanu and prof drd hotia orasanu we deduce as followed

LEIBNIZ INTEGRAL AND LEBESGUE – RIEMANN INTEGRAL. LAGRANGIAN

ABSTRACT

4.3. The logical theory of systems and mappings.

The foregoing may suffice for the reader to obtain a grasp of the sense in which sets were at the basis of ideal theory. Ideal theory was genetical or “constructive” in DEDEKIND’s sense: taking the set of algebraic numbers A as given, it employed set-constructions on it, i.e., subsets of A (the ideals), as its basic objects; and this made it possible to define rigorously the operations on ideals, especially multiplication. In this way, it was perfectly coherent with DEDEKIND’s conception of the number system, to which we now have to turn.

4.3.1. DEDEKIND’s program for the foundations of mathematics. In his works on the foundations of the number system, DEDEKIND expressed repeatedly the idea that this system was obtained from the natural numbers through step-by-step definitions or “constructions” (DEDEKIND 1854, 430-431; 1872, 317-318; 1888, 338). In manuscripts that are still preserved, he developed the well-known idea of defining the integers and their operations on the basis of equivalence classes of pairs of natural numbers; and similarly for the rationals on the basis of equivalence classes of pairs of integers. In 1872 he published the much more sophisticated idea of employing so-called ‘DEDEKIND cuts’ on the set of rational numbers for defining the real numbers and the operations on them. In all of these cases, the procedure is genetical in the above sense: taking a set of numbers as given, the next higher set is defined by means of set-constructions on the former, i.e., it is defined as (isomorphic to) the set of some specified subsets of the former. This, and the operations on the ‘lower’ number-set, suffice to define the operations on the new numbers. Since, according to DEDEKIND’s reliable dating, his theory of real numbers was formulated in 1858, it is a mild assumption that the much simpler theories of the integers and rationals should be traced back, at least, to that same year.

1 INTRODUCTION

possible to consider morphisms between those structures.

A reconstruction of DEDEKIND’s thoughts on analysis is necessarily more tentative, because to the best of my knowledge there is no document recording them. But since the real and complex numbers can be defined within DEDEKIND’s framework, and since we have the general notion of mapping at our disposal, real and complex functions are easily obtained. It might well have been along this line that DEDEKIND saw analysis integrated within the general picture of ‘arithmetic’─meaning his conception of classical mathematics as based on numbers, and ultimately on sets and maps.

DEDEKIND’s foundational masterpiece Was sind und was sollen die Zahlen? (1888) was devoted to a detailed presentation of the elements of the whole edifice of his ‘arithmetic’. It developed the “construction” of the natural numbers on the basis of a careful presentation of the theory of sets and mappings. But above all, it has to be read as presenting all the necessary ingredients for a detailed derivation of arithmetic, algebra, and analysis.

4.3.2. Set and mapping as logical notions. In the preface to his 1888 book, DEDEKIND clearly adopted a logicist viewpoint. The work begins with an statement of the author’s epistemological and methodological views:

In science nothing capable of proof ought to be accepted without proof. As evident as this requirement might seem yet I cannot regard it as having been met even in the simplest science, that part of logic which deals with the theory of numbers […] In speaking of arithmetic (algebra, analysis) as a part of logic I mean to imply that I consider the number-concept entirely independent of the ideas or intuitions of space and time, that I consider it an immediate result from the pure laws of thought. […] It is only through the purely logical construction of the science of numbers and by thus acquiring the continuous number-domain that we are prepared accurately to investigate our notions of space and time by relating them to this number-domain created in our mind. (DEDEKIND 1888, 335; emphases added

LikeLike

in important

situations appear relations for Hermite and Laguerre relations and as look prof dr mircea orasanu and prof drd horia orasanu so that as followed

LEGENDRE AND HERMITE CONFORMAL TRANSFORMS WITH DIRICHLET PROBLEM

ABSTRACT

The appropriateness of the Bayesian approach to inference in the context of economic evaluation has been argued by Jones[13]. A key benefit of the Bayesian method is that it can make use of ‘prior information’ in addition to the sample data. Prior information about the true mean costs *i or the true mean efficacies *i, or about their true differences *c and *e, may be available in the form of experience with related treatments or from clinical judgement of experts. Incorporating such information via a Bayesian analysis can substantially strengthen inferences, as we show in O’Hagan et al[14]. Our purpose in the present paper is to critically review existing literature, to draw attention to deficiencies in existing work and to clarify the similarities and differences between Bayesian and frequentist analyses. For this purpose, it is helpful to ignore prior information and to adopt a Bayesian analysis based on ‘prior ignorance’. This yields simpler results that are easy to apply in practice but which may still offer advantages over frequentist methods.

2. Inference about the C/E acceptability curve

2.1 The C/E ratio and acceptability

Figure 1 was first discussed by Black[15]. It shows the plane of possible pairs of values (*e, *c) of the underlying true mean increments of efficacy and cost. It is divided into four quadrants, and quadrants I and III are further divided by a line of slope K. In quadrant IV, *e > 0 but *c < 0, hence treatment 2 is both more effective and cheaper than treatment 1 and is therefore unconditionally preferred. Conversely, in quadrant II, treatment 2 is less effective and more expensive than treatment 1 and so is unconditionally less acceptable. The situation in quadrants I and III depends on the value of K, which represents a maximum cost per unit of efficacy for a treatment to be acceptable.

LikeLike