PARTIAL DIFFERENTIAL EQUATIONS FOR CALCULUS OF VARIATIONS

ABSTRACT

Tutoring student-athletes on my free time has allowed me to relearn trigonometric functions and their basics. Using Swokowski, E. W., & Cole, J. A. (2008). Precalculus: Functions and Graphs, I introduced students to radian measure and the unit circle before seeing any work done with trigonometric functions. The students were able to understand how the functions relate to the unit circle better from learning radians first. Following this understanding we did work with right triangles. Then found more applications to trig functions and finished with inverse functions.

More Extensive Trigonometric Functions

Trigonometric functions have been used recently in my university levels in both the courses that I tutor for pre-calculus and the Geometry course I am taking. The functions are used as a transformation linear function that will rotate lines about the origin. A transformation that is done by adding the angles and multiplying

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]]>LAGRANGIAN AND CONFORMAL TRANSFORM

ABSTRACT

This lesson uses geometry to introduce students to trigonometric ratios and right triangle trigonometry. In this lesson, students will use their own knowledge of similar triangles to discover the trigonometric ratios sine, cosine and tangent in right triangles. Of course, the students will not recognize these ratios as being trigonometric ratios, but through class discussion will address these definitions. This lesson will also incorporate the ratios found in 45-45-90 and 30-60-90 triangles as well as introduce tan x as the slope of a line. Finally, there will be a short discussion of finding trig values for angles using a calculator.

Objectives:

– introduce trig ratios sine, cosine and tangent

– help students understand ratios in special triangles

– relate tan x as the slope of a line

1 INTRODUCTION

If students are not already in groups, break them into groups of 3-4. Pose the following questions to the groups (using the above diagram)

– Which of the above triangles are similar, assuming right angles at C, E, G, and I? Write proofs for these similarities.

– From your similar triangles, the ratios of which sides are always equal?

– What angle is involved in the similarity of all of the triangles?

– Which of the ratios you found corresponds to th slope of segment AH? Justify your answer.

Discuss students’ results and introduce trig ratios, relating it to the angle question above. All of the ratios involve angle A. Talk about sin A, cos A and tan A in a right triangle.

Talk about the above slope question noting that tan A gives you the slope of AH.

Have students work on a worksheet in groups on finding the ratios for triangles with specific sides. On this worksheet, also have specific angles marked. For example, incorporate the ratios discussed for 45-45-90 and 30-60-90 triangles in Lesson 3.

Finally, come together as a whole class and discuss the worksheet, as well as finding values for specific angles with a calculator.

Assign homework and journal entry.

Students work in groups on the questions. They may have trouble with similar triangles, some may have forgotten what it means, which would require a brief explanation from the teacher or a peer of similar triangles and the angle-angle theorem.

Contribute to class discussion.

Students explain which ratio they found to be the slope of AH, some may not make the immediate connection to tan A, stress this point. It is important for students to know that tan x is a ratio.

Work on worksheet, may have trouble with specific angles. For example, might not be comfortable with writing tan 45 = 1.

Contribute to discussion and work on their calculators.

Possible Accommodations: Most of this lesson will be accessible to most students regardless of their individual needs because there is enough variety and activity to keep students engaged. Also, group work will help students sort out some of the mathematical problems they are having. Students that have trouble writing, would have the option to record their journal assignment on audio tape.

Assessment: For this portion of the lesson, students will be assessed on their class participation as well as their written work. Their worksheets will be collected and graded, partially for completion. Also, their journal will be graded for effort and demonstration of critical thinking.

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]]>FOUNDATIONS AND CONCEPTS HISTORY

ABSTRACT

Almost everyone knows about the concept of electrical resistance. It is the ability of a circuit element to resist the flow of electrical current. Ohm’s law defines resistance in terms of the ratio between voltage E and current I. R = E / I

While this is a well known relationship, it’s use is limited to only one circuit element — the ideal resistor. An ideal resistor has several simplifying properties:

• It follows Ohm’s Law at all current and voltage levels.

• It’s resistance value is independent of frequency.

• AC current and voltage signals keep in phase with each other.

The real world contains circuit elements that exhibit much more complex behavior. These elements force us to abandon the simple concept of resistance. In its place we use impedance, which is a more general circuit parameter. Like resistance, impedance is a measure of the ability of a circuit to resist the flow of electrical current. Unlike resistance, impedance is not limited by the simplifying properties listed above.

Electrochemical impedance is usually measured by applying an AC potential to an electrochemical cell and measuring the current through the cell. Suppose that we apply

1 INTRODUCTION

Impedance definition: concept of complex impedance

a sinusoidal potential excitation. The response to this potential is an AC current signal, containing the excitation frequency and it’s harmonics.

Electrochemical Impedance is normally measured using a small excitation signal of 10 to 50 mV. In a linear (or pseudo-linear) system, the current response to a sinusoidal potential will be a sinusoid at the same frequency but shifted in phase.

Figure 1

Sinusoidal Current Response in a Linear System

The excitation signal, expressed as a function of time, has the form

E(t) is the potential at time tr Eo is the amplitude of the signal, and w is the radial frequency. The relationship between radial frequency ω(expressed in radians/second) and frequency f (expressed in hertz) is:

In a linear system, the response signal, It, is shifted in phase (φ) and has a different amplitude, Io:

An expression analogous to Ohm’s Law allows us to calculate the impedance of the system as:

The impedance is therefore expressed in terms of a magnitude (modulus) │Z│, and a phase shift, φ.

Using Eulers relationship,

it is possible to express the impedance as a complex function. The potential is described as,

and the current response as,

The impedance is then represented as a complex number,

Data Presentation

Look at last equation in the previous section. The expression for Z(ω) is composed of a real and an imaginary part. If the real part is plotted on the Z axis and the imaginary part on the Y axis of a chart, we get a “Nyquist plot”. See Figure 2. Notice that in this plot the y-axis is negative and that each point on the Nyquist plot is the impedance at one frequency.

This semester we have derived the governing equations for a number of different processes including:

1-D Steady ground water flow (Laplace equation; no internal sinks/sources)

( 1 )

1-D Steady ground water flow with internal sources/sinks (Poisson Equation):

( 2 )

1-D Transient ground water flow:

( 3 )

1-D Heat (Diffusion) equation:

( 4 )

1-D CDE for solute transport under steady flow:

( 5 )

We went on to develop algebraic finite difference expressions for each of these equations and solved them for a variety of boundary and initial conditions using either relaxation of a system of equations (the 2-D Laplace and Poisson equations) or explicit finite difference approaches.

There is great value in being able to derive the finite difference expressions; you are in a position to write your own computer code to solve these and similar equations, which arise in many fields of endeavor.

There are also new tools that allow you bypass this step; you provide the PDE and the computer uses (hopefully appropriate) numerical methods to provide a solution.

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]]>HISTORY

ABSTRACT

In 1856, a French hydraulic engineer, Henry Darcy carried out a series of experiments with water flowing through sand under different heads of water. He determined that the flow rate through porous media is proportional to the head loss and inversely proportional to the length of the flow path. Darcy’s Law provides the basis for ground water flow calculations.

He showed that the velocity of water flowing through a porous medium is equal to the hydraulic gradient times a constant that he called permeability. The value of water permeability for a porous medium is called hydraulic conductivity. Discharge is calculated by measuring the velocity through the cross-sectional area of the sample. The parameters of the Law are expressed in Figure 1.3.1.

Figure 1.3.1. Diagramatic representation of Darcy’s experiment showing that the velocity (v) of water flowing through a porous medium is equal to the hydraulic gradient (h/l), times a constant (k) [=permeability]. The value of permeability for a porous medium varies according to hydraulic conductivity. As the amount of flow (Q) is determined by the velocity (v) and the cross-sectional area of the sample (A), therefore Darcy’s Law may be used to calculate discharge.

Source: Brassington (1990:49).

1.3.2. Application of Darcy’s Law.

The driving force of groundwater flow is the hydraulic head, that is, difference in level of the piezometric surface /water table, from recharge to discharge. The flow of the water through the saturated zone of an aquifer may be represented by the equation:

Q = A x K x h/l

Groundwater cross-sectional hydraulic hydraulic

discharge (m3/day) area through conductivity(m/d) gradient

which flow takes

place (m2)

This may be expressed:

V = Ki

Velocity of flow through aquifer = coefficient of permeability x hydraulic gradient

i = Δh

L

Hydraulic gradient = head loss in length of flow path L

Q = AV = Aki

Vol. rate of flow = cross sectional area

of aquifer

[discharge or yield] [width x thickness (wb)]

A = wb

T = Kb

Coefficient of transmissibility of aquifer = coefficient of permeability x thickness

Therefore Q = wbki

discharge or yield = coefficient of perm. x cross sectional area x hydraulic gradient

Therefore Q = Tiw

Discharge/Yield = coefficient of transmissibility x hydraulic gradient x width of aquifer

In aquifers containing large diameter solution openings (e.g. in the karstic Mountain Aquifer of the West Bank) flow is no longer laminar due to high gradients and exhibits non-linear flow between the velocity and hydraulic gradient. This will also be the case with some of the coarser gravels present in the Gaza Coastal Aquifer.

The transmissibility is the flow capacity of an aquifer per unit width under unit hydraulic gradient and is equal to the product of permeability times the saturated thickness of the aquifer.

In a confined aquifer, T = Kb and is independent of the piezometric surface.

In a phreatic aquifer, T = KH, where H is the saturated thickness.

As the water table drops, H decreases and the transmissibility is reduced.

Thus the transmissibility of an unconfined aquifer depends upon the depth of the groundwater table.

Source: Raghunath, H.M.(1985) Hydrology: Principles, Analysis and Design, Wiley Eastern Ltd., New Delhi.

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]]>DEDEKIND PRESENCE IN CONFORMAL TRANSFORMS

ABSTRACT

The final columns of Tables 1 and 2 show the posterior probabilities that * is less than or equal to the given values of K. These have been estimated as the corresponding proportions of simulated values *k in N = 10,000 simulations.

The results emphasise the difference between inference about the C/E ratio * and the C/E acceptability curve. Looking at Table 2, where the proportion of patients discharged is the outcome measure, we see how the probability that anakinra is acceptable for a given threshold cost K is quite different from the probability that * is less than or equal to K. Suppose that the actual value of K for a decision maker (such as a health care provider) is 95000 Dfl per patient surviving and discharged at 28 days. The probability that the incremental cost-effectiveness ratio is less than or equal to 95000 is 0.9, on which basis it might seem that anakinra is preferred to placebo. However, the C/E acceptability curve shows that the probability that it is acceptable (i.e. preferred to placebo) is only 0.74.

In fact, no matter what value of K is appropriate, the probability of acceptability on the discharge outcome never exceeds 0.812. In this example, the maximal probability of acceptability occurs as K tends to infinity, and is therefore the probability that anakinra is more effective (in terms of proportion of patients discharged at 28 days) than placebo, regardless of cost. However, this depends on the sample data, and in other instances the maximal point on the C/E acceptability curve may occur at K = 0 (corresponding to the probability that treatment 2 is less costly than treatment 1) or at some intermediate value.

Willan and O’Brien[6], using the Fieller interval, found a 95% confidence interval from 108,280 to 55,856 for the survival outcome. The corresponding Bayesian 95% interval is from -94,830 to 84,774

1 INTRODUCTION

The Bayesian approach to inference about * is more robust than either the Fieller or Bonferroni intervals, in the sense that it never produces ‘intervals’ covering the whole line. The intervals suggested here are in fact always simple finite intervals, although they are not necessarily the narrowest that could be calculated. Highest posterior density intervals[21] would be narrower, and since the posterior distribution of * can be bimodal they can then take the form of the union of two disjoint, but finite, intervals.

The bootstrap method also produces a sample of simulated values of * and constructs an approximate confidence interval by ordering them in the same way as proposed above. Superficially, therefore, it appears comparable with the Bayesian solution. However, it should be remembered that, given a sufficiently large simulation size N, the Bayesian solution is an interval with exactly the stated probability of containing *. The bootstrap interval has only approximately the stated confidence no matter how large N is; it becomes exact only as the original sample size n becomes very large.

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]]>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.

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]]>HERMITE LEGENDRE TRANSFORMS

ABSTRACT

The increasing burden on the budgets of healthcare providers has led to the need to demonstrate that new healthcare practices provide value for money. Statistical methods for comparing the cost-effectiveness of a new drug or clinical procedure versus a comparative treatment or procedure have received considerable attention in recent years.[1 9] We consider the situation in which both costs and efficacy are obtained on individual patients in a comparative, randomised clinical trial. In this paper we suppose that the number of patients in each treatment group are the same; the case of unequal treatment group sizes is discussed in Section 2.3. We thus suppose that we have 2n patients and observe data cij and eij, where cij is the cost for patient j in treatment group i, and eij is a measure of efficacy for patient j in treatment group i. The subscript i takes values 1 and 2 to identify the two treatment groups and subscript j takes values 1 to n to identify the n patients receiving each treatment. The underlying true mean cost of treatment i is *i and the true mean efficacy of treatment i is *i.

1 INTRODUCTION

Much of the published work has been directed towards making inference about the incremental C/E ratio *. In the classical, frequentist approach to statistical inference, constructing significance tests or confidence intervals for a ratio of means is not straightforward[11]. Willan and O’Brien[6] were the first to apply Fieller’s method of constructing a confidence interval for a ratio to the problem of inference about the C/E ratio (although it was mentioned earlier by Wakker and Klaassen[4]). We discuss this solution, the Bonferroni-based approach and the bootstrap method in Section 3, where we advocate a Bayesian solution that has much more satisfactory behaviour.

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]]>LEGENDRE THEOREMS AND APPLICATIONS

ABSTRACT omputer Networking Machine Learning DevOps Deep Learning Blockchain and Cryptocurrency

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[DOC]Limite de functii – Analiza matematica. MPTAnaliza matematicAnaliza matematica Ia – Universitatea Tehnică de Construcţii …

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]]>LAGRANGIAN OPERATOR AND CONFORM TRANSFORMS

ABSTRACT

. In his reflections on this context Woodrow (1996) has pointed out that the concerns of teachers, following the Swann Report and the tragedy leading up to the MacDonald report, have been dissipated as a result of the introduction of a National Curriculum in which there is ‘no internationalism … no celebration of a pluralist culture and no sense of diversity’. Tooley also creates a false dichotomy between those teachers ‘who prefer to raise the political consciousness of their pupils, rather than their mathematical attainment’. Can it not be the case that teachers of mathematics can do both? Is there not a case to be made for considering the contribution teachers of mathematics might make in terms of citizenship and democracy?

Set against this background it is not surprising that the debate around issues of social justice and equal opportunities in the classroom came to whither on the vine during the last decade in England and Wales

1 INTRODUCTION

Further it is not surprising that schools are now seen, by the African and Caribbean Network of Science and Technology (ACNST), to be failing black pupils in the ‘status and power’ subjects of science, mathematics and technology (Ghouri, 1998b). It does seem that Tooley’s (1990) expectation that the National Curriculum proposals ‘have the potential to tackle that problem’ of underachievement has proved to be unfounded. Rather the ACNST research points towards the lack of role models for young black people e.g. ‘black British scientists’. It does seem that Tooley is also mistaken in his view that the use of exotic stereotypes such as the San people of the Kalahari desert is an appropriate and sufficient level of response to this problem.

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]]>LAGRANGIAN FORMS AND OPERATOR.DEDEKIND CONDITIONS

ABSTRACT

It turns out that for a set to have Jordan measure it should be well-behaved in a certain restrictive sense. For this reason, it is now more common to work with the Lebesgue measure, which is an extension of the Jordan measure to a larger class of sets. Historically speaking, the Jordan measure came first, towards the end of the nineteenth century. For historical reasons, the term Jordan measure is now well-established, despite the fact that it is not a true measure in its modern definition, since Jordan-measurable sets do not form a σ-algebra. For example, singleton sets in each have a Jordan measure of 0, while , a countable union of them, is not Jordan-measurable.[1] For this reason, some authors[2] prefer to use the term Jordan content (see the article on content).

The Peano-Jordan measure is named after its originators, the French mathematician Camille Jordan, and the Italian mathematician Giuseppe Peano.[3

1 INTRODUCTION

Consider the discrete optimization problem (which we refer to as Problem A)

,

where – is a non-decreasing -order-convex function on a partially set .

Let be an optimal solution of Problem A, and let be the point obtained by the following iterative procedure [4]:

which halts on the step if either or is the maximal element of the set (the set contains the zero , as we have stipulated). This point is called the gradient maximum os the function on the set [4].

By a guaranteed error estimate for the gradient algorithm in Problem A we mean a number

.

By perturbations of problem A by means problem B

,

where is a non-decreasing -order-convex function on a partially set and .

Let be a guaranteed error estimate for the gradient algorithm in some unperturbed (perturbed) discrete optimization problem. As usual (see. [3]), we say that the gradient algorithm is stable if , where as .

Theorem. Let and be guaranteed error estimates for the gradient algorithm in Problems A and B, respectively. Then .

To prove Theorem, we need the following lemma.

Lemma. The gradient maximum and the global maximum of any -ordered-convex non-decreasing function on are connected by the following relations:

, (1)

where

– is the set of all maximal elements of the partially ordered set .

Proof of Lemma. By virtue of item of Theorem 4 [4], we have for

Together with the fact that

the last inequality yields

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