Darlington S . DAVID - Profile on Academia.edu (original) (raw)

Papers by Darlington S . DAVID

The Time-Independent Schrödinger Equation

Essential Quantum Mechanics, 2007

Engineering Ethics: An Integrated Approach

xiv 7.8 Additional Problems for Chapter 7 8 DIFFERENTIATION OF MEASURES 8.1 Differentiation of Le... more xiv 7.8 Additional Problems for Chapter 7 8 DIFFERENTIATION OF MEASURES 8.1 Differentiation of Lebesgue-Stieltjes Measures 8.1.1 The ordinary derivative using the cube basis 8.1.2 Mixed partial derivatives 8.1.3 The strong derivative using the interval basis 8.2 The Cube Basis 8.2.1 Vitali's covering theorem for the cube basis 8.2.2 Differentiability of Lebesgue-Stieltjes measures on IR n 8.2.3 A theorem of Fubini 8.2.4 The fundamental theorem of the calculus 8.3 Lebesgue Decomposition Theorem 8.4 The Interval Basis 8.4.1 The Lebesgue density theorem for the interval basis 8.4.2 Approximate continuity 8.4.3 Differentiation of the integral for bounded functions 8.4.4 Mixed partials 8.4.5 Additional remarks 8.5 Net Structures 8.5.1 Differentiation with respect to a net structure 8.5.2 A growth lemma 8.5.3 An analog of de la Vallée Poussin's theorem for net structures 8.5.4 Further remarks xvi 9.5.1 Examples of separable metric spaces 9.6 Complete Spaces 9.6.1 Examples of complete metric spaces 9.6.2 Completion of a metric space 9.7 Contraction Maps 9.8 Applications 9.8.1 Picard's Theorem 9.9 Compactness 9.9.1 Continuous functions on compact metric spaces 9.10 Totally Bounded Spaces 9.11 Compact Sets in C(X) 9.11.1 Arzelà-Ascoli Theorem 9.12 Application of the Arzelà-Ascoli Theorem 9.13 The Stone-Weierstrass Theorem 9.13.1 The Weierstrass approximation theorem 9.14 The Isoperimetric Problem 9.15 More on Convergence 9.16 Additional Problems for Chapter 9 10 BAIRE CATEGORY 10.1 The Banach-Mazur Game on the Real Line 10.2 The Baire Category Theorem 10.2.1 Terminology for applications of the Baire theorem 10.2.2 Typical properties ClassicalRealAnalysis.com Bruckner*Bruckner*Thomson Real Analysis, 2nd Edition (2008) xvii 10.3 The Banach-Mazur Game 10.3.1 The typical continuous function is nowhere monotonic 10.4 The First Classes of Baire and Borel 10.4.1 The identity of B 1 and Bor 1

I) If a nonempty set of real numbers is bounded above, then it has a supremum. Property (I) is ca... more I) If a nonempty set of real numbers is bounded above, then it has a supremum. Property (I) is called completeness, and we say that the real number system is a complete ordered field. It can be shown that the real number system is essentially the only complete ordered field; that is, if an alien from another planet were to construct a mathematical system with properties (A)-(I), the alien's system would differ from the real number system only in that the alien might use different symbols for the real numbers and C, , and <. Theorem 1.1.3 If a nonempty set S of real numbers is bounded above; then sup S is the unique real numberˇsuch that (a) x Äˇfor all x in S I (b) if > 0 .no matter how small/; there is an x 0 in S such that x 0 >ˇ : Section 1.1 The Real Number System 5

In this paper the generalization of the classical mode orthogonality and normalization relationsh... more In this paper the generalization of the classical mode orthogonality and normalization relationships known for undamped systems to non – classical and non – viscously damped systems were established and investigated. Classical mode orthogonality relationships known for undamped systems were generalized to non – viscously damped systems. It was shown that there exists unique relationship which relates the system matrices to the natural frequencies and modes of non – viscously damped systems. These relationships, in return, enable us to reconstruct the system matrices from full set of modal analysis.
The non – viscously damping model is such that the damping forces depends on the past history of motion via convolution integrals over some kernel functions. Classical modal analysis is extended to deal with general non – viscously damped of multiple degree- of- freedom (MDOF) linear dynamic systems. The concept (complex) of non – viscous mode was introduced and further shown that the system response can be obtained exactly in terms of these modes.

Key words: Eigenvectors, eigenvalues, viscously undamped, orthogonality, normalization.

"In this paper, the Schrodinger model was investigated. Our results show that the time-independen... more "In this paper, the Schrodinger model was investigated. Our results show that the time-independent operators correspond to the observables of the quantum system. Also, from the Schrodinger model, it was proven that the model can be used to represent physical quantities such as quanta energy, quanta momentum and harmonic oscillator. Our results also show that operators are very useful tools for the representation of the eigenfunctions of the harmonic oscillator. Eigenfunctions can also be orthogonal basic of unit vector in an n-dimensional vector space that is obtained by solving the Schrodinger equation.

Keywords: Schrodinger equation, wave function, time-independent potential, probability current, probability density."

Talks by Darlington S . DAVID

Years ago, as a young, eager student, I would have told you that a great teacher was someone who ... more Years ago, as a young, eager student, I would have told you that a great teacher was someone who provided classroom entertainment and gave very little homework. Needless to say, after many years of K-12 administrative experience and giving hundreds of teacher evaluations, my perspective has changed.

Have you ever become so frustrated with students and overwhelmed by your workload that you start ... more Have you ever become so frustrated with students and overwhelmed by your workload that you start questioning what you are doing? At times it can feel suffocating. Baruti Kafele, an educator and motivational speaker offers a perspective of being mission oriented to educators and others working with young people in our nation's classrooms. He suggests affirming your goals and motivations to facilitate successes among students. However, in the college classroom, it is also essential that we, as faculty members, remember and affirm our purpose, acknowledge the contributions we make in students' lives and professional pursuits, and respect the call or passion that brought each of us to the teaching

The most effective teachers vary their styles depending on the nature of the subject matter, the ... more The most effective teachers vary their styles depending on the nature of the subject matter, the phase of the course, and other factors. By so doing, they encourage and inspire students to do their best at all times throughout the semester.

It used to be called team teaching, but that term is now used less often to describe the collabor... more It used to be called team teaching, but that term is now used less often to describe the collaboration of colleagues when they jointly teach the same course. Multiple instructors may be involved in the course, each delivering a freestanding module; or two instructors may do the course together, each in class every day with all course activities and assignments integrated. And there are variations of each of these models.

The evidence that students benefit when they talk about course content keeps mounting. In the stu... more The evidence that students benefit when they talk about course content keeps mounting. In the study highlighted below, students in two sections of an introductory zoology course were learning about the physiological mechanisms of RU-486 and about emergency contraception medication. They learned

We regularly tell our students -Don't be afraid to make mistakes. You can learn from your mistake... more We regularly tell our students -Don't be afraid to make mistakes. You can learn from your mistakes.‖ Most of us work hard to create classroom climates where it's okay to make mistakes. We do that because if we're there when the mistake is made, we can expedite the learning, and because we know that everyone else in class can learn from

Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynam... more Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems. When difference equations are employed, the theory is called discrete dynamical systems. When the time variable runs over a set that is discrete over some intervals and continuous over other intervals or is any arbitrary time-set such as a cantor set-one gets dynamic equations on time scales. Some situations may also be modeled by mixed operators, such as differential-difference equations.

DYNAMIC RESPONSE OF KIRCHOFF’S PLATE RESTING ON A NON-LOCAL FOUNDATION SUBJECTED TO UNIFORMLY DIS... more DYNAMIC RESPONSE OF KIRCHOFF’S PLATE RESTING ON A NON-LOCAL FOUNDATION SUBJECTED TO UNIFORMLY DISTRIBUTED MOVING LOADS.

A plate is a two dimensional, initially flat plane, element of a structure that is thin in comparison with its surface dimensions. Loads are forces acting on structures. Uniformly distributed loads which are spread over the entire or part of the surface area of a structure of any shape in such a way that each unit length is located to the same extent. Moving loads are defined as forces/loads acting on a structure and continuously changing position. Dynamics response is the resulting reactions or behavior of a structure due to the loads acting on it. A foundation is a structure that transfers loads to the earth. Foundation can be, in general, classified into two categories namely:
local foundation and
non–local foundation.
A local foundation is one for which the foundation pressure at any point is proportional to the deflection at that point. On the other hand, in the case of non – local foundation, the reaction is obtained as weighted average of state variable over a spatial domain via convolution integrals with spatial kernel functions that depend on a distance measure.
In this proposal, we shall consider moving loads problems, that is the problem of carrying out a dynamic analysis of the reaction of a structure under a moving load. The effects of moving loads on several structures have been the focus of many researchers. This is due to its importance to some engineering constructions. The design of railways and highway bridges under the influence of moving loads is of practical interest to a typical civil, mechanical, structural and transportation engineers as well as applied mathematics and physics.

BACKGROUND TO THE STUDY

Plates are widely used in many branches of modern civil, mechanical and aerospace engineering. In this proposal, we shall consider dynamic moving load applied to plates. In particular, the effects of moving loads on plate known as Kirchoff’s or non–Mindlin plate resting on elastic foundation will be studied. There are various models of foundations will be studied. There are various models of foundations. Example includes:
One–parameter model known as Winkler foundation.
Two–parameter model known as Pasternak foundation.
Three–parameter model known as Kerr foundation.
At this juncture, it is remarked that the local Winkler model is of the form

〖 Q〗_w (x,y,t)=k_1 (x,y) w(x,y,t) (1)

where w(x,y,t) is the lateral deflection of the plate, Q_w (x,y,t) is the foundation reaction, k_1 (x,y) is the stiffness coefficient of the elastic foundation model, x,y and t are the spatial and time variables respectively. The corresponding expressions for local Pasternak and Kerr foundation is

〖 Q〗_p (x,y,t)=-G_1 (x,y)[(∂^2 w(x,y,t))/(∂x^2 )+(∂^2 w(x,y,t))/(∂y^2 )]+k_1 (x,y) w(x,y,t) (2)

where Q_p (x,y,t) is the foundation reaction, k_1 (x,y) and G(x,y,t) are the stiffness coefficients of the spring and shear layers of the Pasternak foundation model respectively and

[1+(k_2 (x,y))/(k_1 (x,y) )] Q_k (x,y,t)-(G_1 (x,y))/(k_1 (x,y) ) [(∂^2 Q_k (x,y,t))/(∂x^2 )+(∂^2 Q_k (x,y,t,))/(∂y^2 )]= k_2 (x,y) w(x,y,t)
-G(x,y)[(∂^2 w(x,y,t))/(∂x^2 )+(∂^2 w(x,y,t))/(∂y^2 )] (3)

where k_1 (x,y),k_2 and G_1 (x,y) are the stiffness coefficients of the two spring layers and the middle shear layer for the Kerr foundation model respectively.
On the other hand, the mathematical expressions describing the non–local Winkler, Pasternak and Kerr foundations are

〖 Q〗_NW (x,y,t)=∫_(x_3)^(x_4)▒∫_(x_1)^(x_2)▒〖k(x,y,η,ξ) w(x,y,t) dξdη〗 (4)

where Q_NW is the foundation reaction of the non–local Winkler foundation model, w(x,y,t) is the bending deflection of the plate, k(x,y,η,ξ) is the elastic spatial kernel and the spatial integrations are over the length of the foundation whose extent is denoted by x_1,x_2,x_3,and x_4 respectively.

〖 Q〗_NP (x,y,t)=-∫_(x_3)^(x_4)▒∫_(x_1)^(x_2)▒〖G_1 (x,y,η,ξ)[(∂^2 w(x,y,t))/(∂x^2 )+(∂^2 w(x,y,t))/(∂y^2 )] 〗 dξdη
+ ∫_(x_3)^(x_4)▒∫_(x_1)^(x_2)▒k_1 (x,y,η,ξ) w(x,y,t)dξdη (5)

where Q_NP is the foundation reaction of the non–local Pasternak model, w(x,y,t) is the lateral deflection of the plate, k_1 (x,y,η,ξ) is the elastic spatial kernel and spatial integration are over the length of the foundation whose extent is denoted by x_1,x_2,x_3 and x_4 respectively.

∫_(x_3)^(x_4)▒∫_(x_1)^(x_2)▒(1+(k_2 (x,y,η,ξ))/(k_1 (x,y,η,ξ) )) Q_NK (x,y,t)dξdη-∫_(x_3)^(x_4)▒∫_(x_1)^(x_2)▒((G_1 (x,y,η,ξ))/(k_1 (x,y,η,ξ) )) [(∂^2 Q_NK (x,y,t))/(∂x^2 )+(∂^2 Q_NK(x,y,t) )/(∂y^2 )]dξdη
= ∫_(x_3)^(x_4)▒〖∫_(x_1)^(x_2)▒〖k_2 (x,y,η,ξ)w(x,y,t)dξdη〗-∫_(x_3)^(x_4)▒〖∫_(x_1)^(x_2)▒〖G_1 (x,y,η,ξ)〗 [(∂^2 w(x,y,t))/(∂x^2 )+(∂^2 w(x,y,t))/(∂x^2 )] 〗〗 dξdη(6)

where Q_NK is the foundation reaction of the non–local Kerr foundation model, w(x,y,t) is the lateral deflection of the plate, k_(1 )and k_2 are the elastic spatial kernels, G_1 is the shear layer for the non–local Kerr foundation respectively.

A comprehensive treatment of the subject on the effect of moving loads on elastic structures can be found in Fryba [5]. In particular, the dynamic analysis of elastic plate resting on local Winkler foundation under the influence of concentrated moving load is well discussed in [5] However, the effects of uniformly distributed moving load as well as a non–local properties of the foundation are not discussed. The dynamic response of a rectangular plate to moving concentrated masses and continuously supported by local elastic Pasternak foundation was studied by Gbadeyan and Oni [6] The authors observed that the values of the modulli increase as the critical speed of the load increases. The authors also neglected the influence of non–local foundation and that of uniformly distributed load. Gbadeyan and Dada [7] extended the work in [6] for local Pasternak foundation by assuming that the plate is acted upon by moving uniformly distributed masses as opposed the moving concentrated mass. It was found that an increase in the area of distribution of the moving masses causes a reduction in the maximum dynamic deflection. Non–local foundation was also not considered in [7].

Earlier, Dowell [8] carried out the dynamics analysis of an elastic plate in thin elastic foundation. The behavior of clamped elastic plate on Pasternak–type elastic foundation was investigated using the boundary element method in Katsikadelis and Kallivokas [9]. There exists a vast amount of work involving elastic structure (plate) resting on local foundation under the influence of moving or non–moving loads. To the best knowledge of the author, the few works involving non – local foundation/damping are those in [1-4]. In particular, in [1] a non–local viscoelastic foundation model was used to carry out the damping analysis of beams with different boundary conditions and finite element technique was used. The eigenvalues and corresponding eigenfunctions of the beam were obtained using numerical technique. A non–local damping model is used in [2] for the dynamic analysis for both beams and plates. A Galerkin method was used. However, we remark at this juncture, that all these researches are about free vibrations. None of them deals with forced vibrational problems. In other word, the effect of load (moving or non – moving) on the elastic structure is neglected.

AIMS AND OBJECTIVES, JUSTIFICATION OF PROBLEM

The following remarks are, therefore, based on the above literature review. Most work done especially in [1],[2],[3] and [4] are on free vibration problems, where the structure vibrates under the action of forces inherent in the structure itself and the external forces/loads are absent. The authors considered free vibration including non–local foundation and non–local damping.

In this work, the main aims and objectives may be enumerated as follows:
to proposed a non–local elastic foundation model and thereby use it to analyse dynamics of plates which is under the influence of moving distributed loads.
to study problem (1) above with different classical boundary conditions.
to consider problem (1) for both one–and–two–parameters non–local foundation.
to investigate the effects of non–local damping on elastic Kirchoff’s plate.
to consider (4) for various classical boundary conditions.

The proposed method of solution is the finite element technique. It shall be remarked at this juncture , that the equation of motion for the plate structure that will be used is an integro–partial–differential equations, rather than the partial–differential equations obtained for a local foundation/damping model. For example, for forced vibration of a rectangular plate resting on a non–local foundation the governing equation is

D(∂^4/(∂x^4 )+(2∂^4)/(∂x^2 ∂y^2 )+∂^4/(∂y^2 ))w(x,y,t)+∫_(x_3)^(x_4)▒∫_(x_1)^(x_2)▒〖k(x,y,η,ξ)w(x,y,t)dξdη+ρh (∂^2 w(x,y,t))/(∂t^2 )〗

=f(x,y,t) (7)

where w(x,y,t) and f(x,y,t) are the traverse deflection and external force, respectively. D=(Eh^3)/(12(1-ν^2)) is the bending stiffness of the plate, h is the thickness, and ν is the Poisson ratio. (D and ρh are assumed constant). The equation (7) above can be compared to the following partial–differential equation

D(∂^4/(∂x^4 )+(2∂^4)/(∂x^2 ∂y^2 )+∂^4/(∂y^4 ))w(x,y,t)+ k(x,y,t)w(x,y,t)+ ρh (∂^2 w(x,y,t))/(∂t^2 ) =f(x,y,t) (8)

where w(x,y,t) and f(x,y,t) are the traverse deflection and external force. D,ρh,and ν are all the same as in (7). Equation (8) holds for the classical local foundation.

Teaching Documents by Darlington S . DAVID

So far, we have made a lot of progress concerning the properties of, and interpretation of the wa... more So far, we have made a lot of progress concerning the properties of, and interpretation of the wave function, but as yet we have had very little to say about how the wave function may be derived in a general situation, that is to say, we do not have on hand a 'wave equation' for the wave function. There is no true derivation of this equation, but its form can be motivated by physical and mathematical arguments at a wide variety of levels of sophistication. Here, we will offer a simple derivation based on what we have learned so far about the wave function.

The Time-Independent Schrödinger Equation

Essential Quantum Mechanics, 2007

Engineering Ethics: An Integrated Approach

Key words: Eigenvectors, eigenvalues, viscously undamped, orthogonality, normalization.

Keywords: Schrodinger equation, wave function, time-independent potential, probability current, probability density."

BACKGROUND TO THE STUDY

〖 Q〗_w (x,y,t)=k_1 (x,y) w(x,y,t) (1)

〖 Q〗_p (x,y,t)=-G_1 (x,y)[(∂^2 w(x,y,t))/(∂x^2 )+(∂^2 w(x,y,t))/(∂y^2 )]+k_1 (x,y) w(x,y,t) (2)

where Q_p (x,y,t) is the foundation reaction, k_1 (x,y) and G(x,y,t) are the stiffness coefficients of the spring and shear layers of the Pasternak foundation model respectively and

〖 Q〗_NW (x,y,t)=∫_(x_3)^(x_4)▒∫_(x_1)^(x_2)▒〖k(x,y,η,ξ) w(x,y,t) dξdη〗 (4)

AIMS AND OBJECTIVES, JUSTIFICATION OF PROBLEM

D(∂^4/(∂x^4 )+(2∂^4)/(∂x^2 ∂y^2 )+∂^4/(∂y^2 ))w(x,y,t)+∫_(x_3)^(x_4)▒∫_(x_1)^(x_2)▒〖k(x,y,η,ξ)w(x,y,t)dξdη+ρh (∂^2 w(x,y,t))/(∂t^2 )〗

=f(x,y,t) (7)

D(∂^4/(∂x^4 )+(2∂^4)/(∂x^2 ∂y^2 )+∂^4/(∂y^4 ))w(x,y,t)+ k(x,y,t)w(x,y,t)+ ρh (∂^2 w(x,y,t))/(∂t^2 ) =f(x,y,t) (8)

where w(x,y,t) and f(x,y,t) are the traverse deflection and external force. D,ρh,and ν are all the same as in (7). Equation (8) holds for the classical local foundation.

Period: is defined as the time taken for one complete oscillation.

A photon is a discrete packet {or quantum} of energy of an electromagnetic radiation/wave. Energy... more A photon is a discrete packet {or quantum} of energy of an electromagnetic radiation/wave. Energy of a photon, E = h f = hc / λ where h: Planck's constant λ violet ≈ 4 x 10 -7 m, λ red ≈ 7 x 10 -7 m Power of electromagnetic radiation, P = Rate of incidence of photon x Energy of a photon = (N/t)(hc/λ) Photoelectric effect refers to the emission of electrons from a cold metal surface when electromagnetic radiation of sufficiently high frequency falls on it. 4 Major Observations: 1. No electrons are emitted if the frequency of the light is below a minimum frequency {called the threshold frequency}, regardless of the intensity of light 2. Rate of electron emission {ie photoelectric current} is proportional to the light intensity. 3. {Emitted electrons have a range of kinetic energy, ranging from zero to a certain maximum value. Increasing the freq increases the kinetic energies of the emitted electrons and in particular, increases the maximum kinetic energy.} This maximum kinetic energy depends only on the frequency and the metal used {ϕ}; the intensity has no effect on the kinetic energy of the electrons.

to biology students. She taught me the strength of phase plane analysis and simple caricature mod... more to biology students. She taught me the strength of phase plane analysis and simple caricature models. Some of the most interesting exercises in this book stem from that course. After I started teaching this course its contents and presentation have evolved, and have adapted to the behavior, the questions, and the comments from numerous students having attended this course. which says that the variable M increases at a rate k per time unit. For instance, this could describe the amount of pesticides in your body when you eat the same amount of fruit sprayed with pesticides every day. Another example is to say that M is the amount of money in your bank account, and that k is the amount of Euros that are deposited in this account on a daily basis. In the latter case the "dimension" of the parameter k is "Euros per day". The ODE formalism assumes that the changes in your bank account are continuous. Although this is evidently wrong, because money is typically deposited on a monthly basis, this makes little difference when one considers time scales longer than one month. 2N (0) = N (0)e rt gives ln 2 = rt or t = ln[2]/r. (2.8) This model also has only one steady state,N = 0, which is unstable because any small perturbation above N = 0 will initiate unlimited growth of the population. To obtain a non-trivial (or non-zero) steady state population size in populations maintaining themselves by reproduction one needs density dependent birth or death rates. This is the subject of the next chapter.