Our “Read to Respond” series addresses the current climate of misinformation by highlighting articles and books that encourage thoughtful, educated debate on today’s most pressing issues. This post focuses on labor, worker’s rights, and neoliberalism. Read, reflect, and share these resources in and out of the classroom to keep these important conversations going.

###### Labor

- “Finance Is Just Another Word for Other People’s Debts: An Interview with David Graeber”

Hannah Chadeayne Appel

*Radical History Review*, number 118

Winter 2014 - Mothering Through Precarity: Women’s Work and Digital Media

Julie A.Wilson and Emily Chivers Yochim

2017 - “
*‘*A Woman’s Place Is in the UMWA’: Women Miners and the Struggle for a Democratic Union in Western Pennsylvania, 1973–1979”

Trish Kahle

*Labor: Studies in Working-Class History*, volume 13, issue 1

February 2016 - “Trump’s Savage Capitalism: The Nightmare is Real”

Enzo Traverso

*World Policy Journal*, volume 34, issue 1

Spring 2017 - Against the Day: “Migrant Workers and New Forms of Exploitation: Europe and Beyond”

Rutvica Andrijasevic and Devi Sacchetto, editors

*South Atlantic Quarterly*, volume 114, issue 1

January 2015 - Give a Man a Fish: Reflections on the New Politics of Distribution

James Ferguson

2015 - “Living in Dose: Nuclear Work and the Politics of Permissible Exposure”

Shannon Cram

*Public Culture*, volume 28, issue 3

September 2016 - Youth, Labor and Politics in East Asia

Gabriella Lukacs, editor

*positions: asia critique*, volume 23, issue 3

August 2015 - Life Interrupted: Trafficking into Forced Labor in the United States

Denise Brennan

2014 - Femininity in Flight: A History of Flight Attendants

Kathleen Barry

2007

*These articles are freely available until December 15, 2017. Follow along with the series over the next several months and share your thoughts with #ReadtoRespond.*

sure these appear say prof dr mircea orasanu

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sure these appear say prof dr mircea orasanu Abstract:

The present paper investigates why Logical Empiricists remained silent about one of the most philosophy-laden matters of theoretical physics of the day, the Principle of Least Action (PLA). In the two decades around 1900, the PLA enjoyed a remarkable renaissance as a formal unification of mechanics, electrodynamics, thermodynamics, and relativity theory. Taking Ernst Mach’s historico-critical stance, it could be liberated from much of its physico-theological dross. Variational calculus, the mathematical discipline on which the PLA was based, obtained a new rigorous basis. These three developments prompted Max Planck to consider the PLA as formal embodiment of his convergent realist methodology. Typically rejecting ontological reductionism, David Hilbert took the PLA as the key concept in his axiomatizations of physical theories. It served one of the main goals of the axiomatic method: ‘deepening the foundations’. Although Moritz Schlick was a student of Planck’s, and Hans Hahn and Philipp Frank enjoyed close ties to Göttingen, the PLA became a veritable Shibboleth to them. Rather than being worried by its historical connections with teleology and determinism, they erroneously identified Hilbert’s axiomatic method tout court with Planck’s metaphysical realism. Logical Empiricists’ strict containment policy against metaphysics required so strict a separation between physics and mathematics to exclude even those features of the PLA and the axiomatic method not tainted with metaphysics.

Keywords:

Principle of Least Action, calculus of variations, Hilbert’s axiomatic method in physics, Mach-Planck controversy, Logical Empiricism, Moritz Schlick, Hans Hahn, Philipp Frank.

Over the centuries, no other principle of classical physics has to a larger extent nourished exalted hopes into a universal theory, has constantly been plagued by mathematical counterexamples, and has ignited metaphysical controversies about causality and teleology than did the Principle of Least Action (henceforth PLA). After some decades of relative neglect, by the end of the 19th century the PLA and its kin enjoyed a remarkable renaissance on three levels.

Since the work of Hermann von Helmholtz, the PLA had become a very successful scheme applicable not only in mechanics, but also in electrodynamics, thermodynamics and relativity theory. Did this spectacular success indicate that physicists possessed – to cite Helmholtz – “a valuable heuristic principle and leitmotif in striving for a formulation of the laws of new classes of phenomena” (Helmholtz, 1886, p. 210), or were these principles – as Ernst Mach held – just useful rules that served the economy of thought in various domains of experience?

A second important reorientation took place in variational calculus, the mathematical discipline on which the PLA was based and which had accompanied it through more than two centuries of philosophical debates. Karl Weierstraß’ critical investigations demonstrated that the precise relationship between the PLA and the differential equations resulting from it was extremely subtle, and that physicists’ customary reasoning in solving important cases only obtained under supplementary conditions. The generations of Euler and Lagrange typically had identified the PLA and the differential equations resulting from it regardless of their metaphysical attitude towards the PLA and the quantity of action. In the 19th century, variational calculus was regarded as a very useful method in analysis the application of which ho

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also here we consider some as say prof dr mircea orasanu and prof horia orasanu as followings

LAGRANGIAN AND CONSTRAINT OPTIMIZATION

Author Horia Orasanu

ABSTRACT

Locomotion of a snake-like structure in accordance with the serpenoid curve, i.e. lateral undulation, is achieved if the joints of the robot move according to the reference joint trajectories in the form of a sinusoidal function with specified amplitude, frequency, and phase shift. In particular, using the foregoing defined new states, we define a constraint function for the ith joint of the snake robot by

1 INTRODUCTION

A fundamental work in the area of snake robots was presented by Hirose [5]. In this work, Hirose considers empirical studies of biological snakes to derive a mathematical approximation of the most common gait pattern among biological snakes, known as lateral undulation. In particular, the shape of a snake conducting lateral undulation can be described by a planar curve (the serpenoid curve) with coordinates in the x-y plane along the curve at arc length s given by

x(s)=∫0scos(acos(bz)+cz)dz

(43)

y(s)=∫0ssin(acos(bz)+cz)dz

(44)

where a, b, and c are positive scalars.

Φi = αsin(η + (i − 1)δ) + ϕo

(45)

where i∈{1,…,N−1}, α denotes the amplitude of the sinusoidal joint motion, and δ is a phase shift that is used to keep the joints out of phase. Moreover, ϕo is an offset value that is identical for all of the joints. It was illustrated in [16] how the offset value ϕo affects the orientation of the snake robot in the plane. Building further on this insight, we consider the second-order time derivative of ϕo in the form of a dynamic compensator, which will be used to control the orientation of the robot. In particular, through this control term, we modify the orientation of the robot in accordance with a reference orientation. This will be done by adding an offset angle to the reference trajectory of each joint. To define the guidance law, without loss of generality, we assign the global coordinate system such that the global x-axis is aligned with the desired path. Consequently, the position of the CM of the robot along the y-axis, denoted by py, defines the shortest distance between the robot and the desired path, often referred to as the cross-track error. In order to solve the path following problem, we use the LOS guidance law as a virtual holonomic constraint, which defines the desired head angle as a function of the cross-track error as

ΦN=−tan−1(pyΔ)

In particular, we use a line-of-sight (LOS) guidance law as the reference angle for the head link. LOS guidance is a much-used method in marine control systems (see, e.g. [27]). In general, guidance-based control strategies are based on defining a reference heading angle for the vehicle through a guidance law and designing a controller to track this angle [27].

If the basis physics is postulated but not known, a computer simulation can relate the theory to observations on complex systems and thus test the theory.

A wide range of numerical techniques are available, from simple searches to sophisticated methods, such as annealing, an algorithm for finding global minima which was inspired by an actual physical process. Some of the techniques are repetitive applications of deterministic equations. Others invoke stochastic processes (using “random” numbers), to focus on the important features.

While most applications of such simulations yield expected results, surprises do occur. This is analogous to an unexpected result from an experiment. Either the simulation/experiment went wrong (usual) or a new aspect of nature has been uncovered (rare). Examples of the latter are the identification of constants of motion in chaotic systems and the discovery of runaway motion in the drift and diffusion of ions in gas. Such discoveries are followed by “proper” theories and “proper” experiments, but the computer plays a vital role in the research.

The research of Professor Gatland involves data analysis and the mathematical modeling and simulation of microscopic physical processes. These activities encompass both research and instruction.

In this lecture we recall the definitions of autonomous and non autonomous Dynamical Systems as well as their different concepts of attractors. After that we introduce the different notions of robustness of attractors under perturbation (Upper semicontinuity, Lower semicontinuity, Topological structural stability and Structural stability) and give conditions on the dynamical systems so that robustness is attained. We show that enforcing the appropriately defined virtual holonomic constraints for the configuration variables implies that the robot converges to and follows a desired geometric path. Numerical simulations and experimental rMethods

In this section, we derive the kinematic model along with the dynamic equations of motion of the snake robot in a Lagrangian framework. Moreover, we use partial feedback linearization to write the model in a simpler form for model-based control design.

In order to perform control design, we need to write the governing equations of the system in an implementable way. This is often done by choosing a local coordinate chart and writing the system equations with respect to (w.r.t.) these coordinates. According to the illustration of the snake robot in Figure Figure1,1, we choose the vector of the generalized coordinates of the N-link snake robot as x = [q1,q2,…,qN−1,θN,px,py]T ∈ ℝN+2, where qi with i∈{1,…,N−1} denotes the ith joint angle, θN denotes the head angle, and the pair (px,py) describes the position of the CM of the robot w.r.t. the global x−y axes. Since the robot is not subject to nonholonomic velocity constraints, the vector of the generalized velocities is defined as x˙=[q˙1,q˙2,…,q˙N−1,θ˙N,p˙x,p˙y]T∈RN+2. Using these coordinates, it is possible to specify the kinematic map of the robot. In this paper, we denote the first N elements of the vector x, i.e. (q1,…,qN−1,θN), as the angular coordinates, and the corresponding dynamics as the angular dynamics of the system.

Figure 1

An illustration of the N -link snake robot. Kinematic parameters of the snake robot.

This is due to the fact that it is usually not straightforward to integrate the anisotropic external dissipative forces, i.e. ground friction forces, acting on these complex robotic structures into their Euler-Lagrange equations of motion. However, ground friction forces have been proved to play a fundamental role in snake robot locomotion (see, e.g. [16]). In this paper, we derive the equations of motion of the snake robot in a Lagrangian framework, i.e. treating the robot as a whole and performing the analysis using a Lagrangian function, which is simple to follow and better suited for studying advanced mechanical phenomena such as elastic link deformations [25], which might be insightful for future research challenges on snake robots. Moreover, we integrate the anisotropic friction forces into these equations using the Jacobian matrices of the links, which gives a straightforward mapping of these forces for the equations of motion.

Snake robots are a class of simple mechanical systems, where the Lagrangian L(qa,x˙) is defined as the difference between the kinetic energy K(qa,x˙) and potential energy 𝒫(x) of the system [26]. Since the planar snake robot is not subject to any potential field, i.e. −∇𝒫(x) = 0, we may write the Lagrangian equal to the kinetic energy, which is the sum of the translational and the rotational kinetic energy of the robot:

L(qa,x˙)=K(qa,x˙)=12m∑i=1N(p˙2x,i+p˙2y,i)+12J∑i=1Nθ˙2i

(9)

where m and J denote the uniformly distributed mass and moment of inertia of the links, respectively. Using the Lagrangian function (9), we write the Euler-Lagrange equations of motion of the controlled system as

ddt[∂L(qa,x˙)∂x˙i]−∂L(qa,x˙)∂xi=(B(x)τ−τf)i

(10)

where i∈{1,…,N+2}, B(x) = [ej] ∈ ℝ(N+2)×(N−1) is the full column rank actuator configuration matrix, where ej denotes the jth standard basis vector in ℝN+2. Moreover, B(x)τ ∈ ℝN+2 with τ = [τ1,…,τN−1]T ∈ ℝN−1 stands for the generalized forces resulting from the control inputs. Furthermore, τf=[τ1f,…,τN+2f]T∈RN+2 denotes viscous and Coulomb friction forces acting on (N+2) DOF of the system. The controlled Euler-Lagrange equations (10) can also be written in the form of a second-order differential equation as

M(qa)x¨+C(x,x˙)x˙=B(x)τ−τf

(11)

where M(qa) ∈ ℝ(N+2)×(N+2) is the positive definite symmetric inertia matrix, C(x,x˙)x˙∈RN+2 denotes the generalized Coriolis and centripetal forces, and the right-hand side terms denote the external forces acting on the system. The fact that the inertia matrix is only a function of the directly actuated shape variables qa is a direct consequence of the invariance of the Lagrangian

The geometry of the problem

The (N+2)-dimensional configuration space of the snake robot is denoted as 𝒬 = 𝒮 × 𝒢, which is composed of the shape space and a Lie group which is freely and properly acting on the configuration space. In particular, the shape variables, i.e. qa=(q1,…,qN−1), which define the internal configuration of the robot and which we have direct control on, take values in . Moreover, the position variables, i.e. qu=(θN,px,py), which are passive DOF of the system, lie in . The velocity space of the robot is defined as the differentiable (2N+4)-dimensional tangent bundle of as T𝒬 = 𝕋N × ℝN+4, where 𝕋N denotes the N-torus in which the angular coordinates live. The free Lagrangian function of the robot ℒ:T𝒬 → ℝ is invariant under the given action of on . The coupling between the shape and the position variables causes the net displacement of the position variables, according to the cyclic motion of the shape variables, i.e. the gait pattern. Note that for simplicity of presentation, in this paper, we consider local representation of T𝒬 embedded in an (2N+4)-dimensional open subset of the Euclidean space ℝ2N+4. To this end, we separate the dynamic equations of the robot given by (11) into two subsets by taking x = [qa,qu]T ∈ ℝN+2, with qa ∈ ℝN−1 and qu ∈ ℝ3 which were defined in the subsection describing the geometry of the problem:

m11(qa)q¨a+m12(qa)q¨u+h1(x,x˙)=ψ∈RN−1

(20)

m21(qa)q¨a+m22(qa)q¨u+h2(x,x˙)=03×1∈R3

(21)

where m11 ∈ ℝ(N−1)×(N−1), m12 ∈ ℝ(N−1)×3, m21 ∈ ℝ3×(N−1), and m22 ∈ ℝ3×3 denote the corresponding submatrices of the inertia matrix, and 03×1 = [0,0,0]T ∈ ℝ3. Furthermore, h1(x,x˙)∈RN−1 and h2(x,x˙)∈R3 include all the contributions of the Coriolis, centripetal, and friction forces. Moreover, ψ ∈ ℝN−1 denotes the non-zero part of the vector of control forces, i.e. B(x)τ = [ψ,03×1]T ∈ ℝN+2. From (21), we have

q¨u=−m−122(h2+m21q¨a)∈R3

(22)

Substituting (22) into (20) yields

(m11−m12m−122m21)q¨a−(m12m−122)h2+h1=ψ

(23)

For linearizing the dynamics of the directly actuated DOF, we apply the global transformation of the vector of control inputs as

ψ=(m11−m12m−122m21)ϑ−(m12m−122)h2+h1

(24)

where 𝜗 = [𝜗1,𝜗2,…,𝜗N−1]T ∈ ℝN−1 is the vector of new control inputs. Consequently, the dynamic model (20)-(21) can be written in the following partially feedback linearized form

q¨a=ϑ∈RN−1

(25)

q¨u=D(x,x˙)+C(qa)ϑ∈R3

(26)

with

D(x,x˙)=−m−122(qa)h2(x,x˙)=[fθN,fx,fy]T∈R3

(27)

C(qa)==−m−122(qa)m21(qa)[βi(qa),0,0]T∈R3×(N−1)

(28)

where βi(qa):𝒬 → ℝ is a smooth scalar-valued function. It can be numerically shown that the value of βi is negative at any configuration qa ∈ 𝒬. Furthermore, fθN, fx, and fy denote the friction forces acting on θN, px, and py, respectively ( fθN also contains Coriolis forces besides the friction forces). For the aim of analysis and model-based control design, we write (25)-(26) in a more detailed form:

q¨a=ϑ∈RN−1

(29)

θ¨N=fθN(x,x˙)+βi(qa)ϑi∈R

(30)

p¨x=fx(x,x˙)∈R

(31)

p¨y=fy(x,x˙)∈R

(32)

where the summation convention is applied in (30), and henceforth, to all the equations which contain repeated upper-lower indices (i.e. whenever an expression contains a repeated index, one as a subscript and the other as a superscript, summation is implied over this index [26]). The dynamical system (29)-(32) is in the form of a control-affine system with drift. In particular, the term

A(x,x˙)=[q˙a,q˙u,0(N−1)×1,D(x,x˙)]T∈R2N+4

(33)

is called the drift vector field, which specifies the dynamics of the robot when the control input is zero. Furthermore, the columns of the matrix

B(qa)=⎡⎣⎢⎢⎢⎢0(N+2)×(N−1)IN−1[β1(qa),…,βN−1(qa)]02×(N−1)⎤⎦⎥⎥⎥⎥∈R(2N+4)×(N−1)

(34)

are called the control vector fields, which enable us to control the internal configuration and consequently the orientation and the position of the robot in the plane.

Remark2.

The last two rows of the control vectors in (34) are composed of zero elements. This implies that the control forces have no direct effect on the dynamics of the position of the CM of the robot, i.e. (31)-(32). Furthermore, the dynamics of the position of the CM are coupled with the dynamics of the directly actuated shape variables qa, i.e. (29), only through the friction forces. Accordingly, in the absence of the friction forces, the linear momentum of the robot is a conserved quantity, and the position of the CM of the robot is not controllable.In this section, we state our control design objectives which will be followed throughout the remaining sections of the paper. In particular, we stress that for a complex mobile multi-link robotic structure such as a snake robot, formulating a pure path following, trajectory tracking, or maneuvering problem is unusual (for definitions of these problem formulations, see [27]). This is due to the fact that for a part of the state variables of the system (particularly the shape variables and the head angle), it is most natural to formulate the control problem as a trajectory tracking problem, while for the other state variables (particularly the position of the CM), we may formulate the problem as a path following or a maneuvering one.

To formulate a combinational track-follow problem for the snake robot, which we define as a trajectory tracking formulation for a subset of the state variables, together with a path following formulation for the remaining subset, we introduce the error variable for the ith joint of the robot as

yi = qi − Φi

(35)

where i∈{1,…,N−1}, and Φi ∈ ℝ denotes a function that defines the reference trajectory for the ith joint which will be chosen through the control design in the next section. The head angle error is defined as

yN = θN − ΦN

(36)

where ΦN ∈ ℝ denotes the reference head angle for the robot.

We divide the control objectives into three main parts. In the first part, the goal is to make the shape variables of the robot track given bounded smooth time-varying references, i.e. asymptotic trajectory tracking problem, such that

limt→∞∥yi(t)∥=0

(37)

for all i∈{1,…,N−1}. Furthermore, we seek to control the head angle of the robot. The second part of the control objective is thus to make the head angle of the robot track a desired head angle such that

limt→∞∥yN(t)∥=0

(38)

Moreover, we define a desired straight path that we want the CM of the snake robot to follow. This is defined as a smooth one-dimensional manifold 𝒫 ⊂ ℝ2, with coordinates in the x-y plane given by the pair (pxd,pyd), which are parameterized by a scalar time-dependent variable Θ(t) as

𝒫 = {(pxd(Θ), pyd(Θ)) ∈ ℝ2:Θ ≥ 0}

(39)

We define the vector of the path following error variables for the position of the CM of the robot as p˜=[px(t)−pxd(Θ),py(t)−pyd(Θ)]T∈R2. Subsequently, the third part of the control objectives is defined as practical convergence (see, e.g. [4]) of the position of the CM of the robot to the desired path such that

limt→∞sup∥p˜(t)∥≤ε

(40)

where ε ∈ ℝ>0 is an arbitrary positive scalar. Moreover, we require that Θ˙(t)≥0 and limt→∞Θ(t) = ∞ (forward motion along the path), and boundedness of the states of the controlled system.

Go toVirtual holonomic constraints are specified through C1 coordinate-dependent functions Φi:𝒬 → ℝ which are called the constraint functions, in the relations of the form Φi(x)=0, which can be enforced through the feedback action. In particular, for the snake robot, we define a vector-valued function

Φ = [Φ1,…,ΦN]T ∈ ℝN

(41)

in which every element defines one constraint function for the corresponding angular coordinate of the system.

At this point, we augment the state vector of the system with three new states that in the following will be used in the control design. The introduction of these new variables to the state vector of the system, which will be used as constraint variables, is inspired by the notion of dynamic virtual holonomic constraints [21], i.e. virtual holonomic constraints which depend on the solutions of a dynamic compensator. The idea is to make the virtual holonomic constraints to depend on the variations of a dynamic parameter, which is used for controlling the system on the constraint manifold. The purpose of these additional states is explained below.

1. We introduce two new states [ϕo,ϕ˙o]T∈R2 where the second-order time derivative of ϕo will be used as an additional control input that drives the snake robot towards the desired path by modifying the orientation of the robot in accordance with a path following guidance law.

2. In the previous section, we defined the control objective for the joints and the head angle of the robot as a trajectory tracking problem. However, it is known that holonomic constraints are coordinate-dependent equality constraints of the form Φi(x)=0, where Φi is a time-independent function [25]. Thus, we remove this explicit time dependency from the reference joint trajectories by augmenting the state vector of the system with a new variable η, with η˙=2π/T and η(0)=0, where T denotes the period of the cyclic motion of the shape variables of the robot.

Subsequently, we denote the augmented coordinate vector of the system by

xˆ=[q1,…,qN−1,θN,px,py,ϕo,η]T∈RN+4

(42)

:

References

1. Liljebäck P, Stavdahl Ø, Beitnes A (2006) SnakeFighter – development of a water hydraulic fire fighting snake robot. In: Proc. IEEE international conference on control, automation, robotics, and vision ICARCV, Singapore.

2. Wang Z, Appleton E (2003) The concept and research of a pipe crawling rescue robot. Adv Robot 17.4: 339–358.

3. Fjerdingen SA, Liljebäck P, Transeth AA (2009) A snake-like robot for internal inspection of complex pipe structures (PIKo). In: Proc. IEEE/RSJ international conference on intelligent robots and systems, St. Louis, MO, USA.

4. Dacic DB, Nesic D, Teel AR, Wang W. Path following for nonlinear systems with unstable zero dynamics: an averaging solution. IEEE Trans Automatic Control. 2011;56:880–886. doi: 10.1109/TAC.2011.2105130. [Cross Ref]

5. Hirose S. Biologically inspired robots: snake-like locomotors and manipulators. Oxford, England: Oxford University Press; 1993.

6. Matsuno F, Sato H (2005) Trajectory tracking control of snake robots based on dynamic model. In: Proc. IEEE international conference on robotics and automation, 3029–3034. 18-22 April 2005.

7. Date H, Hoshi Y, Sampei M (2000) Locomotion control of a snake-like robot based on dynamic manipulability. In: Proc. IEEE/RSJ international conference on intelligent robots and systems, Takamatsu, Japan.

8. Tanaka M, Matsuno F (2008) Control of 3-dimensional snake robots by using redundancy. In: Proc. IEEE international conference on robotics and automation, 1156–1161, Pasadena, CA.

9. Ma S, Ohmameuda Y, Inoue K, Li B (2003) Control of a 3-dimensional snake-like robot. In: Proc. IEEE international conference on robotics and automation, vol. 2, 2067–2072, Taipei, Taiwan.

10. Tanaka M, Matsuno F (2009) A study on sinus-lifting motion of a snake robot with switching constraints. In: Proc. IEEE international conference on robotics and automation, 2270–2275. 12-17 May 2009.

11. Prautsch P, Mita T, Iwasaki T (2000) Analysis and control of a gait of snake robot. Trans IEE J Ind Appl Soc 120-D: 372–381.

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here we consider some as say prof dr mircea orasanu and prof horia orasanu PROBLE<A OPTIMIZARII

Si cum vedem avem constrangeri ale sistemelor neolonome . aceasta sectiune are un caracter

Special original si se refera la comportamentul dinamic al neolonomiei adica la constrangeri care

Au in vedere conservarea legilor a energiei si a vitezelor. Astfel cand intalnim un a,estec de fluide

In raport cu timpul pentru un flux de calcura intr-un mediu omogen sunt considerate relatiile de Forma

Si calculele care conduc efectiv pentru legaturilr neolonome cer in primul rand ca ele sa apara in problema

Bifurcatiei unde ecuatiile lui Boltzman sau Busemann cu consecintele cunoscute. Astfel cateva idei funda

Mentale pot fi tratate pentru miscarile fluidelor in jurul unor profile cand exista forme generale

Si aici mentionam si ecuatii cu parametrii integrale si toate acestea apar in nucleul derivatei de ordin

Superior

Pentru aproape toate abordarile acestor chestiuni exista ,acelea unde se introduc si sistemele dina

Mice cu legaturi neolonome care au la baza consideratii geometrice , ca si convolutia unor functii

Si multe alte consecinte incat aceste problem conduc la problemele la limita de forma cunoscuta incat

Un potential complex care sa satisfaca problem la limita de tip Riemann Hilbert ale carui solutii sa fie

Functie generatoare

Morivarea acestei problem este posibilitatea cand nucleul depinde si de functii arbitrare si se aplica

Unul din principiile ale studiului analytic dar imposibil in cazul omogen rezulta ca legaturile neolo

Nome si a legilor de conservare care sunt parcurse si prezente in sistemele dinamice cu legaturi si in

Virtutea celor de sus putem asocial un potential complex care sa satisfaca problem de tip Riemann

Cele de mai sus sunt conditiile pe care le indeplineste legaturile neolonome in dinamica sistemelor

De puncte materiale la care adaugam sistemele de Lagrangian adica ,iar Lagrangianul poate sa sa

Tisfaca ecuatia lui Laplace si sub anumite conditii in cele de mai sus se poate studia stabilitatea clase

Lor de unde si a oscilatiilor si ale caror solutii sa fie functia generatoare a conservarii legilor , dar si

A conceptului de conservare a legilor care se supun constrangerilot neolonome. Aceasta apar cand nucleul

Si deci rezultatele numerice pentru schema si alte exemple hidrodinamice ca si si cele din teoria elas

Ticitatii apar in acelasi cadru ceea ce se poate vedea pentru problemele la limita de forma unei relatii

Integrale si se aplica unul din principiile ca legaturile neolonome care se mentin daca exista suprafete

In aceste lucrari pot fi considerate conditii asupra legaturilor de neolonomie cu constrangeri optime

Care pot fi inchise in suprafete de discontinuitate si cu o larga diversitate de extremale si deasemenea

Care sunt supuse constrangerilor neolonome. Aceasta pot avea caracterul ecuatiei lui korteweg de vries

Si alte exemple hidromecanice ca si cele ca si cele dinteoria elasticitatii mapar in acelasi cadru ceea ce

In acest caz putem sa be referim la portiuni si este chiar timpul si se ia in considerare o perturbare

A suprafetelor de discontinuitate sau o perturbare a suprafetelor de teren intrucat aceste functii au un

Character haotic ele poseda expresii care rezulta din influenta ale unor ansambluri de multe forte care

Nu pot fi cercetate in amanunt dar astfel ca aici este evident ca se pot aplica si alte modalitati precum

Acum calculand lungimea si elongatia acestui corp la care adaugam si greutatea putem admite

Ca lungimea acestui corp la care adaugam si greutatea putem admite ca lungumea firului este

Este l de asemenea aici mentionam ca analiza instabilitatii miscarilor dfe puncte material sau a parti cu

Lelor fluide care sunt posibile in cadryl legaturilor cinematice cu situatii posibile la miscarile de straturi

Aceste ecuatii se aplica cand solutiile care cu sau al studului analytic care apartine grupului analitic

Deci aceasta sa fie exprimate de functii care evicentiaza aspecrul lor intamplatir iar analiza proprieta

Tilor si ale unor aplicatii sa se faca utilizand si ecuatiile punctelor material sau a particulelor de fluide

Si se obtine o relatie corespunzatoare si integrala fiind luata de la – oo la unde p este o variabila

Complexa/ aceasta permite studiul legaturilor neolonome pentru oscilatii amortizate

Astfel aplicand teorema reziduurilor se completeza domeniul de integrare cu un semicerc

Aici mentionam si rezultatele obtinte de Prof.Dr. Mircea Orasanu si Prof. Drd. Horia Orasanu si deci

In care aceste expresii permit legaturi neolonome cu dezvoltari in serie

Astfel ca apare metoda reducerilor perturbatiilor in cazurile de legaturi neolonome ,si care consista

Mai mult se poate arata ca se poate sa avem indeplinite condirtiile unui Laplacian al functiei. Apoi

Pot aparea dar pe liniile potentiale ,iar cu conditia ca functia de current sa ia valori de forma y=0 si

Astfel ca aici este evident ca se pot aplica si alte modalitati precum teoria ecuatiilor diferentiale

Se crede ca in cazul general al legaturilor neolonome probele si problema studiului oscilatiilor

Si prezinta ecuatia integral si decicare apar la legaturile neolonome pentru analiza suprafetelor

De discontinuitate unde se pune problema conservarii legilor de miscare, si se pare ca pana acum

Cele de sis sunt conditiile pe care le inceplinesc legaqturile neolonome in dinamica sistemelor

Si deci structurile sunt reprezentate prin transformarile de mai sus cand oscilatiile amortizate cand

Variabila din functia f este chiar timpul si se ia in considerare o perturbatie a suprafetelor de disconti

Nuitate sau o perturbare a suprafetelor de teren intrucat aceste functiio au un caracter haoric si in care

Si cele de o variabila complexa. Astfel ca pentru barele subtiri se poate stabili espresia gradului de

Torsiune in functie nu de Solutia incovoierii ci de cea a torsiunii si de se poate arata ca panza elicoidala

Aceste miscari corespund dinamicii sistemelor de puncte material associate legaturilor neolonome

Astfel ca va yrma

In acest caz daca L este un contur inchis compus din mai multe arce ,stunci folosind conditia Holder

Deduce ca pentru un polinom dat are loc egalitatea care comduce la la Solutia este déjà obtinuta si care

Este raportata la linii asimptotice iar ca exemplu avem curbura negative. De aici deduce imediat ca avem

In astfel de cazuri procesele si miscarile in functia de timp si in functie de experienta se introduxc

Studii carora li se aplica metodele de mai sus. Atunci concluziile care permit obtinerea tuturor datelor

Si ca o alta forma a ecuatiilor oscilatiilor cu constrangeri neolonomer care sunt de forma arata ca sunt

De asemeneasi si in astfel de cazuri procesele si miscarile in functie de timp si in functie de experimente

Sau experiente , se introduce studii carora li se aplica metodele de mai sus ,si ca exemplu se considerasi astf

El ajungem la la singularitati integrabile

Conditiile de racordare sunt astfel incat ca si care permite ca o clasa astfel ca de exemplu fybctiile

Cautate sunt aceleasi sunt aceleasi cu Solutia problemei de tip Hilbert si ca aceste rezultate pot fib extinse

La oscilatii in cazul fluidelor grele nemiscibile date de o ecuatie de tip Boussinesq de forma cunoscuta

Avand in vedere cele de mai sus metodele la care ne referimpot fi reprezentate prin integrale speciale

Sau prin transformari integrale , astfel ca aici vom prezenta aceleasi problem si astfel pe conturul dat avem

Apoi termenii si reunite toate arcele lui L, intrucat argumentul de mai sus este o functie continua in puncte

Atunci in mod aproximativ putem scrie ecuatii de forma w de unde rezulta ca w este o constanta

Ce inglobeaza o serie de factori ai miscarii si optimizari si dupa efectuarea calculelor ajungem la

O ecuatie de forma si associate oscilatiilor de diferite tipuri de straturi si unde exista un interes deosebit

In perioada actual si acestea pe domeniul dat si notat d si caracterul acestor metode este de a intelege

Deci aici obtinem un contur inchis compus din mai multe arce folosind conditia Holder si deduce ca pentru

Cazul cand sunt indeplinite conditiile la limita solutiile problemelor la limita fiind functii armonice care

Si astfel cu ajutorul incercarilor integralelor de tip Fourier care sunt necesare la studiul undelor si a

Propagarii lor prin fluide si se pine problema convectiei in straturi subtiri fluide ,dar mai ales in dome

Si astfel ca urmeaza acum aceste rezultate permit sa descriem spatiul numai cu diverse ponderi si care

In acest sens desi transformarea Fourier putem aplica transformarea lui Laplace. Aceste transformare

Este de forma data definite de la – oo la si care coincide unde Laplacianul satisface rcuatii de forma

Fie abcd un paralelogram bisectoarea unghiului dab taie prin centrul cercului circumscris triunghiu

Lui bcd daca si numai daca pab= 90 si deci care si in cazul sau in cele unghiulare cand coincide cu La

Cand nucleul este o functie determinate nedepinzand de functia arbitrara cum ar fi solutiile probleme

Lor dirichlet si Neumann unde se folosesc potentiale dedublu si de strat si in cazul cu Lagrangianul a

Vem un Hamiltonian care depinde de variabilele p in cazul sistemelor cu legaturi liniare si determina

Sa se arate ca ca syma inaltimilor unui triunghi este mai mica decat perimetrul triunghiurlui si sa

Ce unghi trebuie trebuie sa faca cu semiaxa ox dreapta d astfel incat aria delimitata de conturuul

Format din arcele de parabole on , om si segmentul mn , sa fie egala cu 64/3 si deasemenea se pune

Problema ca aceste rezultate pentru aceasta permit sa descriem spatial initial al conditiilor de diferite

Ponderi si si de puncte regulate , care ele asigura legaturile legaturile neolonome in cadrl sistemelor

Mecanice si in acest caz daca L este un contur incat in aceasta relatie sunt posibile unele operatii

Astfel ca se foloseste faptul ca in primul rand noi consideram ca punct de plecare ecuatii de tip Navier

Stokes de tip classic pentru studiul optimizarii cu legaturi care apar in cazul miscarilor stationare plane

Si incompresibile si deci astfel planul complex sunt formate din benzi de convergenta in care legaturile si

Si toate acestea apar din nucleul derivatei de ordin m-1 care va avea [e contur singularitati logaritmice

Ce vor si care sunt ca transformari integrale care pot fi utilizate in problemele mecanicii aplicate a dfi

Namicii

Sa se demonstreze urmatoarele proprietati generale ale conice;pr ,panta tangentelor in dreptul focarelo

R are valoarea e excentricitate. Ordonata punctelor din dreptul focarelor are valoarea parametru

Lui si aici raspunzand problemei de mai sus este aratat ca aceasta poseda cazuri interesante si valabile

Si in care

Aceasta ne permit admiterea ipotezei conservarii energiei si a sectiunilor plane transversal in defini

Rea si deformarea barelor si in acest sens si caz pentru obtinerea legaturilor neolonome trebuie luate

In considerare un domeniu de miscare adica trebuie considerate dinamica sistemelor de puncte mate

Riale sau particule de fluide

Prin varful a al unui patrat abcd se duce o secanta variabila care intersecteaza pe cd in e si pe bs

In f. Sa se arate ca dreapta ce uneste punctul f cu mijlocul m al segmentului ed este tangenta

Cercului inscris in patratul abcd , si astfel ca exista un interes deosebit care care permite conservarea

Legilor pe baza analizei rezultatelor in multe privinte si este dificila datorita conditiilor la limita

Si se poate arata ca subiectele de aceasta maniera in viitor cer numai ipoteze plauzibile si care sa fie

Deci vom gasi de asemenea cele de sus care se reduc la egalitatile echivalente cu H si care ne dau

Functii rationale si acestea sa fie posibile intr-I configuratie dar si in cazul sistemelor de puncte mate

Riale si astfel ca acestea permit admiterea ipotezei conservariii energiei si a sectoimolor plane trans

Versale in deformarea barelor deci ipoteza permite modelarea deformarii unui corp continuu iar a

Plicarea ei reprezinta in multe cazuri o treapta intermediara cand se defines gradele de libertate si astfel

Ca pentru diverse sisteme de puncte material sau dinamica particulelor de fluide si care ne conduc la

Relatiile fundamentale

.

, ; ,

x(t) = t/T X + b t(T-t)

/ \

Deci conchidem ca incat se observa ca si deci ca sa se obtina un system cu legaturi neolonome care

Si ca sa si o noua discutie este posibila asupra rezonantei oscilatiilor si undelor care apar in legaturi

Dreptele be si fm se taie intr-un punct de pe cercul circumscris patratului abcd . sa se arate ca elipsa

X 3rd /2 + y 3rd -1 =0 si hiperbola x 3rd – y 3rd -1 =0 au aceleasi focare

1 X 3rd /2 + y 3rd -1 =0 si hiperbola x 3rd – y 3rd -1 =0 au aceleasi focare

Deci sa se demonstreze ca si se dau cercurile si cercurile construite si laturile unui patrulater conves

Ca diametre trec prin acelasi punct daca su numai daca patrulaterul este ortodiagonal 19 95 91 dar

Ca sa sis a se calculeze si in cazul care urmeaza

Deci aici avem ansambluri de forte reprezentate prin transformarile de sus cand oscilatiile amorti

Zate cand variabila din functia data si astfel satisfacand conditia lui Holder in intregime cu si in afara

Conditia necesara si suficienta ca un triunghi sa fie dreptujghic este ca R(R+r) = ac +ab deci sa si

Astfel sa avem si deci in afara punctelor care au ordinul1 fiind puncte de intrerupere sau puncte de ybg

Aceste rezultate dau optimizarile in cazul legaturilor neolonome in numeroase cazuri sub aspect fizic

Structurile sunt reprezentate si deasemenea deci urmeaza ca indexul are forma pe contur dat ca sa

Apara structurile care sunt reprezentate prin transformarile de mai sus cand oscilatiile amortizate ale

Cazurile de mai sus conduc la solutii analitice de forma data si astfel ca cand ne reunim avem si si

Conditii Holder si de aici urmeaza ca dau optimizarile in cazul legaturilor neolonome in numeroase

Cazuri pe suprafetele de discontinuitate in influente de ,ai multe forte si ele evidentiaza aspectul dat

Cand luam in considerare variaza pentru z=o si mentionam ca pentru potential suntem condusi la un

Numar de constant reale in jurul originii ,incat avem m -1 ecuatii complexe care satisfac ;a constant

Le date si prezente pentru unele valori precum u1 cu ajutorul reprezentarii problemelor la limita, ca

Pentru unele valori functii , sau reprezentarea si solutiei ecuatiei diferentiale din tipurile relevate ,care

Se reduc la problema dezvotarilor potentiale . Si calcul integral in care restrictiile pentru ecuatiile dife

Rentiale in sensul lui Cauchy in sens de distributie este nerezonabil de aceea sunt preferabile ecuatiile

Si aici avem Teorema lui Horia Orasanu . conditia necesara si suficienta ca o functie armonica care

Satisface conditii initiale si la limita de tip Hilbert si apar si ecuatiile ereditare cu principalele sensuri

Deci aici avem

:

Aici pot aparea si termeni diagonali in cazul unor matrice cum ar fi cazul miscarilor de filtratie prin tr

Un mediu poros rectangular cand presupunem mediul poros omogen si miscarea este generate de un po

Tential complex daca miscarea este incompresibila deci conditiile la limita sunt su[use legaturilor

Neolonome sau echivalente. Si astfel ca daca sunt posibile legaturi cu substituiri de solutii in continuare

A consideratiilor si unde formularile diferite de cele ale lui Cauchy pot fi introduce prin cateva expresii

Astfel ca in cazul lucrului mechanic virtual formulat pentru transfotmari ar fi sufficient sa se calculeze

Inversul relatiei functionale dar si sisteme;e dinamice cele mai discutate sunt cele olonome dar cele

In acest caz pot aparea relatii a doua functii cu character aleator de forma m = const,numita valoare

Medie a legaturilor neolonome o matrice de autocorelatie notate b definite , mai sus ,care pune in evi

Denta oscilatiile de forma aleatoare ,in sens vectorial pentru miscari stationare sau stabilitatea acestor

Miscari in care acestea pot fi aplicate ecuatiilor sau sistemelor de ecuatii differentiale stabilite pentru

Miscari stationare sau mestationare incat sa fie indeplinite conditiile care sunt expresiile coeficientilor

Dezvoltarii de mai sus si pe de alta parte folosind prima formula a

Deci o sa avem si este unde f este o functie rationala cu functii dis

Continua deci a lui Green si se gaseste ca are loc incat forma Iacobianu

Lui caruia i se determina valoarea minima. Deci din cele de mai sus

Urmeaza ca

Aceste miscari corespund dinamicii sistemelor de puncte mamateriale

Associate legaturilor neolonome si am ajuns astfel la teorema

Teorema energiei Horia Orasanu. Pe baza celor de mai sus aveem sau sta

Bilitatea acestor miscari in care acestea pot fi aplicate ecuatiiolor

Sau sisteme;pr de ecuatii dferentiale

Astfel ca rezultatele numerice pentru schema si deci urmeaza ca este

Necesar o deformare daca ,atricea de deformabilitate obtinuta prin

Jacobian are forma cunoscuta si cele de mai sus.si astfel ca o functie

Rationala in acest sens in care acestea pot fi aplicate in continuare

Iar sistemul omogen este comparabil este compatibil

Iar sistemul omofen este compatibil totdeauna si ne conduce la rezulta

Tul ca trebuie sa avem 2( m-x ) +1-r ecuatii independente si pe domeni

Ul dat in acest caz putem enunta in cazul problemelor neolonomice ;a pro

Blame izoperimetrice

Astfel in cazul principiului lucrului mechanic virtual deduce si este

Necesar ca din nous a exprimam cele de mai sus prin transformari integra

Le

Aceasta ecuatie fiind de forma rezulta Si ajungem la la o ecuatie

Integral integrala de tip fredholm ,iar problema se rezolva pentru

Doua solutii liniar independente ,si de asemenea ecuatiile fiind de

Forma , si urmeaza ca e ecuatie care admite un numar de solutii si vom

Avea ca Jacobianul conduce la integrantul care trebuie minimizat si

Are forma ,apoi aceste teoreme sunt aplicate in cazul dinamicii siste

Melor cu optimizari

Deasemenea urmeaza ca miscarea intre doi cilindri poate in caresi negli

Jand deformarea axiala si de lunecare a elementelor de deformare axiala

Energia si matricea si urmeaza ca sa in cazul dat unui wsistem dinamic

Cu legaturi neolonome si astfel ca pentru anumite calori apare si dezvol

Tarea in serie

Aceasta ecuatie fiind de forma pentru f = 0 in care aceasta fiind

O solutie care depinde de 2 ,, parametric independent liniari. Mai mult

Se poate arata ca ca printe consecinte pot fi cele cu interpretare fizic

A deosebita si care din punct de vedere mathematic ne dau descrierea

Unui potential complex de forma f

Incat in cazul unei functii analytic generalizata sa putem acea indeplin

Ite conditiile unui Laplacian al functiei cand satisface ecuatia

Deci problemele oscilatiilor cu constrangeri ori neolonomie se reduce la

Rezolvarea ecuatiei Helmholtz ,adica a ecuatiei de forma data care stabi

Leasca diverse ipoteze ale reprezentarilor integrale de asemenea cele de

Mai sus se reduc ;a egalitatile echivalente pentru potentialul complex

Si deci si care sunt date, care sunt deplasari normale

Ori pentru aceste situatii putem folosi ecuatii cu diferente finite side

Unde urmeaza ca este un grup de transformari sisi deasemeneacele de mai

Sus si astfel ca se ia in considerare si in aceste conditii cu o axa

Central

Astfel ca unii autori au aratat ca variatiile pulsatiilor se realozeaza

Cu ajutorul reductorului diferenta de faze a celor doua miscari ,care

Este satisfacuta si modificata prin schimbarea pozitiilor in miscarea re

Lativa a celor doua elemente intr-un mechanism care sunt normale pe plan

Si ca sa se poata fi folosite rigla orizontala care este perfect rigida

Cu axa central

In acest caz avem mai multe trepte intermediare si avem clasa a functiil

Or care satisface

Astfel se considera o masina cu moment redus al fortelor motoare si

Momentul redus al fortelor rezistente pentru ciclu de maqsini , care

Apar intr-o diagrama aici cu un un ghi de rotatie alelementului de re

Ducere de rotatie al elementului de reducere al masinii. Pe baza dia

Gramelor pentru momentele de mai sus variatia energiei cinetice in ori

Ce interval de timp din interiorul unui ciclu de functionare al masinii

In acest caaz aplicand ecuatia energiei cinetice deducedar si in cazuril

E cunoscute de propagare care se face

In cazul miscarilor prin medii neolonome pot aparea matrice de deforma

Re si deci ipoteza permite considerarea unui model de tip bara pentru

Mofelarea deformarii unui corp cand se defines gradele de libertate pent

Tru

Si unde L este valoarea mediana a procesului de lucru mechanic pozitiv

Sau neg

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HELMHOLTZ AND LAGRANGIAN CONSTRAINTS

ABSTRACT

While the above are “black-and-white” descriptions, materials properties truly cover a wide range of possibilities. For example, dielectrics can and do conduct small amounts of electricity while no conductors are perfect; including super conductors. We can begin to understand this “gray scale” reality by considering what happens in a vacuum and a general material. In free space, which is to assume a perfect vacuum, an applied electric field will continually accelerate a free charged particle. Inside of a material, this acceleration does not go on forever. In fact, inside the material, the charged particles regularly collide with the neutral atoms and molecules. See Fig. XX. These collisions slow the charged particle – like a frictional drag. Simple models of friction describe this drag force as

,

where Fdrag is the force and v is the velocity. Thus the force on the charged particle, q, can be described as

.

Here, is the proportionality constant for the drag force and it has units of mass/time, , where m is the charged particle’s mass and is the effective time between collisions. At some point the particle will reach a drift velocity, vdrift, such that the drag force balances the electric field force. Then it is found that

.

The full equation can be solved as follows. First assume that the electric field is zero. We are then left with the equation,

which can be easily solved to find

,

where is an unknown constant. Now we know that the electric field will cause a steady state drift, thus, we can assume that the velocity is of the form,

1 INTRODUCTION

The differences between these types of materials can be understood from solid state theory. [ ]

Conductors

Semiconductors

Dielectrics

In each of these we make assumptions about the materials and describe them.

.

This trial solution can be plugged into our original equation of motion to determine,

.

Adding these equations we find,

.

Because of our original drift solution, we know that,

.

Now what is ? is an effective collision period between energy loss collisions. Copper, a very good conductor, has ~ 10-14 s. This implies that if we

Now we can go back to the drift situation (static). There, we found that the drift velocity was given by

.

From this we can define the mobility, µ,

From this is easy to see that electrons drift in the direction opposite to the electric field while holes and ions drift in the direction of the field. We can use this equation to prove a law that is fundamental to the study of circuits, namely Ohm’s Law. Assuming that we know the drift velocity, we can then calculate the current density from the charge density, n.

where is the conductivity.

Throughout this discussion, we have made a few simplifying assumptions. One of the more important is that the drift is independent of the direction. If one considers a crystalline structure, one might imagine that the drift velocity depends on the direction of travel. This is indeed the case. This results in that both the mobility and the conductivity are matrices. However, for the purposes of this book, we will assume that they are simply material dependent constants.

Now, we return to examine specific properties of our materials. First is an electrically isolated conductor in an electric field. With no continuous source of electrons we quickly attain a state in which electrons are drawn to one side, leaving a net positive on one side of the metal and a net negative charge on the opposite side. These charges produce an electric field internal to the metal, which just exactly cancels the external electric field originating outside the metal. If this were not the case then the charge carriers would be free to move through the metal until the electric field was shielded out. We will return to this later.

Electric field through a electrically floating metal surface.

Dielectrics come in two varieties. Dielectrics with permanent electric dipoles and those without permanent dipoles. Water is the classic example of a material with permanent electric dipoles. (Pure water – known as De-Ionized, DI, water – is a very poor conductor!) Electric dipoles are typified by a pair charges of equal magnitude but opposite sign, q and -q, that are separated by a small distance and direction, l. The dipole vector is defined as . In general, the orientation of these dipoles are mixed such that the average electric field is zero. However, if an external electric field is applied, the dipoles will align, on average, with the external field, so as to reduce the local electric field.

When we have a dielectric in an electric field the dipoles align as such

If we know the sum of all dipoles per unit volume, we get dipole field

where we have assumed that there are N dipoles per unit volume. How susceptible these dipoles are to aligning with the E field is given by the electric susceptibility constant. (This number is experimentally derived.) We define the electric susceptibility constant such that dipole field is

Example: The electric field of a simple dipole. There are several ways in which the electric field strength for a simple dipole can be calculated. One of the simpler is as follows. First, we are going to calculate the field at a point P=(0,y,z), where the dipole is aligned along the z-axis. There is no loss of generality using this coordinate system but it makes solving the problem easier.

Second, we know from earlier that the electric field for a point charge can be written as the negative gradient of the electric potential, , where

.

For the dipole system, the potential is the sum of the potentials and hence,

.

For R>>l, as is typically the case, we can approximate the distance in terms of R and then Taylor expand to get,

.

This gives a potential of

The International Commission on Mathematical Instruction (ICMI), founded in Rome in 1908, has, for the first time in its history, established prizes recognising outstanding achievement in mathematics education research. The Felix Klein Medal, named for the first president of ICMI (1908-1920), honours a lifetime achievement. The Hans Freudenthal Medal, named for the eight president of ICMI (1967-1970), recognizes a major cumulative program of research. These awards are to be made in each odd numbered year, with presentation of the medals, and invited addresses by the medallists at the following International Congress on Mathematical Education (ICME).

These awards, which pay tribute to outstanding scholarship in mathematics education, serve not only to encourage the efforts of others, but also to contribute to the development, through the public recognition of exemplars, of high standards for the field. They represent the judgement of an (anonymous) jury of distinguished scholars of international stature. The jury for the 2003 awards was chaired by Prof. Michèle Artigue of the University Paris 7.

ICMI is proud to announce the first awardees of the Klein and Freudenthal Medals.

The Felix Klein Medal for 2003 is awarded to Guy Brousseau, Professor Emeritus of the University Institute for Teacher Education of Aquitaine in Bordeaux, for his lifetime development of the theory of didactic situations, and its applications to the teaching and learning of mathematics.

The Hans Freudenthal Medal for 2003 is awarded to Celia Hoyles, Professor at the Institute of Education of the University of London, for her seminal research on instructional uses of technology in mathematics education.

Presentation of the medals, and invited addresses of the medallists, will occur at ICME-10 in Copenhagen, July 4-11, 2004.

(Document for a press release issued on April 4, 2004)

Citation for the 2003 ICMI Felix Klein Medal to Guy Brousseau

The first Felix Klein Medal of the Internal Commission on Mathematical Instruction (ICMI) is awarded to Professor Guy Brousseau. This distinction recognises the essential contribution Guy Brousseau has given to the development of mathematics education as a scientific field of research, through his theoretical and experimental work over four decades, and to the sustained effort he has made throughout his professional life to apply the fruits of his research to the mathematics education of both students and teachers.

Born in 1933, Guy Brousseau began his career as an elementary teacher in 1953. In the late sixties, after graduating in mathematics, he entered the University of Bordeaux. In 1986 he earned a ‘doctorat d’état,’ and in 1991 became a full professor at the newly created University Institute for Teacher Education (IUFM) in Bordeaux, where he worked until 1998. He is now Professor Emeritus at the IUFM of Aquitaine. He is also Doctor Honoris Causa of the University of Montréal.

From the early seventies, Guy Brousseau emerged as one of the leading and most original researchers in the new field of mathematics education, convinced on the one hand that this field must be developed as a genuine field of research, with both fundamental and applied dimensions, and on the other hand that it must remain close to the discipline of mathematics. His notable theoretical achievement was the elaboration of the theory of didactic situations, a theory he initiated in the early seventies, and which he has continued to develop with unfailing energy and creativity. At a time when the dominant vision was cognitive, strongly influenced by the Piagetian epistemology, he stressed that what the field needed for its development was not a purely cognitive theory but one allowing us also to understand the social interactions between students, teachers and knowledge that take place in the classroom and condition what is learned by students and how it can be learned. This is the aim of the theory of didactic situations, which has progressively matured, becoming the impressive and complex theory that it is today. To be sure, this was a collective work, but each time there were substantial advances, the critical source was Guy Brousseau.

This theory, visionary in its integration of epistemological, cognitive and social dimensions, has been a constant source of inspiration for many researchers throughout the world. Its main constructs, such as the concepts of adidactic and didactic situations, of didactic contract, of devolution and institutionalization, have been made widely accessible through the translation of Guy Brousseau’s principal texts into many different languages and, more recently, the publication by Kluwer in 1997 of the book, Theory of didactical situations in mathematics — 1970-1990.

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

as

LAGRANGIAN AND CONSTRAINTS

ABSTRACT

The work that a force does on an object is “W=Fr” where r is the distance moved. In reality, the work done depends on the direction of the force. For example gravity does not do work on a box that moves across a flat floor. So in fact . If the force changes with position, ie is a vector field, then the work becomes , where is a vector of infinitesimal length along the direction.

As we will show below, these two operations have physical significance. describes the source of a diverging vector field while describes the source of a twisting vector field. WITH ONE EXCEPTION, ALL VECTOR FIELDS ARE ‘PRODUCED BY’ A TWISTING SOURCE OR A DIVERGING SOURCE OR SOME COMBINATION OF SUCH SOURCES. The one exception is a uniform (non zero) field, but as will be seen later in the semester, this is type of vector field not physically possible for the electric of magnetic fields. (While uniform fields are nice to use in homework problems and as approximations, a truly uniform field across all space results in infinite energy – perhaps we can solve the energy crisis – not.)

The final major operation of the del operator is an operation on a scalar field, which is known as the gradient

We will return to the Del vector shortly.

1.8.1.5. LINE Integrals:

Path 1: so that

Path 2: so that

Path 3: so that

Path 4:

Thus we find that amount of work done by the force in this field depends on the direction of travel. This is not unlike sliding a very heavy object across a floor. Distance and direction, e.g path, do matter.

In some instances – for example if there were no friction – then the path that we took would not matter. This is an example of a conservative force, e.g. the energy is conserved. (The example above is a non-conservative force.) This idea of conservative/non-conservative can be extended to general vector fields. Further, we will come up with generalized rules for conservative/non conservative field. To do this we will start with

1 INTRODUCTION

There are two major operations that the del operator has on vector fields, the Divergence

and the curl

The converse of the above is also true, if

For purely historical reasons in electromagnetism, we have chosen to use rather than to arrive at our potential . In either case, with the exception of the minus sign, the algebra and the results are the same. (In gravity related potentials, the positive sign is used.)

So now let us consider the work of a conservative force when moving from point a to point b

which is clearly independent of path. This was our original definition of a conservative force. Gravity, and electrostatic forces are two examples of conservative forces.

Example 1.3

Is this a conservative field? It is if .The Swedish and the Romanian studies both have an attachment. In the former, particular tasks for evaluating language in mathematics are discussed. The latter gives a rather detailed overview over how language and communication is discursively positioned within the national curriculum in mathematics in Romania.

Finally there are two other texts published separately, one by B. Pepin that compares mathematics education in United Kingdom, Germany and France (Pepin, 2007b), and one by F. M. Singer that discusses the role of cognition in relation to language (Singer, 2007b). A longer paper by S. Ongstad, published separately, will sum up how language and communication is positioned within mathematics education on the curricular level in more general terms (Ongstad, 2007b). This overarching text will even suggest strategies for how LAC in mathematics can contribute to a general framework for language(s) of schooling. The paper will to some extent build on a work published in Educational Studies in Mathematics, Mathematics and

Established at the Fourth International Congress of Mathematicians held in Rome in 1908 with the initial mandate of analysing the similarities and differences in the secondary school teaching of mathematics among various countries, ICMI has expanded its objectives and activities considerably over the years. The Commission aims at offering researchers, practitioners, curriculum designers, decision makers and others interested in mathematical education, a forum for promoting reflection, collaboration, exchange and dissemination of ideas and information on all aspects of the theory and practice of contemporary mathematical education as seen from an international perspective. ICMI thus takes initiatives in inaugurating appropriate programmes designed to further the sound development of mathematical education at all levels, and to secure public appreciation of its importance. The Commission is also charged with the conduct of the activities of IMU bearing on mathematical or scientific education. In the pursuit of its objectives, the Commission cooperates with various groups, regional or thematic, which may be formed within or outside its own structure.

2 FORMULATION

is an example of inner-product differentiation and

is an example of cross-product differentiation. The most common differential vector is the ‘del’ vector, where . Other forms of this vector can be found in other coordinate systems and are derived such that identical results are found independent of coordinate system.

This implies that . So what is ?

Example 1.4

The electric field is given by . What is the potential?

which is our standard potential introduced in earlier classes.

1.8.2. Describing Space

1.8.2.1. Coordinate systems

1.8.2.1.1. Cylindrical

1.8.2.1.2. Spherical

1.8.2.1.3. Generalized curvilinear

1.8.2.2. Points, curves, surfaces and volumes

Understanding how points, curves, surfaces and volumes are described is important for understanding electromagnetism and for solving problems.

Using one of our coordinate systems, we can describe any point in real space. The standard notation is to give the positions as (x,y,z), (r,,z) or (). For example, the point at x=5 on the x-axis would be given as (5,0,0). The point at x=0, y=2, z=6 would be given as (0,2,6). These points can also be described in cylindrical and spherical coordinates as shown in example XX.

A curve or path is a one-dimensional object that can be written as (x(s), y(s), z(s)), where s is a free parameter. You will note that x, y, and z are now functions of this free parameter and thus they are allowed to vary as s changes. While this object might follow a path through our three-dimensional space, the path has only one free parameter and is thus a one-dimensional object. (In a class on topology, one would learn that one can transfer the curve to a space in which only one of the components changes with s. The transformation from Cartesian to cylindrical or spherical is a manifestation of such a topological transformation.)

Surfaces and volumes follow naturally from the definition of a curve. A surface is a two-dimensional object that can be written as (x(s,t),y(s,t),z(s,t)), where both s and t are free parameters. (Here, t does not necessarily stand for time; however, either s or t may represent time.) Finally volumes are three-dimension objects that can be written as (x(s,t,u),y(s,t,u),z(s,t,u)). Like points and curves, surfaces and volumes can be described in several different coordinate systems. An example of a coordinate system translation for a surface is given in example XX.

1.8.2.3. Tangents and Normals

Now that we have developed our basic mathematical operations, we need to return to our study of points, curves, surface and volumes. The first thing to note is that point in space (x,y,z) has a natural vector associated with it, namely the vector (x,y,z) that runs from the origin, (0,0,0), to the point (x,y,z).

____________________

Example 1.2

Let so that

Let us examine 4 paths from (0,0) to (2,1):

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LAGRANGE INTERPOLATION AND EIGENVALUES

ABSTRACT 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)

1 INTRODUCTION

The polynomial is given by

P(x) = Ln,0(x) f(x0) + Ln,1(x) f(x1) + + Ln,n(x)f(xn) = f(xk)

where Ln,k(x) = ; Ln,k(xi) = 0 and Ln,k(xk) = 1

Polynomial approximation constitutes the foundation upon which we shall build the various numerical methods. The approximation P(x) to f(x) is known as a Lagrange interpolation polynomial, and the function Ln,k(x) is called a Lagrange basis polynomial.

Example 3.1-2 _____________________________________________________

Find the Lagrange interpolation polynomial that takes the values prescribed below

xk 0 1 2 4

f(xk) 1 1 2 5

Solution

P(x) = f(xk)

P(x) = (1) + (1)

+ (2) + (5)

When working with grids having large numbers of intervals one typically assigns a set of low degree (n = 1, 2, or 3) basis functions to each adjacent set of n+1 = 2, 3, or 4 nodes.

Example 3.1-3 _____________________________________________________

Use global interpolation by one polynomial and piecewise polynomial interpolation with quadratic for the following nodes.

xk 0 1 2 4 5

f(xk) 0 16 48 88 0

Solution

Global interpolation by one polynomial: P(x) = f(xk)

P(x) = (0) + (16)

+ (48) + (88) + 0

Piecewise polynomial interpolation with quadratic

P(x) = (0) + (16) + (48); 0 x 2

P(x) = (48) + (88) + (0); 2 x 5

The error En(x) associated with the interpolation of f(x) by Pn(x) over the interval [x0, xn] can be estimated as

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) =

E1(x) = | (0.45 0)(0.45 0.6)| = 3.37510-2

Actual error = |ln(1 + 0.45) P1(0.45)| = 1.90610-2

Second degree polynomial

P2(x) = (0) + (0.47)

+ (0.64185)

P2(0.45) = 0.36829

Error bound: E2(x) = | (x x0)(x x1)(x x2)|

E2(x) = | (0.45 0)(0.45 0.6)(0.45 0.9)| = 1.012510-2

Actual error = |ln(1 + 0.45) P2(0.45)| = 3.272910-3

Perspectives on Algebraic Varieties

Levico T. (Trento), September 6-11, 2010

Tentative Timetable

Monday, 6/09 Tuesday, 7/09 Wednesday, 8/09 Thursday, 9/09 Friday, 10/09 Saturday, 11/09

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