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]]>Symbolic view: mathematics is perceived as a collection of numbers and symbols, or rules and procedures to be followed and memorised.

Some examples are mathematics is viewed as comprising or represented by:

‘numbers and equations’ (R005)

‘figures and sums’ (R340)

‘multiply, minus, add, divide’ (R453)

For many of these participants, mathematics is seen as sets of rules and procedures to be followed and memorised. For some people, this is a pleasure because mathematics is

‘formulae, involved and exciting’ (R038)

and some of them just

‘like playing around with numbers, equations, finding solution to problems’ ]]>

robot was controlled to the LOS angle, then also the position of the CM of the robot would converge to the desired path. We will show that a similar guidance-based control strategy can successfully steer the robot towards the desired path. However, we perform the model-based control design based on a more accurate model of the snake robot which does not contain the simplifying assumptions of [17] which are valid for small joint angles

1 INTRODUCTION

. As discussed above, this is probably due to the modification of the joint controller (68) due to the lack of velocity measurements in the lab. Figure Figure1010 shows that the head angle of the robot tracked the reference head angle defined by the constraint function (46) and that the tracking error converged to a neighbourhood of the origin. Figure Figure1111 shows the motion of the CM of the robot in the x-y plane, which converged to and followed the desired path. Figure Figure1212 compares the motion of the CM during the simulation and the experiment, which were performed using the same controller parameters in order to obtain comparable results. In particular, from Figure Figure12,12, it can be seen that the physical snakeThis idea should be familiar from Section 13.2. The new phase space considered here is an example of a symplectic manifold, which has many important properties, such as being orientable and having an even number of dimensions [39].

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, ]]>

Consider a point that is sufficiently close to that the velocity is constant along . Let be the position vector of relative to . It then follows that (see Section A.18)

(4.88)

The previous equation becomes exact in the limit that . Because is arbitrary (provided that it is sufficiently close to ), the direction of the vector is also arbitrary, which implies that

(4.89)

We, thus, conclude that if the motion of a fluid is irrotational then the associated velocity field can always be expressed as minus the gradient of a scalar function of position, . This scalar function is called the velocity potential, and flow which is derived from such a potential is known as potential flow. Note that the velocity potential is undefined to an arbitrary additive constant.

We have demonstrated that a velocity potential necessarily exists in a fluid whose velocity field is irrotational. Conversely, when a velocity potential exists the flow is necessarily irrotational. This follows because [see Equation (A.176)]

(4.90)

Incidentally, the fluid velocity at any given point in an irrotational fluid is normal to the constant- surface that passes through that point.

If a flow pattern is both irrotational and incompressible then we have

(4.91)

and

]]>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

.

Therefore

,

Where

Then, by repeating the scheme of the proof of Theorem 4 [4], we obtain estimates (1). Lemma is proved.

Proof of Theorem. According to Lemma

, (2) . (3)

(2), (3) and the relations bottom follow from theorem

.

Theorem is proved.

Corollary. If , then .

REFERENCES

[1] Emelichev V., Podkapaev D. Quantitative stability analysis for vector problems of 0-1 programming // Discrete Opnimization . – 2010

. In discussing the notion of the approach of a variable magnitude to a fixed limiting value, and especially in proving the theorem that every magnitude which grows continuously but not beyond all limits, must certainly approach a limiting value, I had recourse to geometric evidences. … This feeling of dissatisfaction was so overpowering that I made the fixed resolve to keep mediating on the question till I should find a purely arithmetic and perfectly rigorous foundation for the principles of infinitesimal analysis.

He defined a real number to be a pair (L, R) of sets of rationals which have the following properties.

a. Every rational is in exactly one of the sets

b. Every rational in L is < every rational in R

Such a pair is called a Dedekind cut (Schnitt in German). You can think of it as defining a real number which is the least upper bound of the "Left-hand set" L and also the greatest lower bound of the "right-hand set" R. If the cut defines a rational number then this may be in either of the two sets.

It is rather a rather long (and tedious) task to define the arithmetic operations and order relation on such cuts and to verify that they do then satisfy the axioms for the Reals — including even the Completeness Axiom.

Richard Dedekind, along with Bernhard Riemann was the last research student of Gauss. His arithmetisation of analysis was his most important contribution to mathematics, but was not enthusiastically received by leading mathematicians of his day, notably Kronecker and Weierstrass. His ideas were, however, warmly welcomed by Jordan and especially by Cantor with whom he became firm friends.

I. INTRODUCTION

Introduction initial data of many problem of discrete optimization has the approached character. Therefore the analysis of stability of decisions is actual at fluctuations of parameters of a problem. Numerous publications (see, e.g., [1]) are devoted research of various as pests of stability of scalar and vector problems of discrete optimization. Questions of stability not only decisions of problems of discrete optimization, but also algorithms of their decision (see, e.g., [2, 3]) are actual. One of possible variants of research of stability local (gradient) algorithms is the finding of chance of the guaranteed

]]>(5.16)

Moreover, if the flow is irrotational then is automatically satisfied by writing , where is termed the velocity potential. (See Section 4.15.) Hence,

(5.17)

(5.18)

On the other hand, if the flow is incompressible then is automatically satisfied by writing , where is termed the stream function. (See Section 5.2.) Hence,

(5.19)

(5.20)

Finally, if the flow is both irrotational and incompressible then Equations (5.17)-(5.18) and (5.19)-(5.20) hold simultaneously, which implies that

(5.21)

(5.22)

It immediately follows, from the previous two expressions, that

(5.23)

or

(5.24)

Likewise, it can also be shown that

(5.25)

We conclude that, for two-dimensional, irrotational, incompressible flow, the velocity potential and the stream function both satisfy Laplace’s equation. Equations (5.21) and (5.22) also imply that

(5.26)

In other words, the contours of the velocity potential and the stream function cross at right-angles.

The function is known as the stream function. Moreover, the existence of a stream function is a direct consequence of the assumed incompressible nature of the flow.

Consider two points, and , in addition to the fixed point . (See Figure 5.2.) Let and be the fluxes from right to left across curves and . Using similar arguments to those employed previously, the flux across is equal to the flux across plus the flux across . Thus, the flux across , from right to left, is . If and both lie on the same streamline then the flux across is zero, because the local fluid velocity is directed everywhere parallel to . It follows that . Hence, we conclude that the stream function is constant along a streamline. The equati

.

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

.

Therefore

,

Where

]]>the last inequality yields

.

Therefore

,

Where

Then, by repeating the scheme of the proof of Theorem 4 [4], we obtain estimates (1). Lemma is proved.

Proof of Theorem. According to Lemma

, (2) . (3)

(2), (3) and the relations bottom follow from theorem

.

Theorem is proved.

Corollary. If , then .

REFERENCES

[1] Emelichev V., Podkapaev D. Quantitative stability analysis for vector problems of 0-1 programming // Discrete Opnimization . – 2010.- v.7.- p. 48-63. (in Russian)

[2] Devyaterikova M.V., Kolokolov A.A. Stability Analysis o Some Discrete Optimization Algorithms // Automation and Remote Control, 2004. № 3. pp. 48-54.

[3] Ramazanov A.B. On stability o the gradient algorithm in convex discrete optimization problems and related questions // J. Discrete Mathematics and Applications, 2011, vol. 21, Issue 4, pp. 465-476.

Camille Jordan’s father, Esprit-Alexandre Jordan (1800-1888), was an engineer who had been educated at the École Polytechnique. Camille’s mother, Joséphine Puvis de Chavannes, was the sister of the famous painter Pierre Puvis de Chavannes who was the foremost French mural painter of the second half of the 19th century. Camille’s father’s family were also quite well known; a grand-uncle also called Ennemond-Camille Jordan (1771-1821) achieved a high political position while a cousin Alexis Jordan (1814-1897) was a famous botanist.

Jordan studied at the Lycée de Lyon and at the Collège d’Oullins. He entered the École Polytechnique to study mathematics in 1855. This establishment provided training to be an engineer and Jordan, like many other French mathematicians of his time, qualified as an engineer and took up that profession. Cauchy in particular had been one to take this route and, like Cauchy, Jordan was able to work as an engineer and still devote considerable time to mathematical research. Jordan’s doctoral thesis was in two parts with the first part Sur le nombre des valeurs des fonctions Ⓣ being on algebra. The second part entitled Sur des periodes des fonctions inverses des intégrales des différentielles algebriques Ⓣ was on integrals of the form ∫ u dz where u is a function satisfying an algebraic equation f (u, z) = 0. Jordan was examined on 14 January 1861 by Duhamel, Serret and Puiseux. In fact the topic of the second part of Jordan’s thesis had been proposed by Puiseux and it was this second part which the examiners preferred. After the examination he continued to work as an engineer, first at Privas, then at Chalon-sur-Saône, and finally in Paris.

Jordan married Marie-Isabelle Munet, the daughter of the deputy mayor of Lyon, in 1862. They had eight children, two daughters and six sons.

From 1873 he was an examiner at the École Polytechnique where he became professor of analysis on 25 November 1876. He was also a professor at the Collège de France from 1883 although until 1885 he was at least theoretically still an engineer by profession. It is significant, however, that he found more time to undertake research when he was an engineer. Most of his original research dates from this period.

Jordan was a mathematician who worked in a wide variety of different areas essentially contributing to every mathematical topic which was studied at that time. The references [3], [4], [5], [6] are to the four volumes of his complete works and the range of topics is seen from the contents of these. Volumes 1 and 2 contain Jordan’s papers on finite groups, Volume 3 contains his papers on linear and multilinear algebra and on the theory of numbers, while Volume 4 contains papers on the topology of polyhedra, differential equations, and mechanics.

Topology (called analysis situs at that time) played a major role in some of his first publications which were a combinatorial approach to symmetries. He introduced important topological concepts in 1866 built on his knowledge of Riemann’s work in topology but not the work by Möbius for he was unaware of it. Jordan introduced the notion of homotopy of paths looking at the deformation of paths one into the other. He defined a homotopy group of a surface without explicitly using group terminology.

Jordan was particularly interested in the theory of finite groups. In fact this is not really an accurate statement, for it would be reasonable to argue that before Jordan began his research in this area there was no theory of finite groups. It was Jordan who was the first to develop a systematic approach to the topic. It was not until Liouville republished Galois’s original work in 1846 that its significance was noticed at all. Serret, Bertrand and Hermite had attended Liouville’s lectures on Galois theory and had begun to contribute to the topic but it was Jordan who was the first to formulate the direction the subject would take.

To Jordan a group was what we would call today a permutation group; the concept of an abstract group would only be studied later. To give an illustration of the way he tried to build up groups theory we will say a little about his contributions to finite soluble groups. The standard way to define such groups today would be to say that they are groups whose composition factors are abelian groups. Indeed Jordan introduced the concept of a composition series (a series of subgroups each normal in the preceding with the property that no further terms could be added to the series so that it retains that property). The composition factors of a group G are the groups obtained by computing the factor groups of adjacent groups in the composition series. Jordan proved the Jordan-Hölder theorem, namely that although groups can have different composition series, the set of composition factors is an invariant of the group.

Although the classification of finite abelian groups is straightforward, the classification of finite soluble groups is well beyond mathematicians today and for the foreseeable future. Jordan, however, clearly saw this as an aim of the subject, even if it was not one which might ever be solved. He made some remarkable contributions to how such a classification might proceed setting up a recursive method to determine all soluble groups of order n for a given n.

A second major piece of work on finite groups was the study of the general linear group over the field with p elements, p prime. He applied his work on classical groups to determine the structure of the Galois group of equations whose roots were chosen to be associated with certain geometrical configurations.

His work on group theory done between 1860 and 1870 was written up into a major text Traité des substitutions et des équations algebraic que

To motivate the Lebesgue integral, let us first briefly review two simpler integration concepts. The first is that of an infinite summation

of a sequence of numbers , which can be viewed as a discrete analogue of the Lebesgue integral. Actually, there are two overlapping, but different, notions of summation that we wish to recall hereJust as not every set can be measured by

. The first is that of the unsigned infinite sum, when the lie in the extended non-negative real axis . In this case, the infinite sum can be defined as the limit of the partial sums

or equivalently as a supremum of arbitrary finite partial sums:

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