*Ray Lambert is the Senior Managing Editor for Science, Technology, and Mathematics (STM) journals at Duke University Press. As part of our Journals Publishing Series, we sat down with Ray to talk about some of the unique challenges and workflows of publishing mathematics journals.*

**Tell us a little about your position and math publishing at Duke University Press.**

I first came to the Press in 2005 as an Assistant Managing Editor for Humanities and Social Sciences journals. I left in November of 2007 and came back in February of 2009 as the Editorial Manager for *Duke Mathematical Journal* (*DMJ*) and worked with *DMJ*’s staff. (*DMJ *has had its own dedicated staff for a while, as it is one of our largest journals in terms of number of pages and frequency—currently 3,000 pages over 15 issues, which will increase to 3,600 over 18 issues starting in 2016).

It was later in 2009 that the Press began to publish two additional math journals, *Kyoto Journal of Mathematics* (*KJM*) and *Nagoya Mathematical Journal* (*NMJ*) on behalf of their respective universities.

As part of our agreements, we began to provide editorial and production services for these journals as well, and our *DMJ* group became the STM group in Journals Editorial. We also now provide editorial services for *Notre Dame Journal of Formal Logic* (*NDJFL*) and, beginning this year, *Annals of Functional Analysis* (*AFA*) and *Banach Journal of Mathematical Analysis* (*BJMA*), which are two journals that we publish on behalf of the Tusi Mathematical Research Group of Ferdowsi University of Mashhad, Iran. The STM group includes Marisa Meredith, Editorial Assistant for *DMJ*; Keller Kaufman-Fox, Assistant Managing Editor for STM Journals; and Roy Pattishall, Assistant Managing Editor for STM Journals. We all work closely with our colleagues in Journals Production, our copyeditors and proofreaders, and our typesetter.

*DMJ*, *KJM*, and *NMJ* are general journals and publish research articles in several areas of theoretical mathematics. The Tusi Mathematical Research Group’s journals, *AFA* and *BJMA*, publish short and long articles, respectively, on subjects such as matrix analysis, abstract harmonic analysis, functional analysis, operator theory, and related topics. And *NDJFL *publishes in all areas of logic and the foundations of mathematics.

All of our math journals are hosted in Project Euclid (Project Euclid is a mathematics and statistics publishing platform that Duke University Press co-manages with Cornell University Library). I think that we’ve developed a good working relationship with Project Euclid over the last few years, and we’ve tried to take advantage of the technical features of the Project Euclid platform (features such as reference linking and other searching capabilities) to enhance the content of our journals.

I think that being part of Project Euclid has helped our group learn a lot more about math publishing and about how information is shared within this academic community. I think that as a publisher we’re helping our journals by facilitating that information sharing–for example, we’ve joined other publishers in starting to share bibliographic information with arXiv (also at Cornell University Library), to establish links for earlier draft (preprint) versions of articles to their final published versions on Project Euclid.

### What’s different about publishing a mathematics journal compared to journal editing in general?

Many of the humanities and social sciences journals that we publish are interdisciplinary, so, with copyediting, I think that a big part of our approach is to keep the general reader in mind. I wouldn’t say that this is as big of a concern for the math journals, since we know that our readers are mathematicians and logicians, and we don’t expect too many casual readers. Of course, we want to maintain a high standard of quality and accessibility, so we focus on making sure that the language is clear and that terms and notation are used consistently within articles. Our main goal is to work with our authors to make sure that the final articles are well presented. Our group has a lot of collective experience in editing math articles; and while our approach is thorough, particularly to citations and bibliographies, we do know when to tread carefully!

In terms of editorial tools, though, the biggest difference is that every article that we work with is not in Microsoft Word but in LaTeX, which is both a computer mark-up language and typesetting platform that is widely used in the mathematics community and in other scientific disciplines. It is excellent for writing mathematical notation and formulas and for formatting standard parts of math research papers, such as theorems, proofs, and bibliographies. LaTeX, though, can present challenges to publishers, particularly to editorial and production staff. Everyone in the STM group learned LaTeX on the job, and we’ve trained some of our freelancers as well. With their help, and the help of our typesetter, we’ve become more LaTeX-savvy as a group and, for instance, now we edit TeX files electronically, using the WYSIWYG TeX editing program BaKoMa TeX Word.

**What are the steps in the editorial process, and how does it differ from other editorial workflows?**

All of the editorial work (copyediting and proofreading) that we do occurs after an article has been accepted for publication. Shortly after we receive the TeX and figure files of accepted articles from the journals’ academic editors, we begin working with our typesetter. They facilitate the editing and typesetting process for us by formatting the files according to the journal’s LaTeX style guide (known as a class file), which governs how things like the typefaces, margins, headings, and so on will look; that preparation process allows us to copyedit articles with BaKoMa, the editing tool that I mentioned earlier.

We then start the copyediting process, some of which we do in-house and for some we hire freelancers. Our editors focus on the language and the reference list, as well as on general formatting, and our typesetter applies our style rules for the presentation of all elements in the math formulas. The copyedited manuscript is then typeset and sent to the authors for review. We also provide a marked-up version of the copyedited manuscript, so authors can see what changes were made during the copyediting and typesetting process. I think that authors appreciate having this reference file, as it no doubt makes their chore of reviewing the proofs go much quicker. This also helps us identify any of those (very rare, of course!) instances where we might have introduced some errors.

**What new projects are you working on?**

One new project that we just implemented is an “advance publication” model for *DMJ *and *NDJFL*, in which articles will be published online in Project Euclid before they appear in an assigned print issue. Working with Journals Production, we are also moving *NDJFL *to this workflow and are implementing a “growing issue” model of advance publication for *AFA *and *BJMA*. These models will help keep the time from acceptance to online publication to about three months. These articles will have been copyedited, typeset, reviewed by the authors, corrected, assigned a DOI, and then posted online in PDF.

*All Duke University Press mathematics journals are hosted on **Project Euclid*. *Read more of our Journals Publishing Series here. Stay tuned for our next post featuring an interview with Project Euclid co-director Mira Waller.*

indeed this is very important as say prof dr mircea orasanu and prof horia orasanu

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also here there is an occasion to publish as say prof dr mircea orasanu and prof horia orasanu an article as concerning some foundations and a CURRICULUM VITAE following a themes

LAGRANGIAN AND CONSTRAINTS FOR SIMPLE EXAMPLES

ABSTRACT

In order to introduce the concept of generalized coordinates, let us first consider a few simple examples.

Consider, first, the problem of a simple pendulum moving in the x-y plane. The pendulum has a length l and moves under the action of gravity, so that its potential energy is mgh. The system is illustrated in the figure below.

We could use Cartesian coordinaets x and y to describe the location of the pendulum bob, but x and y are not independent. In fact, since the length of the pendulum is constant, they are related by

displaymath443

This condition would need to be imposed as a constraint on the system, which can be inconvenient. It is more natural to use the angle tex2html_wrap_inline540 that the pendulum makes with respect to the vertical to describe the motion. But what would be the equation of motion for tex2html_wrap_inline540 ? In order to find out what this is, we only need to express the Lagrangian in terms of tex2html_wrap_inline540 . Now, the Lagrangian in terms of x and y is givn by

displaymath445

where we have introduced a general potential function, however, for this example, we know that the potential is given by U(x,y) = -mgy.

The Cartesian coordinates x and y are related to tex2html_wrap_inline540 by a set of transformation equations:

eqnarray142

In order to transform the kinetic energy, we need the time derivatives of the transformation equations:

eqnarray144

Substituting the transformations and their derivatives into the Lagrangian gives

eqnarray150

Now, given the Lagrangian, we just turn the crank on the Euler-Lagrange equation and derive the equation of motion for tex2html_wrap_inline540 :

eqnarray160

so that the equation of motion is

eqnarray168

As another example, consider again a particle moving in the x-y plane subject to a potential U that is a function only of the distance of the particle from the origin of the coordinate system. This distance is tex2html_wrap_inline566 , so that U = U(r). An example would be a radial harmonic potential tex2html_wrap_inline570 , which is a “bowl” potential shown below:

Of course, we can choose to work in Cartesian coordinates, x, and y. In this case, we would write down the Lagrangian

displaymath447

where the potential U(r) is expressed as a function of x and y through the dependence of r on x and y. Then, the equations of motion for x and y can be computed with forces obtained via the chain rule

eqnarray182

which is perfectly correct. However, it obviscates some of the important physics of the problem, which can be revealed by working in polar coordinates r and tex2html_wrap_inline540 , which are more natural for this problem. These are related to x and y via the transformations:

eqnarray194

with time derivatives

eqnarray196

Substituting into the kinetic energy gives, after some algebra

eqnarray204

so that the Lagrangian can be expressed as

displaymath449

! INTRODUCTION The equations of motion of a system can be cast in the generic form

displaymath177

where, for a Hamiltonian system, the vector function tex2html_wrap_inline563 would be

displaymath180

and the incompressibility condition would be a condition on tex2html_wrap_inline563 :

displaymath186

A non-Hamiltonian system, described by a general vector funciton tex2html_wrap_inline563 , will not, in general, satisfy the incompressibility condition. That is:

displaymath191

Non-Hamiltonian dynamical systems are often used to describe open systems, i.e., systems in contact with heat reservoirs or mechanical pistons or particle reservoirs. They are also often used to describe driven systems or systems in contact with external fields.

The fact that the compressibility does not vanish has interesting consequences for the structure of the phase space. The Jacobian, which satisfies

displaymath165

will no longer be 1 for all time. Defining tex2html_wrap_inline569 , the general solution for the Jacobian can be written as

displaymath199

Note that tex2html_wrap_inline571 as before. Also, note that tex2html_wrap_inline573 . Thus, tex2html_wrap_inline575 can be expressed as the total time derivative of some function, which we will denote W, i.e., tex2html_wrap_inline579 . Then, the Jacobian becomes

eqnarray202

Thus, the volume element in phase space now transforms according to

displaymath205

which can be arranged to read as a conservation law:

displaymath207

Thus, we have a conservation law for a modified volume element, involving a “metric factor” tex2html_wrap_inline581 . Introducing the suggestive notation tex2html_wrap_inline583 , the conservation law reads tex2html_wrap_inline585 . This is a generalized version of Liouville’s theorem. Furthermore, a generalized Liouville equation for non-Hamiltonian systems can be derived which incorporates this metric factor. The derivation is beyond the scope of this course, however, the result is

displaymath211

We have called this equation, the generalized Liouville equation Finally, noting that tex2html_wrap_inline587 satisfies the same equation as J, i.e.,

displaymath216

the presence of tex2html_wrap_inline587 in the generalized Liouville equation can be eliminated, resulting in

displaymath219

which is the ordinary Liouville equation from before. Thus, we have derived a modified version of Liouville’s theorem and have shown that it leads to a conservation law for f equivalent to the Hamiltonian case. This, then, supports the generality of the Liouville equation for both Hamiltonian and non-Hamiltonian based ensembles, an important fact considering that this equation is the foundation of statistical mechanics.

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

LAGRANGIAN AND CONSTRAINTS OF SIMPLE HARMONIC EQUATION

ABSTRACT Here’s a simple demonstration of how to solve an energy functional optimization symbolically using Maple.

Suppose we’d like to minimize the 1D Dirichlet energy over the unit line segment:

min 1/2 * f'(t)^2

f

subject to: f(0) = 0, f(1) = 1

we know that the solution is given by solving the differential equation:

f”(t) = 0, f(0) = 0, f(1) = 1

and we know that solution to be

We should emphasize that has dual meaning. It is both a coordinate and the derivative of the position. This traditional abuse of notation should be resolved in favor of one of these interpretations in every particular situation.

1 . INTRODUCTION

It is an importan realization that solutions of the Lagrange equation 5 solve an extreme path problem between two points in the configuration space. The problem can be stated as that of finding the path q(t) , t0 t t1 , such that the integral

S = L(q(t), (t),t) dt (14)

is minimal. The classical variational calculus studies the variation of this integral under perturbations of the path q(t) . We substitute the initial path q(t) with the new path

q (t) = q(t) + q(t) (15)

where q(t) is an arbitrary vector-valued function on the segment [t0,t1] . We define the variation of integral 14 under the perturbation q(t) to be (see Figure 2):

S = = 0 L(q (t), (t),t) dt. (16)

Figure 2: The variation of a path

The classical calculation yields

= 0 L(q (t), (t),t) dt = = 0L(q (t), (t),t) dt

= dt. (17)

Using the fact that

(t) = ( q(t)) (18)

and integration of the second term by parts yield

S = q(t) dt

+ (19)

This equation implies that if the ends of the perturbation path are clamped at the ends, i.e. q(t0) = q(t1) = 0 then the second summand drops out. Moreover, If S = 0 for all perturbations then the Lagrange equations 5 must be satisfied.

The above extreme property of the solutions of the Lagrange equations 5 shows the invariance of these equations under coordinate changes: if we use a time-dependent substitution q = F(Q,t) where F : x is a change of variables then the new Lagrangian with respect to coordinates Q is

K(Q, ,t) = L(F(Q),DF(Q,t) + (Q,t),t) (20)

where DF(Q,t) is the derivative (Jacobi matrix) of F at (Q,t) with respect to Q and = . This formula allowes us to choose coordinates in a convenient manner, for instance, to express the motion of a body in a rotating coordinate system, as it will be done in section 5.

In Hamiltonian mechanics we use generalized momentum in place of velocity as a coordinate. The generalized momentum is defined in terms of the Lagrangian and the coordinates (q, ) :

p = . (21)

2 FORMULATION

For example, for the pendulum we have:

= ml 2 , (6)

= – mglsin . (7)

Thus, the equations of motion are written as

(ml 2 ) = – mglsin . (8)

This equation can be written as second order equation

ml 2 = – mglsin (9)

or in the traditional way

= – sin . (10)

2.5 The meaning of dot

We should emphasize that has dual meaning. It is both a coordinate and the derivative of the position. This traditional abuse of notation should be resolved in favor of one of these interpretations in every particular situation.

2.6 Lagrangian vs. Newtonian mechanics

In Newtonian mechanics we represent the equations of motion in the form of the second Newton’s law:

m = f (q,t) (11)

where f (q,t) is the force applied to the particle.

This equation is identical to the equation obtained from Lagrangian representation if f (q,t) is a conservative field, i.e. it has a potential. A potential is a function U(q,t) such that

f (q,t) = – . (12)

Indeed, the Lagrangian can be written as

L = m( )2 – U(q,t). (13)

According to 5 the equations of motion reduce to 11.

________________________________________

In Hamiltonian mechanics we use generalized momentum in place of velocity as a coordinate. The generalized momentum is defined in terms of the Lagrangian and the coordinates (q, ) :

p = . (21)

In Lagrangian mechanics we start by writing down the Lagrangian of the system

L = T – U (1)

where T is the kinetic energy and U is the potential energy. Both are expressed in terms of coordinates (q, ) where q is the position vector and is the velocity vector.

2.2 The Lagrangian of the pendulum

An example is the physical pendulum (see Figure 1).

Figure 1: The configuration space of the pendulum

The natural configuration space of the pendulum is the circle. The natural coordinate on the configuration space is the angle . If the mass of the ball is m and the length of the rod is l then we have

T = m(l )2 (2)

U = – mglcos (3)

3. SOLUTION. LAGRANGIAN IN MECHANICS

We should emphasize that has dual meaning. It is both a coordinate and the derivative of the position. This traditional abuse of notation should be resolved in favor of one of these interpretations in every particular situation.

2.6 Lagrangian vs. Newtonian mechanics

In Newtonian mechanics we represent the equations of motion in the form of the second Newton’s law:

m = f (q,t) (11)

where f (q,t) is the force applied to the particle.

This equation is identical to the equation obtained from Lagrangian representation if f (q,t) is a conservative field, i.e. it has a potential. A potential is a function U(q,t) such that

f (q,t) = – . (12)

Indeed, the Lagrangian can be written as

L = m( )2 – U(q,t). (13)

According to 5 the equations of motion reduce to 11.

________________________________________

t is an importan realization that solutions of the Lagrange equation 5 solve an extreme path problem between two points in the configuration space. The problem can be stated as that of finding the path q(t) , t0 t t1 , such that the integral

S = L(q(t), (t),t) dt (14)

is minimal. The classical variational calculus studies the variation of this integral under perturbations of the path q(t) . We substitute the initial path q(t) with the new path

q (t) = q(t) + q(t) (15)

where q(t) is an arbitrary vector-valued function on the segment [t0,t1] . We define the variation of integral 14 under the perturbation q(t) to be (see Figure 2):

S = = 0 L(q (t), (t),t) dt. (16)

Figure 2: The variation of a path

The classical calculation yields

= 0 L(q (t), (t),t) dt = = 0L(q (t), (t),t) dt

= dt. (17)

Using the fact that

(t) = ( q(t)) (18)

and integration of the second term by parts yield

S = q(t) dt

+ (19)

This equation implies that if the ends of the perturbation path are clamped at the ends, i.e. q(t0) = q(t1) = 0 then the second summand drops out. Moreover, If S = 0 for all perturbations then the Lagrange equations 5 must be satisfied.

The above extreme property of the solutions of the Lagrange equations 5 shows the invariance of these equations under coordinate changes: if we use a time-dependent substitution q = F(Q,t) where F : x is a change of variables then the new Lagrangian with respect to coordinates Q is

K(Q, ,t) = L(F(Q),DF(Q,t) + (Q,t),t) (20)

where DF(Q,t) is the derivative (Jacobi matrix) of F at (Q,t) with respect to Q and = . This formula allowes us to choose coordinates in a convenient manner, for instance, to express the motion of a body in a rotating coordinate system, as it will be done in section 5.

________________________________________

In Hamiltonian mechanics we use generalized momentum in place of velocity as a coordinate. The generalized momentum is defined in terms of the Lagrangian and the coordinates (q, ) :

p = . (21)

But they must also obey

As we noted briefly in section 3.8, distance in polar coordinates upon making small changes in the variables r and is described by

From this we deduce that

Putting the two equations for ds together, we deduce:

is the partial derivative of f with respect to r, just as is its partial derivative with respect to x.

But because has a factor of r in it, there must be a compensating factor of r in the denominator of the component of f in the direction

and

A similar computation can be made for any orthogonal directions in any dimension, and we can anticipate the result.

The component of f in the direction of any such variable will be the partial derivative of f with respect to that variable, divided by the ratio of distance change in that direction to change in the variable itself.

Using the last equation we can immediately deduce that the gradient of is , except of course at r = 0, where is not differentiable. Similarly we find that the gradient of .

Exercises:

9.5 Use the fact that both angular variables in spherical coordinates are polar variables to express ds2 in 3 dimensions in terms of differentials of the three variables of spherical coordinates. From this deduce the formula for gradient in spherical coordinates.

9.6 Find the gradient of in spherical coordinates by this method and the gradient of in spherical coordinates also.

There is a third way to find the gradient in terms of given coordinates, and that is by using the chain rule.

We can first consider differential change of f in rectangular coordinates, and then relate the differential changes in x and y to differential changes in the other coordinates, say r and . Combining these we can relate the change in f to changes in the latter two variables.

Since we know how to write the gradient in rectangular coordinates and can recognize unit vectors, we can express the resulting expression in terms of components of the gradient in the other coordinate system.

Explicitly we can write

and use the latter two equations to get rid of dx and dy in the first equation. The result is an expression for df in terms of dr and d , the coefficients of which can be described in terms of unit vectors in the various directions, and the gradient in rectangular coordinates.

Comparing that equation with the basic formula defining partial derivatives, Equation (A) above you can read off the components of the gradient.

This approach is useful when f is given in rectangular coordinates but you want to write the gradient in your coordinate system, or if you are unsure of the relation between ds2 and distance in that coordinate system.

Exercises:

9.7 Do this computation out explicitly in polar coordinates.

9.8 Do it as well in spherical coordinates.

What variables should we keep constant in taking partial derivatives?

It is worth noting that when we take the partial derivative with respect to x or y we always mean that we are keeping the other variable, y or x, constant; on the other hand the partials with respect to r and always mean keeping the other one of these, or r, constant. Any other meaning has to be described explicitly.

There are times and places where in a partial derivative one can become confused as to which variable or variables are being kept constant, and under such circumstances it is wise to modify the notation to supply this information explicitly. Thus we can write to mean the partial derivative with respect to x keeping y fixed, and then there can be no confusion as to what is kept constant.

The most important facts to remember about the gradient are:

It is straightforward to compute, in any orthogonal coordinate system

You can use it to determine the directional derivative of the function involved, in any direction.

In rectangular coordinates its components are the respective partial derivatives.

The gradient of the sum of two fields is the sum of their gradients (the gradient is a linear operator).

The gradient of a product can be computed by applying the usual product rule for differentiation.

________________________________________

In calculus of a single variable the definite integral

for f(x)>=0 is the area under the curve f(x) from x=a to x=b. For general f(x) the definite integral is equal to the area above the x-axis minus the area below the x-axis.

The definite integral can be extended to functions of more than one variable. Consider a function of 2 variables z=f(x,y). The definite integral is denoted by

where R is the region of integration in the xy-plane. For positive f(x,y), the definite integral is equal to the volume under the surface z=f(x,y) and above xy-plane for x and y in the region R. This is shown in the figure below.

For general f(x,y), the definite integral is equal to the volume above the xy-plane minus the volume below the xy-plane. This page includes the following sections:

This time we are interested in solving the inhomogeneous wave equation (IWE)

(11.52)

(for example) directly, without doing the Fourier transform(s) we did to convert it into an IHE.

Proceeding as before, we seek a Green’s function that satisfies:

(11.53)

REFERENCES

[1] B. Riemann, “Collected works” , Dover, reprint (1953)

[2] D. Hilbert, “Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen” , Chelsea, reprint (1953)

[3] J. Plemelj, “Riemannsche Funktionenscharen mit gegebenen Monodromiegruppe” Monatsh. Math. Phys. , 19 (1908) pp. 211–245

[4] I.I. Privalov, “On a boundary problem in analytic function theory” Mat. Sb. , 41 : 4 (1934) pp. 519–526 (In Russian) (French abstract)

[5] N.I. Muskhelishvili, “Singular integral equations” , Wolters-Noordhoff (1972) pp. Chapt. 2 (Translated from Russian)

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

LAGRANGIAN AND CONSTRAINTS WITH OPTIMIZATIONS

LAGRANGE NON HOLONOM

ABSTRACT

Instance with annotations:

This example uses xx-patent-document as the model for the creation of an example wo-patent-document. In particular the example is a PCT published application (an A1 document) – this PCT example uses some published parts of patent applications and parts created for example purposes only.

1INTRODUCTION

The example markup does not necessarily reflect PCT practice.

Note: Comments are printed in square brackets, greyed, in [italic bold].

First we start with the bibliographic data required to create the front page (title page) of a PCT patent document:

We have seen that the difference between complex and super-complex systems in based on at least four main issues: levels of reality, chronotopoids, (generalized) interactions, and anticipation. So far, none of them has been adequately formalized. However, considering that chronotopoids and interactions require (and therefore depend on) the theory of levels, and that the issue of anticipation has been advanced by Rosen 1985, subsequent sections will make reference to the issue of levels only. Furthermore, it is apparent that any step towards a proper formalization of the theory of levels (or any other of the mentioned theories) seems to require the development of a non-Abelian framework. Whichever further mathematical property will be required, the first mandatory move is therefore to pass from an Abelian or commutative or natural framework to a non-Abelian one.

6. Organisms Represented as Variable Categories:

Many-Valued LM-logic Algebraic Categories of Functional Biosystems

One of the major road blocks to any successful dynamical theory of complex systems, and also of developing organisms, has been the lack of a flexible enough mathematical structure which could represent the immensely variable and heterogeneous classes of biological and social organisms. In the following subsection we propose to re-examine the representation of organisms in terms of such flexible mathematical structures that can vary in time and/or space, thus providing a natural framework for relational/ theoretical biology, psychology, sociology or global theoretical constructs addressing environmental problems.

We have already mentioned that the problem of time, i.e. the problem of the dynamical nature of reality, is the main problem underling the philosophical theory of categories as reviewed by Poli (2007). This same problem has also been at the center of the mathematical theory of categories over the last six decades, and found a first outcome in the idea of variable category (be it in the form of variable sets, variable classes, etc), albeit in a formal and precise setting. Furthermore, dealing with varying, or variable, objects such as those formalized by the concepts variable sets, variable classes, etc. leads to a further generalization of this categorical approach which is founded in Logic, be it Boolean (as in the Category of Sets), Heyting-intuitionistic (as in “standard” Topos theory; Moerdijk and MacLane, 1994, 2004), or Many-Valued (MV or Łukasiewicz-Moisil (see Georgescu, 2006 for a review of N-valued logics), as in the new theory of Generalized Topoi (Baianu, Brown, Glazebrook and Georgescu, 2005, 2006, 2011). It is worth mentioning that some of the methodological pitfalls of the categorical approach in specific, logical or mathematical, contexts, as for in example in certain areas of Algebraic Topology or Algebraic Logic, were recognized early by topologists, who also branded this approach as “abstract nonsense”, even though it continues to facilitate and be widely employed in the proof of general theorems. Such an objection lies in the fact that the “universal” may, and does, have specific, subtle exceptions and counter-examples as one might, of course, expect it. Things that may appear to be globally “the same”, or “categorically equivalent”, may still differ quite significantly in their specific, local contexts. Such problems with

defining different kinds of equivalence arise not only in the modern theories of Algebraic Topology and Groupoids, Abelian categories, Algebraic Geometry, and so on, but also when attempting to define in precise terms the similarity or analogies between systems that appear physically distinguishable but mathematically equivalent in some selected, specific sense. We are considering in the next section this problem in further detail.

7. Analogous Systems and Dynamic Equivalence

A scientific and/or engineering strategy for dealing with complex systems has long been the analysis of simpler, more readily accessible ‘models’ of a complex system. One often attempts to arrive at computable models with similar dynamic behavior(s) to that of the original, complex system. Computability of such simple models may often involve the use of a super-fast digital computer, and the models can be made indefinitely more and more complicated through iterated attempts at improved computer simulation.

A formal, categorical approach to analogous systems and dynamic equivalence of systems was first reported by Rosen (1968) from a classical standpoint that is, excluding quantum dynamics; subsequently his approach was extended to the development of biological systems and embryology (Baianu and Scripcariu, 1973: BMB, “On Adjoint Dynamical Systems”).

Returning now to the issue of computer simulation, one finds upon careful consideration that there is no recursively computable (either simple or complicated) model of both super-complex biological systems and simpler ‘chaotic’ systems (see for example, Baianu, 1987 and the relevant references cited therein). Therefore, Complex Systems Biology (CSB) cannot be reduced to any finite number of simple(r) mechanistic models that are recursively computable, or accessible to digital computation or numerical simulation. This basic result does not seem, however, to deter the computationally-oriented scientists from publishing a rapidly increasing number of reports on computer simulation of complex biological systems. There is surprising enthusiasm and optimism, not to mention popularity, funding, etc., for computer simulations in both biology and medicine.

Such ‘mechanistic’ approaches to understanding how parts or subsystems of a complex organism work are necessary but insufficient steps towards developing a CSB theory that takes into account what the over-simplified ‘mechanistic’ models have left out—those irreducible interactions that pertain to the essence of an organism’s existence and its integrated physiological functions. Heuristic results are both attractive and stimulating, culminating with the aroused expectation of ‘final answers’ to either biological or medical problems by means of digital super-computers. It seems, however, that even the fastest and best-programmed super-computers are no match for the super-complexity of organisms, or even for the simpler, ‘chaotic’ dynamics, a result which is also widely recognized by many chaotic dynamic theorists.

Fundamentally, the limitations of digital computers that rely upon finite/recursive computations are traced back to the Boolean (or Chrysippian) logic underlying the design of all existing digital computers, and also to the Axiom of Choice upon which set theory is based (Moerdijk and MacLane, 2004). On the other hand, biological, super-complex system dynamics is governed by a many-valued (MV) logic characteristic of biological processes including genetic (Baianu, 1977) and neural ones Baianu et al, 2006, 2011). Such an MV-logic is both non-commutative (unlike the Boolean or the Heyting- intuitionistic logic of standard toposes) and irreducible to Boolean and/or intuitionistic logic (of course, with the exception of the special cases of the category of centred Łukasiewicz-Moisil logic algebras that can be mapped isomorphically onto Boolean Logic algebras (Georgescu, 1974, 2006). Unlike the well known result of von Neumann’s for the Universal Automaton, super-complex biological ‘systems’ are not recursively, or numerically computable. Although this limits severely the usefulness of all digital computers in Complex Systems Biology and Mathematical Medicine, it does not render them completely useless for experimentation, data collection and analysis or graphics and graphical presentation/representation of numerical results. Both the limitations and the advantages of using computers become evident in the final analysis where computer simulations of super-complex ‘system’ dynamics cannot claim a full, or complete, dynamical modelling of organisms as such a result has been formally proven to be unobtainable, in general, through recursive computation with algorithms, universal Turing machines, etc. (Baianu, 1986; Rosen, 1987; Penrose, 1994).

Algebraic computation is still possible, of course, for living systems and their essential subsystems,

such as genetic networks, by employing instead non-commutative, irreducible MV-logics, either in a general context (Georgescu,1974,2006) or in more specific contexts, such as the controlled dynamics of genetic networks in biological organisms (Baianu,1977, 2004a,b, 2005a,b; Baianu et al, 2006). Non-commutative super-complex dynamic modelling has just begun in Biology and Medicine, including diagnostics. A Biostatistics formulation based on LM-logic algebras, but independent of current probability theory, has also become a strong possibility (Georgescu, 2006).

Such recent developments also suggest a paradigm shift occurring now in system theories – from Abelian to non-Abelian/ non-commutative theories. This new paradigm has perhaps already began with the earlier introduction of noncommutative geometric spaces obtained through deformation as models of quantum spaces in attempts by A. Connes et al. (1992, 2004, 2006, and references cited therein) at formalizing a Noncommutative Geometry theory of Quantum Gravity 8. Non-Abelian Systems Theory.

One can formalize the hierarchy of multiple-level relations and structures that are present in super-complex ‘systems’ and meta-systems in terms of the mathematical Theory of Categories, Functors and Natural Transformations (TC-FNT). On the first level of such a hierarchy are the links between the system components represented as morphisms of a structured category which are subject to the axioms/restrictions of Category Theory. Then, on the next, second level of the hierarchy one considers functors or links between such first level categories that compare categories without ‘looking inside’ their objects/system components. On the third level, one compares, or links, functors using natural transformations in a 2-category (meta-category) of categories. At this level, natural transformations not only compare functors but also look inside the first level objects (system components) thus ‘closing’ the structure and establishing ‘the universal links’ between items as an integration of both first and second level links between items. The advantages of this constructive approach in the mathematical theory of categories, functors and natural transformations have been recognized since the beginnings of this mathematical theory in the seminal paper of Mac Lane and Eilenberg (1945). A relevant example from the natural sciences, e.g., neurosciences, would be the higher-dimensional algebra of processes of cognitive processes of still more, linked sub processes (Brown, 2004). Yet another example would be that of groups of groups of item subgroups, 2-groupoids, or double groupoids of groups of items. The hierarchy constructed above, up to level 3, can be further extended to higher, n-levels, always in a consistent, natural manner.

This type of global, natural hierarchy of items inspired by the mathematical TC-FNT has a kind of internal symmetry because at all levels, the link compositions are natural, that is the all link compositions that exist are transitive, i.e., x < y and y x y and g: y –> z ==> h: x –> z, and also h = g o f. The general property of such link composition chains or diagrams involving any number of sequential links is called commutativity, or the naturality condition. This key mathematical property also includes the mirror-like symmetry x * y = y * x; when x and y are operators and the ‘* ‘ represents the operator multiplication. Then, the equality of x * y with y * x implies that the x and y operators ‘commute’; in the case of an eigenvalue problem involving such commuting operators, the two operators would share the same system of eigenvalues, thus leading to ‘equivalent’ numerical results. This is very convenient for both mathematical and physical applications (such as those encountered in quantum mechanics). Unfortunately, not all operators ‘commute’, and not all mathematical structures are commutative. The more general case is the non-commutative one. An example of a non-commutative structure relevant to Quantum Theory is that of the Clifford algebra of quantum observable operators (Dirac, 1962; see also the Appendix); yet another, more recent and popular, example is that of C*-algebras of (quantum) Hilbert spaces. Last but-not least, are the interesting mathematical constructions of non-commutative ‘geometric spaces’ obtained by ‘deformation’ introduced by Allan Connes (1990) as possible models for the physical, quantum space-time. Thus, the microscopic, or quantum, ‘first’ level of physical reality does not appear to be subject to the categorical naturality conditions of Abelian TC-FNT—the ‘standard’ mathematical theory of categories (functors and natural transformations). It would seem therefore that the commutative hierarchy discussed above is not sufficient for the purpose of a General, Categorical Ontology which considers all items, at all levels of reality, including those on the ‘first’, quantum level, which is not commutative. On the other hand, the mathematical, Non-Abelian Algebraic Topology (Brown and Safiro, 2005, 2006), the Non-Abelian Quantum Algebraic Topology (NA-QAT; Baianu et al., 2005-2006), and the physical, non-Abelian gauge theories may provide the ingredients for a proper foundation for non-Abelian, hierarchical multi-level theories of super-complex system dynamics. Furthermore, it was recently pointed out (Baianu et al, 2005, 2006) that the current and future development of both NA-QAT and of Complex Systems Biology theories involve a fortiori non-commutative, many-valued logics of quantum events, such as the Lukasiewicz-Moisil (LM) logic algebra (Georgescu, 2006a), complete with a fully-developed, novel probability/measure theory grounded in the LM-logic algebra (Georgescu, 2006a,b). The latter paves the way to a new projection operator theory founded upon the (non-commutative) quantum logic of events, or dynamic processes, thus opening the possibility of a complete, Non-Abelian Quantum Theory. Furthermore, such recent theoretical developments point towards a paradigm shift in systems theory and to its extension to more general, non-Abelian theories that lie well beyond the bounds of commutative structures/spaces. Non-Abelian theories are also free from the restrictions or basic limitations imposed by the Axiom of Choice and the elementhood, or parthood/subordination relation in Set Theory or Russell’s theory of classes.

References

Albertazzi, L. 2005. Immanent Realism. Introduction to Franz Brentano, Springer, Dordrecht.

Andersen, P. B., Emmeche, C., Finnemann, N.O. and Christiansen, P.V., eds.: 2000. Downward Causation. Minds, Bodies and Matter, Aarhus University Press, Aarhus.

Bertalanffy 1972. The history and status of general systems theory, in G.J. Klir (ed.), Trends in general systems theory, NY, Wiley, 1972, pp. 21-41.

Baianu 1977. Baianu, I.C. A Logical Model of Genetic Activities in Lukasiewicz Algebras: The Non-linear Theory. Bulletin of Mathematical Biophysics, 1977, 39, pp. 249-258.

Atlan, H.: 1972. L’Organisation biologique et la theorie de l’information, Hermann, Paris.

Austin, J. H.: 1998. Zen and the Brain, MIT Press.

Baars, B.: 1988. A Cognitive Theory of Consciousness, Cambridge University Press.

Baars B. and J. Newman: 1994, A neurobiological interpretation of

the Global Workspace theory of consciousness, in “Consciousness in Philosophy and Cognitive Neuroscience”, Erlbaum, Hildale NJ.

Baars B. and S. Franklin: 2003, How conscious experience and working memory interact.Trends in Cognitive Science .7, 166–172.

Baianu, I.C. and M. Marinescu: 1968, Organismic Supercategories: Towards a Unified Theory of Systems. Bulletin of Mathematical Biophysics .30: 148.

Baianu, I.C.: 1970, Organismic Supercategories

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here we mention some as say prof dr mircea orasanu and prof horia orasanu concerning the facts as followed

LAGRANGIAN CONSTRAINTS AND EULERIAN

Bizarre though the above may sound (and there remains of course the usual difficulty of where outside the possible states of the Universe the values of reside!), this is our current most powerful picture of how microscopic physics works. The Big Bang, Quantum Gravity, String Theory …. have all been looked at this (but not only this) way.

Classical Physics follows as just constructive interference. If a lot of nearby trajectories all give the same phase , then through constructive interference this will tend to dominate how amplitude propagates. This happens most where the phase is stationary with respect to variations in the path, and those are the trajectories we are used to seeing:

as we shall confirm later.

Lecture 2.

Examples of Lagrangian:

Newtonian Mechanics

Special Relativity

Electromagnetic Field

Simple check of stationary phase Newtonian Mechanics (skipped 99)

Particle in gravitational field. x =height

, ; ,

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

/ \

gets end points right adds curvature

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