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 Terms of the Second Competition for the Best Research Paper on the Topic "Associative-Commutative Hypercomplex Numbers and Their Applications"
25.10.2004
Summing up the results of the past competition, we are pleased to note that the proposed topic found an interested audience. I would like to express my sincere gratitude to everyone who participated in the competition or showed interest in its content. Although the number of participants was relatively small, it became clear that, on the one hand, the problem is relevant and requires further development, and on the other hand, it requires greater specificity.
The works submitted to the competition touched on a wide variety of aspects of hypercomplex number theory. It should be noted, however, that none of them solved the main goal: to clearly and convincingly demonstrate the right of associative-commutative hypercomplex numbers to occupy a significant place in mathematics on par with real and complex numbers. In this regard, the competition committee decided not to award the first prize in 2002 and to add the remaining undistributed portion of the prize fund to the fund allocated for the next competition on a similar topic.
Second prize was awarded to research that, in the opinion of the competition committee members, best met the competition conditions announced a year ago. However, this does not mean that the papers that were not awarded did not contain interesting ideas, and the authors are welcome to implement them in the Second Competition.
The start of the Second Research Competition on Associative-Commutative Hypercomplex Numbers and Their Applications is hereby announced, as well as the establishment of a prize fund of 500,000 (five hundred thousand) rubles.
Competition Terms:
1. The founder of the Competition is Dmitry Gennadievich Pavlov, Chairman of the Board of Directors of the Antares Group of Companies.
2. The competition accepts works on one of the three topics listed below. The number of topics may be increased during the competition by the founder's decision.
3. To participate in the competition, the work must be written in Russian and sent no later than November 30, 2003, by email to contact@antares.com.ru or by regular mail in hard copy to the following address: 105082, Moscow, Bolshaya Pochtovaya St., Building 2, MOZET OJSC, Dmitry Gennadievich Pavlov.
4. In the event of multiple submissions from different authors covering the same topic in approximately equal terms, priority will be given to the first submitted submission.
5. Submitted submissions will be considered accepted if their authors receive an invitation to present a paper at one of the seminars held as part of the competition. Authors of accepted submissions will not necessarily be recognized as winners.
6. Information media will not be returned to authors.
7. After the submission deadline, but before the end of 2003, a seminar will be held to summarize the competition results, at which decisions will be made on the winners.
8. The founder and members of the competition committee reserve the right to make changes regarding organizational and financial matters.
9. Prizes will be awarded to the authors of the submissions recognized as the best immediately following the final seminar.
Topic 1
Using the numbers H4 as an example, propose rules for constructing functions that generalize the concept of an analytic function of a hypercomplex variable.
Explanations.
Analytic functions of polynumbers (i.e., commutative-associative hypercomplex numbers), as in the case of complex numbers, are characterized by the independence of the derivative with respect to direction. Such functions satisfy a number of partial differential equations, sometimes called generalized Cauchy-Riemann conditions. Thus, for the numbers H4 = a'1 + i a'2 + j a'3 + k a'4, these conditions have the form:
where a'i are the components of H4 in the unit basis 1,i,j,k, and U,V,W,Q are mutually conjugate scalar functions that, if the conditions are satisfied, form an analytic function:
F(H4) = U + i V + j W + k Q.
Analytic functions of a complex variable have an interesting geometric interpretation, which is that the associated transformations do not change the angles between arbitrary lines or, in other words, the corresponding mappings preserve the similarity of infinitesimal figures. Such mappings are called conformal. In polynumeric spaces, each analytic function can also be associated with a conformal transformation that preserves the angles between lines.
However, although every analytic function corresponds to a specific conformal mapping, it seems likely that not every conformal mapping corresponds to an analytic function. Such mappings, if they exist in H4 and other polynumeric spaces, are of independent interest. Furthermore, in multilinear spaces with metric form dimension three or higher, in addition to the lengths of vectors and the angles between them, qualitatively different parameters appear, responsible for the congruence and similarity of figures formed by three or more vectors. The task is to correctly and completely enumerate such parameters using the H4 space as an example, find the type of mappings that preserve them, and analyze the properties associated with these mappings.
It is possible that other ideas may emerge during the research process, allowing for a completely different solution to the problem of generalizing the concept of analytic function.
Topic 2
Using H4 numbers and the associated polyspace as an example, propose an algorithm for constructing three-dimensional fractal objects and implement this algorithm using computer graphics methods, using the fourth coordinate as an evolution parameter.
Explanations.
Using complex numbers and the Euclidean plane, one can construct Mandelbrot and Julia sets, which are typically fractal objects, in a simple and effective way. The beauty and harmonious structure of these fractals is mesmerizing, so it's not surprising that mathematicians have always been interested in finding ways to construct similar three-dimensional objects. Most often, such attempts have been made using quaternions – four-dimensional hypercomplex numbers with non-commutative multiplication. To display four-dimensional fractals associated with quaternions on a computer screen, they are dissected by three-dimensional hyperplanes, and the surfaces of the resulting figures are projected onto a flat surface. By changing the orientation of the projection plane, the image is rendered three-dimensional, and by moving from one hyperplane to another, the effect of fractal evolution over time is achieved.
However, even a cursory examination of the images resulting from this approach reveals their significantly less harmonious appearance compared to their flat counterparts. Of course, the concepts of beauty and harmony are rather vague criteria for science, but any theory has always benefited from aesthetic perfection. Apparently, the main reason for the unsightly nature of three-dimensional fractals is that the quaternions used to obtain them are naturally associated with four-dimensional Euclidean space, whereas real spacetime (whose regularities give rise to our notions of beauty) is much closer to the structure of Minkowski space. Therefore, perhaps more beautiful structures could be obtained by examining fractals directly in Minkowski space. In fact, this grant was conceived to stimulate research in this direction. However, since Minkowski space lacks a four-dimensional numerical analogue, it is proposed to focus on one of its alternatives, namely, the space associated with H4 numbers.
A possible effective approach in the corresponding constructions may be the fact that if an arbitrary non-isotropic direction in H4 space is identified with the proper time of some inertial reference frame, and the concept of physical distance is introduced by identifying it with the time it takes a signal to travel to the world line of the object being studied and back, then the geometry of the three-dimensional space surrounding such a reference frame, under certain restrictions, turns out to be very close to Euclidean. Such a simplifying constraint is the condition that the speed of signals by which three-dimensional space is metrized be much less than the maximum possible. The latter, as in the special theory of relativity, should be associated with isotropic directions. This rather cumbersome construction is justified by the possibility of combining the geometry of H4 space, which is absolutely symmetrical in all its characteristic directions, with the familiar notions of the physical space-time surrounding us, in which one direction is distinguished relative to the other three.
Whether applicants for this grant will succeed in using the above technique, or whether they will propose their own methods, including the use of spaces associated with polynumbers other than H4, is irrelevant. The main criteria for choosing the winner will be the beauty of the resulting fractal images and the naturalness of their use in constructing polynumber spaces.
Topic 3
Investigate the geometries of multilinear Finsler spaces.
Explanation
The geometries of multilinear spaces (i.e., spaces whose metric properties can be expressed through the axioms of a multilinear symmetric form) have been extremely poorly studied. This may be due to the lack of a clear understanding, until recently, that the geometry of an entire class of Finsler spaces is directly related to hypercomplex numbers and multilinear forms. It turns out that for such spaces, the axioms of a symmetric multilinear form in n vectors, or a one-to-one related n-ary metric form, allow us to reduce the study of the geometry of the corresponding space to the study of mathematical objects—numbers and forms. Such support in the study of Finsler spaces is necessary, first and foremost, because our intuition, nurtured by Euclidean notions of space, falters every time the subject of study turns out to be of a slightly different geometric nature. This was the case when pseudo-Euclidean spaces gained a place in physics alongside Euclidean ones. Apparently, the situation is much the same today, as a similar question is being addressed regarding certain Finsler spaces. In both these cases, replacing the clarity of representations and practical experience with mathematical constructs is the most reliable way to understand geometric truths.
January 13, 2003
Topic 4
Investigation of fundamental physical structures associated with geometries and functions of a hypercomplex variable.
Explanation
The conditions for differentiability of functions of a complex variable—the Cauchy-Riemann conditions—are structurally similar to the differential equations of free, i.e., non-interacting, physical fields. Their generalizations to higher-dimensional algebras, both commutative-associative and non-commutative quaternion and biquaternion algebras, have repeatedly been proposed as primary equations of field theory. Most often, generalizations of the Cauchy-Riemann conditions have a form close to Maxwell's linear equations. Moreover, in some approaches (see, for example, the review by K. N. Bystrov and V. D. Zakharov [1]), the biquaternion structure indicates the possible existence of additional components of the electromagnetic field.
However, generalizations of differentiability conditions to non-commutative algebras of quaternion type are typically formal in nature, and the resulting functions lack the properties characteristic of analytic functions of a complex variable (in particular, the derivative of these functions depends on direction; see the notes to Topic 1). Therefore, any new physical predictions based on such generalizations do not appear reliable.
On the other hand, V.V. Kassandrov [2] demonstrated that explicitly accounting for non-commutativity in the case of, for example, biquaternion algebras leads to nonlinearity of the generalized Cauchy-Riemann equations, allowing them to be considered as equations of interacting physical fields. In this case, ordinary equations of physics are obtained as integrability conditions for the original (overdetermined) system of differentiability equations. Overdetermination also leads to certain "selection rules" for solutions of physical equations, including those for the electric charge of sources ("algebraic quantization").
It is interesting to note that the analog of the Laplace equations of the complex-valued complex in the Kassandrov model is the (complexified) eikonal equation. For one of the two classes of general solutions of this equation, obtained in [3], this equation is satisfied simultaneously with the d'Alembert linear wave equation. On the other hand, the approach to analysis on the commutative algebra of polynumbers proposed in the work of G. I. Garas'ko [4] essentially leads to a similar system of equations. This fact once again demonstrates the deep connections existing between various hypercomplex systems and, possibly, between the physical structures corresponding to them.
It is expected that papers submitted for the competition within the framework of this topic will offer an interesting physical interpretation of the functions, mappings, and corresponding geometries arising in hypercomplex analysis, based on already developed or new generalizations of complex analysis to polynumber, non-commutative, and possibly even non-associative hypercomplex structures. Primarily, this refers to applications to fundamental physics, with associated structures including physical fields and interactions, elementary particles, and the geometry of spacetime itself. Regarding the latter, the properties of hypercomplex number systems could serve as a justification for the dimensionality, topology, and metric structure of physical spacetime. However, no deep connection between observable reality and the properties of any hypercomplex system has yet been discovered, so all nontrivial research in this area is welcome.
We would like the papers submitted to the competition to be of a high mathematical standard, with a clear description of their geometric origins, emphasizing the fundamental possibilities of representing physical objects using hypercomplex number systems, and, at the same time, not being overloaded with numerous technical details.
1. Bystrov K.N., Zakharov V.D. // Results of Science and Technology. Classical Field Theory and Theory of Gravitation. Vol. 1. - VINITI, 1991, p. 111.
2. Kassandrov V.V. // Mathematics and Practice. Mathematics and Science. No. 2. - Moscow, "Self-Education", 2000, p. 61. (www.chronos.msu.ru/relectropublications.html);
3. Kassandrov V.V.// Gravitation & Cosmology (Moscow), Vol. 8, Suppl. II, 2002, p. 57.
4. Garas'ko G.I. // www.hypercomplex.ru/worksforprise.html
April 22, 2003
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