V.M. Miklyukov: from dimension 8 to nonassociative algebras

In this short survey we give a background and explain some recent developments in algebraic minimal cones and nonassociative algebras. A good deal of this paper is recollections of my collaboration with my teacher, PhD supervisor and a colleague, Vladimir Miklyukov on minimal surface theory that motivated the present research. This paper is dedicated to his memory.Comment: 19 page

Bifurcation of Fredholm Maps I; The Index Bundle and Bifurcation

We associate to a parametrized family $f$ of nonlinear Fredholm maps possessing a trivial branch of zeroes an {\it index of bifurcation} $\beta(f)$ which provides an algebraic measure for the number of bifurcation points from the trivial branch. The index $\beta(f)$ is derived from the index bundle of the linearization of the family along the trivial branch by means of the generalized $J$-homomorphism. Using the Agranovich reduction and a cohomological form of the Atiyah-Singer family index theorem, due to Fedosov, we compute the bifurcation index of a multiparameter family of nonlinear elliptic boundary value problems from the principal symbol of the linearization along the trivial branch. In this way we obtain criteria for bifurcation of solutions of nonlinear elliptic equations which cannot be achieved using the classical Lyapunov-Schmidt method.Comment: 42 pages. Changes: added Lemma 2.31 and a reference + minor corrections. To appear on TMN

Fourier transform over finite groups for error detection and error correction in computation channels

We consider the methods of error detection and correction in devices and programs calculating functions f: G → K where G is a finite group and K is a field. For error detection and correction we use linear checks generated by convolutions in the field K of the original function f and some checking idempotent function δ: G → , 1 For the construction of the optimal checking function δ we use methods of harmonic analysis over the group G in the field K. Since these methods will be the main tools for the construction of optimal checks, we consider the algorithms for the fast computation of Fourier Transforms over the group G in the field K. We solve the problem of error detecting and correcting capability for our methods for two important classes of decoding procedures (memoryless and memory-aided decoding) and consider the question of syndrome computation for these methods. We describe also properties of error correcting codes generated by convolution checks

Natural Communication

In Natural Communication, the author criticizes the current paradigm of specific goal orientation in the complexity sciences. His model of "natural communication" encapsulates modern theoretical concepts from mathematics and physics, in particular category theory and quantum theory. The author is convinced that only by looking to the past is it possible to establish continuity and coherence in the complexity science