Abstract:
This article is devoted to applications of projection operators to simulate phenomenological properties of the molecular-genetic code system. Oblique projection operators are under consideration, which are connected with matrix representations of the genetic coding system in forms of the Rademacher and Hadamard matrices. Evidences are shown that sums of such projectors give abilities for adequate simulations of ensembles of inherited biological phenomena including ensembles of biological cycles, morphogenetic ensembles of phyllotaxis patterns, etc. For such modeling, the author proposes multidimensional vector spaces, whose subspaces are under a selective control (or coding) by means of a set of matrix operators on base of genetic projectors. Development of genetic biomechanics is discussed. The author proposes and describes special systems of multidimensional numbers under names as tensorcomplex numbers, tensordouble numbers, etc. Described results can be used for developing algebraic biology, biotechnical applications and some other fields of science and technology.

Abstract:
The Truth and Reconciliation Commission (TRC) of Canada was established to uncover and acknowledge the injustices that took place in Indian residential schools and, in doing so, to pave the way to reconciliation. However, the TRC does not define reconciliation or how we would know it when (and if) we get there, thus stirring a debate about what it could mean. This article examines two theories that may potentially be relevant to the TRC’s work: Charles Taylor’s theory of recognition and Nancy Fraser’s tripartite theory of justice. The goal is to discover what each theory contributes to our understanding of the harms that Indigenous peoples suffered in residential schools, as well as in the broader colonial project, and how to address these harms appropriately.

Abstract:
The Canadian government designed Indian residential school (IRS) system to assimilate Indigenous children into European settler society by dispossessing them of their cultures, languages and traditions. By severing the children’s ties to families and communities, and thus integrating them into Euro-Canadian society, the Crown sought to gain control of Indigenous lands (Miller, 2000). In the schools, which were run by church officials, many children died of neglect and diseases and often faced various other injustices perpetrated by staff, including physical, emotional, cultural, and sexual abuse. (Milloy, 1999). Although the last school was closed in 1996, IRS left behind a devastating legacy characterized by sexual and physical abuse in Indigenous communities, substance abuse, loss of Indigenous languages, over-representation of Indigenous people in correctional facilities, and others. Until recently, these were considered to be private issues. However, the growing body of evidence demonstrates that IRS were responsible for the negative impacts and the government and churches were compelled to recognize the damage done. This article explores Michael Burawoy’s (2005) four types of sociology (policy, critical, professional, and public) and assesses the relative contributions of each type in the process of transforming “private troubles” of the IRS legacy into “public issues.” The main thesis of the article is that each type of sociology, with varying degrees of success, promotes the recognition of the injustices inflicted by IRS. The article concludes that Burawoy’s sociology possesses its strengths and weaknesses in identifying private troubles as public issues.Le gouvernement canadien a con u des pensionnats autochtones (PA) pour assimiler les enfants indigènes dans la société des colons européens en les dépossédant de leurs cultures, langues et traditions. En rompant les liens de l'enfant avec ses familles et communautés, et donc en les intégrant dans la société euro-canadienne, la Couronne a tenté de prendre le contr le des terres autochtones (Miller, 2000). Dans les écoles, qui ont été dirigées par les responsables de l'église, plusieurs enfants sont morts suite à des négligences et des maladies et ont souvent fait face à diverses injustices commises par le personnel, y compris des abus physiques, émotionnels, culturels, et sexuels. (Milloy, 1999). Bien que la dernière école ait été fermée en 1996, les PA ont laissé derrière eux un héritage dévastateur caractérisé par des abus physiques et sexuels dans les communautés autochtones, la toxicomanie, la

Abstract:
The set of known dialects of the genetic code (GC) is analyzed from the viewpoint of the genetic octave Yin-Yang-algebra. This algebra was described in the previous author's publications. The algebra was discovered on the basis of structural features of the GC in the matrix form of its presentation ("matrix genetics"). The octave Yin-Yang-algebra is considered as the pre-code or as the model of the GC. From the viewpoint of this algebraic model, for example, the sets of 20 amino acids and of 64 triplets consist of sub-sets of "male", "female" and "androgynous" molecules, etc. This algebra permits to reveal hidden peculiarities of the structure and evolution of the GC and to propose the conception of "sexual" relationships among genetic molecules. The first results of the analysis of the GC systems from such algebraic viewpoint say about the close connection between evolution of the GC and this algebra. They include 8 evolutionary rules of the dialects of the GC. The evolution of the GC is appeared as the struggle between male and female beginnings. The hypothesis about new biophysical factor of "sexual" interactions among genetic molecules is put forward. The matrix forms of presentation of elements of the genetic octave Yin-Yang-algebra are connected with Hadamard matrices by means of the simple U-algorithm. Hadamard matrices play a significant role in the theory of quantum computers, in particular. It gives new opportunities for possible understanding the GC systems as quantum computer systems. Revealed algebraic properties of the GC permit to put forward the problem of algebraization of bioinformatics on the basis of the algebras of the GC. Our investigation is connected with the question: what is life from the viewpoint of algebra? The algebraic version of the origin of the GC is discussed.

Abstract:
Algebraic properties of the genetic code are analyzed. The investigations of the genetic code on the basis of matrix approaches ("matrix genetics") are described. The degeneracy of the vertebrate mitochondria genetic code is reflected in the black-and-white mosaic of the (8*8)-matrix of 64 triplets, 20 amino acids and stop-signals. This mosaic genetic matrix is connected with the matrix form of presentation of the special 8-dimensional Yin-Yang-algebra and of its particular 4-dimensional case. The special algorithm, which is based on features of genetic molecules, exists to transform the mosaic genomatrix into the matrices of these algebras. Two new numeric systems are defined by these 8-dimensional and 4-dimensional algebras: genetic Yin-Yang-octaves and genetic tetrions. Their comparison with quaternions by Hamilton is presented. Elements of new "genovector calculation" and ideas of "genetic mechanics" are discussed. These algebras are considered as models of the genetic code and as its possible pre-code basis. They are related with binary oppositions of the Yin-Yang type and they give new opportunities to investigate evolution of the genetic code. The revealed fact of the relation between the genetic code and these genetic algebras is discussed in connection with the idea by Pythagoras: "All things are numbers". Simultaneously these genetic algebras can be utilized as the algebras of genetic operators in biological organisms. The described results are related with the problem of algebraization of bioinformatics. They take attention to the question: what is life from the viewpoint of algebra?

Abstract:
The article continues an analysis of the genetic 8-dimensional Yin-Yang-algebra. This algebra was revealed in a course of matrix researches of structures of the genetic code and it was described in the author's articles arXiv:0803.3330 and arXiv:0805.4692. The article presents data about many kinds of cyclic permutations of elements of the genetic code in the genetic (8x8)-matrix [C A; U G](3) of 64 triplets, where C, A, U, G are letters of the genetic alphabet. These cyclic permutations lead to such reorganizations of the matrix form of presentation of the initial genetic Yin-Yang-algebra that arisen matrices serve as matrix forms of presentations of new Yin-Yang-algebras as well. They are connected algorithmically with Hadamard matrices. The discovered existence of a hierarchy of the cyclic changes of types of genetic Yin-Yang-algebras allows thinking about new algebraic-genetic models of cyclic processes in inherited biological systems including models of cyclic metamorphoses of animals. These cycles of changes of the genetic 8-dimensional algebras and of their 8-dimensional numeric systems have many analogies with famous facts and doctrines of modern and ancient physiology, medicine, etc. This viewpoint proposes that the famous idea by Pythagoras (about organization of natural systems in accordance with harmony of numerical systems) should be combined with the idea of cyclic changes of Yin-Yang-numeric systems in considered cases. This second idea reminds of the ancient idea of cyclic changes in nature. From such algebraic-genetic viewpoint, the notion of biological time can be considered as a factor of coordinating these hierarchical ensembles of cyclic changes of types of the genetic multi-dimensional algebras.

Abstract:
The Kronecker family of the genetic matrices is investigated, which is based on the genetic matrix [C T; A G], where C, T, A, G are the letters of the genetic alphabet. The matrix [C T; A G] in the second Kronecker power is the (4*4)-matrix of 16 duplets. The matrix [C T; A G] in the third Kronecker power is the (8*8)-matrix of 64 triplets. It is significant that peculiarities of the degeneracy of the genetic code are reflected in the symmetrical black-and-white mosaic of these genetic matrices. The article represents interesting mathematical properties of these mosaic matrices, which are connected with positional permutations inside duplets and triplets; with projector operators; with unitary matrices and cyclic groups, etc. Fractal genetic nets are proposed as a new effective tool to study long nucleotide sequences. Some results about revealing new symmetry principles of long nucleotide sequences are described.

Abstract:
The matrix form of the presentation of the genetic code is described as the cognitive form to analyze structures of the genetic code. A similar matrix form is utilized in the theory of signal processing. The Kronecker family of the genetic matrices is investigated, which is based on the genetic matrix [C A; U G], where C, A, U, G are the letters of the genetic alphabet. This matrix in the third Kronecker power is the (8*8)-matrix, which contains 64 triplets. Peculiarities of the degeneracy of the vertebrate mitochondria genetic code are reflected in the symmetrical black-and-white mosaic of this genetic (8*8)-matrix. This mosaic matrix is connected algorithmically with Hadamard matrices unexpectedly, which are famous in the theory of signal processing, spectral analysis, quantum mechanics and quantum computers. A special decomposition of numeric genetic matrices reveals their close relations with a family of 8-dimensional hypercomplex numbers (not Cayley's octonions). Some hypothesis and thoughts are formulated on the basis of these phenomenological facts.

Abstract:
The article is devoted to phenomena of symmetries and algebras in matrix presentations of the genetic code. The Kronecker family of the genetic matrices is investigated, which is based on the alphabetical matrix [C A; U G], where C, A, U, G are the letters of the genetic alphabet. The matrix P=[C A; U G] in the third Kronecker power is the (8*8)-matrix, which contains 64 triplets. Peculiarities of the degeneracy of the genetic code are reflected in the symmetrical black-and-white mosaic of this genetic (8*8)-matrix of 64 triplets. Phenomena of connections of this mosaic matrix (and many other genetic matrices) with projection operators are revealed. Taking into account an important role of projection operators in quantum mechanics, theory of digital codes, computer science, logic and in many other fields of applied mathematics, we study algebraic properties and biological meanings of these phenomena. Using of notions and formalisms of theory of finite-dimensional vector spaces in bioinformatics and theoretical biology is proposed on the bases of the described results.

Abstract:
Matrix forms of the representation of the multi-level system of molecular-genetic alphabets have revealed algebraic properties of this system. Families of genetic (4*4)- and (8*8)-matrices show an unexpected connections of the genetic system with functions by Rademacher and Walsh and with Hadamard matrices. Dyadic-shift decompositions of such genetic matrices lead to relevant algebras of hypercomplex numbers. It is shown that genetic Hadamard matrices are identical to matrix representations of Hamilton quaternions and its complexification in the case of unit coordinates. The diversity of known dialects of the genetic code is analyzed from the viewpoint of the genetic algebras. An algebraic analogy with Punnett squares for inherited traits is shown. Our results are discussed taking into account the important role of dyadic shifts, Hadamard matrices, fourth roots of unity, Hamilton quaternions and other hypercomplex numbers in mathematics, informatics, physics, etc. These results testify that living matter possesses a profound algebraic essence. They show new promising ways to develop algebraic biology.