Klein Group Research Articles

Gradings and graded linear maps on algebras

Abstract Let A be a superalgebra over a field F of characteristic zero. We prove tight relations between graded automorphisms, pseudoautomorphisms, superautomorphisms and K-gradings on A, where K is the Klein group. Moreover, we investigate the consequences of such connections within the theory of polynomial identities. In the second part we focus on the superalgebra U ⁢ T n ⁢ ( F ) {UT_{n}(F)} of n × n {n\times n} upper triangular matrices by completely classifying the graded-pseudo-super automorphism that one can define on it. Finally, we compute the ideals of identities of U ⁢ T n ⁢ ( F ) {UT_{n}(F)} endowed with a graded or a pseudo automorphism, for any n, and the ideals of identities with superautomorphism in the cases n = 2 {n=2} and n = 3 {n=3} .

Log Canonical Thresholds on Burniat Surfaces with $K^2 = 6$ via Pluricanonical Divisors

Let $S$ be a Burniat surface with $K_S^2 = 6$ and $\varphi$ be the bicanonical map of $S$. In this paper we show optimal lower bounds of log canonical thresholds of members of pluricanonical sublinear systems of $S$ via Klein group $G$ induced by $\varphi$. Indeed, for a positive even integer $m$, the log canonical threshold of members of an invariant (resp. anti-invariant) part of $|mK_S|$ is greater than or equal to $1/(2m)$ (resp. $1/(2m-2)$). For a positive odd integer $m$, the log canonical threshold of members of an invariant (resp. anti-invariant) part of $|mK_S|$ is greater than or equal to $1/(2m-5)$ (resp. $1/(2m)$). The inequalities are all optimal.

Virtual braids and permutations

Let VB_n be the virtual braid group on n strands and let S_n be the symmetric group on n letters. Let n, m ∈ N such that n ≥ 5, m ≥ 2 and n ≥ m. We determine all possible homomorphisms from VB_n to S_m , from S_n to VB_m and from VB_n to VB_m. As corollaries we get that Out(VB_n) is isomorphic to the Klein group and that VB_n is both Hopfian and co-Hofpian.

Lepton and quark mixing from a stepwise breaking of flavor and CP

We explain all features of lepton and quark mixing in a scenario with the flavor symmetry Delta (384) and a CP symmetry, where these are broken in several steps. The residual symmetry in the neutrino and up quark sector is a Klein group and CP, while a Z_3 and a Z_{16} symmetry are preserved among charged leptons and down quarks, respectively. If the Klein group in the neutrino sector is further broken down to a single Z_2 symmetry, we obtain predictions for all lepton mixing parameters in terms of one real quantity, whose size is determined by the value of the reactor mixing angle. The Dirac and Majorana phases are fixed, in particular sin (delta) = -0.936. A sum rule, relating these CP phases and the reactor and atmospheric mixing angles, theta_{13} and theta_{23}, is given. In the quark sector, we have for the Cabibbo angle theta_C= sin (pi/16) = 0.195 after the first step of symmetry breaking. This is brought into full accordance with experimental data with the second step of symmetry breaking, where either the Z_{16} group is broken to a Z_8 symmetry in the down quark sector or the Klein group to one Z_2 symmetry only among up quarks. The other two quark mixing angles are generated in the third and last symmetry breaking step, when the residual symmetries in the up and/or down quark sector are further broken. If this step occurs among both up and down quarks, the amount of CP violation in the quark sector is determined by the lepton sector and explaining the current neutrino oscillation data entails that the Jarlskog invariant J_{CP}^q is in very good agreement with experimental findings. Lastly, a sum rule is derived that contains the CP phase delta^q and theta_C of the quark sector and the lepton mixing parameters theta_{13}, theta_{23} and delta.

Symmetries of the space of solutions to special double confluent Heun equations of integer order

This paper investigates the triplet of linear operators that determines automorphisms of the set of solutions to special double confluent Heun equations of integer order. Their pairwise composition rules are computed in explicit form. It is shown that, under the conditions motivated by physical applications, these operators generate the group of symmetries of the linear space of solutions that is isomorphic to the dihedral group, provided the monodromy equivalence relation is applied. On the corresponding projective space, the symmetry group reduces to the Klein group. The results presented in this paper have implications for the modeling of Josephson junctions.

Hyperelliptic curves on (1, 4)-polarised abelian surfaces

We investigate the number and the geometry of smooth hyperelliptic curves on a general complex abelian surface. We give a short proof for the known fact the only possibilities of genera of such curves are 2, 3, 4 and 5; then we focus on the genus five case. We prove that up to translation, there is a unique hyperelliptic curve in the linear system of a general (1, 4)-polarised abelian surface. Moreover, the curve is invariant with respect to a subgroup of translations isomorphic to the Klein group. Our proof of the existence of hyperelliptic curves on general (1, 4)-polarised abelian surfaces is different from that in the recent paper [10]. We give the decomposition of the Jacobian of such a curve into abelian subvarieties displaying Jacobians of quotient curves and Prym varieties. Motivated by the construction, we prove the statement: every etale Klein covering of a hyperelliptic curve is a hyperelliptic curve, provided that the group of 2-torsion points defining the covering is non-isotropic with respect to the Weil pairing and every element of this group can be written as a difference of two Weierstrass points.

D.W. Winnicott, Melanie Klein, and W.R. Bion: The Controversy Over the Nature of the External Object—Holding and Container/Contained (1941-1967)

The author examines D.W. Winnicott’s multi-layered, dialogue with the London Klein group, most particularly with W.R. Bion and their respective views on the role of the external object in terms of holding and container/contained. When Winnicott earlier on deployed Klein as both a collaborative as well as creative antagonist in evolving a thesis of maternal environment of provision, memorialized in “Transitional Objects,” he expatriated himself from the Klein group by 1953.In the wake of this split, Winnicott continued a dialogic entreaty with W.R. Bion, attempting to interest another Kleinian in the importance of the external object when he commented on Bion’s papers. Bion initially maintained the Klein line of strict focus on the patient’s internal, subjective, and phantasmic experience of the external object during the early period of his “psychosis papers” (1954-1957). He gradually shifted and altered his clinical focus, taking up in metapsychological terms the infant’s normal development of early thinking when he posited the importance of a containing maternal object to metabolize primitive, sensuous elements into rudimentary thought. The author maintains that Bion in part appropriated aspects of Winnicott’s research trajectory, obscuring this appropriation because of Klein’s injunction that Winnicott’s work not be taken up. The separatist existence of the three major schools of analytic thought in the British Psychoanalytical Society also underscored a widespread disinclination towards comparative psychoanalytic theory. As a result, Winnicott’s theory of holding has stood alongside Bion’s theory of container/contained until recent efforts at comparison. By 1964, Winnicott ended his entreaties to Bion and the Klein group when they continued to ignore his many contributions, memorializing it in his 1965 paper, “The Kleinian Development.” The author concludes with comments on how theoretical and clinical differences between analytic theorists can be both generative and still remain antagonistic at the boundaries of passionately maintained group affiliations.

Representations of the Klein Group Determined by Quadruples of Polynomials Associated with the Double Confluent Heun Equation

The canonical representation of the Klein group K4 = ℤ2⊕ℤ2 on the space ℂ* = ℂ {0} induces a representation of this group on the ring L = C[z, z−1], z ∈ ℂ*, of Laurent polynomials and, as a consequence, a representation of the group K4 on the automorphism group of the group G = GL(4,L) by means of the elementwise action. The semidirect product ĜG = GK4 is considered together with a realization of the group Ĝ as a group of semilinear automorphisms of the free 4-dimensional L-module M4. A three-parameter family of representations R of K4 in the group Ĝ and a three-parameter family of elements X ∈ M4 with polynomial coordinates of degrees 2(l − 1), 2l, 2(l − 1), and 2l, where l is an arbitrary positive integer (one of the three parameters), are constructed. It is shown that, for any given family of parameters, the vector X is a fixed point of the corresponding representation R. An algorithm for calculating the polynomials that are the components of X was obtained in a previous paper of the authors, in which it was proved that these polynomials give explicit formulas for automorphisms of the solution space of the doubly confluent Heun equation.

A characterization of Inoue surfaces with $$p_g=0$$ p g = 0 and $$K^2=7$$ K 2 = 7

Inoue constructed the first examples of smooth minimal complex surfaces of general type with \(p_g=0\) and \(K^2=7\). These surfaces are finite Galois covers of the 4-nodal cubic surface with the Galois group, the Klein group \(\mathbb {Z}_2\times \mathbb {Z}_2\). For such a surface S, the bicanonical map of S has degree 2 and it is composed with exactly one involution in the Galois group. The divisorial part of the fixed locus of this involution consists of two irreducible components: one is a genus 3 curve with self-intersection number 0 and the other is a genus 2 curve with self-intersection number \(-\,1\). Conversely, assume that S is a smooth minimal complex surface of general type with \(p_g=0\), \(K^2=7\) and having an involution \(\sigma \). We show that, if the divisorial part of the fixed locus of \(\sigma \) consists of two irreducible components \(R_1\) and \(R_2\), with \(g(R_1)=3, R_1^2=0, g(R_2)=2\) and \(R_2^2=-\,1\), then the Klein group \(\mathbb {Z}_2\times \mathbb {Z}_2\) acts faithfully on S and S is indeed an Inoue surface.

TINJAUAN STRUKTUR ALJABAR PADA KODE GENETIK STANDAR

Genetics is the branch of Biological Sciences that studies about the decline of a living being from parent to the children . Genetics often associated with genes and DNA (deoxyribonucleic acid) which both are one of the genetic material in a living body is very important in terms of inheritance Standard genetic code i as a representation of a gene that is according to needs of the human body protein. Genes in the standard genetic cods presented in the form of code triplet of nitrogen bases and became the language of the gene encoding the living body which produce = 64 different triplet code . All the triplet code of the RNA nucleotide chain translated and produces 20 kinds of amino acids that will be released as a protein in the cell . Set of nitrogen bases in RNA chains compiled and presented in a set N = { C , U , A, G }.This research will be carried out matching N with the cross product Z2 X Z2 {(0,0 ), (0,1 ), (1,0 ), ( 1,1)}. Through this matching then N = { C , U , A, G } forming Klein Group - 4 , the binary vector spaces and The Galois Field . Beside that set of N can be classified into three sets containing partition the set of basic nitrogen based chemical properties of nucleotides that is strong - weak bases, amino - ketonukleotida , and types of bases nitrogen, in the presence of the partition, then N = { C , U , A , G } will form the quotient group. Keywords: DNA, RNA, standard genetic code, quotien group

Automorphism group of Green ring of Sweedler Hopf algebra

Let H2 be Sweedler’s 4-dimensional Hopf algebra and r(H2) be the corresponding Green ring of H2. In this paper, we investigate the automorphism groups of Green ring r(H2) and Green algebra F(H2) = r(H2)⊗ℤF, where F is a field, whose characteristics is not equal to 2. We prove that the automorphism group of r(H2) is isomorphic to K4, where K4 is the Klein group, and the automorphism group of F(H2) is the semidirect product of ℤ2 and G, where G = F {1/2} with multiplication given by a · b = 1− a − b + 2ab.

Rethinking the Role of Small-Group Collaborators and Adversaries in the London Kleinian Development (1914–1968)

The authors historically situate the London Kleinian development in terms of the small-group collaborations and adversaries that arose during the course of Melanie Klein’s career. Some collaborations later became personally adversarial (e.g., those Klein had with Glover and Schmideberg); other adversarial relationships forever remained that way (with A. Freud); while still other long-term collaborations became theoretically contentious (such as with Winnicott and Heimann). After the Controversial Discussions in 1944, Klein marginalized one group of supporters (Heimann, Winnicott, and Riviere) in favor of another group (Rosenfeld, Segal, and Bion). After Klein’s death in 1960, Bion maintained loyalty to Klein’s ideas while quietly distancing his work from the London Klein group, immigrating to the United States in 1968.

MSSM from F-theory SU(5) with Klein monodromy

We revisit a class of $SU(5)$ supersymmetric grand unified theory (SUSY GUT) models which arise in the context of the spectral cover with Klein Group monodromy ${V}_{4}={Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$. We examine the polynomials of the corresponding factorized spectral cover and discuss the constraints imposed on their coefficients for the transitive and nontransitive realization of this monodromy. We show that ${Z}_{2}$ matter parities can be realized via new geometric symmetries respected by the spectral cover.

The Klein Group, Squares of Opposition and the Explanation of Fallacies in Reasoning

During the last decades, the psychology of reasoning has identified experimentally many fallacies committed by spontaneous reasoners. Given these experimental results, some theories have been developed about this phenomenon, mainly algorithmic theories. This paper develops instead a computational modelling of these current fallacies which appear as simplifications in the treatment of information that do not respect the formal rules of classical propositional logic. These fallacies are explained as crushes in the Klein group structure and so, in squares of opposition. These crushes are an effect of not to take into account the dual of the binary operator at work in the major premise of the inference. This analysis allows predictions on fallacies not identified before, which are fallacies produced when reasoning with incompatibilities. The paper concludes with consequences of the analysis on pedagogical strategies for the teaching of logic.

On the Noncommutative Deformation of the Operator Graph Corresponding to the Klein Group

We study the noncommutative operator graph ℒ θ depending on a complex parameter θ recently introduced by M. E. Shirokov to construct channels with positive quantum zero-error capacity having vanishing n-shot capacity. We define a noncommutative group G and an algebra A θ which is the quotient of ℂG by a special algebraic relation depending on θ such that the matrix representation ϕ of A θ results in the algebra ℳ θ generated by ℒ θ . In the case of θ = ±1, the representation ϕ degenerates into a faithful representation of ℂK 4 , where K 4 is the Klein group. Thus, ℒ θ can be regarded as a noncommutative deformation of the graph associated with the Klein group. Bibliography: 16 titles.

Finite Majid Algebras Over the Klein Group

We provide a classification of finite-dimensional connected coradically graded pointed Majid algebras, generated in degrees 0 and 1, over the Klein group on the field of complex numbers.

Do grafo do desejo aos quatro discursos de Lacan

Este trabalho analisa uma etapa do percurso do Lacan pesquisador, envolvendo questões metodológicas na passagem do grafo do desejo à formalização do matema dos quatro discursos. No grafo, a linha em formato de anzol cortando as duas linhas horizontais configurava o efeito de discurso em três etapas, as quais se mostraram equivalentes às três estruturas clínicas. Com os quatro discursos, Lacan estabelece uma estrutura de quatro posições sobre as quais giram quatro elementos. O matema dos quatro discursos nos remete às operações da estrutura do grupo de Klein. Entretanto, duas regras restringem a permutabilidade: o movimento de um quarto de giro dos elementos apenas em um sentido; em segundo lugar, nas quatro posições prévias, há uma disjunção entre as posições de saber e verdade. Podemos então concluir que os quatro discursos consistem em uma apropriação parcial da estrutura do grupo de Klein, de modo a demarcar a diferença entre a psicanálise e o discurso científico.

Further study on algebraic structure of RSA algorithm

By making use of the theory of quadratic residues under the condition of strong prime, a method for studying the algebraic structure of Z*φ( n)of RSA( Rivest-Shamir-Adleman) algorithm was established in this work. A formula to determine the order of element in Z*φ( n)and an expression of maximal order were proposed; in addition, the numbers of quadratic residues and non-residues in the group Z*φ( n)were calculated. This work gave an estimate that the upper bound of maximal order was φ( φ( n)) /4 and obtained a necessary and sufficient condition on maximal order being equal to φ( φ( n)) /4.Furthermore, a sufficient condition for A1being cyclic group was presented, where A1was a subgroup composed of all quadratic residues in Z*φ( n), and a method of the decomposition of Z*φ( n)was also established. Finally, it was proved that the group Z*φ( n)could be generated by seven elements of quadratic non-residues and the quotient group Z*φ( n)/ A1was a Klein group of order 8.

Discrete symmetries and model-independent patterns of lepton mixing

In the context of discrete flavor symmetries, we elaborate a method that allows one to obtain relations between the mixing parameters in a model-independent way. Under very general conditions, we show that flavor groups of the von Dyck type, that are not necessarily finite, determine the absolute values of the entries of one column of the mixing matrix. We apply our formalism to finite subgroups of the infinite von Dyck groups, such as the modular groups, and find cases that yield an excellent agreement with the best fit values for the mixing angles. We explore the Klein group as the residual symmetry of the neutrino sector and explain the permutation property that appears between the elements of the mixing matrix in this case.

Dynamics of (4+1)-Dihedrally Symmetric Nearly Parallel Vortex Filaments

We give a detailed analytical and numerical description of the global dynamics of 4+1 points interacting through the singular logarithmic potential and subject to the following symmetry constraint: at each instant 4 of them form an orbit of the Klein group D 2 of order 4. The main device in order to achieve our results is to use a McGehee-like transformation introduced in (Paparella and Portaluri in Global dynamics of the dihedral singular logarithmic potential and nearly parallel vortex filaments, 2011) for a problem analogous to the present one.After performing this transformation in order to regularize the total collision, we study the rest-points of the flow, the invariant (stable and unstable) manifolds and we derive some interesting information about the global dynamics.