Klein Four-group Research Articles

Automorphisms of the category of free dimonoids

We prove that every automorphism of the category of free finitely generated dimonoids is quasi-inner. Also we show the automorphism group of the endomorphism semigroup of the free dimonoid on a nonsingleton set is isomorphic to the direct product of the Klein four-group and the symmetric group.

8×8 S-boxes over Klein four-group and Galois field $ {GF}\left({2}^{4}\right) $: AES redesign

<abstract> <p>This research paper is supplemented with a unique formation to design state-of-the-art S-boxes. The invented approach is simple but has the capability of creating confusion in our newly proposed algorithm. Our core planned work refined the method of already designed S-boxes to accomplish more compact ones. Various structures were merged here, namely affine transformation, fractional linear transformation, structure of Klein four-group, and the algebraic structures of the Galois fields, $ GF\left({2}^{4}\right) $ and $ GF\left({2}^{8}\right). $ These structures were utilized to synthesize newly $ 1600 $ robust S-boxes. Besides, we discussed encryption steps of AES with these newly generated S-boxes. We highlighted some specific characteristics, performance of parameter's improvement, and their utilization. Nonlinear properties were mainly set to inspect the behavior of I/O bits and could apply image encryption. Then, the performance of proposed S-boxes and newly structured AES was tested in comparison with other prevailing S-boxes.</p> </abstract>

New algorithm for the analysis of nucleotide and amino acid evolutionary relationships based on Klein four-group

Phylogenetics is the study of ancestral relationships among biological species. Such sequence analyses are often represented as phylogenetic trees. The branching pattern of each tree and its topology reflect the evolutionary relatedness between analyzed sequences. We present a Klein four-group algorithm (K4A) for the evolutionary analysis of nucleotide and amino acid sequences. Klein four-group set of operators consists of: identity e (U), and three elements—a = transition (C), b = transversion (G) and c = transition-transversion or complementarity (A). We generated Klein four-group based distance matrices of: 1. Cayley table (CK4), 2. Table rows (K4R), 3. Table columns (K4C), and 4. Euclidean 2D distance (K4E). The performance of the matrices was tested on a dataset of RecA proteins in bacteria, eukaryotes (Rad51 homolog) and archaea (RadA homolog). RecA and its functional homologs are found in all species, and are essential for the repair and maintenance of DNA. Consequently, they represent a good model for the study of evolutionary relationship of protein and nucleotide sequences. The ancestral relationship between the sequences was correctly classified by all K4A matrices concerning general topology. All distance matrices exhibited small variations among species, and overall results of tree classification were in agreement with the general patterns obtained by standard BLOSUM and PAM substitution matrices. During the evolution of a code there is a phase of optimization of system rules, the ambiguity of a code is eliminated, and the system starts producing specific components. Klein four-group algorithm is consistent with the concept of ambiguity reduction. It also enables the use of different genetic code table variants optimized for particular transitions in evolution based on biological specificity.

Kleinian KenKen

Summary We consider the collection of all KenKen puzzles over the Klein four-group with certain symmetry conditions and classify all such puzzles by their number of solutions.

Automata and finite order elements in the Nottingham group

The Nottingham group at 2 is the group of (formal) power series t+a2t2+a3t3+⋯ in the variable t with coefficients ai from the field with two elements, where the group operation is given by composition of power series. The depth of such a series is the largest d⩾1 for which a2=…=ad=0.Only a handful of power series of finite order (forcedly a power of 2) are explicitly known through a formula for their coefficients. We argue in this paper that it is advantageous to describe such series in closed computational form through automata, based on effective versions of proofs of Christol's theorem identifying algebraic and automatic series.Up to conjugation, there are only finitely many series σ of order 2n with fixed break sequence (i.e. the sequence of depths of σ∘2i). Starting from Witt vector or Carlitz module constructions, we give an explicit automaton-theoretic description of: (a) representatives up to conjugation for all series of order 4 with break sequence (1,m) for m<10; (b) representatives up to conjugation for all series of order 8 with minimal break sequence (1,3,11); and (c) an embedding of the Klein four-group into the Nottingham group at 2.We study the complexity of the new examples from the algebro-geometric properties of the equations they satisfy. For this, we generalise the theory of sparseness of power series to a four-step hierarchy of complexity, for which we give both Galois-theoretic and combinatorial descriptions. We identify where our different series fit into this hierarchy. We construct sparse representatives for the conjugacy class of elements of order two and depth 2μ±1(μ⩾1). Series with small state complexity can end up high in the hierarchy. This is true, for example, for a new automaton we found, representing a series of order 4 with 5 states (the minimal possible number for such a series).

Tensor theory for higher-dimensional Chern insulators with large Chern numbers

Recent advances in topological artificial systems open the door to realizing topological states in dimensions higher than the usual three-dimensional space. Here, we present a "tensor product" theory, which offers a method to construct Chern insulators with arbitrarily high dimensions and Chern numbers. Particularly, we show that the tensor product of a $d_A$D Chern insulator $\langle \mathcal{H}_A^{(\kappa_{A})}, C_A\rangle$ with a $d_B$D Chern insulator $\langle \mathcal{H}_B^{(\kappa_B)}, C_B\rangle$ leads to a $(d_A+d_B)$D Chern insulator $\langle \mathcal{H}_{A B}^{(\kappa_A\star \kappa_B)},-2C_AC_B\rangle $, where in the brackets, $\mathcal{H}^{(\kappa)}$ is the $d$D Hamiltonian with $d$ even, $C$ is the corresponding $(d/2)$th Chern number, and $\kappa$ labels the five non-chiral Altland-Zirnbauer symmetry classes A, AI, D, AII and C. The four real classes AI, D, AII and C form a Klein four-group under the multiplication `$\star$' with class AI the identity, and class A is the zero element. Our theory leads to novel higher-dimensional topological physics. (i) The construction can generate large higher-order Chern numbers, e.g., for some cases the resultant classification is $8\mathbb{Z}$. (ii) Fascinatingly, the boundary states feature flat nodal hypersurfaces with nontrivial Chern charges. For the constructed $(d_A+d_B)$D Chern insulator, a boundary perpendicular to a direction of $\mathcal{H}_A$ generically hosts $|C_A|$ $d_B$D nodal hypersurfaces, each of which has topological charge $\pm 2C_B$. Under perturbations, each nodal hypersurface bursts into stable unit nodal points, with the total Chern charge conserved. Examples are given to demonstrate our theory, which can be experimentally realized in artificial systems such as acoustic crystals, electric circuit arrays, ultracold atoms, or mechanical networks.

Klein Four-Group as the Factor Group for Elucidating RS-Stereoisomerism of Cubane Derivatives. The Factor-Group Method for Type-Itemized Enumeration of Stereoisograms

Van't Hoff's way (asymmetry, stereogenicity) and Le Bel's way (dissymmetry, chirality) are compared from the viewpoint of two ways for investigating organic stereochemistry, where cubane derivatives of the point group are selected as probes. For emphasizing Le Bel's way, combinatorial enumerations of 3D structures under point groups are first discussed to develop Fujita's proligand method and Fujita's USCI approach. The foundations of these enumerations are applied to support synthetic studies of stereoisomers for emphasizing van't Hoff's way after the proposal of Fujita's stereoisogram approach based on RS-stereoisomeric groups, e.g., . Importance of the proligand-promolecule model is emphasized in enumerations under point groups (Fujita's proligand method and Fujita's USCI approach) as well as in enumerations of RS-stereoisomeric groups (Fujita's stereoisogram approach). After the five types (type I to type V) of stereoisograms are classified into three categories (i.e., Category 1 (types I/IV), Category 2 (types II/III/V), and the co-existence case), the half-size-subgroup method and the factor-group method have been developed for type-itemized enumerations of stereoisograms. Type-V stereoisograms for characterizing "extended pseudoasymmetry" are discussed by assigning their configuration numbers and CA-descriptors.

The relative Heller operator and relative cohomology for the Klein 4-group

Let G be the Klein four-group and let be an arbitrary field of characteristic 2. A classification of indecomposable -modules is known. We calculate the relative cohomology groups for every indecomposable -module N, where χ is the set of proper subgroups in G. This extends work of Pamuk and Yalcin to cohomology with non-trivial coefficients. We also show that all cup products in strictly positive degree in are trivial.

The group generated by Riordan involutions

We prove that any element in the group generated by the Riordan involutions is the product of at most four of them. We also give a description of this subgroup as a semidirect product of a special subgroup of the commutator subgroup and the Klein four-group.

A Minimal Presentation of a Two-Generator Permutation Group on the Set of Integers

In this paper, we investigate the algebraic structure of certain 2-generator groups of permutations of the integers. The groups fall into two infinite classes: one class terminates with the quaternion group and the other class terminates with the Klein-four group. We show that all the groups are finitely presented and we determine minimal presentations in each case. Finally, we determine the order of each group.

Bifurcations of Periodic Solutions of a Hamiltonian System with a Discrete Symmetry Group

An autonomous Hamiltonian system with two degrees of freedom that is invariant under the Klein four-group of linear canonical automorphisms of the extended phase space of the system is studied. A sequence of symplectic transformations of the monodromy matrix of the symmetric periodic solution of this system is constructed. Using these transformations, the structure and bifurcation of the phase flow in the vicinity of this solution is investigated. It is shown that the bifurcations corresponding to the multiple period increase are different for the solutions with double symmetry and the solutions with a single symmetry. An example of critical periodic solutions of the family of doubly symmetric orbits of the plane circular Hill problem is discussed. The majority of tedious analytical calculations are performed using packages for the computation of Grobner bases and for the work with polynomial ideals in the computer algebra system Maple.

Hopfield neural networks using Klein four-group

Complex-valued Hopfield neural networks have been extended to several high-dimensional Hopfield neural networks (HDHNNs) using hypercomplex numbers. However, the extensions by hypercomplex numbers are limited. For example, the dimensions of Clifford and Cayley-Dickson algebras are power of two. Group theory provides more various algebras since the orders of groups are any. In this work, Klein four-group is introduced to HDHNNs. It is an abelian group whose order is 4. A Klein Hopfield neural network employs the twin-multistate activation function. We compare the noise tolerance with that of quaternion-valued and commutative quaternion-valued twin-multistate Hopfield neural networks by computer simulations.

Klein coverings of genus 2 curves

We investigate the geometry of etale $4:1$ coverings of smooth complex genus 2 curves with the monodromy group isomorphic to the Klein four-group. There are two cases, isotropic and non-isotropic depending on the values of the Weil pairing restricted to the group defining the covering. We recall from our previous work \cite{bo} the results concerning the non-isotropic case and fully describe the isotropic case. We show that the necessary information to construct the Klein coverings is encoded in the 6 points on $\mathbb{P}^1$ defining the genus 2 curve. The main result of the paper is the fact that, in both cases the Prym map associated to these coverings is injective. Additionally, we provide a concrete description of the closure of the image of the Prym map inside the corresponding moduli space of polarised abelian varieties.

Klein four-group and Darboux duality in conformal mechanics

We study the Klein four-group (${K}_{4}$) symmetry of the time-dependent Schr\"odinger equation for the conformal mechanics model of de Alfaro-Fubini-Furlan (AFF) with confining harmonic potential and coupling constant $g=\ensuremath{\nu}(\ensuremath{\nu}+1)\ensuremath{\ge}\ensuremath{-}1/4$. We show that it undergoes a complete or partial (at half-integer $\ensuremath{\nu}$) breaking on eigenstates of the system, and is the automorphism of the $\mathfrak{osp}(2,2)$ superconformal symmetry in super-extensions of the model by inducing a transformation between the exact and spontaneously broken phases of $\mathcal{N}=2$ Poincar\'e supersymmetry. We exploit the ${K}_{4}$ symmetry and its relation with the conformal symmetry to construct the dual Darboux transformations which generate spectrally shifted pairs of the rationally deformed AFF models. Two distinct pairs of intertwining operators originated from Darboux duality allow us to construct complete sets of the spectrum generating ladder operators that identify specific finite-gap structure of a deformed system and generate three distinct related versions of nonlinearly deformed $\mathfrak{sl}(2,\mathbb{R})$ algebra as its symmetry. We show that at half-integer $\ensuremath{\nu}$, the Jordan states associated with confluent Darboux transformations enter the construction, and the spectrum of rationally deformed AFF systems undergoes structural changes.

The indecomposable symplectic and quadratic modules of the Klein-four group

We determine all indecomposable symplectic modules for the Klein-four group K4 over a perfect field of characteristic 2 and classify the symplectic forms up to isometry. We also determine all K4-invariant quadratic forms which polarize to a given symplectic form and classify such quadratic forms up to isometry.

Asymmetric scattering by non-Hermitian potentials

The scattering of quantum particles by non-Hermitian (generally non-local) potentials in one dimension may result in asymmetric transmission and/or reflection from left and right incidence. After extending the concept of symmetry for non-Hermitian potentials, eight generalized symmetries based on the discrete Klein's four-group (formed by parity, time reversal, their product, and unity) are found. Together with generalized unitarity relations they determine selection rules for the possible and/or forbidden scattering asymmetries. Six basic device types are identified when the scattering coefficients (squared moduli of scattering amplitudes) adopt zero/one values, and transmission and/or reflection are asymmetric. They can pictorically be described as a one-way mirror, a one-way barrier (a Maxwell pressure demon), one-way (transmission or reflection) filters, a mirror with unidirectional transmission, and a transparent, one-way reflector. We design potentials for these devices and also demonstrate that the behavior of the scattering coefficients can be extended to a broad range of incident momenta.

Loewy lengths of blocks with abelian defect groups

We consider p p -blocks with abelian defect groups and in the first part prove a relationship between its Loewy length and that for blocks of normal subgroups of index p p . Using this, we show that if B B is a 2 2 -block of a finite group with abelian defect group D ≅ C 2 a 1 × ⋯ × C 2 a r × ( C 2 ) s D \cong C_{2^{a_1}} \times \cdots \times C_{2^{a_r}} \times (C_2)^s , where a i &gt; 1 a_i &gt; 1 for all i i and r ≥ 0 r \geq 0 , then d &gt; LL ⁡ ( B ) ≤ 2 a 1 + ⋯ + 2 a r + 2 s − r + 1 d &gt; \operatorname {LL}(B) \leq 2^{a_1}+\cdots +2^{a_r}+2s-r+1 , where | D | = 2 d |D|=2^d . When s = 1 s=1 the upper bound can be improved to 2 a 1 + ⋯ + 2 a r + 2 − r 2^{a_1}+\cdots +2^{a_r}+2-r . Together these give sharp upper bounds for every isomorphism type of D D . A consequence is that when D D is an abelian 2 2 -group the Loewy length is bounded above by | D | |D| except when D D is a Klein-four group and B B is Morita equivalent to the principal block of A 5 A_5 . We conjecture similar bounds for arbitrary primes and give evidence that it holds for principal 3 3 -blocks.

On coomologies of the Klein four-group

On coomologies of the Klein four-group

The Brauer-Picard group of the Asaeda-Haagerup fusion categories

We prove that the Brauer-Picard group of Morita autoequiv- alences of each of the three fusion categories which arise as an even part of the Asaeda-Haagerup subfactor or of its index 2 extension is the Klein four-group. We describe the 36 bimodule categories which occur in the full subgroupoid of the Brauer-Picard groupoid on these three fusion categories. We also classify all irreducible subfactors both of whose even parts are among these categories, of which there are 111 up to isomorphism of the planar algebra (76 up to duality). Although we identify the entire Brauer-Picard group, there may be additional fusion categories in the groupoid. We prove a partial classification of possible additional fusion categories Morita equivalent to the Asaeda-Haagerup fusion categories and make some conjectures about their existence; we hope to address these conjectures in future work.

The Algebraic Structure of Finite Metabelian Group Algebras

An algorithm for the explicit computation of a complete set of primitive central idempotents, Wedderburn decomposition, and the automorphism group of the semisimple group algebra of a finite metabelian group is developed. The algorithm is illustrated with its application to the semisimple group algebra of an arbitrary metacyclic group, and certain indecomposable groups whose central quotient is the Klein four-group.