Non-free Group Research Articles

Algebraic Morphology of DNA–RNA Transcription and Regulation

Transcription factors (TFs) and microRNAs (miRNAs) are co-actors in genome-scale decoding and regulatory networks, often targeting common genes. To discover the symmetries and invariants of the transcription and regulation at the scale of the genome, in this paper, we introduce tools of infinite group theory and of algebraic geometry to describe both TFs and miRNAs. In TFs, the generator of the group is a DNA-binding domain while, in miRNAs, the generator is the seed of the sequence. For such a generated (infinite) group π, we compute the SL(2,C) character variety, where SL(2,C) is simultaneously a ‘space-time’ (a Lorentz group) and a ‘quantum’ (a spin) group. A noteworthy result of our approach is to recognize that optimal regulation occurs when π looks similar to a free group Fr (r=1 to 3) in the cardinality sequence of its subgroups, a result obtained in our previous papers. A non-free group structure features a potential disease. A second noteworthy result is about the structure of the Groebner basis G of the variety. A surface with simple singularities (such as the well known Cayley cubic) within G is a signature of a potential disease even when π looks similar to a free group Fr in its structure of subgroups. Our methods apply to groups with a generating sequence made of two to four distinct DNA/RNA bases in {A,T/U,G,C}. We produce a few tables of human TFs and miRNAs showing that a disease may occur when either π is away from a free group or G contains surfaces with isolated singularities.

The coloured Tverberg theorem, extensions and new results

AbstractWe prove amultiple coloured Tverberg theoremand abalanced coloured Tverberg theorem, applying different methods, tools and ideas. The proof of the first theorem uses a multiple chessboard complex (as configuration space) and the Eilenberg–Krasnoselskii theory of degrees of equivariant maps for non-free group actions. The proof of the second result relies on the high connectivity of the configuration space, established by using discrete Morse theory.

Orbit configuration spaces of small covers and quasi-toric manifolds

In this article, we investigate the orbit configuration spaces of some equivariant closed manifolds over simple convex polytopes in toric topology, such as small covers, quasi-toric manifolds and (real) moment-angle manifolds; especially for the cases of small covers and quasi-toric manifolds. These kinds of orbit configuration spaces have non-free group actions, and they are all noncompact, but still built via simple convex polytopes. We obtain an explicit formula of the Euler characteristic for orbit configuration spaces of small covers and quasi-toric manifolds in terms of the h-vector of a simple convex polytope. As a by-product of our method, we also obtain a formula of the Euler characteristic for the classical configuration space, which generalizes the Felix-Thomas formula. In addition, we also study the homotopy type of such orbit configuration spaces. In particular, we determine an equivariant strong deformation retraction of the orbit configuration space of 2 distinct orbit-points in a small cover or a quasi-toric manifold, which allows to further study the algebraic topology of such an orbit configuration space by using the Mayer-Vietoris spectral sequence.

Factors affecting parental intention to vaccinate kindergarten children against influenza: A cross-sectional survey in China

BackgroundThe impact of influenza in children under 5 can be severe and fatal. However, the influenza vaccination uptake in China remains suboptimal. The objectives of this study were to investigate parents’ perceptions on influenza vaccination and to assess vaccination promotional factors. MethodsA cross-sectional survey among 1506 parents with children in kindergarten was conducted in two areas with different policies: self-paid vaccination and free vaccination. The questionnaire was based on the structure of the Health Belief Model (HBM). Multiple logistic regression was used to analyze the determinants of parental vaccination intention. Odds ratios (OR) and respective 95% confidence intervals (95% CI) are reported. ResultsWithin the free policy group versus the non-free group, vaccination intention rates were 76.3% versus 83.4%, and vaccination rates were 34.2% versus 3.1%. Results from multivariate analysis showed that parents with high scores for perceived susceptibility (OR = 1.44; 95% CI: 1.09–1.91), perceived benefits (OR = 1.80; 95% CI: 1.30–2.50) and cues to action (OR = 3.32; 95% CI: 2.47–4.46) were more likely to get their children vaccinated, while those perceived more barriers (OR = 0.50; 95% CI: 0.37–0.68) had lower vaccination intention. More knowledge (OR = 1.74; 95% CI: 1.18–2.56) and preferable attitudes (higher perceived necessity: OR = 1.84; 95% CI: 1.53–2.22; less safety worry: OR = 1.35; 95% CI: 1.10–1.66) were associated with significantly higher vaccination intention. Adjusted for parents’ gender, age, education, income and children’s age, the same significant factors were found. Parental intention was found to be influenced by different vaccination policies. Under a free policy, past influenza vaccination uptake (OR = 4.52; 95% CI: 1.07–19.02) greatly promoted parents’ willingness to vaccinate their children. ConclusionParents had high intention to get their kindergarten children vaccinated with the influenza vaccine in spite of the low uptake rate. Our results indicate that offering free influenza vaccines and parental education over the next years may increase the influenza vaccination rate.

On diagonal actions of branch groups and the corresponding characters

We introduce notions of absolutely non-free and perfectly non-free group actions and use them to study the associated unitary representations. We show that every weakly branch group acting on a regular rooted tree acts absolutely non-freely on the boundary of the tree. Using this result and the symmetrized diagonal actions we construct for every countable branch group infinitely many different ergodic perfectly non-free actions, infinitely many II1-factor representations, and infinitely many continuous ergodic invariant random subgroups.

Solving the Conjugacy Decision Problem via Machine Learning

ABSTRACTMachine learning and pattern recognition techniques have been successfully applied to algorithmic problems in free groups. In this paper, we seek to extend these techniques to finitely presented non-free groups, with a particular emphasis on polycyclic and metabelian groups that are of interest to non-commutative cryptography. As a prototypical example, we utilize supervised learning methods to construct classifiers that can solve the conjugacy decision problem, i.e., determine whether or not a pair of elements from a specified group are conjugate. The accuracies of classifiers created using decision trees, random forests, and N-tuple neural network models are evaluated for several non-free groups. The very high accuracy of these classifiers suggests an underlying mathematical relationship with respect to conjugacy in the tested groups.

Splittings of knot groups

Let \(K\) be a knot of genus \(g\). If \(K\) is fibered, then it is well known that the knot group \(\pi (K)\) splits only over a free group of rank \(2g\). We show that if \(K\) is not fibered, then \(\pi (K)\) splits over non-free groups of arbitrarily large rank. Furthermore, if \(K\) is not fibered, then \(\pi (K)\) splits over every free group of rank at least \(2g\). However, \(\pi (K)\) cannot split over a group of rank less than \(2g\). The last statement is proved using recent results of Agol, Przytycki–Wise and Wise.

Magnetic source imaging (MSI) in children with neocortical epilepsy: Surgical outcome association with 3D post-resection analysis

To investigate the validity of magnetic source imaging (MSI) to localize seizure-onset zone using 3D analysis of pre-operative MSI source imaging coregistered to post-resection MRI following neocortical epilepsy surgery. Twenty-two children who had MSI and epilepsy surgery were studied (median age=11 years, 1 year 2 months-22 years). Only seven (31.8%) had localized lesions on pre-operative conventional brain MRIs. Sixteen (72.7%) underwent intracranial EEG monitoring. Mean post-operative follow-up was 4.7 years (1 year 3 months-8 years 2 months). Fifteen patients (68%) were seizure-free. MEG spike dipole sources were superimposed onto post-operative MRIs. The number and proportion of spike dipoles within resection volume were calculated and compared between seizure free and non-free groups. Both number of dipole clusters and proportion of dipoles in resection volume were not associated with seizure-free outcome (p>0.05). In seven cases with MRI lesions, six of these with a ≥70% dipoles within the resection margin were seizure-free, while one with the proportion <70% was not seizure-free. Further, among the 15 cases with non-localized or normal MRI, five with both the proportion <70% and multiple dipoles clusters were post-operatively seizure free. Number and density of clustered spike dipole sources within the surgical resection volume is not associated with postoperative seizure-free outcome. MSI successfully localized the perilesional epileptogenic zone in cases with localized MRI lesions, but not in cases with normal MRI in this study. Even if MEG localizes spikes to a single focal region, confirmation of epilepsy localization with intracranial EEG is still recommended in cases with non-lesional MRI.

K-Free-like groups

The following results are proved. In Theorem 1, it is stated that there exist both finitely presented and not finitely presented 2-generated nonfree groups which are k-free-like for any k ? 2. In Theorem 2, it is claimed that every nonvirtually cyclic (resp., noncyclic and torsion-free) hyperbolic m-generated group is k-free-like for every k ? m + 1 (resp., k ? m). Finally, Theorem 3 asserts that there exists a 2-generated periodic group G which is k-free-like for every k ? 3.

The number of openly generated Boolean algebras

AbstractThis article is devoted to two different generalizations of projective Boolean algebras: openly generated Boolean algebras and tightly σ-filtered Boolean algebras.We show that for every uncountable regular cardinal κ there are 2κ pairwise non-isomorphic openly generated Boolean algebras of size κ &gt; ℵ1 provided there is an almost free non-free abelian group of size κ. The openly generated Boolean algebras constructed here are almost free.Moreover, for every infinite regular cardinal κ we construct 2κ pairwise non-isomorphic Boolean algebras of size κ that are tightly σ-filtered and c.c.c.These two results contrast nicely with Koppelberg's theorem in [12] that for every uncountable regular cardinal κ there are only 2κ isomorphism types of projective Boolean algebras of size κ.

The structure of two-parabolic space: parabolic dust and iteration

A non-elementary Möbius group generated by two-parabolics is determined up to conjugation by one complex parameter and the parameter space, the parameter space for representations of groups generated by two parabolics into $${PSL(2, \mathbb{C})}$$ modulo conjugacy, has been extensively studied. In this paper, we use the results of Gilman and Waterman to obtain an additional structure for the parameter space, which we term the two-parabolic space. This structure allows us to identify groups that contain additional conjugacy classes of primitive parabolics, which we call parabolic dust groups, non-free groups off the real axis, and groups that are both parabolic dust and non-free; some of these contain $${\mathbb{Z} \times \mathbb{Z}}$$ subgroups. The structure theorem also attaches additional geometric structure to discrete and non-discrete groups lying in given regions of the parameter space and allows a new explicit geometric construction of some non-classical T-Schottky groups.

STALLINGS FOLDINGS AND SUBGROUPS OF AMALGAMS OF FINITE GROUPS

Stallings showed that every finitely generated subgroup of a free group is canonically represented by a finite minimal immersion of a bouquet of circles. In terms of the theory of automata, this is a minimal finite inverse automaton. This allows for the deep algorithmic theory of finite automata and finite inverse monoids to be used to answer questions about finitely generated subgroups of free groups. In this paper, we attempt to apply the same methods to other classes of groups. A fundamental new problem is that the Stallings folding algorithm must be modified to allow for "sewing" on relations of non-free groups. We look at the class of groups that are amalgams of finite groups. It is known that these groups are locally quasiconvex and thus, all finitely generated subgroups are represented by finite automata. We present an algorithm to compute such a finite automaton and use it to solve various algorithmic problems.

On the geometry of reduced cotangent bundles at zero momentum

We consider the problem of cotangent bundle reduction for proper non-free group actions at zero momentum. We show that in this context the symplectic stratification obtained by Sjamaar and Lerman refines in two ways: (i) each symplectic stratum admits a stratification which we call the secondary stratification with two distinct types of pieces, one of which is open and dense and symplectomorphic to a cotangent bundle; (ii) the reduced space at zero momentum admits a finer stratification than the symplectic one into pieces that are coisotropic in their respective symplectic strata.

Stability of relative equilibria with singular momentum values in simple mechanical systems

A method for testing the stability of relative equilibria in Hamiltonian systems of the form ‘kinetic + potential energy’ is presented. This method extends previously existing techniques to the case of non-free group actions and singular momentum values. A normal of the symplectic matrix at a relative equilibrium is also obtained.

Applications of the Generalized Fourier Transform in Numerical Linear Algebra

Equivariant matrices, commuting with a group of permutation matrices, are considered. Such matrices typically arise from PDEs and other computational problems where the computational domain exhibits discrete geometrical symmetries. In these cases, group representation theory provides a powerful tool for block diagonalizing the matrix via the Generalized Fourier Transform (GFT). This technique yields substantial computational savings in problems such as solving linear systems, computing eigenvalues and computing analytic matrix functions such as the matrix exponential. The paper is presenting a comprehensive self contained introduction to this field. Building upon the familiar special (finite commutative) case of circulant matrices being diagonalized with the Discrete Fourier Transform, we generalize the classical convolution theorem and diagonalization results to the noncommutative case of block diagonalizing equivariant matrices. Applications of the GFT in problems with domain symmetries have been developed by several authors in a series of papers. In this paper we elaborate upon the results in these papers by emphasizing the connection between equivariant matrices, block group algebras and noncommutative convolutions. Furthermore, we describe the algebraic structure of projections related to non-free group actions. This approach highlights the role of the underlying mathematical structures, and provides insight useful both for software construction and numerical analysis. The theory is illustrated with a selection of numerical examples.

CONNECTIONS FOR GENERAL GROUP ACTIONS

We introduce the notion of a partial connection, which extends to manifolds with nonfree group actions most of the key features of connections on principal bundles. We investigate several properties of curvature in this setting and show that (partial) moving frames determine (partial) connection forms.

Is There a Core Class for Almost Free Groups of Size ℵ 1 ?#

We discuss the prospects for finding a “core class,” i.e., a well-behaved class of non-free abelian groups of cardinality ℵ1 such that every non-free abelian group of cardinality ℵ1 has a subgroup in the core class.

Almost free groups and long Ehrenfeucht–Fraı̈ssé games

An Abelian group G is strongly λ-free iff G is L ∞, λ -equivalent to a free Abelian group iff the isomorphism player has a winning strategy in an Ehrenfeucht–Fraı̈ssé game of length ω between G and a free Abelian group. We study possible longer Ehrenfeucht–Fraı̈ssé games between a nonfree group and a free Abelian group. A group G is called ε-game-free if the isomorphism player has a winning strategy in an Ehrenfeucht–Fraı̈ssé game of length ε between G and a free Abelian group. We prove in ZFC existence of nonfree ε-game-free groups for many successors of regular cardinals. We also show that the length of the game obtained is very close to the optimal length provable in ZFC alone. On the other hand, assuming existence of a Mahlo cardinal, we sketch a proof that it is consistent to have a very highly game-free still nonfree group. First we present an introduction to basic constructions and then we introduce some results concerning the standard tools for building more complicated groups, namely transversals and λ-systems. It follows that all the constructions generalize to algebras in a fixed variety satisfying the strong construction principle.

Persistence of Hamiltonian relative periodic orbits

We prove a persistence result for Hamiltonian relative periodic orbits with generic drift–momentum pairs in the case of non-compact non-free group actions. Our starting point is a relative periodic orbit which is non-degenerate modulo isotropy. We show that the analysis of the persistence problem involves the study of a singular algebraic variety, the space of drift–momentum pairs, which is determined solely by the symmetry group of the problem. We apply our results to relative periodic solutions of deformable bodies in fluids.

Persistence of relative equilibria in Hamiltonian systems with non-compact symmetry

We prove new results on the persistence of Hamiltonian relative equilibria with generic velocity–momentum pairs in the case of non-compact non-free group actions and taking into account time reversibility. Our starting point is a relative equilibrium which is non-degenerate modulo isotropy which, in the case of a generic momentum implies persistence of the given relative equilibrium to all nearby momentum values with the same isotropy. We show that the analysis of the persistence problem involves the study of a singular algebraic variety which is determined solely by the symmetry group of the problem. We present persistence results for relative equilibria with velocity–momentum pairs which are regular points of this variety and give sufficient conditions for a velocity–momentum pair to be regular. We apply our results to relative equilibria of Euclidean equivariant systems, including models of rigid bodies in fluids.