Conjugacy invariant pseudo-norms, representability and RNA secondary structures

To a reasonable approximation, a secondary structures of RNA is determined by Watson-Crick pairing without pseudo-knots in such a way as to minimise the number of unpaired bases: We show that this minimal number is determined by the maximal conjugacy-invariant pseudo-norm on the free group on two generators subject to bounds on the generators. This allows us to construct lower bounds on the minimal number of unpaired bases by constructing conjugacy invariant pseudo-norms. We show that one such construction, based on isometric actions on metric spaces, gives a sharp lower bound. A major goal here is to formulate a purely mathematical question, based on considering orthogonal representations, which we believe is of some interest independent of its biological roots.

Diassociativity in Conjugacy Closed Loops

Let Q be a conjugacy closed loop, and N(Q) its nucleus. Then Z(N(Q)) contains all associators of elements of Q. If in addition Q is diassociative (i.e., an extra loop), then all these associators have order 2. If Q is power-associative and |Q| is finite and relatively prime to 6, then Q is a group. If Q is a finite non-associative extra loop, then 16 ∣ |Q|.

The conjugacy problem for two-by-two matrices over polynomial rings

We give an effective solution of the conjugacy problem for two-by-two matrices over the polynomial ring in one variable over a finite field.

Twisted conjugacy classes in nilpotent groups

A group is said to have the R(infinity) property if every automorphism has an infinite number of twisted conjugacy classes. We study the question whether G has the R(infinity) property when G is a finitely generated torsion-free nilpotent group. As a consequence, we show that for every positive integer n >= 5, there is a compact nilmanifold of dimension n on which every homeomorphism is isotopic to a fixed point free homeomorphism. As a by-product, we give a purely group theoretic proof that the free group on two generators has the R(infinity) property. The R(infinity) property for virtually abelian and for C-nilpotent groups are also discussed.

TWISTED CONJUGACY CLASSES IN R. THOMPSON`S GROUP F

In this article, we prove that any automorphism of R. Thompson`s group F has infinitely many twisted conjugacy classes. The result follows from the work of Brin, together with standard facts about R. Thompson`s group F, and elementary properties of the Reidemeister numbers.

The classification and the conjugacy classes of the finite subgroups of the sphere braid groups

Let n >= 3. We classify the finite groups which are realised as subgroups of the sphere braid group B(n)(S(2)). Such groups must be of cohomological period 2 or 4. Depending on the value of n, we show that the following are the maximal finite subgroups of B(n)(S(2)): Z(2(n-1)); the dicyclic groups of order 4n and 4(n - 2); the binary tetrahedral group T*; the binary octahedral group O*; and the binary icosahedral group I(*). We give geometric as well as some explicit algebraic constructions of these groups in B(n)(S(2)) and determine the number of conjugacy classes of such finite subgroups. We also reprove Murasugi`s classification of the torsion elements of B(n)(S(2)) and explain how the finite subgroups of B(n)(S(2)) are related to this classification, as well as to the lower central and derived series of B(n)(S(2)).

SIGMA THEORY AND TWISTED CONJUGACY CLASSES

Using Sigma theory we show that for large classes of groups G there is a subgroup H of finite index in Aut(G) such that for phi is an element of H the Reidemeister number R(phi) is infinite. This includes all finitely generated nonpolycyclic groups G that fall into one of the following classes: nilpotent-by-abelian groups of type FP(infinity); groups G/G `` of finite Prufer rank; groups G of type FP(2) without free nonabelian subgroups and with nonpolycyclic maximal metabelian quotient; some direct products of groups; or the pure symmetric automorphism group. Using a different argument we show that the result also holds for 1-ended nonabelian nonsurface limit groups. In some cases, such as with the generalized Thompson`s groups F(n,0) and their finite direct products, H = Aut(G).

Twisted conjugacy classes in symplectic groups, mapping class groups and braid groups (with an appendix written jointly with Francois Dahmani)

We prove that the symplectic group Sp(2n, Z) and the mapping class group Mod(S) of a compact surface S satisfy the R(infinity) property. We also show that B(n)(S), the full braid group on n-strings of a surface S, satisfies the R(infinity) property in the cases where S is either the compact disk D, or the sphere S(2). This means that for any automorphism phi of G, where G is one of the above groups, the number of twisted phi-conjugacy classes is infinite.

Divergent diagrams of folds and simultaneous conjugacy of involutions

In this work we show that the smooth classification of divergent diagrams of folds (f(1),..., f(s)) : (R-n, 0) -> (R-n x(...)xR(n), 0) can be reduced to the classification of the s-tuples (p(1)., W) of associated involutions. We apply the result to obtain normal forms when s <= n and {p(1),...,p(s)} is a transversal set of linear involutions. A complete description is given when s = 2 and n >= 2. We also present a brief discussion on applications of our results to the study of discontinuous vector fields and discrete reversible dynamical systems.

LIKELIHOOD ESTIMATION OF CONJUGACY RELATIONSHIPS IN LINEAR MODELS WITH APPLICATIONS TO HIGH-THROUGHPUT GENOMICS

In the simultaneous estimation of a large number of related quantities, multilevel models provide a formal mechanism for efficiently making use of the ensemble of information for deriving individual estimates. In this article we investigate the ability of the likelihood to identify the relationship between signal and noise in multilevel linear mixed models. Specifically, we consider the ability of the likelihood to diagnose conjugacy or independence between the signals and noises. Our work was motivated by the analysis of data from high-throughput experiments in genomics. The proposed model leads to a more flexible family. However, we further demonstrate that adequately capitalizing on the benefits of a well fitting fully-specified likelihood in the terms of gene ranking is difficult.

Groups with Restricted Conjugacy Classes

Let F C 0 be the class of all finite groups, and for each nonnegative integer n define by induction the group class FC^(n+1) consisting of all groups G such that for every element x the factor group G/CG ( ^G ) has the property FC^n . Thus FC^1 -groups are precisely groups with finite conjugacy classes, and the class FC^n obviously contains all finite groups and all nilpotent groups with class at most n. In this paper the known theory of FC-groups is taken as a model, and it is shown that many properties of FC-groups have an analogue in the class of FC^n -groups.

Splittings of free groups, normal forms and partitions of ends

Splittings of a free group correspond to embedded spheres in the 3-manifold M = # (k) S (2) x S (1). These can be represented in a normal form due to Hatcher. In this paper, we determine the normal form in terms of crossings of partitions of ends corresponding to normal spheres, using a graph of trees representation for normal forms. In particular, we give a constructive proof of a criterion determining when a conjugacy class in pi (2)(M) can be represented by an embedded sphere.

电离层行进式扰动的GPS台网监测研究

In this dissertation, we investigated two types of traveling ionospheric disturbances (TIDs)/gravity waves (GWs) triggered separately by auroral energy input during super geomagnetic storms and solar terminator (ST) under quiet geomagnetic conditions (kp<3+) using TEC measurements from the global network of GPS receivers. Research into the generation and propagation of TIDs/GWs during storms greatly enhance our understandings on the evolution processes of energy transportation from the high-latitude’s magnetosphere to the low-latitude ionosphere and the conjugated effect of TIDs propagation between the northern and southern hemispheres. Our results revealed that the conjugacy of propagation direction between the northern and southern hemispheres was subject to the influence of Coriolis force. We also figure out the evolution processes of ionospheric disturbances at the global scale. These are important topics that had not been well addressed previously. In addition, we also obtained thee wave structures of medium scale TIDs excited by the solar terminator (ST) moving over the northern America and physical mechanisms involved. Our observations confirm that the ST is a stable and repetitive source of ionospheric wave disturbances and the evidence of solar terminator generated disturbances has been demonstrated experimentally via the GPS TEC measurement. The main researches and results of this dissertation are as follows. First, the global traveling ionospheric disturbances (TIDs) during the drastic magnetic storms of October 29–31, 2003 were analyzed using the Global Position System (GPS) total electron content (TEC) data observed in the Asian-Australian, European and North American sectors. We collected the most comprehensive set of the TEC data from more than 900 GPS stations on the International GNSS Services (IGS) website and introduce here a strategy that combines polynomial fitting and multi-channel maximum entropy spectral analysis to obtain TID parameters. Moreover, in collaboration with my thesis advisor, I have developed an imaging technique of 2-dimensional map of TIDs structures to obtain spatial and temporal maps of large scale traveling ionospheric disturbances (LSTIDs). The clear structures of TEC perturbations map during the passage of TIDs were displayed. The results of our study are summarized as follows: (1) Large-scale TIDs (LSTIDs) and medium-scale TIDs (MSTIDs) were detected in all three sectors after the sudden commencement (SC) of the magnetic storm, and their features showed longitudinal and latitudinal dependences. The duration of TIDs was longer at higher latitudes than at middle latitudes, with a maximum of about 16 h. The TEC variation amplitude of LSTIDs was larger in the North American sector than in the two other sectors. At the lower latitudes, the ionospheric perturbations were more complicated, and their duration and amplitude were relatively longer and larger. (2) The periods and phase speeds of TIDs were different in these three sectors. In Europe, the TIDs propagated southward; in North America and Asia, the TIDs propagated southwestward; in the near-equator region, the disturbances propagated with the azimuth (the angle of the propagation direction of the LSTIDs measured clockwise from due north with 0°) of 210° showing the influence of Coriolis force; in the Southern Hemisphere, the LSTIDs propagated conjugatedly northwestward. Both the southwestward and northeastward propagating LSTIDs are found in the equatorial region. These results mean that the Coriolis effect cannot be ignored for the wave propagation of LSTIDs and that the propagation direction is correlated with the polar magnetic activity. (3) The day (day of year: 301) before the SC (sudden commencement) of magnetic storm, we observed a sudden TEC skip disturbances (±10 TECU). It should be a response for the high flux of proton during the solar flare event, but not the magnetic storms. Next, the most comprehensive and dense GPS network’s data from North-America region were used in this paper to analyze the medium scale traveling ionospheric disturbances (MSTIDs) which were generated by the moving solar terminator during the quiet days in 2005. We applied the multi-channel maximum entropy spectral analysis to calculated TID parameters, and found that the occurrence of ST-MSTIDs depends on the seasonal variations. The results of our study are summarized as follows: (1) MSTIDs stimulated by the moving ST (ST-MSTIDs) are detected at mid-latitudes after the passage of the solar terminator with the life time of 2～3 hours and the variation amplitude of 0.2～0.8 TECU. Spectral analysis indicated that the horizontal wavelength, average period, horizontal phase velocity of the MSTIDs are around 300±150 km，150±80 m/s and 25±15 min, respectively. In addition, ST-MSTIDs have wave fronts elongating the moving ST direction and almost parallel to ST. (2) The statistical results demonstrate that the dusk MSTIDs stimulated by ST is more obvious than the dawn MSTIDs in summer. On the contrary, the more-pronounced dawn MSTIDs occurs in winter. (3) Further analysis indicates that the seasonal variations of ST-MSTIDs occurrence frequency are most probably related to the seasonal differences of the variations of EUV flux in the ionosphere region and recombination process during sunrise and sunset period at mid-latitudes. Statistical study of occurrence characteristics of TIDs using the GPS network in North-American and European during solar maximum, In conclusion, statistical studies of the propagation characteristics of TIDs, which excited by the two common origins including geomagnetic storms and moving solar terminator, were involved with global GPS TEC databasein this thesis. We employed the multichannel maximum entropy spectral analysis method to diagnose the characteristics of propagation and evolvement of ionospheric disturbances, also, the characteristics of their regional distribution and climatological variations were revealed by the statistic analysis. The results of these studies can improve our knowledge about the energy transfer in the solar-terrestrial system and the coupling process between upper and lower atmosphere (thermosphere-ionosphere-mesosphere). On the other hand, our results of the investigation on TIDs generated by particular linear origin such as ST are important for developing ionospheric irregularity physics and modeling the transionosphere radio wave propagation. Besides, the GPS TEC representation of the ST-generated ionospheric structure suggests a better possibility for investigating this phenomenon. Subsequently, there are scientific meaning of the result of this dissertation to deeply discuss the energy transfer and coupling in the ionosphere, as well as realistic value to space weather forecast in the ionosphere region.

Lognormal and gamma mixed negative binomial regression

In regression analysis of counts, a lack of simple and efficient algorithms for posterior computation has made Bayesian approaches appear unattractive and thus underdeveloped. We propose a lognormal and gamma mixed negative binomial (NB) regression model for counts, and present efficient closed-form Bayesian inference; unlike conventional Poisson models, the proposed approach has two free parameters to include two different kinds of random effects, and allows the incorporation of prior information, such as sparsity in the regression coefficients. By placing a gamma distribution prior on the NB dispersion parameter r, and connecting a log-normal distribution prior with the logit of the NB probability parameter p, efficient Gibbs sampling and variational Bayes inference are both developed. The closed-form updates are obtained by exploiting conditional conjugacy via both a compound Poisson representation and a Polya-Gamma distribution based data augmentation approach. The proposed Bayesian inference can be implemented routinely, while being easily generalizable to more complex settings involving multivariate dependence structures. The algorithms are illustrated using real examples. Copyright 2012 by the author(s)/owner(s).

Cantor exchange systems and renormalization

We prove a one-to-one correspondence between (i) C1+ conjugacy classes of C1+H Cantor exchange systems that are C1+H fixed points of renormalization and (ii) C1+ conjugacy classes of C1+H diffeomorphisms f with a codimension 1 hyperbolic attractor Lambda that admit an invariant measure absolutely continuous with respect to the Hausdorff measure on Lambda. However, we prove that there is no C1+alpha Cantor exchange system, with bounded geometry, that is a C1+alpha fixed point of renormalization with regularity alpha greater than the Hausdorff dimension of its invariant Cantor set.