the index of an eisenstein ideal and multiplicity one mazur s fundamental work on the eisenstein ideal of prime level has a variety of arithmetic applications in this article we generalize some of his work to square free level more specifically we attempt to compute the index of an eisenstein ideal and the dimension of the mathfrak m torsion of the modular jacobian variety where mathfrak m is an eisenstein maximal ideal in many cases the dimension of the mathfrak m torsion is 2 in other words a multiplicity one theorem holds

global dynamics for the simplified lorenz system model in this paper the global attractive sets of the simplified lorenz system are studied according to lyapunov stability theory the method for constructing the lyapunov functions that applied to the former chaotic systems does not work for the simplified lorenz system we overcome this difficulty by adding a cross term to the lyapunov functions of the simplified lorenz system

on differences and similarities in the analysis of lorenz chen and lu systems currently it is being actively discussed the question of the equivalence of various lorenz like systems and the possibility of universal consideration of their behavior algaba et al 2013a b 2014b c chen 2013 chen and yang 2013 leonov 2013a in view of the possibility of reduction of such systems to the same form with the help of various transformations in the present paper the differences and similarities in the analysis of the lorenz the chen and the lu systems are discussed it is shown that the chen and the lu systems stimulate the development of new methods for the analysis of chaotic systems open problems are discussed

modular elliptic curves and fermat s last theorem when andrew john wiles was 10 years old he read eric temple bell s the last problem and was so impressed by it that he decided that he would be the first person to prove fermat s last theorem this theorem states that there are no nonzero integers a b c n with n 2 such that an bn cn the object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular fermat s last theorem follows as a corollary by virtue of previous work by frey serre and ribet

non optimal levels of a reducible mod l modular representation let 3 be a prime and n be a square free integer prime to for each prime p dividing n let ap be either 1 or 1 we give a sufficient criterion for the existence of a newform f of weight 2 for 0 n such that the mod galois representation attached to f is reducible and upf apf for primes p dividing n the main techniques used are level raising methods based on an exact sequence due to ribet

cuspidal class number of the tower of modular curves x1 npn we consider a cuspidal class number which is the order of a subgroup of the full cuspidal divisor class group of x1 npn with p nmid n and n 1 by studying the second generalized bernoulli numbers we obtain results similar to ones ferrero and washington in ann math 2 109 2 377 395 1979 washington in invent math 49 87 97 1978 about the relative class numbers of cyclotomic mathbb z p extension of an abelian number field

ordinary representations and modular forms this paper establishes criteria for a representation gal q q gl2 qp to be modular in the sense that there is a newform f such that f fontaine and mazur have conjectured that is modular provided it is unramified outside of a finite set of primes irreducible has odd determinant and the restriction to a decomposition group dp at p satisfies certain conditions the following conjecture is a special case of fm95 ref 1 conjecture 3c

on prediction and the power transformation family summary the power transformation family is often used for transforming to a normal linear model the variance of the regression parameter estimators can be much larger when the transformation parameter is unknown and must be estimated compared to when the transformation parameter is known we consider prediction of future untransformed observations when the data can be transformed to a linear model when the transformation must be estimated the prediction error is not much larger than when the parameter is known

an analysis of transformations revisited rebutted abstract in a paper written in 1964 box and cox described a method for estimating transformations and showed how in suitable cases valuable increases in simplicity and efficiency were possible since that time this technique has enjoyed wide practical use and considerable success however a recent theoretical paper by bickel and doksum 1981 seems to suggest that serious dangers are associated with the employment of this method and speaks of instability and cost of estimation of the transformation these difficulties seem to be associated with 1 examples which common sense would rule out namely situations where the effect of transformation on the data is almost linear so that it is a matter of indifference which transformation is used 2 the idea that it makes sense to state conclusions in terms of a number measured on an arbitrary scale 3 failure to take proper account of the jacobian of the transformation

the analysis of transformed data abstract recently it was suggested bickel and doksum 1981 that when data are used to select a transformation the post transformation analysis of those data may need to be modified considerably from standard form so as to allow for the selection we argue that common sense and the work of box and cox 1964 point to a contrary conclusion our argument is based on considerations of parameter interpretation and subsequent bayesian analysis within the context of fitting normal error linear models numerical examples are used to illustrate the main points

the influence of variable selection a bayesian diagnostic perspective abstract variable selection is ubiquitous in statistical practice forward selection backwards elimination or a combination thereof are among the most popular techniques statistical texts teach direct interpretation of coefficients after variable selection for scientific purposes the desired interpretation is unconditional on the variable selection in this article an influence analysis of variable selection is performed from a bayesian viewpoint variable selection is shown to be surprisingly influential a new statistic is recommended for routine examination during variable selection

deformations of certain reducible galois representations ii we examine the deformation theory of two dimensional mod p reducible indecomposable galois representations removing or relaxing many of the hypotheses of part i we are able to prescribe local conditions on our deformations by allowing ramification at a set of primes congruent to 1 mod p not 1 mod p 2 and satisfying some other splitting conditions the new hypotheses admit irreducible ordinary characteristic zero lifts of many mod p representations which by skinner wiles allows us to conclude the modularity of such representations

a hyperchaos generated from lorenz system abstract this paper presents a four dimension hyperchaotic lorenz system obtained by adding a nonlinear controller to lorenz chaotic system the hyperchaotic lorenz system is studied by bifurcation diagram lyapunov exponents spectrum and phase diagram numerical simulations show that the new system s behavior can be convergent divergent periodic chaotic and hyperchaotic when the parameter varies

approximate inference in location scale regression models abstract it is shown that 2 and multivariate t distributions provide good approximations to posterior distributions arising in location scale regression models in the case of a linear regression model bayesian inferences based on an improper prior have conditional frequency as well as fiducial structural interpretation so the results are also applicable in a non bayesian context numerical support for the approximations is presented for the case of a weibull regression model and a regression with t distributed errors

treatment effects in a logistic model involving the box cox transformation abstract we consider a logistic model for binary response data that allows the possibility of power transformation of x that is log p 1 p x i where x is a continuous variable x is the box cox transformation and i is a binary variable indicating treatment or group this model is applicable to observational studies or randomized trials when a treatment effect is investigated after controlling for a confounding variable x our focus is on inference concerning the treatment effect in the analysis a common approach might be to treat the estimated value of as fixed and ignore uncertainty associated with its estimation in inference about alternatively we might perform an unconditional analysis in which is regarded as a parameter we show that under the null hypothesis 0 these two approaches are asymptotically equivalent if the two groups have the same distribution of x and the same sample size this result also holds for the situation of multiple covariates each wit

hidden attractor in smooth chua systems abstract the hidden oscillations a basin of attraction of which does not contain neighborhoods of equilibria have been obtained first in the 50 60s of the 20th century in automatic control systems with scalar piecewise linear nonlinearity this brings up the question about the excitation nature of hidden oscillations in the present paper it is shown that hidden oscillations can exist not only in systems with piecewise linear nonlinearity but also in smooth systems here the possibility of the existence of a hidden chaotic attractor in a modified chua s system with a smooth characteristic of nonlinear element is demonstrated

strong modularity of reducible galois representations in this paper we call strongly modular those reducible semi simple odd mod l galois representations for which the conclusion of the strongest form of serre s original modularity conjecture holds under the assumption that the serre weight k satisfies l k 1 we give a precise characterization of strongly modular representations hence generalizing a classical theorem of ribet pertaining to the case of conductor 1 when the representation rho is not strongly modular we give a necessary and sufficient condition on the primes p not dividing nl for which it arises in level np where n denotes the conductor of rho this generalizes a result of mazur on the case n k 1 2

the cost of generalizing logistic regression abstract binary response regression models in which the link function is a family defined by one or more unknown shape parameters are considered detailed attention is given to the two single parameter families proposed by aranda ordaz 1981 that incorporate the logistic linear and complementary log log link functions as special cases one model is a symmetric family of alternatives to the logistic shape and the second model is an asymmetric family the increase in variance of quantities of interest due to the addition of an extra parameter is calculated using asymptotic methods this inflation in variance is interpreted as the cost of adding an additional parameter to the model in considering the seriousness of this cost the reduction in bias due to the estimation of the additional parameter should also be taken into account three quantities of interest predicted probabilities for a fixed value of the explanatory variable predicted value of the explanatory variable to obtain a specific probability

on deformation rings of residually reducible galois representations and r t theorems we introduce a new method of proof for r t theorems in the residually reducible case we study the crystalline universal deformation ring r and its ideal of reducibility i of a mod p galois representation 0 of dimension n whose semisimplification is the direct sum of two absolutely irreducible mutually non isomorphic constituents 1 and 2 under some assumptions on selmer groups associated with 1 and 2 we show that r i is cyclic and often finite using ideas and results of but somewhat different assumptions from bellaiche and chenevier we prove that i is principal for essentially self dual representations and deduce statements about the structure of r using a new commutative algebra criterion we show that given enough information on the hecke side one gets an r t theorem we then apply the technique to modularity problems for 2 dimensional representations over an imaginary quadratic field and a 4 dimensional representation over q

modularity of residual galois extensions and the eisenstein ideal for a totally real field f a finite extension f of fp and a galois character gf f unramified away from a finite set of places p p consider the bloch kato selmer group h h1 f 1 in an earlier paper of the authors it was proved that the number d of isomorphism classes of non semisimple reducible residual representations giving rise to lines in h which are modular by some f also unramified outside satisfies d n dimfh this was proved under the assumption that the order of a congruence module is greater than or equal to that of a divisible selmer group we show here that if in addition the relevant local eisenstein ideal j is non principal then d n when f q we prove the desired bounds on the congruence module and the selmer group we also formulate a congruence condition implying the non principality of j that can be checked in practice allowing us to furnish an example where d n

the retransformed mean after a fitted power transformation abstract an approximate method is given to estimate the mean of a dependent variable after a linear model is fitted to the box cox power transformation of this variable the estimate is accurate except when the transformation power parameter is near zero the properties of the estimate in the one sample and regression cases are considered by both asymptotic calculations and monte carlo simulations and comparisons are made with the smearing estimate duan 1983 it is shown that there can be some cost due to estimating the power transformation as opposed to assuming it is known however this cost is not severe

bayesian criterion based model assessment for categorical data we propose a general bayesian criterion for model assessment for categorical data called the weighted l measure which is constructed from the posterior predictive distribution of the data the measure is based on weighting the observations according to the sampling variance of their future response vector the weight component in the weighted l measure plays the role of a penalty term in the criterion in which a greater weight assigned to covariate values implies a greater penalty term on the dimension of the model a detailed justification is provided for such a weighting procedure and several theoretical properties of the weighted l measure are presented for a wide variety of discrete data models for these models we examine properties of the weighted l measure and show that it can perform better than the unweighted l measure in a variety of settings in addition we show that the weighted quadratic loss l measure is more attractive than the unweighted l measure and the deviance loss l measure for categorical data moreover a calibration for the weighted l measure is motivated and proposed which allows us to compare formally the l measure values of competing models a detailed simulation study is presented to examine the performance of the weighted l measure and it is compared to other established model selection methods finally the method is applied to a real dataset using a bivariate ordinal response model

on modular representations of bar q q arising from modular forms where g is the galois group gai i q and f is a finite field of characteristic i 3 suppose that p is modular of level n i e that it arises from a weight 2 newform of level dividing n and trivial nebentypus character then p is an odd representation the matrix p c where c is a complex conjugation in g has eigenvalues 1 1 since 1 and 1 are distinct in f p is absolutely irreducible and has a model over every subfield of f containing the set trace p we assume that f has been chosen so that it is generated by this set assume that p is a prime which exactly divides n we write p i n and restrict p to a decomposition group of g for the prime p view the restriction as a representation p p of gal t p qp we say that p is finite at p 34 or 35 w if there is a finite flat f vector space scheme h over zp for which the action of gal t p qp on the f vector space h 1 p gives pp i f p 9 is finite at p if and only if pp is unramified it is clear that p is finite at p whenever p is already modular of level n p in w of 34 cf 35 serre conjectured the converse i e that if p is finite then p is modular of level n p soon thereafter mazur 1 20 proved this conjecture in the case

on two families of transformations to additivity for binary response data summary two families of power transformations for probabilities are introduced to model symmetric and asymmetric departures from the logistic model the aim is to provide in a comprehensive way a representation of alternative scales in terms of which to carry out the analysis of binary response data maximum likelihood methods are employed to estimate an appropriate transformation using glim tests for detection of symmetric and asymmetric departures are proposed

the cost of adding parameters to a model summary for a general regression model with n independent observations we consider the variance of the estimate of a quantity of interest under two scenarios one scenario is where all the parameters are estimated from the data the other scenario is where a subset of the parameters is assumed known at their true values and the remaining parameters are estimated we focus on quantities of interest which are defined on the scale of the response variable we show that under certain conditions the ratio of a weighted sum across the design points of the variance of the quantity of interest is given by q p where q and p are the number of free parameters in the two scenarios thus in this average sense the inflation in variance associated with adding parameters also interpreted as the cost of adding parameters to a model is directly proportional to the number of parameters we study models involving power transformations non linear models and exponential family models

measures of location of skew distributions obtained through box cox transformations abstract box cox power transformations are used to transform a set of observations to symmetry a measure of location of the original data is obtained by applying the inverse transformation to the center of the transformed data its properties are discussed and it is compared with other measures of location