on groups of central type a finite group is said to be of central type if it possesses an irreducible complex character which takes the value zero on all noncentral elements equivalently the degree of this character is the square root of the index of the center in 1964 iwahori and matsumoto 15 conjectured that a group of central type must be solvable the paper of liebler and yellen 16 aims to prove this but there is a gap in their proof as explained below nevertheless they do correctly obtain substantial information about a minimal counterexample to the conjecture the aim of this paper is to fill the gap in the liebler yellen proof using as they do the classification of finite simple groups and to provide further information about which solvable groups can have central type we show for instance that every normal subgroup of the central factor group of a solvable group of central type has the property that its system normalizers have square index it is interesting to contrast this restrictive condition with the result of gagola that every solvable group is embeddable in the central factor group of a group of central type see 6 theorem 1 2 the arguments in 16 show that a minimal counterexample g to the iwahori matsumoto conjecture must have the following structure if k z is any minimal normal subgroup of g z where z z g the center then k is abelian k z is a 2 group s cg k is the maximal solvable normal subgroup of g and any minimal normal subgroup his of g s is the direct product of a number l of copies of gl 3 2 indeed the liebler yellen methods can be used to show that the sylow 2 subgroup of k considered as an h s module is a product o f copies of a gl 3 2 module of dimension 4 this module having a unique minimal submodule which is trivial and an irreducible factor module of dimension 3 the minimal submodules are amalgamated in the product

a discussion of testing dmus returns to scale abstract the issue of using banker s most productive scale size mpss concept to characterize dmus returns to scale rts is studied in this paper it is shown that the mpss concept can always be used to estimate rts without any adjustments unless a set of efficient dmus exhibit linear dependency i e it is the dmu itself that causes the mpss concept not to work the mpss concept itself is independent of assuming a linear production function in the ccr model as a result chang and ghu s conclusions are erroneous also several other errors in chang and ghu s proof and illustration of proof are listed the linear dependency condition in the current paper corresponds to the nonunique lambda solution situation in banker and thrall meanwhile a remedy is developed for the mpss concept so that it can still work under linear dependency it is also shown that those dmus which naturally exhibit increasing and decreasing returns to scale have nothing to do with linear dependency

using input output orientation model for determining most productive scale size in dea one of the existing approaches for determining mpss is email protected model proposed in cooper thompson and thrall s article annals of operations research 66 1996 3 the direct solving of that model is rather difficult because the objective function is fractional hence in this paper besides introducing a model with output input orientation the previous model difficulties will be removed because the new objective function is linear and also it is proved that both models are the same in determining mpss furthermore by solving this model it is not necessary to scaling the data as it is necessary in ccr model so that figurative dmu move in the joint frontier of bcc and ccr models see introduction to the theory and applications of data envelopment analysis a foundation text with integrated software 2001 european journal of operational research 17 1984 35 so we can consider this as another merit for this model in determining mpss we obtain the smallest and the most mpss corresponding with the evaluating dmu and also we present necessary and sufficient condition for boundedness of this model by dual model and numerical example confirms validity of this model as a means for determining mpss

moments and distribution of central l values of quadratic twists of elliptic curves we show that if one can compute a little more than a particular moment for some family of l functions then one has upper bounds of the conjectured order of magnitude for all smaller positive real moments and a one sided central limit theorem holds we illustrate our method for the family of quadratic twists of an elliptic curve obtaining sharp upper bounds for all moments below the first we also establish a one sided central limit theorem supporting a conjecture of keating and snaith our work leads to a conjecture on the distribution of the order of the tate shafarevich group for rank zero quadratic twists of an elliptic curve and establishes the upper bound part of this conjecture assuming the birch swinnerton dyer conjecture

counting objects which behave like irreducible brauer characters of finite groups let n 0 g where g is a finite group so that g acts on irr n the set of irreducible ordinary complex characters of n suppose 8 e irr n is g invariant and write irr g 10 to denote the set of those x e irr g such that x involves and hence is a multiple of 8 how large is irr g io the answer to this question is well known and not very hard to prove analogously to the fact that lirr g i is equal to the number of conjugacy classes of g it turns out that iirr gjb l is equal to the number of certain good classes in the group g n there are two equivalent descriptions of which classes of g n these are one characterization depends on a certain complex twisted group algebra of g n and in this formulation the result which counts irr g 1 t9 is due to i schur s the other formulation of this result due to p x gallagher z defines goodness internally to g it depends on the character theory of g and its subgroups specifically gallagher defines an element of 2 e g n to be o good if some extension of 8 to n g is invariant in c where c n c jg the property of being q good is preserved by conjugacy in gin and gallagher shows that lirr g i 13 l is equal to the number of classes of g in which consist of good elements somewhat less well known is the corresponding result for irreducible brauer characters with respect to some prime p if 8 e ibr n is g invariant we intend to count the elements of ibr g t9 i e the irreducible brauer characters of g whose restrictions to n are multiples of 8 in view of the fact that iibr g equals the number of p regular classes of g it is reasonable to guess that iibr gl f l is the number of e good p regular

on fourier coefficients of modular forms of half integral weight at squarefree integers we show that the dirichlet series associated to the fourier coefficients of a half integral weight hecke eigenform at squarefree integers extends analytically to a holomorphic function in the half plane re s tfrac 1 2 this exhibits a high fluctuation of the coefficients at squarefree integers

estimation of returns to scale using data envelopment analysis abstract generalization of the measure of returns to scale from a single number to an interval permits extension of the concept to dea data domains with multiple inputs and multiple outputs the key new approach is a partition of the optimal frontier into three parts corresponding respectively to increasing constant and decreasing returns to scale these parts are characterized in terms of optimal primal solutions and optimal dual solutions for both the original charnes cooper rhodes model 1978 and the later banker charnes cooper model 1984 and relying on concepts developed by r d banker 1984 and r m thrall 1988

measure complexity and mobius disjointness abstract in this paper the notion of measure complexity is introduced for a topological dynamical system and it is shown that sarnak s mobius disjointness conjecture holds for any system for which every invariant borel probability measure has sub polynomial measure complexity we then apply this result to a number of situations including certain systems whose invariant measure don t all have discrete spectrum

on fully ramified brauer characters let z be a normal subgroup of a finite group let p 5 be a prime and let ibr z be an irreducible g invariant p brauer character of z suppose that g e for some ibr g then g z is solvable in other words a twisted group algebra over an algebraically closed field of characteristic not 5 with a unique class of simple modules comes from a solvable group

fourier uniformity of bounded multiplicative functions in short intervals on average let lambda denote the liouville function we show that as x rightarrow infty begin aligned int x 2x sup alpha left sum x 0 fixed but arbitrarily small previously this was only known for theta 5 8 for smaller values of theta this is the first non trivial case of local fourier uniformity on average at this scale we also obtain the analogous statement for non pretentious 1 bounded multiplicative functions we illustrate the strength of the result by obtaining cancellations in the sum of lambda n lambda n h lambda n 2h over the ranges h x theta and n x and where lambda is the von mangoldt function

data envelopment analysis in the absence of convexity the first purpose of this study is to present new dea data envelopment analysis approaches that distinguish the efficiency status efficient or inefficient of decision making units dmus and estimate right and left hand returns to scale rts of efficient dmus in free disposal hull fdh models the second one is to estimate most productive scale size mpss in fdh models the proposed rts method has three preferences with respect to the previous rts methods in this context first one is that it is capable of identifying all fdh efficient dmus whereas the previous methods may not be able to specify all fdh efficient dmus besides the second one is to estimate right and left hand rts of fdh efficient dmus while the former methods are only capable of estimating rts of fdh efficient dmus moreover the third one is that it is always able to estimate right and left hand rts correctly whereas the previous methods are sometimes capable of estimating rts correctly these issues are the motivations for creating this current research finally a numerical example and an empirical application are exhibited for illustrating and comparing the proposed method with the former methods then some consequences are provided based upon the obtained results and also some suggestions for future works are presented proofs of some theorems and lemmas are given in the appendix a dea method for identifying fdh efficient dmus two dea models for the estimation of right and left rts of fdh efficient dmus a dea method for determining mpss in fdh models presentation of two examples to illustrate the use of our proposed approach making comparisons among our rts method and other rts methods in fdh models

on constructing primitive roots in finite fields with advice finding primitive roots in a finite field of p n elements of characteristic p remains to be a hard computational problem with the bottlenecks coming from both locating a small set of possible candidates and also factoring p n 1 in order to test these candidates kopparty et al 2016 have introduced a question of designing a fast algorithm to find primitive roots with a short advice from an oracle trivially for an m bit prime p such a primitive root can be fully described by about mn bits of information received from an all powerful oracle here we have shown that one can achieve this in polynomial time and with about 1 2 o 1 m o log n bits of advice

bombieri vinogradov for multiplicative functions and beyond the x1 2 barrier abstract part and parcel of the study of multiplicative number theory is the study of the distribution of multiplicative functions in arithmetic progressions although appropriate analogies to the bombieri vingradov theorem have been proved for particular examples of multiplicative functions there has not previously been headway on a general theory seemingly none of the different proofs of the bombieri vingradov theorem for primes adapt well to this situation in this article we find out why such a result has been so elusive and discover what can be proved along these lines and develop some limitations for a fixed residue class a we extend such averages out to moduli x 20 39

correlations of the von mangoldt and higher divisor functions ii divisor correlations in short ranges we study the problem of obtaining asymptotic formulas for the sums sum x n le 2x d k n d l n h and sum x n le 2x lambda n d k n h where lambda is the von mangoldt function d k is the k text th divisor function x is large and k ge l ge 2 are integers we show that for almost all h in h h with h log x 10000 k log k the expected asymptotic estimate holds in our previous paper we were able to deal also with the case of lambda n lambda n h and we obtained better estimates for the error terms at the price of having to take h x 8 33 varepsilon

above the glauberman correspondence let s be a finite solvable group and suppose s acts on the finite group n and they have coprime orders then the celebrated glauberman correspondence provides a natural bijection from the set irrs n of irreducible characters of n which are invariant under the action of s to the set irr cn s of all irreducible characters of the centralizer of s in n suppose further that the semidirect product sn is a normal subgroup of a finite group g let irrs n and let irr cn s be its glauberman correspondent we prove that there is a bijection with good compatibility properties from the set irr g of the irreducible characters of g above to irr ng s such that in the case when s is a p group for some prime p it preserves fields of values and schur indices over qp the field of p adic numbers using this result we also prove a strengthening of the mckay conjecture for all p solvable groups

an extension to data envelopment analysis with preference structure for estimating overall inefficiency this paper extends data envelopment analysis dea with preference structure by fully considering the substitution effects among different inputs or outputs when the unit cost and price information on inputs and outputs are available the generalized weighted ccr gwccr models proposed in this paper can provide some scalar values for measuring the overall inefficiency it is found that the gwccr models focus on the relative aspects of overall inefficiency instead of the absolute aspects focused on by the weighted additive dea model

lois locales de la fonction ω dans presque tous les petits intervalles for k 1 an integer and x 1 a real number let k x be the number of integers smaller than x having exactly k distinct prime divisors building on recent work of matomaki and radziwill we investigate the asymptotic behavior of k x h k x for almost all x when h is very small we obtain optimal results for k log2x and close to optimal results for 5 k log2x our method also applies to y friable integers in almost all intervals x x h when logx logy logx 1 6 e