a quadratic convergence yielding iterative method for the implementation of lavrentiev regularization method for ill posed equations george and elmahdy 2012 considered an iterative method which converges quadratically to the unique solution is a nonlinear monotone operator defined on a real hilbert space x the convergence analysis of the method was based on a majorizing sequence in this paper we are concerned with the problem of expanding the applicability of the method considered by george and elmahdy 2012 by weakening the restrictive conditions imposed on the radius of the convergence ball and also by weakening the popular lipschitz type hypotheses considered in earlier studies such as george and elmahdy 2012 mahale and nair 2009 mathe and perverzev 2003 nair and ravishankar 2008 semenova 2010 and tautanhahn 2002 we show that the adaptive scheme considered by perverzev and schock 2005 for choosing the regularization parameter can be effectively used here for obtaining order optimal error estimate in the concluding section the method is applied to numerical solution of the inverse gravimetry problem

some groups generated by transvection subgroups in the 1960 s j e mclaughlin lo 111 determined all irreducible finite dimensional linear groups generated by transvection subgroups a trans section of a k vector space v is linear map t cp x v t given by vt cp x 2 vqj y for all t e v where xe vand pe v are non zero and xcp 0 v is the dual space of v a k transvection subgroup t cp x consists of the identity and all r cp x where q and x range over the l dimensional subspaces spanned by cp and x respectively it is isomorphic to the additive group of k mclaughlin showed that the only such groups are the special linear and symplectic groups and in the case when k is f the field with two elements the orthogonal and symmetric groups it is our purpose to give a new proof of a result which includes mclaughlin s we discard the assumptions of finiteness of the dimension of v over k and also to some extent the irreducibility of the group while still obtaining a result which is recognizably like mclaughlin s our result for irreducible groups is the following unexplained terminol ogy will be defined after the statement of the theorem the more com plicated result under a weaker hypothesis is stated near the end of the introduction

gate elimination circuit size lower bounds and sat upper bounds abstract most of the known lower bounds for binary boolean circuits with unrestricted depth are proved by the gate elimination method the most efficient known algorithms for the sat problem on binary boolean circuits use similar case analyses to the ones in gate elimination chen and kabanets recently showed that the known case analyses can also be used to prove average case circuit lower bounds that is lower bounds on the size of approximations of an explicit function in this paper we provide a general framework for proving worst average case lower bounds for circuits and upper bounds for sat that is built on ideas of chen and kabanets a proof in such a framework goes as follows one starts by fixing three parameters a class of circuits a circuit complexity measure and a set of allowed substitutions the main ingredient of a proof goes as follows by going through a number of cases one shows that for any circuit from the given class one can find an allowed substitution such that the given measure of the circuit reduces by a sufficient amount this case analysis immediately implies an upper bound for sat to obtain worst average case circuit complexity lower bounds one needs to present an explicit construction of a function that is a disperser extractor for the class of sources defined by the set of substitutions under consideration we show that many known proofs of circuit size lower bounds and upper bounds for sat fall into this framework using this framework we prove the following new bounds average case lower bounds of 3 24 n and 2 59 n for circuits over u 2 and b 2 respectively though the lower bound for the basis b 2 is given for a quadratic disperser whose explicit construction is not currently known and faster than 2 n sat algorithms for circuits over u 2 and b 2 of size at most 3 24 n and 2 99 n respectively here by b 2 we mean the set of all bivariate boolean functions and by u 2 the set of all bivariate boolean functions except for parity and its complement

simple irreducible subgroups of exceptional algebraic groups a closed subgroup of a semisimple algebraic group is called irreducible if it lies in no proper parabolic subgroup in this paper we classify all irreducible subgroups of exceptional algebraic groups g which are connected closed and simple of rank at least 2 consequences are given concerning the representations of such subgroups on various g modules for example with one exception the conjugacy classes of irreducible simple connected subgroups of rank at least 2 are determined by their composition factors on the adjoint module of g

ill posed problems with unbounded operators abstract let a be a linear closed densely defined unbounded operator in a hilbert space assume that a is not boundedly invertible if eq 1 a u f is solvable and f f then the following results are provided problem f u a u f 2 u 2 has a unique global minimizer u for any f u a a a i 1 f there is a function lim 0 0 such that lim 0 u y 0 where y is the unique minimal norm solution to 1 a priori and a posteriori choices of are given dynamical systems method dsm is justified for eq 1

correlation bounds and sat algorithms for small linear size circuits we revisit the gate elimination method generalize it to prove correlation bounds of boolean circuits with parity and also derive deterministic satisfiability counting algorithms for small linear size circuits let b2 be the full binary basis and let u2 b2 we prove that for circuits over u2 of size 3nn for any constant 0 5 the correlation with parity is at most 2n 1 and there is a sat algorithm which counts the number of satisfying assignments running in time 2nn 1 for circuit size 3nn for 0 the correlation with parity is at most 2 n and there is a sat algorithm running in time 2n n similar correlation bounds and algorithms are also proved for circuits over b2 of size almost 2 5n

extending morphisms from finite to algebraic groups keywords extending morphisms finite groups of lie type algebraic groups indecomposable representations maximal subgroups reference ctg article 1990 001doi 10 1016 0021 8693 90 90195 t record created on 2008 12 16 modified on 2017 05 12

dynamical systems method for solving nonlinear equations with monotone operators a version of the dynamical systems method dsm for solving ill posed nonlinear equations with monotone operators in a hilbert space is studied in this paper an a posteriori stopping rule based on a discrepancy type principle is proposed and justified mathematically the results of two numerical experiments are presented they show that the proposed version of dsm is numerically efficient the numerical experiments consist of solving nonlinear integral equations

linear transvection groups and embedded polar spaces most classical groups arising from anti hermitian forms or pseudo quadratic forms contain so called isotropic transvections the isotropic transvection subgroups of these classical groups i e the subgroups generated by all isotropic transvections with a fixed axis form a class of abelian subgroups which is a class of abstract transvection groups in the sense of timmesfeld 24 in this paper we give a common characterization of all these classical groups with isotropic transvections as linear groups generated by a class of abstract transvection groups such that the elements of a are transvections

subgroups of lie type groups containing a unipotent radical abstract in this paper we show that if g is a lie type group and r a proper subgroup of g containing some unipotent radical then r has a nilpotent normal subgroup generated by long root subgroups and r is contained in a proper parabolic subgroup of g we also obtain some consequences of this result

algorithms determining finite simple images of finitely presented groups we address the question for which collections of finite simple groups does there exist an algorithm that determines the images of an arbitrary finitely presented group that lie in the collection we prove both positive and negative results for a collection of finite simple groups that contains infinitely many alternating groups or contains classical groups of unbounded dimensions we prove that there is no such algorithm on the other hand for families of simple groups of lie type of bounded rank we obtain positive results for example for any fixed untwisted lie type x there is an algorithm that determines whether or not any given finitely presented group has simple images of the form x q for infinitely many q and if there are finitely many the algorithm determines them

subgroups of exceptional groups of lie type generated by long root elements ii characteristic two let g be an exceptional group of lie type over a finite field f q p and assume the weyl rank of g is at least four let x be the collection of long root subgroups of g and r the collection of long root elements of g in 7 8 j we determined those subgroups l of g generated by long root subgroups with o l 1 this was done by studying the geometry of g acting on s with classification theorems only used in 7 to identify the possible isomorphism classes for l in part i 9 of this series we determined those subgroups l of g generated by long root elements with o l 1 where p 2 in that paper we used aschbacher s powerful result classifying the groups in chev p for p odd 21 in order to identify l in this paper we complete the determination in the case p 2 using the theorems of aschbacher 11 fischer 131 and timmesfeld 181 to identify l the determination of the subgroups of classical groups generated by elements central in a long root subgroup was carried out by kantor 151 in all characteristics using the aforementioned classification theorems for references on earlier attempts at these determinations for the classical groups see the introduction of 15 in another paper 111 we make use of our results to determine the minimal degree of a permutation representation of an exceptional group we also foresee these results having application to the more general problem of determining the maximal subgroups of an exceptional group

the geometry of k transvection groups abstract let k be a commutative field and g a group then a conjugacy class of abelian subgroups of g is called a class of k transvection subgroups in g if and only if it generates g and any two elements of the class either commute or are full unipotent subgroups of the group they generate and which is isomorphic to p sl 2 k in this paper we study the geometry of k transvection groups given a class of k transvection groups we consider a partial linear space whose points are the elements of and whose lines correspond to the groups generated by two noncommuting elements from we derive several properties of this partial linear space these properties are used to give a characterization of the geometries of k transvection groups and provide a classification of groups generated by k transvection subgroups

dynamical systems method dsm for selfadjoint operators abstract let a be a selfadjoint linear operator in a hilbert space h the dsm dynamical systems method for solving equation a v f consists of solving the cauchy problem u t u u 0 u 0 where is a suitable operator and proving that i u t t 0 ii u and iii a u f it is proved that if equation a v f is solvable and u solves the problem u i a i a u i f u 0 u 0 where a 0 is a parameter and u 0 is arbitrary then lim a 0 lim t u t a y where y is the unique minimal norm solution of the equation a v f stable solution of the equation a v f is constructed when the data are noisy i e f is given in place of f f f the case when a a t 0 0 a t d t a t 0 as t is considered it is proved that in this case lim t u t y and if f is given in place of f then lim t u t y where t is properly chosen

groups of lie type generated by long root elements in f4 k abstract let k be a field and g a quasi simple subgroup of the chevalley group f4 k we assume that g is generated by a class of abstract root subgroups such that there are a c with a c and any a is contained in a long root subgroup of f4 k we determine the possibilities for g and describe the embedding of g in f4 k

irreducible a1 subgroups of exceptional algebraic groups abstract a closed subgroup of a semisimple algebraic group is called irreducible if it lies in no proper parabolic subgroup in this paper we classify all irreducible a 1 subgroups of exceptional algebraic groups g consequences are given concerning the representations of such subgroups on various g modules for example the conjugacy classes of irreducible a 1 subgroups are determined by their composition factors on the adjoint module of g

variations on a theme of steinberg abstract the steinberg tensor product theorem is a fundamental tool for studying irreducible representations of simple algebraic groups over fields of positive characteristic this paper is concerned with extending the result replacing the target group sl v by an arbitrary simple algebraic group

black box exceptional groups of lie type if a black box group is known to be isomorphic to an exceptional simple group of lie type of twisted rank 1 other than any f4 q over a field of known size a las vegas algorithm is given to produce a constructive isomorphism in view of its timing this algorithm yields an upgrade of all known nearly linear time monte carlo permutation group algorithms to las vegas algorithms when the input group has no composition factor isomorphic to any group f4 q or g2 q

maximal unipotent subgroups of finitary linear groups finitary linear groups have been in recent years the subject of intensiveinvestigations the structure of these groups has been studied and nowmuch is known about the locally solvable or locally nilpotent finitary lineargroups or about the locally finite ones a remarkable achievement is forinstance the classification of the simple locally finite finitary linear groups recently completed by j hall

periodic simple groups of finitary linear transformations a group is locally finite if every finite subset generates a finite subgroup a group of linear transformations is finitary if each element minus the identity is an endomorphism of finite rank the classification and structure theory for locally finite simple groups splits naturally into two cases those groups that can be faithfully represented as groups of finitary linear transformations and those groups that are not finitary linear this paper completes the finitary case we classify up to isomorphism those infinite locally finite simple groups that are finitary linear but not linear

iterative solution of linear equations with unbounded operators a convergent iterative process is constructed for solving any solvable linear equation in a hilbert space including equations with unbounded closed densely defined linear operators the method is proved to be stable towards small perturbation of the data some abstract results are established and used in an analysis of variational regularization method for equations with unbounded linear operators the dynamical systems method dsm is justified for unbounded closed densely defined linear operators the stopping time is chosen by a discrepancy principle equations with selfadjoint operators are considered separately numerical examples illustrating the efficiency of the proposed method are given

a discrepancy principle for equations with monotone continuous operators a discrepancy principle for solving nonlinear equations with monotone operators given noisy data is formulated the existence and uniqueness of the corresponding regularization parameter a are proved convergence of the solution obtained by the discrepancy principle is justified the results are obtained under natural assumptions on the nonlinear operator

on a conjecture of g e wall abstract we prove that if g is a finite almost simple group having socle of lie type of rank r then the number of maximal subgroups of g is at most c r 2 3 g where c is an absolute constant this verifies a conjecture of wall for groups of sufficiently large rank using this we prove that any finite group g has at most 2 c g 3 2 maximal subgroups

reductions for monotone boolean circuits the large class say nlog of boolean functions including 0 1 sort and 0 1 merge have an upper bound of o nlogn for their monotone circuit size i e they have circuits with o nlogn and or gates of fan in two suppose that we can use besides such normal and or gates any number of more powerful f gates which realize a monotone boolean function f with r 2 inputs and r 1 outputs note that the cost of each and or gate is one and we assume that the cost of each f gate is r now we define a boolean function f in nlog is said to be f easy if f can be constructed by a circuit with and or f gates whose total cost is o nlogn in this paper we show that 0 1 merge is not f easy for an arbitrary monotone function f such that r r logr

dynamical systems method dsm for solving equations with monotone operators without smoothness assumptions on f u abstract a version of the dynamical systems method dsm for solving ill posed nonlinear equations f u f with monotone operators f in a hilbert space is studied in this paper under less restrictive assumptions on the nonlinear operators f than the assumptions used earlier a new method of proof of the basic results is used an a posteriori stopping rule based on a discrepancy type principle is proposed and justified mathematically under weaker assumptions on the nonlinear operator f than the assumptions used earlier

extremal presentations for classical lie algebras abstract the long root elements in lie algebras of chevalley type have been well studied and can be characterized as extremal elements that is elements x such that the image of ad x 2 lies in the subspace spanned by x in this paper assuming an algebraically closed base field of characteristic not 2 we find presentations of the lie algebras of classical chevalley type by means of minimal sets of extremal generators the relations are described by simple graphs on the sets for example for c n the graph is a path of length 2n and for a n the graph is the triangle connected to a path of length n 3

global convergence for ill posed equations with monotone operators the dynamical systems method consider an operator equation f u 0 in a real hilbert space let us call this equation ill posed if the operator f u is not boundedly invertible and well posed otherwise if f is monotone c2loc h operator then we construct a cauchy problem which has the following properties 1 it has a global solution for an arbitrary initial data 2 this solution tends to a limit as time tends to infinity and 3 the limit is the minimum norm solution to the equation f u 0 an example of applications to linear ill posed operator equation is given

factorizations of simple algebraic groups we determine all factorizations of simple algebraic groups as the product of two maximal closed connected subgroups additional results are established which drop the maximality assumption and applications are given to the study of subgroups of classical groups transitive on subspaces of a given dimension

a new discrepancy principle the aim of this note is to prove a new discrepancy principle the advantage of the new discrepancy principle compared with the known one consists of solving a minimization problem approximately rather than exactly and in the proof of a stability result

a 4 n lower bound on the combinational complexity of certain symmetric boolean functions over the basis of unate dyadic boolean functions a simple and easy to check property of a symmetric boolean function is shown to imply a 4n o 1 lower bound on the circuit complexity of the function over u 2 b 2 oplus equiv the basis of unate dyadic boolean functions among the functions to which this lower bound applies are the modular functions operatorname mod k n for any fixed k geqq 3 operatorname mod k n is the function which returns 1 if and only if sum x i bmod k 0 finally a 5n upper bound is obtained on the circuit complexity over u 2 of the function operatorname mod 4 n