makar limanov s conjecture on free subalgebras abstract it is proved that over every countable field k there is a nil algebra r such that the algebra obtained from r by extending the field k contains noncommutative free subalgebras of arbitrarily high rank it is also shown that over every countable field k there is an algebra r without noncommutative free subalgebras of rank two such that the algebra obtained from r by extending the field k contains a noncommutative free subalgebra of rank two this answers a question of makar limanov lenny makar limanov private communication beijing june 2007

trace functions in the ring of fractions of polycyclic group rings let kg be the group ring of a polycyclic by finite group g over a field k of characteristic zero r be the goldie ring of fractions of kg s be an arbitrary subring of r n n we prove that the intersection of the commutator subring s s with the center z s is nilpotent this implies the existence of a nontrivial trace function in r n n

free group algebras in certain division rings abstract let d be a division ring with centre k we show that d contains the k group algebra of the free group on two generators when d is the ring of fractions of a suitable skew polynomial ring or it is generated by a polycyclic by finite group which is not abelian by finite or it is the ring of fractions of the universal enveloping algebra of a finite dimensional lie algebra of characteristic zero

quantifying entanglement resources we present an overview of the quantitative theory of single copy entanglement in finite dimensional quantum systems in particular we emphasize the point of view that different entanglement measures quantify different types of resources which leads to a natural interdependence of entanglement classification and quantification apart from the theoretical basis we outline various methods for obtaining quantitative results on arbitrary mixed states this article is part of a special issue of journal of physics a mathematical and theoretical devoted to 50 years of bell s theorem

the convergence of variable metric matrices in unconstrained optimization it is proved that if the dfp or bfgs algorithm with step lengths of one is applied to a functionf x that has a lipschitz continuous second derivative and if the calculated vectors of variables converge to a point at which f is zero and 2f is positive definite then the sequence of variable metric matrices also converges the limit of this sequence is identified in the case whenf x is a strictly convex quadratic function

spectral decomposition of bell operators for multiqubit systems the spectral decomposition of multipartite bell operators for two dichotomic observables per site as introduced by werner and wolf phys rev a 64 032112 2001 is done implications on the characterization of bell inequalities as criteria of entanglement are discussed

a quasi newton method for estimating the parameter in a nonlinear hyperbolic system a quasi newton method qnm in infinite dimensional spaces for estimating parameters involved in distributed parameter systems is presented in this paper next the global convergence of a sequence generated by the algorithm qnm is also proved we apply the algorithm qnm to an identification problem of a distributed parameter system governed by a nonlinear hyperbolic partial differential equation

on normal subgroups of multiplicative group of skew fields generated by a polycyclic by finite group 1 1 let d be a field and let d be its multiplicative group consider a subgroup g c d and a subring k of the center of d we denote the subfield the term field in this paper is used in the skew field sense generated by k and g by k g when k is the subring generated by the unit elements i e k z or k z we simply say that d is generated by g the main result of the article is the following theorem

the global convergence of partitioned bfgs on problems with convex decompositions and lipschitzian gradients the main purpose of this paper is the extension of powell s 1976 global convergence result to the partitioned bfgs method introduced by griewank and toint 1982 even in the unpartitioned case the original result is strengthened because the search directions need not be computed exactly and the gradient is only required to be lipschitzian rather than differentiable using the functional of byrd and nocedal 1987 a strong form of r superlinear convergence is obtained if the element functions are uniformly convex and their gradients are strictly differentiable at the minimizerx only in order to deal with the possibility of singular functions we utilize a damping of the bfgs update that becomes inactive if the problem turns out to be regular nearx

some skew linear groups associated with lie algebras of characteristic zero let f be a field l a lie f algebra of finite dimension d and u the universal enveloping algebra of l then u is a noetherian domain 3 p 1661 let d denote its division ring of quotients suppose charf 0 lichtman in s 63 considers some group theoretic properties of gl n d and its subgroups in this note we pursue this investigation a little further for fuller statements of both lichtman s results and our own see section 2 below here we merely summarize a few of the more interesting points suppose g is a finitely generated subgroup of gl n d we show for all but a finite number of primes q that g has a normal subgroup of finite index that is residually a finite q group in particular of course g is residually finite further it follows that g is torsion free by finite and hence if g is periodic then g is finite thus s a periodic subgroup p of gl n d is locally finite we also show that such a p is isomorphic to a subgroup of gl n c and hence that s the group p has an abelian normal subgroup with index finite and bounded in terms of n only suppose g is a subgroup of gl n d containing no free subgroup of rank two then the structure of go is described by theorem a of 6 we give an alternative approach yielding some improvement of the bounds involved as a consequence we produce a sharper bound for the derived length of a soluble subgroup of gl n d assume again that g is a finitely generated subgroup of gl n d an easy result is that if every finite image of g is soluble then g is soluble in order to proceed much further with elucidating the structure of g along the lines suggested by the theory of linear groups we would need something like the nilpotence of g given that the finite images of g are nilpotent unfortunately the unipotent elements present problems this is frequently the case where imperfect ground fields are involved to see that is the situation here refer to section 1 below we have only been able to prove under considerably weaker hypotheses that g is nilpotent modulo 155 0021 8693 89 93 00

iterative solution of integral equations by a quasi newton method abstract when broyden s type methods are applied to the solution of integral equations they can be specialized in such a way to generate in the course of the iteration low rank degenerate kernel equations we describe the features of this approach

sequential quadratic programming sqp for optimal control in direct numerical simulation of turbulent flow the current work focuses on the development and application of an efficient algorithm for optimization of three dimensional turbulent flows simulated using direct numerical simulation dns or large eddy simulations and further characterized by large dimensional optimization parameter spaces the optimization algorithm is based on sequential quadratic programming sqp in combination with a damped formulation of the limited memory bfgs method the latter is suitable for solving large scale constrained optimization problems whose hessian matrices cannot be computed and stored at a reasonable cost we combine the algorithm with a line search merit function based on an l 1 norm to enforce the convergence from any remote point it is first shown that the proposed form of the damped l bfgs algorithm is suitable for solving equality constrained rosenbrock type functions then we apply the algorithm to an optimal control test problem that consists of finding the optimal initial perturbations to a turbulent temporal mixing layer such that mixing is improved at the end of a simulation time horizon t the controls are further subject to a non linear equality constraint on the total control energy dnss are used to resolve all turbulent scales of motion and a continuous adjoint formulation is employed to calculate the gradient of the cost functionals we compare the convergence speed of the sqp l bfgs algorithm to a conventional non linear conjugate gradient method i e the current standard in dns based optimal control and find that the sqp algorithm is more than an order of magnitude faster than the conjugate gradient method

second order methods for optimal control of time dependent fluid flow second order methods for open loop optimal control problems governed by the two dimensional instationary navier stokes equations are investigated optimality systems based on a lagrangian formulation and adjoint equations are derived the newton and quasi newton methods as well as various variants of sqp methods are developed for applications to optimal flow control and their complexity in terms of system solves is discussed local convergence and rate of convergence are proved a numerical example illustrates the feasibility of solving optimal control problems for two dimensional instationary navier stokes equations by second order numerical methods in a standard workstation environment

free group algebras in division rings with valuation i abstract let r be an algebra over a commutative ring k suppose that r is endowed with a descending filtration indexed on an ordered group g such that the restriction to k is positive we show that the existence of free algebras on a certain set of generators in the induced graded ring grad r implies the existence of free group algebras in r our best results are obtained for division rings endowed with a valuation

free group algebras in the field of fractions of differential polynomial rings and enveloping algebras abstract we construct free group algebras in the quotient ring of the differential polynomial ring k x for suitable division rings k and nonzero derivations in k

the dynamical mordell lang problem for etale maps we prove a dynamical version of the mordell lang conjecture for etale endomorphisms of quasiprojective varieties we use p adic methods inspired by the work of skolem mahler and lech combined with methods from algebraic geometry as special cases of our result we obtain a new proof of the classical mordell lang conjecture for cyclic subgroups of a semiabelian variety and we also answer positively a question of keeler rogalski stafford for critically dense sequences of closed points of a noetherian integral scheme

free algebras in division rings with an involution abstract some general criteria to produce explicit free algebras inside the division ring of fractions of skew polynomial rings are presented these criteria are applied to some special cases of division rings with natural involutions yielding for instance free subalgebras generated by symmetric elements both in the division ring of fractions of the group algebra of a torsion free nilpotent group and in the division ring of fractions of the first weyl algebra

on universal fields of fractions for free algebras abstract let l be an arbitrary lie algebra then by cohn s theorem its universal enveloping algebra can be embedded in a skew field d l we study this skew field in the case when l is a free lie algebra and prove that in this case d l is isomorphic to the universal field of fractions for the free associative algebra u l we apply this theorem to obtain new results on free fields

superlinear convergence of affine scaling interior point newton methods for infinite dimensional nonlinear problems with pointwise bounds we develop and analyze a superlinearly convergent affine scaling interior point newton method for infinite dimensional problems with pointwise bounds in lp space the problem formulation is motivated by optimal control problems with lp controls and pointwise control constraints the finite dimensional convergence theory by coleman and li siam j optim 6 1996 pp 418 445 makes essential use of the equivalence of norms and the exact identifiability of the active constraints close to an optimizer with strict complementarity since these features are not available in our infinite dimensional framework algorithmic changes are necessary to ensure fast local convergence the main building block is a newton like iteration for an affine scaling formulation of the kkt condition we demonstrate in an example that a stepsize rule to obtain an interior iterate may require very small stepsizes even arbitrarily close to a nondegenerate solution using a pointwise projection instead we prove superlinear convergence under a weak strict complementarity condition and convergence with mbox q rate 1 under a slightly stronger condition if a smoothing step is available we discuss how the algorithm can be embedded in the class of globally convergent trust region interior point methods recently developed by m heinkenschloss and the authors numerical results for the control of a heating process confirm our theoretical findings

free algebras and free groups in ore extensions and free group algebras in division rings abstract let k be a field of characteristic zero let be an automorphism of k and let be a derivation of k we show that the division ring d k x either has the property that every finitely generated subring satisfies a polynomial identity or d contains a free algebra on two generators over its center in the case when k is finitely generated over a subfield k we then see that for a k algebra automorphism of k and a k linear derivation of k k x having a free subalgebra on two generators is equivalent to having infinite order and k x having a free subalgebra is equivalent to being nonzero as an application we show that if d is a division ring with center k of characteristic zero and d contains a solvable subgroup that is not locally abelian by finite then d contains a free k algebra on two generators moreover if we assume that k is uncountable without any restrictions on the characteristic of k then d contains the k group algebra of the free group of rank two

a pde sensitivity equation method for optimal aerodynamic design the use of gradient based optimization algorithms in inverse design is well established as a practical approach to aerodynamic design a typical procedure uses a simulation scheme to evaluate the objective function from the approximate states and its gradient then passes this information to an optimization algorithm once the simulation scheme cfd flow solver has been selected and used to provide approximate function evaluations there are several possible approaches to the problem of computing gradients one popular method is to differentiate the simulation scheme and compute design sensitivities that are then used to obtain gradients although this black box approach has many advantages in shape optimization problems one must compute mesh sensitivities in order to compute the design sensitivity in this paper we present an alternative approach using the pde sensitivity equation to develop algorithms for computing gradients this approach has the advantage that mesh sensitivities need not be computed moreover when it is possible to use the cfd scheme for both the forward problem and the sensitivity equation then there are computational advantages an apparent disadvantage of this approach is that it does not always produce consistent derivatives however for a proper combination of discretization schemes one can showasymptotic consistencyunder mesh refinement which is often sufficient to guarantee convergence of the optimal design algorithm in particular we show that when asymptotically consistent schemes are combined with a trust region optimization algorithm the resulting optimal design method converges we denote this approach as thesensitivity equation method the sensitivity equation method is presented convergence results are given and the approach is illustrated on two optimal design problems involving shocks

a quasi newton method for solving fixed point problems in hilbert spaces we study the convergence properties of a projective variant of newton s method for the approximation of fixed points of completely continuous operators in hilbert spaces we then discuss applications to nonlinear integral equations and we produce some numerical examples

the convergence of matrices generated by rank 2 methods from the restricted β class of broyden it is shown that the matricesb k generated by any method from the restricted class of broyden converge if the method is applied to the unconstrained minimization of a functionf c 2 r n with lipschitz continuous 2 f x and if the method is such that it generates vectorsx k converging sufficiently fast sum limits k x k x infty to a local minimumx off with positive definite 2 f x this result not only holds for constant step sizes k 1 in each iterationx k x k 1 x k k b k 1 f x k of these methods but also for step sizes determined by asymptotically exact line searches the paper generalizes corresponding results of ge ren pu and powell 6 for the dfp and bfgs methods used in conjunction with step sizes k 1

on embedding of group rings of residually torsion free nilpotent groups into skew fields it is proven that the group ring of an amalgamated free product of residually torsion free nilpotent groups is a domain and can be embedded in a skew field this is a generalization of j lewin s theorem proven for the case of free groups our proof is based on the study of the malcev neumann power series ring k g of a residually torsion free nilpotent group g it is shown that its subfield d generated by the group ring kg does not depend on the order of g for many kinds of orders and the study of d can be reduced in some sense to the case when g is nilpotent

translationally invariant multipartite bell inequalities involving only two body correlators bell inequalities are natural tools that allow one to certify the presence of nonlocality in quantum systems the known constructions of multipartite bell inequalities contain however correlation functions involving all observers making their experimental implementation difficult the main purpose of this work is to explore the possibility of witnessing nonlocality in multipartite quantum states from the easiest to measure quantities that is the two body correlations in particular we determine all three and four partite bell inequalities constructed from one and two body expectation values that obey translational symmetry and show that they reveal nonlocality in multipartite states also by providing a particular example of a five partite bell inequality we show that nonlocality can be detected from two body correlators involving only nearest neighbours finally we demonstrate that any translationally invariant bell inequality can be maximally violated by a translationally invariant state and the same set of observables at all sites we provide a numerical algorithm allowing one to seek for maximal violation of a translationally invariant bell inequality this article is part of a special issue of journal of physics a mathematical and theoretical devoted to 50 years of bell s theorem

free pairs of symmetric elements in normal subgroups of division rings abstract let d be a division ring with central subfield k of characteristic 2 let be a k involution of d and let n d be a normal subgroup of the multiplicative group d of d we show that if g is a stable nonabelian subgroup of n that is either torsion free polycyclic by finite but not abelian by finite or finite of odd order then n contains a pair u v of elements such that u u v v and such that u v is a noncyclic free group one aspect of the above proof requires that we extend a theorem of bergman on invariant ideals in commutative group algebras this new result is surely of interest in its own right it appears in the appendix and can be read independently of the remainder of the paper

bounding the set of classical correlations of a many body system we present a method to certify the presence of bell correlations in experimentally observed statistics and to obtain new bell inequalities our approach is based on relaxing the conditions defining the set of correlations obeying a local hidden variable model yielding a convergent hierarchy of semidefinite programs sdp s because the size of these sdp s is independent of the number of parties involved this technique allows us to characterize correlations in many body systems as an example we illustrate our method with the experimental data presented in science 352 441 2016 scieas0036 807510 1126 science aad8665

do all pure entangled states violate bell s inequalities for correlation functions any pure entangled state of two particles violates a bell inequality for two particle correlation functions gisin s theorem we show that there exist pure entangled ng2 qubit states that do not violate any bell inequality for n particle correlation functions for experiments involving two dichotomic observables per local measuring station we also find that mermin ardehali belinskii klyshko inequalities may not always be optimal for refutation of local realistic description

free subalgebras of division algebras over uncountable fields we study the existence of free subalgebras in division algebras and prove the following general result if a is a noetherian domain which is countably generated over an uncountable algebraically closed field k of characteristic 0 then either the quotient division algebra of a contains a free algebra on two generators or it is left algebraic over every maximal subfield as an application we prove that if k is an uncountable algebraically closed field and a is a finitely generated k algebra that is a domain of gk dimension strictly less than 3 then either a satisfies a polynomial identity or the quotient division algebra of a contains a free k algebra on two generators

free products of units in algebras i quaternion algebras abstract letabe a quaternion algebra over a commutative unital ring we find sufficient conditions for pairs of units ofato generate a free group using the well known isomorphism betweenso 3 r and the group of real quaternions of norm 1 we obtain free groups of rotations of the euclidean 3 space specialization techniques allow us to find similar free subgroups in skew polynomial rings a consequence is the following letkgbe the group algebra of a residually torsion free nilpotent groupgover a fieldkwhose characteristic is not 2 ifxandyare any pair of noncommuting elements ofg andc d k then 1 cxand 1 dygenerate a free subgroup of the malcev neumann field of fractions ofkg

composite parameterization and haar measure for all unitary and special unitary groups we adopt the concept of the composite parameterization of the unitary group u d to the special unitary group su d furthermore we also consider the haar measure in terms of the introduced parameters we show that the well defined structure of the parameterization leads to a concise formula for the normalized haar measure on u d and su d with regard to possible applications of our results we consider the computation of high order integrals over unitary groups

a pointwise quasi newton method for integral equations for fully nonlinear integral equations conventional quasi newton methods such as broyden s method may not possess good convergence properties in this paper we present a new method that gives superlinear convergence for such integral equations and provides for savings over newton s method in the cost of solution of the linear equation for the quasi newton step

free groups in quaternion algebras abstract in juriaans et al 2009 9 we constructed pairs of units u v in z orders of a quaternion algebra over q d d a positive and square free integer with d 7 mod 8 such that u n v n is free for some n n here we extend this result to any imaginary quadratic extension of q thus including matrix algebras more precisely we show that u n v n is a free group for all n 1 and d 2 and for d 2 and all n 2 the units we use arise from pell s and gauss equations

generating facets for the cut polytope of a graph by triangular elimination the cut polytope of a graph arises in many fields although much is known about facets of the cut polytope of the complete graph very little is known for general graphs the study of bell inequalities in quantum information science requires knowledge of the facets of the cut polytope of the complete bipartite graph or more generally the complete k partite graph lifting is a central tool to prove certain inequalities are facet inducing for the cut polytope in this paper we introduce a lifting operation named triangular elimination applicable to the cut polytope of a wide range of graphs triangular elimination is a specific combination of zero lifting and fourier motzkin elimination using the triangle inequality we prove sufficient conditions for the triangular elimination of facet inducing inequalities to be facet inducing the proof is based on a variation of the lifting lemma adapted to general graphs the result can be used to derive facet inducing inequalities of the cut polytope of various graphs from those of the complete graph we also investigate the symmetry of facet inducing inequalities of the cut polytope of the complete bipartite graph derived by triangular elimination

free subalgebras of c algebras abstract the aim of this work is to study the existence of free subalgebras in c algebras the kurosh levitzky problem and related conjectures of makar limanov are answered in the context of c algebras in particular we characterize and study the existence of free non abelian subalgebras with two self adjoint contractions in finitely generated c algebras c algebras having linear growth are shown to be subhomogeneous

valuation methods in group rings and in skew fields i abstract in this first in a series of two papers we construct a family of discrete valuations in group rings of residually torsion free nilpotent groups and extend these valuations to the malcev neumann power series skew fields of these group rings we apply these valuations to the study of these skew fields in the second paper we will use our results and methods to study free fields and the universal fields of fractions k x of the magnus power series ring

atlas of finite groups maximal subgroups and ordinary characters for simple groups this atlas covers groups from the families of the classification of finite simple groups recently updated incorporating corrections

a note on the superlinear convergence of gmres in the paper it is shown how the rate of convergence of the gmres method for solving a linear operator equation lambda i k u f in a hilbert space is related to the degree of compactness of k measured by the products of its singular values

bell s theorem for general n qubit states we derive a single general bell inequality which is a sufficient and necessary condition for the correlation function for n particles to be describable in a local and realistic picture for the case in which measurements on each particle can be chosen between two arbitrary dichotomic observables we also derive a necessary and sufficient condition for an arbitrary n qubit mixed state to violate this inequality this condition is a generalization and reformulation of the horodecki family condition for two qubits

matrix rings and linear groups over a field of fractions of enveloping algebras and group rings i 1 1 let h be a lie algebra over a field k u h be its universal envelope it is well known that u h is a domain see 1 section v 31 and the theorem of cohn see 2 states that u h can be imbedded in a field when h is locally finite u h has even a field of fractions see 1 theorem v 3 61 the class of lie algebras whose universal envelope has a field of fractions is however much wider than the class of locally finite lie algebras it follows from proposition 4 1 of the article that it contains for instance all the locally soluble by locally finite algebras this gives a negative answer of the question in 2 page 528 let h satisfy any one of the following three conditions

the local convergence of broyden like methods on lipschitzian problems in hilbert spaces the classical local convergence analysis for broyden s method requires a finite number of real variables holder continuity of the jacobian and a rather accurate initial approximation b 0 here these three conditions are relaxed to allow for lipschitz continuous hilbert space operators that can be reasonably well approximated by a linear operator b that is close to b 0 up to an arbitrary compact perturbation to achieve local and r linear convergence broyden s method is modified by a very mild line search condition in case of a continuously differentiable system in euclidean space we obtain r superlinear convergence without restricting b 0 or requiring the jacobian to be holder continuous

free groups in normal subgroups of the multiplicative group of a division ring abstract let d be a division ring with center z and multiplicative group d 0 d and let n be a normal subgroup of d we investigate various conditions under which n must contain a free noncyclic subgroup in one instance assuming that the transcendence degree of z over its prime field is infinite and that n contains a nonabelian solvable subgroup we use a construction method due to chiba to exhibit free generators of the free subgroup

free products of units in algebras ii crossed products abstract we investigate the problem of explicitly constructing non cyclic free groups in finite dimensional crossed products using valuation criteria the results are applied to produce explicit free groups in division algebras generated by nilpotent groups and symmetric free groups in group rings of finite groups in arbitrary characteristic

on subnormal subgroups of skew fields abstract if d is a skew field with infinite center z and d z then any subnormal subgroup of d which satisfies a laurent generalized polynomial identity is central

spectral decomposition of bell s operators for qubits the spectral decomposition is given for the n qubit bell operators with two observables per qubit it is found that the eigenstates when non degenerate are n qubit ghz states even for those operators that do not allow the maximal violation of the corresponding inequality we present two applications of this analysis in particular we discuss the existence of pure entangled states that do not violate the mermin klyshko inequality for n 3

the approximation of generalized turning points newline by krylov subspace methods certain types of singular solutions of nonlinear parameter dependent operator equations were characterized by griewank and reddien 5 6 as regular solutions of suitable augmented systems for their numerical approximation an approach based on the use of krylov subspaces is here presented the application to boundary value problems is illustrated by numerical examples

bell inequalities for arbitrarily high dimensional systems we develop a novel approach to bell inequalities based on a constraint that the correlations exhibited by local variable theories must satisfy this is used to construct a family of bell inequalities for bipartite quantum systems of arbitrarily high dimensionality which are strongly resistant to noise in particular our work gives an analytic description of previous numerical results and generalizes them to arbitrarily high dimensionality

tight multipartite bell s inequalities involving many measurement settings which quantum states do not allow a local realistic lr description this question still remains open mainly because our present tools to test local realism are not optimal most of bell s inequalities are for the case in which only two measurement settings can be chosen by each observer e g the clauser horneshimony holt chsh inequality 1 inequalities for bipartite higher dimensional systems 2 multipartite bell s inequalities 3 6 one can call such inequalities standard ones one can expect that allowing the observers to choose between more than two observables should give more stringent constraints on lr models thus new inequalities may extend the class of nonseparable states which cannot be described by lr variables violation of local realism is an important ingredient for building quantum information protocols that decrease the communication complexity 7 and is a criterion for the efficient quantum key distribution 8 multisetting bell s inequalities may lead to such novel schemes although various nonstandard bell s inequalities are known 9 13 we still lack an efficient method for their derivation a general way is to define the facets of the correlation polytope 9 13 yet this is a computationally hard np problem 14 recently wu and zong 15 derived an inequality for three parties which involves four local settings for two observers and two settings for the third one and showed that it is stronger than the standard three particle bell inequalities here by generalizing the method of ref 15 we derive a wide class of tight bell s inequalities for n 2 parties and many measurement settings we find a single general bell s inequality that generates all inequalities from this class we also give a necessary and sufficient condition for the violation of the new inequalities by quantum predictions most importantly the inequalities reveal violation of local realism for the classes of states for which all standard bell s inequalities 5 6 are satisfied we start with the case ofn 3 observers suppose that the first two observers can choose between four settings and the third one between two settings we denote such problem as 4 4 2 l etai withi 2f 1 2 3 4g stand for the predetermined local realistic values for the first observer under the local settingi bj withj 2f 1 2 3 4g for similar values for the second observer and ck withk 2 f1 2g for the values for the third observer for the given run of the experiment we assume thata i b j a ndc k can take values 1 or 1 then the lr values satisfy the following algebraic identity a12 s 0 x k l 1 2 s 0 k l a1 1 k a 2 b1 1 l b 2 4

broyden s method in hilbert space broyden s method is formulated for the solution of nonlinear operator equations in hilbert spaces the algorithm is proven to be well defined and a linear rate of convergence is shown under an additional assumption on the initial approximation for the derivative we prove the superlinear rate of convergence