derivations of the parabolic subalgebras of the general linear lie algebra over a commutative ring let r be an arbitrary commutative ring with identity gl n r the general linear lie algebra over r the aim of this paper is to give an explicit description of the derivation algebras of the parabolic subalgebras of gl n r

the calderon problem with partial data in this paper we improve an earlier result by bukhgeim and uhlmann 1 by showing that in dimension n 3 the knowledge of the cauchy data for the schr nodinger equation measured on possibly very small subsets of the boundary determines uniquely the potential we follow the general strategy of 1 but use a richer set of solutions to the dirichlet problem this implies a similar result for the problem of electrical impedance tomography which consists in determining the conductivity of a body by making voltage and current measurements at the boundary

generation of discrete distributions from biased coins the procedure of knuth and yao 1976 to simulate random numbers with specified distribution by parsing sequences of fair coin tosses is generalized to employ discrete distributions of particular form instead of fair coins each probability in these distributions is an integral power of some fixed value t the parse tree for the simulation procedure is closely related to the code trees arising in karp s optimal variable length coding algorithm for code symbols of unequal cost

computing with very weak random sources we give an efficient algorithm to extract randomness from a very weak random source using a small additional number t of truly random bits our work extends that of nisan and zuckerman j comput system sci 52 1996 pp 43 52 in that t remains small even if the entropy rate is well below constant a key application of this is in running randomized algorithms using such a very weak source of randomness for any fixed gamma 0 we show how to simulate rp algorithms in time n o log n using the output of a ds with min entropy r gamma such a weak random source is asked once for r bits it outputs an r bit string according to any probability distribution that places probability at most 2 r gamma on each string if gamma 1 2 our simulation also works for bpp for gamma 1 1 k 1 our simulation takes time n o logk n log k is the logarithm iterated k times we also give a polynomial time bpp simulation using chor goldreich sources of min entropy r omega 1 which is optimal we present applications to time space tradeoffs expander constructions and to the hardness of approximation of independent interest is our randomness efficient leftover hash lemma a key tool for extracting randomness from weak random sources

stability estimates for the calderon problem with partial data this is a follow up of a previous article where we proved local stability estimates for a potential in a schr odinger equation on an open bounded set in dimension n 3 from the dirichlet to neumann map with partial data the region under control was the penumbra delimited by a source of light outside of the convex hull of the open set these local estimates provided stability of log log type corresponding to the uniqueness results in calder on s inverse problem with partial data proved by kenig sj ostrand and uhlmann in this article we prove the corresponding global estimates in all dimensions higher than three the estimates are based on the construction of solutions of the schr odinger equation by complex geometrical optics developed in the anisotropic setting by dos santos ferreira kenig salo and uhlmann to solve the calder on problem in certain admissible geometries

tree algorithms for unbiased coin tossing with a biased coin we give new algorithms for simulating a flip of an unbiased coin by flipping a coin of unknown bias we are interested in efficient algorithms where the expected number of flips is our measure of efficiency other authors have represented algorithms as lattices but by representing them instead as trees we are able to produce an algorithm more efficient than any previously appearing we also prove a conjecture of hoeffding and simons that there is no optimal algorithm further we consider generalizations where the input is a sequence of iid discrete random variables and the output is a uniform random variable with n possible outcomes in this setting we provide an algorithm significantly superior to those previously published

derivations of upper triangular matrix rings we give a description of the derivations in tn r the ringof upper triangular matrices over a ring r assuming only the existence of an identity element we show that every derivation is the sum of an inner derivation and another one induced from r

a note on approximation of uniform distributions from variable to fixed length codes in this correspondence we prove that the probability distribution induced on the leaves of a tunstall parse tree for a given source is a unique lower bound in the partially ordered set of the probability distributions induced by all possible parse trees with a same number of leaves and ordered according to the majorization partial order we apply this result to the problem of optimally approximating a uniform distribution with flips of a biased coin

optimal stability estimate in the inverse boundary value problem for periodic potentials with partial data abstract we consider the inverse boundary value problem for operators of the form q in an infinite domain r r 1 n n 3 with a periodic potential q for dirichlet to neumann data localized on a portion of the boundary of the form 1 r 1 with 1 being the complement either of a flat or spherical portion of we prove that a log type stability estimate holds

determining conductivity by boundary measurements a p calderon pose la question est il possible de determiner la conductivite thermique d un objet a partir de mesures statiques de la temperature et du flux de chaleur a la limite on demontre que ces mesures determinent de facon unique la conductivite thermique et toutes ses derivees a la limite

automorphisms of the lie algebra of strictly upper triangular matrices over a commutative ring let r be an arbitrary commutative ring with identity denote by n r the lie algebra over r consisting of all strictly upper triangular n 1 n 1 matrices over r with n 3 in addition for n 3 assume that the annihilator of 2 in r is zero the aim of this paper is to describe the automorphism group of n r we show that any automorphism of n r can be expressed as where and are graph extremal central and inner automorphisms respectively of n r

efficient generation of fair dice with few biased coins given a random variable x which takes n equiprobable values we consider several algorithmic questions related to the classical problem of simulating the outcomes of x by using a limited number of biased coins

simulating fair dice with biased coins this paper is concerned with simulating a fair die with a bounded number of coin flips and a bounded number of possibly biased coins as the main results this paper shows that for anyn 2 a set ofh n coins is sufficient to simulate a fairn sided die withind lgn coin flips whereh n is the number of 1 s of the binary representation of an integern and that for anyn 2d 1 d 3 a set ofd h n coins is necessary and sufficient to simulate a fairn sided die withind lgn coin flips

unbiased bits from sources of weak randomness and probabilistic communication complexity a new model for weak random physical sources is presented the new model strictly generalizes previous models e g the santha and vazirani model 27 the sources considered output strings according to probability distributions in which no single string is too probable the new model provides a fruitful viewpoint on problems studied previously such as extracting almost perfect bits from sources of weak randomness the question of possibility as well as the question of efficiency of such extraction schemes are addressed probabilistic communication complexity it is shown that most functions have linear communication complexity in a very strong probabilistic sense robustness of bpp with respect to sources of weak randomness generalizing a result of vazirani and vazirani 32 33

multiplicative semigroup automorphisms of the strictly upper triangular matrices over a commutative ring let r be an arbitrary commutative ring with identity and nn r b e the multiplicative semigroup consisting of all n n strictly upper triangular matrices over r t he aim of this paper is to give an explicit description of any automorphism of the semigroup nn r

stability estimates for the radon transform with restricted data and applications in this article we prove a stability estimate going from the radon transform of a function with limited angle distance data to the l p norm of the function itself under some conditions on the support of the function we apply this theorem to obtain stability estimates for an inverse boundary value problem with partial data

streaming universal distortion free entanglement concentration this paper presents a streaming sequential protocol for universal entanglement concentration at the shannon bound alice and bob begin with n identical but unknown two qubit pure states each containing e ebits of entanglement they each run a reversible algorithm on their qubits and end up with y perfect epr pairs where y ne o n our protocol is streaming so the n input systems are fed in one at a time and perfect epr pairs start popping out almost immediately it matches the optimal block protocol exactly at each stage so the average yield after n inputs is y ne o logn so somewhat surprisingly there is no tradeoff between yield and lag our protocol optimizes both in contrast the optimal n qubit block protocol achieves the same yield but since no epr pairs are produced until the entire input block is read its lag is o n finally our algorithm runs in o log n space so a lot of entanglement can be efficiently concentrated using a very small e g current or near future technology quantum processor along the way we find an optimal streaming protocol for extracting randomness from classical i i d sources and a more space efficient implementation of the schur transform

limiting carleman weights and anisotropic inverse problems in this article we consider the anisotropic calderon problem and related inverse problems the approach is based on limiting carleman weights introduced in kenig et al ann math 165 567 591 2007 in the euclidean case we characterize those riemannian manifolds which admit limiting carleman weights and give a complex geometrical optics construction for a class of such manifolds this is used to prove uniqueness results for anisotropic inverse problems via the attenuated geodesic ray transform earlier results in dimension n 3 were restricted to real analytic metrics

the attenuated ray transform on simple surfaces we show that the attenuated geodesic ray transform on two dimensional simple surfaces is injective moreover we give a stability estimate and develop a reconstruction procedure

exact sequences of inner automorphisms of tensors abstract we produce a long exact sequence whose terms are unit groups of associative algebras that behave as inner automorphisms of a given tensor our sequence generalizes known sequences for associative and non associative algebras in a manner similar to those our sequence facilitates inductive reasoning about and calculation of the groups of symmetries of a tensor the new insights these methods afford can be applied to problems ranging from understanding algebraic structures to distinguishing entangled states in particle physics

lie triple derivations of nest algebras on banach spaces abstract let n be a non trivial nest on x alg n be the associated nest algebra and l alg n b x be a linear mapping in this paper it is proved that l is a lie triple derivation if and only if there exist a derivation d alg n b x and a linear mapping h alg n c i with h x y z 0 for any x y z alg n such that l d h on alg n

how to turn loaded dice into fair coins we present a new technique for simulating fair coin flips using a biased stationary source of randomness sequences of random numbers are of pervasive importance in cryptography and vital to many other computing applications many sources of randomness such as radioactive or quantum mechanical sources possess the property of stationarity in other words they produce independent outputs over fixed probability distributions the output of such sources may be viewed as the result of rolling a biased or loaded die while a biased die may be a good source of entropy many applications require input in the form of unbiased bits rather than biased ones for this reason von neumann 1951 presented a now well known and extensively investigated technique for using a biased coin to simulate a fair coin we describe a new generalization of von neumann s algorithm distinguished by its high level of practicality and amenability to analysis in contrast to previous efforts we are able to prove our algorithm optimally efficient in the sense that it simulates the maximum possible number of fair coin flips for a given number of die rolls in fact we are able to prove that in an asymptotic sense our algorithm extracts the full entropy of its input moreover we demonstrate experimentally that our algorithm achieves a high level of computational and output efficiency in a practical setting

derivations of tensor products of nonassociative algebras abstract let r and s be nonassociative unital algebras assuming that either one of them is finite dimensional or both are finitely generated we show that every derivation of r s is the sum of derivations of the following three types a ad u where u belongs to the nucleus of r s b l z f where f is a derivation of s and z lies in the center of r and c g l w where g is a derivation of r and w lies in the center of s

jordan derivations of full matrix algebras abstract let a be a unital associative ring and m be a 2 torsion free a bimodule using an elementary and constructive method we show that every jordan derivation from m n a into m n m is a derivation

skolem noether algebras abstract an algebra s is called a skolem noether algebra sn algebra for short if for every central simple algebra r every homomorphism r r s extends to an inner automorphism of r s one of the important properties of such an algebra is that each automorphism of a matrix algebra over s is the composition of an inner automorphism with an automorphism of s the bulk of the paper is devoted to finding properties and examples of sn algebras the classical skolem noether theorem implies that every central simple algebra is sn in this article it is shown that actually so is every semilocal and hence every finite dimensional algebra not every domain is sn but for instance unique factorization domains polynomial algebras and free algebras are further an algebra s is sn if and only if the power series algebra s is sn

determining a magnetic schrodinger operator from partial cauchy data in this paper we show in dimension n 3 that knowledge of the cauchy data for the schrodinger equation in the presence of a magnetic potential measured on possibly very small subsets of the boundary determines uniquely the magnetic field and the electric potential we follow the general strategy of 7 using a richer set of solutions to the dirichlet problem that has been used in previous works on this problem

extraction of optimally unbiased bits from a biased source we explore the problem of transforming n independent and identically biased 1 1 valued random variables x sub 1 x sub n into a single 1 1 random variable f x sub 1 x sub n so that this result is as unbiased as possible in general no function f produces a completely unbiased result we perform the first study of the relationship between the bias b of these x sub i and the rate at which f x sub 1 x sub n can converge to an unbiased 1 1 random variable as n spl rarr spl infin a 1 1 random variable has bias b if e x sub i b fixing a bias b we explore the rate at which the output bias e f x sub 1 x sub n can tend to zero for a function f 1 1 spl rarr 1 1 this is accomplished by classifying the behavior of the natural normalized quantity spl xi b spl delta inf sub f lim sub n spl rarr spl infin n spl radic e f x sub 1 x sub n this infimum taken over all such f we show that for rational b spl xi b 1 s where 1 b 2 r s r and s relatively prime developing the theory of uniform distribution of sequences to suit our problem we then explore the case where b is irrational we prove a new metrical theorem concerning multidimensional diophantine approximation type from which we show that for lebesgue almost all biases b spl xi b 0 finally we show that algebraic biases exhibit curious boundary behavior falling into two classes class 1 those algebraics b for which spl xi b 0 and furthermore c sub 1 spl les spl xi b spl les c sub 2 where c sub 1 and c sub 2 are positive constants depending only on b s algebraic characteristics class 2 those algebraics b for which there exist n 0 and f 1 1 sup n spl rarr 1 1 so that e f x sub 1 x sub n 0 notice that this classification excludes the possibility that n spl radic e f x sub 1 x sub n limits to zero for algebraics for rational and algebraic biases we also study the computational problem by restricting f to be a polynomial time computable function finally we discuss natural extensions where output distributions other than the uniform distribution on 1 1 are sought