closed embeddings of c in c 2 part i we consider closed curves c c in the affine plane s c2 that admit a good or very good asymptote by this we mean a curve l c in s that in suitable coordinates for s is linear and tangent to c at infinity and meets c once or not at all at finite distance we classify these curves up to automorphism of s relying on the theory of open algebraic surfaces we first determine the possibilities for the singularities of c at infinity and then proceed to give explicit equations

non linearizable algebraic k actions on affine spaces non linearizable faithful algebraic r r actions of torus on the real affine n space r n are given for each r 1 n 3 when n 4

stable tameness of two dimensional polynomial automorphisms over a regular ring abstract in this paper it is established that all two dimensional polynomial automorphisms over a regular ring r are stably tame this results from the main theorem of this paper which asserts that an automorphism in any dimension n is stably tame if said condition holds point wise over spec r a key element in the proof is a theorem which yields the following corollary over an artinian ring a all two dimensional polynomial automorphisms having jacobian determinant one are stably tame and are tame if a is a q algebra another crucial ingredient of interest in itself is that stable tameness is a local property if an automorphism is locally tame then it is stably tame

tomographic reconstruction of three dimensional volumes using the distorted born iterative method although real imaging problems involve objects that have variations in three dimensions a majority of work examining inverse scattering methods for ultrasonic tomography considers 2 d imaging problems therefore the study of 3 d inverse scattering methods is necessary for future applications of ultrasonic tomography in this work 3 d reconstructions using different arrays of rectangular elements focused on elevation were studied when reconstructing spherical imaging targets by producing a series of 2 d image slices using the 2 d distorted born iterative method dbim the effects of focal number f speed of sound contrast deltac and scatterer size were considered for comparison the 3 d wave equation was also inverted using point like transducers to produce fully 3 d dbim image reconstructions in 2 d slicing blurring in the vertical direction was highly correlated with the transmit receive elevation point spread function of the transducers for low deltac the eventual appearance of overshoot artifacts in the vertical direction were observed with increasing deltac these diffraction related artifacts were less severe for smaller focal number values and larger spherical target sizes when using 3 d dbim the overshoot artifacts were not observed and spatial resolution was improved however results indicate that array configuration in 3 d reconstructions is important for good image reconstruction practical arrays were designed and assessed for image reconstruction using 3 d dbim

derivations of r x y z with a slice the question under consideration is whether every locally nilpotent r derivation of r x y z with a slice has kernel a generated by two elements over r where r is a polynomial ring over a field of characteristic zero theorem 1 1 gives a fundamental property of such kernels namely that a is an a2 fibration over r while it is an open question whether every a2 fibration over r is trivial the property of a asserted in the theorem is necessary to the condition that a is a polynomial ring in two variables over r the last section of the paper presents a family of examples n n 1 which are quite simple to define but whose status relative to the kernel question is not known

recovering dielectric constants of explosives via a globally strictly convex cost functional the inverse problem of estimating dielectric constants of explosives using boundary measurements of one component of the scattered electric field is addressed it is formulated as a coefficient inverse problem for a hyperbolic differential equation after applying the laplace transform a new cost functional is constructed and a variational problem is formulated the key feature of this functional is the presence of the carleman weight function for the laplacian the strict convexity of this functional on a bounded set in a hilbert space of an arbitrary size is proven this allows for establishing the global convergence of the gradient descent method some results of numerical experiments are presented

frechet derivatives for some bilinear inverse problems in many inverse problems a functional of u is given by measurements where u solves a partial differential equation of the type l p u su q here q is a known source term and l p s are operators with p as an unknown parameter of the inverse problem for the numerical reconstruction of p the heuristically derived frechet derivative r of the mapping r p rightarrow measurementfunctional of u is often used we show for three problems a transport problem in optical tomography an elliptic equation governing near infrared tomography and the wave equation in moving media that r is the derivative in the strict sense our method is applicable to more general problems than are established methods for similar inverse problems

enhanced multilevel linear sampling methods for inverse scattering problems we develop two enhanced techniques for the multilevel linear sampling method mlsm proposed in 32 for inverse scattering problems under some practical situations the mlsm suffers certain undesirable breakage cells problem we propose to avoid the curse of breakage cells by incorporating expanding and searching techniques the new techniques are shown to significantly improve the robustness of the mlsm and meanwhile they possess the same optimal computational complexity as the mlsm numerical experiments are presented to illustrate the promising features of the enhanced mlsms

on the rigidity of small domains abstract let k be an algebraically closed field of arbitrary characteristic let a be an affine domain over k with transcendence degree 1 which is not isomorphic to k x and let b be a domain over k we show that the ak invariant distributes over the tensor product of a by b as a consequence we obtain a generalization of the cancellation theorem of s abhyankar p eakin and w heinzer

a maximum entropy principle for contractive interpolants abstract in this paper a maximum entropy principle is established for a class of continuous contractive m n matrix valued functions g with preassigned fourier coefficients gj j for j 0 1 the function g in this class which achieves the maximum entropy is characterized in a number of different ways and its entropy is computed in terms of a hankel operator based on the prescribed matrices 0 1 a pair of seemingly new szego type formulas are obtained enroute and some applications to nevanlinna pick interpolation are indicated

globally strictly convex cost functional for a 1 d inverse medium scattering problem with experimental data a new numerical method is proposed for a one dimensional inverse medium scattering problem with multifrequency data this method is based on the construction of a weighted cost functional the weight is a carleman weight function cwf in other words this is the function which is present in the carleman estimate for the undelying differential operator the presence of the cwf makes this functional strictly convex on any a priori chosen ball with the center at 0 in an appropriate hilbert space convergence of the gradient minimization method to the exact solution starting from any point of that ball is proved computational results for both computationally simulated and experimental data show a good accuracy of this method

direct sampling method for diffusive optical tomography in this work we are concerned with the diffusive optical tomography dot problem in the case when only one or two pairs of cauchy data are available we propose a simple and efficient direct sampling method to locate inhomogeneities inside a homogeneous background and solve the dot problem in both full and limited aperture cases this new method is easy to implement and less expensive computationally numerical experiments demonstrate its effectiveness and robustness against noise in the data this provides a new promising numerical strategy for the dot problem

serre s problem for one dimensional ground rings in 131 ming chang kang considered the following question l et r be a commutative noetherian one dimensional ring connected and with finite normalization and let a r x x be the polynomial ring is every bnitely generated projective a module isomorphic to a direct sum of a rank one module and a free module the condition that r be connected is not essential once the result is proved for the connected case it is easily generalized to the case where r is a product of several connected rings if r is a pid this indeed is serre s conjecture and also for r regular i e dedekind the result follows from quillen s localization theorem 4 and the decomposition properties of projective modules over dedekind rings kang solved the question in the affirmative in almost every case the only exception being the case 4 g r and the rank of the projective a module equal to 2 his results extended former partial results of seshadri and murthy see references in 3 for a r x the result follows from plumstead s solution of the first eisenbud evans conjecture 8 here we will till in the gap and prove that the question above always has the answer yes by means of an extension and simplification of kang s arguments which systematically exploits the theory of seminormal rings due to traverso swan and others by standard arguments one may suppose r reduced and connected the case of seminormal r will be handled by means of suslin s results quoting s to perform a reduction to this latter case we seminormalize r and use glueing milnor s theorem to prove the result some questions and remarks conclude the note

on zariski s cancellation problem in positive characteristic abstract in this paper we show that when k is a field of positive characteristic the affine space a k n is not cancellative for any n 3

on the fundamental groups of some open rational surfaces if d is an irreducible curve with ordinary nodes in a relatively minimal smooth projective rational surface m then from the solution of zanski s problem in 12 proposition 3 27 it follows that n m d is abelian on the other hand in 15 o zariski has given an example of an irreducible sextic d in ip 2 with six ordinary cusps lying on a conic such that q f 2 d is an infinite group in view of this the following conjecture seems significant

numerical solution of a coefficient inverse problem with multi frequency experimental raw data by a globally convergent algorithm abstract we analyze in this paper the performance of a newly developed globally convergent numerical method for a coefficient inverse problem for the case of multi frequency experimental backscatter data associated to a single incident wave these data were collected using a microwave scattering facility at the university of north carolina at charlotte the challenges for the inverse problem under the consideration are not only from its high nonlinearity and severe ill posedness but also from the facts that the amount of the measured data is minimal and that these raw data are contaminated by a significant amount of noise due to a non ideal experimental setup this setup is motivated by our target application in detecting and identifying explosives we show in this paper how the raw data can be preprocessed and successfully inverted using our inversion method more precisely we are able to reconstruct the dielectric constants and the locations of the scattering objects with a good accuracy without using any advanced a priori knowledge of their physical and geometrical properties

a globally convergent numerical method for a coefficient inverse problem a new globally convergent numerical method is developed for a multidimensional coefficient inverse problem for a hyperbolic pde with applications in acoustics and electromagnetics on each iterative step the dirichlet boundary value problem for a second order elliptic equation is solved the global convergence is rigorously established and numerical experiments are presented

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

a computational quasi reversiblility method for cauchy problems for laplace s equation this work concerns the use of the method of quasi reversibility for solving cauchy problems for laplace s equation the paper begins with a derivation of an error estimate for this method using car

a method of solving the coefficient inverse problems of wave tomography abstract in this paper the problem of nonlinear wave tomography is formulated as a coefficient inverse problem for a hyperbolic equation in the time domain efficient methods for solving the inverse problems of wave tomography for the case of transparent boundary conditions are presented the algorithms are designed for supercomputers we prove the frechet differentiability theorem for the residual functional and derive an exact expression for the frechet derivative in the case of a transparent boundary in the direct and conjugate problems the expression for the frechet derivative of the residual functional remains valid if the experimental data are provided for only a part of the boundary the effectiveness of the proposed method is illustrated by the numerical solution of a model problem of low frequency wave tomography the model problem is tailored to apply to the differential diagnosis of breast cancer

on double danielewski surfaces and the cancellation problem abstract we study a two dimensional family of affine surfaces which are counter examples to the cancellation problem we describe the makar limanov invariant of these surfaces determine their isomorphism classes and characterize the automorphisms of these surfaces

state space formulae for all stabilizing controllers that satisfy and h norm bound and relations to risk sensitivity abstract given a linear system all stabilizing controllers such that a specified closed loop transfer function has h norm less than a given scalar are parametrized this characterization involves the solution to two algebraic riccati equations each with the same order as the system and further gives gives feasible controllers also with this order the relationship to the risk sensitive lqg stochastic control problem is established giving an equivalence between robust and stochastic control

solutions to the h general distance problem which minimize an entropy integral abstract we pose and solve the problem of minimizing the entropy of an h norm bounded and stabilized closed loop solution proceeds via the equivalent error system distance problem the central member of the admissible class is shown to minimize the entropy at infinity and in that case an explicit state space formula is derived for the minimum value of the entropy links between entropy h 2 norms and h 2 optimal control are given

seminormality projective algebras and invertible algebras unlike module theory not much is known about the projective objects in the category of commutative r algebras where r is a fixed commutative ground ring the similarity between projective r modules and projective ralgebras does not extend much farther to our knowledge than to the following simple facts the free commutative r algebras namely r ti is t are projective and all projectives are retracts of free objects indeed this characterizes the projective r algebras in the definition of projectivity for algebras one must be sure to specify which epimorphisms one wants to admit if all epimorphisms of r alg the category of commutative r algebras are admitted one can show that only r itself is projective it is very reasonable to admit just the surjections which is what we are doing examples nearly as obvious as the free algebras can be found by taking the symmetric r algebra s r over a projective r module p and this algebra is obviously projective but free only if p is a free r module 3 lemma 4 61 besides these there are not many known examples abhyankar eakin and heinzer 1 4 1 and costa 5 3 5 proved some affirmative results if r is a field or more generally a ufd then every projective r algebra in one variable is a symmetric algebra of some rmodule p which must be a rank one projective module in this paper we call a domain a over a domain r an algebra in one variable over r if a is of finite type over r and quot a has transcendency degree 1 over quot r for our definition in case r is not a domain see definition 1 11 below as to the case of two variables we can say something for r a perfect field due to the recent general solution of the cancellation problem any projective kalgebra a of finite type over the perfect field k with tr deg a 2 is isomorphic to k x y for a proof we refer to 10 theorems 3 and 41 a slight change is needed in that author s theorem 4 we do not go into the details here

convergent algorithm based on carleman estimates for the recovery of a potential in the wave equation this article develops the numerical and theoretical study of the reconstruction algorithm of a potential in a wave equation from boundary measurements using a cost functional built on weighted energy terms coming from a carleman estimate more precisely this inverse problem for the wave equation consists in the determination of an unknown time independent potential from a single measurement of the neumann derivative of the solution on a part of the boundary while its uniqueness and stability properties are already well known and studied a constructive and globally convergent algorithm based on carleman estimates for the wave operator was recently proposed in bdbe13 however the numerical implementation of this strategy still presents several challenges that we propose to address here

some results on open algebraic surfaces of logarithmic kodaira dimension zero abstract let s be a smooth open algebraic surface of logarithmic kodaira dimension zero and v d a pair of a smooth projective surface v and a simple normal crossing divisor d on v such that v d s we prove that v d 2 where d is the number of all irreducible components of supp d and determine the surface s with v d 2 and those with v d 1 and p g s 0 furthermore by using these results we determine the affine plane curves whose complements have logarithmic kodaira dimension zero

discounted linear exponential quadratic gaussian control in this note we describe a recursive formulation of discounted costs for a linear quadratic exponential gaussian linear regulator problem which implies time invariant linear decision rules in the infinite horizon case time invariance in the discounted case is attained by surrendering state separability of the risk adjusted costs

risk sensitive estimation and a differential game a large deviation result is employed to solve the state estimation problem of a continuous time gauss markov system with an exponential cost the exponential cost is the expected value of an exponential function of the state estimation error a scalar spl theta appearing in the cost termed as the risk factor determines the penalty on the higher order moments of the error in contrast to the minimum variance estimate penalty on large deviations of the estimation error is possible

on generalized cancellation problem abstract a well known cancellation problem of zariski asks whether for two given domains fields k 1 and k 2 an isomorphism of k 1 t k t and k 2 t k 2 t implies an isomorphism of k 1 and k 2 in this paper we address a related problem whether the ring field embedding of k 1 t k 1 t into k 2 t k 2 t implies the ring field embedding of k 1 into k 2 our main result is affirmative if k 1 and k 2 are arbitrary domains fields of the finite transcendence degree and k 1 t k 1 t can be embedded into k 2 t k 2 t then k 1 can be embedded into k 2 as a consequence we answer a question of abhyankar eakin and heinzer j algebra 23 1972 310 342

zariski cancellation problem for non domain noncommutative algebras we study zariski cancellation problem for noncommutative algebras that are not necessarily domains

an algebraic proof of a cancellation theorem for surfaces let k be an algebraically closed field of arbitrary characteristic we give a self contained algebraic proof of the following statement if v is an affine surface over k such that v k k3 then v k2 this fact which is due to fujita miyanishi sugie and russell solves the zariski cancellation problem for surfaces to achieve our proof we first show that if a is a finitely generated domain with ak a a then ak a x a

convexification of a 3 d coefficient inverse scattering problem abstract a version of the so called convexification numerical method for a coefficient inverse scattering problem for the 3d helmholtz equation is developed analytically and tested numerically backscattering data are used which result from a single direction of the propagation of the incident plane wave on an interval of frequencies the method converges globally the idea is to construct a weighted tikhonov like functional the key element of this functional is the presence of the so called carleman weight function cwf this is the function which is involved in the carleman estimate for the laplace operator this functional is strictly convex on any appropriate ball in a hilbert space for an appropriate choice of the parameters of the cwf thus both the absence of local minima and convergence of minimizers to the exact solution are guaranteed numerical tests demonstrate a good performance of the resulting algorithm unlikeprevious the so called tail functions globally convergent method we neither do not impose the smallness assumption of the interval of wavenumbers nor we do not iterate with respect to the so called tail functions

global optimization methods for multimodal inverse problems abstract global optimization methods such as simulated annealing and genetic algorithms are potentially useful in attacking the multimodal search calculations which arise in a number of geophysical inverse problems in the one dimensional waveform inversion problem considered here the optimization method must find a one dimensional earth structure which produces a seismogram that agrees with an observed seismogram both simulated annealing and genetic algorithms provide satisfactory performance when the earth structure has only 15 free parameters as this number is increased to 22 and then 30 parameters both techniques become more costly genetic algorithms however still yielded accurate solutions for problems with 30 free parameters a point at which simulated annealing was only marginally useful the superior performance of genetic algorithms may reflect the nonproximate search methods used by them or possibly the more complex and capacious memory available to a genetic algorithm for storing its accumulated experience

absolutely integral homomorphisms abstract absolutely integral algebras over a field are the subject of a chapter in field theory they are also known as geometrically integral algebras in algebraic geometry we generalize this concept to arbitrary commutative algebras our first main result is that a ring homomorphism is absolutely integral if and only if it becomes flat after an integral base change and its fibers are geometrically integral these homomorphisms are characterized with the help of absolute integral closedness property and do have some transcendence properties they generally preserve normality regularity and the complete intersection property

a note on retracts of polynomial rings in three variables abstract for retracts of the polynomial ring in 1 costa asks us whether every retract of k n is also the polynomial ring or not where k is a field in this paper we give an affirmative answer in the case where k is a field of characteristic zero and n 3

a risk sensitive maximum principle abstract a stochastic maximum principle is derived which is formally of general application and so not restricted to the diffusion case it moreover differs from the classic deterministic principle only by a modification of the hamiltonian and so requires no solution of stochastic differential equations or calculation of conditional expectations the two essential points in its construction are that one considers the family of criterion functions exponential in an additive cost function of which the conventional expected cost case is a degenerate member and that in the non lqg case appeal to large deviation theory must be valid

on the cancellation problem for the affine space mathbb a 3 in characteristic p we show that the cancellation conjecture does not hold for the affine space mathbb a 3 k over any field k of positive characteristic we prove that an example of t asanuma provides a three dimensional k algebra a for which a is not isomorphic to k x1 x2 x3 although a t is isomorphic to k x1 x2 x3 x4

the structure of a laurent polynomial fibration in n variables abstract bass connell and wright have proved that any finitely presented locally polynomial algebra in n variables over an integral domain r is isomorphic to the symmetric algebra of a finitely generated projective r module of rank n in this paper we prove a corresponding structure theorem for a ring a which is a locally laurent polynomial algebra in n variables over an integral domain r viz we show that a is isomorphic to an r algebra of the form sym r q i 1 where q is a direct sum of n finitely generated projective r modules of rank one and i is a suitable invertible ideal of the symmetric algebra sym r q further we show that any faithfully flat algebra over a noetherian normal domain r whose generic and codimension one fibres are laurent polynomial algebras in n variables is a locally laurent polynomial algebra in n variables over r

entropy minimizing and risk sensitive control rules abstract glover and doyle 4 have shown that the entropy minimising criterion is identical with the infinite horizon form of the leqg optimality criterion leqg linear exponential of quadratic gaussian whittle and kuhn 11 have given a hamiltonian formulation which provides a natural treatment of leqg optimisation this paper contains a direct proof that the control yielded by these hamiltonian methods is entropy minimising the conclusion would follow from the two assertions above but a simple adjoining of the proofs of these two assertions yields an argument so circuitous that a direct proof is of interest

a risk sensitive maximum principle the case of imperfect state observation the risk sensitive maximum principle for optimal stochastic control derived by the author in an earlier work system control letters vol 15 1990 is restated this is an immediate generalization of the classic pontryagin principle to which it reduces in the deterministic case and is expressed immediately in terms of observables it is derived on the assumption that the criterion function is the exponential of an additive cost function and is exact under linear quadratic gaussian assumptions but is otherwise valid as a large deviation approximation the principle is extended to the case of imperfect state observation after preliminary establishment of a certainty equivalence principle the derivation yields as byproduct a large deviation version of the updating equation for nonlinear filtering the development is heuristic it is believed that the mathematical arguments given are the essential ones and provide a self contained treatment at this level

a hamiltonian formulation of risk sensitive linear quadratic gaussian control we consider the optimal control problem with time homogeneous linear and in general non markov dynamics and risk sensitive criterion appeal to the extremal principle prescribed by the risk sensitive certainty equivalence principle rscep yields a symmetric equation system indicating that the extended hamiltonian formulation generalizes naturally to the risk sensitive case the conjugate variable of a hamiltonian formulation now has an interpretation in terms of forecasted process noise and the rscep in fact provides a stochastic maximum principle for which all variables have a clear interpretation and the desired measurability properties in the infinite horizon case meaningful under generalized controllability conditions optimal control is determined explicitly in terms of a canonical factorization for the case of imperfect process observation the rscep leads to coupled equation systems that can again be solved in terms of canonical factorizations in the time invariant stationary case

on affine ruled irrational surfaces let k be an algebraically closed field of characteristic p 0 and let x be a not necessarily projective nonsingular algebraic surface over k according to russell 11 we call x qffine ruled if x contains a non empty open set isomorphic to the product of a curve with a the affine line such an open set is called a cylinderlike open set assume that the logarithmic kodaira dimension x in the sense of iitaka 4 see also kambayashi 5 and russell 11 for the definition in case chark 0 is if x is rational and connected at infinity then x is affine ruled this result was proved in fujita 2 and miyanishi sugie 9 in case c h a r k 0 and proved in 11 in case c h a r k 0 see also miyanishi 8 the proof is valid in case c h a r k 0 without any modification we note that x is not necessarily affine ruled if x is rational and not connected at infinity see sugie 12 8 and 11 in case x is irrational it is shown in 9 that x is also affine ruled provided char k 0 in this article we shall prove first the following