circular planar graphs and resistor networks abstract we consider circular planar graphs and circular planar resistor networks associated with each circular planar graph there is a set p q of pairs of sequences of boundary nodes which are connected through a graph is called critical if removing any edge breaks at least one of the connections p q in we prove that two critical circular planar graphs are y equivalent if and only if they have the same connections if a conductivity is assigned to each edge in there is a linear from boundary voltages to boundary currents called the network response this linear map is represented by a matrix we show that if is any circular planar resistor network whose underlying graph is critical then the values of all the conductors in may be calculated from finally we give an algebraic description of the set of network response matrices that can occur for circular planar resistor networks

modules of finite projective dimension with negative intersection multiplicities let x be the image of x in r the main object of this paper is to study modules of finite length and finite projective dimension over r one consequence of our study will be counterexamples to several conjectures concerning the behavior of intersection multiplicities for modules of finite projective dimension let denote length if s is a noetherian ring m n are finitely generated s modules l m is finite and d pdsm we may define following serre is d zs m n z m n 1 tors m n

lech s inequality the stuckrad vogel conjecture and uniform behavior of koszul homology abstract let r m be a noetherian local ring and let m be a finitely generated r module of dimension d we prove that the set l m i m e i m i m is bounded below by 1 d e r where r r ann m moreover when m is equidimensional this set is bounded above by a finite constant depending only on m the lower bound extends a classical inequality of lech and the upper bound answers a question of stuckrad vogel in the affirmative as an application we obtain results on uniform behavior of the lengths of koszul homology modules

on stabilization and the existence of coprime factorizations shows that any transfer function matrix whose elements belong to the quotient field of h sub infinity and which is stabilizable has a matrix fraction representation over h sub infinity which is coprime in the sense that a matrix bezout identity can be satisfied

on spl hscr sub spl infin control for dead time systems a mixed sensitivity spl hscr sub spl infin problem is solved for dead time systems it is shown that for a given bound on the spl hscr sub spl infin norm causal stabilizing controllers exist that achieve this bound if and only if a related finite dimensional riccati equation has a solution with a certain nonsingularity property in the case of zero time delay the riccati equation is a standard riccati equation and the nonsingularity condition is that the solution be nonnegative definite for nonzero time delay the nonsingularity condition is more involved but still allows us to obtain controllers all suboptimal controllers are parameterized and the central controller is shown to be a feedback interconnection of a finite dimensional system and a finite memory system both of which can be implemented some spl hscr sub spl infin problems are rewritten as pure rational spl hscr sub spl infin problems using a smith predictor parameterization of the controller

a connection between state space and doubly coprime fractional representations explicit formulas for a doubly coprime fractional representation of the transfer function of a lumped lti system are given in terms of a stabilizable and detectable state space realization of the transfer function these formulas allow existing computational algorithms to be utilized for the purpose of computing doubly coprime factorizations several additional implications of this result are briefly discussed as are some immediate extensions and future research directions

lech s conjecture in dimension three abstract let r m s n be a flat local extension of local rings lech conjectured around 1960 that there should be a general inequality e r e s on the hilbert samuel multiplicities 24 this conjecture is known when the base ring r has dimension less than or equal to two 24 and remains open in higher dimensions in this paper we prove lech s conjecture in dimension three when r has equal characteristic in higher dimension our method yields substantial partial estimate e r d 2 d e s where d dim r 4 in equal characteristic

effects of measurement precision and finite numbers of electrodes on linear impedance imaging algorithms impedance imaging systems reconstruct approximations to the conductivity inside a body from electrical measurements made on the boundary the linearized problem was studied by calderon who showed that if the conductivity differs little from a constant then it can be reconstructed approximately in his work calderon assumed that arbitrary current densities can be applied anywhere on the body s surface and the resulting voltages perfectly measured in practice however imaging systems apply currents through a finite number of electrodes on the surface and measure the resulting voltages with a limited precision in this paper linear algorithms are described for approximating the electrical conductivity in a disk from finitely many limited precision measurements the effects of finite precision and electrode number on a two dimensional version of calderon s method are studied it is shown that as the measurement errors go to zero and the electrodes proliferate in a space filling manner the finite electro

hilbert kunz multiplicity and an inequality between multiplicity and colength abstract in this paper we study local rings of small hilbert kunz multiplicity in particular we prove that an unmixed local ring of hilbert kunz multiplicity one is regular and classify two dimensional cohen macaulay local rings whose hilbert kunz multiplicity is 2 or less also we investigate the inequality between the multiplicity and the colength of the tight closure of parameter ideals inverse to the usual inequality between multiplicity and colength

electroid varieties and a compactification of the space of electrical networks abstract we construct a compactification of the space of circular planar electrical networks studied by curtis ingerman morrow 4 and colin de verdiere gitler vertigan 3 using cactus networks we embed this compactification as a linear slice of the totally nonnegative grassmannian and relate kenyon and wilson s grove measurements to postnikov s boundary measurements intersections of the slice with the positroid stratification leads to a class of electroid varieties indexed by matchings the uncrossing partial order on matchings arising from electrical networks is shown to be dual to a subposet of affine bruhat order the analogues of matroids in this setting are certain distinguished collections of non crossing partitions

superregular matrices and applications to convolutional codes abstract the main results of this paper are twofold the first one is a matrix theoretical result we say that a matrix is superregular if all of its minors that are not trivially zero are nonzero given a a b a b superregular matrix over a field we show that if all of its rows are nonzero then any linear combination of its columns with nonzero coefficients has at least a b 1 nonzero entries secondly we make use of this result to construct convolutional codes that attain the maximum possible distance for some fixed parameters of the code namely the rate and the forney indices these results answer some open questions on distances and constructions of convolutional codes posted in the literature 6 9

decoupling of measurable signals via self bounded controlled invariant subspaces minimal unassignable dynamics of feedforward units for prestabilized systems a dynamic feedforward scheme allows measurable signal decoupling to be solved independently of other problems simultaneously present in the design of a control system like e g stabilization robustness insensitivity to disturbances the synthesis procedure based on the properties of self bounded controlled invariant subspaces ensures the minimal complexity of the feedforward unit in terms of minimal unassignable dynamics and minimal dynamic order in the case of left invertible systems and on some specific conditions also in the case of non left invertible systems the output dynamic feedback set up to guarantee stability or more generally robustness or insensitivity properties does not affect the complexity of the feedforward unit since the peculiar layout where the feedback unit receives an additional input from the precompensator preserves in the extended system exactly the same set of unassignable internal eigenvalues of the minimal self bounded controlled invariant subspace as that defined for the original system the overall control structure turns out to be a two degree of freedom controller completely devised in the geometric context

parameter and state model reduction for large scale statistical inverse problems a greedy algorithm for the construction of a reduced model with reduction in both parameter and state is developed for an efficient solution of statistical inverse problems governed by partial differential equations with distributed parameters large scale models are too costly to evaluate repeatedly as is required in the statistical setting furthermore these models often have high dimensional parametric input spaces which compounds the difficulty of effectively exploring the uncertainty space we simultaneously address both challenges by constructing a projection based reduced model that accepts low dimensional parameter inputs and whose model evaluations are inexpensive the associated parameter and state bases are obtained through a greedy procedure that targets the governing equations model outputs and prior information the methodology and results are presented for groundwater inverse problems in one and two dimensions

a uniqueness theorem for an inverse boundary value problem in electrical prospection we show that a near constant conductivity of a two dimensional body can be uniquely determined by steady state direct current measurements at the boundary mathematically we show that the coefficient in the operator is uniquely determined by its dirichlet integrals

omega harmonic functions and inverse conductivity problems on networks in this paper we discuss the inverse problem of identifying the connectivity and the conductivity of the links between adjacent pair of nodes in a network in terms of an input output map to do t

intersection multiplicities over gorenstein rings let r be a complete local ring of dimension d over a perfect field of prime characteristic p and let m be an r module of finite length and finite projective dimension s dutta showed that the equality lim n to infty frac ell f n r m p nd ell m holds when the ring r is a complete intersection or a gorenstein ring of dimension at most 3 we construct a module over a gorenstein ring r of dimension five for which this equality fails to hold this then provides an example of a nonzero todd class tau 3 r and of a bounded free complex whose local chern character does not vanish on this class

brief every stabilizing dead time controller has an observer predictor based structure this paper considers the stabilization problem for linear time invariant systems with a single delay h in the feedback loop two parametrizations of all stabilizing controllers are derived in terms of a state space realization of the rational part of the plant these parametrizations have simple structures and clear interpretations the first is in the form of a generalized dead time compensator smith predictor and the second is in the observer predictor form applications of the proposed parametrization to the h 2 optimization and the robust stabilization problems are discussed

on the construction of graphs with a planar bipartite double cover from boolean formulas and its application to counting satisfying solutions abstract we address the problem to construct a graph g with binary edge weights from the incidence graph of a boolean formula in a way that g has a planar bipartite double cover g this allows one to use the identity between the permanent of g s adjacency matrix a g and the number of perfect matchings of g i e perm a g pm g due to the algorithm of fisher kasteleyn and temperlay fkt the right hand side of the equation can be counted in polynomial time if g is planar we prove a theorem that describes necessary conditions for the gadgets i e small subgraphs with certain properties of g to allow a planar bipartite double cover g for arbitrary 3cnf formulas and the known approaches to build g our theorem shows that gadgets which enable a planar bipartite double cover do not exist however we show that for certain classes of boolean formulas e g 5 pl rtw 2cnf 7 pl rtw 2cnf and 3 forest 3cnf such gadgets exist hence these counting problems are in p the results probably extended to other moduli p and perhaps further counting classes

positive solutions for discrete boundary value problems involving the p laplacian with potential terms in this paper we discuss the existence and the uniqueness of discrete boundary value problems involving the p laplacian with potential terms to this end we deal with the strong comparison principle and the dirichlet principle for the p laplacian with potential terms on weighted graphs with boundary moreover we provide a lower bound of the potential terms which guarantees the existence of solutions of the boundary value problem for the operator finally we also discuss inverse conductivity and potential problems for the operator

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

overdetermined partial boundary value problems on finite networks abstract in this study we define a class of non self adjoint boundary value problems on finite networks associated with schrodinger operators the novel feature of this study is that no data are prescribed on part of the boundary whereas both the values of the function and of its normal derivative are given on another part of the boundary we show that overdetermined partial boundary value problems are crucial for solving inverse boundary value problems on finite networks since they provide the theoretical foundations for the recovery algorithm we analyze the uniqueness and the existence of solution for overdetermined partial boundary value problems based on the nonsingularity of partial dirichlet to neumann maps these maps allow us to determine the value of the solution in the part of the boundary where no data was prescribed we also execute full conductance recovery for spider networks

equimultiplicity in hilbert kunz theory we study further the properties of hilbert kunz multiplicity as a measure of singularity this paper develops a theory of equimultiplicity for hilbert kunz multiplicity and applies it to study the behavior of hilbert kunz multiplicity on the brenner monsky hypersurface a number of applications follows in particular we show that hilbert kunz multiplicity attains infinitely many values and that equimultiple strata may not be locally closed

realizable response matrices of multi terminal electrical acoustic and elastodynamic networks at a given frequency we give a complete characterization of the possible response matrices at a fixed frequency of n terminal electrical networks of inductors capacitors resistors and grounds and of n terminal discrete linear elastodynamic networks of springs and point masses both in three dimensional and two dimensional cases specifically we construct networks that realize any response matrix that is compatible with the known symmetry properties and thermodynamic constraints of response matrices owing to a mathematical equivalence we also obtain a characterization of the response matrices of discrete acoustic networks

asymptotic lech s inequality abstract we explore the classical lech s inequality relating the hilbert samuel multiplicity and colength of an m primary ideal in a noetherian local ring r m we prove optimal versions of lech s inequality for sufficiently deep ideals in characteristic p 0 and we conjecture that they hold in all characteristics our main technical result shows that if r m has characteristic p 0 and r is reduced equidimensional and has an isolated singularity then for any sufficiently deep m primary ideal i the colength and hilbert kunz multiplicity of i are sufficiently close to each other more precisely for all e 0 there exists n 0 such that for any i r with l r i n we have 1 e l r i e hk i 1 e l r i

recovering a quantum graph spectrum from vertex data we study the question to what extent spectral information on a schrodinger operator on a finite compact metric graph subject to standard or type matching conditions can be recovered from a corresponding titchmarsh weyl function on the boundary of the graph in contrast to the case of ordinary or partial differential operators the knowledge of the titchmarsh weyl function is in general not sufficient for recovering the complete spectrum of the operator or the potentials on the edges however it is shown that those eigenvalues with sufficiently high depending on the cyclomatic number of the graph multiplicities can be recovered moreover we prove that under certain additional conditions the titchmarsh weyl function even contains the full spectral information

determining the resistors in a network in this paper it is shown that the resistors in a rectangular network can be determined by measurements at the boundary of the voltages generated by imposed currents an algorithm for using the boundary measurements to compute the resistances is also given

coprime factorization for regular linear systems mild sufficient conditions are given for the existence of a doubly coprime factorization of the transfer function of a regular linear system as well as formulae for such a factorization the results are illustrated by two examples of delay systems one of which has infinitely many unstable poles

a feedback theory of two degree of freedom optimal wiener hopf design the design of linear two degree of freedom stabilizing controllers is treated in a quadratic cost setting the class of all such controllers which give finite cost is established and the tradeoff possible between optimum performance tracking cost sensitivity and stability margins is discussed

a backprojection algorithm for electrical impedance imaging the authors study a two dimensional reconstruction algorithm due to d c barber and b h brown applied to a linearized electrostatic inverse problem first the authors demonstrate how this algorithm fits within the framework of inverses of generalized radon transforms studied by g beylkin second an iterative improvement of the barber brown algorithm is constructed based on the conjugate residual method several numerical results obtained with this iterative algorithm are presented

toric networks geometric r matrices and generalized discrete toda lattices we use the combinatorics of toric networks and the double affine geometric r matrix to define a three parameter family of generalizations of the discrete toda lattice we construct the integrals of motion and a spectral map for this system the family of commuting time evolutions arising from the action of the r matrix is explicitly linearized on the jacobian of the spectral curve the solution to the initial value problem is constructed using riemann theta functions

length approximations for independently generated ideals abstract we study ideals primary to the maximal ideal of a commutative noetherian local ring when such an ideal is generated by elements which are independent in the sense of c lech we prove a lower bound on the length of the quotient ring in terms of the orders of the generators as a corollary we obtain a substantial partial result on lech s conjecture on the multiplicities of a flat couple of local rings

convergence of a reconstruction method for the inverse conductivity problem the inverse conductivity problem is that of reconstructing a spatially varying isotropic conductivity in the interior of some region by means of steady state measurements taken at the boundary reconstruction schemes including least squares type minimization methods have been widely studied and implemented but convergence analysis has been largely ignored this paper establishes the convergence of a well known least squares minimization scheme the levenberg marquardt method on a regularized formulation of the inverse conductivity problem

optimal experimental design for inverse problems with state constraints in this paper we address the challenging problem of optimal experimental design oed of inverse problems with state constraints we consider two oed formulations that allow us to reduce the exper

the positive orthogonal grassmannian and loop amplitudes of abjm in this paper we study the combinatorics associated with the positive orthog onal grassmannian ogk and its connection to abjm scattering amplitudes we present a canonical embedding of ogk into the grassmannian gr k 2k from which we deduce the canonical volume form that is invariant under equivalence moves remarkably the canon ical forms of all reducible graphs can be converted into irreducible ones with products of d log forms unlike n 4 super yang mills here the jacobian plays a crucial role to ensure the d log form of the reduced representation furthermore in the positive region we identify the functional map that arises from the triangle equivalence move as a 3 string scattering s matrix which satises the tetrahedron equations by zamolodchikov implying 2 1 dimensional integrability we study the solution to the bcfw recursion relation for loop amplitudes and demonstrate the presence of all physical singularities as well as the absence of all spurious ones the on shell diagram solution to the loop recursion relation exhibits manifest two site cyclic symmetry and reveals that to all loop four and six point amplitudes only have logarithmic singularities

on spectral properties of signed laplacians with connections to eventual positivity signed graphs have appeared in a broad variety of applications ranging from social networks to biological networks from distributed control and computation to power systems in this paper we investigate spectral properties of signed laplacians for undirected signed graphs we find conditions on the negative weights under which a signed laplacian is positive semidefinite via the kron reduction and multiport network theory for signed laplacians that are indefinite we characterize their inertias with the same framework furthermore we build connections between signed laplacians generalized m matrices and eventually exponentially positive matrices

global uniqueness for a two dimensional inverse boundary value problem we show that the coefficient y x of the elliptic equation vie qyvu 0 in a two dimensional domain is uniquely determined by the corresponding dirichlet to neumann map on the boundary and give a reconstruction pro

the dirichlet to neumann map for a resistor network relations that characterize the dirichlet to neumann map for a square network of resistors are given using these relations a parametrization of the set of dirichlet to neumann maps is given

comment on calderon s paper on an inverse boundary value problem calder6n determined a method to approximate the conductivity a of a con ducting body in rn for n 2 based on measurements of boundary data the approximation is good in the l norm provided that the conductivity is a small perturbation from a constant we calculate the approximation exactly for the case of homogeneous concentric conducting disks in r2 with different conductivities here the difference in the conductivities is the perturbation we show that the approximation yields precise information about the spatial variation of a even when the perturbation is large this ability to distinguish spatial regions with different conductivities is important for clinical monitoring applications

the inverse problem of resistivity sounding the electric potential due to a single point electrode at the surface of a layered conducting medium is calculated by means of a linear combination of the potentials associated with a set of two layer systems this new representation is called the bilayer expansion for the green s function it enables the forward problem of resistivity sounding to be solved very efficiently even for complicated profiles also the bilayer expansion facilitates the solution of the resistivity inverse problem the coefficients in the expansion are linearly related to apparent resistivity as it is measured and they are readily mapped into parameters for a model specifically i consider models comprising uniformly conducting layers of equal thickness for a given finite data set a quadratic program can be used to find the best fitting model in this class for any specified thickness as the thickness is reduced models of this kind can approximate arbitrary profiles with unlimited accuracy if there is a model that satisfies

multifrequency trans admittance scanner mathematical framework and feasibility a trans admittance scanner tas is a device for breast cancer diagnosis based on numerous experimental findings that complex conductivities of breast tumors significantly differ from those of surrounding normal tissues in tas we apply a sinusoidal voltage between a hand held electrode and a scanning probe placed on the breast skin to make current travel through the breast the scanning probe has an array of electrodes at zero voltage we measure exit currents neumann data through the electrodes that provide a map of trans admittance data over the breast surface the inverse problem of tas is to detect a suspicious abnormality underneath the breast skin from the measured neumann data previous anomaly detection methods used the difference between the measured neumann data and a reference neumann data obtained beforehand in the absence of anomaly however in practice the reference data is not available and its computation is not possible since the inhomogeneous complex conductivity of the normal breast is unknown to deal with this problem we propose a frequency difference tas fdtas in which a weighted frequency difference of the trans admittance data measured at a certain moment is used for anomaly detection this paper provides a mathematical framework and the feasibility of fdtas by showing the relationship between the anomaly information and the weighted frequency difference of the neumann data

f signature exists suppose r is a d dimensional reduced f finite noetherian local ring with prime characteristic p 0 and perfect residue field let r 1 p e be the ring of p e th roots of elements of r for e and let a e denote the maximal rank of a free r module appearing in a direct sum decomposition of r 1 p e we show the existence of the limit s r lim e to infty frac a e p ed called the f signature of r this invariant which can be extended to all local rings in prime characteristic was first formally defined by c huneke and g leuschke in math ann 324 2 391 404 2002 and has previously been shown to exist only in special cases the proof of our main result is based on the development of certain uniform hilbert kunz estimates of independent interest additionally we analyze the behavior of the f signature under finite ring extensions and recover explicit formulae for the f signatures of arbitrary finite quotient singularities

bezout factors and l sup 1 optimal controllers for delay systems using a two parameter compensator scheme the authors consider the simultaneous problem of optimal robust stabilization and optimal tracking for single input single output siso systems in an l sup spl infin setting using a two parameter compensator scheme optimal robustness is linked to the work done by georgiou and smith 1992 in the l sup 2 setting optimal tracking involves the resolution of l sup 1 optimization problems the authors consider in particular the robust control of delay systems they determine explicit expressions of the bezout factors for general delay systems which are in the callier desoer class spl bscr spl circ 0 finally they solve several general l sup 1 optimization problems and give an algorithm to solve the optimal robust control problem for a large class of delay systems

overdetermined partial resolvent kernels for finite networks abstract in 2 a study of the existence and uniqueness of solution of partial overdetermined boundary value problems for finite networks was performed these problems involve schrodinger operators and the novel feature is that no data are prescribed on part of the boundary whereas both the values of the function and of its normal derivative are given on another part of the boundary in the present work we study the resolvent kernels associated with overdetermined partial boundary value problems on finite network and we express them in terms of the well known green operator and the dirichlet to robin map moreover we analyze their main properties and we compute them in the case of a generalized cylinder the obtained expression involve polynomials that can be seen as a generalization of chebyshev polynomials and indeed when the conductances along axes are constant the expressions for the overdetermined partial resolvent kernels are given in terms of second kind chebyshev polynomials

numerical equivalence defined on chow groups of noetherian local rings in the present paper we define a notion of numerical equivalence on chow groups or grothendieck groups of noetherian local rings which is an analogue of that on smooth projective varieties under a mild condition it is proved that the chow group modulo numerical equivalence is a finite dimensional vector space as in the case of smooth projective varieties numerical equivalence on local rings is deeply related to that on smooth projective varieties for example if grothendieck s standard conjectures are true then a vanishing of chow group of local rings modulo numerical equivalence can be proven using the theory of numerical equivalence the notion of numerically roberts rings is defined it is proved that a cohen macaulay local ring of positive characteristic is a numerically roberts ring if and only if the hilbert kunz multiplicity of a maximal primary ideal of finite projective dimension is always equal to its colength numerically roberts rings satisfy the vanishing property of intersection multiplicities we shall prove another special case of the vanishing of intersection multiplicities using a vanishing of localized chern characters

loop erased walks and total positivity we consider matrices whose elements enumerate weights of walks in planar directed weighted graphs not necessarily acyclic these matrices are totally nonnegative more precisely all their minors are formal power series in edge weights with nonnegative coefficients a combinatorial explanation of this phenomenon involves loop erased walks applications include total positivity of hitting matrices of brownian motion in planar domains

on the solvability of the discrete conductivity and schroźdinger inverse problems we study the uniqueness question for two inverse problems on graphs both problems consist in finding possibly complex edge or nodal based quantities from boundary measurements of solutions to the dirichlet problem associated with a weighted graph laplacian plus a diagonal perturbation the weights can be thought of as a discrete conductivity and the diagonal perturbation as a discrete schro dinger potential we use a discrete analogue to the complex geometric optics approach to show that if the linearized problem is solvable about some conductivity or schro dinger potential then the linearized problem is solvable for almost all conductivities or schro dinger potentials in a suitable set we show that the conductivities or schro dinger potentials in a certain set are determined uniquely by boundary data except on a zero measure set this criterion for solvability is used in a statistical study of graphs where the conductivity or schro dinger inverse problem is solvable

inverse problem in cylindrical electrical networks in this paper we study the inverse dirichlet to neumann problem for certain cylindrical electrical networks we define and study a birational transformation acting on cylindrical electrical networks called the electrical r matrix we use this transformation to formulate a general conjectural solution to this inverse problem on the cylinder this conjecture extends work of curtis ingerman and morrow linear algebra appl 283 1998 pp 115 150 and of de verdiere gitler and vertigan comment math helv 71 1996 pp 144 167 for circular planar electrical networks we show that our conjectural solution holds for certain purely cylindrical networks here we apply the grove combinatorics introduced by kenyon and wilson trans amer math soc 363 2011 pp 1325 1364

existence and uniqueness for electrode models for electric current computed tomography the following experiment is considered to a body of given conductivity a number of electrodes are attached through which current is sent on the same electrodes the resulting voltages are measured this experiment can be described by a number of mathematical models k s cheng et al ieee transactions on biomedical engineering 36 1989 pp 918 924 these models are discussed and their predictions compared with experiment in particular a model is exhibited that is capable of predicting the experimentally measured voltages to within 0 1 percent for this model existence and uniqueness of the associated electrical potential is proved

on the continuum limit of a discrete inverse spectral problem on optimal finite difference grids we consider finite difference approximations of solutions of inverse stur m liouville problems in bounded intervals using three point finite difference sc hemes we discretize the equations on so called optimal grids constructed as follow s for a staggered grid with 2k points we ask that the finite difference operator a k k jacobi matrix and the sturm liouville differential operator share the k lowest eigenvalues and the values of the orthonormal eigenfunctions at one end of the interval this requirement determines uniquely the entries in the jacobi matrix which are grid cell averages of the coefficients in the continuum problem if these coefficients are known we can find the grid which we call optimal becau se it gives by design a finite difference operator with a prescribed spectr al measure we focus attention on the inverse problem where neither the coefficients n or the grid are known a key question in inversion is how to parametrize the coefficients i e how to choose the grid it is clear that to be successful this grid must be close to the optimal one which is unknown fortunately as we show here the grid dependence on the unknown coefficients is weak so the inversion can be done on a pr ecomputed grid for an a priori guess of the unknown coefficients this obse rvation leads to a simple yet efficient inversion algorithm which gives coefficien ts that converge pointwise to the true solution as the number k of data points tends to infinity the cornerstone of our convergence proof is showing that op timal grids provide an implicit natural regularization of the inverse problem by givin g reconstructions with uniformly bounded total variation the analysis is based on a novel explicit perturbation analysis of lanczos recursions and on a discrete

2dof controller parametrization for systems with a single i o delay this note puts forward a parametrization of all stabilizing two degrees of freedom controllers for possibly unstable processes with dead time the proposed parametrization is based on a doubly coprime factorization of the plant and takes the form of a generalized smith predictor dead time compensator feedback part and a finite dimensional feedforward part prefilter some alternative dead time compensation schemes and disturbance attenuation limitations are also discussed