compact complete minimal immersions in mathbb r 3 in this paper we find for any arbitrary finite topological type a compact riemann surface m an open domain m c m with the fixed topological type and a conformal complete minimal immersion x m 3 which can be extended to a continuous map x m 3 such that x m is an embedding and the hausdorff dimension of x m is 1 we also prove that complete minimal surfaces are dense in the space of minimal surfaces spanning a finite set of closed curves in 3 endowed with the topology of the hausdorff distance

on the distribution of the peak to average power ratio in ofdm signals the distribution of the peak to average power ratio papr in strictly band limited orthogonal frequency division multiplexing ofdm signals is studied assuming that the base band ofdm signal is characterized as a band limited complex gaussian process we first attempt to derive the exact distribution of the papr in the band limited ofdm signals since this distribution cannot be expressed in a closed form we further develop a simple closed form approximation based on the level crossing rate analysis comparisons of the proposed distributions with those obtained by computer simulations show good agreement and convergence with an increase in the number of subcarriers

feedback stabilization over commutative rings the matrix case this paper provides a solution of the feedback stabilization problem over commutative rings for matrix transfer functions stabilizability of a transfer matrix is realised as local stabilizability over the entire spectrum of the ring for stabilizable plants certain modules generated by its fractions and that of the stabilizing controller are shown to be projective compliments of each other in the case of rings with irreducible spectrum this geometric relationship shows that the plant is stabilizable if and only if the above modules of the plant are projective of ranks equal to the number of inputs and the outputs if the maxspectrum of the ring is noetherian and of zero krull dimension then this result shows that the stabilizable plants have doubly coprime fractions over unique factorization domains the above stabilizability condition is interpreted in terms of matrices formed by the fractions of the plant certain invariants of these matrices known as elementary factors are defined and it is shown that the plant is stabilizable if and only if these elementary factors generate the whole ring this condition thus provides a generalization of the coprime factorizability as a condition for stabilizabilty a formula for the class of all stabilizing controllers is then developed that generalizes the previous well known formula in factorization theory for multidimensional transfer functions these results provide concrete conditions for stabilizabilty finally it is shown that the class of polynomial rings over principal ideal domains is an additional class of rings over which stabilizable plants always have doubly coprime fractions

minimal surfaces in minimally convex domains in this paper we prove that every conformal minimal immersion of a compact bordered riemann surface m into a minimally convex domain d r can be approximated uniformly on compacts in m m bm by proper complete conformal minimal immersions m d see theorems 1 1 1 7 and 1 9 we also obtain a rigidity theorem for complete immersed minimal surfaces of finite total curvature contained in a minimally convex domain in r see theorem 1 16 and we characterize the minimal surface hull of a compact set k in r for any n 3 by sequences of conformal minimal disks whose boundaries converge to k in the measure theoretic sense see corollary 5 6

covariance matrix functions of vector χ 2 random fields in space and time this paper introduces vector multivariate or multiple 2 and rayleigh random functions or random fields on a spatial temporal or spatio temporal index domain and explores their basic properties formulated as a sum of squares of independent gaussian random functions a vector 2 random function has an interesting feature that its finite dimensional laplace transforms are not determined by its own covariance matrix but by that of the underlying gaussian one with the conditionally negative definite matrix as an important building block this paper constructs a class of vector 2 random functions from whose covariance matrices one can easily identify those of the underlying gaussian one so that the resulting vector 2 random function can be easily simulated and analyzed

existence of proper minimal surfaces of arbitrary topological type abstract consider a domain d in r 3 which is convex possibly all r 3 or which is smooth and bounded given any open surface m we prove that there exists a complete proper minimal immersion f m d moreover if d is smooth and bounded then we prove that the immersion f m d can be chosen so that the limit sets of distinct ends of m are disjoint connected compact sets in d

on the second derivative of a gaussian process envelope we explore some dynamic characteristics of the envelope of a bandpass gaussian process which are of interest in wireless fading channels specifically we show that unlike the first derivative the second derivative of the envelope which appears in a number of applications does not exist in the traditional mean square sense however we prove that the envelope is twice differentiable almost everywhere with probability one if the power spectrum of the bandpass gaussian process satisfies a certain condition we also derive an integral form for the probability density function pdf of the second derivative of the envelope assuming an arbitrary power spectrum

the calabi yau conjectures for embedded surfaces in this talk i will discuss the proof of the calabi yau conjectures for embedded surfaces this is joint work with bill minicozzi cm9 the calabi yau conjectures about surfaces date back to the 1960s much work has been done on them over the past four decades in particular examples of jorge xavier from 1980 and nadirashvili from 1996 showed that the immersed versions were false we will show here that for embedded surfaces i e injective immersions they are in fact true their original form was given in 1965 in ca where e calabi made the following two conjectures about minimal surfaces see also s s chern page 212 of ch conjecture 1 prove that a complete minimal hypersurface in r must be unbounded calabi continued it is known that there are no compact minimal submanifolds of r or of any simply connected complete riemannian manifold with sectional curvature 0 a more ambitious conjecture is conjecture 2 a complete minimal hypersurface in r has an unbounded projection in every n 2 dimensional flat subspace

holomorphic functions unbounded on curves of finite length given a pseudoconvex domain open image in new window we prove that there is a holomorphic function f on d such that the lengths of paths p 0 1 rightarrow d along which mathfrak r f is bounded above with p 0 fixed grow arbitrarily fast as p 1 rightarrow bd a consequence is the existence of a complete closed complex hypersurface m subset d such that the lengths of paths p 0 1 rightarrow m with p 0 fixed grow arbitrarily fast as p 1 rightarrow bd

on the asymptotic behavior of a complete bounded minimal surface in ℝ in this paper we construct an example of a complete minimal disk which is properly immersed in a ball of r 3

controller parameterization of anantharam s example we consider the controller parameterization of anantharam s example 1985 it is shown that the parameterization is achieved by using only one parameter even though the plant does not have a coprime factorization

null curves in mathbb c 3 and calabi yau conjectures for any open orientable surface m and convex domain omega subset mathbb c 3 there exist a riemann surface n homeomorphic to m and a complete proper null curve f n this result follows from a general existence theorem with many applications among them the followings for any convex domain in mathbb c 2 there exist a riemann surface n homeomorphic to m and a complete proper holomorphic immersion f n furthermore if d subset mathbb r 2 is a convex domain and is the solid right cylinder x in mathbb c 2 mbox re x in d then f can be chosen so that re f n d is proper there exist a riemann surface n homeomorphic to m and a complete bounded holomorphic null immersion f n to rm sl 2 mathbb c there exists a complete bounded cmc 1 immersion x m to mathbb h 3 for any convex domain omega subset mathbb r 3 there exists a complete proper minimal immersion xj j 1 2 3 m with vanishing flux furthermore if d subset mathbb r 2 is a convex domain and omega x j j 1 2 3 in mathbb r 3 x 1 x 2 in d then x can be chosen so that x1 x2 m d is proper any of the above surfaces can be chosen with hyperbolic conformal structure

robustness analysis of uncertain linear singular systems with output feedback control the robustness analysis for a linear singular system with uncertain parameters and static output feedback control is considered the problem is transformed into a robust nonsingularity problem based on the linear fractional transformation lft approach the robustness bounds to preserve regularity impulse immunity and stability are found in terms of the structured singular value spl mu with respect to parametric uncertainties the lft approach provides a unified framework for robustness analysis of both uncertain linear continuous discrete time singular systems

algebraic geometric aspects of feedback stabilization this paper develops a theory of feedback stabilization for siso transfer functions over a general integral domain which extends the well known coprime factorization approach to stabilization necessary and sufficient conditions for stabilizability of a transfer function in this general setting are obtained these conditions are then refined in the special cases of unique factorization domains ufds noetherian rings and rings of fractions it is shown that these conditions can be naturally interpreted geometrically in terms of the prime spectrum of the ring this interpretation provides a natural generalization to the classical notions of the poles and zeros of a plant the set of transfer functions is topologized so as to restrict to the graph topology of vidyasagar ieee trans automatic control ac 29 1984 pp 403 418 when the ring is a bezout domain it is shown that stability of a feedback system is robust in this topology when the ring is a ufd this theory is then applied to the problem of sta

boundary continuity of complete proper holomorphic maps abstract we show that there is no complete proper holomorphic map from the disc to the bidisc 2 which extends continuously through

a foliation of the ball by complete holomorphic discs we show that the open unit ball mathbb b n of mathbb c n n 1 admits a nonsingular holomorphic foliation by complete properly embedded holomorphic discs

feedback stabilization of mimo 3 d linear systems the authors solve the open problem of the existence of double coprime factorizations for a large class of multi input multi output mimo three dimensional 3 d linear systems it is proven that if all the unstable zeros of the contents associated with left and right matrix fraction descriptions of a given feedback stabilizable causal mimo 3 d plant are simple then the plant has a double coprime factorization the authors then give a parameterization of all stabilizing compensators for a mimo 3 d system in this class the key result developed in the paper is a novel and constructive technique of replacing an unstable polynomial with a stable polynomial step by step an illustrative example is also provided

the classical theory of minimal surfaces we present here a survey of recent spectacular successes in classical minimal surface theory we highlight this article with the theorem that the plane the helicoid the catenoid and the one parameter family frtgt2 0 1 of riemann minimal examples are the only complete properly embedded minimal planar domains in r 3 the proof of this result depends primarily on work of colding

feedback stabilization over commutative rings further study of the coordinate free approach this paper is concerned with the coordinate free approach to control systems the coordinate free approach is a factorization approach but does not require the coprime factorizations of the plant we present two criteria for feedback stabilizability for multi input multi output mimo systems in which transfer functions belong to the total rings of fractions of commutative rings both of them are generalizations of sule s results in siam j control optim 32 1994 pp 1675 1695 the first criterion is expressed in terms of modules generated from a causal plant and does not require the plant to be strictly causal it shows that if the plant is stabilizable the modules are projective the other criterion is expressed in terms of ideals called generalized elementary factors this gives the stabilizability of a causal plant in terms of the coprimeness of the generalized elementary factors as an example a discrete finite time delay system is considered

boundary behaviour of inner functions and holomorphic mappings let f be a holomorphic function in the unit disk omitting a set a of values of the complex plane if a has positive logarithmic capacity r nevanlinna proved that f has a radial limit at almost every point of the unit circle if a is any infinite set we show that f has a radial limit at every point of a set of hausdorff dimension 1 a localization technique reduces this result to the following theorem on inner functions if i is an inner function omitting a set of values b in the unit disk then for any accumulation point b of b in the disk there exists a set of hausdorff dimension 1 of points in the circle where i has radial limit b

construction of labyrinths in pseudoconvex domains we build in a given pseudoconvex runge domain d of mathbb c n an mathcal o d convex set gamma every connected component of which is a holomorphically contractible convex compact set enjoying the property that any continuous path gamma 0 1 rightarrow d with lim r rightarrow 1 gamma r in partial d and omitting gamma has infinite length this solves a problem left open in a recent paper by alarcon and forstneric

chi 2 random fields in space and time this paper introduces some 2 random functions which are not homogeneous stationary in general but contain the stationary case as a special case and whose correlation structures are nonparametric and flexible so that many parametric forms can be specified the mean square differentiability of such 2 random function is discussed we also make the comparison of two 2 random functions through their finite dimensional laplace transforms

the oka principle for holomorphic legendrian curves in mathbb c 2n 1 let m be a connected open riemann surface we prove that the space mathscr l m mathbb c 2n 1 of all holomorphic legendrian immersions of m to mathbb c 2n 1 n ge 1 endowed with the standard holomorphic contact structure is weakly homotopy equivalent to the space mathscr c m mathbb s 4n 1 of continuous maps from m to the sphere mathbb s 4n 1 if m has finite topological type then these spaces are homotopy equivalent we determine the homotopy groups of mathscr l m mathbb c 2n 1 in terms of the homotopy groups of mathbb s 4n 1 it follows that mathscr l m mathbb c 2n 1 is 4n 3 connected

every bordered riemann surface is a complete conformal minimal surface bounded by jordan curves in this paper we find approximate solutions of certain riemann hilbert boundary value problems for minimal surfaces in r and null holomorphic curves in c for any n 3 with this tool in hand we construct complete conformally immersed minimal surfaces in r which are normalized by any given bordered riemann surface and have jordan boundaries we also furnish complete conformal proper minimal immersions from any given bordered riemann surface to any smoothly bounded strictly convex domain of r which extend continuously up to the boundary for n 5 we find embeddings with these properties

a jordan curve spanned by a complete minimal surface in this paper we construct complete conformal minimal immersions f mathbb d longrightarrow mathbb r 3 which admit continuous extensions to the closed disk f overline mathbb d longrightarrow mathbb r 3 moreover f mathbb s 1 mathbb s 1 rightarrow f mathbb s 1 is a homeomorphism and f mathbb s 1 is a non rectifiable jordan curve with hausdorff dimension 1

first and second passage times of rayleigh processes corresp the first and second passage times of a stationary rayleigh process r t a are discussed r t a represents the envelope of a stationary random process consisting of a sinusoidal signal of amplitude and frequency f 0 plus stationary gaussian noise of unit variance having a narrow band power spectral density which is symmetrical about f 0 approximate integral equations are developed whose solutions yield approximate probability densities concerning the first and second passage times of r t a the resulting probability functions are presented in graphs for the case when the power spectral density of the noise is gaussian related results concerning the approximate distribution function of the absolute minimum or absolute maximum of r t a in the closed interval 0 tau are also presented the exact probability densities are expressed in the form of an infinite series of multiple integrals

limit sets for complete minimal immersions in this paper we study the behaviour of the limit set of complete proper compact minimal immersions in a domain g subset mathbb r 3 with the boundary partial g subset c 2 we prove that the second fundamental form of the surface g is nonnegatively defined at every point of the limit set of such immersions

frequency response estimation for normalized coprime factors this paper investigates estimating the frequency response of the normalized coprime factors of a possibly unstable transfer function from closed loop frequency domain experimental data a stochastic framework has been adopted the dual youla kucera parametrization approach in closed loop system identification has been developed to guarantee the normality of the estimated frequency response a parametrization of all the normalized coprime factors has been established for transfer functions internally stabilizable by a known controller based on this parametrization interpolation conditions have been derived for the measurement errors being consistent with the stabilizing controller and experimental data an analytic expression has been obtained for the maximum likelihood estimate of the frequency response the effectiveness of the suggested method is confirmed through numerical simulations

a complete complex hypersurface in the ball of c n in 1977 p yang asked whether there exist complete immersed complex submanifolds m k c n with bounded image a positive answer is known for holomorphic curves k 1 and partial answers are known for the case when k 1 the principal result of the present paper is a construction of a holomorphic function on the open unit ball bn of c n whose real part is unbounded on every path in bn of nite length that ends on bbn a consequence is the existence of a complete closed complex hypersurface in bn this gives a positive answer to yang s question in all dimensions k n 1 k n by providing properly embedded complete complex manifolds

bounded domains which are universal for minimal surfaces we construct open domains in 3 which do not admit complete properly immersed minimal surfaces with an annular end these domains can not be smooth by a recent result of martin and morales

origin of rice s formula the origin of s o rice s formula for the average rate of level crossings of a rather general random process is traced to his 1936 private notes entitled singing transmission lines the notes formulate an engineering problem in terms of the probability distribution of the absolute maximum of an envelope process in a closed interval rice showed that if the envelope of the reflected wave on a loaded transmission line were to equal or exceed the maximum value in the closed interval then singing or oscillation could be expected to occur and the analog transmission system would become unstable and useless for the transmission of information the formula and its generalizations are now basic for the analysis of level crossings and first and second passage times of random processes

on stabilization and the existence of coprime factorizations let k be an integral domain and f its quotient field we show there are plants over f which have no stable coprime factorizations but can be stabilized in a stable closed loop this answers a question posed by vidyasagar schneider and francis

asymptotic global confidence regions in parametric shape estimation problems we introduce confidence region techniques for analyzing and visualizing the performance of two dimensional parametric shape estimators assuming an asymptotically normal and efficient estimator for a finite parameterization of the object boundary cramer rao bounds are used to define an asymptotic confidence region centered around the true boundary computation of the probability that an entire boundary estimate lies within the confidence region is a challenging problem because the estimate is a two dimensional nonstationary random process we derive lower bounds on this probability using level crossing statistics the same bounds also apply to asymptotic confidence regions formed around the estimated boundaries lower bounding the probability that the entire true boundary lies within the confidence region the results make it possible to generate asymptotic confidence regions for arbitrary prescribed probabilities these asymptotic global confidence regions conveniently display the uncertainty in various geometric parameters such as shape size orientation and position of the estimated object and facilitate geometric inferences numerical simulations suggest that the new bounds are quite tight

the calabi yau problem null curves and bryant surfaces in this paper we prove that every bordered riemann surface m admits a complete proper null holomorphic embedding into a ball of the complex euclidean 3 space mathbb c 3 the real part of such an embedding is a complete conformal minimal immersion m rightarrow mathbb r 3 with bounded image for any such m we also construct proper null holomorphic embeddings m hookrightarrow mathbb c 3 with a bounded coordinate function these give rise to properly embedded null curves m hookrightarrow sl 2 mathbb c and to properly immersed bryant surfaces m rightarrow mathbb h 3 in the hyperbolic 3 space in particular we provide the first examples of proper bryant surfaces with finite topology and of hyperbolic conformal type the main novelty when compared to the existing results in the literature is that we work with a fixed conformal structure on m this is accomplished by introducing a conceptually new method based on complex analytic techniques one of our main tools is an approximate solution to certain riemann hilbert boundary value problems for null curves in mathbb c 3 developed in sect 3

every bordered riemann surface is a complete proper curve in a ball we prove that every bordered riemann surface admits a complete proper holomorphic immersion into a ball of mathbb c 2 and a complete proper holomorphic embedding into a ball of mathbb c 3